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diff --git a/.gitattributes b/.gitattributes new file mode 100644 index 0000000..6833f05 --- /dev/null +++ b/.gitattributes @@ -0,0 +1,3 @@ +* text=auto +*.txt text +*.md text diff --git a/31061-pdf.pdf b/31061-pdf.pdf Binary files differnew file mode 100644 index 0000000..3aa523e --- /dev/null +++ b/31061-pdf.pdf diff --git a/31061-pdf.zip b/31061-pdf.zip Binary files differnew file mode 100644 index 0000000..f8d963b --- /dev/null +++ b/31061-pdf.zip diff --git a/31061-t.zip b/31061-t.zip Binary files differnew file mode 100644 index 0000000..9b4308a --- /dev/null +++ b/31061-t.zip diff --git a/31061-t/31061-t.tex b/31061-t/31061-t.tex new file mode 100644 index 0000000..d594d3e --- /dev/null +++ b/31061-t/31061-t.tex @@ -0,0 +1,22669 @@ +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % +% % +% The Project Gutenberg EBook of A History of Mathematics, by Florian Cajori +% % +% This eBook is for the use of anyone anywhere at no cost and with % +% almost no restrictions whatsoever. You may copy it, give it away or % +% re-use it under the terms of the Project Gutenberg License included % +% with this eBook or online at www.gutenberg.org % +% % +% % +% Title: A History of Mathematics % +% % +% Author: Florian Cajori % +% % +% Release Date: January 24, 2010 [EBook #31061] % +% Most recently updated: June 11, 2021 % +% % +% Language: English % +% % +% Character set encoding: UTF-8 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK A HISTORY OF MATHEMATICS *** % +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{31061} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Standard DP encoding. Required. %% +%% %% +%% ifthen: Logical conditionals. Required. %% +%% calc: Infix arithmetic for length calculations. Required. %% +%% %% +%% amsmath: AMS mathematics enhancements. Required. %% +%% amssymb: Additional mathematical symbols. Required. %% +%% %% +%% alltt: Fixed-width font environment. Required. %% +%% array: Enhanced tabular features. Required. %% +%% %% +%% yfonts: Gothic font on title page. Optional. %% +%% %% +%% multicol: Multi-column environment for index. Required. %% +%% makeidx: Indexing capabilities. Required. %% +%% %% +%% fancyhdr: Enhanced running headers and footers. Required. %% +%% %% +%% graphicx: Standard interface for graphics inclusion. Required. %% +%% wrapfig: Illustrations surrounded by text. Required. %% +%% %% +%% geometry: Enhanced page layout package. Required. %% +%% hyperref: Hypertext embellishments for pdf output. Required. %% +%% %% +%% %% +%% Producer's Comments: %% +%% %% +%% British spelling is retained. Minor changes to spelling and %% +%% punctuation, and items that may seem questionable to future %% +%% maintainers of this file, are of three types: %% +%% %% +%% \DPtypo{}{}: Denotes a relatively clear typographical mistake. %% +%% %% +%% \Change{}{}: Denotes a change made for consistency, or a less %% +%% clear instance of possible error. For example, Cajori writes %% +%% bi-quadratic equation and biquadratic residue consistently. %% +%% Since the former are indexed as Biquadratic equation, their %% +%% spelling has been \Change{}{}-ed. Instances can be easily %% +%% reverted, see the boolean flag KeepOrigSpelling below. %% +%% %% +%% \DPnote{** PP: <note>}: Remarks signifying a possible point of %% +%% typographical question, placed immediately after the text. %% +%% %% +%% For example, on occasions where a word is hyphenated across a %% +%% line in the original and the word occurs only once, comparison %% +%% with similar words was made to decide whether to hyphenate. %% +%% Such instances are \DPnote{}-ed. %% +%% %% +%% Where possible, hieroglyphs were reproduced from the original %% +%% page scans; see the detailed note in this file at 027.png. %% +%% %% +%% Thw wrapfigures are manually located mid-paragraph, with line %% +%% breaks adjusted for a 4.25 inch text width. %% +%% %% +%% %% +%% Compilation Flags: %% +%% %% +%% The following behaviors may be controlled by boolean flags. %% +%% %% +%% KeepOrigSpelling (false by default): See above. %% +%% %% +%% ForPrinting (false by default): %% +%% Compile a print-optimized PDF file. Set to true for print- %% +%% optimized file (asymmetric layout, reference markers as %% +%% superscripts, matching the original). %% +%% %% +%% %% +%% Things to Check: %% +%% %% +%% %% +%% Spellcheck: .................................. yes %% +%% Smoothreading pool: ......................... xx %% +%% %% +%% lacheck: ..................................... yes %% +%% Numerous false positives %% +%% %% +%% PDF pages: 556 (if ForPrinting set to false) %% +%% PDF page size: 4.5 x 7 inches (if ForPrinting set to false) %% +%% PDF bookmarks: created, point to ToC entries %% +%% PDF document info: filled in %% +%% Images: 33 pdf graphics %% +%% %% +%% Summary of log file: %% +%% * One overfull hbox (1.91pt too wide). %% +%% * One underfull hbox. %% +%% %% +%% %% +%% Compile History: %% +%% %% +%% January, 2010: adhere (Andrew D. Hwang) %% +%% texlive2007, GNU/Linux %% +%% %% +%% Command block: %% +%% %% +%% pdflatex x3 (Run pdflatex three times) %% +%% makeindex %% +%% pdflatex %% +%% %% +%% %% +%% January 2010: pglatex. %% +%% Compile this project with: %% +%% pdflatex 31061-t.tex ..... THREE times %% +%% makeindex 31061-t.idx %% +%% pdflatex 31061-t.tex %% +%% %% +%% pdfTeXk, Version 3.141592-1.40.3 (Web2C 7.5.6) %% +%% %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\listfiles +\documentclass[12pt]{book}[2005/09/16] + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PACKAGES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\usepackage[utf8]{inputenc}[2006/05/05] %% DP standard encoding +\usepackage[french,german,italian,latin,greek,english]{babel}[2005/11/23] + +\usepackage{ifthen}[2001/05/26] %% Logical conditionals +\usepackage{calc}[2005/08/06] %% Infix arithmetic + +\usepackage{amsmath}[2000/07/18] %% Displayed equations +\usepackage{amssymb}[2002/01/22] %% and additional symbols + +\usepackage{alltt}[1997/06/16] %% boilerplate, credits, license + +\usepackage{array}[2005/08/23] %% extended array/tabular features + +% Define \textgoth; document subsequently defines \Mygoth +\IfFileExists{yfonts.sty}% +{\usepackage{yfonts}[2003/01/08]} % fraktur font (titlepage only) +{\providecommand{\textgoth}[1]{\textbf{##1}}} % fallback if no yfonts + + %% extended footnote capabilities + +\usepackage{indentfirst}[1995/11/23] + +\usepackage{multicol}[2006/05/18] +\usepackage{makeidx}[2000/03/29] + +\usepackage{graphicx}[1999/02/16]%% For diagrams +\usepackage{wrapfig}[2003/01/31] %% and wrapping text around them + +% for running heads; no package date available +\usepackage{fancyhdr} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%% Set up PRINTING (default) or SCREEN VIEWING %%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% KeepOrigSpelling = false (default) Make spelling consistent +\newboolean{KeepOrigSpelling} +\setboolean{KeepOrigSpelling}{false} + +% ForPrinting=true false (default) +% Letterpaper Cropped pages +% Asymmetric margins Symmetric margins +% Black hyperlinks Blue hyperlinks +\newboolean{ForPrinting} + +%% UNCOMMENT the next line for a PRINT-OPTIMIZED VERSION of the text %% +%\setboolean{ForPrinting}{true} + +%% Initialize values to ForPrinting=false +\newcommand{\HalfTitleSize}{\Large} +\newcommand{\Margins}{hmarginratio=1:1} % Symmetric margins +\newcommand{\HLinkColor}{blue} % Hyperlink color +\newcommand{\PDFPageLayout}{SinglePage} +\newcommand{\TransNote}{Transcriber's Note} +\newcommand{\TransNoteCommon} +{ + Figures may have been moved with respect to the surrounding text. + Minor typographical corrections and presentational changes have + been made without comment. +} + +\newcommand{\TransNoteText} +{ + \TransNoteCommon + + This PDF file is formatted for screen viewing, but may be easily + formatted for printing. Please consult the preamble of the \LaTeX\ + source file for instructions. +} + +%% Re-set if ForPrinting=true +\ifthenelse{\boolean{ForPrinting}}{% + \renewcommand{\HalfTitleSize}{\LARGE} + \renewcommand{\Margins}{hmarginratio=2:3} % Asymmetric margins + \renewcommand{\HLinkColor}{black} % Hyperlink color + \renewcommand{\PDFPageLayout}{TwoPageRight} + \renewcommand{\TransNote}{Transcriber's Note} + \renewcommand{\TransNoteText}{% + \TransNoteCommon + + This PDF file is formatted for printing, but may be easily formatted + for screen viewing. Please see the preamble of the \LaTeX\ source + file for instructions. + } +}{} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%% End of PRINTING/SCREEN VIEWING code; back to packages %%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%% Set page dimensions +% If text block size is changed, illustrations must be relocated, +% so print and screen versions use the same text block size. +\ifthenelse{\boolean{ForPrinting}}{% + \setlength{\paperwidth}{8.5in} + \setlength{\paperheight}{11in} + \usepackage[body={4.25in,6.75in},\Margins]{geometry}[2002/07/08] +}{% + \setlength{\paperwidth}{4.5in} + \setlength{\paperheight}{8in} + \usepackage[body={4.25in,6.75in},\Margins,includeheadfoot]{geometry}[2002/07/08] +} + +\raggedbottom + +\providecommand{\ebook}{00000} % Overridden during white-washing +\usepackage[pdftex, + hyperfootnotes=false, + pdftitle={The Project Gutenberg eBook \#\ebook: A History of Mathematics}, + pdfauthor={Florian Cajori}, + pdfkeywords={Andrew D. Hwang, Peter Vachuska, Carl Hudkins, + Project Gutenberg Online Distributed Proofreading Team, + The Internet Archive/American Libraries}, + pdfstartview=Fit, % default value + pdfstartpage=1, % default value + pdfpagemode=UseNone, % default value + bookmarks=true, % default value + linktocpage=false, % default value + pdfpagelayout=\PDFPageLayout, + pdfdisplaydoctitle, + pdfpagelabels=true, + bookmarksopen=true, + bookmarksopenlevel=1, + colorlinks=true, + linkcolor=\HLinkColor]{hyperref}[2007/02/07] + +%%%% Re-crop screen-formatted version, omit blank verso pages %%%% +\ifthenelse{\boolean{ForPrinting}} + {} +% {\hypersetup{pdfpagescrop = 100 90 512 780} + {\hypersetup{pdfpagescrop = 0 0 324 576} + % If ForPrinting=false, don't skip to recto + \renewcommand{\cleardoublepage}{\clearpage} +} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%% COMMANDS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%%%% Fixed-width environment to format PG boilerplate %%%% +% 8pt more-or-less evenly fits 72 chars into 4.25in +\newenvironment{PGtext}{% +\begin{alltt} +\fontsize{8}{9.6}\ttfamily\selectfont}% +{\end{alltt}} + + +%%%% Illustrations and decorations %%%% +\newcommand{\Graphic}[2]{\includegraphics[width=#1]{./images/#2.pdf}} + +\newcommand{\Glyph}[1] + {\smash[b]{\raisebox{-2pt}{\includegraphics[height=12pt]{./images/#1.pdf}}}} + +\newcommand{\TallGlyph}[1] + {\raisebox{-8pt}{\includegraphics[height=24pt]{./images/#1.pdf}}} + +% Thought breaks +\newsavebox{\tbreakglyph} +\savebox{\tbreakglyph}[0.875in]{\hbox{\Graphic{0.75in}{tbreak1}}} + +\newsavebox{\fbreakglyph} % fancy break +\savebox{\fbreakglyph}[1.5in]{\hbox{\Graphic{1.25in}{tbreak2}}} + +\newcommand{\fancybreak}[1]{% + \ifthenelse{\equal{#1}{}}{% + \usebox{\tbreakglyph} + }{% + \usebox{\fbreakglyph} + }% +} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% SECTIONING %%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% Each chapter starts unindented, with smallcaps +\newcommand{\First}[1]{\textsc{#1}} + +\newcommand{\Heading}{\centering\normalfont\normalsize} + +\newcommand{\ChapRule}[1]{\fancybreak{#1}} + +% Running heads +\newcommand{\SetRunningHeads}[1]{% + \fancyhf{} + \fancyhead[CE]{\small A HISTORY OF MATHEMATICS.} + \fancyhead[CO]{\small \MakeUppercase{#1}.} + + \ifthenelse{\boolean{ForPrinting}} + {\fancyhead[RO,LE]{\thepage}} + {\fancyhead[R]{\thepage}} +} + +\newcommand{\Chapter}[2][]{% + \cleardoublepage + \phantomsection + \section*{\centering\Heading\large \MakeUppercase{#2}.} + \begin{center}\ChapRule{#1}\end{center} + + \ToCPageMark + \addcontentsline{toc}{chapter}{\texorpdfstring{\protect\MakeUppercase{#2}}{#2}} + \SetRunningHeads{#2} + \thispagestyle{plain} +} + +% Special handling for Introduction +\newcommand{\IntroChapter}[2]{% + \phantomsection % Already cleared to recto + + \section*{\centering\Heading\LARGE \MakeUppercase{#1}.} + \medskip + \begin{center}\ChapRule{2}\end{center} + \vspace*{-2\Titleskip} + \subsection*{\centering\Heading\large #2.} + + % Set up ToC anchor to detect when a new page is issued + \ToCPageMark + \addtocontents{toc}% + {\protect\renewcommand{\protect\ToCAnchor}{\protect\pageref{toc:1}}} + + % ToC entry proper + \addcontentsline{toc}{chapter}{\texorpdfstring{\protect\MakeUppercase{#2}}{#2}} + + \SetRunningHeads{#2} + \thispagestyle{plain} + + % Set wider line spacing document-wide + \linespread{\BLS}\selectfont% +} + +% PDF bookmark, but no ToC entry +\newcommand{\NChapter}[2][\BLS]{% + \cleardoublepage + \phantomsection + \pdfbookmark[0]{#2}{#2} + + \section*{\centering\Heading\large \MakeUppercase{#2}.} + \ifthenelse{\not\equal{#2}{Addenda}}{% + \begin{center}\ChapRule{}\end{center}% + }{}% + + \linespread{#1}\selectfont% + + \SetRunningHeads{#2} + \thispagestyle{plain} +} + + +\newcommand{\Section}[1]{% + \phantomsection + \subsection*{\Heading\MakeUppercase{#1}.} + + \ToCPageMark + \addcontentsline{toc}{section}{\texorpdfstring{{\scshape#1}}{#1}} + \SetRunningHeads{#1} +} + +\newcommand{\SpecialSection}[1]{% + \phantomsection + \subsubsection*{\Heading\footnotesize\MakeUppercase{#1}.} + + \ToCPageMark + \addcontentsline{toc}{section}{\texorpdfstring{\quad{\itshape#1}}{#1}} +} + +\newcommand{\Subsection}[1]{% + \phantomsection + \subsection*{\Heading\textit{#1}.} + + \ToCPageMark + \addcontentsline{toc}{subsection}{#1} +} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%% CROSS-REFERENCES %%%%%%%%%%%%%%%%%%%%%%%%% +% Anchors +\newcommand{\Addendlabel}[1] + {\phantomsection\label{addend:#1}} + +\newcommand{\Tag}[1]% + {\tag{#1}\phantomsection\label{eqn:#1}} + +% Provides its own trailing space; following text must abut braces. +\newcommand{\BookRef}[1]{% + \settowidth{\TmpLen}{100. }% + \smallskip\par\phantomsection\label{book:#1}% + \noindent\hangindent3.5em\makebox[\TmpLen][r]{#1. }} + +% and links +\newcommand{\Addendref}[1] + {\hyperref[addend:#1]{\textsc{Page~\pageref{addend:#1}}}} + +% References; use empty optional argument to suppress end-of-sentence space +% In screen version use [xx] instead of superscript +\newcommand{\Reference}[2][.]{% + \ifthenelse{\boolean{ForPrinting}}{% + \hyperref[book:#2]{\textsuperscript{#2}}% + }{\,% Else: not ForPrinting, add thin space + \hyperref[book:#2]{\texorpdfstring{[#2]}{#2}}% + }%** Add fake end of sentence space by default + \ifthenelse{\equal{#1}{.}}{\ }{#1}% +} + +% Only needed once, for double reference +\newcommand{\BiReference}[2]{% + \ifthenelse{\boolean{ForPrinting}}{% + \hyperref[book:#1]{\textsuperscript{#1}}\textsuperscript{,}% + \hyperref[book:#2]{\textsuperscript{#2}}% + }{\,% Else: not ForPrinting, add thin space + \hyperref[book:#1]{\texorpdfstring{[#1}{#1}}, % + \hyperref[book:#2]{\texorpdfstring{#2]}{#2}}% + }\ %** Add fake end-of-sentence space +} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%% TABLE OF CONTENTS %%%%%%%%%%%%%%%%%%%%%%%% +\newcommand{\TableofContents}{% + \linespread{1}% + \small + \tableofcontents +} + +\AtBeginDocument{% + \renewcommand{\contentsname}{% + \thispagestyle{plain}% + \fancyhead[C]{\small TABLE OF CONTENTS.} + \phantomsection + \pdfbookmark[0]{Contents}{Table of Contents} + \vspace*{-72pt} + \centering\normalfont\large\scshape TABLE OF CONTENTS\\[12pt] + \fancybreak{}\\[-36pt] + } +} + +% The table of contents may be set across two pages; we'll ensure the +% word ``page'' is printed at the top of the page number column, +% centered on the first page (to match the scan) and right-justified +% on subsequent pages (better visual appearance). +% +% Each sectional unit puts a \ToCLine into the toc file. The IntroChapter +% command defines a \ToCAnchor macro, which expands to the current +% page of the table of contents. +% +% Subsequent units put down a label and get the \pageref. If this has +% changed, the word ``page'' is written at the right margin, and the +% \ToCAnchor command is updated. +\newcommand{\ToCLine}[1]{% + \label{toc:#1}% + \ifthenelse{\not\equal{\pageref{toc:#1}}{\ToCAnchor}}{% + \renewcommand{\ToCAnchor}{\pageref{toc:#1}}% + % And write Page + \null\hfill\mbox{\footnotesize\scshape Page}% + }{}% Else do nothing +} + +% Null defintion; will be redefined usefully in the toc file +\newcommand{\ToCAnchor}{} + +\newcounter{ToCUnit} +\newcommand{\ToCPageMark}{% + \stepcounter{ToCUnit}% + \addtocontents{toc}{\protect\filbreak}% + \addtocontents{toc}{\protect\ToCLine{\theToCUnit}}% +} + +% Chapters in toc with dotted line to page number +\makeatletter +\renewcommand\l@chapter{\@dottedtocline{0}{0em}{3.5em}} +\makeatother + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% INDEX %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\makeatletter +\renewcommand{\@idxitem}{\par\hangindent 24\p@\global\let\idxbrk\nobreak} +\renewcommand{\subitem}{\idxbrk\@idxitem \hspace*{12\p@}\let\idxbrk\relax} +\renewcommand{\indexspace}{\par\penalty-3000 \vskip 10pt plus5pt minus3pt\relax} + +\renewenvironment{theindex}{% + \setlength\columnseprule{0.5pt}% + \setlength\columnsep{12pt}% + \begin{multicols}{2}[\begin{center}\Large INDEX\\\fancybreak{}\end{center}]% + \footnotesize% + \setlength\parindent{0pt}% + \setlength\parskip{0pt plus 0.3pt}% + \thispagestyle{plain}% + \let\item\@idxitem\raggedright } + {\end{multicols}\clearpage\normalsize\fancyhead{}\cleardoublepage} +\makeatother +\newcommand\IndexBookmark{\pdfbookmark[0]{Index}{Index}} + +% Miscellaneous extra formatting for individual entries +\newcommand{\etseq}[1]{\hyperpage{#1} \protect\textit{et~seq.}} +\renewcommand{\see}[2]{\textit{See} #1} + +\makeindex + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LICENSING %%%%%%%%%%%%%%%%%%%%%%%%%%%% +\newcommand{\Licensing}{% + % Clear heading in previous blank verso page, if any + \clearpage + \fancyhead{} + \renewcommand{\headrulewidth}{0pt} + + % And set up running heads + \cleardoublepage + \renewcommand{\headrulewidth}{0.5pt} + + \phantomsection + \pdfbookmark[0]{PG License}{Project Gutenberg License} + \ifthenelse{\boolean{ForPrinting}}{% + \fancyhead[RO,LE]{\thepage} + }{% + \fancyhead[R]{\thepage}% + } + \fancyhead[C]{\textsc{LICENSE}} +} + +%%%%%%%%%%%%%%%%%%%%%%% GLOBAL STYLE PARAMETERS %%%%%%%%%%%%%%%%%%%%%% +\hyphenation{Anax-a-go-ras Arch-i-medes evan-es-cent fa-vour-ite know-ledge man-u-scripts Ni-co-medes Pythag-o-ras Pythag-o-re-an Pythag-o-re-ans quad-ra-tic trig-ono-metry} + +\DeclareMathSizes{12}{11}{9}{7} + +\renewcommand{\headrulewidth}{0pt} +\setlength{\parindent}{1em} + +% Loosen up the spacing +\setlength{\parsep}{0.5ex plus 0.25ex minus 0.5ex} +\setlength{\parskip}{0.375ex plus .625ex minus 0.125ex} +\setlength{\partopsep}{0.25ex plus .75ex minus 0.125ex} + +% Issue this just before the Preface +\newcommand{\stretchyspace}{\spaceskip0.25em plus 0.375em minus 0.125em} + +% Set this at the Introduction +\newcommand{\BLS}{1.2} %Default \linespread + +\newlength{\Titleskip} + +%%%%%%%%%%%%%%%%%%%% MISCELLANEOUS FORMATTING %%%%%%%%%%%%%%%%%%%%%%%% + +% Table Entry, fits to \linewidth +\newcommand{\TE}[1]{% + \makebox[\TmpLen][c]{#1}% +} + +% Crossed-out digits on 137.png +\newcommand{\cancel}[1]{% + \makebox[\TmpLen][c]{\ensuremath{#1\llap{$/$}}}% +} + +\newcommand{\Neg}{\phantom{-}} +% Fraction strut, for use in numerator of inline \dfrac, in +% combination with \smash, when \dfracs appear on consecutive lines +\newcommand{\Fstr}[1][8pt]{\rule{0pt}{#1}} +\newcommand{\Eta}{H} + +% Text markings +\newcommand{\DPnote}[1]{} +\newcommand{\DPtypo}[2]{#2} + +% To effect changes made solely for consistency +\newcommand{\Change}[2]{% + \ifthenelse{\boolean{KeepOrigSpelling}}{#1}{#2}% +} + +\DeclareMathOperator{\cn}{\textit{cn}} +\DeclareMathOperator{\dn}{\textit{dn}} +\DeclareMathOperator{\sn}{\textit{sn}} + +\DeclareMathOperator{\omn}{omn.} +\DeclareMathOperator{\versin}{ver\,sin} +\DeclareMathOperator{\natlog}{nat.\,log} +\DeclareMathOperator{\Naplog}{Nap.\,log} +\newcommand{\nnobrkdash}[1]{$n$\nobreakdash-\hspace{0pt}} + +\newcommand\olddecimal[2]{% + {% + \renewcommand\arraystretch{0.5}% anything < 0.7 will do + \begin{tabular}{@{}r@{}|@{}l@{}}% + \mathstrut#1\,&\,#2\\\cline{2-2}% + \end{tabular} + }% +} + +\newlength{\TmpLen} +\newlength{\CharW} +\setlength{\CharW}{12pt} +\newcommand{\CircleDigit}[1]{% + \raisebox{1pt}{\makebox[\CharW][c]{\ensuremath\bigcirc}\kern-\CharW + \makebox[\CharW][c]{{\footnotesize#1}}}% +} + +% For small caps A.D. and B.C. +\newcommand{\SCAbbrev}[3]{% + \ifthenelse{\equal{#3}{.}}% + {\textsc{\MakeLowercase{#1.#2}.}}% + {\textsc{\MakeLowercase{#1.#2}.}\@#3}% +} + +\newcommand{\AD}[1]{\SCAbbrev{A}{D}{#1}} +\newcommand{\BC}[1]{\SCAbbrev{B}{C}{#1}} + +\newcommand{\ie}{\textit{i.e.}} + +\DeclareUnicodeCharacter{00A3}{\pounds} +\DeclareInputText{183}{\ifmmode{\cdot}\else\textperiodcentered\fi} + +%% Language conveniences +\newcommand{\Eng}[1]{\foreignlanguage{english}{#1}} +\newcommand{\Fre}[1]{\foreignlanguage{french}{#1}} +\newcommand{\Ger}[1]{\foreignlanguage{german}{#1}} +\newcommand{\Grk}[1]{\foreignlanguage{greek}{#1}} +\newcommand{\Ita}[1]{\foreignlanguage{italian}{#1}} +\newcommand{\Lat}[1]{\foreignlanguage{latin}{#1}} + +%%%%%%%%%%%%%%%%%%%%% CATALOGUE FORMATTING MACROS %%%%%%%%%%%%%%%%%%%%% +\newlength{\QUAD} +\newlength{\QQUAD} + +\newcommand{\Catalog}{% + \cleardoublepage + \phantomsection + \pdfbookmark[0]{Catalogue}{Catalogue} + + \begin{center} + {\Large THE\quad MACMILLAN\quad COMPANY'S}\\[2\Titleskip] + {\scriptsize PUBLICATIONS ON}\\[2\Titleskip] + \makebox[0pt][c]{\Huge \scshape Mathematics \ and \ Physics.}\\[2\Titleskip] + \Graphic{2.5in}{tbreak3} + \end{center} + +% \pagenumbering{arabic} + \renewcommand{\headrulewidth}{0.5pt} + \fancyhf{} + \fancyhead[CE]{\textit{\small MACMILLAN \& CO'S PUBLICATIONS ON}} + \fancyhead[CO]{\textit{\small MATHEMATICS AND PHYSICS.}} + + \ifthenelse{\boolean{ForPrinting}} + {\fancyhead[RO,LE]{\thepage}} + {\fancyhead[R]{\thepage}} + + \thispagestyle{plain} + + %% Tighten spacing + \setlength{\parsep}{0.25ex plus 0.125ex minus 0.25ex} + \setlength{\parskip}{0.1875ex plus .3125ex minus 0.0625ex} + \setlength{\partopsep}{0.125ex plus .375ex minus 0.0625ex} + \footnotesize + \setlength{\QUAD}{1.5em} + \setlength{\QQUAD}{2\QUAD} +} + +\newcommand{\CatalogSection}[1]{% + \subsection*{\centering\normalsize\bfseries\sffamily #1} +} + +% For snippets of non-bold in \Entry headings +\newcommand{\Rm}[1]{{\normalfont #1}} + +\newcommand{\Entry}[2][]{% + \par\noindent\hangindent\QQUAD\textbf{#2}% + \ifthenelse{\equal{#1}{}}{\quad}{}% +} +\newcommand{\Subentry}{\par\noindent\hspace*{\QUAD}\hangindent\QQUAD} + +% Author, Title, and Price +\newcommand{\Au}[1]{{\scshape #1}} +\newcommand{\Ti}[1]{{\scshape #1}} +\renewcommand{\Pr}[1]{#1} +\newcommand{\Date}[1]{\oldstylenums{#1}} +\newcommand{\Degr}[1]{{\normalfont\scshape #1}} + +% For very last page +\newcommand{\SizeA}{\large} +\newcommand{\SizeB}{\normalsize} +\newcommand{\SizeC}{\small} +\newcommand{\SizeD}{\footnotesize} +\newcommand{\SizeE}{\scriptsize} + + +%%%% DPalign %%%% +\makeatletter +\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 + 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+\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 + +%%%%%%%%%%%%%%%%%%%%%%%% START OF DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{document} + +\pagestyle{empty} +\pagenumbering{Alph} +\phantomsection +\pdfbookmark[-1]{Front Matter}{Front Matter} + +%%%% PG BOILERPLATE %%%% +\phantomsection +\pdfbookmark[0]{PG Boilerplate}{Project Gutenberg Boilerplate} + +\begin{center} +\begin{minipage}{\textwidth} +\small +\begin{PGtext} +The Project Gutenberg EBook of A History of Mathematics, by Florian Cajori + +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: A History of Mathematics + +Author: Florian Cajori + +Release Date: January 24, 2010 [EBook #31061] +Most recently updated: June 11, 2021 + +Language: English + +Character set encoding: UTF-8 + +*** START OF THIS PROJECT GUTENBERG EBOOK A HISTORY OF MATHEMATICS *** +\end{PGtext} +\end{minipage} +\end{center} + +\clearpage + + +%%%% Credits and transcriber's note %%%% +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by Andrew D. Hwang, Peter Vachuska, Carl Hudkins +and the Online Distributed Proofreading Team at +http://www.pgdp.net +\end{PGtext} +\end{minipage} +\end{center} +\vfill + +\begin{minipage}{0.85\textwidth} +\small +\pdfbookmark[0]{Transcriber's Note}{Transcriber's Note} +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% + +\raggedright +\TransNoteText +\end{minipage} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% + +\frontmatter + +\setlength{\headheight}{15pt} +\fancyfoot{} +\ifthenelse{\boolean{ForPrinting}} + {\fancyhead[RO,LE]{\thepage}} + {\fancyhead[R]{\thepage}} + +\normalsize +\cleardoublepage + +%% -----File: 001.png---Folio i------- +%Half-Title Page +\thispagestyle{empty} +\null\vfil +\begin{center} + \makebox[0pt][c]{\textbf{\HalfTitleSize A HISTORY OF MATHEMATICS}}% +\end{center} +\vfil +\clearpage + +%% -----File: 002.png---Folio ii------- +% Publisher's device +\thispagestyle{empty} +\null\vfil +\begin{center} +\Graphic{1.5in}{macmillan} +\end{center} +\vfil +\clearpage + +%% -----File: 003.png---Folio iii------- +% title page +\thispagestyle{empty} +\ifthenelse{\boolean{ForPrinting}}{% + \setlength{\Titleskip}{8pt}% + \enlargethispage{96pt}% +}{% + \setlength{\Titleskip}{5.75pt}% +} + +\begin{center} +{\LARGE A HISTORY OF}\\[6\Titleskip] +% [** PP: Poor man's gesperrt for one-off use] +{\Huge M\,A\,T\,H\,E\,M\,A\,T\,I\,C\,S}\\[6\Titleskip] +{\footnotesize BY}\\[2\Titleskip] +{\normalsize\scshape FLORIAN CAJORI, Ph.D.}\\[\Titleskip] +\scriptsize\scshape% +Formerly Professor of Applied Mathematics in the Tulane University \\ +of Louisiana; now Professor of Physics \\ +in Colorado College\normalfont\\[6\Titleskip] +\begin{minipage}{3in} +\scriptsize +\null\hspace*{1em}% +``I am sure that no subject loses more than mathematics +by any attempt to dissociate it from its history.''---\textsc{J.~W.~L. +Glaisher} +\end{minipage} +\vspace*{12\Titleskip} + +\normalsize\textgoth{New York}\\[\Titleskip] +THE MACMILLAN COMPANY\\[\Titleskip] +\scriptsize\scshape LONDON: MACMILLAN \&~CO., Ltd.\\[\Titleskip] +\normalsize 1909\\[2\Titleskip] +\scriptsize\itshape All rights reserved +\end{center} +\normalsize\normalfont +\clearpage + +%% -----File: 004.png---Folio iv------- +\thispagestyle{empty} +\null\vfil +\begin{center} +\scriptsize\scshape Copyright, 1893,\\[\Titleskip] +By MACMILLAN AND CO. + +\rule{1in}{0.5pt}\\[\Titleskip] +\upshape Set up and electrotyped January, \Date{1894}.\quad Reprinted March, \\ +\Date{1895}; October, \Date{1897}; November, \Date{1901}; January, \Date{1906}; July, \Date{1909}. +\vfil\vfil + +{\footnotesize\textgoth{Norwood Press:}}\\ +J.~S. Cushing \& Co.---Berwick \& Smith.\\ +Norwood, Mass., U.S.A. +\end{center} +\normalsize\normalfont +\clearpage + +%% -----File: 005.png---Folio v------- + +% Globally apply looser inter-word spacing +\stretchyspace + +\NChapter{Preface} + +\First{An} increased interest in the history of the exact sciences +manifested in recent years by teachers everywhere, and the +attention given to historical inquiry in the mathematical +class-rooms and seminaries of our leading universities, cause +me to believe that a brief general History of Mathematics +will be found acceptable to teachers and students. + +The pages treating---necessarily in a very condensed +form---of the progress made during the present century, +are put forth with great diffidence, although I have spent +much time in the effort to render them accurate and +reasonably complete. Many valuable suggestions and criticisms +on the chapter on ``Recent Times'' have been made +by Dr.~E.~W. Davis, of the University of Nebraska. The +proof-sheets of this chapter have also been submitted to +Dr.~J.~E. Davies and Professor C.~A. Van~Velzer, both of the +University of Wisconsin; to Dr.~G.~B. Halsted, of the +University of Texas; Professor L.~M. Hoskins, of the Leland +Stanford~Jr.\ University; and Professor G.~D.~Olds, of Amherst +College,---all of whom have afforded valuable assistance. +I am specially indebted to Professor F.~H.~Loud, of Colorado +College, who has read the proof-sheets throughout. To all +the gentlemen above named, as well as to Dr.~Carlo Veneziani +%% -----File: 006.png---Folio vi------- +of Salt Lake City, who read the first part of my work in +manuscript, I desire to express my hearty thanks. But in +acknowledging their kindness, I trust that I shall not seem +to lay upon them any share in the responsibility for errors +which I may have introduced in subsequent revision of the +text. + +\null\hfill{\footnotesize FLORIAN CAJORI.}\hspace*{1em} +\vspace*{2\Titleskip} + +\footnotesize\textsc{Colorado College}, December, 1893. +%% -----File: 007.png---Folio vii------- + +% Resets font size to \normalsize +\TableofContents + +\iffalse +TABLE OF CONTENTS. + + Page +INTRODUCTION ............ 1 +ANTIQUITY.............. 5 + The Babylonians........... 5 + The Egyptians............ 9 + The Greeks ............. 16 + Greek Geometry........... 16 + The Ionic School.......... 17 + The School of Pythagoras...... 19 + The Sophist School........ 23 + The Platonic School........ 29 + The First Alexandrian School .... 34 + The Second Alexandrian School . . . 54 + Greek Arithmetic.......... 63 + The Romans .... ..... 77 +MIDDLE AGES ... ..... 84 + The Hindoos ... ...... 84 + The Arabs.............. 100 + Europe during the Middle Ages .... 117 + Introduction of Roman Mathematics . . 117 + Translation of Arabic Manuscripts .... 124 + The First Awakening and its Sequel . . . 128 +MODERN EUROPE .......... 138 + The Renaissance ... ..... 139 + Vieta to Descartes . . ..... 156 + Descartes to Newton . . ..... 183 + Newton to Euler.......... 199 + +%% -----File: 008.png---Folio viii------- + +EULER, LAGRANGE, AND LAPLACE 246 +The Origin of Modern Geometry 285 + +RECENT TIMES 291 + +SYNTHETIC Geometry 293 + +ANALYTIC GEOMETRY 307 + +ALGEBRA 315 + +ANALYSIS 331 + +THEORY of FUNCTIONS 347 + +THEORY of NUMBERS 362 + +APPLIED MATHEMATICS 373 + +INDEX 405 +\fi +%% -----File: 009.png---Folio IX------- + +\pagestyle{fancy} + +\NChapter[1]{Books of Reference}%[** PP: `Normal' line spacing] + +\Change{The}{\First{The}} following books, pamphlets, and articles have been used +in the preparation of this history. Reference to any of them +is made in the text by giving the respective number. Histories +marked with a star are the only ones of which extensive use +has been made. +\bigskip + +\footnotesize +\BookRef{1}\textsc{Günther, S\@.} \Ger{\textit{Ziele und Resultate der neueren Mathe\-ma\-tisch-historischen +Forschung.} Erlangen, 1876.} +\index{Gunther@{Günther, S.}}% + +\BookRef{2}\textsc{Cajori, F\@.} \textit{The Teaching and History of Mathematics in the U.~S\@.} +Washington, 1890. + +\BookRef{3}*\textsc{Cantor, Moritz.} \Ger{\textit{Vorlesungen über Geschichte der Mathematik.} +Leipzig. Bd.~I., 1880; Bd.~II., 1892.} +\index{Cantor, M.}% + +\BookRef{4}\textsc{Epping, J\@.} \Ger{\textit{Astronomisches aus Babylon. Unter Mitwirkung von} +\textsc{P.~J.~R. Strassmaier}. Freiburg, 1889.} +\index{Epping}% +\index{Strassmaier}% + +\BookRef{5}\textsc{Bretschneider, C.~A\@.} \Ger{\textit{Die Geometrie und die Geometer vor Euklides.} +Leipzig, 1870.} +\index{Bretschneider}% + +\BookRef{6}*\textsc{Gow, James.} \textit{A Short History of Greek Mathematics.} Cambridge, +1884. +\index{Gow}% + +\BookRef{7}*\textsc{Hankel, Hermann.} \Ger{\textit{Zur Geschichte der Mathematik im Alterthum +und Mittelalter.} Leipzig, 1874.} + +\BookRef{8}*\textsc{Allman, G.~J\@.} \textit{Greek Geometry from Thales to Euclid.} Dublin, +1889. +\index{Allman}% + +\BookRef{9}\textsc{De Morgan, A\@.} ``Euclides'' in \textit{Smith's Dictionary of Greek and +Roman Biography and Mythology}. + +\BookRef{10}\textsc{Hankel, Hermann.} \Ger{\textit{Theorie der Complexen Zahlensysteme.} Leipzig, +1867.} +\index{Hankel!ref.\ to}% + +\BookRef{11}\textsc{Whewell, William.} \textit{History of the Inductive Sciences.} +\index{Whewell}% + +\BookRef{12}\textsc{Zeuthen, H.~G\@.} \Ger{\textit{Die Lehre von den Kegelschnitten im Alterthum.} +Kopenhagen, 1886.} +\index{Zeuthen!ref.\ to}% +%% -----File: 010.png---Folio X------- + +\BookRef{13}*\textsc{Chasles, M\@.} \Ger{\textit{Geschichte der Geometrie.} Aus dem Fran\-zösi\-schen +übertragen durch \textsc{Dr.~L.~A. Sohncke}. Halle, 1839.} +\index{Chasles}% + +\BookRef{14}\textsc{Marie, Maximilien.} \Fre{\textit{Histoire des Sciences Mathématiques et Physiques.} +Tome~I.--XII\@. Paris, 1883--1888.} +\index{Marie, M.}% + +\BookRef{15}\textsc{Comte, A\@.} \textit{Philosophy of Mathematics}, translated by \textsc{W.~M. Gillespie}. +\index{Comte}% + +\BookRef{16}\textsc{Hankel, Hermann.} \Ger{\textit{Die Entwickelung der Mathematik in den letzten +Jahrhunderten.} Tübingen, 1884.} +\index{Hankel!ref.\ to}% + +\BookRef{17}\textsc{Günther, Siegmund} und \textsc{Windelband, W\@.} \Ger{\textit{Geschichte der antiken +Naturwissenschaft und Philosophie.} Nördlingen, 1888.} +\index{Gunther@{Günther, S.}}% + +\BookRef{18}\textsc{Arneth, A\@.} \Ger{\textit{Geschichte der reinen Mathematik.} Stuttgart, 1852.} +\index{Arneth,}% + +\BookRef{19}\textsc{Cantor, Moritz.} \Ger{\textit{Mathematische Beiträge zum Kulturleben der +Völker.} Halle, 1863.} +\index{Cantor, M.}% + +\BookRef{20}\textsc{Matthiessen, Ludwig.} \Ger{\textit{Grundzüge der Antiken und Modernen +Algebra der Litteralen Gleichungen.} Leipzig, 1878.} +\index{Matthiessen}% + +\BookRef{21}\textsc{Ohrtmann} und \textsc{Müller}. \Ger{\textit{Fortschritte der Mathematik.}} +\index{Muller@{Müller}}% +\index{Ohrtmann}% + +\BookRef{22}\textsc{Peacock, George.} Article ``Arithmetic,'' in \textit{The Encyclopædia of +Pure Mathematics}. London, 1847. + +\BookRef{23}\textsc{Herschel, J.~F.~W\@.} Article ``Mathematics,'' in \textit{Edinburgh Encyclopædia}. +\index{Herschel, J.~F.~W.!ref.\ to}% + +\BookRef{24}\textsc{Suter, Heinrich.} \Ger{\textit{Geschichte der Mathematischen Wissenschaften.} +Zürich, 1873--75.} +\index{Suter}% + +\BookRef{25}\textsc{Quetelet, A\@.} \Fre{\textit{Sciences Mathématiques et Physiques chez les Belges.} +Bruxelles, 1866.} +\index{Quetelet!ref.\ to}% + +\BookRef{26}\textsc{Playfair, John.} Article ``Progress of the Mathematical and Physical +Sciences,'' in \textit{Encyclopædia Britannica}, 7th~edition, continued +in the 8th~edition by \textsc{Sir John Leslie}. +\index{Leslie}% +\index{Playfair}% + +\BookRef{27}\textsc{De Morgan, A\@.} \textit{Arithmetical Books from the Invention of Printing +to the Present Time.} + +\BookRef{28}\textsc{Napier, Mark.} \textit{Memoirs of John Napier of Merchiston.} Edinburgh, +1834. +\index{Napier, M.}% + +\BookRef{29}\textsc{Halsted, G.~B\@.} ``Note on the First English Euclid,'' \textit{American +Journal of Mathematics}, Vol.~II., 1879. +\index{Halsted}% + +\BookRef{30}\textsc{Madame Perier.} \textit{The Life of Mr.~Paschal.} Translated into +English by W.~A., London, 1744. +\index{Perier, Madame}% + +\BookRef{31}\textsc{Montucla, J.~F\@.} \Fre{\textit{Histoire des Mathématiques.} Paris, 1802.} +\index{Montucla}% + +\BookRef{32}\textsc{Dühring E\@.} \Ger{\textit{Kritische Geschichte der allgemeinen Principien der +Mechanik.} Leipzig, 1887.} +\index{Duhring@{Dühring, E.}}% + +\BookRef{33}\textsc{Brewster, D\@.} \textit{The Memoirs of Newton.} Edinburgh, 1860. + +\BookRef{34}\textsc{Ball, W.~W.~R\@.} \textit{A Short Account of the History of Mathematics.} +London, 1888, 2nd~edition, 1893. +\index{Ball, W.~W.~R.}% + +\BookRef{35}\textsc{De Morgan, A\@.} ``On the Early History of Infinitesimals,'' in the +\textit{Philosophical Magazine}, November, 1852. +\index{Demorgan@{De Morgan}!ref.\ to}% +%% -----File: 011.png---Folio XI------- + +\BookRef{36}\Ger{\textit{Bibliotheca Mathematica}, herausgegeben von \textsc{Gustaf Eneström}}, +Stockholm. +\index{Eneström}% + +\BookRef{37}\textsc{Günther, Siegmund.} \Ger{\textit{Vermischte Untersuchungen zur Geschichte +der mathematischen Wissenschaften.} Leipzig, 1876.}\index{Gunther@{Günther, S.}}% +%[** PP: Prev. line exactly full; placing \index{} on next line adds vspace.] + +\BookRef{38}*\textsc{Gerhardt, C.~I\@.} \Ger{\textit{Geschichte der Mathematik in Deutschland.} +München, 1877.} +\index{Gerhardt}% + +\BookRef{39}\textsc{Gerhardt, C.~I\@.} \Ger{\textit{Entdeckung der Differenzialrechnung durch Leibniz.} +Halle, 1848.} + +%[** PP: Same Gerhardt, but retaining legitimate variant from original.] +\BookRef{40}\textsc{Gerhardt, K.~I\@.} ``Leibniz in London,'' in \Ger{\textit{Sitzungsberichte der +Königlich Preussischen Academie der Wissenschaften zu Berlin}, +Februar, 1891.} + +\BookRef{41}\textsc{De Morgan, A\@.} Articles ``Fluxions'' and ``Commercium Epistolicum,'' +in the \textit{Penny Cyclopædia}. +\index{Demorgan@{De Morgan}!ref.\ to}% + +\BookRef{42}*\textsc{Todhunter, I\@.} \textit{A History of the Mathematical Theory of Probability +from the Time of Pascal to that of Laplace.} Cambridge and +London, 1865. +\index{Todhunter!ref.\ to}% + +\BookRef{43}*\textsc{Todhunter, I\@.} \textit{A History of the Theory of Elasticity and of the +Strength of Materials.} Edited and completed by \textsc{Karl Pearson}. +Cambridge, 1886. + +\BookRef{44}\textsc{Todhunter, I\@.} ``Note on the History of Certain Formulæ in Spherical +Trigonometry,'' \textit{Philosophical Magazine}, February, 1873. + +\BookRef{45}\Ger{\textit{Die Basler Mathematiker, Daniel Bernoulli und Leonhard Euler.} +Basel, 1884.} + +\BookRef{46}\textsc{Reiff, R\@.} \Ger{\textit{Geschichte der Unendlichen Reihen.} Tübingen, 1889.} +\index{Reiff}% + +\BookRef{47}\textsc{Waltershausen, W.~Sartorius.} \Ger{\textit{Gauss, zum Gedächtniss.} Leipzig, +1856.} +\index{Waltershausen}% + +\BookRef{48}\textsc{Baumgart, Oswald.} \Ger{\textit{Ueber das Quadratische Reciprocitätsgesetz.} +Leipzig, 1885.} +\index{Baumgart}% + +\BookRef{49}\textsc{Hathaway, A.~S\@.} ``Early History of the Potential,'' \textit{Bulletin of +the N.~Y. Mathematical Society}, I.~3. +\index{Hathaway}% + +\BookRef{50}\textsc{Wolf, Rudolf.} \Ger{\textit{Geschichte der Astronomie.} München, 1877.} +\index{Wolf, R.}% + +\BookRef{51}\textsc{Arago, D.~F.~J\@.} ``Eulogy on Laplace.'' Translated by \textsc{B.~Powell}, +\textit{Smithsonian Report}, 1874. +\index{Arago}% + +\BookRef{52}\textsc{Beaumont, M.~Élie~De.} ``Memoir of Legendre.'' Translated by +\textsc{C.~A. Alexander}, \textit{Smithsonian Report}, 1867. +\index{Beaumont}% + +\BookRef{53}\textsc{Arago, D.~F.~J\@.} ``Joseph Fourier.'' \textit{Smithsonian Report}, +1871. + +\BookRef{54}\textsc{Wiener, Christian.} \Ger{\textit{Lehrbuch der Darstellenden Geometrie.} Leipzig, +1884.} +\index{Wiener}% + +\BookRef{55}*\textsc{Loria, Gino.} \Ger{\textit{Die Hauptsächlichsten Theorien der Geometrie in +ihrer früheren und heutigen Entwickelung}, ins deutsche über\-tragen +von \textsc{Fritz Schütte}. Leipzig, 1888.} +\index{Loria}% +%% -----File: 012.png---Folio XII------- + +\BookRef{56}\textsc{Cayley, Arthur.} Inaugural Address before the British Association, +1883. +\index{Cayley}% + +\BookRef{57}\textsc{Spottiswoode, William.} Inaugural Address before the British +Association, 1878. +\index{Spottiswoode!ref.\ to}% + +\BookRef{58}\textsc{Gibbs, J.~Willard.} ``Multiple Algebra,'' \textit{Proceedings of the +American Association for the Advancement of Science}, 1886. +\index{Gibbs!ref.\ to}% + +\BookRef{59}\textsc{Fink, Karl.} \Ger{\textit{Geschichte der Elementar-Mathematik.} Tübingen, +1890.} +\index{Fink}% + +\BookRef{60}\textsc{Wittstein, Armin.} \Ger{\textit{Zur Geschichte des Malfatti'schen Problems.} +Nördlingen, 1878.} +\index{Wittstein}% + +\BookRef{61}\textsc{Klein, Felix.} \Ger{\textit{Vergleichende Betrachtungen über neuere geometrische +Forschungen.} Erlangen, 1872.} + +\BookRef{62}\textsc{Forsyth, A.~R\@.} \textit{Theory of Functions of a Complex Variable.} +Cambridge, 1893. +\index{Forsyth}% + +\BookRef{63}\textsc{Graham, R.~H\@.} \textit{Geometry of Position.} London, 1891. +\index{Graham}% + +\BookRef{64}\textsc{Schmidt, Franz.} \Ger{``Aus dem Leben zweier ungarischer Mathematiker +Johann und Wolfgang Bolyai von Bolya.'' \textit{Grunert's +Archiv}, 48:2, 1868.} +\index{Schmidt}% + +\BookRef{65}\textsc{Favaro, Anton.} \Ger{``Justus Bellavitis,'' \textit{Zeitschrift für Mathematik +und Physik}, 26:5, 1881.} +\index{Favaro}% + +\BookRef{66}\textsc{Dronke, Ad.} \textit{Julius Plücker.} Bonn, 1871. +\index{Dronke}% + +\BookRef{67}\textsc{Bauer, Gustav.} \Ger{\textit{Gedächtnissrede auf Otto Hesse.} München, +1882.} +\index{Bauer}% + +\BookRef{68}\textsc{Alfred Clebsch.} \Ger{\textit{Versuch einer Darlegung und Würdigung seiner +wissenschaftlichen Leistungen von einigen seiner Freunde.} Leipzig, 1873.} +\index{Clebsch!ref.\ to}% + +\BookRef{69}\textsc{Haas, August.} \Ger{\textit{Versuch einer Darstellung der Geschichte des +Krümmungsmasses.} Tübingen, 1881.} +\index{Haas}% + +\BookRef{70}\textsc{Fine, Henry~B\@.} \textit{The Number-System of Algebra.} Boston and +New York, 1890. +\index{Fine}% + +\BookRef{71}\textsc{Schlegel, Victor.} \Ger{\textit{Hermann Grassmann, sein Leben und seine +Werke.} Leipzig, 1878.} + +\BookRef{72}\textsc{Zahn, W.~v.} \Ger{``Einige Worte zum Andenken an Hermann Hankel,'' +\textit{Mathematische Annalen}, VII.~4, 1874.} +\index{Zahn}% + +\BookRef{73}\textsc{Muir, Thomas.} \textit{A Treatise on Determinants.} 1882. +\index{Muir}% + +\BookRef{74}\textsc{Salmon, George.} ``Arthur Cayley,'' \textit{Nature}, 28:21, September, +1883. +\index{Salmon}% + +\BookRef{75}\textsc{Cayley, A\@.} ``James Joseph Sylvester,'' \textit{Nature}, 39:10, January, +1889. + +\BookRef{76}\textsc{Burkhardt, Heinrich.} \Ger{``Die Anfänge der Gruppentheorie und +Paolo Ruffini,'' \textit{Zeitschrift für Mathematik und Physik}, Supplement, +1892.} +\index{Burkhardt, H.}% +%% -----File: 013.png---Folio XIII------- + +\BookRef{77}\textsc{Sylvester, J.~J\@.} \textit{Inaugural Presidential Address to the Mathematical +and Physical Section of the British Association at Exeter.} 1869. +\index{Sylvester!ref.\ to}% + +\BookRef{78}\textsc{Valson, C.~A\@.} \Fre{\textit{La Vie et les travaux du Baron Cauchy.} Tome~I., +II., Paris, 1868.} +\index{Valson}% + +\BookRef{79}\textsc{Sachse, Arnold.} \Ger{\textit{Versuch einer Geschichte der Darstellung willkürlicher +Funktionen einer variablen durch trigonometrische +Reihen.} Göttingen, 1879.} +\index{Sachse}% + +\BookRef{80}\textsc{Bois-Reymond, Paul du.} \Ger{\textit{Zur Geschichte der Trigonometrischen +Reihen, Eine Entgegnung.} Tübingen.} +\index{Bois-Reymond, P.~du}% + +\BookRef{81}\textsc{Poincaré, Henri.} \Fre{\textit{Notice sur les Travaux Scientifiques de Henri +Poincaré.} Paris, 1886.} +\index{Poincaré!ref.\ to}% + +\BookRef{82}\textsc{Bjerknes, C.~A\@.} \Fre{\textit{Niels-Henrik Abel, Tableau de sa vie et de son +action scientifique.} Paris, 1885.} +\index{Bjerknes, C. A.}% + +\BookRef{83}\textsc{Tucker, R\@.} ``Carl Friedrich Gauss,'' \textit{Nature}, April, 1877. +\index{Tucker}% + +\BookRef{84}\textsc{Dirichlet, Lejeune.} \Ger{\textit{Gedächtnissrede auf Carl Gustav Jacob +Jacobi.} 1852.} +\index{Dirichlet!ref.\ to}% + +\BookRef{85}\textsc{Enneper, Alfred.} \Ger{\textit{Elliptische Funktionen. Theorie und Geschichte.} +Halle~a/S., 1876.} +\index{Enneper!ref.\ to}% + +\BookRef{86}\textsc{Henrici, O\@.} ``Theory of Functions,'' \textit{Nature}, 43:14~and~15, 1891. +\index{Henrici}% + +\BookRef{87}\textsc{Darboux, Gaston.} \Fre{\textit{Notice sur les Travaux Scientifiques de M. Gaston +Darboux.} Paris, 1884.} +\index{Darboux}% + +\BookRef{88}\textsc{Kummer, E.~E\@.} \Ger{\textit{Gedächtnissrede auf Gustav Peter Lejeune-Diri\-chlet.} +Berlin, 1860.} +\index{Kummer!ref.\ to}% + +\BookRef{89}\textsc{Smith, H.~J. Stephen.} ``On the Present State and Prospects of +Some Branches of Pure Mathematics,'' \textit{Proceedings of the London +Mathematical Society}, Vol.~VIII., Nos.\ 104,~105, 1876. +\index{Smith, H.!ref.\ to}% + +\BookRef{90}\textsc{Glaisher, J.~W.~L\@.} ``Henry John Stephen Smith,'' \textit{Monthly Notices +of the Royal Astronomical Society}, XLIV.,~4, 1884. + +\BookRef{91}\Ger{\textit{Bessel als Bremer Handlungslehrling.} Bremen, 1890.} + +\BookRef{92}\textsc{Frantz, J\@.} \Ger{\textit{Festrede aus Veranlassung von Bessel's hundert\-jähr\-igem +Geburtstag.} Königsberg, 1884.} +\index{Frantz}% + +\BookRef{93}\textsc{Dziobek, O\@.} \textit{Mathematical Theories of Planetary Motions.} +Translated into English by M.~W. Harrington and W.~J. Hussey.\index{Dziobek}% +%[** PP: Prev. line exactly full; placing \index{} on next line adds vspace.] + +\BookRef{94}\textsc{Hermite, Ch.} \Fre{``Discours prononcé devant le président de la République,'' +\textit{Bulletin des Sciences Mathématiques}, XIV., Janvier, +1890.} +\index{Hermite!ref.\ to}% + +\BookRef{95}\textsc{Schuster, Arthur.} ``The Influence of Mathematics on the Progress +of Physics,'' \textit{Nature}, 25:17, 1882. +\index{Schuster}% + +\BookRef{96}\textsc{Kerbedz, E.~de.} \Ita{``Sophie de Kowalevski,'' \textit{Rendiconti del Circolo +Matematico di Palermo}, V., 1891.} +\index{Kerbedz}% + +\BookRef{97}\textsc{Voigt, W\@.} \Ger{\textit{Zum Gedächtniss von G.~Kirchhoff.} Göttingen, 1888.} +\index{Voigt}% +%% -----File: 014.png---Folio XIV------- + +\BookRef{98}\textsc{Bôcher, Maxime.} ``A Bit of Mathematical History,'' \textit{Bulletin of +the N.~Y. Math.\ Soc.}, Vol.~II., No.~5. +\index{Bocher@{Bôcher}}% + +\BookRef{99}\textsc{Cayley, Arthur.} \textit{Report on the Recent Progress of Theoretical +Dynamics.} 1857. +\index{Cayley}% + +\BookRef{100}\textsc{Glazebrook, R.~T\@.} \textit{Report on Optical Theories.} 1885. +\index{Glazebrook!ref.\ to}% + +\BookRef{101}\textsc{Rosenberger, F\@.} \Ger{\textit{Geschichte der Physik.} Braunschweig, 1887--1890.} +\index{Rosenberger}% + +\normalsize + +%% -----File: 015.png---Folio 1------- +%% Index cross-references +\index{Alembert, D'|see{D'Alembert}}% +\index{Algebra|see{Notation}}% +\index{Applied mathematics|see{Astronomy, Mechanics}}% +\index{Arithmetic|see{Numbers, Notation}}% +\index{Assumption, tentative|see{Regula falsa}}% +\index{Astronomy|see{Mechanics}}% +\index{Attraction|see{Gravitation, Ellipsoid}}% +\index{Bachet de Méziriac|see{Méziriac}}% +\index{Baune, De|see{De Baune}}% +\index{Byrgius|see{Bürgi}}% +\index{Calculus|see{Differential Calculus}}% +\index{Ceulen, van|see{Ludolph}}% +\index{Complex quantities|see{Imaginaries}}% +\index{Conic sections|see{Geometry}}% +\index{Cube, duplication of|see{Duplication of the cube}}% +\index{Cubic equations|see{Algebra}}% +\index{Curves|see{Cubic curves, Rectification, Geometry, Conic sections}}% +\index{Delian problem|see{Duplication of the cube}}% +\index{Differences, finite|see{Finite differences}}% +\index{Differential calculus|see{Bernoullis, Euler, Lagrange, Laplace, etc}}% +\index{Dyck|see{Groups}}% +\index{Elements (Euclid's)|see{Euclid}}% +\index{Elimination|see{Equations}}% +\index{Elliptic geometry|see{Non-Euclidean geometry}}% +\index{Equations|see{Cubic equations, Algebra, Theory of numbers}}% +\index{Errors, theory of|see{Least squares}}% +\index{Euclidean space|see{Non-Euclidean geometry}}% +\index{Fibonacci|see{Leonardo of Pisa}}% +\index{Force-function|see{Potential}}% +\index{Fractions|see{Arithmetic}}% +\index{Functions|see{Elliptic functions, Abelian functions, Hyperelliptic functions, Theta functions, Beta function, Gamma function, Omega function, Sigma function, Bessel's function, Potential}}% +\index{Geber|see{Gabir ben Aflah}}% +\index{Geometry|see{Curves, Surfaces, Curvature, Quadrature, Rectification, Circle}}% +\index{Guldinus|see{Guldin}}% +\index{Holder@{Hölder, O.}|see{Groups}}% +\index{Hydrodynamics|see{Mechanics}}% +\index{Hydrostatics|see{Mechanics}}% +\index{Hyperbolic geometry|see{Non-Euclidean geometry}}% +\index{Incommensurables|see{Irrationals}}% +\index{Indeterminate analysis|see{Theory of numbers}}% +\index{Indeterminate equations|see{Theory of numbers}}% +\index{Indian mathematics|see{Hindoos}}% +\index{Indian numerals|see{Arabic numerals}}% +\index{Indices|see{Exponents}}% +\index{Infinitesimal calculus|see{Differential calculus}}% +\index{Irrationals|see{Incommensurables}}% +\index{Isoperimetrical figures|see{Calculus of variations}}% +\index{Joachim|see{Rhæticus}}% +\index{Kauffmann|see{Mercator, N.}}% +\index{Kelvin, Lord|see{Thomson, W.}}% +\index{Lucas de Burgo|see{Pacioli}}% +\index{Mechanics|see{Dynamics, Hydrodynamics, Hydrostatics, Graphic statics, Laws of motion, Astronomy, D'Alembert's principle}}% +\index{Moon|see{Astronomy}}% +\index{Muller@{Müller, J.}|see{Regiomontanus}}% +\index{Negative quantities|see{Algebra}}% +\index{Negative roots|see{Algebra}}% +\index{Nicolo of Brescia|see{Tartaglia}}% +\index{Notation|see{Exponents, Algebra}}% +\index{Numerals|see{Apices}}% +\index{Parabola|see{Geometry}}% +\index{Parabolic geometry|see{Non-Euclidean geometry}}% +\index{Perspective|see{Geometry}}% +\index{Physics, mathematical|see{Applied mathematics}}% +\index{Plato Tiburtinus|see{Plato of Tivoli}}% +\index{Ptolemæus|see{Ptolemy}}% +\index{Quadratic equations|see{Algebra, Equations}}% +\index{Quadrature of the circle|see{Circle; also see Circle-squarers, $\pi$}}% +\index{Rectification of curves|see{Curves}}% +\index{Regula aurea|see{Falsa positio}}% +\index{Rheticus|see{Rhæticus}}% +\index{Sacro Bosco|see{Halifax}}% +\index{Series|see{Infinite series, Trigonometric series, Divergent series, Absolutely convergent series, Semi-convergent series, Fourier's series, Uniformly convergent series}}% +\index{Sound, velocity of|see{Acoustics}}% +\index{Squaring the circle|see{Quadrature of the circle}}% +\index{Statics|see{Mechanics}}% +\index{Staudt, von|see{Von Staudt}}% +\index{Stevinus|see{Stevin}}% +\index{Strutt, J.~W.|see{Rayleigh}}% +\index{Tentative assumption|see{Regula falsa}}% +\index{Theory of equations|see{Equations}}% +\index{Theory of functions|see{Functions}}% +\index{Thomson, Sir William|see{Kelvin (Lord)}}% +\index{Trigonometric series|see{Fourier's series}}% +\index{Van Ceulen|see{Ludolph}}% +\index{Von Helmholtz|see{Helmholtz}}% +\index{Wave theory|see{Undulatory theory}}% + +\clearpage +\fancyhf{} +\cleardoublepage +\phantomsection +\pdfbookmark[-1]{Main Matter}{Main Matter} +\mainmatter + +\IntroChapter{A History of Mathematics}{Introduction} +\index{History of mathematics, its value|(}% +\pagestyle{fancy} +\pagenumbering{arabic} + +\First{The} contemplation of the various steps by which mankind +has come into possession of the vast stock of mathematical +knowledge can hardly fail to interest the mathematician. He +takes pride in the fact that his science, more than any other, +is an \emph{exact} science, and that hardly anything ever done in +mathematics has proved to be useless. The chemist smiles +at the childish efforts of alchemists, but the mathematician +finds the geometry of the Greeks and the arithmetic of the +Hindoos as useful and admirable as any research of to-day. +He is pleased to notice that though, in course of its development, +mathematics has had periods of slow growth, yet in +the main it has been pre-eminently a \emph{progressive} science. + +The history of mathematics may be instructive as well as +agreeable; it may not only remind us of what we have, but +may also teach us how to increase our store. Says De Morgan, +\index{Demorgan@{De Morgan}!ref.\ to}% +``The early history of the mind of men with regard to mathematics +leads us to point out our own errors; and in this +respect it is well to pay attention to the history of mathematics.'' +It warns us against hasty conclusions; it points out +the importance of a good notation upon the progress of the +science; it discourages excessive specialisation on the part of +%% -----File: 016.png---Folio 2------- +\index{p@{$\pi$}: values for!proved to be transcendental}% +investigators, by showing how apparently distinct branches +have been found to possess unexpected connecting links; it +saves the student from wasting time and energy upon problems +which were, perhaps, solved long since; it discourages +him from attacking an unsolved problem by the same method +which has led other mathematicians to failure; it teaches that +fortifications can be taken in other ways than by direct attack, +that when repulsed from a direct assault it is well to reconnoitre +and occupy the surrounding ground and to discover the +secret paths by which the apparently unconquerable position +can be taken.\Reference{1} The importance of this strategic rule may +be emphasised by citing a case in which it has been violated. +An untold amount of intellectual energy has been expended +on the quadrature of the circle, yet no conquest has been made +by direct assault. The circle-squarers have existed in crowds +\index{Circle-squarers}% +ever since the period of Archimedes. After innumerable failures +\index{Archimedes!ref.\ to}% +to solve the problem at a time, even, when investigators +possessed that most powerful tool, the differential calculus, +persons versed in mathematics dropped the subject, while +those who still persisted were completely ignorant of its history +and generally misunderstood the conditions of the problem. +``Our problem,'' says De Morgan, ``is to square the +\index{Demorgan@{De Morgan}!ref.\ to}% +circle with the \emph{old allowance of means}: Euclid's postulates +and nothing more. We cannot remember an instance in which +a question to be solved by a \emph{definite method} was tried by the +best heads, and answered at last, \emph{by that method}, after thousands +of complete failures.'' But progress was made on this +problem by approaching it from a different direction and by +newly discovered paths. Lambert proved in 1761 that the +\index{Lambert!ref.\ to}% +ratio of the circumference of a circle to its diameter is incommensurable. +Some years ago, Lindemann demonstrated that +\index{Lindemann!ref.\ to}% +this ratio is also transcendental and that the quadrature of +the circle, by means of the ruler and compass only, is \emph{impossible}. +%% -----File: 017.png---Folio 3------- +\index{Arabic numerals and notation}% +He thus showed by actual proof that which keen-minded +mathematicians had long suspected; namely, that the +great army of circle-squarers have, for two thousand years, +been assaulting a fortification which is as indestructible as +the firmament of heaven. + +Another reason for the desirability of historical study is +the value of historical knowledge to the teacher of mathematics. +The interest which pupils take in their studies may +be greatly increased if the solution of problems and the cold +logic of geometrical demonstrations are interspersed with +historical remarks and anecdotes. A class in arithmetic will +be pleased to hear about the Hindoos and their invention of +\index{Hindoos!ref.\ to}% +the ``Arabic notation''; they will marvel at the thousands +\index{Notation!Arabic notation}% +of years which elapsed before people had even thought of +introducing into the numeral notation that Columbus-egg---the +zero; they will find it astounding that it should have +taken so long to \emph{invent} a notation which they themselves can +now \emph{learn} in a month. After the pupils have learned how to +bisect a given angle, surprise them by telling of the many +futile attempts which have been made to solve, by elementary +geometry, the apparently very simple problem of the trisection +of an angle. When they know how to construct a square +whose area is double the area of a given square, tell them +about the duplication of the cube---how the wrath of Apollo +could be appeased only by the construction of a cubical altar +double the given altar, and how mathematicians long wrestled +with this problem. After the class have exhausted their energies +on the theorem of the right triangle, tell them the legend +about its discoverer---how Pythagoras, jubilant over his great +\index{Pythagoras!ref.\ to}% +accomplishment, sacrificed a hecatomb to the Muses who inspired +him. When the value of mathematical training is +called in question, quote the inscription over the entrance into +the academy of Plato, the philosopher: ``Let no one who is +\index{Plato!ref.\ to}% +%% -----File: 018.png---Folio 4------- +unacquainted with geometry enter here.'' Students in analytical +geometry should know something of Descartes, and, after +\index{Descartes!ref.\ to}% +taking up the differential and integral calculus, they should +become familiar with the parts that Newton, Leibniz, and +\index{Leibniz!ref.\ to}% +\index{Newton!ref.\ to}% +Lagrange played in creating that science. In his historical +\index{Lagrange!ref.\ to}% +talk it is possible for the teacher to make it plain to the +student that mathematics is not a dead science, but a living +one in which steady progress is made.\Reference{2} + +The history of mathematics is important also as a valuable +contribution to the history of civilisation. Human progress +is closely identified with scientific thought. Mathematical +and physical researches are a reliable record of intellectual +progress. The history of mathematics is one of the large +\index{History of mathematics, its value|)}% +windows through which the philosophic eye looks into past +ages and traces the line of intellectual development. +%% -----File: 019.png---Folio 5------- +\index{Numerals!Babylonian|(}% + + +\Chapter{Antiquity} + +\Section{The Babylonians} +\index{Babylonians|(}% + +\First{The} fertile valley of the Euphrates and Tigris was one of +the primeval seats of human society. Authentic history of +the peoples inhabiting this region begins only with the foundation, +in Chaldæa and Babylonia, of a united kingdom out +of the previously disunited tribes. Much light has been +thrown on their history by the discovery of the art of reading +the \emph{cuneiform} or wedge-shaped system of writing. + +In the study of Babylonian mathematics we begin with the +notation of numbers. A vertical wedge~\Glyph{cune1} stood for~$1$, while +\index{Notation!Babylonian numbers|(}% +the characters \Glyph{cune10}~and~\Glyph{cune100} signified $10$ and $100$ respectively. +Grotefend believes the character for~$10$ originally to +have been the picture of two hands, as held in prayer, the +palms being pressed together, the fingers close to each other, +but the thumbs thrust out. In the Babylonian notation two +principles were employed---the additive and multiplicative. +Numbers below 100 were expressed by symbols whose +respective values had to be \emph{added}. Thus, \Glyph{cune2}~stood for~$2$, +\Glyph{cune3}~for~$3$, \Glyph{cune4}~for~$4$, \Glyph{cune23}~for~$23$, \Glyph{cune30}~for~$30$. Here the +symbols of higher order appear always to the left of those of +lower order. In writing the hundreds, on the other hand, a +\emph{smaller} symbol was placed to the left of the~$100$, and was, in +that case, to be \emph{multiplied} by~$100$. Thus, \Glyph{cune1000}~signified +%% -----File: 020.png---Folio 6------- +$10$~times~$100$, or~$1000$. But this symbol for~$1000$ was itself +taken for a new unit, which could take smaller coefficients to +its left. Thus, \Glyph{cune2000} denoted, not $20$~times~$100$, but +$10$~times~$1000$. Of the largest numbers written in cuneiform +symbols, which have hitherto been found, none go as high as +a million.\Reference{3} + +If, as is believed by most specialists, the early Sumerians +were the inventors of the cuneiform writing, then they were, +in all probability, also familiar with the notation of numbers. +Most surprising, in this connection, is the fact that Sumerian +inscriptions disclose the use, not only of the above \emph{decimal} +system, but also of a \emph{sexagesimal} one. The latter was used +chiefly in constructing tables for weights and measures. It is +full of historical interest. Its consequential development, +both for integers and fractions, reveals a high degree of +mathematical insight. We possess two Babylonian tablets +which exhibit its use. One of them, probably written between +$2300$ and $1600$~\BC, contains a table of square numbers up to~$60^{2}$. +The numbers $1$,~$4$,~$9$, $16$,~$25$, $36$,~$49$, are given as the +squares of the first seven integers respectively. We have next +$1.4 = 8^{2}$, $1.21 = 9^{2}$, $1.40 = 10^{2}$, $2.1 = 11^{2}$, etc. This remains +unintelligible, unless we assume the sexagesimal scale, which +makes $1.4 = 60 + 4$, $1.21 = 60 + 21$, $2.1 = 2.60 + 1$. The second +tablet records the magnitude of the illuminated portion of the +moon's disc for every day from new to full moon, the whole disc +being assumed to consist of $240$~parts. The illuminated parts +during the first five days are the series $5$,~$10$, $20$,~$40$, $1.20 +(=80)$, which is a geometrical progression. From here on +the series becomes an arithmetical progression, the numbers +from the fifth to the fifteenth day being respectively $1.20$, $1.36$, +$1.52$, $\DPtypo{2.8}{1.8}$, $2.24$, $2.40$, $2.56$, $3.12$, $3.28$, $3.44$, $4$. This table not +only exhibits the use of the sexagesimal system, but also indicates +the acquaintance of the Babylonians with progressions. +%% -----File: 021.png---Folio 7------- +\index{Numerals!Babylonian|)}% +\index{Ptolemy!ref.\ to}% +Not to be overlooked is the fact that in the sexagesimal notation +of integers the ``principle of position'' was employed. +Thus, in $1.4$~($=64$), the $1$~is made to stand for~$60$, the unit +of the second order, by virtue of its position with respect to +the~$4$. The introduction of this principle at so early a date +is the more remarkable, because in the decimal notation it +was not introduced till about the fifth or sixth century after +Christ. The principle of position, in its general and systematic +application, requires a symbol for zero. We ask, Did +\index{Zero!(symbol for)}% +the Babylonians possess one? Had they already taken the +gigantic step of representing by a symbol the \emph{absence} of +units? Neither of the above tables answers this question, +for they happen to contain no number in which there was +occasion to use a zero. The sexagesimal system was used also +\index{Sexagesimal system}% +in fractions. Thus, in the Babylonian inscriptions, $\frac{1}{2}$ and~$\frac{1}{3}$ +\index{Fractions!Babylonian}% +\index{Fractions!sexagesimal}% +are designated by $30$ and~$20$, the reader being expected, in +his mind, to supply the word ``sixtieths.'' The Greek geometer +Hypsicles and the Alexandrian astronomer Ptolemæus +\index{Hypsicles!ref.\ to}% +borrowed the sexagesimal notation of fractions from the +\index{Notation!Babylonian numbers|)}% +Babylonians and introduced it into Greece. From that time +sexagesimal fractions held almost full sway in astronomical +and mathematical calculations until the sixteenth century, +when they finally yielded their place to the decimal fractions. +It may be asked, What led to the invention of the sexagesimal +system? Why was it that $60$~parts were selected? To +this we have no positive answer. \emph{Ten} was chosen, in the +decimal system, because it represents the number of fingers. +But nothing of the human body could have suggested~$60$. +Cantor offers the following theory: At first the Babylonians +reckoned the year at $360$~days. This led to the division of +\index{Division of the circle}% +the circle into $360$~degrees, each degree representing the daily +\index{Circle!degrees of}% +amount of the supposed yearly revolution of the sun around +the earth. Now they were, very probably, familiar with the +%% -----File: 022.png---Folio 8------- +\index{Musical proportion}% +\index{p@{$\pi$}: values for!Babylonian and Hebrew}% +fact that the radius can be applied to its circumference as a +chord $6$~times, and that each of these chords subtends an arc +measuring exactly $60$~degrees. Fixing their attention upon +these degrees, the division into $60$~parts may have suggested +itself to them. Thus, when greater precision necessitated a +subdivision of the degree, it was partitioned into $60$~minutes. +In this way the sexagesimal notation may have originated. +The division of the day into $24$~hours, and of the hour +into minutes and seconds on the scale of~$60$, is due to the +Babylonians. + +It appears that the people in the Tigro-Euphrates basin had +made very creditable advance in arithmetic. Their knowledge +of arithmetical and geometrical progressions has already been +\index{Progressions, first appearance of arithmetical and geometrical}% +alluded to. Iamblichus attributes to them also a knowledge +of proportion, and even the invention of the so-called \emph{musical} +proportion. Though we possess no conclusive proof, we have +nevertheless reason to believe that in practical calculation +they used the \emph{abacus}. Among the races of middle Asia, even +\index{Abacus}% +as far as China, the abacus is as old as fable. Now, Babylon +was once a great commercial centre,---the metropolis of many +nations,---and it is, therefore, not unreasonable to suppose that +her merchants employed this most improved aid to calculation. + +In geometry the Babylonians accomplished almost nothing. +\index{Geometry!Babylonian}% +Besides the division of the circumference into $6$~parts by its +radius, and into $360$~degrees, they had some knowledge of +geometrical figures, such as the triangle and quadrangle, which +they used in their auguries. Like the Hebrews (1~Kin.~7:23), +they took $\pi = 3$. Of geometrical demonstrations there is, of +course, no trace. ``As a rule, in the Oriental mind the intuitive +powers eclipse the severely rational and logical.'' + +The astronomy of the Babylonians has attracted much +\index{Astronomy!Babylonian}% +attention. They worshipped the heavenly bodies from the +earliest historic times. When Alexander the Great, after +%% -----File: 023.png---Folio 9------- +\index{Babylonians|)}% +\index{Calendar}% +the battle of Arbela (331~\BC), took possession of Babylon, +Callisthenes found there on burned brick astronomical records +\index{Callisthenes}% +reaching back as far as 2234~\BC\ Porphyrius says that these +were sent to Aristotle. Ptolemy, the Alexandrian astronomer, +\index{Aristotle!ref.\ to}% +\index{Ptolemy!ref.\ to}% +possessed a Babylonian record of eclipses going back to +747~\BC. Recently Epping and Strassmaier\Reference[]{4} threw considerable +\index{Epping}% +light on Babylonian chronology and astronomy by explaining +two calendars of the years 123~\BC\ and 111~\BC, taken +from cuneiform tablets coming, presumably, from an old +observatory. These scholars have succeeded in giving an +account of the Babylonian calculation of the new and full +moon, and have identified by calculations the Babylonian +names of the planets, and of the twelve zodiacal signs and +twenty-eight normal stars which correspond to some extent +with the twenty-eight \textit{nakshatras} of the Hindoos. We append +part of an Assyrian astronomical report, as translated by +Oppert:--- +\medskip + +\footnotesize\noindent +``To the King, my lord, thy faithful servant, Mar-Istar.'' + +``\ldots On the first day, as the new moon's day of the month Thammuz +declined, the moon was again visible over the planet Mercury, as +I had already predicted to my master the King. I erred not.'' + +\normalsize + + +\Section{The Egyptians} +\index{Egyptians|(}% + +Though there is great difference of opinion regarding the +antiquity of Egyptian civilisation, yet all authorities agree in +the statement that, however far back they go, they find no +uncivilised state of society. ``Menes, the first king, changes +the course of the Nile, makes a great reservoir, and builds the +temple of Phthah at Memphis.'' The Egyptians built the +pyramids at a very early period. Surely a people engaging in +%% -----File: 024.png---Folio 10------- +\index{Rhind papyrus|(}% +enterprises of such magnitude must have known something of +mathematics---at least of practical mathematics. + +All Greek writers are unanimous in ascribing, without +envy, to Egypt the priority of invention in the mathematical +sciences. Plato in \textit{Phædrus} says: ``At the Egyptian city +\index{Plato!ref.\ to}% +of Naucratis there was a famous old god whose name was +Theuth; the bird which is called the Ibis was sacred to +him, and he was the inventor of many arts, such as arithmetic +and calculation and geometry and astronomy and draughts +\index{Astronomy!Egyptian}% +\index{Geometry!Egyptian|(}% +and dice, but his great discovery was the use of letters.'' + +Aristotle says that mathematics had its birth in Egypt, +because there the priestly class had the leisure needful for +the study of it. Geometry, in particular, is said by Herodotus, +Diodorus, Diogenes Laertius, Iamblichus, and other ancient +\index{Diodorus}% +\index{Iamblichus!ref.\ to}% +\index{Laertius}% +writers to have originated in Egypt.\Reference{5} In Herodotus we find +this (II.~c.~109): ``They said also that this king [Sesostris] +divided the land among all Egyptians so as to give each one a +quadrangle of equal size and to draw from each his revenues, +by imposing a tax to be levied yearly. But every one from +whose part the river tore away anything, had to go to him +and notify what had happened; he then sent the overseers, +who had to measure out by how much the land had become +smaller, in order that the owner might pay on what was left, +in proportion to the entire tax imposed. In this way, it +appears to me, geometry originated, which passed thence to +Hellas.'' + +We abstain from introducing additional Greek opinion +regarding Egyptian mathematics, or from indulging in wild +conjectures. We rest our account on documentary evidence. +A hieratic papyrus, included in the Rhind collection of the +British Museum, was deciphered by Eisenlohr in 1877, and +found to be a mathematical manual containing problems in +arithmetic and geometry. It was written by \textbf{Ahmes} some +\index{Ahmes|(}% +%% -----File: 025.png---Folio 11------- +time before 1700~\BC, and was founded on an older work +believed by Birch to date back as far as 3400~\BC! This +curious papyrus---the most ancient mathematical handbook +known to us---puts us at once in contact with the mathematical +thought in Egypt of three or five thousand years +ago. It is entitled ``Directions for obtaining the Knowledge +of all Dark Things.'' We see from it that the Egyptians +cared but little for theoretical results. Theorems are not +found in it at all. It contains ``hardly any general rules of +procedure, but chiefly mere statements of results intended +possibly to be explained by a teacher to his pupils.''\Reference{6} In +geometry the forte of the Egyptians lay in making constructions +and determining areas. The area of an isosceles +triangle, of which the sides measure $10$~\textit{ruths} and the base +$4$~\textit{ruths}, was erroneously given as $20$ square \DPtypo{ruths}{\textit{ruths}}, or half +the product of the base by one side. The area of an isosceles +trapezoid is found, similarly, by multiplying half the sum +of the parallel sides by one of the non-parallel sides. The +area of a circle is found by deducting from the diameter +$\frac{1}{9}$ of its length and squaring the remainder. Here $\pi$ is +\index{p@{$\pi$}: values for!Egyptian}% +taken $= (\frac{16}{9})^2 = 3.1604 \ldots$, a very fair approximation.\Reference{6} The +papyrus explains also such problems as these,---To mark +out in the field a right triangle whose sides are $10$ and $4$~units; +or a trapezoid whose parallel sides are $6$ and~$4$, and +the non-parallel sides each $20$~units. + +Some problems in this papyrus seem to imply a rudimentary +knowledge of proportion. + +The base-lines of the pyramids run north and south, and +east and west, but probably only the lines running north and +south were determined by astronomical observations. This, +coupled with the fact that the word \textit{harpedonaptæ}, applied to +Egyptian geometers, means ``rope-stretchers,'' would point to +the conclusion that the Egyptian, like the Indian and Chinese +%% -----File: 026.png---Folio 12------- +geometers, constructed a right triangle upon a given line, by +stretching around three pegs a rope consisting of three parts +in the ratios $3:4:5$, and thus forming a right triangle.\Reference{3} If +this explanation is correct, then the Egyptians were familiar, +2000~years~\BC, with the well-known property of the right +triangle, for the special case at least when the sides are +in the ratio $3:4:5$. + +On the walls of the celebrated temple of Horus at Edfu +\index{Edfu}% +have been found hieroglyphics, written about 100~\BC, which +enumerate the pieces of land owned by the priesthood, and +give their areas. The area of any quadrilateral, however +irregular, is there found by the formula $\dfrac{a+b\Fstr}{2} · \dfrac{c+d}{2}$. Thus, +for a quadrangle whose opposite sides are $5$~and~$8$, $20$~and~$15$, +%[** PP: Namely, $113\frac{3}{4}$; see note in text on 028.png below.] +is given the area $113 \frac{1}{2}\, \frac{1}{4}$.\Reference{7} The incorrect formulæ of +Ahmes of 3000~years~\BC\ yield generally closer approximations +than those of the Edfu inscriptions, written 200~years +after Euclid! + +The fact that the geometry of the Egyptians consists +chiefly of constructions, goes far to explain certain of its +great defects. The Egyptians failed in two essential points +without which a \emph{science} of geometry, in the true sense of +the word, cannot exist. In the first place, they failed to +construct a rigorously logical system of geometry, resting +upon a few axioms and postulates. A great many of their +rules, especially those in solid geometry, had probably not +been proved at all, but were known to be true merely from +observation or as matters of fact. The second great defect +was their inability to bring the numerous special cases under +a more general view, and thereby to arrive at broader and +more fundamental theorems. Some of the simplest geometrical +truths were divided into numberless special cases +of which each was supposed to require separate treatment. +%% -----File: 027.png---Folio 13------- +\index{Abacus}% +\index{Notation!Egyptian numbers}% + +Some particulars about Egyptian geometry can be mentioned +\index{Geometry!Egyptian|)}% +more advantageously in connection with the early +Greek mathematicians who came to the Egyptian priests for +instruction. + +%% [** PP: Notes on hieroglyphs: +%% The glyphs for 1, 10, 100, 10,000, 100,000, and 10,000,000 were +%% reproduced from the scanned image. The glyphs for 1000 and 1,000,000 +%% were unclear in the scanned image, and were re-created based on +%% images obtained from http://en.wikipedia.org/wiki/Egyptian_numerals +%% on December 30, 2009.] +An insight into Egyptian methods of numeration was obtained +\index{Numerals!Egyptian}% +through the ingenious deciphering of the hieroglyphics +by Champollion, Young, and their successors. The symbols +used were the following: \Glyph{hiero1}~for~$1$, \Glyph{hiero10}~for~$10$, \Glyph{hiero100}~for~$100$, +\Glyph{hierothou}~for~$1000$, \Glyph{hiero10thou}~for~$10,000$, \Glyph{hiero100thou}~for~$100,000$, \Glyph{hieromil}~for~$1,000,000$, +\Glyph{hiero10mil}\ for~$10,000,000$.\Reference{3} The symbol for~$1$ represents a vertical +staff; that for~$10,000$ a pointing finger; that for~$100,000$ a +burbot; that for~$1,000,000$, a man in astonishment. The +significance of the remaining symbols is very doubtful. +The writing of numbers with these hieroglyphics was very +cumbrous. The unit symbol of each order was repeated as +many times as there were units in that order. The principle +employed was the \emph{additive}. Thus, $23$~was written \Glyph{hiero10}\,\Glyph{hiero10}\Glyph{hiero1}\Glyph{hiero1}\Glyph{hiero1}. + +Besides the hieroglyphics, Egypt possesses the \emph{hieratic} and +\emph{demotic} writings, but for want of space we pass them by. + +Herodotus makes an important statement concerning the +mode of computing among the Egyptians. He says that +they ``\emph{calculate with pebbles} by moving the hand from right +to left, while the Hellenes move it from left to right.'' +Herein we recognise again that \emph{instrumental} method of +figuring so extensively used by peoples of antiquity. The +Egyptians used the decimal scale. Since, in figuring, they +moved their hands horizontally, it seems probable that they +used ciphering-boards with vertical columns. In each column +there must have been not more than nine pebbles, for ten +pebbles would be equal to one pebble in the column next +to the left. + +The \textit{Ahmes papyrus} contains interesting information on +the way in which the Egyptians employed fractions. Their +\index{Fractions!Egyptian}% +%% -----File: 028.png---Folio 14------- +methods of operation were, of course, radically different from +ours. Fractions were a subject of very great difficulty with +the ancients. Simultaneous changes in both numerator and +denominator were usually avoided. In manipulating fractions +the Babylonians kept the denominators ($60$) constant. The +Romans likewise kept them constant, but equal to~$12$. The +Egyptians and Greeks, on the other hand, kept the numerators +\Addendlabel{14}% +constant, and dealt with variable denominators. Ahmes used +the term ``fraction'' in a restricted sense, for he applied it +only to \emph{unit-fractions}, or fractions having unity for the numerator. +It was designated by writing the denominator and then +placing over it a dot. Fractional values which could not be +expressed by any one unit-fraction were expressed as the +\emph{sum} of two or more of them. Thus, he wrote $\frac{1}{3}\, \frac{1}{15}$ in place +of~$\frac{2}{5}$. The first important problem naturally arising was, how +to represent any fractional value as the sum of unit-fractions. +This was solved by aid of a table, given in the papyrus, in +which all fractions of the form $\dfrac{2\Fstr}{2n+1}$ (where $n$ designates +successively all the numbers up to~$49$) are reduced to the +sum of unit-fractions. Thus, $\frac{2}{7} = \frac{1}{4}\, \frac{1}{28}$; $\frac{2}{99} = \frac{1}{66}\, \frac{1}{198}$. When, +by whom, and how this table was calculated, we do not know. +Probably it was compiled empirically at different times, by +different persons. It will be seen that by repeated application +of this table, a fraction whose numerator exceeds two +can be expressed in the desired form, provided that there +is a fraction in the table having the same denominator that +\emph{it} has. Take, for example, the problem, to divide $5$ by~$21$. +In the first place, $5 = 1 + 2 + 2$. From the table we get +$\frac{2}{21} = \frac{1}{14}\, \frac{1}{42}$. Then $\frac{5}{21} = \frac{1}{21} + (\frac{1}{14}\, \frac{1}{42}) + (\frac{1}{14}\, \frac{1}{42}) = \frac{1}{21} + (\frac{2}{14}\, \frac{2}{42}) += \frac{1}{21}\, \frac{1}{7}\, \frac{1}{21} = \frac{1}{7}\, \frac{2}{21} = \frac{1}{7}\, \frac{1}{14}\, \frac{1}{42}$. The papyrus contains problems +in which it is required that fractions be raised by addition +or multiplication to given whole numbers or to other fractions. +%% -----File: 029.png---Folio 15------- +\index{Notation!in algebra}% +\index{Rhind papyrus|)}% +For example, it is required to increase $\frac{1}{4}\, \frac{1}{8}\, \frac{1}{10}\, \frac{1}{30}\, \frac{1}{45}$ to~$1$. +The common denominator taken appears to be~$45$, for the +numbers are stated as $11 \frac{1}{4}$, $5 \frac{1}{2}\, \frac{1}{8}$, $4 \frac{1}{2}$, $1 \frac{1}{2}$,~$1$. The sum of these +is $23 \frac{1}{2}\, \frac{1}{4}\, \frac{1}{8}$ forty-fifths. Add to this $\frac{1}{9}\, \frac{1}{40}$, and the sum is~$\frac{2}{3}$. +Add~$\frac{1}{3}$, and we have~$1$. Hence the quantity to be added to +the given fraction is $\frac{1}{3}\, \frac{1}{9}\, \frac{1}{40}$. + +Having finished the subject of fractions, Ahmes proceeds to +the solution of equations of one unknown quantity. The unknown +\index{Equations!solution of}% +quantity is called `hau' or heap. Thus the problem, +``heap, its~$\frac{1}{7}$, its whole, it makes~$19$,'' \ie~$\dfrac{x}{7} + x = 19$. In this +case, the solution is as follows: $\dfrac{8x}{7} = 19$; $\dfrac{x}{7} = 2 \frac{1}{4}\, \frac{1}{8}$; $x = 16 \frac{1}{2}\, \frac{1}{8}$. +But in other problems, the solutions are effected by various +other methods. It thus appears that the beginnings of algebra +\index{Algebra!Beginnings in Egypt}% +are as ancient as those of geometry. + +The principal defect of Egyptian arithmetic was the lack of +a simple, comprehensive symbolism---a defect which not even +the Greeks were able to remove. + +The Ahmes papyrus doubtless represents the most advanced +\index{Ahmes|)}% +attainments of the Egyptians in arithmetic and geometry. It +is remarkable that they should have reached so great proficiency +in mathematics at so remote a period of antiquity. But +strange, indeed, is the fact that, during the next two thousand +years, they should have made no progress whatsoever in it. +The conclusion forces itself upon us, that they resemble the +Chinese in the \emph{stationary character}, not only of their government, +but also of their learning. All the knowledge of geometry +which they possessed when Greek scholars visited them, +six centuries~\BC, was doubtless known to them two thousand +years earlier, when they built those stupendous and gigantic +structures---the pyramids. An explanation for this stagnation +of learning has been sought in the fact that their early discoveries +in mathematics and medicine had the misfortune of +%% -----File: 030.png---Folio 16------- +being entered upon their sacred books and that, in after ages, +it was considered heretical to augment or modify anything +therein. Thus the books themselves closed the gates to +progress. + +\Section{The Greeks} +\index{Greeks|(}% + +%[** PP: A unit with this heading is present in ToC, but not in the text] +\SpecialSection{Greek Geometry} + +About the seventh century~\BC\ an active commercial intercourse +sprang up between Greece and Egypt. Naturally there +arose an interchange of ideas as well as of merchandise. +Greeks, thirsting for knowledge, sought the Egyptian priests +for instruction. Thales, Pythagoras, \OE{}nopides, Plato, Democritus, +\index{Democritus!ref.\ to}% +\index{OEnopides@{\OE{}nopides}!ref.\ to}% +\index{Plato!ref.\ to}% +\index{Pythagoras!ref.\ to}% +\index{Thales!ref.\ to}% +Eudoxus, all visited the land of the pyramids. +\index{Eudoxus!ref.\ to}% +Egyptian ideas were thus transplanted across the sea and +there stimulated Greek thought, directed it into new lines, +and gave to it a basis to work upon. Greek culture, therefore, +is not primitive. Not only in mathematics, but also in mythology +and art, Hellas owes a debt to older countries. To +Egypt Greece is indebted, among other things, for its elementary +geometry. But this does not lessen our admiration for +the Greek mind. From the moment that Hellenic philosophers +applied themselves to the study of Egyptian geometry, +this science assumed a radically different aspect. ``Whatever +we Greeks receive, we improve and perfect,'' says Plato. The +Egyptians carried geometry no further than was absolutely +\index{Egyptians|)}% +necessary for their practical wants. The Greeks, on the other +hand, had within them a strong speculative tendency. They +felt a craving to discover the reasons for things. They found +pleasure in the contemplation of \emph{ideal} relations, and loved +science \emph{as} science. + +Our sources of information on the history of Greek geometry +before Euclid consist merely of scattered notices in ancient +writers. The early mathematicians, Thales and Pythagoras, +%% -----File: 031.png---Folio 17------- +left behind no written records of their discoveries. A full +history of Greek geometry and astronomy during this period, +\index{Geometry!Greek|(}% +written by Eudemus, a pupil of Aristotle, has been lost. It +\index{Aristotle!ref.\ to}% +\index{Eudemus}% +was well known to Proclus, who, in his commentaries on +\index{Proclus!ref.\ to}% +Euclid, gives a brief account of it. This abstract constitutes +\index{Euclid!ref.\ to}% +our most reliable information. We shall quote it frequently +under the name of \textit{Eudemian Summary}. +\index{Eudemian Summary}% + +\Subsection{The Ionic School} +\index{Ionic School|(}% + +To \textbf{Thales} of Miletus (640--546~\BC), one of the ``seven wise +\index{Thales}% +men,'' and the founder of the Ionic school, falls the honour of +having introduced the study of geometry into Greece. During +middle life he engaged in commercial pursuits, which took +him to Egypt. He is said to have resided there, and to have +studied the physical sciences and mathematics with the Egyptian +priests. Plutarch declares that Thales soon excelled his +masters, and amazed King Amasis by measuring the heights +of the pyramids from their shadows. According to Plutarch, +this was done by considering that the shadow cast by a vertical +staff of known length bears the same ratio to the shadow +of the pyramid as the height of the staff bears to the height +of the pyramid. This solution presupposes a knowledge of +proportion, and the Ahmes papyrus actually shows that the +\index{Ahmes!ref.\ to}% +\index{Proportion}% +rudiments of proportion were known to the Egyptians. According +to Diogenes Laertius, the pyramids were measured by +\index{Diogenes Laertius}% +Thales in a different way; viz.~by finding the length of the +shadow of the pyramid at the moment when the shadow of a +staff was equal to its own length. + +The \textit{Eudemian Summary} ascribes to Thales the invention +of the theorems on the equality of vertical angles, the equality +of the angles at the base of an isosceles triangle, the bisection +of a circle by any diameter, and the congruence of two +%% -----File: 032.png---Folio 18------- +triangles having a side and the two adjacent angles equal respectively. +The last theorem he applied to the measurement +of the distances of ships from the shore. Thus Thales was +\index{Thales}% +the first to apply theoretical geometry to practical uses. The +theorem that all angles inscribed in a semicircle are right +angles is attributed by some ancient writers to Thales, by +others to Pythagoras. Thales was doubtless familiar with +\index{Pythagoras!ref.\ to}% +other theorems, not recorded by the ancients. It has been +inferred that he knew the sum of the three angles of a triangle +to be equal to two right angles, and the sides of equiangular +triangles to be proportional.\Reference{8} The Egyptians must +have made use of the above theorems on the straight line, in +some of their constructions found in the Ahmes papyrus, but +\index{Ahmes!ref.\ to}% +it was left for the Greek philosopher to give these truths, +which others saw, but did not formulate into words, an +explicit, abstract expression, and to put into scientific language +and subject to proof that which others merely felt to +be true. Thales may be said to have created the geometry +of lines, essentially abstract in its character, while the Egyptians +studied only the geometry of surfaces and the rudiments +of solid geometry, empirical in their character.\Reference{8} + +With Thales begins also the study of scientific astronomy. +\index{Astronomy!Greek}% +He acquired great celebrity by the prediction of a solar eclipse +in 585~\BC. Whether he predicted the day of the occurrence, +or simply the year, is not known. It is told of him that +while contemplating the stars during an evening walk, he fell +into a ditch. The good old woman attending him exclaimed, +``How canst thou know what is doing in the heavens, when +thou seest not what is at thy feet?'' + +The two most prominent pupils of Thales were \textbf{Anaximander} +\index{Anaximander}% +(b.~611~\BC) and \textbf{Anaximenes} (b.~570~\BC). They studied +\index{Anaximenes}% +chiefly astronomy and physical philosophy. Of \textbf{Anaxagoras}, a +\index{Anaxagoras}% +pupil of Anaximenes, and the last philosopher of the Ionic +%% -----File: 033.png---Folio 19------- +\index{Circle-squarers}% +\index{Pythagorean School|(}% +school, we know little, except that, while in prison, he passed +his time attempting to square the circle. This is the first +\index{Circle}% +time, in the history of mathematics, that we find mention of +the famous problem of the quadrature of the circle, that rock +upon which so many reputations have been destroyed. It +turns upon the determination of the exact value of~$\pi$. Approximations +to~$\pi$ had been made by the Chinese, Babylonians, +\index{Babylonians!ref.\ to}% +\index{Chinese}% +Hebrews, and Egyptians. But the invention of a method to +\index{Egyptians}% +\index{Hebrews}% +find its \emph{exact} value, is the knotty problem which has engaged +the attention of many minds from the time of Anaxagoras +down to our own. Anaxagoras did not offer any solution of +it, and seems to have luckily escaped paralogisms. + +About the time of Anaxagoras, but isolated from the Ionic +school, flourished \textbf{\OE{}nopides} of Chios. Proclus ascribes to him +\index{OEnopides@{\OE{}nopides}}% +\index{Proclus!ref.\ to}% +the solution of the following problems: From a point without, +to draw a perpendicular to a given line, and to draw an angle +on a line equal to a given angle. That a man could gain a +reputation by solving problems so elementary as these, indicates +that geometry was still in its infancy, and that the +Greeks had not yet gotten far beyond the Egyptian constructions. + +The Ionic school lasted over one hundred years. The +\index{Ionic School|)}% +progress of mathematics during that period was slow, as +compared with its growth in a later epoch of Greek history. +A new impetus to its progress was given by Pythagoras. +\index{Pythagoras|(}% + +\Subsection{The School of Pythagoras} + +\textbf{Pythagoras} (580?--500?~\BC) was one of those figures which +impressed the imagination of succeeding times to such an +extent that their real histories have become difficult to be +discerned through the mythical haze that envelops them. The +following account of Pythagoras excludes the most doubtful +%% -----File: 034.png---Folio 20------- +statements. He was a native of Samos, and was drawn by +the fame of Pherecydes to the island of Syros. He then +\index{Pherecydes}% +visited the ancient Thales, who incited him to study in Egypt. +\index{Thales!ref.\ to}% +He sojourned in Egypt many years, and may have visited +Babylon. On his return to Samos, he found it under the +tyranny of Polycrates. Failing in an attempt to found a +school there, he quitted home again and, following the current +of civilisation, removed to Magna Græcia in South Italy. He +settled at Croton, and founded the famous Pythagorean school. +This was not merely an academy for the teaching of philosophy, +mathematics, and natural science, but it was a brotherhood, +the members of which were united for life. This brotherhood +had observances approaching masonic peculiarity. They were +forbidden to divulge the discoveries and doctrines of their +school. Hence we are obliged to speak of the Pythagoreans +as a body, and find it difficult to determine to whom each +particular discovery is to be ascribed. The Pythagoreans +themselves were in the habit of referring every discovery back +to the great founder of the sect. + +This school grew rapidly and gained considerable political +ascendency\DPnote{** PP: [sic]}. But the mystic and secret observances, introduced +in imitation of Egyptian usages, and the aristocratic +tendencies of the school, caused it to become an object of +suspicion. The democratic party in Lower Italy revolted and +destroyed the buildings of the Pythagorean school. Pythagoras +fled to Tarentum and thence to Metapontum, where he was +murdered. + +Pythagoras has left behind no mathematical treatises, and +our sources of information are rather scanty. Certain it is +that, in the Pythagorean school, mathematics was the principal +study. Pythagoras raised mathematics to the rank of a science. +Arithmetic was courted by him as fervently as geometry. In +\index{Arithmetic!Pythagoreans}% +fact, arithmetic is the foundation of his philosophic system. +%% -----File: 035.png---Folio 21------- +\index{Regular solids}% + +The \textit{Eudemian Summary} says that ``Pythagoras changed +\index{Eudemian Summary}% +the study of geometry into the form of a liberal education, +for he examined its principles to the bottom, and investigated +its theorems in an immaterial and intellectual manner.'' His +geometry was connected closely with his arithmetic. He was +especially fond of those geometrical relations which admitted +of arithmetical expression. + +Like Egyptian geometry, the geometry of the Pythagoreans +is much concerned with areas. To Pythagoras is ascribed the +important theorem that the square on the hypotenuse of a +right triangle is equal to the sum of the squares on the other +two sides. He had probably learned from the Egyptians the +truth of the theorem in the special case when the sides are +$3$,~$4$,~$5$, respectively. The story goes, that Pythagoras was so +jubilant over this discovery that he sacrificed a hecatomb. Its +authenticity is doubted, because the Pythagoreans believed in +the transmigration of the soul and opposed, therefore, the +shedding of blood. In the later traditions of the Neo-Pythagoreans +this objection is removed by replacing this bloody sacrifice +by that of ``an ox made of flour''! The proof of the law +of three squares, given in Euclid's \textit{Elements}, I.~47, is due to +\index{Euclid!ref.\ to}% +Euclid himself, and not to the Pythagoreans. What the Pythagorean +method of proof was has been a favourite topic for +conjecture. + +The theorem on the sum of the three angles of a triangle, +presumably known to Thales, was proved by the Pythagoreans +\index{Thales!ref.\ to}% +after the manner of Euclid. They demonstrated also that the +plane about a point is completely filled by six equilateral +triangles, four squares, or three regular hexagons, so that it +is possible to divide up a plane into figures of either kind. + +From the equilateral triangle and the square arise the solids, +namely the tetraedron, octaedron, icosaedron,\DPnote{[** PP: Original spellings]} and the cube. +These solids were, in all probability, known to the Egyptians, +%% -----File: 036.png---Folio 22------- +\index{Irrationals}% +\index{Star-polygons}% +excepting, perhaps, the icosaedron. In Pythagorean philosophy, +they represent respectively the four elements of the +physical world; namely, fire, air, water, and earth. Later +another regular solid was discovered, namely the dodecaedron, +which, in absence of a fifth element, was made to represent +the universe itself. Iamblichus states that Hippasus, a Pythagorean, +\index{Hippasus}% +\index{Iamblichus!ref.\ to}% +perished in the sea, because he boasted that he first +divulged ``the sphere with the twelve pentagons.'' The star-shaped +pentagram was used as a symbol of recognition by the +Pythagoreans, and was called by them Health. + +Pythagoras called the sphere the most beautiful of all solids, +and the circle the most beautiful of all plane figures. The +treatment of the subjects of proportion and of irrational +\index{Proportion}% +quantities by him and his school will be taken up under the +head of arithmetic. + +According to Eudemus, the Pythagoreans invented the problems +\index{Eudemus}% +concerning the application of areas, including the cases +of defect and excess, as in Euclid, VI.~28,~29. +\index{Euclid!ref.\ to}% + +They were also familiar with the construction of a polygon +equal in area to a given polygon and similar to another given +polygon. This problem depends upon several important and +somewhat advanced theorems, and testifies to the fact that +the Pythagoreans made no mean progress in geometry. + +Of the theorems generally ascribed to the Italian school, +some cannot be attributed to Pythagoras himself, nor to his +earliest successors. The progress from empirical to reasoned +solutions must, of necessity, have been slow. It is worth +noticing that on the circle no theorem of any importance was +discovered by this school. + +Though politics broke up the Pythagorean fraternity, yet +the school continued to exist at least two centuries longer. +Among the later Pythagoreans, Philolaus and Archytas are +\index{Philolaus}% +the most prominent. \textbf{Philolaus} wrote a book on the Pythagorean +%% -----File: 037.png---Folio 23------- +\index{Pythagoras|)}% +\index{Pythagorean School|)}% +doctrines. By him were first given to the world the +teachings of the Italian school, which had been kept secret +for a whole century. The brilliant \textbf{Archytas} of Tarentum +\index{Archytas}% +(428--347~\BC), known as a great statesman and general, and +universally admired for his virtues, was the only great geometer +among the Greeks when Plato opened his school. Archytas +\index{Plato!ref.\ to}% +was the first to apply geometry to mechanics and to treat +\index{Mechanics!Greek}% +the latter subject methodically. He also found a very ingenious +mechanical solution to the problem of the duplication of +\index{Duplication of the cube|(}% +the cube. His solution involves clear notions on the generation +of cones and cylinders. This problem reduces itself to +finding two mean proportionals between two given lines. +These mean proportionals were obtained by Archytas from +the section of a half-cylinder. The doctrine of proportion +\index{Proportion}% +was advanced through him. + +There is every reason to believe that the later Pythagoreans +exercised a strong influence on the study and development of +mathematics at Athens. The Sophists acquired geometry from +Pythagorean sources. Plato bought the works of Philolaus, +and had a warm friend in Archytas. + +\Subsection{The Sophist School} +\index{Sophist School|(}% + +After the defeat of the Persians under Xerxes at the battle +of Salamis, 480~\BC, a league was formed among the Greeks +to preserve the freedom of the now liberated Greek cities on +the islands and coast of the Ægæan Sea. Of this league +Athens soon became leader and dictator. She caused the +separate treasury of the league to be merged into that of +Athens, and then spent the money of her allies for her own +aggrandisement. Athens was also a great commercial centre. +Thus she became the richest and most beautiful city of antiquity. +All menial work was performed by slaves. The +%% -----File: 038.png---Folio 24------- +\index{Pythagoras!ref.\ to}% +\index{Trisection of angles}% +citizen of Athens was well-to-do and enjoyed a large amount +of leisure. The government being purely democratic, every +citizen was a politician. To make his influence felt among +his fellow-men he must, first of all, be educated. Thus there +arose a demand for teachers. The supply came principally +from Sicily, where Pythagorean doctrines had spread. These +teachers were called \emph{Sophists}, or ``wise men.'' Unlike the +Pythagoreans, they accepted pay for their teaching. Although +rhetoric was the principal feature of their instruction, they +also taught geometry, astronomy, and philosophy. Athens +\index{Astronomy!Greek}% +soon became the headquarters of Grecian men of letters, and +of mathematicians in particular. The home of mathematics +among the Greeks was first in the Ionian Islands, then in +Lower Italy, and during the time now under consideration, +at Athens. + +The geometry of the circle, which had been entirely +\index{Circle|(}% +neglected by the Pythagoreans, was taken up by the Sophists. +Nearly all their discoveries were made in connection with +their innumerable attempts to solve the following three +famous problems:--- + +(1)~To trisect an arc or an angle. + +(2)~To ``double the cube,'' \ie~to find a cube whose \emph{volume} +is double that of a given cube. + +(3)~To ``square the circle,'' \ie~to find a square or some +other rectilinear figure exactly equal in area to a given circle. + +These problems have probably been the subject of more +discussion and research than any other problems in mathematics. +The bisection of an angle was one of the easiest +problems in geometry. The trisection of an angle, on the +other hand, presented unexpected difficulties. A right angle +had been divided into three equal parts by the Pythagoreans. +But the general problem, though easy in appearance, transcended +the power of \emph{elementary} geometry. Among the first +%% -----File: 039.png---Folio 25------- +\index{Dinostratus!ref.\ to}% +to wrestle with it was \textbf{Hippias of Elis}, a contemporary of +\index{Hippias of Elis}% +Socrates, and born about 460~\BC. Like all the later geometers, +he failed in effecting the trisection by means of a ruler +and compass only. Proclus mentions a man, Hippias, presumably +Hippias of Elis, as the inventor of a transcendental curve +which served to divide an angle not only into three, but into +any number of equal parts. This same curve was used later +by Deinostratus and others for the quadrature of the circle. +On this account it is called the \emph{quadratrix}. +\index{Quadratrix}% + +The Pythagoreans had shown that the diagonal of a square +is the side of another square having double the area of the +original one. This probably suggested the problem of the +duplication of the cube, \ie~to find the edge of a cube having +double the volume of a given cube. Eratosthenes ascribes to +\index{Eratosthenes!ref.\ to}% +this problem a different origin. The Delians were once suffering +from a pestilence and were ordered by the oracle to +double a certain cubical altar. Thoughtless workmen simply +constructed a cube with edges twice as long, but this did not +pacify the gods. The error being discovered, Plato was consulted +on the matter. He and his disciples searched eagerly +for a solution to this ``Delian Problem.'' \textbf{Hippocrates of Chios} +\index{Hippocrates of Chios}% +(about 430~\BC), a talented mathematician, but otherwise slow +and stupid, was the first to show that the problem could be +reduced to finding two mean proportionals between a given +line and another twice as long. For, in the proportion $a:x +=x:y=y:2a$, since $x^2 = ay$ and $y^2 = 2ax$ and $x^4 = a^2y^2$, we +have $x^4 = 2a^3x$ and $x^3 = 2a^3$. But he failed to find the two +mean proportionals. His attempt to square the circle was +also a failure; for though he made himself celebrated by +squaring a lune, he committed an error in attempting to apply +\index{Lune, squaring of}% +this result to the squaring of the circle. + +In his study of the quadrature and duplication-problems, +\index{Duplication of the cube|)}% +Hippocrates contributed much to the geometry of the circle. +%% -----File: 040.png---Folio 26------- +\index{Irrationals}% + +The subject of similar figures was studied and partly +developed by Hippocrates. This involved the theory of +proportion. Proportion had, thus far, been used by the +\index{Proportion}% +Greeks only in numbers. They never succeeded in uniting +the notions of numbers and magnitudes. The term ``number'' +was used by them in a restricted sense. What we call +irrational numbers was not included under this notion. Not +even rational fractions were called numbers. They used the +\index{Fractions!Greek}% +word in the same sense as we use ``integers.'' Hence numbers +were conceived as \emph{discontinuous}, while magnitudes were +\emph{continuous}. The two notions appeared, therefore, entirely +distinct. The chasm between them is exposed to full view +in the statement of Euclid that ``incommensurable magnitudes +\index{Euclid!ref.\ to}% +do not have the same ratio as numbers.'' In Euclid's +\textit{Elements} we find the theory of proportion of magnitudes +developed and treated independent of that of numbers. The +transfer of the theory of proportion from numbers to magnitudes +(and to lengths in particular) was a difficult and +important step. + +Hippocrates added to his fame by writing a geometrical +text-book, called the \textit{Elements}. This publication shows that +the Pythagorean habit of secrecy was being abandoned; +secrecy was contrary to the spirit of Athenian life. + +The Sophist \textbf{Antiphon}, a contemporary of Hippocrates, introduced +\index{Antiphon}% +the \emph{process} of exhaustion for the purpose of solving +\index{Exhaustion, method of}% +the problem of the quadrature. He did himself credit by +remarking that by inscribing in a circle a square, and on its +sides erecting isosceles triangles with their vertices in the +circumference, and on the sides of these triangles erecting +new triangles, etc., one could obtain a succession of regular +polygons of $8$, $16$, $32$, $64$~sides, and so on, of which each +approaches nearer to the circle than the previous one, until +the circle is finally \emph{exhausted}. Thus is obtained an inscribed +%% -----File: 041.png---Folio 27------- +polygon whose sides coincide with the circumference. Since +there can be found squares equal in area to any polygon, +there also can be found a square equal to the last polygon +inscribed, and therefore equal to the circle itself. \textbf{Bryson +\index{Bryson of Heraclea}% +of Heraclea}, a contemporary of Antiphon, advanced the problem +\index{Antiphon!ref.\ to}% +of the quadrature considerably by circumscribing polygons +at the same time that he inscribed polygons. He erred, +however, in assuming that the area of a circle was the arithmetical +mean between circumscribed and inscribed polygons. +Unlike Bryson and the rest of Greek geometers, Antiphon +seems to have believed it possible, by continually doubling +the sides of an inscribed polygon, to obtain a polygon coinciding +with the circle. This question gave rise to lively +disputes in Athens. If a polygon can coincide with the +circle, then, says Simplicius, we must put aside the notion +that magnitudes are divisible \textit{ad infinitum}. Aristotle always +\index{Aristotle!ref.\ to}% +supported the theory of the infinite divisibility, while Zeno, +\index{Zeno}% +the Stoic, attempted to show its absurdity by proving that +if magnitudes are infinitely divisible, motion is impossible. +Zeno argues that Achilles could not overtake a tortoise; for +\index{Achilles and tortoise, paradox of}% +while he hastened to the place where the tortoise had been +when he started, the tortoise crept some distance ahead, and +while Achilles reached that second spot, the tortoise again +moved forward a little, and so on. Thus the tortoise was +always in advance of Achilles. Such arguments greatly confounded +Greek geometers. No wonder they were deterred +by such paradoxes from introducing the idea of infinity into +\index{Infinity}% +their geometry. It did not suit the rigour of their proofs. + +The process of Antiphon and Bryson gave rise to the cumbrous +but perfectly rigorous ``method of exhaustion.'' In +determining the ratio of the areas between two curvilinear +plane figures, say two circles, geometers first inscribed or +circumscribed similar polygons, and then by increasing indefinitely +%% -----File: 042.png---Folio 28------- +the number of sides, nearly exhausted the spaces +between the polygons and circumferences. From the theorem +that similar polygons inscribed in circles are to each +other as the squares on their diameters, geometers may have +divined the theorem attributed to Hippocrates of Chios that +\index{Hippocrates of Chios}% +the circles, which differ but little from the last drawn polygons, +must be to each other as the squares on their diameters. +But in order to exclude all vagueness and possibility of doubt, +later Greek geometers applied reasoning like that in Euclid, +XII.~2, as follows: Let $C$~and~$c$, $D$~and~$d$ be respectively the +circles and diameters in question. Then if the proportion +$D^2 : d^2 = C : c$ is not true, suppose that $D^2 : d^2 = C : c'$. If~$c' < c$, +then a polygon~$p$ can be inscribed in the circle~$c$ which comes +\index{Circle|)}% +nearer to it in area than does~$c'$. If~$P$ be the corresponding +polygon in~$C$, then $P : p = D^2 : d^2 = C : c'$, and $P : C = p : c'$. +Since~$p > c'$, we have~$P > C$, which is absurd. Next they +proved by this same method of \textit{reductio ad absurdum} the +\index{Reductio ad absurdum}% +falsity of the supposition that~$c' > c$. Since~$c'$ can be neither +larger nor smaller than~$c$, it must be equal to it,~\textsc{q.e.d.} +Hankel refers this Method of Exhaustion back to Hippocrates +\index{Exhaustion, method of}% +\index{Hankel!ref.\ to}% +\index{Method of exhaustion}% +of Chios, but the reasons for assigning it to this early +writer, rather than to Eudoxus, seem insufficient. +\index{Eudoxus!ref.\ to}% + +Though progress in geometry at this period is traceable only +at Athens, yet Ionia, Sicily, Abdera in Thrace, and Cyrene +produced mathematicians who made creditable contributions +to the science. We can mention here only \textbf{Democritus of +\index{Democritus}% +Abdera} (about 460--370~\BC), a pupil of Anaxagoras, a friend +\index{Anaxagoras!ref.\ to}% +of Philolaus, and an admirer of the Pythagoreans. He +\index{Philolaus!ref.\ to}% +visited Egypt and perhaps even Persia. He was a successful +geometer and wrote on incommensurable lines, on geometry, +on numbers, and on perspective. None of these works are +extant. He used to boast that in the construction of plane +figures with proof no one had yet surpassed him, not even +%% -----File: 043.png---Folio 29------- +\index{Pythagoras!ref.\ to}% +\index{Sophist School|)}% +the so-called harpedonaptæ (``rope-stretchers'') of Egypt. By +this assertion he pays a flattering compliment to the skill +and ability of the Egyptians. + +\Subsection{The Platonic School} +\index{Platonic School|(}% + +During the Peloponnesian War (431--404~\BC) the progress +of geometry was checked. After the war, Athens sank into +the background as a minor political power, but advanced more +and more to the front as the leader in philosophy, literature, +and science. Plato was born at Athens in 429~\BC, the year +\index{Plato|(}% +of the great plague, and died \Change{in~348.}{in 348~\BC.} He was a pupil and +near friend of Socrates, but it was not from him that he +acquired his taste for mathematics. After the death of Socrates, +Plato travelled extensively. In Cyrene he studied +mathematics under Theodorus. He went to Egypt, then to +\index{Theodorus!ref.\ to}% +Lower Italy and Sicily, where he came in contact with the +Pythagoreans. Archytas of Tarentum and Timæus of Locri +\index{Archytas!ref.\ to}% +\index{Timæus of Locri}% +became his intimate friends. On his return to Athens, about +389~\BC, he founded his school in the groves of the \textit{Academia}, +and devoted the remainder of his life to teaching and writing. + +Plato's physical philosophy is partly based on that of the +Pythagoreans. Like them, he sought in arithmetic and +\index{Arithmetic!Platonists}% +geometry the key to the universe. When questioned about +the occupation of the Deity, Plato answered that ``He geometrises +continually.'' Accordingly, a knowledge of geometry +is a necessary preparation for the study of philosophy. To +show how great a value he put on mathematics and how +necessary it is for higher speculation, Plato placed the inscription +over his porch, ``Let no one who is unacquainted with +geometry enter here.'' Xenocrates, a successor of Plato as +\index{Xenocrates}% +teacher in the Academy, followed in his master's footsteps, by +declining to admit a pupil who had no mathematical training, +%% -----File: 044.png---Folio 30------- +with the remark, ``Depart, for thou hast not the grip of +philosophy.'' Plato observed that geometry trained the mind +for correct and vigorous thinking. Hence it was that the +\textit{Eudemian Summary} says, ``He filled his writings with mathematical +\index{Eudemian Summary}% +discoveries, and exhibited on every occasion the remarkable +connection between mathematics and philosophy.'' + +With Plato as the head-master, we need not wonder that +the Platonic school produced so large a number of mathematicians. +Plato did little real original work, but he made +valuable improvements in the logic and methods employed +in geometry. It is true that the Sophist geometers of the +previous century were rigorous in their proofs, but as a rule +they did not reflect on the inward nature of their methods. +They used the axioms without giving them explicit expression, +\index{Axioms (of geometry)}% +and the geometrical concepts, such as the point, line, surface, +etc., without assigning to them formal definitions. The Pythagoreans +called a point ``unity in position,'' but this is a +statement of a philosophical theory rather than a definition. +Plato objected to calling a point a ``geometrical fiction.'' He +defined a point as the ``beginning of a line'' or as ``an indivisible +line,'' and a line as ``length without breadth.'' He called +the point, line, surface, the `boundaries' of the line, surface, +solid, respectively. Many of the definitions in Euclid are to +\index{Euclid!ref.\ to}% +be ascribed to the Platonic school. The same is probably +true of Euclid's axioms. Aristotle refers to Plato the axiom +that ``equals subtracted from equals leave equals.'' + +One of the greatest achievements of Plato and his school is +the invention of \emph{analysis} as a method of proof. To be sure, +\index{Analysis!(in synthetic geometry)}% +this method had been used unconsciously by Hippocrates and +\index{Hippocrates of Chios}% +others; but Plato, like a true philosopher, turned the instinctive +logic into a conscious, legitimate method. + +The terms \emph{synthesis} and \emph{analysis} are used in mathematics +\index{Synthesis}% +in a more special sense than in logic. In ancient mathematics +%% -----File: 045.png---Folio 31------- +\index{Circle}% +\index{Trisection of angles}% +they had a different meaning from what they now have. The +oldest definition of mathematical analysis as opposed to synthesis +\index{Synthesis}% +is that given in Euclid, XIII.~5, which in all probability +\index{Euclid!ref.\ to}% +was framed by Eudoxus: ``Analysis is the obtaining of the +\index{Eudoxus!ref.\ to}% +thing sought by assuming it and so reasoning up to an +admitted truth; synthesis is the obtaining of the thing +sought by reasoning up to the inference and proof of it.'' +The analytic method is not conclusive, unless all operations +involved in it are known to be reversible. To remove all +doubt, the Greeks, as a rule, added to the analytic process +a synthetic one, consisting of a reversion of all operations +occurring in the analysis. Thus the aim of analysis was to +aid in the discovery of synthetic proofs or solutions. + +Plato is said to have solved the problem of the duplication +\index{Duplication of the cube}% +of the cube. But the solution is open to the very same objection +which he made to the solutions by Archytas, Eudoxus, +\index{Archytas!ref.\ to}% +and Menæchmus. He called their solutions not geometrical, +\index{Menæchmus!ref.\ to}% +but mechanical, for they required the use of other instruments +than the ruler and compasses. He said that thereby ``the good +of geometry is set aside and destroyed, for we again reduce it +to the world of sense, instead of elevating and imbuing it with +the eternal and incorporeal images of thought, even as it is +employed by God, for which reason He always is God.'' These +objections indicate either that the solution is wrongly attributed +to Plato or that he wished to show how easily non-geometric +solutions of that character can be found. It is now +generally admitted that the duplication problem, as well as +the trisection and quadrature problems, cannot be solved by +means of the ruler and compass only. + +Plato gave a healthful stimulus to the study of stereometry, +\index{Plato|)}% +\index{Stereometry}% +which until his time had been entirely neglected. The sphere +and the regular solids had been studied to some extent, but +\index{Regular solids}% +the prism, pyramid, cylinder, and cone were hardly known to +%% -----File: 046.png---Folio 32------- +\index{Astronomy!Greek}% +\index{Deinostratus.!see{Dinostratus}}% +\index{Duplication of the cube}% +exist. All these solids became the subjects of investigation +by the Platonic school. One result of these inquiries was +epoch-making. \textbf{Menæchmus}, an associate of Plato and pupil +\index{Menæchmus}% +\index{Plato!ref.\ to}% +of Eudoxus, invented the conic sections, which, in course of +\index{Conic sections!Greek}% +\index{Eudoxus}% +\index{Eudoxus!ref.\ to}% +only a century, raised geometry to the loftiest height which +it was destined to reach during antiquity. Menæchmus cut +three kinds of cones, the `right-angled,' `acute-angled,' and +`obtuse-angled,' by planes at right angles to a side of the +cones, and thus obtained the three sections which we now call +the parabola, ellipse, and hyperbola. Judging from the two +very elegant solutions of the ``Delian Problem'' by means of +intersections of these curves, Menæchmus must have succeeded +well in investigating their properties. + +Another great geometer was \textbf{Dinostratus}, the brother of +\index{Dinostratus}% +Menæchmus and pupil of Plato. Celebrated is his mechanical +solution of the quadrature of the circle, by means of the \emph{quadratrix} +\index{Quadratrix}% +of Hippias. + +Perhaps the most brilliant mathematician of this period was +\textbf{Eudoxus}. He was born at Cnidus about 408~\BC, studied under +Archytas, and later, for two months, under Plato. He was +\index{Archytas!ref.\ to}% +imbued with a true spirit of scientific inquiry, and has been +called the father of scientific astronomical observation. From +the fragmentary notices of his astronomical researches, found +in later writers, Ideler and Schiaparelli succeeded in reconstructing +\index{Ideler}% +\index{Schiaparelli}% +the system of Eudoxus with its celebrated representation +of planetary motions by ``concentric spheres.'' Eudoxus +\index{Concentric spheres of Eudoxus}% +had a school at Cyzicus, went with his pupils to Athens, visiting +Plato, and then returned to Cyzicus, where he died 355~\BC. +The fame of the academy of Plato is to a large extent +due to Eudoxus's pupils of the school at Cyzicus, among +whom are Menæchmus, Dinostratus, Athenæus, and Helicon. +\index{Athenæus}% +\index{Helicon}% +Diogenes Laertius describes Eudoxus as astronomer, physician, +\index{Diogenes Laertius}% +legislator, as well as geometer. The \textit{Eudemian Summary} +\index{Eudemian Summary}% +%% -----File: 047.png---Folio 33------- +\index{Method of exhaustion!ref.\ to}% +\index{Proportion}% +\index{Stereometry}% +says that Eudoxus ``first increased the number of general +\index{Eudoxus}% +theorems, added to the three proportions three more, and +raised to a considerable quantity the learning, begun by Plato, +\index{Plato!ref.\ to}% +on the subject of the section, to which he applied the analytical +method.'' By this `section' is meant, no doubt, the +``golden section'' (\textit{sectio aurea}), which cuts a line in extreme +\index{Golden section}% +\index{Sectio aurea}% +\index{Section, the golden}% +and mean ratio. The first five propositions in Euclid~XIII. relate +\index{Euclid!ref.\ to}% +to lines cut by this section, and are generally attributed +to Eudoxus. Eudoxus added much to the knowledge of solid +geometry. He proved, says Archimedes, that a pyramid is +exactly one-third of a prism, and a cone one-third of a cylinder, +having equal base and altitude. The proof that spheres are +to each other as the cubes of their radii is probably due to +him. He made frequent and skilful use of the method of +exhaustion, of which he was in all probability the inventor. +\index{Exhaustion, method of}% +A scholiast on Euclid, thought to be Proclus, says further that +\index{Proclus!ref.\ to}% +Eudoxus practically invented the whole of Euclid's fifth book. +Eudoxus also found two mean proportionals between two +given lines, but the method of solution is not known. + +Plato has been called a maker of mathematicians. Besides +the pupils already named, the \textit{Eudemian Summary} mentions +\index{Eudemian Summary}% +the following: \textbf{Theætetus} of Athens, a man of great +\index{Theætetus}% +natural gifts, to whom, no doubt, Euclid was greatly indebted +in the composition of the 10th~book,\Reference[]{8} treating of incommensurables; +\textbf{Leodamas} of Thasos; \textbf{Neocleides} and his pupil \textbf{Leon}, +\index{Leodamas}% +\index{Leon}% +\index{Neocleides}% +who added much to the work of their predecessors, for Leon +wrote an \textit{Elements} carefully designed, both in number and +utility of its proofs; \textbf{Theudius of Magnesia}, who composed a +\index{Theudius}% +very good book of \textit{Elements} and generalised propositions, +which had been confined to particular cases; \textbf{Hermotimus of +\index{Hermotimus}% +Colophon}, who discovered many propositions of the \textit{Elements} +and composed some on \emph{loci}; and, finally, the names of \textbf{Amyclas +\index{Amyclas}% +of Heraclea}, \textbf{Cyzicenus of Athens}, and \textbf{Philippus of Mende}. +\index{Cyzicenus}% +\index{Philippus}% +%% -----File: 048.png---Folio 34------- +\index{Conic sections!Greek}% + +A skilful mathematician of whose life and works we have +no details is \textbf{Aristæus}, the elder, probably a senior contemporary +\index{Aristæus}% +of Euclid. The fact that he wrote a work on conic +\index{Euclid!ref.\ to}% +sections tends to show that much progress had been made in +their study during the time of Menæchmus. Aristæus wrote +\index{Menæchmus!ref.\ to}% +also on regular solids and cultivated the analytic method. +\index{Regular solids}% +His works contained probably a summary of the researches +of the Platonic school.\Reference{8} + +\textbf{Aristotle} (384--322~\BC), the systematiser of deductive logic, +\index{Aristotle}% +though not a professed mathematician, promoted the science +of geometry by improving some of the most difficult definitions. +His \textit{Physics} contains passages with suggestive hints +of the principle of virtual velocities. About his time there +\index{Virtual velocities}% +appeared a work called \textit{Mechanica}, of which he is regarded +by some as the author. Mechanics was totally neglected by +\index{Mechanics!Greek}% +the Platonic school. +\index{Platonic School|)}% + +\Subsection{The First Alexandrian School} +\index{Alexandrian School!(first)|(}% + +In the previous pages we have seen the birth of geometry +in Egypt, its transference to the Ionian Islands, thence to +Lower Italy and to Athens. We have witnessed its growth +in Greece from feeble childhood to vigorous manhood, and +now we shall see it return to the land of its birth and there +derive new vigour. + +During her declining years, immediately following the +Peloponnesian War, Athens produced the greatest scientists +and philosophers of antiquity. It was the time of Plato +\index{Plato!ref.\ to}% +and Aristotle. In 338~\BC, at the battle of Chæronea, Athens +was beaten by Philip of Macedon, and her power was broken +forever. Soon after, Alexander the Great, the son of Philip, +started out to conquer the world. In eleven years he built +up a great empire which broke to pieces in a day. Egypt +%% -----File: 049.png---Folio 35------- +fell to the lot of Ptolemy Soter. Alexander had founded +the seaport of Alexandria, which soon became ``the noblest +of all cities.'' Ptolemy made Alexandria the capital. The +history of Egypt during the next three centuries is mainly +the history of Alexandria. Literature, philosophy, and art +were diligently cultivated. Ptolemy created the university +of Alexandria. He founded the great Library and built laboratories, +museums, a zoölogical garden, and promenades. Alexandria +soon became the great centre of learning. + +Demetrius Phalereus was invited from Athens to take +charge of the Library, and it is probable, says Gow, that +\index{Gow}% +\textbf{Euclid} was invited with him to open the mathematical school. +\index{Euclid|(}% +Euclid's greatest activity was during the time of the first +Ptolemy, who reigned from 306 to 283~\BC. Of the life of +Euclid, little is known, except what is added by Proclus to +\index{Proclus!ref.\ to}% +the \textit{Eudemian Summary}. Euclid, says Proclus, was younger +\index{Eudemian Summary}% +than Plato and older than Eratosthenes and Archimedes, the +\index{Archimedes!ref.\ to}% +\index{Eratosthenes!ref.\ to}% +\index{Plato!ref.\ to}% +latter of whom mentions him. He was of the Platonic sect, and +well read in its doctrines. He collected the \textit{Elements}, put +in order much that Eudoxus had prepared, completed many +\index{Eudoxus!ref.\ to}% +things of Theætetus, and was the first who reduced to unobjectionable +\index{Theætetus!ref.\ to}% +demonstration the imperfect attempts of his predecessors. +When Ptolemy once asked him if geometry could +not be mastered by an easier process than by studying the +\textit{Elements}, Euclid returned the answer, ``There is no royal +road to geometry.'' Pappus states that Euclid was distinguished +\index{Pappus!ref.\ to}% +by the fairness and kindness of his disposition, particularly +toward those who could do anything to advance +the mathematical sciences. Pappus is evidently making a +contrast to Apollonius, of whom he more than insinuates the +\index{Apollonius!ref.\ to}% +opposite character.\Reference{9} A pretty little story is related by Stobæus:\Reference[]{6} +``A youth who had begun to read geometry with +Euclid, when he had learnt the first proposition, inquired, +%% -----File: 050.png---Folio 36------- +\index{Exhaustion, method of}% +\index{Method of exhaustion!ref.\ to}% +`What do I get by learning these things?' So Euclid called +his slave and said, `Give him threepence, since he must +make gain out of what he learns.'\,'' These are about all the +personal details preserved by Greek writers. Syrian and +Arabian writers claim to know much more, but they are unreliable. +At one time Euclid of Alexandria was universally +confounded with Euclid of Megara, who lived a century +earlier. + +The fame of Euclid has at all times rested mainly upon his +book on geometry, called the \textit{Elements}. This book was so far +\index{Elements (Euclid's)|(}% +superior to the \textit{Elements} written by Hippocrates, Leon, and +Theudius, that the latter works soon perished in the struggle +for existence. The Greeks gave Euclid the special title of +``the author of the \textit{Elements}.'' It is a remarkable fact in the +history of geometry, that the \textit{Elements} of Euclid, written two +thousand years ago, are still regarded by many as the best +introduction to the mathematical sciences. In England they +are used at the present time extensively as a text-book in +schools. Some editors of Euclid have, however, been inclined +to credit him with more than is his due. They would have +us believe that a finished and unassailable system of geometry +sprang at once from the brain of Euclid, ``an armed Minerva +from the head of Jupiter.'' They fail to mention the earlier +eminent mathematicians from whom Euclid got his material. +Comparatively few of the propositions and proofs in the +\textit{Elements} are his own discoveries. In fact, the proof of the +``Theorem of Pythagoras'' is the only one directly ascribed to +\index{Pythagoras!ref.\ to}% +him. Allman conjectures that the substance of Books~I., II.,~IV. +\index{Allman}% +comes from the Pythagoreans, that the substance of Book~VI. +is due to the Pythagoreans and Eudoxus, the latter contributing +\index{Eudoxus!ref.\ to}% +the doctrine of proportion as applicable to incommensurables +\index{Incommensurables}% +\index{Proportion}% +and also the Method of Exhaustions (Book~XII.), +that Theætetus contributed much toward Books X.~and~XIII., +\index{Theætetus!ref.\ to}% +%% -----File: 051.png---Folio 37------- +that the principal part of the original work of Euclid himself +is to be found in Book~X\@.\Reference{8} Euclid was the greatest systematiser +of his time. By careful selection from the material +before him, and by logical arrangement of the propositions +selected, he built up, from a few definitions and axioms, a +\index{Axioms (of geometry)}% +proud and lofty structure. It would be erroneous to believe +that he incorporated into his \textit{Elements} all the elementary +theorems known at his time. Archimedes, Apollonius, and +\index{Apollonius!ref.\ to}% +\index{Archimedes!ref.\ to}% +even he himself refer to theorems not included in his \textit{Elements}, +as being well-known truths. + +The text of the \textit{Elements} now commonly used is Theon's +\index{Theon of Alexandria!ref.\ to}% +edition. Theon of Alexandria, the father of Hypatia, brought +\index{Hypatia!ref.\ to}% +out an edition, about 700~years after Euclid, with some alterations +in the text. As a consequence, later commentators, +especially Robert Simson, who laboured under the idea that +\index{Simson!ref.\ to}% +Euclid must be absolutely perfect, made Theon the scapegoat +for all the defects which they thought they could discover +in the text as they knew it. But among the manuscripts sent +by Napoleon~I. from the Vatican to Paris was found a copy of +the \textit{Elements} believed to be anterior to Theon's recension. +Many variations from Theon's version were noticed therein, +but they were not at all important, and showed that Theon +generally made only verbal changes. The defects in the +\textit{Elements} for which Theon was blamed must, therefore, be +due to Euclid himself. The \textit{Elements} has been considered as +offering models of scrupulously rigorous demonstrations. It +is certainly true that in point of rigour it compares favourably +with its modern rivals; but when examined in the light of +strict mathematical logic, it has been pronounced by C.~S. +\index{Logic}% +Peirce to be ``riddled with fallacies.'' The results are correct +\index{Peirce, C.~S.!ref.\ to}% +only because the writer's experience keeps him on his guard. + +At the beginning of our editions of the \textit{Elements}, under +the head of definitions, are given the assumptions of such +%% -----File: 052.png---Folio 38------- +\index{Non-Euclidean geometry}% +\index{Prime numbers}% +notions as the point, line, etc., and some verbal explanations. +Then follow three postulates or demands, and twelve axioms. +\index{Axioms (of geometry)}% +The term `axiom' was used by Proclus, but not by Euclid. +\index{Proclus!ref.\ to}% +He speaks, instead, of `common notions'---common either +to all men or to all sciences. There has been much controversy +among ancient and modern critics on the postulates and +axioms. An immense preponderance of manuscripts and the +testimony of Proclus place the `axioms' about \emph{right angles} +and \emph{parallels} (Axioms 11 and~12) among the postulates.\BiReference{9}{10} +\index{Parallels}% +This is indeed their proper place, for they are really \emph{assumptions}, +and not \emph{common notions} or axioms. The postulate +about \emph{parallels} plays an important rôle in the history of non-Euclidean +geometry. The only postulate which Euclid missed +was the one of superposition, according to which figures +can be moved about in space without any alteration in form +or magnitude. + +The \textit{Elements} contains thirteen books by Euclid, and two, +of which it is supposed that Hypsicles and Damascius are +\index{Damascius!ref.\ to}% +\index{Hypsicles!ref.\ to}% +the authors. The first four books are on plane geometry. +The fifth book treats of the theory of proportion as applied +\index{Proportion}% +to magnitudes in general. The sixth book develops the +geometry of similar figures. The seventh, eighth, ninth +books are on the theory of numbers, or on arithmetic. In the +\index{Arithmetic!Euclid}% +ninth book is found the proof to the theorem that the number +of primes is infinite. The tenth book treats of the theory of +incommensurables. The next three books are on stereometry. +\index{Incommensurables}% +\index{Stereometry}% +The eleventh contains its more elementary theorems; the +twelfth, the metrical relations of the pyramid, prism, cone, +cylinder, and sphere. The thirteenth treats of the regular +polygons, especially of the triangle and pentagon, and then uses +them as faces of the five regular solids; namely, the tetraedron, +\index{Regular solids}% +octaedron, icosaedron, cube, and dodecaedron. The regular +solids were studied so extensively by the Platonists that they +%% -----File: 053.png---Folio 39------- +received the name of ``Platonic figures.'' The statement of +\index{Platonic figures}% +Proclus that the whole aim of Euclid in writing the \textit{Elements} +\index{Proclus!ref.\ to}% +was to arrive at the construction of the regular solids, is +obviously wrong. The fourteenth and fifteenth books, treating +of solid geometry, are apocryphal. + +A remarkable feature of Euclid's, and of all Greek geometry +before Archimedes is that it eschews mensuration. Thus the +\index{Archimedes!ref.\ to}% +theorem that the area of a triangle equals half the product +of its base and its altitude is foreign to Euclid. + +Another extant book of Euclid is the \textit{Data}. It seems to +\index{Data (Euclid's)}% +have been written for those who, having completed the \textit{Elements}, +wish to acquire the power of solving new problems +proposed to them. The \textit{Data} is a course of practice in \emph{analysis}. +\index{Analysis!(in synthetic geometry)}% +It contains little or nothing that an intelligent student +could not pick up from the \textit{Elements} itself. Hence it contributes +\index{Elements (Euclid's)|)}% +little to the stock of scientific knowledge. The following +are the other extant works generally attributed to Euclid: +\textit{Phænomena}, a work on spherical geometry and astronomy; +\index{Astronomy!Greek}% +\textit{Optics}, which develops the hypothesis that light proceeds +\index{Optics}% +from the eye, and not from the object seen; \textit{Catoptrica}, containing +propositions on reflections from mirrors; \textit{De~Divisionibus}, +a treatise on the division of plane figures into parts +having to one another a given ratio; \textit{Sectio Canonis}, a work +on musical intervals. His treatise on \textit{Porisms} is lost; but +\index{Porisms}% +much learning has been expended by Robert Simson and +\index{Simson!ref.\ to}% +M.~Chasles in restoring it from numerous notes found in the +\index{Chasles!ref.\ to}% +writings of Pappus. The term `porism' is vague in meaning. +\index{Pappus!ref.\ to}% +\Addendlabel{39}% +The aim of a porism is not to state some property or truth, +like a theorem, nor to effect a construction, like a problem, +but to find and bring to view a thing which necessarily exists +with given numbers or a given construction, as, to find the +centre of a given circle, or to find the G.C.D. of two given +numbers.\Reference{6} His other lost works are \textit{Fallacies}, containing +%% -----File: 054.png---Folio 40------- +exercises in detection of fallacies; \textit{Conic Sections}, in four +\index{Conic sections!Greek}% +books, which are the foundation of a work on the same subject +by Apollonius; and \textit{Loci on a Surface}, the meaning of +\index{Apollonius!ref.\ to}% +which title is not understood. Heiberg believes it to mean +``loci which are surfaces.'' + +The immediate successors of Euclid in the mathematical +\index{Euclid|)}% +school at Alexandria were probably \textbf{Conon}, \textbf{Dositheus}, and +\index{Conon}% +\index{Dositheus}% +\textbf{Zeuxippus}, but little is known of them. +\index{Zeuxippus}% + +\textbf{Archimedes} (287?--212~\BC), the greatest mathematician of +\index{Archimedes|(}% +antiquity, was born in Syracuse. Plutarch calls him a relation +of King Hieron; but more reliable is the statement of +Cicero, who tells us he was of low birth. Diodorus says he +\index{Diodorus}% +visited Egypt, and, since he was a great friend of Conon and +Eratosthenes, it is highly probable that he studied in Alexandria. +\index{Eratosthenes!ref.\ to}% +This belief is strengthened by the fact that he had +the most thorough acquaintance with all the work previously +done in mathematics. He returned, however, to Syracuse, +where he made himself useful to his admiring friend and +patron, King Hieron, by applying his extraordinary inventive +genius to the construction of various war-engines, by which +he inflicted much loss on the Romans during the siege of +Marcellus. The story that, by the use of mirrors reflecting +the sun's rays, he set on fire the Roman ships, when they +came within bow-shot of the walls, is probably a fiction. The +city was taken at length by the Romans, and Archimedes +perished in the indiscriminate slaughter which followed. According +to tradition, he was, at the time, studying the diagram +to some problem drawn in the sand. As a Roman soldier +approached him, he called out, ``Don't spoil my circles.'' +The soldier, feeling insulted, rushed upon him and killed +him. No blame attaches to the Roman general Marcellus, +who admired his genius, and raised in his honour a tomb +bearing the figure of a sphere inscribed in a cylinder. When +%% -----File: 055.png---Folio 41------- +\index{Conic sections!Greek}% +Cicero was in Syracuse, he found the tomb buried under +rubbish. + +Archimedes was admired by his fellow-citizens chiefly for +his mechanical inventions; he himself prized far more highly +his discoveries in pure science. He declared that ``every kind +of art which was connected with daily needs was ignoble and +vulgar.'' Some of his works have been lost. The following +are the extant books, arranged approximately in chronological +order: 1.~Two books on \textit{Equiponderance of Planes} or \textit{Centres +of Plane Gravities}, between which is inserted his treatise on +the \textit{Quadrature of the Parabola}; 2.~Two books on the \textit{Sphere} +and \textit{Cylinder}; 3.~The \textit{Measurement of the Circle}; 4.~\textit{On Spirals}; +\index{Circle}% +5.~\textit{Conoids} and \textit{Spheroids}; 6.~The \textit{Sand-Counter}; 7.~Two books +on \textit{Floating Bodies}; 8.~Fifteen \textit{Lemmas}. + +In the book on the \textit{Measurement of the Circle}, Archimedes +proves first that the area of a circle is equal to that of a +right triangle having the length of the circumference for its +base, and the radius for its altitude. In this he assumes that +there exists a straight line equal in length to the circumference---an +assumption objected to by some ancient critics, on +the ground that it is not evident that a straight line can equal +a curved one. The finding of such a line was the next problem. +He first finds an upper limit to the ratio of the circumference +to the diameter, or~$\pi$. To do this, he starts with an +\index{p@{$\pi$}: values for!Archimedean}% +equilateral triangle of which the base is a tangent and the +vertex is the centre of the circle. By successively bisecting +the angle at the centre, by comparing ratios, and by taking the +irrational square roots always a little too small, he finally +arrived at the conclusion that $\pi < 3 \frac{1}{7}$. Next he finds a lower +limit by inscribing in the circle regular polygons of $6$, $12$, $24$, +$48$, $96$~sides, finding for each successive polygon its perimeter, +which is, of course, always less than the circumference. Thus +he finally concludes that ``the circumference of a circle exceeds +%% -----File: 056.png---Folio 42------- +\index{Curves!quadrature of}% +\index{Quadrature of curves}% +three times its diameter by a part which is less than $\frac{1}{7}$ +but more than $\frac{10}{71}$ of the diameter.'' This approximation is +exact enough for most purposes. + +The \textit{Quadrature of the Parabola} contains two solutions to +\index{Parabola}% +the problem---one mechanical, the other geometrical. The +method of exhaustion is used in both. +\index{Exhaustion, method of}% +\index{Method of exhaustion!ref.\ to}% + +Archimedes studied also the ellipse and accomplished its +quadrature, but to the hyperbola he seems to have paid less attention. +It is believed that he wrote a book on conic sections. + +Of all his discoveries Archimedes prized most highly those +in his \textit{Sphere} and \textit{Cylinder}. In it are proved the new +theorems, that the surface of a sphere is equal to four times +a great circle; that the surface of a segment of a sphere is +equal to a circle whose radius is the straight line drawn from +the vertex of the segment to the circumference of its basal +circle; that the volume and the surface of a sphere are $\frac{2}{3}$~of +the volume and surface, respectively, of the cylinder circumscribed +about the sphere. Archimedes desired that the figure +to the last proposition be inscribed on his tomb. This was +ordered done by Marcellus. + +The spiral now called the ``spiral of Archimedes,'' and +described in the book \textit{On Spirals}, was discovered by Archimedes, +\index{Spirals}% +and not, as some believe, by his friend Conon.\Reference{3} His +\index{Conon!ref.\ to}% +treatise thereon is, perhaps, the most wonderful of all his +works. Nowadays, subjects of this kind are made easy by +the use of the infinitesimal calculus. In its stead the ancients +used the method of exhaustion. Nowhere is the fertility of +his genius more grandly displayed than in his masterly use of +this method. With Euclid and his predecessors the method +\index{Euclid!ref.\ to}% +of exhaustion was only the means of proving propositions +which must have been seen and believed before they were +proved. But in the hands of Archimedes it became an instrument +of discovery.\Reference{9} +%% -----File: 057.png---Folio 43------- + +By the word `conoid,' in his book on \textit{Conoids and +Spheroids}, is meant the solid produced by the revolution +of a parabola or a hyperbola about its axis. Spheroids +are produced by the revolution of an ellipse, and are long +or flat, according as the ellipse revolves around the major +or minor axis. The book leads up to the cubature of these +solids. + +We have now reviewed briefly all his extant works on geometry. +His arithmetical treatise and problems will be considered +later. We shall now notice his works on mechanics. +\index{Mechanics!Greek}% +Archimedes is the author of the first sound knowledge on this +subject. Archytas, Aristotle, and others attempted to form +\index{Archytas!ref.\ to}% +\index{Aristotle!ref.\ to}% +the known mechanical truths into a science, but failed. Aristotle +knew the property of the lever, but could not establish +its true mathematical theory. The radical and fatal defect +in the speculations of the Greeks, says Whewell, was ``that +\index{Whewell}% +though they had in their possession facts and ideas, \emph{the ideas +were not distinct and appropriate to the facts}.'' For instance, +Aristotle asserted that when a body at the end of a lever is +moving, it may be considered as having two motions; one in +the direction of the tangent and one in the direction of the +radius; the former motion is, he says, \emph{according to nature}, the +latter \emph{contrary to nature}. These inappropriate notions of +`natural' and `unnatural' motions, together with the habits +of thought which dictated these speculations, made the perception +of the true grounds of mechanical properties impossible.\Reference{11} +It seems strange that even after Archimedes had +entered upon the right path, this science should have remained +absolutely stationary till the time of Galileo---a period of +\index{Galileo!ref.\ to}% +nearly two thousand years. + +The proof of the property of the lever, given in his \textit{Equiponderance +of Planes}, holds its place in text-books to this day. +His estimate of the efficiency of the lever is expressed in the +%% -----File: 058.png---Folio 44------- +\index{Statics}% +saying attributed to him, ``Give me a fulcrum on which to +rest, and I will move the earth.'' + +While the \textit{Equiponderance} treats of solids, or the equilibrium +of solids, the book on \textit{Floating Bodies} treats of hydrostatics. +\index{Hydrostatics}% +His attention was first drawn to the subject of +specific gravity when King Hieron asked him to test whether +a crown, professed by the maker to be pure gold, was not +alloyed with silver. The story goes that our philosopher was +in a bath when the true method of solution flashed on his +mind. He immediately ran home, naked, shouting, ``I have +found it!'' To solve the problem, he took a piece of gold and +a piece of silver, each weighing the same as the crown. According +to one author, he determined the volume of water +displaced by the gold, silver, and crown respectively, and +calculated from that the amount of gold and silver in the +crown. According to another writer, he weighed separately +the gold, silver, and crown, while immersed in water, thereby +determining their loss of weight in water. From these data +he easily found the solution. It is possible that Archimedes +solved the problem by both methods. + +After examining the writings of Archimedes, one can well +understand how, in ancient times, an `Archimedean problem' +came to mean a problem too deep for ordinary minds to solve, +and how an `Archimedean proof' came to be the synonym for +unquestionable certainty. Archimedes wrote on a very wide +range of subjects, and displayed great profundity in each. He +is the Newton of antiquity. + +\textbf{Eratosthenes}, eleven years younger than Archimedes, was a +\index{Eratosthenes}% +native of Cyrene. He was educated in Alexandria under +Callimachus the poet, whom he succeeded as custodian of +the Alexandrian Library. His many-sided activity may be +inferred from his works. He wrote on \textit{Good and Evil}, \textit{Measurement +of the Earth}, \textit{Comedy}, \textit{Geography}, \textit{Chronology}, \textit{Constellations}, +%% -----File: 059.png---Folio 45------- +and the \textit{Duplication of the Cube}. He was also a +\index{Duplication of the cube}% +philologian and a poet. He measured the obliquity of the +ecliptic and invented a device for finding prime numbers. +\index{Prime numbers}% +Of his geometrical writings we possess only a letter to +Ptolemy Euergetes, giving a history of the duplication problem +and also the description of a very ingenious mechanical +contrivance of his own to solve it. In his old age he lost +his eyesight, and on that account is said to have committed +suicide by voluntary starvation. + +About forty years after Archimedes flourished \textbf{Apollonius of +\index{Apollonius|(}% +\index{Archimedes!ref.\ to}% +Perga}, whose genius nearly equalled that of his great predecessor. +He incontestably occupies the second place in distinction +among ancient mathematicians. Apollonius was +born in the reign of Ptolemy Euergetes and died under +Ptolemy Philopator, who reigned 222--205~\BC. He studied at +Alexandria under the successors of Euclid, and for some time, +also, at Pergamum, where he made the acquaintance of that +Eudemus to whom he dedicated the first three books of his +\index{Eudemus}% +\textit{Conic Sections}. The brilliancy of his great work brought him +\index{Conic sections!Greek|(}% +the title of the ``Great Geometer.'' This is all that is known +of his life. + +His \textit{Conic Sections} were in eight books, of which the first +four only have come down to us in the original Greek. The +next three books were unknown in Europe till the middle of +the seventeenth century, when an Arabic translation, made +about~1250, was discovered. The eighth book has never been +found. In~1710 Halley of Oxford published the Greek text +\index{Halley}% +of the first four books and a Latin translation of the remaining +three, together with his conjectural restoration of the +eighth book, founded on the introductory lemmas of Pappus. +\index{Pappus!ref.\ to}% +The first four books contain little more than the substance +of what earlier geometers had done. Eutocius tells us that +\index{Eutocius!ref.\ to}% +Heraclides, in his life of Archimedes, accused Apollonius of +\index{Archimedes|)}% +\index{Heraclides}% +%% -----File: 060.png---Folio 46------- +having appropriated, in his \textit{Conic Sections}, the unpublished +discoveries of that great mathematician. It is difficult to +believe that this charge rests upon good foundation. Eutocius +\index{Eutocius!ref.\ to}% +quotes Geminus as replying that neither Archimedes nor +\index{Geminus!ref.\ to}% +Apollonius claimed to have invented the conic sections, but +that Apollonius had introduced a real improvement. While +the first three or four books were founded on the works of +Menæchmus, Aristæus, Euclid, and Archimedes, the remaining +\index{Aristæus!ref.\ to}% +\index{Euclid!ref.\ to}% +\index{Menæchmus!ref.\ to}% +ones consisted almost entirely of new matter. The first three +books were sent to Eudemus at intervals, the other books +\index{Eudemus}% +(after Eudemus's death) to one Attalus. The preface of the +\index{Attalus}% +second book is interesting as showing the mode in which +Greek books were `published' at this time. It reads thus: +``I have sent my son Apollonius to bring you (Eudemus) the +second book of my Conics. Read it carefully and communicate +it to such others as are worthy of it. If Philonides, the +\index{Philonides}% +geometer, whom I introduced to you at Ephesus, comes into +the neighbourhood of Pergamum, give it to him also.''\Reference{12} + +The first book, says Apollonius in his preface to it, ``contains +the mode of producing the three sections and the conjugate +hyperbolas and their principal characteristics, more fully +and generally worked out than in the writings of other +authors.'' We remember that Menæchmus, and all his successors +down to Apollonius, considered only sections of \emph{right} +cones by a plane perpendicular to their sides, and that the +three sections were obtained each from a different cone. +Apollonius introduced an important generalisation. He produced +all the sections from one and the same cone, whether +right or scalene, and by sections which may or may not be +perpendicular to its sides. The old names for the three curves +were now no longer applicable. Instead of calling the three +curves, sections of the `acute-angled,' `right-angled,' and +`obtuse-angled' cone, he called them \emph{ellipse}, \emph{parabola}, and +%% -----File: 061.png---Folio 47------- +\emph{hyperbola}, respectively. To be sure, we find the words `parabola' +and `ellipse' in the works of Archimedes, but they are +\index{Archimedes!ref.\ to}% +probably only interpolations. The word `ellipse' was applied +because $y^2 < px$, $p$~being the parameter; the word `parabola' +was introduced because $y^2 = px$, and the term `hyperbola' +because $y^2 > px$. + +The treatise of Apollonius rests on a unique property of +conic sections, which is derived directly from the nature of +the cone in which these sections are found. How this property +forms the key to the system of the ancients is told in a masterly +way by M.~Chasles.\Reference{13} ``Conceive,'' says he, ``an oblique +\index{Chasles!ref.\ to}% +cone on a circular base; the straight line drawn from its +summit to the centre of the circle forming its base is called +the \emph{axis} of the cone. The plane passing through the axis, +perpendicular to its base, cuts the cone along two lines and +determines in the circle a diameter; the triangle having this +diameter for its base and the two lines for its sides, is called +\emph{the triangle through the axis}. In the formation of his conic +sections, Apollonius supposed the cutting plane to be perpendicular +to the plane of the triangle through the axis. The +points in which this plane meets the two sides of this triangle +are the \emph{vertices} of the curve; and the straight line which joins +these two points is a diameter of it. Apollonius called this +diameter \textit{latus transversum}. At one of the two vertices of the +curve erect a perpendicular (\textit{latus rectum}) to the plane of the +triangle through the axis, of a certain length, to be determined +as we shall specify later, and from the extremity of this perpendicular +draw a straight line to the other vertex of the +curve; now, through any point whatever of the diameter of +the curve, draw at right angles an \emph{ordinate}: the square of this +ordinate, comprehended between the diameter and the curve, +will be equal to the rectangle constructed on the portion of +the ordinate comprised between the diameter and the straight +%% -----File: 062.png---Folio 48-------%[** Quote continues] +line, and the part of the diameter comprised between the first +vertex and the foot of the ordinate. Such is the characteristic +property which Apollonius recognises in his conic sections and +which he uses for the purpose of inferring from it, by adroit +transformations and deductions, nearly all the rest. It plays, +as we shall see, in his hands, almost the same rôle as the +equation of the second degree with two variables (abscissa and +ordinate) in the system of analytic geometry of Descartes. +\index{Descartes!ref.\ to}% + +``It will be observed from this that the diameter of the +curve and the perpendicular erected at one of its extremities +suffice to construct the curve. These are the two elements +which the ancients used, with which to establish their theory +of conics. The perpendicular in question was called by them +\textit{latus erectum}; the moderns changed this name first to that of +\textit{latus rectum}, and afterwards to that of \emph{parameter}.'' +\index{Latus rectum}% +\index{Parameter}% + +The first book of the \textit{Conic Sections} of Apollonius is almost +wholly devoted to the generation of the three principal conic +sections. + +The second book treats mainly of asymptotes, axes, and +diameters. + +The third book treats of the equality or proportionality +of triangles, rectangles, or squares, of which the component +parts are determined by portions of transversals, chords, +asymptotes, or tangents, which are frequently subject to a +great number of conditions. It also touches the subject of +foci of the ellipse and hyperbola. + +In the fourth book, Apollonius discusses the harmonic division +of straight lines. He also examines a system of two +conics, and shows that they cannot cut each other in more +than four points. He investigates the various possible relative +positions of two conics, as, for instance, when they have one +or two points of contact with each other. + +The fifth book reveals better than any other the giant +%% -----File: 063.png---Folio 49------- +\index{Maxima and minima}% +\index{Quadrature of curves}% +intellect of its author. Difficult questions of \emph{maxima and +minima}, of which few examples are found in earlier works, are +here treated most exhaustively. The subject investigated is, +to find the longest and shortest lines that can be drawn from +a given point to a conic. Here are also found the germs of +the subject of \emph{evolutes} and \emph{centres of osculation}. +\index{Centres of osculation}% +\index{Evolutes}% + +The sixth book is on the similarity of conics. + +The seventh book is on conjugate diameters. + +The eighth book, as restored by Halley, continues the subject +of conjugate diameters. + +It is worthy of notice that Apollonius nowhere introduces +the notion of \emph{directrix} for a conic, and that, though he incidentally +\index{Directrix}% +discovered the \emph{focus} of an ellipse and hyperbola, he did +\index{Focus}% +not discover the focus of a parabola.\Reference{6} Conspicuous in his +geometry is also the absence of technical terms and symbols, +which renders the proofs long and cumbrous. + +The discoveries of Archimedes and Apollonius, says M. Chasles,\Reference[]{13} +\index{Archimedes!ref.\ to}% +\index{Chasles!ref.\ to}% +marked the most brilliant epoch of ancient geometry. +Two questions which have occupied geometers of all periods +may be regarded as having originated with them. The first +of these is the quadrature of curvilinear figures, which gave +birth to the infinitesimal calculus. The second is the theory +of conic sections, which was the prelude to the theory of +geometrical curves of all degrees, and to that portion of +\index{Curves!quadrature of}% +geometry which considers only the forms and situations +of figures, and uses only the intersection of lines and surfaces +and the ratios of rectilineal distances. These two great +divisions of geometry may be designated by the names of +\emph{Geometry of Measurements} and \emph{Geometry of Forms and Situations}, +or, Geometry of Archimedes and of Apollonius. + +Besides the \textit{Conic Sections}, Pappus ascribes to Apollonius +\index{Conic sections!Greek|)}% +\index{Pappus!ref.\ to}% +the following works: \textit{On Contacts}, \textit{Plane Loci}, \textit{Inclinations}, +\textit{Section of an Area}, \textit{Determinate Section}, and gives lemmas +%% -----File: 064.png---Folio 50------- +\index{Duplication of the cube}% +\index{Trisection of angles}% +from which attempts have been made to restore the lost +originals. Two books on \textit{De Sectione Rationis} have been +found in the Arabic. The book on \textit{Contacts}, as restored by +Vieta, contains the so-called ``Apollonian Problem'': Given +\index{Apollonian Problem}% +\index{Vieta!ref.\ to}% +three circles, to find a fourth which shall touch the three. + +Euclid, Archimedes, and Apollonius brought geometry to +\index{Archimedes!ref.\ to}% +\index{Euclid!ref.\ to}% +as high a state of perfection as it perhaps could be brought +without first introducing some more general and more powerful +method than the old method of exhaustion. A briefer symbolism, +a Cartesian geometry, an infinitesimal calculus, were +needed. The Greek mind was not adapted to the invention of +general methods. Instead of a climb to still loftier heights +we observe, therefore, on the part of later Greek geometers, a +descent, during which they paused here and there to look +around for details which had been passed by in the hasty +ascent.\Reference{3} + +Among the earliest successors of Apollonius was \textbf{Nicomedes}. +\index{Apollonius|)}% +\index{Nicomedes}% +Nothing definite is known of him, except that he invented the +\emph{conchoid} (``mussel-like''). He devised a little machine by +\index{Conchoid}% +which the curve could be easily described. With aid of the +conchoid he duplicated the cube. The curve can also be used +for trisecting angles in a way much resembling that in the +eighth lemma of Archimedes. Proclus ascribes this mode of +\index{Proclus!ref.\ to}% +trisection to Nicomedes, but Pappus, on the other hand, claims +\index{Pappus!ref.\ to}% +it as his own. The conchoid was used by Newton in constructing +\index{Newton!ref.\ to}% +curves of the third degree. + +About the time of Nicomedes, flourished also \textbf{Diocles}, the +\index{Diocles}% +inventor of the \emph{cissoid} (``ivy-like''). This curve he used for +\index{Cissoid}% +finding two mean proportionals between two given straight +lines. + +About the life of \textbf{Perseus} we know as little as about that of +\index{Perseus}% +Nicomedes and Diocles. He lived some time between 200 and +100~\BC. From Heron and Geminus we learn that he wrote a +\index{Geminus!ref.\ to}% +\index{Heron the Elder!ref.\ to}% +%% -----File: 065.png---Folio 51------- +\index{Astronomy!Greek}% +\index{Isoperimetrical figures}% +work on the \emph{spire}, a sort of anchor-ring surface described by +Heron as being produced by the revolution of a circle around +one of its chords as an axis. The sections of this surface +yield peculiar curves called \emph{spiral sections}, which, according to +Geminus, were thought out by Perseus. These curves appear +to be the same as the \textit{Hippopede} of Eudoxus. +\index{Eudoxus!ref.\ to}% +\index{Hippopede}% + +Probably somewhat later than Perseus lived \textbf{Zenodorus}. He +\index{Zenodorus}% +wrote an interesting treatise on a new subject; namely, \emph{isoperimetrical +figures}. Fourteen propositions are preserved by +Pappus and Theon. Here are a few of them: Of isoperimetrical, +\index{Theon of Alexandria!ref.\ to}% +regular polygons, the one having the largest number of +angles has the greatest area; the circle has a greater area than +any regular polygon of equal periphery; of all isoperimetrical +polygons of $n$~sides, the regular is the greatest; of all solids +having surfaces equal in area, the sphere has the greatest +volume. + +\textbf{Hypsicles} (between 200 and 100~\BC) was supposed to be +\index{Hypsicles}% +the author of both the fourteenth and fifteenth books of +Euclid, but recent critics are of opinion that the fifteenth +book was written by an author who lived several centuries +after Christ. The fourteenth book contains seven elegant +theorems on \emph{regular solids}. A treatise of Hypsicles on \textit{Risings} +\index{Regular solids}% +is of interest because it is the first Greek work giving the +division of the circumference into $360$~degrees after the fashion +of the Babylonians. +\index{Babylonians!ref.\ to}% + +\textbf{Hipparchus} of Nicæa in Bithynia was the greatest astronomer +\index{Hipparchus}% +of antiquity. He established inductively the famous +theory of epicycles and eccentrics. As might be expected, he +\index{Epicycles}% +was interested in mathematics, not \textit{per se}, but only as an aid +to astronomical inquiry. No mathematical writings of his +are extant, but Theon of Alexandria informs us that Hipparchus +originated the science of \emph{trigonometry}, and that he calculated +\index{Trigonometry}% +a ``table of chords'' in twelve books. Such calculations +%% -----File: 066.png---Folio 52------- +\index{Circle}% +must have required a ready knowledge of arithmetical and +algebraical operations. + +About 100~\BC\ flourished \textbf{Heron the Elder} of Alexandria. +\index{Heron the Elder}% +He was the pupil of Ctesibius, who was celebrated for his +\index{Ctesibius}% +ingenious mechanical inventions, such as the hydraulic organ, +the water-clock, and catapult. It is believed by some that +Heron was a son of Ctesibius. He exhibited talent of the +same order as did his master by the invention of the eolipile +and a curious mechanism known as ``Heron's fountain.'' +Great uncertainty exists concerning his writings. Most authorities +believe him to be the author of an important \textit{Treatise +on the Dioptra}, of which there exist three manuscript copies, +quite dissimilar. But M.~Marie\Reference[]{14} thinks that the \textit{Dioptra} is +\index{Marie, M.}% +the work of \textit{Heron the Younger}, who lived in the seventh or +eighth century after Christ, and that \textit{Geodesy}, another book +supposed to be by Heron, is only a corrupt and defective copy +of the former work. \textit{Dioptra} contains the important formula +for finding the area of a triangle expressed in terms of its +sides; its derivation is quite laborious and yet exceedingly +ingenious. ``It seems to me difficult to believe,'' says Chasles, +\index{Chasles!ref.\ to}% +``that so beautiful a theorem should be found in a work so +ancient as that of Heron the Elder, without that some Greek +geometer should have thought to cite it.'' Marie lays great +stress on this silence of the ancient writers, and argues from +it that the true author must be Heron the Younger or some +writer much more recent than Heron the Elder. But no reliable +evidence has been found that there actually existed a +second mathematician by the name of Heron. + +``Dioptra,'' says Venturi, were instruments which had great +\index{Venturi}% +resemblance to our modern theodolites. The book \textit{Dioptra} is +a treatise on geodesy containing solutions, with aid of these +instruments, of a large number of questions in geometry, such +as to find the distance between two points, of which one only +%% -----File: 067.png---Folio 53------- +is accessible, or between two points which are visible but both +inaccessible; from a given point to draw a perpendicular to a +line which cannot be approached; to find the difference of +level between two points; to measure the area of a field without +entering it. + +Heron was a practical surveyor. This may account for the +fact that his writings bear so little resemblance to those of +the Greek authors, who considered it degrading the science +to apply geometry to surveying. The character of his geometry +is not Grecian, but decidedly Egyptian. This fact is the +more surprising when we consider that Heron demonstrated +his familiarity with Euclid by writing a commentary on the +\index{Euclid!ref.\ to}% +\textit{Elements}.\Reference{21} Some of Heron's formulas point to an old Egyptian +origin. Thus, besides the above exact formula for the +area of a triangle in terms of its sides, Heron gives the formula +$\dfrac{a_1 + a_2\Fstr}{2} × \dfrac{b}{2}$, which bears a striking likeness to the formula +$\dfrac{a_1 + a_2}{2} × \dfrac{b_1 + b_2}{2}$ for finding the area of a quadrangle, +found in the Edfu inscriptions. There are, moreover, points +\index{Edfu}% +of resemblance between Heron's writings and the ancient +Ahmes papyrus. Thus Ahmes used unit-fractions exclusively; +\index{Ahmes!ref.\ to}% +Heron uses them oftener than other fractions. Like Ahmes +and the priests at Edfu, Heron divides complicated figures +into simpler ones by drawing auxiliary lines; like them, +he shows, throughout, a special fondness for the isosceles +trapezoid. + +The writings of Heron satisfied a practical want, and for +that reason were borrowed extensively by other peoples. We +find traces of them in Rome, in the Occident during the Middle +Ages, and even in India. + +\textbf{Geminus} of Rhodes (about 70~\BC) published an astronomical +\index{Geminus}% +work still extant. He wrote also a book, now lost, on the +\textit{Arrangement of Mathematics}, which contained many valuable +%% -----File: 068.png---Folio 54------- +notices of the early history of Greek mathematics. Proclus +\index{Proclus!ref.\ to}% +and Eutocius quote it frequently. \textbf{Theodosius} of Tripolis is +\index{Eutocius!ref.\ to}% +\index{Theodosius}% +the author of a book of little merit on the geometry of the +sphere. \textbf{Dionysodorus} of Amisus in Pontus applied the intersection +\index{Dionysodorus}% +of a parabola and hyperbola to the solution of a problem +which Archimedes, in his \textit{Sphere and Cylinder}, had left +\index{Archimedes!ref.\ to}% +incomplete. The problem is ``to cut a sphere so that its segments +shall be in a given ratio.'' + +We have now sketched the progress of geometry down to +the time of Christ. Unfortunately, very little is known of +the history of geometry between the time of Apollonius and +\index{Apollonius!ref.\ to}% +the beginning of the Christian era. The names of quite a +number of geometers have been mentioned, but very few of +their works are now extant. It is certain, however, that there +were no mathematicians of real genius from Apollonius to +Ptolemy, excepting Hipparchus and perhaps Heron. +\index{Heron the Elder!ref.\ to}% +\index{Hipparchus!ref.\ to}% +\index{Ptolemy!ref.\ to}% + +\Subsection{The Second Alexandrian School} +\index{Alexandrian School!(first)|)}% +\index{Alexandrian School!(second)|(}% + +The close of the dynasty of the Lagides which ruled Egypt +from the time of Ptolemy Soter, the builder of Alexandria, +for 300~years; the absorption of Egypt into the Roman Empire; +the closer commercial relations between peoples of the +East and of the West; the gradual decline of paganism and +spread of Christianity,---these events were of far-reaching +influence on the progress of the sciences, which then had their +home in Alexandria. Alexandria became a commercial and +intellectual emporium. Traders of all nations met in her +busy streets, and in her magnificent Library, museums, lecture-halls, +scholars from the East mingled with those of the +West; Greeks began to study older literatures and to compare +them with their own. In consequence of this interchange +of ideas the Greek philosophy became fused with Oriental +%% -----File: 069.png---Folio 55------- +\index{Conic sections!Greek}% +\index{Ptolemy!ref.\ to}% +philosophy. Neo-Pythagoreanism and Neo-Platonism were the +names of the modified systems. These stood, for a time, in +opposition to Christianity. The study of Platonism and +Pythagorean mysticism led to the revival of the theory of +numbers. Perhaps the dispersion of the Jews and their +\index{Numbers!theory of numbers}% +introduction to Greek learning helped in bringing about this +revival. The theory of numbers became a favourite study. +\index{Theory of numbers}% +This new line of mathematical inquiry ushered in what we +may call a new school. There is no doubt that even now +geometry continued to be one of the most important studies +in the Alexandrian course. This Second Alexandrian School +may be said to begin with the Christian era. It was made +famous by the names of Claudius Ptolemæus, Diophantus, +\index{Diophantus!ref.\ to}% +Pappus, Theon of Smyrna, Theon of Alexandria, Iamblichus, +\index{Pappus!ref.\ to}% +\index{Theon of Alexandria!ref.\ to}% +\index{Theon of Smyrna}% +Porphyrius, and others. +\index{Porphyrius}% + +By the side of these we may place \textbf{Serenus} of Antissa, as +\index{Serenus}% +having been connected more or less with this new school. +He wrote on sections of the cone and cylinder, in two books, +one of which treated only of the triangular section of the cone +through the apex. He solved the problem, ``given a cone +(cylinder), to find a cylinder (cone), so that the section of +both by the same plane gives similar ellipses.'' Of particular +interest is the following theorem, which is the foundation +of the modern theory of +harmonics: If from $D$ we +\index{Harmonics}% +% [Illustration] +\begin{wrapfigure}{r}{2.25in} %[* PP: Width-dependent line break] +\hfill\Graphic{2.25in}{069a} +\end{wrapfigure} +draw~$DF$, cutting the triangle +$ABC$, and choose +$H$ on it, so that $DE : DF += EH : HF$, and if we draw +the line~$AH$, then every transversal through $D$, such as +$DG$, will be divided by $AH$ so that $DK : DG = KJ : JG$. +\textbf{Menelaus} of Alexandria (about 98~\AD) was the author of +\index{Menelaus}% +\textit{Sphærica}, a work extant in Hebrew and Arabic, but not +%% -----File: 070.png---Folio 56------- +\index{Astronomy!Greek}% +\index{Ptolemaic System}% +\index{Spherical trigonometry}% +in Greek. In it he proves the theorems on the congruence +of spherical triangles, and describes their properties in +much the same way as Euclid treats plane triangles. In +it are also found the theorems that the sum of the three +sides of a spherical triangle is less than a great circle, and +that the sum of the three angles exceeds two right angles. +Celebrated are two theorems of his on plane and spherical +triangles. The one on plane triangles is that, ``if the three +sides be cut by a straight line, the product of the three segments +which have no common extremity is equal to the +product of the other three.'' The illustrious Carnot makes +\index{Carnot, Lazare!ref.\ to}% +this proposition, known as the `lemma of Menelaus,' the base +of his theory of transversals. The corresponding theorem for +spherical triangles, the so-called `regula sex quantitatum,' is +obtained from the above by reading ``chords of three segments +doubled,'' in place of ``three segments.'' + +\textbf{Claudius Ptolemæus}, a celebrated astronomer, was a native of +Egypt. Nothing is known of his personal history except that +he flourished in Alexandria in 139~\AD\ and that he made the +earliest astronomical observations recorded in his works, in +125~\AD, the latest in 151~\AD. The chief of his works are +the \textit{Syntaxis Mathematica} (or the \textit{Almagest}, as the Arabs call +\index{Almagest|(}% +it) and the \textit{Geographica}, both of which are extant. The +former work is based partly on his own researches, but mainly +on those of Hipparchus. Ptolemy seems to have been not +\index{Hipparchus!ref.\ to}% +\index{Ptolemy|(}% +so much of an independent investigator, as a corrector and +improver of the work of his great predecessors. The \textit{Almagest} +forms the foundation of all astronomical science down to +Copernicus. The fundamental idea of his system, the ``Ptolemaic +\index{Copernicus}% +System,'' is that the earth is in the centre of the universe, +and that the sun and planets revolve around the earth. +Ptolemy did considerable for mathematics. He created, for +astronomical use, a \emph{trigonometry} remarkably perfect in form. +\index{Trigonometry}% +%% -----File: 071.png---Folio 57------- +\index{Fractions!sexagesimal}% +\index{Sexagesimal system}% +The foundation of this science was laid by the illustrious +Hipparchus. + +The \textit{Almagest} is in 13~books. Chapter~9 of the first book +shows how to calculate tables of chords. The circle is divided +into $360$~degrees, each of which is halved. The diameter is +divided into $120$~divisions; each of these into $60$~parts, which +are again subdivided into $60$~smaller parts. In Latin, these +parts were called \textit{partes minutæ primæ} and \textit{partes minutæ +secundæ}. Hence our names, `minutes' and `seconds.'\Reference{3} The +sexagesimal method of dividing the circle is of Babylonian +origin, and was known to Geminus and Hipparchus. But +\index{Geminus!ref.\ to}% +Ptolemy's method of calculating chords seems original with +him. He first proved the proposition, now appended to +Euclid~VI.~(D), that ``the rectangle contained by the diagonals +\index{Euclid!ref.\ to}% +of a quadrilateral figure inscribed in a circle is equal +to both the rectangles contained by its opposite sides.'' He +then shows how to find from the chords of two arcs the +chords of their sum and difference, and from the chord of any +arc that of its half. These theorems he applied to the calculation +of his tables of chords. The proofs of these theorems +are very pretty. + +Another chapter of the first book in the \textit{Almagest} is devoted +to \emph{trigonometry}, and to \emph{spherical} trigonometry in particular. +\index{Trigonometry!spherical}% +Ptolemy proved the `lemma of Menelaus,' and also the `regula +\index{Menelaus!ref.\ to}% +sex quantitatum.' Upon these propositions he built up his +trigonometry. The fundamental theorem of plane trigonometry, +that two sides of a triangle are to each other as the chords +of double the arcs measuring the angles opposite the two +sides, was not stated explicitly by him, but was contained +implicitly in other theorems. More complete are the propositions +in spherical trigonometry. + +The fact that trigonometry was cultivated not for its own +sake, but to aid astronomical inquiry, explains the rather +%% -----File: 072.png---Folio 58------- +startling fact that spherical trigonometry came to exist in a +developed state earlier than plane trigonometry. + +The remaining books of the \textit{Almagest} are on astronomy. +Ptolemy has written other works which have little or no bearing +on mathematics, except one on geometry. Extracts from +this book, made by Proclus, indicate that Ptolemy did not +\index{Proclus!ref.\ to}% +regard the parallel-axiom of Euclid as self-evident, and that +\index{Euclid!ref.\ to}% +Ptolemy was the first of the long line of geometers from +ancient time down to our own who toiled in the vain attempt +to prove it. + +Two prominent mathematicians of this time were Nicomachus +\index{Nicomachus!ref.\ to}% +and Theon of Smyrna. Their favourite study was theory +\index{Theon of Smyrna}% +of numbers. The investigations in this science culminated +later in the algebra of Diophantus. But no important geometer +appeared after Ptolemy for 150~years. The only occupant +\index{Ptolemy|)}% +of this long gap was \textbf{Sextus Julius Africanus}, who wrote an +\index{Sextus Julius Africanus}% +unimportant work on geometry applied to the art of war, +entitled \textit{Cestes}. + +\textbf{Pappus}, probably born about 340~\AD, in Alexandria, was +\index{Pappus|(}% +the last great mathematician of the Alexandrian school. His +genius was inferior to that of Archimedes, Apollonius, and +Euclid, who flourished over 500~years earlier. But living, +as he did, at a period when interest in geometry was declining, +he towered above his contemporaries ``like the peak +of Teneriffa above the Atlantic.'' He is the author of a \textit{Commentary +on the Almagest}, a \textit{Commentary on Euclid's Elements}, +\index{Almagest|)}% +a \textit{Commentary on the Analemma of Diodorus},---a writer of +\index{Diodorus}% +whom nothing is known. All these works are lost. Proclus, +probably quoting from the \textit{Commentary on Euclid}, says that +Pappus objected to the statement that an angle equal to a +right angle is always itself a right angle. + +The only work of Pappus still extant is his \textit{Mathematical +Collections}. This was originally in eight books, but the first +%% -----File: 073.png---Folio 59------- +and portions of the second are now missing. The \textit{Mathematical +Collections} seems to have been written by Pappus to supply +the geometers of his time with a succinct analysis of the most +difficult mathematical works and to facilitate the study of +them by explanatory lemmas. But these lemmas are selected +very freely, and frequently have little or no connection with the +subject on hand. However, he gives very accurate summaries +of the works of which he treats. The \textit{Mathematical Collections} +is invaluable to us on account of the rich information it +gives on various treatises by the foremost Greek mathematicians, +which are now lost. Mathematicians of the last century +considered it possible to restore lost works from the \textit{résumé} +by Pappus alone. + +We shall now cite the more important of those theorems in +the \textit{Mathematical Collections} which are supposed to be original +with Pappus. First of all ranks the elegant theorem re-discovered +by \textit{Guldin},\DPnote{** PP: [sic], not boldface} over 1000~years later, that the volume +\index{Guldin!ref.\ to}% +generated by the revolution of a plane curve which lies wholly +on one side of the axis, equals the area of the curve multiplied +by the circumference described by its centre of gravity. +Pappus proved also that the centre of gravity of a triangle is +that of another triangle whose vertices lie upon the sides of +the first and divide its three sides in the same ratio. In the +fourth book are new and brilliant propositions on the quadratrix +\index{Quadratrix}% +which indicate an intimate acquaintance with curved +surfaces. He generates the quadratrix as follows: Let a +spiral line be drawn upon a right circular cylinder; then the +perpendiculars to the axis of the cylinder drawn from each +point of the spiral line form the surface of a screw. A plane +passed through one of these perpendiculars, making any convenient +angle with the base of the cylinder, cuts the screw-surface +in a curve, the orthogonal projection of which upon +the base is the \emph{quadratrix}. A second mode of generation is +%% -----File: 074.png---Folio 60------- +\index{Problem of Pappus}% +\index{Spirals}% +no less admirable: If we make the spiral of Archimedes the +base of a right cylinder, and imagine a cone of revolution +having for its axis the side of the cylinder passing through +the initial point of the spiral, then this cone cuts the cylinder +in a curve of double curvature. The perpendiculars to the +axis drawn through every point in this curve form the surface +of a screw which Pappus here calls the \emph{plectoidal surface}. A +\index{Plectoidal surface}% +plane passed through one of the perpendiculars at any convenient +angle cuts that surface in a curve whose orthogonal +projection upon the plane of the spiral is the required \emph{quadratrix}. +\index{Quadratrix}% +Pappus considers curves of double curvature still further. +He produces a \emph{spherical spiral} by a point moving uniformly +along the circumference of a great circle of a sphere, while +the great circle itself revolves uniformly around its diameter. +He then finds the area of that portion of the surface of the +sphere determined by the spherical spiral, ``a complanation +which claims the more lively admiration, if we consider that, +although the entire surface of the sphere was known since +Archimedes' time, to measure portions thereof, such as spherical +triangles, was then and for a long time afterwards an +unsolved problem.''\Reference{3} A question which was brought into +prominence by Descartes and Newton is the ``problem of +\index{Descartes!ref.\ to}% +\index{Newton!ref.\ to}% +Pappus.'' Given several straight lines in a plane, to find the +locus of a point such that when perpendiculars (or, more +generally, straight lines at given angles) are drawn from it to +the given lines, the product of certain ones of them shall be in +a given ratio to the product of the remaining ones. It is +worth noticing that it was Pappus who first found the focus +\index{Focus}% +of the parabola, suggested the use of the directrix, and propounded +\index{Directrix}% +the theory of the involution of points. He solved +\index{Involution of points}% +the problem to draw through three points lying in the same +straight line, three straight lines which shall form a triangle +inscribed in a given circle.\Reference{3} From the \textit{Mathematical Collections} +%% -----File: 075.png---Folio 61------- +many more equally difficult theorems might be quoted which +are original with Pappus as far as we know. It ought to be +remarked, however, that he is known in three instances to +have copied theorems without giving due credit, and that he +may have done the same thing in other cases in which we +have no data by which to ascertain the real discoverer. + +About the time of Pappus lived \textbf{Theon} of Alexandria. He +\index{Pappus|)}% +\index{Theon of Alexandria}% +brought out an edition of Euclid's \textit{Elements} with notes, which +\index{Elements (Euclid's)}% +\index{Euclid!ref.\ to}% +he probably used as a text-book in his classes. His commentary +on the \textit{Almagest} is valuable for the many historical notices, +and especially for the specimens of Greek arithmetic which it +contains. Theon's daughter \textbf{Hypatia}, a woman celebrated for +\index{Hypatia}% +her beauty and modesty, was the last Alexandrian teacher of +reputation, and is said to have been an abler philosopher and +mathematician than her father. Her notes on the works of +Diophantus and Apollonius have been lost. Her tragic death +\index{Apollonius!ref.\ to}% +\index{Diophantus!ref.\ to}% +in 415~\AD\ is vividly described in Kingsley's \textit{Hypatia}. + +From now on, mathematics ceased to be cultivated in +Alexandria. The leading subject of men's thoughts was +Christian theology. Paganism disappeared, and with it pagan +learning. The Neo-Platonic school at Athens struggled on a +century longer. Proclus, Isidorus, and others kept up the +\index{Isidorus of Seville!ref.\ to}% +\index{Proclus}% +``golden chain of Platonic succession.'' \textbf{Proclus}, the successor +of Syrianus, at the Athenian school, wrote a commentary on +Euclid's \textit{Elements}. We possess only that on the first book, +which is valuable for the information it contains on the +history of geometry. \textbf{Damascius} of Damascus, the pupil of +\index{Damascius}% +Isidorus, is now believed to be the author of the fifteenth +book of Euclid. Another pupil of Isidorus was \textbf{Eutocius} of +\index{Eutocius}% +Ascalon, the commentator of Apollonius and Archimedes. +\index{Archimedes!ref.\ to}% +\textbf{Simplicius} wrote a commentary on Aristotle's \textbf{De C\oe{}lo}. In +\index{Aristotle!ref.\ to}% +\index{Simplicius}% +the year~529, Justinian, disapproving heathen learning, finally +closed by imperial edict the schools at Athens. +%% -----File: 076.png---Folio 62------- +\index{Alexandrian School!(second)|)}% + +As a rule, the geometers of the last 500~years showed +a lack of creative power. They were commentators rather +than discoverers. + +The principal characteristics of ancient geometry are:--- + +(1)~A wonderful clearness and definiteness of its concepts +and an almost perfect logical rigour of its conclusions. + +(2)~A complete want of general principles and methods. +Ancient geometry is decidedly \emph{special}. Thus the Greeks +\index{Geometry!Greek|)}% +possessed no general method of drawing tangents. ``The +\index{Tangents!in geometry}% +determination of the tangents to the three conic sections did +not furnish any rational assistance for drawing the tangent to +any other new curve, such as the conchoid, the cissoid, etc.''\Reference{15} +In the demonstration of a theorem, there were, for the ancient +geometers, as many different cases requiring separate proof +as there were different positions for the lines. The greatest +geometers considered it necessary to treat all possible cases +independently of each other, and to prove each with equal +fulness. To devise methods by which the various cases could +all be disposed of by one stroke, was beyond the power of the +ancients. ``If we compare a mathematical problem with a +huge rock, into the interior of which we desire to penetrate, +then the work of the Greek mathematicians appears to us like +that of a vigorous stonecutter who, with chisel and hammer, +begins with indefatigable perseverance, from without, to +crumble the rock slowly into fragments; the modern mathematician +appears like an excellent miner, who first bores +through the rock some few passages, from which he then bursts +it into pieces with one powerful blast, and brings to light the +treasures within.''\Reference{16} +%% -----File: 077.png---Folio 63------- +\index{Finger-reckoning}% + +\SpecialSection{Greek Arithmetic} +\index{Arithmetic!Greek|(}% + +Greek mathematicians were in the habit of discriminating +between the \emph{science} of numbers and the \emph{art} of calculation. +The former they called \textit{arithmetica}, the latter \textit{logistica}. The +drawing of this distinction between the two was very natural +and proper. The difference between them is as marked as +that between theory and practice. Among the Sophists the +art of calculation was a favourite study. Plato, on the other +\index{Plato!ref.\ to}% +hand, gave considerable attention to philosophical arithmetic, +but pronounced calculation a vulgar and childish art. + +In sketching the history of Greek calculation, we shall first +give a brief account of the Greek mode of counting and of +writing numbers. Like the Egyptians and Eastern nations, +the earliest Greeks counted on their fingers or with pebbles. +In case of large numbers, the pebbles were probably arranged +in parallel vertical lines. Pebbles on the first line +represented units, those on the second tens, those on the third +hundreds, and so on. Later, frames came into use, in which +strings or wires took the place of lines. According to tradition, +Pythagoras, who travelled in Egypt and, perhaps, in +\index{Pythagoras!ref.\ to}% +India, first introduced this valuable instrument into Greece. +The \emph{abacus}, as it is called, existed among different peoples and +\index{Abacus}% +at different times, in various stages of perfection. An abacus +is still employed by the Chinese under the name of \emph{Swan-pan}. +We possess no specific information as to how the Greek abacus +looked or how it was used. Boethius says that the Pythagoreans +\index{Boethius!ref.\ to}% +used with the abacus certain nine signs called \emph{apices}, +\index{Apices of Boethius!ref.\ to}% +which resembled in form the nine ``Arabic numerals.'' But +the correctness of this assertion is subject to grave doubts. + +The oldest Grecian numerical symbols were the so-called +\emph{Herodianic signs} (after Herodianus, a Byzantine grammarian of +\index{Herodianic signs}% +about 200~\AD, who describes them). These signs occur frequently +%% -----File: 078.png---Folio 64------- +\index{Notation!Greek numbers}% +in Athenian inscriptions and are, on that account, now +generally called \emph{Attic}. For some unknown reason these symbols +were afterwards replaced by the \emph{alphabetic numerals}, in +\index{Numerals!Greek}% +which the letters of the Greek alphabet were used, together +with three strange and antique letters \textgreek{\stigma},~\Glyph{qoppa}, and~\Glyph{sampi}, and the +symbol~$\mathbf{M}$. This change was decidedly for the worse, for the +old Attic numerals were less burdensome on the memory, inasmuch +as they contained fewer symbols and were better adapted +to show forth analogies in numerical operations. The following +table shows the Greek alphabetic numerals and their +respective values:--- +\medskip + +\footnotesize +\setlength{\TmpLen}{0.055555\linewidth}% +\noindent\begin{tabular}{*{18}{@{}c@{}}} +\TE{$\alpha$} & \TE{$\beta$} & \TE{$\gamma$} & \TE{$\delta$} & +\TE{$\epsilon$} & \TE{\textgreek{\stigma}} & +\TE{$\zeta$} & \TE{$\eta$} & \TE{$\theta$} & \TE{$\iota$} & +\TE{$\kappa$} & \TE{$\lambda$} & \TE{$\mu$} & +\TE{$\nu$} & \TE{$\xi$} & \TE{o} & \TE{$\pi$} & \TE{\Glyph{qoppa}} \\ +1&2&3&4&5&6&7&8&9&10&20&30&40&50&60&70&80&90 +\end{tabular} + +\setlength{\TmpLen}{0.076923\linewidth}% +\noindent\begin{tabular}{*{13}{@{}c@{}}} +\TE{$\rho$} & \TE{$\sigma$} & \TE{$\tau$} & \TE{$\upsilon$} & +\TE{$\phi$} & \TE{$\chi$} & \TE{$\psi$} & \TE{$\omega$} & +\TE{\Glyph{sampi}} & +\TE{\textgreek{\katwtonos}$\alpha$} & +\TE{\textgreek{\katwtonos}$\beta$ } & +\TE{\textgreek{\katwtonos}$\gamma$} & \TE{etc.} \\ +100&200&300&400&500&600&700&800&900&1000&2000&3000 +\end{tabular} + +\noindent\begin{tabular}{@{}*{4}{c}@{}} +$\mathbf{M}$&$\overset{\beta}{\mathbf{M}}$&$\overset{\gamma}{\mathbf{M}}$&etc.\\ +10,000&20,000&30,000 +\end{tabular} +\medskip + +\normalsize +It will be noticed that at $1000$, the alphabet is begun over +again, but, to prevent confusion, a stroke is now placed before +the letter and generally somewhat below it. A horizontal line +drawn over a number served to distinguish it more readily +from words. The coefficient for~$\mathbf{M}$ was sometimes placed +before or behind instead of over the~$\mathbf{M}$. Thus $43,678$ was +written $\overline{\delta\text{M\textgreek{\katwtonos}}\gamma\chi o\eta}$. It is to be observed that the Greeks had no +zero. + +Fractions were denoted by first writing the numerator +\index{Fractions!Greek}% +marked with an accent, then the denominator marked with +two accents and written twice. Thus, $\iota \gamma' \kappa \theta'' \kappa \theta'' = \frac{13}{29}$. In case +of fractions having unity for the numerator, the $\alpha'$ was omitted +and the denominator was written only once. Thus $\mu \delta'' = \frac{1}{44}$. +%% -----File: 079.png---Folio 65------- +\index{Sexagesimal system}% + +Greek writers seldom refer to calculation with alphabetic +numerals. Addition, subtraction, and even multiplication were +probably performed on the abacus. Expert mathematicians +may have used the symbols. Thus Eutocius, a commentator +\index{Eutocius!ref.\ to}% +of the sixth century after Christ, gives a great many multiplications +of which the following is a specimen:\Reference[]{6}--- + +\begin{wrapfigure}[8]{l}{2.125in}\footnotesize +\vspace*{-0.0625in}\begin{tabular}{@{}c@{\;}c@{\;}c@{\;}c@{}c@{}r@{\;}r@{\;}r@{}} +\multicolumn{4}{l}{$\quad\overline{\sigma\,\xi\,\epsilon}$} & + $\mspace{18mu}$ & \multicolumn{2}{l}{\quad 2 6 5} \\ +\multicolumn{4}{l}{$\quad\overline{\sigma\,\xi\,\epsilon}$} & + & \multicolumn{2}{l}{\quad 2 6 5} \\\cline{1-4}\cline{6-8} +$\overset{\delta}{\mathbf{M}}$ & $\overset{\alpha}{\mathbf{M}}$ & \textgreek{\katwtonos}$\beta$ & \textgreek{\katwtonos}$\alpha$ & + & 40000, & 12000, & 1000 \\ +$\overset{\alpha}{\mathbf{M}}$ & \textgreek{\katwtonos}$\beta$ & \textgreek{\katwtonos}$\overline{\gamma\chi}$ & $\overline{\tau}$ & + & 12000, & 3600, & 300 \\ +\phantom{$\overset{\alpha}{\mathbf{M}}$} & \textgreek{\katwtonos}$\alpha$ & $\overline{\tau}$ & $\overline{\kappa\epsilon}$ & + & 1000, & 300, & 25 \\\cline{1-4}\cline{6-8} +\multicolumn{4}{c}{$\overset{\zeta}{\mathbf{M}}\enskip\overline{\sigma\,\kappa\,\epsilon}$} & + & 70225\phantom{,} +\end{tabular} +\end{wrapfigure} +The operation is explained sufficiently by the modern numerals appended. +In case of mixed numbers, the process was still more clumsy. Divisions +are found in Theon of Alexandria's commentary on the \textit{Almagest}. As +\index{Theon of Alexandria!ref.\ to}% +might be expected, the process is long and tedious. + +We have seen in geometry that the more advanced mathematicians +frequently had occasion to extract the square root. +\index{Square root}% +Thus Archimedes in his \textit{Mensuration of the Circle} gives a +\index{Archimedes!ref.\ to}% +large number of square roots. He states, for instance, that +$\sqrt{3} < \frac{1351}{780}$ and $\sqrt{3} > \frac{265}{153}$, but he gives no clue to the method +by which he obtained these approximations. It is not improbable +that the earlier Greek mathematicians found the +square root by trial only. Eutocius says that the method of +extracting it was given by Heron, Pappus, Theon, and other +\index{Heron the Elder!ref.\ to}% +\index{Pappus!ref.\ to}% +commentators on the \textit{Almagest}. Theon's is the only ancient +method known to us. It is the same as the one used nowadays, +except that sexagesimal fractions are employed in place +\index{Fractions!Greek}% +\index{Fractions!sexagesimal}% +of our decimals. What the mode of procedure actually was +when sexagesimal fractions were not used, has been the subject +of conjecture on the part of numerous modern writers.\Reference{17} + +Of interest, in connection with arithmetical symbolism, is +the \textit{Sand-Counter} (Arenarius), an essay addressed by \textbf{Archimedes} +\index{Arenarius}% +\index{Sand-counter}% +%% -----File: 080.png---Folio 66------- +to Gelon, king of Syracuse. In it Archimedes shows +that people are in error who think the sand cannot be counted, +or that if it can be counted, the number cannot be expressed +by arithmetical symbols. He shows that the number of grains +in a heap of sand not only as large as the whole earth, but as +large as the entire universe, can be arithmetically expressed. +Assuming that $10,000$ grains of sand suffice to make a little +solid of the magnitude of a poppy-seed, and that the diameter +of a poppy-seed be not smaller than $\frac{1}{40}$~part of a finger's +breadth; assuming further, that the diameter of the universe +(supposed to extend to the sun) be less than $10,000$ diameters +of the earth, and that the latter be less than $1,000,000$ stadia, +Archimedes finds a number which would exceed the number +of grains of sand in the sphere of the universe. He goes on +even further. Supposing the universe to reach out to the fixed +stars, he finds that the sphere, having the distance from the +earth's centre to the fixed stars for its radius, would contain +a number of grains of sand less than $1000$ myriads of the +eighth octad. In our notation, this number would be $10^{63}$ or +$1$~with $63$~ciphers after it. It can hardly be doubted that one +object which Archimedes had in view in making this calculation +was the improvement of the Greek symbolism. It is not +known whether he invented some short notation by which to +represent the above number or not. + +We judge from fragments in the second book of Pappus that +\index{Pappus!ref.\ to}% +Apollonius proposed an improvement in the Greek method of +\index{Apollonius!ref.\ to}% +writing numbers, but its nature we do not know. Thus we +see that the Greeks never possessed the boon of a clear, comprehensive +symbolism. The honour of giving such to the world, +once for all, was reserved by the irony of fate for a nameless +Indian of an unknown time, and we know not whom to thank +for an invention of such importance to the general progress of +intelligence.\Reference{6} +%% -----File: 081.png---Folio 67------- +\index{Arithmetic!Pythagoreans|(}% +\index{Fractions!sexagesimal}% +\index{Sexagesimal system}% + +Passing from the subject of \textit{logistica} to that of \textit{arithmetica}, +our attention is first drawn to the science of numbers of +\textbf{Pythagoras}. Before founding his school, Pythagoras studied +\index{Pythagoras|(}% +for many years under the Egyptian priests and familiarised +himself with Egyptian mathematics and mysticism. If he +ever was in Babylon, as some authorities claim, he may have +learned the sexagesimal notation in use there; he may have +picked up considerable knowledge on the theory of proportion, +\index{Proportion}% +and may have found a large number of interesting astronomical +observations. Saturated with that speculative spirit then +pervading the Greek mind, he endeavoured to discover some +principle of homogeneity in the universe. Before him, the +philosophers of the Ionic school had sought it in the matter +of things; Pythagoras looked for it in the structure of things. +He observed various numerical relations or analogies between +numbers and the phenomena of the universe. Being convinced +that it was in numbers and their relations that he was to find +the foundation to true philosophy, he proceeded to trace the +origin of all things to numbers. Thus he observed that musical +strings of equal length stretched by weights having the +proportion of $\frac{1}{2}$, $\frac{2}{3}$, $\frac{3}{4}$, produced intervals which were an octave, +a fifth, and a fourth. Harmony, therefore, depends on musical +proportion; it is nothing but a mysterious numerical relation. +Where harmony is, there are numbers. Hence the +order and beauty of the universe have their origin in numbers. +There are seven intervals in the musical scale, and also seven +planets crossing the heavens. The same numerical relations +which underlie the former must underlie the latter. But where +numbers are, there is harmony. Hence his spiritual ear discerned +in the planetary motions a wonderful `harmony of the +spheres.' The Pythagoreans invested particular numbers with +extraordinary attributes. Thus \emph{one} is the essence of things; +it is an absolute number; hence the origin of all numbers and +%% -----File: 082.png---Folio 68------- +so of all things. \emph{Four} is the most perfect number, and was in +some mystic way conceived to correspond to the human soul. +Philolaus believed that $5$ is the cause of color, $6$ of cold, $7$ of +\index{Philolaus!ref.\ to}% +mind and health and light, $8$ of love and friendship.\Reference{6} In +Plato's works are evidences of a similar belief in religious +\index{Plato!ref.\ to}% +relations of numbers. Even Aristotle referred the virtues to +\index{Aristotle!ref.\ to}% +\index{Numbers!amicable}% +\index{Numbers!excessive}% +\index{Numbers!heteromecic}% +\index{Numbers!perfect}% +\index{Numbers!defective}% +numbers. + +Enough has been said about these mystic speculations to +show what lively interest in mathematics they must have +created and maintained. Avenues of mathematical inquiry +were opened up by them which otherwise would probably +have remained closed at that time. + +The Pythagoreans classified numbers into odd and even. +They observed that the sum of the series of odd numbers +from $1$ to $2n + 1$ was always a complete square, and that +by addition of the even numbers arises the series $2$, $6$, $12$, $20$, +in which every number can be decomposed into two factors +differing from each other by unity. Thus, $6 = 2·3$, $12 = 3·4$, +etc. These latter numbers were considered of sufficient importance +to receive the separate name of \emph{heteromecic} (not +equilateral).\Reference{7} Numbers of the form $\dfrac{n(n+1)}{2}$ were called \emph{triangular}, +because they could always be arranged thus,~\smash[t]{\TallGlyph{tendots}}. +Numbers which were equal to the sum of all their possible +factors, such as $6$, $28$, $496$, were called \emph{perfect}; those exceeding +that sum, \emph{excessive}; and those which were less, \emph{defective}. +\emph{Amicable} numbers were those of which each was the sum of +the factors in the other. Much attention was paid by the +Pythagoreans to the subject of proportion. The quantities +\index{Proportion}% +$a$, $b$, $c$, $d$ were said to be in \emph{arithmetical} proportion when +$a - b = c - d$; in \emph{geometrical} proportion, when $a : b = c : d$; in +\emph{harmonic} proportion, when $a - b : b - c = a : c$. It is probable +that the Pythagoreans were also familiar with the \emph{musical} +%% -----File: 083.png---Folio 69------- +\index{Irrationals}% +proportion $a : \dfrac{a + b\Fstr}{2} = \dfrac{2ab}{a+b} : b$. Iamblichus says that Pythagoras +\index{Iamblichus!ref.\ to}% +introduced it from Babylon. + +In connection with arithmetic, Pythagoras made extensive +investigations into geometry. He believed that an arithmetical +\index{Geometry!Greek}% +fact had its analogue in geometry, and \textit{vice versa}. In +connection with his theorem on the right triangle he devised +a rule by which integral numbers could be found, such +that the sum of the squares of two of them equalled the +square of the third. Thus, take for one side an odd number +$(2n + 1)$; then $\dfrac{(2n + 1)^{2} - 1}{2} = 2n^{2} + 2n =$ the other side, and +$(2n^2 + 2n + 1) =$ hypotenuse. If $2n + 1 = 9$, then the other +two numbers are $40$~and~$41$. But this rule only applies to +cases in which the hypotenuse differs from one of the sides +by~$1$. In the study of the right triangle there doubtless arose +questions of puzzling subtlety. Thus, given a number equal +to the side of an isosceles right triangle, to find the number +which the hypotenuse is equal to. The side may have been +taken equal to $1$, $2$, $\frac{3}{2}$, $\frac{6}{5}$, or any other number, yet in every +instance all efforts to find a number exactly equal to the +hypotenuse must have remained fruitless. The problem may +have been attacked again and again, until finally ``some rare +genius, to whom it is granted, during some happy moments, +to soar with eagle's flight above the level of human thinking,'' +grasped the happy thought that this problem cannot be +solved. In some such manner probably arose the theory of +\emph{irrational quantities}, which is attributed by Eudemus to the +\index{Eudemus}% +Pythagoreans. It was indeed a thought of extraordinary +boldness, to assume that straight lines could exist, differing +from one another not only in length,---that is, in quantity,---but +also in a quality, which, though real, was absolutely +invisible.\Reference{7} Need we wonder that the Pythagoreans saw in +%% -----File: 084.png---Folio 70------- +\index{Parabola}% +irrationals a deep mystery, a symbol of the unspeakable? +We are told that the one who first divulged the theory of +irrationals, which the Pythagoreans kept secret, perished in +consequence in a shipwreck. Its discovery is ascribed to +Pythagoras, but we must remember that all important Pythagorean +\index{Pythagoras|)}% +discoveries were, according to Pythagorean custom, +referred back to him. The first incommensurable ratio known +seems to have been that of the side of a square to its diagonal, +as $1 : \sqrt {2}$. \textbf{Theodorus of Cyrene} added to this the fact that the +\index{Theodorus}% +sides of squares represented in length by $\sqrt {3}$, $\sqrt {5}$, etc., up to +$\sqrt {17}$, and Theætetus, that the sides of any square, represented +\index{Theætetus!ref.\ to}% +by a surd, are incommensurable with the linear unit. \textbf{Euclid} +\index{Euclid}% +(about 300~\BC), in his \textit{Elements}, X.~9, generalised still further: +Two magnitudes whose squares are (or are not) to one another +as a square number to a square number are commensurable +(or incommensurable), and conversely. In the tenth book, he +treats of incommensurable quantities at length. He investigates +every possible variety of lines which can be represented +by $\sqrt { \sqrt {\vphantom{b}a} ± \sqrt {b}}$, $a$~and~$b$ representing two commensurable lines, +and obtains 25~species. Every individual of every species +is incommensurable with all the individuals of every other +species. ``This book,'' says De~Morgan, ``has a completeness +\index{Demorgan@{De Morgan}!ref.\ to}% +which none of the others (not even the fifth) can boast of; +and we could almost suspect that Euclid, having arranged his +materials in his own mind, and having completely elaborated +the tenth book, wrote the preceding books after it, and did not +live to revise them thoroughly.''\Reference{9} The theory of incommensurables +\index{Incommensurables}% +remained where Euclid left it, till the fifteenth +century. + +Euclid devotes the seventh, eighth, and ninth books of his +\textit{Elements} to arithmetic. Exactly how much contained in +\index{Arithmetic!Pythagoreans|)}% +\index{Arithmetic!Euclid}% +these books is Euclid's own invention, and how much is +borrowed from his predecessors, we have no means of knowing. +%% -----File: 085.png---Folio 71------- +Without doubt, much is original with Euclid. The \textit{seventh +\index{Euclid}% +book} begins with twenty-one definitions. All except that for +`prime' numbers are known to have been given by the +\index{Prime numbers}% +Pythagoreans. Next follows a process for finding the +G.C.D.~of two or more numbers. The \textit{eighth book} deals with +numbers in continued proportion, and with the mutual relations +of squares, cubes, and plane numbers. Thus,~XXII., if +three numbers are in continued proportion, and the first is a +square, so is the third. In the \textit{ninth book}, the same subject is +continued. It contains the proposition that the number of +primes is greater than any given number. + +After the death of Euclid, the theory of numbers remained +almost stationary for 400~years. Geometry monopolised +the attention of all Greek mathematicians. Only two are +known to have done work in arithmetic worthy of mention. +\textbf{Eratosthenes} (275--194~\BC) invented a `sieve' for finding prime +\index{Eratosthenes!ref.\ to}% +numbers. All composite numbers are `sifted' out in the +following manner: Write down the odd numbers from $3$~up, +in succession. By striking out every third number after the~$3$, +we remove all multiples of~$3$. By striking out every fifth +number after the~$5$, we remove all multiples of~$5$. In this way, +by rejecting multiples of $7$,~$11$, $13$,~etc., we have left prime +numbers only. \textbf{Hypsicles} (between 200 and 100~\BC) worked +\index{Hypsicles!ref.\ to}% +at the subjects of polygonal numbers and arithmetical progressions, +which Euclid entirely neglected. In his work on +`risings of the stars,' he showed (1)~that in an arithmetical +series of $2n$~terms, the sum of the last~$n$ terms exceeds the sum +of the first~$n$ by a multiple of~$n^2$; (2)~that in such a series of +$2n + 1$~terms, the sum of the series is the number of terms +multiplied by the middle term; (3)~that in such a series of +$2n$~terms, the sum is half the number of terms multiplied by +the two middle terms.\Reference{6} + +For two centuries after the time of Hypsicles, arithmetic +%% -----File: 086.png---Folio 72------- +\index{Cube numbers}% +disappears from history. It is brought to light again about +100~\AD\ by \textbf{Nicomachus}, a Neo-Pythagorean, who inaugurated +\index{Nicomachus}% +the final era of Greek mathematics. From now on, arithmetic +was a favourite study, while geometry was neglected. Nicomachus +wrote a work entitled \textit{Introductio Arithmetica}, which +was very famous in its day. The great number of commentators +it has received vouch for its popularity. Boethius +\index{Boethius!ref.\ to}% +translated it into Latin. Lucian could pay no higher +compliment to a calculator than this: ``You reckon like +Nicomachus of Gerasa.'' The \textit{Introductio Arithmetica} was +the first exhaustive work in which arithmetic was treated +quite independently of geometry. Instead of drawing lines, +like Euclid, he illustrates things by real numbers. To be +\index{Euclid!ref.\ to}% +sure, in his book the old geometrical nomenclature is retained, +but the method is inductive instead of deductive. ``Its sole +business is classification, and all its classes are derived from, +and exhibited by, actual numbers.'' The work contains few +results that are really original. We mention one important +proposition which is probably the author's own. He states +that cubical numbers are always equal to the sum of successive +odd numbers. Thus, $8 = 2^{3} = 3 + 5$, $27 = 3^{3} = 7 + 9 + 11$, $64 += 4^{3} = 13 + 15 + 17 + 19$, and so on. This theorem was used +later for finding the sum of the cubical numbers themselves. +\textbf{Theon} of Smyrna is the author of a treatise on ``the mathematical +\index{Theon of Smyrna}% +rules necessary for the study of Plato.'' The work is +ill arranged and of little merit. Of interest is the theorem, +that every square number, or that number minus~$1$, is +divisible by $3$ or $4$ or both. A remarkable discovery is a +proposition given by \textbf{Iamblichus} in his treatise on Pythagorean +\index{Iamblichus}% +philosophy. It is founded on the observation that the +Pythagoreans called $1$, $10$, $100$, $1000$, units of the first, second, +third, fourth `course' respectively. The theorem is this: +If we add any three consecutive numbers, of which the highest +%% -----File: 087.png---Folio 73------- +\index{Arabic numerals and notation}% +is divisible by~$3$, then add the digits of that sum, then, again, +the digits of \emph{that} sum, and so on, the final sum will be~$6$. +Thus, $61 + 62 + 63 = 186$, $1 + 8 + 6 = 15$, $1 + 5 = 6$. This +discovery was the more remarkable, because the ordinary +Greek numerical symbolism was much less likely to suggest +any such property of numbers than our ``Arabic'' notation +\index{Notation!Arabic notation}% +would have been. + +The works of Nicomachus, Theon of Smyrna, Thymaridas, +\index{Thymaridas}% +and others contain at times investigations of subjects which +are really algebraic in their nature. Thymaridas in one place +uses the Greek word meaning ``unknown quantity'' in a way +which would lead one to believe that algebra was not far +\index{Algebra!early Greek}% +distant. Of interest in tracing the invention of algebra are +the arithmetical epigrams in the \textit{Palatine Anthology}, which contain +\index{Anthology, Palatine}% +\index{Palatine anthology}% +about fifty problems leading to linear equations. Before +the introduction of algebra these problems were propounded +as puzzles. A riddle attributed to Euclid and contained in +\index{Euclid!ref.\ to}% +the \textit{Anthology} is to this effect: A mule and a donkey were +walking along, laden with corn. The mule says to the +donkey, ``If you gave me one measure, I should carry twice +as much as you. If I gave you one, we should both carry +equal burdens. Tell me their burdens, O~most learned master +of geometry.''\Reference{6} + +It will be allowed, says Gow, that this problem, if authentic, +was not beyond Euclid, and the appeal to geometry smacks of +antiquity. A far more difficult puzzle was the famous `cattle-problem,' +\index{Cattle-problem}% +which Archimedes propounded to the Alexandrian +\index{Archimedes!ref.\ to}% +mathematicians. The problem is indeterminate, for from only +seven equations, eight unknown quantities in integral numbers +are to be found. It may be stated thus: The sun had a herd +of bulls and cows, of different colours. (1)~Of Bulls, the +white~($W$) were, in number, $(\frac{1}{2} + \frac{1}{3})$ of the blue~($B$) and yellow~($Y$): +the $B$ were $(\frac{1}{4} + \frac{1}{5})$ of the~$Y$ and piebald~($P$): the +%% -----File: 088.png---Folio 74------- +$P$ were $(\frac{1}{6} + \frac{1}{7})$ of the~$W$ and~$Y$. (2)~Of Cows, which had the +same colours ($w$, $b$, $y$, $p$), +\iffalse +\begin{align*}% [** PP: This commented code visually matches the original] +w=(\tfrac{1}{3} + \tfrac{1}{4})(B + b) + &: b = (\tfrac{1}{4} + \tfrac{1}{5})(P + p) : p = (\tfrac{1}{5} + \tfrac{1}{6})(Y + y) \\ + &: y = (\tfrac{1}{6} + \tfrac{1}{7})(W + w). +\end{align*} +\fi +%[** PP: Re-breaking and aligning on =, but otherwise retaining verbatim] +\begin{align*} +w = (\tfrac{1}{3} + \tfrac{1}{4})(B + b) : b + &= (\tfrac{1}{4} + \tfrac{1}{5})(P + p) : p \\ + &= (\tfrac{1}{5} + \tfrac{1}{6})(Y + y) : y + = (\tfrac{1}{6} + \tfrac{1}{7})(W + w). +\end{align*} +Find the number of bulls and cows.\Reference{6} Another problem in the +\textit{Anthology} is quite familiar to school-boys: ``Of four pipes, +one fills the cistern in one day, the next in two days, the third +in three days, the fourth in four days: if all run together, how +soon will they fill the cistern?'' A great many of these +problems, puzzling to an arithmetician, would have been solved +easily by an algebraist. They became very popular about the +time of Diophantus, and doubtless acted as a powerful stimulus +\index{Diophantus|(}% +on his mind. + +\textbf{Diophantus} was one of the last and most fertile mathematicians +of the second Alexandrian school. He died about +330~\AD. His age was eighty-four, as is known from an epitaph +to this effect: Diophantus passed $\frac{1}{6}$~of his life in childhood, $\frac{1}{12}$~in +youth, and $\frac{1}{7}$~more as a bachelor; five years after his marriage +was born a son who died four years before his father, at half +his father's age. The place of nativity and parentage of +Diophantus are unknown. If his works were not written in +Greek, no one would think for a moment that they were the +product of Greek mind. There is nothing in his works that +reminds us of the classic period of Greek mathematics. His +were almost entirely new ideas on a new subject. In the +circle of Greek mathematicians he stands alone in his specialty. +Except for him, we should be constrained to say that +among the Greeks \emph{algebra} was always an unknown science. +\index{Algebra!Diophantus|(}% + +Of his works we have lost the \textit{Porisms}, but possess a fragment +of \textit{Polygonal Numbers}, and seven books of his great work +on \textit{Arithmetica}, said to have been written in 13~books. + +If we except the Ahmes papyrus, which contains the first +\index{Ahmes!ref.\ to}% +%% -----File: 089.png---Folio 75------- +\index{Tentative assumption}% +suggestions of algebraic notation, and of the solution of +\index{Notation!in algebra}% +equations, then his \textit{Arithmetica} is the earliest treatise on +\index{Equations!theory of}% +algebra now extant. In this work is introduced the idea of +an algebraic equation expressed in algebraic symbols. His +treatment is purely analytical and completely divorced from +geometrical methods. He is, as far as we know, the first to +state that ``a negative number multiplied by a negative number +gives a positive number.'' This is applied to the multiplication +of differences, such as $(x - 1)(x - 2)$. It must be +remarked, however, that Diophantus had no notion whatever +of negative numbers standing by themselves. All he knew +were differences, such as $(2x - 10)$, in which $2x$ could not be +smaller than~$10$ without leading to an absurdity. He appears +to be the first who could perform such operations as $(x - 1) +× (x - 2)$ without reference to geometry. Such identities as +$(a + b)^2 = a^2 + 2ab + b^2$, which with Euclid appear in the elevated +rank of geometric theorems, are with Diophantus the +simplest consequences of the algebraic laws of operation. His +sign for subtraction was~\Glyph{subtract}, for equality~$\iota$. For unknown +quantities he had only one symbol,~$\varsigma$. He had no sign for +addition except juxtaposition. Diophantus used but few symbols, +and sometimes ignored even these by describing an operation +in words when the symbol would have answered just +as well. + +In the solution of simultaneous equations Diophantus adroitly +managed with only one symbol for the unknown quantities and +arrived at answers, most commonly, by the method of \emph{tentative +assumption}, which consists in assigning to some of the unknown +\index{Assumption, tentative}% +quantities preliminary values, that satisfy only one or two of +the conditions. These values lead to expressions palpably +wrong, but which generally suggest some stratagem by which +values can be secured satisfying all the conditions of the +problem. +%% -----File: 090.png---Folio 76------- +\index{Numbers!theory of numbers}% +\index{Theory of numbers}% + +Diophantus also solved determinate equations of the second +degree. We are ignorant of his method, for he nowhere goes +through with the whole process of solution, but merely states +the result. Thus, ``$84 x^2 + 7x = 7$, whence $x$ is found $= \frac{1}{4}$.'' +Notice he gives only one root. His failure to observe that a +quadratic equation has two roots, even when both roots are +positive, rather surprises us. It must be remembered, however, +that this same inability to perceive more than one out of +the several solutions to which a problem may point is common +to all Greek mathematicians. Another point to be observed +is that he never accepts as an answer a quantity which is +negative or irrational. + +Diophantus devotes only the first book of his \textit{Arithmetica} to +the solution of determinate equations. The remaining books +extant treat mainly of \emph{indeterminate quadratic equations} of the +\index{Quadratic equations}% +form $Ax^2 + Bx + C = y^2$, or of two simultaneous equations of the +same form. He considers several but not all the possible +cases which may arise in these equations. The opinion of +Nesselmann on the method of Diophantus, as stated by Gow, +\index{Nesselmann}% +is as follows: ``(1)~Indeterminate equations of the second +degree are treated completely only when the quadratic or +the absolute term is wanting: his solution of the equations +$Ax^{2} + C = y^{2}$ and $Ax^{2} + Bx + C = y^{2}$ is in many respects cramped. +(2)~For the `double equation' of the second degree he has a +definite rule only when the quadratic term is wanting in both +expressions: even then his solution is not general. More complicated +expressions occur only under specially favourable +circumstances.'' Thus, he solves $Bx + C^{2} = y^{2}$, $B_{1}x + C_1^2 = y_1^2$. + +The extraordinary ability of Diophantus lies rather in +another direction, namely, in his wonderful ingenuity to reduce +all sorts of equations to particular forms which he knows +how to solve. Very great is the variety of problems considered. +The 130~problems found in the great work of Diophantus contain +%% -----File: 091.png---Folio 77------- +\index{Arithmetic!Greek|)}% +\index{Greeks|)}% +over 50~different classes of problems, which are strung +together without any attempt at classification. But still more +multifarious than the problems are the solutions. General +methods are unknown to Diophantus. Each problem has its +own distinct method, which is often useless for the most +closely related problems. ``It is, therefore, difficult for a +modern, after studying 100~Diophantine solutions, to solve +the~101st.''\Reference{7} + +That which robs his work of much of its scientific value is +the fact that he always feels satisfied with one solution, though +his equation may admit of an indefinite number of values. +Another great defect is the absence of general methods. Modern +mathematicians, such as Euler, \DPtypo{La~Grange}{Lagrange}, Gauss, had to +\index{Euler!ref.\ to}% +\index{Gauss!ref.\ to}% +\index{Lagrange!ref.\ to}% +begin the study of indeterminate analysis anew and received +no direct aid from Diophantus in the formulation of methods. +In spite of these defects we cannot fail to admire the work +for the wonderful ingenuity exhibited therein in the solution +of particular equations. + +It is still an open question and one of great difficulty +whether Diophantus derived portions of his algebra from +\index{Algebra!Diophantus|)}% +\index{Diophantus|)}% +Hindoo sources or not. + +\Section{The Romans} +\index{Romans|(}% + +Nowhere is the contrast between the Greek and Roman +mind shown forth more distinctly than in their attitude toward +the mathematical science. The sway of the Greek was a +flowering time for mathematics, but that of the Roman a +period of sterility. In philosophy, poetry, and art the Roman +was an imitator. But in mathematics he did not even rise to +the desire for imitation. The mathematical fruits of Greek +genius lay before him untasted. In him a science which had +%% -----File: 092.png---Folio 78------- +no direct bearing on practical life could awake no interest. +As a consequence, not only the higher geometry of Archimedes +\index{Archimedes!ref.\ to}% +and Apollonius, but even the \textit{Elements} of Euclid, were entirely +\index{Apollonius!ref.\ to}% +\index{Euclid!ref.\ to}% +neglected. What little mathematics the Romans possessed +did not come from the Greeks, but from more ancient +sources. Exactly where and how it originated is a matter of +doubt. It seems most probable that the ``Roman notation,'' +\index{Notation!Roman}% +as well as the practical geometry of the Romans, came from +the old Etruscans, who, at the earliest period to which our +knowledge of them extends, inhabited the district between the +Arno and Tiber. + +Livy tells us that the Etruscans were in the habit of representing +the number of years elapsed, by driving yearly a nail +into the sanctuary of Minerva, and that the Romans continued +this practice. A less primitive mode of designating numbers, +presumably of Etruscan origin, was a notation resembling the +present ``Roman notation.'' This system is noteworthy from +the fact that a principle is involved in it which is not met +with in any other; namely, the principle of subtraction. If a +letter be placed before another of greater value, its value is +not to be added to, but subtracted from, that of the greater. +In the designation of large numbers a horizontal bar placed +over a letter was made to increase its value one thousand fold. +In fractions the Romans used the duodecimal system. +\index{Fractions!Roman}% + +Of arithmetical calculations, the Romans employed three +different kinds: Reckoning on the fingers, upon the abacus, +and by tables prepared for the purpose.\Reference{3} Finger-symbolism +was known as early as the time of King Numa, for he had +erected, says Pliny, a statue of the double-faced Janus, of +which the fingers indicated $365$ ($355$?), the number of days in +a year. Many other passages from Roman authors point out +the use of the fingers as aids to calculation. In fact, a finger-symbolism +of practically the same form was in use not only in +%% -----File: 093.png---Folio 79------- +\index{Canon paschalis}% +Rome, but also in Greece and throughout the East, certainly +as early as the beginning of the Christian era, and continued +to be used in Europe during the Middle Ages. We possess no +knowledge as to where or when it was invented. The second +mode of calculation, by the abacus, was a subject of elementary +\index{Abacus}% +\index{Calculation, origin of word}% +instruction in Rome. Passages in Roman writers indicate +that the kind of abacus most commonly used was covered with +dust and then divided into columns by drawing straight lines. +Each column was supplied with pebbles (calculi, whence `calculare' +and `calculate') which served for calculation. Additions +and subtractions could be performed on the abacus quite +easily, but in multiplication the abacus could be used only for +adding the particular products, and in division for performing +the subtractions occurring in the process. Doubtless at this +point recourse was made to mental operations and to the multiplication +table. Possibly finger-multiplication may also have +been used. But the multiplication of large numbers must, by +either method, have been beyond the power of the ordinary +arithmetician. To obviate this difficulty, the arithmetical +tables mentioned above were used, from which the desired +products could be copied at once. Tables of this kind were +prepared by \textit{Victorius} of Aquitania. His tables contain a +\index{Victorius}% +peculiar notation for fractions, which continued in use throughout +the Middle Ages. Victorius is best known for his \textit{canon +paschalis}, a rule for finding the correct date for Easter, which +he published in 457~\AD. + +Payments of interest and problems in interest were very old +among the Romans. The Roman laws of inheritance gave +rise to numerous arithmetical examples. Especially unique is +the following: A dying man wills that, if his wife, being with +child, gives birth to a son, the son shall receive~$\frac{2}{3}$ and she $\frac{1}{3}$~of +his estates; but if a daughter is born, she shall receive~$\frac{1}{3}$ and +his wife~$\frac{2}{3}$. It happens that twins are born, a boy and a girl. +%% -----File: 094.png---Folio 80------- +How shall the estates be divided so as to satisfy the will? +The celebrated Roman jurist, Salvianus Julianus, decided that +the estates shall be divided into seven equal parts, of which +the son receives four, the wife two, the daughter one. + +We next consider Roman geometry. He who expects to +\index{Geometry!Roman}% +find in Rome a science of geometry, with definitions, axioms, +theorems, and proofs arranged in logical order, will be disappointed. +The only geometry known was a \emph{practical} geometry, +which, like the old Egyptian, consisted only of empirical rules. +This practical geometry was employed in surveying. Treatises +thereon have come down to us, compiled by the Roman surveyors, +called \textit{agrimensores} or \textit{gromatici}. One would naturally +\index{Agrimensores}% +\index{Gromatici}% +expect rules to be clearly formulated. But no; they are left +to be abstracted by the reader from a mass of numerical examples. +``The total impression is as though the Roman gromatic +were thousands of years older than Greek geometry, and as +though a deluge were lying between the two.'' Some of their +rules were probably inherited from the Etruscans, but others +are identical with those of Heron. Among the latter is that +\index{Heron the Elder!ref.\ to}% +for finding the area of a triangle from its sides and the approximate +formula,~$\frac{13}{30}a^{2}$, for the area of equilateral triangles ($a$ +being one of the sides). But the latter area was also calculated +by the formulas $\frac{1}{2}(a^{2} + a)$ and $\frac{1}{2} a^{2}$, the first of which was +unknown to Heron. Probably the expression $\frac{1}{2}a^{2}$ was derived +from the Egyptian formula $\dfrac{a + b\Fstr}{2} · \dfrac{c + d}{2}$ for the determination +of the surface of a quadrilateral. This Egyptian formula +was used by the Romans for finding the area, not only of rectangles, +but of any quadrilaterals whatever. Indeed, the gromatici +considered it even sufficiently accurate to determine the +areas of cities, laid out irregularly, simply by measuring their +circumferences.\Reference{7} Whatever Egyptian geometry the Romans +possessed was transplanted across the Mediterranean at the +%% -----File: 095.png---Folio 81------- +\index{Julian calendar}% +time of \textit{Julius Cæsar}, who ordered a survey of the whole +\index{Caesar@{Cæsar}, Julius}% +empire to secure an equitable mode of taxation. Cæsar also +reformed the calendar, and, for that purpose, drew from +\index{Calendar}% +Egyptian learning. He secured the services of the Alexandrian +astronomer, \textit{Sosigenes}. +\index{Sosigenes}% + +In the fifth century, the Western Roman Empire was fast +falling to pieces. Three great branches---Spain, Gaul, and +the province of Africa---broke off from the decaying trunk. +In~476, the Western Empire passed away, and the Visigothic +chief, Odoacer, became king. Soon after, Italy was conquered +by the Ostrogoths under Theodoric. It is remarkable that +this very period of political humiliation should be the one +during which Greek science was studied in Italy most zealously. +School-books began to be compiled from the elements +of Greek authors. These compilations are very deficient, but +are of absorbing interest, from the fact that, down to the +twelfth century, they were the only sources of mathematical +knowledge in the Occident. Foremost among these writers is +\textbf{Boethius} (died~524). At first he was a great favourite of King +\index{Boethius}% +Theodoric, but later, being charged by envious courtiers with +treason, he was imprisoned, and at last decapitated. While +in prison he wrote \textit{On the Consolations of Philosophy}. As a +mathematician, Boethius was a Brobdingnagian among Roman +scholars, but a Liliputian by the side of Greek masters. He +wrote an \textit{Institutis Arithmetica}, which is essentially a translation +of the arithmetic of Nicomachus, and a \textit{Geometry} in +\index{Nicomachus!ref.\ to}% +several books. Some of the most beautiful results of Nicomachus +are omitted in Boethius' arithmetic. The first book +on geometry is an extract from Euclid's \textit{Elements}, which contains, +\index{Euclid!ref.\ to}% +in addition to definitions, postulates, and axioms, the +theorems in the first three books, without proofs. How can +this omission of proofs be accounted for? It has been argued +by some that Boethius possessed an incomplete Greek copy of +%% -----File: 096.png---Folio 82------- +\index{Apices of Boethius}% +\index{Gobar numerals}% +the \textit{Elements}; by others, that he had Theon's edition before +\index{Theon of Alexandria!ref.\ to}% +him, and believed that only the theorems came from Euclid, +while the proofs were supplied by Theon. The second book, +as also other books on geometry attributed to Boethius, +teaches, from numerical examples, the mensuration of plane +figures after the fashion of the agrimensores. + +A celebrated portion in the geometry of Boethius is that +pertaining to an abacus, which he attributes to the Pythagoreans. +\index{Abacus}% +A considerable improvement on the old abacus is +there introduced. Pebbles are discarded, and \textit{apices} (probably +small cones) are used. Upon each of these apices is drawn +a numeral giving it some value below~$10$. The names of +these numerals are pure Arabic, or nearly so, but are added, +apparently, by a later hand. These figures are obviously the +parents of our modern ``Arabic'' numerals. The~$0$~is not +mentioned by Boethius in the text. These numerals bear +striking resemblance to the Gubar-numerals of the West-Arabs, +\index{Gubar-numerals}% +which are admittedly of Indian origin. These facts +have given rise to an endless controversy. Some contended +that Pythagoras was in India, and from there brought the +\index{Pythagoras!ref.\ to}% +nine numerals to Greece, where the Pythagoreans used them +secretly. This hypothesis has been generally abandoned, for +it is not certain that Pythagoras or any disciple of his ever +was in India, nor is there any evidence in any Greek author, +that the apices were known to the Greeks, or that numeral +signs of any sort were used by them with the abacus. It is +improbable, moreover, that the Indian signs, from which the +apices are derived, are so old as the time of Pythagoras. +A second theory is that the \textit{Geometry} attributed to Boethius +is a forgery; that it is not older than the tenth, or possibly +the ninth, century, and that the apices are derived from the +Arabs. This theory is based on contradictions between passages +in the \textit{Arithmetica} and others in the \textit{Geometry}. But +%% -----File: 097.png---Folio 83------- +there is an Encyclopædia written by \textit{Cassiodorius} (died about +\index{Cassiodorius}% +570) in which both the arithmetic and geometry of Boethius +are mentioned. There appears to be no good reason for doubting +the trustworthiness of this passage in the Encyclopædia. +A third theory (Woepcke's) is that the Alexandrians either +\index{Woepcke}% +directly or indirectly obtained the nine numerals from the +Hindoos, about the second century~\AD, and gave them to +the Romans on the one hand, and to the Western Arabs +\index{Romans|)}% +on the other. This explanation is the most plausible. +%% -----File: 098.png---Folio 84------- + + +\Chapter{Middle Ages} + +\Section{The Hindoos} +\index{Hindoos|(}% + +\First{The} first people who distinguished themselves in mathematical +research, after the time of the ancient Greeks, belonged, +like them, to the Aryan race. It was, however, not a European, +but an Asiatic nation, and had its seat in far-off India. + +Unlike the Greek, Indian society was fixed into castes. The +only castes enjoying the privilege and leisure for advanced +study and thinking were the \emph{Brahmins}, whose prime business +was religion and philosophy, and the \emph{Kshatriyas}, who attended +to war and government. + +Of the development of Hindoo mathematics we know but +little. A few manuscripts bear testimony that the Indians +had climbed to a lofty height, but their path of ascent is no +longer traceable. It would seem that Greek mathematics grew +up under more favourable conditions than the Hindoo, for in +Greece it attained an independent existence, and was studied +for its own sake, while Hindoo mathematics always remained +merely a servant to astronomy. Furthermore, in Greece +mathematics was a science of the people, free to be cultivated +by all who had a liking for it; in India, as in Egypt, it was in +the hands chiefly of the priests. Again, the Indians were in +the habit of putting into verse all mathematical results they +obtained, and of clothing them in obscure and mystic language, +%% -----File: 099.png---Folio 85------- +which, though well adapted to aid the memory of him who +already understood the subject, was often unintelligible to the +uninitiated. Although the great Hindoo mathematicians +doubtless reasoned out most or all of their discoveries, yet +they were not in the habit of preserving the proofs, so that +the naked theorems and processes of operation are all that +have come down to our time. Very different in these respects +were the Greeks. Obscurity of language was generally +avoided, and proofs belonged to the stock of knowledge quite +as much as the theorems themselves. Very striking was the +difference in the bent of mind of the Hindoo and Greek; for, +while the Greek mind was pre-eminently \emph{geometrical}, the +Indian was first of all \emph{arithmetical}. The Hindoo dealt with +number, the Greek with form. Numerical symbolism, the +science of numbers, and algebra attained in India far greater +perfection than they had previously reached in Greece. On +the other hand, we believe that there was little or no geometry +in India of which the source may not be traced back to +Greece. Hindoo trigonometry might possibly be mentioned +as an exception, but it rested on arithmetic more than on +geometry. + +An interesting but difficult task is the tracing of the relation +between Hindoo and Greek mathematics. It is well +known that more or less trade was carried on between Greece +and India from early times. After Egypt had become a +Roman province, a more lively commercial intercourse sprang +up between Rome and India, by way of Alexandria. \textit{A~priori}, +it does not seem improbable, that with the traffic of merchandise +there should also be an interchange of ideas. That +communications of thought from the Hindoos to the Alexandrians +actually did take place, is evident from the fact that +certain philosophic and theologic teachings of the Manicheans, +Neo-Platonists, Gnostics, show unmistakable likeness to +%% -----File: 100.png---Folio 86------- +Indian tenets. Scientific facts passed also from Alexandria +to India. This is shown plainly by the Greek origin of some +of the technical terms used by the Hindoos. Hindoo astronomy +\index{Astronomy!Hindoo}% +was influenced by Greek astronomy. Most of the geometrical +knowledge which they possessed is traceable to +Alexandria, and to the writings of Heron in particular. In +algebra there was, probably, a mutual giving and receiving. +We suspect that Diophantus got the first glimpses of algebraic +\index{Diophantus!ref.\ to}% +knowledge from India. On the other hand, evidences have +been found of Greek algebra among the Brahmins. The +earliest knowledge of algebra in India may possibly have been +of Babylonian origin. When we consider that Hindoo scientists +looked upon arithmetic and algebra merely as tools +useful in astronomical research, there appears deep irony in +the fact that these secondary branches were after all the only +ones in which they won real distinction, while in their pet +science of astronomy they displayed an inaptitude to observe, +to collect facts, and to make inductive investigations. + +We shall now proceed to enumerate the names of the +leading Hindoo mathematicians, and then to review briefly +Indian mathematics. We shall consider the science only in +its complete state, for our data are not sufficient to trace the +history of the development of methods. Of the great Indian +mathematicians, or rather, astronomers,---for India had no +mathematicians proper,---\textbf{Aryabhatta} is the earliest. He was +\index{Aryabhatta}% +born 476~\AD, at Pataliputra, on the upper Ganges. His +celebrity rests on a work entitled \textit{Aryabhattiyam}, of which +the third chapter is devoted to mathematics. About one +hundred years later, mathematics in India reached the highest +mark. At that time flourished \textbf{Brahmagupta} (born 598). In +\index{Brahmagupta}% +628 he wrote his \textit{Brahma-sphuta-siddhanta} (``The Revised System +of Brahma''), of which the twelfth and eighteenth chapters +belong to mathematics. To the fourth or fifth century belongs +%% -----File: 101.png---Folio 87------- +\index{Arabic numerals and notation}% +\index{Numerals!Arabic}% +an anonymous astronomical work, called \textit{Surya-siddhanta} +(``Knowledge from the Sun''), which by native authorities +was ranked second only to the \textit{Brahma-siddhanta}, but is of interest +to us merely as furnishing evidence that Greek science +influenced Indian science even before the time of Aryabhatta. +The following centuries produced only two names of importance; +namely, \textbf{Cridhara}, who wrote a \textit{Ganita-sara} (``Quintessence +\index{Cridhara}% +of Calculation''), and \textbf{Padmanabha}, the author of an +\index{Padmanabha}% +algebra. The science seems to have made but little progress +at this time; for a work entitled \textit{Siddhantaciromani} (``Diadem +of an Astronomical System''), written by \textbf{Bhaskara Acarya} in +\index{Bhaskara}% +1150, stands little higher than that of Brahmagupta, written +over 500~years earlier. The two most important mathematical +chapters in this work are the \textit{Lilavati} ($=$~``the beautiful,'' \ie~the +noble science) and \textit{Viga-ganita} ($=$~``root-extraction''), devoted +to arithmetic and algebra. From now on, the Hindoos +in the Brahmin schools seemed to content themselves with +studying the masterpieces of their predecessors. Scientific +intelligence decreases continually, and in modern times a very +deficient Arabic work of the sixteenth century has been held +in great authority.\Reference{7} + +The mathematical chapters of the \textit{Brahma-siddhanta} and +\textit{Siddhantaciromani} were translated into English by H.~T. +Colebrooke, London, 1817. The \textit{Surya-siddhanta} was translated +\index{Colebrooke}% +by E.~Burgess, and annotated by W.~D. Whitney, New +\index{Whitney}% +Haven, Conn., 1860. + +The grandest achievement of the Hindoos and the one +which, of all mathematical inventions, has contributed most +to the general progress of intelligence, is the invention of +the principle of position in writing numbers. Generally we +speak of our notation as the ``Arabic'' notation, but it should +\index{Notation!Arabic notation}% +be called the ``Hindoo'' notation, for the Arabs borrowed it +from the Hindoos. That the invention of this notation was +%% -----File: 102.png---Folio 88------- +not so easy as we might suppose at first thought, may be +inferred from the fact that, of other nations, not even the +keen-minded Greeks possessed one like it. We inquire, who +invented this ideal symbolism, and when? But we know +neither the inventor nor the time of invention. That our +system of notation is of Indian origin is the only point of +which we are certain. From the evolution of ideas in general +we may safely infer that our notation did not spring into +existence a completely armed Minerva from the head of +Jupiter. The nine figures for writing the units are supposed +to have been introduced earliest, and the sign of zero and the +\index{Zero!(symbol for)}% +principle of position to be of later origin. This view receives +support from the fact that on the island of Ceylon a notation +resembling the Hindoo, but without the zero has been preserved. +We know that Buddhism and Indian culture were +transplanted to Ceylon about the third century after Christ, +and that this culture remained stationary there, while it made +progress on the continent. It seems highly probable, then, +that the numerals of Ceylon are the old, imperfect numerals +of India. In Ceylon, nine figures were used for the units, +nine others for the tens, one for~$100$, and also one for~$1000$. +These 20~characters enabled them to write all the numbers up +to~$9999$. Thus, $8725$ would have been written with six signs, +representing the following numbers: $8$,~$1000$, $7$,~$100$, $20$,~$5$. +These Singhalesian signs, like the old Hindoo numerals, are +supposed originally to have been the initial letters of the corresponding +numeral adjectives. There is a marked resemblance +between the notation of Ceylon and the one used by Aryabhatta +\index{Aryabhatta!ref.\ to}% +in the first chapter of his work, and there only. Although the +zero and the principle of position were unknown to the scholars +of Ceylon, they were probably known to Aryabhatta; for, in +the second chapter, he gives directions for extracting the square +and cube roots, which seem to indicate a knowledge of them. +%% -----File: 103.png---Folio 89------- +It would appear that the zero and the accompanying principle +of position were introduced about the time of Aryabhatta. +These are the inventions which give the Hindoo system its +great superiority, its admirable perfection. + +There appear to have been several notations in use in +different parts of India, which differed, not in principle, but +merely in the forms of the signs employed. Of interest is +also a \emph{symbolical system of position}, in which the figures +generally were not expressed by numerical adjectives, but by +objects suggesting the particular numbers in question. Thus, +for~$1$ were used the words \emph{moon}, \emph{Brahma}, \emph{Creator}, or \emph{form}; +for~$4$, the words \emph{Veda}, (because it is divided into four parts) +or \emph{ocean}, etc. The following example, taken from the \textit{Surya-siddhanta}, +illustrates the idea. The number $1,577,917,828$ is +expressed from right to left as follows: Vasu (a class of $8$~gods)% +${}+{}$ two${}+{}$eight${}+{}$mountains (the $7$~mountain-chains) +${}+{}$form${}+{}$digits (the $9$~digits)${}+{}$seven${}+{}$mountains${}+{}$lunar +days (half of which equal~$15$). The use of such notations +made it possible to represent a number in several different +ways. This greatly facilitated the framing of verses containing +arithmetical rules or scientific constants, which could +thus be more easily remembered. + +At an early period the Hindoos exhibited great skill in +calculating, even with large numbers. Thus, they tell us of +an examination to which Buddha, the reformer of the Indian +\index{Buddha}% +religion, had to submit, when a youth, in order to win the +maiden he loved. In arithmetic, after having astonished his +examiners by naming all the periods of numbers up to the~53d, +he was asked whether he could determine the number +of primary atoms which, when placed one against the other, +would form a line one mile in length. Buddha found the +required answer in this way: $7$~primary atoms make a very +minute grain of dust, $7$~of these make a minute grain of dust, +%% -----File: 104.png---Folio 90------- +\index{Arithmetic!Hindoo|(}% +$7$~of \emph{these} a grain of dust whirled up by the wind, and so on. +Thus he proceeded, step by step, until he finally reached the +length of a mile. The multiplication of all the factors gave +for the multitude of primary atoms in a mile a number consisting +of 15~digits. This problem reminds one of the `Sand-Counter' +\index{Sand-counter}% +of Archimedes. +\index{Archimedes!ref.\ to}% + +After the numerical symbolism had been perfected, figuring +was made much easier. Many of the Indian modes of +operation differ from ours. The Hindoos were generally +inclined to follow the motion from left to right, as in writing. +Thus, they \emph{added} the left-hand columns first, and made the +necessary corrections as they proceeded. For instance, they +would have added $254$ and $663$ thus: $2 + 6 = 8$, $5 + 6=11$, +which changes $8$ into~$9$, $4 + 3 = 7$. Hence the sum~$917$. In +\emph{subtraction} they had two methods. Thus in $821 - 348$ they +would say, $8\text{ from }11 = 3$, $4\text{ from }11 = 7$, $3\text{ from }7 = 4$. Or +they would say, $8\text{ from }11 = 3$, $5\text{ from }12 = 7$, $4\text{ from }8 = 4$. +In \emph{multiplication} of a number by another of only one digit, say +$569$ by~$5$, they generally said, $5·5 = 25$, $5·6 = 30$, which +changes $25$ into~$28$, $5·9 = 45$, hence the $0$ must be increased by~$4$. +The product is~$2845$. In the multiplication with each +other of many-figured numbers, they first multiplied, in the +manner just indicated, with the left-hand digit of the multiplier, +which was written above the multiplicand, and placed +the product above the multiplier. On multiplying with the +next digit of the multiplier, the product was not placed in +a new row, as with us, but the first product obtained was +corrected, as the process continued, by erasing, whenever +necessary, the old digits, and replacing them by new ones, +until finally the whole product was obtained. We who possess +the modern luxuries of pencil and paper, would not be likely +to fall in love with this Hindoo method. But the Indians +wrote ``with a cane-pen upon a small blackboard with a white, +%% -----File: 105.png---Folio 91------- +\index{Casting out the 9's}% +thinly liquid paint which made marks that could be easily +erased, or upon a white tablet, less than a foot square, strewn +with red flour, on which they wrote the figures with a small +stick, so that the figures appeared white on a red ground.''\Reference{7} +Since the digits had to be quite large to be distinctly legible, +and since the boards were small, it was desirable to have a +method which would not require much space. Such a one +was the above method of multiplication. Figures could be +easily erased and replaced by others without sacrificing neatness. +But the Hindoos had also other +ways of multiplying, of which we mention +the following: The %[** PP: Width-dependent line break] +%[Illustration] +\begin{wrapfigure}{r}{1.6in} +\hfill\Graphic{1.6in}{105a} +\end{wrapfigure} +tablet was +divided into squares like a chess-board. +Diagonals were also drawn, as seen in +the figure. The multiplication of $12 × +735 = 8820$ is exhibited in the adjoining diagram.\Reference{3} +The manuscripts +extant give no information of how \emph{divisions} were +executed. The correctness of their additions, subtractions, +and multiplications was tested ``by excess of $9$'s.'' In writing +fractions, the numerator was placed above the denominator, +but no line was drawn between them. + +We shall now proceed to the consideration of some arithmetical +problems and the Indian modes of solution. A favourite +method was that of \emph{inversion}. With laconic brevity, +Aryabhatta describes it thus: ``Multiplication becomes division, +\index{Aryabhatta!ref.\ to}% +division becomes multiplication; what was gain becomes +loss, what loss, gain; inversion.'' Quite different from this +quotation in style is the following problem from Aryabhatta, +which illustrates the method:\Reference[]{3} ``Beautiful maiden with beaming +eyes, tell me, as thou understandst the right method of +inversion, which is the number which multiplied by~$3$, then +increased by $\frac{3}{4}$ of the product, divided by~$7$, diminished by $\frac{1}{3}$ of +the quotient, multiplied by itself, diminished by~$52$, the square +%% -----File: 106.png---Folio 92------- +\index{Arithmetic!Hindoo|)}% +root extracted, addition of~$8$, and division by~$10$, gives the +number~$2$?'' The process consists in beginning with $2$ and +working backwards. Thus, $(2·10-8)^{2}+52 = 196$, $\sqrt {196}=14$, +and $14·\frac{3}{2}·7·\frac{4}{7} ÷ 3 = 28$, the answer. + +Here is another example taken from \textit{Lilavati}, a chapter in +Bhaskara's great work: ``The square root of half the number +\index{Bhaskara!ref.\ to|(}% +of bees in a swarm has flown out upon a jessamine-bush, $\frac{8}{9}$ of +the whole swarm has remained behind; one female bee flies +about a male that is buzzing within a lotus-flower into which +he was allured in the night by its sweet odour, but is now +imprisoned in it. Tell me the number of bees.'' Answer,~$72$. +The pleasing poetic garb in which all arithmetical problems +are clothed is due to the Indian practice of writing all school-books +in verse, and especially to the fact that these problems, +propounded as puzzles, were a favourite social amusement. +Says Brahmagupta: ``These problems are proposed simply for +\index{Brahmagupta!ref.\ to}% +pleasure; the wise man can invent a thousand others, or he +can solve the problems of others by the rules given here. As +the sun eclipses the stars by his brilliancy, so the man of +knowledge will eclipse the fame of others in assemblies of the +people if he proposes algebraic problems, and still more if he +solves them.'' + +The Hindoos solved problems in interest, discount, partnership, +alligation, summation of arithmetical and geometric +series, devised rules for determining the numbers of combinations +and permutations, and invented magic squares. It may +\index{Magic squares}% +here be added that chess, the profoundest of all games, had its +\index{Chess}% +origin in India. + +The Hindoos made frequent use of the ``rule of three,'' and +\index{Rule of three}% +also of the method of ``falsa positio,'' which is almost identical +\index{Falsa positio}% +with that of the ``tentative assumption'' of Diophantus. +\index{Assumption, tentative}% +\index{Tentative assumption}% +These and other rules were applied to a large number of +problems. +%% -----File: 107.png---Folio 93------- +\index{Notation!in algebra}% +\index{Quadratic equations}% + +Passing now to \emph{algebra}, we shall first take up the symbols +\index{Algebra!Hindoo|(}% +of operation. Addition was indicated simply by juxtaposition +as in Diophantine algebra; subtraction, by placing a dot over +the subtrahend; multiplication, by putting after the factors, +\emph{bha}, the abbreviation of the word \emph{bhavita}, ``the product''; +division, by placing the divisor beneath the dividend; square-root, +by writing \emph{ka}, from the word \emph{karana} (irrational), before +the quantity. The unknown quantity was called by Brahmagupta +\emph{yâvattâvat} (\emph{quantum tantum}). When several unknown +quantities occurred, he gave, unlike Diophantus, to each a +\index{Diophantus!ref.\ to}% +distinct name and symbol. The first unknown was designated +by the general term ``unknown quantity.'' The rest were +distinguished by names of colours, as the black, blue, yellow, +red, or green unknown. The initial syllable of each word +constituted the symbol for the respective unknown quantity. +Thus \emph{yâ} meant~$x$; \emph{kâ} (from $\text{\emph{kâlaka}} = \text{black}$) meant~$y$; \emph{yâ kâ +bha}, ``$x$~times~$y$''; \emph{ka}~$15$ \emph{ka}~$10$, ``$\sqrt{15} - \sqrt{10}$.'' + +The Indians were the first to recognise the existence of +absolutely negative quantities. They brought out the difference +\index{Negative quantities}% +between positive and negative quantities by attaching to +the one the idea of `possession,' to the other that of `debts.' +The conception also of opposite directions on a line, as an +interpretation of $+$~and~$-$ quantities, was not foreign to them. +They advanced beyond Diophantus in observing that a quadratic +has always two roots. Thus Bhaskara gives $x = 50$ and +$x = -5$ for the roots of $x^{2} - 45 x = 250$. ``But,'' says he, +``the second value is in this case not to be taken, for it is +inadequate; people do not approve of negative roots.'' Commentators +\index{Negative roots}% +speak of this as if negative roots were seen, but not +admitted. + +Another important generalisation, says Hankel, was this, +\index{Hankel!ref.\ to}% +that the Hindoos never confined their arithmetical operations +to rational numbers. For instance, Bhaskara showed how, +%% -----File: 108.png---Folio 94------- +\index{Irrationals}% +by the formula %[** PP: Displaying] +\[ +\sqrt{a + \sqrt{b}} + = \sqrt{\frac{a + \sqrt{a^{2} - b}}{2}} + + \sqrt{\frac{a - \sqrt{a^{2} - b}}{2}} +\] +the square root of the sum of rational and irrational numbers +\index{Square root}% +could be found. The Hindoos never discerned the dividing +line between numbers and magnitudes, set up by the Greeks, +which, though the product of a scientific spirit, greatly retarded +the progress of mathematics. They passed from magnitudes +to numbers and from numbers to magnitudes without +anticipating that gap which to a sharply discriminating mind +exists between the continuous and discontinuous. Yet by +doing so the Indians greatly aided the general progress of +mathematics. ``Indeed, if one understands by algebra the +application of arithmetical operations to complex magnitudes +of all sorts, whether rational or irrational numbers or space-magnitudes, +then the learned Brahmins of Hindostan are the +real inventors of algebra.''\Reference{7} + +Let us now examine more closely the Indian algebra. In +extracting the square and cube roots they used the formulas +$(a + b)^{2} = a^{2} + 2ab + b^{2}$ and $(a + b)^{3}= a^{3} + 3a^{2}b + 3ab^{2} + b^{3}$. +In this connection Aryabhatta speaks of dividing a number +into periods of two and three digits. From this we infer that +the principle of position and the zero in the numeral notation +were already known to him. In figuring with zeros, a statement +of Bhaskara is interesting. A fraction whose denominator +is zero, says he, admits of no alteration, though much be +added or subtracted. Indeed, in the same way, no change +takes place in the infinite and immutable Deity when worlds +are destroyed or created, even though numerous orders of beings +be taken up or brought forth. Though in this he apparently +evinces clear mathematical notions, yet in other places he +makes a complete failure in figuring with fractions of zero +\index{Fractions!Hindoo}% +denominator. + +In the Hindoo solutions of determinate equations, Cantor +%% -----File: 109.png---Folio 95------- +\index{Numbers!theory of numbers}% +\index{Theory of numbers}% +thinks he can see traces of Diophantine methods. Some +technical terms betray their Greek origin. Even if it be true +that the Indians borrowed from the Greeks, they deserve great +credit for improving and generalising the solutions of linear +and quadratic equations. Bhaskara advances far beyond the +\index{Bhaskara!ref.\ to|)}% +Greeks and even beyond Brahmagupta when he says that +\index{Brahmagupta!ref.\ to}% +``the square of a positive, as also of a negative number, +is positive; that the square root of a positive number is +twofold, positive and negative. There is no square root +of a negative number, for it is not a square.'' Of equations +of higher degrees, the Indians succeeded in solving +only some special cases in which both sides of the equation +could be made perfect powers by the addition of certain +terms to each. + +Incomparably greater progress than in the solution of determinate +equations was made by the Hindoos in the treatment +of \emph{indeterminate equations}. Indeterminate analysis was a +\index{Indeterminate analysis}% +\index{Indeterminate equations}% +subject to which the Hindoo mind showed a happy adaptation. +We have seen that this very subject was a favourite with Diophantus, +\index{Diophantus!ref.\ to}% +and that his ingenuity was almost inexhaustible in +devising solutions for particular cases. But the glory of +having invented \emph{general} methods in this most subtle branch +of mathematics belongs to the Indians. The Hindoo indeterminate +analysis differs from the Greek not only in method, +but also in aim. The object of the former was to find all +possible integral solutions. Greek analysis, on the other hand, +demanded not necessarily integral, but simply rational answers. +Diophantus was content with a single solution; the Hindoos +endeavoured to find all solutions possible. Aryabhatta gives +solutions in integers to linear equations of the form $ax ± by=c$, +where $a$,~$b$,~$c$ are integers. The rule employed is called the +\emph{pulveriser}. For this, as for most other rules, the Indians give +\index{Pulveriser}% +no proof. Their solution is essentially the same as the one of +%% -----File: 110.png---Folio 96------- +\index{Algebra!Hindoo|)}% +Euler. Euler's process of reducing $\dfrac{a}{b}$ to a continued fraction +\index{Euler!ref.\ to}% +amounts to the same as the Hindoo process of finding the +greatest common divisor of $a$~and~$b$ by division. This is frequently +called the Diophantine method. Hankel protests +\index{Hankel!ref.\ to}% +against this name, on the ground that Diophantus not only +\index{Diophantus!ref.\ to}% +never knew the method, but did not even aim at solutions +purely integral.\Reference{7} These equations probably grew out of problems +in astronomy. They were applied, for instance, to +determine the time when a certain constellation of the planets +would occur in the heavens. + +Passing by the subject of linear equations with more than +two unknown quantities, we come to indeterminate quadratic +equations. In the solution of $xy = ax + by + c$, they applied +the method re-invented later by Euler, of decomposing $(ab + c)$ +into the product of two integers $m·n$ and of placing $x = m + b$ +and $y = n + a$. + +Remarkable is the Hindoo solution of the quadratic equation +$cy^{2} = ax^{2} + b$. With great keenness of intellect they +recognised in the special case $y^{2} = ax^{2} + 1$ a fundamental +problem in indeterminate quadratics. They solved it by the +\emph{cyclic method}. ``It consists,'' says De~Morgan, ``in a rule for +\index{Cyclic method}% +\index{Demorgan@{De Morgan}!ref.\ to}% +finding an indefinite number of solutions of $y^{2} = ax^{2} +1$ ($a$~being +an integer which is not a square), by means of one solution +given or found, and of feeling for one solution by making a +solution of $y^2 = ax^{2} + b$ give a solution of $y^{2} = ax^{2} + b^{2}$. It +amounts to the following theorem: If $p$ and~$q$ be one set of +values of $x$ and~$y$ in $y^{2} = ax^{2} + b$ and $p'$ and~$q'$ the same or +another set, then $qp + pq'$ and $app' + qq'$ are values of $x$ and~$y$ +in $y^{2} = ax^{2} + b^{2}$. From this it is obvious that one solution of +$y^{2} = ax^{2} + 1$ may be made to give any number, and that if, +taking $b$ at pleasure, $y^{2} = ax^{2} + b^{2}$ can be solved so that $x$ and~$y$ +are divisible by~$b$, then one preliminary solution of $y^{2} = ax^{2} + 1$ +%% -----File: 111.png---Folio 97------- +\index{Pell's problem}% +\index{Pythagoras!ref.\ to}% +can be found. Another mode of trying for solutions is a +combination of the preceding with the \emph{cuttaca} (pulveriser).'' +These calculations were used in astronomy. + +Doubtless this ``cyclic method'' constitutes the greatest +\index{Cyclic method}% +invention in the theory of numbers before the time of Lagrange. +The perversity of fate has willed it, that the equation +$y^{2} = ax^{2} + 1$ should now be called \emph{Pell's} problem, while in +recognition of Brahmin scholarship it ought to be called the +``Hindoo problem.'' It is a problem that has exercised the +highest faculties of some of our greatest modern analysts. +By them the work of the Hindoos was done over again; for, +unfortunately, the Arabs transmitted to Europe only a small +part of Indian algebra and the original Hindoo manuscripts, +which we now possess, were unknown in the Occident. + +Hindoo \emph{geometry} is far inferior to the Greek. In it are +\index{Geometry!Hindoo}% +found no definitions, no postulates, no axioms, no logical +chain of reasoning or rigid form of demonstration, as with +Euclid. Each theorem stands by itself as an independent +\index{Euclid!ref.\ to}% +truth. Like the early Egyptian, it is empirical. Thus, in the +proof of the theorem of the right triangle, Bhaskara draws +\index{Bhaskara!ref.\ to}% +the %[** PP: Width-dependent line break] +% *[Illustration] +\begin{wrapfigure}[5]{r}{2.25in} +\hfill\Graphic{2.25in}{111a} +\end{wrapfigure} +right triangle four times in the square of the hypotenuse, +so that in the middle +there remains a square +whose side equals the +difference between the +two sides of the right +triangle. Arranging this square and the four triangles in a +different way, they are seen, together, to make up the sum +of the square of the two sides. ``Behold!'' says Bhaskara, +without adding another word of explanation. Bretschneider +\index{Bretschneider}% +conjectures that the Pythagorean proof was substantially the +same as this. In another place, Bhaskara gives a second +demonstration of this theorem by drawing from the vertex of +%% -----File: 112.png---Folio 98------- +\index{Ptolemy!ref.\ to}% +the right angle a perpendicular to the hypotenuse, and comparing +the two triangles thus obtained with the given triangle +to which they are similar. This proof was unknown in Europe +till Wallis \Change{rediscovered}{re-discovered} it. The Brahmins never inquired +\index{Wallis!ref.\ to}% +into the properties of figures. They considered only metrical +relations applicable in practical life. In the Greek sense, the +Brahmins never had a science of geometry. Of interest is +\index{Geometry!Hindoo}% +the formula given by Brahmagupta for the area of a triangle +\index{Brahmagupta!ref.\ to}% +in terms of its sides. In the great work attributed to Heron +\index{Heron the Elder!ref.\ to}% +the Elder this formula is first found. Whether the Indians +themselves invented it, or whether they borrowed it from +Heron, is a disputed question. Several theorems are given +by Brahmagupta on quadrilaterals which are true only of +those which can be inscribed on a circle---a limitation which +he omits to state. Among these is the proposition of Ptolemæus, +that the product of the diagonals is equal to the sum +of the products of the opposite sides. The Hindoos were +familiar with the calculation of the areas of circles and their +segments, of the length of chords and perimeters of regular +inscribed polygons. An old Indian tradition makes $\pi = 3$, +also $= \sqrt{10}$; but Aryabhatta gives the value $\frac{31416}{10000}$. Bhaskara +\index{Aryabhatta!ref.\ to}% +gives two values,---the `accurate,' $\frac{3927}{1250}$, and the `inaccurate,' +Archimedean value,~$\frac{22}{7}$. A commentator on \textit{Lilavati} says +that these values were calculated by beginning with a regular +inscribed hexagon, and applying repeatedly the formula +$AD = \sqrt{2 - \sqrt{\vphantom{\rule{0pt}{12pt}}4 - \smash[t]{\overline{AB}^{2}}}}$, wherein~$AB$ is the side of the given +polygon, and $AD$ that of one with double the number of +sides. In this way were obtained the perimeters of the +inscribed polygons of $12$, $24$, $48$, $96$, $192$, $384$~sides. Taking +the radius $= 100$, the perimeter of the last one gives the value +which Aryabhatta used for~$\pi$. +\index{p@{$\pi$}: values for!Hindoo}% + +Greater taste than for geometry was shown by the Hindoos +for \emph{trigonometry}. Like the Babylonians and Greeks, they +\index{Trigonometry|(}% +%% -----File: 113.png---Folio 99------- +divided the circle into quadrants, each quadrant into $90$~degrees +and $5400$ minutes. The whole circle was therefore +made up of $21,600$ equal parts. From Bhaskara's `accurate' +value for~$\pi$ it was found that the radius contained $3438$ of +these circular parts. This last step was not Grecian. The +Greeks might have had scruples about taking a part of a +\emph{curve} as the measure of a \emph{straight line}. Each quadrant was +divided into $24$~equal parts, so that each part embraced $225$ +units of the whole circumference, and corresponds to $3 \frac{3}{4}$ degrees. +Notable is the fact that the Indians never reckoned, like the +Greeks, with the whole chord of double the arc, but always +with the \emph{sine} (\emph{joa}) and \emph{versed sine}. Their mode of calculating +\index{Sine}% +\index{Versed sine}% +tables was theoretically very simple. The sine of~$90°$ was +equal to the radius, or~$3438$; the sine of~$30°$ was evidently +half that, or~$1719$. Applying the formula $\sin^{2}a + \cos^{2}a = r^{2}$, +they obtained $\sin 45° = \sqrt{\dfrac{r^2}{2}} = 2431$. Substituting for $\cos a$ +its equal $\sin(90 - a)$, and making $a = 60°$, they obtained +$\sin 60° = \dfrac{\sqrt{3r^2}}{2} = 2978$. With the sines of $90$, $60$, $45$, and~$30$ +as starting-points, they reckoned the sines of half the angles +by the formula $\versin 2a = 2 \sin^{2}a$, thus obtaining the sines +of $22°\ 30'$, $11°\ 15'$, $7°\ 30'$, $3°\ 45'$. They now figured out the +sines of the complements of these angles, namely, the sines +of $86°\ 15'$, $82°\ 30'$, $78°\ 45'$, $75°$, $67°\ 30'$; then they calculated +the sines of half these angles; then of their complements; +then, again, of half their complements; and so on. By this +very simple process they got the sines of angles at intervals of +$3°\ 45'$. In this table they discovered the unique law that if +$a$,~$b$,~$c$ be three successive arcs such that $a - b = b - c = 3°\ 45'$, +then $\sin a - \sin b = (\sin b - \sin c) - \dfrac{\sin b}{225}$. This formula was +afterwards used whenever a re-calculation of tables had to +be made. No Indian trigonometrical treatise on the triangle +%% -----File: 114.png---Folio 100------- +\index{Hindoos|)}% +\index{Trigonometry|)}% +is extant. In astronomy they solved plane and spherical +\index{Astronomy!Arabic}% +right triangles.\Reference{18} + +It is remarkable to what extent Indian mathematics enters +into the science of our time. Both the form and the spirit +of the arithmetic and algebra of modern times are essentially +Indian and not Grecian. Think of that most perfect of +mathematical symbolisms---the Hindoo notation, think of the +Indian arithmetical operations nearly as perfect as our own, +think of their elegant algebraical methods, and then judge +whether the Brahmins on the banks of the Ganges are not +entitled to some credit. Unfortunately, some of the most +brilliant of Hindoo discoveries in indeterminate analysis +reached Europe too late to exert the influence they would +have exerted, had they come two or three centuries earlier. + +\Section{The Arabs} +\index{Arabs|(}% + +After the flight of Mohammed from Mecca to Medina in +622~\AD, an obscure people of Semitic race began to play an +important part in the drama of history. Before the lapse of +ten years, the scattered tribes of the Arabian peninsula were +fused by the furnace blast of religious enthusiasm into a +powerful nation. With sword in hand the united Arabs +subdued Syria and Mesopotamia. Distant Persia and the +lands beyond, even unto India, were added to the dominions +of the Saracens. They conquered Northern Africa, and nearly +the whole Spanish peninsula, but were finally checked from +further progress in Western Europe by the firm hand of +Charles Martel (732~\AD). The Moslem dominion extended +now from India to Spain; but a war of succession to the +caliphate ensued, and in 755 the Mohammedan empire was +divided,---one caliph reigning at Bagdad, the other at Cordova +%% -----File: 115.png---Folio 101------- +\index{Indeterminate analysis}% +\index{Indeterminate equations}% +in Spain. Astounding as was the grand march of conquest by +the Arabs, still more so was the ease with which they put +aside their former nomadic life, adopted a higher civilisation, +and assumed the sovereignty over cultivated peoples. Arabic +was made the written language throughout the conquered +lands. With the rule of the Abbasides in the East began a +new period in the history of learning. The capital, Bagdad, +situated on the Euphrates, lay half-way between two old +centres of scientific thought,---India in the East, and Greece +in the West. The Arabs were destined to be the custodians +of the torch of Greek and Indian science, to keep it ablaze +during the period of confusion and chaos in the Occident, and +afterwards to pass it over to the Europeans. Thus science +passed from Aryan to Semitic races, and then back again +to the Aryan. The Mohammedans have added but little to +the knowledge in mathematics which they received. They +now and then explored a small region to which the path had +been previously pointed out, but they were quite incapable of +discovering new fields. Even the more elevated regions in +which the Hellenes and Hindoos delighted to wander---namely, +the Greek conic sections and the Indian indeterminate +\index{Conic sections!Arabs}% +analysis---were seldom entered upon by the Arabs. They +were less of a speculative, and more of a practical turn of +mind. + +The Abbasides at Bagdad encouraged the introduction of +the sciences by inviting able specialists to their court, irrespective +of nationality or religious belief. Medicine and +astronomy were their favourite sciences. Thus Haroun-al-Raschid, +\index{Astronomy!Arabic}% +the most distinguished Saracen ruler, drew Indian +physicians to Bagdad. In the year~772 there came to the +court of Caliph Almansur a Hindoo astronomer with astronomical +tables which were ordered to be translated into Arabic. +These tables, known by the Arabs as the \textit{Sindhind}, and +%% -----File: 116.png---Folio 102------- +\index{Sine}% +probably taken from the \textit{Brahma-sphuta-siddhanta} of Brahmagupta, +\index{Brahmagupta!ref.\ to}% +stood in great authority. They contained the important +Hindoo table of sines. + +Doubtless at this time, and along with these astronomical +tables, the Hindoo numerals, with the zero and the principle +\index{Numerals!Arabic}% +of position, were introduced among the Saracens. Before the +time of Mohammed the Arabs had no numerals. Numbers +were written out in words. Later, the numerous computations +connected with the financial administration over the conquered +lands made a short symbolism indispensable. In some localities, +the numerals of the more civilised conquered nations +were used for a time. Thus in Syria, the Greek notation was +\index{Notation!Arabic notation}% +retained; in Egypt, the Coptic. In some cases, the numeral +adjectives may have been abbreviated in writing. The \textit{Diwani-numerals}, +\index{Diwani-numerals}% +found in an Arabic-Persian dictionary, are supposed +to be such abbreviations. Gradually it became the practice to +employ the 28~Arabic letters of the alphabet for numerals, in +analogy to the Greek system. This notation was in turn +superseded by the Hindoo notation, which quite early was +adopted by merchants, and also by writers on arithmetic. Its +superiority was so universally recognised, that it had no rival, +except in astronomy, where the alphabetic notation continued +to be used. Here the alphabetic notation offered no great +disadvantage, since in the sexagesimal arithmetic, taken from +the \textit{Almagest}, numbers of generally only one or two places +had to be written.\Reference{7} + +As regards the form of the so-called Arabic numerals, the +\index{Arabic numerals and notation}% +statement of the Arabic writer \textit{Albiruni} (died~1039), who +\index{Albiruni!ref.\ to}% +spent many years in India, is of interest. He says that the +shape of the numerals, as also of the letters in India, differed +in different localities, and that the Arabs selected from the +various forms the most suitable. An Arabian astronomer +says there was among people much difference in the use of +%% -----File: 117.png---Folio 103------- +\index{Gobar numerals}% +symbols, especially of those for $5$, $6$, $7$, and~$8$. The symbols +used by the Arabs can be traced back to the tenth century. +We find material differences between those used by the +Saracens in the East and those used in the West. But +most surprising is the fact that the symbols of both the East +and of the West Arabs deviate so extraordinarily from the +Hindoo \textit{Devanagari} numerals ($=$~divine numerals) of to-day, +\index{Devanagari-numerals}% +\index{Numerals!Arabic}% +and that they resemble much more closely the apices of +\index{Apices of Boethius!ref.\ to}% +the Roman writer Boethius. This strange similarity on the +\index{Boethius!ref.\ to}% +one hand, and dissimilarity on the other, is difficult to explain. +The most plausible theory is the one of Woepcke: (1)~that +\index{Woepcke}% +about the second century after Christ, before the zero had +been invented, the Indian numerals were brought to Alexandria, +whence they spread to Rome and also to West Africa; +(2)~that in the eighth century, after the notation in India had +been already much modified and perfected by the invention of +the zero, the Arabs at Bagdad got it from the Hindoos; (3)~that +the Arabs of the West borrowed the Columbus-egg, the zero, +from those in the East, but retained the old forms of the nine +numerals, if for no other reason, simply to be contrary to their +political enemies of the East; (4)~that the old forms were +remembered by the West-Arabs to be of Indian origin, and +were hence called \textit{Gubar-numerals} ($=$~dust-numerals, in memory +\index{Gubar-numerals}% +of the Brahmin practice of reckoning on tablets strewn +with dust or sand; (5)~that, since the eighth century, the +numerals in India underwent further changes, and assumed +the greatly modified forms of the modern Devanagari-numerals.\Reference{3} +This is rather a bold theory, but, whether true or not, +it explains better than any other yet propounded, the relations +between the apices, the Gubar, the East-Arabic, and Devanagari +numerals. + +It has been mentioned that in 772 the Indian \textit{Siddhanta} was +brought to Bagdad and there translated into Arabic. There +%% -----File: 118.png---Folio 104------- +\index{Geometry!Arabic}% +\index{Ptolemy!ref.\ to}% +is no evidence that any intercourse existed between Arabic +and Indian astronomers either before or after this time, excepting +the travels of Albiruni. But we should be very slow +\index{Albiruni!ref.\ to}% +to deny the probability that more extended communications +actually did take place. + +Better informed are we regarding the way in which Greek +science, in successive waves, dashed upon and penetrated Arabic +soil. In Syria the sciences, especially philosophy and medicine, +were cultivated by Greek Christians. Celebrated were the +schools at Antioch and Emesa, and, first of all, the flourishing +Nestorian school at Edessa. From Syria, Greek physicians +and scholars were called to Bagdad. Translations of works +from the Greek began to be made. A large number of Greek +manuscripts were secured by Caliph \textit{Al Mamun} (813--833) from +the emperor in Constantinople and were turned over to Syria. +The successors of Al Mamun continued the work so auspiciously +begun, until, at the beginning of the tenth century, the +more important philosophic, medical, mathematical, and astronomical +works of the Greeks could all be read in the Arabic +tongue. The translations of mathematical works must have +been very deficient at first, as it was evidently difficult to +secure translators who were masters of both the Greek and +Arabic and at the same time proficient in mathematics. The +translations had to be revised again and again before they +were satisfactory. The first Greek authors made to speak in +Arabic were Euclid and Ptolemæus. This was accomplished +\index{Euclid!ref.\ to}% +during the reign of the famous Haroun-al-Raschid. A revised +\index{Haroun-al-Raschid}% +translation of Euclid's \textit{Elements} was ordered by Al Mamun. +\index{Elements (Euclid's)}% +As this revision still contained numerous errors, a new translation +was made, either by the learned Honein ben Ishak, or +\index{Honein ben Ishak}% +by his son, Ishak ben Honein. To the thirteen books of the +\index{Ishak ben Honein}% +\textit{Elements} were added the fourteenth, written by Hypsicles, +\index{Hypsicles!ref.\ to}% +and the fifteenth by Damascius. But it remained for Tabit +\index{Damascius!ref.\ to}% +%% -----File: 119.png---Folio 105------- +ben Korra to bring forth an Arabic Euclid satisfying every +\index{Tabit ben Korra!ref.\ to}% +need. Still greater difficulty was experienced in securing an +intelligible translation of the \textit{Almagest}. Among other important +\index{Almagest!ref.\ to}% +translations into Arabic were the works of Apollonius, +\index{Apollonius!ref.\ to}% +Archimedes, Heron, and Diophantus. Thus we see that in +\index{Archimedes!ref.\ to}% +\index{Diophantus!ref.\ to}% +\index{Heron the Elder!ref.\ to}% +the course of one century the Arabs gained access to the vast +treasures of Greek science. Having been little accustomed to +abstract thought, we need not marvel if, during the ninth century, +all their energy was exhausted merely in appropriating +the foreign material. No attempts were made at original +work in mathematics until the next century. + +In astronomy, on the other hand, great activity in original +\index{Astronomy!Arabic}% +research existed as early as the ninth century. The religious +observances demanded by Mohammedanism presented to astronomers +several practical problems. The Moslem dominions +being of such enormous extent, it remained in some localities +for the astronomer to determine which way the ``Believer'' +must turn during prayer that he may be facing Mecca. The +prayers and ablutions had to take place at definite hours during +the day and night. This led to more accurate determinations +of time. To fix the exact date for the Mohammedan +feasts it became necessary to observe more closely the motions +of the moon. In addition to all this, the old Oriental superstition +that extraordinary occurrences in the heavens in some +mysterious way affect the progress of human affairs added +increased interest to the prediction of eclipses.\Reference{7} + +For these reasons considerable progress was made. Astronomical +tables and instruments were perfected, observatories +erected, and a connected series of observations instituted. This +intense love for astronomy and astrology continued during the +whole Arabic scientific period. As in India, so here, we hardly +ever find a man exclusively devoted to pure mathematics. Most +of the so-called mathematicians were first of all astronomers. +%% -----File: 120.png---Folio 106------- +\index{Ptolemy!ref.\ to}% + +The first notable author of mathematical books was \textbf{Mohammed +ben Musa Al Hovarezmi}, who lived during the reign of Caliph +\index{Hovarezmi}% +\index{Mohammed ben Musa Hovarezmi}% +Al Mamun (813--833). He was engaged by the caliph in making +extracts from the \textit{Sindhind}, in revising the tablets of Ptolemæus, +in taking observations at Bagdad and Damascus, and in +measuring a degree of the earth's meridian. Important to us +is his work on algebra and arithmetic. The portion on arithmetic +\index{Arithmetic!Arabic}% +is not extant in the original, and it was not till 1857 +that a Latin translation of it was found. It begins thus: +``Spoken has Algoritmi. Let us give deserved praise to God, +our leader and defender.'' Here the name of the author, \textit{Al +Hovarezmi}, has passed into \textit{Algoritmi}, from which comes our +modern word, \emph{algorithm}, signifying the art of computing in +\index{Algorithm!origin of term}% +any particular way. The arithmetic of Hovarezmi, being +based on the principle of position and the Hindoo method of +calculation, ``excels,'' says an Arabic writer, ``all others in +brevity and easiness, and exhibits the Hindoo intellect and +sagacity in the grandest inventions.'' This book was followed +by a large number of arithmetics by later authors, which differed +from the earlier ones chiefly in the greater variety of +methods. Arabian arithmetics generally contained the four +operations with integers and fractions, modelled after the +Indian processes. They explained the operation of \emph{casting out +\index{Casting out the 9's}% +\index{Nines, casting out the}% +the~9's}, which was sometimes called the ``Hindoo proof.'' They +contained also the \textit{regula falsa} and the \textit{regula duorum falsorum}, +\index{Regula duorum falsorum}% +\index{Regula falsa}% +by which algebraical examples could be solved without algebra. +Both these methods were known to the Indians. The \textit{regula +falsa} or \textit{falsa positio} was the assigning of an assumed value to +the unknown quantity, which value, if wrong, was corrected +by some process like the ``rule of three.'' Diophantus used a +\index{Diophantus!ref.\ to}% +\index{Rule of three}% +method almost identical with this. The \textit{regula duorum falsorum} +was as follows:\Reference[]{7} To solve an equation $f(x) = V$, assume, +for the moment, two values for $x$; namely, $x = a$ and $x = b$. +%% -----File: 121.png---Folio 107------- +\index{Irrationals}% +Then form $f(a) = A$ and $f(b) = B$, and determine the errors +$V - A = E_a$ and $V - B = E_b$; then the required +$x = \dfrac{bE_a - aE_b}{E_a - E_b}$ +is generally a close approximation, but is absolutely accurate +whenever $f(x)$ is a linear function of~$x$. + +We now return to Hovarezmi, and consider the other part +\index{Hovarezmi!ref.\ to}% +\index{Mohammed ben Musa Hovarezmi!ref.\ to}% +of his work,---the \emph{algebra}. This is the first book known to +\index{Algebra!Arabic}% +\index{Algebra!origin of terms}% +contain this word itself as title. Really the title consists of +two words, \textit{aldshebr walmukabala}, the nearest English translation +of which is ``restoration'' and ``reduction.'' By ``restoration'' +was meant the transposing of negative terms to the +other side of the equation; by ``reduction,'' the uniting of +similar terms. Thus, $x^2 - 2x = 5x + 6$ passes by aldshebr into +$x^2 = 5x + 2x + 6$; and this, by walmukabala, into $x^2 = 7x + 6$. +The work on algebra, like the arithmetic, by the same author, +contains nothing original. It explains the elementary operations +and the solutions of linear and quadratic equations. +\index{Quadratic equations}% +From whom did the author borrow his knowledge of algebra? +That it came entirely from Indian sources is impossible, for +the Hindoos had no such rules like the ``restoration'' and +``reduction.'' They were, for instance, never in the habit of +making all terms in an equation positive, as is done by the +process of ``restoration.'' Diophantus gives two rules which +\index{Diophantus!ref.\ to}% +resemble somewhat those of our Arabic author, but the probability +that the Arab got all his algebra from Diophantus is +lessened by the considerations that he recognised both roots +of a quadratic, while Diophantus noticed only one; and that +the Greek algebraist, unlike the Arab, habitually rejected irrational +solutions. It would seem, therefore, that the algebra +of Hovarezmi was neither purely Indian nor purely Greek, +but was a hybrid of the two, with the Greek element predominating. + +The algebra of Hovarezmi contains also a few meagre +%% -----File: 122.png---Folio 108------- +fragments on \emph{geometry}. He gives the theorem of the right +\index{Geometry!Arabic}% +triangle, but proves it after Hindoo fashion and only for the +simplest case, when the right triangle is isosceles. He then +calculates the areas of the triangle, parallelogram, and circle. +For $\pi$ he uses the value~$3 \frac{1}{7}$, and also the two Indian, $\pi = \sqrt{10}$ +\index{p@{$\pi$}: values for!Arabic}% +and $\pi = \frac{62832}{20000}$. Strange to say, the last value was afterwards +forgotten by the Arabs, and replaced by others less +accurate. This bit of geometry doubtless came from India. +Later Arabic writers got their geometry almost entirely from +Greece. + +Next to be noticed are the three sons of \textbf{Musa ben Sakir}, +\index{Musa ben Sakir}% +who lived in Bagdad at the court of the Caliph Al Mamun. +They wrote several works, of which we mention a geometry +in which is also contained the well-known formula for the +area of a triangle expressed in terms of its sides. We are +told that one of the sons travelled to Greece, probably to +collect astronomical and mathematical manuscripts, and that +on his way back he made acquaintance with Tabit ben Korra. +\index{Tabit ben Korra}% +Recognising in him a talented and learned astronomer, Mohammed +procured for him a place among the astronomers at +the court in Bagdad. \textbf{Tabit ben Korra} (836--901) was born +at Harran in Mesopotamia. He was proficient not only in +astronomy and mathematics, but also in the Greek, Arabic, +and Syrian languages. His translations of Apollonius, Archimedes, +\index{Apollonius!ref.\ to}% +\index{Archimedes!ref.\ to}% +Euclid, Ptolemy, Theodosius, rank among the best. +\index{Euclid!ref.\ to}% +\index{Ptolemy!ref.\ to}% +\index{Theodosius!ref.\ to}% +His dissertation on \emph{amicable numbers} (of which each is the +\index{Numbers!amicable}% +\index{Numbers!theory of numbers}% +sum of the factors of the other) is the first known specimen +of original work in mathematics on Arabic soil. It shows +that he was familiar with the Pythagorean theory of numbers. +\index{Theory of numbers}% +Tabit invented the following rule for finding amicable numbers: +If $p = 3 · 2^{n} - 1$, $q = 3 · 2^{n-1} - 1$, $r = 9 · 2^{2n-1} - 1$ ($n$~being +a whole number) are three primes, then $a = 2^{n} pq$, $b = 2^{n} r$ are +a pair of amicable numbers. Thus, if $n = 2$, then $p = 11$, +%% -----File: 123.png---Folio 109------- +$q = 5$, $r = 71$, and $a = 220$, $b = 284$. Tabit also trisected an +angle. + +Foremost among the astronomers of the ninth century +ranked \textbf{Al Battani}, called \textit{Albategnius} by the Latins. Battan +\index{Albattani@{Al Battani}}% +in Syria was his birthplace. His observations were celebrated +for great precision. His work, \textit{De scientia stellarum}, was translated +into Latin by Plato Tiburtinus, in the twelfth century. +\index{Plato of Tivoli}% +Out of this translation sprang the word `sinus,' as the name +of a trigonometric function. The Arabic word for ``sine,'' +\index{Sine}% +\index{Sine!origin of term}% +\textit{dschiba}, was derived from the Sanscrit \textit{jiva}, and resembled +the Arabic word \textit{dschaib}, meaning an indentation or gulf. +Hence the Latin ``sinus.''\Reference{3} Al Battani was a close student +of Ptolemy, but did not follow him altogether. He took an +\index{Ptolemy!ref.\ to}% +important step for the better, when he introduced the Indian +``sine'' or \emph{half} the chord, in place of the \emph{whole} chord of +Ptolemy. Another improvement on Greek trigonometry made +\index{Trigonometry}% +by the Arabs points likewise to Indian influences. Propositions +and operations which were treated by the Greeks geometrically +are expressed by the Arabs algebraically. Thus, +\textit{Al Battani} at once gets from an equation +$\smash[b]{\dfrac{\sin \theta}{\cos \theta}} = D$, the value +of $\theta$ by means of +$\sin \theta = \dfrac{D\Fstr}{\sqrt{1 + D^2}}$,---a process unknown to +the ancients. He knows, of course, all the formulas for +spherical triangles given in the \textit{Almagest}, but goes further, +\index{Almagest!ref.\ to}% +and adds an important one of his own for oblique-angled +triangles; namely, $\cos a = \cos b \cos c + \sin b \sin c \cos A$. + +At the beginning of the tenth century political troubles +arose in the East, and as a result the house of the Abbasides +lost power. One province after another was taken, till, in~945, +all possessions were wrested from them. Fortunately, +the new rulers at Bagdad, the Persian Buyides, were as much +interested in astronomy as their predecessors. The progress +of the sciences was not only unchecked, but the conditions +%% -----File: 124.png---Folio 110------- +\index{Geometry!Arabic}% +\index{Regular solids}% +\index{Tycho Brahe}% +for it became even more favourable. The Emir \textit{Adud-ed-daula} +(978--983) gloried in having studied astronomy himself. His +son \textit{Saraf-ed-daula} erected an observatory in the garden of his +palace, and called thither a whole group of scholars.\Reference{7} Among +them were \textit{Abul Wefa}, \textit{Al Kuhi}, \textit{Al Sagani}. +\index{Abul Wefa}% + +\textbf{Abul Wefa} (940--998) was born at Buzshan in Chorassan, a +region among the Persian mountains, which has brought forth +many Arabic astronomers. He forms an important exception +to the unprogressive spirit of Arabian scientists by his brilliant +discovery of the \emph{variation} of the moon, an inequality +usually supposed to have been first discovered by Tycho +Brahe.\Reference{11} Abul Wefa translated Diophantus. He is one of the +\index{Brahe, Tycho}% +\index{Diophantus!ref.\ to}% +last Arabic translators and commentators of Greek authors. +The fact that he esteemed the algebra of Mohammed ben +Musa Hovarezmi worthy of his commentary indicates that +\index{Hovarezmi!ref.\ to}% +\index{Mohammed ben Musa Hovarezmi!ref.\ to}% +thus far algebra had made little or no progress on Arabic soil. +Abul Wefa invented a method for computing tables of sines +which gives the sine of half a degree correct to nine decimal +places. He did himself credit by introducing the \emph{tangent} into +trigonometry and by calculating a table of tangents. The +\index{Tangents!in trigonometry}% +\index{Trigonometry}% +first step toward this had been taken by Al Battani. Unfortunately, +\index{Albattani@{Al Battani}!ref.\ to}% +this innovation and the discovery of the moon's +variation excited apparently no notice among his contemporaries +and followers. ``We can hardly help looking upon this +circumstance as an evidence of a servility of intellect belonging +to the Arabian period.'' A treatise by Abul Wefa on +``geometric constructions'' indicates that efforts were being +made at that time to improve draughting. It contains a neat +construction of the corners of the regular \DPtypo{polyedrons}{polyhedrons} on the +circumscribed sphere. Here, for the first time, appears the +condition which afterwards became very famous in the Occident, +that the construction be effected with a single opening +of the compass. +%% -----File: 125.png---Folio 111------- +\index{Albiruni}% +\index{Alhogendi@{Al Hogendi}}% +\index{Cube numbers}% +\index{Indeterminate equations}% +\index{Quadratic equations}% + +\textbf{Al Kuhi}, the second astronomer at the observatory of the +\index{Alkuhi@{Al Kuhi}}% +emir at Bagdad, was a close student of Archimedes and +Apollonius. He solved the problem, to construct a segment +of a sphere equal in volume to a given segment and having +a curved surface equal in area to that of another given segment. +He, \textbf{Al Sagani}, and \textbf{Al Biruni} made a study of the +\index{Alsagani@{Al Sagani}}% +trisection of angles. \textbf{Abul Gud}, an able geometer, solved the +\index{Abul Gud}% +problem by the intersection of a parabola with an equilateral +hyperbola. + +The Arabs had already discovered the theorem that the +sum of two cubes can never be a cube. \textbf{Abu Mohammed Al +Hogendi} of Chorassan thought he had proved this, but we are +told that the demonstration was defective. Creditable work +in theory of numbers and algebra was done by \textbf{Al Karhi} of +\index{Algebra!Arabic}% +\index{Alkarhi@{Al Karhi}}% +\index{Fahri des Al Karhi}% +Bagdad, who lived at the beginning of the eleventh century. +His treatise on algebra is the greatest algebraic work of the +Arabs. In it he appears as a disciple of Diophantus. He +\index{Diophantus!ref.\ to}% +was the first to operate with higher roots and to solve equations +of the form $x^{2n} + ax^{n} = b$. For the solution of quadratic +equations he gives both arithmetical and geometric proofs. +He was the first Arabic author to give and prove the theorems +on the summation of the series:--- +\index{Series}% +\begin{align*} +1^2 + 2^2 + 3^2 + \dotsb + n^2 + &= (1 + 2 + \dotsb + n)\DPtypo{^}{\,}\frac{2n+1}{3}, \\ +1^3 + 2^3 + 3^3 + \dotsb + n^3 + &= (1 + 2 + \dotsb + n)^2. +\end{align*} + +Al Karhi also busied himself with indeterminate analysis. +\index{Indeterminate analysis}% +He showed skill in handling the methods of Diophantus, but +added nothing whatever to the stock of knowledge already +on hand. As a subject for original research, indeterminate +analysis was too subtle for even the most gifted of Arabian +minds. Rather surprising is the fact that Al Karhi's algebra +shows no traces whatever of Hindoo indeterminate analysis. +%% -----File: 126.png---Folio 112------- +\index{Alhayyami@{Al Hayyami}}% +\index{Arabic numerals and notation}% +\index{Cubic equations}% +\index{Notation!Arabic notation}% +But most astonishing it is, that an arithmetic by the same +author completely excludes the Hindoo numerals. It is constructed +\index{Numerals!Arabic}% +wholly after Greek pattern. Abul Wefa also, in the +\index{Abul Wefa!ref.\ to}% +second half of the tenth century, wrote an arithmetic in which +Hindoo numerals find no place. This practice is the very +opposite to that of other Arabian authors. The question, +why the Hindoo numerals were ignored by so eminent authors, +is certainly a puzzle. Cantor suggests that at one time there +\index{Cantor, M.}% +may have been rival schools, of which one followed almost +exclusively Greek mathematics, the other Indian. + +The Arabs were familiar with geometric solutions of quadratic +equations. Attempts were now made to solve cubic +equations geometrically. They were led to such solutions by +the study of questions like the Archimedean problem, demanding +the section of a sphere by a plane so that the two segments +shall be in a prescribed ratio. The first to state this +problem in form of a cubic equation was \textbf{Al Mahani} of Bagdad, +\index{Almahani@{Al Mahani}}% +while \textbf{Abu Gafar Al Hazin} was the first Arab to solve the +\index{Alhazin@{Al Hazin}}% +equation by conic sections. Solutions were given also by +\index{Conic sections!Arabs}% +Al Kuhi, Al Hasan ben Al Haitam, and others.\Reference{20} Another +\index{Alhaitam@{Al Haitam}!ref.\ to}% +\index{Alkuhi@{Al Kuhi}!ref.\ to}% +difficult problem, to determine the side of a regular heptagon, +required the construction of the side from the equation +$x^3 - x^2 - 2x + 1 = 0$. It was attempted by many and at last +solved by Abul Gud. + +The one who did most to elevate to a \emph{method} the solution +of algebraic equations by intersecting conics, was \textbf{Omar al +Hayyami} of Chorassan, about 1079~\AD\ He divides cubics into +two classes, the trinomial and quadrinomial, and each class +into families and species. Each species is treated separately +but according to a general plan. He believed that cubics +could not be solved by calculation, nor \Change{bi-quadratics}{biquadratics} by geometry. +He rejected negative roots and often failed to discover +\index{Negative roots}% +all the positive ones. Attempts at \Change{bi-quadratic}{biquadratic} equations +\index{Biquadratic equation}% +\index{Quadratic equations}% +%% -----File: 127.png---Folio 113------- +\index{Geometry!Arabic}% +were made by Abul Wefa,\Reference[]{20} who solved geometrically +\index{Abul Wefa!ref.\ to}% +$x^4 = a$ and $x^4 + ax^3 = b$. + +The solution of cubic equations by intersecting conics was +\index{Cubic equations}% +the greatest achievement of the Arabs in algebra. The foundation +to this work had been laid by the Greeks, for it was +Menæchmus who first constructed the roots of $x^3 - a = 0$ or +\index{Menæchmus!ref.\ to}% +$x^3 - 2 a^3 = 0$. It was not his aim to find the number corresponding +to~$x$, but simply to determine the side~$x$ of a cube +double another cube of side~$a$. The Arabs, on the other +hand, had another object in view: to find the roots of given +numerical equations. In the Occident, the Arabic solutions +of cubics remained unknown until quite recently. Descartes +\index{Descartes!ref.\ to}% +and Thomas Baker invented these constructions anew. The +\index{Baker, Th}% +works of Al Hayyami, Al Karhi, Abul Gud, show how the +\index{Abul Gud!ref.\ to}% +\index{Alhayyami@{Al Hayyami}!ref.\ to}% +\index{Alkarhi@{Al Karhi}}% +Arabs departed further and further from the Indian methods, +and placed themselves more immediately under Greek influences. +In this way they barred the road of progress against +themselves. The Greeks had advanced to a point where +material progress became difficult with their methods; but the +Hindoos furnished new ideas, many of which the Arabs now +rejected. + +With Al Karhi and Omar Al Hayyami, mathematics among +the Arabs of the East reached flood-mark, and now it begins to +ebb. Between 1100 and 1300~\AD\ come the crusades with +war and bloodshed, during which European Christians profited +much by their contact with Arabian culture, then far superior +to their own; but the Arabs got no science from the Christians +in return. The crusaders were not the only adversaries of the +Arabs. During the first half of the thirteenth century, they +had to encounter the wild Mongolian hordes, and, in~1256, were +conquered by them under the leadership of \textit{Hulagu}. The +caliphate at Bagdad now ceased to exist. At the close of the +fourteenth century still another empire was formed by Timur +%% -----File: 128.png---Folio 114------- +or \textit{Tamerlane}, the Tartar. During such sweeping turmoil, it +is not surprising that science declined. Indeed, it is a marvel +that it existed at all. During the supremacy of Hulagu, lived +\textbf{Nasir Eddin} (1201--1274), a man of broad culture and an able +\index{Nasir Eddin}% +\Addendlabel{114}% +astronomer. He persuaded Hulagu to build him and his associates +a large observatory at Maraga. Treatises on algebra, +geometry, arithmetic, and a translation of Euclid's \textit{Elements}, +\index{Elements (Euclid's)}% +\index{Euclid!ref.\ to}% +\index{Geometry!Arabic}% +were prepared by him. Even at the court of Tamerlane in +Samarkand, the sciences were by no means neglected. A +group of astronomers was drawn to this court. \textbf{Ulug Beg} +\index{Ulug Beg}% +(1393--1449), a grandson of Tamerlane, was himself an +astronomer. Most prominent at this time was \textbf{Al Kaschi}, the +\index{Alkaschi@{Al Kaschi}}% +author of an arithmetic. Thus, during intervals of peace, +science continued to be cultivated in the East for several +centuries. The last Oriental writer was \textit{Beha Eddin} (1547--1622). +\index{Beha Eddin}% +His \textit{Essence of Arithmetic} stands on about the same +level as the work of Mohammed ben Musa Hovarezmi, written +\index{Hovarezmi!ref.\ to}% +\index{Mohammed ben Musa Hovarezmi!ref.\ to}% +nearly 800 years before. + +``Wonderful is the expansive power of Oriental peoples, +with which upon the wings of the wind they conquer half +the world, but more wonderful the energy with which, in +less than two generations, they raise themselves from the +lowest stages of cultivation to scientific efforts.'' During +all these centuries, astronomy and mathematics in the Orient +greatly excel these sciences in the Occident. + +Thus far we have spoken only of the Arabs in the East. +Between the Arabs of the East and of the West, which were +under separate governments, there generally existed considerable +political animosity. In consequence of this, and of the +enormous distance between the two great centres of learning, +Bagdad and Cordova, there was less scientific intercourse +among them than might be expected to exist between peoples +having the same religion and written language. Thus the +%% -----File: 129.png---Folio 115------- +\index{Moors}% +course of science in Spain was quite independent of that in +Persia. While wending our way westward to Cordova, we +must stop in Egypt long enough to observe that there, too, +scientific activity was rekindled. Not Alexandria, but Cairo +with its library and observatory, was now the home of learning. +Foremost among her scientists ranked \textbf{Ben Junus} (died +\index{Ben Junus}% +1008), a contemporary of Abul Wefa. He solved some difficult +problems in spherical trigonometry. Another Egyptian astronomer +\index{Spherical trigonometry}% +\index{Trigonometry}% +\index{Trigonometry!spherical}% +was \textbf{Ibn Al Haitam} (died 1038), who wrote on geometric +\index{Alhaitam@{Al Haitam}}% +loci. Travelling westward, we meet in Morocco \textbf{Abul Hasan +\index{Abul Hasan}% +Ali}, whose treatise `on astronomical instruments' discloses a +thorough knowledge of the \textit{Conics} of Apollonius. Arriving +\index{Apollonius!ref.\ to}% +finally in Spain at the capital, Cordova, we are struck by the +magnificent splendour of her architecture. At this renowned +seat of learning, schools and libraries were founded during the +tenth century. + +Little is known of the progress of mathematics in Spain. +The earliest name that has come down to us is \textbf{Al Madshriti} +\index{Almadshriti@{Al Madshriti}}% +(died 1007), the author of a mystic paper on `amicable numbers.' +\index{Numbers!amicable}% +His pupils founded schools at Cordova, Dania, and +Granada. But the only great astronomer among the Saracens +in Spain is \textbf{Gabir ben Aflah} of Sevilla, frequently called \textit{Geber}. +\index{Gabir ben Aflah}% +He lived in the second half of the eleventh century. It was +formerly believed that he was the inventor of algebra, and that +\index{Algebra!Arabic}% +\index{Algebra!origin of terms}% +the word \emph{algebra} came from `Gabir' or `Geber.' He ranks +among the most eminent astronomers of this time, but, like so +many of his contemporaries, his writings contain a great deal +of mysticism. His chief work is an astronomy in nine books, of +\index{Astronomy!Arabic}% +which the first is devoted to trigonometry. In his treatment +of spherical trigonometry, he exercises great independence of +thought. He makes war against the time-honoured procedure +adopted by Ptolemy of applying ``the rule of six quantities,'' +\index{Ptolemy!ref.\ to}% +and gives a new way of his own, based on the `rule of four +%% -----File: 130.png---Folio 116------- +\index{Geber's theorem}% +quantities.' This is: If $PP_1$~and~$QQ_1$ be two arcs of great +circles intersecting in~$A$, and if $PQ$~and~$P_1Q_1$ be arcs of great +circles drawn perpendicular to~$QQ_1$, then we have the proportion +\[ +\sin AP : \sin PQ = \sin AP_1 : \sin P_1Q_1. +\] +From this he derives the formulas for spherical right triangles. +To the four fundamental formulas already given by Ptolemy, +he added a fifth, discovered by himself. If $a$,~$b$,~$c$, be the sides, +and $A$,~$B$,~$C$, the angles of a spherical triangle, right-angled at~$A$, +then $\cos B = \cos b \sin C$. This is frequently called ``Geber's +Theorem.'' Radical and bold as were his innovations in +\Addendlabel{116}% +spherical trigonometry, in plane trigonometry he followed +slavishly the old beaten path of the Greeks. Not even did he +adopt the Indian `sine' and `cosine,' but still used the Greek +\index{Sine}% +`chord of double the angle.' So painful was the departure +from old ideas, even to an independent Arab! After the time +of Gabir ben Aflah there was no mathematician among the +Spanish Saracens of any reputation. In the year in which +Columbus discovered America, the Moors lost their last foothold +\index{Moors}% +on Spanish soil. + +We have witnessed a laudable intellectual activity among +the Arabs. They had the good fortune to possess rulers +who, by their munificence, furthered scientific research. At +the courts of the caliphs, scientists were supplied with libraries +and observatories. A large number of astronomical and +mathematical works were written by Arabic authors. Yet +we fail to find a single important principle in mathematics +brought forth by the Arabic mind. Whatever discoveries +they made, were in fields previously traversed by the Greeks +or the Indians, and consisted of objects which the latter had +overlooked in their rapid march. The Arabic mind did not +possess that penetrative insight and invention by which mathematicians +in Europe afterwards revolutionised the science. +%% -----File: 131.png---Folio 117------- +The Arabs were learned, but not original. Their chief service +\index{Arabs|)}% +to science consists in this, that they adopted the learning of +Greece and India, and kept what they received with scrupulous +care. When the love for science began to grow in the +Occident, they transmitted to the Europeans the valuable +treasures of antiquity. Thus a Semitic race was, during the +Dark Ages, the custodian of the Aryan intellectual possessions. + + +\Section{Europe During the Middle Ages} +\index{Middle Ages|(}% + +With the third century after Christ begins an era of migration +of nations in Europe. The powerful Goths quit their +swamps and forests in the North and sweep onward in steady +southwestern current, dislodging the Vandals, Sueves, and +Burgundians, crossing the Roman territory, and stopping and +recoiling only when reaching the shores of the Mediterranean. +From the Ural Mountains wild hordes sweep down on the +Danube. The Roman Empire falls to pieces, and the Dark +Ages begin. But dark though they seem, they are the germinating +season of the institutions and nations of modern Europe. +The Teutonic element, partly pure, partly intermixed with the +Celtic and Latin, produces that strong and luxuriant growth, +the modern civilisation of Europe. Almost all the various +nations of Europe belong to the Aryan stock. As the Greeks +and the Hindoos---both Aryan races---were the great thinkers +of antiquity, so the nations north of the Alps became the great +intellectual leaders of modern times. + +\Subsection{Introduction of Roman Mathematics} +\index{Roman mathematics in Occident|(}% + +We shall now consider how these as yet barbaric nations of +the North gradually came in possession of the intellectual +%% -----File: 132.png---Folio 118------- +treasures of antiquity. With the spread of Christianity the +Latin language was introduced not only in ecclesiastical but +also in scientific and all important worldly transactions. Naturally +the science of the Middle Ages was drawn largely from +Latin sources. In fact, during the earlier of these ages Roman +authors were the only ones read in the Occident. Though +Greek was not wholly unknown, yet before the thirteenth +century not a single Greek scientific work had been read or +translated into Latin. Meagre indeed was the science which +could be gotten from Roman writers, and we must wait several +centuries before any substantial progress is made in mathematics. + +After the time of Boethius and Cassiodorius mathematical +\index{Boethius!ref.\ to}% +\index{Cassiodorius}% +activity in Italy died out. The first slender blossom of science +among tribes that came from the North was an encyclopædia +entitled \textit{Origines}, written by \textbf{Isidorus} (died 636 as bishop of +\index{Isidorus of Seville}% +Seville). This work is modelled after the Roman encyclopædias +of Martianus Capella of Carthage and of Cassiodorius. +Part of it is devoted to the quadrivium, arithmetic, music, +geometry, and astronomy. He gives definitions and grammatical +explications of technical terms, but does not describe the +modes of computation then in vogue. After Isidorus there +follows a century of darkness which is at last dissipated by +the appearance of \textbf{Bede the Venerable} (672--735), the most +\index{Bede, the Venerable}% +learned man of his time. He was a native of Ireland, then +the home of learning in the Occident. His works contain +treatises on the \textit{Computus}, or the computation of Easter-time, +\index{Computus}% +and on finger-reckoning. It appears that a finger-symbolism +\index{Finger-reckoning}% +was then widely used for calculation. The correct determination +of the time of Easter was a problem which in those days +greatly agitated the Church. It became desirable to have at +least one monk at each monastery who could determine the +day of religious festivals and could compute the calendar. +%% -----File: 133.png---Folio 119------- +Such determinations required some knowledge of arithmetic. +\index{Arithmetic!Middle Ages}% +Hence we find that the art of calculating always found some +little corner in the curriculum for the education of monks. + +The year in which Bede died is also the year in which +\textbf{Alcuin} (735--804) was born. Alcuin was educated in Ireland, +\index{Alcuin}% +and was called to the court of Charlemagne to direct the progress +of education in the great Frankish Empire. Charlemagne +was a great patron of learning and of learned men. In the +great sees and monasteries he founded schools in which were +taught the psalms, writing, singing, computation (\textit{computus}), +\index{Computus}% +and grammar. By \textit{computus} was here meant, probably, not +merely the determination of Easter-time, but the art of computation +in general. Exactly what modes of reckoning were +then employed we have no means of knowing. It is not likely +that Alcuin was familiar with the apices of Boethius or with +\index{Apices of Boethius!ref.\ to}% +the Roman method of reckoning on the abacus. He belongs +\index{Abacus}% +to that long list of scholars who dragged the theory of numbers +\index{Numbers!theory of numbers}% +\index{Theory of numbers}% +into theology. Thus the number of beings created by +God, who created all things well, is~$6$, because $6$~is a perfect +number (the sum of its divisors being $1 + 2 + 3 = 6$); $8$,~on +the other hand, is an imperfect number ($1 + 2 + 4 < 8$); hence +the second origin of mankind emanated from the number~$8$, +which is the number of souls said to have been in Noah's ark. + +There is a collection of ``Problems for Quickening the +Mind'' (\textit{propositiones ad acuendos iuvenes}), which are certainly +\index{Propositiones ad acuendos iuvenes}% +as old as 1000~\AD\ and possibly older. Cantor is of the opinion +that they were written much earlier and by Alcuin. The +following is a specimen of these ``Problems'': A dog chasing +a rabbit, which has a start of $150$~feet, jumps $9$~feet every time +the rabbit jumps~$7$. In order to determine in how many leaps +the dog overtakes the rabbit, $150$~is to be divided by~$2$. In +this collection of problems, the areas of triangular and quadrangular +pieces of land are found by the same formulas of +%% -----File: 134.png---Folio 120------- +\index{Anthology, Palatine}% +\index{Palatine anthology}% +\index{Propositiones ad acuendos iuvenes}% +\index{Sylvester II. (Gerbert)|(}% +approximation as those used by the Egyptians and given by +Boethius in his geometry. An old problem is the ``cistern-problem'' +(given the time in which several pipes can fill a +cistern singly, to find the time in which they fill it jointly), +which has been found previously in Heron, in the Greek \textit{Anthology}, +and in Hindoo works. Many of the problems show that +the collection was compiled chiefly from Roman sources. The +problem which, on account of its uniqueness, gives the most +positive testimony regarding the Roman origin is that on the +interpretation of a will in a case where twins are born. The +problem is identical with the Roman, except that different +ratios are chosen. Of the exercises for recreation, we mention +the one of the wolf, goat, and cabbage, to be rowed across a +river in a boat holding only one besides the ferry-man. Query: +How must he carry them across so that the goat shall not eat +the cabbage, nor the wolf the goat? The solutions of the +``problems for quickening the mind'' require no further knowledge +than the recollection of some few formulas used in +surveying, the ability to solve linear equations and to perform +the four fundamental operations with integers. Extraction +of roots was nowhere demanded; fractions hardly +\index{Fractions!Middle Ages}% +ever occur.\Reference{3} + +The great empire of Charlemagne tottered and fell almost +immediately after his death. War and confusion ensued. +Scientific pursuits were abandoned, not to be resumed until +the close of the tenth century, when under Saxon rule in Germany +and Capetian in France, more peaceful times began. +The thick gloom of ignorance commenced to disappear. The +zeal with which the study of mathematics was now taken up +by the monks is due principally to the energy and influence +of one man,---\textbf{Gerbert}. He was born in Aurillac in Auvergne. +\index{Gerbert|(}% +After receiving a monastic education, he engaged in study, +chiefly of mathematics, in Spain. On his return he taught +%% -----File: 135.png---Folio 121------- +school at Rheims for ten years and became distinguished for +his profound scholarship. By King Otto~I. and his successors +Gerbert was held in highest esteem. He was elected bishop +of Rheims, then of Ravenna, and finally was made Pope under +the name of Sylvester~II. by his former pupil Emperor Otho~III. +He died in~1003, after a life intricately involved in many +political and ecclesiastical quarrels. Such was the career of +the greatest mathematician of the tenth century in Europe. +By his contemporaries his mathematical knowledge was considered +wonderful. Many even accused him of criminal intercourse +with evil spirits. + +Gerbert enlarged the stock of his knowledge by procuring +copies of rare books. Thus in Mantua he found the geometry +\index{Geometry!Middle Ages}% +of Boethius. Though this is of small scientific value, yet it +\index{Boethius!ref.\ to}% +is of great importance in history. It was at that time the +only book from which European scholars could learn the elements +of geometry. Gerbert studied it with zeal, and is +generally believed himself to be the author of a geometry. +H.~Weissenborn denies his authorship, and claims that the +book in question consists of three parts which cannot come +from one and the same author.\Reference{21} This geometry contains +nothing more than the one of Boethius, but the fact that +occasional errors in the latter are herein corrected shows that +the author had mastered the subject. ``The first mathematical +paper of the Middle Ages which deserves this name,'' +says Hankel, ``is a letter of Gerbert to Adalbold, bishop of +Utrecht,'' in which is explained the reason why the area of a +triangle, obtained ``geometrically'' by taking the product of +the base by half its altitude, differs from the area calculated +``arithmetically,'' according to the formula +$\frac{1}{2} a (a + 1)$, used +by surveyors, where $a$ stands for a side of an equilateral triangle. +He gives the correct explanation that in the latter +formula all the small squares, in which the triangle is supposed +%% -----File: 136.png---Folio 122------- +to be divided, are counted in wholly, even though parts +of them project beyond it. + +Gerbert made a careful study of the arithmetical works of +Boethius. He himself published two works,---\textit{Rule of Computation +on the Abacus}, and \textit{A Small Book on the Division of +\index{Abacus}% +Numbers}. They give an insight into the methods of calculation +practised in Europe before the introduction of the +Hindoo numerals. Gerbert used the abacus, which was probably +unknown to Alcuin. \textbf{Bernelinus}, a pupil of Gerbert, +\index{Bernelinus}% +describes it as consisting of a smooth board upon which geometricians +were accustomed to strew blue sand, and then to draw +their diagrams. For arithmetical purposes the board was +divided into 30 columns, of which 3 were reserved for fractions, +while the remaining 27 were divided into groups with +3 columns in each. In every group the columns were marked +respectively by the letters C (\textit{centum}), D (\textit{decem}), and +S (\textit{singularis}) or M (\textit{monas}). Bernelinus gives the nine +numerals used, which are the apices of Boethius, and then +remarks that the Greek letters may be used in their place.\Reference{3} +By the use of these columns any number can be written +without introducing a zero, and all operations in arithmetic +\index{Arithmetic!Middle Ages}% +can be performed in the same way as we execute ours without +the columns, but with the symbol for zero. Indeed, the +methods of adding, subtracting, and multiplying in vogue +among the abacists agree substantially with those of to-day. +But in a division there is very great difference. The early rules +for division appear to have been framed to satisfy the following +three conditions: (1)~The use of the multiplication table shall +be restricted as far as possible; at least, it shall never be +required to multiply mentally a figure of two digits by another +of one digit. (2)~Subtractions shall be avoided as much as +possible and replaced by additions. (3)~The operation shall +proceed in a purely mechanical way, without requiring trials.\Reference{7} +%% -----File: 137.png---Folio 123------- +\index{Arithmetic!Middle Ages}% +That it should be necessary to make such conditions seems +strange to us; but it must be remembered that the monks of +the Middle Ages did not attend school during childhood and +learn the multiplication table while the memory was fresh. +Gerbert's rules for division are the oldest extant. They are +so brief as to be very obscure to the uninitiated. They were +probably intended simply to aid the memory by calling to +mind the successive steps in the work. In later manuscripts +they are stated more fully. In dividing any number by another +of one digit, say $668$~by~$6$, the divisor was first increased to~10 +by adding~4. The process is exhibited in the adjoining figure.\Reference{3} +As it continues, we must imagine the digits +which are crossed out, to be erased and then +replaced by the ones beneath. It is as follows: +$600 ÷ 10 = 60$, but, to rectify the error, $4 × 60$, +or~$240$, must be added; $200 ÷ 10 = 20$, but $4 × 20$, +or~$80$, must be added. We now write for +$60 + 40 + 80$, %[** PP: Width-dependent line break] +\begin{wrapfigure}[21]{r}{0.875in} +\setlength{\TmpLen}{1.75em}%[** PP: Sets width of \cancel{digit}] +\hfill\begin{tabular}{|*{3}{@{}c@{}|}} +\multicolumn{3}{@{}c@{}}{\smash[b]{\makebox[0pt][c]{\Graphic{3\TmpLen}{arc}}}} \\ +\multicolumn{2}{|@{}c@{}}{\smash{\Graphic{2\TmpLen}{arc}}} & \Graphic{\TmpLen}{arc} \\ + \sffamily C & \sffamily D & \sffamily S \\ + & & $6$ \\ + & & $4$ \\ +\hline + $6$ & $6$ & $8$ \\ +\hline + \cancel{6} & \cancel{6} & \cancel{8} \\ + \cancel{2} & \cancel{4} & \cancel{4} \\ + \cancel{1} & \cancel{8} & \cancel{8} \\ + \cancel{1} & \cancel{4} & \cancel{8} \\ + & \cancel{2} & $2$ \\ + & \cancel{4} & \\ + & \cancel{6} & \\ + & \cancel{2} & \\ + & \cancel{2} & \\ +\hline + & \cancel{6} & \cancel{6} \\ + & \cancel{2} & \cancel{2} \\ + & \cancel{1} & \cancel{2} \\ + & $1$ & $1$ +\end{tabular} +\end{wrapfigure} +its sum~$180$, and continue thus: +$100 ÷ 10 = 10$; the correction necessary is $4 × 10$, +or~$40$, which, added to~$80$, gives~$120$. Now +$100 ÷ 10 = 10$, and the correction $4 × 10$, together +with the~$20$, gives~$60$. Proceeding as +before, $60 ÷ 10 = 6$; the correction is $4 × 6 = 24$. +Now $20 ÷ 10 = 2$, the correction being $4 × 2 = 8$. +In the column of units we have now $8 + 4 + 8$, +or~$20$. As before, $20 ÷ 10 = 2$; the correction +is $2 × 4 = 8$, which is not divisible by~$10$, but +only by~$6$, giving the quotient~$1$ and the remainder~$2$. +All the partial quotients taken +together give $60 + 20 + 10 + 10 + 6 + 2 + 2 + 1 = 111$, and +the remainder~$2$. + +Similar but more complicated, is the process when the +divisor contains two or more digits. Were the divisor~27, +%% -----File: 138.png---Folio 124------- +\index{Roman mathematics in Occident|)}% +\index{Sylvester II. (Gerbert)|)}% +then the next higher multiple of~$10$, or~$30$, would be taken +for the divisor, but corrections would be required for the~$3$. +He who has the patience to carry such a division through +to the end, will understand why it has been said of Gerbert +that ``Regulas dedit, quæ a sudantibus abacistis vix intelliguntur.'' +He will also perceive why the Arabic method of +division, when first introduced, was called the \textit{divisio aurea}, +but the one on the abacus, the \textit{divisio ferrea}. + +In his book on the abacus, Bernelinus devotes a chapter to +fractions. These are, of course, the \textit{duodecimals}, first used +\index{Duodecimals}% +\index{Fractions!Middle Ages}% +\index{Fractions!duodecimal}% +by the Romans. For want of a suitable notation, calculation +with them was exceedingly difficult. It would be so even to +us, were we accustomed, like the early abacists, to express +them, not by a numerator or denominator, but by the application +of names, such as \textit{uncia} for~$\frac{1}{12}$, \textit{quincunx} for~$\frac{5}{12}$, \textit{dodrans} +for~$\frac{9}{12}$. + +In the tenth century, Gerbert was the central figure among +the learned. In his time the Occident came into secure possession +of all mathematical knowledge of the Romans. During +the eleventh century it was studied assiduously. Though +numerous works were written on arithmetic and geometry, +mathematical knowledge in the Occident was still very insignificant. +Scanty indeed were the mathematical treasures +obtained from Roman sources. + +\Subsection{Translation of Arabic Manuscripts} +\index{Arabic manuscripts|(}% + +By his great erudition and phenomenal activity, Gerbert +\index{Gerbert|)}% +infused new life into the study not only of mathematics, but +also of philosophy. Pupils from France, Germany, and Italy +gathered at Rheims to enjoy his instruction. When they +themselves became teachers, they taught of course not only +the use of the abacus and geometry, but also what they had +%% -----File: 139.png---Folio 125------- +\index{Geometry!Middle Ages}% +\index{Moors}% +\index{Sine}% +learned of the philosophy of Aristotle. His philosophy was +\index{Aristotle!ref.\ to}% +known, at first, only through the writings of Boethius. But +the growing enthusiasm for it created a demand for his complete +works. Greek texts were wanting. But the Latins +heard that the Arabs, too, were great admirers of Peripatetism, +and that they possessed translations of Aristotle's works and +commentaries thereon. This led them finally to search for +and translate Arabic manuscripts. During this search, mathematical +works also came to their notice, and were translated +into Latin. Though some few unimportant works may have +been translated earlier, yet the period of greatest activity +began about~1100. The zeal displayed in acquiring the +Mohammedan treasures of knowledge excelled even that of +the Arabs themselves, when, in the eighth century, they +plundered the rich coffers of Greek and Hindoo science. + +Among the earliest scholars engaged in translating manuscripts +into Latin was \textbf{Athelard of Bath}. The period of his +\index{Athelard of Bath}% +\Addendlabel{125}% +activity is the first quarter of the twelfth century. He +travelled extensively in Asia Minor, Egypt, and Spain, and +braved a thousand perils, that he might acquire the language +and science of the Mohammedans. He made the earliest +translations, from the Arabic, of Euclid's \textit{Elements} and of +\index{Elements (Euclid's)}% +\index{Euclid!ref.\ to}% +the astronomical tables of Mohammed ben Musa Hovarezmi. +\index{Hovarezmi!ref.\ to}% +\index{Mohammed ben Musa Hovarezmi!ref.\ to}% +In~1857, a manuscript was found in the library at Cambridge, +which proved to be the arithmetic by Mohammed ben Musa +in Latin. This translation also is very probably due to +Athelard. + +At about the same time flourished \textit{Plato of Tivoli} or \textit{Plato +\index{Plato of Tivoli}% +Tiburtinus}. He effected a translation of the astronomy of +Al Battani and of the \textit{Sphærica} of Theodosius. Through the +\index{Albattani@{Al Battani}!ref.\ to}% +\index{Theodosius!ref.\ to}% +former, the term \textit{sinus} was introduced into trigonometry. + +About the middle of the twelfth century there was a group +of Christian scholars busily at work at Toledo, under the +%% -----File: 140.png---Folio 126------- +\index{Algorithm!Middle Ages}% +\index{Arithmetic!Middle Ages}% +\index{Gerard of Cremona}% +\index{Sexagesimal system}% +leadership of Raymond, then archbishop of Toledo. Among +those who worked under his direction, \textbf{John of Seville} was +\index{John of Seville}% +most prominent. He translated works chiefly on Aristotelian +philosophy. Of importance to us is a \textit{liber algorismi}, compiled +by him from Arabic authors. On comparing works like +this with those of the abacists, we notice at once the most +\index{Abacists}% +striking difference, which shows that the two parties drew +from independent sources. It is argued by some that Gerbert +got his apices and his arithmetical knowledge, not from +\index{Apices of Boethius!ref.\ to}% +Boethius, but from the Arabs in Spain, and that part or the +whole of the geometry of Boethius is a forgery, dating from +the time of Gerbert. If this were the case, then the writings +of Gerbert would betray Arabic sources, as do those of John +of Seville. But no points of resemblance are found. Gerbert +could not have learned from the Arabs the use of the abacus, +\index{Abacus}% +because all evidence we have goes to show that they did not +employ it. Nor is it probable that he borrowed from the +Arabs the apices, because they were never used in Europe +except on the abacus. In illustrating an example in division, +mathematicians of the tenth and eleventh centuries state an +example in Roman numerals, then draw an abacus and insert +in it the necessary numbers with the apices. Hence it seems +probable that the abacus and apices were borrowed from the +same source. The contrast between authors like John of +Seville, drawing from Arabic works, and the abacists, consists +in this, that, unlike the latter, the former mention the Hindoos, +use the term \emph{algorism}, calculate with the zero, and do +not employ the abacus. The former teach the extraction of +roots, the abacists do not; they teach the sexagesimal fractions +\index{Fractions!sexagesimal}% +\index{Fractions!duodecimal}% +used by the Arabs, while the abacists employ the duodecimals +\index{Duodecimals}% +of the Romans.\Reference{3} + +A little later than John of Seville flourished \textbf{Gerard of +Cremona} in Lombardy. Being desirous to gain possession of +%% -----File: 141.png---Folio 127------- +\index{Alfonso's tables}% +\index{Alphonso's tables}% +\index{Arabic numerals and notation|(}% +\index{Elements (Euclid's)}% +\index{Geometry!Middle Ages}% +\index{Girard!ref.\ to}% +the \textit{Almagest}, he went to Toledo, and there, in~1175, translated +\index{Almagest!ref.\ to}% +this great work of Ptolemy. Inspired by the richness of +Mohammedan literature, he gave himself up to its study. He +translated into Latin over 70~Arabic works. Of mathematical +treatises, there were among these, besides the \textit{Almagest}, the +15~books of Euclid, the \textit{Sphærica} of Theodosius, a work of +\index{Euclid!ref.\ to}% +\index{Theodosius!ref.\ to}% +Menelaus, the algebra of Mohammed ben Musa Hovarezmi, the +\index{Hovarezmi!ref.\ to}% +\index{Mohammed ben Musa Hovarezmi!ref.\ to}% +astronomy of Dshabir ben Aflah, and others less important. +\index{Astronomy!Middle Ages}% +\index{Gabir ben Aflah!ref.\ to}% + +In the thirteenth century, the zeal for the acquisition of +Arabic learning continued. Foremost among the patrons of +science at this time ranked Emperor Frederick~II. of Hohenstaufen +(died 1250). Through frequent contact with Mohammedan +scholars, he became familiar with Arabic science. +He employed a number of scholars in translating Arabic +manuscripts, and it was through him that we came in possession +of a new translation of the \textit{Almagest}. Another royal +head deserving mention as a zealous promoter of Arabic +science was Alfonso~X. of Castile (died 1284). He gathered +around him a number of Jewish and Christian scholars, who +translated and compiled astronomical works from Arabic +sources. \textbf{Rabbi Zag} and \textbf{Iehuda ben Mose Cohen} were the +\index{Iehuda ben Mose Cohen}% +\index{Zag}% +most prominent among them. Astronomical tables prepared +by these two Jews spread rapidly in the Occident, and constituted +the basis of all astronomical calculation till the +sixteenth century.\Reference{7} The number of scholars who aided in +transplanting Arabic science upon Christian soil was large. +But we mention only one more. \textbf{Giovanni Campano} of Novara +\index{Giovanni Campano}% +(about 1260) brought out a new translation of Euclid, which +drove the earlier ones from the field, and which formed the +basis of the printed editions.\Reference{7} + +At the close of the twelfth century, the Occident was in +possession of the so-called Arabic notation. The Hindoo +\index{Notation!Arabic notation|(}% +methods of calculation began to supersede the cumbrous methods +%% -----File: 142.png---Folio 128------- +\index{Arabic manuscripts|)}% +\index{Elements (Euclid's)}% +inherited from Rome. Algebra, with its rules for solving +linear and quadratic equations, had been made accessible to +the Latins. The geometry of Euclid, the \textit{Sphærica} of Theodosius, +\index{Geometry!Middle Ages}% +the astronomy of Ptolemy, and other works were now +accessible in the Latin tongue. Thus a great amount of new +scientific material had come into the hands of the Christians. +The talent necessary to digest this heterogeneous mass of +knowledge was not wanting. The figure of Leonardo of Pisa +\index{Leonardo of Pisa}% +adorns the vestibule of the thirteenth century. + +It is important to notice that no work either on mathematics +or astronomy was translated directly from the Greek previous +to the fifteenth century. + +\Subsection{The First Awakening and its Sequel} + +Thus far, France and the British Isles have been the headquarters +of mathematics in Christian Europe. But at the +beginning of the thirteenth century the talent and activity +of one man was sufficient to assign the mathematical science +a new home in Italy. This man was not a monk, like Bede, +Alcuin, or Gerbert, but a merchant, who in the midst of +business pursuits found time for scientific study. \textbf{Leonardo +of Pisa} is the man to whom we owe the first renaissance of +mathematics on Christian soil. He is also called \textit{Fibonacci}, +\ie~son of Bonaccio. His father was secretary at one of the +numerous factories erected on the south and east coast of the +Mediterranean by the enterprising merchants of Pisa. He +made Leonardo, when a boy, learn the use of the abacus. The +boy acquired a strong taste for mathematics, and, in later years, +during his extensive business travels in Egypt, Syria, Greece, +and Sicily, collected from the various peoples all the knowledge +he could get on this subject. Of all the methods of +calculation, he found the Hindoo to be unquestionably the +%% -----File: 143.png---Folio 129------- +\index{Algorithm!Middle Ages}% +\index{Arabic numerals and notation|)}% +best. Returning to Pisa, he published, in~1202, his great +work, the \textit{Liber Abaci}. A revised edition of this appeared in~1228. +This work contains about all the knowledge the Arabs +possessed in arithmetic and algebra, and treats the subject in +a free and independent way. This, together with the other +books of Leonardo, shows that he was not merely a compiler, +or, like other writers of the Middle Ages, a slavish imitator +of the form in which the subject had been previously presented, +but that he was an original worker of exceptional +power. + +He was the first great mathematician to advocate the adoption +of the ``Arabic notation.'' The calculation with the zero +\index{Zero!origin of term}% +was the portion of Arabic mathematics earliest adopted by +the Christians. The minds of men had been prepared for the +reception of this by the use of the abacus and the apices. +\index{Abacus}% +\index{Apices of Boethius!ref.\ to}% +The reckoning with columns was gradually abandoned, and +the very word \emph{abacus} changed its meaning and became a +synonym for \emph{algorism}. For the zero, the Latins adopted +the name \emph{zephirum}, from the Arabic \textit{sifr} (\textit{sifra}=empty); +hence our English word \emph{cipher}. The new notation was +\index{Notation!Arabic notation|)}% +accepted readily by the enlightened masses, but, at first, +rejected by the learned circles. The merchants of Italy used +it as early as the thirteenth century, while the monks in the +monasteries adhered to the old forms. In~1299, nearly 100~years +after the publication of Leonardo's \textit{Liber Abaci}, the +Florentine merchants were forbidden the use of the Arabic +numerals in book-keeping, and ordered either to employ the +Roman numerals or to write the numeral adjectives out in +full. In the fifteenth century the abacus with its counters +ceased to be used in Spain and Italy. In France it was used +later, and it did not disappear in England and Germany before +the middle of the seventeenth century.\Reference{22} Thus, in the \textit{Winter's +Tale} (iv.~3), Shakespeare lets the clown be embarrassed by +%% -----File: 144.png---Folio 130------- +a problem which he could not do without counters. Iago +(in \textit{Othello}, i.~1) expresses his contempt for Michael Cassio, +``forsooth a great mathematician,'' by calling him a ``counter-caster.'' +So general, indeed, says Peacock, appears to have +\index{Peacock!ref.\ to, X}% +been the practice of this species of arithmetic, that its rules +\index{Arithmetic!Middle Ages}% +and principles form an essential part of the arithmetical +treatises of that day. The real fact seems to be that the old +methods were used long after the Hindoo numerals were in +common and general use. With such dogged persistency does +man cling to the old! + +The \textit{Liber Abaci} was, for centuries, the storehouse from +which authors got material for works on arithmetic and +algebra. In it are set forth the most perfect methods of +calculation with integers and fractions, known at that time; +the square and cube root are explained; equations of the first +and second degree leading to problems, either determinate +or indeterminate, are solved by the methods of `single' or +`double position,' and also by real algebra. The book contains +a large number of problems. The following was proposed +to Leonardo of Pisa by a magister in Constantinople, +as a difficult problem: If A gets from B $7$~denare, then A's +sum is five-fold B's; if B gets from A $5$~denare, then B's sum +is seven-fold A's. How much has each? The \textit{Liber Abaci} +contains another problem, which is of historical interest, +because it was given with some variations by Ahmes, 3000~years +\index{Ahmes!ref.\ to}% +earlier: $7$~old women go to Rome; each woman has +$7$~mules, each mule carries $7$~sacks, each sack contains $7$~loaves, +with each loaf are $7$~knives, each knife is put up in $7$~sheaths. +What is the sum total of all named? \textit{Ans.}~$137,256$.\Reference{3} + +In~1220, Leonardo of Pisa published his \textit{Practica Geometriæ}, +which contains all the knowledge of geometry and trigonometry +\index{Geometry!Middle Ages}% +transmitted to him. The writings of Euclid and of some +other Greek masters were known to him, either from Arabic +%% -----File: 145.png---Folio 131------- +\index{Theory of numbers}% +manuscripts directly or from the translations made by his +countrymen, Gerard of Cremona and Plato of Tivoli. Leonardo's +\textit{Geometry} contains an elegant geometrical demonstration +\index{Geometry!Middle Ages}% +of Heron's formula for the area of a triangle, as a +\index{Heron the Elder!ref.\ to}% +function of its three sides. Leonardo treats the rich material +before him with skill and Euclidean rigour. + +Of still greater interest than the preceding works are those +containing Fibonacci's original investigations. We must here +preface that after the publication of the \textit{Liber Abaci}, Leonardo +was presented by the astronomer Dominicus to Emperor +Frederick~II. of Hohenstaufen. On that occasion, John of +Palermo, an imperial notary, proposed several problems, which +Leonardo solved promptly. The first problem was to find a +number $x$, such that $x^2 + 5$ and $x^2 - 5$ are each square numbers. +\index{Numbers!theory of numbers}% +The answer is $x = 3\frac{5}{12}$; for $(3 \frac{5}{12})^2 + 5 = (4\frac{1}{12})^2$, $(3\frac{5}{12})^2 - 5 += (2\frac{7}{12})^2$. His masterly solution of this is given in his \textit{liber +quadratorum}, a copy of which work was sent by him to +Frederick~II\@. The problem was not original with John of +Palermo, since the Arabs had already solved similar ones. +Some parts of Leonardo's solution may have been borrowed +from the Arabs, but the method which he employed of building +squares by the summation of odd numbers is original with him. + +The second problem proposed to Leonardo at the famous +scientific tournament which accompanied the presentation of +this celebrated algebraist to that great patron of learning, +Emperor Frederick~II., was the solving of the equation +$x^3 + 2x^2 + 10x = 20$. As yet cubic equations had not been +solved algebraically. Instead of brooding stubbornly over +this knotty problem, and after many failures still entertaining +new hopes of success, he changed his method of inquiry and +showed by clear and rigorous demonstration that the roots +of this equation could not be represented by the Euclidean +irrational quantities, or, in other words, that they could not be +%% -----File: 146.png---Folio 132------- +constructed with the ruler and compass only. He contented +himself with finding a very close approximation to the +required root. His work on this cubic is found in the \textit{Flos}, +together with the solution of the following third problem +given him by John of Palermo: Three men possess in common +an unknown sum of money~$t$; the share of the first +is~$\dfrac{t}{2}$; that of the second,~$\dfrac{t}{3\Fstr}$; that of the third,~$\smash[t]{\dfrac{t}{6}}$. Desirous of +depositing the sum at a safer place, each takes at hazard a +certain amount; the first takes~$x$, but deposits only~$\dfrac{x}{2}$; the second +carries~$y$, but deposits only~$\smash{\dfrac{y}{3}}$; the third takes~$z$, and +deposits~$\smash[t]{\dfrac{z}{6}}$. Of the amount deposited each one must receive +exactly~$\frac{1}{3}$, in order to possess his share of the whole sum. +Find $x$, $y$,~$z$. Leonardo shows the problem to be indeterminate. +Assuming 7 for the sum drawn by each from the +deposit, he finds $t = 47$, $x = 33$, $y = 13$, $z = 1$. + +One would have thought that after so brilliant a beginning, +the sciences transplanted from Mohammedan to Christian soil +would have enjoyed a steady and vigorous development. But +this was not the case. During the fourteenth and fifteenth +centuries, the mathematical science was almost stationary. +Long wars absorbed the energies of the people and thereby +kept back the growth of the sciences. The death of Frederick~II. +in 1254 was followed by a period of confusion in Germany. +The German emperors and the popes were continually quarrelling, +and Italy was inevitably drawn into the struggles between +the Guelphs and the Ghibellines. France and England were +engaged in the Hundred Years' War (1338--1453). Then +followed in England the Wars of the Roses. The growth of +science was retarded not only by war, but also by the injurious +influence of scholastic philosophy. The intellectual +leaders of those times quarrelled over subtle subjects in metaphysics +%% -----File: 147.png---Folio 133------- +and theology. Frivolous questions, such as ``How +many angels can stand on the point of a needle?'' were discussed +with great interest. Indistinctness and confusion of +ideas characterised the reasoning during this period. Among +the mathematical productions of the Middle Ages, the works +of Leonardo of Pisa appear to us like jewels among quarry-rubbish. +\index{Leonardo of Pisa!ref.\ to}% +The writers on mathematics during this period were +not few in number, but their scientific efforts were vitiated +by the method of scholastic thinking. Though they possessed +the \textit{Elements} of Euclid, yet the true nature of a mathematical +\index{Elements (Euclid's)}% +proof was so little understood, that Hankel believes it no +exaggeration to say that ``since Fibonacci, not a single proof, +not borrowed from Euclid, can be found in the whole literature +of these ages, which fulfils all necessary conditions.'' + +The only noticeable advance is a simplification of numerical +operations and a more extended application of them. Among +the Italians are evidences of an early maturity of arithmetic. +\index{Arithmetic!Middle Ages}% +Peacock\Reference[]{22} says: The Tuscans generally, and the Florentines +\index{Peacock!ref.\ to, X}% +in particular, whose city was the cradle of the literature and +arts of the thirteenth and fourteenth centuries, were celebrated +for their knowledge of arithmetic and book-keeping, which +were so necessary for their extensive commerce; the Italians +were in familiar possession of commercial arithmetic long +before the other nations of Europe; to them we are indebted +for the formal introduction into books of arithmetic, under +distinct heads, of questions in the single and double rule of +three, loss and gain, fellowship, exchange, simple and compound +interest, discount, and so on. + +There was also a slow improvement in the algebraic notation. +\index{Notation!in algebra}% +The Hindoo algebra possessed a tolerable symbolic +\index{Algebra!Middle Ages}% +notation, which was, however, completely ignored by the Mohammedans. +In this respect, Arabic algebra approached +much more closely to that of Diophantus, which can scarcely +%% -----File: 148.png---Folio 134------- +\index{Albertus Magnus}% +\index{Exponents}% +\index{Stevin!ref.\ to}% +be said to employ symbols in a systematic way. Leonardo of +Pisa possessed no algebraic symbolism. Like the Arabs, he +expressed the relations of magnitudes to each other by lines +or in words. But in the mathematical writings of the monk +\textit{Luca Pacioli} (also called Lucas de Burgo sepulchri) symbols +\index{Pacioli!ref.\ to}% +began to appear. They consisted merely in abbreviations of +Italian words, such as \textit{p} for \textit{piu} (more), \textit{m} for \textit{meno} (less), \textit{co} +for \textit{cosa} (the thing or unknown quantity). ``Our present +notation has arisen by almost insensible degrees as convenience +\index{Notation!in algebra}% +suggested different marks of abbreviation to different +authors; and that perfect symbolic language which addresses +itself solely to the eye, and enables us to take in at a glance +the most complicated relations of quantity, is the result of a +large series of small improvements.''\Reference{23} + +We shall now mention a few authors who lived during the +thirteenth and fourteenth and the first half of the fifteenth +centuries. About the time of Leonardo of Pisa (1200~\AD), +lived the German monk \textbf{Jordanus Nemorarius}, who wrote a once +\index{Jordanus Nemorarius}% +famous work on the properties of numbers (1496), modelled +after the arithmetic of Boethius. The most trifling numeral +\index{Arithmetic!Middle Ages}% +\index{Boethius!ref.\ to}% +properties are treated with nauseating pedantry and prolixity. +A practical arithmetic based on the Hindoo notation was +also written by him. \textbf{John Halifax} (Sacro Bosco, died 1256) +\index{Halifax}% +taught in Paris and made an extract from the \textit{Almagest} containing +\index{Almagest!ref.\ to}% +only the most elementary parts of that work. This +extract was for nearly 400~years a work of great popularity +and standard authority. Other prominent writers are \textbf{Albertus +Magnus} and \textbf{George Purbach} in Germany, and \textbf{Roger Bacon} in +\index{Bacon, R.}% +\index{Purbach}% +England. It appears that here and there some of our modern +ideas were anticipated by writers of the Middle Ages. Thus, +\textbf{Nicole Oresme}, a bishop in Normandy (died 1382), first conceived +\index{Oresme}% +a notation of fractional powers, afterwards re-discovered +by Stevinus, and gave rules for operating with them. +%% -----File: 149.png---Folio 135------- +\index{Elements (Euclid's)}% +\index{Infinitesimals}% +\index{Infinity}% +His notation was totally different from ours. \textbf{Thomas Bradwardine}, +\index{Bradwardine}% +archbishop of Canterbury, studied star-polygons,---a +\index{Star-polygons}% +subject which has recently received renewed attention. The +first appearance of such polygons was with Pythagoras and +\index{Pythagoras!ref.\ to}% +his school. We next meet with such polygons in the geometry +of Boethius and also in the translation of Euclid from +\index{Boethius!ref.\ to}% +the Arabic by Athelard of Bath. Bradwardine's philosophic +\index{Athelard of Bath!ref.\ to}% +writings contain discussions on the infinite and the infinitesimal---subjects +never since lost sight of. To England +falls the honour of having produced the earliest European +writers on trigonometry. The writings of Bradwardine, of +\index{Trigonometry}% +Richard of Wallingford, and John Maudith, both professors +\index{Maudith}% +\index{Richard of Wallingford}% +at Oxford, and of Simon Bredon of Winchecombe, contain +\index{Bredon}% +trigonometry drawn from Arabic sources. + +The works of the Greek monk \textbf{Maximus Planudes}, who lived +\index{Planudes, M.}% +in the first half of the fourteenth century, are of interest only +as showing that the Hindoo numerals were then known in +Greece. A writer belonging, like Planudes, to the Byzantine +school, was \textbf{Moschopulus}, who lived in Constantinople in the +\index{Moschopulus}% +early part of the fifteenth century. To him appears to be +due the introduction into Europe of magic squares. He wrote +\index{Magic squares}% +a treatise on this subject. Magic squares were known to the +Arabs, and perhaps to the Hindoos. Mediæval astrologers +and physicians believed them to possess mystical properties +and to be a charm against plague, when engraved on silver +plate. + +In 1494 was \emph{printed} the \textit{Summa de Arithmetica, Geometria, +Proportione et Proportionalita}, written by the Tuscan monk +\textbf{Lucas Pacioli}, who, as we remarked, first introduced symbols +\index{Pacioli}% +in algebra. This contains all the knowledge of his day on +\index{Algebra!Middle Ages}% +arithmetic, algebra, and trigonometry, and is the first comprehensive +work which appeared after the \textit{Liber Abaci} of +Fibonacci. It contains little of importance which cannot be +%% -----File: 150.png---Folio 136------- +\index{Elements (Euclid's)}% +\index{Halifax!ref.\ to}% +found in Fibonacci's great work, published three centuries +earlier.\Reference{1} + +Perhaps the greatest result of the influx of Arabic learning +was the establishment of universities. What was their +\index{Universities of Cologne, Leipzig, Oxford, Paris, and Prague}% +attitude toward mathematics? The \textit{University of Paris}, so +famous at the beginning of the twelfth century under the +teachings of Abelard, paid but little attention to this science +during the Middle Ages. Geometry was neglected, and Aristotle's +logic was the favourite study. In~1336, a rule was +introduced that no student should take a degree without +attending lectures on mathematics, and from a commentary +on the first six books of Euclid, dated~1536, it appears that +\index{Euclid!ref.\ to}% +candidates for the degree of A.M. had to give an oath that +they had attended lectures on these books.\Reference{7} Examinations, +when held at all, probably did not extend beyond the first +book, as is shown by the nickname ``magister matheseos,'' +\index{Magister matheseos}% +applied to the Theorem of Pythagoras, the last in the first +book. More attention was paid to mathematics at the \textit{University +of Prague}, founded 1384. For the Baccalaureate degree, +students were required to take lectures on Sacro Bosco's +famous work on astronomy. Of candidates for the A.M. were +required not only the six books of Euclid, but an additional +knowledge of applied mathematics. Lectures were given on +the \textit{Almagest}. At the \textit{University of Leipzig}, the daughter of +\index{Almagest!ref.\ to}% +Prague, and at \textit{Cologne}, less work was required, and, as late +as the sixteenth century, the same requirements were made at +these as at Prague in the fourteenth. The universities of +Bologna, Padua, Pisa, occupied similar positions to the ones +in Germany, only that purely astrological lectures were given +in place of lectures on the \textit{Almagest}. At Oxford, in the +middle of the fifteenth century, the first two books of Euclid +were read.\Reference{6} + +Thus it will be seen that the study of mathematics was +%% -----File: 151.png---Folio 137------- +\index{Middle Ages|)}% +maintained at the universities only in a half-hearted manner. +No great mathematician and teacher appeared, to inspire the +students. The best energies of the schoolmen were expended +upon the stupid subtleties of their philosophy. The genius +of Leonardo of Pisa left no permanent impress upon the age, +\index{Leonardo of Pisa!ref.\ to}% +and another Renaissance of mathematics was wanted. +%% -----File: 152.png---Folio 138------- +\index{Elements (Euclid's)}% +\index{Geometry!Renaissance}% + + +\Chapter{Modern Europe} +\index{Modern Europe|etseq}% + +\First{We} find it convenient to choose the time of the capture of +Constantinople by the Turks as the date at which the Middle +Ages ended and Modern Times began. In~1453, the Turks +battered the walls of this celebrated metropolis with cannon, +and finally captured the city; the Byzantine Empire fell, to +rise no more. Calamitous as was this event to the East, it +acted favourably upon the progress of learning in the West. +A great number of learned Greeks fled into Italy, bringing +with them precious manuscripts of Greek literature. This +contributed vastly to the reviving of classic learning. Up +to this time, Greek masters were known only through the +often very corrupt Arabic manuscripts, but now they began +to be studied from original sources and in their own language. +The first English translation of Euclid was made in 1570 from +\index{Euclid!ref.\ to}% +the Greek by \textit{Sir Henry Billingsley}, assisted by \textit{John Dee}.\Reference{29} +\index{Billingsley}% +\index{Dee}% +About the middle of the fifteenth century, printing was invented; +books became cheap and plentiful; the printing-press +transformed Europe into an audience-room. Near the close of +the fifteenth century, America was discovered, and, soon after, +the earth was circumnavigated. The pulse and pace of the +world began to quicken. Men's minds became less servile; +they became clearer and stronger. The indistinctness of +thought, which was the characteristic feature of mediæval +learning, began to be remedied chiefly by the steady cultivation +%% -----File: 153.png---Folio 139------- +\index{Ptolemy!ref.\ to}% +of Pure Mathematics and Astronomy. Dogmatism was +attacked; there arose a long struggle with the authority of +the Church and the established schools of philosophy. The +Copernican System was set up in opposition to the time-honoured +\index{Copernican System}% +Ptolemaic System. The long and eager contest between +the two culminated in a crisis at the time of Galileo, and +\index{Galileo!ref.\ to}% +resulted in the victory of the new system. Thus, by slow +degrees, the minds of men were cut adrift from their old +scholastic moorings and sent forth on the wide sea of scientific +inquiry, to discover new islands and continents of truth. + +\Section{The Renaissance} +\index{Renaissance|(}% + +With the sixteenth century began a period of increased +intellectual activity. The human mind made a vast effort to +achieve its freedom. Attempts at its emancipation from +Church authority had been made before, but they were stifled +and rendered abortive. The first great and successful revolt +against ecclesiastical authority was made in Germany. The +new desire for judging freely and independently in matters +of religion was preceded and accompanied by a growing spirit +of scientific inquiry. Thus it was that, for a time, Germany +led the van in science. She produced \textit{Regiomontanus}, \textit{Copernicus, +\index{Copernicus}% +\index{Regiomontanus!ref.\ to}% +Rhæticus}, \textit{Kepler}, and \textit{Tycho Brahe}, at a period when +\index{Brahe, Tycho}% +\index{Kepler!ref.\ to}% +\index{Rhæticus!ref.\ to}% +\index{Tycho Brahe}% +France and England had, as yet, brought forth hardly any +great scientific thinkers. This remarkable scientific productiveness +was no doubt due, to a great extent, to the commercial +prosperity of Germany. Material prosperity is an essential +condition for the progress of knowledge. As long as every +individual is obliged to collect the necessaries for his subsistence, +there can be no leisure for higher pursuits. At this +time, Germany had accumulated considerable wealth. The +%% -----File: 154.png---Folio 140------- +Hanseatic League commanded the trade of the North. Close +commercial relations existed between Germany and Italy. +Italy, too, excelled in commercial activity and enterprise. +We need only mention Venice, whose glory began with the crusades, +and Florence, with her bankers and her manufacturers +of silk and wool. These two cities became great intellectual +centres. Thus, Italy, too, produced men in art, literature, and +science, who shone forth in fullest splendour. In fact, Italy +was the fatherland of what is termed the Renaissance. + +For the first great contributions to the mathematical sciences +we must, therefore, look to Italy and Germany. In Italy +brilliant accessions were made to algebra, in Germany to +\index{Algebra!Renaissance}% +astronomy and trigonometry. +\index{Trigonometry}% + +On the threshold of this new era we meet in Germany with +the figure of John Mueller, more generally called \textbf{Regiomontanus} +\index{Regiomontanus}% +(1436--1476). Chiefly to him we owe the revival of +trigonometry. He studied astronomy and trigonometry at +Vienna under the celebrated George Purbach. The latter +\index{Purbach!ref.\ to}% +perceived that the existing Latin translations of the \textit{Almagest} +\index{Almagest!ref.\ to}% +were full of errors, and that Arabic authors had not remained +true to the Greek original. Purbach therefore began to make +a translation directly from the Greek. But he did not live to +finish it. His work was continued by Regiomontanus, who +went beyond his master. Regiomontanus learned the Greek +language from Cardinal Bessarion, whom he followed to Italy, +where he remained eight years collecting manuscripts from +Greeks who had fled thither from the Turks. In addition to +the translation of and the commentary on the \textit{Almagest}, he +prepared translations of the \textit{Conics} of Apollonius, of Archimedes, +\index{Apollonius!ref.\ to}% +\index{Archimedes!ref.\ to}% +and of the mechanical works of Heron. Regiomontanus +\index{Heron the Elder!ref.\ to}% +and Purbach adopted the Hindoo \emph{sine} in place of the Greek +\index{Sine}% +\emph{chord of double the arc}. The Greeks and afterwards the Arabs +divided the radius into $60$~equal parts, and each of these again +%% -----File: 155.png---Folio 141------- +\index{Sine}% +into $60$~smaller ones. The Hindoos expressed the length of +the radius by parts of the circumference, saying that of the +$21,600$ equal divisions of the latter, it took $3438$ to measure +the radius. Regiomontanus, to secure greater precision, constructed +\index{Regiomontanus}% +one table of sines on a radius divided into $600,000$ +parts, and another on a radius divided decimally into $10,000,000$ +divisions. He emphasised the use of the \emph{tangent} in trigonometry. +\index{Trigonometry}% +Following out some ideas of his master, he calculated +a table of tangents. German mathematicians were not the +\index{Tangents!in trigonometry}% +first Europeans to use this function. In England it was known +a century earlier to Bradwardine, who speaks of tangent (\textit{umbra +\index{Bradwardine!ref.\ to}% +recta}) and cotangent (\textit{umbra versa}), and to John Maudith. +\index{Cotangent}% +\index{Maudith!ref.\ to}% +Regiomontanus was the author of an arithmetic and also of +a complete treatise on trigonometry, containing solutions of +both plane and spherical triangles. The form which he gave +to trigonometry has been retained, in its main features, to the +present day. + +Regiomontanus ranks among the greatest men that Germany +has ever produced. His complete mastery of astronomy and +mathematics, and his enthusiasm for them, were of far-reaching +influence throughout Germany. So great was his +reputation, that Pope Sixtus~IV. called him to Italy to +improve the calendar. Regiomontanus left his beloved city +\index{Calendar}% +of Nürnberg for Rome, where he died in the following year. + +After the time of Purbach and Regiomontanus, trigonometry +and especially the calculation of tables continued to occupy +German scholars. More refined astronomical instruments were +made, which gave observations of greater precision; but these +would have been useless without trigonometrical tables of corresponding +accuracy. Of the several tables calculated, that +by \textit{Georg Joachim} of Feldkirch in Tyrol, generally called +\textbf{Rhæticus}, deserves special mention. He calculated a table of +\index{Rhæticus}% +sines with the radius $= 10,000,000,000$ and from $10''$ to~$10''$; +%% -----File: 156.png---Folio 142------- +\index{Cubic equations|(}% +and, later on, another with the radius $= 1,000,000,000,000,000$, +and proceeding from $10''$ to~$10''$. He began also the construction +of tables of tangents and secants, to be carried to +\index{Secants}% +\index{Tangents!in trigonometry}% +the same degree of accuracy; but he died before finishing them. +For twelve years he had had in continual employment several +calculators. The work was completed by his pupil, \textbf{Valentine +Otho}, in~1596. This was indeed a gigantic work,---a monument +\index{Otho}% +of German diligence and indefatigable perseverance. +The tables were republished in 1613 by \textbf{Pitiscus}, who spared +\index{Pitiscus}% +no pains to free them of errors. Astronomical tables of +so great a degree of accuracy had never been dreamed of +by the Greeks, Hindoos, or Arabs. That Rhæticus was not a +\index{Rhæticus!ref.\ to}% +ready calculator only, is indicated by his views on \DPtypo{trignometrical}{trigonometrical} +lines. Up to his time, the trigonometric functions had +been considered always with relation to the arc; he was the +first to construct the right triangle and to make them depend +directly upon its angles. It was from the right triangle that +Rhæticus got his idea of calculating the hypotenuse; \ie~he +was the first to plan a table of secants. Good work in trigonometry +was done also by Vieta and Romanus. +\index{Romanus!ref.\ to}% +\index{Vieta!ref.\ to}% + +We shall now leave the subject of trigonometry to witness +the progress in the solution of algebraical equations. To do +so, we must quit Germany for Italy. The first comprehensive +algebra printed was that of Lucas Pacioli. He closes his +\index{Algebra!Renaissance|(}% +\index{Pacioli!ref.\ to}% +book by saying that the solution of the equations $x^{3} + mx = n$, +$x^{3} + n = mx$ is as impossible at the present state of science as +the quadrature of the circle. This remark doubtless stimulated +thought. The first step in the algebraic solution of +cubics was taken by \textbf{Scipio Ferro} (died 1526), a professor of +\index{Ferro, Scipio}% +mathematics at Bologna, who solved the equation $x^{3} + mx = n$. +Nothing more is known of his discovery than that he imparted +it to his pupil, \textit{Floridas}, in~1505. It was the practice in those +\index{Floridas}% +days and for two centuries afterwards to keep discoveries +%% -----File: 157.png---Folio 143------- +secret, in order to secure by that means an advantage over +rivals by proposing problems beyond their reach. This practice +gave rise to numberless disputes regarding the priority of +inventions. A second solution of cubics was given by \textit{Nicolo} +of Brescia (1506(?)--1557). When a boy of six, Nicolo was +so badly cut by a French soldier that he never again gained +the free use of his tongue. Hence he was called \textbf{Tartaglia}, +\index{Tartaglia|(}% +\ie~the stammerer. His widowed mother being too poor to +pay his tuition in school, he learned to read and picked up a +knowledge of Latin, Greek, and mathematics by himself. +Possessing a mind of extraordinary power, he was able to +appear as teacher of mathematics at an early age. In~1530, +one Colla proposed him several problems, one leading to the +\index{Colla}% +equation $x^3 + px^{2} = q$. Tartaglia found an imperfect method +for solving this, but kept it secret. He spoke about his secret +in public and thus caused Ferro's pupil, Floridas, to proclaim +his own knowledge of the form $x^{3} + mx = n$. Tartaglia, believing +him to be a mediocrist and braggart, challenged him to a +public discussion, to take place on the 22d of February, 1535. +Hearing, meanwhile, that his rival had gotten the method +from a deceased master, and fearing that he would be beaten +in the contest, Tartaglia put in all the zeal, industry, and +skill to find the rule for the equations, and he succeeded in it +ten days before the appointed date, as he himself modestly +says.\Reference{7} The most difficult step was, no doubt, the passing from +quadratic irrationals, used in operating from time of old, to +cubic irrationals. Placing $x = \sqrt[3]{t} - \sqrt[3]{u}$, Tartaglia perceived +that the irrationals disappeared from the equation $x^3 + mx = n$, +making $n = t - u$. But this last equality, together with +$(\frac{1}{3}m)^3 = tu$, gives at once +\[ +t = \sqrt{\left( \frac{n}{2} \right)^3 + \left( \frac{m}{3} \right)^3} + \frac{n}{2}, \quad +u = \sqrt{\left( \frac{n}{2} \right)^2 + \left( \frac{m}{2} \right)^3} - \frac{n}{2}. +\] +%% -----File: 158.png---Folio 144------- +This is Tartaglia's solution of $x^{3} + mx = n$. On the 13th~of +February, he found a similar solution for $x^{3} = mx + n$. The +contest began on the~22d. Each contestant proposed thirty +problems. The one who could solve the greatest number within +fifty days should be the victor. Tartaglia solved the thirty +problems proposed by Floridas in two hours; Floridas could +\index{Floridas}% +not solve any of Tartaglia's. From now on, Tartaglia studied +cubic equations with a will. In 1541 he discovered a general +solution for the cubic $x^{3} ± px^{2} = ± q$, by transforming it into +the form $x^{3} ± mx = ± n$. The news of Tartaglia's victory +spread all over Italy. Tartaglia was entreated to make known +his method, but he declined to do so, saying that after his +completion of the translation from the Greek of Euclid and +\index{Euclid!ref.\ to}% +Archimedes, he would publish a large algebra containing his +\index{Archimedes!ref.\ to}% +method. But a scholar from Milan, named \textbf{Hieronimo Cardano} +(1501--1576), after many solicitations, and after giving the +most solemn and sacred promises of secrecy, succeeded in +obtaining from Tartaglia a knowledge of his rules. + +At this time Cardan was writing his \textit{Ars Magna}, and he +\index{Cardan}% +knew no better way to crown his work than by inserting the +much sought for rules for solving cubics. Thus Cardan broke +his most solemn vows, and published in 1545 in his \textit{Ars Magna} +Tartaglia's solution of cubics. Tartaglia became desperate. +His most cherished hope, of giving to the world an immortal +work which should be the monument of his deep learning and +power for original research, was suddenly destroyed; for the +crown intended for his work had been snatched away. His +first step was to write a history of his invention; but, to completely +annihilate his enemies, he challenged Cardan and his +pupil Lodovico Ferrari to a contest: each party should propose +\index{Ferrari!ref.\ to}% +thirty-one questions to be solved by the other within fifteen +days. Tartaglia solved most questions in seven days, but the +other party did not send in their solution before the expiration +%% -----File: 159.png---Folio 145------- +of the fifth month; moreover, all their solutions except one +were wrong. A replication and a rejoinder followed. Endless +were the problems proposed and solved on both sides. The +dispute produced much chagrin and heart-burnings to the parties, +and to Tartaglia especially, who met with many other +disappointments. After having recovered himself again, Tartaglia +\index{Tartaglia|)}% +began, in 1556, the publication of the work which he +had had in his mind for so long; but he died before he reached +the consideration of cubic equations. Thus the fondest wish +\index{Cubic equations|)}% +of his life remained unfulfilled; the man to whom we owe the +greatest contribution to algebra made in the sixteenth century +was forgotten, and his method came to be regarded as the discovery +of Cardan and to be called Cardan's solution. + +Remarkable is the great interest that the solution of cubics +excited throughout Italy. It is but natural that after this +great conquest mathematicians should attack \Change{bi-quadratic}{biquadratic} equations. +As in the case of cubics, so here, the first impulse was +given by Colla, who, in 1540, proposed for solution the equation +\index{Colla}% +$x^{4} + 6x^{2} + 36 = 60x$. To be sure, Cardan had studied +particular cases as early as~1539. Thus he solved the equation +$13x^{2} = x^{4} + 2x^{3} + 2x + 1$ by a process similar to that employed +by Diophantus and the Hindoos; namely, by adding +to both sides $3x^{2}$ and thereby rendering both numbers +complete squares. But Cardan failed to find a general solution; +it remained for his pupil \textbf{Ferrari} to prop the reputation +\index{Ferrari}% +of his master by the brilliant discovery of the general +solution of \Change{bi-quadratic}{biquadratic} equations. Ferrari reduced Colla's +equation to the form $(x^{2}+ 6)^{2} = 60x + 6x^{2}$. In order to +give also the right member the form of a complete square +he added to both members the expression $2 (x^{2} + 6) y + y^{2}$, +containing a new unknown quantity~$y$. This gave him $(x^{2} + 6 ++ y)^{2} = (6 + 2y) x^{2} + 60x + (12y + y^{2})$. The condition that +the right member be a complete square is expressed by the +%% -----File: 160.png---Folio 146------- +\index{Biquadratic equation}% +cubic equation $(2y + 6) (12y + y^{2}) = 900$. Extracting the +square root of the \Change{bi-quadratic}{biquadratic}, he got $x^{2} + 6 + y = x \sqrt{2y + 6} ++ \dfrac{900\Fstr}{\sqrt{2y + 6}}$. Solving the cubic for~$y$ and substituting, it remained +only to determine~$x$ from the resulting quadratic. +Ferrari pursued a similar method with other numerical \Change{bi-quadratic}{biquadratic} +equations.\Reference{7} Cardan had the pleasure of publishing +this discovery in his \textit{Ars Magna} in~1545. Ferrari's solution +is sometimes ascribed to \textit{Bombelli}, but he is no more the discoverer +\index{Bombelli}% +of it than Cardan is of the solution called by his +name. + +To Cardan algebra is much indebted. In his \textit{Ars Magna} +he takes notice of negative roots of an equation, calling them +\index{Negative roots}% +\emph{fictitious}, while the positive roots are called \emph{real}. Imaginary +\index{Imaginary quantities}% +roots he does not consider; cases where they appear he calls +impossible. Cardan also observed the difficulty in the irreducible +case in the cubics, which, like the quadrature of the +circle, has since ``so much tormented the perverse ingenuity of +mathematicians.'' But he did not understand its nature. It remained +for \textbf{Raphael Bombelli} of Bologna, who published in 1572 +an algebra of great merit, to point out the reality of the apparently +imaginary expression which the root assumes, and thus +to lay the foundation of a more intimate knowledge of imaginary +quantities. + +After this brilliant success in solving equations of the third +and fourth degrees, there was probably no one who doubted, +that with aid of irrationals of higher degrees, the solution of +equations of any degree whatever could be found. But all +attempts at the algebraic solution of the quintic were fruitless, +and, finally, Abel demonstrated that all hopes of finding algebraic +\index{Abel!ref.\ to}% +solutions to equations of higher than the fourth degree +were purely Utopian. + +Since no solution by radicals of equations of higher degrees +%% -----File: 161.png---Folio 147------- +\index{Equations!numerical}% +\index{Falsa positio}% +could be found, there remained nothing else to be done than +the devising of rules by which at least the numerical values of +the roots could be ascertained. Cardan applied the Hindoo +rule of ``false position'' (called by him \textit{regula aurea}) to the +cubic, but this mode of approximating was exceedingly rough. +An incomparably better method was invented by \textbf{Franciscus +Vieta}, a French mathematician, whose transcendent genius +\index{Vieta}% +enriched mathematics with several important innovations. +Taking the equation $f(x) = Q$, wherein $f(x)$ is a polynomial +containing different powers of~$x$, with numerical coefficients, +and $Q$ is a given number, Vieta first substitutes in~$f(x)$ a +known approximate value of the root, and then shows that +another figure of the root can be obtained by division. A repetition +of the same process gives the next figure of the root, +and so on. Thus, in $x^2 + 14x = 7929$, taking $80$ for the approximate +root, and placing $x = 80 + b$, we get +\begin{DPalign*} +&(80 + b)^2 + 14 (80 + b) = 7929,\\ +\lintertext{or} &174b + b^2 = 409. +\end{DPalign*} +Since $174b$ is much greater than~$b^2$, we place $174b = 409$, and +obtain thereby $b = 2$. Hence the second approximation is~$82$. +Put $x = 82 + c$, then $(82 + c)^2 + 14(82 + c) = 7929$, or $178 c + c^2 += 57$. As before, place $178c = 57$, then $c = .3$, and the third +approximation gives $82.3$. Assuming $x = 82.3 + d$, and substituting, +gives $178.6 d + d^2 = 3.51$, and $178.6d = 3.51$, $\therefore d = .01$; +giving for the fourth approximation $82.31$. In the same way, +$e = .009$, and the value for the root of the given equation is +$82.319 \dots$. For this process, Vieta was greatly admired by his +contemporaries. It was employed by Harriot, Oughtred, Pell, +\index{Harriot!ref.\ to}% +\index{Oughtred!ref.\ to}% +\index{Pell}% +and others. Its principle is identical with the main principle +involved in the methods of approximation of Newton and +\index{Newton!ref.\ to}% +Horner. The only change lies in the arrangement of the +\index{Horner}% +%% -----File: 162.png---Folio 148------- +work. This alteration was made to afford facility and security +in the process of evolution of the root. + +We pause a moment to sketch the life of Vieta, the most +eminent French mathematician of the sixteenth century. He +was born in Poitou in 1540, and died in 1603 at Paris. He +was employed throughout life in the service of the state, +under Henry~III. and Henry~IV\@. He was, therefore, not a +mathematician by profession, but his love for the science was +so great that he remained in his chamber studying, sometimes +several days in succession, without eating and sleeping more +than was necessary to sustain himself. So great devotion to +abstract science is the more remarkable, because he lived at a +time of incessant political and religious turmoil. During the +war against Spain, Vieta rendered service to Henry~IV. by +deciphering intercepted letters written in a species of cipher, +and addressed by the Spanish Court to their governor of Netherlands. +The Spaniards attributed the discovery of the key +to magic. + +An ambassador from Netherlands once told Henry~IV. that +France did not possess a single geometer capable of solving a +problem propounded to geometers by a Belgian mathematician, +Adrianus Romanus. It was the solution of the equation of +\index{Romanus!ref.\ to}% +the forty-fifth degree:--- +\[ +45y - 3795y^3 + 95634y^5 - \dotsb + 945y^{41} - 45y^{43} + y^{45} = C. +\] +Henry~IV. called Vieta, who, having already pursued similar +investigations, saw at once that this awe-inspiring problem +was simply the equation by which $C=2\sin\phi$ was expressed +in terms of $y = 2 \sin \frac{1}{45}\phi$; that, since $45 = 3·3·5$, it was +necessary only to divide an angle once into 5~equal parts, and +then twice into~3,---a division which could be effected by corresponding +equations of the fifth and third degrees. Brilliant +was the discovery by Vieta of 23~roots to this equation, instead +%% -----File: 163.png---Folio 149------- +\index{Biquadratic equation}% +\index{Cubic equations}% +\index{Negative roots}% +\index{Notation!in algebra}% +of only one. The reason why he did not find 45~solutions, is +that the remaining ones involve negative sines, which were +unintelligible to him. Detailed investigations on the famous +old problem of the section of an angle into an odd number of +equal parts, led Vieta to the discovery of a trigonometrical +solution of Cardan's irreducible case in cubics. He applied +\index{Cardan!ref.\ to}% +the equation $\bigl(2 \cos \frac{1}{3} \phi\bigr)^3 - 3 \bigl(2 \cos \frac{1}{3} \phi\bigr) = 2 \cos \phi$ to the solution +of $x^3 - 3a^{2}x = a^{2}b$, when $a > \frac{1}{2} b$, by placing $x = 2a \cos \frac{1}{3}\phi$, and +determining~$\phi$ from $b = 2 a \cos \phi$. + +The main principle employed by him in the solution of +equations is that of \emph{reduction}. He solves the quadratic by +\index{Equations!solution of}% +making a suitable substitution which will remove the term +containing $x$ to the first degree. Like Cardan, he reduces the +general expression of the cubic to the form $x^{3}+ mx + n = 0$; +then, assuming $x = \bigl(\frac{1}{3}a - z^2\bigr) ÷ z$ and substituting, he gets +$z^6 - bz^{3} - \frac{1}{27} a^3 = 0$. Putting $z^3 = y$, he has a quadratic. In +the solution of \Change{bi-quadratics}{biquadratics}, Vieta still remains true to his +principle of reduction. This gives him the well-known cubic +resolvent. He thus adheres throughout to his favourite principle, +and thereby introduces into algebra a uniformity of +method which claims our lively admiration. In Vieta's algebra +we discover a partial knowledge of the relations existing +between the coefficients and the roots of an equation. He +shows that if the coefficient of the second term in an equation +of the second degree is minus the sum of two numbers whose +product is the third term, then the two numbers are roots of +the equation. Vieta rejected all except positive roots; hence +it was impossible for him to fully perceive the relations in +question. + +The most epoch-making innovation in algebra due to Vieta +is the denoting of general or indefinite quantities by letters +of the alphabet. To be sure, Regiomontanus and Stifel in +\index{Regiomontanus!ref.\ to}% +\index{Stifel!ref.\ to}% +Germany, and Cardan in Italy, used letters before him, but +%% -----File: 164.png---Folio 150------- +\index{Plus and minus, signs for}% +Vieta extended the idea and first made it an essential part of +algebra. The new algebra was called by him \textit{logistica speciosa} +in distinction to the old \textit{logistica numerosa}. Vieta's formalism +differed considerably from that of to-day. The equation +$a^3 + 3 a^2b + 3 ab^2 + b^3 = (a + b)^3$ was written by him ``$a \text{ cubus } + b +\text{ in } a \text{ quadr.\ } 3 + a \text{ in } b \text{ quadr.\ } 3 + b \text{ cubo æqualia } \overline{a + b} \text{ cubo}$.'' +In numerical equations the unknown quantity was denoted by~$N$, +its square by~$Q$, and its cube by~$C$. Thus the equation +$x^3 - 8x^2 +16x = 40$ was written $1C - 8Q + 16N \textit{ æqual.\ } 40$. +Observe that exponents and our symbol~($=$) for equality were +not yet in use; but that Vieta employed the Maltese cross~($+$) +as the short-hand symbol for addition, and the~($-$) for +subtraction. These two characters had not been in general +use before the time of Vieta. ``It is very singular,'' says +Hallam, ``that discoveries of the greatest convenience, and, +apparently, not above the ingenuity of a village schoolmaster, +should have been overlooked by men of extraordinary acuteness +like Tartaglia, Cardan, and Ferrari; and hardly less so +that, by dint of that acuteness, they dispensed with the aid of +these contrivances in which we suppose that so much of the +utility of algebraic expression consists.'' Even after improvements +in notation were once proposed, it was with extreme +\index{Notation!in algebra}% +slowness that they were admitted into general use. They +were made oftener by accident than design, and their authors +had little notion of the effect of the change which they were +making. The introduction of the $+$ and $-$ symbols seems +to be due to the Germans, who, although they did not enrich +algebra during the Renaissance with great inventions, as did +\index{Algebra!Renaissance|)}% +the Italians, still cultivated it with great zeal. The arithmetic +\index{Arithmetic!Renaissance}% +of \textbf{John Widmann}, printed \AD~1489 in Leipzig, is the +\index{Widmann}% +earliest book in which the $+$ and $-$ symbols have been found. +There are indications leading us to surmise that they were in +use first among merchants. They occur again in the arithmetic +%% -----File: 165.png---Folio 151------- +\index{Arithmetic!Renaissance}% +\index{Notation!in algebra}% +of Grammateus, a teacher at the University of Vienna. +\index{Grammateus}% +His pupil, Christoff Rudolff, the writer of the first text-book +\index{Rudolff}% +on algebra in the German language (printed in~1525), employs +these symbols also. So did Stifel, who brought out a second +\index{Stifel!ref.\ to}% +edition of Rudolff's \textit{Coss} in~1553. Thus, by slow degrees, +their adoption became universal. There is another short-hand +symbol of which we owe the origin to the Germans. In a +manuscript published sometime in the fifteenth century, a dot +placed before a number is made to signify the extraction of a +root of that number. This dot is the embryo of our present +symbol for the square root. Christoff Rudolff, in his algebra, +remarks that ``the radix quadrata is, for brevity, designated +in his algorithm with the character $\sqrt{\vphantom{4}}$, as~$\sqrt{4}$.'' Here the dot +has grown into a symbol much like our own. This same +symbol was used by \textit{Michael Stifel}. Our sign of equality is due +to \textbf{Robert Recorde} (1510--1558), the author of \textit{The Whetstone of +\index{Recorde}% +Witte} (1557), which is the first English treatise on algebra. +He selected this symbol because no two things could be more +equal than two parallel lines~$=$. The sign~$÷$ for division was +first used by \textit{Johann Heinrich Rahn}, a Swiss, in~1659, and +\index{Rahn}% +was introduced in England by \textit{John Pell} in~1668. +\index{Pell}% + +\textbf{Michael Stifel} (1486?--1567), the greatest German algebraist +\index{Stifel}% +of the sixteenth century, was born in Esslingen, and died in +Jena. He was educated in the monastery of his native place, +and afterwards became Protestant minister. The study of the +significance of mystic numbers in Revelation and in Daniel +drew him to mathematics. He studied German and Italian +works, and published in~1544, in Latin, a book entitled +\textit{Arithmetica integra}. Melanchthon wrote a preface to it. Its +three parts treat respectively of rational numbers, irrational +numbers, and algebra. Stifel gives a table containing the numerical +values of the binomial coefficients for powers below the~18th. +He observes an advantage in letting a geometric progression +%% -----File: 166.png---Folio 152------- +\index{Tartaglia!ref.\ to}% +correspond to an arithmetical progression, and arrives at +the designation of integral powers by numbers. Here are the +germs of the theory of exponents. In 1545 Stifel published +\index{Exponents}% +an arithmetic in German. His edition of Rudolff's \textit{Coss} contains +\index{Coss, term for algebra}% +rules for solving cubic equations, derived from the\DPnote{** PP: Removed duplicate `the'.} +\index{Cubic equations}% +writings of Cardan. +\index{Cardan!ref.\ to}% + +We remarked above that Vieta discarded negative roots of +\index{Negative roots}% +\index{Vieta!ref.\ to}% +equations. Indeed, we find few algebraists before and during +the Renaissance who understood the significance even of +negative quantities. Fibonacci seldom uses them. Pacioli +\index{Negative quantities}% +\index{Pacioli!ref.\ to}% +states the rule that ``minus times minus gives plus,'' but +applies it really only to the development of the product of +$(a - b) (c - d)$; purely negative quantities do not appear in +his work. The great German ``Cossist'' (algebraist), \textit{Michael +Stifel}, speaks as early as 1544 of numbers which are ``absurd'' +or ``fictitious below zero,'' and which arise when ``real numbers +above zero'' are subtracted from zero. Cardan, at last, speaks +of a ``pure minus''; ``but these ideas,'' says Hankel, ``remained +sparsely, and until the beginning of the seventeenth century, +mathematicians dealt exclusively with absolute positive quantities.'' +The first algebraist who occasionally places a purely +negative quantity by itself on one side of an equation, is +\textit{Harriot} in England. As regards the recognition of negative +\index{Harriot!ref.\ to}% +roots, Cardan and Bombelli were far in advance of all writers +\index{Bombelli!ref.\ to}% +of the Renaissance, including Vieta. Yet even they mentioned +these so-called false or fictitious roots only in passing, and +without grasping their real significance and importance. On +this subject Cardan and Bombelli had advanced to about the +same point as had the Hindoo Bhaskara, who saw negative roots, +\index{Bhaskara!ref.\ to}% +but did not approve of them. The generalisation of the conception +of quantity so as to include the negative, was an +exceedingly slow and difficult process in the development of +algebra. +\index{Algebra!Renaissance}% +%% -----File: 167.png---Folio 153------- +\index{Cubic equations}% +\index{Equations!solution of}% +\index{Trisection of angles}% + +We shall now consider the history of geometry during the +\index{Geometry!Renaissance}% +Renaissance. Unlike algebra, it made hardly any progress. +The greatest gain was a more intimate knowledge of Greek +geometry. No essential progress was made before the time of +Descartes. Regiomontanus, Xylander of Augsburg, Tartaglia, +\index{Regiomontanus!ref.\ to}% +\index{Tartaglia!ref.\ to}% +\index{Xylander}% +Commandinus of Urbino in Italy, Maurolycus, and others, +\index{Commandinus}% +\index{Maurolycus}% +made translations of geometrical works from the Greek. \textbf{John +Werner} of Nürnberg published in 1522 the first work on +\index{Werner}% +conics which appeared in Christian Europe. Unlike the +geometers of old, he studied the sections in relation with the +cone, and derived their properties directly from it. This mode +of studying the conics was followed by \textbf{Maurolycus} of Messina +(1494--1575). The latter is, doubtless, the greatest geometer +of the sixteenth century. From the notes of Pappus, he +\index{Pappus!ref.\ to}% +attempted to restore the missing fifth book of Apollonius +\index{Apollonius!ref.\ to}% +on \textit{maxima} and \textit{minima}. His chief work is his masterly +and original treatment of the conic sections, wherein he discusses +\index{Conic sections!Renaissance}% +tangents and asymptotes more fully than Apollonius +had done, and applies them to various physical and astronomical +problems. + +The foremost geometrician of Portugal was \textbf{Nonius}; of +\index{Nonius}% +France, before Vieta, was \textbf{Peter Ramus}, who perished in the +\index{Ramus}% +\index{Vieta!ref.\ to}% +massacre of St.~Bartholomew. \textit{Vieta} possessed great familiarity +with ancient geometry. The new form which he gave +to algebra, by representing general quantities by letters, enabled +him to point out more easily how the construction of +the roots of cubics depended upon the celebrated ancient problems +of the duplication of the cube and the trisection of an +\index{Duplication of the cube}% +angle. He reached the interesting conclusion that the former +problem includes the solutions of all cubics in which the radical +in Tartaglia's formula is real, but that the latter problem +includes only those leading to the irreducible case. + +The problem of the quadrature of the circle was revived in +%% -----File: 168.png---Folio 154------- +\index{Apollonius!ref.\ to}% +\index{Gregorian Calendar}% +this age, and was zealously studied even by men of eminence +and mathematical ability. The army of circle-squarers became +\index{Circle}% +most formidable during the seventeenth century. Among the +first to revive this problem was the German Cardinal \textbf{Nicolaus +Cusanus} (died 1464), who had the reputation of being a great +\index{Cusanus}% +logician. His fallacies were exposed to full view by Regiomontanus. +\index{Regiomontanus!ref.\ to}% +As in this case, so in others, every quadrator of +note raised up an opposing mathematician: Orontius was met +\index{Orontius}% +by Buteo and Nonius; Joseph Scaliger by Vieta, Adrianus +\index{Buteo}% +\index{Nonius!ref.\ to}% +\index{Scaliger}% +\index{Vieta!ref.\ to}% +Romanus, and Clavius; A. Quercu by Peter Metius. Two +\index{Clavius!ref.\ to}% +\index{Metius}% +\index{Quercu, a}% +\index{Romanus}% +\index{Romanus!ref.\ to}% +mathematicians of Netherlands, \textbf{Adrianus Romanus} and \textbf{Ludolph +\index{Ludolph}% +van Ceulen}, occupied themselves with approximating to the +ratio between the circumference and the diameter. The former +carried the value~$\pi$ to~$15$, the latter to~$35$, places. The +\index{p@{$\pi$}: values for!Ludolph's}% +value of~$\pi$ is therefore often named ``Ludolph's number.'' His +\index{Ludolph's number}% +performance was considered so extraordinary, that the numbers +were cut on his tomb-stone in St.~Peter's church-yard, at +Leyden. Romanus was the one who propounded for solution +that equation of the forty-fifth degree solved by Vieta. On +receiving Vieta's solution, he at once departed for Paris, to +make his acquaintance with so great a master. Vieta proposed +to him the Apollonian problem, to draw a circle touching +\index{Apollonian Problem}% +three given circles. ``Adrianus Romanus solved the problem +by the intersection of two hyperbolas; but this solution did not +possess the rigour of the ancient geometry. Vieta caused him +\index{Geometry!Renaissance}% +to see this, and then, in his turn, presented a solution which +had all the rigour desirable.''\Reference{25} Romanus did much toward +simplifying spherical trigonometry by reducing, by means of +\index{Trigonometry}% +certain projections, the $28$~cases in triangles then considered +to only six. + +Mention must here be made of the improvements of the +Julian calendar. The yearly determination of the movable +\index{Calendar}% +feasts had for a long time been connected with an untold +%% -----File: 169.png---Folio 155------- +amount of confusion. The rapid progress of astronomy led +to the consideration of this subject, and many new calendars +were proposed. Pope Gregory~XIII. convoked a large number +of mathematicians, astronomers, and prelates, who decided +upon the adoption of the calendar proposed by the Jesuit +\textbf{Lilius Clavius}. To rectify the errors of the Julian calendar +\index{Clavius}% +it was agreed to write in the new calendar the 15th~of +October immediately after the 4th~of October of the year +1582. The Gregorian calendar met with a great deal of opposition +both among scientists and among Protestants. Clavius, +who ranked high as a geometer, met the objections of the +former most ably and effectively; the prejudices of the latter +passed away with time. + +The passion for the study of mystical properties of numbers +descended from the ancients to the moderns. Much was +written on numerical mysticism even by such eminent men +as Pacioli and Stifel. The \textit{Numerorum Mysteria} of Peter +\index{Pacioli!ref.\ to}% +\index{Stifel!ref.\ to}% +Bungus covered $700$~quarto pages. He worked with great +\index{Bungus}% +industry and satisfaction on~$666$, which is the number of the +beast in Revelation (xiii.~18), the symbol of Antichrist. He +reduced the name of the `impious' Martin Luther to a form +which may express this formidable number. Placing $a = 1$, +$b = 2$,~etc, $k = 10$, $l = 20$,~etc., he finds, after misspelling the +name, that +$\text{M}_{(30)} \text{A}_{(1)} \text{R}_{(80)} \text{T}_{(100)} \text{I}_{(9)} \text{N}_{(40)}$ +$\text{L}_{(20)} \text{V}_{(200)} \text{T}_{(100)} \text{E}_{(5)} \text{R}_{(80)} \text{A}_{(1)}$ +constitutes the number required. These attacks on the great +reformer were not unprovoked, for his friend, Michael Stifel, +the most acute and original of the early mathematicians of +Germany, exercised an equal ingenuity in showing that the +above number referred to Pope Leo~X.,---a demonstration +which gave Stifel unspeakable comfort.\Reference{22} + +Astrology also was still a favourite study. It is well +\index{Astrology}% +known that Cardan, Maurolycus, Regiomontanus, and many +\index{Cardan!ref.\ to}% +\index{Maurolycus!ref.\ to}% +\index{Regiomontanus!ref.\ to}% +other eminent scientists who lived at a period even later than +%% -----File: 170.png---Folio 156------- +\index{Renaissance|)}% +this, engaged in deep astrological study; but it is not so generally +known that besides the occult sciences already named, +men engaged in the mystic study of star-polygons and magic +\index{Star-polygons}% +squares. ``The pentagramma gives you pain,'' says Faust to +Mephistopheles. It is of deep psychological interest to see +scientists, like the great Kepler, demonstrate on one page a +\index{Kepler!ref.\ to}% +theorem on star-polygons, with strict geometric rigour, while +on the next page, perhaps, he explains their use as amulets +or in conjurations.\Reference{1} Playfair, speaking of Cardan as an astrologer, +\index{Cardan!ref.\ to}% +\index{Playfair}% +calls him ``a melancholy proof that there is no folly +or weakness too great to be united to high intellectual attainments.''\Reference{26} +Let our judgment not be too harsh. The period +under consideration is too near the Middle Ages to admit of +complete emancipation from mysticism even among scientists. +Scholars like Kepler, Napier, Albrecht \Change{Duerer}{Dürer}, while in the +\index{Durer@{Dürer, A.}}% +\index{Napier, J.!ref.\ to}% +van of progress and planting one foot upon the firm ground +of truly scientific inquiry, were still resting with the other +foot upon the scholastic ideas of preceding ages. + + +\Section{Vieta to Descartes} + +The ecclesiastical power, which in the ignorant ages was an +unmixed benefit, in more enlightened ages became a serious +evil. Thus, in France, during the reigns preceding that of +Henry~IV., the theological spirit predominated. This is painfully +shown by the massacres of Vassy and of St.~Bartholomew. +Being engaged in religious disputes, people had no +leisure for science and for secular literature. Hence, down +to the time of Henry~IV., the French ``had not put forth a +single work, the destruction of which would now be a loss to +Europe.'' In England, on the other hand, no religious wars +were waged. The people were comparatively indifferent about +%% -----File: 171.png---Folio 157------- +\index{Menelaus!ref.\ to}% +religious strifes; they concentrated their ability upon secular +matters, and acquired, in the sixteenth century, a literature +which is immortalised by the genius of Shakespeare and +Spenser. This great literary age in England was followed +by a great scientific age. At the close of the sixteenth century, +the shackles of ecclesiastical authority were thrown off +by France. The ascension of Henry~IV. to the throne was +followed in 1598 by the Edict of Nantes, granting freedom +of worship to the Huguenots, and thereby terminating religious +wars. The genius of the French nation now began to blossom. +Cardinal Richelieu, during the reign of Louis~XIII., pursued +the broad policy of not favouring the opinions of any sect, but +of promoting the interests of the nation. His age was remarkable +for the progress of knowledge. It produced that +great secular literature, the counterpart of which was found +in England in the sixteenth century. The seventeenth century +was made illustrious also by the great French mathematicians, +Roberval, Descartes, Desargues, Fermat, and Pascal. + +More gloomy is the picture in Germany. The great changes +which revolutionised the world in the sixteenth century, and +which led England to national greatness, led Germany to +degradation. The first effects of the Reformation there were +salutary. At the close of the fifteenth and during the sixteenth +century, Germany had been conspicuous for her scientific +pursuits. She had been the leader in astronomy and +trigonometry. Algebra also, excepting for the discoveries in +cubic equations, was, before the time of Vieta, in a more +advanced state there than elsewhere. But at the beginning +of the seventeenth century, when the sun of science began to +rise in France, it set in Germany. Theologic disputes and +religious strife ensued. The Thirty Years' War (1618--1648) +proved ruinous. The German empire was shattered, and +became a mere lax confederation of petty despotisms. Commerce +%% -----File: 172.png---Folio 158------- +was destroyed; national feeling died out. Art disappeared, +and in literature there was only a slavish imitation +of French artificiality. Nor did Germany recover from this +low state for $200$~years; for in 1756 began another struggle, +the Seven Years' War, which turned Prussia into a wasted +land. Thus it followed that at the beginning of the seventeenth +century, the great Kepler was the only German mathematician +\index{Kepler!ref.\ to}% +of eminence, and that in the interval of $200$~years +between Kepler and Gauss, there arose no great mathematician +\index{Gauss!ref.\ to}% +in Germany excepting Leibniz. +\index{Leibniz!ref.\ to}% + +Up to the seventeenth century, mathematics was cultivated +but little in Great Britain. During the sixteenth century, she +brought forth no mathematician comparable with Vieta, Stifel, +or Tartaglia. But with the time of Recorde, the English +\index{Recorde!ref.\ to}% +became conspicuous for numerical skill. The first important +arithmetical work of English authorship was published in +Latin in 1522 by \textbf{Cuthbert Tonstall} (1474--1559). He had +\index{Tonstall}% +studied at Oxford, Cambridge, and Padua, and drew freely +from the works of Pacioli and Regiomontanus. Reprints of +\index{Pacioli!ref.\ to}% +\index{Regiomontanus!ref.\ to}% +his arithmetic appeared in England and France. After +\index{Arithmetic!Renaissance|(}% +Recorde the higher branches of mathematics began to be +studied. Later, Scotland brought forth Napier, the inventor +of logarithms. The instantaneous appreciation of their value +\index{Logarithms}% +is doubtless the result of superiority in calculation. In Italy, +and especially in France, geometry, which for a long time had +\index{Geometry!Renaissance}% +been an almost stationary science, began to be studied with +success. Galileo, Torricelli, Roberval, Fermat, Desargues, +Pascal, Descartes, and the English Wallis are the great revolutioners +of this science. Theoretical mechanics began to be +\index{Mechanics!Stevin and Galileo}% +studied. The foundations were laid by Fermat and Pascal +for the theory of numbers and the theory of probability. +\index{Probability}% + +We shall first consider the improvements made in the art +of calculating. The nations of antiquity experimented thousands +%% -----File: 173.png---Folio 159------- +\index{Arabic numerals and notation}% +of years upon numeral notations before they happened +to strike upon the so-called ``Arabic notation.'' In the simple +\index{Notation!Arabic notation}% +expedient of the cipher, which was introduced by the Hindoos +about the fifth or sixth century after Christ, mathematics received +one of the most powerful impulses. It would seem that +after the ``Arabic notation'' was once thoroughly understood, +decimal fractions would occur at once as an obvious extension +\index{Decimal fractions|(}% +\index{Fractions!decimal}% +\index{Fractions!continued}% +of it. But ``it is curious to think how much science had +attempted in physical research and how deeply numbers had +been pondered, before it was perceived that the all-powerful +simplicity of the `Arabic notation' was as valuable and as +manageable in an infinitely descending as in an infinitely +ascending progression.''\Reference{28} Simple as decimal fractions appear +to us, the invention of them is not the result of one mind or +even of one age. They came into use by almost imperceptible +degrees. The first mathematicians identified with their history +did not perceive their true nature and importance, and +failed to invent a suitable notation. The idea of decimal +fractions makes its first appearance in methods for approximating +to the square roots of numbers. Thus John of Seville, +\index{John of Seville}% +presumably in imitation of Hindoo rules, adds $2~n$~ciphers +to the number, then finds the square root, and takes this +\index{Square root}% +as the numerator of a fraction whose denominator is 1 followed +by $n$~ciphers. The same method was followed by +\textit{Cardan}, but it failed to be generally adopted even by his +\index{Cardan!ref.\ to}% +Italian contemporaries; for otherwise it would certainly have +been at least mentioned by \textit{Cataldi} (died 1626) in a work +\index{Cataldi}% +devoted exclusively to the extraction of roots. Cataldi finds +the square root by means of continued fractions---a method +\index{Continued fractions}% +ingenious and novel, but for practical purposes inferior to +Cardan's. \textbf{Orontius Finaeus} (died 1555) in France, and \textbf{William +\index{Finæus}% +Buckley} (died about 1550) in England extracted the +\index{Buckley}% +square root in the same way as Cardan and John of Seville. +%% -----File: 174.png---Folio 160------- +The invention of decimals is frequently attributed to Regiomontanus, +\index{Regiomontanus!ref.\ to}% +on the ground that instead of placing the sinus +totus, in trigonometry, equal to a multiple of~$60$, like the +\index{Trigonometry}% +Greeks, he put it $= 100,000$. But here the trigonometrical +lines were expressed in \emph{integers}, and not in fractions. Though +\index{Fractions!decimal}% +he adopted a decimal division of the radius, he and his successors +did not apply the idea outside of trigonometry and, +indeed, had no notion whatever of decimal \emph{fractions}. To +\textbf{Simon Stevin} of Bruges in Belgium (1548--1620), a man who +\index{Stevin}% +did a great deal of work in most diverse fields of science, we +owe the first systematic treatment of decimal fractions. In +his \textit{La Disme} (1585) he describes in very express terms the +advantages, not only of decimal fractions, but also of the +decimal division in systems of weights and measures. Stevin +applied the new fractions ``to all the operations of ordinary +arithmetic.''\Reference{25} What he lacked was a suitable notation. In +\index{Notation!in algebra}% +\index{Notation!decimal fractions}% +place of our decimal point, he used a cipher; to each place in +the fraction was attached the corresponding index. Thus, in +his notation, the number $5.912$ would be +$\overset{0}{5}\overset{1}{9}\overset{2}{1}\overset{3}{2}$ or +$5\CircleDigit{0}9\CircleDigit{1}1\CircleDigit{2}2\CircleDigit{3}$. +These indices, though cumbrous in practice, are of interest, +because they are the germ of an important innovation. To +Stevin belongs the honour of inventing our present mode of +designating powers and also of introducing fractional exponents +\index{Exponents}% +into algebra. Strictly speaking, this had been done much +earlier by \textit{Oresme}, but it remained wholly unnoticed. Not even +\index{Oresme!ref.\ to}% +Stevin's innovations were immediately appreciated or at once +accepted, but, unlike Oresme's, they remained a secure possession. +No improvement was made in the notation of decimals +till the beginning of the seventeenth century. After Stevin, +decimals were used by \textbf{Joost Bürgi}, a Swiss by birth, who prepared +\index{Burgi@{Bürgi}}% +a manuscript on arithmetic soon after 1592, and by +\textbf{Johann Hartmann Beyer}, who assumes the invention as his own. +\index{Beyer}% +In~1603, he published at Frankfurt on the Main a \textit{Logistica} +%% -----File: 175.png---Folio 161------- +\index{Arithmetic!Renaissance|)}% +\index{Trigonometry}% +\textit{Decimalis}. With Bürgi, a zero placed underneath the digit +in unit's place answers as sign of separation. Beyer's notation +resembles Stevin's. The decimal point, says Peacock, is +\index{Decimal point}% +\index{Peacock!ref.\ to, X}% +due to Napier, who in 1617 published his \textit{Rabdologia}, containing +\index{Napier, J.!ref.\ to}% +a treatise on decimals, wherein the decimal point is used +in one or two instances. In the English translation of Napier's +\textit{Mirifici logarithmorum canonis descriptio}, executed by Edward +Wright in 1616, and corrected by the author, the decimal +point occurs in the tables. There is no mention of decimals +in English arithmetics between 1619 and~1631. \textit{Oughtred} in +\index{Oughtred!ref.\ to}% +1631 designates the fraction $.56$ thus, $\olddecimal{0}{56}$. \textit{Albert Girard}, +\index{Girard!ref.\ to}% +a pupil of Stevin, in 1629 uses the point on one occasion. +John Wallis in 1657 writes \olddecimal{12}{345}, but afterwards in his +\index{Wallis!ref.\ to}% +algebra adopts the usual point. De Morgan says that ``to the +\index{Demorgan@{De Morgan}!ref.\ to}% +first quarter of the eighteenth century we must refer not only +the complete and final victory of the decimal point, but also +that of the now universal method of performing the operations +of division and extraction of the square root.\Reference{27} We have +dwelt at some length on the progress of the decimal notation, +because ``the history of language \ldots is of the highest order +of interest, as well as utility: its suggestions are the best +lesson for the future which a reflecting mind can have.''\Reference{27} + +The miraculous powers of modern calculation are due to +three inventions: the Arabic Notation, Decimal Fractions, and +\index{Decimal fractions|)}% +Logarithms. The invention of logarithms in the first quarter +\index{Logarithms|(}% +of the seventeenth century was admirably timed, for Kepler +\index{Kepler!ref.\ to}% +was then examining planetary orbits, and Galileo had just +\index{Galileo!ref.\ to}% +turned the telescope to the stars. During the Renaissance +German mathematicians had constructed trigonometrical +tables of great accuracy, but this greater precision enormously +increased the work of the calculator. It is no exaggeration +to say that the invention of logarithms ``by shortening the +labours doubled the life of the astronomer.'' Logarithms were +%% -----File: 176.png---Folio 162------- +\index{Euclid!ref.\ to}% +invented by \textbf{John Napier}, Baron of Merchiston, in Scotland +\index{Napier, J.}% +(1550--1617). It is one of the greatest curiosities of the +history of science that Napier constructed logarithms before +exponents were used. To be sure, Stifel and Stevin made +\index{Exponents}% +\index{Stevin!ref.\ to}% +\index{Stifel!ref.\ to}% +some attempts to denote powers by indices, but this notation +was not generally known,---not even to \textit{Harriot}, whose algebra +\index{Harriot!ref.\ to}% +appeared long after Napier's death. That logarithms flow +naturally from the exponential symbol was not observed until +much later. It was Euler who first considered logarithms as +being indices of powers. What, then, was Napier's line of +thought? + +Let $AB$ be a definite line, $DE$ a line extending from $D$ indefinitely. +Imagine two points starting at the same %[** PP: Width-dependent line break] +%[**Illustration: ] +\begin{wrapfigure}{l}{2.5in} +\Graphic{2.5in}{176a} +\end{wrapfigure} +moment; +the one moving from +$A$ toward~$B$, the other +from $D$ toward~$E$. Let +the velocity during the +first moment be the +same for both: let that of the point on line~$DE$ be uniform; +but the velocity of the point on $AB$ decreasing in such a way +that when it arrives at any point~$C$, its velocity is proportional +to the remaining distance~$BC$. While the first point moves +over a distance~$AC$, the second one moves over a distance~$DF$. +Napier calls $DF$ the logarithm of~$BC$. + +Napier's process is so unique and so different from all other +modes of presenting the subject that there cannot be the +shadow of a doubt that this invention is entirely his own; it +is the result of unaided, isolated speculation. He first sought +the logarithms only of sines; the line~$AB$ was the sine of~$90°$ +and was taken $= 10^7$; $BC$ was the sine of the arc, and $DF$ its +logarithm. We notice that as the motion proceeds, $BC$ decreases +in geometrical progression, while $DF$ increases in +arithmetical progression. Let $AB = a = 10^7$, let $x = DF$, +%% -----File: 177.png---Folio 163------- +$y = BC$, then $AC = a - y$. The velocity of the point~$C$ is +$\dfrac{d(a - y)}{dt} = y$; this gives $-\natlog y = t + c$. When $t = 0$, +then $y = a$ and $c = -\natlog a$. Again, let $\smash[t]{\dfrac{dx}{dt}} = a$ be the +velocity of the point~$F$, then $x = at$. Substituting for $t$~and~$c$ +their values and remembering that $a = 10^7$ and that by definition +$x = \Naplog y$, we get +\[ +\Naplog y = 10^7 \natlog \frac{10^7}{y}. +\] + +It is evident from this formula that Napier's logarithms are +\index{Napier, J.}% +not the same as the natural logarithms. Napier's logarithms +increase as the number itself decreases. He took the logarithm +of $\sin 90 = 0$; \ie~the logarithm of $10^7 = 0$. The logarithm +of $\sin \alpha$ increased from zero as $\alpha$ decreased from~$90°$. +Napier's genesis of logarithms from the conception of two +flowing points reminds us of Newton's doctrine of fluxions. +The relation between geometric and arithmetical progressions, +so skilfully utilised by Napier, had been observed by Archimedes, +Stifel, and others. Napier did not determine the base +to his system of logarithms. The notion of a ``base'' in fact +never suggested itself to him. The one demanded by his +reasoning is the reciprocal of that of the natural system, but +such a base would not reproduce accurately all of Napier's +figures, owing to slight inaccuracies in the calculation of the +tables. Napier's great invention was given to the world in +1614 in a work entitled \textit{Mirifici logarithmorum canonis descriptio}. +In it he explained the nature of his logarithms, and +gave a logarithmic table of the natural sines of a quadrant +from minute to minute. + +\textbf{Henry Briggs} (1556--1631), in Napier's time professor of +\index{Briggs}% +geometry at Gresham College, London, and afterwards +professor at Oxford, was so struck with admiration of +Napier's book, that he left his studies in London to do +%% -----File: 178.png---Folio 164------- +homage to the Scottish philosopher. Briggs was delayed in +his journey, and Napier complained to a common friend, ``Ah, +\index{Napier, J.!ref.\ to}% +John, Mr. Briggs will not come.'' At that very moment +knocks were heard at the gate, and Briggs was brought into +the lord's chamber. Almost one-quarter of an hour was +spent, each beholding the other without speaking a word. At +last Briggs began: ``My lord, I have undertaken this long +journey purposely to see your person, and to know by what +engine of wit or ingenuity you came first to think of this most +excellent help in astronomy, viz.~the logarithms; but, my +lord, being by you found out, I wonder nobody found it out +before, when now known it is so easy.''\Reference{28} Briggs suggested +to Napier the advantage that would result from retaining zero +for the logarithm of the whole sine, but choosing $10,000,000,000$ +for the logarithm of the $10$th part of that same sine, \ie~of +$5°\ 44' 22''$. Napier said that he had already thought of the +change, and he pointed out a slight improvement on Briggs' +idea; viz.~that zero should be the logarithm of~$1$, and +$10,000,000,000$ that of the whole sine, thereby making the +characteristic of numbers greater than unity positive and +not negative, as suggested by Briggs. Briggs admitted this +to be more convenient. The invention of ``Briggian logarithms'' +occurred, therefore, to Briggs and Napier independently. +The great practical advantage of the new system +was that its fundamental progression was accommodated to +the base, $10$, of our numerical scale. Briggs devoted all +his energies to the construction of tables upon the new plan. +Napier died in 1617, with the satisfaction of having found +in Briggs an able friend to bring to completion his unfinished +plans. In~1624 Briggs published his \textit{Arithmetica logarithmica}, +containing the logarithms to $14$~places of numbers, +from $1$ to~$20,000$ and from $90,000$ to~$100,000$. The gap from +$20,000$ to~$90,000$ was filled up by that illustrious successor of +%% -----File: 179.png---Folio 165------- +\index{Trigonometry}% +Napier and Briggs, \textbf{Adrian Vlacq} of Gouda in Holland. He +\index{Napier, J.!ref.\ to}% +\index{Vlacq}% +published in 1628 a table of logarithms from 1 to~$100,000$, of +which $70,000$ were calculated by himself. The first publication +of Briggian logarithms of trigonometric functions was made +in 1620 by \textbf{Gunter}, a colleague of Briggs, who found the logarithmic +\index{Gunter, E.}% +sines and tangents for every minute to seven places. +Gunter was the inventor of the words \emph{cosine} and \emph{cotangent}. +\index{Cosine}% +\index{Cotangent}% +Briggs devoted the last years of his life to calculating more +extensive Briggian logarithms of trigonometric functions, but +he died in~1631, leaving his work unfinished. It was carried +on by the English \textbf{Henry Gellibrand}, and then published by +\index{Gellibrand}% +Vlacq at his own expense. Briggs divided a degree into $100$~parts, +but owing to the publication by Vlacq of trigonometrical +tables constructed on the old sexagesimal division, Briggs' +innovation remained unrecognised. Briggs and Vlacq published +four fundamental works, the results of which ``have never been +superseded by any subsequent calculations.'' + +The first logarithms upon the \emph{natural} base $e$ were published +by \textbf{John Speidell} in his \textit{New Logarithmes} (London, 1619), which +\index{Speidell}% +contains the natural logarithms of sines, tangents, and secants. + +The only possible rival of John Napier in the invention of +logarithms was the Swiss \textbf{Justus Byrgius} (Joost Bürgi). He +\index{Burgi@{Bürgi}!ref.\ to}% +published a rude table of logarithms six years after the +appearance of the \textit{Canon Mirificus}, but it appears that he +conceived the idea and constructed that table as early, if not +earlier, than Napier did his. But he neglected to have the +results published until Napier's logarithms were known and +\index{Logarithms|)}% +admired throughout Europe. + +Among the various inventions of Napier to assist the +memory of the student or calculator, is ``Napier's rule of +circular parts'' for the solution of spherical right triangles. +\index{Napier's rule of circular parts}% +It is, perhaps, ``the happiest example of artificial memory +that is known.'' +%% -----File: 180.png---Folio 166------- + +The most brilliant conquest in algebra during the sixteenth +\index{Algebra!seventeenth century}% +century had been the solution of cubic and \Change{bi-quadratic}{biquadratic} equations. +\index{Equations!theory of}% +All attempts at solving algebraically equations of higher +degrees remaining fruitless, a new line of inquiry---the properties +of equations and their roots---was gradually opened up. +We have seen that Vieta had attained a partial knowledge of +the relations between roots and coefficients. \textbf{Peletarius}, a +\index{Peletarius}% +Frenchman, had observed as early as~1558, that the root of an +equation is a divisor of the last term. One who extended the +theory of equations somewhat further than Vieta, was \textbf{Albert +Girard} (1590--1634), a Flemish mathematician. Like Vieta, +\index{Girard}% +this ingenious author applied algebra to geometry, and was +the first who understood the use of negative roots in the +\index{Negative roots}% +solution of geometric problems. He spoke of imaginary quantities; +\index{Imaginary quantities}% +inferred by induction that every equation has as many +roots as there are units in the number expressing its degree; +and first showed how to express the sums of their powers in +terms of the coefficients. Another algebraist of considerable +power was the English \textbf{Thomas Harriot} (1560--1621). He +\index{Harriot}% +accompanied the first colony sent out by Sir Walter Raleigh +to Virginia. After having surveyed that country he returned +to England. As a mathematician, he was the boast of his +country. He brought the theory of equations under one +comprehensive point of view by grasping that truth in its +full extent to which Vieta and Girard only approximated; viz.~that +in an equation in its simplest form, the coefficient of +the second term with its sign changed is equal to the sum of +the roots; the coefficient of the third is equal to the sum +of the products of every two of the roots; etc. He was the +first to decompose equations into their simple factors; but, +since he failed to recognise imaginary and even negative roots, +he failed also to prove that every equation could be thus +decomposed. Harriot made some changes in algebraic notation, +%% -----File: 181.png---Folio 167------- +\index{Notation!in algebra}% +adopting small letters of the alphabet in place of the +capitals used by Vieta. The symbols of inequality $>$ and $<$ +\index{Vieta!ref.\ to}% +were introduced by him. Harriot's work, \textit{Artis Analyticæ +praxis}, was published in 1631, ten years after his death. +\textbf{William Oughtred} (1574--1660) contributed vastly to the propagation +\index{Oughtred}% +of mathematical knowledge in England by his treatises, +which were long used in the universities. He introduced $×$ +as symbol of multiplication, and $::$ as that of proportion. By +him ratio was expressed by only one dot. In the eighteenth +century \textit{Christian Wolf} secured the general adoption of the +\index{Wolf, C.!ref.\ to}% +dot as a symbol of multiplication, and the sign for ratio was +thereupon changed to two dots. Oughtred's ministerial duties +left him but little time for the pursuit of mathematics during +daytime, and evenings his economical wife denied him the +use of a light. + +Algebra was now in a state of sufficient perfection to enable +Descartes to take that important step which forms one of the +\index{Descartes!ref.\ to}% +grand epochs in the history of mathematics,---the application +of algebraic analysis to define the nature and investigate the +properties of algebraic curves. + +In geometry, the determination of the areas of curvilinear +\index{Geometry!Renaissance}% +figures was diligently studied at this period. \textbf{Paul Guldin} +\index{Guldin}% +(1577--1643), a Swiss mathematician of considerable note, +\Change{rediscovered}{re-discovered} the following theorem, published in his \textit{Centrobaryca}, +which has been named after him, though first found +in the \textit{Mathematical Collections} of Pappus: The volume of a +solid of revolution is equal to the area of the generating +figure, multiplied by the circumference described by the centre +of gravity. We shall see that this method excels that of +Kepler and Cavalieri in following a more exact and natural +\index{Cavalieri!ref.\ to}% +\index{Kepler!ref.\ to}% +course; but it has the disadvantage of necessitating the determination +of the centre of gravity, which in itself may be a +more difficult problem than the original one of finding the +%% -----File: 182.png---Folio 168------- +\index{Stereometry}% +volume. Guldin made some attempts to prove his theorem, +but Cavalieri pointed out the weakness of his demonstration. + +\textbf{Johannes Kepler} (1571--1630) was a native of Würtemberg +\index{Kepler|(}% +and imbibed Copernican principles while at the University of +Tübingen. His pursuit of science was repeatedly interrupted +by war, religious persecution, pecuniary embarrassments, frequent +changes of residence, and family troubles. In~1600 he +became for one year assistant to the Danish astronomer, Tycho +Brahe, in the observatory near Prague. The relation between +\index{Brahe, Tycho}% +the two great astronomers was not always of an agreeable +character. Kepler's publications are voluminous. His first +attempt to explain the solar system was made in~1596, when he +thought he had discovered a curious relation between the +five regular solids and the number and distance of the planets. +\index{Regular solids}% +The publication of this pseudo-discovery brought him much +fame. Maturer reflection and intercourse with Tycho Brahe +\index{Tycho Brahe}% +and Galileo led him to investigations and results more worthy of +\index{Galileo!ref.\ to}% +his genius---``Kepler's laws.'' He enriched pure mathematics +\index{Kepler's laws}% +as well as astronomy. It is not strange that he was interested +in the mathematical science which had done him so much +service; for ``if the Greeks had not cultivated conic sections, +\index{Conic sections!Kepler}% +Kepler could not have superseded Ptolemy.''\Reference{11} The Greeks +never dreamed that these curves would ever be of practical +use; Aristæus and Apollonius studied them merely to satisfy +their intellectual cravings after the ideal; yet the conic +sections assisted Kepler in tracing the march of the planets in +their elliptic orbits. Kepler made also extended use of logarithms +\index{Logarithms}% +and decimal fractions, and was enthusiastic in diffusing +a knowledge of them. At one time, while purchasing wine, he +was struck by the inaccuracy of the ordinary modes of determining +the contents of kegs. This led him to the study of +the volumes of solids of revolution and to the publication of +the \textit{Stereometria Doliorum} in~1615. In it he deals first with the +%% -----File: 183.png---Folio 169------- +\index{Definite integrals}% +\index{Infinitesimals}% +\index{Method of exhaustion!ref.\ to}% +\index{Rectification of curves}% +solids known to Archimedes and then takes up others. Kepler +\index{Archimedes!ref.\ to}% +introduced a new idea into geometry; namely, that of infinitely +great and infinitely small quantities. Greek mathematicians +always shunned this notion, but with it modern mathematicians +have completely revolutionised the science. In comparing +rectilinear figures, the method of superposition was employed +by the ancients, but in comparing rectilinear and curvilinear +figures with each other, this method failed because no addition +or subtraction of rectilinear figures could ever produce curvilinear +ones. To meet this case, they devised the Method of +Exhaustion, which was long and difficult; it was purely +\index{Exhaustion, method of}% +synthetical, and in general required that the conclusion should +be known at the outset. The new notion of infinity led +\index{Infinity}% +gradually to the invention of methods immeasurably more +powerful. Kepler conceived the circle to be composed of an +infinite number of triangles having their common vertices at +the centre, and their bases in the circumference; and the +sphere to consist of an infinite number of pyramids. He +applied conceptions of this kind to the determination of the +areas and volumes of figures generated by curves revolving +about any line as axis, but succeeded in solving only a few of +the simplest out of the 84~problems which he proposed for +investigation in his \textit{Stereometria}. + +Other points of mathematical interest in Kepler's works are +(1)~the statement of the earliest problem of inverse tangents; +\index{Inverse tangents (problem of)}% +\index{Tangents!inverse problem of}% +(2)~an investigation which amounts to the evaluation of +the definite integral $\displaystyle \int_{0}^{\phi} \sin \phi\, d\phi = 1 - \cos \phi$; (3)~the assertion +that the circumference of an ellipse, whose axes are $2a$ and +$2b$, is nearly $\pi (a + b)$; (4)~a passage from which it has been +inferred that Kepler knew the variation of a function near its +maximum value to disappear; (5)~the assumption of the principle +of continuity (which differentiates modern from ancient +\index{Continuity}% +geometry), when he shows that a parabola has a focus at +\index{Focus}% +%% -----File: 184.png---Folio 170------- +\index{Kepler|)}% +infinity, that lines radiating from this ``cæcus focus'' are +\index{Focus}% +parallel and have no other point at infinity. + +The \textit{Stereometria} led Cavalieri, an Italian Jesuit, to the +\index{Cavalieri}% +consideration of infinitely small quantities. \textbf{Bonaventura +Cavalieri} (1598--1647), a pupil of Galileo and professor at +\index{Galileo!ref.\ to}% +Bologna, is celebrated for his \textit{Geometria indivisibilibus continuorum +nova quadam ratione promota}, 1635. This work +expounds his method of Indivisibles, which occupies an intermediate +\index{Indivisibles|(}% +place between the method of exhaustion of the Greeks +and the methods of Newton and Leibniz. He considers lines +as composed of an infinite number of points, surfaces as composed +of an infinite number of lines, and solids of an infinite +number of planes. The relative magnitude of two solids or +surfaces could then be found simply by the summation of +series of planes or lines. For example, he finds the sum of the +squares of all lines making up a triangle equal to one-third +the sum of the squares of all lines of a parallelogram of equal +base and altitude; for if in a triangle, the first line at the apex +be~1, then the second is~2, the third is~3, and so on; and the +sum of their squares is +\[ +1^2 + 2^2 + 3^2 + \dotsb + n^2 = n(n + 1)(2n + 1) ÷ 6. +\] +In the parallelogram, each of the lines is $n$ and their number is +$n$; hence the total sum of their squares is $n^3$. The ratio +between the two sums is therefore +\[ +n(n + 1)(2n + 1) ÷ 6n^3 = \tfrac{1}{3}, +\] +since $n$ is infinite. From this he concludes that the pyramid +or cone is respectively $\frac{1}{3}$ of a prism or cylinder of equal base +and altitude, since the polygons or circles composing the former +decrease from the base to the apex in the same way as the +squares of the lines parallel to the base in a triangle decrease +from base to apex. By the Method of Indivisibles, Cavalieri +%% -----File: 185.png---Folio 171------- +solved the majority of the problems proposed by Kepler. +\index{Kepler!ref.\ to}% +Though expeditious and yielding correct results, Cavalieri's +method lacks a scientific foundation. If a line has absolutely +no width, then no number, however great, of lines can ever +make up an area; if a plane has no thickness whatever, then +even an infinite number of planes cannot form a solid. The +reason why this method led to correct conclusions is that one +area is to another area in the same ratio as the sum of the +series of lines in the one is to the sum of the series of lines in +the other. Though unscientific, Cavalieri's method was used +for fifty years as a sort of integral calculus. It yielded solutions +\index{Integral calculus}% +to some difficult problems. Guldin made a severe attack +\index{Guldin!ref.\ to}% +on Cavalieri and his method. The latter published in~1647, +after the death of Guldin, a treatise entitled \textit{Exercitationes +geometricæ sex}, in which he replied to the objections of his +opponent and attempted to give a clearer explanation of his +method. Guldin had never been able to demonstrate the +theorem named after him, except by metaphysical reasoning, +but Cavalieri proved it by the method of indivisibles. A +revised edition of the \textit{Geometry of Indivisibles} appeared in~1653. + +There is an important curve, not known to the ancients, +which now began to be studied with great zeal. Roberval +gave it the name of ``trochoid,'' Pascal the name of ``roulette,'' +\index{Roulette}% +\index{Trochoid}% +Galileo the name of ``cycloid.'' The invention of this curve +\index{Cycloid}% +\index{Galileo!ref.\ to}% +seems to be due to Galileo, who valued it for the graceful form +it would give to arches in architecture. He ascertained its +area by weighing paper figures of the cycloid against that of +the generating circle, and found thereby the first area to be +nearly but not exactly thrice the latter. A mathematical +determination was made by his pupil, \textbf{Evangelista Torricelli} +\index{Torricelli}% +(1608--1647), who is more widely known as a physicist than +as a mathematician. +%% -----File: 186.png---Folio 172------- + +By the Method of Indivisibles he demonstrated its area to +be triple that of the revolving circle, and published his +solution. This same quadrature had been effected a few years +earlier by Roberval in France, but his solution was not known +\index{Roberval}% +\index{Roberval!ref.\ to}% +to the Italians. Roberval, being a man of irritable and +violent disposition, unjustly accused the mild and amiable +Torricelli of stealing the proof. This accusation of plagiarism +created so much chagrin with Torricelli that it is considered +to have been the cause of his early death. \textbf{Vincenzo Viviani}, +\index{Viviani}% +another prominent pupil of Galileo, determined the tangent +to the cycloid. This was accomplished in France by Descartes +and Fermat. +\index{Fermat!ref.\ to}% + +In France, where geometry began to be cultivated with +greatest success, Roberval, Fermat, Pascal, employed the +Method of Indivisibles and made new improvements in it. +\textbf{Giles Persone de Roberval} (1602--1675), for forty years professor +of mathematics at the College of France in Paris, +claimed for himself the invention of the Method of Indivisibles. +Since his complete works were not published until +after his death, it is difficult to settle questions of priority. +Montucla and Chasles are of the opinion that he invented the +\index{Chasles!ref.\ to}% +\index{Montucla}% +method independent of and earlier than the Italian geometer, +though the work of the latter was published much earlier +than Roberval's. Marie finds it difficult to believe that the +\index{Marie, M.}% +Frenchman borrowed nothing whatever from the Italian, for +both could not have hit independently upon the word \textit{Indivisibles}, +which is applicable to infinitely small quantities, as +conceived by Cavalieri, but not as conceived by Roberval. +Roberval and Pascal improved the rational basis of the +Method of Indivisibles, by considering an area as made up +of an indefinite number of rectangles instead of lines, and +a solid as composed of indefinitely small solids instead of +surfaces. Roberval applied the method to the finding of +%% -----File: 187.png---Folio 173------- +\index{Indivisibles|)}% +areas, volumes, and centres of gravity. He effected the quadrature +of a parabola of any degree $y^m = a^{m-1} x$, and also of a +parabola $y^m = a^{m-n} x^n$. We have already mentioned his quadrature +of the cycloid. Roberval is best known for his method +\index{Cycloid}% +of drawing tangents. He was the first to apply motion to +\index{Tangents!in geometry}% +the resolution of this important problem. His method is +allied to Newton's principle of fluxions. Archimedes conceived +\index{Archimedes!ref.\ to}% +\index{Newton!ref.\ to}% +his spiral to be generated by a double motion. This +idea Roberval extended to all curves. Plane curves, as for +instance the conic sections, may be generated by a point +acted upon by two forces, and are the resultant of two +motions. If at any point of the curve the resultant be +resolved into its components, then the diagonal of the parallelogram +determined by them is the tangent to the curve +at that point. The greatest difficulty connected with this +ingenious method consisted in resolving the resultant into +components having the proper lengths and directions. Roberval +did not always succeed in doing this, yet his new idea was +a great step in advance. He broke off from the ancient +definition of a tangent as a straight line having only one point +in common with a curve,---a definition not valid for curves of +higher degrees, nor apt even in curves of the second degree to +bring out the properties of tangents and the parts they may +be made to play in the generation of the curves. The subject +of tangents received special attention also from Fermat, +\index{Fermat}% +\index{Fermat!ref.\ to}% +Descartes, and Barrow, and reached its highest development +\index{Barrow!ref.\ to}% +\index{Descartes!ref.\ to}% +after the invention of the differential calculus. Fermat and +Descartes defined tangents as secants whose two points of +intersection with the curve coincide; Barrow considered a +curve a polygon, and called one of its sides produced a tangent. + +A profound scholar in all branches of learning and a mathematician +of exceptional powers was \textbf{Pierre de Fermat} (1601--1665). +He studied law at Toulouse, and in 1631 was made +%% -----File: 188.png---Folio 174------- +\index{Midorge}% +councillor for the parliament of Toulouse. His leisure time +was mostly devoted to mathematics, which he studied with +irresistible passion. Unlike Descartes and Pascal, he led a +\index{Descartes!ref.\ to}% +quiet and unaggressive life. Fermat has left the impress of +his genius upon all branches of mathematics then known. A +great contribution to geometry was his \textit{De maximis et minimis}. +About twenty years earlier, Kepler had first observed that the +\index{Kepler!ref.\ to}% +increment of a variable, as, for instance, the ordinate of a +curve, is evanescent for values very near a maximum or a +minimum value of the variable. Developing this idea, Fermat +obtained his rule for maxima and minima. He substituted +\index{Maxima and minima}% +$x + e$ for $x$ in the given function of $x$ and then equated to each +other the two consecutive values of the function and divided +the equation by~$e$. If $e$ be taken~0, then the roots of this +equation are the values of~$x$, making the function a maximum +or a minimum. Fermat was in possession of this rule in~1629. +The main difference between it and the rule of the differential +\index{Differential calculus!alleged invention by Pascal}% +calculus is that it introduces the indefinite quantity~$e$ instead +of the infinitely small~$dx$. Fermat made it the basis for his +method of drawing tangents. + +Owing to a want of explicitness in statement, Fermat's +method of maxima and minima, and of tangents, was severely +attacked by his great contemporary, Descartes, who could +never be brought to render due justice to his merit. In the +ensuing dispute, Fermat found two zealous defenders in Roberval +and Pascal, the father; while Mydorge, Desargues, and +\index{Desargues!ref.\ to}% +Hardy supported Descartes. +\index{Hardy}% + +Since Fermat introduced the conception of infinitely small +differences between consecutive values of a function and arrived +at the principle for finding the maxima and minima, +it was maintained by Lagrange, Laplace, and Fourier, that +\index{Fourier!ref.\ to}% +\index{Lagrange!ref.\ to}% +\index{Laplace!ref.\ to}% +Fermat may be regarded as the first inventor of the differential +calculus. This point is not well taken, as will be seen +%% -----File: 189.png---Folio 175------- +from the words of Poisson, himself a Frenchman, who rightly +\index{Poisson!ref.\ to}% +says that the differential calculus ``consists in a system of +rules proper for finding the differentials of all functions, +rather than in the use which may be made of these infinitely +small variations in the solution of one or two isolated problems.'' + +A contemporary mathematician, whose genius excelled even +that of the great Fermat, was \textbf{Blaise Pascal} (1623--1662). He +\index{Pascal|(}% +was born at Clermont in Auvergne. In~1626 his father retired +to Paris, where he devoted himself to teaching his son, for he +would not trust his education to others. Blaise Pascal's genius +for geometry showed itself when he was but twelve years +old. His father was well skilled in mathematics, but did not +wish his son to study it until he was perfectly acquainted with +Latin and Greek. All mathematical books were hidden out of +his sight. The boy once asked his father what mathematics +treated of, and was answered, in general, ``that it was the +method of making figures with exactness, and of finding out +what proportions they relatively had to one another.'' He +was at the same time forbidden to talk any more about it, +or ever to think of it. But his genius could not submit to be +confined within these bounds. Starting with the bare fact +that mathematics taught the means of making figures infallibly +exact, he employed his thoughts about it and with a piece +of charcoal drew figures upon the tiles of the pavement, trying +the methods of drawing, for example, an exact circle or equilateral +triangle. He gave names of his own to these figures +and then formed axioms, and, in short, came to make perfect +demonstrations. In this way he arrived unaided at the theorem +that the sum of the three angles of a triangle is equal to +two right angles. His father caught him in the act of studying +this theorem, and was so astonished at the sublimity and +force of his genius as to weep for joy. The father now gave +%% -----File: 190.png---Folio 176------- +\index{Conic sections!more recent researches|(}% +him Euclid's \textit{Elements}, which he, without assistance, mastered +easily. His regular studies being languages, the boy employed +only his hours of amusement on the study of geometry, yet he +had so ready and lively a penetration that, at the age of sixteen, +he wrote a treatise upon conics, which passed for such a +surprising effort of genius, that it was said nothing equal to it +in strength had been produced since the time of Archimedes. +Descartes refused to believe that it was written by one so +young as Pascal. This treatise was never published, and is +now lost. Leibniz saw it in Paris and reported on a portion +\index{Leibniz!ref.\ to}% +of its contents. The precocious youth made vast progress in +all the sciences, but the constant application at so tender an +age greatly impaired his health. Yet he continued working, +and at nineteen invented his famous machine for performing +arithmetical operations mechanically. This continued strain +from overwork resulted in a permanent indisposition, and he +would sometimes say that from the time he was eighteen, +he never passed a day free from pain. At the age of +twenty-four he resolved to lay aside the study of the human +sciences and to consecrate his talents to religion. His Provincial +Letters against the Jesuits are celebrated. But at +times he returned to the favourite study of his youth. Being +kept awake one night by a toothache, some thoughts undesignedly +came into his head concerning the roulette or +cycloid; one idea followed another; and he thus discovered +\index{Cycloid}% +properties of this curve even to demonstration. A correspondence +between him and Fermat on certain problems was +the beginning of the theory of probability. Pascal's illness +increased, and he died at Paris at the early age of thirty-nine +years.\Reference{30} By him the answer to the objection to Cavalieri's +Method of Indivisibles was put in the clearest form. Like +\index{Indivisibles}% +Roberval, he explained ``the sum of right lines'' to mean ``the +sum of infinitely small rectangles.'' Pascal greatly advanced +%% -----File: 191.png---Folio 177------- +\index{Centre!of gravity}% +\index{Curves!quadrature of}% +\index{Involution of points}% +\index{Quadrature of curves}% +the knowledge of the cycloid. He determined the area of a +section produced by any line parallel to the base; the volume +generated by it revolving around its base or around the axis; +and, finally, the centres of gravity of these volumes, and also +of half these volumes cut by planes of symmetry. Before +publishing his results, he sent, in~1658, to all mathematicians +that famous challenge offering prizes for the first two solutions +of these problems. Only Wallis and A.~La~Louère competed +\index{Lalouere@{La Louère}}% +\index{Wallis!ref.\ to}% +for them. The latter was quite unequal to the task; +the former, being pressed for time, made numerous mistakes: +neither got a prize. Pascal then published his own solutions, +which produced a great sensation among scientific men. Wallis, +too, published his, with the errors corrected. Though not +competing for the prizes, Huygens, Wren, and Fermat solved +\index{Fermat!ref.\ to}% +\index{Huygens!ref.\ to}% +\index{Wren}% +some of the questions. The chief discoveries of \textbf{Christopher +Wren} (1632--1723), the celebrated architect of St.~Paul's +Cathedral in London, were the rectification of a cycloidal arc +\index{Rectification of curves}% +and the determination of its centre of gravity. Fermat +found the area generated by an arc of the cycloid. Huygens +invented the cycloidal pendulum. + +The beginning of the seventeenth century witnessed also +a revival of synthetic geometry. One who treated conics still +by ancient methods, but who succeeded in greatly simplifying +many prolix proofs of Apollonius, was \textbf{Claude Mydorge} in +\index{Mydorge}% +Paris (1585--1647), a friend of Descartes. But it remained +for \textbf{Girard Desargues} (1593--1662) of Lyons, and for Pascal, to +\index{Desargues}% +leave the beaten track and cut out fresh paths. They introduced +the important method of Perspective. All conics on +\index{Perspective}% +a cone with circular base appear circular to an eye at the apex. +Hence Desargues and Pascal conceived the treatment of the +\index{Pascal|)}% +conic sections as projections of circles. Two important and +beautiful theorems were given by Desargues: The one is on +the ``involution of the six points,'' in which a transversal +%% -----File: 192.png---Folio 178------- +\index{Brianchion}% +\index{Conic sections!more recent researches|)}% +\index{Hexagrammum mysticum}% +\index{Homological figures}% +meets a conic and an inscribed quadrangle; the other is that, +if the vertices of two triangles, situated either in space or in +a plane, lie on three lines meeting in a point, then their sides +meet in three points lying on a line; and conversely. This +last theorem has been employed in recent times by Brianchon, +Sturm, Gergonne, and Poncelet. Poncelet made it the basis +\index{Gergonne!ref.\ to}% +\index{Poncelet!ref.\ to}% +\index{Sturm, J.~C.~F.!ref.\ to}% +of his beautiful theory of homoligical figures. We owe to +Desargues the theory of involution and of transversals; also +the beautiful conception that the two extremities of a straight +line may be considered as meeting at infinity, and that parallels +\index{Infinity}% +differ from other pairs of lines only in having their points +of intersection at infinity. Pascal greatly admired Desargues' +\index{Pascal!ref.\ to}% +results, saying (in his \textit{Essais pour les Coniques}), ``I wish to +acknowledge that I owe the little that I have discovered on +this subject, to his writings.'' Pascal's and Desargues' writings +contained the fundamental ideas of modern synthetic +geometry. In Pascal's wonderful work on conics, written +at the age of sixteen and now lost, were given the theorem +on the anharmonic ratio, first found in Pappus, and also that +\index{Anharmonic ratio}% +\index{Pappus!ref.\ to}% +celebrated proposition on the mystic hexagon, known as +``Pascal's theorem,'' viz.~that the opposite sides of a hexagon +\index{Pascal's theorem}% +inscribed in a conic intersect in three points which are collinear. +This theorem formed the keystone to his theory. He +himself said that from this alone he deduced over 400~corollaries, +embracing the conics of Apollonius and many other +results. Thus the genius of Desargues and Pascal uncovered +several of the rich treasures of modern synthetic geometry; +but owing to the absorbing interest taken in the analytical +geometry of Descartes and later in the differential calculus, +the subject was almost entirely neglected until the present +century. + +In the theory of numbers no new results of scientific value +\index{Numbers!theory of numbers|(}% +\index{Theory of numbers|(}% +had been reached for over 1000~years, extending from the +%% -----File: 193.png---Folio 179------- +times of Diophantus and the Hindoos until the beginning of +\index{Diophantus!ref.\ to}% +the seventeenth century. But the illustrious period we are +now considering produced men who rescued this science from +the realm of mysticism and superstition, in which it had been +so long imprisoned; the properties of numbers began again +to be studied scientifically. Not being in possession of the +Hindoo indeterminate analysis, many beautiful results of the +Brahmins had to be re-discovered by the Europeans. Thus +a solution in integers of linear indeterminate equations was +re-discovered by the Frenchman \textbf{Bachet de Méziriac} (1581--1638), +\index{Meziriac@{Méziriac}}% +who was the earliest noteworthy European Diophantist. +In~1612 he published \textit{Problèmes plaisants et délectables qui se +font par les nombres}, and in 1621 a Greek edition of \textit{Diophantus} +with notes. The father of the modern theory of numbers is +\textbf{Fermat}. He was so uncommunicative in disposition, that he +\index{Fermat|(}% +generally concealed his methods and made known his results +only. In some cases later analysts have been greatly puzzled +in the attempt of supplying the proofs. Fermat owned a copy +of Bachet's \textit{Diophantus}, in which he entered numerous marginal +notes. In~1670 these notes were incorporated in a new +edition of \textit{Diophantus}, brought out by his son. Other theorems +on numbers, due to Fermat, were published in his \textit{Opera varia} +(edited by his son) and in Wallis's \textit{Commercium epistolicum} +\index{Wallis!ref.\ to}% +of~1658. Of the following theorems, the first seven are found +in the marginal notes:--- + +(1)~$x^n + y^n = z^n$ is impossible for integral values of $x$,~$y$, +and~$z$, when $n > 2$. Remark: ``I have found for this a truly +wonderful proof, but the margin is too small to hold it.'' +Repeatedly was this theorem made the prize question of +learned societies. It has given rise to investigations of +great interest and difficulty on the part of Euler, Lagrange, +\index{Euler!ref.\ to}% +\index{Lagrange!ref.\ to}% +Dirichlet, and Kummer. +\index{Dirichlet!ref.\ to}% +\index{Kummer!ref.\ to}% + +(2)~A prime of the form $4n + 1$ is only once the hypothenuse +\index{Prime numbers}% +%% -----File: 194.png---Folio 180------- +\index{Cube numbers}% +of a right triangle; its square is twice; its cube is three +times, etc. Example: $5^2 = 3^2 + 4^2$; $25^2 = 15^2 + 20^2 = 7^2 + 24^2$; +$125^2 = 75^2 + 100^2 = 35^2 +120^2 = 44^2 + 117^2$. + +(3)~A prime of the form $4n + 1$ can be expressed once, and +\index{Prime numbers}% +only once, as the sum of two squares. Proved by Euler. +\index{Euler!ref.\ to}% + +(4)~A number composed of two cubes can be resolved into +two other cubes in an infinite multiplicity of ways. + +(5)~Every number is either a triangular number or the sum +of two or three triangular numbers; either a square or the +\index{Numbers!triangular}% +sum of two, three, or four squares; either a pentagonal number +or the sum of two, three, four, or five pentagonal numbers; +similarly for polygonal numbers in general. The proof of this +and other theorems is promised by Fermat in a future work +which never appeared. This theorem is also given, with +others, in a letter of 1637(?) addressed to \textit{Pater Mersenne}. +\index{Mersenne}% + +(6)~As many numbers as you please may be found, such +that the square of each remains a square on the addition to +or subtraction from it of the sum of all the numbers. + +(7)~$x^4 + y^4 = z^2$ is impossible. + +(8)~In a letter of 1640 he gives the celebrated theorem +generally known as ``Fermat's theorem,'' which we state in +\index{Fermat's theorem}% +Gauss's notation: If $p$ is prime, and $a$ is prime to~$p$, then $a^{p-1} \equiv 1 \pmod{p}$. It was proved by Euler. + +(9)~Fermat died with the belief that he had found a long-sought-for +law of prime numbers in the formula $2^{2^n} + 1 =$ a +prime, but he admitted that he was unable to prove it rigorously. +The law is not true, as was pointed out by Euler in the +example $2^{2^5} + 1 = 4,294,967,297 = 6,700,417 \text{ times } 641$. The +American lightning calculator \textit{Zerah Colburn}, when a boy, +\index{Colburn, Z}% +readily found the factors, but was unable to explain the +method by which he made his marvellous mental computation. + +(10)~An odd prime number can be expressed as the difference +of two squares in one, and only one, way. This theorem, +%% -----File: 195.png---Folio 181------- +given in the \textit{Relation}, was used by Fermat for the decomposition +of large numbers into prime factors. + +(11)~If the integers $a$, $b$, $c$ represent the sides of a right +triangle, then its area cannot be a square number. This was +proved by Lagrange. + +(12)~Fermat's solution of $ax^2 + 1 = y^2$, where $a$ is integral +but not a square, has come down in only the broadest outline, +as given in the \textit{Relation}. He proposed the problem to the +Frenchman, \textit{Bernhard Frenicle de Bessy}, and in 1657 to all +\index{Bessy}% +living mathematicians. In England, Wallis and Lord Brounker +conjointly found a laborious solution, which was published in~1658, +and also in~1668, in an algebraical work brought out by +\textit{John Pell}. Though Pell had no other connection with the +\index{Pell}% +problem, it went by the name of ``Pell's problem.'' The first +\index{Pell's problem}% +solution was given by the Hindoos. + +We are not sure that Fermat subjected all his theorems to +rigorous proof. His methods of proof were entirely lost +until~1879, when a document was found buried among the +manuscripts of Huygens in the library of Leyden, entitled +\textit{Relation des découvertes en la science des nombres}. It appears +from it that he used an inductive method, called by him \textit{la +descente infinie ou indefinie}. He says that this was particularly +applicable in proving the impossibility of certain relations, as, +for instance, Theorem~11, given above, but that he succeeded +in using the method also in proving affirmative statements. +Thus he proved Theorem~3 by showing that if we suppose +there be a prime~$4n+1$ which does not possess this property, +then there will be a smaller prime of the form~$4n+1$ not +possessing it; and a third one smaller than the second, not +possessing it; and so on. Thus descending indefinitely, he +arrives at the number~$5$, which is the smallest prime factor +of the form $4n+1$. From the above supposition it would +follow that $5$ is not the sum of two squares---a conclusion +%% -----File: 196.png---Folio 182------- +\index{Mechanics!Stevin and Galileo}% +\index{Numbers!theory of numbers|)}% +\index{Theory of numbers|)}% +contrary to fact. Hence the supposition is false, and the +theorem is established. Fermat applied this method of descent +with success in a large number of theorems. By this +method Euler, Legendre, Dirichlet, proved several of his +enunciations and many other numerical propositions. + +A correspondence between \textit{Pascal} and \textit{Fermat} relating to a +\index{Pascal!ref.\ to}% +certain game of chance was the germ of the theory of probabilities, +which has since attained a vast growth. Chevalier de +Méré proposed to Pascal the fundamental problem, to determine +\index{Mere@{Méré}}% +the probability which each player has, at any given +\index{Probability}% +stage of the game, of winning the game. Pascal and Fermat +supposed that the players have equal chances of winning a +single point. + +The former communicated this problem to Fermat, who +\index{Fermat|)}% +studied it with lively interest and solved it by the theory of +combinations, a theory which was diligently studied both by +him and Pascal. The calculus of probabilities engaged the +attention also of Huygens. The most important theorem +\index{Huygens!ref.\ to}% +reached by him was that, if A has $p$~chances of winning a sum~$a$, +and $q$~chances of winning a sum~$b$, then he may expect to +win the sum $\dfrac{ap+bq}{p+q}$. The next great work on the theory of +probability was the \textit{Ars conjectandi} of Jakob Bernoulli. +\index{Bernoulli, James (born 1654)!ref.\ to}% + +Among the ancients, Archimedes was the only one who +\index{Archimedes!ref.\ to}% +attained clear and correct notions on theoretical statics. He +\index{Statics}% +had acquired firm possession of the idea of pressure, which +lies at the root of mechanical science. But his ideas slept +nearly twenty centuries, until the time of \textbf{Stevin} and \textbf{Galileo}. +\index{Galileo}% +\index{Stevin!ref.\ to}% +Stevin determined accurately the force necessary to sustain +a body on a plane inclined at any angle to the horizon. He +was in possession of a complete doctrine of equilibrium. While +Stevin investigated statics, Galileo pursued principally dynamics. +Galileo was the first to abandon the Aristotelian idea +that bodies descend more quickly in proportion as they are +%% -----File: 197.png---Folio 183------- +\index{Centrifugal force}% +\index{Laws of motion}% +heavier; he established the first law of motion; determined +\index{Motion, laws of}% +the laws of falling bodies; and, having obtained a clear notion +of acceleration and of the independence of different motions, +was able to prove that projectiles move in parabolic curves. +Up to his time it was believed that a cannon-ball moved +forward at first in a straight line and then suddenly fell +vertically to the ground. Galileo had an understanding of +\emph{centrifugal forces}, and gave a correct definition of \emph{momentum}. +Though he formulated the fundamental principle of statics, +known as the \emph{parallelogram of forces}, yet he did not fully +\index{Parallelogram of forces}% +recognise its scope. The principle of virtual velocities was +partly conceived by \textbf{Guido Ubaldo} (died 1607), and afterwards +\index{Ubaldo}% +more fully by Galileo. + +Galileo is the founder of the science of dynamics. Among +his contemporaries it was chiefly the novelties he detected in +the sky that made him celebrated, but Lagrange claims that +\index{Lagrange!ref.\ to}% +his astronomical discoveries required only a telescope and +perseverance, while it took an extraordinary genius to discover +laws from phenomena, which we see constantly and of +which the true explanation escaped all earlier philosophers. +The first contributor to the science of mechanics after Galileo +was Descartes. +\index{Descartes|(}% + +\Section{Descartes to Newton} + +Among the earliest thinkers of the seventeenth and eighteenth +centuries, who employed their mental powers toward the +destruction of old ideas and the up-building of new ones, ranks +\textbf{René Descartes} (1596--1650). Though he professed orthodoxy +in faith all his life, yet in science he was a profound sceptic. +He found that the world's brightest thinkers had been long +exercised in metaphysics, yet they had discovered nothing +%% -----File: 198.png---Folio 184------- +certain; nay, had even flatly contradicted each other. This +led him to the gigantic resolution of taking nothing whatever +on authority, but of subjecting everything to scrutinous examination, +according to new methods of inquiry. The certainty +of the conclusions in geometry and arithmetic brought out in +his mind the contrast between the true and false ways of +seeking the truth. He thereupon attempted to apply mathematical +reasoning to all sciences. ``Comparing the mysteries +of nature with the laws of mathematics, he dared to hope that +the secrets of both could be unlocked with the same key.'' +Thus he built up a system of philosophy called Cartesianism. + +Great as was Descartes' celebrity as a metaphysician, it +may be fairly questioned whether his claim to be remembered +by posterity as a mathematician is not greater. His philosophy +has long since been superseded by other systems, but the analytical +geometry of Descartes will remain a valuable possession +forever. At the age of twenty-one, Descartes enlisted in the +army of Prince Maurice of Orange. His years of soldiering +were years of leisure, in which he had time to pursue his +studies. At that time mathematics was his favourite science. +But in 1625 he ceased to devote himself to pure mathematics. +Sir William Hamilton is in error when he states that Descartes +\index{Hamilton, W.}% +considered mathematical studies absolutely pernicious as a +means of internal culture. In a letter to Mersenne, Descartes +says: ``M.~Desargues puts me under obligations on account of +\index{Desargues!ref.\ to}% +the pains that it has pleased him to have in me, in that he +shows that he is sorry that I do not wish to study more in geometry, +but I have resolved to quit only abstract geometry, that +is to say, the consideration of questions which \textit{serve only to +exercise the mind}, and this, in order to study another kind of +geometry, which has for its object the explanation of the +phenomena of nature\ldots. You know that all my physics is +nothing else than geometry.'' The years between 1629 and +%% -----File: 199.png---Folio 185------- +\index{Analytic geometry|(}% +\index{Co-ordinates}% +\index{Debaune@{De Baune}!ref.\ to}% +1649 were passed by him in Holland in the study, principally, +of physics and metaphysics. His residence in Holland was +during the most brilliant days of the Dutch state. In 1637 he +published his \textit{Discours de la Méthode}, containing among others +an essay of 106~pages on geometry. His \textit{Geometry} is not easy +reading. An edition appeared subsequently with notes by his +friend \textit{De Beaune}, which were intended to remove the difficulties. + +It is frequently stated that Descartes was the first to apply +algebra to geometry. This statement is inaccurate, for Vieta +and others had done this before him. Even the Arabs sometimes +used algebra in connection with geometry. The new +step that Descartes did take was the introduction into geometry +of an analytical method based on the notion of variables +and constants, which enabled him to represent curves by algebraic +equations. In the Greek geometry, the idea of motion +was wanting, but with Descartes it became a very fruitful +conception. By him a point on a plane was determined in +position by its distances from two fixed right lines or axes. +These distances varied with every change of position in +the point. This geometric idea of \emph{co-ordinate representation}, +together with the algebraic idea of \emph{two variables in one equation} +having an indefinite number of simultaneous values, furnished +a method for the study of loci, which is admirable for +the generality of its solutions. Thus the entire conic sections +of Apollonius is wrapped up and contained in a single equation +of the second degree. + +The Latin term for ``ordinate'' used by Descartes comes +from the expression \textit{lineæ ordinatæ}, employed by Roman surveyors +for parallel lines. The term \emph{abscissa} occurs for the +first time in a Latin work of 1659, written by \textit{Stefano degli +Angeli} (1623--1697), a professor of mathematics in Rome.\Reference{3} +\index{Angeli}% +Descartes' geometry was called ``analytical geometry,'' partly +%% -----File: 200.png---Folio 186------- +because, unlike the synthetic geometry of the ancients, it is +\index{Geometry!analytic|(}% +actually \emph{analytical}, in the sense that the word is used in +logic; and partly because the practice had then already +arisen, of designating by the term \emph{analysis} the calculus with +\index{Analysis!Descartes'}% +general quantities. + +The first important example solved by Descartes in his +geometry is the ``problem of Pappus''; viz.~``Given several +\index{Pappus!ref.\ to}% +straight lines in a plane, to find the locus of a point such that +the perpendiculars, or more generally, straight lines at given +angles, drawn from the point to the given lines, shall satisfy +the condition that the product of certain of them shall be in +a given ratio to the product of the rest.'' Of this celebrated +problem, the Greeks solved only the special case when the +number of given lines is four, in which case the locus of the +point turns out to be a conic section. By Descartes it was +solved completely, and it afforded an excellent example of +the use which can be made of his analytical method in the +study of loci. Another solution was given later by Newton +\index{Newton!ref.\ to}% +in the \textit{Principia}. + +The methods of drawing tangents invented by Roberval +\index{Tangents!in geometry}% +and Fermat were noticed earlier. Descartes gave a third +method. Of all the problems which he solved by his geometry, +none gave him as great pleasure as his mode of constructing +tangents. It is profound but operose, and, on that account, +inferior to Fermat's. His solution rests on the method of +\emph{Indeterminate Coefficients}, of which he bears the honour of +\index{Indeterminate coefficients}% +invention. Indeterminate coefficients were employed by him +also in solving \Change{bi-quadratic}{biquadratic} equations. +\index{Equations!solution of}% + +The essays of Descartes on dioptrics and geometry were +sharply criticised by Fermat, who wrote objections to the +former, and sent his own treatise on ``maxima and minima'' +\index{Maxima and minima}% +to show that there were omissions in the geometry. Descartes +thereupon made an attack on Fermat's method of tangents. +%% -----File: 201.png---Folio 187------- +Descartes was in the wrong in this attack, yet he continued +\index{Descartes!rule of signs}% +the controversy with obstinacy. He had a controversy also +with Roberval on the cycloid. This curve has been called +\index{Cycloid}% +\index{Roberval!ref.\ to}% +the ``Helen of geometers,'' on account of its beautiful properties +\index{Helen of geometers}% +and the controversies which their discovery occasioned. +Its quadrature by Roberval was generally considered a brilliant +achievement, but Descartes commented on it by saying +that any one moderately well versed in geometry might have +done this. He then sent a short demonstration of his own. +On Roberval's intimating that he had been assisted by a +knowledge of the solution, Descartes constructed the tangent +to the curve, and challenged Roberval and Fermat to do the +same. Fermat accomplished it, but Roberval never succeeded +in solving this problem, which had cost the genius of Descartes +but a moderate degree of attention. + +He studied some new curves, now called ``ovals of Descartes,'' +\index{Ovals of Descartes}% +which were intended by him to serve in the construction +of converging lenses, but which yielded no results +of practical value. + +The application of algebra to the doctrine of curved lines +\index{Algebra!seventeenth century}% +reacted favourably upon algebra. As an abstract science, +Descartes improved it by the systematic use of exponents and +\index{Exponents}% +by the full interpretation and construction of negative quantities. +\index{Negative quantities}% +Descartes also established some theorems on the theory of +equations. Celebrated is his ``rule of signs'' for determining +\index{Rule of signs}% +\index{Signs, rule of}% +the number of positive and negative roots; viz.~an equation +may have as many $+$~roots as there are variations of signs, and +as many $-$~roots as there are permanencies of signs. Descartes +was charged by Wallis with availing himself, without acknowledgment, +\index{Wallis!ref.\ to}% +of Harriot's theory of equations, particularly his mode +\index{Harriot!ref.\ to}% +of generating equations; but there seems to be no good ground +for the charge. Wallis also claimed that Descartes failed to +observe that the above rule of signs is not true whenever the +%% -----File: 202.png---Folio 188------- +\index{Apollonian Problem}% +equation has imaginary roots; but Descartes does not say that +the equation \emph{always has}, but that it \emph{may have} so many roots. +It is true that Descartes does not consider the case of imaginaries +directly, but further on in his \textit{Geometry} he gives +incontestable evidence of being able to handle this case +also. + +In mechanics, Descartes can hardly be said to have advanced +\index{Mechanics!Descartes, Wallis, Wren, Huygens, Newton}% +beyond Galileo. The latter had overthrown the ideas of +\index{Galileo!ref.\ to}% +Aristotle on this subject, and Descartes simply ``threw himself +upon the enemy'' that had already been ``put to the rout.'' +His statement of the first and second laws of motion was an +\index{Laws of motion}% +\index{Motion, laws of}% +improvement in form, but his third law is false in substance. +The motions of bodies in their direct impact was imperfectly +understood by Galileo, erroneously given by Descartes, and +first correctly stated by Wren, Wallis, and Huygens. +\index{Huygens!ref.\ to}% +\index{Wallis!ref.\ to}% +\index{Wren!ref.\ to}% + +One of the most devoted pupils of Descartes was the learned +\textit{Princess Elizabeth}, daughter of Frederick~V\@. She applied the +\index{Elizabeth, Princess}% +\index{Princess Elizabeth}% +new analytical geometry to the solution of the ``Apollonian +problem.'' His second royal follower was \textit{Queen Christina}, +the daughter of Gustavus Adolphus. She urged upon Descartes +to come to the Swedish court. After much hesitation +he accepted the invitation in~1649. He died at Stockholm one +year later. His life had been one long warfare against the +prejudices of men. + +It is most remarkable that the mathematics and philosophy +of Descartes should at first have been appreciated less by his +countrymen than by foreigners. The indiscreet temper of +Descartes alienated the great contemporary French mathematicians, +Roberval, Fermat, Pascal. They continued in investigations +of their own, and on some points strongly opposed +Descartes. The universities of France were under strict +ecclesiastical control and did nothing to introduce his mathematics +and philosophy. It was in the youthful universities of +%% -----File: 203.png---Folio 189------- +\index{Analytic geometry|)}% +\index{Debaune@{De Baune}}% +\index{Inverse tangents (problem of)}% +\index{Maxima and minima}% +Holland that the effect of Cartesian teachings was most +immediate and strongest. + +The only prominent Frenchman who immediately followed +in the footsteps of the great master was \textbf{De Beaune} (1601--1652). +He was one of the first to point out that the properties of a +curve can be deduced from the properties of its tangent. This +mode of inquiry has been called the \emph{inverse method of tangents}. +\index{Tangents!inverse problem of}% +He contributed to the theory of equations by considering for +\index{Equations!theory of}% +the first time the upper and lower limits of the roots of +numerical equations. + +In the Netherlands a large number of distinguished mathematicians +were at once struck with admiration for the Cartesian +geometry. Foremost among these are \textit{van Schooten}, \textit{John de +\index{Schooten, van}% +\index{Vanschooten@{Van Schooten}}% +Witt}, \textit{van Heuraet}, \textit{Sluze}, and \textit{Hudde}. \textbf{Van Schooten} (died 1660), +\index{Hudde}% +\index{Sluze}% +professor of mathematics at Leyden, brought out an edition +of Descartes' geometry, together with the notes thereon by +\index{Descartes!ref.\ to}% +De~Beaune. His chief work is his \textit{Exercitationes Mathematicæ}, +in which he applies the analytical geometry to the solution of +many interesting and difficult problems. The noble-hearted +\textbf{Johann de Witt}, grand-pensioner of Holland, celebrated as a +statesman and for his tragical end, was an ardent geometrician. +He conceived a new and ingenious way of generating conics, +which is essentially the same as that by projective pencils of +rays in modern synthetic geometry. He treated the subject +not synthetically, but with aid of the Cartesian analysis. +\textbf{René François de Sluze} (1622--1685) and \textbf{Johann Hudde} (1633--1704) +made some improvements on Descartes' and Fermat's +\index{Descartes|)}% +methods of drawing tangents, and on the theory of maxima and +minima. With Hudde, we find the first use of three variables +in analytical geometry. He is the author of an ingenious rule +\index{Geometry!analytic|)}% +for finding equal roots. We illustrate it by the equation +$x^3 - x^2 - 8x + 12 = 0$. Taking an arithmetical progression +3,~2, 1,~0, of which the highest term is equal to the degree of +%% -----File: 204.png---Folio 190------- +\index{Circle-squarers}% +\index{Quadrature of curves}% +the equation, and multiplying each term of the equation respectively +by the corresponding term of the progression, we get +$3x^3-2x^2-8x=0$, or $3x^2-2x-8=0$. This last equation +is by one degree lower than the original one. Find the G.C.D. +of the two equations. This is $x - 2$; hence 2~is one of the two +equal roots. Had there been no common divisor, then the +original equation would not have possessed equal roots. Hudde +gave a demonstration for this rule.\Reference{24} + +\textbf{Heinrich van Heuraet} must be mentioned as one of the earliest +\index{Heuraet}% +geometers who occupied themselves with success in the +rectification of curves. He observed in a general way that the +\index{Curves!quadrature of}% +\index{Rectification of curves}% +two problems of quadrature and of rectification are really +identical, and that the one can be reduced to the other. Thus +he carried the rectification of the hyperbola back to the +quadrature of the hyperbola. The semi-cubical parabola +\index{Parabola!semi-cubical}% +\index{Semi-cubical parabola}% +$y^3 = ax^2$ was the first curve that was ever rectified absolutely. +This appears to have been accomplished independently by Van +Heuraet in Holland and by \textbf{William Neil} (1637--1670) in England. +\index{Neil}% +According to Wallis the priority belongs to Neil. Soon +after, the cycloid was rectified by Wren and Fermat. +\index{Cycloid}% + +The prince of philosophers in Holland, and one of the +greatest scientists of the seventeenth century, was \textbf{Christian +Huygens} (1629--1695), a native of the\DPnote{** Uncapitalized in original} Hague. Eminent as a +\index{Huygens|(}% +physicist and astronomer, as well as mathematician, he was +a worthy predecessor of Sir Isaac Newton. He studied at +Leyden under the younger \textit{Van Schooten}. The perusal of +\index{Schooten, van!ref.\ to}% +\index{Vanschooten@{Van Schooten}!ref.\ to}% +some of his earliest theorems led Descartes to predict his +future greatness. In 1651 Huygens wrote a treatise in which +he pointed out the fallacies of Gregory St.~Vincent (1584--1667) +\index{Stvincent@{St.\ Vincent, Gregory}}% +\index{Vincent, Gregory St.}% +on the subject of quadratures. He himself gave a remarkably +close and convenient approximation to the length of a circular +arc. In 1660 and 1663 he went to Paris and to London. In +1666 he was appointed by Louis~XIV. member of the French +%% -----File: 205.png---Folio 191------- +\index{Analytic geometry}% +\index{Centre!of gravity}% +\index{Centre!of oscillation}% +\index{Mechanics!Descartes, Wallis, Wren, Huygens, Newton}% +Academy of Sciences. He was induced to remain in Paris +from that time until~1681, when he returned to his native +city, partly for consideration of his health and partly on +account of the revocation of the Edict of Nantes. + +The majority of his profound discoveries were made with +aid of the ancient geometry, though at times he used the +\index{Geometry!analytic}% +geometry of Descartes or of Cavalieri and Fermat. Thus, +\index{Descartes!ref.\ to}% +like his illustrious friend, Sir Isaac Newton, he always showed +\index{Newton!ref.\ to}% +partiality for the Greek geometry. Newton and Huygens +were kindred minds, and had the greatest admiration for each +other. Newton always speaks of him as the ``Summus +Hugenius.'' + +To the two curves (cubical parabola and cycloid) previously +\index{Cycloid}% +rectified he added a third,---the cissoid. He solved the +\index{Cissoid}% +problem of the catenary, determined the surface of the +\index{Catenary}% +parabolic and hyperbolic conoid, and discovered the properties +of the logarithmic curve and the solids generated by it. +Huygens' \textit{De horologio oscillatorio} (Paris, 1673) is a work that +ranks second only to the \textit{Principia} of Newton and constitutes +\index{Newton's Principia}% +\index{Principia (Newton's)}% +historically a necessary introduction to it.\Reference{13} The book opens +with a description of pendulum clocks, of which Huygens +\index{Pendulum}% +is the inventor. Then follows a treatment of accelerated +motion of bodies falling free, or sliding on inclined planes, or +on given curves,---culminating in the brilliant discovery that +the cycloid is the tautochronous curve. To the theory of +\index{Tautochronous curve}% +curves he added the important theory of ``evolutes.'' After +\index{Evolutes}% +explaining that the tangent of the evolute is normal to the +involute, he applied the theory to the cycloid, and showed by +simple reasoning that the evolute of this curve is an equal +cycloid. Then comes the complete general discussion of the +centre of oscillation. This subject had been proposed for +\index{Oscillation, centre of}% +investigation by Mersenne and discussed by Descartes and +\index{Mersenne}% +Roberval. In Huygens' assumption that the common centre +\index{Roberval!ref.\ to}% +%% -----File: 206.png---Folio 192------- +\index{Mechanics!Descartes, Wallis, Wren, Huygens, Newton}% +\index{Saturn's rings}% +\index{Undulatory theory of light}% +of gravity of a group of bodies, oscillating about a horizontal +axis, rises to its original height, but no higher, is expressed +for the first time one of the most beautiful principles of +dynamics, afterwards called the principle of the conservation +\index{Conservation!of \textit{vis viva}}% +of \textit{vis viva}.\Reference{32} The thirteen theorems at the close of the work +relate to the theory of centrifugal force in circular motion. +\index{Centrifugal force}% +This theory aided Newton in discovering the law of gravitation. +\index{Newton!ref.\ to}% + +Huygens wrote the first formal treatise on probability. He +\index{Probability}% +proposed the wave-theory of light and with great skill applied +geometry to its development. This theory was long neglected, +but was revived and successfully worked out by Young and +Fresnel a century later. Huygens and his brother improved +the telescope by devising a better way of grinding and polishing +lenses. With more efficient instruments he determined +the nature of Saturn's appendage and solved other astronomical +questions. Huygens' \textit{Opuscula posthuma} appeared +\index{Huygens|)}% +in~1703. + +Passing now from Holland to England, we meet there one +of the most original mathematicians of his day---\textbf{John Wallis} +\index{Wallis|(}% +(1616--1703). He was educated for the Church at Cambridge +and entered Holy Orders. But his genius was employed +chiefly in the study of mathematics. In 1649 he was appointed +Savilian professor of geometry at Oxford. He was one of +the original members of the Royal Society, which was founded +in~1663. Wallis thoroughly grasped the mathematical methods +both of Cavalieri and Descartes. His \textit{Conic Sections} is the +\index{Conic sections!more recent researches}% +\index{Descartes!ref.\ to}% +earliest work in which these curves are no longer considered +as sections of a cone, but as curves of the second degree, and +are treated analytically by the Cartesian method of co-ordinates. +In this work Wallis speaks of Descartes in the +highest terms, but in his \textit{Algebra} he, without good reason, +\index{Algebra!seventeenth century}% +accuses Descartes of plagiarising from Harriot. We have +\index{Harriot!ref.\ to}% +%% -----File: 207.png---Folio 193------- +\index{Analytic geometry}% +\index{Curves!quadrature of}% +\index{Descartes!rule of signs}% +\index{Equations!theory of}% +\index{Geometry!analytic}% +\index{Quadrature of curves}% +\index{Rule of signs}% +\index{Signs, rule of}% +already mentioned elsewhere Wallis's solution of the prize +questions on the cycloid, which were proposed by Pascal. + +The \textit{Arithmetic of Infinites}, published in 1655, is his greatest +work. By the application of analysis to the Method of Indivisibles, +\index{Indivisibles}% +he greatly increased the power of this instrument for +effecting quadratures. He advanced beyond Kepler by making +more extended use of the ``law of continuity'' and placing +\index{Continuity}% +full reliance in it. By this law he was led to regard the +denominators of fractions as powers with negative exponents. +Thus, the descending geometrical progression $x^3$,~$x^2$, $x^1$, $x^0$, if +continued, gives $x^{-1}$,~$x^{-2}$, $x^{-3}$, etc.; which is the same thing +as $\dfrac{1\Fstr}{x}$,~$\dfrac{1}{x^2}$,~$\dfrac{1}{x^3}$. The exponents of this geometric series are in +continued arithmetical progression, $3$,~$2$, $1$, $0$, $-1$, $-2$,~$-3$. +He also used fractional exponents, which, like the negative, +had been invented long before, but had failed to be generally +introduced. The symbol~$\infty$ for infinity is due to him. +\index{Infinity}% +\index{Infinity!symbol for}% + +Cavalieri and the French geometers had ascertained the +\index{Cavalieri!ref.\ to}% +formula for squaring the parabola of any degree, $y = x^m$, $m$~being +a positive integer. By the summation of the powers +of the terms of infinite arithmetical series, it was found that +the curve $y = x^m$ is to the area of the parallelogram having +the same base and altitude as $1$ is to $m + 1$. Aided by the law +of continuity, Wallis arrived at the result that this formula +holds true not only when $m$ is positive and integral, but also +when it is fractional or negative. Thus, in the parabola +$y = \sqrt{px\mathstrut}$, $m = \frac{1}{2}$; hence the area of the parabolic segment +is to that of the circumscribed rectangle as $1 : 1\frac{1}{2}$, or as $2 : 3$. +Again, suppose that in $y = x^m$, $m = -\frac{1}{2}$; then the curve is a +kind of hyperbola referred to its asymptotes, and the hyperbolic +space between the curve and its asymptotes is to the +corresponding parallelogram as $1 : \frac{1}{2}$. If $m = -1$, as in the +common equilateral hyperbola $y = x^{-1}$ or $xy = 1$, then this +ratio is $1 : -1 + 1$, or $1 : 0$, showing that its asymptotic space +%% -----File: 208.png---Folio 194------- +is infinite. But in the case when $m$~is greater than unity and +negative, Wallis was unable to interpret correctly his results. +For example, if $m = -3$, then the ratio becomes $1 : -2$, or +as unity to a negative number. What is the meaning of this? +Wallis reasoned thus: If the denominator is only zero, then +the area is already infinite; but if it is less than zero, then the +area must be more than infinite. It was pointed out later by +Varignon, that this space, supposed to exceed infinity, is really +finite, but taken negatively; that is, measured in a contrary +direction.\Reference{31} The method of Wallis was easily extended to +cases such as $y = ax^\frac{m}{n} + bx^\frac{p}{q}$ by performing the quadrature +for each term separately, and then adding the results. + +The manner in which Wallis studied the quadrature of the +circle and arrived at his expression for the value of $\pi$ is +\index{Circle}% +\index{p@{$\pi$}: values for!Wallis'}% +extraordinary. He found that the areas comprised between +the axes, the ordinate corresponding to $x$, and the curves +represented by the equations $y = (1 - x^2)^0$, $y = (1 - x^2)^1$, $y = +(1 - x^2)^2$, $y = (1 - x^2)^3$, etc., are expressed in functions of the +circumscribed rectangles having $x$ and $y$ for their sides, by the +quantities forming the series +\begin{align*} +&x,\\ +&x - \tfrac{1}{3}x^3,\\ +&x - \tfrac{2}{3}x^3 + \tfrac{1}{5}x^5,\\ +&x - \tfrac{3}{3}x^3 + \tfrac{3}{5}x^5 - \tfrac{1}{7}x^7, etc. +\end{align*} +When $x = 1$, these values become respectively $1$,~$\frac{2}{3}$, $\frac{8}{15}$, $\frac{48}{105}$, +etc. Now since the ordinate of the circle is $y = (1 - x^2)^\frac{1}{2}$, the +exponent of which is $\frac{1}{2}$ or the mean value between $0$ and $1$, the +question of this quadrature reduced itself to this: If $0$,~$1$, $2$, +$3$, etc., operated upon by a certain law, give $1$,~$\frac{2}{3}$, $\frac{8}{15}$, $\frac{48}{105}$, +what will $\frac{1}{2}$ give, when operated upon by the same law? He +attempted to solve this by \emph{interpolation}, a method first brought +\index{Interpolation}% +into prominence by him, and arrived by a highly complicated +%% -----File: 209.png---Folio 195------- +\index{Binomial formula}% +\index{p@{$\pi$}: values for!Wallis'}% +\index{Wallis|)}% +and difficult analysis at the following very remarkable expression: +\[ +\frac{\pi}{2} = \frac{2 · 2 · 4 · 4 · 6 · 6 · 8 · 8 \dotsm} + {1 · 3 · 3 · 5 · 5 · 7 · 7 · 9 \dotsm} +\] + +He did not succeed in making the interpolation itself, +because he did not employ literal or general exponents, and +could not conceive a series with more than one term and less +than two, which it seemed to him the interpolated series must +have. The consideration of this difficulty led Newton to the +\index{Newton!ref.\ to}% +discovery of the Binomial Theorem. This is the best place to +\index{Newton's discovery of binomial theorem}% +speak of that discovery. Newton virtually assumed that the +same conditions which underlie the general expressions for +the areas given above must also hold for the expression to be +interpolated. In the first place, he observed that in each +expression the first term is~$x$, that $x$~increases in odd powers, +that the signs alternate $+$~and~$-$, and that the second terms +$\frac{0}{3}x^3$,~$\frac{1}{3}x^3$, $\frac{2}{3}x^3$, $\frac{3}{3}x^3$, are in arithmetical progression. Hence the +first two terms of the interpolated series must be $x - \dfrac{\frac{1}{2}x^3}{3}$. +He next considered that the denominators $1$,~$3$, $5$, $7$, etc., are in +arithmetical progression, and that the coefficients in the numerators +in each expression are the digits of some power of +the number~$11$; namely, for the first expression, $11^0$ or~$1$; for +the second, $11^1$ or $1$,~$1$; for the third, $11^2$ or $1$, $2$,~$1$; for the +fourth, $11^3$ or $1$, $3$, $3$,~$1$; etc. He then discovered that, having +given the second digit (call it~$m$), the remaining digits can +be found by continual multiplication of the terms of the series +$\dfrac{m - 0}{1} · \dfrac{m - 1}{2} · \dfrac{m - 2}{3} · \dfrac{m - 3}{4} ·$ etc. Thus, if $m = 4$, then +$4 · \smash[b]{\dfrac{m - 1}{2}}$ gives~$6$; $6 · \smash[b]{\dfrac{m - 2}{3}}$ gives~$4$; $4 · \smash[b]{\dfrac{m - 3}{4}}$ gives~$1$. Applying +this rule to the required series, since the second term +is $\dfrac{\frac{1}{2}x^3\Fstr[12pt]}{3}$, we have $m = \frac{1}{2}$, and then get for the succeeding coefficients +%% -----File: 210.png---Folio 196------- +\index{Pacioli!ref.\ to}% +in the numerators respectively $-\frac{1}{8}\Fstr$, $+\frac{1}{16}$, $-\frac{5}{128}$, +etc.; hence the required area for the circular segment is +$x - \dfrac{\frac{1}{2}x^3\Fstr[12pt]}{3} - \dfrac{\frac{1}{8}x^5}{5} - \dfrac{\frac{1}{16}x^7}{7} -{}$ etc. Thus he found the interpolated +expression to be an infinite series, instead of one having more +than one term and less than two, as Wallis believed it must +be. This interpolation suggested to Newton a mode of expanding +$(1 - x^2)^\frac{1}{2}$, or, more generally, $(1-x^2)^m$, into a series. +He observed that he had only to omit from the expression +just found the denominators $1$, $3$, $5$, $7$, etc., and to lower +each power of $x$ by unity, and he had the desired expression. +In a letter to Oldenburg (June~13, 1676), Newton states the +theorem as follows: The extraction of roots is much shortened +by the theorem +\index{Binomial formula}% +\[ +(P + PQ)^\frac{m}{n} = P^\frac{m}{n} + \frac{m}{n} AQ + \frac{m-n}{2n} BQ + \frac{m-2n}{3n} CQ + \text{etc.}, +\] +where $A$ means the first term, $P^\frac{m}{n}$, $B$ the second term, $C$ the +third term, etc. He verified it by actual multiplication, but +gave no regular proof of it. He gave it for any exponent +whatever, but made no distinction between the case when the +exponent is positive and integral, and the others. + +It should here be mentioned that very rude beginnings of +the binomial theorem are found very early. The Hindoos and +\index{Newton's discovery of binomial theorem}% +Arabs used the expansions of $(a + b)^2$ and $(a + b)^3$ for extracting +roots; Vieta knew the expansion of $(a + b)^4$; but these +\index{Vieta!ref.\ to}% +were the results of simple multiplication without the discovery +of any law. The binomial coefficients for positive whole exponents +were known to some Arabic and European mathematicians. +Pascal derived the coefficients from the method of +\index{Pascal!ref.\ to}% +what is called the ``arithmetical triangle.'' Lucas de Burgo, +\index{Arithmetical triangle}% +Stifel, Stevinus, Briggs, and others, all possessed something +from which one would think the binomial theorem could have +%% -----File: 211.png---Folio 197------- +\index{Infinite series}% +been gotten with a little attention, ``if we did not know that +such simple relations were difficult to discover.'' + +Though Wallis had obtained an entirely new expression for~$\pi$, +\index{p@{$\pi$}: values for!Brouncker's}% +\index{Wallis!ref.\ to}% +he was not satisfied with it; for instead of a finite number +of terms yielding an absolute value, it contained merely an +infinite number, approaching nearer and nearer to that value. +He therefore induced his friend, \textbf{Lord Brouncker} (1620?-1684), +\index{Brouncker}% +the first president of the Royal Society, to investigate this +subject. Of course Lord Brouncker did not find what they +were after, but he obtained the following beautiful equality:--- +\[ +%[** PP: \cfrac leaves a bit too much vertical space] +\pi = \dfrac{4}{1 + + \dfrac{1\Fstr[10pt]}{2 + + \dfrac{9\Fstr[10pt]}{2 + + \dfrac{25\Fstr[10pt]}{2 + + \dfrac{49\Fstr[10pt]}{2 + \text{etc.}\Fstr[10pt]} + } + } + } +} +\] +Continued fractions, both ascending and descending, appear to +\index{Continued fractions}% +\index{Fractions!continued}% +have been known already to the Greeks and Hindoos, though +not in our present notation. Brouncker's expression gave +birth to the theory of continued fractions. + +Wallis' method of quadratures was diligently studied by +his disciples. Lord Brouncker obtained the first infinite +series for the area of an equilateral hyperbola between its +asymptotes. \textbf{Nicolaus Mercator} of Holstein, who had settled +\index{Mercator, N.}% +in England, gave, in his \textit{Logarithmotechnia} (London, 1668), a +similar series. He started with the grand property of the +equilateral hyperbola, discovered in 1647 by \textit{Gregory St.~Vincent}, +\index{Stvincent@{St.\ Vincent, Gregory}}% +\index{Vincent, Gregory St.}% +which connected the hyperbolic space between the +asymptotes with the natural logarithms and led to these +\index{Logarithms}% +logarithms being called hyperbolic. By it Mercator arrived +at the logarithmic series, which Wallis had attempted but +\index{Logarithmic series}% +failed to obtain. He showed how the construction of logarithmic +%% -----File: 212.png---Folio 198------- +\index{Prime and ultimate ratios}% +\index{Rectification of curves}% +tables could be reduced to the quadrature of hyperbolic +spaces. Following up some suggestions of Wallis, \textit{William +Neil} succeeded in rectifying the cubical parabola, and \textit{Wren} in +\index{Neil!ref.\ to}% +\index{Parabola}% +\index{Wren!ref.\ to}% +rectifying any cycloidal arc. + +A prominent English mathematician and contemporary of +Wallis was \textbf{Isaac Barrow} (1630-1677). He was professor of +\index{Barrow}% +mathematics in London, and then in Cambridge, but in 1669 +he resigned his chair to his illustrious pupil, Isaac Newton, +and renounced the study of mathematics for that of divinity. +As a mathematician, he is most celebrated for his method of +tangents. He simplified the method of Fermat by introducing +\index{Fermat!ref.\ to}% +\index{Tangents!direct problem of}% +two infinitesimals instead of one, and approximated to the +course of reasoning afterwards followed by Newton in his doctrine +on Ultimate Ratios. + +He considered the infinitesimal right triangle $ABB'$ having +%[** PP: Width-dependent line break] +%[Illustration] +\begin{wrapfigure}[8]{l}{2.125in} +\Graphic{2.125in}{212a} +\end{wrapfigure} +for its sides the difference between two successive ordinates, +the distance between them, and the portion of the curve intercepted +by them. This triangle is similar to $BPT$, formed by +the ordinate, the tangent, and the sub-tangent. Hence, if we +know the ratio of $B'A$ to $BA$, then we know the ratio of the +ordinate and the sub-tangent, and the tangent can be constructed +at once. For any curve, say $y^2 =px$, the ratio of $B'A$ +to $BA$ is determined from its +equation as follows: If $x$ receives +an infinitesimal increment +$PP'= e$, then $y$ receives +an increment $B'A = a$, and the +equation for the ordinate $B'P'$ +becomes $y^2+2ay+a^2=px+pe$. +Since $y^2 =px$, we get $2ay + a^2 = pe$; neglecting higher powers +of the infinitesimals, we have $2ay = pe$, which gives +\[ +a:e=p:2y=p:2\sqrt{px}. +\] +%% -----File: 213.png---Folio 199------- +But $a : e = \text{the ordinate} : \text{the sub-tangent}$; hence +\[ +p : 2\sqrt{px} = \sqrt{px} : \text{sub-tangent}, +\] +giving $2x$ for the value of the sub-tangent. This method differs +from that of the differential calculus only in notation.\Reference{31} + +\Section{Newton to Euler} + +It has been seen that in France prodigious scientific progress +was made during the beginning and middle of the seventeenth +century. The toleration which marked the reign of Henry IV. +and Louis~XIII. was accompanied by intense intellectual +activity. Extraordinary confidence came to be placed in the +power of the human mind. The bold intellectual conquests +of Descartes, Fermat, and Pascal enriched mathematics with +imperishable treasures. During the early part of the reign +of Louis~XIV. we behold the sunset splendour of this glorious +period. Then followed a night of mental effeminacy. This +lack of great scientific thinkers during the reign of Louis~XIV. +may be due to the simple fact that no great minds were born; +but, according to Buckle, it was due to the paternalism, to +the spirit of dependence and subordination, and to the lack +of toleration, which marked the policy of Louis~XIV\@. + +In the absence of great French thinkers, Louis~XIV. surrounded +himself by eminent foreigners. Römer from Denmark, +\index{Romer@{Römer}}% +Huygens from Holland, Dominic Cassini from Italy, +were the mathematicians and astronomers adorning his court. +They were in possession of a brilliant reputation before going +to Paris. Simply because they performed scientific work in +Paris, that work belongs no more to France than the discoveries +of Descartes belong to Holland, or those of Lagrange +to Germany, or those of Euler and Poncelet to Russia. We +%% -----File: 214.png---Folio 200------- +\index{Fluxions}% +must look to other countries than France for the great scientific +men of the latter part of the seventeenth century. + +About the time when Louis~XIV. assumed the direction +of the French government Charles II. became king of England. +At this time England was extending her commerce +and navigation, and advancing considerably in material prosperity. +A strong intellectual movement took place, which +was unwittingly supported by the king. The age of poetry +was soon followed by an age of science and philosophy. In +two successive centuries England produced Shakespeare and +Newton! +\index{Newton!ref.\ to}% + +Germany still continued in a state of national degradation. +The Thirty Years' War had dismembered the empire and +brutalised the people. Yet this darkest period of Germany's +history produced Leibniz, one of the greatest geniuses of +\index{Leibniz!ref.\ to}% +modern times. + +There are certain focal points in history toward which +the lines of past progress converge, and from which radiate +the advances of the future. Such was the age of Newton +and Leibniz in the history of mathematics. During fifty +years preceding this era several of the brightest and acutest +mathematicians bent the force of their genius in a direction +which finally led to the discovery of the infinitesimal calculus +by Newton and Leibniz. Cavalieri, Roberval, Fermat, Descartes, +Wallis, and others had each contributed to the new +geometry. So great was the advance made, and so near +was their approach toward the invention of the infinitesimal +analysis, that both Lagrange and Laplace pronounced their +countryman, Fermat, to be the true inventor of it. The differential +\index{Differential calculus}% +calculus, therefore, was not so much an individual +discovery as the grand result of a succession of discoveries +by different minds. Indeed, no great discovery ever flashed +upon the mind at once, and though those of Newton will +%% -----File: 215.png---Folio 201------- +influence mankind to the end of the world, yet it must be +admitted that Pope's lines are only a ``poetic fancy'':--- +\begin{center} +\footnotesize% +\settowidth{\TmpLen}{``\,God said, `Let Newton be,' and all was light.''}% +\parbox{\TmpLen}{% + ``\,Nature and Nature's laws lay hid in night; \\ + \settowidth{\TmpLen}{``\,}% + \makebox[\TmpLen][c]{}God said, `Let Newton be,' and all was light.''}% +\end{center} + +\textbf{Isaac Newton} (1642-1727) was born at Woolsthorpe, in +\index{Newton|(}% +Lincolnshire, the same year in which Galileo died. At his +birth he was so small and weak that his life was despaired of. +His mother sent him at an early age to a village school, and +in his twelfth year to the public school at Grantham. At +first he seems to have been very inattentive to his studies +and very low in the school; but when, one day, the little +Isaac received a severe kick upon his stomach from a boy +who was above him, he laboured hard till he ranked higher +in school than his antagonist. From that time he continued +to rise until he was the head boy.\Reference{33} At Grantham, Isaac +showed a decided taste for mechanical inventions. He constructed +a water-clock, a wind-mill, a carriage moved by the +person who sat in it, and other toys. When he had attained +his fifteenth year his mother took him home to assist her in +the management of the farm, but his great dislike for farm-work %[** PP: Only instance] +and his irresistible passion for study, induced her to +send him back to Grantham, where he remained till his +eighteenth year, when he entered Trinity College, Cambridge +(1660). Cambridge was the real birthplace of Newton's +genius. Some idea of his strong intuitive powers may be +drawn from the fact that he regarded the theorems of ancient +geometry as self-evident truths, and that, without any preliminary +study, he made himself master of Descartes' \textit{Geometry}. +He afterwards regarded this neglect of elementary geometry +a mistake in his mathematical studies, and he expressed to +Dr. Pemberton his regret that ``he had applied himself to the +\index{Pemberton}% +works of Descartes and other algebraic writers before he had +%% -----File: 216.png---Folio 202------- +\index{Binomial formula}% +\index{Exponents}% +considered the \textit{Elements} of Euclid with that attention which +so excellent a writer deserves.'' Besides Descartes' \textit{Geometry}, +he studied Oughtred's \textit{Clavis}, Kepler's \textit{Optics}, the works of +\index{Kepler!ref.\ to}% +\index{Oughtred!ref.\ to}% +Vieta, Schooten's \textit{Miscellanies}, Barrow's \textit{Lectures}, and the +\index{Barrow!ref.\ to}% +\index{Schooten, van!ref.\ to}% +\index{Vanschooten@{Van Schooten}!ref.\ to}% +\index{Vieta!ref.\ to}% +works of Wallis. He was particularly delighted with Wallis' +\index{Wallis!ref.\ to}% +\textit{Arithmetic of Infinites}, a treatise fraught with rich and varied +suggestions. Newton had the good fortune of having for +a teacher and fast friend the celebrated Dr. Barrow, who +had been elected professor of Greek in~1660, and was made +Lucasian professor of mathematics in~1663. The mathematics +of Barrow and of Wallis were the starting-points +from which Newton, with a higher power than his masters', +moved onward into wider fields. Wallis had effected the +quadrature of curves whose ordinates are expressed by any +\index{Curves!quadrature of}% +integral and positive power of $(1-x^2)$. We have seen how +Wallis attempted but failed to interpolate between the areas +thus calculated, the areas of other curves, such as that of +the circle; how Newton attacked the problem, effected the +interpolation, and discovered the Binomial Theorem, which +afforded a much easier and direct access to the quadrature +of curves than did the method of interpolation; for even +though the binomial expression for the ordinate be raised +to a fractional or negative power, the binomial could at once +be expanded into a series, and the quadrature of each separate +term of that series could be effected by the method of Wallis. +Newton introduced the system of literal indices. + +Newton's study of quadratures soon led him to another +and most profound invention. He himself says that in 1665 +and 1666 he conceived the method of fluxions and applied +\index{Fluxions|(}% +them to the quadrature of curves. Newton did not communicate +the invention to any of his friends till 1669, when +he placed in the hands of Barrow a tract, entitled \textit{De Analysi +per Æquationes Numero Terminorum Infinitas}, which was sent +%% -----File: 217.png---Folio 203------- +\index{Hudde!ref.\ to}% +by Barrow to Collins, who greatly admired it. In this treatise +\index{Barrow!ref.\ to}% +\index{Collins}% +the principle of fluxions, though distinctly pointed out, is only +partially developed and explained. Supposing the abscissa +to increase uniformly in proportion to the time, he looked +upon the area of a curve as a nascent quantity increasing +by continued fluxion in the proportion of the length of the +ordinate. The expression which was obtained for the fluxion +he expanded into a finite or infinite series of monomial terms, +\index{Infinite series}% +to which Wallis' rule was applicable. Barrow urged Newton +to publish this treatise; ``but the modesty of the author, of +which the excess, if not culpable, was certainly in the present +instance very unfortunate, prevented his compliance.''\Reference{26} Had +this tract been published then, instead of forty-two years +later, there would probably have been no occasion for that +long and deplorable controversy between Newton and Leibniz. + +For a long time Newton's method remained unknown, except +to his friends and their correspondents. In a letter +to Collins, dated December~10th, 1672, Newton states the fact +of his invention with one example, and then says: ``This +is one particular, or rather corollary, of a general method, +which extends itself, without any troublesome calculation, not +only to the drawing of tangents to any curve lines, whether +geometrical or mechanical, or anyhow respecting right lines +or other curves, but also to the resolving other abstruser +kinds of problems about the crookedness, areas, lengths, +centres of gravity of curves, etc.; nor is it (as Hudden's +method of Maximis and Minimis) limited to equations which +are free from surd quantities. This method I have interwoven +with that other of working in equations, by reducing +them to infinite series.'' + +These last words relate to a treatise he composed in the +year 1671, entitled \textit{Method of Fluxions}, in which he aimed +to represent his method as an independent calculus and as +%% -----File: 218.png---Folio 204------- +a complete system. This tract was intended as an introduction +to an edition of Kinckhuysen's \textit{Algebra}, which he had +\index{Kinckhuysen}% +undertaken to publish. ``But the fear of being involved in +disputes about this new discovery, or perhaps the wish to +render it more complete, or to have the sole advantage of +employing it in his physical researches, induced him to abandon +this design.''\Reference{33} + +Excepting two papers on optics, all of his works appear +to have been published only after the most pressing solicitations +of his friends and against his own wishes.\Reference{34} His researches +on light were severely criticised, and he wrote in~1675: +``I was so persecuted with discussions arising out of +my theory of light that I blamed my own imprudence for +parting with so substantial a blessing as my quiet to run +after a shadow.'' + +The \textit{Method of Fluxions}, translated by J. Colson from Newton's +\index{Colson}% +Latin, was first published in~1736, or sixty-five years +after it was written. In it he explains first the expansion +into series of fractional and irrational quantities,---a subject +which, in his first years of study, received the most careful +attention. He then proceeds to the solution of the two following +mechanical problems, which constitute the pillars, so +to speak, of the abstract calculus:--- + +``I\@. The length of the space described being continually +(\ie~at all times) given; to find the velocity of the motion at +any time proposed. + +``II\@. The velocity of the motion being continually given; to +find the length of the space described at any time proposed.'' + +Preparatory to the solution, Newton says: ``Thus, in the +equation $y = x^2$, if $y$ represents the length of the space at any +time described, which (time) another space~$x$, by increasing +with an uniform celerity~$\dot{x}$, measures and exhibits as described: +then $2x\dot{x}$ will represent the celerity by which the space~$y$, +%% -----File: 219.png---Folio 205------- +\index{Notation!differential calculus}% +at the same moment of time, proceeds to be described; and +contrarywise.'' + +``But whereas we need not consider the time here, any farther +than it is expounded and measured by an equable local +motion; and besides, whereas only quantities of the same +kind can be compared together, and also their velocities of +increase and decrease; therefore, in what follows I shall have +no regard to time formally considered, but I shall suppose +some one of the quantities proposed, being of the same kind, +to be increased by an equable fluxion, to which the rest may be +referred, as it were to time; and, therefore, by way of analogy, +it may not improperly receive the name of time.'' In this +statement of Newton there is contained a satisfactory answer +to the objection which has been raised against his method, +that it introduces into analysis the foreign idea of motion. A +quantity thus increasing by uniform fluxion, is what we now +call an independent variable. + +Newton continues: ``Now those quantities which I consider +as gradually and indefinitely increasing, I shall hereafter call +\emph{fluents}, or \emph{flowing quantities}, and shall represent them by the +\index{Fluents}% +final letters of the alphabet, $v$,~$x$, $y$, and~$z$; \dots and the velocities +by which every fluent is increased by its generating motion +(which I may call \emph{fluxions}, or simply velocities, or celerities), +I shall represent by the same letters pointed, thus, $\dot{v}$,~$\dot{x}$, $\dot{y}$,~$\dot{z}$. +That is, for the celerity of the quantity~$v$ I shall put~$\dot{v}$, and so +for the celerities of the other quantities $x$,~$y$, and~$z$, I shall put +$\dot{x}$,~$\dot{y}$, and~$\dot{z}$, respectively.'' It must here be observed that Newton +does not take the fluxions themselves infinitely small. +The ``moments of fluxions,'' a term introduced further on, are +\index{Moments in fluxionary calculus}% +infinitely small quantities. These ``moments,'' as defined and +used in the \textit{Method of Fluxions}, are substantially the differentials +of Leibniz. De Morgan points out that no small amount of +\index{Demorgan@{De Morgan}!ref.\ to}% +confusion has arisen from the use of the word \emph{fluxion} and the +%% -----File: 220.png---Folio 206------- +\index{Fluents}% +notation~$\dot{x}$ by all the English writers previous to~1704, excepting +Newton and Cheyne, in the sense of an infinitely small increment.\Reference{35} +\index{Cheyne}% +Strange to say, even in the \textit{Commercium Epistolicum} +\index{Commercium epistolicum}% +the words \emph{moment} and \emph{fluxion} appear to be used as synonymous. + +After showing by examples how to solve the first problem, +Newton proceeds to the demonstration of his solution:--- + +``The moments of flowing quantities (that is, their indefinitely +\index{Moments in fluxionary calculus}% +small parts, by the accession of which, in infinitely +small portions of time, they are continually increased) are as +the velocities of their flowing or increasing. + +``Wherefore, if the moment of any one (as~$x$) be represented +by the product of its celerity~$\dot{x}$ into an infinitely small quantity~$0$ +(\ie by~$\dot{x}0$), the moments of the others, $v$, $y$, $z$, will be represented +by $\dot{v}0$, $\dot{y}0$,~$\dot{z}0$; because $\dot{v}0$,~$\dot{x}0$, $\dot{y}0$, and~$\dot{z}0$ are to each other +as $\dot{v}$,~$\dot{x}$, $\dot{y}$, and~$\dot{z}$. + +``Now since the moments, as $\dot{x}0$ and~$\dot{y}0$, are the indefinitely +little accessions of the flowing quantities $x$~and~$y$, by which +those quantities are increased through the several indefinitely +little intervals of time, it follows that those quantities, $x$~and~$y$, +after any indefinitely small interval of time, become +$x + \dot{x}0$ and $y + \dot{y}0$, and therefore the equation, which at all +times indifferently expresses the relation of the flowing quantities, +will as well express the relation between $x + \dot{x}0$ and +$y + \dot{y}0$, as between $x$~and~$y$; so that $x + \dot{x}0$ and $y + \dot{y}0$ may +be substituted in the same equation for those quantities, instead +of $x$~and~$y$. Thus let any equation $x^3 - ax^2 + axy - y^3 = 0$ +be given, and substitute $x + \dot{x}0$ for~$x$, and $y + \dot{y}0$ for~$y$, and +there will arise +\[ +\left. +\begin{alignedat}{4} +&\phantom{a}x^3 &&+ 3x^2\dot{x}0 &&+ 3x\dot{x}0\dot{x}0 &&+ \dot{x}^3 0^3 \\ +-{}&ax^2 &&- 2ax\dot{x}0 &&- a\dot{x}0\dot{x}0 \\ ++{}&axy &&+ ay\dot{x}0 &&+ a\dot{x}0\dot{y}0 \\ +& &&+ ax\dot{y}0 \\ +-{}&y^3 &&- 3y^2\dot{y}0 &&- 3y\dot{y}0\dot{y}0 &&- \dot{y}^3 0^3 +\end{alignedat} +\right\} = 0. +\] +%% -----File: 221.png---Folio 207------- + +``Now, by supposition, $x^3-ax^2+axy-y^3=0$, which therefore, +being expunged and the remaining terms being divided +by 0, there will remain +\begin{align*} +3x^2\dot{x} &- 2ax\dot{x} + ay\dot{x} + ax\dot{y} - 3y^2\dot{y} + 3x\dot{x}\dot{x}0 - a\dot{x}\dot{x}0 + a\dot{x}\dot{y}0 \\ + &- 3y\dot{y}\dot{y}0 + \dot{x}^3 00 - \dot{y}^300 = 0. +\end{align*} +But whereas zero is supposed to be infinitely little, that it +may represent the moments of quantities, the terms that are +multiplied by it will be nothing in respect of the rest (\textit{termini +in eam ducti pro nihilo possunt haberi cum aliis collati}); therefore +I reject them, and there remains +\[ +3x^2\dot{x} - 2ax\dot{x} + ay\dot{x} + ax\dot{y} - 3y^2\dot{y} = 0, +\] +as above in Example~I.'' Newton here uses infinitesimals. +\index{Infinitesimals}% + +Much greater than in the first problem were the difficulties +encountered in the solution of the second problem, involving, +as it does, inverse operations which have been taxing the skill +of the best analysts since his time. Newton gives first a +special solution to the second problem in which he resorts +to a rule for which he has given no proof. + +In the general solution of his second problem, Newton +assumed homogeneity with respect to the fluxions and then +considered three cases: (1)~when the equation contains two +fluxions of quantities and but one of the fluents; (2)~when +the equation involves both the fluents as well as both the fluxions; +(3)~when the equation contains the fluents and the fluxions +of three or more quantities. The first case is the easiest +since it requires simply the integration of $\dfrac{dy}{dx}=f(x)$, to which +his ``special solution'' is applicable. The second case demanded +nothing less than the general solution of a differential +equation of the first order. Those who know what +efforts were afterwards needed for the complete exploration +of this field in analysis, will not depreciate Newton's work +%% -----File: 222.png---Folio 208------- +\index{Partial differential equations}% +even though he resorted to solutions in form of infinite series. +\index{Infinite series}% +Newton's third case comes now under the solution of partial +differential equations. He took the equation $2\dot{x} - \dot{z} + x\dot{y} = 0$ +and succeeded in finding a particular integral of it. + +The rest of the treatise is devoted to the determination of +maxima and minima, the radius of curvature of curves, and +\index{Maxima and minima}% +other geometrical applications of his fluxionary calculus. All +this was done previous to the year~1672. + +It must be observed that in the \textit{Method of Fluxions} (as well +as in his \textit{De Analysi} and all earlier papers) the method +employed by Newton is strictly infinitesimal, and in substance +like that of Leibniz. Thus, the original conception of the +\index{Leibniz!ref.\ to}% +calculus in England, as well as on the Continent, was based +on infinitesimals. The fundamental principles of the fluxionary +\index{Infinitesimals}% +calculus were first given to the world in the \textit{Principia}; but its +\index{Newton's Principia}% +\index{Principia (Newton's)}% +peculiar notation did not appear until published in the second +volume of Wallis' \textit{Algebra} in 1693. The exposition given in +the \textit{Algebra} was substantially a contribution of Newton; it +rests on infinitesimals. In the first edition of the \textit{Principia} +(1687) the description of fluxions is likewise founded on +infinitesimals, but in the second (1713) the foundation is +somewhat altered. In Book~II. Lemma~II. of the first edition +we read: ``Cave tamen intellexeris particulas finitas. \textit{Momenta +quam primum finitæ sunt magnitudinis, desinunt esse momenta. +Finiri enim repugnat aliquatenus perpetuo eorum incremento vel +decremento.} Intelligenda sunt principia jamjam nascentia +finitorum magnitudinum.'' In the second edition the two +sentences which we print in italics are replaced by the +following: ``Particulæ finitæ non sunt momenta sed quantitates +ipsæ ex momentis genitæ.'' Through the difficulty of +the phrases in both extracts, this much distinctly appears, +that in the first, moments are infinitely small quantities. +What else they are in the second is not clear.\Reference{35} In the +%% -----File: 223.png---Folio 209------- +\index{Fresnel's wave-surface}% +\textit{Quadrature of Curves} of 1704, the infinitely small quantity is +completely abandoned. It has been shown that in the \textit{Method +of Fluxions} Newton rejected terms involving the quantity 0, +because they are infinitely small compared with other terms. +This reasoning is evidently erroneous; for as long as 0 is a +quantity, though ever so small, this rejection cannot be made +without affecting the result. Newton seems to have felt this, +for in the \textit{Quadrature of Curves} he remarked that ``in mathematics +the minutest errors are not to be neglected" (errores +quam minimi in rebus mathematicis non sunt contemnendi). + +The early distinction between the system of Newton and +Leibniz lies in this, that Newton, holding to the conception +\index{Leibniz!ref.\ to}% +of velocity or fluxion, used the infinitely small increment as a +means of determining it, while with Leibniz the relation of the +infinitely small increments is itself the object of determination. +The difference between the two rests mainly upon a difference +in the mode of generating quantities.\Reference{35} + +We give Newton's statement of the method of fluxions or +rates, as given in the introduction to his \textit{Quadrature of Curves}. +``I consider mathematical quantities in this place not as +consisting of very small parts, but as described by a continued +motion. Lines are described, and thereby generated, not by +the apposition of parts, but by the continued motion of points; +superficies by the motion of lines; solids by the motion of +superficies; angles by the rotation of the sides; portions of +time by continual flux: and so on in other quantities. These +geneses really take place in the nature of things, and are +daily seen in the motion of bodies\dots. + +``Fluxions are, as near as we please (\textit{quam proxime}), as the +increments of fluents generated in times, equal and as small as +possible, and to speak accurately, they are in the prime ratio +of nascent increments; yet they can be expressed by any lines +whatever, which are proportional to them." +%% -----File: 224.png---Folio 210------- + +Newton exemplifies this last assertion by the problem of +tangency: Let $AB$ be the abscissa, $BC$~the ordinate, $VCH$~the +tangent, $Ec$~the increment of the ordinate, which produced +meets $VH$ at~$T$, and $Cc$~the increment of the curve. +The right line~$Cc$ being produced to~$K$, there are formed +three small triangles, the rectilinear~$CEc$, the mixtilinear~$CEc$, +and the rectilinear~$CET$. Of these, the first is evidently +the smallest, and the last the greatest. Now suppose +the ordinate~$bc$ to move into the place~$BC$, so that the +point~$c$ exactly coincides +with %[** PP: Width-dependent line break] +%[Illustration:] +\begin{wrapfigure}[8]{l}{2.75in} +\Graphic{2.75in}{224a} +\end{wrapfigure} +the +point $C$; $CK$, and +therefore the curve +$Cc$, is coincident +with the tangent~$CH$, +$Ec$ is absolutely +equal to $ET$, +and the mixtilinear +evanescent triangle $CEc$ is, in the last form, similar to the +triangle $CET$; and its evanescent sides $CE$, $Ec$, $Cc$, will be +proportional to $CE$, $ET$, and~$CT$, the sides of the triangle~$CET$. +Hence it follows that the fluxions of the lines $AB$, +$BC$, $AC$, being in the last ratio of their evanescent increments, +are proportional to the sides of the triangle~$CET$, or, which is +all one, of the triangle~$VBC$ similar thereunto. As long as +the points $C$~and~$c$ are distant from each other by an interval, +however small, the line~$CK$ will stand apart by a small angle +from the tangent~$CH$. But when $CK$ coincides with~$CH$, and +the lines $CE$, $Ec$, $cC$ reach their ultimate ratios, then the +points $C$~and~$c$ accurately coincide and are one and the same. +Newton then adds that ``in mathematics the minutest errors +are not to be neglected.'' This is plainly a rejection of the +postulates of Leibniz. The doctrine of infinitely small quantities +\index{Leibniz!ref.\ to}% +%% -----File: 225.png---Folio 211------- +is here renounced in a manner which would lead one +to suppose that Newton had never held it himself. Thus it +appears that Newton's doctrine was different in different +periods. Though, in the above reasoning, the Charybdis of +infinitesimals is safely avoided, the dangers of a Scylla stare +\index{Infinitesimals}% +us in the face. We are required to believe that a point may +be considered a triangle, or that a triangle can be inscribed +in a point; nay, that three dissimilar triangles become similar +and equal when they have reached their ultimate form in one +and the same point. + +In the introduction to the \textit{Quadrature of Curves} the fluxion +of $x^n$ is determined as follows:--- + +``In the same time that~$x$, by flowing, becomes~$x + 0$, the +power~$x^n$ becomes $(x+0)^n$, \ie~by the method of infinite series +\[ +x^n + n0x^{n-1} + \frac{n^2-n}{2} 0^{2} x^{n-2} + \text{etc.}, +\] +and the increments +\[ +0 \text{ and } n0 x^{n-1} + \frac{n^2-n}{2} 0^{2} x^{n-2} + \text{etc.}, +\] +are to one another as +\[ +1 \text{ to } nx^{n-1} + \frac{n^2-n}{2} 0 x^{n-2} + \text{etc.} +\] + +``Let now the increments vanish, and their last proportion +will be $1$~to~$nx^{n-1}$: hence the fluxion of the quantity~$x$ is to +the fluxion of the quantity~$x^n$ as~$1 : nx^{n-1}$. + +``The fluxion of lines, straight or curved, in all cases whatever, +as also the fluxions of superficies, angles, and other +quantities, can be obtained in the same manner by the method +of prime and ultimate ratios. But to establish in this way +the analysis of infinite quantities, and to investigate prime +and ultimate ratios of finite quantities, nascent or evanescent, +is in harmony with the geometry of the ancients; and I have +endeavoured to show that, in the method of fluxions, it is not +%% -----File: 226.png---Folio 212------- +\index{Astronomy!Newton|(}% +\index{Mechanics!Descartes, Wallis, Wren, Huygens, Newton|(}% +necessary to introduce into geometry infinitely small quantities.'' +This mode of differentiating does not remove all the +difficulties connected with the subject. When $0$~becomes +nothing, then we get the ratio $\dfrac{0}{0} = nx^{n-1}$, which needs further +elucidation. Indeed, the method of Newton, as delivered by +himself, is encumbered with difficulties and objections. Among +the ablest admirers of Newton, there have been obstinate disputes +respecting his explanation of his method of ``prime and +\index{Prime and ultimate ratios}% +ultimate ratios.'' + +The so-called ``method of limits'' is frequently attributed +\index{Limits, method of}% +to Newton, but the pure method of limits was never adopted +by him as his method of constructing the calculus. All he +did was to establish in his \textit{Principia} certain principles which +\index{Newton's Principia|(}% +\index{Principia (Newton's)|(}% +are applicable to that method, but which he used for a different +purpose. The first lemma of the first book has been made the +foundation of the method of limits:--- + +``Quantities and the ratios of quantities, which in any finite +time converge continually to equality, and before the end of +that time approach nearer the one to the other than by any +given difference, become ultimately equal.'' + +In this, as well as in the lemmas following this, there are +obscurities and difficulties. Newton appears to teach that a +variable quantity and its limit will ultimately coincide and be +equal. But it is now generally agreed that in the clearest +statements which have been made of the theory of limits, the +variable does not actually reach its limit, though the variable +may approach it as near as we please. + +The full title of Newton's \textit{Principia} is \textit{Philosophiæ Naturalis +Principia Mathematica}. It was printed in 1687 under +the direction, and at the expense, of Dr.~Edmund Halley. +A second edition was brought out in 1713 with many alterations +and improvements, and accompanied by a preface from +%% -----File: 227.png---Folio 213------- +\index{Fluxions|)}% +\index{Kepler's laws}% +\index{Laws of motion}% +\index{Motion, laws of}% +Mr.~Cotes. It was sold out in a few months, but a pirated +edition published in Amsterdam supplied the demand.\Reference{34} The +third and last edition which appeared in England during +Newton's lifetime was published in 1726 by Henry Pemberton. +The \textit{Principia} consists of three books, of which the first two, +constituting the great bulk of the work, treat of the mathematical +principles of natural philosophy, namely, the laws and +conditions of motions and forces. In the third book is drawn +up the constitution of the universe as deduced from the foregoing +principles. The great principle underlying this memorable +work is that of universal gravitation. The first book +\index{Gravitation, theory of}% +\index{Newton's discovery of universal gravitation}% +was completed on April~28, 1686. After the remarkably short +period of three months, the second book was finished. The +third book is the result of the next nine or ten months' +labours. It is only a sketch of a much more extended elaboration +of the subject which he had planned, but which was never +brought to completion. + +The law of gravitation is enunciated in the first book. Its +discovery envelops the name of Newton in a halo of perpetual +glory. The current version of the discovery is as follows: it +was conjectured by Hooke, Huygens, Halley, Wren, Newton, +\index{Halley}% +\index{Hooke}% +\index{Huygens!ref.\ to}% +\index{Wren!ref.\ to}% +and others, that, if Kepler's third law was true (its absolute +\index{Kepler!ref.\ to}% +accuracy was doubted at that time), then the attraction +between the earth and other members of the solar system +varied inversely as the square of the distance. But the proof +of the truth or falsity of the guess was wanting. In~1666 +Newton reasoned, in substance, that if $g$~represent the acceleration +of gravity on the surface of the earth, $r$~be the earth's +radius, $R$~the distance of the moon from the earth, $T$~the time +of lunar revolution, and $a$~a degree at the equator, then, if the +law is true, +\[ +g \frac{r^2}{R^2} = 4 \pi^2 \frac{R}{T^2}, \text{ or } g = \frac{4\pi}{T^2}\left(\frac{R}{r}\right)^3 · 180a. +\] +%% -----File: 228.png---Folio 214------- +\index{Centrifugal force}% +The data at Newton's command gave $R = 60.4r$, $T = 2,360,628$ +seconds, but $a$ only $60$ instead of $69\frac{1}{2}$ English miles. This +wrong value of~$a$ rendered the calculated value of~$g$ smaller +than its true value, as known from actual measurement. It +looked as though the law of inverse squares were not the true +law, and Newton laid the calculation aside. In~1684 he casually +ascertained at a meeting of the Royal Society that Jean +Picard had measured an arc of the meridian, and obtained a +\index{Picard, J.}% +more accurate value for the earth's radius. Taking the corrected +\index{Earth!size of}% +value for~$a$, he found a figure for~$g$ which corresponded +to the known value. Thus the law of inverse squares was +verified. In a scholium in the \textit{Principia}, Newton acknowledged +his indebtedness to Huygens for the laws on centrifugal +\index{Huygens!ref.\ to}% +force employed in his calculation. + +The perusal by the astronomer Adams of a great mass of +\index{Adams!ref.\ to}% +unpublished letters and manuscripts of Newton forming the +Portsmouth collection (which remained private property +until~1872, when its owner placed it in the hands of the +University of Cambridge) seems to indicate that the difficulties +encountered by Newton in the above calculation were of a +different nature. According to Adams, Newton's numerical +verification was fairly complete in~1666, but Newton had not +been able to determine what the attraction of a spherical shell +upon an external point would be. His letters to Halley show +\index{Halley}% +that he did not suppose the earth to attract as though all its +mass were concentrated into a point at the centre. He could +not have asserted, therefore, that the assumed law of gravity +was verified by the figures, though for long distances he might +have claimed that it yielded close approximations. When +Halley visited Newton in~1684, he requested Newton to determine +what the orbit of a planet would be if the law of attraction +were that of inverse squares. Newton had solved a +similar problem for Hooke in~1679, and replied at once that it +%% -----File: 229.png---Folio 215------- +\index{Ellipsoid!(attraction of)}% +\index{Solid of least resistance}% +was an ellipse. After Halley's visit, Newton, with Picard's +\index{Picard, J.}% +new value for the earth's radius, reviewed his early calculation, +\index{Earth!size of}% +and was able to show that if the distances between the +bodies in the solar system were so great that the bodies might +be considered as points, then their motions were in accordance +with the assumed law of gravitation. In 1685 he completed +his discovery by showing that a sphere whose density at any +point depends only on the distance from the centre attracts +an external point as though its whole mass were concentrated +at the centre.\Reference{34} + +Newton's unpublished manuscripts in the Portsmouth collection +show that he had worked out, by means of fluxions and +fluents, his lunar calculations to a higher degree of approximation +than that given in the \textit{Principia}, but that he was unable +to interpret his results geometrically. The papers in that collection +throw light upon the mode by which Newton arrived +at some of the results in the \textit{Principia}, as, for instance, the +famous construction in Book~II., Prop.~25, which is unproved +in the \textit{Principia}, but is demonstrated by him twice in a draft +of a letter to David Gregory, of Oxford.\Reference{34} +\index{Gregory, David F}% + +It is chiefly upon the \textit{Principia} that the fame of Newton +\index{Newton's Principia|)}% +\index{Principia (Newton's)|)}% +rests. Brewster calls it ``the brightest page in the records of +human reason.'' Let us listen, for a moment, to the comments +of Laplace, the foremost among those followers of Newton who +\index{Laplace!ref.\ to}% +grappled with the subtle problems of the motions of planets +under the influence of gravitation: ``Newton has well established +the existence of the principle which he had the merit +of discovering, but the development of its consequences and +advantages has been the work of the successors of this great +mathematician. The imperfection of the infinitesimal calculus, +when first discovered, did not allow him completely to +resolve the difficult problems which the theory of the universe +offers; and he was oftentimes forced to give mere hints, which +%% -----File: 230.png---Folio 216------- +\index{Astronomy!Newton|)}% +\index{Mechanics!Descartes, Wallis, Wren, Huygens, Newton|)}% +were always uncertain till confirmed by rigorous analysis. +Notwithstanding these unavoidable defects, the importance +and the generality of his discoveries respecting the system of +the universe, and the most interesting points of natural philosophy, +the great number of profound and original views, +which have been the origin of the most brilliant discoveries of +the mathematicians of the last century, which were all presented +with much elegance, will insure to the \textit{Principia} a lasting +pre-eminence over all other productions of the human +mind." + +Newton's \textit{Arithmetica Universalis}, consisting of algebraical +lectures delivered by him during the first nine years he was +professor at Cambridge, were published in~1707, or more than +thirty years after they were written. This work was published +by Mr.~Whiston. We are not accurately informed how +\index{Whiston}% +Mr.~Whiston came in possession of it, but according to some +authorities its publication was a breach of confidence on his +part. + +The \textit{Arithmetica Universalis} contains new and important +results on the theory of equations. His theorem on the +\index{Equations!theory of}% +sums of powers of roots is well known. Newton showed +that in equations with real coefficients, imaginary roots always +occur in pairs. His inventive genius is grandly displayed +in his rule for determining the inferior limit of the number +of imaginary roots, and the superior limits for the number +of positive and negative roots. Though less expeditious than +Descartes', Newton's rule always gives as close, and generally +\index{Descartes!ref.\ to}% +closer, limits to the number of positive and negative roots. +Newton did not prove his rule. It awaited demonstration +for a century and a half, until, at last, Sylvester established +\index{Sylvester~ref.\ to}% +a remarkable general theorem which includes Newton's rule +as a special case. + +The treatise on \textit{Method of Fluxions} contains Newton's method +%% -----File: 231.png---Folio 217------- +\index{Cubic curves}% +of approximating to the roots of numerical equations. This +is simply the method of Vieta improved. The same treatise +\index{Vieta!ref.\ to}% +contains ``Newton's parallelogram," which enabled him, in an +\index{Newton's parallelogram}% +equation, $f(x, y) = 0$, to find a series in powers of~$x$ equal to +the variable~$y$. The great utility of this rule lay in its determining +the \emph{form} of the series; for, as soon as the law was +known by which the exponents in the series vary, then the +expansion could be effected by the method of indeterminate +coefficients. The rule is still used in determining the infinite +branches to curves, or their figure at multiple points. Newton +gave no proof for it, nor any clue as to how he discovered it. +The proof was supplied half a century later, by Kaestner and +\index{Kaestner!ref.\ to}% +Cramer, independently.\Reference{37} +\index{Cramer}% + +In 1704 was published, as an appendix to the \textit{Opticks}, the +\textit{Enumeratio linearum tertii ordinis}, which contains theorems +on the theory of curves. Newton divides cubics into seventy-two +species, arranged in larger groups, for which his commentators +have supplied the names ``genera'' and ``classes,'' +recognising fourteen of the former and seven (or four) of the +latter. He overlooked six species demanded by his principles +of classification, and afterwards added by Stirling, Murdoch, +and Cramer. He enunciates the remarkable theorem that the +five species which he names ``divergent parabolas'' give by +\index{Divergent parabolas}% +their projection every cubic curve whatever. As a rule, the +tract contains no proofs. It has been the subject of frequent +conjecture how Newton deduced his results. Recently we have +gotten at the facts, since much of the analysis used by Newton +and a few additional theorems have been discovered among the +Portsmouth papers. An account of the four holograph manuscripts +on this subject has been published by W.~W. Rouse +Ball, in the \textit{Transactions of the London Mathematical Society} +\index{Ball, W.~W.~R.}% +(vol.~xx., pp.~104--143). It is interesting to observe how +Newton begins his research on the classification of cubic +%% -----File: 232.png---Folio 218------- +curves by the algebraic method, but, finding it laborious, +attacks the problem geometrically, and afterwards returns +again to analysis.\Reference{36} + +Space does not permit us to do more than merely mention +Newton's prolonged researches in other departments of science. +He conducted a long series of experiments in optics and is the +author of the corpuscular theory of light. The last of a +\index{Light, theory of}% +number of papers on optics, which he contributed to the Royal +Society, 1687, elaborates the theory of ``fits.'' He explained +the decomposition of light and the theory of the rainbow. By +him were invented the reflecting telescope and the sextant +\index{Sextant}% +(afterwards re-discovered by Thomas Godfrey of Philadelphia\Reference[]{2} +\index{Godfrey}% +and by John Hadley). He deduced a theoretical expression +\index{Hadley}% +for the velocity of sound in air, engaged in experiments on +chemistry, elasticity, magnetism, and the law of cooling, and +entered upon geological speculations. + +During the two years following the close of~1692, Newton +\index{Newton|)}% +suffered from insomnia and nervous irritability. Some thought +that he laboured under temporary mental aberration. Though +he recovered his tranquillity and strength of mind, the time +of great discoveries was over; he would study out questions +propounded to him, but no longer did he by his own accord +enter upon new fields of research. The most noted investigation +after his sickness was the testing of his lunar theory +by the observations of Flamsteed, the astronomer royal. In~1695 +\index{Flamsteed}% +he was appointed warden, and in~1699 master, of the +mint, which office he held until his death. His body was +interred in Westminster Abbey, where in~1731 a magnificent +monument was erected, bearing an inscription ending with, +``Sibi gratulentur mortales tale tantumque exstitisse humani +generis decus.'' It is not true that the Binomial Theorem is +also engraved on it. + +We pass to Leibniz, the second and independent inventor +%% -----File: 233.png---Folio 219------- +\index{Kuhn, J.}% +of the calculus. \textbf{Gottfried Wilhelm Leibniz} (1646--1716) was +\index{Leibniz|(}% +born in Leipzig. No period in the history of any civilised +nation could have been less favourable for literary and scientific +pursuits than the middle of the seventeenth century in Germany. +Yet circumstances seem to have happily combined +to bestow on the youthful genius an education hardly otherwise +obtainable during this darkest period of German history. +He was brought early in contact with the best of the culture +then existing. In his fifteenth year he entered the University +of Leipzig. Though law was his principal study, he applied +himself with great diligence to every branch of knowledge. +Instruction in German universities was then very low. The +higher mathematics was not taught at all. We are told that +a certain John Kuhn lectured on Euclid's \textit{Elements}, but that +his lectures were so obscure that none except Leibniz could +understand them. Later on, Leibniz attended, for a half-year, +at Jena, the lectures of Erhard Weigel, a philosopher and +\index{Weigel}% +mathematician of local reputation. In~1666 Leibniz published +a treatise, \textit{De Arte Combinatoria}, in which he does not pass +beyond the rudiments of mathematics. Other theses written +by him at this time were metaphysical and juristical in +character. A fortunate circumstance led Leibniz abroad. In~1672 +he was sent by Baron Boineburg on a political mission +to Paris. He there formed the acquaintance of the most +distinguished men of the age. Among these was Huygens, +\index{Huygens!ref.\ to}% +who presented a copy of his work on the oscillation of the +pendulum to Leibniz, and first led the gifted young German +to the study of higher mathematics. In~1673 Leibniz went +to London, and remained there from January till March. He +there became incidentally acquainted with the mathematician +Pell, to whom he explained a method he had found on the +\index{Pell}% +summation of series of numbers by their differences. Pell +told him that a similar formula had been published by Mouton +\index{Mouton}% +%% -----File: 234.png---Folio 220------- +\index{Calculating machines}% +\index{Inverse tangents (problem of)}% +\index{p@{$\pi$}: values for!Leibniz's}% +as early as~1670, and then called his attention to Mercator's +\index{Mercator, N.!ref.\ to}% +work on the rectification of the parabola. While in London, +Leibniz exhibited to the Royal Society his arithmetical machine, +\index{Arithmetical machine}% +\index{Machine, arithmetical}% +which was similar to Pascal's, but more efficient and +\index{Pascal!ref.\ to}% +perfect. After his return to Paris, he had the leisure to study +mathematics more systematically. With indomitable energy +he set about removing his ignorance of higher mathematics. +Huygens was his principal master. He studied the geometric +works of Descartes, Honorarius Fabri, Gregory St.~Vincent, +\index{Descartes!ref.\ to}% +and Pascal. A careful study of infinite series led him to the +\index{Infinite series}% +discovery of the following expression for the ratio of the +circumference to the diameter of the circle, previously discovered +by James Gregory:--- +\[ +\frac{\pi}{4} = 1 - \tfrac{1}{3} + \tfrac{1}{5} - \tfrac{1}{7} + \tfrac{1}{9} - \text{etc.} +\] +This elegant series was found in the same way as Mercator's +on the hyperbola. Huygens was highly pleased with it and +urged him on to new investigations. Leibniz entered into a +detailed study of the quadrature of curves and thereby became +\index{Curves!quadrature of}% +\index{Quadrature of curves}% +intimately acquainted with the higher mathematics. Among +the papers of Leibniz is still found a manuscript on quadratures, +written before he left Paris in~1676, but which was +never printed by him. The more important parts of it were +embodied in articles published later in the \textit{Acta Eruditorum}. + +In the study of Cartesian geometry the attention of Leibniz +was drawn early to the direct and inverse problems of tangents. +\index{Tangents!inverse problem of}% +The direct problem had been solved by Descartes for +the simplest curves only; while the inverse had completely +transcended the power of his analysis. Leibniz investigated +both problems for any curve; he constructed what he called +the \textit{triangulum characteristicum}---an infinitely small triangle +\index{Triangulum characteristicum}% +between the infinitely small part of the curve coinciding with +the tangent, and the differences of the ordinates and abscissas. +%% -----File: 235.png---Folio 221------- +\index{Differential calculus|(}% +A curve is here considered to be a polygon. The \textit{triangulum +characteristicum} is similar to the triangle formed by the tangent, +the ordinate of the point of contact, and the \DPtypo{subtangent}{sub-tangent}, +as well as to that between the ordinate, normal, +and sub-normal. It was first employed by Barrow in England, +\index{Barrow!ref.\ to}% +but appears to have been \DPtypo{reinvented}{re-invented} by Leibniz. From +it Leibniz observed the connection existing between the direct +and inverse problems of tangents. He saw also that the latter +could be carried back to the quadrature of curves. All these +results are contained in a manuscript of Leibniz, written in~1673. +One mode used by him in effecting quadratures was +as follows: The rectangle formed by a sub-tangent~$p$ and +an element~$a$ (\ie~infinitely small part of the abscissa) is +equal to the rectangle formed by the ordinate~$y$ and the element~$l$ +of that ordinate; or in symbols, $pa = yl$. But the +summation of these rectangles from zero on gives a right +triangle equal to half the square of the ordinate. Thus, +using Cavalieri's notation, he gets +\index{Cavalieri!ref.\ to}% +\index{Notation!differential calculus}% +\begin{align*} +&\omn pa = \omn yl = \frac{y^2}{2} + \quad \text{(\textit{omn.}\ meaning \textit{omnia}, all)}. +\intertext{But $y = \omn l$; hence} +&\overline{\omn \overline{\omn l}\, \frac{l}{a}} + = \overline{\frac{\omn l^2}{2a}}. +\end{align*} +This equation is especially interesting, since it is here that +Leibniz first introduces a new notation. He says: ``It will +be useful to write~$\int$ for \textit{omn.}, as $\int l$ for $\textit{omn.}\,l$, that is, the +sum of the~$l$'s''; he then writes the equation thus:--- +\[ +\frac{\int \overline{l^2}}{2a} + = \int \overline{{\textstyle\int \bar{l}} \frac{l}{a}}. +\] +From this he deduced the simplest integrals, such as +\[ +\int x = \frac{x^2}{2},\quad \int (x + y) = \int x + \int y. +\] +%% -----File: 236.png---Folio 222------- +\index{Inverse tangents (problem of)}% +\index{Quadrature of curves}% +Since the symbol of summation~$\int$ raises the dimensions, he +concluded that the opposite calculus, or that of differences~$d$, +would lower them. Thus, if $\int l = ya$, then $l = \dfrac{ya}{d}$. The +symbol~$d$ was at first placed by Leibniz in the denominator, +because the lowering of the power of a term was brought +about in ordinary calculation by division. The manuscript +giving the above is dated October~29th, 1675.\Reference{39} This, then, +was the memorable day on which the notation of the new +\index{Notation!differential calculus}% +calculus came to be,---a notation which contributed enormously +to the rapid growth and perfect development of the +calculus. + +Leibniz proceeded to apply his new calculus to the solution +of certain problems then grouped together under the name +of the Inverse Problems of Tangents. He found the cubical +\index{Tangents!inverse problem of}% +parabola to be the solution to the following: To find the +curve in which the sub-normal is reciprocally proportional +to the ordinate. The correctness of his solution was tested +by him by applying to the result Sluze's method of tangents +\index{Sluze!ref.\ to}% +and reasoning backwards to the original supposition. In the +solution of the third problem he changes his notation from +$\dfrac{x}{d}$ to the now usual notation~$dx$. It is worthy of remark +that in these investigations, Leibniz nowhere explains the +significance of $dx$ and~$dy$, except at one place in a marginal +note: ``Idem est $dx$~et~$\dfrac{x}{d}$, id est, differentia inter duas $x$~proximas.'' +Nor does he use the term \emph{differential}, but +always \emph{difference}. Not till ten years later, in the \textit{Acta +Eruditorum}, did he give further explanations of these symbols. +What he aimed at principally was to determine the +change an expression undergoes when the symbol $\int$~or~$d$ is +placed before it. It may be a consolation to students wrestling +with the elements of the differential calculus to know +that it required Leibniz considerable thought and attention\Reference[]{39} +%% -----File: 237.png---Folio 223------- +\index{Debaune@{De Baune}!ref.\ to}% +\index{Inverse tangents (problem of)}% +to determine whether $dx\,dy$ is the same as~$d(xy)$, and +$\dfrac{dx}{dy}$ the same as~$d \smash[t]{\dfrac{x}{y}}$. After considering these questions at +the close of one of his manuscripts, he concluded that the +expressions were not the same, though he could not give the +true value for each. Ten days later, in a manuscript dated +November~21, 1675, he found the equation $y\, d\overline{x} = d\,\overline{xy} - x\,d\overline{y}$, +giving an expression for~$d(xy)$, which he observed to be true +for all curves. He succeeded also in eliminating~$dx$ from +a differential equation, so that it contained only~$dy$, and +thereby led to the solution of the problem under consideration. +``Behold, a most elegant way by which the problems +of the inverse methods of tangents are solved, or at least +\index{Tangents!direct problem of}% +\index{Tangents!inverse problem of}% +are reduced to quadratures!'' Thus he saw clearly that the +inverse problems of tangents could be solved by quadratures, +or, in other words, by the integral calculus. In course of a +\index{Integral calculus}% +half-year he discovered that the direct problem of tangents, +too, yielded to the power of his new calculus, and that thereby +a more general solution than that of Descartes could be +\index{Descartes!ref.\ to}% +obtained. He succeeded in solving all the special problems +of this kind, which had been left unsolved by Descartes. +Of these we mention only the celebrated problem proposed +to Descartes by De~Beaune, viz.~to find the curve whose +ordinate is to its sub-tangent as a given line is to that part +of the ordinate which lies between the curve and a line drawn +from the vertex of the curve at a given inclination to the axis. + +Such was, in brief, the progress in the evolution of the new +calculus made by Leibniz during his stay in Paris. Before +his departure, in October, 1676, he found himself in possession +of the most elementary rules and formulæ of the infinitesimal +calculus. + +From Paris, Leibniz returned to Hanover by way of London +and Amsterdam. In London he met Collins, who showed him +\index{Collins}% +%% -----File: 238.png---Folio 224------- +\index{Tchirnhausen!ref.\ to}% +a part of his scientific correspondence. Of this we shall speak +later. In Amsterdam he discussed mathematics with Sluze, +\index{Sluze!ref.\ to}% +and became satisfied that his own method of constructing +tangents not only accomplished all that Sluze's did, but even +more, since it could be extended to three variables, by which +tangent planes to surfaces could be found; and especially, +since neither irrationals nor fractions prevented the immediate +application of his method. + +In a paper of July~11, 1677, Leibniz gave correct rules for +the differentiation of sums, products, quotients, powers, and +roots. He had given the differentials of a few negative and +fractional powers, as early as November, 1676, but had made +some mistakes. For $d\sqrt{x}$ he had given the erroneous value +$\smash[b]{\dfrac{1}{\sqrt{x}}}$, and in another place the value $-\frac{1}{2} x^{-\frac{1}{2}}$; for $d \dfrac{1}{\DPtypo{x^2}{x^3}}$\DPnote{** PP: Typo presumed on mathematical context.} occurs in +one place the wrong value, $-\smash{\dfrac{2}{x^2}}$, while a few lines lower is +given $-\dfrac{3}{x^4}$, its correct value. + +In 1682 was founded in Berlin the \textit{Acta Eruditorum}, a +journal usually known by the name of \textit{Leipzig Acts}. It +was a partial imitation of the French \textit{Journal des Savans} +(founded in 1665), and the literary and scientific review +published in Germany. Leibniz was a frequent contributor. +Tschirnhaus, who had studied mathematics in Paris with +Leibniz, and who was familiar with the new analysis of +Leibniz, published in the \textit{Acta \DPtypo{Eroditorum}{Eruditorum}} a paper on quadratures, +which consists principally of subject-matter communicated +by Leibniz to Tschirnhaus during a controversy +which they had had on this subject. Fearing that Tschirnhaus +might claim as his own and publish the notation and rules of +the differential calculus, Leibniz decided, at last, to make +public the fruits of his inventions. In~1684, or nine years +after the new calculus first dawned upon the mind of Leibniz, +and nineteen years after Newton first worked at fluxions, +%% -----File: 239.png---Folio 225------- +\index{Debaune@{De Baune}!ref.\ to}% +and three years before the publication of Newton's \textit{Principia}, +Leibniz published, in the \textit{Leipzig Acts}, his first paper on the +differential calculus. He was unwilling to give to the world +all his treasures, but chose those parts of his work which were +most abstruse and least perspicuous. This epoch-making paper +of only six pages bears the title: ``Nova methodus pro maximis +et minimis, itemque tangentibus, \Change{quae}{quæ} nec fractas nec irrationales +quantitates moratur, et singulare pro illis calculi +genus.'' The rules of calculation are briefly stated without +proof, and the meaning of $dx$~and~$dy$ is not made clear. It has +been inferred from this that Leibniz himself had no definite +and settled ideas on this subject. Are $dy$~and~$dx$ finite or +infinitesimal quantities? At first they appear, indeed, to have +been taken as finite, when he says: ``We now call any line +selected at random~$dx$, then we designate the line which is to~$dx$ +as $y$~is to the sub-tangent, by~$dy$, which is the difference of~$y$.'' +Leibniz then ascertains, by his calculus, in what way a +ray of light passing through two differently refracting media, +can travel easiest from one point to another; and then closes +his article by giving his solution, in a few words, of De +Beaune's problem. Two years later (1686) Leibniz published +in the \textit{Acta Eruditorum} a paper containing the rudiments of +the integral calculus. The quantities $dx$~and~$dy$ are there +treated as infinitely small. He showed that by the use of his +notation, the properties of curves could be fully expressed by +equations. Thus the equation +\[ +y = \sqrt{2x - x^2} + \int \frac{dx}{\sqrt{2x - x^2}} +\] +characterises the cycloid.\Reference{38} +\index{Cycloid}% + +The great invention of Leibniz, now made public by his +articles in the \textit{Leipzig Acts}, made little impression upon the +mass of mathematicians. In Germany no one comprehended +%% -----File: 240.png---Folio 226------- +\index{Analysis situs}% +\index{Singular solutions}% +\index{Tchirnhausen!ref.\ to}% +the new calculus except Tschirnhaus, who remained indifferent +to it. The author's statements were too short and succinct +to make the calculus generally understood. The first +to recognise its importance and to take up the study of it +were two foreigners,---the Scotchman \textit{John Craig}, and +\index{Craig, J.}% +the Swiss \textit{James Bernoulli}. The latter wrote Leibniz a +\index{Bernoulli, James (born 1654)!ref.\ to}% +\index{Bernoulli, John (born 1667)!ref.\ to}% +letter in~1687, wishing to be initiated into the mysteries +of the new analysis. Leibniz was then travelling abroad, +so that this letter remained unanswered till~\DPtypo{1790}{1690}. James +Bernoulli succeeded, meanwhile, by close application, in uncovering +the secrets of the differential calculus without assistance. +He and his brother John proved to be mathematicians +of exceptional power. They applied themselves to the new +science with a success and to an extent which made Leibniz +declare that it was as much theirs as his. Leibniz carried on +an extensive correspondence with them, as well as with other +mathematicians. In a letter to John Bernoulli he suggests, +among other things, that the integral calculus be improved by +reducing integrals back to certain fundamental irreducible +forms. The integration of logarithmic expressions was then +studied. The writings of Leibniz contain many innovations, +and anticipations of since prominent methods. Thus he made +use of variable parameters, laid the foundation of \textit{analysis in +situ}, introduced the first notion of determinants in his effort +\index{Determinants}% +to simplify the expression arising in the elimination of the +unknown quantities from a set of linear equations. He +resorted to the device of breaking up certain fractions into +the sum of other fractions for the purpose of easier integration; +he explicitly assumed the principle of continuity; he gave the +\index{Continuity}% +first instance of a ``singular solution,'' and laid the foundation +to the theory of envelopes in two papers, one of which contains +for the first time the terms \emph{co-ordinate} and \emph{axes of co-ordinates}. +\index{Co-ordinates!first use of term}% +He wrote on osculating curves, but his paper contained the +\index{Curves!osculating}% +\index{Curves!theory of}% +%% -----File: 241.png---Folio 227------- +\index{Fluxional controversy|(}% +error (pointed out by John Bernoulli, but not admitted by +him) that an osculating circle will necessarily cut a curve in +four consecutive points. Well known is his theorem on the +$n$th differential coefficient of the product of two functions of a +\index{Differential calculus|)}% +\index{Differential calculus!controversy between Newton and Leibniz|(}% +variable. Of his many papers on mechanics, some are valuable, +\index{Mechanics!Leibniz}% +while others contain grave errors. + +Before tracing the further development of the calculus we +shall sketch the history of that long and bitter controversy +between English and Continental mathematicians on the invention +of the calculus. The question was, did Leibniz invent it +independently of Newton, or was he a plagiarist? +\index{Newton, controversy with Leibniz|(}% + +We must begin with the early correspondence between the +parties appearing in this dispute. Newton had begun using +his notation of fluxions in 1666.\Reference{41} In~1669 Barrow sent Collins +\index{Barrow!ref.\ to}% +\index{Collins}% +Newton's tract, \textit{De Analysi per Equationes}, etc. + +The first visit of Leibniz to London extended from the 11th +of January until March,~1673. He was in the habit of committing +to writing important scientific communications received +from others. In~1890 Gerhardt discovered in the royal library +\index{Gerhardt}% +at Hanover a sheet of manuscript with notes taken by Leibniz +during this journey.\Reference{40} They are headed ``Observata Philosophica +in itinere Anglicano sub initium anni~1673.'' The sheet +is divided by horizontal lines into sections. The sections +given to Chymica, Mechanica, Magnetica, Botanica, Anatomica, +Medica, Miscellanea, contain extensive memoranda, while those +devoted to mathematics have very few notes. Under Geometrica +he says only this: ``Tangentes omnium figurarum. +Figurarum geometricarum explicatio per motum puncti in +moto lati.'' We suspect from this that Leibniz had read +Barrow's lectures. Newton is referred to only under Optica. +Evidently Leibniz did not obtain a knowledge of fluxions +during this visit to London, nor is it claimed that he did by +his opponents. +%% -----File: 242.png---Folio 228------- + +Various letters of Newton, Collins, and others, up to the +\index{Collins}% +beginning of~1676, state that Newton invented a method by +which tangents could be drawn without the necessity of freeing +their equations from irrational terms. Leibniz announced in +1674 to Oldenburg, then secretary of the Royal Society, that +\index{Oldenburg}% +he possessed very general analytical methods, by which he had +found theorems of great importance on the quadrature of the +circle by means of series. In answer, Oldenburg stated +Newton and James Gregory had also discovered methods of +\index{Gregory, James}% +quadratures, which extended to the circle. Leibniz desired to +have these methods communicated to him; and Newton, at +the request of Oldenburg and Collins, wrote to the former the +celebrated letters of June~13 and October~24, 1676. The first +contained the Binomial Theorem and a variety of other matters +relating to infinite series and quadratures; but nothing +directly on the method of fluxions. Leibniz in reply speaks +in the highest terms of what Newton had done, and requests +further explanation. Newton in his second letter just mentioned +explains the way in which he found the Binomial +Theorem, and also communicates his method of fluxions and +fluents in form of an anagram in which all the letters in the +sentence communicated were placed in alphabetical order. +Thus Newton says that his method of drawing tangents was +\[ +6\,\textit{a\,cc\,d\,æ }\ +13\,\textit{e\,ff }\ +7\,\textit{i }\ +3\,\textit{l }\ +9\,\textit{n }\ +4\,\textit{o }\ +4\,\textit{q\,rr }\ +4\,\textit{s }\ +9\,\textit{t }\ +12\,\textit{v\,x}. +\] +The sentence was, \Lat{``Data æquatione quotcunque fluentes +quantitates involvente fluxiones invenire, et vice versa.''} +(``Having any given equation involving never so many flowing +quantities, to find the fluxions, and vice versa.'') Surely this +anagram afforded no hint. Leibniz wrote a reply to Collins, +in which, without any desire of concealment, he explained the +principle, notation, and the use of the differential calculus. + +The death of Oldenburg brought this correspondence to a +%% -----File: 243.png---Folio 229------- +close. Nothing material happened till~1684, when Leibniz +published his first paper on the differential calculus in the +\textit{Leipzig Acts}, so that while Newton's claim to the priority of +invention must be admitted by all, it must also be granted +that Leibniz was the first to give the full benefit of the calculus +to the world. Thus, while Newton's invention remained a +secret, communicated only to a few friends, the calculus of +Leibniz was spreading over the Continent. No rivalry or +hostility existed, as yet, between the illustrious scientists. +Newton expressed a very favourable opinion of Leibniz's +inventions, known to him through the above correspondence +with Oldenburg, in the following celebrated scholium (\textit{Principia}, +\index{Newton's Principia}% +\index{Principia (Newton's)}% +first edition, 1687, Book~II., Prop.~7, scholium):--- + +``In letters which went between me and that most excellent +geometer, G.~G. Leibniz, ten years ago, when I signified that +I was in the knowledge of a method of determining maxima +and minima, of drawing tangents, and the like, and when I +concealed it in transposed letters involving this sentence (Data +æquatione, etc., above cited), that most distinguished man +wrote back that he had also fallen upon a method of the same +kind, and communicated his method, which hardly differed +from mine, except in his forms of words and symbols.'' + +As regards this passage, we shall see that Newton was afterwards +weak enough, as De Morgan says: ``First, to deny the +\index{Demorgan@{De Morgan}!ref.\ to}% +plain and obvious meaning, and secondly, to omit it entirely +from the third edition of the \textit{Principia}.'' On the Continent, +great progress was made in the calculus by Leibniz and his +coadjutors, the brothers James and John Bernoulli, and +\index{Bernoulli, James (born 1654)!ref.\ to}% +\index{Bernoulli, John (born 1667)!ref.\ to}% +Marquis de l'Hospital. In~1695 Wallis informed Newton by +\index{Lhospital@{L'Hospital}!ref.\ to}% +\index{Wallis!ref.\ to}% +letter that ``he had heard that his notions of fluxions passed +in Holland with great applause by the name of `Leibniz's +Calculus Differentialis.'\,'' Accordingly Wallis stated in the +preface to a volume of his works that the calculus differentialis +%% -----File: 244.png---Folio 230------- +was Newton's method of fluxions which had been +communicated to Leibniz in the Oldenburg letters. A review +of Wallis' works, in the \textit{Leipzig Acts} for~1696, reminded the +reader of Newton's own admission in the scholium above +cited. + +For fifteen years Leibniz had enjoyed unchallenged the +honour of being the inventor of his calculus. But in 1699 Fato +de Duillier, a Swiss, who had settled in England, stated in a +\index{Duillier}% +mathematical paper, presented to the Royal Society, his conviction +that Newton was the first inventor; adding that, +whether Leibniz, the second inventor, had borrowed anything +from the other, he would leave to the judgment of those who +had seen the letters and manuscripts of Newton. This was +the first distinct insinuation of plagiarism. It would seem that +the English mathematicians had for some time been cherishing +suspicions unfavourable to Leibniz. A feeling had doubtless +long prevailed that Leibniz, during his second visit to London +in~1676, had or might have seen among the papers of Collins, +\index{Collins}% +Newton's \textit{Analysis per æquationes}, etc., which contained applications +of the fluxionary method, but no systematic development +or explanation of it. Leibniz certainly did see at least +part of this tract. During the week spent in London, he took +note of whatever interested him among the letters and papers +of Collins. His memoranda discovered by Gerhardt in 1849 in +\index{Gerhardt}% +the Hanover library fill two sheets.\Reference{40} The one bearing on our +question is headed ``Excerpta ex tractatu Newtoni Msc.~de +Analysi per æquationes numero terminorum infinitas.'' The +notes are very brief, excepting those \textit{De Resolutione æquationum +affectarum}, of which there is an almost complete copy. +This part was evidently new to him. If he examined +Newton's entire tract, the other parts did not particularly +impress him. From it he seems to have gained nothing pertaining +to the infinitesimal calculus. By the previous introduction +%% -----File: 245.png---Folio 231------- +of his own algorithm he had made greater progress +than by what came to his knowledge in London. Nothing +mathematical that he had received engaged his thoughts in +the immediate future, for on his way back to Holland he composed +a lengthy dialogue on mechanical subjects. + +Duillier's insinuations lighted up a flame of discord which a +whole century was hardly sufficient to extinguish. Leibniz, +who had never contested the priority of Newton's discovery, +and who appeared to be quite satisfied with Newton's admission +in his scholium, now appears for the first time in the +controversy. He made an animated reply in the \textit{Leipzig Acts}, +and complained to the Royal Society of the injustice done him. + +Here the affair rested for some time. In the \textit{Quadrature of +Curves}, published~1704, for the first time, a formal exposition +of the method and notation of fluxions was made public. In~1705 +appeared an unfavourable review of this in the \textit{Leipzig +Acts}, stating that Newton uses and always has used fluxions +for the differences of Leibniz. This was considered by Newton's +friends an imputation of plagiarism on the part of their +chief, but this interpretation was always strenuously resisted +by Leibniz. Keill, professor of astronomy at Oxford, undertook +\index{Keill}% +with more zeal than judgment the defence of Newton. +In a paper inserted in the \textit{Philosophical Transactions} of~1708, +he claimed that Newton was the first inventor of fluxions and +``that the same calculus was afterward published by Leibniz, +the name and the mode of notation being changed.'' Leibniz +complained to the secretary of the Royal Society of bad treatment +and requested the interference of that body to induce +Keill to disavow the intention of imputing fraud. Keill was +not made to retract his accusation; on the contrary, was +authorised by Newton and the Royal Society to explain and +defend his statement. This he did in a long letter. Leibniz +thereupon complained that the charge was now more open than +%% -----File: 246.png---Folio 232------- +before, and appealed for justice to the Royal Society and to +Newton himself. The Royal Society, thus appealed to as a +judge, appointed a committee which collected and reported +upon a large mass of documents---mostly letters from and to +Newton, Leibniz, Wallis, Collins, etc. This report, called the +\index{Collins}% +\textit{Commercium Epistolicum}, appeared in the year~1712 and again +\index{Commercium epistolicum}% +in~1725, with a Recensio prefixed, and additional notes by Keill. +\index{Keill}% +The final conclusion in the \textit{Commercium Epistolicum} was +that Newton was the first inventor. But this was not to the +point. The question was not whether Newton was the first +inventor, but whether Leibniz had stolen the method. The +committee had not formally ventured to assert their belief +that Leibniz was a plagiarist. Yet there runs throughout the +document a desire of proving Leibniz guilty of more than +they meant positively to affirm. Leibniz protested only in +private letters against the proceeding of the Royal Society, +declaring that he would not answer an argument so weak. +John Bernoulli, in a letter to Leibniz, which was published +\index{Bernoulli, John (born 1667)!ref.\ to}% +later in an anonymous tract, is as decidedly unfair towards +Newton as the friends of the latter had been towards Leibniz. +Keill replied, and then Newton and Leibniz appear as mutual +accusers in several letters addressed to third parties. In a +letter to Conti, April~9, 1716, Leibniz again reminded Newton +of the admission he had made in the scholium, which he was +now desirous of disavowing; Leibniz also states that he +always believed Newton, but that, seeing him connive at +accusations which he must have known to be false, it was +natural that he (Leibniz) should begin to doubt. Newton +did not reply to this letter, but circulated some remarks among +his friends which he published immediately after hearing +of the death of Leibniz, November~14, 1716. This paper +of Newton gives the following explanation pertaining to the +scholium in question: ``He [Leibniz] pretends that in my +%% -----File: 247.png---Folio 233------- +\index{Differential calculus!controversy between Newton and Leibniz|)}% +\index{Fluxional controversy|)}% +book of principles I allowed him the invention of the calculus +differentialis, independently of my own; and that to attribute +this invention to myself is contrary to my knowledge there +avowed. But in the paragraph there referred unto I do not +find one word to this purpose.'' In the third edition of the +\textit{Principia}, 1726, Newton omitted the scholium and substituted +\index{Newton's Principia}% +\index{Principia (Newton's)}% +in its place another, in which the name of Leibniz does not +appear. + +National pride and party feeling long prevented the adoption +of impartial opinions in England, but now it is generally admitted +by nearly all familiar with the matter, that Leibniz +really was an independent inventor. Perhaps the most telling +evidence to show that Leibniz was an independent inventor +is found in the study of his mathematical papers (collected +and edited by C.~I. Gerhardt, in six volumes, Berlin, 1849--1860), +\index{Gerhardt}% +which point out a gradual and natural evolution of the +rules of the calculus in his own mind. ``There was throughout +the whole dispute,'' says De~Morgan, ``a confusion between +\index{Demorgan@{De Morgan}!ref.\ to}% +the knowledge of fluxions or differentials and that of a \emph{calculus} +of fluxions or differentials; that is, a digested method with +general rules.'' + +This controversy is to be regretted on account of the long +and bitter alienation which it produced between English and +Continental mathematicians. It stopped almost completely +all interchange of ideas on scientific subjects. The English +adhered closely to Newton's methods and, until about~1820, +\index{Newton, controversy with Leibniz|)}% +remained, in most cases, ignorant of the brilliant mathematical +discoveries that were being made on the Continent. The loss +in point of scientific advantage was almost entirely on the +side of Britain. The only way in which this dispute may be +said, in a small measure, to have furthered the progress of +mathematics, is through the challenge problems by which +each side attempted to annoy its adversaries. +%% -----File: 248.png---Folio 234------- +\index{Brachistochrone (line of swiftest descent)}% +\index{Curve of swiftest descent}% + +The recurring practice of issuing challenge problems was +inaugurated at this time by Leibniz. They were, at first, not +intended as defiances, but merely as exercises in the new calculus. +Such was the problem of the isochronous curve (to +\index{Isochronous curve}% +find the curve along which a body falls with uniform velocity), +proposed by him to the Cartesians in~1687, and solved by +James Bernoulli, himself, and John Bernoulli. James Bernoulli +\index{Bernoulli, John (born 1667)!ref.\ to}% +proposed in the Leipzig \textit{Journal} the question to find the +curve (the catenary) formed by a chain of uniform weight +\index{Catenary}% +suspended freely from its ends. It was resolved by Huygens, +\index{Huygens!ref.\ to}% +Leibniz, and himself. In~1697 John Bernoulli challenged the +best mathematicians in Europe to solve the difficult problem, +to find the curve (the cycloid) along which a body falls from +\index{Cycloid}% +one point to another in the shortest possible time. Leibniz +solved it the day he received it. Newton, de~l'Hospital, and +\index{Lhospital@{L'Hospital}!ref.\ to}% +the two Bernoullis gave solutions. Newton's appeared anonymously +in the \textit{Philosophical Transactions}, but John Bernoulli +recognised in it his powerful mind, ``anquam,'' he says, ``ex +ungue leonem.'' The problem of orthogonal trajectories (a +\index{Trajectories}% +system of curves described by a known law being given, to +describe a curve which shall cut them all at right angles) had +been long proposed in the \textit{Acta Eruditorum}, but failed at +first to receive much attention. It was again proposed in~1716 +by Leibniz, to feel the pulse of the English mathematicians. + +This may be considered as the first defiance problem professedly +aimed at the English. Newton solved it the same +evening on which it was delivered to him, although he was +much fatigued by the day's work at the mint. His solution, +as published, was a general plan of an investigation rather +than an actual solution, and was, on that account, criticised by +Bernoulli as being of no value. Brook Taylor undertook the +\index{Taylor, B.!ref.\ to}% +defence of it, but ended by using very reprehensible language. +%% -----File: 249.png---Folio 235------- +Bernoulli was not to be outdone in incivility, and made a +bitter reply. Not long afterwards Taylor sent an open defiance +to Continental mathematicians of a problem on the +integration of a fluxion of complicated form which was known +to very few geometers in England and supposed to be beyond +the power of their adversaries. The selection was injudicious, +for Bernoulli had long before explained the method of this +and similar integrations. It served only to display the skill +and augment the triumph of the followers of Leibniz. The +last and most unskilful challenge was by John Keill. The +\index{Keill}% +problem was to find the path of a projectile in a medium +which resists proportionally to the square of the velocity. +Without first making sure that he himself could solve it, +Keill boldly challenged Bernoulli to produce a solution. The +latter resolved the question in very short time, not only for a +resistance proportional to the square, but to any power of the +velocity. Suspecting the weakness of the adversary, he repeatedly +offered to send his solution to a confidential person +in London, provided Keill would do the same. Keill never +made a reply, and Bernoulli abused him and cruelly exulted +over him.\Reference{26} + +The explanations of the fundamental principles of the calculus, +as given by Newton and Leibniz, lacked clearness and +rigour. For that reason it met with opposition from several +quarters. In~1694 Bernard Nieuwentyt of Holland denied +\index{Nieuwentyt}% +the existence of differentials of higher orders and objected to +the practice of neglecting infinitely small quantities. These +objections Leibniz was not able to meet satisfactorily. In his +reply he said the value of $\dfrac{dy}{dx}$ in geometry could be expressed +as the ratio of finite quantities. In the interpretation of $dx$~and~$dy$ +Leibniz vacillated. At one time they appear in his +\index{Leibniz|)}% +writings as finite lines; then they are called infinitely small +%% -----File: 250.png---Folio 236------- +quantities, and again, \emph{quantitates inassignabiles}, which spring +from \emph{quantitates assignabiles} by the law of continuity. In this +last presentation Leibniz approached nearest to Newton. + +In England the principles of fluxions were boldly attacked +by Bishop Berkeley, the eminent metaphysician, who argued +\index{Berkeley}% +with great acuteness, contending, among other things, that +the fundamental idea of supposing a finite ratio to exist +between terms absolutely evanescent---``the ghosts of departed +quantities,'' as he called them---was absurd and unintelligible. +The reply made by Jurin failed to remove all the +\index{Jurin}% +objections. Berkeley was the first to point out what was +again shown later by Lazare Carnot, that correct answers were +\index{Carnot, Lazare!ref.\ to}% +reached by a ``compensation of errors.'' Berkeley's attack +was not devoid of good results, for it was the immediate cause +of the work on fluxions by Maclaurin. In France \textit{Michel +\index{Maclaurin!ref.\ to}% +Rolle} rejected the differential calculus and had a controversy +\index{Differential calculus|(}% +\index{Differential calculus!philosophy of}% +\index{Rolle!ref.\ to}% +with \textit{Varignon} on the subject. +\index{Varignon!ref.\ to}% + +Among the most vigorous promoters of the calculus on the +Continent were the Bernoullis. They and Euler made Basel +in Switzerland famous as the cradle of great mathematicians. +The family of Bernoullis furnished in course of a century +eight members who distinguished themselves in mathematics. +We subjoin the following genealogical table:--- +\begin{center} +\begin{minipage}{\linewidth} +\footnotesize +\noindent$\overbrace{ +\begin{tabular}{@{}lll@{}} +\textbf{Jacob}, 1654--1705 & \textit{Nicolaus} & \textbf{Johann}, 1667--1748 \\ +& \qquad$|$ & \qquad$|$ \\ +& \textit{Nicolaus}, 1687--1759 & \textit{Nicolaus}, 1695--1726 \\ +& & \textbf{Daniel}, 1700--1782 \\ +& & \textit{Johann}, 1710--1790 \\ +& \multicolumn{2}{c}{% +\qquad\makebox[0pt][c]{$\overbrace{ +\begin{tabular}{@{}ccc@{}} +\textit{Daniel} & \textit{Johann}, 1744--1807 & \textit{Jacob}, 1758--1789 +\end{tabular}}^{}$}} +\end{tabular}}^{\mbox{\small\textit{Nicolaus Bernoulli}, the Father}}$ +\end{minipage} +\end{center} +\index{Bernoullis, genealogical table of}% +Most celebrated were the two brothers Jacob (James) and +Johann (John), and Daniel, the son of John. James and +%% -----File: 251.png---Folio 237------- +\index{Integral calculus!origin of term}% +\index{Isoperimetrical figures}% +\index{Mechanics!Bernoullis}% +\index{Probability}% +John were staunch friends of Leibniz and worked hand in +\index{Leibniz!ref.\ to}% +hand with him. \textbf{James Bernoulli} (1654--1705) was born in +\index{Bernoulli, James (born 1654)}% +\index{Bernoulli, John (born 1667)!ref.\ to}% +Basel. Becoming interested in the calculus, he mastered it +without aid from a teacher. From 1687 until his death he +occupied the mathematical chair at the University of Basel. +He was the first to give a solution to Leibniz's problem of the +isochronous curve. In his solution, published in the \textit{Acta +Eruditorum}, 1690, we meet for the first time with the word +\emph{integral}. Leibniz had called the integral calculus \textit{calculus +summatorius}, but in 1696 the term \textit{calculus integralis} was +agreed upon between Leibniz and John Bernoulli. James +proposed the problem of the catenary, then proved the correctness +\index{Catenary}% +of Leibniz's construction of this curve, and solved the +more complicated problems, supposing the string to be (1)~of +variable density, (2)~extensible, (3)~acted upon at each point +by a force directed to a fixed centre. Of these problems he +published answers without explanations, while his brother +John gave in addition their theory. He determined the shape +of the ``elastic curve'' formed by an elastic plate or rod fixed +\index{Elastic curve}% +at one end and bent by a weight applied to the other end; of +the ``lintearia,'' a flexible rectangular plate with two sides +\index{Lintearia}% +fixed horizontally at the same height, filled with a liquid; of +the ``volaria,'' a rectangular sail filled with wind. He studied +\index{Volaria}% +the loxodromic and logarithmic spirals, in the last of which +\index{Spirals}% +he took particular delight from its remarkable property of +reproducing itself under a variety of conditions. Following +the example of Archimedes, he willed that the curve be engraved +upon his \Change{tombstone}{tomb-stone} with the inscription ``\textit{eadem mutata +resurgo}.'' In~1696 he proposed the famous problem of isoperimetrical +figures, and in 1701 published his own solution. He +wrote a work on \textit{Ars Conjectandi}, which is a development of +the calculus of probabilities and contains the investigation +now called ``Bernoulli's theorem'' and the so-called ``numbers +\index{Bernoulli's theorem}% +%% -----File: 252.png---Folio 238------- +\index{Brachistochrone (line of swiftest descent)}% +\index{Caustics}% +\index{Mechanics!Bernoullis}% +\index{Numbers of Bernoulli}% +of Bernoulli,'' which are in fact (though not so considered by +\index{Bernoulli, Daniel}% +\index{Bernoulli, Nicolaus (born 1695)}% +\index{Bernoulli, James (born 1654)}% +\index{Bernoulli, John (born 1667)}% +him) the coefficients of $\dfrac{x^n}{n!}$ in the expansion of $(e^x - 1)^{-1}$. Of +his collected works, in three volumes, one was printed in~1713, +the other two in~1744. + +\textbf{John Bernoulli} (1667--1748) was initiated into mathematics +by his brother. He afterwards visited France, where he met +Malebranche, Cassini, De Lahire, Varignon, and de l'Hospital. +For ten years he occupied the mathematical chair at Gröningen +and then succeeded his brother at Basel. He was one of the +most enthusiastic teachers and most successful original investigators +of his time. He was a member of almost every learned +society in Europe. His controversies were almost as numerous +as his discoveries. He was ardent in his friendships, but +unfair, mean, and violent toward all who incurred his dislike---even +his own brother and son. He had a bitter dispute +with James on the isoperimetrical problem. James convicted +him of several paralogisms. After his brother's death he +attempted to substitute a disguised solution of the former for +an incorrect one of his own. John admired the merits of +Leibniz and Euler, but was blind to those of Newton. He +\index{Newton!ref.\ to}% +immensely enriched the integral calculus by his labours. +Among his discoveries are the exponential calculus, the line +of swiftest descent, and its beautiful relation to the path +\index{Curve of swiftest descent}% +described by a ray passing through strata of variable density. +He treated trigonometry by the analytical method, studied +\index{Trigonometry}% +caustic curves and trajectories. Several times he was given +\index{Trajectories}% +prizes by the Academy of Science in Paris. + +Of his sons, \textbf{Nicholas} and \textbf{Daniel} were appointed professors +of mathematics at the same time in the Academy of St.~Petersburg. +The former soon died in the prime of life; the +latter returned to Basel in~1733, where he assumed the chair +of experimental philosophy. His first mathematical publication +%% -----File: 253.png---Folio 239------- +\index{Differential equations}% +was the solution of a differential equation proposed by +Riccati. He wrote a work on hydrodynamics. His investigations +\index{Hydrodynamics}% +\index{Riccati!ref.\ to}% +on probability are remarkable for their boldness and +\index{Probability}% +originality. He proposed the theory of \emph{moral expectation}, +\index{Moral expectation}% +which he thought would give results more in accordance with +our ordinary notions than the theory of \emph{mathematical probability}. +His ``moral expectation'' has become classic, but no +one ever makes use of it. He applies the theory of probability +to insurance; to determine the mortality caused by small-pox +\index{Insurance}% +at various stages of life; to determine the number of survivors +at a given age from a given number of births; to determine +how much inoculation lengthens the average duration of life. +He showed how the differential calculus could be used in the +theory of probability. He and Euler enjoyed the honour of +\index{Euler!ref.\ to}% +having gained or shared no less than ten prizes from the +Academy of Sciences in Paris. + +\textbf{Johann Bernoulli} (born 1710) succeeded his father in the +\index{Bernoulli, Nicolaus (born 1687)}% +\index{Bernoulli, James (born 1758)}% +\index{Bernoulli, John (born 1710)}% +\index{Bernoulli, John (born 1744)}% +professorship of mathematics at Basel. He captured three +prizes (on the capstan, the propagation of light, and the +magnet) from the Academy of Sciences at Paris. \textbf{Nicolaus +Bernoulli} (born 1687) held for a time the mathematical chair +at Padua which Galileo had once filled. \textbf{Johann Bernoulli} +(born 1744) at the age of nineteen was appointed astronomer +royal at Berlin, and afterwards director of the mathematical +department of the Academy. His brother \textit{Jacob} took upon +himself the duties of the chair of experimental physics at +Basel, previously performed by his uncle Jacob, and later +was appointed mathematical professor in the Academy at St.\ +Petersburg. + +Brief mention will now be made of some other mathematicians +belonging to the period of Newton, Leibniz, and the +elder Bernoullis. + +\textbf{Guillaume François Antoine l'Hospital} (1661--1704), a pupil +\index{Hospital,~l'}% +\index{Lhospital@{L'Hospital}}% +%% -----File: 254.png---Folio 240------- +\index{Analytic geometry}% +\index{Cycloid}% +\index{Equations!theory of}% +\index{Hospital,~l'}% +\index{Laplace!ref.\ to}% +\index{Lhospital@{L'Hospital}}% +of John Bernoulli, has already been mentioned as taking +part in the challenges issued by Leibniz and the Bernoullis. +He helped powerfully in making the calculus of Leibniz better +known to the mass of mathematicians by the publication of a +treatise thereon in 1696. This contains for the first time the +method of finding the limiting value of a fraction whose two +terms tend toward zero at the same time. + +Another zealous French advocate of the calculus was \textbf{Pierre +Varignon} (1654--1722). \textbf{Joseph Saurin} (1659--1737) solved the +\index{Saurin}% +\index{Varignon}% +delicate problem of how to determine the tangents at the +multiple points of algebraic curves. \textbf{François Nicole} (1683--1758) +\index{Curves!theory of}% +\index{Nicole}% +in 1717 issued the first systematic treatise on finite +differences, in which he finds the sums of a considerable +\Addendlabel{240}% +number of interesting series. He wrote also on roulettes, +particularly spherical epicycloids, and their rectification. Also +interested in finite differences was \textbf{Pierre Raymond de~Montmort} +\index{Finite differences}% +\index{Montmort, de}% +(1678--1719). His chief writings, on the theory of probability, +\index{Probability}% +served to stimulate his more distinguished successor, De~Moivre. +\index{Demoivre@{De Moivre}}% +\index{Moivre, de}% +\textbf{Jean Paul de~Gua} (1713--1785) gave the demonstration +\index{Gua, de}% +of Descartes' rule of signs, now given in books. This skilful +\index{Descartes!ref.\ to}% +geometer wrote in 1740 a work on analytical geometry, the +\index{Geometry!modern synthetic}% +object of which was to show that most investigations on curves +could be carried on with the analysis of Descartes quite as +easily as with the calculus. He shows how to find the tangents, +asymptotes, and various singular points of curves of all +degrees, and proved by perspective that several of these points +can be at infinity. A mathematician who clung to the methods +of the ancients was \textbf{Philippe de~Lahire} (1640--1718), a pupil of +\index{Lahire, de}% +Desargues. His work on conic sections is purely synthetic, +\index{Desargues!ref.\ to}% +but differs from ancient treatises in deducing the properties of +conics from those of the circle in the same manner as did +Desargues and Pascal. His innovations stand in close relation +\index{Pascal!ref.\ to}% +with modern synthetic geometry. He wrote on roulettes, on +%% -----File: 255.png---Folio 241------- +\index{Caustics}% +\index{Magic squares}% +\index{p@{$\pi$}: values for!Fagnano's}% +\index{Tchirnhausen}% +graphical methods, epicycloids, conchoids, and on magic +squares. \textbf{Michel Rolle} (1652--1719) is the author of a theorem +\index{Rolle}% +named after him. + +Of Italian mathematicians, Riccati and Fagnano must not +\index{Fagnano}% +\index{Riccati}% +remain unmentioned. \textbf{Jacopo Francesco, Count Riccati} (1676--1754) +is best known in connection with his problem, called +Riccati's equation, published in the \textit{Acta Eruditorum} in~1724. +He succeeded in integrating this differential equation for some +special cases. A geometrician of remarkable power was \textbf{Giulio +Carlo, Count de~Fagnano} (1682--1766). He discovered the following +formula, $\pi = 2i \log \dfrac{1 - i}{1 + i}$, in which he anticipated Euler +\index{Euler!ref.\ to}% +in the use of imaginary exponents and logarithms. His studies +\index{Exponents}% +\index{Imaginary quantities}% +on the rectification of the ellipse and hyperbola are the starting-points +of the theory of elliptic functions. He showed, for +\index{Elliptic functions}% +instance, that two arcs of an ellipse can be found in an indefinite +number of ways, whose difference is expressible by a +right line. + +In Germany the only noted contemporary of Leibniz is +\index{Leibniz!ref.\ to}% +\textbf{Ehrenfried Walter Tschirnhausen} (1651--1708), who discovered +the caustic of reflection, experimented on metallic reflectors +and large burning-glasses, and gave us a method of transforming +equations named after him. Believing that the most +\index{Equations!theory of}% +simple methods (like those of the ancients) are the most +correct, he concluded that in the researches relating to the +properties of curves the calculus might as well be dispensed +with. + +After the death of Leibniz there was in Germany not a +single mathematician of note. \textbf{Christian Wolf} (1679--1754), +\index{Wolf, C.}% +professor at Halle, was ambitious to figure as successor of +Leibniz, but he ``forced the ingenious ideas of Leibniz into a +pedantic scholasticism, and had the unenviable reputation of +having presented the elements of the arithmetic, algebra, and +%% -----File: 256.png---Folio 242------- +\index{Curves!theory of}% +\index{Differential calculus|)}% +\index{Strings, vibrating}% +\index{Vibrating strings}% +analysis developed since the time of the Renaissance in the +form of Euclid,---of course only in outward form, for into the +spirit of them he was quite unable to penetrate.''\Reference{16} + +The contemporaries and immediate successors of Newton in +Great Britain were men of no mean merit. We have reference +to Cotes, Taylor, Maclaurin, and De~Moivre. We are +\index{Cotes}% +\index{Demoivre@{De Moivre}}% +\index{Moivre, de}% +\index{Taylor, B.}% +told that at the death of \textbf{Roger Cotes} (1682--1716), Newton +exclaimed, ``If Cotes had lived, we might have known something.'' +It was at the request of Dr.~Bentley that Cotes +undertook the publication of the second edition of Newton's +\textit{Principia}. His mathematical papers were published after his +\index{Newton's Principia}% +\index{Principia (Newton's)}% +death by Robert Smith, his successor in the Plumbian professorship +\index{Smith, R.}% +at Trinity College. The title of the work, \textit{Harmonia +Mensurarum}, was suggested by the following theorem +contained in it: If on each radius vector, through a fixed point~$O$, +there be taken a point~$R$, such that the reciprocal of $OR$ be +the arithmetic mean of the reciprocals of $OR_1, OR_2, \ldots OR_n$, +then the locus of~$R$ will be a straight line. In this work +progress was made in the application of logarithms and the +\index{Logarithms}% +properties of the circle to the calculus of fluents. To Cotes +we owe a theorem in trigonometry which depends on the +\index{Trigonometry}% +forming of factors of~$x^n - 1$. Chief among the admirers of +Newton were Taylor and Maclaurin. The quarrel between +English and Continental mathematicians caused them to work +quite independently of their great contemporaries across the +Channel. + +\textbf{Brook Taylor} (1685--1731) was interested in many branches +of learning, and in the latter part of his life engaged mainly in +religious and philosophic speculations. His principal work, +\textit{Methodus incrementorum directa et inversa}, London, 1715--1717, +added a new branch to mathematics, now called ``finite differences.'' +\index{Finite differences}% +He made many important applications of it, particularly +to the study of the form of movement of vibrating +%% -----File: 257.png---Folio 243------- +\index{Centre!of oscillation}% +\index{Mechanics!Taylor}% +strings, first reduced to mechanical principles by him. This +work contains also ``Taylor's theorem,'' the importance of +\index{Taylor's theorem}% +which was not recognised by analysts for over fifty years, +until Lagrange pointed out its power. His proof of it does not +consider the question of convergency, and is quite worthless. +The first rigorous proof was given a century later by Cauchy. +\index{Cauchy!ref.\ to}% +Taylor's work contains the first correct explanation of astronomical +refraction. He wrote also a work on linear perspective, +a treatise which, like his other writings, suffers for +want of fulness and clearness of expression. At the age of +twenty-three he gave a remarkable solution of the problem of +the centre of oscillation, published in~1714. His claim to +\index{Oscillation, centre of}% +priority was unjustly disputed by John Bernoulli. +\index{Bernoulli, John (born 1667)!ref.\ to}% + +\textbf{Colin Maclaurin} (1698--1746) was elected professor of mathematics +\index{Maclaurin}% +at Aberdeen at the age of nineteen by competitive +examination, and in 1725 succeeded James Gregory at the University +\index{Gregory, James}% +of Edinburgh. He enjoyed the friendship of Newton, +\index{Newton!ref.\ to}% +and, inspired by Newton's discoveries, he published in 1719 his +\textit{Geometria Organica}, containing a new and remarkable mode +of generating conics, known by his name. A second tract, +\textit{De Linearum geometricarum Proprietatibus}, 1720, is remarkable +for the elegance of its demonstrations. It is based upon two +theorems: the first is the theorem of Cotes; the second is +\index{Cotes!ref.\ to}% +Maclaurin's: If through any point~$O$ a line be drawn meeting +the curve in $n$~points, and at these points tangents be drawn, +and if any other line through~$O$ cut the curve in $R_1$, $R_2$, etc., +and the system of $n$~tangents in $r_1$, $r_2$, etc., then $\sum \dfrac{1}{OR} = \sum \dfrac{1}{Or}$. +This and Cotes' theorem are generalisations of theorems of +Newton. Maclaurin uses these in his treatment of curves of +\index{Curves!theory of}% +the second and third degree, culminating in the remarkable +theorem that if a quadrangle has its vertices and the two +points of intersection of its opposite sides upon a curve of the +%% -----File: 258.png---Folio 244------- +third degree, then the tangents drawn at two opposite vertices +cut each other on the curve. He deduced independently +Pascal's theorem on the hexagram. The following is his extension +of this theorem (\textit{Phil.\ Trans.}, 1735): If a polygon +move so that each of its sides passes through a fixed point, +and if all its summits except one describe curves of the degrees +$m$,~$n$, $p$,~etc., respectively, then the free summit moves on a +curve of the degree $2 mnp\cdots$,\DPnote{** PP: Why dots?} which reduces to $mnp \cdots$ when +the fixed points all lie on a straight line. Maclaurin wrote on +\index{Maclaurin!ref.\ to}% +pedal curves. He is the author of an \textit{Algebra}. The object of +his treatise on \textit{Fluxions} was to found the doctrine of fluxions +on geometric demonstrations after the manner of the ancients, +and thus, by rigorous exposition, answer such attacks as Berkeley's +that the doctrine rested on false reasoning. The \textit{Fluxions} +contained for the first time the correct way of distinguishing +between maxima and minima, and explained their use in the +\index{Maxima and minima}% +theory of multiple points. ``Maclaurin's theorem'' was previously +given by James Stirling, and is but a particular case +\index{Stirling}% +of ``Taylor's theorem.'' Appended to the treatise on \textit{Fluxions} +is the solution of a number of beautiful geometric, mechanical, +and astronomical problems, in which he employs ancient +methods with such consummate skill as to induce Clairaut to +\index{Clairaut!ref.\ to}% +abandon analytic methods and to attack the problem of the +figure of the earth by pure geometry. His solutions commanded +the liveliest admiration of Lagrange. Maclaurin investigated +\index{Lagrange!ref.\ to}% +the attraction of the ellipsoid of revolution, and +showed that a homogeneous liquid mass revolving uniformly +around an axis under the action of gravity must assume the +form of an ellipsoid of revolution. Newton had given this +\index{Newton!ref.\ to}% +theorem without proof. Notwithstanding the genius of Maclaurin, +his influence on the progress of mathematics in Great +Britain was unfortunate; for, by his example, he induced his +countrymen to neglect analysis and to be indifferent to the +%% -----File: 259.png---Folio 245------- +wonderful progress in the higher analysis made on the Continent. + +It remains for us to speak of \textbf{Abraham de~Moivre} (1667--1754), +\index{Demoivre@{De Moivre}}% +\index{Moivre, de}% +who was of French descent, but was compelled to leave France +at the age of eighteen, on the Revocation of the Edict of Nantes. +He settled in London, where he gave lessons in mathematics. +He lived to the advanced age of eighty-seven and sank into a +state of almost total lethargy. His subsistence was latterly +dependent on the solution of questions on games of chance and +problems on probabilities, which he was in the habit of giving +at a tavern in St.~Martin's Lane. Shortly before his death he +declared that it was necessary for him to sleep ten or twenty +minutes longer every day. The day after he had reached the +total of over twenty-three hours, he slept exactly twenty-four +hours and then passed away in his sleep. De~Moivre enjoyed +the friendship of Newton and Halley. His power as a mathematician +lay in analytic rather than geometric investigation. +He revolutionised higher trigonometry by the discovery of the +\index{Trigonometry}% +theorem known by his name and by extending the theorems on +the multiplication and division of sectors from the circle to the +hyperbola. His work on the theory of probability surpasses +\index{Probability}% +anything done by any other mathematician except Laplace. +\index{Laplace!ref.\ to}% +His principal contributions are his investigations respecting +the Duration of Play, his Theory of Recurring Series, and +\index{Series}% +his extension of the value of Bernoulli's theorem by the aid +of Stirling's theorem.\Reference{42} His chief works are the \textit{Doctrine of +Chances}, 1716, the \textit{Miscellanea Analytica}, 1730, and his papers +in the \textit{Philosophical Transactions}. +%% -----File: 260.png---Folio 246------- + + +\Section{Euler, Lagrange, and Laplace} +\index{Euler!ref.\ to}% +\index{Lagrange!ref.\ to}% + +During the epoch of ninety years from 1730 to 1820 the French +and Swiss cultivated mathematics with most brilliant success. +No previous period had shown such an array of illustrious +names. At this time Switzerland had her Euler; France, her +Lagrange, Laplace, Legendre, and Monge. The mediocrity of +French mathematics which marked the time of Louis~XIV. +was now followed by one of the very brightest periods of all +history. England and Germany, on the other hand, which +during the unproductive period in France had their Newton +and Leibniz, could now boast of no great mathematician. +France now waved the mathematical sceptre. Mathematical +studies among the English and German people had sunk to +the lowest ebb. Among them the direction of original research +was ill-chosen. The former adhered with excessive partiality +to ancient geometrical methods; the latter produced the combinatorial +school, which brought forth nothing of value. + +The labours of Euler, Lagrange, and Laplace lay in higher +analysis, and this they developed to a wonderful degree. By +them analysis came to be completely severed from geometry. +During the preceding period the effort of mathematicians +not only in England, but, to some extent, even on the continent, +had been directed toward the solution of problems +clothed in geometric garb, and the results of calculation +were usually reduced to geometric form. A change now +took place. Euler brought about an emancipation of the +analytical calculus from geometry and established it as an +independent science. Lagrange and Laplace scrupulously +adhered to this separation. Building on the broad foundation +laid for higher analysis and mechanics by Newton +and Leibniz, Euler, with matchless fertility of mind, erected +%% -----File: 261.png---Folio 247------- +an elaborate structure. There are few great ideas pursued +by succeeding analysts which were not suggested by Euler, +or of which he did not share the honour of invention. +With, perhaps, less exuberance of invention, but with more +comprehensive genius and profounder reasoning, Lagrange +\index{Lagrange!ref.\ to}% +developed the infinitesimal calculus and put analytical +\index{Calculus!of variations}% +mechanics into the form in which we now know it. Laplace +applied the calculus and mechanics to the elaboration +of the theory of universal gravitation, and thus, largely extending +and supplementing the labours of Newton, gave a full +analytical discussion of the solar system. He also wrote an +epoch-marking work on Probability. Among the analytical +branches created during this period are the calculus of Variations +by Euler and Lagrange, Spherical Harmonics by Laplace +\index{Spherical Harmonics}% +and Legendre, and Elliptic Integrals by Legendre. +\index{Elliptic integrals}% +\index{Legendre!ref.\ to}% + +Comparing the growth of analysis at this time with the +growth during the time of Gauss, Cauchy, and recent mathematicians, +\index{Cauchy!ref.\ to}% +\index{Gauss!ref.\ to}% +we observe an important difference. During the +former period we witness mainly a development with reference +to \emph{form}. Placing almost implicit confidence in results of +calculation, mathematicians did not always pause to discover +rigorous proofs, and were thus led to general propositions, +some of which have since been found to be true in only special +cases. The Combinatorial School in Germany carried this +\index{Combinatorial School}% +tendency to the greatest extreme; they worshipped formalism +and paid no attention to the actual contents of formulæ. But +in recent times there has been added to the dexterity in the +formal treatment of problems, a much-needed rigour of demonstration. +A good example of this increased rigour is seen in +the present use of infinite series as compared to that of Euler, +\index{Infinite series}% +and of Lagrange in his earlier works. + +The ostracism of geometry, brought about by the master-minds %[** PP: Only instance] +of this period, could not last permanently. Indeed, a +%% -----File: 262.png---Folio 248------- +new geometric school sprang into existence in France before +the close of this period. Lagrange would not permit a single +\index{Lagrange!ref.\ to}% +diagram to appear in his \textit{Mécanique analytique}, but thirteen +years before his death, Monge published his epoch-making +\index{Monge!ref.\ to}% +\textit{Géometrie descriptive}. + +\textbf{Leonhard Euler} (1707--1783) was born in Basel. His father, +\index{Euler|(}% +a minister, gave him his first instruction in mathematics and +then sent him to the University of Basel, where he became a +favourite pupil of John Bernoulli. In his nineteenth year he +composed a dissertation on the masting of ships, which received +the second prize from the French Academy of Sciences. +When John Bernoulli's two sons, Daniel and Nicolaus, went to +Russia, they induced Catharine~I., in 1727, to invite their friend +Euler to St.~Petersburg, where Daniel, in 1733, was assigned to +the chair of mathematics. In 1735 the solving of an astronomical +problem, proposed by the Academy, for which several +eminent mathematicians had demanded some months' time, +was achieved in three days by Euler with aid of improved +methods of his own. But the effort threw him into a fever +and deprived him of the use of his right eye. With still +superior methods this same problem was solved later by the +illustrious Gauss in one hour!\Reference{47} The despotism of Anne~I. +\index{Gauss!ref.\ to}% +caused the gentle Euler to shrink from public affairs and to +devote all his time to science. After his call to Berlin by +Frederick the Great in 1747, the queen of Prussia, who +received him kindly, wondered how so distinguished a scholar +should be so timid and reticent. Euler naïvely replied, +``Madam, it is because I come from a country where, when one +speaks, one is hanged.'' In 1766 he with difficulty obtained +permission to depart from Berlin to accept a call by Catharine~II. +to St.~Petersburg. Soon after his return to Russia he +became blind, but this did not stop his wonderful literary +productiveness, which continued for seventeen years, until the +%% -----File: 263.png---Folio 249------- +\index{Beta function}% +\index{Gamma function}% +\index{Isoperimetrical figures}% +\index{Notation!trigonometry}% +day of his death.\Reference{45} He dictated to his servant his \textit{Anleitung +zur Algebra}, 1770, which, though purely elementary, is meritorious +as one of the earliest attempts to put the fundamental +processes on a sound basis. + +Euler wrote an immense number of works, chief of which +are the following: \textit{Introductio in analysin infinitorum}, 1748, +a work that caused a revolution in analytical mathematics, a +subject which had hitherto never been presented in so general +and systematic manner; \textit{Institutiones calculi differentialis}, 1755, +and \textit{Institutiones calculi integralis}, 1768--1770, which were the +most complete and accurate works on the calculus of that time, +\index{Calculus!of variations}% +and contained not only a full summary of everything then +known on this subject, but also the Beta and Gamma Functions +and other original investigations; \textit{Methodus inveniendi +lineas curvas maximi minimive proprietate gaudentes}, 1744, +which, displaying an amount of mathematical genius seldom +rivalled, contained his researches on the calculus of variations +(a subject afterwards improved by Lagrange), to the invention +of which Euler was led by the study of isoperimetrical curves, +the brachistochrone in a resisting medium, and the theory of +geodesics (subjects which had previously engaged the attention +\index{Geodesics}% +of the elder Bernoullis and others); the \textit{Theoria motuum planetarum +et cometarum}, 1744, \textit{Theoria motus lunæ}, 1753, \textit{Theoria +motuum lunæ}, 1772, are his chief works on astronomy; \textit{Ses +lettres à une princesse d'Allemagne sur quelques sujets de +Physique et de Philosophie}, 1770, was a work which enjoyed +great popularity. + +We proceed to mention the principal innovations and inventions +of Euler. He treated trigonometry as a branch of +\index{Trigonometry}% +analysis, introduced (simultaneously with Thomas Simpson in +\index{Simpson}% +England) the now current abbreviations for trigonometric +functions, and simplified formulæ by the simple expedient +of designating the angles of a triangle by $A$,~$B$,~$C$, and the +%% -----File: 264.png---Folio 250------- +\index{Bezout@{Bézout}!ref.\ to}% +opposite sides by $a$,~$b$,~$c$, respectively. He pointed out the +relation between trigonometric and exponential functions. In +a paper of 1737 we first meet the symbol~$\pi$ to denote $3.14159\ldots$.\Reference{21} +\index{p@{$\pi$}: values for!selection of letter~$\pi$}% +\Addendlabel{250}% +Euler laid down the rules for the transformation of co-ordinates +\index{Euler!ref.\ to}% +in space, gave a methodic analytic treatment of plane curves +and of surfaces of the second order. He was the first to +\index{Surfaces, theory of}% +discuss the equation of the second degree in three variables, +and to classify the surfaces represented by it. By criteria +analogous to those used in the classification of conics he +obtained five species. He devised a method of solving \Change{bi-quadratic}{biquadratic} +equations by assuming $x = \sqrt{\smash[b]{p}} + \sqrt{\smash[b]{q}} + \sqrt{r}$, with the +\index{Equations!solution of}% +\index{Equations!theory of}% +hope that it would lead him to a general solution of algebraic +equations. The method of elimination by solving a series of +\index{Elimination}% +linear equations (invented independently by \DPtypo{Bezout}{Bézout}) and the +method of elimination by symmetric functions, are due to him.\Reference{20} +\index{Symmetric functions}% +Far reaching are Euler's researches on logarithms. Leibniz +\index{Leibniz!ref.\ to}% +\index{Logarithms}% +and John Bernoulli once argued the question whether a +\index{Bernoulli, John (born 1667)!ref.\ to}% +negative number has a logarithm. Bernoulli claimed that +since $(-a)^2 = (+a)^2$, we have $\log(-a)^2= \log(+a)^2$ and +$2 \log(-a) = 2 \log(+a)$, and finally $\log(-a) = \log(+a)$. +Euler proved that $a$ has really an infinite number of logarithms, +all of which are imaginary when $a$ is negative, and all +except one when $a$ is positive. He then explained how +$\log(-a)^2$ might equal $\log(+a)^2$, and yet $\log(-a)$ not +equal $\log(+a)$. + +The subject of infinite series received new life from him. +\index{Infinite series}% +To his researches on series we owe the creation of the theory of +definite integrals by the development of the so-called \emph{Eulerian +integrals}. He warns his readers occasionally against the use +of divergent series, but is nevertheless very careless himself. +The rigid treatment to which infinite series are subjected now +was then undreamed of. No clear notions existed as to what +constitutes a convergent series. Neither Leibniz nor Jacob +%% -----File: 265.png---Folio 251------- +and John Bernoulli had entertained any serious doubt of the +\index{Bernoulli, Nicolaus (born 1687)}% +\index{Bernoulli, James (born 1654)!ref.\ to}% +\index{Bernoulli, John (born 1667)!ref.\ to}% +correctness of the expression $\frac{1}{2} = 1 - 1 + 1 - 1 +\dotsb$. Guido +Grandi went so far as to conclude from this that $\frac{1}{2} = 0 + 0 + +\index{Grandi}% +0 +\dotsb$. In the treatment of series Leibniz advanced a metaphysical +\index{Leibniz!ref.\ to}% +method of proof which held sway over the minds of +the elder Bernoullis, and even of Euler.\Reference{46} The tendency of +that reasoning was to justify results which seem to us now +highly absurd. The looseness of treatment can best be seen +from examples. The very paper in which Euler cautions +against divergent series contains the proof that +\begin{gather*} +\dotsb \frac{1}{n^2} + \frac{1}{n} + 1 + n + n^2 +\dotsb = 0 \text{ as follows:} \\ +n+n^2+\dotsb = \frac{n}{1-n},\quad 1+\frac{1}{n}+\frac{1}{n^2}+\dotsb=\frac{n}{n-1}; +\end{gather*} +these added give zero. Euler has no hesitation to write +$1-3+5-7+\dotsb=0$, and no one objected to such results +excepting Nicolaus Bernoulli, the nephew of John and Jacob. +Strange to say, Euler finally succeeded in converting Nicolaus +Bernoulli to his own erroneous views. At the present time +it is difficult to believe that Euler should have confidently +written $\sin\phi - 2 \sin 2\phi + 3 \sin 3\phi - 4 \sin 4\phi+\dotsb = 0$, but +such examples afford striking illustrations of the want of +scientific basis of certain parts of analysis at that time. +Euler's proof of the binomial formula for negative and +\index{Binomial formula}% +fractional exponents, which has been reproduced in elementary +text-books of even recent years, is faulty. A remarkable +development, due to Euler, is what he named the hypergeometric +series, the summation of which he observed to be +dependent upon the integration of a linear differential equation +of the second order, but it remained for Gauss to point +\index{Gauss!ref.\ to}% +out that for special values of its letters, this series represented +nearly all functions then known. + +Euler developed the calculus of finite differences in the first +\index{Finite differences}% +%% -----File: 266.png---Folio 252------- +\index{Numbers!theory of numbers}% +\index{Theory of numbers}% +chapters of his \textit{Institutiones calculi differentialis}, and then +deduced the differential calculus from it. He established a +theorem on homogeneous functions, known by his name, and +\index{Addition theorem of elliptic integrals}% +contributed largely to the theory of differential equations, a +\index{Differential equations}% +subject which had received the attention of Newton, Leibniz, +\index{Leibniz!ref.\ to}% +\index{Newton!ref.\ to}% +and the Bernoullis, but was still undeveloped. Clairaut, +\index{Clairaut!ref.\ to}% +Fontaine, and Euler about the same time observed criteria of +\index{Fontaine}% +integrability, but Euler in addition showed how to employ +them to determine integrating factors. The principles on +which the criteria rested involved some degree of obscurity. +The celebrated addition-theorem for elliptic integrals was first +\index{Elliptic integrals}% +established by Euler. He invented a new algorithm for +continued fractions, which he employed in the solution of +\index{Continued fractions}% +\index{Fractions!continued}% +the indeterminate equation $ax + by = c$. We now know that +substantially the same solution of this equation was given +$1000$ years earlier, by the Hindoos. By giving the factors of +the number $2^{2^n} + 1$ when $n = 5$, he pointed out that this expression +did not always represent primes, as was supposed by +Fermat. He first supplied the proof to ``Fermat's theorem,'' +\index{Fermat!ref.\ to}% +\index{Fermat's theorem}% +and to a second theorem of Fermat, which states that every +prime of the form $4n + 1$ is expressible as the sum of two +squares in one and only one way. A third theorem of Fermat, +that $x^n + y^n = z^n$, has no integral solution for values of $n$ +greater than~$2$, was proved by Euler to be correct when $n = 3$. +Euler discovered four theorems which taken together make +out the great law of quadratic reciprocity, a law independently +\index{Quadratic reciprocity}% +discovered by Legendre.\Reference{48} Euler enunciated and proved a +\index{Legendre!ref.\ to}% +well-known theorem, giving the relation between the number +of vertices, faces, and edges of certain polyhedra, which, +however, appears to have been known to Descartes. The +powers of Euler were directed also towards the fascinating +subject of the theory of probability, in which he solved some +\index{Probability}% +difficult problems. +%% -----File: 267.png---Folio 253------- +\index{Action, least}% + +Of no little importance are Euler's labours in analytical +mechanics. Says Whewell: ``The person who did most to +\index{Mechanics!Euler}% +\index{Whewell}% +give to analysis the generality and symmetry which are now +its pride, was also the person who made mechanics analytical; +I mean Euler.''\Reference{11} He worked out the theory of the rotation of +a body around a fixed point, established the general equations +of motion of a free body, and the general equation of hydrodynamics. +He solved an immense number and variety of mechanical +problems, which arose in his mind on all occasions. Thus, +on reading Virgil's lines, ``The anchor drops, the rushing keel +is staid,'' he could not help inquiring what would be the +ship's motion in such a case. About the same time as Daniel +Bernoulli he published the \textit{Principle of the Conservation of +\index{Conservation!of areas}% +Areas} and defended the principle of ``least action,'' advanced +\index{Areas, conservation of}% +\index{Least action}% +by Maupertius. He wrote also on tides and on sound. +\index{Maupertius}% + +Astronomy owes to Euler the method of the variation of +\index{Astronomy!more recent researches}% +arbitrary constants. By it he attacked the problem of perturbations, +explaining, in case of two planets, the secular variations +of eccentricities, nodes, etc. He was one of the first +to take up with success the theory of the moon's motion by +giving approximate solutions to the ``problem of three bodies.'' +\index{Problem of three bodies}% +\index{Three bodies, problem of}% +He laid a sound basis for the calculation of tables of the moon. +These researches on the moon's motion, which captured two +prizes, were carried on while he was blind, with the assistance +of his sons and two of his pupils. + +Most of his memoirs are contained in the transactions of +the Academy of Sciences at St.~Petersburg, and in those of +the Academy at Berlin. From 1728 to 1783 a large portion +of the Petropolitan transactions were filled by his writings. +He had engaged to furnish the Petersburg Academy with +memoirs in sufficient number to enrich its acts for twenty +years---a promise more than fulfilled, for down to 1818 the +volumes usually contained one or more papers of his. It has +%% -----File: 268.png---Folio 254------- +been said that an edition of Euler's complete works would fill +$16,000$ quarto pages. His mode of working was, first to concentrate +his powers upon a special problem, then to solve +separately all problems growing out of the first. No one +excelled him in dexterity of accommodating methods to special +problems. It is easy to see that mathematicians could not +long continue in Euler's habit of writing and publishing. The +material would soon grow to such enormous proportions as to +be unmanageable. We are not surprised to see almost the +opposite in Lagrange, his great successor. The great Frenchman +\index{Lagrange!ref.\ to}% +delighted in the general and abstract, rather than, like +Euler, in the special and concrete. His writings are condensed +and give in a nutshell what Euler narrates at great +\index{Euler|)}% +length. + +\textbf{Jean-le-Rond D'Alembert} (1717--1783) was exposed, when +\index{D'Alembert|(}% +\index{D'Alembert!ref.\ to}% +\index{D'Alembert's principle}% +an infant, by his mother in a market by the church of St.\ +Jean-le-Rond, near the Nôtre-Dame in Paris, from which he +derived his Christian name. He was brought up by the wife +of a poor glazier. It is said that when he began to show signs +of great talent, his mother sent for him, but received the +reply, ``You are only my step-mother; the glazier's wife is +my mother.'' His father provided him with a yearly income. +D'Alembert entered upon the study of law, but such was his +love for mathematics, that law was soon abandoned. At the +age of twenty-four his reputation as a mathematician secured +for him admission to the Academy of Sciences. In 1743 +appeared his \textit{Traité de dynamique}, founded upon the important +general principle bearing his name: The impressed forces are +equivalent to the effective forces. D'Alembert's principle +seems to have been recognised before him by Fontaine, and +\index{Fontaine}% +in some measure by John Bernoulli and Newton. D'Alembert +\index{Newton!ref.\ to}% +gave it a clear mathematical form and made numerous applications +of it. It enabled the laws of motion and the reasonings +%% -----File: 269.png---Folio 255------- +\index{Hydrodynamics}% +\index{Hydrostatics}% +\index{Infinite series}% +\index{Strings, vibrating}% +\index{Vibrating strings}% +depending on them to be represented in the most general +form, in analytical language. D'Alembert applied it in 1744 +in a treatise on the equilibrium and motion of fluids, in 1746 +to a treatise on the general causes of winds, which obtained +a prize from the Berlin Academy. In both these treatises, as +also in one of 1747, discussing the famous problem of vibrating +chords, he was led to partial differential equations. He was +\index{Partial differential equations}% +a leader among the pioneers in the study of such equations. +To the equation $\dfrac{\partial^2 y}{\partial t^2} = a^2 \dfrac{\partial^2 y}{\partial x^2}$, arising in the problem of vibrating +chords, he gave as the general solution, +\[ +y = f(x + at) + \phi(x - at), +\] +and showed that there is only one arbitrary function, if $y$ be +supposed to vanish for $x = 0$ and~$x=l$. Daniel Bernoulli, +\index{Bernoulli, Daniel!ref.\ to}% +starting with a particular integral given by Brook Taylor, +\index{Taylor, B.!ref.\ to}% +showed that this differential equation is satisfied by the +trigonometric series +\[ +y = \alpha \sin \frac{\pi x}{l} · \cos \frac{\pi t}{l} + + \beta \sin \frac{2 \pi x}{l}· \cos \frac{2 \pi t}{l} + \dotsb, +\] +and claimed this expression to be the most general solution. +Euler denied its generality, on the ground that, if true, the +doubtful conclusion would follow that the above series represents +any arbitrary function of a variable. These doubts were +dispelled by Fourier. Lagrange proceeded to find the sum +\index{Fourier!ref.\ to}% +\index{Lagrange!ref.\ to}% +of the above series, but D'Alembert rightly objected to his +process, on the ground that it involved divergent series.\Reference{46} +\index{Divergent series}% + +A most beautiful result reached by D'Alembert, with aid +of his principle, was the complete solution of the problem of +the precession of the equinoxes, which had baffled the talents +of the best minds. He sent to the French Academy in 1747, +on the same day with Clairaut, a solution of the problem of +\index{Clairaut!ref.\ to}% +three bodies. This had become a question of universal interest +%% -----File: 270.png---Folio 256------- +\index{Problem of three bodies}% +to mathematicians, in which each vied to outdo all others. +The problem of two bodies, requiring the determination of +their motion when they attract each other with forces inversely +proportional to the square of the distance between +them, had been completely solved by Newton. The ``problem +of three bodies'' asks for the motion of three bodies attracting +\index{Three bodies, problem of}% +each other according to the law of gravitation. Thus far, +the complete solution of this has transcended the power of +analysis. The general differential equations of motion were +\index{Differential calculus!philosophy of}% +stated by Laplace, but the difficulty arises in their integration. +\index{Laplace!ref.\ to}% +The ``solutions'' hitherto given are merely convenient methods +of approximation in special cases when one body is the sun, +disturbing the motion of the moon around the earth, or where +a planet moves under the influence of the sun and another +planet. + +In the discussion of the meaning of negative quantities, of +\index{Negative quantities}% +the fundamental processes of the calculus, and of the theory of +probability, D'Alembert paid some attention to the philosophy +of mathematics. His criticisms were not always happy. In +1754 he was made permanent secretary of the French Academy. +During the last years of his life he was mainly occupied with +the great French encyclopædia, which was begun by Diderot +and himself. D'Alembert declined, in 1762, an invitation of +\index{D'Alembert|)}% +Catharine~II. to undertake the education of her son. Frederick +the~Great pressed him to go to Berlin. He made a visit, but +declined a permanent residence there. + +\textbf{Alexis Claude Clairaut} (1713--1765) was a youthful prodigy. +\index{Clairaut|(}% +He read l'Hospital's works on the infinitesimal calculus and on +conic sections at the age of ten. In 1731 was published his +\textit{Recherches sur les courbes à double courbure}, which he had ready +for the press when he was sixteen. It was a work of remarkable +elegance and secured his admission to the Academy of +Sciences when still under legal age. In 1731 he gave a proof of +%% -----File: 271.png---Folio 257------- +\index{Astronomy!more recent researches}% +\index{Cubic curves}% +the theorem enunciated by Newton, that every cubic is a projection +\index{Newton!ref.\ to}% +of one of five divergent parabolas. Clairaut formed the +\index{Divergent parabolas}% +acquaintance of Maupertius, whom he accompanied on an expedition +\index{Maupertius}% +to Lapland to measure the length of a degree of the +meridian. At that time the shape of the earth was a subject +\index{Earth!figure of}% +of serious disagreement. Newton and Huygens had concluded +\index{Huygens!ref.\ to}% +from theory that the earth was flattened at the poles. About +1713 Dominico Cassini measured an arc extending from Dunkirk +\index{Cassini, D}% +to Perpignan and arrived at the startling result that the earth +is elongated at the poles. To decide between the conflicting +opinions, measurements were renewed. Maupertius earned by +his work in Lapland the title of ``earth flattener'' by disproving +the Cassinian tenet that the earth was elongated at the +poles, and showing that Newton was right. On his return, in +1743, Clairaut published a work, \textit{Théorie de la figure de la Terre}, +\index{Figure of the earth}% +which was based on the results of Maclaurin on homogeneous +ellipsoids. It contains a remarkable theorem, named after +Clairaut, that the sum of the fractions expressing the ellipticity +and the increase of gravity at the pole is equal to $2\frac{1}{2}$~times the +fraction expressing the centrifugal force at the equator, the +unit of force being represented by the force of gravity at the +equator. This theorem is independent of any hypothesis with +respect to the law of densities of the successive strata of the +earth. It embodies most of Clairaut's researches. Todhunter +says that ``in the figure of the earth no other person has +accomplished so much as Clairaut, and the subject remains at +present substantially as he left it, though the form is different. +The splendid analysis which Laplace supplied, adorned but did +not really alter the theory which started from the creative +hands of Clairaut.'' + +In 1752 he gained a prize of the St.~Petersburg Academy +for his paper on \textit{Théorie de la Lune}, in which for the first time +modern analysis is applied to lunar motion. This contained +%% -----File: 272.png---Folio 258------- +the explanation of the motion of the lunar apsides. This +motion, left unexplained by Newton, seemed to him at first +\index{Newton!ref.\ to}% +inexplicable by Newton's law, and he was on the point of +advancing a new hypothesis regarding gravitation, when, taking +\index{Gravitation, theory of}% +the precaution to carry his calculation to a higher degree +of approximation, he reached results agreeing with observation. +The motion of the moon was studied about the same +time by Euler and D'Alembert. Clairaut predicted that +\index{D'Alembert!ref.\ to}% +\index{Euler!ref.\ to}% +``Halley's Comet,'' then expected to return, would arrive at +\index{Halley's Comet}% +its nearest point to the sun on April~13, 1759, a date which +turned out to be one month too late. He was the first to +detect singular solutions in differential equations of the first +order but of higher degree than the first. + +In their scientific labours there was between Clairaut and +D'Alembert great rivalry, often far from friendly. The growing +ambition of Clairaut to shine in society, where he was a +\index{Clairaut|)}% +great favourite, hindered his scientific work in the latter part +of his life. + +\textbf{Johann Heinrich Lambert} (1728--1777), born at Mühl\-hausen +\index{Lambert|(}% +in Alsace, was the son of a poor tailor. While working at his +father's trade, he acquired through his own unaided efforts a +knowledge of elementary mathematics. At the age of thirty he +became tutor in a Swiss family and secured leisure to continue +his studies. In his travels with his pupils through Europe he +became acquainted with the leading mathematicians. In 1764 +he settled in Berlin, where he became member of the Academy, +and enjoyed the society of Euler and Lagrange. He received +a small pension, and later became editor of the Berlin \textit{Ephemeris}. +His many-sided scholarship reminds one of Leibniz. +In his \textit{Cosmological Letters} he made some remarkable prophecies +regarding the stellar system. In mathematics he made +several discoveries which were extended and overshadowed by +his great contemporaries. His first research on pure mathematics +%% -----File: 273.png---Folio 259------- +\index{Differential calculus!philosophy of}% +developed in an infinite series the root~$x$ of the equation +\index{Infinite series}% +$x^m + px = q$. Since each equation of the form $ax^r + bx^s = d$ +can be reduced to $x^m + px = q$ in two ways, one or the other of +the two resulting series was always found to be convergent, +and to give a value of~$x$. Lambert's results stimulated Euler, +\index{Euler!ref.\ to}% +who extended the method to an equation of four terms, and +particularly Lagrange, who found that a function of a root of +\index{Lagrange!ref.\ to}% +$a - x + \phi(x) = 0$ can be expressed by the series bearing his +name. In 1761 Lambert communicated to the Berlin Academy +a memoir, in which he proves that $\pi$ is irrational. This proof +\index{p@{$\pi$}: values for!proved to be irrational}% +is given in Note~IV. of Legendre's \textit{Géometrie}, where it is +\index{Legendre!ref.\ to}% +extended to~$\pi^2$. To the genius of Lambert we owe the introduction +\index{Lambert|)}% +into trigonometry of hyperbolic functions, which he +\index{Trigonometry}% +designated by $\textit{sinh}\,x$, $\textit{cosh}\,x$,~etc. His \textit{Freye Perspective}, 1759 +and 1773, contains researches on descriptive geometry, and +entitle him to the honour of being the forerunner of Monge. +\index{Monge!ref.\ to}% +In his effort to simplify the calculation of cometary orbits, he +was led geometrically to some remarkable theorems on conics, +for instance this: ``If in two ellipses having a common major +axis we take two such arcs that their chords are equal, and +that also the sums of the radii vectores, drawn respectively +from the foci to the extremities of these arcs, are equal to +each other, then the sectors formed in each ellipse by the arc +and the two radii vectores are to each other as the square +roots of the parameters of the ellipses.''\Reference{13} + +\textbf{John Landen} (1719--1790) was an English mathematician +\index{Landen}% +whose writings served as the starting-point of investigations +by Euler, Lagrange, and Legendre. Landen's capital discovery, +contained in a memoir of 1755, was that every arc of the +hyperbola is immediately rectified by means of two arcs of an +ellipse. In his ``residual analysis'' he attempted to obviate +the metaphysical difficulties of fluxions by adopting a purely +algebraic method. Lagrange's \textit{Calcul des Fonctions} is based +%% -----File: 274.png---Folio 260------- +\index{Witch of Agnesi}% +upon this idea. Landen showed how the algebraic expression +for the roots of a cubic equation could be derived by application +of the differential and integral calculus. Most of the +time of this suggestive writer was spent in the pursuits of +active life. + +\textbf{Étienne Bézout} (1730--1783) was a French writer of popular +\index{Bezout@{Bézout}}% +mathematical school-books. In his \textit{Théorie générale des Équations +Algébriques}, 1779, he gave the method of elimination by +\index{Bezout's@{Bézout's method of elimination}}% +linear equations (invented also by Euler). This method was +\index{Equations!solution of}% +first published by him in a memoir of 1764, in which he uses +determinants, without, however, entering upon their theory. +A beautiful theorem as to the degree of the resultant goes by +his name. + +\textbf{Louis Arbogaste} (1759--1803) of Alsace was professor of +\index{Arbogaste}% +mathematics at Strasburg. His chief work, the \textit{Calcul des +Dérivations}, 1800, gives the method known by his name, by +which the successive coefficients of a development are derived +from one another when the expression is complicated. De~Morgan +\index{Demorgan@{De Morgan}!ref.\ to}% +has pointed out that the true nature of derivation +is differentiation accompanied by integration. In this book +for the first time are the symbols of operation separated from +those of quantity. The notation $D_x y$ for $dy/dx$ is due to him. +\index{Notation!differential calculus}% + +\textbf{Maria Gaetana Agnesi} (1718--1799) of Milan, distinguished as +\index{Agnesi}% +a linguist, mathematician, and philosopher, filled the mathematical +chair at the University of Bologna during her father's +sickness. In 1748 she published her \textit{Instituzioni Analitiche}, +which was translated into English in 1801. The ``witch of +Agnesi'' or ``versiera'' is a plane curve containing a straight +line, $x = 0$, and a cubic $\left(\dfrac{y}{c}\right)^2 + 1 = \dfrac{c}{x}$. + +\textbf{Joseph Louis Lagrange} (1736--1813), one of the greatest +\index{Lagrange|(}% +mathematicians of all times, was born at Turin and died at +Paris. He was of French extraction. His father, who had +%% -----File: 275.png---Folio 261------- +\index{Isoperimetrical figures}% +charge of the Sardinian military chest, was once wealthy, but +lost all he had in speculation. Lagrange considered this loss +his good fortune, for otherwise he might not have made mathematics +the pursuit of his life. While at the college in Turin +his genius did not at once take its true bent. Cicero and Virgil +at first attracted him more than Archimedes and Newton. +He soon came to admire the geometry of the ancients, but the +perusal of a tract of Halley roused his enthusiasm for the +\index{Halley}% +analytical method, in the development of which he was destined +to reap undying glory. He now applied himself to +mathematics, and in his seventeenth year he became professor +of mathematics in the royal military academy at Turin. +Without assistance or guidance he entered upon a course of +study which in two years placed him on a level with the +greatest of his contemporaries. With aid of his pupils he +established a society which subsequently developed into the +Turin Academy. In the first five volumes of its transactions +appear most of his earlier papers. At the age of nineteen he +communicated to Euler a general method of dealing with +\index{Euler!ref.\ to}% +``isoperimetrical problems,'' known now as the Calculus of +\index{Calculus!of variations}% +Variations. This commanded Euler's lively admiration, and +he courteously withheld for a time from publication some +researches of his own on this subject, so that the youthful +Lagrange might complete \emph{his} investigations and claim the +invention. Lagrange did quite as much as Euler towards the +creation of the Calculus of Variations. As it came from Euler +it lacked an analytic foundation, and this Lagrange supplied. +He separated the principles of this calculus from geometric +considerations by which his predecessor had derived them. +Euler had assumed as fixed the limits of the integral, \ie~the +extremities of the curve to be determined, but Lagrange +removed this restriction and allowed all co-ordinates of the +curve to vary at the same time. Euler introduced in 1766 the +%% -----File: 276.png---Folio 262------- +\index{Acoustics}% +\index{Astronomy!more recent researches}% +name ``calculus of variations,'' and did much to improve this +science along the lines marked out by Lagrange. + +Another subject engaging the attention of Lagrange at +Turin was the propagation of sound. In his papers on this +subject in the \textit{Miscellanea Taurinensia}, the young mathematician +appears as the critic of Newton, and the arbiter between +\index{Newton!ref.\ to}% +Euler and D'Alembert. By considering only the particles +\index{D'Alembert!ref.\ to}% +\index{Euler!ref.\ to}% +which are in a straight line, he reduced the problem to the +same partial differential equation that represents the motions +of vibrating strings. The general integral of this was found by +\index{Strings, vibrating}% +\index{Vibrating strings}% +D'Alembert to contain two arbitrary functions, and the question +\index{Functions!arbitrary}% +now came to be discussed whether an arbitrary function +may be discontinuous. D'Alembert maintained the negative +against Euler, Daniel Bernoulli, and finally Lagrange,---arguing +\index{Bernoulli, Daniel!ref.\ to}% +that in order to determine the position of a point of the +chord at a time~$t$, the initial position of the chord must be +continuous. Lagrange settled the question in the affirmative. + +By constant application during nine years, Lagrange, at the +age of twenty-six, stood at the summit of European fame. +But his intense studies had seriously weakened a constitution +never robust, and though his physicians induced him to take +rest and exercise, his nervous system never fully recovered its +tone, and he was thenceforth subject to fits of melancholy. + +In 1764 the French Academy proposed as the subject of +a prize the theory of the libration of the moon. It demanded +an explanation, on the principle of universal gravitation, why +the moon always turns, with but slight variations, the same +\DPtypo{phase}{face} to the earth. Lagrange secured the prize. This success +encouraged the Academy to propose as a prize the theory +of the four satellites of Jupiter,---a problem of six bodies, +more difficult than the one of three bodies previously solved +by Clairaut, D'Alembert, and Euler. Lagrange overcame the +\index{Clairaut!ref.\ to}% +difficulties, but the shortness of time did not permit him to +%% -----File: 277.png---Folio 263------- +exhaust the subject. Twenty-four years afterwards it was +completed by Laplace. Later astronomical investigations of +\index{Laplace!ref.\ to}% +Lagrange are on cometary perturbations (1778 and 1783), on +Kepler's problem, and on a new method of solving the problem +\index{Kepler!ref.\ to}% +of three bodies. + +Being anxious to make the personal acquaintance of leading +mathematicians, Lagrange visited Paris, where he enjoyed the +stimulating delight of conversing with Clairaut, D'Alembert, +Condorcet, the Abbé Marie, and others. He had planned a +visit to London, but he fell dangerously ill after a dinner in +Paris, and was compelled to return to Turin. In 1766 Euler +left Berlin for St.~Petersburg, and he pointed out Lagrange as +the only man capable of filling the place. D'Alembert recommended +him at the same time. Frederick the Great thereupon +sent a message to Turin, expressing the wish of ``the +greatest king of Europe'' to have ``the greatest mathematician'' +at his court. Lagrange went to Berlin, and staid\DPnote{** PP: [sic], archaic form} there +twenty years. Finding all his colleagues married, and being +assured by their wives that the marital state alone is happy, +he married. The union was not a happy one. His wife +soon died. Frederick the Great held him in high esteem, +and frequently conversed with him on the advantages of perfect +regularity of life. This led Lagrange to cultivate regular +habits. He worked no longer each day than experience taught +him he could without breaking down. His papers were carefully +thought out before he began writing, and when he wrote +he did so without a single correction. + +During the twenty years in Berlin he crowded the transactions +of the Berlin Academy with memoirs, and wrote also +the epoch-making work called the \textit{Mécanique Analytique}. He +enriched algebra by researches on the solution of equations. +\index{Equations!solution of}% +There are two methods of solving directly algebraic equations,---that +of substitution and that of combination. The +%% -----File: 278.png---Folio 264------- +\index{Warring}% +former method was developed by Ferrari, Vieta, Tchirnhausen, +\index{Ferrari!ref.\ to}% +\index{Tchirnhausen!ref.\ to}% +\index{Vieta!ref.\ to}% +Euler, Bézout, and Lagrange; the latter by Vandermonde and +\index{Bezout@{Bézout}!ref.\ to}% +\index{Euler!ref.\ to}% +\index{Vandermonde!ref.\ to}% +Lagrange.\Reference{20} In the method of substitution the original forms +are so transformed that the determination of the roots is made +to depend upon simpler functions (resolvents). In the method +of combination auxiliary quantities are substituted for certain +simple combinations (``types'') of the unknown roots of the +equation, and auxiliary equations (resolvents) are obtained for +\index{Equations!numerical}% +these quantities with aid of the coefficients of the given equation. +Lagrange traced all known algebraic solutions of equations +to the uniform principle consisting in the formation and +solution of equations of lower degree whose roots are linear +functions of the required roots, and of the roots of unity. He +showed that the quintic cannot be reduced in this way, its +resolvent being of the sixth degree. His researches on the +theory of equations were continued after he left Berlin. In +the \textit{Résolution des équations numériques} (1798) he gave a +method of approximating to the real roots of numerical equations +by continued fractions. Among other things, it contains +also a proof that every equation must have a root,---a theorem +which appears before this to have been considered self-evident. +Other proofs of this were given by Argand, Gauss, and Cauchy. +\index{Argand!ref.\ to}% +\index{Cauchy!ref.\ to}% +\index{Gauss!ref.\ to}% +In a note to the above work Lagrange uses Fermat's theorem +\index{Fermat!ref.\ to}% +and certain suggestions of Gauss in effecting a complete algebraic +solution of any binomial equation. + +While in Berlin Lagrange published several papers on the +theory of numbers. In~1769 he gave a solution in integers of +\index{Numbers!theory of numbers}% +\index{Theory of numbers}% +indeterminate equations of the second degree, which resembles +the Hindoo cyclic method; he was the first to prove, in~1771, +``Wilson's theorem,'' enunciated by an Englishman, John +\index{Wilson}% +\index{Wilson's theorem}% +Wilson, and first published by Waring in his \textit{Meditationes +Algebraicæ}; he investigated in 1775 under what conditions +$±2$~and~$±5$ ($-1$~and~$±3$ having been discussed by Euler) +%% -----File: 279.png---Folio 265------- +are quadratic residues, or non-residues of odd prime numbers,~$q$; +he proved in 1770 Méziriac's theorem that every integer is +\index{Meziriac@{Méziriac}!ref.\ to}% +equal to the sum of four, or a less number, of squares. He +proved Fermat's theorem on $x^n + y^n = z^n$, for the case $n = 4$, +\index{Fermat!ref.\ to}% +also Fermat's theorem that, if $a^2 + b^2 = c^2$, then $ab$ is not a +square. + +In his memoir on Pyramids, 1773, Lagrange made considerable +use of determinants of the third order, and demonstrated +\index{Determinants}% +that the square of a determinant is itself a determinant. He +never, however, dealt explicitly and directly with determinants; +he simply obtained accidentally identities which are +now recognised as relations between determinants. + +Lagrange wrote much on differential equations. Though +\index{Differential equations}% +the subject of contemplation by the greatest mathematicians +(Euler, D'Alembert, Clairaut, Lagrange, Laplace), yet more +\index{D'Alembert!ref.\ to}% +\index{Euler!ref.\ to}% +than other branches of mathematics did they resist the systematic +application of fixed methods and principles. Lagrange +established criteria for singular solutions (\textit{Calcul des Fonctions}, +\index{Singular solutions}% +Lessons~14--17), which are, however, erroneous. He was the +first to point out the geometrical significance of such solutions. +He generalised Euler's researches on total differential equations +of two variables, and of the ninth order; he gave a solution +of partial differential equations of the first order (\textit{Berlin +Memoirs}, 1772 and 1774), and spoke of their singular solutions, +extending their solution in \textit{Memoirs} of 1779 and 1785 to equations +of any number of variables. The discussion on partial +differential equations of the second order, carried on by +D'Alembert, Euler, and Lagrange, has already been referred +to in our account of D'Alembert. + +While in Berlin, Lagrange wrote the ``\textit{Mécanique Analytique},'' +the greatest of his works (Paris, 1788). From the principle +of virtual velocities he deduced, with aid of the calculus of +\index{Calculus!of variations}% +\index{Virtual velocities}% +variations, the whole system of mechanics so elegantly and +%% -----File: 280.png---Folio 266------- +harmoniously that it may fitly be called, in Sir William +Rowan Hamilton's words, ``a kind of scientific poem.'' It is a +\index{Hamilton, W.~R.!ref.\ to}% +most consummate example of analytic generality. Geometrical +figures are nowhere allowed. ``On ne trouvera point de figures +dans cet ouvrage'' (Preface). The two divisions of mechanics---statics +\index{Mechanics!Lagrange}% +and dynamics---are in the first four sections of each +carried out analogously, and each is prefaced by a historic +sketch of principles. Lagrange formulated the principle of +least action. In their original form, the equations of motion +\index{Least action}% +involve the co-ordinates $x$,~$y$,~$z$, of the different particles $m$ or +$dm$ of the system. But $x$,~$y$,~$z$, are in general not independent, +and Lagrange introduced in place of them any variables $\xi$,~$\psi$,~$\phi$, +whatever, determining the position of the point at the time. +These may be taken to be independent. The equations of +motion may now assume the form +\[ +\frac{d}{dt}\, \frac{dT}{d\xi'} - \frac{dT}{d\xi} + \Xi = 0; +\] +or when $\Xi$,~$\Change{\psi}{\Psi}$, $\Change{\phi}{\Phi},\ldots$ are the partial differential coefficients +with respect to $\xi$,~$\psi$, $\phi,\ldots$ of one and the same function~$V$, +then the form +\[ +\frac{d}{dt}\, \frac{dT}{d\xi'} - \frac{dT}{d\xi} + \frac{dV}{d\xi} = 0. +\] +The latter is \textit{par excellence} the Lagrangian form of the equations +of motion. With Lagrange originated the remark that +mechanics may be regarded as a geometry of four dimensions. +To him falls the honour of the introduction of the potential +into dynamics.\Reference{49} Lagrange was anxious to have his \textit{Mécanique +Analytique} published in Paris. The work was ready for print +in 1786, but not till 1788 could he find a publisher, and then +only with the condition that after a few years he would purchase +all the unsold copies. The work was edited by +Legendre. +\index{Legendre!ref.\ to}% +%% -----File: 281.png---Folio 267------- + +After the death of Frederick the Great, men of science +were no longer respected in Germany, and Lagrange accepted +an invitation of Louis~XVI. to migrate to Paris. The French +queen treated him with regard, and lodging was procured for +him in the Louvre. But he was seized with a long attack of +melancholy which destroyed his taste for mathematics. For +two years his printed copy of the \textit{Mécanique}, fresh from the +press,---the work of a quarter of a century,---lay unopened on +his desk. Through Lavoisier he became interested in chemistry, +which he found ``as easy as algebra.'' The disastrous +\index{Algebra!Lagrange}% +crisis of the French Revolution aroused him again to activity. +About this time the young and accomplished daughter of the +astronomer Lemonnier took compassion on the sad, lonely +\index{Lemonnier}% +Lagrange, and insisted upon marrying him. Her devotion to +him constituted the one tie to life which at the approach of +death he found it hard to break. + +He was made one of the commissioners to establish weights +and measures having units founded on nature. Lagrange +strongly favoured the decimal subdivision, the general idea of +which was obtained from a work of Thomas Williams, London, +\index{Williams}% +1788. Such was the moderation of Lagrange's character, and +such the universal respect for him, that he was retained as president +of the commission on weights and measures even after it +had been \emph{purified} by the Jacobins by striking out the names +of Lavoisier, Laplace, and others. Lagrange took alarm at the +fate of Lavoisier, and planned to return to Berlin, but at the +establishment of the \textit{École Normale} in 1795 in Paris, he was +induced to accept a professorship. Scarcely had he time to +elucidate the foundations of arithmetic and algebra to young +pupils, when the school was closed. His additions to the +algebra of Euler were prepared at this time. In 1797 the +\index{Euler!ref.\ to}% +\textit{École Polytechnique} was founded, with Lagrange as one of +the professors. The earliest triumph of this institution was +%% -----File: 282.png---Folio 268------- +\index{Functions!theory of}% +\index{Theory of functions}% +the restoration of Lagrange to analysis. His mathematical +activity burst out anew. He brought forth the \textit{Théorie des +fonctions analytiques} (1797), \textit{Leçons sur le calcul des fonctions}, +a treatise on the same lines as the preceding (1801), and the +\textit{Résolution des équations \DPtypo{numeriques}{numériques}} (1798). In 1810 he +began a thorough revision of his \textit{Mécanique analytique}, but +he died before its completion. + +The \textit{Théorie des fonctions}, the germ of which is found in a +memoir of his of 1772, aimed to place the principles of the +calculus upon a sound foundation by relieving the mind of the +difficult conception of a limit or infinitesimal. John Landen's +\index{Landen!ref.\ to}% +residual calculus, professing a similar object, was unknown to +him. Lagrange attempted to prove Taylor's theorem (the +\index{Taylor's theorem}% +power of which he was the first to point out) by simple algebra, +and then to develop the entire calculus from that theorem. +The principles of the calculus were in his day involved in +philosophic difficulties of a serious nature. The infinitesimals +of Leibniz had no satisfactory metaphysical basis. In the +\index{Leibniz!ref.\ to}% +differential calculus of Euler they were treated as absolute +\index{Differential calculus!philosophy of}% +\index{Euler!ref.\ to}% +zeros. In Newton's limiting ratio, the magnitudes of which it +\index{Newton!ref.\ to}% +is the ratio cannot be found, for at the moment when they +should be caught and equated, there is neither arc nor chord. +The chord and arc were not taken by Newton as equal before +vanishing, nor after vanishing, but \emph{when} they vanish. ``That +method,'' said Lagrange, ``has the great inconvenience of considering +quantities in the state in which they cease, so to +speak, to be quantities; for though we can always well conceive +the ratios of two quantities, as long as they remain +finite, that ratio offers to the mind no clear and precise idea, +as soon as its terms become both nothing at the same time.'' +D'Alembert's method of limits was much the same as the +\index{D'Alembert!ref.\ to|(}% +\index{Limits, method of}% +method of prime and ultimate ratios. D'Alembert taught +\index{Prime and ultimate ratios}% +that a variable actually reached its limit. When Lagrange +%% -----File: 283.png---Folio 269------- +endeavoured to free the calculus of its metaphysical difficulties, +by resorting to common algebra, he avoided the whirlpool of +Charybdis only to suffer wreck against the rocks of Scylla. +The algebra of his day, as handed down to him by Euler, was +founded on a false view of infinity. No correct theory of +\index{Infinity}% +infinite series had then been established. Lagrange proposed +\index{Infinite series}% +to define the differential coefficient of $f(x)$ with respect to~$x$ +as the coefficient of~$h$ in the expansion of $f(x + h)$ by Taylor's +theorem, and thus to avoid all reference to limits. But he +used infinite series without ascertaining that they were convergent, +and his proof that $f(x + h)$ can always be expanded +in a series of ascending powers of~$h$, labours under serious +defects. Though Lagrange's method of developing the calculus +was at first greatly applauded, its defects were fatal, and to-day +his ``method of derivatives,'' as it was called, has been generally +\index{Derivatives, method of}% +abandoned. He introduced a notation of his own, but +\index{Notation!differential calculus}% +it was inconvenient, and was abandoned by him in the second +edition of his \textit{Mécanique}, in which he used infinitesimals. The +primary object of the \textit{Théorie des fonctions} was not attained, +but its secondary results were far-reaching. It was a purely +abstract mode of regarding functions, apart from geometrical +\index{Functions!theory of}% +or mechanical considerations. In the further development +of higher analysis a function became the leading idea, and +Lagrange's work may be regarded as the starting-point of the +theory of functions as developed by Cauchy, Riemann, Weierstrass, +\index{Theory of functions}% +and others. + +In the treatment of infinite series Lagrange displayed in +his earlier writings that laxity common to all mathematicians +of his time, excepting Nicolaus Bernoulli~II. and D'Alembert. +\index{Bernoulli, Nicolaus (born 1687)}% +But his later articles mark the beginning of a period of greater +rigour. Thus, in the \textit{Calcul de fonctions} he gives his theorem +on the limits of Taylor's theorem. Lagrange's mathematical +\index{Taylor's theorem}% +researches extended to subjects which have not been mentioned +%% -----File: 284.png---Folio 270------- +\index{Acoustics}% +\index{Probability}% +here---such as probabilities, finite differences, ascending +\index{Finite differences}% +continued fractions, elliptic integrals. Everywhere his +\index{Continued fractions}% +\index{Fractions!continued}% +wonderful powers of generalisation and abstraction are made +manifest. In that respect he stood without a peer, but +his great contemporary, Laplace, surpassed him in practical +\index{Laplace|(}% +sagacity. Lagrange was content to leave the application of +his general results to others, and some of the most important +researches of Laplace (particularly those on the velocity of +sound and on the secular acceleration of the moon) are implicitly +\index{Sound, velocity of}% +contained in Lagrange's works. + +Lagrange was an extremely modest man, eager to avoid +\index{Lagrange|)}% +controversy, and even timid in conversation. He spoke in +tones of doubt, and his first words generally were, ``Je ne +sais pas.'' He would never allow his portrait to be taken, +and the only ones that were secured were sketched without +his knowledge by persons attending the meetings of the +Institute. + +\textbf{Pierre Simon Laplace} (1749--1827) was born at Beau\-mont-en-Auge +in Normandy. Very little is known of his early +life. When at the height of his fame he was loath to speak +of his boyhood, spent in poverty. His father was a small +farmer. Some rich neighbours who recognised the boy's +talent assisted him in securing an education. As an extern +he attended the military school in Beaumont, where at an +early age he became teacher of mathematics. At eighteen +he went to Paris, armed with letters of recommendation to +D'Alembert, who was then at the height of his fame. The +letters remained unnoticed, but young Laplace, undaunted, +wrote the great geometer a letter on the principles of mechanics, +which brought the following enthusiastic response: +``You needed no introduction; you have recommended yourself; +my support is your due.'' D'Alembert secured him a +\index{D'Alembert!ref.\ to|)}% +position at the \textit{École Militaire} of Paris as professor of mathematics. +%% -----File: 285.png---Folio 271------- +\index{Circle!degrees of}% +His future was now assured, and he entered upon +those profound researches which brought him the title of ``the +Newton of France.'' With wonderful mastery of analysis, +Laplace attacked the pending problems in the application +of the law of gravitation to celestial motions. During the +\index{Gravitation, theory of}% +succeeding fifteen years appeared most of his original contributions +to astronomy. His career was one of almost uninterrupted +\index{Astronomy!more recent researches|(}% +prosperity. In 1784 he succeeded Bézout as examiner +to the royal artillery, and the following year he became member +of the Academy of Sciences. He was made president of +the Bureau of Longitude; he aided in the introduction of the +decimal system, and taught, with Lagrange, mathematics in +the \textit{École Normale}. When, during the Revolution, there arose +a cry for the reform of everything, even of the calendar, +\index{Calendar}% +Laplace suggested the adoption of an era beginning with the +year~1250, when, according to his calculation, the major axis +of the earth's orbit had been perpendicular to the equinoctial +line. The year was to begin with the vernal equinox, and the +zero meridian was to be located east of Paris by $185.30$~degrees +of the centesimal division of the quadrant, for by this meridian +\index{Division of the circle}% +the beginning of his proposed era fell at midnight. But the +revolutionists rejected this scheme, and made the start of the +new era coincide with the beginning of the glorious French +Republic.\Reference{50} + +Laplace was justly admired throughout Europe as a most +sagacious and profound scientist, but, unhappily for his reputation, +he strove not only after greatness in science, but also +after political honours. The political career of this eminent +scientist was stained by servility and suppleness. After the +18th of Brumaire, the day when Napoleon was made emperor, +Laplace's ardour for republican principles suddenly gave way +to a great devotion to the emperor. Napoleon rewarded this +devotion by giving him the post of minister of the interior, +%% -----File: 286.png---Folio 272------- +but dismissed him after six months for incapacity. Said +Napoleon, "Laplace ne saisissait aucune question sous son +véritable point de vue; il cherchait des subtilités partout, +n'avait que des idées problematiques, et portait enfin l'esprit +des infiniment petits jusque dans l'administration." Desirous +to retain his allegiance, Napoleon elevated him to the Senate +and bestowed various other honours upon him. Nevertheless, +he cheerfully gave his voice in 1814 to the dethronement of +his patron and hastened to tender his services to the Bourbons, +thereby earning the title of marquis. This pettiness of his +character is seen in his writings. The first edition of the +\textit{Système du monde} was dedicated to the Council of Five Hundred. +To the third volume of the \textit{Mécanique Céleste} is prefixed +a note that of all the truths contained in the book, that most +precious to the author was the declaration he thus made of +gratitude and devotion to the peace-maker of Europe. After +this outburst of affection, we are surprised to find in the editions +of the \textit{Théorie analytique des probabilités}, which appeared after +the Restoration, that the original dedication to the emperor is +suppressed. + +Though supple and servile in politics, it must be said that +in religion and science Laplace never misrepresented or concealed +his own convictions however distasteful they might be +to others. In mathematics and astronomy his genius shines +with a lustre excelled by few. Three great works did he give +to the scientific world,---the \textit{Mécanique Céleste}, the \textit{Exposition +du système du monde}, and the \textit{Théorie analytique des \DPtypo{probabilities}{probabilités}}. +Besides these he contributed important memoirs to the +French Academy. + +We first pass in brief review his astronomical researches. +In~1773 he brought out a paper in which he proved that the +mean motions or mean distances of planets are invariable or +merely subject to small periodic changes. This was the first +%% -----File: 287.png---Folio 273------- +and most important step in establishing the stability of the +solar system.\Reference{51} To Newton and also to Euler it had seemed +\index{Euler!ref.\ to}% +doubtful whether forces so numerous, so variable in position, +so different in intensity, as those in the solar system, could be +capable of maintaining permanently a condition of equilibrium. +Newton was of the opinion that a powerful hand must intervene +from time to time to repair the derangements occasioned +by the mutual action of the different bodies. This +paper was the beginning of a series of profound researches by +Lagrange and Laplace on the limits of variation of the various +\index{Lagrange!ref.\ to}% +elements of planetary orbits, in which the two great mathematicians +alternately surpassed and supplemented each other. +Laplace's first paper really grew out of researches on the +theory of Jupiter and Saturn. The behaviour of these planets +had been studied by Euler and Lagrange without receiving +satisfactory explanation. Observation revealed the existence +of a steady acceleration of the mean motions of our moon and +of Jupiter and an equally strange diminution of the mean motion +of Saturn. It looked as though Saturn might eventually +leave the planetary system, while Jupiter would fall into the +sun, and the moon upon the earth. Laplace finally succeeded +in showing, in a paper of 1784--1786, that these variations +(called the ``great inequality'') belonged to the class of ordinary +periodic perturbations, depending upon the law of attraction. +\index{Perturbations}% +The cause of so influential a perturbation was found in +the commensurability of the mean motion of the two planets. + +In the study of the Jovian system, Laplace was enabled to +determine the masses of the moons. He also discovered certain +very remarkable, simple relations between the movements +of those bodies, known as ``Laws of Laplace.'' His theory of +\index{Laws of Laplace}% +these bodies was completed in papers of 1788 and~1789. +These, as well as the other papers here mentioned, were published +in the \textit{\DPtypo{Mémoirs}{Mémoires} présentés par divers savans}. +The year +%% -----File: 288.png---Folio 274------- +\index{Nebular hypothesis}% +1787 was made memorable by Laplace's announcement that +the lunar acceleration depended upon the secular changes in +the eccentricity of the earth's orbit. This removed all doubt +then existing as to the stability of the solar system. The universal +validity of the law of gravitation to explain all motion +in the solar system was established. That system, as then +known, was at last found to be a complete machine. + +In 1796 Laplace published his \textit{Exposition du système du +monde}, a non-mathematical popular treatise on astronomy, +\index{Astronomy!more recent researches|)}% +ending with a sketch of the history of the science. In this +work he enunciates for the first time his celebrated nebular +hypothesis. A similar theory had been previously proposed +by Kant in~1755, and by Swedenborg; but Laplace does not +\index{Kant}% +\index{Swedenborg}% +appear to have been aware of this. + +Laplace conceived the idea of writing a work which should +contain a complete analytical solution of the mechanical problem +presented by the solar system, without deriving from +observation any but indispensable data. The result was the +\textit{Mécanique Céleste}, which is a systematic presentation embracing +all the discoveries of Newton, Clairaut, D'Alembert, Euler, +Lagrange, and of Laplace himself, on celestial mechanics. +\index{Mechanics!Laplace}% +The first and second volumes of this work were published in~1799; +the third appeared in~1802, the fourth in~1805. Of the +fifth volume, Books XI.~and~XII. were published in~1823; +Books XIII.,~XIV.,~XV. in~1824, and Book~XVI. in~1825. The +first two volumes contain the general theory of the motions +and figure of celestial bodies. The third and fourth volumes +give special theories of celestial motions,---treating particularly +of motions of comets, of our moon, and of other satellites. +The fifth volume opens with a brief history of celestial +mechanics, and then gives in appendices the results of the +author's later researches. The \textit{Mécanique Céleste} was such a +master-piece, and so complete, that Laplace's successors have +%% -----File: 289.png---Folio 275------- +been able to add comparatively little. The general part of +the work was translated into German by Joh.\ Karl Burkhardt, +\index{Burkhardt, J.~K.}% +and appeared in Berlin, 1800--1802. Nathaniel Bowditch +\index{Bowditch}% +brought out an edition in English, with an extensive commentary, +in Boston, 1829--1839. The \textit{Mécanique Céleste} is not +easy reading. The difficulties lie, as a rule, not so much in the +subject itself as in the want of verbal explanation. A complicated +chain of reasoning receives often no explanation whatever. +Biot, who assisted Laplace in revising the work for +\index{Biot}% +the press, tells that he once asked Laplace some explanation +of a passage in the book which had been written not long +before, and that Laplace spent an hour endeavouring to recover +the reasoning which had been carelessly suppressed with the +remark, ``Il est facile de voir.'' Notwithstanding the important +researches in the work, which are due to Laplace himself, +it naturally contains a great deal that is drawn from his predecessors. +It is, in fact, the organised result of a century of +patient toil. But Laplace frequently neglects to properly +acknowledge the source from which he draws, and lets the +reader infer that theorems and formulæ due to a predecessor +are really his own. + +We are told that when Laplace presented Napoleon with a +copy of the \textit{Mécanique Céleste}, the latter made the remark, +``M.~Laplace, they tell me you have written this large book on +the system of the universe, and have never even mentioned +its Creator.'' Laplace is said to have replied bluntly, ``Je +n'avais pas besoin de cette hypothèse-la.'' This assertion, +taken literally, is impious, but may it not have been intended +to convey a meaning somewhat different from its literal one? +Newton was not able to explain by his law of gravitation all +\index{Gravitation, theory of}% +questions arising in the mechanics of the heavens. Thus, +being unable to show that the solar system was stable, and +suspecting in fact that it was unstable, Newton expressed the +%% -----File: 290.png---Folio 276------- +opinion that the special intervention, from time to time, of a +powerful hand was necessary to preserve order. Now Laplace +was able to prove by the law of gravitation that the solar +system is stable, and in that sense may be said to have felt +no necessity for reference to the Almighty. + +We now proceed to researches which belong more properly +to pure mathematics. Of these the most conspicuous are on +the theory of probability. Laplace has done more towards +\index{Probability}% +advancing this subject than any one other investigator. He +published a series of papers, the main results of which were +collected in his \textit{Théorie analytique des probabilités}, 1812. The +third edition (1820) consists of an introduction and two books. +The introduction was published separately under the title, +\textit{Essai philosophique sur les probabilités}, and is an admirable +and masterly exposition without the aid of analytical formulæ +of the principles and applications of the science. The first +book contains the theory of generating functions, which are +applied, in the second book, to the theory of probability. +Laplace gives in his work on probability his method of +approximation to the values of definite integrals. The solution +of linear differential equations was reduced by him to +definite integrals. One of the most important parts of the +work is the application of probability to the method of least +squares, which is shown to give the most probable as well as +the most convenient results. + +The first printed statement of the principle of least squares +\index{Least squares}% +was made in 1806 by Legendre, without demonstration. Gauss +\index{Gauss!ref.\ to}% +\index{Legendre!ref.\ to}% +had used it still earlier, but did not publish it until 1809. +The first deduction of the law of probability of error that +appeared in print was given in 1808 by Robert Adrain in the +\index{Adrain}% +\textit{Analyst}, a journal published by himself in Philadelphia.\Reference{2} +Proofs of this law have since been given by Gauss, Ivory, +\index{Ivory!ref.\ to}% +Herschel, Hagen, and others; but all proofs contain some +\index{Hagen}% +\index{Herschel, J.~F.~W.!ref.\ to}% +%% -----File: 291.png---Folio 277------- +point of difficulty. Laplace's proof is perhaps the most satisfactory. + +Laplace's work on probability is very difficult reading, particularly +the part on the method of least squares. The +analytical processes are by no means clearly established or +free from error. ``No one was more sure of giving the +result of analytical processes correctly, and no one ever took +so little care to point out the various small considerations on +which correctness depends'' (De~Morgan). +\index{Demorgan@{De Morgan}!ref.\ to}% + +Of Laplace's papers on the attraction of ellipsoids, the most +\index{Attraction}% +important is the one published in 1785, and to a great extent +reprinted in the third volume of the \textit{Mécanique Céleste}. It +gives an exhaustive treatment of the general problem of +attraction of any ellipsoid upon a particle situated outside +\index{Ellipsoid!(attraction of)}% +or upon its surface. Spherical harmonics, or the so-called +``Laplace's coefficients,'' constitute a powerful analytic engine +\index{Laplace's coefficients}% +in the theory of attraction, in electricity, and magnetism. +The theory of spherical harmonics for two dimensions had +been previously given by Legendre. Laplace failed to make +due acknowledgment of this, and there existed, in consequence, +between the two great men, ``a feeling more +than coldness.'' The potential function,~$V$, is much used by +\index{Potential}% +Laplace, and is shown by him to satisfy the partial differential +equation $\dfrac{\partial^2V}{\partial x^2} + \dfrac{\partial^2V}{\partial y^2} + \dfrac{\partial^2V}{\partial z^2} = 0$. This is known as Laplace's +equation, and was first given by him in the more complicated +form which it assumes in polar co-ordinates. The notion +of potential was, however, not introduced into analysis by +Laplace. The honour of that achievement belongs to Lagrange.\Reference{49} +\index{Lagrange!ref.\ to}% + +Among the minor discoveries of Laplace are his method of +solving equations of the second, third, and fourth degrees, +\index{Equations!solution of}% +his memoir on singular solutions of differential equations, his +\index{Singular solutions}% +%% -----File: 292.png---Folio 278------- +\index{Acoustics}% +\index{Differential equations}% +researches in finite differences and in determinants, the establishment +\index{Determinants}% +\index{Finite differences}% +of the expansion theorem in determinants which had +been previously given by Vandermonde for a special case, the +\index{Vandermonde}% +\index{Vandermonde!ref.\ to}% +determination of the complete integral of the linear differential +equation of the second order. In the \textit{Mécanique Céleste} he +made a generalisation of Lagrange's theorem on the development +\index{Lagrange!ref.\ to}% +of functions in series known as Laplace's theorem. + +Laplace's investigations in physics were quite extensive. +We mention here his correction of Newton's formula on the +velocity of sound in gases by taking into account the changes +\index{Sound, velocity of}% +of elasticity due to the heat of compression and cold of rarefaction; +\index{Elasticity}% +his researches on the theory of tides; his mathematical +\index{Tides}% +theory of capillarity; his explanation of astronomical refraction; +\index{Capillarity}% +his formulæ for measuring heights by the barometer. + +Laplace's writings stand out in bold contrast to those of +Lagrange in their lack of elegance and symmetry. Laplace +\index{Laplace|)}% +looked upon mathematics as the tool for the solution of physical +problems. The true result being once reached, he spent +little time in explaining the various steps of his analysis, or +in polishing his work. The last years of his life were spent +mostly at Arcueil in peaceful retirement on a country-place, +where he pursued his studies with his usual vigour until his +death. He was a great admirer of Euler, and would often +\index{Euler!ref.\ to}% +say, ``Lisez Euler, lisez Euler, c'est notre maître à tous.'' + +\textbf{Abnit-Théophile Vandermonde} (1735--1796) studied music +during his youth in Paris and advocated the theory that all +art rested upon one general law, through which any one could +become a composer with the aid of mathematics. He was the +first to give a connected and logical exposition of the theory +of determinants, and may, therefore, almost be regarded as +the founder of that theory. He and Lagrange originated the +method of combinations in solving equations.\Reference{20} + +\textbf{Adrien Marie Legendre} (1752--1833) was educated at the +\index{Legendre|(}% +%% -----File: 293.png---Folio 279------- +\index{Elliptic functions}% +Collège Mazarin in Paris, where he began the study of mathematics +under Abbé Marie. His mathematical genius secured +\index{Marie, Abbé}% +for him the position of professor of mathematics at the military +school of Paris. While there he prepared an essay on +the curve described by projectiles thrown into resisting media +(ballistic curve), which captured a prize offered by the Royal +\index{Ballistic curve}% +Academy of Berlin. In 1780 he resigned his position in order +to reserve more time for the study of higher mathematics. +He was then made member of several public commissions. +In 1795 he was elected professor at the Normal School and +later was appointed to some minor government positions. +Owing to his timidity and to Laplace's unfriendliness toward +\index{Laplace!ref.\ to}% +him, but few important public offices commensurate with his +ability were tendered to him. + +As an analyst, second only to Laplace and Lagrange, Legendre +\index{Lagrange!ref.\ to}% +enriched mathematics by important contributions, mainly +on elliptic integrals, theory of numbers, attraction of ellipsoids, +and least squares. The most important of Legendre's +works is his \textit{Fonctions elliptiques}, issued in two volumes in +1825 and~1826. He took up the subject where Euler, Landen, +\index{Euler!ref.\ to}% +\index{Landen!ref.\ to}% +and Lagrange had left it, and for forty years was the only one +to cultivate this new branch of analysis, until at last Jacobi and +\index{Jacobi!ref.\ to}% +Abel stepped in with admirable new discoveries.\Reference{52} Legendre +\index{Abel!ref.\ to}% +imparted to the subject that connection and arrangement +which belongs to an independent science. Starting with an +integral depending upon the square root of a polynomial of +the fourth degree in~$x$, he showed that such integrals can be +brought back to three canonical forms, designated by $F(\phi)$, +$E(\phi)$, and~$\Pi(\phi)$, the radical being expressed in the form +$\Delta(\phi) = \sqrt{1 - k^2 \sin^2\phi}$. He also undertook the prodigious task +of calculating tables of arcs of the ellipse for different degrees +of amplitude and eccentricity, which supply the means of +integrating a large number of differentials. +%% -----File: 294.png---Folio 280------- +\index{Ellipsoid!(attraction of)}% +\index{Eulerian integrals}% +\index{Legendre's function}% +\index{Spherical trigonometry}% +\index{Trigonometry!spherical}% + +An earlier publication which contained part of his researches +on elliptic functions was his \textit{Calcul intégral} in three volumes +\index{Elliptic functions}% +(1811, 1816, 1817), in which he treats also at length of the +two classes of definite integrals named by him \emph{Eulerian}. He +tabulated the values of $\log \Gamma(p)$ for values of~$p$ between +$1$~and~$2$. + +One of the earliest subjects of research was the attraction +of spheroids, which suggested to Legendre the function~$P_n$, +named after him. His memoir was presented to the Academy +of Sciences in 1783. The researches of Maclaurin and Lagrange +\index{Lagrange!ref.\ to}% +\index{Maclaurin!ref.\ to}% +suppose the point attracted by a spheroid to be at the surface +or within the spheroid, but Legendre showed that in order to +determine the attraction of a spheroid on any external point +it suffices to cause the surface of another spheroid described +upon the same foci to pass through that point. Other memoirs +on ellipsoids appeared later. + +The two household gods to which Legendre sacrificed with +ever-renewed pleasure in the silence of his closet were the +elliptic functions and the theory of numbers. His researches +\index{Numbers!theory of numbers}% +\index{Theory of numbers}% +on the latter subject, together with the numerous scattered +fragments on the theory of numbers due to his predecessors +in this line, were arranged as far as possible into a systematic +whole, and published in two large quarto volumes, entitled +\textit{Théorie des nombres}, 1830. Before the publication of this +work Legendre had issued at divers times preliminary articles. +Its crowning pinnacle is the theorem of quadratic reciprocity, +\index{Quadratic reciprocity}% +previously indistinctly given by Euler without proof, but for the +\index{Euler!ref.\ to}% +first time clearly enunciated and partly proved by Legendre.\Reference{48} + +While acting as one of the commissioners to connect Greenwich +and Paris geodetically, Legendre calculated all the triangles +in France. This furnished the occasion of establishing +formulæ and theorems on geodesics, on the treatment of the +spherical triangle as if it were a plane triangle, by applying +%% -----File: 295.png---Folio 281------- +\index{Axioms (of geometry)}% +\index{Least squares}% +certain corrections to the angles, and on the method of least +squares, published for the first time by him without demonstration +in~1806. + +Legendre wrote an \textit{Éléments de Géométrie}, 1794, which +enjoyed great popularity, being generally adopted on the +Continent and in the United States as a substitute for Euclid. +\index{Euclid!ref.\ to}% +This great modern rival of Euclid passed through numerous +editions; the later ones containing the elements of trigonometry +and a proof of the irrationality of $\pi$ and~$\pi^2$. Much +\index{p@{$\pi$}: values for!proved to be irrational}% +attention was given by Legendre to the subject of parallel +lines. In the earlier editions of the \textit{Éléments}, he made direct +appeal to the senses for the correctness of the ``parallel-axiom.'' +He then attempted to demonstrate that ``axiom,'' but his +proofs did not satisfy even himself. In Vol.~XII. of the +Memoirs of the Institute is a paper by Legendre, containing +\index{Legendre|)}% +his last attempt at a solution of the problem. Assuming +space to be infinite, he proved satisfactorily that it is impossible +for the sum of the three angles of a triangle to exceed two +right angles; and that if there be any triangle the sum of +whose angles is two right angles, then the same must be true +of all triangles. But in the next step, to show that this sum +cannot be less than two right angles, his demonstration necessarily +failed. If it could be granted that the sum of the three +angles is always equal to two right angles, then the theory of +parallels could be strictly deduced. +\index{Parallels}% + +\textbf{Joseph Fourier} (1768--1830) was born at Auxerre, in central +\index{Fourier|(}% +France. He became an orphan in his eighth year. Through +the influence of friends he was admitted into the military +school in his native place, then conducted by the Benedictines +of the Convent of St.~Mark. He there prosecuted his studies, +particularly mathematics, with surprising success. He wished +to enter the artillery, but, being of low birth (the son of a +tailor), his application was answered thus: ``Fourier, not +%% -----File: 296.png---Folio 282------- +being noble, could not enter the artillery, although he were +a second Newton.''\Reference{53} He was soon appointed to the mathematical +\index{Newton!ref.\ to}% +chair in the military school. At the age of twenty-one +he went to Paris to read before the Academy of Sciences +a memoir on the resolution of numerical equations, which +\index{Equations!numerical}% +was an improvement on Newton's method of approximation. +This investigation of his early youth he never lost sight of. +He lectured upon it in the Polytechnic School; he developed +it on the banks of the Nile; it constituted a part of a work +entitled \textit{Analyse des equationes determines} (1831), which was +in press when death overtook him. This work contained +``Fourier's theorem'' on the number of real roots between +\index{Fourier's theorem}% +two chosen limits. Budan had published this result as early +\index{Budan}% +as 1807, but there is evidence to show that Fourier had established +it before Budan's publication. These brilliant results +were eclipsed by the theorem of Sturm, published in~1835. +\index{Sturm, J.~C.~F.!ref.\ to}% + +Fourier took a prominent part at his home in promoting +the Revolution. Under the French Revolution the arts and +sciences seemed for a time to flourish. The reformation of +the weights and measures was planned with grandeur of conception. +The Normal School was created in 1795, of which +Fourier became at first pupil, then lecturer. His brilliant +success secured him a chair in the Polytechnic School, the +duties of which he afterwards quitted, along with Monge and +\index{Monge!ref.\ to}% +Berthollet, to accompany Napoleon on his campaign to Egypt. +Napoleon founded the Institute of Egypt, of which Fourier +became secretary. In Egypt he engaged not only in scientific +work, but discharged important political functions. After +his return to France he held for fourteen years the prefecture +of Grenoble. During this period he carried on his elaborate +investigations on the propagation of heat in solid bodies, +published in 1822 in his work entitled \textit{La Theorie Analytique +de la Chaleur}. This work marks an epoch in the history of +%% -----File: 297.png---Folio 283------- +\index{Functions!arbitrary}% +\index{Infinite series}% +\index{Kelvin, Lord!ref.\ to}% +mathematical physics. ``Fourier's series'' constitutes its +\index{Fourier's series}% +gem. By this research a long controversy was brought to a +close, and the fact established that any arbitrary function +can be represented by a trigonometric series. The first +\index{Trigonometric series}% +announcement of this great discovery was made by Fourier +in 1807, before the French Academy. The trigonometric +series $\smash[b]{\sum\limits_{n=0}^{n=\infty} (a_n \sin nx + b_n \cos nx)}$ represents the function~$\phi(x)$ +for every value of~$x$, if the coefficients $a_n = \displaystyle\frac{1}{\pi} \int_{-\pi}^\pi \phi(x) \sin nx\, dx$, +and $b_n$ be equal to a similar integral. The weak point in +Fourier's analysis lies in his failure to prove generally that +the trigonometric series actually converges to the value of +the function. In 1827 Fourier succeeded Laplace as president +of the council of the Polytechnic School. + +Before proceeding to the origin of modern geometry we shall +speak briefly of the introduction of higher analysis into Great +Britain. This took place during the first quarter of this century. +The British began to deplore the very small progress +that science was making in England as compared with its +racing progress on the Continent. In 1813 the ``Analytical +\index{Analytical Society (in Cambridge)}% +Society'' was formed at Cambridge. This was a small club +established by George Peacock, John Herschel, Charles Babbage, +\index{Babbage}% +\index{Herschel, J.~F.~W.!ref.\ to}% +\index{Peacock!ref.\ to, X}% +and a few other Cambridge students, to promote, as it +was humorously expressed, the principles of pure ``$D$-ism,'' +that is, the Leibnizian notation in the calculus against those +\index{Notation!differential calculus}% +of ``dot-age,'' or of the Newtonian notation. This struggle +ended in the introduction into Cambridge of the notation +$\dfrac{dy}{dx}$, to the exclusion of the fluxional notation~$\dot{y}$. This +was a great step in advance, not on account of any great +superiority of the Leibnizian over the Newtonian notation, +but because the adoption of the former opened up to English +students the vast storehouses of continental discoveries. Sir +William Thomson, Tait, and some other modern writers find +\index{Tait}% +\index{Thomson, Sir William!ref.\ to}% +%% -----File: 298.png---Folio 284------- +\index{Arithmetical machine}% +\index{Calculating machines}% +\index{Fourier|)}% +\index{Machine, arithmetical}% +it frequently convenient to use both notations. Herschel, +\index{Herschel, J.~F.~W.!ref.\ to}% +Peacock, and Babbage translated, in 1816, from the French, +\index{Peacock}% +Lacroix's treatise on the differential and integral calculus, and +\index{Lacroix}% +added in 1820 two volumes of examples. Lacroix's was one +of the best and most extensive works on the calculus of that +time. Of the three founders of the ``Analytical Society,'' +Peacock afterwards did most work in pure mathematics. +Babbage became famous for his invention of a calculating +engine superior to Pascal's. It was never finished, owing +\index{Pascal!ref.\ to}% +to a misunderstanding with the government, and a consequent +failure to secure funds. John Herschel, the eminent +astronomer, displayed his mastery over higher analysis in +memoirs communicated to the Royal Society on new applications +of mathematical analysis, and in articles contributed +to cyclopædias on light, on meteorology, and on the history +of mathematics. + +\textbf{George Peacock} (1791--1858) was educated at Trinity College, +Cambridge, became Lowndean professor there, and later, dean +of Ely. His chief publications are his \textit{Algebra}, 1830 and~1842, +\index{Algebra!Peacock}% +and his \textit{Report on Recent Progress in Analysis}, which was the +first of several valuable summaries of scientific progress printed +in the volumes of the British Association. He was one of the +first to study seriously the fundamental principles of algebra, +and to fully recognise its purely symbolic character. He +advances, though somewhat imperfectly, the ``principle of the +permanence of equivalent forms.'' It assumes that the rules +applying to the symbols of arithmetical algebra apply also +in symbolical algebra. About this time D.~F. Gregory wrote +\index{Gregory, David F}% +a paper ``on the real nature of symbolical algebra,'' which +brought out clearly the commutative and distributive laws. +These laws had been noticed years before by the inventors +of symbolic methods in the calculus. It was Servois who +\index{Servois}% +introduced the names \emph{commutative} and \emph{distributive} in~1813. +%% -----File: 299.png---Folio 285------- +Peacock's investigations on the foundation of algebra were +considerably advanced by De~Morgan and Hankel. +\index{Demorgan@{De Morgan}!ref.\ to}% +\index{Hankel!ref.\ to}% + +\textbf{James Ivory} (1765--1842) was a Scotch mathematician who +\index{Ivory}% +for twelve years, beginning in 1804, held the mathematical +chair in the Royal Military College at Marlow (now at Sandhurst). +He was essentially a self-trained mathematician, and +almost the only one in Great Britain previous to the organisation +of the Analytical Society who was well versed in continental +mathematics. Of importance is his memoir (\textit{Phil.\ +Trans.}, 1809) in which the problem of the attraction of a +homogeneous ellipsoid upon an external point is reduced to +\index{Ellipsoid!(attraction of)}% +the simpler problem of the attraction of a related ellipsoid +upon a corresponding point interior to it. This is known as +``Ivory's theorem.'' He criticised with undue severity Laplace's +\index{Ivory's theorem}% +\index{Laplace!ref.\ to}% +solution of the method of least squares, and gave three proofs +\index{Least squares}% +of the principle without recourse to probability; but they are +\index{Probability}% +far from being satisfactory. + +\Subsection{The Origin of Modern Geometry} +\index{Geometry!modern synthetic|(}% + +By the researches of Descartes and the invention of the calculus, +the analytical treatment of geometry was brought into +great prominence for over a century. Notwithstanding the +efforts to revive synthetic methods made by Desargues, Pascal, +\index{Desargues!ref.\ to}% +\index{Pascal!ref.\ to}% +De~Lahire, Newton, and Maclaurin, the analytical method +\index{Delahire@{De Lahire}}% +\index{Maclaurin!ref.\ to}% +\index{Newton!ref.\ to}% +retained almost undisputed supremacy. It was reserved for +the genius of Monge to bring synthetic geometry in the +foreground, and to open up new avenues of progress. His +\textit{Géométrie descriptive} marks the beginning of a wonderful +development of modern geometry. + +Of the two leading problems of descriptive geometry, the +one---to represent by drawings geometrical magnitudes---was +brought to a high degree of perfection before the time of +%% -----File: 300.png---Folio 286------- +Monge; the other---to solve problems on figures in space +\index{Monge|(}% +by constructions in a plane---had received considerable attention +before his time. His most noteworthy predecessor +in descriptive geometry was the Frenchman Frézier (1682--1773). +\index{Descriptive geometry|(}% +\index{Frezier@{Frézier}}% +\index{Geometry!descriptive|(}% +But it remained for Monge to create descriptive +geometry as a \emph{distinct} branch of science by imparting to it +geometric generality and elegance. All problems previously +treated in a special and uncertain manner were referred +back to a few general principles. He introduced the line +of intersection of the horizontal and the vertical plane as +the axis of projection. By revolving one plane into the +other around this axis or ground-line, many advantages were +gained.\Reference{54} + +\textbf{Gaspard Monge} (1746--1818) was born at Beaune. The construction +of a plan of his native town brought the boy under +the notice of a colonel of engineers, who procured for him an +appointment in the college of engineers at Mézières. Being +of low birth, he could not receive a commission in the army, +but he was permitted to enter the annex of the school, where +surveying and drawing were taught. Observing that all the +operations connected with the construction of plans of fortification +were conducted by long arithmetical processes, he substituted +a geometrical method, which the commandant at first +refused even to look at, so short was the time in which it +could be practised; when once examined, it was received with +avidity. Monge developed these methods further and thus +created his descriptive geometry. Owing to the rivalry +between the French military schools of that time, he was not +permitted to divulge his new methods to any one outside of +this institution. In 1768 he was made professor of mathematics +at Mézières. In 1780, when conversing with two of his +pupils, S.~F. Lacroix and Gayvernon in Paris, he was obliged +\index{Lacroix}% +to say, ``All that I have here done by calculation, I could have +%% -----File: 301.png---Folio 287------- +\index{Imaginary quantities}% +done with the ruler and compass, but I am not allowed to +reveal these secrets to you.'' But Lacroix set himself to +\index{Lacroix}% +examine what the secret could be, discovered the processes, and +published them in 1795. The method was published by Monge +himself in the same year, first in the form in which the short-hand +writers took down his lessons given at the Normal School, +where he had been elected professor, and then again, in revised +form, in the \textit{Journal des écoles normales}. The next edition +occurred in 1798--1799. After an ephemeral existence of only +four months the Normal School was closed in~1795. In the +same year the Polytechnic School was opened, in the establishing +of which Monge took active part. He taught there +descriptive geometry until his departure from France to accompany +\index{Geometry!analytic}% +Napoleon on the Egyptian campaign. He was the first +president of the Institute of Egypt. Monge was a zealous +partisan of Napoleon and was, for that reason, deprived of all +his honours by Louis~XVIII\@. This and the destruction of the +Polytechnic School preyed heavily upon his mind. He did +not long survive this insult. + +Monge's numerous papers were by no means confined to descriptive +geometry. His analytical discoveries are hardly less +remarkable. He introduced into analytic geometry the methodic +\index{Analytic geometry}% +use of the equation of a line. He made important +contributions to surfaces of the second degree (previously +\index{Surfaces, theory of}% +studied by Wren and Euler) and discovered between the +\index{Euler!ref.\ to}% +\index{Wren!ref.\ to}% +theory of surfaces and the integration of partial differential +\index{Partial differential equations}% +equations, a hidden relation which threw new light upon both +subjects. He gave the differential of curves of curvature, +established a general theory of curvature, and applied it to the +ellipsoid. He found that the validity of solutions was not +impaired when imaginaries are involved among subsidiary +quantities. Monge published the following books: \textit{Statics}, +1786; \textit{Applications de l'algèbre à la géométrie}, 1805; \textit{Application +%% -----File: 302.png---Folio 288------- +\index{Brianchion}% +\index{Carnot, Lazare}% +de l'analyse à la géométrie}. The last two contain most of +his miscellaneous papers. + +Monge was an inspiring teacher, and he gathered around +him a large circle of pupils, among which were Dupin, Servois, +\index{Dupin}% +\index{Servois}% +Brianchon, Hachette, Biot, and Poncelet. +\index{Biot}% +\index{Hachette}% +\index{Poncelet!ref.\ to}% + +\textbf{Charles Dupin} (1784--1873), for many years professor of +mechanics in the Conservatoire des Arts et Métiers in Paris, +published in 1813 an important work on \textit{Développements de +géométrie}, in which is introduced the conception of conjugate +tangents of a point of a surface, and of the indicatrix.\Reference{53} It +contains also the theorem known as ``Dupin's theorem.'' +Surfaces of the second degree and descriptive geometry were +successfully studied by \textit{Jean Nicolas Pierre Hachette} (1769--1834), +who became professor of descriptive geometry at the +Polytechnic School after the departure of Monge for Rome and +Egypt. In 1822 he published his \textit{Traité de géométrie descriptive}. + +Descriptive geometry, which arose, as we have seen, in +technical schools in France, was transferred to Germany at +the foundation of technical schools there. G.~Schreiber, +\index{Schreiber}% +professor in Karlsruhe, was the first to spread Monge's +\index{Monge|)}% +geometry in Germany by the publication of a work thereon +in 1828--1829.\Reference{54} In the United States descriptive geometry was +\index{Descriptive geometry|)}% +\index{Geometry!descriptive|)}% +introduced in 1816 at the Military Academy in West Point +by Claude Crozet, once a pupil at the Polytechnic School in +\index{Crozet}% +Paris. Crozet wrote the first English work on the subject.\Reference{2} + +\textbf{Lazare Nicholas Marguerite Carnot} (1753--1823) was born at +Nolay in Burgundy, and educated in his native province. +He entered the army, but continued his mathematical studies, +and wrote in 1784 a work on machines, containing the earliest +proof that kinetic energy is lost in collisions of bodies. With +the advent of the Revolution he threw himself into politics, +and when coalesced Europe, in 1793, launched against France +a million soldiers, the gigantic task of organising fourteen +%% -----File: 303.png---Folio 289------- +\index{Brianchion}% +\index{Carnot, Lazare}% +\index{Differential calculus!philosophy of}% +\index{Dupin}% +armies to meet the enemy was achieved by him. He was +banished in 1796 for opposing Napoleon's \textit{coup d'état}. The +refugee went to Geneva, where he issued, in 1797, a work +still frequently quoted, entitled, \textit{Réflexions sur la Métaphysique +du Calcul Infinitésimal}. He declared himself as an ``irreconcilable +enemy of kings.'' After the Russian campaign he +offered to fight for France, though not for the empire. On +the restoration he was exiled. He died in Magdeburg. His +\textit{Géométrie de position}, 1803, and his \textit{Essay on Transversals}, +1806, are important contributions to modern geometry. While +Monge revelled mainly in three-dimensional geometry, Carnot +confined himself to that of two. By his effort to explain +the meaning of the negative sign in geometry he established +a ``geometry of position,'' which, however, is different from +the ``Geometrie der Lage'' of to-day. He invented a class +of general theorems on projective properties of figures, which +have since been pushed to great extent by Poncelet, Chasles, +\index{Chasles!ref.\ to}% +\index{Poncelet}% +and others. + +\textbf{Jean Victor Poncelet} (1788--1867), a native of Metz, took +part in the Russian campaign, was abandoned as dead on the +bloody field of Krasnoi, and taken prisoner to Saratoff. Deprived +there of all books, and reduced to the remembrance +of what he had learned at the Lyceum at Metz and the Polytechnic +School, where he had studied with predilection the +works of Monge, Carnot, and Brianchon, he began to study +mathematics from its elements. He entered upon original +researches which afterwards made him illustrious. While +in prison he did for mathematics what Bunyan did for +literature,---produced a much-read work, which has remained +of great value down to the present time. He returned to +France in 1814, and in 1822 published the work in question, +entitled, \textit{Traité des Propriétés projectives des figures}. In it +he investigated the properties of figures which remain unaltered +%% -----File: 304.png---Folio 290------- +by projection of the figures. The projection is not +effected here by parallel rays of prescribed direction, as with +Monge, but by central projection. Thus perspective projection, +used before him by Desargues, Pascal, Newton, and Lambert, +\index{Desargues!ref.\ to}% +\index{Lambert!ref.\ to}% +\index{Newton!ref.\ to}% +\index{Pascal!ref.\ to}% +was elevated by him into a fruitful geometric method. +In the same way he elaborated some ideas of De~Lahire, +\index{Delahire@{De Lahire}}% +Servois, and Gergonne into a regular method---the method +\index{Gergonne!ref.\ to}% +\index{Servois}% +of ``reciprocal polars.'' To him we owe the Law of Duality +\index{Duality}% +\index{Reciprocal polars}% +as a consequence of reciprocal polars. As an independent +principle it is due to Gergonne. Poncelet wrote much on +\index{Poncelet}% +applied mechanics. In 1838 the Faculty of Sciences was +\index{Mechanics!more recent work}% +enlarged by his election to the chair of mechanics. + +While in France the school of Monge was creating modern +geometry, efforts were made in England to revive Greek +geometry by \textbf{Robert Simson} (1687--1768) and \textbf{Matthew Stewart} +\index{Simson}% +\index{Stewart}% +(1717--1785). Stewart was a pupil of Simson and Maclaurin, +\index{Maclaurin!ref.\ to}% +and succeeded the latter in the chair at Edinburgh. During +the eighteenth century he and Maclaurin were the only prominent +mathematicians in Great Britain. His genius was ill-directed +by the fashion then prevalent in England to ignore +higher analysis. In his \textit{Four Tracts, Physical and Mathematical}, +1761, he applied geometry to the solution of difficult +\index{Geometry!modern synthetic|)}% +astronomical problems, which on the Continent were approached +analytically with greater success. He published, in +1746, \textit{General Theorems}, and in 1763, his \textit{Propositiones geometricæ +more veterum demonstratæ}. The former work contains +sixty-nine theorems, of which only five are accompanied +by demonstrations. It gives many interesting new results +on the circle and the straight line. Stewart extended some +theorems on transversals due to Giovanni Ceva (1648--1737), +\index{Ceva}% +an Italian, who published in 1678 at Mediolani a work containing +the theorem now known by his name. +%% -----File: 305.png---Folio 291------- +\index{Bolyai, Johann!ref.\ to}% +\index{Bolyai, Wolfgang!ref.\ to}% + + +\Chapter{Recent Times} + +\First{Never} more zealously and successfully has mathematics +been cultivated than in this century. Nor has progress, as +in previous periods, been confined to one or two countries. +While the French and Swiss, who alone during the preceding +epoch carried the torch of progress, have continued to develop +mathematics with great success, from other countries whole +armies of enthusiastic workers have wheeled into the front +rank. Germany awoke from her lethargy by bringing forward +Gauss, Jacobi, Dirichlet, and hosts of more recent men; +\index{Dirichlet!ref.\ to}% +\index{Gauss!ref.\ to}% +\index{Jacobi!ref.\ to}% +Great Britain produced her De~Morgan, Boole, Hamilton, +\index{Boole!ref.\ to}% +\index{Demorgan@{De Morgan}!ref.\ to}% +\index{Hamilton, W.~R.!ref.\ to}% +besides champions who are still living; Russia entered the +arena with her Lobatchewsky; Norway with Abel; Italy with +\index{Abel!ref.\ to}% +\index{Lobatchewsky!ref.\ to}% +Cremona; Hungary with her two Bolyais; the United States +\index{Cremona!ref.\ to}% +with Benjamin Peirce. +\index{Peirce, B.!ref.\ to}% + +The productiveness of modern writers has been enormous. +``It is difficult,'' says Professor Cayley,\Reference[]{56} ``to give an idea of +\index{Cayley!ref.\ to}% +the vast extent of modern mathematics. This word `extent' +is not the right one: I mean extent crowded with beautiful +detail,---not an extent of mere uniformity such as an objectless +plain, but of a tract of beautiful country seen at first in +the distance, but which will bear to be rambled through and +studied in every detail of hillside and valley, stream, rock, +wood, and flower.'' It is pleasant to the mathematician to +think that in his, as in no other science, the achievements of +%% -----File: 306.png---Folio 292------- +every age remain possessions forever; new discoveries seldom +disprove older tenets; seldom is anything lost or wasted. + +If it be asked wherein the utility of some modern extensions +of mathematics lies, it must be acknowledged that it is +at present difficult to see how they are ever to become applicable +to questions of common life or physical science. But +our inability to do this should not be urged as an argument +against the pursuit of such studies. In the first place, we +know neither the day nor the hour when these abstract +developments will find application in the mechanic arts, in +physical science, or in other branches of mathematics. For +example, the whole subject of graphical statics, so useful +\index{Graphical statics}% +to the practical engineer, was made to rest upon von Staudt's +\index{Von Staudt!ref.\ to}% +\textit{Geometrie der Lage}; Hamilton's ``principle of varying action'' +\index{Action!varying}% +\index{Hamilton, W.~R.!ref.\ to}% +\index{Varying action, principle of}% +has its use in astronomy; complex quantities, general integrals, +\index{Complex quantities}% +and general theorems in integration offer advantages in +the study of electricity and magnetism. ``The utility of such +researches,'' says Spottiswoode,\Reference[]{57} ``can in no case be discounted, +\index{Spottiswoode!ref.\ to}% +or even imagined beforehand. Who, for instance, would have +supposed that the calculus of forms or the theory of substitutions +\index{Calculus!of operations}% +\index{Substitutions, theory of}% +would have thrown much light upon ordinary equations; +or that Abelian functions and hyperelliptic transcendents +\index{Abelian functions}% +\index{Hyperelliptic functions}% +would have told us anything about the properties of curves; +\index{Curves!theory of}% +or that the calculus of operations would have helped us in +\index{Operations, calculus of}% +any way towards the figure of the earth?'' A second reason +\index{Earth!figure of}% +\index{Figure of the earth}% +in favour of the pursuit of advanced mathematics, even when +there is no promise of practical application, is this, that mathematics, +like poetry and music, deserves cultivation for its +own sake. + +The great characteristic of modern mathematics is its generalising +tendency. Nowadays little weight is given to isolated +theorems, ``except as affording hints of an unsuspected +new sphere of thought, like meteorites detached from some +%% -----File: 307.png---Folio 293------- +\index{Invariant}% +undiscovered planetary orb of speculation.'' In mathematics, +as in all true sciences, no subject is considered in itself alone, +but always as related to, or an outgrowth of, other things. +The development of the notion of continuity plays a leading +\index{Continuity}% +part in modern research. In geometry the principle of continuity, +\index{Geometry!modern synthetic|(}% +the idea of correspondence, and the theory of projection +\index{Correspondence, principle of}% +constitute the fundamental modern notions. Continuity +asserts itself in a most striking way in relation to the circular +points at infinity in a plane. In algebra the modern idea finds +\index{Infinity}% +expression in the theory of linear transformations and invariants, +and in the recognition of the value of homogeneity and +\index{Homogeneity}% +symmetry. + + +\Section{Synthetic Geometry} +\index{Synthetic geometry|(}% + +The conflict between geometry and analysis which arose +near the close of the last century and the beginning of the +present has now come to an end. Neither side has come +out victorious. The greatest strength is found to lie, not in +the suppression of either, but in the friendly rivalry between +the two, and in the stimulating influence of the one upon the +other. Lagrange prided himself that in his \textit{\DPtypo{Mecanique}{Mécanique} Analytique} +\index{Lagrange!ref.\ to}% +he had succeeded in avoiding all figures; but since his +time mechanics has received much help from geometry. + +Modern synthetic geometry was created by several investigators +about the same time. It seemed to be the outgrowth +of a desire for general methods which should serve as threads +of Ariadne to guide the student through the labyrinth of theorems, +corollaries, porisms, and problems. Synthetic geometry +was first cultivated by Monge, Carnot, and Poncelet in France; +\index{Carnot, Lazare!ref.\ to}% +\index{Monge!ref.\ to}% +\index{Poncelet!ref.\ to}% +it then bore rich fruits at the hands of Möbius and Steiner in +\index{Mobius@{Möbius}!ref.\ to}% +\index{Steiner!ref.\ to}% +Germany and Switzerland, and was finally developed to still +%% -----File: 308.png---Folio 294------- +\index{Anharmonic ratio}% +higher perfection by Chasles in France, von Staudt in Germany, +\index{Chasles!ref.\ to}% +\index{Von Staudt!ref.\ to}% +and Cremona in Italy. +\index{Cremona!ref.\ to|(}% + +\textbf{Augustus Ferdinand Möbius} (1790--1868) was a native of +\index{Mobius@{Möbius}}% +Schulpforta in Prussia. He studied at Göttingen under +Gauss, also at Leipzig and Halle. In Leipzig he became, in +\index{Gauss!ref.\ to}% +1815, privat-docent, the next year extraordinary professor of +astronomy, and in 1844 ordinary professor. This position he +held till his death. The most important of his researches +are on geometry. They appeared in \textit{Crelle's Journal}, and in +his celebrated work entitled \textit{Der Barycentrische Calcul}, Leipzig, +1827. As the name indicates, this calculus is based upon +properties of the centre of gravity.\Reference{58} Thus, that the point~$S$ +is the centre of gravity of weights $a$,~$b$, $c$,~$d$ placed at the +points $A$,~$B$, $C$,~$D$ respectively, is expressed by the equation +\[ +(a + b + c + d)S = aA + bB + cC + dD. +\] +His calculus is the beginning of a quadruple algebra, and contains +the germs of Grassmann's marvellous system. In designating +\index{Grassmann!ref.\ to}% +segments of lines we find throughout this work for the +first time consistency in the distinction of positive and negative +by the order of letters $AB$,~$BA$. Similarly for triangles +and tetrahedra. The remark that it is always possible to give +three points $A$,~$B$,~$C$ such weights $\alpha$,~$\beta$,~$\gamma$ that any fourth point~$M$ +in their plane will become a centre of mass, led Möbius +to a new system of co-ordinates in which the position of a +\index{Co-ordinates}% +point was indicated by an equation, and that of a line by +co-ordinates. By this algorithm he found by algebra many +geometric theorems expressing mainly invariantal properties,---for +example, the theorems on the anharmonic relation. +Möbius wrote also on statics and astronomy. He generalised +spherical trigonometry by letting the sides or angles of triangles +\index{Spherical trigonometry}% +\index{Trigonometry!spherical}% +exceed~$180°$. +%% -----File: 309.png---Folio 295------- +\index{Salmon}% + +\textbf{Jacob Steiner} (1796--1863), ``the greatest geometrician since +\index{Steiner}% +the time of Euclid,'' was born in Utzendorf in the Canton of +Bern. He did not learn to write till he was fourteen. At +eighteen he became a pupil of Pestalozzi. Later he studied +at Heidelberg and Berlin. When Crelle started, in 1826, the +celebrated mathematical journal bearing his name, Steiner and +Abel became leading contributors. In 1832 Steiner published +his \textit{Systematische Entwickelung der Abhängigkeit geometrischer +Gestalten von einander}, ``in which is uncovered the organism +by which the most diverse phenomena (\textit{Erscheinungen}) in +the world of space are united to each other.'' Through the +influence of Jacobi and others, the chair of geometry was +\index{Jacobi!ref.\ to}% +founded for him at Berlin in~1834. This position he occupied +until his death, which occurred after years of bad health. In +his \textit{Systematische Entwickelungen}, for the first time, is the +principle of duality introduced at the outset. This book and +von Staudt's lay the foundation on which synthetic geometry +\index{Von Staudt!ref.\ to}% +in its present form rests. Not only did he fairly complete the +theory of curves and surfaces of the second degree, but he +\index{Surfaces, theory of}% +made great advances in the theory of those of higher degrees. +In his hands synthetic geometry made prodigious progress. +New discoveries followed each other so rapidly that he often +did not take time to record their demonstrations. In an +article in \textit{Crelle's Journal} on \textit{Allgemeine Eigenschaften Algebraischer +\index{Crelle's Journal}% +Curven} he gives without proof theorems which were +declared by Hesse to be ``like Fermat's theorems, riddles to +\index{Hesse!ref.\ to}% +the present and future generations.'' Analytical proofs of +some of them have been given since by others, but Cremona +finally proved them all by a synthetic method. Steiner discovered +synthetically the two prominent properties of a surface +of the third order; viz.~that it contains twenty-seven +straight lines and a pentahedron which has the double points +for its vertices and the lines of the Hessian of the given surface +\index{Hessian}% +%% -----File: 310.png---Folio 296------- +for its edges.\Reference{55} The first property was discovered analytically +somewhat earlier in England by Cayley and Salmon, +\index{Cayley!ref.\ to}% +and the second by Sylvester. Steiner's work on this subject +\index{Steiner}% +\index{Sylvester!ref.\ to}% +was the starting-point of important researches by H.~Schröter, +\index{SchroterH@{Schröter, H.}!ref.\ to}% +F.~August, L.~Cremona, and R.~Sturm. Steiner made investigations +\index{August}% +\index{Cremona!ref.\ to|)}% +\index{Sturm, R.}% +by synthetic methods on maxima and minima, and +arrived at the solution of problems which at that time altogether +surpassed the analytic power of the calculus of variations. +\index{Calculus!of variations}% +He generalised the \textit{hexagrammum mysticum} and also +\index{Hexagrammum mysticum}% +Malfatti's problem.\Reference{59} Malfatti, in 1803, proposed the problem, +\index{Malfatti}% +\index{Malfatti's problem}% +to cut three cylindrical holes out of a three-sided prism in +such a way that the cylinders and the prism have the same +altitude and that the volume of the cylinders be a maximum. +This problem was reduced to another, now generally known +as Malfatti's problem: to inscribe three circles in a triangle +that each circle will be tangent to two sides of a triangle and +to the other two circles. Malfatti gave an analytical solution, +but Steiner gave without proof a construction, remarked that +there were thirty-two solutions, generalised the problem by +replacing the three lines by three circles, and solved the +analogous problem for three dimensions. This general problem +was solved analytically by C.~H. Schellbach (1809--1892) +\index{Schellbach}% +and Cayley, and by Clebsch with the aid of the addition +\index{Clebsch!ref.\ to}% +theorem of elliptic functions.\Reference{60} +\index{Elliptic functions}% + +Steiner's researches are confined to synthetic geometry. He +hated analysis as thoroughly as Lagrange disliked geometry. +\index{Lagrange!ref.\ to}% +Steiner's \textit{Gesammelte Werke} were published in Berlin in 1881 +and 1882. + +\textbf{Michel Chasles} (1793--1880) was born at Epernon, entered +\index{Chasles|(}% +the Polytechnic School of Paris in 1812, engaged afterwards +in business, which he later gave up that he might devote all +his time to scientific pursuits. In 1841 he became professor of +geodesy and mechanics at the Polytechnic School; later, +%% -----File: 311.png---Folio 297------- +\index{Cubic curves}% +``Professeur de Géométrie supérieure à la Faculté des Sciences +de Paris.'' He was a voluminous writer on geometrical subjects. +In 1837 he published his admirable \textit{Aperçu historique +sur l'origine et le développement des méthodes en géométrie}, containing +a history of geometry and, as an appendix, a treatise +``sur deux principes généraux de la Science.'' The \textit{Aperçu +historique} is still a standard historical work; the appendix +contains the general theory of Homography (Collineation) and +of duality (Reciprocity). The name \emph{duality} is due to Joseph +\index{Duality}% +Diaz Gergonne (1771--1859). Chasles introduced the term +\index{Gergonne}% +\emph{anharmonic ratio}, corresponding to the German \textit{Doppelverhältniss} +\index{Anharmonic ratio}% +and to Clifford's \emph{cross-ratio}. Chasles and Steiner +\index{Clifford!ref.\ to}% +\index{Steiner!ref.\ to}% +elaborated independently the modern synthetic or projective +geometry. Numerous original memoirs of Chasles were published +later in the \textit{Journal de l'École Polytechnique}. He gave +a reduction of cubics, different from Newton's in this, that the +\index{Newton!ref.\ to}% +five curves from which all others can be projected are symmetrical +with respect to a centre. In 1864 he began the publication, +in the \textit{Comptes rendus}, of articles in which he solves +by his ``method of characteristics'' and the ``principle of correspondence'' +\index{Characteristics, method of}% +\index{Correspondence, principle of}% +\index{Method of characteristics}% +an immense number of problems. He determined, +for instance, the number of intersections of two curves +in a plane. The method of characteristics contains the basis +of enumerative geometry. The application of the principle of +\index{Enumerative geometry}% +correspondence was extended by Cayley, A.~Brill, H.~G. Zeuthen, +\index{Brill, A.}% +\index{Cayley!ref.\ to}% +\index{Zeuthen!ref.\ to}% +H.~A. Schwarz, G.~H. Halphen (1844--1889), and others. +\index{Halphen!ref.\ to}% +\index{Schwarz!ref.\ to}% +The full value of these principles of Chasles was not brought +out until the appearance, in 1879, of the \textit{Kalkül der Abzählenden +Geometrie} by Hermann Schubert of Hamburg. This +\index{Schubert}% +work contains a masterly discussion of the problem of enumerative +geometry, viz.~to determine how many geometric figures +of given definition satisfy a sufficient number of conditions. +Schubert extended his enumerative geometry to $n$-dimensional +space.\Reference{55} +%% -----File: 312.png---Folio 298------- + +To Chasles we owe the introduction into projective geometry +of non-projective properties of figures by means of the infinitely +distant imaginary sphero-circle.\Reference{61} Remarkable is his +\index{Imaginary points, lines, etc}% +complete solution, in 1846, by synthetic geometry, of the +difficult question of the attraction of an ellipsoid on an external +\index{Ellipsoid!(attraction of)}% +point. This was accomplished analytically by Poisson in +\index{Poisson!ref.\ to}% +1835. The labours of Chasles and Steiner raised synthetic +\index{Steiner!ref.\ to}% +geometry to an honoured and respected position by the side +of analysis. + +\textbf{Karl Georg Christian von Staudt} (1798--1867) was born in +\index{Von Staudt}% +Rothenburg on the Tauber, and, at his death, was professor +in Erlangen. His great works are the \textit{Geometrie der Lage}, +Nürnberg, 1847, and his \textit{Beiträge zur Geometrie der Lage}, 1856--1860. +The author cut loose from algebraic formulæ and from +metrical relations, particularly the anharmonic ratio of Steiner +and Chasles, and then created a geometry of position, which +\index{Chasles|)}% +is a complete science in itself, independent of all measurements. +He shows that projective properties of figures have +no dependence whatever on measurements, and can be established +without any mention of them. In his theory of what +he calls ``Würfe,'' he even gives a geometrical definition of +a number in its relation to geometry as determining the position +of a point. The \textit{Beiträge} contains the first complete and +general theory of imaginary points, lines, and planes in projective +geometry. Representation of an imaginary point is +sought in the combination of an involution with a determinate +direction, both on the real line through the point. +While purely projective, von Staudt's method is intimately +related to the problem of representing by actual points and +lines the imaginaries of analytical geometry. This was systematically +undertaken by C.~F. Maximilien Marie, who worked, +\index{Marie, C. F. M.}% +however, on entirely different lines. An independent attempt +has been made recently (1893) by F.~H. Loud of Colorado +\index{Loud}% +%% -----File: 313.png---Folio 299------- +\index{Funicular polygons}% +College. Von~Staudt's geometry of position was for a long +\index{Von Staudt}% +time disregarded, mainly, no doubt, because his book is +extremely condensed. An impulse to the study of this subject +was given by Culmann, who rests his graphical statics upon +\index{Culmann}% +\index{Graphical statics}% +the work of von~Staudt. An interpreter of von~Staudt was +at last found in Theodor Reye of Strassburg, who wrote a +\index{Reye}% +\textit{Geometrie der Lage} in 1868. + +Synthetic geometry has been studied with much success by +\textbf{Luigi Cremona}, professor in the University of Rome. In +\index{Cremona}% +his \textit{Introduzione ad una teoria geometrica delle curve piane} +he developed by a uniform method many new results and +proved synthetically all important results reached before that +time by analysis. His writings have been translated into +German by M.~Curtze, professor at the gymnasium in Thorn. +\index{Curtze, M}% +The theory of the transformation of curves and of the correspondence +of points on curves was extended by him to three +dimensions. Ruled surfaces, surfaces of the second order, +\index{Surfaces, theory of}% +space-curves of the third order, and the general theory of +surfaces have received much attention at his hands. + +\textbf{Karl Culmann}, professor at the Polytechnicum in Zürich, +published an epoch-making work on \textit{Die graphische Statik}, +Zürich, 1864, which has rendered graphical statics a great +rival of analytical statics. Before Culmann, \textit{B.~E. Cousinery} +\index{Cousinery}% +had turned his attention to the graphical calculus, but he +made use of perspective, and not of modern geometry.\Reference{62} Culmann +is the first to undertake to present the graphical calculus +as a symmetrical whole, holding the same relation to the new +geometry that analytical mechanics does to higher analysis. +He makes use of the polar theory of reciprocal figures as +expressing the relation between the force and the funicular +polygons. He deduces this relation without leaving the plane +of the two figures. But if the polygons be regarded as projections +of lines in space, these lines may be treated as reciprocal +%% -----File: 314.png---Folio 300------- +\index{Culmann}% +\index{Non-Euclidean geometry|(}% +\index{Parallels}% +elements of a ``Nullsystem.'' This was done by \textit{Clerk +Maxwell} in 1864, and elaborated further by \textit{Cremona}.\Reference{63} The +\index{Cremona!ref.\ to}% +\index{Maxwell!ref.\ to}% +graphical calculus has been applied by \textit{O.~Mohr} of Dresden +\index{Mohr}% +to the elastic line for continuous spans. \textit{Henry~T. Eddy}, of +\index{Eddy}% +the Rose Polytechnic Institute, gives graphical solutions of +problems on the maximum stresses in bridges under concentrated +loads, with aid of what he calls ``reaction polygons.'' +\index{Reaction polygons}% +A standard work, \textit{La Statique graphique}, 1874, was issued by +Maurice Levy of Paris. +\index{Levy}% + +Descriptive geometry (reduced to a science by Monge in +\index{Descriptive geometry}% +\index{Geometry!descriptive}% +\index{Monge!ref.\ to}% +France, and elaborated further by his successors, \textit{Hachette}, +\index{Hachette}% +\textit{Dupin}, \textit{Olivier}, \textit{J.~de la Gournerie}) was soon studied also in +\index{Dupin!ref.\ to}% +\index{Gournerie}% +\index{Olivier}% +other countries. The French directed their attention mainly +to the theory of surfaces and their curvature; the Germans +and Swiss, through Schreiber, Pohlke, Schlessinger, and particularly +\index{Pohlke}% +\index{Schlessinger}% +\index{Schreiber}% +Fiedler, interwove projective and descriptive geometry. +\index{Fiedler}% +Bellavitis in Italy worked along the same line. The +\index{Bellavitis!ref.\ to}% +theory of shades and shadows was first investigated by the +French writers just quoted, and in Germany treated most +exhaustively by Burmester.\Reference{62} +\index{Burmester}% + +During the present century very remarkable generalisations +have been made, which reach to the very root of two of the +oldest branches of mathematics,---elementary algebra and +geometry. In algebra the laws of operation have been extended; +in geometry the axioms have been searched to the +\index{Axioms (of geometry)}% +bottom, and the conclusion has been reached that the space +defined by Euclid's axioms is not the only possible non-contradictory +space. Euclid proved (I.~27) that ``if a straight +line falling on two other straight lines make the alternate +angles equal to one another, the two straight lines shall be +parallel to one another.'' Being unable to prove that in every +other case the two lines are not parallel, he \emph{assumed} this to +be true in what is generally called the 12th ``axiom,'' by some +%% -----File: 315.png---Folio 301------- +the 11th ``axiom.'' But this so-called axiom is far from +axiomatic. After centuries of desperate but fruitless attempts +to prove Euclid's assumption, the bold idea dawned upon +the minds of several mathematicians that a geometry might +be built up without assuming the parallel-axiom. While +Legendre still endeavoured to establish the axiom by rigid +\index{Legendre!ref.\ to}% +proof, Lobatchewsky brought out a publication which assumed +\index{Lobatchewsky}% +the contradictory of that axiom, and which was the first of +a series of articles destined to clear up obscurities in the +fundamental concepts, and to greatly extend the field of +geometry. + +\textbf{Nicholaus Ivanovitch Lobatchewsky} (1793--1856) was born at +Makarief, in Nischni-Nowgorod, Russia, studied at Kasan, and +from 1827 to 1846 was professor and rector of the University +of Kasan. His views on the foundation of geometry were +first made public in a discourse before the physical and mathematical +faculty at Kasan, and first printed in the Kasan +\textit{Messenger} for 1829, and then in the \textit{Gelehrte Schriften der +Universität Kasan}, 1836--1838, under the title, ``New Elements +of Geometry, with a complete theory of Parallels.'' Being +\index{Parallels}% +in the Russian language, the work remained unknown to +foreigners, but even at home it attracted no notice. In 1840 +he published a brief statement of his researches in Berlin. +Lobatchewsky constructed an ``imaginary geometry,'' as he +\index{Imaginary geometry}% +called it, which has been described by Clifford as ``quite +simple, merely Euclid without the vicious assumption.'' A +remarkable part of this geometry is this, that through a +point an indefinite number of lines can be drawn in a plane, +none of which cut a given line in the same plane. A similar +system of geometry was deduced independently by the Bolyais +in Hungary, who called it ``absolute geometry.'' +\index{Absolute geometry}% + +\textbf{Wolfgang Bolyai de~Bolya} (1775--1856) was born in Szekler-Land, +\index{Bolyai, Wolfgang}% +Transylvania. After studying at Jena, he went to +%% -----File: 316.png---Folio 302------- +Göttingen, where he became intimate with Gauss, then nineteen +\index{Gauss!ref.\ to}% +years old. Gauss used to say that Bolyai was the only +\index{Bolyai, Johann}% +\index{Bolyai, Wolfgang}% +man who fully understood his views on the metaphysics of +mathematics. Bolyai became professor at the Reformed College +of Maros-Vásárhely, where for forty-seven years he had +for his pupils most of the present professors of Transylvania. +The first publications of this remarkable genius were dramas +and poetry. Clad in old-time planter's garb, he was truly +original in his private life as well as in his mode of thinking. +He was extremely modest. No monument, said he, should +stand over his grave, only an apple-tree, in memory of the +three apples; the two of Eve and Paris, which made hell out +of earth, and that of Newton, which elevated the earth again +\index{Newton!ref.\ to}% +into the circle of heavenly bodies.\Reference{64} His son, \textbf{Johann Bolyai} +(1802--1860), was educated for the army, and distinguished +himself as a profound mathematician, an impassioned violin-player, +and an expert fencer. He once accepted the challenge +of thirteen officers on condition that after each duel he might +play a piece on his violin, and he vanquished them all. + +The chief mathematical work of Wolfgang Bolyai appeared +in two volumes, 1832--1833, entitled \textit{Tentamen juventutem +studiosam in elementa matheseos puræ\ldots introducendi}. It +is followed by an appendix composed by his son Johann on +\textit{The Science Absolute of Space}. Its twenty-six pages make the +name of Johann Bolyai immortal. He published nothing else, +but he left behind one thousand pages of manuscript which +have never been read by a competent mathematician! His +father seems to have been the only person in Hungary who +really appreciated the merits of his son's work. For thirty-five +years this appendix, as also Lobatchewsky's researches, +remained in almost entire oblivion. Finally Richard Baltzer +\index{Baltzer, R.!ref.\ to}% +of the University of Giessen, in 1867, called attention to the +wonderful researches. Johann Bolyai's \textit{Science Absolute of} +%% -----File: 317.png---Folio 303------- +\textit{Space} and Lobatchewsky's \textit{Geometrical Researches on the +\index{Lobatchewsky!ref.\ to}% +Theory of Parallels} (1840) were rendered easily accessible to +\index{Parallels}% +American readers by translations into English made in 1891 +by George Bruce Halsted of the University of Texas. +\index{Halsted}% + +The Russian and Hungarian mathematicians were not the +only ones to whom pangeometry suggested itself. A copy of +the \textit{Tentamen} reached Gauss, the elder Bolyai's former room-mate %[** PP: Only instance] +\index{Gauss!ref.\ to}% +at Göttingen, and this Nestor of German mathematicians +was surprised to discover in it worked out what he himself +had begun long before, only to leave it after him in his +papers. As early as 1792 he had started on researches of that +character. His letters show that in 1799 he was trying to +prove \textit{a~priori} the reality of Euclid's system; but some time +\index{Euclid!ref.\ to}% +within the next thirty years he arrived at the conclusion +reached by Lobatchewsky and Bolyai. In 1829 he wrote to +Bessel, stating that his ``conviction that we cannot found +\index{Bessel!ref.\ to}% +geometry completely a~priori has become, if possible, still +firmer,'' and that ``if number is merely a product of our +mind, space has also a \emph{reality beyond} our mind of which we +cannot fully foreordain the laws \textit{a~priori}.'' The term \emph{non-Euclidean +geometry} is due to Gauss. It has recently been +brought to notice that \textit{Geronimo Saccheri}, a Jesuit father of +\index{Saccheri}% +Milan, in 1733 anticipated Lobatchewsky's doctrine of the +parallel angle. Moreover, G.~B. Halsted has pointed out +that in 1766 Lambert wrote a paper ``Zur Theorie der Parallellinien,'' +\index{Lambert!ref.\ to}% +published in the \textit{Leipziger Magazin für reine und +angewandte Mathematik}, 1786, in which: (1)~The failure of +the parallel-axiom in surface-spherics gives a geometry with +angle-sum $> 2$ right angles; (2)~In order to make intuitive +a geometry with angle-sum $< 2$ right angles we need the aid +of an ``imaginary sphere" (pseudo-sphere); (3)~In a space +with the angle-sum differing from 2 right angles, there is +an absolute measure (Bolyai's natural unit for length). +%% -----File: 318.png---Folio 304------- + +In 1854, nearly twenty years later, Gauss heard from his +\index{Gauss!ref.\ to}% +pupil, \textit{Riemann}, a marvellous dissertation carrying the discussion +\index{Riemann!ref.\ to}% +one step further by developing the notion of \nnobrkdash-ply +extended magnitude, and the measure-relations of which a +manifoldness of $n$ dimensions is capable, on the assumption +that every line may be measured by every other. Riemann +applied his ideas to space. He taught us to distinguish +between ``unboundedness'' and ``infinite extent.'' According +\index{Infinity}% +to him we have in our mind a more general notion of space, +\ie~a notion of non-Euclidean space; but we learn \emph{by experience} +that our physical space is, if not exactly, at least to high +degree of approximation, Euclidean space. Riemann's profound +dissertation was not published until 1867, when it +appeared in the \textit{Göttingen Abhandlungen}. Before this the +idea of \nnobrkdash-dimensions had suggested itself under various +\index{Hyperspace}% +aspects to Lagrange, Plücker, and H.~Grassmann. About the +\index{Grassmann!ref.\ to}% +\index{Lagrange!ref.\ to}% +\index{Plücker!ref.\ to}% +same time with Riemann's paper, others were published from +the pens of \textit{Helmholtz} and \textit{Beltrami}. These contributed powerfully +\index{Beltrami}% +\index{Helmholtz!ref.\ to}% +to the victory of logic over excessive empiricism. This +period marks the beginning of lively discussions upon this subject. +Some writers---Bellavitis, for example---were able to +\index{Bellavitis!ref.\ to}% +see in non-Euclidean geometry and \nnobrkdash-dimensional space nothing +but huge caricatures, or diseased outgrowths of mathematics. +Helmholtz's article was entitled \textit{Thatsachen, welche +der Geometrie zu Grunde liegen}, 1868, and contained many of +the ideas of Riemann. Helmholtz popularised the subject in +lectures, and in articles for various magazines. + +\textbf{Eugenio Beltrami}, born at Cremona, Italy, in 1835, and now +professor at Rome, wrote the classical paper \textit{Saggio di interpretazione +della geometria non-euclidea} (Giorn.\ di Matem.,~6), +which is analytical (and, like several other papers, should be +mentioned elsewhere were we to adhere to a strict separation +between synthesis and analysis). He reached the brilliant +%% -----File: 319.png---Folio 305------- +and surprising conclusion that the theorems of non-Euclidean +geometry find their realisation upon surfaces of constant negative +curvature. He studied, also, surfaces of constant positive +curvature, and ended with the interesting theorem that the +space of constant positive curvature is contained in the space +of constant negative curvature. These researches of Beltrami, +\index{Beltrami}% +Helmholtz, and Riemann culminated in the conclusion that +\index{Helmholtz!ref.\ to}% +\index{Riemann!ref.\ to}% +on surfaces of constant curvature we may have three geometries,---the +non-Euclidean on a surface of constant negative +curvature, the spherical on a surface of constant positive curvature, +and the Euclidean geometry on a surface of zero curvature. +The three geometries do not contradict each other, but +are members of a system,---a geometrical trinity. The ideas +of \DPtypo{hyper-space}{hyperspace} were brilliantly expounded and popularised in +\index{Hyperspace}% +England by Clifford. +\index{Clifford}% + +\textbf{William Kingdon Clifford} (1845--1879) was born at Exeter, +educated at Trinity College, Cambridge, and from 1871 until +his death professor of applied mathematics in University College, +London. His premature death left incomplete several +brilliant researches which he had entered upon. Among these +are his paper \textit{On Classification of Loci} and his \textit{Theory of +Graphs}. He wrote articles \textit{On the Canonical Form and +Dissection of a Riemann's Surface}, on \textit{Biquaternions}, and +an incomplete work on the \textit{Elements of Dynamic}. The +theory of polars of curves and surfaces was generalised by +him and by Reye. His classification of loci, 1878, being a +\index{Reye!ref.\ to}% +general study of curves, was an introduction to the study +of \nnobrkdash-dimensional space in a direction mainly projective. +This study has been continued since chiefly by G.~Veronese +\index{Veronese}% +of Padua, C.~Segre of Turin, E.~Bertini, F.~Aschieri, P.~Del~Pezzo +\index{Aschieri}% +\index{Bertini}% +\index{Delpezzo@{Del Pezzo}}% +\index{Segre}% +of Naples. + +Beltrami's researches on non-Euclidean geometry were followed, +in 1871, by important investigations of Felix Klein, +\index{Klein!ref.\ to}% +%% -----File: 320.png---Folio 306------- +\index{Clifford}% +\index{Parallels}% +resting upon Cayley's \textit{Sixth Memoir on Quantics}, 1859. The +\index{Cayley!ref.\ to}% +question whether it is not possible to so express the metrical +properties of figures that they will not vary by projection (or +linear transformation) had been solved for special projections +by Chasles, Poncelet, and E.~Laguerre (1834--1886) of Paris, +\index{Chasles!ref.\ to}% +\index{Laguerre}% +\index{Poncelet!ref.\ to}% +but it remained for Cayley to give a general solution by defining +the distance between two points as an arbitrary constant +multiplied by the logarithm of the anharmonic ratio in which +\index{Anharmonic ratio}% +the line joining the two points is divided by the fundamental +quadric. Enlarging upon this notion, Klein showed the independence +\index{Klein!ref.\ to}% +of projective geometry from the parallel-axiom, and +by properly choosing the law of the measurement of distance +deduced from projective geometry the spherical, Euclidean, +and pseudospherical geometries, named by him respectively +the elliptic, parabolic, and hyperbolic geometries. This suggestive +investigation was followed up by numerous writers, +particularly by G.~Battaglini of Naples, E.~d'Ovidio of Turin, +\index{Battaglini}% +\index{Dovidio@{D'Ovidio}}% +R.~de~Paolis of Pisa, F.~Aschieri, A.~Cayley, F.~Lindemann +\index{Aschieri}% +\index{Depaolis@{De Paolis}}% +\index{Lindemann!ref.\ to}% +of Munich, E.~Schering of Göttingen, W.~Story of Clark +\index{Schering}% +\index{Story}% +University, H.~Stahl of Tübingen, A.~Voss of Würzburg, +\index{Stahl}% +\index{Voss}% +Homersham Cox, A.~Buchheim.\Reference{55} The geometry of $n$ dimensions +\index{Buchheim!ref.\ to}% +\index{Cox}% +was studied along a line mainly metrical by a host of +writers, among whom may be mentioned Simon Newcomb of +\index{Newcomb!ref.\ to}% +the Johns Hopkins University, L.~Schläfli of Bern, W.~I. +\index{Schlafli@{Schläfli}}% +Stringham of the University of California, W.~Killing of +\index{Killing}% +\index{Stringham}% +Münster, T.~Craig of the Johns Hopkins, R.~Lipschitz of +\index{Craig, T.}% +\index{Lipschitz}% +Bonn. R.~S. Heath and Killing investigated the kinematics +\index{Heath}% +and mechanics of such a space. Regular solids in \nnobrkdash-dimensional +space were studied by Stringham, Ellery~W. Davis +\index{Davis, E.~W.}% +of the University of Nebraska, R.~Hoppe of Berlin, and +\index{Hoppe}% +others. Stringham gave pictures of projections upon our +space of regular solids in four dimensions, and Schlegel at +\index{Schlegel!ref.\ to, XII}% +Hagen constructed models of such projections. These are +%% -----File: 321.png---Folio 307------- +\index{Non-Euclidean geometry|)}% +among the most curious of a series of models published by +L.~Brill in Darmstadt. It has been pointed out that if a +\index{Brill, L.}% +fourth dimension existed, certain motions could take place +which we hold to be impossible. Thus Newcomb showed the +\index{Newcomb!ref.\ to}% +possibility of turning a closed material shell inside out by simple +flexure without either stretching or tearing; Klein pointed +\index{Klein!ref.\ to}% +out that knots could not be tied; Veronese showed that a +\index{Veronese!ref.\ to}% +body could be removed from a closed room without breaking +the walls; C.~S. Peirce proved that a body in four-fold space +\index{Peirce, C.~S.!ref.\ to}% +either rotates about two axes at once, or cannot rotate without +losing one of its dimensions. + +\Section{Analytic Geometry} +\index{Analytic geometry|(}% +\index{Geometry!analytic|(}% + +In the preceding chapter we endeavoured to give a flash-light %[** PP: Only instance] +view of the rapid advance of synthetic geometry. In +connection with hyperspace we also mentioned analytical +treatises. Modern synthetic and modern analytical geometry +have much in common, and may be grouped together +under the common name ``projective geometry.'' Each has +\index{Projective geometry}% +advantages over the other. The continual direct viewing of +figures as existing in space adds exceptional charm to the +study of the former, but the latter has the advantage in this, +that a well-established routine in a certain degree may outrun +thought itself, and thereby aid original research. While in +Germany Steiner and von Staudt developed synthetic geometry, +\index{Synthetic geometry|)}% +Plücker laid the foundation of modern analytic geometry. +\index{Geometry!modern synthetic|)}% +\index{Plücker|(}% + +\textbf{Julius Plücker} (1801--1868) was born at Elberfeld, in Prussia. +After studying at Bonn, Berlin, and Heidelberg, he spent +a short time in Paris attending lectures of Monge and his +pupils. Between 1826 and 1836 he held positions successively +at Bonn, Berlin, and Halle. He then became professor of +%% -----File: 322.png---Folio 308------- +physics at Bonn. Until 1846 his original researches were on +geometry. In 1828 and in 1831 he published his \textit{Analytisch-Geometrische +Entwicklungen} in two volumes. Therein he +adopted the abbreviated notation (used before him in a more +restricted way by Bobillier), and avoided the tedious process +\index{Bobillier}% +of algebraic elimination by a geometric consideration. In the +\index{Elimination}% +second volume the principle of duality is formulated analytically. +\index{Duality}% +With him duality and homogeneity found expression +\index{Homogeneity}% +already in his system of co-ordinates. The homogenous or +\index{Co-ordinates}% +tri-linear system used by him is much the same as the co-ordinates +of Möbius. In the identity of analytical operation +and geometric construction Plücker looked for the source of +\index{Plücker!ref.\ to}% +his proofs. The \textit{System der Analytischen Geometrie}, 1835, contains +a complete classification of plane curves of the third +order, based on the nature of the points at infinity. The +\index{Infinity}% +\textit{Theorie der Algebraischen Curven}, 1839, contains, besides an +enumeration of curves of the fourth order, the analytic relations +between the ordinary singularities of plane curves +known as ``Plücker's equations,'' by which he was able to +explain ``Poncelet's paradox.'' The discovery of these relations +\index{Poncelet!ref.\ to}% +\index{Poncelet's paradox}% +is, says Cayley, ``the most important one beyond all +\index{Cayley!ref.\ to}% +comparison in the entire subject of modern geometry.'' But +in Germany Plücker's researches met with no favour. His +method was declared to be unproductive as compared with +the synthetic method of Steiner and Poncelet! His relations +\index{Steiner!ref.\ to}% +with Jacobi were not altogether friendly. Steiner once +\index{Jacobi!ref.\ to}% +declared that he would stop writing for \textit{Crelle's Journal} if +Plücker continued to contribute to it.\Reference{66} The result was that +many of Plücker's researches were published in foreign journals, +and that his work came to be better known in France +and England than in his native country. The charge was +also brought against Plücker that, though occupying the chair +of physics, he was no physicist. This induced him to relinquish +%% -----File: 323.png---Folio 309------- +\index{Congruency of lines}% +mathematics, and for nearly twenty years to devote +his energies to physics. Important discoveries on Fresnel's +wave-surface, magnetism, spectrum-analysis were made by +him. But towards the close of his life he returned to his +first love,---mathematics,---and enriched it with new discoveries. +By considering space as made up of lines he created +a ``new geometry of space.'' Regarding a right line as a +curve involving four arbitrary parameters, one has the whole +system of lines in space. By connecting them by a single +relation, he got a ``complex'' of lines; by connecting them +\index{Complex of lines}% +with a twofold relation, he got a ``congruency'' of lines. His +first researches on this subject were laid before the Royal +Society in~1865. His further investigations thereon appeared +in 1868 in a posthumous work entitled \textit{Neue Geometrie des +Raumes gegründet auf die Betrachtung der geraden Linie als +Raumelement}, edited by Felix Klein. Plücker's analysis lacks +\index{Klein!ref.\ to}% +the elegance found in Lagrange, Jacobi, Hesse, and Clebsch. +\index{Clebsch!ref.\ to}% +\index{Hesse|(}% +\index{Hesse!ref.\ to}% +\index{Jacobi!ref.\ to}% +\index{Lagrange!ref.\ to}% +For many years he had not kept up with the progress of +geometry, so that many investigations in his last work had +already received more general treatment on the part of others. +The work contained, nevertheless, much that was fresh and +original. The theory of complexes of the second degree, left +unfinished by Plücker, was continued by Felix Klein, who +\index{Plücker|)}% +greatly extended and supplemented the ideas of his master. + +\textbf{Ludwig Otto Hesse} (1811--1874) was born at Königsberg, and +studied at the university of his native place under Bessel, +\index{Bessel!ref.\ to}% +Jacobi, Richelot, and F.~Neumann. Having taken the doctor's +\index{Neumann, C.!ref.\ to}% +\index{Neumann, F.~E.!ref.\ to}% +\index{Richelot!ref.\ to}% +degree in 1840, he became docent at Königsberg, and in 1845 +extraordinary professor there. Among his pupils at that time +were Durège, Carl Neumann, Clebsch, Kirchhoff. The Königsberg +\index{Durège!ref.\ to}% +\index{Kirchhoff!ref.\ to}% +period was one of great activity for Hesse. Every new +discovery increased his zeal for still greater achievement. +His earliest researches were on surfaces of the second order, +\index{Surfaces, theory of}% +%% -----File: 324.png---Folio 310------- +\index{Invariant}% +and were partly synthetic. He solved the problem to construct +any tenth point of such a surface when nine points are given. +The analogous problem for a conic had been solved by Pascal +\index{Pascal!ref.\ to}% +by means of the hexagram. A difficult problem confronting +mathematicians of this time was that of elimination. Plücker +\index{Elimination}% +had seen that the main advantage of his special method in +analytic geometry lay in the avoidance of algebraic elimination. +Hesse, however, showed how by determinants to make +algebraic elimination easy. In his earlier results he was +anticipated by Sylvester, who published his dialytic method +\index{Sylvester!ref.\ to}% +of elimination in 1840. These advances in algebra Hesse +applied to the analytic study of curves of the third order. By +linear substitutions, he reduced a form of the third degree in +three variables to one of only four terms, and was led to an +important determinant involving the second differential coefficient +of a form of the third degree, called the ``Hessian.'' +\index{Hessian}% +The ``Hessian'' plays a leading part in the theory of invariants, +a subject first studied by Cayley. Hesse showed that +his determinant gives for every curve another curve, such that +the double points of the first are points on the second, or +``Hessian." Similarly for surfaces (Crelle, 1844). Many of +\index{Surfaces, theory of}% +the most important theorems on curves of the third order are +due to Hesse. He determined the curve of the 14th order, +which passes through the 56~points of contact of the 28~bi-tangents +of a curve of the fourth order. His great memoir on +this subject (Crelle, 1855) was published at the same time as +was a paper by Steiner treating of the same subject. + +Hesse's income at Königsberg had not kept pace with his +growing reputation. Hardly was he able to support himself +and family. In 1855 he accepted a more lucrative position at +Halle, and in 1856 one at Heidelberg. Here he remained until +1868, when he accepted a position at a technic school in +Munich.\Reference{67} At Heidelberg he revised and enlarged upon his +%% -----File: 325.png---Folio 311------- +\index{Gournerie}% +\index{McCullagh}% +previous researches, and published in 1861 his \textit{Vorlesungen +über die Analytische Geometrie des Raumes, insbesondere über +Flächen 2.~Ordnung}. More elementary works soon followed. +While in Heidelberg he elaborated a principle, his ``Uebertragungsprincip." +According to this, there corresponds to every +point in a plane a pair of points in a line, and the projective +geometry of the plane can be carried back to the geometry of +points in a line. + +The researches of Plücker and Hesse were continued in England +\index{Hesse|)}% +by Cayley, Salmon, and Sylvester. It may be premised here +\index{Cayley!ref.\ to}% +\index{Salmon|(}% +\index{Sylvester!ref.\ to}% +that among the early writers on analytical geometry in England +was \textbf{James Booth} (1806--1878), whose chief results are embodied +\index{Booth}% +in his \textit{Treatise on Some New Geometrical Methods}; and \textbf{James +MacCullagh} (1809--1846), who was professor of natural philosophy +\index{MacCullagh}% +at Dublin, and made some valuable discoveries on the +theory of quadrics. The influence of these men on the +progress of geometry was insignificant, for the interchange of +scientific results between different nations was not so complete +at that time as might have been desired. In further illustration +of this, we mention that Chasles in France elaborated +\index{Chasles!ref.\ to}% +subjects which had previously been disposed of by Steiner in +\index{Steiner!ref.\ to}% +Germany, and Steiner published researches which had been +given by Cayley, Sylvester, and Salmon nearly five years +earlier. Cayley and Salmon in 1849 determined the straight +lines in a cubic surface, and studied its principal properties, +while Sylvester in 1851 discovered the pentahedron of such a +surface. Cayley extended Plücker's equations to curves of +higher singularities. Cayley's own investigations, and those +of M.~Nöther of Erlangen, G.~H. Halphen (1844--1889) of the +\index{Halphen}% +\index{Nother@{Nöther}}% +Polytechnic School in Paris, De~La~\DPtypo{Gournérie}{Gournerie} of Paris, A.~Brill +\index{Brill, A.}% +of Tübingen, lead to the conclusion that each higher singularity +of a curve is equivalent to a certain number of simple +singularities,---the node, the ordinary cusp, the double tangent, +%% -----File: 326.png---Folio 312------- +\index{Deficiency of curves}% +and the inflection. Sylvester studied the ``twisted Cartesian,'' +\index{Sylvester!ref.\ to}% +\index{Twisted Cartesian}% +a curve of the fourth order. Salmon helped powerfully +towards the spreading of a knowledge of the new algebraic and +geometric methods by the publication of an excellent series of +text-books (\textit{Conic Sections}, \textit{Modern Higher Algebra}, \textit{Higher +Plane Curves}, \textit{Geometry of Three Dimensions}), which have +been placed within easy reach of German readers by a free +translation, with additions, made by Wilhelm Fiedler of the +\index{Fiedler}% +Polytechnicum in Zürich. The next great worker in the field +of analytic geometry was Clebsch. +\index{Clebsch}% + +\textbf{Rudolf Friedrich Alfred Clebsch} (1833--1872) was born at +Königsberg in Prussia, studied at the university of that place +under Hesse, Richelot, F.~Neumann. From 1858 to 1863 he +\index{Hesse!ref.\ to}% +\index{Neumann, F.~E.!ref.\ to}% +\index{Richelot!ref.\ to}% +held the chair of theoretical mechanics at the Polytechnicum +in Carlsruhe. The study of Salmon's works led him into +algebra and geometry. In 1863 he accepted a position at the +University of \DPtypo{Giesen}{Giessen}, where he worked in conjunction with +Paul Gordan (now of Erlangen). In 1868 Clebsch went to +\index{Gordan}% +Göttingen, and remained there until his death. He worked +successively at the following subjects: Mathematical physics, +the calculus of variations and partial differential equations of +the first order, the general theory of curves and surfaces, +Abelian functions and their use in geometry, the theory of +\index{Abelian functions}% +invariants, and ``Flächenabbildung.''\Reference{68} He proved theorems +on the pentahedron enunciated by Sylvester and Steiner; he +\index{Steiner!ref.\ to}% +made systematic use of ``deficiency'' (\textit{Geschlecht}) as a fundamental +principle in the classification of algebraic curves. The +notion of deficiency was known before him to Abel and Riemann. +\index{Abel!ref.\ to}% +\index{Riemann!ref.\ to}% +At the beginning of his career, Clebsch had shown +how elliptic functions could be advantageously applied to +Malfatti's problem. The idea involved therein, viz.\ the use +\index{Malfatti's problem}% +of higher transcendentals in the study of geometry, led him +to his greatest discoveries. Not only did he apply Abelian +%% -----File: 327.png---Folio 313------- +\index{Armemante}% +\index{Ptolemy!ref.\ to}% +functions to geometry, but conversely, he drew geometry into +the service of Abelian functions. + +Clebsch made liberal use of determinants. His study of +\index{Clebsch}% +\index{Determinants}% +curves and surfaces began with the determination of the points +of contact of lines which meet a surface in four consecutive +points. Salmon had proved that these points lie on the intersection +\index{Salmon|)}% +of the surface with a derived surface of the degree +$11n - 24$, but his solution was given in inconvenient form. +Clebsch's investigation thereon is a most beautiful piece of +analysis. + +The representation of one surface upon another (\textit{Flächenabbildung}), +\index{Flächenabbildung}% +so that they have a $(1, 1)$ correspondence, was +thoroughly studied for the first time by Clebsch. The representation +of a sphere on a plane is an old problem which +drew the attention of Ptolemæus, Gerard Mercator, Lambert, +\index{Lambert!ref.\ to}% +\index{Mercator, G.}% +Gauss, Lagrange. Its importance in the construction of maps +\index{Gauss!ref.\ to}% +\index{Lagrange!ref.\ to}% +is obvious. Gauss was the first to represent a surface upon +another with a view of more easily arriving at its properties. +Plücker, Chasles, Cayley, thus represented on a plane the +\index{Cayley!ref.\ to}% +\index{Chasles!ref.\ to}% +\index{Plücker!ref.\ to}% +geometry of quadric surfaces; Clebsch and Cremona, that of +\index{Cremona!ref.\ to}% +cubic surfaces. Other surfaces have been studied in the same +way by recent writers, particularly M.~Nöther of Erlangen, +\index{Nother@{Nöther}}% +Armenante, Felix Klein, Korndörfer, Caporali, H.~G. Zeuthen +\index{Caporali}% +\index{Klein!ref.\ to}% +\index{Korndörfer}% +\index{Zeuthen}% +of Copenhagen. A fundamental question which has as yet +received only a partial answer is this: What surfaces can be +represented by a $(1, 1)$ correspondence upon a given surface? +This and the analogous question for curves was studied by +Clebsch. Higher correspondences between surfaces have been +investigated by Cayley and Nöther. The theory of surfaces +has been studied also by \textbf{Joseph Alfred Serret} (1819--1885), professor +\index{Serret}% +at the Sorbonne in Paris, \textbf{Jean Gaston Darboux} of Paris, +\index{Darboux}% +\textit{John Casey} of Dublin (died 1891), \textit{W.~R.~W. Roberts} of Dublin, +\index{Casey}% +\index{Roberts}% +\textit{H.~Schröter} (1829--1892) of Breslau. Surfaces of the +\index{SchroterH@{Schröter, H.}}% +%% -----File: 328.png---Folio 314------- +\index{Differential equations}% +\index{Fresnel's wave-surface}% +fourth order were investigated by Kummer, and Fresnel's +\index{Kummer!ref.\ to}% +wave-surface, studied by Hamilton, is a particular case of +\index{Hamilton, W.~R.!ref.\ to}% +Kummer's quartic surface, with sixteen canonical points and +sixteen singular tangent planes.\Reference{56} + +The infinitesimal calculus was first applied to the determination +of the measure of curvature of surfaces by Lagrange, +\index{Curvature, measure of}% +\index{Lagrange!ref.\ to}% +\index{Surfaces, theory of}% +Euler, and \DPtypo{Meunier}{Meusnier} (1754--1793) of Paris. Then followed the +\index{Euler!ref.\ to}% +\index{Meunier}% +researches of Monge and Dupin, but they were eclipsed by +\index{Dupin!ref.\ to}% +\index{Monge!ref.\ to}% +the work of Gauss, who disposed of this difficult subject in a +\index{Gauss!ref.\ to}% +way that opened new vistas to geometricians. His treatment +is embodied in the \textit{Disquisitiones generales circa superficies +curvas} (1827) and \textit{Untersuchungen über gegenstände\DPnote{** Non-capitalization seems to be a legitimate variant} der +höheren Geodäsie} of 1843 and~1846. He defined the measure +of curvature at a point to be the reciprocal of the product +of the two principal radii of curvature at that point. From +this flows the theorem of \textit{Johann August Grunert} (1797--1872; +\index{Grunert}% +professor in Greifswald), that the arithmetical mean of the +radii of curvature of all normal sections through a point is the +radius of a sphere which has the same measure of curvature +as has the surface at that point. Gauss's deduction of the +formula of curvature was simplified through the use of determinants +by \textit{Heinrich Richard Baltzer} (1818--1887) of Giessen.\Reference{69} +\index{Baltzer, R.}% +Gauss obtained an interesting theorem that if one surface be +developed (\textit{abgewickelt}) upon another, the measure of curvature +remains unaltered at each point. The question whether +two surfaces having the same curvature in corresponding +points can be unwound, one upon the other, was answered +by F.~Minding in the affirmative only when the curvature is +\index{Minding}% +constant. The case of variable curvature is difficult, and was +studied by Minding, J.~Liouville (1806--1882) of the Polytechnic +\index{Liouville!ref.\ to}% +School in Paris, Ossian Bonnet of Paris (died 1892). +\index{Bonnet, O.}% +Gauss's measure of curvature, expressed as a function of curvilinear +co-ordinates, gave an impetus to the study of differential-invariants, +\index{Co-ordinates}% +%% -----File: 329.png---Folio 315------- +\index{Analytic geometry|)}% +\index{Kelvin, Lord!ref.\ to}% +or differential-parameters, which have been +investigated by Jacobi, C.~Neumann, Sir James Cockle, +\index{Cockle}% +\index{Jacobi!ref.\ to}% +\index{Neumann, C.!ref.\ to}% +Halphen, and elaborated into a general theory by Beltrami, +\index{Beltrami!ref.\ to}% +\index{Halphen!ref.\ to}% +S.~Lie, and others. Beltrami showed also the connection +between the measure of curvature and the geometric axioms. +\index{Axioms (of geometry)}% + +Various researches have been brought under the head of +``analysis situs.'' The subject was first investigated by +\index{Analysis situs}% +Leibniz, and was later treated by Gauss, whose theory of +\index{Gauss!ref.\ to}% +\index{Leibniz!ref.\ to}% +knots (\textit{Verschlingungen}) has been employed recently by J.~B. +Listing, O.~Simony, F.~Dingeldey, and others in their ``topologic +\index{Dingeldey}% +\index{Listing}% +\index{Simony}% +studies.'' Tait was led to the study of knots by Sir +William Thomson's theory of vortex atoms. In the hands +\index{Thomson, Sir William!ref.\ to}% +of Riemann the analysis situs had for its object the determination +\index{Riemann!ref.\ to}% +of what remains unchanged under transformations +brought about by a combination of infinitesimal distortions. +In continuation of his work, Walter Dyck of Munich wrote on +\index{Dyck}% +the analysis situs of three-dimensional spaces. + +Of geometrical text-books not yet mentioned, reference +should be made to \textbf{Alfred Clebsch's} \textit{Vorlesungen über Geometrie}, +\index{Clebsch!ref.\ to}% +edited by Ferdinand Lindemann, now of Munich; \textbf{Frost's} +\index{Frost}% +\index{Lindemann}% +\textit{Solid Geometry}; \textbf{Durège's} \textit{Ebene Curven dritter Ordnung}. +\index{Durège!ref.\ to}% +\index{Geometry!analytic|)}% + + +\Section{Algebra} +\index{Algebra!recent|(}% + +The progress of algebra in recent times may be considered +under three principal heads: the study of fundamental laws +and the birth of new algebras, the growth of the theory of +equations, and the development of what is called modern +higher algebra. + +We have already spoken of George Peacock and D.~F. +\index{Peacock!ref.\ to, X}% +Gregory in connection with the fundamental laws of algebra. +\index{Gregory, David F}% +Much was done in this line by De~Morgan. +%% -----File: 330.png---Folio 316------- + +\textbf{Augustus De~Morgan} (1806--1871) was born at Madura (Madras), +\index{Demorgan@{De Morgan}}% +and educated at Trinity College, Cambridge. His scruples +about the doctrines of the established church prevented +him from proceeding to the M.A.~degree, and from sitting +for a fellowship. In 1828 he became professor at the newly +established University of London, and taught there until +1867, except for five years, from 1831--1835. De~Morgan was +a unique, manly character, and pre-eminent as a teacher. The +value of his original work lies not so much in increasing our +stock of mathematical knowledge as in putting it all upon a +thoroughly logical basis. He felt keenly the lack of close +reasoning in mathematics as he received it. He said once: +``We know that mathematicians care no more for logic than +\index{Logic}% +logicians for mathematics. The two eyes of exact science are +mathematics and logic: the mathematical sect puts out the +logical eye, the logical sect puts out the mathematical eye; +each believing that it can see better with one eye than with +two.'' De~Morgan saw with both eyes. He analysed logic +mathematically, and studied the logical analysis of the laws, +symbols, and operations of mathematics; he wrote a \textit{Formal +Logic} as well as a \textit{Double Algebra}, and corresponded both with +Sir William Hamilton, the metaphysician, and Sir William +\index{Hamilton, W.}% +\index{Hamilton, W.~R.!ref.\ to}% +Rowan Hamilton, the mathematician. Few contemporaries +were as profoundly read in the history of mathematics as +was De~Morgan. No subject was too insignificant to receive +his attention. The authorship of ``Cocker's Arithmetic'' and +the work of circle-squarers was investigated as minutely as was +\index{Circle-squarers}% +the history of the invention of the calculus. Numerous articles +of his lie scattered in the volumes of the \textit{Penny} and \textit{English +Cyclopædias}. His \textit{Differential Calculus}, 1842, is still a +standard work, and contains much that is original with the +author. For the \textit{Encyclopædia Metropolitana} he wrote on the +calculus of functions (giving principles of symbolic reasoning) +%% -----File: 331.png---Folio 317------- +and on the theory of probability. Celebrated is his \textit{Budget of +Paradoxes}, 1872. He published memoirs ``On the Foundation +of Algebra'' (\textit{Trans.\ of Cam.\ Phil.\ Soc.}, 1841, 1842, 1844, and~1847). + +In Germany symbolical algebra was studied by Martin Ohm, +\index{Ohm, M}% +who wrote a \textit{System der Mathematik} in~1822. The ideas of +Peacock and De~Morgan recognise the possibility of algebras +which differ from ordinary algebra. Such algebras were +indeed not slow in forthcoming, but, like non-Euclidean +geometry, some of them were slow in finding recognition. +This is true of Grassmann's, Bellavitis's, and Peirce's discoveries, +\index{Bellavitis!ref.\ to}% +\index{Grassmann!ref.\ to}% +\index{Peirce, B.!ref.\ to}% +but Hamilton's quaternions met with immediate +\index{Hamilton, W.~R.!ref.\ to}% +\index{Quaternions!ref.\ to}% +appreciation in England. These algebras offer a geometrical +interpretation of imaginaries. During the times of Descartes, +\index{Descartes!ref.\ to}% +Newton, and Euler, we have seen the negative and the imaginary, +\index{Euler!ref.\ to}% +\index{Newton!ref.\ to}% +$\sqrt{-1}$, accepted as numbers, but the latter was still +regarded as an algebraic fiction. The first to give it a geometric +picture, analogous to the geometric interpretation of the +negative, was \textit{H.~Kühn}, a teacher in Danzig, in a publication of +\index{Kuhn@{Kühn, H.}}% +1750--1751. He represented $a\sqrt{-1}$ by a line perpendicular +to the line~$a$, and equal to~$a$ in length, and construed $\sqrt{-1}$ as +the mean proportional between $+1$~and~$-1$. This same idea +was developed further, so as to give a geometric interpretation +of $a + \sqrt{-b}$, by \textit{Jean-Robert Argand} (1768--?) of Geneva, +\index{Argand}% +in a remarkable \textit{Essai} (1806).\Reference{70} The writings of Kühn and +Argand were little noticed, and it remained for Gauss to break +\index{Gauss!ref.\ to}% +down the last opposition to the imaginary. He introduced~$i$ as +an independent unit co-ordinate to~$1$, and $a + ib$ as a ``complex +\index{Complex quantities}% +number.'' The connection between complex numbers and +points on a plane, though artificial, constituted a powerful +aid in the further study of symbolic algebra. The mind +required a visual representation to aid it. The notion of +what we now call vectors was growing upon mathematicians, +%% -----File: 332.png---Folio 318------- +and the geometric addition of vectors in space was discovered +independently by Hamilton, Grassmann, and others, about the +\index{Grassmann!ref.\ to}% +\index{Hamilton, W.~R.}% +same time. + +\textbf{William Rowan Hamilton} (1805--1865) was born of Scotch +parents in Dublin. His early education, carried on at home, +was mainly in languages. At the age of thirteen he is said to +have been familiar with as many languages as he had lived +years. About this time he came across a copy of Newton's +\textit{Universal Arithmetic}. After reading that, he took up successively +analytical geometry, the calculus, Newton's \textit{Principia}, +Laplace's \textit{Mécanique Céleste}. At the age of eighteen he +published a paper correcting a mistake in Laplace's work. In +1824 he entered Trinity College, Dublin, and in 1827, while he +was still an undergraduate, he was appointed to the chair of +astronomy. His early papers were on optics. In 1832 he +predicted conical refraction, a discovery by aid of mathematics +which ranks with the discovery of Neptune by +Le~Verrier and Adams. Then followed papers on the \textit{Principle +of Varying Action} (1827) and a general method of +\index{Action!varying}% +\index{Varying action, principle of}% +dynamics (1834--1835). He wrote also on the solution of +\index{Dynamics}% +equations of the fifth degree, the hodograph, fluctuating +functions, the numerical solution of differential equations. +\index{Differential equations}% + +The capital discovery of Hamilton is his quaternions, in +\index{Quaternions}% +which his study of algebra culminated. In 1835 he published +in the \textit{Transactions of the Royal Irish Academy} his Theory of +Algebraic Couples. He regarded algebra ``as being no mere +art, nor language, nor primarily a science of quantity, but +rather as the science of order of progression.'' Time appeared +to him as the picture of such a progression. Hence his definition +of algebra as ``the science of pure time.'' It was the +subject of years' meditation for him to determine what he +should regard as the product of each pair of a system of perpendicular +directed lines. At last, on the 16th~of October, +%% -----File: 333.png---Folio 319------- +1843, while walking with his wife one evening, along the +Royal Canal in Dublin, the discovery of quaternions flashed +\index{Quaternions}% +upon him, and he then engraved with his knife on a stone in +Brougham Bridge the fundamental formula $i^2 = j^2 = k^2 = ijk = +-1$. At the general meeting of the Irish Academy, a month +later, he made the first communication on quaternions. An +account of the discovery was given the following year in the +\textit{Philosophical Magazine}. Hamilton displayed wonderful fertility +\index{Hamilton, W.~R.}% +in their development. His \textit{Lectures on Quaternions}, +delivered in Dublin, were printed in 1852. His \textit{Elements of +Quaternions} appeared in 1866. Quaternions were greatly +admired in England from the start, but on the Continent they +received less \DPtypo{attenttion}{attention}. P.~G. Tait's \textit{Elementary Treatise} +\index{Tait}% +helped powerfully to spread a knowledge of them in England. +Cayley, Clifford, and Tait advanced the subject somewhat by +\index{Cayley!ref.\ to}% +\index{Clifford!ref.\ to}% +original contributions. But there has been little progress in +recent years, except that made by Sylvester in the solution of +\index{Sylvester!ref.\ to}% +quaternion equations, nor has the application of quaternions +to physics been as extended as was predicted. The change +in notation made in France by Hoüel and by Laisant has been +\index{Houel@{Hoüel}}% +\index{Laisant}% +considered in England as a wrong step, but the true cause for +the lack of progress is perhaps more deep-seated. There is +indeed great doubt as to whether the quaternionic product can +claim a necessary and fundamental place in a system of vector +analysis. Physicists claim that there is a loss of naturalness +in taking the square of a vector to be negative. In order to +meet more adequately their wants, \textit{J.~W. Gibbs} of Yale University +\index{Gibbs!ref.\ to}% +and \textit{A.~Macfarlane} of the University of Texas, have +\index{Macfarlane}% +each suggested an algebra of vectors with a new notation. +Each gives a definition of his own for the product of two +vectors, but in such a way that the square of a vector is +positive. A third system of vector analysis has been used by +\textit{Oliver Heaviside} in his electrical researches. +\index{Heaviside}% +%% -----File: 334.png---Folio 320------- + +\textbf{Hermann Grassmann} (1809--1877) was born at Stettin, +\index{Grassmann|(}% +attended a gymnasium at his native place (where his father +was teacher of mathematics and physics), and studied theology +in Berlin for three years. In 1834 he succeeded Steiner as +\index{Steiner!ref.\ to}% +teacher of mathematics in an industrial school in Berlin, but +returned to Stettin in 1836 to assume the duties of teacher of +mathematics, the sciences, and of religion in a school there.\Reference{71} +Up to this time his knowledge of mathematics was pretty +much confined to what he had learned from his father, who +had written two books on ``Raumlehre'' and ``Grössenlehre.'' +But now he made his acquaintance with the works of Lacroix, +\index{Lacroix}% +Lagrange, and Laplace. He noticed that Laplace's results +\index{Laplace!ref.\ to}% +could be reached in a shorter way by some new ideas advanced +in his father's books, and he proceeded to elaborate this +abridged method, and to apply it in the study of tides. He +was thus led to a new geometric analysis. In 1840 he had +made considerable progress in its development, but a new +book of Schleiermacher drew him again to theology. In +1842 he resumed mathematical research, and becoming thoroughly +convinced of the importance of his new analysis, +decided to devote himself to it. It now became his ambition +to secure a mathematical chair at a university, but in this he +never succeeded. In 1844 appeared his great classical work, +the \textit{Lineale Ausdehnungslehre}, which was full of new and +\index{Ausdehnungslehre}% +strange matter, and so general, abstract, and out of fashion in +its mode of exposition, that it could hardly have had less +influence on European mathematics during its first twenty +years, had it been published in China.\DPnote{ouch} Gauss, Grunert, and +\index{Gauss!ref.\ to}% +\index{Grunert!ref.\ to}% +Möbius glanced over it, praised it, but complained of the +\index{Mobius@{Möbius}!ref.\ to}% +strange terminology and its ``philosophische Allgemeinheit.'' +Eight years afterwards, Bretschneider of Gotha was said to be +\index{Bretschneider}% +the only man who had read it through. An article in \textit{Crelle's +Journal}, in which Grassmann eclipsed the geometers of that +%% -----File: 335.png---Folio 321------- +\index{Curves!theory of}% +\index{Differential equations}% +\index{Matrices}% +time by constructing, with aid of his method, geometrically +any algebraic curve, remained again unnoticed. Need we marvel +if Grassmann turned his attention to other subjects,---to +Schleiermacher's philosophy, to politics, to philology? Still, +articles by him continued to appear in \textit{Crelle's Journal}, and in +1862 came out the second part of his \textit{Ausdehnungslehre}. It +\index{Ausdehnungslehre}% +was intended to show better than the first part the broad +scope of the Ausdehnungslehre, by considering not only geometric +applications, but by treating also of algebraic functions, +infinite series, and the differential and integral calculus. But +the second part was no more appreciated than the first. At the +age of fifty-three, this wonderful man, with heavy heart, gave +up mathematics, and directed his energies to the study of Sanskrit, +achieving in philology results which were better appreciated, +and which vie in splendour with those in mathematics. + +Common to the Ausdehnungslehre and to quaternions are geometric +addition, the function of two vectors represented in quaternions +by $S\alpha\beta$~and~$V\alpha\beta$, and the linear vector functions. The +quaternion is peculiar to Hamilton, while with Grassmann we +\index{Hamilton, W.~R.!ref.\ to}% +find in addition to the algebra of vectors a geometrical algebra +of wide application, and resembling Möbius's \textit{Barycentrische +\index{Mobius@{Möbius}!ref.\ to}% +Calcul}, in which the point is the fundamental element. Grassmann +developed the idea of the ``external product,'' the ``internal +product,'' and the ``open product.'' The last we now call +a matrix. His Ausdehnungslehre has very great extension, +having no limitation to any particular number of dimensions. +Only in recent years has the wonderful richness of +his discoveries begun to be appreciated. A second edition of +the \textit{Ausdehnungslehre} of 1844 was printed in 1877. C.~S. +Peirce gave a representation of Grassmann's system in the +\index{Peirce, C.~S.!ref.\ to}% +logical notation, and E.~W. Hyde of the University of Cincinnati +\index{Hyde}% +wrote the first text-book on Grassmann's calculus in +\index{Grassmann|)}% +the English language. +%% -----File: 336.png---Folio 322------- +\index{Aequipollences@{Æquipollences}}% + +Discoveries of less value, which in part covered those of +Grassmann and Hamilton, were made by \textit{Saint-Venant} (1797--1886), +\index{Saint-Venant!ref.\ to}% +who described the multiplication of vectors, and the +addition of vectors and oriented areas; by \textit{Cauchy}, whose +\index{Cauchy!ref.\ to}% +``clefs algébriques'' were units subject to combinatorial multiplication, +and were applied by the author to the theory of +elimination in the same way as had been done earlier by +Grassmann; by \textbf{Justus Bellavitis} (1803--1880), who published +\index{Bellavitis}% +in 1835 and 1837 in the \textit{Annali delle Scienze} his calculus of +æquipollences. Bellavitis, for many years professor at Padua, +was a self-taught mathematician of much power, who in his +thirty-eighth year laid down a city office in his native place, +Bassano, that he might give his time to science.\Reference{65} + +The first impression of Grassmann's ideas is marked in the +writings of \textbf{Hermann Hankel} (1839--1873), who published in +\index{Hankel}% +1867 his \textit{Vorlesungen über die Complexen Zahlen}. Hankel, +then docent in Leipzig, had been in correspondence with +Grassmann. The ``alternate numbers'' of Hankel are subject +\index{Alternate numbers}% +to his law of combinatorial multiplication. In considering +the foundations of algebra Hankel affirms the principle of +the permanence of formal laws previously enunciated incompletely +by Peacock. Hankel was a close student of mathematical +history, and left behind an unfinished work thereon. +Before his death he was professor at Tübingen. His \textit{\DPtypo{Complexe}{Complexen} +Zahlen} was at first little read, and we must turn to +\textbf{Victor Schlegel} of Hagen as the successful interpreter of Grassmann. +\index{Schlegel}% +Schlegel was at one time a young colleague of Grassmann +at the Marienstifts-Gymnasium in Stettin. Encouraged +by Clebsch, Schlegel wrote a \textit{System der Raumlehre} which +\index{Clebsch!ref.\ to}% +explained the essential conceptions and operations of the +Ausdehnungslehre. + +Multiple algebra was powerfully advanced by Peirce, whose +theory is not geometrical, as are those of Hamilton and Grassmann. +%% -----File: 337.png---Folio 323------- +\textbf{Benjamin Peirce} (1809--1880) was born at Salem, Mass., +\index{Peirce, B.}% +\index{Peirce, C.~S.}% +and graduated at Harvard College, having as undergraduate +carried the study of mathematics far beyond the limits of the +college course.\Reference{2} When Bowditch was preparing his translation +\index{Bowditch}% +and commentary of the \textit{Mécanique Céleste}, young Peirce +helped in reading the proof-sheets. He was made professor at +Harvard in 1833, a position which he retained until his death. +For some years he was in charge of the \textit{Nautical Almanac} +and superintendent of the United States Coast Survey. He +published a series of college text-books on mathematics, an +\textit{Analytical Mechanics}, 1855, and calculated, together with Sears~C. +Walker of Washington, the orbit of Neptune. Profound +\index{Walker}% +are his researches on \textit{Linear Associative Algebra}. The first of +\index{Linear associative algebra}% +several papers thereon was read at the first meeting of the +American Association for the Advancement of Science in 1864. +Lithographed copies of a memoir were distributed among friends +in 1870, but so small seemed to be the interest taken in this +subject that the memoir was not printed until 1881 (\textit{Am.\ Jour.\ +Math.}, Vol.~IV., No.~2). Peirce works out the multiplication +tables, first of \emph{single} algebras, then of \emph{double} algebras, and so +on up to sextuple, making in all 162~algebras, which he shows +to be possible on the consideration of symbols $A$,~$B$,~etc., +which are linear functions of a determinate number of letters +or units $i$,~$j$, $k$,~$l$,~etc., with coefficients which are ordinary analytical +magnitudes, real or imaginary,---the letters $i$,~$j$,~etc., +being such that every binary combination $i^2$,~$ij$, $ji$,~etc., is equal +to a linear function of the letters, but under the restriction of +satisfying the associative law.\Reference{56} \textit{Charles~S. Peirce}, a son of +Benjamin Peirce, and one of the foremost writers on mathematical +logic, showed that these algebras were all defective +\index{Logic}% +forms of quadrate algebras which he had previously discovered +by logical analysis, and for which he had devised a simple +notation. Of these quadrate algebras quaternions is a simple +%% -----File: 338.png---Folio 324------- +example; nonions is another. C.~S. Peirce showed that of all +linear associative algebras there are only three in which division +is unambiguous. These are ordinary single algebra, +ordinary double algebra, and quaternions, from which the +imaginary scalar is excluded. He showed that his father's +algebras are operational and matricular. Lectures on multiple +algebra were delivered by J.~J. Sylvester at the Johns Hopkins +\index{Sylvester!ref.\ to}% +University, and published in various journals. They treat +largely of the algebra of matrices. The theory of matrices +\index{Matrices}% +was developed as early as 1858 by Cayley in an important +\index{Cayley!ref.\ to}% +memoir which, in the opinion of Sylvester, ushered in the +reign of Algebra the Second. Clifford, Sylvester, H.~Taber, +\index{Clifford!ref.\ to}% +\index{Taber}% +C.~H. Chapman, carried the investigations much further. The +\index{Chapman}% +originator of matrices is really Hamilton, but his theory, published +\index{Hamilton, W.~R.!ref.\ to}% +in his \textit{Lectures on Quaternions}, is less general than that +of Cayley. The latter makes no reference to Hamilton. + +The theory of determinants\Reference[]{73} was studied by Hoëné Wronski +\index{Determinants}% +\index{Wronski}% +in Italy and J.~Binet in France; but they were forestalled by +\index{Binet}% +the great master of this subject, Cauchy. In a paper (\textit{Jour.\ +\index{Cauchy!ref.\ to}% +de l'ecole Polyt.}, IX.,~16) Cauchy developed several general +theorems. He introduced the name \emph{determinant}, a term +previously used by Gauss in the functions considered by him. +\index{Gauss!ref.\ to}% +In 1826 Jacobi began using this calculus, and he gave brilliant +\index{Jacobi!ref.\ to}% +proof of its power. In 1841 he wrote extended memoirs on +determinants in \textit{Crelle's Journal}, which rendered the theory +easily accessible. In England the study of linear transformations +of quantics gave a powerful impulse. Cayley developed +skew-determinants and Pfaffians, and introduced the use of +determinant brackets, or the familiar pair of upright lines. +More recent researches on determinants appertain to special +forms. ``Continuants'' are due to Sylvester; ``alternants,'' +originated by Cauchy, have been developed by Jacobi, N.~Trudi, +\index{Trudi}% +H.~Nägelbach, and G.~Garbieri; ``axisymmetric determinants,'' +\index{Garbieri}% +\index{Nagelbach@{Nägelbach}}% +%% -----File: 339.png---Folio 325------- +\index{Invariant}% +first used by Jacobi, have been studied by V.~A. Lebesgue, +\index{Lebesgue}% +Sylvester, and Hesse; ``circulants'' are due to E.~Catalan of +\index{Catalan, E.}% +\index{Hesse!ref.\ to}% +\index{Sylvester!ref.\ to}% +Liège, W.~Spottiswoode (1825--1883), J.~W.~L. Glaisher, and +\index{Glaisher!ref.\ to}% +\index{Spottiswoode}% +R.~F. Scott; for ``centro-symmetric determinants'' we are +\index{Determinants}% +\index{Scott}% +indebted to G.~Zehfuss. E.~B. Christoffel of Strassburg and +\index{Christoffel}% +\index{Zehfuss}% +G.~Frobenius discovered the properties of ``Wronskians,'' first +\index{Frobenius}% +used by Wronski. V.~Nachreiner and S.~Günther, both of +\index{Nachreiner}% +Munich, pointed out relations between determinants and continued +fractions; Scott uses Hankel's alternate numbers in his +\index{Hankel!ref.\ to}% +treatise. Text-books on determinants were written by Spottiswoode +(1851), Brioschi (1854), Baltzer (1857), Günther +\index{Baltzer, R.!ref.\ to}% +\index{Brioschi!ref.\ to}% +\index{Gunther@{Günther, S.}}% +(1875), Dostor (1877), Scott (1880), Muir (1882), Hanus +\index{Dostor}% +\index{Hanus}% +\index{Muir}% +(1886). + +Modern higher algebra is especially occupied with the +theory of linear transformations. Its development is mainly +the work of Cayley and Sylvester. +\index{Cayley}% + +\textbf{Arthur Cayley}, born at Richmond, in Surrey, in 1821, was +educated at Trinity College, Cambridge.\Reference{74} He came out Senior +Wrangler in 1842. He then devoted some years to the study +and practice of law. On the foundation of the Sadlerian professorship +at Cambridge, he accepted the offer of that chair, +thus giving up a profession promising wealth for a very +modest provision, but which would enable him to give all his +time to mathematics. Cayley began his mathematical publications +in the \textit{Cambridge Mathematical Journal} while he was +still an undergraduate. Some of his most brilliant discoveries +were made during the time of his legal practice. There is +hardly any subject in pure mathematics which the genius of +Cayley has not enriched, but most important is his creation +of a new branch of analysis by his theory of invariants. +Germs of the principle of invariants are found in the writings +of Lagrange, Gauss, and particularly of Boole, who +\index{Boole!ref.\ to}% +\index{Gauss!ref.\ to}% +\index{Lagrange!ref.\ to}% +showed, in 1841, that invariance is a property of discriminants +%% -----File: 340.png---Folio 326------- +generally, and who applied it to the theory of orthogonal +substitution. Cayley set himself the problem to determine +\index{Cayley}% +\textit{a~priori} what functions of the coefficients of a given equation +possess this property of invariance, and found, to begin with, +in 1845, that the so-called ``hyper-determinants'' possessed it. +Boole made a number of additional discoveries. Then Sylvester +\index{Sylvester}% +began his papers in the \textit{Cambridge and Dublin Mathematical +Journal} on the Calculus of Forms. After this, +discoveries followed in rapid succession. At that time Cayley +and Sylvester were both residents of London, and they +stimulated each other by frequent oral communications. It +has often been difficult to determine how much really belongs +to each. + +\textbf{James Joseph Sylvester} was born in London in 1814, and +educated at St.~Johns College, Cambridge. He came out +Second Wrangler in 1837. His Jewish origin incapacitated +him from taking a degree. In 1846 he became a student at +the Inner Temple, and was called to the bar in 1850. He +became professor of natural philosophy at University College, +London; then, successively, professor of mathematics at the +University of Virginia, at the Royal Military Academy in +Woolwich, at the Johns Hopkins University in Baltimore, +and is, since 1883, professor of geometry at Oxford. His +first printed paper was on Fresnel's optic theory, 1837. Then +followed his researches on invariants, the theory of equations, +theory of partitions, multiple algebra, the theory of numbers, +and other subjects mentioned elsewhere. About 1874 he took +part in the development of the geometrical theory of link-work %[** PP: Only instance] +movements, originated by the beautiful discovery of +A.~Peaucellier, Capitaine du Génie à Nice (published in +\index{Peaucellier}% +\textit{Nouvelles Annales}, 1864 and 1873), and made the subject +of close study by A.~B. Kempe. To Sylvester is ascribed the +\index{Kempe}% +general statement of the theory of contravariants, the discovery +\index{Contravariants}% +%% -----File: 341.png---Folio 327------- +\index{Macmahon}% +of the partial differential equations satisfied by the +invariants and covariants of binary quantics, and the subject +\index{Covariants}% +of mixed concomitants. In the \textit{American Journal of Mathematics} +are memoirs on binary and ternary quantics, elaborated +partly with aid of \textit{F.~Franklin}, now professor at the Johns +\index{Franklin}% +Hopkins University. At Oxford, Sylvester has opened up a +new subject, the theory of reciprocants, treating of the functions +\index{Reciprocants}% +of a dependent variable~$y$ and the functions of its differential +coefficients in regard to~$x$, which remain unaltered by +the interchange of $x$~and~$y$. This theory is more general than +one on differential invariants by Halphen (1878), and has +\index{Differential invariants}% +\index{Halphen!ref.\ to}% +been developed further by J.~Hammond of Oxford, McMahon +\index{Hammond, J}% +of Woolwich, A.~R. Forsyth of Cambridge, and others. Sylvester +\index{Forsyth}% +playfully lays claim to the appellation of the Mathematical +Adam, for the many names he has introduced into +mathematics. Thus the terms \emph{invariant}, \emph{discriminant}, \emph{Hessian}, +\index{Hessian}% +\emph{Jacobian}, are his. + +The great theory of invariants, developed in England mainly +by Cayley and Sylvester, came to be studied earnestly in Germany, +France, and Italy. One of the earliest in the field was +\textbf{Siegfried Heinrich Aronhold} (1819--1884), who demonstrated +\index{Aronhold}% +the existence of invariants, $S$~and~$T$, of the ternary cubic. +Hermite discovered evectants and the theorem of reciprocity +named after him. Paul Gordan showed, with the aid of +\index{Gordan}% +symbolic methods, that the number of distinct forms for a +binary quantic is finite. Clebsch proved this to be true for +\index{Clebsch!ref.\ to}% +quantics with any number of variables. A very much simpler +proof of this was given in 1891, by David Hilbert of Königsberg. +\index{Hilbert}% +In Italy, F.~Brioschi of Milan and \textit{Faà de Bruno} +\index{Brioschi}% +\index{Bruno, Faà de}% +(1825--1888) contributed to the theory of invariants, the +latter writing a text-book on binary forms, which ranks by +the side of Salmon's treatise and those of Clebsch and +Gordan. Among other writers on invariants are E.~B. Christoffel, +\index{Christoffel}% +%% -----File: 342.png---Folio 328------- +\index{Burkhardt, H.}% +\index{Invariant}% +\index{Sylvester!ref.\ to}% +\index{Tchirnhausen!ref.\ to}% +Wilhelm Fiedler, P.~A. McMahon, J.~W.~L. Glaisher of +\index{Fiedler}% +\index{Glaisher!ref.\ to}% +\index{McMahon}% +Cambridge, Emory McClintock of New York. McMahon discovered +\index{McClintock}% +that the theory of semi-invariants is a part of that of +\index{Semi-invariants}% +symmetric functions. The modern higher algebra has reached +\index{Symmetric functions}% +out and indissolubly connected itself with several other +branches of mathematics---geometry, calculus of variations, +\index{Calculus!of variations}% +mechanics. Clebsch extended the theory of binary forms to +\index{Clebsch!ref.\ to}% +\index{Mechanics!more recent work}% +ternary, and applied the results to geometry. Clebsch, Klein, +\index{Klein!ref.\ to}% +Weierstrass, Burckhardt, and Bianchi have used the theory of +\index{Bianchi}% +\index{Weierstrass!ref.\ to}% +invariants in hyperelliptic and Abelian functions. +\index{Abelian functions}% +\index{Hyperelliptic functions}% + +In the theory of equations Lagrange, Argand, and Gauss +\index{Equations!theory of|(}% +furnished proof to the important theorem that every algebraic +equation has a real or a complex root. Abel proved rigorously +\index{Abel!ref.\ to}% +that the general algebraic equation of the fifth or of higher +degrees cannot be solved by radicals (\textit{Crelle},~I., 1826). A modification +of Abel's proof was given by Wantzel. Before Abel, +\index{Wantzel}% +an Italian physician, \textit{Paolo Ruffini} (1765--1822), had printed +\index{Ruffini}% +proofs of the insolvability, which were criticised by his +countryman Malfatti. Though inconclusive, Ruffini's papers +\index{Malfatti}% +are remarkable as containing anticipations of Cauchy's theory +\index{Cauchy!ref.\ to}% +of groups.\Reference{76} A transcendental solution of the quintic involving +\index{Groups, theory of|(}% +elliptic integrals was given by Hermite (\textit{Compt.\ Rend.}, 1858, +\index{Elliptic integrals}% +\index{Hermite!ref.\ to}% +1865, 1866). After Hermite's first publication, Kronecker, in +\index{Kronecker!ref.\ to}% +1858, in a letter to Hermite, gave a second solution in which +was obtained a simple resolvent of the sixth degree. \textit{Jerrard}, in +\index{Jerrard}% +his \textit{Mathematical Researches} (1832--1835), reduced the quintic +to the trinomial form by an extension of the method of +Tschirnhausen. This important reduction had been effected +as early as 1786 by \textit{E.~S. Bring}, a Swede, and brought out +\index{Bring}% +in a publication of the University of Lund. Jerrard, like +Tschirnhausen, believed that his method furnished a general +algebraic solution of equations of any degree. In 1836 William~R. +Hamilton made a report on the validity of Jerrard's +\index{Hamilton, W.~R.!ref.\ to}% +%% -----File: 343.png---Folio 329------- +\index{Elliptic functions}% +\index{Hamilton's numbers}% +method, and showed that by his process the quintic could be +transformed to any one of the four trinomial forms. Hamilton +defined the limits of its applicability to higher equations. +Sylvester investigated this question, What is the lowest degree +an equation can have in order that it may admit of being +deprived of $i$~consecutive terms by aid of equations not higher +than $i$th~degree. He carried the investigation as far as $i = 8$, +and was led to a series of numbers which he named ``Hamilton's +numbers.'' A transformation of equal importance to Jerrard's +is that of Sylvester, who expressed the quintic as the sum of +three fifth-powers. The covariants and invariants of higher +equations have been studied much in recent years. + +Abel's proof that higher equations cannot always be solved +algebraically led to the inquiry as to what equations of a given +degree can be solved by radicals. Such equations are the +ones discussed by Gauss in considering the division of the +\index{Division of the circle}% +\index{Gauss!ref.\ to}% +circle. Abel advanced one step further by proving that an +\index{Circle!division of}% +irreducible equation can always be solved in radicals, if, of +two of its roots, the one can be expressed rationally in terms +of the other, provided that the degree of the equation is +prime; if it is not prime, then the solution depends upon +that of equations of lower degree. Through geometrical considerations, +Hesse came upon algebraically solvable equations +\index{Hesse!ref.\ to}% +of the ninth degree, not included in the previous groups. +The subject was powerfully advanced in Paris by the youthful +\textit{Evariste Galois} (born, 1811; killed in a duel, 1832), who +\index{Galois}% +introduced the notion of a group of substitutions. To him are +\index{Substitutions, theory of}% +due also some valuable results in relation to another set of +equations, presenting themselves in the theory of elliptic +functions, viz.\ the modular equations. Galois's labours gave +\index{Modular equations}% +birth to the important theory of substitutions, which has been +\index{Theory of substitutions}% +greatly advanced by \textit{C.~Jordan} of Paris, \textit{J.~A. Serret} (1819--1885) +\index{Jordan}% +of the Sorbonne in Paris, L.~Kronecker (1823--1891) of +\index{Kronecker}% +%% -----File: 344.png---Folio 330------- +\index{Sturm's theorem}% +\index{Warring}% +Berlin, Klein of Göttingen, M.~Nöther of Erlangen, C.~Hermite +\index{Hermite!ref.\ to}% +\index{Klein!ref.\ to}% +\index{Nother@{Nöther}}% +of Paris, A.~Capelli of Naples, L.~Sylow of Friedrichshald, +\index{Capelli}% +\index{Sylow}% +E.~Netto of Giessen. Netto's book, the \textit{Substitutionstheorie}, +\index{Netto}% +has been translated into English by F.~N. Cole of +\index{Cole}% +the University of Michigan, who contributed to the theory. +A simple group of 504~substitutions of nine letters, discovered +by Cole, has been shown by E.~H. Moore of the University of +\index{Moore}% +Chicago to belong to a doubly-infinite system of simple groups. +\index{Groups, theory of|)}% +The theory of substitutions has important applications in the +theory of differential equations. Kronecker published, in 1882, +\index{Kronecker!ref.\ to}% +his \textit{Grundzüge einer Arithmetischen Theorie der Algebraischen +Grössen}. + +Since Fourier and Budan, the solution of numerical equations +has been advanced by \textit{W.~G. Horner} of Bath, who gave +\index{Horner}% +an improved method of approximation (\textit{Philosophical Transactions}, +1819). \textbf{Jacques Charles François Sturm} (1803--1855), a +\index{Sturm, J.~C.~F.}% +native of Geneva, Switzerland, and the successor of Poisson +\index{Poisson!ref.\ to}% +in the chair of mechanics at the Sorbonne, published in 1829 +his celebrated theorem determining the number and situation +of roots of an equation comprised between given limits. Sturm +tells us that his theorem stared him in the face in the midst +of some mechanical investigations connected with the motion +of a compound pendulum.\Reference{77} This theorem, and Horner's +method, offer together sure and ready means of finding the +real roots of a numerical equation. + +The symmetric functions of the sums of powers of the roots +\index{Symmetric functions}% +of an equation, studied by Newton and Waring, was considered +\index{Newton!ref.\ to}% +more recently by Gauss, Cayley, Sylvester, Brioschi. Cayley +\index{Brioschi!ref.\ to}% +\index{Cayley!ref.\ to}% +\index{Gauss!ref.\ to}% +\index{Sylvester!ref.\ to}% +gives rules for the ``weight'' and ``order'' of symmetric functions. + +The theory of elimination was greatly advanced by Sylvester, +\index{Elimination}% +Cayley, Salmon, Jacobi, Hesse, Cauchy, Brioschi, and +\index{Cauchy!ref.\ to}% +\index{Hesse!ref.\ to}% +\index{Jacobi!ref.\ to}% +\index{Salmon}% +Gordan. Sylvester gave the dialytic method (\textit{Philosophical +\index{Dialytic method of elimination}% +\index{Gordan}% +%% -----File: 345.png---Folio 331------- +\index{Algebra!recent|)}% +Magazine}, 1840), and in 1852 established a theorem relating +to the expression of an eliminant as a determinant. Cayley +made a new statement of Bézout's method of elimination and +\index{Bezout's@{Bézout's method of elimination}}% +\index{Elimination}% +established a general theory of elimination (1852). + +\Section{Analysis} +\index{Analysis!modern|(}% + +Under this head we find it convenient to consider the subjects +of the differential and integral calculus, the calculus of +variations, infinite series, probability, and differential equations. +\index{Equations!theory of|)}% +Prominent in the development of these subjects was +Cauchy. +\index{Cauchy|(}% + +\textbf{Augustin-Louis Cauchy}\Reference[]{78} (1789--1857) was born in Paris, +and received his early education from his father. Lagrange +and Laplace, with whom the father came in frequent contact, +foretold the future greatness of the young boy. At the +École Centrale du Panthéon he excelled in ancient classical +studies. In 1805 he entered the Polytechnic School, and two +years later the École des Ponts et Chaussées. Cauchy left for +Cherbourg in 1810, in the capacity of engineer. Laplace's +\textit{Mécanique Céleste and} Lagrange's \textit{Fonctions Analytiques} were +among his book companions there. Considerations of health +induced him to return to Paris after three years. Yielding to +the persuasions of Lagrange and Laplace, he renounced engineering +in favour of pure science. We find him next holding +a professorship at the Polytechnic School. On the expulsion +of Charles~X., and the accession to the throne of Louis Philippe +in 1830, Cauchy, being exceedingly conscientious, found himself +unable to take the oath demanded of him. Being, in +consequence, deprived of his positions, he went into voluntary +exile. At Fribourg in Switzerland, Cauchy resumed his +studies, and in 1831 was induced by the king of Piedmont to +%% -----File: 346.png---Folio 332------- +accept the chair of mathematical physics, especially created +for him at the university of Turin. In 1833 he obeyed the +call of his exiled king, Charles~X., to undertake the education +of a grandson, the Duke of Bordeaux. This gave Cauchy an +opportunity to visit various parts of Europe, and to learn how +extensively his works were being read. Charles~X. bestowed +upon him the title of Baron. On his return to Paris in 1838, +a chair in the College de France was offered to him, but the +oath demanded of him prevented his acceptance. He was +nominated member of the Bureau of Longitude, but declared +ineligible by the ruling power. During the political events of +1848 the oath was suspended, and Cauchy at last became +professor at the Polytechnic School. On the establishment of +the second empire, the oath was re-instated, but Cauchy and +Arago were exempt from it. Cauchy was a man of great +\index{Arago}% +piety, and in two of his publications staunchly defended the +Jesuits. + +Cauchy was a prolific and profound mathematician. By a +prompt publication of his results, and the preparation of +standard text-books, he exercised a more immediate and +beneficial influence upon the great mass of mathematicians +than any contemporary writer. He was one of the leaders +in infusing rigour into analysis. His researches extended +over the field of series, of imaginaries, theory of numbers, +differential equations, theory of substitutions, theory of functions, +determinants, mathematical astronomy, light, elasticity, +etc.,---covering pretty much the whole realm of mathematics, +pure and applied. + +Encouraged by Laplace and Poisson, Cauchy published in +1821 his \textit{Cours d'Analyse de l'École Royale Polytechnique}, a +work of great merit. Had it been studied more diligently by +writers of text-books in England and the United States, many +a lax and loose method of analysis hardly as yet eradicated +%% -----File: 347.png---Folio 333------- +\index{Differential equations}% +from elementary text-books would have been discarded over +half a century ago. Cauchy was the first to publish a +rigorous proof of Taylor's theorem. He greatly improved +\index{Taylor's theorem}% +the exposition of fundamental principles of the differential +\index{Differential calculus!philosophy of}% +calculus by his mode of considering limits and his new theory +\index{Calculus!of variations|(}% +on the continuity of functions. The method of Cauchy and +\index{Continuity}% +Duhamel was accepted with favour by Hoüel and others. In +\index{Duhamel}% +England special attention to the clear exposition of fundamental +principles was given by De~Morgan. Recent American +\index{Demorgan@{De Morgan}!ref.\ to}% +treatises on the calculus introduce time as an independent variable, +and the allied notions of velocity and acceleration---thus +virtually returning to the method of fluxions. +\index{Fluxions}% + +Cauchy made some researches on the calculus of variations. +\index{Cauchy|)}% +This subject is now in its essential principles the same as +when it came from the hands of Lagrange. Recent studies +pertain to the variation of a double integral when the limits +are also variable, and to variations of multiple integrals in +general. Memoirs were published by Gauss in 1829, Poisson +\index{Gauss!ref.\ to}% +\index{Poisson!ref.\ to}% +in 1831, and Ostrogradsky of St.~Petersburg in 1834, without, +\index{Ostrogradsky}% +however, determining in a general manner the number and +form of the equations which must subsist at the limits in case +of a double or triple integral. In 1837 Jacobi published a +\index{Jacobi!ref.\ to}% +memoir, showing that the difficult integrations demanded by +the discussion of the second variation, by which the existence +of a maximum or minimum can be ascertained, are included +in the integrations of the first variation, and thus are superfluous. +This important theorem, presented with great brevity +by Jacobi, was elucidated and extended by V.~A. Lebesgue, +\index{Lebesgue}% +C.~E. Delaunay, Eisenlohr, S.~Spitzer, Hesse, and Clebsch. An +\index{Clebsch!ref.\ to}% +\index{Delaunay!ref.\ to}% +\index{Eisenlohr}% +\index{Hesse!ref.\ to}% +\index{Spitzer}% +important memoir by Sarrus on the question of determining +\index{Sarrus}% +the limiting equations which must be combined with the +indefinite equations in order to determine completely the +maxima and minima of multiple integrals, was awarded a +\index{Maxima and minima}% +%% -----File: 348.png---Folio 334------- +\index{Criteria of convergence|(}% +\index{Jellet}% +prize by the French Academy in 1845, honourable mention +being made of a paper by Delaunay. Sarrus's method was +\index{Delaunay!ref.\ to}% +simplified by Cauchy. In 1852 G.~Mainardi attempted to +\index{Mainardi}% +exhibit a new method of discriminating maxima and minima, +\index{Maxima and minima}% +and extended Jacobi's theorem to double integrals. Mainardi +and F.~Brioschi showed the value of determinants in exhibiting +\index{Brioschi!ref.\ to}% +\index{Determinants}% +the terms of the second variation. In 1861 \textit{Isaac Todhunter} +\index{Todhunter}% +(1820--1884) of St.~John's College, Cambridge, published +his valuable work on the \textit{History of the Progress of the Calculus +of Variations}, which contains researches of his own. In 1866 +he published a most important research, developing the theory +of discontinuous solutions (discussed in particular cases by +Legendre), and doing for this subject what Sarrus had done +for multiple integrals. + +The following are the more important authors of systematic +treatises on the calculus of variations, and the dates of publication: +\index{Calculus!of variations|)}% +Robert Woodhouse, Fellow of Caius College, Cambridge, +\index{Woodhouse}% +1810; Richard Abbatt in London, 1837; John Hewitt +\index{Abbatt}% +Jellett (1817--1888), once Provost of Trinity College, Dublin, +1850; G.~W. Strauch in \DPtypo{Zurich}{Zürich}, 1849; Moigno and Lindelöf, +\index{Lindelöf}% +\index{Moigno}% +\index{Strauch}% +1861; Lewis Buffett Carll of Flushing in New York, 1881. +\index{Carll}% + +The lectures on definite integrals, delivered by Dirichlet in +\index{Definite integrals}% +\index{Dirichlet!ref.\ to}% +1858, have been elaborated into a standard work by G.~F. +Meyer. The subject has been treated most exhaustively by +\index{Meyer, G.~F.}% +D.~Bierens de Haan of Leiden in his \textit{Exposé de la théorie des +\index{Haan}% +intégrals définies}, Amsterdam, 1862. + +The history of infinite series illustrates vividly the salient +\index{Infinite series|(}% +feature of the new era which analysis entered upon during the +\index{Analysis!modern|)}% +first quarter of this century. Newton and Leibniz felt the +\index{Leibniz!ref.\ to}% +\index{Newton!ref.\ to}% +necessity of inquiring into the convergence of infinite series, +\index{Convergence of series|(}% +but they had no proper criteria, excepting the test advanced +by Leibniz for alternating series. By Euler and his contemporaries +\index{Euler!ref.\ to}% +the \emph{formal} treatment of series was greatly extended, +%% -----File: 349.png---Folio 335------- +\index{Multiplication of series}% +while the necessity for determining the convergence was generally +lost sight of. Euler reached some very pretty results +on infinite series, now well known, and also some very +\index{Absolutely convergent series}% +absurd results, now quite forgotten. The faults of his time +found their culmination in the Combinatorial School in Germany, +\index{Combinatorial School}% +which has now passed into deserved oblivion. At the +beginning of the period now under consideration, the doubtful, +or plainly absurd, results obtained from infinite series stimulated +profounder inquiries into the validity of operations with +them. Their \emph{actual contents} came to be the primary, \emph{form} a +secondary, consideration. The first important and strictly +rigorous investigation of series was made by Gauss in connection +\index{Gauss!ref.\ to}% +with the hypergeometric series. The criterion developed +\index{Hypergeometric series}% +by him settles the question of convergence in every case +which it is intended to cover, and thus bears the stamp of +generality so characteristic of Gauss's writings. Owing to the +strangeness of treatment and unusual rigour, Gauss's paper +excited little interest among the mathematicians of that time. + +More fortunate in reaching the public was Cauchy, whose +\index{Cauchy!ref.\ to}% +\textit{Analyse Algébrique} of 1821 contains a rigorous treatment of +series. All series whose sum does not approach a fixed limit +as the number of terms increases indefinitely are called divergent. +Like Gauss, he institutes comparisons with geometric +series, and finds that series with positive terms are convergent +or not, according as the $n$th root of the $n$th term, or the ratio +of the $(n + 1)$th term and the $n$th term, is ultimately less +or greater than unity. To reach some of the cases where +these expressions become ultimately unity and fail, Cauchy +established two other tests. He showed that series with negative +\Addendlabel{335}% +terms converge when the absolute values of the terms +converge, and then deduces Leibniz's test for alternating +series. The product of two convergent series was not found +to be necessarily convergent. Cauchy's theorem that the +%% -----File: 350.png---Folio 336------- +product of two absolutely convergent series converges to the +product of the sums of the two series was shown half a century +later by F.~Mertens of Graz to be still true if, of the two +\index{Mertens}% +convergent series to be multiplied together, only one is absolutely +convergent. + +The most outspoken critic of the old methods in series +was Abel. His letter to his friend Holmboe (1826) contains +\index{Abel!ref.\ to}% +\index{Holmboe}% +severe criticisms. It is very interesting reading, even to +modern students. In his demonstration of the binomial theorem +he established the theorem that if two series and their +product series are all convergent, then the product series will +converge towards the product of the sums of the two given +series. This remarkable result would dispose of the whole +problem of multiplication of series if we had a universal +\index{Multiplication of series}% +practical criterion of convergency for semi-convergent series. +\index{Semi-convergent series}% +Since we do not possess such a criterion, theorems have been +recently established by A.~Pringsheim of Munich and A.~Voss +\index{Pringsheim|(}% +\index{Voss!ref.\ to}% +of Würzburg which remove in certain cases the necessity of +applying tests of convergency to the product series by the +application of tests to easier related expressions. Pringsheim +reaches the following interesting conclusions: The product +of two semi-convergent series can never converge absolutely, +but a semi-convergent series, or even a divergent series, multiplied +by an absolutely convergent series, \emph{may} yield an absolutely +convergent product. + +The researches of Abel and Cauchy caused a considerable +stir. We are told that after a scientific meeting in which +Cauchy had presented his first researches on series, Laplace +\index{Laplace!ref.\ to}% +hastened home and remained there in seclusion until he had +examined the series in his \textit{Mécanique Céleste}. Luckily, every +one was found to be convergent! We must not conclude, +however, that the new ideas at once displaced the old. On the +contrary, the new views were generally accepted only after a +%% -----File: 351.png---Folio 337------- +severe and long struggle. As late as 1844 De~Morgan began a +\index{Demorgan@{De Morgan}!ref.\ to}% +paper on ``divergent series'' in this style: ``I believe it will +\index{Absolutely convergent series}% +\index{Divergent series}% +be generally admitted that the heading of this paper describes +the only subject yet remaining, of an elementary character, on +which a serious schism exists among mathematicians as to the +absolute correctness or incorrectness of results.'' + +First in time in the evolution of more delicate criteria of +convergence and divergence come the researches of Josef Ludwig +Raabe (\textit{Crelle}, Vol.~IX.); then follow those of De~Morgan +\index{Raabe}% +as given in his calculus. De~Morgan established the logarithmic +criteria which were discovered in part independently +by J.~Bertrand. The forms of these criteria, as given by +\index{Bertrand}% +Bertrand and by Ossian Bonnet, are more convenient than +\index{Bonnet, O.!ref.\ to}% +De~Morgan's. It appears from Abel's posthumous papers +\index{Abel!ref.\ to}% +that he had anticipated the above-named writers in establishing +logarithmic criteria. It was the opinion of Bonnet +\index{Logarithmic criteria of convergence}% +that the logarithmic criteria never fail; but Du~Bois-Reymond +\index{Bois-Reymond, P.~du|(}% +and Pringsheim have each discovered series demonstrably +convergent in which these criteria fail to determine +the convergence. The criteria thus far alluded to have +been called by Pringsheim \emph{special} criteria, because they all +depend upon a comparison of the $n$th term of the series +with special functions $a^n$, $n^x$, $n(\log n)^x$, etc. Among the +first to suggest \emph{general} criteria, and to consider the subject +from a still wider point of view, culminating in a regular +mathematical theory, was Kummer. He established a theorem +\index{Kummer!ref.\ to}% +yielding a test consisting of two parts, the first part of +which was afterwards found to be superfluous. The study +of general criteria was continued by U.~Dini of Pisa, Paul +\index{Dini}% +Du~Bois-Reymond, G.~Kohn of Minden, and Pringsheim. +\index{Kohn}% +Du~Bois-Reymond divides criteria into two classes: criteria +of the \emph{first kind} and criteria of the \emph{second kind}, according as +the general $n$th term, or the ratio of the $(n + 1)$th term and +%% -----File: 352.png---Folio 338------- +the $n$th~term, is made the basis of research. Kummer's is a +\index{Kummer!ref.\ to}% +criterion of the second kind. A criterion of the first kind, +analogous to this, was invented by Pringsheim. From the +general criteria established by Du~Bois-Reymond and Pringsheim +respectively, all the special criteria can be derived. The +theory of Pringsheim is very complete, and offers, in addition +\index{Pringsheim|)}% +to the criteria of the first kind and second kind, entirely new +criteria of a \emph{third kind}, and also generalised criteria of the +second kind, which apply, however, only to series with never +\index{Absolutely convergent series}% +increasing terms. Those of the third kind rest mainly on the +consideration of the limit of the difference either of consecutive +terms or of their reciprocals. In the generalised criteria +of the second kind he does not consider the ratio of two consecutive +terms, but the ratio of any two terms however far +apart, and deduces, among others, two criteria previously given +by Kohn and Ermakoff respectively. + +Difficult questions arose in the study of Fourier's series.\Reference{79} +\index{Fourier's series}% +Cauchy was the first who felt the necessity of inquiring into +\index{Cauchy!ref.\ to}% +its convergence. But his mode of proceeding was found +by Dirichlet to be unsatisfactory. Dirichlet made the first +\index{Dirichlet!ref.\ to}% +thorough researches on this subject (\textit{Crelle}, Vol.~IV.). They +culminate in the result that whenever the function does not +become infinite, does not have an infinite number of discontinuities, +and does not possess an infinite number of +maxima and minima, then Fourier's series converges toward +the value of that function at all places, except points of +discontinuity, and there it converges toward the mean of +the two boundary values. Schläfli of Bern and Du~Bois-Reymond +\index{Schlafli@{Schläfli}!ref.\ to}% +expressed doubts as to the correctness of the mean +value, which were, however, not well founded. Dirichlet's +conditions are sufficient, but not necessary. Lipschitz, of +\index{Lipschitz!ref.\ to}% +Bonn, proved that Fourier's series still represents the function +when the number of discontinuities is infinite, and +%% -----File: 353.png---Folio 339------- +\index{Criteria of convergence|)}% +\index{Definite integrals}% +\index{Undulatory theory of light}% +established a condition on which it represents a function +having an infinite number of maxima and minima. Dirichlet's +\index{Dirichlet!ref.\ to}% +\index{Maxima and minima}% +belief that all continuous functions can be represented +by Fourier's series at all points was shared by Riemann and +\index{Fourier's series}% +\index{Riemann!ref.\ to}% +H.~Hankel, but was proved to be false by Du~Bois-Reymond +\index{Hankel!ref.\ to}% +and H.~A. Schwarz. +\index{Schwarz!ref.\ to}% + +Riemann inquired what properties a function must have, +so that there may be a trigonometric series which, whenever +\index{Trigonometric series}% +it is convergent, converges toward the value of the function. +He found necessary and sufficient conditions for this. They +do not decide, however, whether such a series actually represents +the function or not. Riemann rejected Cauchy's definition +\index{Cauchy!ref.\ to}% +of a definite integral on account of its arbitrariness, gave +a new definition, and then inquired when a function has an +integral. His researches brought to light the fact that continuous +functions need not always have a differential coefficient. +But this property, which was shown by Weierstrass to +\index{Weierstrass!ref.\ to}% +belong to large classes of functions, was not found necessarily +to exclude them from being represented by Fourier's series. +Doubts on some of the conclusions about Fourier's series were +thrown by the observation, made by Weierstrass, that the +integral of an infinite series can be shown to be equal to the +\index{Infinite series|)}% +sum of the integrals of the separate terms only when the series +converges \emph{uniformly} within the region in question. The subject +of uniform convergence was investigated by Philipp Ludwig +\index{Convergence of series|)}% +Seidel (1848) and G.~G. Stokes (1847), and has assumed +\index{Seidel}% +\index{Stokes!ref.\ to}% +great importance in Weierstrass' theory of functions. It +became necessary to prove that a trigonometric series representing +a continuous function converges uniformly. This was +done by Heinrich Eduard Heine (1821--1881), of Halle. Later +\index{Heine}% +researches on Fourier's series were made by G.~Cantor and +\index{Cantor, G.}% +Du~Bois-Reymond. +\index{Bois-Reymond, P.~du|)}% + +As compared with the vast development of other mathematical +%% -----File: 354.png---Folio 340------- +branches, the theory of probability has made very +\index{Probability}% +insignificant progress since the time of Laplace. Improvements +\index{Laplace!ref.\ to}% +and \DPtypo{simplications}{simplifications} in the mode of exposition have been +made by A.~De~Morgan, G.~Boole, A.~Meyer (edited by E.~Czuber), +\index{Boole!ref.\ to}% +\index{Czuber}% +\index{Demorgan@{De Morgan}!ref.\ to}% +\index{Meyer, A.}% +J.~Bertrand. Cournot's and Westergaard's treatment +\index{Bertrand}% +\index{Cournot}% +\index{Westergaard}% +of insurance and the theory of life-tables are classical. Applications +\index{Insurance}% +of the calculus to statistics have been made by L.~A.~J. +\index{Statistics}% +Quetelet (1796--1874), director of the observatory at Brussels; +\index{Quetelet}% +by Lexis; Harald Westergaard, of Copenhagen; and Düsing. +\index{Dusing@{Düsing}}% +\index{Lexis}% + +Worthy of note is the rejection of inverse probability by the +\index{Inverse probability}% +best authorities of our time. This branch of probability had +been worked out by Thomas Bayes (died 1761) and by Laplace +\index{Bayes}% +(Bk.~II., Ch.~VI. of his \textit{Théorie Analytique}). By it some +logicians have explained induction. For example, if a man, +\index{Induction}% +who has never heard of the tides, were to go to the shore of the +Atlantic Ocean and witness on $m$~successive days the rise of the +sea, then, says Quetelet, he would be entitled to conclude that +there was a probability equal to $\dfrac{m+1}{m+2}$ that the sea would rise +next day. Putting $m = 0$, it is seen that this view rests upon +the unwarrantable assumption that the probability of a totally +unknown event is~$\frac{1}{2}$, or that of all theories proposed for investigation +one-half are true. W.~S. Jevons in his \textit{Principles of +\index{Jevons}% +Science} founds induction upon the theory of inverse probability, +and F.~Y. Edgeworth also accepts it in his \textit{Mathematical +\index{Edgeworth}% +Psychics}. + +The only noteworthy recent addition to probability is the +subject of ``local probability,'' developed by several English +\index{Local probability}% +and a few American and French mathematicians. The earliest +problem on this subject dates back to the time of Buffon, the +\index{Buffon}% +naturalist, who proposed the problem, solved by himself and +Laplace, to determine the probability that a short needle, +thrown at random upon a floor ruled with equidistant parallel +%% -----File: 355.png---Folio 341------- +lines, will fall on one of the lines. Then came Sylvester's +\index{Sylvester!ref.\ to}% +four-point problem: to find the probability that four points, +\index{Four-point problem}% +\index{Probability}% +taken at random within a given boundary, shall form a re-entrant %[** PP: Only instance] +quadrilateral. Local probability has been studied in +England by A.~R. Clarke, H.~McColl, S.~Watson, J.~Wolstenholme, +\index{Clarke}% +\index{McColl}% +\index{Watson, S.}% +\index{Wolstenholme}% +but with greatest success by \textit{M.~W. Crofton} of the +\index{Crofton}% +military school at Woolwich. It was pursued in America by +E.~B. Seitz; in France by C.~Jordan, E.~Lemoine, E.~Barbier, +\index{Barbier}% +\index{Jordan!ref.\ to}% +\index{Lemoine}% +\index{Seitz}% +and others. Through considerations of local probability, +Crofton was led to the evaluation of certain definite integrals. +\index{Definite integrals}% + +The first full scientific treatment of differential equations +\index{Differential equations|(}% +was given by Lagrange and Laplace. This remark is especially +true of partial differential equations. The latter were investigated +\index{Partial differential equations|etseq}% +in more recent time by Monge, Pfaff, Jacobi, Émile Bour +\index{Bour}% +\index{Jacobi!ref.\ to}% +\index{Monge!ref.\ to}% +\index{Pfaff}% +(1831--1866) of Paris, A.~Weiler, Clebsch, A.~N. Korkine of St.~Petersburg, +\index{Clebsch!ref.\ to}% +\index{Korkine!ref.\ to}% +\index{Weiler}% +G.~Boole, A.~Meyer, Cauchy, Serret, Sophus Lie, +\index{Boole!ref.\ to}% +\index{Cauchy!ref.\ to}% +\index{Lie!ref.\ to}% +\index{Meyer, A.}% +\index{Serret!ref.\ to}% +and others. In 1873 their \DPtypo{reseaches}{researches}, on partial differential +equations of the first order, were presented in text-book form +by Paul Mansion, of the University of Gand. The keen +\index{Mansion}% +researches of \textbf{Johann Friedrich Pfaff} (1795--1825) marked a +decided advance. He was an intimate friend of young Gauss +at Göttingen. Afterwards he was with the astronomer Bode. +\index{Bode}% +Later he became professor at Helmstädt, then at Halle. By a +peculiar method, Pfaff found the general integration of partial +differential equations of the first order for any number +of variables. Starting from the theory of ordinary differential +equations of the first order in $n$~variables, he gives first their +general integration, and then considers the integration of the +partial differential equations as a particular case of the former, +assuming, however, as known, the general integration of differential +equations of any order between two variables. His +researches led Jacobi to introduce the name ``Pfaffian problem.'' +\index{Pfaffian problem}% +From the connection, observed by Hamilton, between +\index{Hamilton, W.~R.!ref.\ to}% +%% -----File: 356.png---Folio 342------- +\index{Maxima and minima}% +\index{Pfaffian problem}% +a system of ordinary differential equations (in analytical +mechanics) and a partial differential equation, Jacobi drew +the conclusion that, of the series of systems whose successive +integration Pfaff's method demanded, all but the first system +\index{Pfaff}% +were entirely superfluous. Clebsch considered Pfaff's problem +\index{Clebsch!ref.\ to}% +from a new point of view, and reduced it to systems of simultaneous +linear partial differential equations, which can be +established independently of each other without any integration. +Jacobi materially advanced the theory of differential +equations of the first order. The problem to determine unknown +functions in such a way that an integral containing +these functions and their differential coefficients, in a prescribed +manner, shall reach a maximum or minimum value, +demands, in the first place, the vanishing of the first variation +of the integral. This condition leads to differential equations, +the integration of which determines the functions. To ascertain +whether the value is a maximum or a minimum, the +second variation must be examined. This leads to new and +difficult differential equations, the integration of which, for +the simpler cases, was ingeniously deduced by Jacobi from +the integration of the differential equations of the first variation. +Jacobi's solution was perfected by Hesse, while Clebsch +\index{Hesse!ref.\ to}% +extended to the general case Jacobi's results on the second +variation. Cauchy gave a method of solving partial differential +\index{Cauchy!ref.\ to}% +equations of the first order having any number of +variables, which was corrected and extended by Serret, J.~Bertrand, +\index{Bertrand}% +\index{Serret!ref.\ to}% +O.~Bonnet in France, and Imschenetzky in Russia. +\index{Bonnet, O.!ref.\ to}% +\index{Imschenetzky}% +Fundamental is the proposition of Cauchy that every ordinary +differential equation admits in the vicinity of any non-singular +point of an integral, which is synectic within a certain circle +of convergence, and is developable by Taylor's theorem. +\index{Taylor's theorem}% +Allied to the point of view indicated by this theorem is that +of Riemann, who regards a function of a single variable as +\index{Riemann!ref.\ to}% +%% -----File: 357.png---Folio 343------- +\index{Logic}% +defined by the position and nature of its singularities, and +who has applied this conception to that linear differential +equation of the second order, which is satisfied by the hypergeometric +series. This equation was studied also by Gauss +\index{Gauss!ref.\ to}% +and Kummer. Its general theory, when no restriction is +\index{Kummer!ref.\ to}% +imposed upon the value of the variable, has been considered +by J.~Tannery, of Paris, who employed Fuchs' method of +\index{Fuchs}% +\index{Tannery}% +linear differential equations and found all of Kummer's +twenty-four integrals of this equation. This study has been +continued by Édouard Goursat of Paris. +\index{Goursat}% + +A standard text-book on \textit{Differential Equations}, including +original matter on integrating factors, singular solutions, and +especially on symbolical methods, was prepared in 1859 by +\textbf{George Boole} (1815--1864), at one time professor in Queen's +\index{Boole}% +University, Cork, Ireland. He was a native of Lincoln, and a +self-educated mathematician of great power. His treatise on +\textit{Finite Differences} (1860) and his \textit{Laws of Thought} (1854) are +\index{Finite differences}% +works of high merit. + +The fertility of the conceptions of Cauchy and Riemann +with regard to differential equations is attested by the +researches to which they have given rise on the part of +\textbf{Lazarus Fuchs} of Berlin (born 1835), \textbf{Felix Klein} of Göttingen +\index{Klein}% +(born 1849), \textbf{Henri Poincaré} of Paris (born 1854), and others. +\index{Poincaré}% +The study of linear differential equations entered a new +period with the publication of Fuchs' memoirs of 1866 +and~1868. Before this, linear equations with \emph{constant} coefficients +were almost the only ones for which general methods +of integration were known. While the general theory of +these equations has recently been presented in a new light +by Hermite, Darboux, and Jordan, Fuchs began the study +\index{Darboux}% +\index{Hermite!ref.\ to}% +\index{Jordan!ref.\ to}% +from the more general standpoint of the linear differential +equations whose coefficients are not constant. He directed +his attention mainly to those whose integrals are all regular. +%% -----File: 358.png---Folio 344------- +\index{Invariant}% +If the variable be made to describe all possible paths enclosing +one or more of the critical points of the equation, we +have a certain substitution corresponding to each of the +paths; the aggregate of all these substitutions being called +a \emph{group}. The forms of integrals of such equations were +examined by Fuchs and by G.~Frobenius by independent +\index{Frobenius}% +\index{Fuchs!ref.\ to}% +methods. Logarithms generally appear in the integrals of +a group, and Fuchs and Frobenius investigated the conditions +under which no logarithms shall appear. Through the study +of groups the reducibility or irreducibility of linear differential +\index{Groups, theory of|(}% +equations has been examined by Frobenius and Leo +Königsberger. The subject of linear differential equations, +\index{Konigsberger@{Königsberger}!ref.\ to}% +not all of whose integrals are regular, has been attacked by +G.~Frobenius of Berlin, W.~Thomé of Greifswald (born 1841), +\index{Thome@{Thomé}}% +and Poincaré, but the resulting theory of irregular integrals +\index{Irregular integrals}% +is as yet in very incomplete form. + +The theory of invariants associated with linear differential +equations has been developed by Halphen and by A.~R. Forsyth. +\index{Forsyth}% +\index{Halphen!ref.\ to}% + +The researches above referred to are closely connected with +the theory of functions and of groups. Endeavours have thus +\index{Theory of functions}% +been made to determine the nature of the function defined by +a differential equation from the differential equation itself, +and not from any analytical expression of the function, obtained +first by solving the differential equation. Instead of studying +the properties of the integrals of a differential equation for all +the values of the variable, investigators at first contented themselves +with the study of the properties in the vicinity of a +given point. The nature of the integrals at singular points +and at ordinary points is entirely different. \textit{Albert Briot} +\index{Briot}% +(1817--1882) and \textit{Jean Claude Bouquet} (1819--1885), both of +\index{Bouquet}% +Paris, studied the case when, near a singular point, the differential +equations take the form $(x-x_0) \dfrac{dy}{dx} = \displaystyle\int (xy)$. Fuchs +%% -----File: 359.png---Folio 345------- +\index{Kleinian groups}% +\index{Theory of functions}% +gave the development in series of the integrals for the particular +case of linear equations. Poincaré did the same for the +\index{Poincaré!ref.\ to}% +case when the equations are not linear, as also for partial +differential equations of the first order. The developments for +ordinary points were given by Cauchy and Madame Kowalevsky. +\index{Cauchy!ref.\ to}% +\index{Kowalevsky!ref.\ to}% + +The attempt to express the integrals by developments that +are always convergent and not limited to particular points in +a plane necessitates the introduction of new transcendents, for +the old functions permit the integration of only a small number +\index{Functions!theory of}% +of differential equations. Poincaré tried this plan with +linear equations, which were then the best known, having +been studied in the vicinity of given points by Fuchs, Thomé, +\index{Fuchs!ref.\ to}% +\index{Thome@{Thomé}!ref.\ to}% +Frobenius, Schwarz, Klein, and Halphen. Confining himself to +\index{Frobenius}% +\index{Halphen!ref.\ to}% +\index{Klein!ref.\ to}% +\index{Schwarz!ref.\ to}% +those with rational algebraical coefficients, Poincaré was able +to integrate them by the use of functions named by him \emph{Fuchsians}.\Reference{81} +He divided these equations into ``families.'' If the +integral of such an equation be subjected to a certain transformation, +the result will be the integral of an equation +belonging to the same family. The new transcendents have a +great analogy to elliptic functions; while the region of the +\index{Elliptic functions}% +latter may be divided into parallelograms, each representing a +group, the former may be divided into curvilinear polygons, +so that the knowledge of the function inside of one polygon +carries with it the knowledge of it inside the others. Thus +Poincaré arrives at what he calls \emph{Fuchsian groups}. He found, +\index{Fuchsian groups}% +moreover, that Fuchsian functions can be expressed as the +\index{Fuchsian functions}% +ratio of two transcendents (theta-fuchsians) in the same way +\index{Theta-fuchsians}% +that elliptic functions can be. If, instead of linear substitutions +with real coefficients, as employed in the above groups, +imaginary coefficients be used, then discontinuous groups are +obtained, which he called \emph{Kleinians}. The extension to non-linear %[** PP: Only instance] +equations of the method thus applied to linear equations +has been begun by Fuchs and Poincaré. +%% -----File: 360.png---Folio 346------- +\index{Abelian functions}% +\index{Algebraic functions}% +\index{Elliptic functions}% + +We have seen that among the earliest of the several kinds +of ``groups'' are the finite discontinuous groups (groups in +the theory of substitution), which since the time of Galois +have become the leading concept in the theory of algebraic +equations; that since 1876 Felix Klein, H.~Poincaré, and +\index{Klein!ref.\ to}% +\index{Poincaré!ref.\ to}% +others have applied the theory of finite and infinite discontinuous +groups to the theory of functions and of differential +\index{Theory of functions}% +equations. The finite continuous groups were first made +\index{Groups, theory of|)}% +the subject of general research in 1873 by Sophus Lie, now +\index{Lie}% +of Leipzig, and applied by him to the integration of ordinary +linear partial differential equations. + +Much interest attaches to the determination of those linear +differential equations which can be integrated by simpler +functions, such as algebraic, elliptic, or Abelian. This has +been studied by C.~Jordan, P.~Appel of Paris (born 1858), and +\index{Appel}% +\index{Jordan!ref.\ to}% +Poincaré. + +The mode of integration above referred to, which makes +known the properties of equations from the standpoint of the +theory of functions, does not suffice in the application of +differential equations to questions of mechanics. If we consider +\index{Mechanics!more recent work}% +the function as defining a plane curve, then the general +form of the curve does not appear from the above mode of +investigation. It is, however, often desirable to construct +the curves defined by differential equations. Studies having +this end in view have been carried on by Briot and Bouquet, +\index{Bouquet!ref.\ to}% +\index{Briot!ref.\ to}% +and by Poincaré.\Reference{81} + +The subject of singular solutions of differential equations +has been materially advanced since the time of Boole by G.~Darboux +\index{Boole!ref.\ to}% +\index{Darboux}% +and Cayley. The papers prepared by these mathematicians +\index{Cayley!ref.\ to}% +point out a difficulty as yet unsurmounted: whereas +a singular solution, from the point of view of the integrated +equation, ought to be a phenomenon of universal, or at least of +general occurrence, it is, on the other hand, a very special and +%% -----File: 361.png---Folio 347------- +exceptional phenomenon from the point of view of the differential +equation.\Reference{89} A geometrical theory of singular solutions +resembling the one used by Cayley was previously employed +by W.~W. Johnson of Annapolis. +\index{Johnson}% + +An advanced \textit{Treatise on Linear Differential Equations} +(1889) was brought out by Thomas Craig of the Johns Hopkins +\index{Craig, T.}% +University. He chose the algebraic method of presentation +followed by Hermite and Poincaré, instead of the geometric +\index{Hermite!ref.\ to}% +\index{Poincaré!ref.\ to}% +method preferred by Klein and Schwarz. A notable work, the +\index{Klein!ref.\ to}% +\index{Schwarz!ref.\ to}% +\textit{Traité d'Analyse}, is now being published by Émile Picard of +\index{Picard, E.}% +Paris, the interest of which is made to centre in the subject of +differential equations. +\index{Differential equations|)}% + + +\Section{Theory of Functions} +\index{Theory of functions|(}% + +We begin our sketch of the vast progress in the theory of +functions by considering the special class called elliptic functions. +\index{Elliptic functions|(}% +These were richly developed by Abel and Jacobi. +\index{Abel}% +\index{Jacobi!ref.\ to}% + +\textbf{Niels Henrick Abel} (1802--1829) was born at Findoë in Norway, +and was prepared for the university at the cathedral +school in Christiania. He exhibited no interest in mathematics +until 1818, when B.~Holmboe became lecturer there, +\index{Holmboe}% +and aroused Abel's interest by assigning original problems +to the class. Like Jacobi and many other young men who +became eminent mathematicians, Abel found the first exercise +of his talent in the attempt to solve by algebra the general +equation of the fifth degree. In 1821 he entered the University +in Christiania. The works of Euler, Lagrange, and +Legendre were closely studied by him. The idea of the inversion +of elliptic functions dates back to this time. His extraordinary +success in mathematical study led to the offer of a +stipend by the government, that he might continue his studies +%% -----File: 362.png---Folio 348------- +\index{Abelian functions}% +\index{Binomial formula}% +\index{Integral calculus}% +in Germany and France. Leaving Norway in 1825, Abel visited +\index{Abel}% +the astronomer, Schumacher, in Hamburg, and spent six +\index{Schumacher!ref.\ to}% +months in Berlin, where he became intimate with \textbf{August +Leopold Crelle} (1780--1855), and met Steiner. Encouraged by +\index{Crelle}% +\index{Steiner!ref.\ to}% +Abel and Steiner, Crelle started his journal in 1826. Abel +began to put some of his work in shape for print. His proof +of the impossibility of solving the general equation of the fifth +degree by radicals,---first printed in 1824 in a very concise +form, and difficult of apprehension,---was elaborated in greater +detail, and published in the first volume. He entered also +upon the subject of infinite series (particularly the binomial +\index{Infinite series}% +theorem, of which he gave in \textit{Crelle's Journal} a rigid general +investigation), the study of functions, and of the integral +calculus. The obscurities everywhere encountered by him +owing to the prevailing loose methods of analysis he endeavoured +to clear up. For a short time he left Berlin for Freiberg, +where he had fewer interruptions to work, and it was +there that he made researches on hyperelliptic and Abelian +\index{Hyperelliptic functions}% +functions. In July, 1826, Abel left Germany for Paris without +having met Gauss! Abel had sent to Gauss his proof of +\index{Gauss!ref.\ to}% +1824 of the impossibility of solving equations of the fifth +\index{Equations!solution of}% +degree, to which Gauss never paid any attention. This slight, +and a haughtiness of spirit which he associated with Gauss, +prevented the genial Abel from going to Göttingen. A similar +feeling was entertained by him later against Cauchy. Abel +\index{Cauchy!ref.\ to}% +remained ten months in Paris. He met there Dirichlet, +\index{Dirichlet!ref.\ to}% +Legendre, Cauchy, and others; but was little appreciated. +He had already published several important memoirs in +\textit{Crelle's Journal}, but by the French this new periodical was +as yet hardly known to exist, and Abel was too modest to +speak of his own work. Pecuniary embarrassments induced +him to return home after a second short stay in Berlin. At +Christiania he for some time gave private lessons, and served +%% -----File: 363.png---Folio 349------- +\index{Elliptic integrals}% +\index{Periodicity of functions}% +as docent. Crelle secured at last an appointment for him at +\index{Crelle!ref.\ to}% +Berlin; but the news of it did not reach Norway until after +the death of Abel at Froland.\Reference{82} + +At nearly the same time with Abel, Jacobi published articles +\index{Jacobi!ref.\ to}% +on elliptic functions. Legendre's favourite subject, so long +\index{Legendre!ref.\ to}% +neglected, was at last to be enriched by some extraordinary +discoveries. The advantage to be derived by inverting the +elliptic integral of the first kind and treating it as a function +of its amplitude (now called elliptic function) was recognised +by Abel, and a few months later also by Jacobi. A second +fruitful idea, also arrived at independently by both, is the +introduction of imaginaries leading to the observation that +the new functions simulated at once trigonometric and exponential +functions. For it was shown that while trigonometric +functions had only a real period, and exponential only an imaginary, +\index{Imaginary quantities}% +elliptic functions had both sorts of periods. These two +discoveries were the foundations upon which Abel and Jacobi, +each in his own way, erected beautiful new structures. Abel +developed the curious expressions representing elliptic functions +by infinite series or quotients of infinite products. +\index{Infinite products}% +\index{Infinite series}% +Great as were the achievements of Abel in elliptic functions, +they were eclipsed by his researches on what are now called +Abelian functions. Abel's theorem on these functions was +\index{Abelian functions}% +given by him in several forms, the most general of these +being that in his \textit{Mémoire sur une propriété générale d'une +classe très-étendue de fonctions transcendentes} (1826). The history +of this memoir is interesting. A few months after his +arrival in Paris, Abel submitted it to the French Academy. +Cauchy and Legendre were appointed to examine it; but said +\index{Cauchy!ref.\ to}% +nothing about it until after Abel's death. In a brief statement +of the discoveries in question, published by Abel in \textit{Crelle's +Journal}, 1829, reference is made to that memoir. This led +Jacobi to inquire of Legendre what had become of it. Legendre +%% -----File: 364.png---Folio 350------- +says that the manuscript was so badly written as to be +illegible, and that Abel was asked to hand in a better copy, +\index{Abel!ref.\ to}% +which he neglected to do. The memoir remained in Cauchy's +\index{Cauchy!ref.\ to}% +hands. It was not published until 1841. By a singular mishap, +the manuscript was lost before the proof-sheets were +read. + +In its form, the contents of the memoir belongs to the integral +calculus. Abelian integrals depend upon an irrational +\index{Abelian integrals}% +function $y$ which is connected with $x$ by an algebraic equation +$F(x, y) = 0$. Abel's theorem asserts that a sum of such +\index{Addition theorem of elliptic integrals}% +integrals can be expressed by a definite number $p$ of similar +integrals, where $p$ depends merely on the properties of the +equation $F(x, y) = 0$. It was shown later that $p$ is the deficiency +of the curve $F(x, y) = 0$. The addition theorems of +elliptic integrals are deducible from Abel's theorem. The +\index{Elliptic integrals}% +hyperelliptic integrals introduced by Abel, and proved by him +to possess multiple periodicity, are special cases of Abelian +\index{Periodicity of functions}% +integrals whenever $p= \text{or} > 3$. The reduction of Abelian to +elliptic integrals has been studied mainly by Jacobi, Hermite, +\index{Hermite!ref.\ to}% +\index{Jacobi!ref.\ to}% +Königsberger, Brioschi, Goursat, E.~Picard, and O.~Bolza of +\index{Bolza}% +\index{Brioschi!ref.\ to}% +\index{Goursat!ref.\ to}% +\index{Konigsberger@{Königsberger}!ref.\ to}% +\index{Picard, E.}% +the University of Chicago. + +Two editions of Abel's works have been published: the first +by Holmboe in 1839, and the second by Sylow and Lie in +\index{Holmboe}% +\index{Lie!ref.\ to}% +\index{Sylow!ref.\ to}% +1881. + +Abel's theorem was pronounced by Jacobi the greatest discovery +of our century on the integral calculus. The aged +\index{Integral calculus}% +Legendre, who greatly admired Abel's genius, called it ``\textit{monumentum +\index{Legendre!ref.\ to}% +aere perennius}.'' During the few years of work +allotted to the young Norwegian, he penetrated new fields of +research, the development of which has kept mathematicians +busy for over half a century. + +Some of the discoveries of Abel and Jacobi were anticipated +by Gauss. In the \textit{Disquisitiones Arithmeticæ} he observed +\index{Gauss!ref.\ to}% +%% -----File: 365.png---Folio 351------- +that the principles which he used in the division of the circle +were applicable to many other functions, besides the circular, +and particularly to the transcendents dependent on the integral +$\displaystyle\int \frac{dx}{\sqrt{1-x^4}}$. From this Jacobi\Reference[]{83} concluded that Gauss had +\index{Gauss!ref.\ to}% +\index{Jacobi|(}% +thirty years earlier considered the nature and properties of +elliptic functions and had discovered their double periodicity. +The papers in the collected works of Gauss confirm this conclusion. + +\textbf{Carl Gustav Jacob Jacobi}\Reference[]{84} (1804--1851) was born of Jewish +parents at Potsdam. Like many other mathematicians he was +initiated into mathematics by reading Euler. At the University +of Berlin, where he pursued his mathematical studies +independently of the lecture courses, he took the degree of +Ph.D. in 1825. After giving lectures in Berlin for two years, +he was elected extraordinary professor at Königsberg, and two +years later to the ordinary professorship there. After the +publication of his \textit{Fundamenta Nova} he spent some time in +travel, meeting Gauss in Göttingen, and Legendre, Fourier, +\index{Fourier!ref.\ to}% +\index{Legendre!ref.\ to}% +Poisson, in Paris. In 1842 he and his colleague, Bessel, attended +\index{Bessel!ref.\ to}% +\index{Poisson!ref.\ to}% +the meetings of the British Association, where they +made the acquaintance of English mathematicians. + +His early researches were on Gauss' approximation to the +value of definite integrals, partial differential equations, Legendre's +\index{Definite integrals}% +coefficients, and cubic residues. He read Legendre's +\textit{Exercises}, which give an account of elliptic integrals. When +he returned the book to the library, he was depressed in spirits +and said that important books generally excited in him new +ideas, but that this time he had not been led to a single original +thought. Though slow at first, his ideas flowed all the +richer afterwards. Many of his discoveries in elliptic functions +were made independently by Abel. Jacobi communicated +his first researches to \textit{Crelle's Journal}. In 1829, at the age +%% -----File: 366.png---Folio 352------- +\index{Kowalevsky!ref.\ to}% +of twenty-five, he published his \textit{Fundamenta Nova Theoriæ +Functionum Ellipticarum}, which contains in condensed form +the main results in elliptic functions. This work at once +secured for him a wide reputation. He then made a closer +study of theta-functions and lectured to his pupils on a new +\index{Theta-functions}% +theory of elliptic functions based on the theta-functions. He +developed a theory of transformation which led him to a multitude +of formulæ containing~$q$, a transcendental function of +the modulus, defined by the equation $q = e^{-\pi k'/k}$. He was also +led by it to consider the two new functions $\Eta$~and~$\Theta$, which +taken each separately with two different arguments are the +four (single) theta-functions designated by the $\Theta_1$, $\Theta_2$, $\Theta_3$, $\Theta_4$.\Reference{56} +In a short but very important memoir of 1832, he shows that +for the hyperelliptic integral of any class the direct functions +\index{Hyperelliptic integrals}% +to which Abel's theorem has reference are not functions of a +\index{Abel's theorem}% +single variable, such as the elliptic $\sn$,~$\cn$,~$\dn$, but functions of +$p$~variables.\Reference{56} Thus in the case $p = 2$, which Jacobi especially +considers, it is shown that Abel's theorem has reference to +two functions $\lambda(u, v)$, $\lambda_1(u, v)$, each of two variables, and +gives in effect an addition-theorem for the expression of the +functions $\lambda(u + u', v + v')$, $\lambda_1(u + u', v + v')$ algebraically in +terms of the functions $\lambda(u, v)$, $\lambda_1(u, v)$, $\lambda(u', v')$, $\lambda_1(u', v')$. By +the memoirs of Abel and Jacobi it may be considered that the +notion of the Abelian function of $p$~variables was established +and the addition-theorem for these functions given. Recent +studies touching Abelian functions have been made by Weierstrass, +\index{Abelian functions}% +\index{Weierstrass!ref.\ to}% +E.~Picard, Madame Kowalevski, and Poincaré. Jacobi's +\index{Poincaré!ref.\ to}% +work on differential equations, determinants, dynamics, and +the theory of numbers is mentioned elsewhere. + +In 1842 Jacobi visited Italy for a few months to recuperate +\index{Jacobi|)}% +his health. At this time the Prussian government gave him +a pension, and he moved to Berlin, where the last years of his +life were spent. +%% -----File: 367.png---Folio 353------- + +The researches on functions mentioned thus far have been +greatly extended. In 1858 \textbf{Charles Hermite} of Paris (born 1822), +\index{Hermite}% +introduced in place of the variable $q$ of Jacobi a new variable $\omega$ +\index{Jacobi!ref.\ to}% +connected with it by the equation $q = e^{i\pi\omega}$, so that $\omega = ik'/k$, and +was led to consider the functions $\phi(\omega)$, $\psi(\omega)$, $\chi(\omega)$.\Reference{56} Henry +Smith regarded a theta-function with the argument equal to +\index{Smith, H.!ref.\ to}% +zero, as a function of $\omega$. This he called an omega-function, +\index{Omega-function}% +while the three functions $\phi(\omega)$, $\psi(\omega)$, $\chi(\omega)$, are his modular +functions. Researches on theta-functions with respect to real +\index{Theta-functions}% +and imaginary arguments have been made by Meissel of Kiel, +\index{Meissel}% +J.~Thomae of Jena, Alfred Enneper of Göttingen (1830--1885). +\index{Enneper}% +\index{Thomae}% +A general formula for the product of two theta-functions was +given in 1854 by H.~Schröter of Breslau (1829--1892). These +\index{SchroterH@{Schröter, H.}!ref.\ to}% +functions have been studied also by Cauchy, Königsberger of +\index{Cauchy!ref.\ to}% +\index{Konigsberger@{Königsberger}}% +Heidelberg (born 1837), F.~S. Richelot of Königsberg (1808--1875), +\index{Richelot}% +Johann Georg Rosenhain of Königsberg (1816--1887), +\index{Rosenhain!ref.\ to}% +L.~Schläfli of Bern (born 1818).\Reference{85} +\index{Schlafli@{Schläfli}!ref.\ to}% + +Legendre's method of reducing an elliptic differential to its +\index{Legendre!ref.\ to}% +normal form has called forth many investigations, most important +of which are those of Richelot and of Weierstrass of +\index{Weierstrass!ref.\ to}% +Berlin. + +The algebraic transformations of elliptic functions involve +a relation between the old modulus and the new one which +Jacobi expressed by a differential equation of the third order, +and also by an algebraic equation, called by him ``modular +equation.'' The notion of modular equations was familiar to +\index{Modular equations}% +Abel, but the development of this subject devolved upon later +\index{Abel!ref.\ to}% +investigators. These equations have become of importance in +the theory of algebraic equations, and have been studied by +Sohnke, E.~Mathieu, L.~Königsberger, E.~Betti of Pisa (died +\index{Betti}% +\index{Mathieu!ref.\ to}% +\index{Sohnke}% +1892), C.~Hermite of Paris, Joubert of Angers, Francesco +\index{Joubert}% +Brioschi of Milan, Schläfli, H.~Schröter, M.~Gudermann of +\index{Brioschi!ref.\ to}% +\index{Gudermann}% +Cleve, Gützlaff. +\index{Gutzlaff@{Gützlaff}}% +%% -----File: 368.png---Folio 354------- +\index{Invariant}% +\index{Primary factors, Weierstrass' theory of}% + +Felix Klein of Göttingen has made an extensive study of +\index{Klein!ref.\ to}% +modular functions, dealing with a type of operations lying +\index{Modular functions}% +between the two extreme types, known as the theory of substitutions +\index{Theory of substitutions}% +and the theory of invariants and covariants. Klein's +\index{Covariants}% +theory has been presented in book-form by his pupil, Robert +Fricke. The bolder features of it were first published in his +\index{Fricke}% +\textit{Ikosaeder}, 1884. His researches embrace the theory of modular +functions as a specific class of elliptic functions, the +statement of a more general problem as based on the doctrine +of groups of operations, and the further development of the +subject in connection with a class of Riemann's surfaces. +\index{Riemann!ref.\ to}% + +The elliptic functions were expressed by Abel as quotients +of doubly infinite products. He did not, however, inquire +\index{Infinite products}% +rigorously into the convergency of the products. In 1845 +Cayley studied these products, and found for them a complete +\index{Cayley!ref.\ to}% +theory, based in part upon geometrical interpretation, which +he made the basis of the whole theory of elliptic functions. +Eisenstein discussed by purely analytical methods the general +\index{Eisenstein!ref.\ to}% +doubly infinite product, and arrived at results which have +been greatly simplified in form by the theory of primary +factors, due to Weierstrass. A certain function involving a +\index{Weierstrass!ref.\ to}% +doubly infinite product has been called by Weierstrass the +sigma-function, and is the basis of his beautiful theory of +\index{Sigma-function}% +elliptic functions. The first systematic presentation of Weierstrass' +theory of elliptic functions was published in 1886 by +G.~H. Halphen in his \textit{Théorie des fonctions elliptiques et des +\index{Halphen!ref.\ to}% +leurs applications}. Applications of these functions have been +given also by A.~G. Greenhill. Generalisations analogous to +\index{Greenhill}% +those of Weierstrass on elliptic functions have been made +by Felix Klein on hyperelliptic functions. +\index{Hyperelliptic functions}% + +Standard works on elliptic functions have been published by +\index{Elliptic functions|)}% +\textit{Briot and Bouquet} (1859), by \textit{Königsberger}, \textit{Cayley}, \textit{Heinrich +\index{Bouquet!ref.\ to}% +\index{Briot!ref.\ to}% +\index{Konigsberger@{Königsberger}!ref.\ to}% +Durège} of Prague (1821--1893), and others. +\index{Durège}% +%% -----File: 369.png---Folio 355------- + +Jacobi's work on Abelian and theta-functions was greatly +\index{Theta-functions}% +extended by \textbf{Adolph Göpel} (1812--1847), professor in a gymnasium +\index{Gopel@{Göpel}}% +near Potsdam, and \textbf{Johann Georg Rosenhain} of Königsberg +\index{Rosenhain}% +(1816--1887). Göpel in his \Lat{\textit{Theoriæ transcendentium primi +ordinis adumbratio levis}} (\textit{Crelle},~35, 1847) and Rosenhain in +several memoirs established each independently, on the analogy +of the single theta-functions, the functions of two variables, +called double theta-functions, and worked out in connection +with them the theory of the Abelian functions of two variables. +\index{Abelian functions|(}% +The theta-relations established by Göpel and Rosenhain received +for thirty years no further development, notwithstanding +the fact that the double theta series came to be of increasing +importance in analytical, geometrical, and mechanical problems, +and that Hermite and Königsberger had considered the +\index{Hermite!ref.\ to}% +\index{Konigsberger@{Königsberger}!ref.\ to}% +subject of transformation. Finally, the investigations of C.~W. +Borchardt of Berlin (1817--1880), treating of the representation +\index{Borchardt}% +of Kummer's surface by Göpel's biquadratic relation between +\index{Kummer!ref.\ to}% +four theta-functions of two variables, and researches of H.~H. +Weber of Marburg, F.~Prym of Würzburg, Adolf Krazer, and +\index{Krazer}% +\index{Prym}% +\index{Weber, H.~H.}% +Martin Krause of Dresden led to broader views. Researches +\index{Krause}% +on double theta-functions, made by Cayley, were extended to +\index{Cayley!ref.\ to}% +quadruple theta-functions by Thomas Craig of the Johns +\index{Craig, T.}% +Hopkins University. + +Starting with the integrals of the most general form and +considering the inverse functions corresponding to these integrals +(the Abelian functions of $p$~variables), \textit{Riemann} +\index{Riemann!ref.\ to}% +defined the theta-functions of $p$~variables as the sum of a +$p$\nobreakdash-tuply infinite series of exponentials, the general term depending +on $p$~variables. Riemann shows that the Abelian +functions are algebraically connected with theta-functions of +the proper arguments, and presents the theory in the broadest +form.\Reference{56} He rests the theory of the multiple theta-functions +upon the general principles of the theory of functions of a +complex variable. +%% -----File: 370.png---Folio 356------- +\index{Bernoulli, James (born 1758)}% +\index{Bernoulli, John (born 1667)!ref.\ to}% +\index{Riemann's surfaces!ref.\ to}% + +Through the researches of A.~Brill of Tübingen, M.~Nöther +\index{Brill, A.}% +\index{Nother@{Nöther}}% +of Erlangen, and Ferdinand Lindemann of Munich, made +\index{Lindemann!ref.\ to}% +in connection with Riemann-Roch's theorem and the theory +\index{Riemann!ref.\ to}% +of residuation, there has grown out of the theory of Abelian +functions a theory of algebraic functions and point-groups on +\index{Functions!definition of}% +\index{Functions!theory of|(}% +algebraic curves. + +Before proceeding to the general theory of functions, we +make mention of the ``calculus of functions,'' studied chiefly +\index{Calculus!of variations}% +by C.~Babbage, J.~F.~W. Herschel, and De~Morgan, which was +\index{Babbage}% +\index{Demorgan@{De Morgan}!ref.\ to}% +\index{Herschel, J.~F.~W.!ref.\ to}% +not so much a theory of functions as a theory of the solution +of functional equations by means of known functions or +symbols. + +The history of the general theory of functions begins with +the adoption of new definitions of a function. With the +Bernoullis and Leibniz, $y$ was called a function of~$x$, if there +\index{Leibniz!ref.\ to}% +existed an equation between these variables which made it +possible to calculate~$y$ for any given value of~$x$ lying anywhere +between $-\infty$~and~$+\infty$. The study of Fourier's theory +\index{Fourier!ref.\ to}% +of heat led Dirichlet to a new definition: $y$ is called a function +\index{Dirichlet!ref.\ to}% +of~$x$, if $y$ possess one or more definite values for each of certain +values that $x$ is assumed to take in an interval $x_0$~to~$x_1$. In +functions thus defined, there need be no analytical connection +between $y$~and~$x$, and it becomes necessary to look for possible +discontinuities. A great revolution in the ideas of a function +was brought about by Cauchy when, in a function as defined +\index{Cauchy!ref.\ to}% +by Dirichlet, he gave the variables imaginary values, and when +he extended the notion of a definite integral by letting the +variable pass from one limit to the other by a succession of +imaginary values along arbitrary paths. Cauchy established +several fundamental theorems, and gave the first great impulse +to the study of the general theory of functions. His researches +were continued in France by Puiseux and Liouville. But more +\index{Liouville!ref.\ to}% +\index{Puiseux}% +profound investigations were made in Germany by Riemann. +%% -----File: 371.png---Folio 357------- + +\textbf{Georg Friedrich Bernhard Riemann} (1826--1866) was born at +\index{Riemann|(}% +Breselenz in Hanover. His father wished him to study +theology, and he accordingly entered upon philological and +theological studies at Göttingen. He attended also some lectures +on mathematics. Such was his predilection for this +science that he abandoned theology. After studying for a +time under Gauss and Stern, he was drawn, in 1847, to Berlin +\index{Gauss!ref.\ to}% +\index{Stern}% +by a galaxy of mathematicians, in which shone Dirichlet, +\index{Dirichlet!ref.\ to}% +Jacobi, Steiner, and Eisenstein. Returning to Göttingen in +\index{Eisenstein!ref.\ to}% +\index{Jacobi!ref.\ to}% +\index{Steiner!ref.\ to}% +1850, he studied physics under Weber, and obtained the +\index{Weber, W.~E.!ref.\ to}% +doctorate the following year. The thesis presented on that +occasion, \textit{Grundlagen für eine allgemeine Theorie der Funktionen +einer veränderlichen complexen Grösse}, excited the admiration of +Gauss to a very unusual degree, as did also Riemann's trial +lecture, \textit{Ueber die Hypothesen welche der Geometrie zu Grunde +liegen}. Riemann's Habilitationsschrift was on the Representation +of a Function by means of a Trigonometric Series, in +\index{Trigonometric series}% +which he advanced materially beyond the position of Dirichlet. +Our hearts are drawn to this extraordinarily gifted but +shy genius when we read of the timidity and nervousness +displayed when he began to lecture at Göttingen, and of his +jubilation over the unexpectedly large audience of eight students +at his first lecture on differential equations. + +Later he lectured on Abelian functions to a class of three +\index{Abelian functions|)}% +only,---Schering, Bjerknes, and Dedekind. Gauss died in 1855, +\index{Bjerknes, C. A.}% +\index{Dedekind!ref.\ to}% +\index{Schering!ref.\ to}% +and was succeeded by Dirichlet. On the death of the latter, +in 1859, Riemann was made ordinary professor. In 1860 he +visited Paris, where he made the acquaintance of French +mathematicians. The delicate state of his health induced +him to go to Italy three times. He died on his last trip at +Selasca, and was buried at Biganzolo. + +Like all of Riemann's researches, those on functions were +profound and far-reaching. He laid the foundation for a +%% -----File: 372.png---Folio 358------- +\index{Kelvin, Lord!ref.\ to}% +general theory of functions of a complex variable. The theory +of potential, which up to that time had been used only in +\index{Potential}% +mathematical physics, was applied by him in pure mathematics. +He accordingly based his theory of functions on the +partial differential equation, $\dfrac{\partial^2u}{\partial x^2} + \dfrac{\partial^2 u}{\partial y^2} = \Delta u = 0$, which must +hold for the analytical function $w = u + iv$ of $z = x + iy$. It +had been proved by Dirichlet that (for a plane) there is always +one, and only one, function of $x$~and~$y$, which satisfies $\Delta u = 0$, +and which, together with its differential quotients of the first +two orders, is for all values of $x$~and~$y$ within a given area +one-valued and continuous, and which has for points on the +boundary of the area arbitrarily given values.\Reference{86} Riemann +called this ``Dirichlet's principle,'' but the same theorem was +stated by Green and proved analytically by Sir William +\index{Green!ref.\ to}% +Thomson. It follows then that $w$ is uniquely determined for +\index{Thomson, Sir William!ref.\ to}% +all points within a closed surface, if $u$ is arbitrarily given +for all points on the curve, whilst $v$ is given for one point +within the curve. In order to treat the more complicated +case where $w$ has $n$~values for one value of~$z$, and to observe +the conditions about continuity, Riemann invented the celebrated +surfaces, known as ``Riemann's surfaces,'' consisting +\index{Riemann's surfaces}% +of $n$~coincident planes or sheets, such that the passage from +one sheet to another is made at the branch-points, and that the +$n$~sheets form together a multiply-connected surface, which +can be dissected by cross-cuts into a singly-connected surface. +The $n$-valued function~$w$ becomes thus a one-valued function. +Aided by researches of J.~Lüroth of Freiburg and of Clebsch, +\index{Clebsch!ref.\ to}% +\index{Luroth@{Lüroth}}% +W.~K. Clifford brought Riemann's surface for algebraic functions +\index{Clifford!ref.\ to}% +\Addendlabel{358}% +to a canonical form, in which only the two last of the $n$~leaves +are multiply-connected, and then transformed the surface into +the surface of a solid with $p$~holes. A.~Hurwitz of Zürich +\index{Hurwitz}% +discussed the question, how far a Riemann's surface is determinate +%% -----File: 373.png---Folio 359------- +\index{Thomson's theorem}% +by the assignment of its number of sheets, its branch-points +and branch-lines.\Reference{62} + +Riemann's theory ascertains the criteria which will determine +an analytical function by aid of its discontinuities and +boundary conditions, and thus defines a function independently +of a mathematical expression. In order to show that +two different expressions are identical, it is not necessary to +transform one into the other, but it is sufficient to prove the +agreement to a far less extent, merely in certain critical points. + +Riemann's theory, as based on Dirichlet's principle (Thomson's +\index{Dirichlet!ref.\ to}% +theorem), is not free from objections. It has become +evident that the existence of a derived function is not a consequence +of continuity, and that a function may be integrable +\index{Continuity}% +without being differentiable. It is not known how far the +methods of the infinitesimal calculus and the calculus of +variations (by which Dirichlet's principle is established) can +be applied to an unknown analytical function in its generality. +Hence the use of these methods will endow the functions with +properties which themselves require proof. Objections of this +kind to Riemann's theory have been raised by Kronecker, +\index{Kronecker!ref.\ to}% +Weierstrass, and others, and it has become doubtful whether +\index{Weierstrass}% +\index{Weierstrass!ref.\ to}% +his most important theorems are actually proved. In consequence +of this, attempts have been made to graft Riemann's +speculations on the more strongly rooted methods of Weierstrass. +The latter developed a theory of functions by starting, +not with the theory of potential, but with analytical +expressions and operations. Both applied their theories to +Abelian functions, but there Riemann's work is more general.\Reference{86} +\index{Abelian functions}% + +The theory of functions of one complex variable has been +studied since Riemann's time mainly by \textbf{Karl Weierstrass} of +\index{Riemann|)}% +Berlin (born 1815), \textbf{Gustaf Mittag-Leffler} of Stockholm (born +\index{Mittag-Leffler}% +1846), and Poincaré of Paris. Of the three classes of such +\index{Poincaré!ref.\ to}% +%% -----File: 374.png---Folio 360------- +\index{Fuchsian functions}% +\index{Hyperelliptic functions}% +functions (viz.\ functions uniform throughout, functions uniform +only in lacunary spaces, and non-uniform functions) +Weierstrass showed that those functions of the first class +which can be developed according to ascending powers of~$x$ +into converging series, can be decomposed into a product of an +infinite number of primary factors. A primary factor of the +\index{Primary factors, Weierstrass' theory of}% +species~$n$ is the product $\left(1- \dfrac{x}{a}\right)e^{\Change{P_{(x)}}{P(x)}}$, $\Change{P_{(x)}}{P(x)}$ being an entire polynomial +of the $n$th~degree. A function of the species~$n$ is one, +all the primary factors of which are of species~$n$. This classification +gave rise to many interesting problems studied also +by Poincaré. + +The first of the three classes of functions of a complex +variable embraces, among others, functions having an infinite +number of singular points, but no singular lines, and at the +same time no isolated singular points. These are Fuchsian +functions, existing throughout the whole extent. Poincaré +first gave an example of such a function. + +Uniform functions of two variables, unaltered by certain +linear substitutions, called hyperfuchsian functions, have been +studied by E.~Picard of Paris, and by Poincaré.\Reference{81} +\index{Picard, E.}% + +Functions of the second class, uniform only in lacunary +spaces, were first pointed out by Weierstrass. The Fuchsian +and the Kleinian functions do not generally exist, except in +\index{Kleinian functions}% +the interior of a circle or of a domain otherwise bounded, and +are therefore examples of functions of the second class. +Poincaré has shown how to generate functions of this class, +and has studied them along the lines marked out by Weierstrass. +Important is his proof that there is no way of +generalising them so as to get rid of the lacunæ. + +Non-uniform functions are much less developed than the +preceding classes, even though their properties in the vicinity +of a given point have been diligently studied, and though +%% -----File: 375.png---Folio 361------- +\index{Conform representation of surfaces}% +\index{Infinite series}% +much light has been thrown on them by the use of Riemann's +surfaces. With the view of reducing their study to that of +uniform transcendents, Poincaré proved that if $y$ is any +analytical non-uniform function of~$x$, one can always find a +variable~$z$, such that $x$~and~$y$ are uniform functions of~$z$. + +Weierstrass and Darboux have each given examples of continuous +\index{Darboux}% +\index{Weierstrass!ref.\ to}% +functions having no derivatives. Formerly it had +been generally assumed that every function had a derivative. +\Addendlabel{361}% +Ampère was the first who attempted to prove analytically +\index{Ampère!ref.\ to}% +(1806) the existence of a derivative, but the demonstration +is not valid. In treating of discontinuous functions, Darboux +established rigorously the necessary and sufficient condition +that a continuous or discontinuous function be susceptible of +integration. He gave fresh evidence of the care that must +be exercised in the use of series by giving an example of a +series always convergent and continuous, such that the series +formed by the integrals of the terms is always convergent, and +yet does not represent the integral of the first series.\Reference{87} + +The general theory of functions of two variables has been +investigated to some extent by Weierstrass and Poincaré. + +\textbf{H.~A. Schwarz} of Berlin (born 1845), a pupil of Weierstrass, +\index{Schwarz}% +has given the conform\DPnote{** PP: [sic]} representation (\emph{Abbildung}) of various +surfaces on a circle. In transforming by aid of certain +substitutions a polygon bounded by circular arcs into another +also bounded by circular arcs, he was led to a remarkable +differential equation $\psi(u', t) = \psi(u, t)$, where $\psi(u, t)$ is the +expression which Cayley calls the ``Schwarzian derivative,'' +\index{Schwarzian derivative}% +and which led Sylvester to the theory of reciprocants. +\index{Reciprocants}% +\index{Sylvester!ref.\ to}% +Schwarz's developments on minimum surfaces, his work on +hypergeometric series, his inquiries on the existence of solutions +\index{Hypergeometric series}% +to important partial differential equations under prescribed +conditions, have secured a prominent place in mathematical +literature. +%% -----File: 376.png---Folio 362------- +\index{Irrationals}% + +The modern theory of functions of one real variable was +first worked out by H.~Hankel, Dedekind, G.~Cantor, Dini, and +\index{Cantor, G.}% +\index{Dedekind!ref.\ to}% +\index{Dini!ref.\ to}% +\index{Hankel!ref.\ to}% +Heine, and then carried further, principally, by Weierstrass, +\index{Heine!ref.\ to}% +\index{Weierstrass!ref.\ to}% +Schwarz, Du~Bois-Reymond, Thomae, and Darboux. Hankel +\index{Bois-Reymond, P.~du}% +\index{Darboux}% +\index{Schwarz!ref.\ to}% +\index{Thomae}% +established the principle of the condensation of singularities; +\index{Condensation of singularities}% +Dedekind and Cantor gave definitions for irrational numbers; +\index{Numbers!theory of numbers|(}% +definite integrals were studied by Thomae, Du~Bois-Reymond, +\index{Definite integrals}% +and Darboux along the lines indicated by the definitions of +such integrals given by Cauchy, Dirichlet, and Riemann. Dini +\index{Cauchy!ref.\ to}% +\index{Dirichlet!ref.\ to}% +\index{Riemann!ref.\ to}% +wrote a text-book on functions of a real variable (1878), which +was translated into German, with additions, by J.~Lüroth and +\index{Luroth@{Lüroth}!ref.\ to}% +A.~Schepp. Important works on the theory of functions are +\index{Schepp}% +the \textit{Cours de M.~Hermite}, \textit{Tannery's Théorie des Fonctions +\index{Hermite!ref.\ to}% +\index{Tannery!ref.\ to}% +d'une variable seule}, \textit{A Treatise on the Theory of Functions} by +\textit{James Harkness} and \textit{Frank Morley}, and \textit{Theory of Functions of +\index{Functions!theory of|)}% +\index{Harkness}% +\index{Morley}% +\index{Theory of functions|)}% +a Complex Variable} by \textit{A.~R. Forsyth}. +\index{Forsyth}% + +\Section{Theory of Numbers} +\index{Theory of numbers|(}% + +``Mathematics, the queen of the sciences, and arithmetic, +the queen of mathematics.'' Such was the dictum of Gauss, +\index{Gauss!ref.\ to}% +who was destined to revolutionise the theory of numbers. +When asked who was the greatest mathematician in Germany, +Laplace answered, Pfaff. When the questioner said +\index{Laplace!ref.\ to}% +\index{Pfaff!ref.\ to}% +he should have thought Gauss was, Laplace replied, ``Pfaff +is by far the greatest mathematician in Germany; but Gauss +is the greatest in all Europe.''\Reference{83} Gauss is one of the three +greatest masters of modern analysis,---Lagrange, Laplace, +\index{Lagrange!ref.\ to}% +Gauss. Of these three contemporaries he was the youngest. +While the first two belong to the period in mathematical history +preceding the one now under consideration, Gauss is the +one whose writings may truly be said to mark the beginning +%% -----File: 377.png---Folio 363------- +\index{Imaginary quantities}% +\index{Infinite series}% +\index{Kaestner}% +of our own epoch. In him that abundant fertility of invention, +displayed by mathematicians of the preceding period, +is combined with an absolute rigorousness in demonstration +which is too often wanting in their writings, and which the +ancient Greeks might have envied. Unlike Laplace, Gauss +\index{Gauss|(}% +\index{Laplace!ref.\ to}% +strove in his writings after perfection of form. He rivals +Lagrange in elegance, and surpasses this great Frenchman in +\index{Lagrange!ref.\ to}% +rigour. Wonderful was his richness of ideas; one thought followed +another so quickly that he had hardly time to write +down even the most meagre outline. At the age of twenty +Gauss had overturned old theories and old methods in all +branches of higher mathematics; but little pains did he take +to publish his results, and thereby to establish his priority. +He was the first to observe rigour in the treatment of infinite +series, the first to fully recognise and emphasise the importance, +and to make systematic use of determinants and of +\index{Determinants}% +imaginaries, the first to arrive at the method of least squares, +\index{Least squares}% +the first to observe the double periodicity of elliptic functions. +\index{Elliptic functions}% +He invented the heliotrope and, together with Weber, the +\index{Heliotrope}% +\index{Weber, W.~E.!ref.\ to}% +bifilar magnetometer and the declination instrument. He +reconstructed the whole of magnetic science. + +\textbf{Carl Friedrich Gauss}\Reference[]{47} (1777--1855), the son of a bricklayer, +was born at Brunswick. He used to say, jokingly, that he +could reckon before he could talk. The marvellous aptitude +for calculation of the young boy attracted the attention of +Bartels, afterwards professor of mathematics at Dorpat, who +brought him under the notice of Charles William, Duke of +Brunswick. The duke undertook to educate the boy, and sent +him to the Collegium Carolinum. His progress in languages +there was quite equal to that in mathematics. In 1795 he +went to Göttingen, as yet undecided whether to pursue philology +or mathematics. Abraham Gotthelf Kästner, then professor +of mathematics there, and now chiefly remembered for +%% -----File: 378.png---Folio 364------- +his \textit{Geschichte der Mathematik} (1796), was not an inspiring +teacher. At the age of nineteen Gauss discovered a method of +inscribing in a circle a regular polygon of seventeen sides, and +this success encouraged him to pursue mathematics. He worked +quite independently of his teachers, and while a student at +Göttingen made several of his greatest discoveries. Higher +arithmetic was his favourite study. Among his small circle +of intimate friends was Wolfgang Bolyai. After completing +\index{Bolyai, Wolfgang!ref.\ to}% +his course he returned to Brunswick. In 1798 and 1799 he +repaired to the university at \Change{Helmstadt}{Helmstädt} to consult the library, +and there made the acquaintance of Pfaff, a mathematician of +much power. In 1807 the Emperor of Russia offered Gauss a +chair in the Academy at St.~Petersburg, but by the advice of +the astronomer Olbers, who desired to secure him as director +\index{Olbers}% +of a proposed new observatory at Göttingen, he declined the +offer, and accepted the place at Göttingen. Gauss had a +marked objection to a mathematical chair, and preferred the +post of astronomer, that he might give all his time to science. +He spent his life in Göttingen in the midst of continuous +work. In 1828 he went to Berlin to attend a meeting of +scientists, but after this he never again left Göttingen, except +in 1854, when a railroad was opened between Göttingen and +Hanover. He had a strong will, and his character showed +a curious mixture of self-conscious dignity and child-like +simplicity. He was little communicative, and at times +morose. + +A new epoch in the theory of numbers dates from the publication +of his \textit{Disquisitiones Arithmeticæ}, Leipzig, 1801. The +beginning of this work dates back as far as 1795. Some of its +results had been previously given by Lagrange and Euler, but +\index{Euler!ref.\ to}% +\index{Lagrange!ref.\ to}% +were reached independently by Gauss, who had gone deeply +into the subject before he became acquainted with the writings +of his great predecessors. The \textit{Disquisitiones Arithmeticæ} +%% -----File: 379.png---Folio 365------- +\index{Congruencies, theory of}% +was already in print when Legendre's \textit{Théorie des Nombres} +\index{Legendre!ref.\ to}% +appeared. The great law of quadratic reciprocity, given in +\index{Quadratic reciprocity}% +the fourth section of Gauss' work, a law which involves the +whole theory of quadratic residues, was discovered by him by +induction before he was eighteen, and was proved by him one +year later. Afterwards he learned that Euler had imperfectly +\index{Euler!ref.\ to}% +enunciated that theorem, and that Legendre had attempted +to prove it, but met with apparently insuperable difficulties. +In the fifth section Gauss gave a second proof of this ``gem'' +of higher arithmetic. In 1808 followed a third and fourth +demonstration; in 1817, a fifth and sixth. No wonder that +he felt a personal attachment to this theorem. Proofs were +given also by Jacobi, Eisenstein, Liouville, Lebesgue, A.~Genocchi, +\index{Eisenstein!ref.\ to}% +\index{Genocchi}% +\index{Jacobi!ref.\ to}% +\index{Lebesgue}% +\index{Liouville!ref.\ to}% +Kummer, M.~A. Stern, Chr.\ Zeller, Kronecker, +\index{Kronecker!ref.\ to}% +\index{Kummer!ref.\ to}% +\index{Stern}% +\index{Zeller}% +Bouniakowsky, E.~Schering, J.~Petersen, Voigt, E.~Busche, +\index{Bouniakowsky}% +\index{Busche}% +\index{Petersen}% +\index{Schering!ref.\ to}% +\index{Voigt}% +and Th.\ Pepin.\Reference{48} The solution of the problem of the representation +\index{Pepin}% +of numbers by binary quadratic forms is one of the +great achievements of Gauss. He created a new algorithm by +introducing the theory of congruences. The fourth section +of the \textit{Disquisitiones Arithmeticæ}, treating of congruences of +the second degree, and the fifth section, treating of quadratic +forms, were, until the time of Jacobi, passed over with universal +neglect, but they have since been the starting-point of a long +series of important researches. The seventh or last section, +developing the theory of the division of the circle, was received +\index{Circle!division of}% +\index{Division of the circle}% +from the start with deserved enthusiasm, and has since been +repeatedly elaborated for students. A standard work on +\textit{Kreistheilung} was published in 1872 by Paul Bachmann, then +\index{Bachmann!ref.\ to}% +of Breslau. Gauss had planned an eighth section, which was +omitted to lessen the expense of publication. His papers on +the theory of numbers were not all included in his great treatise. +Some of them were published for the first time after his death +in his collected works (1863--1871). He wrote two memoirs on +%% -----File: 380.png---Folio 366------- +\index{Action, least}% +\index{Capillarity}% +\index{Ellipsoid!(attraction of)}% +\index{Geodesy}% +the theory of biquadratic residues (1825 and 1831), the second +\index{Biquadratic residues}% +of which contains a theorem of biquadratic reciprocity. + +Gauss was led to astronomy by the discovery of the planet +\index{Astronomy!more recent researches}% +Ceres at Palermo in 1801. His determination of the elements +of its orbit with sufficient accuracy to enable Olbers to \Change{rediscover}{re-discover} +it, made the name of Gauss generally known. In 1809 +he published the \textit{Theoria motus corporum coelestium}, which +contains a discussion of the problems arising in the determination +of the movements of planets and comets from +observations made on them under any circumstances. In it +are found four formulæ in spherical trigonometry, now usually +called ``Gauss' Analogies,'' but which were published somewhat +\index{Gauss' Analogies}% +earlier by Karl Brandon Mollweide of Leipzig (1774--1825), +\index{Mollweide}% +and earlier still by Jean Baptiste Joseph Delambre +\index{Delambre}% +(1749--1822).\Reference{44} Many years of hard work were spent in the +astronomical and magnetic observatory. He founded the +German Magnetic Union, with the object of securing continuous +\index{German Magnetic Union}% +observations at fixed times. He took part in geodetic +observations, and in 1843 and 1846 wrote two memoirs, \textit{Ueber +Gegenstände der höheren Geodesie}. He wrote on the attraction +of homogeneous ellipsoids, 1813. In a memoir on capillary +attraction, 1833, he solves a problem in the calculus of +\index{Calculus!of variations}% +variations involving the variation of a certain double integral, +the limits of integration being also variable; it is the earliest +example of the solution of such a problem. He discussed the +problem of rays of light passing through a system of lenses. + +Among Gauss' pupils were Christian Heinrich Schumacher, +\index{Schumacher}% +Christian Gerling, Friedrich Nicolai, August Ferdinand +\index{Gerling}% +\index{Nicolai}% +Möbius, Georg Wilhelm Struve,\DPnote{** PP: [sic] no `and'} Johann Frantz Encke. +\index{Encke}% +\index{Mobius@{Möbius}!ref.\ to}% +\index{Struve}% + +Gauss' researches on the theory of numbers were the starting-point +for a school of writers, among the earliest of whom +was Jacobi. The latter contributed to \textit{Crelle's Journal} an article +on cubic residues, giving theorems without proofs. After the +\index{Cubic residues}% +%% -----File: 381.png---Folio 367------- +\index{Elliptic functions}% +publication of Gauss' paper on biquadratic residues, giving +the law of biquadratic reciprocity, and his treatment of complex +numbers, Jacobi found a similar law for cubic residues. +\index{Jacobi!ref.\ to}% +By the theory of elliptical functions, he was led to beautiful +theorems on the representation of numbers by $2$,~$4$,~$6$, and~$8$ +squares. Next come the researches of Dirichlet, the expounder +\index{Dirichlet|(}% +of Gauss, and a contributor of rich results of his own. + +\textbf{Peter Gustav Lejeune Dirichlet}\Reference[]{88} (1805--1859) was born in +Düren, attended the gymnasium in Bonn, and then the +Jesuit gymnasium in Cologne. In 1822 he was attracted to +Paris by the names of Laplace, Legendre, Fourier, Poisson, +\index{Fourier!ref.\ to}% +\index{Legendre!ref.\ to}% +Cauchy. The facilities for a mathematical education there +were far better than in Germany, where Gauss was the only +great figure. He read in Paris Gauss' \textit{Disquisitiones Arithmeticæ}, +a work which he never ceased to admire and study. +Much in it was simplified by Dirichlet, and thereby placed +within easier reach of mathematicians. His first memoir on +the impossibility of certain indeterminate equations of the +fifth degree was presented to the French Academy in 1825. +He showed that Fermat's equation, $x^n + y^n = z^n$, cannot exist +\index{Fermat!ref.\ to}% +when $n = 5$. Some parts of the analysis are, however, +Legendre's. Euler and Lagrange had proved this when $n$~is~$3$ +\index{Euler!ref.\ to}% +\index{Lagrange!ref.\ to}% +and~$4$, and Lamé proved it when $n = 7$. Dirichlet's acquaintance +\index{Lame@{Lamé}!ref.\ to}% +with Fourier led him to investigate Fourier's series. He +\index{Fourier's series}% +became docent in Breslau in 1827. In 1828 he accepted a +position in Berlin, and finally succeeded Gauss at Göttingen +in 1855. The general principles on which depends the average +number of classes of binary quadratic forms of positive +and negative determinant (a subject first investigated by +Gauss) were given by Dirichlet in a memoir, \textit{Ueber die Bestimmung +\index{Gauss|)}% +der mittleren Werthe in der Zahlentheorie}, 1849. More +recently F.~Mertens of Graz has determined the asymptotic +\index{Mertens}% +values of several numerical functions. Dirichlet gave some +%% -----File: 382.png---Folio 368------- +attention to prime numbers. Gauss and Legendre had given +\index{Prime numbers}% +expressions denoting approximately the asymptotic value of +the number of primes inferior to a given limit, but it remained +for Riemann in his memoir, \textit{Ueber die Anzahl der Primzahlen +\index{Riemann!ref.\ to}% +unter einer gegebenen Grösse}, 1859, to give an investigation of +the asymptotic frequency of primes which is rigorous. Approaching +the problem from a different direction, \textit{Patnutij +Tchebycheff}, formerly professor in the University of St.~Petersburg +\index{Tchebycheff}% +(born 1821), established, in a celebrated memoir, \textit{Sur les +Nombres Premiers}, 1850, the existence of limits within which +the sum of the logarithms of the primes~$P$, inferior to a given +number~$x$, must be comprised.\Reference{89} This paper depends on very +elementary considerations, and, in that respect, contrasts +strongly with Riemann's, which involves abstruse theorems +of the integral calculus. Poincaré's papers, Sylvester's contraction +\index{Integral calculus}% +\index{Poincaré!ref.\ to}% +\index{Sylvester!ref.\ to}% +of Tchebycheff's limits, with reference to the distribution +of primes, and researches of J.~Hadamard (awarded the +\index{Hadamard}% +\textit{Grand prix} of 1892), are among the latest researches in this +line. The enumeration of prime numbers has been undertaken +at different times by various mathematicians. In 1877 the +British Association began the preparation of factor-tables, +\index{Factor-tables}% +under the direction of J.~W.~L. Glaisher. The printing, by +\index{Glaisher!ref.\ to}% +the Association, of tables for the sixth million marked the +completion of tables, to the preparation of which Germany, +France, and England contributed, and which enable us to +resolve into prime factors every composite number less than +$9,000,000$. + +Miscellaneous contributions to the theory of numbers were +made by \textit{Cauchy}. He showed, for instance, how to find all +\index{Cauchy!ref.\ to}% +the infinite solutions of a homogeneous indeterminate equation +of the second degree in three variables when one solution is +given. He established the theorem that if two congruences, +which have the same modulus, admit of a common solution, +%% -----File: 383.png---Folio 369------- +\index{Covariants}% +\index{Dirichlet|)}% +the modulus is a divisor of their resultant. \textbf{Joseph Liouville} +\index{Liouville}% +(1809--1882), professor at the \DPtypo{Collége}{Collège} de France, investigated +mainly questions on the theory of quadratic forms of two, and +of a greater number of variables. Profound researches were +instituted by \textbf{Ferdinand Gotthold Eisenstein} (1823--1852), of +\index{Eisenstein}% +Berlin. Ternary quadratic forms had been studied somewhat +by Gauss, but the extension from two to three indeterminates +was the work of Eisenstein who, in his memoir, \textit{Neue Theoreme +der höheren Arithmetik}, defined the ordinal and generic +characters of ternary quadratic forms of uneven determinant; +and, in case of definite forms, assigned the weight of any order +or genus. But he did not publish demonstrations of his results. +In inspecting the theory of binary cubic forms, he was +led to the discovery of the first covariant ever considered in +analysis. He showed that the series of theorems, relating to +the presentation of numbers by sums of squares, ceases when +the number of squares surpasses eight. Many of the proofs +omitted by Eisenstein were supplied by Henry Smith, who +\index{Smith, H.}% +was one of the few Englishmen who devoted themselves to the +study of higher arithmetic. + +\textbf{Henry John Stephen Smith}\Reference[]{90} (1826--1883) was born in London, +and educated at Rugby and at Balliol College, Oxford. +Before 1847 he travelled much in Europe for his health, +and at one time attended lectures of Arago in Paris, but +after that year he was never absent from Oxford for a +single term. In 1861 he was elected Savilian professor of +geometry. His first paper on the theory of numbers appeared +in 1855. The results of ten years' study of everything published +on the theory of numbers are contained in his Reports +which appeared in the British Association volumes from 1859 +to 1865. These reports are a model of clear and precise +exposition and perfection of form. They contain much original +matter, but the chief results of his own discoveries were +%% -----File: 384.png---Folio 370------- +printed in the \textit{Philosophical Transactions} for 1861 and 1867. +They treat of linear indeterminate equations and congruences, +and of the orders and genera of ternary quadratic forms. He +established the principles on which the extension to the general +case of $n$ indeterminates of quadratic forms depends. +He contributed also two memoirs to the \textit{Proceedings of the +Royal Society} of 1864 and 1868, in the second of which he +remarks that the theorems of Jacobi, Eisenstein, and Liouville, +\index{Eisenstein!ref.\ to}% +\index{Jacobi!ref.\ to}% +\index{Liouville!ref.\ to}% +relating to the representation of numbers by $4$,~$6$,~$8$ +squares, and other simple quadratic forms are deducible by a +uniform method from the principles indicated in his paper. +Theorems relating to the case of $5$~squares were given by +Eisenstein, but Smith completed the enunciation of them, and +\index{Smith, H.}% +added the corresponding theorems for $7$~squares. The solution +of the cases of $2$,~$4$,~$6$ squares may be obtained by elliptic +functions, but when the number of squares is odd, it involves +processes peculiar to the theory of numbers. This class of +theorems is limited to $8$~squares, and Smith completed the +group. In ignorance of Smith's investigations, the French +Academy offered a prize for the demonstration and completion +of Eisenstein's theorems for $5$~squares. This Smith had +accomplished fifteen years earlier. He sent in a dissertation +in 1882, and next year, a month after his death, the prize was +awarded to him, another prize being also awarded to H.~Minkowsky +\index{Minkowsky}% +of Bonn. The theory of numbers led Smith to the +study of elliptic functions. He wrote also on modern geometry. +\index{Elliptic functions}% +His successor at Oxford was J.~J. Sylvester. +\index{Sylvester!ref.\ to}% + +\textbf{Ernst Eduard Kummer} (1810--1893), professor in the University +\index{Kummer}% +of Berlin, is closely identified with the theory of numbers. +Dirichlet's work on complex numbers of the form $a + ib$, +introduced by Gauss, was extended by him, by Eisenstein, +and Dedekind. Instead of the equation $x^4 - 1 = 0$, the roots +of which yield Gauss' units, Eisenstein used the equation +%% -----File: 385.png---Folio 371------- +$x^3 - 1 = 0$ and complex numbers $a + b\rho$ ($\rho$~being a cube root +of unity), the theory of which resembles that of Gauss' numbers. +Kummer passed to the general case $x^n - 1 = 0$ and got +\index{Kummer}% +complex numbers of the form $\alpha = a_1A_1 + a_2A_2 + a_3A_3 + \dotsb$, +where $a_i$~are whole real numbers, and $A_i$~roots of the above +equation.\Reference{59} Euclid's theory of the greatest common divisor +is not applicable to such complex numbers, and their prime +factors cannot be defined in the same way as prime factors of +common integers are defined. In the effort to overcome this +difficulty, Kummer was led to introduce the conception of +``ideal numbers.'' These ideal numbers have been applied by +\index{Ideal numbers}% +G.~Zolotareff of St.~Petersburg to the solution of a problem +\index{Zolotareff}% +of the integral calculus, left unfinished by Abel (\textit{Liouville's +\index{Abel!ref.\ to}% +\index{Integral calculus}% +Journal}, Second Series, 1864, Vol.~IX.). \textbf{Julius Wilhelm Richard +Dedekind} of Braunschweig (born 1831) has given in the second +\index{Dedekind}% +edition of Dirichlet's \textit{Vorlesungen über Zahlentheorie} a new +\index{Dirichlet!ref.\ to}% +theory of complex numbers, in which he to some extent +deviates from the course of Kummer, and avoids the use of +ideal numbers. Dedekind has taken the roots of any irreducible +equation with integral coefficients as the units for his complex +numbers. Attracted by Kummer's investigations, his +pupil, \textbf{Leopold Kronecker} (1823--1891) made researches which +he applied to algebraic equations. + +On the other hand, efforts have been made to utilise in the +theory of numbers the results of the modern higher algebra. +Following up researches of Hermite, \textbf{Paul Bachmann} of Münster +\index{Bachmann}% +investigated the arithmetical formula which gives the automorphics +of a ternary quadratic form.\Reference{89} The problem of the +equivalence of two positive or definite ternary quadratic forms +was solved by L.~Seeber; and that of the arithmetical automorphics +\index{Seeber}% +of such forms, by Eisenstein. The more difficult problem +\index{Eisenstein!ref.\ to}% +of the equivalence for indefinite ternary forms has been +investigated by Edward Selling of Würzburg. On quadratic +\index{Selling}% +%% -----File: 386.png---Folio 372------- +\index{Negative quantities}% +\index{Partition of numbers}% +\index{Theory of numbers|)}% +forms of four or more indeterminates little has yet been done. +Hermite showed that the number of non-equivalent classes of +\index{Hermite!ref.\ to}% +quadratic forms having integral coefficients and a given discriminant +is finite, while Zolotareff and A.~N. Korkine, both +\index{Korkine}% +\index{Zolotareff!ref.\ to}% +of St.~Petersburg, investigated the minima of positive quadratic +forms. In connection with binary quadratic forms, Smith +\index{Smith, H.!ref.\ to}% +established the theorem that if the joint invariant of two +properly primitive forms vanishes, the determinant of either +of them is represented primitively by the duplicate of the +other. + +The interchange of theorems between arithmetic and algebra +is displayed in the recent researches of J.~W.~L. Glaisher +\index{Glaisher}% +of Trinity College (born 1848) and Sylvester. Sylvester gave +a Constructive Theory of Partitions, which received additions +from his pupils, F.~Franklin and G.~S. Ely. +\index{Ely}% +\index{Franklin}% + +The conception of ``number'' has been much extended in +our time. With the Greeks it included only the ordinary +positive whole numbers; Diophantus added rational fractions +\index{Diophantus!ref.\ to}% +\index{Numbers!definitions of numbers}% +to the domain of numbers. Later negative numbers and +imaginaries came gradually to be recognised. Descartes fully +grasped the notion of the negative; Gauss, that of the imaginary. +\index{Imaginary quantities}% +With Euclid, a ratio, whether rational or irrational, was +not a number. The recognition of ratios and irrationals as +\index{Irrationals}% +\index{Ratios}% +numbers took place in the sixteenth century, and found expression +with Newton. By the ratio method, the continuity of the +\index{Continuity}% +\index{Newton!ref.\ to}% +real number system has been based on the continuity of space, +but in recent time three theories of irrationals have been +advanced by Weierstrass, J.~W.~R. Dedekind, G.~Cantor, and +\index{Cantor, G.}% +\index{Dedekind!ref.\ to}% +\index{Weierstrass!ref.\ to}% +Heine, which prove the continuity of numbers without borrowing +\index{Heine!ref.\ to}% +it from space. They are based on the definition of numbers +\index{Numbers!theory of numbers|)}% +by regular sequences, the use of series and limits, and some +new mathematical conceptions. +%% -----File: 387.png---Folio 373------- +\index{Astronomy!more recent researches|(}% + + +\Section{Applied Mathematics} +\index{Applied mathematics|(}% + +Notwithstanding the beautiful developments of celestial +mechanics reached by Laplace at the close of the eighteenth +\index{Laplace!ref.\ to}% +century, there was made a discovery on the first day of the +present century which presented a problem seemingly beyond +the power of that analysis. We refer to the discovery of Ceres +by Piazzi in Italy, which became known in Germany just after +\index{Piazzi}% +the philosopher Hegel had published a dissertation proving \textit{a~priori} +\index{Hegel}% +that such a discovery could not be made. From the +positions of the planet observed by Piazzi its orbit could not +be satisfactorily calculated by the old methods, and it remained +for the genius of Gauss to devise a method of calculating +\index{Gauss!ref.\ to}% +elliptic orbits which was free from the assumption of a small +eccentricity and inclination. Gauss' method was developed +further in his \textit{Theoria Motus}. The new planet was re-discovered +with aid of Gauss' data by Olbers, an astronomer +\index{Olbers}% +who promoted science not only by his own astronomical +studies, but also by discerning and directing towards astronomical +pursuits the genius of Bessel. +\index{Bessel|(}% + +\textbf{Friedrich Wilhelm Bessel}\Reference[]{91} (1784--1846) was a native of +Minden in Westphalia. Fondness for figures, and a distaste +for Latin grammar led him to the choice of a mercantile +career. In his fifteenth year he became an apprenticed clerk +in Bremen, and for nearly seven years he devoted his days to +mastering the details of his business, and part of his nights to +study. Hoping some day to become a supercargo on trading +expeditions, he became interested in observations at sea. With +a sextant constructed by him and an ordinary clock he determined +the latitude of Bremen. His success in this inspired +him for astronomical study. One work after another was +mastered by him, unaided, during the hours snatched from +%% -----File: 388.png---Folio 374------- +sleep. From old observations he calculated the orbit of +Halley's comet. Bessel introduced himself to Olbers, and +\index{Halley's Comet}% +submitted to him the calculation, which Olbers immediately +sent for publication. Encouraged by Olbers, Bessel turned +his back to the prospect of affluence, chose poverty and the +stars, and became assistant in J.~H. Schröter's observatory at +\index{SchroterJH@{Schröter, J.~H.}}% +Lilienthal. Four years later he was chosen to superintend +the construction of the new observatory at Königsberg.\Reference{92} In +the absence of an adequate mathematical teaching force, Bessel +was obliged to lecture on mathematics to prepare students for +astronomy. He was relieved of this work in 1825 by the +arrival of Jacobi. We shall not recount the labours by which +\index{Jacobi!ref.\ to}% +Bessel earned the title of founder of modern practical astronomy +and geodesy. As an observer he towered far above +Gauss, but as a mathematician he reverently bowed before the +genius of his great contemporary. Of Bessel's papers, the one +of greatest mathematical interest is an ``\textit{Untersuchung des +Theils der planetarischen Störungen, welcher aus der Bewegung +der Sonne ensteht}'' (1824), in which he introduces a class of +transcendental functions, $J_n(x)$, much used in applied mathematics, +and known as ``Bessel's functions.'' He gave their +\index{Bessel's functions}% +principal properties, and constructed tables for their evaluation. +Recently it has been observed that Bessel's functions +appear much earlier in mathematical literature.\Reference{98} +Such functions of the zero order occur in papers of Daniel +Bernoulli (1732) and Euler on vibration of heavy strings suspended +\index{Euler!ref.\ to}% +from one end. All of Bessel's functions of the first +kind and of integral orders occur in a paper by Euler (1764) on +the vibration of a stretched elastic membrane. In 1878 Lord +Rayleigh proved that Bessel's functions are merely particular +\index{Rayleigh, Lord!ref.\ to}% +cases of Laplace's functions. J.~W.~L. Glaisher illustrates +\index{Glaisher!ref.\ to}% +by Bessel's functions his assertion that mathematical branches +growing out of physical inquiries as a rule ``lack the easy flow +%% -----File: 389.png---Folio 375------- +\index{Bessel|)}% +or homogeneity of form which is characteristic of a mathematical +theory properly so called.'' These functions have been +studied by C.~Th.\ Anger of Danzig, O.~Schlömilch of Dresden, +\index{Anger}% +\index{Schlomilch@{Schlömilch}}% +R.~Lipschitz of Bonn (born 1832), Carl Neumann of Leipzig +\index{Lipschitz!ref.\ to}% +\index{Neumann, C.}% +(born 1832), Eugen Lommel of Leipzig, I.~Todhunter of St.~John's +\index{Lommel}% +\index{Todhunter!ref.\ to}% +College, Cambridge. + +Prominent among the successors of Laplace are the following: +\index{Laplace!ref.\ to}% +\textit{Siméon Denis Poisson} (1781--1840), who wrote in 1808 +\index{Poisson!ref.\ to}% +a classic \textit{Mémoire sur les inégalités séculaires des moyens mouvements +des planètes}. \textit{Giovanni Antonio Amadeo Plana} (1781--1864) +\index{Plana}% +of Turin, a nephew of Lagrange, who published in 1811 +a \Ita{\textit{Memoria sulla teoria dell'\DPnote{** PP: [sic] space} attrazione degli sferoidi ellitici}}, and +contributed to the theory of the moon. \textbf{Peter Andreas Hansen} +\index{Hansen}% +(1795--1874) of Gotha, at one time a clockmaker in Tondern, +then Schumacher's assistant at Altona, and finally director of +the observatory at Gotha, wrote on various astronomical subjects, +but mainly on the lunar theory, which he elaborated in +his work \textit{Fundamenta nova investigationes orbitæ veræ quam +Luna perlustrat} (1838), and in subsequent investigations +embracing extensive lunar tables. \textbf{George Biddel Airy} (1801--1892), +\index{Airy}% +royal astronomer at Greenwich, published in 1826 his +\textit{Mathematical Tracts on the Lunar and Planetary Theories}. +These researches have since been greatly extended by him. +\textbf{August Ferdinand Möbius} (1790--1868) of Leipzig wrote, in 1842, +\index{Mobius@{Möbius}!ref.\ to}% +\textit{Elemente der Mechanik des Himmels}. \textbf{Urbain Jean Joseph Le +Verrier} (1811--1877) of Paris wrote\DPtypo{,}{} the \textit{Recherches Astronomiques}, +\index{Leverrier@{Le Verrier}}% +constituting in part a new elaboration of celestial +mechanics, and is famous for his theoretical discovery of +Neptune. \textbf{John Couch Adams} (1819--1892) of Cambridge +\index{Adams}% +\index{Neptune, discovery of}% +divided with Le~Verrier the honour of the mathematical discovery +of Neptune, and pointed out in 1853 that Laplace's +explanation of the secular acceleration of the moon's mean +motion accounted for only half the observed acceleration. +%% -----File: 390.png---Folio 376------- +\textbf{Charles Eugène Delaunay} (born 1816, and drowned off Cherbourg +\index{Delaunay}% +in 1872), professor of mechanics at the Sorbonne in +Paris, explained most of the remaining acceleration of the +moon, unaccounted for by Laplace's theory as corrected by +\index{Laplace!ref.\ to}% +Adams, by tracing the effect of tidal friction, a theory +previously suggested independently by Kant, Robert Mayer, +\index{Kant}% +\index{Mayer!ref.\ to}% +and William Ferrel of Kentucky. \textbf{George Howard Darwin} of +\index{Darwin}% +\index{Ferrel!ref.\ to}% +Cambridge (born 1845) made some very remarkable investigations +in 1879 on tidal friction, which trace with great +certainty the history of the moon from its origin. He has +since studied also the effects of tidal friction upon other +bodies in the solar system. Criticisms on some parts of his +researches have been made by James Nolan of Victoria. \textbf{Simon +\index{Nolan}% +Newcomb} (born 1835), superintendent of the \textit{Nautical Almanac} +\index{Nautical almanac, United States}% +\index{Newcomb}% +at Washington, and professor of mathematics at the Johns +Hopkins University, investigated the errors in Hansen's tables +of the moon. For the last twelve years the main work of the +\textit{U.~S. Nautical Almanac} office has been to collect and discuss +data for new tables of the planets which will supplant the +tables of Le~Verrier. \textit{G.~W. Hill} of that office has contributed +\index{Hill}% +\index{Leverrier@{Le Verrier}!ref.\ to}% +an elegant paper on certain possible abbreviations in the computation +of the long-period of the moon's motion due to the +direct action of the planets, and has made the most elaborate +determination yet undertaken of the inequalities of the moon's +motion due to the figure of the earth. He has also computed +certain lunar inequalities due to the action of Jupiter. + +The mathematical discussion of Saturn's rings was taken up +\index{Saturn's rings}% +first by Laplace, who demonstrated that a homogeneous solid +ring could not be in equilibrium, and in 1851 by B.~Peirce, +\index{Peirce, B.!ref.\ to}% +who proved their non-solidity by showing that even an irregular +solid ring could not be in equilibrium about Saturn. The +mechanism of these rings was investigated by James Clerk +Maxwell in an essay to which the Adams prize was awarded. +\index{Maxwell!ref.\ to}% +%% -----File: 391.png---Folio 377------- +\index{Algebraic functions!integrals}% +\index{Mechanics!more recent work|(}% +He concluded that they consisted of an aggregate of unconnected +particles. + +The problem of three bodies has been treated in various +\index{Problem of three bodies}% +\index{Three bodies, problem of}% +ways since the time of Lagrange, but no decided advance +towards a more complete algebraic solution has been made, +and the problem stands substantially where it was left by him. +He had made a reduction in the differential equations to the +seventh order. This was elegantly accomplished in a different +way by Jacobi in 1843. \textit{R.~Radau} (\textit{Comptes Rendus}, LXVII., +\index{Jacobi!ref.\ to}% +\index{Radau}% +1868, p.~841) and \textit{Allégret} (\textit{Journal de Mathématiques}, 1875, +\index{Allégret}% +p.~277) showed that the reduction can be performed on the +equations in their original form. Noteworthy transformations +and discussions of the problem have been given by J.~L.~F. +Bertrand, by Émile Bour (1831--1866) of the Polytechnic School +\index{Bertrand}% +\index{Bour}% +in Paris, by Mathieu, Hesse, J.~A. Serret. H.~Bruns of Leipzig +\index{Bruns}% +\index{Hesse!ref.\ to}% +\index{Mathieu!ref.\ to}% +\index{Serret!ref.\ to}% +has shown that no advance in the problem of three or of $n$ +bodies may be expected by algebraic integrals, and that we +must look to the modern theory of functions for a complete +solution (\textit{Acta Math.}, XI., p.~43).\Reference{93} + +Among valuable text-books on mathematical astronomy rank +the following works: \textit{Manual of Spherical and Practical Astronomy} +by \textit{Chauvenet} (1863), \textit{Practical and Spherical Astronomy} +\index{Chauvenet}% +by \textit{Robert Main} of Cambridge, \textit{Theoretical Astronomy} by \textit{James~C. +\index{Astronomy!more recent researches|)}% +\index{Main}% +Watson} of Ann Arbor (1868), \textit{Traité élémentaire de Mécanique +\index{Watson, J.~C.}% +Céleste} of \textit{H.~Resal} of the Polytechnic School in Paris, +\index{Resal}% +\textit{Cours d'Astronomie de l'École Polytechnique} by \textit{Faye}, \textit{Traité +\index{Faye}% +de Mécanique Céleste} by \textit{Tisserand}, \textit{Lehrbuch der Bahnbestimmung} +\index{Tisserand}% +by \textit{T.~Oppolzer}, \textit{Mathematische Theorien der Planetenbewegung} +\index{Oppolzer}% +by \textit{O.~Dziobek}, translated into English by M.~W. +\index{Dziobek}% +Harrington and W.~J. Hussey. +\index{Harrington}% +\index{Hussey}% + +During the present century we have come to recognise the +advantages frequently arising from a geometrical treatment of +mechanical problems. To Poinsot, Chasles, and Möbius we +\index{Chasles!ref.\ to}% +\index{Mobius@{Möbius}!ref.\ to}% +\index{Poinsot!ref.\ to}% +%% -----File: 392.png---Folio 378------- +\index{Dynamics|(}% +\index{Kowalevsky!ref.\ to}% +\index{Variation of arbitrary consonants}% +owe the most important developments made in geometrical +mechanics. \textbf{Louis Poinsot} (1777--1859), a graduate of the +\index{Poinsot}% +Polytechnic School in Paris, and for many years member of +the superior council of public instruction, published in 1804 +his \textit{Éléments de Statique}. This work is remarkable not only +as being the earliest introduction to synthetic mechanics, but +also as containing for the first time the idea of couples, which +was applied by Poinsot in a publication of 1834 to the theory +of rotation. A clear conception of the nature of rotary +motion was conveyed by Poinsot's elegant geometrical representation +by means of an ellipsoid rolling on a certain fixed +\index{Ellipsoid!(attraction of)}% +plane. This construction was extended by Sylvester so as +\index{Sylvester!ref.\ to}% +to measure the rate of rotation of the ellipsoid on the plane. + +A particular class of dynamical problems has recently been +treated geometrically by \textbf{Sir Robert Stawell Ball}, formerly +\index{Ball, Sir~R.~S.}% +astronomer royal of Ireland, now Lowndean Professor of +Astronomy and Geometry at Cambridge. His method is given +in a work entitled \textit{Theory of Screws}, Dublin, 1876, and in +\index{Screws, theory of}% +subsequent articles. Modern geometry is here drawn upon, +as was done also by Clifford in the related subject of Biquaternions. +\index{Biquaternions}% +\index{Clifford!ref.\ to}% +Arthur Buchheim of Manchester (1859--1888), +\index{Buchheim}% +showed that Grassmann's Ausdehnungslehre supplies all the +\index{Ausdehnungslehre}% +\index{Grassmann!ref.\ to}% +necessary materials for a simple calculus of screws in elliptic +space. Horace Lamb applied the theory of screws to the question +\index{Lamb}% +of the steady motion of any solid in a fluid. + +Advances in theoretical mechanics, bearing on the integration +and the alteration in form of dynamical equations, +were made since Lagrange by Poisson, William Rowan Hamilton, +\index{Hamilton, W.~R.!ref.\ to}% +\index{Lagrange!ref.\ to}% +\index{Poisson!ref.\ to}% +Jacobi, Madame Kowalevski, and others. Lagrange had +\index{Jacobi!ref.\ to}% +established the ``Lagrangian form'' of the equations of +motion. He had given a theory of the variation of the +arbitrary constants which, however, turned out to be less +fruitful in results than a theory advanced by Poisson.\Reference{99} Poisson's +%% -----File: 393.png---Folio 379------- +\index{Geodesics}% +\index{Ultimate multiplier, theory of}% +theory of the variation of the arbitrary constants and +the method of integration thereby afforded marked the first +onward step since Lagrange. Then came the researches of +Sir William Rowan Hamilton. His discovery that the integration +\index{Hamilton, W.~R.!ref.\ to}% +of the dynamic differential equations is connected with +the integration of a certain partial differential equation of the +\index{Partial differential equations}% +first order and second degree, grew out of an attempt to deduce, +by the undulatory theory, results in geometrical optics previously +\index{Undulatory theory of light}% +based on the conceptions of the emission theory. The +\textit{Philosophical Transactions} of 1833 and 1834 contain Hamilton's +papers, in which appear the first applications to mechanics +of the principle of varying action and the characteristic +\index{Action!varying}% +\index{Varying action, principle of}% +function, established by him some years previously. The +object which Hamilton proposed to himself is indicated by +the title of his first paper, viz.\ the discovery of a function +by means of which all integral equations can be actually +represented. The new form obtained by him for the equation +of motion is a result of no less importance than that which +was the professed object of the memoir. Hamilton's method +of integration was freed by Jacobi of an unnecessary complication, +\index{Jacobi!ref.\ to}% +and was then applied by him to the determination of a +geodetic line on the general ellipsoid. With aid of elliptic \DPtypo{coordinates}{co-ordinates} +\index{Co-ordinates}% +\index{Elliptic co-ordinates}% +\index{Ellipsoid!(attraction of)}% +Jacobi integrated the partial differential equation +and expressed the equation of the geodetic in form of a +relation between two Abelian integrals. Jacobi applied to +\index{Abelian integrals}% +differential equations of dynamics the theory of the ultimate +multiplier. The differential equations of dynamics are only +one of the classes of differential equations considered by +Jacobi. Dynamic investigations along the lines of Lagrange, +Hamilton, and Jacobi were made by Liouville, A.~Desboves, +\index{Desboves}% +\index{Liouville!ref.\ to}% +Serret, J.~C.~F. Sturm, Ostrogradsky, J.~Bertrand, Donkin, +\index{Bertrand}% +\index{Donkin}% +\index{Ostrogradsky}% +\index{Serret!ref.\ to}% +\index{Sturm, J.~C.~F.!ref.\ to}% +Brioschi, leading up to the development of the theory of a +\index{Brioschi!ref.\ to}% +system of canonical integrals. +%% -----File: 394.png---Folio 380------- +\index{Hydrodynamics}% +\index{Ignoration of co-ordinates}% +\index{Kowalevsky}% + +An important addition to the theory of the motion of a solid +body about a fixed point was made by Madame \textbf{Sophie de +Kowalevski}\Reference[]{96} (1853--1891), who discovered a new case in which +the differential equations of motion can be integrated. By +the use of theta-functions of two independent variables she +\index{Theta-functions}% +furnished a remarkable example of how the modern theory of +functions may become useful in mechanical problems. She +was a native of Moscow, studied under Weierstrass, obtained +the doctor's degree at Göttingen, and from 1884 until her +death was professor of higher mathematics at the University +of Stockholm. The research above mentioned received the +Bordin prize of the French Academy in 1888, which was +doubled on account of the exceptional merit of the paper. + +There are in vogue three forms for the expression of the +kinetic energy of a dynamical system: the Lagrangian, the +Hamiltonian, and a modified form of Lagrange's equations in +which certain velocities are omitted. The kinetic energy +is expressed in the first form as a homogeneous quadratic +function of the velocities, which are the time-variations of the +co-ordinates of the system; in the second form, as a homogeneous +quadratic function of the momenta of the system; +the third form, elaborated recently by Edward John Routh +\index{Routh}% +of Cambridge, in connection with his theory of ``ignoration of +co-ordinates,'' and by A.~B. Basset, is of importance in hydrodynamical +\index{Basset}% +problems relating to the motion of perforated solids +in a liquid, and in other branches of physics. + +In recent time great practical importance has come to be +attached to the principle of mechanical similitude. By it one +\index{Similitude (mechanical)}% +can determine from the performance of a model the action of +the machine constructed on a larger scale. The principle was +first enunciated by Newton (\textit{Principia}, Bk.~II., Sec.~VIII., +\index{Newton!ref.\ to}% +Prop.~32), and was derived by Bertrand from the principle +\index{Bertrand}% +of virtual velocities. A corollary to it, applied in ship-building, +%% -----File: 395.png---Folio 381------- +\index{Hydrodynamics|(}% +goes by the name of William Froude's law, but was enunciated +\index{Froude}% +also by Reech. +\index{Reech}% + +The present problems of dynamics differ materially from +those of the last century. The explanation of the orbital and +axial motions of the heavenly bodies by the law of universal +gravitation was the great problem solved by Clairaut, Euler, +D'Alembert, Lagrange, and Laplace. It did not involve the +consideration of frictional resistances. In the present time +the aid of dynamics has been invoked by the physical +sciences. The problems there arising are often complicated +by the presence of friction. Unlike astronomical problems of +a century ago, they refer to phenomena of matter and motion +that are usually concealed from direct observation. The great +pioneer in such problems is Lord Kelvin. While yet an +\index{Kelvin, Lord!ref.\ to}% +undergraduate at Cambridge, during holidays spent at the +seaside, he entered upon researches of this kind by working +out the theory of spinning tops, which previously had been +only partially explained by Jellet in his \textit{Treatise on the Theory +\index{Jellet!ref.\ to}% +of Friction} (1872), and by Archibald Smith. +\index{Smith, A.}% + +Among standard works on mechanics are \textbf{Jacobi's} \textit{Vorlesungen +\index{Jacobi!ref.\ to}% +über Dynamik}, edited by Clebsch, 1866; \textbf{Kirchhoff's} \textit{Vorlesungen +\index{Clebsch!ref.\ to}% +\index{Kirchhoff!ref.\ to}% +über mathematische Physik}, 1876; \textbf{Benjamin Peirce's} +\index{Peirce, B.!ref.\ to}% +\textit{Analytic Mechanics}, 1855; \textbf{Somoff's} \textit{Theoretische Mechanik}, +\index{Mechanics!more recent work|)}% +\index{Somoff}% +1879; \textbf{Tait and Steele's} \textit{Dynamics of a Particle}, 1856; \textbf{Minchin's} +\index{Minchin}% +\index{Steele}% +\index{Tait}% +\textit{Treatise on Statics}; \textbf{Routh's} \textit{Dynamics of a System of Rigid +\index{Dynamics|)}% +\index{Routh!ref.\ to}% +Bodies}; \textbf{Sturm's} \textit{Cours de Mécanique de l'École Polytechnique}. +\index{Sturm, J.~C.~F.!ref.\ to}% + +The equations which constitute the foundation of the theory +of fluid motion were fully laid down at the time of Lagrange, +but the solutions actually worked out were few and mainly +of the irrotational type. A powerful method of attacking +problems in fluid motion is that of images, introduced in 1843 +\index{Images, theory of}% +by George Gabriel Stokes of Pembroke College, Cambridge. +\index{Stokes}% +It received little attention until Sir William Thomson's discovery +\index{Thomson, Sir William!ref.\ to}% +%% -----File: 396.png---Folio 382------- +\index{Kelvin, Lord!ref.\ to}% +\index{Vortex rings}% +of electrical images, whereupon the theory was extended +by Stokes, Hicks, and Lewis. In 1849, Thomson gave the +\index{Hicks}% +\index{Lewis}% +\index{Stokes!ref.\ to}% +\index{Thomson, J.~J.}% +\index{Thomson, Sir William!ref.\ to}% +maximum and minimum theorem peculiar to hydrodynamics, +which was afterwards extended to dynamical problems in +general. + +A new epoch in the progress of hydrodynamics was created, +in 1856, by Helmholtz, who worked out remarkable properties +\index{Helmholtz!ref.\ to}% +of rotational motion in a homogeneous, incompressible fluid, +devoid of viscosity. He showed that the vortex filaments in +such a medium may possess any number of knottings and twistings, +but are either endless or the ends are in the free surface +of the medium; they are indivisible. These results suggested +to Sir William Thomson the possibility of founding on them a +new form of the atomic theory, according to which every atom +\index{Atomic theory}% +is a vortex ring in a non-frictional ether, and as such must be +absolutely permanent in substance and duration. The vortex-atom +theory is discussed by J.~J. Thomson of Cambridge +(born 1856) in his classical treatise on the \textit{Motion of Vortex +Rings}, to which the Adams Prize was awarded in 1882. +Papers on vortex motion have been published also by Horace +\index{Vortex motion}% +Lamb, Thomas Craig, Henry~A. Rowland, and Charles Chree. +\index{Chree}% +\index{Craig, T.}% +\index{Lamb}% +\index{Rowland}% + +The subject of jets was investigated by Helmholtz, Kirchhoff, +\index{Jets}% +\index{Kirchhoff!ref.\ to}% +Plateau, and Rayleigh; the motion of fluids in a fluid by +\index{Plateau}% +Stokes, Sir W.~Thomson, Köpcke, Greenhill, and Lamb; the +\index{Greenhill}% +\index{Kopcke@{Köpcke}}% +theory of viscous fluids by Navier, Poisson, Saint-Venant, +\index{Navier!ref.\ to}% +\index{Poisson!ref.\ to}% +\index{Saint-Venant!ref.\ to}% +Stokes, O.~E. Meyer, Stefano, Maxwell, Lipschitz, Craig, +\index{Lipschitz!ref.\ to}% +\index{Maxwell!ref.\ to}% +\index{Meyer, O.~E.}% +\index{Stefano}% +Helmholtz, and A.~B. Basset. Viscous fluids present great +\index{Basset}% +difficulties, because the equations of motion have not the same +degree of certainty as in perfect fluids, on account of a deficient +theory of friction, and of the difficulty of connecting +\index{Friction, theory of}% +oblique pressures on a small area with the differentials of the +velocities. + +Waves in liquids have been a favourite subject with English +\index{Waves|(}% +%% -----File: 397.png---Folio 383------- +\index{Kelvin, Lord!ref.\ to}% +\index{Solitary wave}% +mathematicians. The early inquiries of Poisson and +\index{Poisson!ref.\ to}% +Cauchy were directed to the investigation of waves produced +\index{Cauchy!ref.\ to}% +by disturbing causes acting arbitrarily on a small portion +of the fluid. The velocity of the long wave was given +\index{Long wave}% +approximately by Lagrange in 1786 in case of a channel of +\index{Lagrange!ref.\ to}% +rectangular cross-section, by Green in 1839 for a channel of +\index{Green!ref.\ to}% +triangular section, and by P.~Kelland for a channel of any +\index{Kelland}% +uniform section. Sir George B. Airy, in his treatise on \textit{Tides +\index{Airy!ref.\ to}% +\index{Tides}% +and Waves}, discarded mere approximations, and gave the exact +equation on which the theory of the long wave in a channel of +uniform rectangular section depends. But he gave no general +solutions. J.~McCowan of University College at Dundee +\index{McCowan}% +discusses this topic more fully, and arrives at exact and +complete solutions for certain cases. The most important +application of the theory of the long wave is to the explanation +of tidal phenomena in rivers and estuaries. + +The mathematical treatment of solitary waves was first +taken up by S.~Earnshaw in 1845, then by Stokes; but the first +\index{Earnshaw}% +\index{Stokes!ref.\ to}% +sound approximate theory was given by J.~Boussinesq in 1871, +\index{Boussinesq}% +who obtained an equation for their form, and a value for the +velocity in agreement with experiment. Other methods of +approximation were given by Rayleigh and J.~McCowan. In +\index{Rayleigh, Lord!ref.\ to}% +connection with deep-water waves, Osborne Reynolds gave in +\index{Reynolds}% +1877 the dynamical explanation for the fact that a group +of such waves advances with only half the rapidity of the +individual waves. + +The solution of the problem of the general motion of an +ellipsoid in a fluid is due to the successive labours of Green +\index{Ellipsoid!motion of}% +(1833), Clebsch (1856), and Bjerknes (1873). The free +\index{Bjerknes, C. A.}% +\index{Clebsch!ref.\ to}% +motion of a solid in a liquid has been investigated by W.~Thomson, +\index{Thomson, Sir William!ref.\ to}% +Kirchhoff, and Horace Lamb. By these labours, the +\index{Kirchhoff!ref.\ to}% +\index{Lamb}% +motion of a single solid in a fluid has come to be pretty well +understood, but the case of two solids in a fluid is not developed +%% -----File: 398.png---Folio 384------- +\index{Hydrodynamics|)}% +\index{Spheroid (liquid)}% +so fully. The problem has been attacked by W.~M. +Hicks. +\index{Hicks}% + +The determination of the period of oscillation of a rotating +liquid spheroid has important bearings on the question of the +origin of the moon. G.~H. Darwin's investigations thereon, +\index{Darwin!ref.\ to}% +viewed in the light of Riemann's and Poincaré's researches, +\index{Poincaré!ref.\ to}% +\index{Riemann!ref.\ to}% +seem to disprove Laplace's hypothesis that the moon separated +\index{Laplace!ref.\ to}% +from the earth as a ring, because the angular velocity was too +great for stability; Darwin finds no instability. + +The explanation of the contracted vein has been a point of +\index{Contracted vein}% +much controversy, but has been put in a much better light by +the application of the principle of momentum, originated by +Froude and Rayleigh. Rayleigh considered also the reflection +\index{Froude}% +\index{Rayleigh, Lord!ref.\ to}% +of waves, not at the surface of separation of two uniform +media, where the transition is abrupt, but at the confines of +two media between which the transition is gradual. + +The first serious study of the circulation of winds on the +\index{Winds|(}% +earth's surface was instituted at the beginning of the second +quarter of this century by \textit{H.~W. Dové}, \textit{William~C. Redfield}, and +\index{Dové}% +\index{Redfield}% +\textit{James~P. Espy}, followed by researches of \textit{W.~Reid}, \textit{Piddington}, +\index{Espy}% +\index{Piddington}% +\index{Reid}% +and \textit{Elias Loomis}. But the deepest insight into the wonderful +\index{Loomis}% +correlations that exist among the varied motions of the +atmosphere was obtained by \textbf{William Ferrel} (1817--1891). He +\index{Ferrel}% +was born in Fulton County, Pa., and brought up on a farm. +Though in unfavourable surroundings, a burning thirst for +knowledge spurred the boy to the mastery of one branch after +another. He attended Marshall College, Pa., and graduated +in 1844 from Bethany College. While teaching school he +became interested in meteorology and in the subject of tides. +\index{Meteorology|(}% +In 1856 he wrote an article on ``the winds and currents of the +ocean.'' The following year he became connected with the +\textit{Nautical Almanac}. A mathematical paper followed in 1858 +on ``the motion of fluids and solids relative to the earth's +%% -----File: 399.png---Folio 385------- +surface.'' The subject was extended afterwards so as to +embrace the mathematical theory of cyclones, tornadoes, +water-spouts, etc. In 1885 appeared his \textit{Recent Advances in +Meteorology}. In the opinion of a leading European meteorologist +(\textit{Julius Hann} of Vienna), Ferrel has ``contributed more +\index{Hann}% +to the advance of the physics of the atmosphere than any +other living physicist or meteorologist.'' + +Ferrel teaches that the air flows in great spirals toward the +poles, both in the upper strata of the atmosphere and on the +earth's surface beyond the 30th~degree of latitude; while +the return current blows at nearly right angles to the above +spirals, in the middle strata as well as on the earth's surface, +in a zone comprised between the parallels $30°$~N. and $30°$~S. The +idea of three superposed currents blowing spirals was first +advanced by James Thomson, but was published in very +\index{Thomson, J.}% +meagre abstract. + +Ferrel's views have given a strong impulse to theoretical +research in America, Austria, and Germany. Several objections +raised against his argument have been abandoned, or +have been answered by W.~M. Davis of Harvard. The mathematical +\index{Davis, W.~M.}% +analysis of F.~Waldo of Washington, and of others, +\index{Waldo}% +has further confirmed the accuracy of the theory. The transport +of Krakatoa dust and observations made on clouds point +toward the existence of an upper east current on the equator, +and Pernter has mathematically deduced from Ferrel's theory +\index{Pernter, J.~M.}% +the existence of such a current. + +Another theory of the general circulation of the atmosphere +was propounded by Werner Siemens of Berlin, in which an +\index{Siemens}% +attempt is made to apply thermodynamics to aërial currents. +\index{Thermodynamics}% +Important new points of view have been introduced recently +by Helmholtz, who concludes that when two air currents blow +\index{Helmholtz!ref.\ to}% +one above the other in different directions, a system of air +waves must arise in the same way as waves are formed on the +\index{Waves|)}% +%% -----File: 400.png---Folio 386------- +\index{Meteorology|)}% +\index{Winds|)}% +sea. He and A.~Oberbeck showed that when the waves on the +\index{Oberbeck}% +sea attain lengths of from $16$ to $33$~feet, the air waves must +attain lengths of from $10$ to $20$~miles, and proportional depths. +Superposed strata would thus mix more thoroughly, and their +energy would be partly dissipated. From hydrodynamical +equations of rotation Helmholtz established the reason why +\index{Helmholtz!ref.\ to}% +the observed velocity from equatorial regions is much less in +a latitude of, say, $20°$ or~$30°$, than it would be were the movements +unchecked. + +About 1860 acoustics began to be studied with renewed +\index{Acoustics}% +zeal. The mathematical theory of pipes and vibrating strings +had been elaborated in the eighteenth century by Daniel Bernoulli, +\index{Bernoulli, Daniel!ref.\ to}% +D'Alembert, Euler, and Lagrange. In the first part of +\index{D'Alembert!ref.\ to}% +\index{Euler!ref.\ to}% +\index{Lagrange!ref.\ to}% +the present century Laplace corrected Newton's theory on the +\index{Laplace!ref.\ to}% +\index{Newton!ref.\ to}% +velocity of sound in gases, Poisson gave a mathematical discussion +\index{Poisson!ref.\ to}% +of torsional vibrations; Poisson, Sophie Germain, and +\index{Germain, Sophie!ref.\ to}% +Wheatstone studied Chladni's figures; Thomas Young and the +\index{Chladni's figures}% +\index{Wheatstone}% +\index{Young!ref.\ to}% +brothers Weber developed the wave-theory of sound. \textbf{Sir J.~F.~W. +Herschel} wrote on the mathematical theory of sound for +\index{Herschel, J.~F.~W.}% +the \textit{Encyclopædia Metropolitana}, 1845. Epoch-making were +Helmholtz's experimental and mathematical researches. In +his hands and Rayleigh's, Fourier's series received due +\index{Fourier's series}% +\index{Rayleigh, Lord}% +attention. Helmholtz gave the mathematical theory of beats, +difference tones, and summation tones. \textbf{Lord Rayleigh} (John +William Strutt) of Cambridge (born 1842) made extensive +\index{Strutt, J.~W.}% +mathematical researches in acoustics as a part of the theory of +vibration in general. Particular mention may be made of his +discussion of the disturbance produced by a spherical obstacle +on the waves of sound, and of phenomena, such as sensitive +flames, connected with the instability of jets of fluid. In 1877 +\index{Jets}% +and 1878 he published in two volumes a treatise on \textit{The Theory +of Sound}. Other mathematical researches on this subject have +been made in England by Donkin and Stokes. +\index{Stokes!ref.\ to}% +%% -----File: 401.png---Folio 387------- + +The theory of elasticity\Reference[]{42} belongs to this century. Before +\index{Elasticity|(}% +1800 no attempt had been made to form general equations for +the motion or equilibrium of an elastic solid. Particular problems +had been solved by special hypotheses. Thus, James +Bernoulli considered elastic laminæ; Daniel Bernoulli and +\index{Bernoulli, James (born 1758)}% +Euler investigated vibrating rods; Lagrange and Euler, the +\index{Euler!ref.\ to}% +\index{Lagrange!ref.\ to}% +\index{Vibrating rods}% +equilibrium of springs and columns. The earliest investigations +of this century, by Thomas Young (``Young's modulus of +\index{Young!ref.\ to}% +elasticity'') in England, J.~Binet in France, and G.~A.~A. Plana +\index{Binet}% +\index{Plana}% +in Italy, were chiefly occupied in extending and correcting the +earlier labours. Between 1830 and 1840 the broad outline of the +modern theory of elasticity was established. This was accomplished +almost exclusively by French writers,---Louis-Marie-Henri +Navier (1785--1836), Poisson, Cauchy, Mademoiselle +\index{Cauchy!ref.\ to}% +\index{Navier}% +\index{Poisson}% +\index{Poisson!ref.\ to}% +Sophie Germain (1776--1831), Félix Savart (1791--1841). +\index{Germain, Sophie}% +\index{Savart}% + +\textbf{Siméon Denis Poisson}\Reference[]{94} (1781--1840) was born at Pithiviers. +The boy was put out to a nurse, and he used to tell that when +his father (a common soldier) came to see him one day, the +nurse had gone out and left him suspended by a thin cord to a +nail in the wall in order to protect him from perishing under +the teeth of the carnivorous and unclean animals that roamed +on the floor. Poisson used to add that his gymnastic efforts +when thus suspended caused him to swing back and forth, and +thus to gain an early familiarity with the pendulum, the study +of which occupied him much in his maturer life. His father +destined him for the medical profession, but so repugnant was +this to him that he was permitted to enter the Polytechnic +School at the age of seventeen. His talents excited the interest +of Lagrange and Laplace. At eighteen he wrote a memoir +on finite differences which was printed on the recommendation +of Legendre. He soon became a lecturer at the school, and +continued through life to hold various government scientific +posts and professorships. He prepared some 400 publications, +%% -----File: 402.png---Folio 388------- +\index{Capillarity}% +\index{Kelvin, Lord!ref.\ to}% +mainly on applied mathematics. His \textit{Traité de Mécanique}, +2~vols., 1811 and 1833, was long a standard work. He wrote +on the mathematical theory of heat, capillary action, probability +of judgment, the mathematical theory of electricity and +magnetism, physical astronomy, the attraction of ellipsoids, +definite integrals, series, and the theory of elasticity. He was +considered one of the leading analysts of his time. + +His work on elasticity is hardly excelled by that of Cauchy, +\index{Cauchy!ref.\ to}% +and second only to that of Saint-Venant. There is hardly a +problem in elasticity to which he has not contributed, while +many of his inquiries were new. The equilibrium and motion +of a circular plate was first successfully treated by him. +Instead of the definite integrals of earlier writers, he used +preferably finite summations. Poisson's contour conditions +for elastic plates were objected to by Gustav Kirchhoff of +\index{Kirchhoff!ref.\ to}% +Berlin, who established new conditions. But Thomson and +\index{Thomson, Sir William!ref.\ to}% +Tait in their \textit{Treatise on Natural Philosophy} have explained +\index{Tait}% +the discrepancy between Poisson's and Kirchhoff's boundary +conditions, and established a reconciliation between them. + +Important contributions to the theory of elasticity were +made by Cauchy. To him we owe the origin of the theory +of stress, and the transition from the consideration of the +force upon a molecule exerted by its neighbours to the consideration +of the stress upon a small plane at a point. He +anticipated Green and Stokes in giving the equations of isotropic +\index{Green!ref.\ to}% +\index{Stokes!ref.\ to}% +elasticity with two constants. The theory of elasticity +was presented by Gabrio Piola of Italy according to the principles +\index{Piola}% +of Lagrange's \textit{Mécanique Analytique}, but the superiority +of this method over that of Poisson and Cauchy is far from +evident. The influence of temperature on stress was first +investigated experimentally by Wilhelm Weber of Göttingen, +\index{Weber, W.~E.!ref.\ to}% +and afterwards mathematically by Duhamel, who, assuming +\index{Duhamel}% +Poisson's theory of elasticity, examined the alterations of +%% -----File: 403.png---Folio 389------- +form which the formulæ undergo when we allow for changes +of temperature. Weber was also the first to experiment +on elastic after-strain. Other important experiments were +made by different scientists, which disclosed a wider range +of phenomena, and demanded a more comprehensive theory. +Set was investigated by Gerstner (1756--1832) and Eaton +\index{Gerstner}% +Hodgkinson, while the latter physicist in England and Vicat +\index{Hodgkinson}% +\index{Vicat}% +(1786--1861) in France experimented extensively on absolute +strength. Vicat boldly attacked the mathematical theories of +flexure because they failed to consider shear and the time-element. +\index{Flexure, theory of}% +As a result, a truer theory of flexure was soon propounded +by Saint-Venant. Poncelet advanced the theories of +\index{Poncelet!ref.\ to}% +\index{Saint-Venant!ref.\ to}% +resilience and cohesion. + +\textbf{Gabriel Lamé}\Reference[]{94} (1795--1870) was born at Tours, and graduated +\index{Lame@{Lamé}}% +\index{Lame@{Lamé}!ref.\ to}% +at the Polytechnic School. He was called to Russia +with Clapeyron and others to superintend the construction of +bridges and roads. On his return, in 1832, he was elected +professor of physics at the Polytechnic School. Subsequently +he held various engineering posts and professorships in Paris. +As engineer he took an active part in the construction of the +first railroads in France. Lamé devoted his fine mathematical +talents mainly to mathematical physics. In four works: +\textit{Leçons sur les fonctions inverses des transcendantes et les surfaces +isothermes}; \textit{Sur les coordonnées curvilignes et leurs diverses +applications}; \textit{Sur la théorie analytique de la chaleur}; \textit{Sur la +théorie mathématique de l'élasticité des corps solides} (1852), and +in various memoirs he displays fine analytical powers; but a +certain want of physical touch sometimes reduces the value of +his contributions to elasticity and other physical subjects. In +considering the temperature in the interior of an ellipsoid +under certain conditions, he employed functions analogous to +Laplace's functions, and known by the name of ``Lamé's functions.'' +\index{Lame's@{Lamé's functions}}% +A problem in elasticity called by Lamé's name, viz.\ +%% -----File: 404.png---Folio 390------- +\index{Jellet!ref.\ to}% +to investigate the conditions for equilibrium of a spherical +elastic envelope subject to a given distribution of load on +the bounding spherical surfaces, and the determination of the +resulting shifts is the only completely general problem on +elasticity which can be said to be completely solved. He +deserves much credit for his derivation and transformation +of the general elastic equations, and for his application of +them to double refraction. Rectangular and triangular membranes +were shown by him to be connected with questions in +the theory of numbers. The field of photo-elasticity was +entered upon by Lamé, F.~E. Neumann, Clerk Maxwell. +\index{Maxwell!ref.\ to}% +\index{Neumann, F.~E.!ref.\ to}% +Stokes, Wertheim, R.~Clausius, Jellett, threw new light upon +\index{Clausius!ref.\ to}% +\index{Light, theory of}% +\index{Stokes!ref.\ to}% +\index{Wertheim}% +the subject of ``rari-constancy'' and ``multi-constancy,'' which +\index{Multi-constancy}% +\index{Rari-constancy}% +has long divided elasticians into two opposing factions. The +uni-constant isotropy of Navier and Poisson had been questioned +\index{Navier!ref.\ to}% +\index{Poisson!ref.\ to}% +by Cauchy, and was now severely criticised by Green +\index{Cauchy!ref.\ to}% +\index{Green!ref.\ to}% +and Stokes. + +\textbf{Barré de Saint-Venant} (1797--1886), ingénieur des ponts et +\index{Saint-Venant}% +chaussées, made it his life-work to render the theory of +elasticity of practical value. The charge brought by practical +engineers, like Vicat, against the theorists led Saint-Venant to +\index{Vicat!ref.\ to}% +place the theory in its true place as a guide to the practical +man. Numerous errors committed by his predecessors were +removed. He corrected the theory of flexure by the consideration +of slide, the theory of elastic rods of double curvature by +the introduction of the third moment, and the theory of torsion +by the discovery of the distortion of the primitively +plane section. His results on torsion abound in beautiful +graphic illustrations. In case of a rod, upon the side surfaces +of which no forces act, he showed that the problems of flexure +and torsion can be solved, if the end-forces are distributed +over the end-surfaces by a definite law. Clebsch, in his +\index{Clebsch!ref.\ to|(}% +\textit{Lehrbuch der Elasticität}, 1862, showed that this problem is +%% -----File: 405.png---Folio 391------- +reversible to the case of side-forces without end-forces. +Clebsch\Reference[]{68} extended the research to very thin rods and to very +thin plates. Saint-Venant considered problems arising in the +scientific design of built-up artillery, and his solution of them +differs considerably from Lamé's solution, which was popularised +by Rankine, and much used by gun-designers. In Saint-Venant's +translation into French of Clebsch's \textit{Elasticität}, he +develops extensively a double-suffix notation for strain and +stresses. Though often advantageous, this notation is cumbrous, +and has not been generally adopted. \textit{Karl Pearson}, +\index{Pearson}% +professor in University College, London, has recently examined +mathematically the permissible limits of the application +of the ordinary theory of flexure of a beam. + +The mathematical theory of elasticity is still in an unsettled +condition. Not only are scientists still divided into two +schools of ``rari-constancy'' and ``multi-constancy,'' but difference +\index{Multi-constancy}% +of opinion exists on other vital questions. Among the +numerous modern writers on elasticity may be mentioned +Émile Mathieu (1835--1891), professor at Besançon, Maurice +\index{Mathieu}% +Levy of Paris, Charles Chree, superintendent of the Kew Observatory, +\index{Chree}% +\index{Levy}% +A.~B. Basset, Sir William Thomson (Lord Kelvin) +\index{Kelvin, Lord!ref.\ to}% +\index{Thomson, Sir William!ref.\ to}% +of Glasgow, J.~Boussinesq of Paris, and others. Sir William +\index{Boussinesq}% +Thomson applied the laws of elasticity of solids to the investigation +of the earth's elasticity, which is an important element +\index{Earth!rigidity of}% +in the theory of ocean-tides. If the earth is a solid, then its +elasticity co-operates with gravity in opposing deformation +due to the attraction of the sun and moon. Laplace had +shown how the earth would behave if it resisted deformation +only by gravity. Lamé had investigated how a solid sphere +would change if its elasticity only came into play. Sir +William Thomson combined the two results, and compared +them with the actual deformation. Thomson, and afterwards +G.~H. Darwin, computed that the resistance of the earth to +\index{Darwin!ref.\ to}% +%% -----File: 406.png---Folio 392------- +tidal deformation is nearly as great as though it were of steel. +This conclusion has been confirmed recently by Simon Newcomb, +\index{Newcomb!ref.\ to}% +from the study of the observed periodic changes in +latitude. For an ideally rigid earth the period would be $360$~days, +\index{Latitude, periodic changes in}% +but if as rigid as steel, it would be~$441$, the observed +period being $430$~days. + +Among text-books on elasticity may be mentioned the works +\index{Elasticity|)}% +of Lamé, Clebsch, Winkler, Beer, Mathieu, W.~J. Ibbetson, and +\index{Beer}% +\index{Clebsch!ref.\ to|)}% +\index{Ibbetson}% +\index{Lame@{Lamé}!ref.\ to}% +\index{Mathieu!ref.\ to}% +\index{Winkler}% +F.~Neumann, edited by O.~E. Meyer. +\index{Meyer, O.~E.}% +\index{Neumann, F.~E.!ref.\ to}% + +Riemann's opinion that a \emph{science} of physics only exists since +\index{Riemann!ref.\ to}% +the invention of differential equations finds corroboration even +in this brief and fragmentary outline of the progress of mathematical +physics. The undulatory theory of light, first advanced +\index{Undulatory theory of light|(}% +by Huygens, owes much to the power of mathematics: +\index{Huygens!ref.\ to}% +by mathematical analysis its assumptions were worked out +to their last consequences. \textbf{Thomas Young}\Reference[]{95} (1773--1829) was +\index{Young}% +the first to explain the principle of interference, both of +light and sound, and the first to bring forward the idea +of transverse vibrations in light waves. Young's explanations, +not being verified by him by extensive numerical calculations, +attracted little notice, and it was not until \textbf{Augustin +Fresnel} (1788--1827) applied mathematical analysis to a much +\index{Fresnel}% +greater extent than Young had done, that the undulatory +theory began to carry conviction. Some of Fresnel's mathematical +assumptions were not satisfactory; hence Laplace, +\index{Laplace!ref.\ to}% +Poisson, and others belonging to the strictly mathematical +\index{Poisson!ref.\ to}% +school, at first disdained to consider the theory. By their +opposition Fresnel was spurred to greater exertion. Arago +\index{Arago}% +was the first great convert made by Fresnel. When polarisation +and double refraction were explained by Young and +Fresnel, then Laplace was at last won over. Poisson drew +from Fresnel's formulæ the seemingly paradoxical deduction +that a small circular disc, illuminated by a luminous point, +%% -----File: 407.png---Folio 393------- +\index{Kelvin, Lord!ref.\ to}% +\index{MacCullagh!ref.\ to}% +must cast a shadow with a bright spot in the centre. But +this was found to be in accordance with fact. The theory +was taken up by another great mathematician, Hamilton, who +\index{Hamilton, W.~R.!ref.\ to}% +from his formulæ predicted conical refraction, verified experimentally +by Lloyd. These predictions do not prove, however, +\index{Lloyd}% +that Fresnel's formulæ are correct, for these prophecies might +have been made by other forms of the wave-theory. The +theory was placed on a sounder dynamical basis by the writings +of Cauchy, Biot, Green, C.~Neumann, Kirchhoff, McCullagh, +\index{Biot}% +\index{Cauchy!ref.\ to}% +\index{Green!ref.\ to}% +\index{Kirchhoff!ref.\ to}% +\index{McCullagh}% +\index{Neumann, C.!ref.\ to}% +Stokes, Saint-Venant, Sarrau, Lorenz, and Sir William Thomson. +\index{Lorenz}% +\index{Saint-Venant!ref.\ to}% +\index{Sarrau}% +\index{Stokes!ref.\ to}% +\index{Thomson, Sir William!ref.\ to}% +In the wave-theory, as taught by Green and others, the +luminiferous ether was an incompressible elastic solid, for +\index{Ether, luminiferous}% +the reason that fluids could not propagate transverse vibrations. +But, according to Green, such an elastic solid would +transmit a longitudinal disturbance with infinite velocity. +Stokes remarked, however, that the ether might act like +a fluid in case of finite disturbances, and like an elastic solid +in case of the infinitesimal disturbances in light propagation. + +Fresnel postulated the density of ether to be different in +different media, but the elasticity the same, while C.~Neumann +and McCullagh assume the density uniform and the elasticity +different in all substances. On the latter assumption the +direction of vibration lies in the plane of polarisation, and not +perpendicular to it, as in the theory of Fresnel. + +While the above writers endeavoured to explain all optical +properties of a medium on the supposition that they arise +entirely from difference in rigidity or density of the ether in +the medium, there is another school advancing theories in +which the mutual action between the molecules of the body +and the ether is considered the main cause of refraction and +dispersion.\Reference{100} The chief workers in this field are J.~Boussinesq, +\index{Boussinesq}% +W.~Sellmeyer, Helmholtz, E.~Lommel, E.~Ketteler, W.~Voigt, +\index{Helmholtz!ref.\ to}% +\index{Ketteler}% +\index{Lommel}% +\index{Sellmeyer}% +\index{Voigt}% +and Sir William Thomson in his lectures delivered at the +%% -----File: 408.png---Folio 394------- +\index{Kelvin, Lord!ref.\ to}% +\index{Undulatory theory of light|)}% +Johns Hopkins University in 1884. Neither this nor the +first-named school succeeded in explaining all the phenomena. +A third school was founded by Maxwell. He proposed the +\index{Maxwell!ref.\ to}% +electro-magnetic theory, which has received extensive development +\index{Electro-magnetic theory of light}% +recently. It will be mentioned again later. According +to Maxwell's theory, the direction of vibration does not lie +exclusively in the plane of polarisation, nor in a plane perpendicular +to it, but something occurs in both planes---a magnetic +vibration in one, and an electric in the other. Fitzgerald and +\index{Fitzgerald}% +Trouton in Dublin verified this conclusion of Maxwell by +\index{Trouton}% +experiments on electro-magnetic waves. + +Of recent mathematical and experimental contributions to +optics, mention must be made of H.~A. Rowland's theory of +\index{Rowland}% +concave gratings, and of A.~A. Michelson's work on interference, +\index{Michelson}% +and his application of interference methods to astronomical +measurements. + +In electricity the mathematical theory and the measurements +\index{Electricity|(}% +of \textbf{Henry Cavendish} (1731--1810), and in magnetism +\index{Cavendish}% +the measurements of \textbf{Charles Augustin Coulomb} (1736--1806), +\index{Coulomb}% +became the foundations for a system of measurement. For +electro-magnetism the same thing was done by \textbf{Andrè Marie +Ampère} (1775--1836). The first complete method of measurement +\index{Ampère}% +was the system of absolute measurements of terrestrial +magnetism introduced by \textit{Gauss} and \textbf{Wilhelm Weber} (1804--1891) +\index{Weber, W.~E.}% +and afterwards extended by Wilhelm Weber and F.~Kohlrausch +\index{Kohlrausch}% +to electro-magnetism and electro-statics. In 1861 +the British Association and the Royal Society appointed a +special commission with Sir William Thomson at the head, to +\index{Thomson, Sir William!ref.\ to}% +consider the unit of electrical resistance. The commission +recommended a unit in principle like W. Weber's, but greater +than Weber's by a factor of~$10^7$.\Reference{101} The discussions and labours +on this subject continued for twenty years, until in 1881 a +general agreement was reached at an electrical congress in Paris. +%% -----File: 409.png---Folio 395------- +\index{Kelvin, Lord|(}% +\index{Kelvin, Lord!ref.\ to}% + +A function of fundamental importance in the mathematical +theories of electricity and magnetism is the ``potential.'' It +\index{Potential}% +was first used by Lagrange in the determination of gravitational +attractions in 1773. Soon after, Laplace gave the +\index{Laplace!ref.\ to}% +celebrated differential equation, +\[ +\frac{\partial^2V}{\DPtypo{d}{\partial}x^2} + +\frac{\partial^2V}{\DPtypo{d}{\partial}y^2} + +\frac{\partial^2V}{\DPtypo{d}{\partial}z^2} = 0, +\] +which was extended by Poisson by writing $-4\pi k$ in place of +\index{Poisson!ref.\ to}% +zero in the right-hand member of the equation, so that it +applies not only to a point external to the attracting mass, +but to any point whatever. The first to apply the potential +function to other than gravitation problems was \textbf{George Green} +\index{Green}% +\index{Green!ref.\ to}% +(1793--1841). He introduced it into the mathematical theory +of electricity and magnetism. Green was a self-educated man +who started out as a baker, and at his death was fellow of +Caius College, Cambridge. In 1828 he published by subscription +at Nottingham a paper entitled \textit{Essay on the application +of mathematical analysis to the theory of electricity and magnetism}. +It escaped the notice even of English mathematicians +until 1846, when Sir William Thomson had it reprinted in +\index{Thomson, Sir William}% +\index{Thomson, Sir William!ref.\ to}% +\textit{Crelle's Journal}, vols.~xliv.\ and xlv. It contained what is now +known as ``Green's theorem'' for the treatment of potential. +Meanwhile all of Green's general theorems had been re-discovered +by Sir William Thomson, Chasles, Sturm, and Gauss. +The term \emph{potential function} is due to Green. Hamilton used +the word \emph{force-function}, while Gauss, who about 1840 secured +\index{Force-function}% +the general adoption of the function, called it simply \emph{potential}. + +Large contributions to electricity and magnetism have been +made by \textbf{William Thomson}. He was born in 1824 at Belfast, +Ireland, but is of Scotch descent. He and his brother James +studied in Glasgow. From there he entered Cambridge, and +was graduated as Second Wrangler in 1845. William Thomson, +%% -----File: 410.png---Folio 396------- +\index{Addition theorem of elliptic integrals}% +Sylvester, Maxwell, Clifford, and J.~J. Thomson are a group +\index{Clifford!ref.\ to}% +\index{Maxwell!ref.\ to}% +\index{Sylvester!ref.\ to}% +\index{Thomson, J.~J.!ref.\ to}% +\index{Thomson, Sir William}% +of great men who were Second Wranglers at Cambridge. At +the age of twenty-two W.~Thomson was elected professor of +natural philosophy in the University of Glasgow, a position +which he has held ever since. For his brilliant mathematical +and physical achievements he was knighted, and in 1892 was +made Lord Kelvin. His researches on the theory of potential +\index{Kelvin, Lord|)}% +are epoch-making. What is called ``Dirichlet's principle'' +\index{Dirichlet!ref.\ to}% +was discovered by him in 1848, somewhat earlier than by +Dirichlet. We owe to Sir William Thomson new synthetical +methods of great elegance, viz.\ the theory of electric images +and the method of electric inversion founded thereon. By +them he determined the distribution of electricity on a bowl, +a problem previously considered insolvable. The distribution +of static electricity on conductors had been studied before this +mainly by Poisson and Plana. In 1845 F.~E. Neumann of +\index{Neumann, F.~E.!ref.\ to}% +\index{Plana}% +\index{Poisson!ref.\ to}% +Königsberg developed from the experimental laws of Lenz the +mathematical theory of magneto-electric induction. In 1855 +W.~Thomson predicted by mathematical analysis that the discharge +of a Leyden jar through a linear conductor would in +\index{Leyden jar}% +certain cases consist of a series of decaying oscillations. This +was first established experimentally by Joseph Henry of +\index{Henry}% +Washington. William Thomson worked out the electro-static +induction in submarine cables. The subject of the screening +effect against induction, due to sheets of different metals, was +worked out mathematically by Horace Lamb and also by +\index{Lamb}% +Charles Niven. W.~Weber's chief researches were on electro-dynamics. +\index{Niven}% +\index{Weber, W.~E.!ref.\ to}% +Helmholtz in 1851 gave the mathematical theory +\index{Helmholtz!ref.\ to}% +of the course of induced currents in various cases. \textbf{Gustav +Robert Kirchhoff}\Reference[]{97} (1824--1887) investigated the distribution of +\index{Kirchhoff}% +\index{Kirchhoff!ref.\ to}% +a current over a flat conductor, and also the strength of current +in each branch of a network of linear conductors. + +The entire subject of electro-magnetism was revolutionised +%% -----File: 411.png---Folio 397------- +\index{Conservation!of energy}% +\index{Energy, conservation of}% +by \textbf{James Clerk Maxwell} (1831--1879). He was born near +\index{Maxwell}% +Edinburgh, entered the University of Edinburgh, and became +a pupil of Kelland and Forbes. In 1850 he went to Trinity +\index{Forbes}% +\index{Kelland}% +College, Cambridge, and came out Second Wrangler, E.~Routh +\index{Routh!ref.\ to}% +being Senior Wrangler. Maxwell then became lecturer at +Cambridge, in 1856 professor at Aberdeen, and in 1860 +professor at King's College, London. In 1865 he retired to +private life until 1871, when he became professor of physics +at Cambridge. Maxwell not only translated into mathematical +language the experimental results of Faraday, but established +the electro-magnetic theory of light, since verified experimentally +by Hertz. His first researches thereon were published +in 1864. In 1871 appeared his great \textit{Treatise on Electricity +and Magnetism}. He constructed the electro-magnetic theory +from general equations, which are established upon purely +dynamical principles, and which determine the state of the +electric field. It is a mathematical discussion of the stresses +and strains in a dielectric medium subjected to electro-magnetic +forces. The electro-magnetic theory has received developments +from Lord Rayleigh, J.~J. Thomson, H.~A. Rowland, R.~T. +\index{Rayleigh, Lord!ref.\ to}% +\index{Rowland}% +\index{Thomson, J.~J.!ref.\ to}% +Glazebrook, H.~Helmholtz, L.~Boltzmann, O.~Heaviside, J.~H. +\index{Boltzmann}% +\index{Glazebrook}% +\index{Heaviside}% +\index{Helmholtz}% +Poynting, and others. \textbf{Hermann von~Helmholtz} turned his +\index{Poynting}% +attention to this part of the subject in 1871. He was born +in 1821 at Potsdam, studied at the University of Berlin, and +published in 1847 his pamphlet \textit{Ueber die Erhaltung der Kraft}. +He became teacher of anatomy in the Academy of Art in +Berlin. He was elected professor of physiology at Königsberg +in 1849, at Bonn in 1855, at Heidelberg in 1858. It was +at Heidelberg that he produced his work on \textit{Tonempfindung}. +In 1871 he accepted the chair of physics at the University of +Berlin. From this time on he has been engaged chiefly on +inquiries in electricity and hydrodynamics. Helmholtz aimed +to determine in what direction experiments should be made to +%% -----File: 412.png---Folio 398------- +\index{Electricity|)}% +\index{Helmholtz}% +\index{Thermodynamics|(}% +decide between the theories of W.~Weber, F.~E. Neumann, +\index{Neumann, F.~E.}% +\index{Weber, W.~E.!ref.\ to}% +Riemann, and Clausius, who had attempted to explain electro-dynamic +\index{Clausius!ref.\ to}% +\index{Riemann!ref.\ to}% +phenomena by the assumption of forces acting at a distance +between two portions of the hypothetical electrical fluid,---the +intensity being dependent not only on the distance, but also +on the velocity and acceleration,---and the theory of Faraday +\index{Faraday}% +and Maxwell, which discarded action at a distance and assumed +\index{Maxwell!ref.\ to}% +stresses and strains in the dielectric. His experiments favoured +the British theory. He wrote on abnormal dispersion, and +created analogies between electro-dynamics and hydrodynamics. +Lord Rayleigh compared electro-magnetic problems +\index{Rayleigh, Lord!ref.\ to}% +with their mechanical analogues, gave a dynamical theory of +diffraction, and applied Laplace's coefficients to the theory of +\index{Laplace!ref.\ to}% +radiation. Rowland made some emendations on Stokes' paper +\index{Rowland}% +\index{Stokes!ref.\ to}% +on diffraction and considered the propagation of an arbitrary +electro-magnetic disturbance and spherical waves of light. +Electro-magnetic induction has been investigated mathematically +by Oliver Heaviside, and he showed that in a cable it is +\index{Heaviside}% +an actual benefit. Heaviside and Poynting have reached +\index{Poynting}% +remarkable mathematical results in their interpretation and +development of Maxwell's theory. Most of Heaviside's papers +have been published since 1882; they cover a wide field. + +One part of the theory of capillary attraction, left defective +by Laplace, namely, the action of a solid upon a liquid, and +the mutual action between two liquids, was made dynamically +perfect by Gauss. He stated the rule for angles of contact +\index{Gauss!ref.\ to}% +between liquids and solids. A similar rule for liquids was +established by Ernst Franz Neumann. Chief among recent +workers on the mathematical theory of capillarity are Lord +\index{Capillarity}% +Rayleigh and E.~Mathieu. +\index{Mathieu!ref.\ to}% + +The great principle of the conservation of energy was +\index{Conservation!of energy}% +\index{Energy, conservation of}% +established by \textbf{Robert Mayer} (1814--1878), a physician in +\index{Mayer}% +Heilbronn, and again independently by Colding of Copenhagen, +\index{Colding}% +%% -----File: 413.png---Folio 399------- +\index{Kelvin, Lord!ref.\ to}% +Joule, and Helmholtz. \textbf{James Prescott Joule} (1818--1889) +\index{Joule}% +determined experimentally the mechanical equivalent +of heat. Helmholtz in 1847 applied the conceptions of the +\index{Heat, theory of|(}% +transformation and conservation of energy to the various +branches of physics, and thereby linked together many well-known +phenomena. These labours led to the abandonment +of the corpuscular theory of heat. The mathematical treatment +of thermic problems was demanded by practical considerations. +Thermodynamics grew out of the attempt to +determine mathematically how much work can be gotten out +of a steam engine. \textbf{Sadi-Carnot}, an adherent of the corpuscular +\index{Carnot, Sadi}% +theory, gave the first impulse to this. The principle known +by his name was published in 1824. Though the importance +of his work was emphasised by \textit{B.~P.~E. Clapeyron}, it did not +\index{Clapeyron}% +meet with general recognition until it was brought forward +by William Thomson. The latter pointed out the necessity +\index{Thomson, Sir William!ref.\ to}% +of modifying Carnot's reasoning so as to bring it into accord +with the new theory of heat. William Thomson showed in +1848 that Carnot's principle led to the conception of an +absolute scale of temperature. In 1849 he published ``an +account of Carnot's theory of the motive power of heat, with +numerical results deduced from Regnault's experiments.'' In +February, 1850, \textbf{Rudolph Clausius} (1822--1888), then in Zürich +\index{Clausius}% +(afterwards professor in Bonn), communicated to the Berlin +Academy a paper on the same subject which contains the +Protean second law of thermodynamics. In the same month +\textbf{William John~M. Rankine} (1820--1872), professor of engineering +\index{Rankine}% +and mechanics at Glasgow, read before the Royal Society +of Edinburgh a paper in which he declares the nature of +heat to consist in the rotational motion of molecules, and +arrives at some of the results reached previously by Clausius. +He does not mention the second law of thermodynamics, but +in a subsequent paper he declares that it could be derived +%% -----File: 414.png---Folio 400------- +\index{Kelvin, Lord!ref.\ to}% +from equations contained in his first paper. His proof of +the second law is not free from objections. In March, 1851, +appeared a paper of William Thomson which contained a +\index{Thomson, Sir William!ref.\ to}% +perfectly rigorous proof of the second law. He obtained it +before he had seen the researches of Clausius. The statement +\index{Clausius!ref.\ to|(}% +of this law, as given by Clausius, has been much +criticised, particularly by Rankine, Theodor Wand, P.~G. +\index{Rankine!ref.\ to}% +\index{Wand}% +Tait, and Tolver Preston. Repeated efforts to deduce it from +\index{Preston}% +\index{Tait}% +general mechanical principles have remained fruitless. The +science of thermodynamics was developed with great success +by Thomson, Clausius, and Rankine. As early as 1852 +Thomson discovered the law of the dissipation of energy, +\index{Dissipation of energy}% +deduced at a later period also by Clausius. The latter designated +the non-transformable energy by the name \emph{entropy}, +\index{Entropy}% +and then stated that the entropy of the universe tends +toward a maximum. For entropy Rankine used the term +\emph{thermodynamic function}. Thermodynamic investigations have +been carried on also by G.~Ad.\ Hirn of Colmar, and Helmholtz +\index{Helmholtz!ref.\ to}% +\index{Hirn}% +(monocyclic and polycyclic systems). Valuable graphic +methods for the study of thermodynamic relations were devised +in 1873--1878 by \textbf{J.~Willard Gibbs} of Yale College. +\index{Gibbs}% +Gibbs first gives an account of the advantages of using +various pairs of the five fundamental thermodynamic quantities +for graphical representation, then discusses the entropy-temperature +and entropy-volume diagrams, and the volume-energy-entropy +surface (described in Maxwell's \textit{Theory of +\index{Maxwell!ref.\ to}% +Heat}). Gibbs formulated the energy-entropy criterion of +equilibrium and stability, and expressed it in a form applicable +to complicated problems of dissociation. Important +works on thermodynamics have been prepared by Clausius +in 1875, by R.~Rühlmann in 1875, and by Poincaré in 1892. +\index{Poincaré!ref.\ to}% +\index{Ruhlmann@{Rühlmann}}% + +In the study of the law of dissipation of energy and the +principle of least action, mathematics and metaphysics met on +%% -----File: 415.png---Folio 401------- +\index{Action, least}% +\index{Thermodynamics|)}% +common ground. The doctrine of least action was first propounded +\index{Least action}% +by Maupertius in 1744. Two years later he proclaimed +\index{Maupertius}% +it to be a universal law of nature, and the first +scientific proof of the existence of God. It was weakly supported +by him, violently attacked by König of Leipzig, and +\index{Konig@{König}}% +keenly defended by Euler. Lagrange's conception of the principle +\index{Euler!ref.\ to}% +\index{Lagrange!ref.\ to}% +of least action became the mother of analytic mechanics, +\index{Mechanics!more recent work}% +but his statement of it was inaccurate, as has been remarked +by Josef Bertrand in the third edition of the \textit{Mécanique Analytique}. +\index{Bertrand}% +The form of the principle of least action, as it now +exists, was given by Hamilton, and was extended to electro-dynamics +\index{Hamilton, W.~R.!ref.\ to}% +by F.~E. Neumann, Clausius, Maxwell, and Helmholtz. +\index{Helmholtz!ref.\ to}% +\index{Maxwell!ref.\ to}% +\index{Neumann, F.~E.!ref.\ to}% +To subordinate the principle to all reversible processes, +Helmholtz introduced into it the conception of the ``kinetic +potential.'' In this form the principle has universal validity. + +An offshoot of the mechanical theory of heat is the modern +kinetic theory of gases, developed mathematically by \textit{Clausius}, +\index{Gases!Kinetic theory of|(}% +\index{Kinetic theory of gases|(}% +\textit{Maxwell}, \textit{Ludwig Boltzmann} of Munich, and others. The first +suggestions of a kinetic theory of matter go back as far as the +time of the Greeks. The earliest work to be mentioned here is +that of Daniel Bernoulli, 1738. He attributed to gas-molecules +\index{Bernoulli, Daniel!ref.\ to}% +great velocity, explained the pressure of a gas by molecular +bombardment, and deduced Boyle's law as a consequence of +\index{Boyle's law}% +his assumptions. Over a century later his ideas were taken +up by Joule (in~1846), A.~K. Krönig (in~1856), and Clausius +\index{Joule!ref.\ to}% +\index{Krönig}% +(in~1857). Joule dropped his speculations on this subject +when he began his experimental work on heat. Krönig +\index{Heat, theory of|)}% +explained by the kinetic theory the fact determined experimentally +by Joule that the internal energy of a gas is not +altered by expansion when no external work is done. Clausius +took an important step in supposing that molecules may have +rotary motion, and that atoms in a molecule may move relatively +to each other. He assumed that the force acting +%% -----File: 416.png---Folio 402------- +between molecules is a function of their distances, that temperature +depends solely upon the kinetic energy of molecular +motions, and that the number of molecules which at any +moment are so near to each other that they perceptibly influence +each other is comparatively so small that it may be +neglected. He calculated the average velocities of molecules, +and explained evaporation. Objections to his theory, raised +by Buy's-Ballot and by Jochmann, were satisfactorily answered +\index{Buy's-Ballot}% +\index{Jochmann}% +by Clausius and Maxwell, except in one case where an additional +\index{Maxwell!ref.\ to}% +hypothesis had to be made. Maxwell proposed to himself +the problem to determine the average number of molecules, +the velocities of which lie between given limits. His expression +therefor constitutes the important law of distribution of +velocities named after him. By this law the distribution of +molecules according to their velocities is determined by the +same formula (given in the theory of probability) as the distribution +of empirical observations according to the magnitude +of their errors. The average molecular velocity as deduced +by Maxwell differs from that of Clausius by a constant factor. +\index{Clausius!ref.\ to|)}% +Maxwell's first deduction of this average from his law of distribution +was not rigorous. A sound derivation was given by +O.~E. Meyer in 1866. Maxwell predicted that so long as +\index{Meyer, O.~E.}% +Boyle's law is true, the coefficient of viscosity and the coefficient +of thermal conductivity remain independent of the pressure. +His deduction that the coefficient of viscosity should +be proportional to the square root of the absolute temperature +appeared to be at variance with results obtained from pendulum +experiments. This induced him to alter the very foundation +of his kinetic theory of gases by assuming between +the molecules a repelling force varying inversely as the fifth +power of their distances. The founders of the kinetic theory +had assumed the molecules of a gas to be hard elastic spheres; +but Maxwell, in his second presentation of the theory in 1866, +%% -----File: 417.png---Folio 403------- +\index{Applied mathematics|)}% +\index{Thomson, Sir William!ref.\ to}% +went on the assumption that the molecules behave like centres +of forces. He demonstrated anew the law of distribution +of velocities; but the proof had a flaw in argument, pointed +out by Boltzmann, and recognised by Maxwell, who adopted +\index{Boltzmann}% +\index{Maxwell!ref.\ to}% +a somewhat different form of the distributive function in a +paper of 1879, intended to explain mathematically the effects +observed in Crookes' radiometer. Boltzmann gave a rigorous +\index{Radiometer}% +general proof of Maxwell's law of the distribution of velocities. + +None of the fundamental assumptions in the kinetic theory +of gases leads by the laws of probability to results in very +close agreement with observation. Boltzmann tried to establish +kinetic theories of gases by assuming the forces between +molecules to act according to different laws from those previously +assumed. Clausius, Maxwell, and their predecessors +took the mutual action of molecules in collision as repulsive, +but Boltzmann assumed that they may be attractive. \DPtypo{Experiment}{Experiments} +of Joule and Lord Kelvin seem to support the latter +\index{Joule!ref.\ to}% +\index{Kelvin, Lord!ref.\ to}% +assumption. + +Among the latest researches on the kinetic theory is Lord +\index{Gases!Kinetic theory of|)}% +\index{Kinetic theory of gases|)}% +Kelvin's disproof of a general theorem of Maxwell and Boltzmann, +asserting that the average kinetic energy of two given +portions of a system must be in the ratio of the number of +degrees of freedom of those portions. +%% -----File: 418.png---Folio 404------- + +\clearpage +\phantomsection +\pdfbookmark[-1]{Back Matter}{Back Matter} + +\NChapter[1]{Addenda} + +\footnotesize +\Addendref{14}. The new \textit{Akhmim papyrus}, written in Greek, is probably the +copy of an older papyrus, antedating Heron's works, and is the oldest +extant text-book on practical Greek arithmetic. It contains, besides +arithmetical examples, a table for finding ``unit-fractions,'' identical in +scope with that of Ahmes, and, like Ahmes's, without a clue as to its mode +of construction. See \textsc{Biblioth.\ Math.}, 1893, p.~79--89. The papyrus is +edited by J.~Baillet (\textit{Mémoires publiés par les membres de la mission +archéologique française au Caire}, T.~IX., 1\textsuperscript{r}~\textit{fascicule}, Paris, 1892, p.~1--88). + +\Addendref{39}. Chasles's or Simson's definition of a Porism is preferable to +Proclus's, given in the text. See Gow, p.~217--221. + +\Addendref{114}. Nasir Eddin for the first time elaborated trigonometry independently +of astronomy and to such great perfection that, had his work +been known, Europeans of the 15th~century might have spared their +labours. See \textsc{Biblioth.\ Math.}, 1893, p.~6. + +\Addendref{116}. This law of sines was probably known before Gabir ben +Aflah to Tabit ben Korra and others. See \textsc{Biblioth.\ Math.}, 1893, p.~7. + +\Addendref{125}. Athelard was probably not the first to translate Euclid's +\textit{Elements} from the Arabic. See M.~Cantor's \textsc{Vorlesungen}, Vol.~II., +p.~91,~92. + +\Addendref{240}. G.~Eneström argues that Taylor and not Nicole is the real +inventor of finite differences. See \textsc{Biblioth.\ Math.}, 1893, p.~91. + +\Addendref{250}. An earlier publication in which $3.14159\ldots$ is designated by~$\pi$, +is W.~Jones's \textit{Synopsis palmariorum matheseos}, London, 1706, p.~243, +263~\textit{et~seq.} See \textsc{Biblioth.\ Math.}, 1894, p.~106. + +\Addendref{335}. Before Gauss a theorem on convergence, usually attributed +to Cauchy, was given by Maclaurin (\textit{Fluxions}, §~350). A rule of convergence +was deduced also by Stirling. See \textit{Bull.\ N.~Y. Math.\ Soc.}, Vol.~III., +p.~186. + +\Addendref{358}. The surface of a solid with $p$~holes was considered before +Clifford by Tonelli, and was probably used by Riemann himself. See +\textsc{Math.\ Annalen}, Vol.~45, p.~142. + +\Addendref{361}. As early as 1835, Lobachevsky showed in a memoir the +necessity of distinguishing between continuity and differentiability. See +G.~B. Halsted's transl.\ of A.~Vasiliev's \textit{Address} on Lobachevsky, p.~23. + +\textit{Recent deaths.} Johann Rudolf Wolf, Dec.~6, 1893; Heinrich Hertz, +Jan.~1, 1894; Eugène Catalan, Feb.~14, 1894; Hermann von~Helmholtz, +Sept.~8, 1894; Arthur Cayley, Jan.~26, 1895. +%% -----File: 419.png---Folio 405------- + +\normalsize +\cleardoublepage + +\IndexBookmark +\fancyhead[C]{\textsc{INDEX}} +\printindex + + +% **** Need to replace index text + +%% -----File: 437.png---Folio 423------- + +\Catalog + +\CatalogSection{ALGEBRA.} + +\Entry{ALDIS: A Text-book of Algebra.} By W.~S.~\Au{Aldis}. 12mo. \$\Pr{1.90}. + +\Entry{{BALL} \Rm{(W.~W.~R.):} Elementary Algebra.} 16mo. \$\Pr{1.25}. + +\Entry{CHRYSTAL: Algebra.} An Elementary Text-book for the Higher Classes + of Secondary Schools and Colleges. 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Paper. \Pr{75}~cents each. + +\Entry{MURDOCK: Notes on Electricity and Magnetism.} Designed as a companion +to Silvanus~P. Thompson's ``Elementary Lessons in Electricity +and Magnetism.'' \Au{By J.~B. Murdock}, Lieut.\ U.S.N\@. 18mo. New +Edition. \Pr{60}~cents. + +\Entry{POOLE: The Practical Telephone Handbook.} By \Au{Joseph Poole}. With +227~Illustrations. Small crown 8vo. \$\Pr{1.00}. +%% -----File: 457.png---Folio 443------- + +\Entry{PREECE \Rm{and} STUBBS: A Manual of Telephony.} By \Au{Wm.~Henry +Preece} and \Au{Arthur~J. Stubbs}. \$\Pr{4.50}. + +\Entry{RUSSELL: Electric Light Cables and the Distribution of Electricity.} +By \Au{Stuart~A. Russell}, \Degr{A.M.}, \Degr{I.C.E\@.} With over 100~Illustrations. +12mo. \$2.25. + +\Entry{STEWART \Rm{and} GEE: Practical Physics for Schools and Junior Students +of Colleges.} By \Au{Balfour Stewart}, \Degr{M.A.}, \Degr{LL.D.}, \Degr{F.R.S.}, and +\Au{W.~W. 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Murdock}. \Pr{60}~cents. + +\Entry{WALKER: How to Light a Colliery by Electricity.} 4to. Limp. \Pr{75}~cents. + +\Subentry\textbf{Town Lighting by Electricity.} (\emph{In the Press.}) + +\Entry{WATSON \Rm{and} BURBURY: The Mathematical Theory of Electricity +and Magnetism.} By \Au{H.~W. Watson}, \Degr{D.Sc.}, \Degr{F.R.S.}, and \Au{S.~H. Burbury}, +\Degr{M.A\@.} + +\Subentry\Ti{Vol.~I\@.} Electrostatics. 8vo. \$\Pr{2.75}. + +\settowidth{\TmpLen}{\Ti{Vol.~I\@.}} +\Subentry\makebox[\TmpLen][r]{\Ti{II\@.}} Magnetism and Electrodynamics. 8vo. \$\Pr{2.60}. + + +\CatalogSection{HISTORICAL.} + +\Entry{BALL:} Works by \Au{Walter~W. Rouse Ball}. + +\Subentry\textbf{A Short Account of the History of Mathematics.} 12mo. \$\Pr{2.60}. + +\Subentry\textbf{History of the Study of Mathematics at Cambridge.} 12mo. \$\Pr{1.90}. + +\Entry{BARROW: Mathematical Works.} Edited by \Au{W.~Whewell}. \$\Pr{2.25}. + +\Entry{CAYLEY \Rm{(Arthur):} The Collected Mathematical Papers of.} To be +completed in ten volumes. Vols.~I.--VI. published. \$\Pr{5.50}~each. + +\Entry{GOW: A Short History of Greek Mathematics.} By \Au{J.~Gow}. 8vo. \$\Pr{3.00}. + +\Entry{HEATH: Diophantos of Alexandria.} A Study in the History of Greek +Algebra. By \Au{T.~L. Heath}, \Degr{B.A\@.} 8vo. \$\Pr{2.25}. + +\Entry{KLEIN: Lectures on Mathematics.} \textit{The Evanston Colloquium.} Reported +by \Au{Alexander Ziwet}. \$\Pr{1.50}. + +\Entry{SMITH \Rm{(Henry J. S.):} Mathematical Papers.} With Portrait and Memoir. +2~vols. (\emph{In the Press.}) + +\Entry{WOOLWICH: Mathematical Papers.} Edited by \Au{E.~J. Brooksmith}, \Degr{B.A\@.} +\$\Pr{1.75}. +%% -----File: 458.png---Folio 444------- +%[** PP: Page of special ads; adding hrule] + +\setlength{\Titleskip}{2pt} +\begin{center}\rule{1.5in}{0.5pt}\end{center} + +\begin{center} +{\SizeB \bfseries A SHORT ACCOUNT}\\[\Titleskip] +{\SizeD OF}\\[\Titleskip] +{\SizeA \bfseries\sffamily THE\quad HISTORY\quad OF\quad MATHEMATICS.}\\[\Titleskip] +{\SizeC \Au{By WALTER W. ROUSE BALL},}\\[\Titleskip] +{\SizeE \textsc{Fellow and Tutor of Trinity College, Cambridge.}}\\[\Titleskip] +{\SizeD Second Edition, Revised.\quad \$\Pr{3.25}.} +\end{center} + +\SizeE% +``While technical and exact enough to be of value to the specialist in mathematics as a +handy book of reference, it is so clearly and familiarly written, that it is the best work on this +subject for the general reader that we know of\ldots. From this history, or historical sketch, +the intelligent reader can gain a very complete view of the progress of mathematical science +from its beginnings until its contemporary differentiation into numerous specialties,---each +of them important and difficult enough to detain for a lifetime a brilliant mind,---all of which +are fruitful in their applications to the various phases of modern science and modern +industry.''---\textit{Science}. + +\normalsize +\ifthenelse{\boolean{ForPrinting}}{% + \vfil\clearpage +}{% +\begin{center}\rule{1.5in}{0.5pt}\end{center} +} + +\begin{center} +{\SizeB \bfseries A HISTORY}\\[\Titleskip] +{\SizeD OF}\\[\Titleskip] +\ifthenelse{\boolean{ForPrinting}}{% + \makebox[0pt][c]{\SizeA \bfseries\sffamily THE STUDY OF MATHEMATICS AT CAMBRIDGE.}\\[\Titleskip] +}{% + \makebox[0pt][c]{\SizeB \bfseries\sffamily THE STUDY OF MATHEMATICS AT CAMBRIDGE.}\\[\Titleskip] +} +{\SizeC \Au{By WALTER W. ROUSE BALL},}\\[\Titleskip] +{\SizeD 12mo.\quad \$\Pr{1.90}.} +\end{center} + +\begin{center}\rule{1.5in}{0.5pt}\end{center} + +\begin{center} +{\SizeB \bfseries A SHORT HISTORY}\\[\Titleskip] +{\SizeD OF}\\[\Titleskip] +{\SizeA \bfseries\sffamily GREEK\quad MATHEMATICS.}\\[\Titleskip] +{\SizeC \Au{By JAMES GOW},}\\[\Titleskip] +{\SizeD 8vo.\quad \$\Pr{3.00}.} +\end{center} + +\SizeE +``\ldots Evidently the production of a scholar, and the result of years of laborious research. +Mr.~Gow divides his history into three parts. The first treats of the decimal scale and Egyptian +arithmetic; the second and third parts are concerned with Greek arithmetic and geometry\ldots. +The largest part of Mr.~Gow's history, and that which will probably be the most interesting +to the general mathematical reader, is justly devoted to geometry; for it is in this +department of mathematics that the acuteness of the Greek mind is most conspicuously seen, +and that the continuity of mathematical discovery can be more fully traced\ldots. The interesting +character of the notes is quite a feature of the book, which is in this respect distinguished +from almost all histories of mathematics\ldots. It must be to all students of +mathematics a most welcome and instructive volume.''---\Au{J.~S. Mackay}, in \textit{The Academy}. + +\normalsize +\begin{center}\rule{1.5in}{0.5pt}\end{center} + +\begin{center} +{\SizeA \bfseries\sffamily DIOPHANTOS\quad OF\quad ALEXANDRIA:}\\[\Titleskip] +{\SizeD A STUDY IN THE HISTORY OF}\\[\Titleskip] +{\SizeA \bfseries\sffamily GREEK\quad ALGEBRA.}\\[\Titleskip] +{\SizeC \Au{By T.~S. HEATH}, \Degr{B.A.},}\\[\Titleskip] +{\SizeE \textsc{Scholar of Trinity College, Cambridge.}}\\[\Titleskip] +{\SizeD 8vo.\quad \$\Pr{2.00}.} +\end{center} + +\begin{center}\rule{1.5in}{0.5pt}\end{center} + +\begin{center} +{\Large THE\quad MACMILLAN\quad COMPANY,}\\[\Titleskip] +{\large 66 \ FIFTH \ AVENUE, \ NEW \ YORK.} +\end{center} + + +%%%%%%%%%%%%%%%%%%%%%%%%% GUTENBERG LICENSE %%%%%%%%%%%%%%%%%%%%%%%%%% + +\Licensing + +\begin{PGtext} +End of Project Gutenberg's A History of Mathematics, by Florian Cajori + +*** END OF THIS PROJECT GUTENBERG EBOOK A HISTORY OF MATHEMATICS *** + +***** This file should be named 31061-pdf.pdf or 31061-pdf.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/3/1/0/6/31061/ + +Produced by Andrew D. 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+\draw (0.75in,1in)--(0in,0in)--(1.8in,0in)--(0.75in,1in)--cycle; +\draw (2.25in,0.3in)--(0.3in,0.4in); +\draw (2.25in,0.3in)--(0.075in,0.1in); +\draw (0.75in,1in)--(1.30645in,0in); +\pgftext[at={\pgfpoint{0.75in}{1.05535in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}\footnotesize $A$}}} +\pgftext[at={\pgfpoint{-0.027674in}{0.027674in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}\footnotesize $B$}}} +\pgftext[at={\pgfpoint{1.82767in}{0.027674in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}\footnotesize $C$}}} +\pgftext[at={\pgfpoint{2.27767in}{0.3in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}\footnotesize $D$}}} +\pgftext[at={\pgfpoint{1.49681in}{0.369137in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}\footnotesize $E$}}} +\pgftext[at={\pgfpoint{0.244652in}{0.427674in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}\footnotesize $F$}}} +\pgftext[at={\pgfpoint{0.019652in}{0.127674in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}\footnotesize $G$}}} +\pgftext[at={\pgfpoint{1.16224in}{0.386295in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}\footnotesize $H$}}} +\pgftext[at={\pgfpoint{1.13822in}{0.175183in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\footnotesize $J$}}} +\pgftext[at={\pgfpoint{1.49701in}{0.208175in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\footnotesize $K$}}} +\end{tikzpicture} diff --git a/31061-t/images/sources/069a.xp b/31061-t/images/sources/069a.xp new file mode 100644 index 0000000..fd2702a --- /dev/null +++ b/31061-t/images/sources/069a.xp @@ -0,0 +1,47 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +int main() +{ + picture(P(0,0), P(2.25,1), "2.25 x 1in"); + + begin(); + P ptA(0.75,1), ptB(0,0), ptC(1.8,0), ptD(2.25,0.3); + Segment AB(ptA, ptB), AC(ptA, ptC), BC(ptB, ptC); + + P ptF(AB.midpoint(0.6)), ptG(AB.midpoint(0.9)); + Segment DF(ptD, ptF), DG(ptD, ptG); + + P ptE(AC*DF), ptK(AC*DG); + double t1(norm(ptD - ptE)/norm(ptD - ptF)), t2(t1/(1+t1)); + + P ptH((1-t2)*ptE + t2*ptF); + + Segment AH(ptA,ptH); + + P ptJ(DG*AH), ptL(BC*AH); + + triangle(ptA, ptB, ptC); + line(ptD, ptF); + line(ptD, ptG); + line(ptA, ptL); + + font_size("footnotesize"); + label(ptA, P(0,4), "$A$", t); + label(ptB, P(-2,2), "$B$", l); + label(ptC, P( 2,2), "$C$", r); + + label(ptD, P( 2,0), "$D$", r); + label(ptE, P( 4,2), "$E$", t); + + label(ptF, P(-4,2), "$F$", t); + label(ptG, P(-4,2), "$G$", t); + + label(ptH, P( 4,2), "$H$", t); + label(ptJ, P(-4,-2), "$J$", b); + label(ptK, P(-4,-2), "$K$", b); + + tikz_format(); + end(); +} diff --git a/31061-t/images/sources/105a.eepic b/31061-t/images/sources/105a.eepic new file mode 100644 index 0000000..4e7db48 --- /dev/null +++ b/31061-t/images/sources/105a.eepic @@ -0,0 +1,39 @@ +%% Generated from 105a.xp on Wed Dec 30 21:23:15 EST 2009 by +%% ePiX-1.2.6 +%% +%% Cartesian bounding box: [-1,3] x [-0.5,2] +%% Actual size: 1.6 x 1in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (1.6in,1in); +\draw (0.4in,0.2in)--(1.6in,0.2in)--(1.6in,1in)--(0.4in,1in)--(0.4in,0.2in)--cycle; +\draw (0.8in,0.2in)--(0.8in,1in); +\draw (1.2in,0.2in)--(1.2in,1in); +\draw (0.4in,0.6in)--(1.6in,0.6in); +\draw (0.4in,0.6in)--(0.8in,1in); +\draw (0.4in,0.2in)--(1.2in,1in); +\draw (0.8in,0.2in)--(1.6in,1in); +\draw (1.2in,0.2in)--(1.6in,0.6in); +\pgftext[at={\pgfpoint{0.344652in}{0.4in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}\footnotesize $2$}}} +\pgftext[at={\pgfpoint{0.344652in}{0.8in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}\footnotesize $1$}}} +\pgftext[at={\pgfpoint{0.6in}{1.02767in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}\footnotesize $7$}}} +\pgftext[at={\pgfpoint{1in}{1.02767in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}\footnotesize $3$}}} +\pgftext[at={\pgfpoint{1.4in}{1.02767in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}\footnotesize $5$}}} +\pgftext[at={\pgfpoint{0.68in}{0.32in}}] {\makebox(0,0)[c]{\hbox{\color{rgb_000000}\footnotesize $4$}}} +\pgftext[at={\pgfpoint{1.08in}{0.32in}}] {\makebox(0,0)[c]{\hbox{\color{rgb_000000}\footnotesize $6$}}} +\pgftext[at={\pgfpoint{1.48in}{0.32in}}] {\makebox(0,0)[c]{\hbox{\color{rgb_000000}\footnotesize $0$}}} +\pgftext[at={\pgfpoint{0.68in}{0.72in}}] {\makebox(0,0)[c]{\hbox{\color{rgb_000000}\footnotesize $7$}}} +\pgftext[at={\pgfpoint{1.08in}{0.72in}}] {\makebox(0,0)[c]{\hbox{\color{rgb_000000}\footnotesize $3$}}} +\pgftext[at={\pgfpoint{1.48in}{0.72in}}] {\makebox(0,0)[c]{\hbox{\color{rgb_000000}\footnotesize $5$}}} +\pgftext[at={\pgfpoint{0.52in}{0.48in}}] {\makebox(0,0)[c]{\hbox{\color{rgb_000000}\footnotesize $1$}}} +\pgftext[at={\pgfpoint{1.32in}{0.48in}}] {\makebox(0,0)[c]{\hbox{\color{rgb_000000}\footnotesize $1$}}} +\pgftext[at={\pgfpoint{0.027674in}{0.1in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}\footnotesize $8$}}} +\pgftext[at={\pgfpoint{0.427674in}{0.1in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}\footnotesize $8$}}} +\pgftext[at={\pgfpoint{0.827674in}{0.1in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}\footnotesize $2$}}} +\pgftext[at={\pgfpoint{1.22767in}{0.1in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}\footnotesize $0$}}} +\end{tikzpicture} diff --git a/31061-t/images/sources/105a.xp b/31061-t/images/sources/105a.xp new file mode 100644 index 0000000..501b09f --- /dev/null +++ b/31061-t/images/sources/105a.xp @@ -0,0 +1,54 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +const double dX(0.3); + +void ldigit(const P& loc, const std::string& d) +{ + label(loc+P(1-dX, dX), d); +} + +void udigit(const P& loc, const std::string& d) +{ + label(loc+P(dX, 1-dX), d); +} + +int main() +{ + picture(P(-1,-0.5), P(3,2), "1.6 x 1in"); + + begin(); + grid(P(0,0), P(3,2), 3, 2); + line(P(0,1), P(1,2)); + line(P(0,0), P(2,2)); + line(P(1,0), P(3,2)); + line(P(2,0), P(3,1)); + + font_size("footnotesize"); + label(P(0,0.5), P(-4,0), "$2$", l); + label(P(0,1.5), P(-4,0), "$1$", l); + + label(P(0.5, 2), P(0,2), "$7$", t); + label(P(1.5, 2), P(0,2), "$3$", t); + label(P(2.5, 2), P(0,2), "$5$", t); + + ldigit(P(0,0), "$4$"); + ldigit(P(1,0), "$6$"); + ldigit(P(2,0), "$0$"); + + ldigit(P(0,1), "$7$"); + ldigit(P(1,1), "$3$"); + ldigit(P(2,1), "$5$"); + + udigit(P(0,0), "$1$"); + udigit(P(2,0), "$1$"); + + label(P(-1, -0.25), P(2,0), "$8$", r); + label(P( 0, -0.25), P(2,0), "$8$", r); + label(P( 1, -0.25), P(2,0), "$2$", r); + label(P( 2, -0.25), P(2,0), "$0$", r); + + tikz_format(); + end(); +} diff --git a/31061-t/images/sources/111a.eepic b/31061-t/images/sources/111a.eepic new file mode 100644 index 0000000..fbec8ee --- /dev/null +++ b/31061-t/images/sources/111a.eepic @@ -0,0 +1,25 @@ +%% Generated from 111a.xp on Wed Dec 30 22:01:19 EST 2009 by +%% ePiX-1.2.6 +%% +%% Cartesian bounding box: [0,2.75] x [0,1] +%% Actual size: 2.0625 x 0.75in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (2.0625in,0.75in); +\draw (0in,0in)--(0.75in,0in)--(0.75in,0.75in)--(0in,0.75in)--(0in,0in)--cycle; +\draw (0in,0in)--(0.565385in,0.323077in); +\draw (0.75in,0in)--(0.426923in,0.565385in); +\draw (0.75in,0.75in)--(0.184615in,0.426923in); +\draw (0in,0.75in)--(0.323077in,0.184615in); +\draw (1.03921in,0in)--(1.6904in,0in)--(1.6904in,0.372104in)-- + (1.03921in,0.372104in)--(1.03921in,0in)--cycle; +\draw (1.6904in,0in)--(2.0625in,0in)--(2.0625in,0.651182in)-- + (1.6904in,0.651182in)--(1.6904in,0in)--cycle; +\draw (1.41132in,0.372104in)--(1.6904in,0.372104in)-- + (1.6904in,0.651182in)--(1.41132in,0.651182in)--(1.41132in,0.372104in)--cycle; +\end{tikzpicture} diff --git a/31061-t/images/sources/111a.xp b/31061-t/images/sources/111a.xp new file mode 100644 index 0000000..2043052 --- /dev/null +++ b/31061-t/images/sources/111a.xp @@ -0,0 +1,34 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +double lena(1), lenb(1.75), lenc(hypot(lena, lenb)), th(Atan(lena/lenb)); + +int main() +{ + picture(P(0,0), P(2.75,1), "2.0625 x 0.75in"); + + begin(); + + screen scr1(P(0,0), P(lenc, lenc)); + activate(scr1); + grid(1,1); + line(P(0,0), polar(lenb, th)); + line(P(lenc,0), P(lenc,0) + polar(lenb, M_PI_2+th)); + line(P(lenc,lenc), P(lenc,lenc) - polar(lenb, th)); + line(P(0,lenc), P(0,lenc) - polar(lenb, M_PI_2+th)); + deactivate(scr1); + + screen scr2(P(0,0), P(lena + lenb, lenb)); + activate(scr2); + rect(P(0,0), P(lenb, lena)); + rect(P(lenb,0), P(lena+lenb, lenb)); + rect(P(lena, lena), P(lenb, lenb)); + deactivate(scr2); + + inset(scr1, P(0,0), P(1,1)); + inset(scr2, P(2.75, 0), P(2.75 - (lena + lenb)/lenc, lenb/lenc)); + + tikz_format(); + end(); +} diff --git a/31061-t/images/sources/176a.eepic b/31061-t/images/sources/176a.eepic new file mode 100644 index 0000000..1fe4bc6 --- /dev/null +++ b/31061-t/images/sources/176a.eepic @@ -0,0 +1,28 @@ +%% Generated from 176a.xp on Fri Jan 1 13:42:34 EST 2010 by +%% ePiX-1.2.6 +%% +%% Cartesian bounding box: [0,1.4] x [0,1] +%% Actual size: 3 x 0.375in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (3in,0.375in); +\draw (0in,0in)--(3in,0in); +\draw (0in,0.375in)--(2.14286in,0.375in); +\pgftext[at={\pgfpoint{0in}{0.458022in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}$A$}}} +\pgftext[at={\pgfpoint{0in}{0.375in}}] {\makebox(0,0){\hbox{\color{rgb_000000}$\rule{0.5pt}{6pt}$}}} +\pgftext[at={\pgfpoint{2.14286in}{0.458022in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}$B$}}} +\pgftext[at={\pgfpoint{2.14286in}{0.375in}}] {\makebox(0,0){\hbox{\color{rgb_000000}$\rule{0.5pt}{6pt}$}}} +\pgftext[at={\pgfpoint{0.857143in}{0.458022in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}$C$}}} +\pgftext[at={\pgfpoint{0.857143in}{0.375in}}] {\makebox(0,0){\hbox{\color{rgb_000000}$\rule{0.5pt}{6pt}$}}} +\pgftext[at={\pgfpoint{0in}{-0.083022in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}$D$}}} +\pgftext[at={\pgfpoint{0in}{0in}}] {\makebox(0,0){\hbox{\color{rgb_000000}$\rule{0.5pt}{6pt}$}}} +\pgftext[at={\pgfpoint{3in}{-0.083022in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}$E$}}} +\pgftext[at={\pgfpoint{3in}{0in}}] {\makebox(0,0){\hbox{\color{rgb_000000}$\rule{0.5pt}{6pt}$}}} +\pgftext[at={\pgfpoint{1.09463in}{-0.083022in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}$F$}}} +\pgftext[at={\pgfpoint{1.09463in}{0in}}] {\makebox(0,0){\hbox{\color{rgb_000000}$\rule{0.5pt}{6pt}$}}} +\end{tikzpicture} diff --git a/31061-t/images/sources/176a.xp b/31061-t/images/sources/176a.xp new file mode 100644 index 0000000..b9c7ed8 --- /dev/null +++ b/31061-t/images/sources/176a.xp @@ -0,0 +1,33 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +void tick(const P& loc, const std::string& msg, epix_label_posn T) +{ + double dY(T == b? -6 : 6); + + label(loc, P(0,dY), msg, T); + marker(loc, HTICK); +} + +int main() +{ + picture(P(0,0), P(1.4,1), "3 x 0.375in"); + + begin(); + + const double cx(0.4); + line(P(0,0), P(xmax(),0)); + line(P(0,1), P(1,1)); + + tick(P(0,1), "$A$", t); + tick(P(1,1), "$B$", t); + tick(P(cx,1), "$C$", t); + + tick(P(0,0), "$D$", b); + tick(P(xmax(),0), "$E$", b); + tick(P(-log(1-cx),0), "$F$", b); + + tikz_format(); + end(); +} diff --git a/31061-t/images/sources/212a.eepic b/31061-t/images/sources/212a.eepic new file mode 100644 index 0000000..b0f981a --- /dev/null +++ b/31061-t/images/sources/212a.eepic @@ -0,0 +1,43 @@ +%% Generated from 212a.xp on Wed Dec 30 22:33:53 EST 2009 by +%% ePiX-1.2.6 +%% +%% Cartesian bounding box: [-1,2] x [0,1.5] +%% Actual size: 2.5 x 1.5in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (2.5in,1.5in); +\draw (0.833333in,0in)--(0.834375in,0.0353553in)-- + (0.8375in,0.0707107in)--(0.842708in,0.106066in)-- + (0.85in,0.141421in)--(0.859375in,0.176777in)-- + (0.870833in,0.212132in)--(0.884375in,0.247487in)-- + (0.9in,0.282843in)--(0.917708in,0.318198in)-- + (0.9375in,0.353553in)--(0.959375in,0.388909in)-- + (0.983333in,0.424264in)--(1.00938in,0.459619in)-- + (1.0375in,0.494975in)--(1.06771in,0.53033in)--(1.1in,0.565685in)-- + (1.13438in,0.601041in)--(1.17083in,0.636396in)-- + (1.20938in,0.671751in)--(1.25in,0.707107in)-- + (1.29271in,0.742462in)--(1.3375in,0.777817in)-- + (1.38437in,0.813173in)--(1.43333in,0.848528in)-- + (1.48438in,0.883883in)--(1.5375in,0.919239in)-- + (1.59271in,0.954594in)--(1.65in,0.989949in)--(1.70938in,1.0253in)-- + (1.77083in,1.06066in)--(1.83438in,1.09602in)--(1.9in,1.13137in)-- + (1.96771in,1.16673in)--(2.0375in,1.20208in)-- + (2.10938in,1.23744in)--(2.18333in,1.27279in)-- + (2.25938in,1.30815in)--(2.3375in,1.3435in)--(2.41771in,1.37886in)--(2.5in,1.41421in); +\draw (1.66667in,0in)--(1.66667in,1in); +\draw (2.03333in,0in)--(2.03333in,1.2in); +\draw (2.03333in,1in)--(1.66667in,1in); +\draw (0in,0in)--(2.5in,0in); +\pgftext[at={\pgfpoint{0in}{-0.027674in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\footnotesize $T$}}} +\pgftext[at={\pgfpoint{1.66667in}{-0.027674in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\footnotesize $P$}}} +\pgftext[at={\pgfpoint{2.03333in}{-0.027674in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\footnotesize $P'$}}} +\pgftext[at={\pgfpoint{1.6136in}{1.03184in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}\footnotesize $B$}}} +\pgftext[at={\pgfpoint{1.98027in}{1.23184in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}\footnotesize $B'$}}} +\pgftext[at={\pgfpoint{2.06101in}{1in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}\footnotesize $A$}}} +\draw (0in,0in)--(2.5in,1.5in); +\end{tikzpicture} diff --git a/31061-t/images/sources/212a.xp b/31061-t/images/sources/212a.xp new file mode 100644 index 0000000..89b30d8 --- /dev/null +++ b/31061-t/images/sources/212a.xp @@ -0,0 +1,41 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +P f(double y) +{ + return P(y*y, y); +} + +int main() +{ + picture(P(-1,0), P(2,1.5), "2.5 x 1.5in"); + + begin(); + + double p1(1), p2(1.2); + P ptB1(f(p1)), ptB2(f(p2)), ptA(ptB2.x1(), ptB1.x2()), pT(-p1,0); + + plot(f, 0, sqrt(2), 40); + + line(P(p1*p1, 0), f(p1)); + line(P(p2*p2, 0), f(p2)); + line(ptA, ptB1); + line(xmin(), xmax()); + font_size("footnotesize"); + + label(pT, P(0,-2), "$T$", b); + label(p1*p1, P(0,-2), "$P$", b); + label(p2*p2, P(0,-2), "$P'$", b); + + label(ptB1, P(-4,2), "$B$", t); + label(ptB2, P(-4,2), "$B'$", t); + + label(ptA, P(2,0), "$A$", r); + + set_crop(); + Line(pT, ptB1); + + tikz_format(); + end(); +} diff --git a/31061-t/images/sources/224a.eepic b/31061-t/images/sources/224a.eepic new file mode 100644 index 0000000..f4df22d --- /dev/null +++ b/31061-t/images/sources/224a.eepic @@ -0,0 +1,51 @@ +%% Generated from 224a.xp on Sat Jan 2 17:23:16 EST 2010 by +%% ePiX-1.2.6 +%% +%% Cartesian bounding box: [-0.75,2] x [0,1.75] +%% Actual size: 3.75 x 2.25in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (3.75in,2.25in); +\draw (1.02273in,0in)--(1.0336in,0.0408821in)-- + (1.04467in,0.0817641in)--(1.05613in,0.122646in)-- + (1.06819in,0.163528in)--(1.08104in,0.20441in)-- + (1.09487in,0.245292in)--(1.10988in,0.286174in)-- + (1.12628in,0.327057in)--(1.14426in,0.367939in)-- + (1.16401in,0.408821in)--(1.18573in,0.449703in)-- + (1.20962in,0.490585in)--(1.23588in,0.531467in)-- + (1.26471in,0.572349in)--(1.29629in,0.613231in)-- + (1.33084in,0.654113in)--(1.36854in,0.694995in)-- + (1.4096in,0.735877in)--(1.45421in,0.776759in)-- + (1.50256in,0.817641in)--(1.55486in,0.858523in)-- + (1.61131in,0.899405in)--(1.67209in,0.940287in)-- + (1.73741in,0.98117in)--(1.80747in,1.02205in)-- + (1.88246in,1.06293in)--(1.96257in,1.10382in)-- + (2.04802in,1.1447in)--(2.13898in,1.18558in)-- + (2.23567in,1.22646in)--(2.33828in,1.26734in)--(2.447in,1.30823in)-- + (2.56203in,1.34911in)--(2.68358in,1.38999in)-- + (2.81183in,1.43087in)--(2.94699in,1.47175in)-- + (3.08925in,1.51264in)--(3.23881in,1.55352in)-- + (3.39586in,1.5944in)--(3.56061in,1.63528in); +\draw (1.81909in,0in)--(1.81909in,1.02857in); +\draw (3.03027in,0in)--(3.03027in,1.49594in); +\draw (3.03027in,1.02857in)--(1.81909in,1.02857in); +\draw (1.81909in,1.02857in)--(3.03027in,1.49594in); +\draw (-0.0245455in,0in)--(3.03027in,0in); +\draw (-0.0245455in,0in)--(3.33576in,1.87472in); +\pgftext[at={\pgfpoint{-0.0245455in}{-0.055348in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}$V$}}} +\pgftext[at={\pgfpoint{1.81909in}{-0.055348in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}$B$}}} +\pgftext[at={\pgfpoint{3.08562in}{0in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}$b$}}} +\pgftext[at={\pgfpoint{3.08562in}{1.02857in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}$E$}}} +\pgftext[at={\pgfpoint{1.76356in}{1.04166in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}$C$}}} +\pgftext[at={\pgfpoint{3.08809in}{1.59134in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}$c$}}} +\pgftext[at={\pgfpoint{3.36343in}{1.87472in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}$H$}}} +\pgftext[at={\pgfpoint{3.56061in}{1.66296in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}$K$}}} +\pgftext[at={\pgfpoint{1.02273in}{-0.055348in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}$A$}}} +\pgftext[at={\pgfpoint{2.98804in}{1.73084in}}] {\makebox(0,0)[br]{\hbox{\color{rgb_000000}$T$}}} +\draw (3.03027in,0in)--(3.03027in,1.87472in); +\end{tikzpicture} diff --git a/31061-t/images/sources/224a.xp b/31061-t/images/sources/224a.xp new file mode 100644 index 0000000..bf7a5cf --- /dev/null +++ b/31061-t/images/sources/224a.xp @@ -0,0 +1,77 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +const double mult(0.75); + +double f(double y) +{ + return y*(0.25 + mult*y*y); +} + +P F(double y) +{ + return P(f(y), y); +} + +double slope(double y) +{ + return 0.25 + 3*mult*y*y; +} + +void ray(const P& tail, const P& head, double t) +{ + line(tail, tail + (1+t)*(head - tail)); +} + +const double XMAX(1.5); +double g(double y) { return XMAX; } + +const double YMAX(newton(f, g, 1)); + +int main() +{ + picture(P(-0.75,0), P(2,1.75), "3.75 x 2.25in"); + + begin(); + + double p1(0.8), p2(0.95*sqrt(XMAX)), dY(0.1); + + P ptC(F(p1)), ptc(F(p2)), ptE(ptc.x1(), ptC.x2()); + P ptB(f(p1), 0), ptb(f(p2), 0); + + P ptV(ptC - ptC.x2()*P(slope(p1), 1)), ptK(F(YMAX+dY)); + + plot(F, 0, YMAX+dY, 40); + + line(ptB, F(p1)); + line(ptb, F(p2)); + line(ptE, ptC); + line(ptC, ptc); + + Segment bc(ptb, ptc), VC(ptV, ptC); + + P ptT(bc*VC); + line(ptV, ptb); + ray(ptV, ptT, 0.1); + + //font_size("footnotesize"); + + label(ptV, P(0,-4), "$V$", b); + label(ptB, P(0,-4), "$B$", b); + label(ptb, P(4,0), "$b$", r); + label(ptE, P(4,0), "$E$", r); + + label(ptC, P(-4,1), "$C$", t); + label(ptc, P(4,7), "$c$", r); + + label(ptV + 1.1*(ptT - ptV), P(2,0), "$H$", r); + label(ptK, P(0,2), "$K$", t); + + label(F(0), P(0,-4), "$A$", b); + label(ptT, P(-3,2), "$T$", tl); + ray(ptb, ptT, 0.1); + + tikz_format(); + end(); +} diff --git a/31061-t/images/subtract.pdf b/31061-t/images/subtract.pdf new file mode 100644 index 0000000..c5514c3 --- /dev/null +++ b/31061-t/images/subtract.pdf @@ -0,0 +1,80 @@ +%PDF-1.4 +%Çì¢ +5 0 obj +<</Length 6 0 R/Filter /FlateDecode>> +stream +xœMRKn AÜÏ)8á}Œä +#Å+/"ß_r1²Ÿ£^´ ›¢ªà +Éž¯û~¿~ýizû¸„Þ.óJQa7z¿t*9ƒbÔXÔ#Ù…žä)RñÃeäQ¸‹îkNñqruá.ªTns²Žfm@{Yùp& +, Hø1,xÉa“F „¢tÃ?%s nEÁÑbé!Ó.®!5j¡`’-ÀqŒë€‚Ær»/ÍÕ£äˆtòp …Íá“@*{â•€V¾b"ª¹ÁÌE¡¸ÏÖšTì£kùÒ]ô”ývy“¦Ã›¦;à7€ËóŠÑ@D–ÙwÆTÑ0áæivÄð‚ …añx¤Ã6o+‡G
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You may copy it, give it away or % +% re-use it under the terms of the Project Gutenberg License included % +% with this eBook or online at www.gutenberg.org % +% % +% % +% Title: A History of Mathematics % +% % +% Author: Florian Cajori % +% % +% Release Date: January 24, 2010 [EBook #31061] % +% % +% Language: English % +% % +% Character set encoding: ISO-8859-1 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK A HISTORY OF MATHEMATICS *** % +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{31061} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Standard DP encoding. Required. %% +%% %% +%% ifthen: Logical conditionals. Required. %% +%% calc: Infix arithmetic for length calculations. Required. %% +%% %% +%% amsmath: AMS mathematics enhancements. Required. %% +%% amssymb: Additional mathematical symbols. Required. %% +%% %% +%% alltt: Fixed-width font environment. Required. %% +%% array: Enhanced tabular features. Required. %% +%% %% +%% yfonts: Gothic font on title page. Optional. %% +%% %% +%% multicol: Multi-column environment for index. Required. %% +%% makeidx: Indexing capabilities. Required. %% +%% %% +%% fancyhdr: Enhanced running headers and footers. Required. %% +%% %% +%% graphicx: Standard interface for graphics inclusion. Required. %% +%% wrapfig: Illustrations surrounded by text. Required. %% +%% %% +%% geometry: Enhanced page layout package. Required. %% +%% hyperref: Hypertext embellishments for pdf output. Required. %% +%% %% +%% %% +%% Producer's Comments: %% +%% %% +%% British spelling is retained. Minor changes to spelling and %% +%% punctuation, and items that may seem questionable to future %% +%% maintainers of this file, are of three types: %% +%% %% +%% \DPtypo{}{}: Denotes a relatively clear typographical mistake. %% +%% %% +%% \Change{}{}: Denotes a change made for consistency, or a less %% +%% clear instance of possible error. For example, Cajori writes %% +%% bi-quadratic equation and biquadratic residue consistently. %% +%% Since the former are indexed as Biquadratic equation, their %% +%% spelling has been \Change{}{}-ed. Instances can be easily %% +%% reverted, see the boolean flag KeepOrigSpelling below. %% +%% %% +%% \DPnote{** PP: <note>}: Remarks signifying a possible point of %% +%% typographical question, placed immediately after the text. %% +%% %% +%% For example, on occasions where a word is hyphenated across a %% +%% line in the original and the word occurs only once, comparison %% +%% with similar words was made to decide whether to hyphenate. %% +%% Such instances are \DPnote{}-ed. %% +%% %% +%% Where possible, hieroglyphs were reproduced from the original %% +%% page scans; see the detailed note in this file at 027.png. %% +%% %% +%% Thw wrapfigures are manually located mid-paragraph, with line %% +%% breaks adjusted for a 4.25 inch text width. %% +%% %% +%% %% +%% Compilation Flags: %% +%% %% +%% The following behaviors may be controlled by boolean flags. %% +%% %% +%% KeepOrigSpelling (false by default): See above. %% +%% %% +%% ForPrinting (false by default): %% +%% Compile a print-optimized PDF file. Set to true for print- %% +%% optimized file (asymmetric layout, reference markers as %% +%% superscripts, matching the original). %% +%% %% +%% %% +%% Things to Check: %% +%% %% +%% %% +%% Spellcheck: .................................. yes %% +%% Smoothreading pool: ......................... xx %% +%% %% +%% lacheck: ..................................... yes %% +%% Numerous false positives %% +%% %% +%% PDF pages: 556 (if ForPrinting set to false) %% +%% PDF page size: 4.5 x 7 inches (if ForPrinting set to false) %% +%% PDF bookmarks: created, point to ToC entries %% +%% PDF document info: filled in %% +%% Images: 33 pdf graphics %% +%% %% +%% Summary of log file: %% +%% * One overfull hbox (1.91pt too wide). %% +%% * One underfull hbox. %% +%% %% +%% %% +%% Compile History: %% +%% %% +%% January, 2010: adhere (Andrew D. Hwang) %% +%% texlive2007, GNU/Linux %% +%% %% +%% Command block: %% +%% %% +%% pdflatex x3 (Run pdflatex three times) %% +%% makeindex %% +%% pdflatex %% +%% %% +%% %% +%% January 2010: pglatex. %% +%% Compile this project with: %% +%% pdflatex 31061-t.tex ..... THREE times %% +%% makeindex 31061-t.idx %% +%% pdflatex 31061-t.tex %% +%% %% +%% pdfTeXk, Version 3.141592-1.40.3 (Web2C 7.5.6) %% +%% %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\listfiles +\documentclass[12pt]{book}[2005/09/16] + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PACKAGES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\usepackage[latin1]{inputenc}[2006/05/05] %% DP standard encoding +\usepackage[french,german,italian,latin,greek,english]{babel}[2005/11/23] + +\usepackage{ifthen}[2001/05/26] %% Logical conditionals +\usepackage{calc}[2005/08/06] %% Infix arithmetic + +\usepackage{amsmath}[2000/07/18] %% Displayed equations +\usepackage{amssymb}[2002/01/22] %% and additional symbols + +\usepackage{alltt}[1997/06/16] %% boilerplate, credits, license + +\usepackage{array}[2005/08/23] %% extended array/tabular features + +% Define \textgoth; document subsequently defines \Mygoth +\IfFileExists{yfonts.sty}% +{\usepackage{yfonts}[2003/01/08]} % fraktur font (titlepage only) +{\providecommand{\textgoth}[1]{\textbf{##1}}} % fallback if no yfonts + + %% extended footnote capabilities + +\usepackage{indentfirst}[1995/11/23] + +\usepackage{multicol}[2006/05/18] +\usepackage{makeidx}[2000/03/29] + +\usepackage{graphicx}[1999/02/16]%% For diagrams +\usepackage{wrapfig}[2003/01/31] %% and wrapping text around them + +% for running heads; no package date available +\usepackage{fancyhdr} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%% Set up PRINTING (default) or SCREEN VIEWING %%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% KeepOrigSpelling = false (default) Make spelling consistent +\newboolean{KeepOrigSpelling} +\setboolean{KeepOrigSpelling}{false} + +% ForPrinting=true false (default) +% Letterpaper Cropped pages +% Asymmetric margins Symmetric margins +% Black hyperlinks Blue hyperlinks +\newboolean{ForPrinting} + +%% UNCOMMENT the next line for a PRINT-OPTIMIZED VERSION of the text %% +%\setboolean{ForPrinting}{true} + +%% Initialize values to ForPrinting=false +\newcommand{\HalfTitleSize}{\Large} +\newcommand{\Margins}{hmarginratio=1:1} % Symmetric margins +\newcommand{\HLinkColor}{blue} % Hyperlink color +\newcommand{\PDFPageLayout}{SinglePage} +\newcommand{\TransNote}{Transcriber's Note} +\newcommand{\TransNoteCommon} +{ + Figures may have been moved with respect to the surrounding text. + Minor typographical corrections and presentational changes have + been made without comment. +} + +\newcommand{\TransNoteText} +{ + \TransNoteCommon + + This PDF file is formatted for screen viewing, but may be easily + formatted for printing. Please consult the preamble of the \LaTeX\ + source file for instructions. +} + +%% Re-set if ForPrinting=true +\ifthenelse{\boolean{ForPrinting}}{% + \renewcommand{\HalfTitleSize}{\LARGE} + \renewcommand{\Margins}{hmarginratio=2:3} % Asymmetric margins + \renewcommand{\HLinkColor}{black} % Hyperlink color + \renewcommand{\PDFPageLayout}{TwoPageRight} + \renewcommand{\TransNote}{Transcriber's Note} + \renewcommand{\TransNoteText}{% + \TransNoteCommon + + This PDF file is formatted for printing, but may be easily formatted + for screen viewing. Please see the preamble of the \LaTeX\ source + file for instructions. + } +}{} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%% End of PRINTING/SCREEN VIEWING code; back to packages %%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%% Set page dimensions +% If text block size is changed, illustrations must be relocated, +% so print and screen versions use the same text block size. +\ifthenelse{\boolean{ForPrinting}}{% + \setlength{\paperwidth}{8.5in} + \setlength{\paperheight}{11in} + \usepackage[body={4.25in,6.75in},\Margins]{geometry}[2002/07/08] +}{% + \setlength{\paperwidth}{4.5in} + \setlength{\paperheight}{8in} + \usepackage[body={4.25in,6.75in},\Margins,includeheadfoot]{geometry}[2002/07/08] +} + +\raggedbottom + +\providecommand{\ebook}{00000} % Overridden during white-washing +\usepackage[pdftex, + hyperfootnotes=false, + pdftitle={The Project Gutenberg eBook \#\ebook: A History of Mathematics}, + pdfauthor={Florian Cajori}, + pdfkeywords={Andrew D. Hwang, Peter Vachuska, Carl Hudkins, + Project Gutenberg Online Distributed Proofreading Team, + The Internet Archive/American Libraries}, + pdfstartview=Fit, % default value + pdfstartpage=1, % default value + pdfpagemode=UseNone, % default value + bookmarks=true, % default value + linktocpage=false, % default value + pdfpagelayout=\PDFPageLayout, + pdfdisplaydoctitle, + pdfpagelabels=true, + bookmarksopen=true, + bookmarksopenlevel=1, + colorlinks=true, + linkcolor=\HLinkColor]{hyperref}[2007/02/07] + +%%%% Re-crop screen-formatted version, omit blank verso pages %%%% +\ifthenelse{\boolean{ForPrinting}} + {} +% {\hypersetup{pdfpagescrop = 100 90 512 780} + {\hypersetup{pdfpagescrop = 0 0 324 576} + % If ForPrinting=false, don't skip to recto + \renewcommand{\cleardoublepage}{\clearpage} +} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%% COMMANDS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%%%% Fixed-width environment to format PG boilerplate %%%% +% 8pt more-or-less evenly fits 72 chars into 4.25in +\newenvironment{PGtext}{% +\begin{alltt} +\fontsize{8}{9.6}\ttfamily\selectfont}% +{\end{alltt}} + + +%%%% Illustrations and decorations %%%% +\newcommand{\Graphic}[2]{\includegraphics[width=#1]{./images/#2.pdf}} + +\newcommand{\Glyph}[1] + {\smash[b]{\raisebox{-2pt}{\includegraphics[height=12pt]{./images/#1.pdf}}}} + +\newcommand{\TallGlyph}[1] + {\raisebox{-8pt}{\includegraphics[height=24pt]{./images/#1.pdf}}} + +% Thought breaks +\newsavebox{\tbreakglyph} +\savebox{\tbreakglyph}[0.875in]{\hbox{\Graphic{0.75in}{tbreak1}}} + +\newsavebox{\fbreakglyph} % fancy break +\savebox{\fbreakglyph}[1.5in]{\hbox{\Graphic{1.25in}{tbreak2}}} + +\newcommand{\fancybreak}[1]{% + \ifthenelse{\equal{#1}{}}{% + \usebox{\tbreakglyph} + }{% + \usebox{\fbreakglyph} + }% +} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% SECTIONING %%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% Each chapter starts unindented, with smallcaps +\newcommand{\First}[1]{\textsc{#1}} + +\newcommand{\Heading}{\centering\normalfont\normalsize} + +\newcommand{\ChapRule}[1]{\fancybreak{#1}} + +% Running heads +\newcommand{\SetRunningHeads}[1]{% + \fancyhf{} + \fancyhead[CE]{\small A HISTORY OF MATHEMATICS.} + \fancyhead[CO]{\small \MakeUppercase{#1}.} + + \ifthenelse{\boolean{ForPrinting}} + {\fancyhead[RO,LE]{\thepage}} + {\fancyhead[R]{\thepage}} +} + +\newcommand{\Chapter}[2][]{% + \cleardoublepage + \phantomsection + \section*{\centering\Heading\large \MakeUppercase{#2}.} + \begin{center}\ChapRule{#1}\end{center} + + \ToCPageMark + \addcontentsline{toc}{chapter}{\texorpdfstring{\protect\MakeUppercase{#2}}{#2}} + \SetRunningHeads{#2} + \thispagestyle{plain} +} + +% Special handling for Introduction +\newcommand{\IntroChapter}[2]{% + \phantomsection % Already cleared to recto + + \section*{\centering\Heading\LARGE \MakeUppercase{#1}.} + \medskip + \begin{center}\ChapRule{2}\end{center} + \vspace*{-2\Titleskip} + \subsection*{\centering\Heading\large #2.} + + % Set up ToC anchor to detect when a new page is issued + \ToCPageMark + \addtocontents{toc}% + {\protect\renewcommand{\protect\ToCAnchor}{\protect\pageref{toc:1}}} + + % ToC entry proper + \addcontentsline{toc}{chapter}{\texorpdfstring{\protect\MakeUppercase{#2}}{#2}} + + \SetRunningHeads{#2} + \thispagestyle{plain} + + % Set wider line spacing document-wide + \linespread{\BLS}\selectfont% +} + +% PDF bookmark, but no ToC entry +\newcommand{\NChapter}[2][\BLS]{% + \cleardoublepage + \phantomsection + \pdfbookmark[0]{#2}{#2} + + \section*{\centering\Heading\large \MakeUppercase{#2}.} + \ifthenelse{\not\equal{#2}{Addenda}}{% + \begin{center}\ChapRule{}\end{center}% + }{}% + + \linespread{#1}\selectfont% + + \SetRunningHeads{#2} + \thispagestyle{plain} +} + + +\newcommand{\Section}[1]{% + \phantomsection + \subsection*{\Heading\MakeUppercase{#1}.} + + \ToCPageMark + \addcontentsline{toc}{section}{\texorpdfstring{{\scshape#1}}{#1}} + \SetRunningHeads{#1} +} + +\newcommand{\SpecialSection}[1]{% + \phantomsection + \subsubsection*{\Heading\footnotesize\MakeUppercase{#1}.} + + \ToCPageMark + \addcontentsline{toc}{section}{\texorpdfstring{\quad{\itshape#1}}{#1}} +} + +\newcommand{\Subsection}[1]{% + \phantomsection + \subsection*{\Heading\textit{#1}.} + + \ToCPageMark + \addcontentsline{toc}{subsection}{#1} +} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%% CROSS-REFERENCES %%%%%%%%%%%%%%%%%%%%%%%%% +% Anchors +\newcommand{\Addendlabel}[1] + {\phantomsection\label{addend:#1}} + +\newcommand{\Tag}[1]% + {\tag{#1}\phantomsection\label{eqn:#1}} + +% Provides its own trailing space; following text must abut braces. +\newcommand{\BookRef}[1]{% + \settowidth{\TmpLen}{100. }% + \smallskip\par\phantomsection\label{book:#1}% + \noindent\hangindent3.5em\makebox[\TmpLen][r]{#1. }} + +% and links +\newcommand{\Addendref}[1] + {\hyperref[addend:#1]{\textsc{Page~\pageref{addend:#1}}}} + +% References; use empty optional argument to suppress end-of-sentence space +% In screen version use [xx] instead of superscript +\newcommand{\Reference}[2][.]{% + \ifthenelse{\boolean{ForPrinting}}{% + \hyperref[book:#2]{\textsuperscript{#2}}% + }{\,% Else: not ForPrinting, add thin space + \hyperref[book:#2]{\texorpdfstring{[#2]}{#2}}% + }%** Add fake end of sentence space by default + \ifthenelse{\equal{#1}{.}}{\ }{#1}% +} + +% Only needed once, for double reference +\newcommand{\BiReference}[2]{% + \ifthenelse{\boolean{ForPrinting}}{% + \hyperref[book:#1]{\textsuperscript{#1}}\textsuperscript{,}% + \hyperref[book:#2]{\textsuperscript{#2}}% + }{\,% Else: not ForPrinting, add thin space + \hyperref[book:#1]{\texorpdfstring{[#1}{#1}}, % + \hyperref[book:#2]{\texorpdfstring{#2]}{#2}}% + }\ %** Add fake end-of-sentence space +} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%% TABLE OF CONTENTS %%%%%%%%%%%%%%%%%%%%%%%% +\newcommand{\TableofContents}{% + \linespread{1}% + \small + \tableofcontents +} + +\AtBeginDocument{% + \renewcommand{\contentsname}{% + \thispagestyle{plain}% + \fancyhead[C]{\small TABLE OF CONTENTS.} + \phantomsection + \pdfbookmark[0]{Contents}{Table of Contents} + \vspace*{-72pt} + \centering\normalfont\large\scshape TABLE OF CONTENTS\\[12pt] + \fancybreak{}\\[-36pt] + } +} + +% The table of contents may be set across two pages; we'll ensure the +% word ``page'' is printed at the top of the page number column, +% centered on the first page (to match the scan) and right-justified +% on subsequent pages (better visual appearance). +% +% Each sectional unit puts a \ToCLine into the toc file. The IntroChapter +% command defines a \ToCAnchor macro, which expands to the current +% page of the table of contents. +% +% Subsequent units put down a label and get the \pageref. If this has +% changed, the word ``page'' is written at the right margin, and the +% \ToCAnchor command is updated. +\newcommand{\ToCLine}[1]{% + \label{toc:#1}% + \ifthenelse{\not\equal{\pageref{toc:#1}}{\ToCAnchor}}{% + \renewcommand{\ToCAnchor}{\pageref{toc:#1}}% + % And write Page + \null\hfill\mbox{\footnotesize\scshape Page}% + }{}% Else do nothing +} + +% Null defintion; will be redefined usefully in the toc file +\newcommand{\ToCAnchor}{} + +\newcounter{ToCUnit} +\newcommand{\ToCPageMark}{% + \stepcounter{ToCUnit}% + \addtocontents{toc}{\protect\filbreak}% + \addtocontents{toc}{\protect\ToCLine{\theToCUnit}}% +} + +% Chapters in toc with dotted line to page number +\makeatletter +\renewcommand\l@chapter{\@dottedtocline{0}{0em}{3.5em}} +\makeatother + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% INDEX %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\makeatletter +\renewcommand{\@idxitem}{\par\hangindent 24\p@\global\let\idxbrk\nobreak} +\renewcommand{\subitem}{\idxbrk\@idxitem \hspace*{12\p@}\let\idxbrk\relax} +\renewcommand{\indexspace}{\par\penalty-3000 \vskip 10pt plus5pt minus3pt\relax} + +\renewenvironment{theindex}{% + \setlength\columnseprule{0.5pt}% + \setlength\columnsep{12pt}% + \begin{multicols}{2}[\begin{center}\Large INDEX\\\fancybreak{}\end{center}]% + \footnotesize% + \setlength\parindent{0pt}% + \setlength\parskip{0pt plus 0.3pt}% + \thispagestyle{plain}% + \let\item\@idxitem\raggedright } + {\end{multicols}\clearpage\normalsize\fancyhead{}\cleardoublepage} +\makeatother +\newcommand\IndexBookmark{\pdfbookmark[0]{Index}{Index}} + +% Miscellaneous extra formatting for individual entries +\newcommand{\etseq}[1]{\hyperpage{#1} \protect\textit{et~seq.}} +\renewcommand{\see}[2]{\textit{See} #1} + +\makeindex + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LICENSING %%%%%%%%%%%%%%%%%%%%%%%%%%%% +\newcommand{\Licensing}{% + % Clear heading in previous blank verso page, if any + \clearpage + \fancyhead{} + \renewcommand{\headrulewidth}{0pt} + + % And set up running heads + \cleardoublepage + \renewcommand{\headrulewidth}{0.5pt} + + \phantomsection + \pdfbookmark[0]{PG License}{Project Gutenberg License} + \ifthenelse{\boolean{ForPrinting}}{% + \fancyhead[RO,LE]{\thepage} + }{% + \fancyhead[R]{\thepage}% + } + \fancyhead[C]{\textsc{LICENSE}} +} + +%%%%%%%%%%%%%%%%%%%%%%% GLOBAL STYLE PARAMETERS %%%%%%%%%%%%%%%%%%%%%% +\hyphenation{Anax-a-go-ras Arch-i-medes evan-es-cent fa-vour-ite know-ledge man-u-scripts Ni-co-medes Pythag-o-ras Pythag-o-re-an Pythag-o-re-ans quad-ra-tic trig-ono-metry} + +\DeclareMathSizes{12}{11}{9}{7} + +\renewcommand{\headrulewidth}{0pt} +\setlength{\parindent}{1em} + +% Loosen up the spacing +\setlength{\parsep}{0.5ex plus 0.25ex minus 0.5ex} +\setlength{\parskip}{0.375ex plus .625ex minus 0.125ex} +\setlength{\partopsep}{0.25ex plus .75ex minus 0.125ex} + +% Issue this just before the Preface +\newcommand{\stretchyspace}{\spaceskip0.25em plus 0.375em minus 0.125em} + +% Set this at the Introduction +\newcommand{\BLS}{1.2} %Default \linespread + +\newlength{\Titleskip} + +%%%%%%%%%%%%%%%%%%%% MISCELLANEOUS FORMATTING %%%%%%%%%%%%%%%%%%%%%%%% + +% Table Entry, fits to \linewidth +\newcommand{\TE}[1]{% + \makebox[\TmpLen][c]{#1}% +} + +% Crossed-out digits on 137.png +\newcommand{\cancel}[1]{% + \makebox[\TmpLen][c]{\ensuremath{#1\llap{$/$}}}% +} + +\newcommand{\Neg}{\phantom{-}} +% Fraction strut, for use in numerator of inline \dfrac, in +% combination with \smash, when \dfracs appear on consecutive lines +\newcommand{\Fstr}[1][8pt]{\rule{0pt}{#1}} +\newcommand{\Eta}{H} + +% Text markings +\newcommand{\DPnote}[1]{} +\newcommand{\DPtypo}[2]{#2} + +% To effect changes made solely for consistency +\newcommand{\Change}[2]{% + \ifthenelse{\boolean{KeepOrigSpelling}}{#1}{#2}% +} + +\DeclareMathOperator{\cn}{\textit{cn}} +\DeclareMathOperator{\dn}{\textit{dn}} +\DeclareMathOperator{\sn}{\textit{sn}} + +\DeclareMathOperator{\omn}{omn.} +\DeclareMathOperator{\versin}{ver\,sin} +\DeclareMathOperator{\natlog}{nat.\,log} +\DeclareMathOperator{\Naplog}{Nap.\,log} +\newcommand{\nnobrkdash}[1]{$n$\nobreakdash-\hspace{0pt}} + +\newcommand\olddecimal[2]{% + {% + \renewcommand\arraystretch{0.5}% anything < 0.7 will do + \begin{tabular}{@{}r@{}|@{}l@{}}% + \mathstrut#1\,&\,#2\\\cline{2-2}% + \end{tabular} + }% +} + +\newlength{\TmpLen} +\newlength{\CharW} +\setlength{\CharW}{12pt} +\newcommand{\CircleDigit}[1]{% + \raisebox{1pt}{\makebox[\CharW][c]{\ensuremath\bigcirc}\kern-\CharW + \makebox[\CharW][c]{{\footnotesize#1}}}% +} + +% For small caps A.D. and B.C. +\newcommand{\SCAbbrev}[3]{% + \ifthenelse{\equal{#3}{.}}% + {\textsc{\MakeLowercase{#1.#2}.}}% + {\textsc{\MakeLowercase{#1.#2}.}\@#3}% +} + +\newcommand{\AD}[1]{\SCAbbrev{A}{D}{#1}} +\newcommand{\BC}[1]{\SCAbbrev{B}{C}{#1}} + +\newcommand{\ie}{\textit{i.e.}} + +\DeclareInputText{176}{\ifmmode{{}^\circ}\else\textdegree\fi} +\DeclareInputText{183}{\ifmmode{\cdot}\else\textperiodcentered\fi} + +%% Language conveniences +\newcommand{\Eng}[1]{\foreignlanguage{english}{#1}} +\newcommand{\Fre}[1]{\foreignlanguage{french}{#1}} +\newcommand{\Ger}[1]{\foreignlanguage{german}{#1}} +\newcommand{\Grk}[1]{\foreignlanguage{greek}{#1}} +\newcommand{\Ita}[1]{\foreignlanguage{italian}{#1}} +\newcommand{\Lat}[1]{\foreignlanguage{latin}{#1}} + +%%%%%%%%%%%%%%%%%%%%% CATALOGUE FORMATTING MACROS %%%%%%%%%%%%%%%%%%%%% +\newlength{\QUAD} +\newlength{\QQUAD} + +\newcommand{\Catalog}{% + \cleardoublepage + \phantomsection + \pdfbookmark[0]{Catalogue}{Catalogue} + + \begin{center} + {\Large THE\quad MACMILLAN\quad COMPANY'S}\\[2\Titleskip] + {\scriptsize PUBLICATIONS ON}\\[2\Titleskip] + \makebox[0pt][c]{\Huge \scshape Mathematics \ and \ Physics.}\\[2\Titleskip] + \Graphic{2.5in}{tbreak3} + \end{center} + +% \pagenumbering{arabic} + \renewcommand{\headrulewidth}{0.5pt} + \fancyhf{} + \fancyhead[CE]{\textit{\small MACMILLAN \& CO'S PUBLICATIONS ON}} + \fancyhead[CO]{\textit{\small MATHEMATICS AND PHYSICS.}} + + \ifthenelse{\boolean{ForPrinting}} + {\fancyhead[RO,LE]{\thepage}} + {\fancyhead[R]{\thepage}} + + \thispagestyle{plain} + + %% Tighten spacing + \setlength{\parsep}{0.25ex plus 0.125ex minus 0.25ex} + \setlength{\parskip}{0.1875ex plus .3125ex minus 0.0625ex} + \setlength{\partopsep}{0.125ex plus .375ex minus 0.0625ex} + \footnotesize + \setlength{\QUAD}{1.5em} + \setlength{\QQUAD}{2\QUAD} +} + +\newcommand{\CatalogSection}[1]{% + \subsection*{\centering\normalsize\bfseries\sffamily #1} +} + +% For snippets of non-bold in \Entry headings +\newcommand{\Rm}[1]{{\normalfont #1}} + +\newcommand{\Entry}[2][]{% + \par\noindent\hangindent\QQUAD\textbf{#2}% + \ifthenelse{\equal{#1}{}}{\quad}{}% +} +\newcommand{\Subentry}{\par\noindent\hspace*{\QUAD}\hangindent\QQUAD} + +% Author, Title, and Price +\newcommand{\Au}[1]{{\scshape #1}} +\newcommand{\Ti}[1]{{\scshape #1}} +\renewcommand{\Pr}[1]{#1} +\newcommand{\Date}[1]{\oldstylenums{#1}} +\newcommand{\Degr}[1]{{\normalfont\scshape #1}} + +% For very last page +\newcommand{\SizeA}{\large} +\newcommand{\SizeB}{\normalsize} +\newcommand{\SizeC}{\small} +\newcommand{\SizeD}{\footnotesize} +\newcommand{\SizeE}{\scriptsize} + + +%%%% DPalign %%%% +\makeatletter +\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 + +%%%%%%%%%%%%%%%%%%%%%%%% START OF DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{document} + +\pagestyle{empty} +\pagenumbering{Alph} +\phantomsection +\pdfbookmark[-1]{Front Matter}{Front Matter} + +%%%% PG BOILERPLATE %%%% +\phantomsection +\pdfbookmark[0]{PG Boilerplate}{Project Gutenberg Boilerplate} + +\begin{center} +\begin{minipage}{\textwidth} +\small +\begin{PGtext} +The Project Gutenberg EBook of A History of Mathematics, by Florian Cajori + +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: A History of Mathematics + +Author: Florian Cajori + +Release Date: January 24, 2010 [EBook #31061] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK A HISTORY OF MATHEMATICS *** +\end{PGtext} +\end{minipage} +\end{center} + +\clearpage + + +%%%% Credits and transcriber's note %%%% +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by Andrew D. Hwang, Peter Vachuska, Carl Hudkins +and the Online Distributed Proofreading Team at +http://www.pgdp.net +\end{PGtext} +\end{minipage} +\end{center} +\vfill + +\begin{minipage}{0.85\textwidth} +\small +\pdfbookmark[0]{Transcriber's Note}{Transcriber's Note} +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% + +\raggedright +\TransNoteText +\end{minipage} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% + +\frontmatter + +\setlength{\headheight}{15pt} +\fancyfoot{} +\ifthenelse{\boolean{ForPrinting}} + {\fancyhead[RO,LE]{\thepage}} + {\fancyhead[R]{\thepage}} + +\normalsize +\cleardoublepage + +%% -----File: 001.png---Folio i------- +%Half-Title Page +\thispagestyle{empty} +\null\vfil +\begin{center} + \makebox[0pt][c]{\textbf{\HalfTitleSize A HISTORY OF MATHEMATICS}}% +\end{center} +\vfil +\clearpage + +%% -----File: 002.png---Folio ii------- +% Publisher's device +\thispagestyle{empty} +\null\vfil +\begin{center} +\Graphic{1.5in}{macmillan} +\end{center} +\vfil +\clearpage + +%% -----File: 003.png---Folio iii------- +% title page +\thispagestyle{empty} +\ifthenelse{\boolean{ForPrinting}}{% + \setlength{\Titleskip}{8pt}% + \enlargethispage{96pt}% +}{% + \setlength{\Titleskip}{5.75pt}% +} + +\begin{center} +{\LARGE A HISTORY OF}\\[6\Titleskip] +% [** PP: Poor man's gesperrt for one-off use] +{\Huge M\,A\,T\,H\,E\,M\,A\,T\,I\,C\,S}\\[6\Titleskip] +{\footnotesize BY}\\[2\Titleskip] +{\normalsize\scshape FLORIAN CAJORI, Ph.D.}\\[\Titleskip] +\scriptsize\scshape% +Formerly Professor of Applied Mathematics in the Tulane University \\ +of Louisiana; now Professor of Physics \\ +in Colorado College\normalfont\\[6\Titleskip] +\begin{minipage}{3in} +\scriptsize +\null\hspace*{1em}% +``I am sure that no subject loses more than mathematics +by any attempt to dissociate it from its history.''---\textsc{J.~W.~L. +Glaisher} +\end{minipage} +\vspace*{12\Titleskip} + +\normalsize\textgoth{New York}\\[\Titleskip] +THE MACMILLAN COMPANY\\[\Titleskip] +\scriptsize\scshape LONDON: MACMILLAN \&~CO., Ltd.\\[\Titleskip] +\normalsize 1909\\[2\Titleskip] +\scriptsize\itshape All rights reserved +\end{center} +\normalsize\normalfont +\clearpage + +%% -----File: 004.png---Folio iv------- +\thispagestyle{empty} +\null\vfil +\begin{center} +\scriptsize\scshape Copyright, 1893,\\[\Titleskip] +By MACMILLAN AND CO. + +\rule{1in}{0.5pt}\\[\Titleskip] +\upshape Set up and electrotyped January, \Date{1894}.\quad Reprinted March, \\ +\Date{1895}; October, \Date{1897}; November, \Date{1901}; January, \Date{1906}; July, \Date{1909}. +\vfil\vfil + +{\footnotesize\textgoth{Norwood Press:}}\\ +J.~S. Cushing \& Co.---Berwick \& Smith.\\ +Norwood, Mass., U.S.A. +\end{center} +\normalsize\normalfont +\clearpage + +%% -----File: 005.png---Folio v------- + +% Globally apply looser inter-word spacing +\stretchyspace + +\NChapter{Preface} + +\First{An} increased interest in the history of the exact sciences +manifested in recent years by teachers everywhere, and the +attention given to historical inquiry in the mathematical +class-rooms and seminaries of our leading universities, cause +me to believe that a brief general History of Mathematics +will be found acceptable to teachers and students. + +The pages treating---necessarily in a very condensed +form---of the progress made during the present century, +are put forth with great diffidence, although I have spent +much time in the effort to render them accurate and +reasonably complete. Many valuable suggestions and criticisms +on the chapter on ``Recent Times'' have been made +by Dr.~E.~W. Davis, of the University of Nebraska. The +proof-sheets of this chapter have also been submitted to +Dr.~J.~E. Davies and Professor C.~A. Van~Velzer, both of the +University of Wisconsin; to Dr.~G.~B. Halsted, of the +University of Texas; Professor L.~M. Hoskins, of the Leland +Stanford~Jr.\ University; and Professor G.~D.~Olds, of Amherst +College,---all of whom have afforded valuable assistance. +I am specially indebted to Professor F.~H.~Loud, of Colorado +College, who has read the proof-sheets throughout. To all +the gentlemen above named, as well as to Dr.~Carlo Veneziani +%% -----File: 006.png---Folio vi------- +of Salt Lake City, who read the first part of my work in +manuscript, I desire to express my hearty thanks. But in +acknowledging their kindness, I trust that I shall not seem +to lay upon them any share in the responsibility for errors +which I may have introduced in subsequent revision of the +text. + +\null\hfill{\footnotesize FLORIAN CAJORI.}\hspace*{1em} +\vspace*{2\Titleskip} + +\footnotesize\textsc{Colorado College}, December, 1893. +%% -----File: 007.png---Folio vii------- + +% Resets font size to \normalsize +\TableofContents + +\iffalse +TABLE OF CONTENTS. + + Page +INTRODUCTION ............ 1 +ANTIQUITY.............. 5 + The Babylonians........... 5 + The Egyptians............ 9 + The Greeks ............. 16 + Greek Geometry........... 16 + The Ionic School.......... 17 + The School of Pythagoras...... 19 + The Sophist School........ 23 + The Platonic School........ 29 + The First Alexandrian School .... 34 + The Second Alexandrian School . . . 54 + Greek Arithmetic.......... 63 + The Romans .... ..... 77 +MIDDLE AGES ... ..... 84 + The Hindoos ... ...... 84 + The Arabs.............. 100 + Europe during the Middle Ages .... 117 + Introduction of Roman Mathematics . . 117 + Translation of Arabic Manuscripts .... 124 + The First Awakening and its Sequel . . . 128 +MODERN EUROPE .......... 138 + The Renaissance ... ..... 139 + Vieta to Descartes . . ..... 156 + Descartes to Newton . . ..... 183 + Newton to Euler.......... 199 + +%% -----File: 008.png---Folio viii------- + +EULER, LAGRANGE, AND LAPLACE 246 +The Origin of Modern Geometry 285 + +RECENT TIMES 291 + +SYNTHETIC Geometry 293 + +ANALYTIC GEOMETRY 307 + +ALGEBRA 315 + +ANALYSIS 331 + +THEORY of FUNCTIONS 347 + +THEORY of NUMBERS 362 + +APPLIED MATHEMATICS 373 + +INDEX 405 +\fi +%% -----File: 009.png---Folio IX------- + +\pagestyle{fancy} + +\NChapter[1]{Books of Reference}%[** PP: `Normal' line spacing] + +\Change{The}{\First{The}} following books, pamphlets, and articles have been used +in the preparation of this history. Reference to any of them +is made in the text by giving the respective number. Histories +marked with a star are the only ones of which extensive use +has been made. +\bigskip + +\footnotesize +\BookRef{1}\textsc{Günther, S\@.} \Ger{\textit{Ziele und Resultate der neueren Mathe\-ma\-tisch-historischen +Forschung.} Erlangen, 1876.} +\index{Gunther@{Günther, S.}}% + +\BookRef{2}\textsc{Cajori, F\@.} \textit{The Teaching and History of Mathematics in the U.~S\@.} +Washington, 1890. + +\BookRef{3}*\textsc{Cantor, Moritz.} \Ger{\textit{Vorlesungen über Geschichte der Mathematik.} +Leipzig. Bd.~I., 1880; Bd.~II., 1892.} +\index{Cantor, M.}% + +\BookRef{4}\textsc{Epping, J\@.} \Ger{\textit{Astronomisches aus Babylon. Unter Mitwirkung von} +\textsc{P.~J.~R. Strassmaier}. Freiburg, 1889.} +\index{Epping}% +\index{Strassmaier}% + +\BookRef{5}\textsc{Bretschneider, C.~A\@.} \Ger{\textit{Die Geometrie und die Geometer vor Euklides.} +Leipzig, 1870.} +\index{Bretschneider}% + +\BookRef{6}*\textsc{Gow, James.} \textit{A Short History of Greek Mathematics.} Cambridge, +1884. +\index{Gow}% + +\BookRef{7}*\textsc{Hankel, Hermann.} \Ger{\textit{Zur Geschichte der Mathematik im Alterthum +und Mittelalter.} Leipzig, 1874.} + +\BookRef{8}*\textsc{Allman, G.~J\@.} \textit{Greek Geometry from Thales to Euclid.} Dublin, +1889. +\index{Allman}% + +\BookRef{9}\textsc{De Morgan, A\@.} ``Euclides'' in \textit{Smith's Dictionary of Greek and +Roman Biography and Mythology}. + +\BookRef{10}\textsc{Hankel, Hermann.} \Ger{\textit{Theorie der Complexen Zahlensysteme.} Leipzig, +1867.} +\index{Hankel!ref.\ to}% + +\BookRef{11}\textsc{Whewell, William.} \textit{History of the Inductive Sciences.} +\index{Whewell}% + +\BookRef{12}\textsc{Zeuthen, H.~G\@.} \Ger{\textit{Die Lehre von den Kegelschnitten im Alterthum.} +Kopenhagen, 1886.} +\index{Zeuthen!ref.\ to}% +%% -----File: 010.png---Folio X------- + +\BookRef{13}*\textsc{Chasles, M\@.} \Ger{\textit{Geschichte der Geometrie.} Aus dem Fran\-zösi\-schen +übertragen durch \textsc{Dr.~L.~A. Sohncke}. Halle, 1839.} +\index{Chasles}% + +\BookRef{14}\textsc{Marie, Maximilien.} \Fre{\textit{Histoire des Sciences Mathématiques et Physiques.} +Tome~I.--XII\@. Paris, 1883--1888.} +\index{Marie, M.}% + +\BookRef{15}\textsc{Comte, A\@.} \textit{Philosophy of Mathematics}, translated by \textsc{W.~M. Gillespie}. +\index{Comte}% + +\BookRef{16}\textsc{Hankel, Hermann.} \Ger{\textit{Die Entwickelung der Mathematik in den letzten +Jahrhunderten.} Tübingen, 1884.} +\index{Hankel!ref.\ to}% + +\BookRef{17}\textsc{Günther, Siegmund} und \textsc{Windelband, W\@.} \Ger{\textit{Geschichte der antiken +Naturwissenschaft und Philosophie.} Nördlingen, 1888.} +\index{Gunther@{Günther, S.}}% + +\BookRef{18}\textsc{Arneth, A\@.} \Ger{\textit{Geschichte der reinen Mathematik.} Stuttgart, 1852.} +\index{Arneth,}% + +\BookRef{19}\textsc{Cantor, Moritz.} \Ger{\textit{Mathematische Beiträge zum Kulturleben der +Völker.} Halle, 1863.} +\index{Cantor, M.}% + +\BookRef{20}\textsc{Matthiessen, Ludwig.} \Ger{\textit{Grundzüge der Antiken und Modernen +Algebra der Litteralen Gleichungen.} Leipzig, 1878.} +\index{Matthiessen}% + +\BookRef{21}\textsc{Ohrtmann} und \textsc{Müller}. \Ger{\textit{Fortschritte der Mathematik.}} +\index{Muller@{Müller}}% +\index{Ohrtmann}% + +\BookRef{22}\textsc{Peacock, George.} Article ``Arithmetic,'' in \textit{The Encyclopædia of +Pure Mathematics}. London, 1847. + +\BookRef{23}\textsc{Herschel, J.~F.~W\@.} Article ``Mathematics,'' in \textit{Edinburgh Encyclopædia}. +\index{Herschel, J.~F.~W.!ref.\ to}% + +\BookRef{24}\textsc{Suter, Heinrich.} \Ger{\textit{Geschichte der Mathematischen Wissenschaften.} +Zürich, 1873--75.} +\index{Suter}% + +\BookRef{25}\textsc{Quetelet, A\@.} \Fre{\textit{Sciences Mathématiques et Physiques chez les Belges.} +Bruxelles, 1866.} +\index{Quetelet!ref.\ to}% + +\BookRef{26}\textsc{Playfair, John.} Article ``Progress of the Mathematical and Physical +Sciences,'' in \textit{Encyclopædia Britannica}, 7th~edition, continued +in the 8th~edition by \textsc{Sir John Leslie}. +\index{Leslie}% +\index{Playfair}% + +\BookRef{27}\textsc{De Morgan, A\@.} \textit{Arithmetical Books from the Invention of Printing +to the Present Time.} + +\BookRef{28}\textsc{Napier, Mark.} \textit{Memoirs of John Napier of Merchiston.} Edinburgh, +1834. +\index{Napier, M.}% + +\BookRef{29}\textsc{Halsted, G.~B\@.} ``Note on the First English Euclid,'' \textit{American +Journal of Mathematics}, Vol.~II., 1879. +\index{Halsted}% + +\BookRef{30}\textsc{Madame Perier.} \textit{The Life of Mr.~Paschal.} Translated into +English by W.~A., London, 1744. +\index{Perier, Madame}% + +\BookRef{31}\textsc{Montucla, J.~F\@.} \Fre{\textit{Histoire des Mathématiques.} Paris, 1802.} +\index{Montucla}% + +\BookRef{32}\textsc{Dühring E\@.} \Ger{\textit{Kritische Geschichte der allgemeinen Principien der +Mechanik.} Leipzig, 1887.} +\index{Duhring@{Dühring, E.}}% + +\BookRef{33}\textsc{Brewster, D\@.} \textit{The Memoirs of Newton.} Edinburgh, 1860. + +\BookRef{34}\textsc{Ball, W.~W.~R\@.} \textit{A Short Account of the History of Mathematics.} +London, 1888, 2nd~edition, 1893. +\index{Ball, W.~W.~R.}% + +\BookRef{35}\textsc{De Morgan, A\@.} ``On the Early History of Infinitesimals,'' in the +\textit{Philosophical Magazine}, November, 1852. +\index{Demorgan@{De Morgan}!ref.\ to}% +%% -----File: 011.png---Folio XI------- + +\BookRef{36}\Ger{\textit{Bibliotheca Mathematica}, herausgegeben von \textsc{Gustaf Eneström}}, +Stockholm. +\index{Eneström}% + +\BookRef{37}\textsc{Günther, Siegmund.} \Ger{\textit{Vermischte Untersuchungen zur Geschichte +der mathematischen Wissenschaften.} Leipzig, 1876.}\index{Gunther@{Günther, S.}}% +%[** PP: Prev. line exactly full; placing \index{} on next line adds vspace.] + +\BookRef{38}*\textsc{Gerhardt, C.~I\@.} \Ger{\textit{Geschichte der Mathematik in Deutschland.} +München, 1877.} +\index{Gerhardt}% + +\BookRef{39}\textsc{Gerhardt, C.~I\@.} \Ger{\textit{Entdeckung der Differenzialrechnung durch Leibniz.} +Halle, 1848.} + +%[** PP: Same Gerhardt, but retaining legitimate variant from original.] +\BookRef{40}\textsc{Gerhardt, K.~I\@.} ``Leibniz in London,'' in \Ger{\textit{Sitzungsberichte der +Königlich Preussischen Academie der Wissenschaften zu Berlin}, +Februar, 1891.} + +\BookRef{41}\textsc{De Morgan, A\@.} Articles ``Fluxions'' and ``Commercium Epistolicum,'' +in the \textit{Penny Cyclopædia}. +\index{Demorgan@{De Morgan}!ref.\ to}% + +\BookRef{42}*\textsc{Todhunter, I\@.} \textit{A History of the Mathematical Theory of Probability +from the Time of Pascal to that of Laplace.} Cambridge and +London, 1865. +\index{Todhunter!ref.\ to}% + +\BookRef{43}*\textsc{Todhunter, I\@.} \textit{A History of the Theory of Elasticity and of the +Strength of Materials.} Edited and completed by \textsc{Karl Pearson}. +Cambridge, 1886. + +\BookRef{44}\textsc{Todhunter, I\@.} ``Note on the History of Certain Formulæ in Spherical +Trigonometry,'' \textit{Philosophical Magazine}, February, 1873. + +\BookRef{45}\Ger{\textit{Die Basler Mathematiker, Daniel Bernoulli und Leonhard Euler.} +Basel, 1884.} + +\BookRef{46}\textsc{Reiff, R\@.} \Ger{\textit{Geschichte der Unendlichen Reihen.} Tübingen, 1889.} +\index{Reiff}% + +\BookRef{47}\textsc{Waltershausen, W.~Sartorius.} \Ger{\textit{Gauss, zum Gedächtniss.} Leipzig, +1856.} +\index{Waltershausen}% + +\BookRef{48}\textsc{Baumgart, Oswald.} \Ger{\textit{Ueber das Quadratische Reciprocitätsgesetz.} +Leipzig, 1885.} +\index{Baumgart}% + +\BookRef{49}\textsc{Hathaway, A.~S\@.} ``Early History of the Potential,'' \textit{Bulletin of +the N.~Y. Mathematical Society}, I.~3. +\index{Hathaway}% + +\BookRef{50}\textsc{Wolf, Rudolf.} \Ger{\textit{Geschichte der Astronomie.} München, 1877.} +\index{Wolf, R.}% + +\BookRef{51}\textsc{Arago, D.~F.~J\@.} ``Eulogy on Laplace.'' Translated by \textsc{B.~Powell}, +\textit{Smithsonian Report}, 1874. +\index{Arago}% + +\BookRef{52}\textsc{Beaumont, M.~Élie~De.} ``Memoir of Legendre.'' Translated by +\textsc{C.~A. Alexander}, \textit{Smithsonian Report}, 1867. +\index{Beaumont}% + +\BookRef{53}\textsc{Arago, D.~F.~J\@.} ``Joseph Fourier.'' \textit{Smithsonian Report}, +1871. + +\BookRef{54}\textsc{Wiener, Christian.} \Ger{\textit{Lehrbuch der Darstellenden Geometrie.} Leipzig, +1884.} +\index{Wiener}% + +\BookRef{55}*\textsc{Loria, Gino.} \Ger{\textit{Die Hauptsächlichsten Theorien der Geometrie in +ihrer früheren und heutigen Entwickelung}, ins deutsche über\-tragen +von \textsc{Fritz Schütte}. Leipzig, 1888.} +\index{Loria}% +%% -----File: 012.png---Folio XII------- + +\BookRef{56}\textsc{Cayley, Arthur.} Inaugural Address before the British Association, +1883. +\index{Cayley}% + +\BookRef{57}\textsc{Spottiswoode, William.} Inaugural Address before the British +Association, 1878. +\index{Spottiswoode!ref.\ to}% + +\BookRef{58}\textsc{Gibbs, J.~Willard.} ``Multiple Algebra,'' \textit{Proceedings of the +American Association for the Advancement of Science}, 1886. +\index{Gibbs!ref.\ to}% + +\BookRef{59}\textsc{Fink, Karl.} \Ger{\textit{Geschichte der Elementar-Mathematik.} Tübingen, +1890.} +\index{Fink}% + +\BookRef{60}\textsc{Wittstein, Armin.} \Ger{\textit{Zur Geschichte des Malfatti'schen Problems.} +Nördlingen, 1878.} +\index{Wittstein}% + +\BookRef{61}\textsc{Klein, Felix.} \Ger{\textit{Vergleichende Betrachtungen über neuere geometrische +Forschungen.} Erlangen, 1872.} + +\BookRef{62}\textsc{Forsyth, A.~R\@.} \textit{Theory of Functions of a Complex Variable.} +Cambridge, 1893. +\index{Forsyth}% + +\BookRef{63}\textsc{Graham, R.~H\@.} \textit{Geometry of Position.} London, 1891. +\index{Graham}% + +\BookRef{64}\textsc{Schmidt, Franz.} \Ger{``Aus dem Leben zweier ungarischer Mathematiker +Johann und Wolfgang Bolyai von Bolya.'' \textit{Grunert's +Archiv}, 48:2, 1868.} +\index{Schmidt}% + +\BookRef{65}\textsc{Favaro, Anton.} \Ger{``Justus Bellavitis,'' \textit{Zeitschrift für Mathematik +und Physik}, 26:5, 1881.} +\index{Favaro}% + +\BookRef{66}\textsc{Dronke, Ad.} \textit{Julius Plücker.} Bonn, 1871. +\index{Dronke}% + +\BookRef{67}\textsc{Bauer, Gustav.} \Ger{\textit{Gedächtnissrede auf Otto Hesse.} München, +1882.} +\index{Bauer}% + +\BookRef{68}\textsc{Alfred Clebsch.} \Ger{\textit{Versuch einer Darlegung und Würdigung seiner +wissenschaftlichen Leistungen von einigen seiner Freunde.} Leipzig, 1873.} +\index{Clebsch!ref.\ to}% + +\BookRef{69}\textsc{Haas, August.} \Ger{\textit{Versuch einer Darstellung der Geschichte des +Krümmungsmasses.} Tübingen, 1881.} +\index{Haas}% + +\BookRef{70}\textsc{Fine, Henry~B\@.} \textit{The Number-System of Algebra.} Boston and +New York, 1890. +\index{Fine}% + +\BookRef{71}\textsc{Schlegel, Victor.} \Ger{\textit{Hermann Grassmann, sein Leben und seine +Werke.} Leipzig, 1878.} + +\BookRef{72}\textsc{Zahn, W.~v.} \Ger{``Einige Worte zum Andenken an Hermann Hankel,'' +\textit{Mathematische Annalen}, VII.~4, 1874.} +\index{Zahn}% + +\BookRef{73}\textsc{Muir, Thomas.} \textit{A Treatise on Determinants.} 1882. +\index{Muir}% + +\BookRef{74}\textsc{Salmon, George.} ``Arthur Cayley,'' \textit{Nature}, 28:21, September, +1883. +\index{Salmon}% + +\BookRef{75}\textsc{Cayley, A\@.} ``James Joseph Sylvester,'' \textit{Nature}, 39:10, January, +1889. + +\BookRef{76}\textsc{Burkhardt, Heinrich.} \Ger{``Die Anfänge der Gruppentheorie und +Paolo Ruffini,'' \textit{Zeitschrift für Mathematik und Physik}, Supplement, +1892.} +\index{Burkhardt, H.}% +%% -----File: 013.png---Folio XIII------- + +\BookRef{77}\textsc{Sylvester, J.~J\@.} \textit{Inaugural Presidential Address to the Mathematical +and Physical Section of the British Association at Exeter.} 1869. +\index{Sylvester!ref.\ to}% + +\BookRef{78}\textsc{Valson, C.~A\@.} \Fre{\textit{La Vie et les travaux du Baron Cauchy.} Tome~I., +II., Paris, 1868.} +\index{Valson}% + +\BookRef{79}\textsc{Sachse, Arnold.} \Ger{\textit{Versuch einer Geschichte der Darstellung willkürlicher +Funktionen einer variablen durch trigonometrische +Reihen.} Göttingen, 1879.} +\index{Sachse}% + +\BookRef{80}\textsc{Bois-Reymond, Paul du.} \Ger{\textit{Zur Geschichte der Trigonometrischen +Reihen, Eine Entgegnung.} Tübingen.} +\index{Bois-Reymond, P.~du}% + +\BookRef{81}\textsc{Poincaré, Henri.} \Fre{\textit{Notice sur les Travaux Scientifiques de Henri +Poincaré.} Paris, 1886.} +\index{Poincaré!ref.\ to}% + +\BookRef{82}\textsc{Bjerknes, C.~A\@.} \Fre{\textit{Niels-Henrik Abel, Tableau de sa vie et de son +action scientifique.} Paris, 1885.} +\index{Bjerknes, C. A.}% + +\BookRef{83}\textsc{Tucker, R\@.} ``Carl Friedrich Gauss,'' \textit{Nature}, April, 1877. +\index{Tucker}% + +\BookRef{84}\textsc{Dirichlet, Lejeune.} \Ger{\textit{Gedächtnissrede auf Carl Gustav Jacob +Jacobi.} 1852.} +\index{Dirichlet!ref.\ to}% + +\BookRef{85}\textsc{Enneper, Alfred.} \Ger{\textit{Elliptische Funktionen. Theorie und Geschichte.} +Halle~a/S., 1876.} +\index{Enneper!ref.\ to}% + +\BookRef{86}\textsc{Henrici, O\@.} ``Theory of Functions,'' \textit{Nature}, 43:14~and~15, 1891. +\index{Henrici}% + +\BookRef{87}\textsc{Darboux, Gaston.} \Fre{\textit{Notice sur les Travaux Scientifiques de M. Gaston +Darboux.} Paris, 1884.} +\index{Darboux}% + +\BookRef{88}\textsc{Kummer, E.~E\@.} \Ger{\textit{Gedächtnissrede auf Gustav Peter Lejeune-Diri\-chlet.} +Berlin, 1860.} +\index{Kummer!ref.\ to}% + +\BookRef{89}\textsc{Smith, H.~J. Stephen.} ``On the Present State and Prospects of +Some Branches of Pure Mathematics,'' \textit{Proceedings of the London +Mathematical Society}, Vol.~VIII., Nos.\ 104,~105, 1876. +\index{Smith, H.!ref.\ to}% + +\BookRef{90}\textsc{Glaisher, J.~W.~L\@.} ``Henry John Stephen Smith,'' \textit{Monthly Notices +of the Royal Astronomical Society}, XLIV.,~4, 1884. + +\BookRef{91}\Ger{\textit{Bessel als Bremer Handlungslehrling.} Bremen, 1890.} + +\BookRef{92}\textsc{Frantz, J\@.} \Ger{\textit{Festrede aus Veranlassung von Bessel's hundert\-jähr\-igem +Geburtstag.} Königsberg, 1884.} +\index{Frantz}% + +\BookRef{93}\textsc{Dziobek, O\@.} \textit{Mathematical Theories of Planetary Motions.} +Translated into English by M.~W. Harrington and W.~J. Hussey.\index{Dziobek}% +%[** PP: Prev. line exactly full; placing \index{} on next line adds vspace.] + +\BookRef{94}\textsc{Hermite, Ch.} \Fre{``Discours prononcé devant le président de la République,'' +\textit{Bulletin des Sciences Mathématiques}, XIV., Janvier, +1890.} +\index{Hermite!ref.\ to}% + +\BookRef{95}\textsc{Schuster, Arthur.} ``The Influence of Mathematics on the Progress +of Physics,'' \textit{Nature}, 25:17, 1882. +\index{Schuster}% + +\BookRef{96}\textsc{Kerbedz, E.~de.} \Ita{``Sophie de Kowalevski,'' \textit{Rendiconti del Circolo +Matematico di Palermo}, V., 1891.} +\index{Kerbedz}% + +\BookRef{97}\textsc{Voigt, W\@.} \Ger{\textit{Zum Gedächtniss von G.~Kirchhoff.} Göttingen, 1888.} +\index{Voigt}% +%% -----File: 014.png---Folio XIV------- + +\BookRef{98}\textsc{Bôcher, Maxime.} ``A Bit of Mathematical History,'' \textit{Bulletin of +the N.~Y. Math.\ Soc.}, Vol.~II., No.~5. +\index{Bocher@{Bôcher}}% + +\BookRef{99}\textsc{Cayley, Arthur.} \textit{Report on the Recent Progress of Theoretical +Dynamics.} 1857. +\index{Cayley}% + +\BookRef{100}\textsc{Glazebrook, R.~T\@.} \textit{Report on Optical Theories.} 1885. +\index{Glazebrook!ref.\ to}% + +\BookRef{101}\textsc{Rosenberger, F\@.} \Ger{\textit{Geschichte der Physik.} Braunschweig, 1887--1890.} +\index{Rosenberger}% + +\normalsize + +%% -----File: 015.png---Folio 1------- +%% Index cross-references +\index{Alembert, D'|see{D'Alembert}}% +\index{Algebra|see{Notation}}% +\index{Applied mathematics|see{Astronomy, Mechanics}}% +\index{Arithmetic|see{Numbers, Notation}}% +\index{Assumption, tentative|see{Regula falsa}}% +\index{Astronomy|see{Mechanics}}% +\index{Attraction|see{Gravitation, Ellipsoid}}% +\index{Bachet de Méziriac|see{Méziriac}}% +\index{Baune, De|see{De Baune}}% +\index{Byrgius|see{Bürgi}}% +\index{Calculus|see{Differential Calculus}}% +\index{Ceulen, van|see{Ludolph}}% +\index{Complex quantities|see{Imaginaries}}% +\index{Conic sections|see{Geometry}}% +\index{Cube, duplication of|see{Duplication of the cube}}% +\index{Cubic equations|see{Algebra}}% +\index{Curves|see{Cubic curves, Rectification, Geometry, Conic sections}}% +\index{Delian problem|see{Duplication of the cube}}% +\index{Differences, finite|see{Finite differences}}% +\index{Differential calculus|see{Bernoullis, Euler, Lagrange, Laplace, etc}}% +\index{Dyck|see{Groups}}% +\index{Elements (Euclid's)|see{Euclid}}% +\index{Elimination|see{Equations}}% +\index{Elliptic geometry|see{Non-Euclidean geometry}}% +\index{Equations|see{Cubic equations, Algebra, Theory of numbers}}% +\index{Errors, theory of|see{Least squares}}% +\index{Euclidean space|see{Non-Euclidean geometry}}% +\index{Fibonacci|see{Leonardo of Pisa}}% +\index{Force-function|see{Potential}}% +\index{Fractions|see{Arithmetic}}% +\index{Functions|see{Elliptic functions, Abelian functions, Hyperelliptic functions, Theta functions, Beta function, Gamma function, Omega function, Sigma function, Bessel's function, Potential}}% +\index{Geber|see{Gabir ben Aflah}}% +\index{Geometry|see{Curves, Surfaces, Curvature, Quadrature, Rectification, Circle}}% +\index{Guldinus|see{Guldin}}% +\index{Holder@{Hölder, O.}|see{Groups}}% +\index{Hydrodynamics|see{Mechanics}}% +\index{Hydrostatics|see{Mechanics}}% +\index{Hyperbolic geometry|see{Non-Euclidean geometry}}% +\index{Incommensurables|see{Irrationals}}% +\index{Indeterminate analysis|see{Theory of numbers}}% +\index{Indeterminate equations|see{Theory of numbers}}% +\index{Indian mathematics|see{Hindoos}}% +\index{Indian numerals|see{Arabic numerals}}% +\index{Indices|see{Exponents}}% +\index{Infinitesimal calculus|see{Differential calculus}}% +\index{Irrationals|see{Incommensurables}}% +\index{Isoperimetrical figures|see{Calculus of variations}}% +\index{Joachim|see{Rhæticus}}% +\index{Kauffmann|see{Mercator, N.}}% +\index{Kelvin, Lord|see{Thomson, W.}}% +\index{Lucas de Burgo|see{Pacioli}}% +\index{Mechanics|see{Dynamics, Hydrodynamics, Hydrostatics, Graphic statics, Laws of motion, Astronomy, D'Alembert's principle}}% +\index{Moon|see{Astronomy}}% +\index{Muller@{Müller, J.}|see{Regiomontanus}}% +\index{Negative quantities|see{Algebra}}% +\index{Negative roots|see{Algebra}}% +\index{Nicolo of Brescia|see{Tartaglia}}% +\index{Notation|see{Exponents, Algebra}}% +\index{Numerals|see{Apices}}% +\index{Parabola|see{Geometry}}% +\index{Parabolic geometry|see{Non-Euclidean geometry}}% +\index{Perspective|see{Geometry}}% +\index{Physics, mathematical|see{Applied mathematics}}% +\index{Plato Tiburtinus|see{Plato of Tivoli}}% +\index{Ptolemæus|see{Ptolemy}}% +\index{Quadratic equations|see{Algebra, Equations}}% +\index{Quadrature of the circle|see{Circle; also see Circle-squarers, $\pi$}}% +\index{Rectification of curves|see{Curves}}% +\index{Regula aurea|see{Falsa positio}}% +\index{Rheticus|see{Rhæticus}}% +\index{Sacro Bosco|see{Halifax}}% +\index{Series|see{Infinite series, Trigonometric series, Divergent series, Absolutely convergent series, Semi-convergent series, Fourier's series, Uniformly convergent series}}% +\index{Sound, velocity of|see{Acoustics}}% +\index{Squaring the circle|see{Quadrature of the circle}}% +\index{Statics|see{Mechanics}}% +\index{Staudt, von|see{Von Staudt}}% +\index{Stevinus|see{Stevin}}% +\index{Strutt, J.~W.|see{Rayleigh}}% +\index{Tentative assumption|see{Regula falsa}}% +\index{Theory of equations|see{Equations}}% +\index{Theory of functions|see{Functions}}% +\index{Thomson, Sir William|see{Kelvin (Lord)}}% +\index{Trigonometric series|see{Fourier's series}}% +\index{Van Ceulen|see{Ludolph}}% +\index{Von Helmholtz|see{Helmholtz}}% +\index{Wave theory|see{Undulatory theory}}% + +\clearpage +\fancyhf{} +\cleardoublepage +\phantomsection +\pdfbookmark[-1]{Main Matter}{Main Matter} +\mainmatter + +\IntroChapter{A History of Mathematics}{Introduction} +\index{History of mathematics, its value|(}% +\pagestyle{fancy} +\pagenumbering{arabic} + +\First{The} contemplation of the various steps by which mankind +has come into possession of the vast stock of mathematical +knowledge can hardly fail to interest the mathematician. He +takes pride in the fact that his science, more than any other, +is an \emph{exact} science, and that hardly anything ever done in +mathematics has proved to be useless. The chemist smiles +at the childish efforts of alchemists, but the mathematician +finds the geometry of the Greeks and the arithmetic of the +Hindoos as useful and admirable as any research of to-day. +He is pleased to notice that though, in course of its development, +mathematics has had periods of slow growth, yet in +the main it has been pre-eminently a \emph{progressive} science. + +The history of mathematics may be instructive as well as +agreeable; it may not only remind us of what we have, but +may also teach us how to increase our store. Says De Morgan, +\index{Demorgan@{De Morgan}!ref.\ to}% +``The early history of the mind of men with regard to mathematics +leads us to point out our own errors; and in this +respect it is well to pay attention to the history of mathematics.'' +It warns us against hasty conclusions; it points out +the importance of a good notation upon the progress of the +science; it discourages excessive specialisation on the part of +%% -----File: 016.png---Folio 2------- +\index{p@{$\pi$}: values for!proved to be transcendental}% +investigators, by showing how apparently distinct branches +have been found to possess unexpected connecting links; it +saves the student from wasting time and energy upon problems +which were, perhaps, solved long since; it discourages +him from attacking an unsolved problem by the same method +which has led other mathematicians to failure; it teaches that +fortifications can be taken in other ways than by direct attack, +that when repulsed from a direct assault it is well to reconnoitre +and occupy the surrounding ground and to discover the +secret paths by which the apparently unconquerable position +can be taken.\Reference{1} The importance of this strategic rule may +be emphasised by citing a case in which it has been violated. +An untold amount of intellectual energy has been expended +on the quadrature of the circle, yet no conquest has been made +by direct assault. The circle-squarers have existed in crowds +\index{Circle-squarers}% +ever since the period of Archimedes. After innumerable failures +\index{Archimedes!ref.\ to}% +to solve the problem at a time, even, when investigators +possessed that most powerful tool, the differential calculus, +persons versed in mathematics dropped the subject, while +those who still persisted were completely ignorant of its history +and generally misunderstood the conditions of the problem. +``Our problem,'' says De Morgan, ``is to square the +\index{Demorgan@{De Morgan}!ref.\ to}% +circle with the \emph{old allowance of means}: Euclid's postulates +and nothing more. We cannot remember an instance in which +a question to be solved by a \emph{definite method} was tried by the +best heads, and answered at last, \emph{by that method}, after thousands +of complete failures.'' But progress was made on this +problem by approaching it from a different direction and by +newly discovered paths. Lambert proved in 1761 that the +\index{Lambert!ref.\ to}% +ratio of the circumference of a circle to its diameter is incommensurable. +Some years ago, Lindemann demonstrated that +\index{Lindemann!ref.\ to}% +this ratio is also transcendental and that the quadrature of +the circle, by means of the ruler and compass only, is \emph{impossible}. +%% -----File: 017.png---Folio 3------- +\index{Arabic numerals and notation}% +He thus showed by actual proof that which keen-minded +mathematicians had long suspected; namely, that the +great army of circle-squarers have, for two thousand years, +been assaulting a fortification which is as indestructible as +the firmament of heaven. + +Another reason for the desirability of historical study is +the value of historical knowledge to the teacher of mathematics. +The interest which pupils take in their studies may +be greatly increased if the solution of problems and the cold +logic of geometrical demonstrations are interspersed with +historical remarks and anecdotes. A class in arithmetic will +be pleased to hear about the Hindoos and their invention of +\index{Hindoos!ref.\ to}% +the ``Arabic notation''; they will marvel at the thousands +\index{Notation!Arabic notation}% +of years which elapsed before people had even thought of +introducing into the numeral notation that Columbus-egg---the +zero; they will find it astounding that it should have +taken so long to \emph{invent} a notation which they themselves can +now \emph{learn} in a month. After the pupils have learned how to +bisect a given angle, surprise them by telling of the many +futile attempts which have been made to solve, by elementary +geometry, the apparently very simple problem of the trisection +of an angle. When they know how to construct a square +whose area is double the area of a given square, tell them +about the duplication of the cube---how the wrath of Apollo +could be appeased only by the construction of a cubical altar +double the given altar, and how mathematicians long wrestled +with this problem. After the class have exhausted their energies +on the theorem of the right triangle, tell them the legend +about its discoverer---how Pythagoras, jubilant over his great +\index{Pythagoras!ref.\ to}% +accomplishment, sacrificed a hecatomb to the Muses who inspired +him. When the value of mathematical training is +called in question, quote the inscription over the entrance into +the academy of Plato, the philosopher: ``Let no one who is +\index{Plato!ref.\ to}% +%% -----File: 018.png---Folio 4------- +unacquainted with geometry enter here.'' Students in analytical +geometry should know something of Descartes, and, after +\index{Descartes!ref.\ to}% +taking up the differential and integral calculus, they should +become familiar with the parts that Newton, Leibniz, and +\index{Leibniz!ref.\ to}% +\index{Newton!ref.\ to}% +Lagrange played in creating that science. In his historical +\index{Lagrange!ref.\ to}% +talk it is possible for the teacher to make it plain to the +student that mathematics is not a dead science, but a living +one in which steady progress is made.\Reference{2} + +The history of mathematics is important also as a valuable +contribution to the history of civilisation. Human progress +is closely identified with scientific thought. Mathematical +and physical researches are a reliable record of intellectual +progress. The history of mathematics is one of the large +\index{History of mathematics, its value|)}% +windows through which the philosophic eye looks into past +ages and traces the line of intellectual development. +%% -----File: 019.png---Folio 5------- +\index{Numerals!Babylonian|(}% + + +\Chapter{Antiquity} + +\Section{The Babylonians} +\index{Babylonians|(}% + +\First{The} fertile valley of the Euphrates and Tigris was one of +the primeval seats of human society. Authentic history of +the peoples inhabiting this region begins only with the foundation, +in Chaldæa and Babylonia, of a united kingdom out +of the previously disunited tribes. Much light has been +thrown on their history by the discovery of the art of reading +the \emph{cuneiform} or wedge-shaped system of writing. + +In the study of Babylonian mathematics we begin with the +notation of numbers. A vertical wedge~\Glyph{cune1} stood for~$1$, while +\index{Notation!Babylonian numbers|(}% +the characters \Glyph{cune10}~and~\Glyph{cune100} signified $10$ and $100$ respectively. +Grotefend believes the character for~$10$ originally to +have been the picture of two hands, as held in prayer, the +palms being pressed together, the fingers close to each other, +but the thumbs thrust out. In the Babylonian notation two +principles were employed---the additive and multiplicative. +Numbers below 100 were expressed by symbols whose +respective values had to be \emph{added}. Thus, \Glyph{cune2}~stood for~$2$, +\Glyph{cune3}~for~$3$, \Glyph{cune4}~for~$4$, \Glyph{cune23}~for~$23$, \Glyph{cune30}~for~$30$. Here the +symbols of higher order appear always to the left of those of +lower order. In writing the hundreds, on the other hand, a +\emph{smaller} symbol was placed to the left of the~$100$, and was, in +that case, to be \emph{multiplied} by~$100$. Thus, \Glyph{cune1000}~signified +%% -----File: 020.png---Folio 6------- +$10$~times~$100$, or~$1000$. But this symbol for~$1000$ was itself +taken for a new unit, which could take smaller coefficients to +its left. Thus, \Glyph{cune2000} denoted, not $20$~times~$100$, but +$10$~times~$1000$. Of the largest numbers written in cuneiform +symbols, which have hitherto been found, none go as high as +a million.\Reference{3} + +If, as is believed by most specialists, the early Sumerians +were the inventors of the cuneiform writing, then they were, +in all probability, also familiar with the notation of numbers. +Most surprising, in this connection, is the fact that Sumerian +inscriptions disclose the use, not only of the above \emph{decimal} +system, but also of a \emph{sexagesimal} one. The latter was used +chiefly in constructing tables for weights and measures. It is +full of historical interest. Its consequential development, +both for integers and fractions, reveals a high degree of +mathematical insight. We possess two Babylonian tablets +which exhibit its use. One of them, probably written between +$2300$ and $1600$~\BC, contains a table of square numbers up to~$60^{2}$. +The numbers $1$,~$4$,~$9$, $16$,~$25$, $36$,~$49$, are given as the +squares of the first seven integers respectively. We have next +$1.4 = 8^{2}$, $1.21 = 9^{2}$, $1.40 = 10^{2}$, $2.1 = 11^{2}$, etc. This remains +unintelligible, unless we assume the sexagesimal scale, which +makes $1.4 = 60 + 4$, $1.21 = 60 + 21$, $2.1 = 2.60 + 1$. The second +tablet records the magnitude of the illuminated portion of the +moon's disc for every day from new to full moon, the whole disc +being assumed to consist of $240$~parts. The illuminated parts +during the first five days are the series $5$,~$10$, $20$,~$40$, $1.20 +(=80)$, which is a geometrical progression. From here on +the series becomes an arithmetical progression, the numbers +from the fifth to the fifteenth day being respectively $1.20$, $1.36$, +$1.52$, $\DPtypo{2.8}{1.8}$, $2.24$, $2.40$, $2.56$, $3.12$, $3.28$, $3.44$, $4$. This table not +only exhibits the use of the sexagesimal system, but also indicates +the acquaintance of the Babylonians with progressions. +%% -----File: 021.png---Folio 7------- +\index{Numerals!Babylonian|)}% +\index{Ptolemy!ref.\ to}% +Not to be overlooked is the fact that in the sexagesimal notation +of integers the ``principle of position'' was employed. +Thus, in $1.4$~($=64$), the $1$~is made to stand for~$60$, the unit +of the second order, by virtue of its position with respect to +the~$4$. The introduction of this principle at so early a date +is the more remarkable, because in the decimal notation it +was not introduced till about the fifth or sixth century after +Christ. The principle of position, in its general and systematic +application, requires a symbol for zero. We ask, Did +\index{Zero!(symbol for)}% +the Babylonians possess one? Had they already taken the +gigantic step of representing by a symbol the \emph{absence} of +units? Neither of the above tables answers this question, +for they happen to contain no number in which there was +occasion to use a zero. The sexagesimal system was used also +\index{Sexagesimal system}% +in fractions. Thus, in the Babylonian inscriptions, $\frac{1}{2}$ and~$\frac{1}{3}$ +\index{Fractions!Babylonian}% +\index{Fractions!sexagesimal}% +are designated by $30$ and~$20$, the reader being expected, in +his mind, to supply the word ``sixtieths.'' The Greek geometer +Hypsicles and the Alexandrian astronomer Ptolemæus +\index{Hypsicles!ref.\ to}% +borrowed the sexagesimal notation of fractions from the +\index{Notation!Babylonian numbers|)}% +Babylonians and introduced it into Greece. From that time +sexagesimal fractions held almost full sway in astronomical +and mathematical calculations until the sixteenth century, +when they finally yielded their place to the decimal fractions. +It may be asked, What led to the invention of the sexagesimal +system? Why was it that $60$~parts were selected? To +this we have no positive answer. \emph{Ten} was chosen, in the +decimal system, because it represents the number of fingers. +But nothing of the human body could have suggested~$60$. +Cantor offers the following theory: At first the Babylonians +reckoned the year at $360$~days. This led to the division of +\index{Division of the circle}% +the circle into $360$~degrees, each degree representing the daily +\index{Circle!degrees of}% +amount of the supposed yearly revolution of the sun around +the earth. Now they were, very probably, familiar with the +%% -----File: 022.png---Folio 8------- +\index{Musical proportion}% +\index{p@{$\pi$}: values for!Babylonian and Hebrew}% +fact that the radius can be applied to its circumference as a +chord $6$~times, and that each of these chords subtends an arc +measuring exactly $60$~degrees. Fixing their attention upon +these degrees, the division into $60$~parts may have suggested +itself to them. Thus, when greater precision necessitated a +subdivision of the degree, it was partitioned into $60$~minutes. +In this way the sexagesimal notation may have originated. +The division of the day into $24$~hours, and of the hour +into minutes and seconds on the scale of~$60$, is due to the +Babylonians. + +It appears that the people in the Tigro-Euphrates basin had +made very creditable advance in arithmetic. Their knowledge +of arithmetical and geometrical progressions has already been +\index{Progressions, first appearance of arithmetical and geometrical}% +alluded to. Iamblichus attributes to them also a knowledge +of proportion, and even the invention of the so-called \emph{musical} +proportion. Though we possess no conclusive proof, we have +nevertheless reason to believe that in practical calculation +they used the \emph{abacus}. Among the races of middle Asia, even +\index{Abacus}% +as far as China, the abacus is as old as fable. Now, Babylon +was once a great commercial centre,---the metropolis of many +nations,---and it is, therefore, not unreasonable to suppose that +her merchants employed this most improved aid to calculation. + +In geometry the Babylonians accomplished almost nothing. +\index{Geometry!Babylonian}% +Besides the division of the circumference into $6$~parts by its +radius, and into $360$~degrees, they had some knowledge of +geometrical figures, such as the triangle and quadrangle, which +they used in their auguries. Like the Hebrews (1~Kin.~7:23), +they took $\pi = 3$. Of geometrical demonstrations there is, of +course, no trace. ``As a rule, in the Oriental mind the intuitive +powers eclipse the severely rational and logical.'' + +The astronomy of the Babylonians has attracted much +\index{Astronomy!Babylonian}% +attention. They worshipped the heavenly bodies from the +earliest historic times. When Alexander the Great, after +%% -----File: 023.png---Folio 9------- +\index{Babylonians|)}% +\index{Calendar}% +the battle of Arbela (331~\BC), took possession of Babylon, +Callisthenes found there on burned brick astronomical records +\index{Callisthenes}% +reaching back as far as 2234~\BC\ Porphyrius says that these +were sent to Aristotle. Ptolemy, the Alexandrian astronomer, +\index{Aristotle!ref.\ to}% +\index{Ptolemy!ref.\ to}% +possessed a Babylonian record of eclipses going back to +747~\BC. Recently Epping and Strassmaier\Reference[]{4} threw considerable +\index{Epping}% +light on Babylonian chronology and astronomy by explaining +two calendars of the years 123~\BC\ and 111~\BC, taken +from cuneiform tablets coming, presumably, from an old +observatory. These scholars have succeeded in giving an +account of the Babylonian calculation of the new and full +moon, and have identified by calculations the Babylonian +names of the planets, and of the twelve zodiacal signs and +twenty-eight normal stars which correspond to some extent +with the twenty-eight \textit{nakshatras} of the Hindoos. We append +part of an Assyrian astronomical report, as translated by +Oppert:--- +\medskip + +\footnotesize\noindent +``To the King, my lord, thy faithful servant, Mar-Istar.'' + +``\ldots On the first day, as the new moon's day of the month Thammuz +declined, the moon was again visible over the planet Mercury, as +I had already predicted to my master the King. I erred not.'' + +\normalsize + + +\Section{The Egyptians} +\index{Egyptians|(}% + +Though there is great difference of opinion regarding the +antiquity of Egyptian civilisation, yet all authorities agree in +the statement that, however far back they go, they find no +uncivilised state of society. ``Menes, the first king, changes +the course of the Nile, makes a great reservoir, and builds the +temple of Phthah at Memphis.'' The Egyptians built the +pyramids at a very early period. Surely a people engaging in +%% -----File: 024.png---Folio 10------- +\index{Rhind papyrus|(}% +enterprises of such magnitude must have known something of +mathematics---at least of practical mathematics. + +All Greek writers are unanimous in ascribing, without +envy, to Egypt the priority of invention in the mathematical +sciences. Plato in \textit{Phædrus} says: ``At the Egyptian city +\index{Plato!ref.\ to}% +of Naucratis there was a famous old god whose name was +Theuth; the bird which is called the Ibis was sacred to +him, and he was the inventor of many arts, such as arithmetic +and calculation and geometry and astronomy and draughts +\index{Astronomy!Egyptian}% +\index{Geometry!Egyptian|(}% +and dice, but his great discovery was the use of letters.'' + +Aristotle says that mathematics had its birth in Egypt, +because there the priestly class had the leisure needful for +the study of it. Geometry, in particular, is said by Herodotus, +Diodorus, Diogenes Laertius, Iamblichus, and other ancient +\index{Diodorus}% +\index{Iamblichus!ref.\ to}% +\index{Laertius}% +writers to have originated in Egypt.\Reference{5} In Herodotus we find +this (II.~c.~109): ``They said also that this king [Sesostris] +divided the land among all Egyptians so as to give each one a +quadrangle of equal size and to draw from each his revenues, +by imposing a tax to be levied yearly. But every one from +whose part the river tore away anything, had to go to him +and notify what had happened; he then sent the overseers, +who had to measure out by how much the land had become +smaller, in order that the owner might pay on what was left, +in proportion to the entire tax imposed. In this way, it +appears to me, geometry originated, which passed thence to +Hellas.'' + +We abstain from introducing additional Greek opinion +regarding Egyptian mathematics, or from indulging in wild +conjectures. We rest our account on documentary evidence. +A hieratic papyrus, included in the Rhind collection of the +British Museum, was deciphered by Eisenlohr in 1877, and +found to be a mathematical manual containing problems in +arithmetic and geometry. It was written by \textbf{Ahmes} some +\index{Ahmes|(}% +%% -----File: 025.png---Folio 11------- +time before 1700~\BC, and was founded on an older work +believed by Birch to date back as far as 3400~\BC! This +curious papyrus---the most ancient mathematical handbook +known to us---puts us at once in contact with the mathematical +thought in Egypt of three or five thousand years +ago. It is entitled ``Directions for obtaining the Knowledge +of all Dark Things.'' We see from it that the Egyptians +cared but little for theoretical results. Theorems are not +found in it at all. It contains ``hardly any general rules of +procedure, but chiefly mere statements of results intended +possibly to be explained by a teacher to his pupils.''\Reference{6} In +geometry the forte of the Egyptians lay in making constructions +and determining areas. The area of an isosceles +triangle, of which the sides measure $10$~\textit{ruths} and the base +$4$~\textit{ruths}, was erroneously given as $20$ square \DPtypo{ruths}{\textit{ruths}}, or half +the product of the base by one side. The area of an isosceles +trapezoid is found, similarly, by multiplying half the sum +of the parallel sides by one of the non-parallel sides. The +area of a circle is found by deducting from the diameter +$\frac{1}{9}$ of its length and squaring the remainder. Here $\pi$ is +\index{p@{$\pi$}: values for!Egyptian}% +taken $= (\frac{16}{9})^2 = 3.1604 \ldots$, a very fair approximation.\Reference{6} The +papyrus explains also such problems as these,---To mark +out in the field a right triangle whose sides are $10$ and $4$~units; +or a trapezoid whose parallel sides are $6$ and~$4$, and +the non-parallel sides each $20$~units. + +Some problems in this papyrus seem to imply a rudimentary +knowledge of proportion. + +The base-lines of the pyramids run north and south, and +east and west, but probably only the lines running north and +south were determined by astronomical observations. This, +coupled with the fact that the word \textit{harpedonaptæ}, applied to +Egyptian geometers, means ``rope-stretchers,'' would point to +the conclusion that the Egyptian, like the Indian and Chinese +%% -----File: 026.png---Folio 12------- +geometers, constructed a right triangle upon a given line, by +stretching around three pegs a rope consisting of three parts +in the ratios $3:4:5$, and thus forming a right triangle.\Reference{3} If +this explanation is correct, then the Egyptians were familiar, +2000~years~\BC, with the well-known property of the right +triangle, for the special case at least when the sides are +in the ratio $3:4:5$. + +On the walls of the celebrated temple of Horus at Edfu +\index{Edfu}% +have been found hieroglyphics, written about 100~\BC, which +enumerate the pieces of land owned by the priesthood, and +give their areas. The area of any quadrilateral, however +irregular, is there found by the formula $\dfrac{a+b\Fstr}{2} · \dfrac{c+d}{2}$. Thus, +for a quadrangle whose opposite sides are $5$~and~$8$, $20$~and~$15$, +%[** PP: Namely, $113\frac{3}{4}$; see note in text on 028.png below.] +is given the area $113 \frac{1}{2}\, \frac{1}{4}$.\Reference{7} The incorrect formulæ of +Ahmes of 3000~years~\BC\ yield generally closer approximations +than those of the Edfu inscriptions, written 200~years +after Euclid! + +The fact that the geometry of the Egyptians consists +chiefly of constructions, goes far to explain certain of its +great defects. The Egyptians failed in two essential points +without which a \emph{science} of geometry, in the true sense of +the word, cannot exist. In the first place, they failed to +construct a rigorously logical system of geometry, resting +upon a few axioms and postulates. A great many of their +rules, especially those in solid geometry, had probably not +been proved at all, but were known to be true merely from +observation or as matters of fact. The second great defect +was their inability to bring the numerous special cases under +a more general view, and thereby to arrive at broader and +more fundamental theorems. Some of the simplest geometrical +truths were divided into numberless special cases +of which each was supposed to require separate treatment. +%% -----File: 027.png---Folio 13------- +\index{Abacus}% +\index{Notation!Egyptian numbers}% + +Some particulars about Egyptian geometry can be mentioned +\index{Geometry!Egyptian|)}% +more advantageously in connection with the early +Greek mathematicians who came to the Egyptian priests for +instruction. + +%% [** PP: Notes on hieroglyphs: +%% The glyphs for 1, 10, 100, 10,000, 100,000, and 10,000,000 were +%% reproduced from the scanned image. The glyphs for 1000 and 1,000,000 +%% were unclear in the scanned image, and were re-created based on +%% images obtained from http://en.wikipedia.org/wiki/Egyptian_numerals +%% on December 30, 2009.] +An insight into Egyptian methods of numeration was obtained +\index{Numerals!Egyptian}% +through the ingenious deciphering of the hieroglyphics +by Champollion, Young, and their successors. The symbols +used were the following: \Glyph{hiero1}~for~$1$, \Glyph{hiero10}~for~$10$, \Glyph{hiero100}~for~$100$, +\Glyph{hierothou}~for~$1000$, \Glyph{hiero10thou}~for~$10,000$, \Glyph{hiero100thou}~for~$100,000$, \Glyph{hieromil}~for~$1,000,000$, +\Glyph{hiero10mil}\ for~$10,000,000$.\Reference{3} The symbol for~$1$ represents a vertical +staff; that for~$10,000$ a pointing finger; that for~$100,000$ a +burbot; that for~$1,000,000$, a man in astonishment. The +significance of the remaining symbols is very doubtful. +The writing of numbers with these hieroglyphics was very +cumbrous. The unit symbol of each order was repeated as +many times as there were units in that order. The principle +employed was the \emph{additive}. Thus, $23$~was written \Glyph{hiero10}\,\Glyph{hiero10}\Glyph{hiero1}\Glyph{hiero1}\Glyph{hiero1}. + +Besides the hieroglyphics, Egypt possesses the \emph{hieratic} and +\emph{demotic} writings, but for want of space we pass them by. + +Herodotus makes an important statement concerning the +mode of computing among the Egyptians. He says that +they ``\emph{calculate with pebbles} by moving the hand from right +to left, while the Hellenes move it from left to right.'' +Herein we recognise again that \emph{instrumental} method of +figuring so extensively used by peoples of antiquity. The +Egyptians used the decimal scale. Since, in figuring, they +moved their hands horizontally, it seems probable that they +used ciphering-boards with vertical columns. In each column +there must have been not more than nine pebbles, for ten +pebbles would be equal to one pebble in the column next +to the left. + +The \textit{Ahmes papyrus} contains interesting information on +the way in which the Egyptians employed fractions. Their +\index{Fractions!Egyptian}% +%% -----File: 028.png---Folio 14------- +methods of operation were, of course, radically different from +ours. Fractions were a subject of very great difficulty with +the ancients. Simultaneous changes in both numerator and +denominator were usually avoided. In manipulating fractions +the Babylonians kept the denominators ($60$) constant. The +Romans likewise kept them constant, but equal to~$12$. The +Egyptians and Greeks, on the other hand, kept the numerators +\Addendlabel{14}% +constant, and dealt with variable denominators. Ahmes used +the term ``fraction'' in a restricted sense, for he applied it +only to \emph{unit-fractions}, or fractions having unity for the numerator. +It was designated by writing the denominator and then +placing over it a dot. Fractional values which could not be +expressed by any one unit-fraction were expressed as the +\emph{sum} of two or more of them. Thus, he wrote $\frac{1}{3}\, \frac{1}{15}$ in place +of~$\frac{2}{5}$. The first important problem naturally arising was, how +to represent any fractional value as the sum of unit-fractions. +This was solved by aid of a table, given in the papyrus, in +which all fractions of the form $\dfrac{2\Fstr}{2n+1}$ (where $n$ designates +successively all the numbers up to~$49$) are reduced to the +sum of unit-fractions. Thus, $\frac{2}{7} = \frac{1}{4}\, \frac{1}{28}$; $\frac{2}{99} = \frac{1}{66}\, \frac{1}{198}$. When, +by whom, and how this table was calculated, we do not know. +Probably it was compiled empirically at different times, by +different persons. It will be seen that by repeated application +of this table, a fraction whose numerator exceeds two +can be expressed in the desired form, provided that there +is a fraction in the table having the same denominator that +\emph{it} has. Take, for example, the problem, to divide $5$ by~$21$. +In the first place, $5 = 1 + 2 + 2$. From the table we get +$\frac{2}{21} = \frac{1}{14}\, \frac{1}{42}$. Then $\frac{5}{21} = \frac{1}{21} + (\frac{1}{14}\, \frac{1}{42}) + (\frac{1}{14}\, \frac{1}{42}) = \frac{1}{21} + (\frac{2}{14}\, \frac{2}{42}) += \frac{1}{21}\, \frac{1}{7}\, \frac{1}{21} = \frac{1}{7}\, \frac{2}{21} = \frac{1}{7}\, \frac{1}{14}\, \frac{1}{42}$. The papyrus contains problems +in which it is required that fractions be raised by addition +or multiplication to given whole numbers or to other fractions. +%% -----File: 029.png---Folio 15------- +\index{Notation!in algebra}% +\index{Rhind papyrus|)}% +For example, it is required to increase $\frac{1}{4}\, \frac{1}{8}\, \frac{1}{10}\, \frac{1}{30}\, \frac{1}{45}$ to~$1$. +The common denominator taken appears to be~$45$, for the +numbers are stated as $11 \frac{1}{4}$, $5 \frac{1}{2}\, \frac{1}{8}$, $4 \frac{1}{2}$, $1 \frac{1}{2}$,~$1$. The sum of these +is $23 \frac{1}{2}\, \frac{1}{4}\, \frac{1}{8}$ forty-fifths. Add to this $\frac{1}{9}\, \frac{1}{40}$, and the sum is~$\frac{2}{3}$. +Add~$\frac{1}{3}$, and we have~$1$. Hence the quantity to be added to +the given fraction is $\frac{1}{3}\, \frac{1}{9}\, \frac{1}{40}$. + +Having finished the subject of fractions, Ahmes proceeds to +the solution of equations of one unknown quantity. The unknown +\index{Equations!solution of}% +quantity is called `hau' or heap. Thus the problem, +``heap, its~$\frac{1}{7}$, its whole, it makes~$19$,'' \ie~$\dfrac{x}{7} + x = 19$. In this +case, the solution is as follows: $\dfrac{8x}{7} = 19$; $\dfrac{x}{7} = 2 \frac{1}{4}\, \frac{1}{8}$; $x = 16 \frac{1}{2}\, \frac{1}{8}$. +But in other problems, the solutions are effected by various +other methods. It thus appears that the beginnings of algebra +\index{Algebra!Beginnings in Egypt}% +are as ancient as those of geometry. + +The principal defect of Egyptian arithmetic was the lack of +a simple, comprehensive symbolism---a defect which not even +the Greeks were able to remove. + +The Ahmes papyrus doubtless represents the most advanced +\index{Ahmes|)}% +attainments of the Egyptians in arithmetic and geometry. It +is remarkable that they should have reached so great proficiency +in mathematics at so remote a period of antiquity. But +strange, indeed, is the fact that, during the next two thousand +years, they should have made no progress whatsoever in it. +The conclusion forces itself upon us, that they resemble the +Chinese in the \emph{stationary character}, not only of their government, +but also of their learning. All the knowledge of geometry +which they possessed when Greek scholars visited them, +six centuries~\BC, was doubtless known to them two thousand +years earlier, when they built those stupendous and gigantic +structures---the pyramids. An explanation for this stagnation +of learning has been sought in the fact that their early discoveries +in mathematics and medicine had the misfortune of +%% -----File: 030.png---Folio 16------- +being entered upon their sacred books and that, in after ages, +it was considered heretical to augment or modify anything +therein. Thus the books themselves closed the gates to +progress. + +\Section{The Greeks} +\index{Greeks|(}% + +%[** PP: A unit with this heading is present in ToC, but not in the text] +\SpecialSection{Greek Geometry} + +About the seventh century~\BC\ an active commercial intercourse +sprang up between Greece and Egypt. Naturally there +arose an interchange of ideas as well as of merchandise. +Greeks, thirsting for knowledge, sought the Egyptian priests +for instruction. Thales, Pythagoras, \OE{}nopides, Plato, Democritus, +\index{Democritus!ref.\ to}% +\index{OEnopides@{\OE{}nopides}!ref.\ to}% +\index{Plato!ref.\ to}% +\index{Pythagoras!ref.\ to}% +\index{Thales!ref.\ to}% +Eudoxus, all visited the land of the pyramids. +\index{Eudoxus!ref.\ to}% +Egyptian ideas were thus transplanted across the sea and +there stimulated Greek thought, directed it into new lines, +and gave to it a basis to work upon. Greek culture, therefore, +is not primitive. Not only in mathematics, but also in mythology +and art, Hellas owes a debt to older countries. To +Egypt Greece is indebted, among other things, for its elementary +geometry. But this does not lessen our admiration for +the Greek mind. From the moment that Hellenic philosophers +applied themselves to the study of Egyptian geometry, +this science assumed a radically different aspect. ``Whatever +we Greeks receive, we improve and perfect,'' says Plato. The +Egyptians carried geometry no further than was absolutely +\index{Egyptians|)}% +necessary for their practical wants. The Greeks, on the other +hand, had within them a strong speculative tendency. They +felt a craving to discover the reasons for things. They found +pleasure in the contemplation of \emph{ideal} relations, and loved +science \emph{as} science. + +Our sources of information on the history of Greek geometry +before Euclid consist merely of scattered notices in ancient +writers. The early mathematicians, Thales and Pythagoras, +%% -----File: 031.png---Folio 17------- +left behind no written records of their discoveries. A full +history of Greek geometry and astronomy during this period, +\index{Geometry!Greek|(}% +written by Eudemus, a pupil of Aristotle, has been lost. It +\index{Aristotle!ref.\ to}% +\index{Eudemus}% +was well known to Proclus, who, in his commentaries on +\index{Proclus!ref.\ to}% +Euclid, gives a brief account of it. This abstract constitutes +\index{Euclid!ref.\ to}% +our most reliable information. We shall quote it frequently +under the name of \textit{Eudemian Summary}. +\index{Eudemian Summary}% + +\Subsection{The Ionic School} +\index{Ionic School|(}% + +To \textbf{Thales} of Miletus (640--546~\BC), one of the ``seven wise +\index{Thales}% +men,'' and the founder of the Ionic school, falls the honour of +having introduced the study of geometry into Greece. During +middle life he engaged in commercial pursuits, which took +him to Egypt. He is said to have resided there, and to have +studied the physical sciences and mathematics with the Egyptian +priests. Plutarch declares that Thales soon excelled his +masters, and amazed King Amasis by measuring the heights +of the pyramids from their shadows. According to Plutarch, +this was done by considering that the shadow cast by a vertical +staff of known length bears the same ratio to the shadow +of the pyramid as the height of the staff bears to the height +of the pyramid. This solution presupposes a knowledge of +proportion, and the Ahmes papyrus actually shows that the +\index{Ahmes!ref.\ to}% +\index{Proportion}% +rudiments of proportion were known to the Egyptians. According +to Diogenes Laertius, the pyramids were measured by +\index{Diogenes Laertius}% +Thales in a different way; viz.~by finding the length of the +shadow of the pyramid at the moment when the shadow of a +staff was equal to its own length. + +The \textit{Eudemian Summary} ascribes to Thales the invention +of the theorems on the equality of vertical angles, the equality +of the angles at the base of an isosceles triangle, the bisection +of a circle by any diameter, and the congruence of two +%% -----File: 032.png---Folio 18------- +triangles having a side and the two adjacent angles equal respectively. +The last theorem he applied to the measurement +of the distances of ships from the shore. Thus Thales was +\index{Thales}% +the first to apply theoretical geometry to practical uses. The +theorem that all angles inscribed in a semicircle are right +angles is attributed by some ancient writers to Thales, by +others to Pythagoras. Thales was doubtless familiar with +\index{Pythagoras!ref.\ to}% +other theorems, not recorded by the ancients. It has been +inferred that he knew the sum of the three angles of a triangle +to be equal to two right angles, and the sides of equiangular +triangles to be proportional.\Reference{8} The Egyptians must +have made use of the above theorems on the straight line, in +some of their constructions found in the Ahmes papyrus, but +\index{Ahmes!ref.\ to}% +it was left for the Greek philosopher to give these truths, +which others saw, but did not formulate into words, an +explicit, abstract expression, and to put into scientific language +and subject to proof that which others merely felt to +be true. Thales may be said to have created the geometry +of lines, essentially abstract in its character, while the Egyptians +studied only the geometry of surfaces and the rudiments +of solid geometry, empirical in their character.\Reference{8} + +With Thales begins also the study of scientific astronomy. +\index{Astronomy!Greek}% +He acquired great celebrity by the prediction of a solar eclipse +in 585~\BC. Whether he predicted the day of the occurrence, +or simply the year, is not known. It is told of him that +while contemplating the stars during an evening walk, he fell +into a ditch. The good old woman attending him exclaimed, +``How canst thou know what is doing in the heavens, when +thou seest not what is at thy feet?'' + +The two most prominent pupils of Thales were \textbf{Anaximander} +\index{Anaximander}% +(b.~611~\BC) and \textbf{Anaximenes} (b.~570~\BC). They studied +\index{Anaximenes}% +chiefly astronomy and physical philosophy. Of \textbf{Anaxagoras}, a +\index{Anaxagoras}% +pupil of Anaximenes, and the last philosopher of the Ionic +%% -----File: 033.png---Folio 19------- +\index{Circle-squarers}% +\index{Pythagorean School|(}% +school, we know little, except that, while in prison, he passed +his time attempting to square the circle. This is the first +\index{Circle}% +time, in the history of mathematics, that we find mention of +the famous problem of the quadrature of the circle, that rock +upon which so many reputations have been destroyed. It +turns upon the determination of the exact value of~$\pi$. Approximations +to~$\pi$ had been made by the Chinese, Babylonians, +\index{Babylonians!ref.\ to}% +\index{Chinese}% +Hebrews, and Egyptians. But the invention of a method to +\index{Egyptians}% +\index{Hebrews}% +find its \emph{exact} value, is the knotty problem which has engaged +the attention of many minds from the time of Anaxagoras +down to our own. Anaxagoras did not offer any solution of +it, and seems to have luckily escaped paralogisms. + +About the time of Anaxagoras, but isolated from the Ionic +school, flourished \textbf{\OE{}nopides} of Chios. Proclus ascribes to him +\index{OEnopides@{\OE{}nopides}}% +\index{Proclus!ref.\ to}% +the solution of the following problems: From a point without, +to draw a perpendicular to a given line, and to draw an angle +on a line equal to a given angle. That a man could gain a +reputation by solving problems so elementary as these, indicates +that geometry was still in its infancy, and that the +Greeks had not yet gotten far beyond the Egyptian constructions. + +The Ionic school lasted over one hundred years. The +\index{Ionic School|)}% +progress of mathematics during that period was slow, as +compared with its growth in a later epoch of Greek history. +A new impetus to its progress was given by Pythagoras. +\index{Pythagoras|(}% + +\Subsection{The School of Pythagoras} + +\textbf{Pythagoras} (580?--500?~\BC) was one of those figures which +impressed the imagination of succeeding times to such an +extent that their real histories have become difficult to be +discerned through the mythical haze that envelops them. The +following account of Pythagoras excludes the most doubtful +%% -----File: 034.png---Folio 20------- +statements. He was a native of Samos, and was drawn by +the fame of Pherecydes to the island of Syros. He then +\index{Pherecydes}% +visited the ancient Thales, who incited him to study in Egypt. +\index{Thales!ref.\ to}% +He sojourned in Egypt many years, and may have visited +Babylon. On his return to Samos, he found it under the +tyranny of Polycrates. Failing in an attempt to found a +school there, he quitted home again and, following the current +of civilisation, removed to Magna Græcia in South Italy. He +settled at Croton, and founded the famous Pythagorean school. +This was not merely an academy for the teaching of philosophy, +mathematics, and natural science, but it was a brotherhood, +the members of which were united for life. This brotherhood +had observances approaching masonic peculiarity. They were +forbidden to divulge the discoveries and doctrines of their +school. Hence we are obliged to speak of the Pythagoreans +as a body, and find it difficult to determine to whom each +particular discovery is to be ascribed. The Pythagoreans +themselves were in the habit of referring every discovery back +to the great founder of the sect. + +This school grew rapidly and gained considerable political +ascendency\DPnote{** PP: [sic]}. But the mystic and secret observances, introduced +in imitation of Egyptian usages, and the aristocratic +tendencies of the school, caused it to become an object of +suspicion. The democratic party in Lower Italy revolted and +destroyed the buildings of the Pythagorean school. Pythagoras +fled to Tarentum and thence to Metapontum, where he was +murdered. + +Pythagoras has left behind no mathematical treatises, and +our sources of information are rather scanty. Certain it is +that, in the Pythagorean school, mathematics was the principal +study. Pythagoras raised mathematics to the rank of a science. +Arithmetic was courted by him as fervently as geometry. In +\index{Arithmetic!Pythagoreans}% +fact, arithmetic is the foundation of his philosophic system. +%% -----File: 035.png---Folio 21------- +\index{Regular solids}% + +The \textit{Eudemian Summary} says that ``Pythagoras changed +\index{Eudemian Summary}% +the study of geometry into the form of a liberal education, +for he examined its principles to the bottom, and investigated +its theorems in an immaterial and intellectual manner.'' His +geometry was connected closely with his arithmetic. He was +especially fond of those geometrical relations which admitted +of arithmetical expression. + +Like Egyptian geometry, the geometry of the Pythagoreans +is much concerned with areas. To Pythagoras is ascribed the +important theorem that the square on the hypotenuse of a +right triangle is equal to the sum of the squares on the other +two sides. He had probably learned from the Egyptians the +truth of the theorem in the special case when the sides are +$3$,~$4$,~$5$, respectively. The story goes, that Pythagoras was so +jubilant over this discovery that he sacrificed a hecatomb. Its +authenticity is doubted, because the Pythagoreans believed in +the transmigration of the soul and opposed, therefore, the +shedding of blood. In the later traditions of the Neo-Pythagoreans +this objection is removed by replacing this bloody sacrifice +by that of ``an ox made of flour''! The proof of the law +of three squares, given in Euclid's \textit{Elements}, I.~47, is due to +\index{Euclid!ref.\ to}% +Euclid himself, and not to the Pythagoreans. What the Pythagorean +method of proof was has been a favourite topic for +conjecture. + +The theorem on the sum of the three angles of a triangle, +presumably known to Thales, was proved by the Pythagoreans +\index{Thales!ref.\ to}% +after the manner of Euclid. They demonstrated also that the +plane about a point is completely filled by six equilateral +triangles, four squares, or three regular hexagons, so that it +is possible to divide up a plane into figures of either kind. + +From the equilateral triangle and the square arise the solids, +namely the tetraedron, octaedron, icosaedron,\DPnote{[** PP: Original spellings]} and the cube. +These solids were, in all probability, known to the Egyptians, +%% -----File: 036.png---Folio 22------- +\index{Irrationals}% +\index{Star-polygons}% +excepting, perhaps, the icosaedron. In Pythagorean philosophy, +they represent respectively the four elements of the +physical world; namely, fire, air, water, and earth. Later +another regular solid was discovered, namely the dodecaedron, +which, in absence of a fifth element, was made to represent +the universe itself. Iamblichus states that Hippasus, a Pythagorean, +\index{Hippasus}% +\index{Iamblichus!ref.\ to}% +perished in the sea, because he boasted that he first +divulged ``the sphere with the twelve pentagons.'' The star-shaped +pentagram was used as a symbol of recognition by the +Pythagoreans, and was called by them Health. + +Pythagoras called the sphere the most beautiful of all solids, +and the circle the most beautiful of all plane figures. The +treatment of the subjects of proportion and of irrational +\index{Proportion}% +quantities by him and his school will be taken up under the +head of arithmetic. + +According to Eudemus, the Pythagoreans invented the problems +\index{Eudemus}% +concerning the application of areas, including the cases +of defect and excess, as in Euclid, VI.~28,~29. +\index{Euclid!ref.\ to}% + +They were also familiar with the construction of a polygon +equal in area to a given polygon and similar to another given +polygon. This problem depends upon several important and +somewhat advanced theorems, and testifies to the fact that +the Pythagoreans made no mean progress in geometry. + +Of the theorems generally ascribed to the Italian school, +some cannot be attributed to Pythagoras himself, nor to his +earliest successors. The progress from empirical to reasoned +solutions must, of necessity, have been slow. It is worth +noticing that on the circle no theorem of any importance was +discovered by this school. + +Though politics broke up the Pythagorean fraternity, yet +the school continued to exist at least two centuries longer. +Among the later Pythagoreans, Philolaus and Archytas are +\index{Philolaus}% +the most prominent. \textbf{Philolaus} wrote a book on the Pythagorean +%% -----File: 037.png---Folio 23------- +\index{Pythagoras|)}% +\index{Pythagorean School|)}% +doctrines. By him were first given to the world the +teachings of the Italian school, which had been kept secret +for a whole century. The brilliant \textbf{Archytas} of Tarentum +\index{Archytas}% +(428--347~\BC), known as a great statesman and general, and +universally admired for his virtues, was the only great geometer +among the Greeks when Plato opened his school. Archytas +\index{Plato!ref.\ to}% +was the first to apply geometry to mechanics and to treat +\index{Mechanics!Greek}% +the latter subject methodically. He also found a very ingenious +mechanical solution to the problem of the duplication of +\index{Duplication of the cube|(}% +the cube. His solution involves clear notions on the generation +of cones and cylinders. This problem reduces itself to +finding two mean proportionals between two given lines. +These mean proportionals were obtained by Archytas from +the section of a half-cylinder. The doctrine of proportion +\index{Proportion}% +was advanced through him. + +There is every reason to believe that the later Pythagoreans +exercised a strong influence on the study and development of +mathematics at Athens. The Sophists acquired geometry from +Pythagorean sources. Plato bought the works of Philolaus, +and had a warm friend in Archytas. + +\Subsection{The Sophist School} +\index{Sophist School|(}% + +After the defeat of the Persians under Xerxes at the battle +of Salamis, 480~\BC, a league was formed among the Greeks +to preserve the freedom of the now liberated Greek cities on +the islands and coast of the Ægæan Sea. Of this league +Athens soon became leader and dictator. She caused the +separate treasury of the league to be merged into that of +Athens, and then spent the money of her allies for her own +aggrandisement. Athens was also a great commercial centre. +Thus she became the richest and most beautiful city of antiquity. +All menial work was performed by slaves. The +%% -----File: 038.png---Folio 24------- +\index{Pythagoras!ref.\ to}% +\index{Trisection of angles}% +citizen of Athens was well-to-do and enjoyed a large amount +of leisure. The government being purely democratic, every +citizen was a politician. To make his influence felt among +his fellow-men he must, first of all, be educated. Thus there +arose a demand for teachers. The supply came principally +from Sicily, where Pythagorean doctrines had spread. These +teachers were called \emph{Sophists}, or ``wise men.'' Unlike the +Pythagoreans, they accepted pay for their teaching. Although +rhetoric was the principal feature of their instruction, they +also taught geometry, astronomy, and philosophy. Athens +\index{Astronomy!Greek}% +soon became the headquarters of Grecian men of letters, and +of mathematicians in particular. The home of mathematics +among the Greeks was first in the Ionian Islands, then in +Lower Italy, and during the time now under consideration, +at Athens. + +The geometry of the circle, which had been entirely +\index{Circle|(}% +neglected by the Pythagoreans, was taken up by the Sophists. +Nearly all their discoveries were made in connection with +their innumerable attempts to solve the following three +famous problems:--- + +(1)~To trisect an arc or an angle. + +(2)~To ``double the cube,'' \ie~to find a cube whose \emph{volume} +is double that of a given cube. + +(3)~To ``square the circle,'' \ie~to find a square or some +other rectilinear figure exactly equal in area to a given circle. + +These problems have probably been the subject of more +discussion and research than any other problems in mathematics. +The bisection of an angle was one of the easiest +problems in geometry. The trisection of an angle, on the +other hand, presented unexpected difficulties. A right angle +had been divided into three equal parts by the Pythagoreans. +But the general problem, though easy in appearance, transcended +the power of \emph{elementary} geometry. Among the first +%% -----File: 039.png---Folio 25------- +\index{Dinostratus!ref.\ to}% +to wrestle with it was \textbf{Hippias of Elis}, a contemporary of +\index{Hippias of Elis}% +Socrates, and born about 460~\BC. Like all the later geometers, +he failed in effecting the trisection by means of a ruler +and compass only. Proclus mentions a man, Hippias, presumably +Hippias of Elis, as the inventor of a transcendental curve +which served to divide an angle not only into three, but into +any number of equal parts. This same curve was used later +by Deinostratus and others for the quadrature of the circle. +On this account it is called the \emph{quadratrix}. +\index{Quadratrix}% + +The Pythagoreans had shown that the diagonal of a square +is the side of another square having double the area of the +original one. This probably suggested the problem of the +duplication of the cube, \ie~to find the edge of a cube having +double the volume of a given cube. Eratosthenes ascribes to +\index{Eratosthenes!ref.\ to}% +this problem a different origin. The Delians were once suffering +from a pestilence and were ordered by the oracle to +double a certain cubical altar. Thoughtless workmen simply +constructed a cube with edges twice as long, but this did not +pacify the gods. The error being discovered, Plato was consulted +on the matter. He and his disciples searched eagerly +for a solution to this ``Delian Problem.'' \textbf{Hippocrates of Chios} +\index{Hippocrates of Chios}% +(about 430~\BC), a talented mathematician, but otherwise slow +and stupid, was the first to show that the problem could be +reduced to finding two mean proportionals between a given +line and another twice as long. For, in the proportion $a:x +=x:y=y:2a$, since $x^2 = ay$ and $y^2 = 2ax$ and $x^4 = a^2y^2$, we +have $x^4 = 2a^3x$ and $x^3 = 2a^3$. But he failed to find the two +mean proportionals. His attempt to square the circle was +also a failure; for though he made himself celebrated by +squaring a lune, he committed an error in attempting to apply +\index{Lune, squaring of}% +this result to the squaring of the circle. + +In his study of the quadrature and duplication-problems, +\index{Duplication of the cube|)}% +Hippocrates contributed much to the geometry of the circle. +%% -----File: 040.png---Folio 26------- +\index{Irrationals}% + +The subject of similar figures was studied and partly +developed by Hippocrates. This involved the theory of +proportion. Proportion had, thus far, been used by the +\index{Proportion}% +Greeks only in numbers. They never succeeded in uniting +the notions of numbers and magnitudes. The term ``number'' +was used by them in a restricted sense. What we call +irrational numbers was not included under this notion. Not +even rational fractions were called numbers. They used the +\index{Fractions!Greek}% +word in the same sense as we use ``integers.'' Hence numbers +were conceived as \emph{discontinuous}, while magnitudes were +\emph{continuous}. The two notions appeared, therefore, entirely +distinct. The chasm between them is exposed to full view +in the statement of Euclid that ``incommensurable magnitudes +\index{Euclid!ref.\ to}% +do not have the same ratio as numbers.'' In Euclid's +\textit{Elements} we find the theory of proportion of magnitudes +developed and treated independent of that of numbers. The +transfer of the theory of proportion from numbers to magnitudes +(and to lengths in particular) was a difficult and +important step. + +Hippocrates added to his fame by writing a geometrical +text-book, called the \textit{Elements}. This publication shows that +the Pythagorean habit of secrecy was being abandoned; +secrecy was contrary to the spirit of Athenian life. + +The Sophist \textbf{Antiphon}, a contemporary of Hippocrates, introduced +\index{Antiphon}% +the \emph{process} of exhaustion for the purpose of solving +\index{Exhaustion, method of}% +the problem of the quadrature. He did himself credit by +remarking that by inscribing in a circle a square, and on its +sides erecting isosceles triangles with their vertices in the +circumference, and on the sides of these triangles erecting +new triangles, etc., one could obtain a succession of regular +polygons of $8$, $16$, $32$, $64$~sides, and so on, of which each +approaches nearer to the circle than the previous one, until +the circle is finally \emph{exhausted}. Thus is obtained an inscribed +%% -----File: 041.png---Folio 27------- +polygon whose sides coincide with the circumference. Since +there can be found squares equal in area to any polygon, +there also can be found a square equal to the last polygon +inscribed, and therefore equal to the circle itself. \textbf{Bryson +\index{Bryson of Heraclea}% +of Heraclea}, a contemporary of Antiphon, advanced the problem +\index{Antiphon!ref.\ to}% +of the quadrature considerably by circumscribing polygons +at the same time that he inscribed polygons. He erred, +however, in assuming that the area of a circle was the arithmetical +mean between circumscribed and inscribed polygons. +Unlike Bryson and the rest of Greek geometers, Antiphon +seems to have believed it possible, by continually doubling +the sides of an inscribed polygon, to obtain a polygon coinciding +with the circle. This question gave rise to lively +disputes in Athens. If a polygon can coincide with the +circle, then, says Simplicius, we must put aside the notion +that magnitudes are divisible \textit{ad infinitum}. Aristotle always +\index{Aristotle!ref.\ to}% +supported the theory of the infinite divisibility, while Zeno, +\index{Zeno}% +the Stoic, attempted to show its absurdity by proving that +if magnitudes are infinitely divisible, motion is impossible. +Zeno argues that Achilles could not overtake a tortoise; for +\index{Achilles and tortoise, paradox of}% +while he hastened to the place where the tortoise had been +when he started, the tortoise crept some distance ahead, and +while Achilles reached that second spot, the tortoise again +moved forward a little, and so on. Thus the tortoise was +always in advance of Achilles. Such arguments greatly confounded +Greek geometers. No wonder they were deterred +by such paradoxes from introducing the idea of infinity into +\index{Infinity}% +their geometry. It did not suit the rigour of their proofs. + +The process of Antiphon and Bryson gave rise to the cumbrous +but perfectly rigorous ``method of exhaustion.'' In +determining the ratio of the areas between two curvilinear +plane figures, say two circles, geometers first inscribed or +circumscribed similar polygons, and then by increasing indefinitely +%% -----File: 042.png---Folio 28------- +the number of sides, nearly exhausted the spaces +between the polygons and circumferences. From the theorem +that similar polygons inscribed in circles are to each +other as the squares on their diameters, geometers may have +divined the theorem attributed to Hippocrates of Chios that +\index{Hippocrates of Chios}% +the circles, which differ but little from the last drawn polygons, +must be to each other as the squares on their diameters. +But in order to exclude all vagueness and possibility of doubt, +later Greek geometers applied reasoning like that in Euclid, +XII.~2, as follows: Let $C$~and~$c$, $D$~and~$d$ be respectively the +circles and diameters in question. Then if the proportion +$D^2 : d^2 = C : c$ is not true, suppose that $D^2 : d^2 = C : c'$. If~$c' < c$, +then a polygon~$p$ can be inscribed in the circle~$c$ which comes +\index{Circle|)}% +nearer to it in area than does~$c'$. If~$P$ be the corresponding +polygon in~$C$, then $P : p = D^2 : d^2 = C : c'$, and $P : C = p : c'$. +Since~$p > c'$, we have~$P > C$, which is absurd. Next they +proved by this same method of \textit{reductio ad absurdum} the +\index{Reductio ad absurdum}% +falsity of the supposition that~$c' > c$. Since~$c'$ can be neither +larger nor smaller than~$c$, it must be equal to it,~\textsc{q.e.d.} +Hankel refers this Method of Exhaustion back to Hippocrates +\index{Exhaustion, method of}% +\index{Hankel!ref.\ to}% +\index{Method of exhaustion}% +of Chios, but the reasons for assigning it to this early +writer, rather than to Eudoxus, seem insufficient. +\index{Eudoxus!ref.\ to}% + +Though progress in geometry at this period is traceable only +at Athens, yet Ionia, Sicily, Abdera in Thrace, and Cyrene +produced mathematicians who made creditable contributions +to the science. We can mention here only \textbf{Democritus of +\index{Democritus}% +Abdera} (about 460--370~\BC), a pupil of Anaxagoras, a friend +\index{Anaxagoras!ref.\ to}% +of Philolaus, and an admirer of the Pythagoreans. He +\index{Philolaus!ref.\ to}% +visited Egypt and perhaps even Persia. He was a successful +geometer and wrote on incommensurable lines, on geometry, +on numbers, and on perspective. None of these works are +extant. He used to boast that in the construction of plane +figures with proof no one had yet surpassed him, not even +%% -----File: 043.png---Folio 29------- +\index{Pythagoras!ref.\ to}% +\index{Sophist School|)}% +the so-called harpedonaptæ (``rope-stretchers'') of Egypt. By +this assertion he pays a flattering compliment to the skill +and ability of the Egyptians. + +\Subsection{The Platonic School} +\index{Platonic School|(}% + +During the Peloponnesian War (431--404~\BC) the progress +of geometry was checked. After the war, Athens sank into +the background as a minor political power, but advanced more +and more to the front as the leader in philosophy, literature, +and science. Plato was born at Athens in 429~\BC, the year +\index{Plato|(}% +of the great plague, and died \Change{in~348.}{in 348~\BC.} He was a pupil and +near friend of Socrates, but it was not from him that he +acquired his taste for mathematics. After the death of Socrates, +Plato travelled extensively. In Cyrene he studied +mathematics under Theodorus. He went to Egypt, then to +\index{Theodorus!ref.\ to}% +Lower Italy and Sicily, where he came in contact with the +Pythagoreans. Archytas of Tarentum and Timæus of Locri +\index{Archytas!ref.\ to}% +\index{Timæus of Locri}% +became his intimate friends. On his return to Athens, about +389~\BC, he founded his school in the groves of the \textit{Academia}, +and devoted the remainder of his life to teaching and writing. + +Plato's physical philosophy is partly based on that of the +Pythagoreans. Like them, he sought in arithmetic and +\index{Arithmetic!Platonists}% +geometry the key to the universe. When questioned about +the occupation of the Deity, Plato answered that ``He geometrises +continually.'' Accordingly, a knowledge of geometry +is a necessary preparation for the study of philosophy. To +show how great a value he put on mathematics and how +necessary it is for higher speculation, Plato placed the inscription +over his porch, ``Let no one who is unacquainted with +geometry enter here.'' Xenocrates, a successor of Plato as +\index{Xenocrates}% +teacher in the Academy, followed in his master's footsteps, by +declining to admit a pupil who had no mathematical training, +%% -----File: 044.png---Folio 30------- +with the remark, ``Depart, for thou hast not the grip of +philosophy.'' Plato observed that geometry trained the mind +for correct and vigorous thinking. Hence it was that the +\textit{Eudemian Summary} says, ``He filled his writings with mathematical +\index{Eudemian Summary}% +discoveries, and exhibited on every occasion the remarkable +connection between mathematics and philosophy.'' + +With Plato as the head-master, we need not wonder that +the Platonic school produced so large a number of mathematicians. +Plato did little real original work, but he made +valuable improvements in the logic and methods employed +in geometry. It is true that the Sophist geometers of the +previous century were rigorous in their proofs, but as a rule +they did not reflect on the inward nature of their methods. +They used the axioms without giving them explicit expression, +\index{Axioms (of geometry)}% +and the geometrical concepts, such as the point, line, surface, +etc., without assigning to them formal definitions. The Pythagoreans +called a point ``unity in position,'' but this is a +statement of a philosophical theory rather than a definition. +Plato objected to calling a point a ``geometrical fiction.'' He +defined a point as the ``beginning of a line'' or as ``an indivisible +line,'' and a line as ``length without breadth.'' He called +the point, line, surface, the `boundaries' of the line, surface, +solid, respectively. Many of the definitions in Euclid are to +\index{Euclid!ref.\ to}% +be ascribed to the Platonic school. The same is probably +true of Euclid's axioms. Aristotle refers to Plato the axiom +that ``equals subtracted from equals leave equals.'' + +One of the greatest achievements of Plato and his school is +the invention of \emph{analysis} as a method of proof. To be sure, +\index{Analysis!(in synthetic geometry)}% +this method had been used unconsciously by Hippocrates and +\index{Hippocrates of Chios}% +others; but Plato, like a true philosopher, turned the instinctive +logic into a conscious, legitimate method. + +The terms \emph{synthesis} and \emph{analysis} are used in mathematics +\index{Synthesis}% +in a more special sense than in logic. In ancient mathematics +%% -----File: 045.png---Folio 31------- +\index{Circle}% +\index{Trisection of angles}% +they had a different meaning from what they now have. The +oldest definition of mathematical analysis as opposed to synthesis +\index{Synthesis}% +is that given in Euclid, XIII.~5, which in all probability +\index{Euclid!ref.\ to}% +was framed by Eudoxus: ``Analysis is the obtaining of the +\index{Eudoxus!ref.\ to}% +thing sought by assuming it and so reasoning up to an +admitted truth; synthesis is the obtaining of the thing +sought by reasoning up to the inference and proof of it.'' +The analytic method is not conclusive, unless all operations +involved in it are known to be reversible. To remove all +doubt, the Greeks, as a rule, added to the analytic process +a synthetic one, consisting of a reversion of all operations +occurring in the analysis. Thus the aim of analysis was to +aid in the discovery of synthetic proofs or solutions. + +Plato is said to have solved the problem of the duplication +\index{Duplication of the cube}% +of the cube. But the solution is open to the very same objection +which he made to the solutions by Archytas, Eudoxus, +\index{Archytas!ref.\ to}% +and Menæchmus. He called their solutions not geometrical, +\index{Menæchmus!ref.\ to}% +but mechanical, for they required the use of other instruments +than the ruler and compasses. He said that thereby ``the good +of geometry is set aside and destroyed, for we again reduce it +to the world of sense, instead of elevating and imbuing it with +the eternal and incorporeal images of thought, even as it is +employed by God, for which reason He always is God.'' These +objections indicate either that the solution is wrongly attributed +to Plato or that he wished to show how easily non-geometric +solutions of that character can be found. It is now +generally admitted that the duplication problem, as well as +the trisection and quadrature problems, cannot be solved by +means of the ruler and compass only. + +Plato gave a healthful stimulus to the study of stereometry, +\index{Plato|)}% +\index{Stereometry}% +which until his time had been entirely neglected. The sphere +and the regular solids had been studied to some extent, but +\index{Regular solids}% +the prism, pyramid, cylinder, and cone were hardly known to +%% -----File: 046.png---Folio 32------- +\index{Astronomy!Greek}% +\index{Deinostratus.!see{Dinostratus}}% +\index{Duplication of the cube}% +exist. All these solids became the subjects of investigation +by the Platonic school. One result of these inquiries was +epoch-making. \textbf{Menæchmus}, an associate of Plato and pupil +\index{Menæchmus}% +\index{Plato!ref.\ to}% +of Eudoxus, invented the conic sections, which, in course of +\index{Conic sections!Greek}% +\index{Eudoxus}% +\index{Eudoxus!ref.\ to}% +only a century, raised geometry to the loftiest height which +it was destined to reach during antiquity. Menæchmus cut +three kinds of cones, the `right-angled,' `acute-angled,' and +`obtuse-angled,' by planes at right angles to a side of the +cones, and thus obtained the three sections which we now call +the parabola, ellipse, and hyperbola. Judging from the two +very elegant solutions of the ``Delian Problem'' by means of +intersections of these curves, Menæchmus must have succeeded +well in investigating their properties. + +Another great geometer was \textbf{Dinostratus}, the brother of +\index{Dinostratus}% +Menæchmus and pupil of Plato. Celebrated is his mechanical +solution of the quadrature of the circle, by means of the \emph{quadratrix} +\index{Quadratrix}% +of Hippias. + +Perhaps the most brilliant mathematician of this period was +\textbf{Eudoxus}. He was born at Cnidus about 408~\BC, studied under +Archytas, and later, for two months, under Plato. He was +\index{Archytas!ref.\ to}% +imbued with a true spirit of scientific inquiry, and has been +called the father of scientific astronomical observation. From +the fragmentary notices of his astronomical researches, found +in later writers, Ideler and Schiaparelli succeeded in reconstructing +\index{Ideler}% +\index{Schiaparelli}% +the system of Eudoxus with its celebrated representation +of planetary motions by ``concentric spheres.'' Eudoxus +\index{Concentric spheres of Eudoxus}% +had a school at Cyzicus, went with his pupils to Athens, visiting +Plato, and then returned to Cyzicus, where he died 355~\BC. +The fame of the academy of Plato is to a large extent +due to Eudoxus's pupils of the school at Cyzicus, among +whom are Menæchmus, Dinostratus, Athenæus, and Helicon. +\index{Athenæus}% +\index{Helicon}% +Diogenes Laertius describes Eudoxus as astronomer, physician, +\index{Diogenes Laertius}% +legislator, as well as geometer. The \textit{Eudemian Summary} +\index{Eudemian Summary}% +%% -----File: 047.png---Folio 33------- +\index{Method of exhaustion!ref.\ to}% +\index{Proportion}% +\index{Stereometry}% +says that Eudoxus ``first increased the number of general +\index{Eudoxus}% +theorems, added to the three proportions three more, and +raised to a considerable quantity the learning, begun by Plato, +\index{Plato!ref.\ to}% +on the subject of the section, to which he applied the analytical +method.'' By this `section' is meant, no doubt, the +``golden section'' (\textit{sectio aurea}), which cuts a line in extreme +\index{Golden section}% +\index{Sectio aurea}% +\index{Section, the golden}% +and mean ratio. The first five propositions in Euclid~XIII. relate +\index{Euclid!ref.\ to}% +to lines cut by this section, and are generally attributed +to Eudoxus. Eudoxus added much to the knowledge of solid +geometry. He proved, says Archimedes, that a pyramid is +exactly one-third of a prism, and a cone one-third of a cylinder, +having equal base and altitude. The proof that spheres are +to each other as the cubes of their radii is probably due to +him. He made frequent and skilful use of the method of +exhaustion, of which he was in all probability the inventor. +\index{Exhaustion, method of}% +A scholiast on Euclid, thought to be Proclus, says further that +\index{Proclus!ref.\ to}% +Eudoxus practically invented the whole of Euclid's fifth book. +Eudoxus also found two mean proportionals between two +given lines, but the method of solution is not known. + +Plato has been called a maker of mathematicians. Besides +the pupils already named, the \textit{Eudemian Summary} mentions +\index{Eudemian Summary}% +the following: \textbf{Theætetus} of Athens, a man of great +\index{Theætetus}% +natural gifts, to whom, no doubt, Euclid was greatly indebted +in the composition of the 10th~book,\Reference[]{8} treating of incommensurables; +\textbf{Leodamas} of Thasos; \textbf{Neocleides} and his pupil \textbf{Leon}, +\index{Leodamas}% +\index{Leon}% +\index{Neocleides}% +who added much to the work of their predecessors, for Leon +wrote an \textit{Elements} carefully designed, both in number and +utility of its proofs; \textbf{Theudius of Magnesia}, who composed a +\index{Theudius}% +very good book of \textit{Elements} and generalised propositions, +which had been confined to particular cases; \textbf{Hermotimus of +\index{Hermotimus}% +Colophon}, who discovered many propositions of the \textit{Elements} +and composed some on \emph{loci}; and, finally, the names of \textbf{Amyclas +\index{Amyclas}% +of Heraclea}, \textbf{Cyzicenus of Athens}, and \textbf{Philippus of Mende}. +\index{Cyzicenus}% +\index{Philippus}% +%% -----File: 048.png---Folio 34------- +\index{Conic sections!Greek}% + +A skilful mathematician of whose life and works we have +no details is \textbf{Aristæus}, the elder, probably a senior contemporary +\index{Aristæus}% +of Euclid. The fact that he wrote a work on conic +\index{Euclid!ref.\ to}% +sections tends to show that much progress had been made in +their study during the time of Menæchmus. Aristæus wrote +\index{Menæchmus!ref.\ to}% +also on regular solids and cultivated the analytic method. +\index{Regular solids}% +His works contained probably a summary of the researches +of the Platonic school.\Reference{8} + +\textbf{Aristotle} (384--322~\BC), the systematiser of deductive logic, +\index{Aristotle}% +though not a professed mathematician, promoted the science +of geometry by improving some of the most difficult definitions. +His \textit{Physics} contains passages with suggestive hints +of the principle of virtual velocities. About his time there +\index{Virtual velocities}% +appeared a work called \textit{Mechanica}, of which he is regarded +by some as the author. Mechanics was totally neglected by +\index{Mechanics!Greek}% +the Platonic school. +\index{Platonic School|)}% + +\Subsection{The First Alexandrian School} +\index{Alexandrian School!(first)|(}% + +In the previous pages we have seen the birth of geometry +in Egypt, its transference to the Ionian Islands, thence to +Lower Italy and to Athens. We have witnessed its growth +in Greece from feeble childhood to vigorous manhood, and +now we shall see it return to the land of its birth and there +derive new vigour. + +During her declining years, immediately following the +Peloponnesian War, Athens produced the greatest scientists +and philosophers of antiquity. It was the time of Plato +\index{Plato!ref.\ to}% +and Aristotle. In 338~\BC, at the battle of Chæronea, Athens +was beaten by Philip of Macedon, and her power was broken +forever. Soon after, Alexander the Great, the son of Philip, +started out to conquer the world. In eleven years he built +up a great empire which broke to pieces in a day. Egypt +%% -----File: 049.png---Folio 35------- +fell to the lot of Ptolemy Soter. Alexander had founded +the seaport of Alexandria, which soon became ``the noblest +of all cities.'' Ptolemy made Alexandria the capital. The +history of Egypt during the next three centuries is mainly +the history of Alexandria. Literature, philosophy, and art +were diligently cultivated. Ptolemy created the university +of Alexandria. He founded the great Library and built laboratories, +museums, a zoölogical garden, and promenades. Alexandria +soon became the great centre of learning. + +Demetrius Phalereus was invited from Athens to take +charge of the Library, and it is probable, says Gow, that +\index{Gow}% +\textbf{Euclid} was invited with him to open the mathematical school. +\index{Euclid|(}% +Euclid's greatest activity was during the time of the first +Ptolemy, who reigned from 306 to 283~\BC. Of the life of +Euclid, little is known, except what is added by Proclus to +\index{Proclus!ref.\ to}% +the \textit{Eudemian Summary}. Euclid, says Proclus, was younger +\index{Eudemian Summary}% +than Plato and older than Eratosthenes and Archimedes, the +\index{Archimedes!ref.\ to}% +\index{Eratosthenes!ref.\ to}% +\index{Plato!ref.\ to}% +latter of whom mentions him. He was of the Platonic sect, and +well read in its doctrines. He collected the \textit{Elements}, put +in order much that Eudoxus had prepared, completed many +\index{Eudoxus!ref.\ to}% +things of Theætetus, and was the first who reduced to unobjectionable +\index{Theætetus!ref.\ to}% +demonstration the imperfect attempts of his predecessors. +When Ptolemy once asked him if geometry could +not be mastered by an easier process than by studying the +\textit{Elements}, Euclid returned the answer, ``There is no royal +road to geometry.'' Pappus states that Euclid was distinguished +\index{Pappus!ref.\ to}% +by the fairness and kindness of his disposition, particularly +toward those who could do anything to advance +the mathematical sciences. Pappus is evidently making a +contrast to Apollonius, of whom he more than insinuates the +\index{Apollonius!ref.\ to}% +opposite character.\Reference{9} A pretty little story is related by Stobæus:\Reference[]{6} +``A youth who had begun to read geometry with +Euclid, when he had learnt the first proposition, inquired, +%% -----File: 050.png---Folio 36------- +\index{Exhaustion, method of}% +\index{Method of exhaustion!ref.\ to}% +`What do I get by learning these things?' So Euclid called +his slave and said, `Give him threepence, since he must +make gain out of what he learns.'\,'' These are about all the +personal details preserved by Greek writers. Syrian and +Arabian writers claim to know much more, but they are unreliable. +At one time Euclid of Alexandria was universally +confounded with Euclid of Megara, who lived a century +earlier. + +The fame of Euclid has at all times rested mainly upon his +book on geometry, called the \textit{Elements}. This book was so far +\index{Elements (Euclid's)|(}% +superior to the \textit{Elements} written by Hippocrates, Leon, and +Theudius, that the latter works soon perished in the struggle +for existence. The Greeks gave Euclid the special title of +``the author of the \textit{Elements}.'' It is a remarkable fact in the +history of geometry, that the \textit{Elements} of Euclid, written two +thousand years ago, are still regarded by many as the best +introduction to the mathematical sciences. In England they +are used at the present time extensively as a text-book in +schools. Some editors of Euclid have, however, been inclined +to credit him with more than is his due. They would have +us believe that a finished and unassailable system of geometry +sprang at once from the brain of Euclid, ``an armed Minerva +from the head of Jupiter.'' They fail to mention the earlier +eminent mathematicians from whom Euclid got his material. +Comparatively few of the propositions and proofs in the +\textit{Elements} are his own discoveries. In fact, the proof of the +``Theorem of Pythagoras'' is the only one directly ascribed to +\index{Pythagoras!ref.\ to}% +him. Allman conjectures that the substance of Books~I., II.,~IV. +\index{Allman}% +comes from the Pythagoreans, that the substance of Book~VI. +is due to the Pythagoreans and Eudoxus, the latter contributing +\index{Eudoxus!ref.\ to}% +the doctrine of proportion as applicable to incommensurables +\index{Incommensurables}% +\index{Proportion}% +and also the Method of Exhaustions (Book~XII.), +that Theætetus contributed much toward Books X.~and~XIII., +\index{Theætetus!ref.\ to}% +%% -----File: 051.png---Folio 37------- +that the principal part of the original work of Euclid himself +is to be found in Book~X\@.\Reference{8} Euclid was the greatest systematiser +of his time. By careful selection from the material +before him, and by logical arrangement of the propositions +selected, he built up, from a few definitions and axioms, a +\index{Axioms (of geometry)}% +proud and lofty structure. It would be erroneous to believe +that he incorporated into his \textit{Elements} all the elementary +theorems known at his time. Archimedes, Apollonius, and +\index{Apollonius!ref.\ to}% +\index{Archimedes!ref.\ to}% +even he himself refer to theorems not included in his \textit{Elements}, +as being well-known truths. + +The text of the \textit{Elements} now commonly used is Theon's +\index{Theon of Alexandria!ref.\ to}% +edition. Theon of Alexandria, the father of Hypatia, brought +\index{Hypatia!ref.\ to}% +out an edition, about 700~years after Euclid, with some alterations +in the text. As a consequence, later commentators, +especially Robert Simson, who laboured under the idea that +\index{Simson!ref.\ to}% +Euclid must be absolutely perfect, made Theon the scapegoat +for all the defects which they thought they could discover +in the text as they knew it. But among the manuscripts sent +by Napoleon~I. from the Vatican to Paris was found a copy of +the \textit{Elements} believed to be anterior to Theon's recension. +Many variations from Theon's version were noticed therein, +but they were not at all important, and showed that Theon +generally made only verbal changes. The defects in the +\textit{Elements} for which Theon was blamed must, therefore, be +due to Euclid himself. The \textit{Elements} has been considered as +offering models of scrupulously rigorous demonstrations. It +is certainly true that in point of rigour it compares favourably +with its modern rivals; but when examined in the light of +strict mathematical logic, it has been pronounced by C.~S. +\index{Logic}% +Peirce to be ``riddled with fallacies.'' The results are correct +\index{Peirce, C.~S.!ref.\ to}% +only because the writer's experience keeps him on his guard. + +At the beginning of our editions of the \textit{Elements}, under +the head of definitions, are given the assumptions of such +%% -----File: 052.png---Folio 38------- +\index{Non-Euclidean geometry}% +\index{Prime numbers}% +notions as the point, line, etc., and some verbal explanations. +Then follow three postulates or demands, and twelve axioms. +\index{Axioms (of geometry)}% +The term `axiom' was used by Proclus, but not by Euclid. +\index{Proclus!ref.\ to}% +He speaks, instead, of `common notions'---common either +to all men or to all sciences. There has been much controversy +among ancient and modern critics on the postulates and +axioms. An immense preponderance of manuscripts and the +testimony of Proclus place the `axioms' about \emph{right angles} +and \emph{parallels} (Axioms 11 and~12) among the postulates.\BiReference{9}{10} +\index{Parallels}% +This is indeed their proper place, for they are really \emph{assumptions}, +and not \emph{common notions} or axioms. The postulate +about \emph{parallels} plays an important rôle in the history of non-Euclidean +geometry. The only postulate which Euclid missed +was the one of superposition, according to which figures +can be moved about in space without any alteration in form +or magnitude. + +The \textit{Elements} contains thirteen books by Euclid, and two, +of which it is supposed that Hypsicles and Damascius are +\index{Damascius!ref.\ to}% +\index{Hypsicles!ref.\ to}% +the authors. The first four books are on plane geometry. +The fifth book treats of the theory of proportion as applied +\index{Proportion}% +to magnitudes in general. The sixth book develops the +geometry of similar figures. The seventh, eighth, ninth +books are on the theory of numbers, or on arithmetic. In the +\index{Arithmetic!Euclid}% +ninth book is found the proof to the theorem that the number +of primes is infinite. The tenth book treats of the theory of +incommensurables. The next three books are on stereometry. +\index{Incommensurables}% +\index{Stereometry}% +The eleventh contains its more elementary theorems; the +twelfth, the metrical relations of the pyramid, prism, cone, +cylinder, and sphere. The thirteenth treats of the regular +polygons, especially of the triangle and pentagon, and then uses +them as faces of the five regular solids; namely, the tetraedron, +\index{Regular solids}% +octaedron, icosaedron, cube, and dodecaedron. The regular +solids were studied so extensively by the Platonists that they +%% -----File: 053.png---Folio 39------- +received the name of ``Platonic figures.'' The statement of +\index{Platonic figures}% +Proclus that the whole aim of Euclid in writing the \textit{Elements} +\index{Proclus!ref.\ to}% +was to arrive at the construction of the regular solids, is +obviously wrong. The fourteenth and fifteenth books, treating +of solid geometry, are apocryphal. + +A remarkable feature of Euclid's, and of all Greek geometry +before Archimedes is that it eschews mensuration. Thus the +\index{Archimedes!ref.\ to}% +theorem that the area of a triangle equals half the product +of its base and its altitude is foreign to Euclid. + +Another extant book of Euclid is the \textit{Data}. It seems to +\index{Data (Euclid's)}% +have been written for those who, having completed the \textit{Elements}, +wish to acquire the power of solving new problems +proposed to them. The \textit{Data} is a course of practice in \emph{analysis}. +\index{Analysis!(in synthetic geometry)}% +It contains little or nothing that an intelligent student +could not pick up from the \textit{Elements} itself. Hence it contributes +\index{Elements (Euclid's)|)}% +little to the stock of scientific knowledge. The following +are the other extant works generally attributed to Euclid: +\textit{Phænomena}, a work on spherical geometry and astronomy; +\index{Astronomy!Greek}% +\textit{Optics}, which develops the hypothesis that light proceeds +\index{Optics}% +from the eye, and not from the object seen; \textit{Catoptrica}, containing +propositions on reflections from mirrors; \textit{De~Divisionibus}, +a treatise on the division of plane figures into parts +having to one another a given ratio; \textit{Sectio Canonis}, a work +on musical intervals. His treatise on \textit{Porisms} is lost; but +\index{Porisms}% +much learning has been expended by Robert Simson and +\index{Simson!ref.\ to}% +M.~Chasles in restoring it from numerous notes found in the +\index{Chasles!ref.\ to}% +writings of Pappus. The term `porism' is vague in meaning. +\index{Pappus!ref.\ to}% +\Addendlabel{39}% +The aim of a porism is not to state some property or truth, +like a theorem, nor to effect a construction, like a problem, +but to find and bring to view a thing which necessarily exists +with given numbers or a given construction, as, to find the +centre of a given circle, or to find the G.C.D. of two given +numbers.\Reference{6} His other lost works are \textit{Fallacies}, containing +%% -----File: 054.png---Folio 40------- +exercises in detection of fallacies; \textit{Conic Sections}, in four +\index{Conic sections!Greek}% +books, which are the foundation of a work on the same subject +by Apollonius; and \textit{Loci on a Surface}, the meaning of +\index{Apollonius!ref.\ to}% +which title is not understood. Heiberg believes it to mean +``loci which are surfaces.'' + +The immediate successors of Euclid in the mathematical +\index{Euclid|)}% +school at Alexandria were probably \textbf{Conon}, \textbf{Dositheus}, and +\index{Conon}% +\index{Dositheus}% +\textbf{Zeuxippus}, but little is known of them. +\index{Zeuxippus}% + +\textbf{Archimedes} (287?--212~\BC), the greatest mathematician of +\index{Archimedes|(}% +antiquity, was born in Syracuse. Plutarch calls him a relation +of King Hieron; but more reliable is the statement of +Cicero, who tells us he was of low birth. Diodorus says he +\index{Diodorus}% +visited Egypt, and, since he was a great friend of Conon and +Eratosthenes, it is highly probable that he studied in Alexandria. +\index{Eratosthenes!ref.\ to}% +This belief is strengthened by the fact that he had +the most thorough acquaintance with all the work previously +done in mathematics. He returned, however, to Syracuse, +where he made himself useful to his admiring friend and +patron, King Hieron, by applying his extraordinary inventive +genius to the construction of various war-engines, by which +he inflicted much loss on the Romans during the siege of +Marcellus. The story that, by the use of mirrors reflecting +the sun's rays, he set on fire the Roman ships, when they +came within bow-shot of the walls, is probably a fiction. The +city was taken at length by the Romans, and Archimedes +perished in the indiscriminate slaughter which followed. According +to tradition, he was, at the time, studying the diagram +to some problem drawn in the sand. As a Roman soldier +approached him, he called out, ``Don't spoil my circles.'' +The soldier, feeling insulted, rushed upon him and killed +him. No blame attaches to the Roman general Marcellus, +who admired his genius, and raised in his honour a tomb +bearing the figure of a sphere inscribed in a cylinder. When +%% -----File: 055.png---Folio 41------- +\index{Conic sections!Greek}% +Cicero was in Syracuse, he found the tomb buried under +rubbish. + +Archimedes was admired by his fellow-citizens chiefly for +his mechanical inventions; he himself prized far more highly +his discoveries in pure science. He declared that ``every kind +of art which was connected with daily needs was ignoble and +vulgar.'' Some of his works have been lost. The following +are the extant books, arranged approximately in chronological +order: 1.~Two books on \textit{Equiponderance of Planes} or \textit{Centres +of Plane Gravities}, between which is inserted his treatise on +the \textit{Quadrature of the Parabola}; 2.~Two books on the \textit{Sphere} +and \textit{Cylinder}; 3.~The \textit{Measurement of the Circle}; 4.~\textit{On Spirals}; +\index{Circle}% +5.~\textit{Conoids} and \textit{Spheroids}; 6.~The \textit{Sand-Counter}; 7.~Two books +on \textit{Floating Bodies}; 8.~Fifteen \textit{Lemmas}. + +In the book on the \textit{Measurement of the Circle}, Archimedes +proves first that the area of a circle is equal to that of a +right triangle having the length of the circumference for its +base, and the radius for its altitude. In this he assumes that +there exists a straight line equal in length to the circumference---an +assumption objected to by some ancient critics, on +the ground that it is not evident that a straight line can equal +a curved one. The finding of such a line was the next problem. +He first finds an upper limit to the ratio of the circumference +to the diameter, or~$\pi$. To do this, he starts with an +\index{p@{$\pi$}: values for!Archimedean}% +equilateral triangle of which the base is a tangent and the +vertex is the centre of the circle. By successively bisecting +the angle at the centre, by comparing ratios, and by taking the +irrational square roots always a little too small, he finally +arrived at the conclusion that $\pi < 3 \frac{1}{7}$. Next he finds a lower +limit by inscribing in the circle regular polygons of $6$, $12$, $24$, +$48$, $96$~sides, finding for each successive polygon its perimeter, +which is, of course, always less than the circumference. Thus +he finally concludes that ``the circumference of a circle exceeds +%% -----File: 056.png---Folio 42------- +\index{Curves!quadrature of}% +\index{Quadrature of curves}% +three times its diameter by a part which is less than $\frac{1}{7}$ +but more than $\frac{10}{71}$ of the diameter.'' This approximation is +exact enough for most purposes. + +The \textit{Quadrature of the Parabola} contains two solutions to +\index{Parabola}% +the problem---one mechanical, the other geometrical. The +method of exhaustion is used in both. +\index{Exhaustion, method of}% +\index{Method of exhaustion!ref.\ to}% + +Archimedes studied also the ellipse and accomplished its +quadrature, but to the hyperbola he seems to have paid less attention. +It is believed that he wrote a book on conic sections. + +Of all his discoveries Archimedes prized most highly those +in his \textit{Sphere} and \textit{Cylinder}. In it are proved the new +theorems, that the surface of a sphere is equal to four times +a great circle; that the surface of a segment of a sphere is +equal to a circle whose radius is the straight line drawn from +the vertex of the segment to the circumference of its basal +circle; that the volume and the surface of a sphere are $\frac{2}{3}$~of +the volume and surface, respectively, of the cylinder circumscribed +about the sphere. Archimedes desired that the figure +to the last proposition be inscribed on his tomb. This was +ordered done by Marcellus. + +The spiral now called the ``spiral of Archimedes,'' and +described in the book \textit{On Spirals}, was discovered by Archimedes, +\index{Spirals}% +and not, as some believe, by his friend Conon.\Reference{3} His +\index{Conon!ref.\ to}% +treatise thereon is, perhaps, the most wonderful of all his +works. Nowadays, subjects of this kind are made easy by +the use of the infinitesimal calculus. In its stead the ancients +used the method of exhaustion. Nowhere is the fertility of +his genius more grandly displayed than in his masterly use of +this method. With Euclid and his predecessors the method +\index{Euclid!ref.\ to}% +of exhaustion was only the means of proving propositions +which must have been seen and believed before they were +proved. But in the hands of Archimedes it became an instrument +of discovery.\Reference{9} +%% -----File: 057.png---Folio 43------- + +By the word `conoid,' in his book on \textit{Conoids and +Spheroids}, is meant the solid produced by the revolution +of a parabola or a hyperbola about its axis. Spheroids +are produced by the revolution of an ellipse, and are long +or flat, according as the ellipse revolves around the major +or minor axis. The book leads up to the cubature of these +solids. + +We have now reviewed briefly all his extant works on geometry. +His arithmetical treatise and problems will be considered +later. We shall now notice his works on mechanics. +\index{Mechanics!Greek}% +Archimedes is the author of the first sound knowledge on this +subject. Archytas, Aristotle, and others attempted to form +\index{Archytas!ref.\ to}% +\index{Aristotle!ref.\ to}% +the known mechanical truths into a science, but failed. Aristotle +knew the property of the lever, but could not establish +its true mathematical theory. The radical and fatal defect +in the speculations of the Greeks, says Whewell, was ``that +\index{Whewell}% +though they had in their possession facts and ideas, \emph{the ideas +were not distinct and appropriate to the facts}.'' For instance, +Aristotle asserted that when a body at the end of a lever is +moving, it may be considered as having two motions; one in +the direction of the tangent and one in the direction of the +radius; the former motion is, he says, \emph{according to nature}, the +latter \emph{contrary to nature}. These inappropriate notions of +`natural' and `unnatural' motions, together with the habits +of thought which dictated these speculations, made the perception +of the true grounds of mechanical properties impossible.\Reference{11} +It seems strange that even after Archimedes had +entered upon the right path, this science should have remained +absolutely stationary till the time of Galileo---a period of +\index{Galileo!ref.\ to}% +nearly two thousand years. + +The proof of the property of the lever, given in his \textit{Equiponderance +of Planes}, holds its place in text-books to this day. +His estimate of the efficiency of the lever is expressed in the +%% -----File: 058.png---Folio 44------- +\index{Statics}% +saying attributed to him, ``Give me a fulcrum on which to +rest, and I will move the earth.'' + +While the \textit{Equiponderance} treats of solids, or the equilibrium +of solids, the book on \textit{Floating Bodies} treats of hydrostatics. +\index{Hydrostatics}% +His attention was first drawn to the subject of +specific gravity when King Hieron asked him to test whether +a crown, professed by the maker to be pure gold, was not +alloyed with silver. The story goes that our philosopher was +in a bath when the true method of solution flashed on his +mind. He immediately ran home, naked, shouting, ``I have +found it!'' To solve the problem, he took a piece of gold and +a piece of silver, each weighing the same as the crown. According +to one author, he determined the volume of water +displaced by the gold, silver, and crown respectively, and +calculated from that the amount of gold and silver in the +crown. According to another writer, he weighed separately +the gold, silver, and crown, while immersed in water, thereby +determining their loss of weight in water. From these data +he easily found the solution. It is possible that Archimedes +solved the problem by both methods. + +After examining the writings of Archimedes, one can well +understand how, in ancient times, an `Archimedean problem' +came to mean a problem too deep for ordinary minds to solve, +and how an `Archimedean proof' came to be the synonym for +unquestionable certainty. Archimedes wrote on a very wide +range of subjects, and displayed great profundity in each. He +is the Newton of antiquity. + +\textbf{Eratosthenes}, eleven years younger than Archimedes, was a +\index{Eratosthenes}% +native of Cyrene. He was educated in Alexandria under +Callimachus the poet, whom he succeeded as custodian of +the Alexandrian Library. His many-sided activity may be +inferred from his works. He wrote on \textit{Good and Evil}, \textit{Measurement +of the Earth}, \textit{Comedy}, \textit{Geography}, \textit{Chronology}, \textit{Constellations}, +%% -----File: 059.png---Folio 45------- +and the \textit{Duplication of the Cube}. He was also a +\index{Duplication of the cube}% +philologian and a poet. He measured the obliquity of the +ecliptic and invented a device for finding prime numbers. +\index{Prime numbers}% +Of his geometrical writings we possess only a letter to +Ptolemy Euergetes, giving a history of the duplication problem +and also the description of a very ingenious mechanical +contrivance of his own to solve it. In his old age he lost +his eyesight, and on that account is said to have committed +suicide by voluntary starvation. + +About forty years after Archimedes flourished \textbf{Apollonius of +\index{Apollonius|(}% +\index{Archimedes!ref.\ to}% +Perga}, whose genius nearly equalled that of his great predecessor. +He incontestably occupies the second place in distinction +among ancient mathematicians. Apollonius was +born in the reign of Ptolemy Euergetes and died under +Ptolemy Philopator, who reigned 222--205~\BC. He studied at +Alexandria under the successors of Euclid, and for some time, +also, at Pergamum, where he made the acquaintance of that +Eudemus to whom he dedicated the first three books of his +\index{Eudemus}% +\textit{Conic Sections}. The brilliancy of his great work brought him +\index{Conic sections!Greek|(}% +the title of the ``Great Geometer.'' This is all that is known +of his life. + +His \textit{Conic Sections} were in eight books, of which the first +four only have come down to us in the original Greek. The +next three books were unknown in Europe till the middle of +the seventeenth century, when an Arabic translation, made +about~1250, was discovered. The eighth book has never been +found. In~1710 Halley of Oxford published the Greek text +\index{Halley}% +of the first four books and a Latin translation of the remaining +three, together with his conjectural restoration of the +eighth book, founded on the introductory lemmas of Pappus. +\index{Pappus!ref.\ to}% +The first four books contain little more than the substance +of what earlier geometers had done. Eutocius tells us that +\index{Eutocius!ref.\ to}% +Heraclides, in his life of Archimedes, accused Apollonius of +\index{Archimedes|)}% +\index{Heraclides}% +%% -----File: 060.png---Folio 46------- +having appropriated, in his \textit{Conic Sections}, the unpublished +discoveries of that great mathematician. It is difficult to +believe that this charge rests upon good foundation. Eutocius +\index{Eutocius!ref.\ to}% +quotes Geminus as replying that neither Archimedes nor +\index{Geminus!ref.\ to}% +Apollonius claimed to have invented the conic sections, but +that Apollonius had introduced a real improvement. While +the first three or four books were founded on the works of +Menæchmus, Aristæus, Euclid, and Archimedes, the remaining +\index{Aristæus!ref.\ to}% +\index{Euclid!ref.\ to}% +\index{Menæchmus!ref.\ to}% +ones consisted almost entirely of new matter. The first three +books were sent to Eudemus at intervals, the other books +\index{Eudemus}% +(after Eudemus's death) to one Attalus. The preface of the +\index{Attalus}% +second book is interesting as showing the mode in which +Greek books were `published' at this time. It reads thus: +``I have sent my son Apollonius to bring you (Eudemus) the +second book of my Conics. Read it carefully and communicate +it to such others as are worthy of it. If Philonides, the +\index{Philonides}% +geometer, whom I introduced to you at Ephesus, comes into +the neighbourhood of Pergamum, give it to him also.''\Reference{12} + +The first book, says Apollonius in his preface to it, ``contains +the mode of producing the three sections and the conjugate +hyperbolas and their principal characteristics, more fully +and generally worked out than in the writings of other +authors.'' We remember that Menæchmus, and all his successors +down to Apollonius, considered only sections of \emph{right} +cones by a plane perpendicular to their sides, and that the +three sections were obtained each from a different cone. +Apollonius introduced an important generalisation. He produced +all the sections from one and the same cone, whether +right or scalene, and by sections which may or may not be +perpendicular to its sides. The old names for the three curves +were now no longer applicable. Instead of calling the three +curves, sections of the `acute-angled,' `right-angled,' and +`obtuse-angled' cone, he called them \emph{ellipse}, \emph{parabola}, and +%% -----File: 061.png---Folio 47------- +\emph{hyperbola}, respectively. To be sure, we find the words `parabola' +and `ellipse' in the works of Archimedes, but they are +\index{Archimedes!ref.\ to}% +probably only interpolations. The word `ellipse' was applied +because $y^2 < px$, $p$~being the parameter; the word `parabola' +was introduced because $y^2 = px$, and the term `hyperbola' +because $y^2 > px$. + +The treatise of Apollonius rests on a unique property of +conic sections, which is derived directly from the nature of +the cone in which these sections are found. How this property +forms the key to the system of the ancients is told in a masterly +way by M.~Chasles.\Reference{13} ``Conceive,'' says he, ``an oblique +\index{Chasles!ref.\ to}% +cone on a circular base; the straight line drawn from its +summit to the centre of the circle forming its base is called +the \emph{axis} of the cone. The plane passing through the axis, +perpendicular to its base, cuts the cone along two lines and +determines in the circle a diameter; the triangle having this +diameter for its base and the two lines for its sides, is called +\emph{the triangle through the axis}. In the formation of his conic +sections, Apollonius supposed the cutting plane to be perpendicular +to the plane of the triangle through the axis. The +points in which this plane meets the two sides of this triangle +are the \emph{vertices} of the curve; and the straight line which joins +these two points is a diameter of it. Apollonius called this +diameter \textit{latus transversum}. At one of the two vertices of the +curve erect a perpendicular (\textit{latus rectum}) to the plane of the +triangle through the axis, of a certain length, to be determined +as we shall specify later, and from the extremity of this perpendicular +draw a straight line to the other vertex of the +curve; now, through any point whatever of the diameter of +the curve, draw at right angles an \emph{ordinate}: the square of this +ordinate, comprehended between the diameter and the curve, +will be equal to the rectangle constructed on the portion of +the ordinate comprised between the diameter and the straight +%% -----File: 062.png---Folio 48-------%[** Quote continues] +line, and the part of the diameter comprised between the first +vertex and the foot of the ordinate. Such is the characteristic +property which Apollonius recognises in his conic sections and +which he uses for the purpose of inferring from it, by adroit +transformations and deductions, nearly all the rest. It plays, +as we shall see, in his hands, almost the same rôle as the +equation of the second degree with two variables (abscissa and +ordinate) in the system of analytic geometry of Descartes. +\index{Descartes!ref.\ to}% + +``It will be observed from this that the diameter of the +curve and the perpendicular erected at one of its extremities +suffice to construct the curve. These are the two elements +which the ancients used, with which to establish their theory +of conics. The perpendicular in question was called by them +\textit{latus erectum}; the moderns changed this name first to that of +\textit{latus rectum}, and afterwards to that of \emph{parameter}.'' +\index{Latus rectum}% +\index{Parameter}% + +The first book of the \textit{Conic Sections} of Apollonius is almost +wholly devoted to the generation of the three principal conic +sections. + +The second book treats mainly of asymptotes, axes, and +diameters. + +The third book treats of the equality or proportionality +of triangles, rectangles, or squares, of which the component +parts are determined by portions of transversals, chords, +asymptotes, or tangents, which are frequently subject to a +great number of conditions. It also touches the subject of +foci of the ellipse and hyperbola. + +In the fourth book, Apollonius discusses the harmonic division +of straight lines. He also examines a system of two +conics, and shows that they cannot cut each other in more +than four points. He investigates the various possible relative +positions of two conics, as, for instance, when they have one +or two points of contact with each other. + +The fifth book reveals better than any other the giant +%% -----File: 063.png---Folio 49------- +\index{Maxima and minima}% +\index{Quadrature of curves}% +intellect of its author. Difficult questions of \emph{maxima and +minima}, of which few examples are found in earlier works, are +here treated most exhaustively. The subject investigated is, +to find the longest and shortest lines that can be drawn from +a given point to a conic. Here are also found the germs of +the subject of \emph{evolutes} and \emph{centres of osculation}. +\index{Centres of osculation}% +\index{Evolutes}% + +The sixth book is on the similarity of conics. + +The seventh book is on conjugate diameters. + +The eighth book, as restored by Halley, continues the subject +of conjugate diameters. + +It is worthy of notice that Apollonius nowhere introduces +the notion of \emph{directrix} for a conic, and that, though he incidentally +\index{Directrix}% +discovered the \emph{focus} of an ellipse and hyperbola, he did +\index{Focus}% +not discover the focus of a parabola.\Reference{6} Conspicuous in his +geometry is also the absence of technical terms and symbols, +which renders the proofs long and cumbrous. + +The discoveries of Archimedes and Apollonius, says M. Chasles,\Reference[]{13} +\index{Archimedes!ref.\ to}% +\index{Chasles!ref.\ to}% +marked the most brilliant epoch of ancient geometry. +Two questions which have occupied geometers of all periods +may be regarded as having originated with them. The first +of these is the quadrature of curvilinear figures, which gave +birth to the infinitesimal calculus. The second is the theory +of conic sections, which was the prelude to the theory of +geometrical curves of all degrees, and to that portion of +\index{Curves!quadrature of}% +geometry which considers only the forms and situations +of figures, and uses only the intersection of lines and surfaces +and the ratios of rectilineal distances. These two great +divisions of geometry may be designated by the names of +\emph{Geometry of Measurements} and \emph{Geometry of Forms and Situations}, +or, Geometry of Archimedes and of Apollonius. + +Besides the \textit{Conic Sections}, Pappus ascribes to Apollonius +\index{Conic sections!Greek|)}% +\index{Pappus!ref.\ to}% +the following works: \textit{On Contacts}, \textit{Plane Loci}, \textit{Inclinations}, +\textit{Section of an Area}, \textit{Determinate Section}, and gives lemmas +%% -----File: 064.png---Folio 50------- +\index{Duplication of the cube}% +\index{Trisection of angles}% +from which attempts have been made to restore the lost +originals. Two books on \textit{De Sectione Rationis} have been +found in the Arabic. The book on \textit{Contacts}, as restored by +Vieta, contains the so-called ``Apollonian Problem'': Given +\index{Apollonian Problem}% +\index{Vieta!ref.\ to}% +three circles, to find a fourth which shall touch the three. + +Euclid, Archimedes, and Apollonius brought geometry to +\index{Archimedes!ref.\ to}% +\index{Euclid!ref.\ to}% +as high a state of perfection as it perhaps could be brought +without first introducing some more general and more powerful +method than the old method of exhaustion. A briefer symbolism, +a Cartesian geometry, an infinitesimal calculus, were +needed. The Greek mind was not adapted to the invention of +general methods. Instead of a climb to still loftier heights +we observe, therefore, on the part of later Greek geometers, a +descent, during which they paused here and there to look +around for details which had been passed by in the hasty +ascent.\Reference{3} + +Among the earliest successors of Apollonius was \textbf{Nicomedes}. +\index{Apollonius|)}% +\index{Nicomedes}% +Nothing definite is known of him, except that he invented the +\emph{conchoid} (``mussel-like''). He devised a little machine by +\index{Conchoid}% +which the curve could be easily described. With aid of the +conchoid he duplicated the cube. The curve can also be used +for trisecting angles in a way much resembling that in the +eighth lemma of Archimedes. Proclus ascribes this mode of +\index{Proclus!ref.\ to}% +trisection to Nicomedes, but Pappus, on the other hand, claims +\index{Pappus!ref.\ to}% +it as his own. The conchoid was used by Newton in constructing +\index{Newton!ref.\ to}% +curves of the third degree. + +About the time of Nicomedes, flourished also \textbf{Diocles}, the +\index{Diocles}% +inventor of the \emph{cissoid} (``ivy-like''). This curve he used for +\index{Cissoid}% +finding two mean proportionals between two given straight +lines. + +About the life of \textbf{Perseus} we know as little as about that of +\index{Perseus}% +Nicomedes and Diocles. He lived some time between 200 and +100~\BC. From Heron and Geminus we learn that he wrote a +\index{Geminus!ref.\ to}% +\index{Heron the Elder!ref.\ to}% +%% -----File: 065.png---Folio 51------- +\index{Astronomy!Greek}% +\index{Isoperimetrical figures}% +work on the \emph{spire}, a sort of anchor-ring surface described by +Heron as being produced by the revolution of a circle around +one of its chords as an axis. The sections of this surface +yield peculiar curves called \emph{spiral sections}, which, according to +Geminus, were thought out by Perseus. These curves appear +to be the same as the \textit{Hippopede} of Eudoxus. +\index{Eudoxus!ref.\ to}% +\index{Hippopede}% + +Probably somewhat later than Perseus lived \textbf{Zenodorus}. He +\index{Zenodorus}% +wrote an interesting treatise on a new subject; namely, \emph{isoperimetrical +figures}. Fourteen propositions are preserved by +Pappus and Theon. Here are a few of them: Of isoperimetrical, +\index{Theon of Alexandria!ref.\ to}% +regular polygons, the one having the largest number of +angles has the greatest area; the circle has a greater area than +any regular polygon of equal periphery; of all isoperimetrical +polygons of $n$~sides, the regular is the greatest; of all solids +having surfaces equal in area, the sphere has the greatest +volume. + +\textbf{Hypsicles} (between 200 and 100~\BC) was supposed to be +\index{Hypsicles}% +the author of both the fourteenth and fifteenth books of +Euclid, but recent critics are of opinion that the fifteenth +book was written by an author who lived several centuries +after Christ. The fourteenth book contains seven elegant +theorems on \emph{regular solids}. A treatise of Hypsicles on \textit{Risings} +\index{Regular solids}% +is of interest because it is the first Greek work giving the +division of the circumference into $360$~degrees after the fashion +of the Babylonians. +\index{Babylonians!ref.\ to}% + +\textbf{Hipparchus} of Nicæa in Bithynia was the greatest astronomer +\index{Hipparchus}% +of antiquity. He established inductively the famous +theory of epicycles and eccentrics. As might be expected, he +\index{Epicycles}% +was interested in mathematics, not \textit{per se}, but only as an aid +to astronomical inquiry. No mathematical writings of his +are extant, but Theon of Alexandria informs us that Hipparchus +originated the science of \emph{trigonometry}, and that he calculated +\index{Trigonometry}% +a ``table of chords'' in twelve books. Such calculations +%% -----File: 066.png---Folio 52------- +\index{Circle}% +must have required a ready knowledge of arithmetical and +algebraical operations. + +About 100~\BC\ flourished \textbf{Heron the Elder} of Alexandria. +\index{Heron the Elder}% +He was the pupil of Ctesibius, who was celebrated for his +\index{Ctesibius}% +ingenious mechanical inventions, such as the hydraulic organ, +the water-clock, and catapult. It is believed by some that +Heron was a son of Ctesibius. He exhibited talent of the +same order as did his master by the invention of the eolipile +and a curious mechanism known as ``Heron's fountain.'' +Great uncertainty exists concerning his writings. Most authorities +believe him to be the author of an important \textit{Treatise +on the Dioptra}, of which there exist three manuscript copies, +quite dissimilar. But M.~Marie\Reference[]{14} thinks that the \textit{Dioptra} is +\index{Marie, M.}% +the work of \textit{Heron the Younger}, who lived in the seventh or +eighth century after Christ, and that \textit{Geodesy}, another book +supposed to be by Heron, is only a corrupt and defective copy +of the former work. \textit{Dioptra} contains the important formula +for finding the area of a triangle expressed in terms of its +sides; its derivation is quite laborious and yet exceedingly +ingenious. ``It seems to me difficult to believe,'' says Chasles, +\index{Chasles!ref.\ to}% +``that so beautiful a theorem should be found in a work so +ancient as that of Heron the Elder, without that some Greek +geometer should have thought to cite it.'' Marie lays great +stress on this silence of the ancient writers, and argues from +it that the true author must be Heron the Younger or some +writer much more recent than Heron the Elder. But no reliable +evidence has been found that there actually existed a +second mathematician by the name of Heron. + +``Dioptra,'' says Venturi, were instruments which had great +\index{Venturi}% +resemblance to our modern theodolites. The book \textit{Dioptra} is +a treatise on geodesy containing solutions, with aid of these +instruments, of a large number of questions in geometry, such +as to find the distance between two points, of which one only +%% -----File: 067.png---Folio 53------- +is accessible, or between two points which are visible but both +inaccessible; from a given point to draw a perpendicular to a +line which cannot be approached; to find the difference of +level between two points; to measure the area of a field without +entering it. + +Heron was a practical surveyor. This may account for the +fact that his writings bear so little resemblance to those of +the Greek authors, who considered it degrading the science +to apply geometry to surveying. The character of his geometry +is not Grecian, but decidedly Egyptian. This fact is the +more surprising when we consider that Heron demonstrated +his familiarity with Euclid by writing a commentary on the +\index{Euclid!ref.\ to}% +\textit{Elements}.\Reference{21} Some of Heron's formulas point to an old Egyptian +origin. Thus, besides the above exact formula for the +area of a triangle in terms of its sides, Heron gives the formula +$\dfrac{a_1 + a_2\Fstr}{2} × \dfrac{b}{2}$, which bears a striking likeness to the formula +$\dfrac{a_1 + a_2}{2} × \dfrac{b_1 + b_2}{2}$ for finding the area of a quadrangle, +found in the Edfu inscriptions. There are, moreover, points +\index{Edfu}% +of resemblance between Heron's writings and the ancient +Ahmes papyrus. Thus Ahmes used unit-fractions exclusively; +\index{Ahmes!ref.\ to}% +Heron uses them oftener than other fractions. Like Ahmes +and the priests at Edfu, Heron divides complicated figures +into simpler ones by drawing auxiliary lines; like them, +he shows, throughout, a special fondness for the isosceles +trapezoid. + +The writings of Heron satisfied a practical want, and for +that reason were borrowed extensively by other peoples. We +find traces of them in Rome, in the Occident during the Middle +Ages, and even in India. + +\textbf{Geminus} of Rhodes (about 70~\BC) published an astronomical +\index{Geminus}% +work still extant. He wrote also a book, now lost, on the +\textit{Arrangement of Mathematics}, which contained many valuable +%% -----File: 068.png---Folio 54------- +notices of the early history of Greek mathematics. Proclus +\index{Proclus!ref.\ to}% +and Eutocius quote it frequently. \textbf{Theodosius} of Tripolis is +\index{Eutocius!ref.\ to}% +\index{Theodosius}% +the author of a book of little merit on the geometry of the +sphere. \textbf{Dionysodorus} of Amisus in Pontus applied the intersection +\index{Dionysodorus}% +of a parabola and hyperbola to the solution of a problem +which Archimedes, in his \textit{Sphere and Cylinder}, had left +\index{Archimedes!ref.\ to}% +incomplete. The problem is ``to cut a sphere so that its segments +shall be in a given ratio.'' + +We have now sketched the progress of geometry down to +the time of Christ. Unfortunately, very little is known of +the history of geometry between the time of Apollonius and +\index{Apollonius!ref.\ to}% +the beginning of the Christian era. The names of quite a +number of geometers have been mentioned, but very few of +their works are now extant. It is certain, however, that there +were no mathematicians of real genius from Apollonius to +Ptolemy, excepting Hipparchus and perhaps Heron. +\index{Heron the Elder!ref.\ to}% +\index{Hipparchus!ref.\ to}% +\index{Ptolemy!ref.\ to}% + +\Subsection{The Second Alexandrian School} +\index{Alexandrian School!(first)|)}% +\index{Alexandrian School!(second)|(}% + +The close of the dynasty of the Lagides which ruled Egypt +from the time of Ptolemy Soter, the builder of Alexandria, +for 300~years; the absorption of Egypt into the Roman Empire; +the closer commercial relations between peoples of the +East and of the West; the gradual decline of paganism and +spread of Christianity,---these events were of far-reaching +influence on the progress of the sciences, which then had their +home in Alexandria. Alexandria became a commercial and +intellectual emporium. Traders of all nations met in her +busy streets, and in her magnificent Library, museums, lecture-halls, +scholars from the East mingled with those of the +West; Greeks began to study older literatures and to compare +them with their own. In consequence of this interchange +of ideas the Greek philosophy became fused with Oriental +%% -----File: 069.png---Folio 55------- +\index{Conic sections!Greek}% +\index{Ptolemy!ref.\ to}% +philosophy. Neo-Pythagoreanism and Neo-Platonism were the +names of the modified systems. These stood, for a time, in +opposition to Christianity. The study of Platonism and +Pythagorean mysticism led to the revival of the theory of +numbers. Perhaps the dispersion of the Jews and their +\index{Numbers!theory of numbers}% +introduction to Greek learning helped in bringing about this +revival. The theory of numbers became a favourite study. +\index{Theory of numbers}% +This new line of mathematical inquiry ushered in what we +may call a new school. There is no doubt that even now +geometry continued to be one of the most important studies +in the Alexandrian course. This Second Alexandrian School +may be said to begin with the Christian era. It was made +famous by the names of Claudius Ptolemæus, Diophantus, +\index{Diophantus!ref.\ to}% +Pappus, Theon of Smyrna, Theon of Alexandria, Iamblichus, +\index{Pappus!ref.\ to}% +\index{Theon of Alexandria!ref.\ to}% +\index{Theon of Smyrna}% +Porphyrius, and others. +\index{Porphyrius}% + +By the side of these we may place \textbf{Serenus} of Antissa, as +\index{Serenus}% +having been connected more or less with this new school. +He wrote on sections of the cone and cylinder, in two books, +one of which treated only of the triangular section of the cone +through the apex. He solved the problem, ``given a cone +(cylinder), to find a cylinder (cone), so that the section of +both by the same plane gives similar ellipses.'' Of particular +interest is the following theorem, which is the foundation +of the modern theory of +harmonics: If from $D$ we +\index{Harmonics}% +% [Illustration] +\begin{wrapfigure}{r}{2.25in} %[* PP: Width-dependent line break] +\hfill\Graphic{2.25in}{069a} +\end{wrapfigure} +draw~$DF$, cutting the triangle +$ABC$, and choose +$H$ on it, so that $DE : DF += EH : HF$, and if we draw +the line~$AH$, then every transversal through $D$, such as +$DG$, will be divided by $AH$ so that $DK : DG = KJ : JG$. +\textbf{Menelaus} of Alexandria (about 98~\AD) was the author of +\index{Menelaus}% +\textit{Sphærica}, a work extant in Hebrew and Arabic, but not +%% -----File: 070.png---Folio 56------- +\index{Astronomy!Greek}% +\index{Ptolemaic System}% +\index{Spherical trigonometry}% +in Greek. In it he proves the theorems on the congruence +of spherical triangles, and describes their properties in +much the same way as Euclid treats plane triangles. In +it are also found the theorems that the sum of the three +sides of a spherical triangle is less than a great circle, and +that the sum of the three angles exceeds two right angles. +Celebrated are two theorems of his on plane and spherical +triangles. The one on plane triangles is that, ``if the three +sides be cut by a straight line, the product of the three segments +which have no common extremity is equal to the +product of the other three.'' The illustrious Carnot makes +\index{Carnot, Lazare!ref.\ to}% +this proposition, known as the `lemma of Menelaus,' the base +of his theory of transversals. The corresponding theorem for +spherical triangles, the so-called `regula sex quantitatum,' is +obtained from the above by reading ``chords of three segments +doubled,'' in place of ``three segments.'' + +\textbf{Claudius Ptolemæus}, a celebrated astronomer, was a native of +Egypt. Nothing is known of his personal history except that +he flourished in Alexandria in 139~\AD\ and that he made the +earliest astronomical observations recorded in his works, in +125~\AD, the latest in 151~\AD. The chief of his works are +the \textit{Syntaxis Mathematica} (or the \textit{Almagest}, as the Arabs call +\index{Almagest|(}% +it) and the \textit{Geographica}, both of which are extant. The +former work is based partly on his own researches, but mainly +on those of Hipparchus. Ptolemy seems to have been not +\index{Hipparchus!ref.\ to}% +\index{Ptolemy|(}% +so much of an independent investigator, as a corrector and +improver of the work of his great predecessors. The \textit{Almagest} +forms the foundation of all astronomical science down to +Copernicus. The fundamental idea of his system, the ``Ptolemaic +\index{Copernicus}% +System,'' is that the earth is in the centre of the universe, +and that the sun and planets revolve around the earth. +Ptolemy did considerable for mathematics. He created, for +astronomical use, a \emph{trigonometry} remarkably perfect in form. +\index{Trigonometry}% +%% -----File: 071.png---Folio 57------- +\index{Fractions!sexagesimal}% +\index{Sexagesimal system}% +The foundation of this science was laid by the illustrious +Hipparchus. + +The \textit{Almagest} is in 13~books. Chapter~9 of the first book +shows how to calculate tables of chords. The circle is divided +into $360$~degrees, each of which is halved. The diameter is +divided into $120$~divisions; each of these into $60$~parts, which +are again subdivided into $60$~smaller parts. In Latin, these +parts were called \textit{partes minutæ primæ} and \textit{partes minutæ +secundæ}. Hence our names, `minutes' and `seconds.'\Reference{3} The +sexagesimal method of dividing the circle is of Babylonian +origin, and was known to Geminus and Hipparchus. But +\index{Geminus!ref.\ to}% +Ptolemy's method of calculating chords seems original with +him. He first proved the proposition, now appended to +Euclid~VI.~(D), that ``the rectangle contained by the diagonals +\index{Euclid!ref.\ to}% +of a quadrilateral figure inscribed in a circle is equal +to both the rectangles contained by its opposite sides.'' He +then shows how to find from the chords of two arcs the +chords of their sum and difference, and from the chord of any +arc that of its half. These theorems he applied to the calculation +of his tables of chords. The proofs of these theorems +are very pretty. + +Another chapter of the first book in the \textit{Almagest} is devoted +to \emph{trigonometry}, and to \emph{spherical} trigonometry in particular. +\index{Trigonometry!spherical}% +Ptolemy proved the `lemma of Menelaus,' and also the `regula +\index{Menelaus!ref.\ to}% +sex quantitatum.' Upon these propositions he built up his +trigonometry. The fundamental theorem of plane trigonometry, +that two sides of a triangle are to each other as the chords +of double the arcs measuring the angles opposite the two +sides, was not stated explicitly by him, but was contained +implicitly in other theorems. More complete are the propositions +in spherical trigonometry. + +The fact that trigonometry was cultivated not for its own +sake, but to aid astronomical inquiry, explains the rather +%% -----File: 072.png---Folio 58------- +startling fact that spherical trigonometry came to exist in a +developed state earlier than plane trigonometry. + +The remaining books of the \textit{Almagest} are on astronomy. +Ptolemy has written other works which have little or no bearing +on mathematics, except one on geometry. Extracts from +this book, made by Proclus, indicate that Ptolemy did not +\index{Proclus!ref.\ to}% +regard the parallel-axiom of Euclid as self-evident, and that +\index{Euclid!ref.\ to}% +Ptolemy was the first of the long line of geometers from +ancient time down to our own who toiled in the vain attempt +to prove it. + +Two prominent mathematicians of this time were Nicomachus +\index{Nicomachus!ref.\ to}% +and Theon of Smyrna. Their favourite study was theory +\index{Theon of Smyrna}% +of numbers. The investigations in this science culminated +later in the algebra of Diophantus. But no important geometer +appeared after Ptolemy for 150~years. The only occupant +\index{Ptolemy|)}% +of this long gap was \textbf{Sextus Julius Africanus}, who wrote an +\index{Sextus Julius Africanus}% +unimportant work on geometry applied to the art of war, +entitled \textit{Cestes}. + +\textbf{Pappus}, probably born about 340~\AD, in Alexandria, was +\index{Pappus|(}% +the last great mathematician of the Alexandrian school. His +genius was inferior to that of Archimedes, Apollonius, and +Euclid, who flourished over 500~years earlier. But living, +as he did, at a period when interest in geometry was declining, +he towered above his contemporaries ``like the peak +of Teneriffa above the Atlantic.'' He is the author of a \textit{Commentary +on the Almagest}, a \textit{Commentary on Euclid's Elements}, +\index{Almagest|)}% +a \textit{Commentary on the Analemma of Diodorus},---a writer of +\index{Diodorus}% +whom nothing is known. All these works are lost. Proclus, +probably quoting from the \textit{Commentary on Euclid}, says that +Pappus objected to the statement that an angle equal to a +right angle is always itself a right angle. + +The only work of Pappus still extant is his \textit{Mathematical +Collections}. This was originally in eight books, but the first +%% -----File: 073.png---Folio 59------- +and portions of the second are now missing. The \textit{Mathematical +Collections} seems to have been written by Pappus to supply +the geometers of his time with a succinct analysis of the most +difficult mathematical works and to facilitate the study of +them by explanatory lemmas. But these lemmas are selected +very freely, and frequently have little or no connection with the +subject on hand. However, he gives very accurate summaries +of the works of which he treats. The \textit{Mathematical Collections} +is invaluable to us on account of the rich information it +gives on various treatises by the foremost Greek mathematicians, +which are now lost. Mathematicians of the last century +considered it possible to restore lost works from the \textit{résumé} +by Pappus alone. + +We shall now cite the more important of those theorems in +the \textit{Mathematical Collections} which are supposed to be original +with Pappus. First of all ranks the elegant theorem re-discovered +by \textit{Guldin},\DPnote{** PP: [sic], not boldface} over 1000~years later, that the volume +\index{Guldin!ref.\ to}% +generated by the revolution of a plane curve which lies wholly +on one side of the axis, equals the area of the curve multiplied +by the circumference described by its centre of gravity. +Pappus proved also that the centre of gravity of a triangle is +that of another triangle whose vertices lie upon the sides of +the first and divide its three sides in the same ratio. In the +fourth book are new and brilliant propositions on the quadratrix +\index{Quadratrix}% +which indicate an intimate acquaintance with curved +surfaces. He generates the quadratrix as follows: Let a +spiral line be drawn upon a right circular cylinder; then the +perpendiculars to the axis of the cylinder drawn from each +point of the spiral line form the surface of a screw. A plane +passed through one of these perpendiculars, making any convenient +angle with the base of the cylinder, cuts the screw-surface +in a curve, the orthogonal projection of which upon +the base is the \emph{quadratrix}. A second mode of generation is +%% -----File: 074.png---Folio 60------- +\index{Problem of Pappus}% +\index{Spirals}% +no less admirable: If we make the spiral of Archimedes the +base of a right cylinder, and imagine a cone of revolution +having for its axis the side of the cylinder passing through +the initial point of the spiral, then this cone cuts the cylinder +in a curve of double curvature. The perpendiculars to the +axis drawn through every point in this curve form the surface +of a screw which Pappus here calls the \emph{plectoidal surface}. A +\index{Plectoidal surface}% +plane passed through one of the perpendiculars at any convenient +angle cuts that surface in a curve whose orthogonal +projection upon the plane of the spiral is the required \emph{quadratrix}. +\index{Quadratrix}% +Pappus considers curves of double curvature still further. +He produces a \emph{spherical spiral} by a point moving uniformly +along the circumference of a great circle of a sphere, while +the great circle itself revolves uniformly around its diameter. +He then finds the area of that portion of the surface of the +sphere determined by the spherical spiral, ``a complanation +which claims the more lively admiration, if we consider that, +although the entire surface of the sphere was known since +Archimedes' time, to measure portions thereof, such as spherical +triangles, was then and for a long time afterwards an +unsolved problem.''\Reference{3} A question which was brought into +prominence by Descartes and Newton is the ``problem of +\index{Descartes!ref.\ to}% +\index{Newton!ref.\ to}% +Pappus.'' Given several straight lines in a plane, to find the +locus of a point such that when perpendiculars (or, more +generally, straight lines at given angles) are drawn from it to +the given lines, the product of certain ones of them shall be in +a given ratio to the product of the remaining ones. It is +worth noticing that it was Pappus who first found the focus +\index{Focus}% +of the parabola, suggested the use of the directrix, and propounded +\index{Directrix}% +the theory of the involution of points. He solved +\index{Involution of points}% +the problem to draw through three points lying in the same +straight line, three straight lines which shall form a triangle +inscribed in a given circle.\Reference{3} From the \textit{Mathematical Collections} +%% -----File: 075.png---Folio 61------- +many more equally difficult theorems might be quoted which +are original with Pappus as far as we know. It ought to be +remarked, however, that he is known in three instances to +have copied theorems without giving due credit, and that he +may have done the same thing in other cases in which we +have no data by which to ascertain the real discoverer. + +About the time of Pappus lived \textbf{Theon} of Alexandria. He +\index{Pappus|)}% +\index{Theon of Alexandria}% +brought out an edition of Euclid's \textit{Elements} with notes, which +\index{Elements (Euclid's)}% +\index{Euclid!ref.\ to}% +he probably used as a text-book in his classes. His commentary +on the \textit{Almagest} is valuable for the many historical notices, +and especially for the specimens of Greek arithmetic which it +contains. Theon's daughter \textbf{Hypatia}, a woman celebrated for +\index{Hypatia}% +her beauty and modesty, was the last Alexandrian teacher of +reputation, and is said to have been an abler philosopher and +mathematician than her father. Her notes on the works of +Diophantus and Apollonius have been lost. Her tragic death +\index{Apollonius!ref.\ to}% +\index{Diophantus!ref.\ to}% +in 415~\AD\ is vividly described in Kingsley's \textit{Hypatia}. + +From now on, mathematics ceased to be cultivated in +Alexandria. The leading subject of men's thoughts was +Christian theology. Paganism disappeared, and with it pagan +learning. The Neo-Platonic school at Athens struggled on a +century longer. Proclus, Isidorus, and others kept up the +\index{Isidorus of Seville!ref.\ to}% +\index{Proclus}% +``golden chain of Platonic succession.'' \textbf{Proclus}, the successor +of Syrianus, at the Athenian school, wrote a commentary on +Euclid's \textit{Elements}. We possess only that on the first book, +which is valuable for the information it contains on the +history of geometry. \textbf{Damascius} of Damascus, the pupil of +\index{Damascius}% +Isidorus, is now believed to be the author of the fifteenth +book of Euclid. Another pupil of Isidorus was \textbf{Eutocius} of +\index{Eutocius}% +Ascalon, the commentator of Apollonius and Archimedes. +\index{Archimedes!ref.\ to}% +\textbf{Simplicius} wrote a commentary on Aristotle's \textbf{De C\oe{}lo}. In +\index{Aristotle!ref.\ to}% +\index{Simplicius}% +the year~529, Justinian, disapproving heathen learning, finally +closed by imperial edict the schools at Athens. +%% -----File: 076.png---Folio 62------- +\index{Alexandrian School!(second)|)}% + +As a rule, the geometers of the last 500~years showed +a lack of creative power. They were commentators rather +than discoverers. + +The principal characteristics of ancient geometry are:--- + +(1)~A wonderful clearness and definiteness of its concepts +and an almost perfect logical rigour of its conclusions. + +(2)~A complete want of general principles and methods. +Ancient geometry is decidedly \emph{special}. Thus the Greeks +\index{Geometry!Greek|)}% +possessed no general method of drawing tangents. ``The +\index{Tangents!in geometry}% +determination of the tangents to the three conic sections did +not furnish any rational assistance for drawing the tangent to +any other new curve, such as the conchoid, the cissoid, etc.''\Reference{15} +In the demonstration of a theorem, there were, for the ancient +geometers, as many different cases requiring separate proof +as there were different positions for the lines. The greatest +geometers considered it necessary to treat all possible cases +independently of each other, and to prove each with equal +fulness. To devise methods by which the various cases could +all be disposed of by one stroke, was beyond the power of the +ancients. ``If we compare a mathematical problem with a +huge rock, into the interior of which we desire to penetrate, +then the work of the Greek mathematicians appears to us like +that of a vigorous stonecutter who, with chisel and hammer, +begins with indefatigable perseverance, from without, to +crumble the rock slowly into fragments; the modern mathematician +appears like an excellent miner, who first bores +through the rock some few passages, from which he then bursts +it into pieces with one powerful blast, and brings to light the +treasures within.''\Reference{16} +%% -----File: 077.png---Folio 63------- +\index{Finger-reckoning}% + +\SpecialSection{Greek Arithmetic} +\index{Arithmetic!Greek|(}% + +Greek mathematicians were in the habit of discriminating +between the \emph{science} of numbers and the \emph{art} of calculation. +The former they called \textit{arithmetica}, the latter \textit{logistica}. The +drawing of this distinction between the two was very natural +and proper. The difference between them is as marked as +that between theory and practice. Among the Sophists the +art of calculation was a favourite study. Plato, on the other +\index{Plato!ref.\ to}% +hand, gave considerable attention to philosophical arithmetic, +but pronounced calculation a vulgar and childish art. + +In sketching the history of Greek calculation, we shall first +give a brief account of the Greek mode of counting and of +writing numbers. Like the Egyptians and Eastern nations, +the earliest Greeks counted on their fingers or with pebbles. +In case of large numbers, the pebbles were probably arranged +in parallel vertical lines. Pebbles on the first line +represented units, those on the second tens, those on the third +hundreds, and so on. Later, frames came into use, in which +strings or wires took the place of lines. According to tradition, +Pythagoras, who travelled in Egypt and, perhaps, in +\index{Pythagoras!ref.\ to}% +India, first introduced this valuable instrument into Greece. +The \emph{abacus}, as it is called, existed among different peoples and +\index{Abacus}% +at different times, in various stages of perfection. An abacus +is still employed by the Chinese under the name of \emph{Swan-pan}. +We possess no specific information as to how the Greek abacus +looked or how it was used. Boethius says that the Pythagoreans +\index{Boethius!ref.\ to}% +used with the abacus certain nine signs called \emph{apices}, +\index{Apices of Boethius!ref.\ to}% +which resembled in form the nine ``Arabic numerals.'' But +the correctness of this assertion is subject to grave doubts. + +The oldest Grecian numerical symbols were the so-called +\emph{Herodianic signs} (after Herodianus, a Byzantine grammarian of +\index{Herodianic signs}% +about 200~\AD, who describes them). These signs occur frequently +%% -----File: 078.png---Folio 64------- +\index{Notation!Greek numbers}% +in Athenian inscriptions and are, on that account, now +generally called \emph{Attic}. For some unknown reason these symbols +were afterwards replaced by the \emph{alphabetic numerals}, in +\index{Numerals!Greek}% +which the letters of the Greek alphabet were used, together +with three strange and antique letters \textgreek{\stigma},~\Glyph{qoppa}, and~\Glyph{sampi}, and the +symbol~$\mathbf{M}$. This change was decidedly for the worse, for the +old Attic numerals were less burdensome on the memory, inasmuch +as they contained fewer symbols and were better adapted +to show forth analogies in numerical operations. The following +table shows the Greek alphabetic numerals and their +respective values:--- +\medskip + +\footnotesize +\setlength{\TmpLen}{0.055555\linewidth}% +\noindent\begin{tabular}{*{18}{@{}c@{}}} +\TE{$\alpha$} & \TE{$\beta$} & \TE{$\gamma$} & \TE{$\delta$} & +\TE{$\epsilon$} & \TE{\textgreek{\stigma}} & +\TE{$\zeta$} & \TE{$\eta$} & \TE{$\theta$} & \TE{$\iota$} & +\TE{$\kappa$} & \TE{$\lambda$} & \TE{$\mu$} & +\TE{$\nu$} & \TE{$\xi$} & \TE{o} & \TE{$\pi$} & \TE{\Glyph{qoppa}} \\ +1&2&3&4&5&6&7&8&9&10&20&30&40&50&60&70&80&90 +\end{tabular} + +\setlength{\TmpLen}{0.076923\linewidth}% +\noindent\begin{tabular}{*{13}{@{}c@{}}} +\TE{$\rho$} & \TE{$\sigma$} & \TE{$\tau$} & \TE{$\upsilon$} & +\TE{$\phi$} & \TE{$\chi$} & \TE{$\psi$} & \TE{$\omega$} & +\TE{\Glyph{sampi}} & +\TE{\textgreek{\katwtonos}$\alpha$} & +\TE{\textgreek{\katwtonos}$\beta$ } & +\TE{\textgreek{\katwtonos}$\gamma$} & \TE{etc.} \\ +100&200&300&400&500&600&700&800&900&1000&2000&3000 +\end{tabular} + +\noindent\begin{tabular}{@{}*{4}{c}@{}} +$\mathbf{M}$&$\overset{\beta}{\mathbf{M}}$&$\overset{\gamma}{\mathbf{M}}$&etc.\\ +10,000&20,000&30,000 +\end{tabular} +\medskip + +\normalsize +It will be noticed that at $1000$, the alphabet is begun over +again, but, to prevent confusion, a stroke is now placed before +the letter and generally somewhat below it. A horizontal line +drawn over a number served to distinguish it more readily +from words. The coefficient for~$\mathbf{M}$ was sometimes placed +before or behind instead of over the~$\mathbf{M}$. Thus $43,678$ was +written $\overline{\delta\text{M\textgreek{\katwtonos}}\gamma\chi o\eta}$. It is to be observed that the Greeks had no +zero. + +Fractions were denoted by first writing the numerator +\index{Fractions!Greek}% +marked with an accent, then the denominator marked with +two accents and written twice. Thus, $\iota \gamma' \kappa \theta'' \kappa \theta'' = \frac{13}{29}$. In case +of fractions having unity for the numerator, the $\alpha'$ was omitted +and the denominator was written only once. Thus $\mu \delta'' = \frac{1}{44}$. +%% -----File: 079.png---Folio 65------- +\index{Sexagesimal system}% + +Greek writers seldom refer to calculation with alphabetic +numerals. Addition, subtraction, and even multiplication were +probably performed on the abacus. Expert mathematicians +may have used the symbols. Thus Eutocius, a commentator +\index{Eutocius!ref.\ to}% +of the sixth century after Christ, gives a great many multiplications +of which the following is a specimen:\Reference[]{6}--- + +\begin{wrapfigure}[8]{l}{2.125in}\footnotesize +\vspace*{-0.0625in}\begin{tabular}{@{}c@{\;}c@{\;}c@{\;}c@{}c@{}r@{\;}r@{\;}r@{}} +\multicolumn{4}{l}{$\quad\overline{\sigma\,\xi\,\epsilon}$} & + $\mspace{18mu}$ & \multicolumn{2}{l}{\quad 2 6 5} \\ +\multicolumn{4}{l}{$\quad\overline{\sigma\,\xi\,\epsilon}$} & + & \multicolumn{2}{l}{\quad 2 6 5} \\\cline{1-4}\cline{6-8} +$\overset{\delta}{\mathbf{M}}$ & $\overset{\alpha}{\mathbf{M}}$ & \textgreek{\katwtonos}$\beta$ & \textgreek{\katwtonos}$\alpha$ & + & 40000, & 12000, & 1000 \\ +$\overset{\alpha}{\mathbf{M}}$ & \textgreek{\katwtonos}$\beta$ & \textgreek{\katwtonos}$\overline{\gamma\chi}$ & $\overline{\tau}$ & + & 12000, & 3600, & 300 \\ +\phantom{$\overset{\alpha}{\mathbf{M}}$} & \textgreek{\katwtonos}$\alpha$ & $\overline{\tau}$ & $\overline{\kappa\epsilon}$ & + & 1000, & 300, & 25 \\\cline{1-4}\cline{6-8} +\multicolumn{4}{c}{$\overset{\zeta}{\mathbf{M}}\enskip\overline{\sigma\,\kappa\,\epsilon}$} & + & 70225\phantom{,} +\end{tabular} +\end{wrapfigure} +The operation is explained sufficiently by the modern numerals appended. +In case of mixed numbers, the process was still more clumsy. Divisions +are found in Theon of Alexandria's commentary on the \textit{Almagest}. As +\index{Theon of Alexandria!ref.\ to}% +might be expected, the process is long and tedious. + +We have seen in geometry that the more advanced mathematicians +frequently had occasion to extract the square root. +\index{Square root}% +Thus Archimedes in his \textit{Mensuration of the Circle} gives a +\index{Archimedes!ref.\ to}% +large number of square roots. He states, for instance, that +$\sqrt{3} < \frac{1351}{780}$ and $\sqrt{3} > \frac{265}{153}$, but he gives no clue to the method +by which he obtained these approximations. It is not improbable +that the earlier Greek mathematicians found the +square root by trial only. Eutocius says that the method of +extracting it was given by Heron, Pappus, Theon, and other +\index{Heron the Elder!ref.\ to}% +\index{Pappus!ref.\ to}% +commentators on the \textit{Almagest}. Theon's is the only ancient +method known to us. It is the same as the one used nowadays, +except that sexagesimal fractions are employed in place +\index{Fractions!Greek}% +\index{Fractions!sexagesimal}% +of our decimals. What the mode of procedure actually was +when sexagesimal fractions were not used, has been the subject +of conjecture on the part of numerous modern writers.\Reference{17} + +Of interest, in connection with arithmetical symbolism, is +the \textit{Sand-Counter} (Arenarius), an essay addressed by \textbf{Archimedes} +\index{Arenarius}% +\index{Sand-counter}% +%% -----File: 080.png---Folio 66------- +to Gelon, king of Syracuse. In it Archimedes shows +that people are in error who think the sand cannot be counted, +or that if it can be counted, the number cannot be expressed +by arithmetical symbols. He shows that the number of grains +in a heap of sand not only as large as the whole earth, but as +large as the entire universe, can be arithmetically expressed. +Assuming that $10,000$ grains of sand suffice to make a little +solid of the magnitude of a poppy-seed, and that the diameter +of a poppy-seed be not smaller than $\frac{1}{40}$~part of a finger's +breadth; assuming further, that the diameter of the universe +(supposed to extend to the sun) be less than $10,000$ diameters +of the earth, and that the latter be less than $1,000,000$ stadia, +Archimedes finds a number which would exceed the number +of grains of sand in the sphere of the universe. He goes on +even further. Supposing the universe to reach out to the fixed +stars, he finds that the sphere, having the distance from the +earth's centre to the fixed stars for its radius, would contain +a number of grains of sand less than $1000$ myriads of the +eighth octad. In our notation, this number would be $10^{63}$ or +$1$~with $63$~ciphers after it. It can hardly be doubted that one +object which Archimedes had in view in making this calculation +was the improvement of the Greek symbolism. It is not +known whether he invented some short notation by which to +represent the above number or not. + +We judge from fragments in the second book of Pappus that +\index{Pappus!ref.\ to}% +Apollonius proposed an improvement in the Greek method of +\index{Apollonius!ref.\ to}% +writing numbers, but its nature we do not know. Thus we +see that the Greeks never possessed the boon of a clear, comprehensive +symbolism. The honour of giving such to the world, +once for all, was reserved by the irony of fate for a nameless +Indian of an unknown time, and we know not whom to thank +for an invention of such importance to the general progress of +intelligence.\Reference{6} +%% -----File: 081.png---Folio 67------- +\index{Arithmetic!Pythagoreans|(}% +\index{Fractions!sexagesimal}% +\index{Sexagesimal system}% + +Passing from the subject of \textit{logistica} to that of \textit{arithmetica}, +our attention is first drawn to the science of numbers of +\textbf{Pythagoras}. Before founding his school, Pythagoras studied +\index{Pythagoras|(}% +for many years under the Egyptian priests and familiarised +himself with Egyptian mathematics and mysticism. If he +ever was in Babylon, as some authorities claim, he may have +learned the sexagesimal notation in use there; he may have +picked up considerable knowledge on the theory of proportion, +\index{Proportion}% +and may have found a large number of interesting astronomical +observations. Saturated with that speculative spirit then +pervading the Greek mind, he endeavoured to discover some +principle of homogeneity in the universe. Before him, the +philosophers of the Ionic school had sought it in the matter +of things; Pythagoras looked for it in the structure of things. +He observed various numerical relations or analogies between +numbers and the phenomena of the universe. Being convinced +that it was in numbers and their relations that he was to find +the foundation to true philosophy, he proceeded to trace the +origin of all things to numbers. Thus he observed that musical +strings of equal length stretched by weights having the +proportion of $\frac{1}{2}$, $\frac{2}{3}$, $\frac{3}{4}$, produced intervals which were an octave, +a fifth, and a fourth. Harmony, therefore, depends on musical +proportion; it is nothing but a mysterious numerical relation. +Where harmony is, there are numbers. Hence the +order and beauty of the universe have their origin in numbers. +There are seven intervals in the musical scale, and also seven +planets crossing the heavens. The same numerical relations +which underlie the former must underlie the latter. But where +numbers are, there is harmony. Hence his spiritual ear discerned +in the planetary motions a wonderful `harmony of the +spheres.' The Pythagoreans invested particular numbers with +extraordinary attributes. Thus \emph{one} is the essence of things; +it is an absolute number; hence the origin of all numbers and +%% -----File: 082.png---Folio 68------- +so of all things. \emph{Four} is the most perfect number, and was in +some mystic way conceived to correspond to the human soul. +Philolaus believed that $5$ is the cause of color, $6$ of cold, $7$ of +\index{Philolaus!ref.\ to}% +mind and health and light, $8$ of love and friendship.\Reference{6} In +Plato's works are evidences of a similar belief in religious +\index{Plato!ref.\ to}% +relations of numbers. Even Aristotle referred the virtues to +\index{Aristotle!ref.\ to}% +\index{Numbers!amicable}% +\index{Numbers!excessive}% +\index{Numbers!heteromecic}% +\index{Numbers!perfect}% +\index{Numbers!defective}% +numbers. + +Enough has been said about these mystic speculations to +show what lively interest in mathematics they must have +created and maintained. Avenues of mathematical inquiry +were opened up by them which otherwise would probably +have remained closed at that time. + +The Pythagoreans classified numbers into odd and even. +They observed that the sum of the series of odd numbers +from $1$ to $2n + 1$ was always a complete square, and that +by addition of the even numbers arises the series $2$, $6$, $12$, $20$, +in which every number can be decomposed into two factors +differing from each other by unity. Thus, $6 = 2·3$, $12 = 3·4$, +etc. These latter numbers were considered of sufficient importance +to receive the separate name of \emph{heteromecic} (not +equilateral).\Reference{7} Numbers of the form $\dfrac{n(n+1)}{2}$ were called \emph{triangular}, +because they could always be arranged thus,~\smash[t]{\TallGlyph{tendots}}. +Numbers which were equal to the sum of all their possible +factors, such as $6$, $28$, $496$, were called \emph{perfect}; those exceeding +that sum, \emph{excessive}; and those which were less, \emph{defective}. +\emph{Amicable} numbers were those of which each was the sum of +the factors in the other. Much attention was paid by the +Pythagoreans to the subject of proportion. The quantities +\index{Proportion}% +$a$, $b$, $c$, $d$ were said to be in \emph{arithmetical} proportion when +$a - b = c - d$; in \emph{geometrical} proportion, when $a : b = c : d$; in +\emph{harmonic} proportion, when $a - b : b - c = a : c$. It is probable +that the Pythagoreans were also familiar with the \emph{musical} +%% -----File: 083.png---Folio 69------- +\index{Irrationals}% +proportion $a : \dfrac{a + b\Fstr}{2} = \dfrac{2ab}{a+b} : b$. Iamblichus says that Pythagoras +\index{Iamblichus!ref.\ to}% +introduced it from Babylon. + +In connection with arithmetic, Pythagoras made extensive +investigations into geometry. He believed that an arithmetical +\index{Geometry!Greek}% +fact had its analogue in geometry, and \textit{vice versa}. In +connection with his theorem on the right triangle he devised +a rule by which integral numbers could be found, such +that the sum of the squares of two of them equalled the +square of the third. Thus, take for one side an odd number +$(2n + 1)$; then $\dfrac{(2n + 1)^{2} - 1}{2} = 2n^{2} + 2n =$ the other side, and +$(2n^2 + 2n + 1) =$ hypotenuse. If $2n + 1 = 9$, then the other +two numbers are $40$~and~$41$. But this rule only applies to +cases in which the hypotenuse differs from one of the sides +by~$1$. In the study of the right triangle there doubtless arose +questions of puzzling subtlety. Thus, given a number equal +to the side of an isosceles right triangle, to find the number +which the hypotenuse is equal to. The side may have been +taken equal to $1$, $2$, $\frac{3}{2}$, $\frac{6}{5}$, or any other number, yet in every +instance all efforts to find a number exactly equal to the +hypotenuse must have remained fruitless. The problem may +have been attacked again and again, until finally ``some rare +genius, to whom it is granted, during some happy moments, +to soar with eagle's flight above the level of human thinking,'' +grasped the happy thought that this problem cannot be +solved. In some such manner probably arose the theory of +\emph{irrational quantities}, which is attributed by Eudemus to the +\index{Eudemus}% +Pythagoreans. It was indeed a thought of extraordinary +boldness, to assume that straight lines could exist, differing +from one another not only in length,---that is, in quantity,---but +also in a quality, which, though real, was absolutely +invisible.\Reference{7} Need we wonder that the Pythagoreans saw in +%% -----File: 084.png---Folio 70------- +\index{Parabola}% +irrationals a deep mystery, a symbol of the unspeakable? +We are told that the one who first divulged the theory of +irrationals, which the Pythagoreans kept secret, perished in +consequence in a shipwreck. Its discovery is ascribed to +Pythagoras, but we must remember that all important Pythagorean +\index{Pythagoras|)}% +discoveries were, according to Pythagorean custom, +referred back to him. The first incommensurable ratio known +seems to have been that of the side of a square to its diagonal, +as $1 : \sqrt {2}$. \textbf{Theodorus of Cyrene} added to this the fact that the +\index{Theodorus}% +sides of squares represented in length by $\sqrt {3}$, $\sqrt {5}$, etc., up to +$\sqrt {17}$, and Theætetus, that the sides of any square, represented +\index{Theætetus!ref.\ to}% +by a surd, are incommensurable with the linear unit. \textbf{Euclid} +\index{Euclid}% +(about 300~\BC), in his \textit{Elements}, X.~9, generalised still further: +Two magnitudes whose squares are (or are not) to one another +as a square number to a square number are commensurable +(or incommensurable), and conversely. In the tenth book, he +treats of incommensurable quantities at length. He investigates +every possible variety of lines which can be represented +by $\sqrt { \sqrt {\vphantom{b}a} ± \sqrt {b}}$, $a$~and~$b$ representing two commensurable lines, +and obtains 25~species. Every individual of every species +is incommensurable with all the individuals of every other +species. ``This book,'' says De~Morgan, ``has a completeness +\index{Demorgan@{De Morgan}!ref.\ to}% +which none of the others (not even the fifth) can boast of; +and we could almost suspect that Euclid, having arranged his +materials in his own mind, and having completely elaborated +the tenth book, wrote the preceding books after it, and did not +live to revise them thoroughly.''\Reference{9} The theory of incommensurables +\index{Incommensurables}% +remained where Euclid left it, till the fifteenth +century. + +Euclid devotes the seventh, eighth, and ninth books of his +\textit{Elements} to arithmetic. Exactly how much contained in +\index{Arithmetic!Pythagoreans|)}% +\index{Arithmetic!Euclid}% +these books is Euclid's own invention, and how much is +borrowed from his predecessors, we have no means of knowing. +%% -----File: 085.png---Folio 71------- +Without doubt, much is original with Euclid. The \textit{seventh +\index{Euclid}% +book} begins with twenty-one definitions. All except that for +`prime' numbers are known to have been given by the +\index{Prime numbers}% +Pythagoreans. Next follows a process for finding the +G.C.D.~of two or more numbers. The \textit{eighth book} deals with +numbers in continued proportion, and with the mutual relations +of squares, cubes, and plane numbers. Thus,~XXII., if +three numbers are in continued proportion, and the first is a +square, so is the third. In the \textit{ninth book}, the same subject is +continued. It contains the proposition that the number of +primes is greater than any given number. + +After the death of Euclid, the theory of numbers remained +almost stationary for 400~years. Geometry monopolised +the attention of all Greek mathematicians. Only two are +known to have done work in arithmetic worthy of mention. +\textbf{Eratosthenes} (275--194~\BC) invented a `sieve' for finding prime +\index{Eratosthenes!ref.\ to}% +numbers. All composite numbers are `sifted' out in the +following manner: Write down the odd numbers from $3$~up, +in succession. By striking out every third number after the~$3$, +we remove all multiples of~$3$. By striking out every fifth +number after the~$5$, we remove all multiples of~$5$. In this way, +by rejecting multiples of $7$,~$11$, $13$,~etc., we have left prime +numbers only. \textbf{Hypsicles} (between 200 and 100~\BC) worked +\index{Hypsicles!ref.\ to}% +at the subjects of polygonal numbers and arithmetical progressions, +which Euclid entirely neglected. In his work on +`risings of the stars,' he showed (1)~that in an arithmetical +series of $2n$~terms, the sum of the last~$n$ terms exceeds the sum +of the first~$n$ by a multiple of~$n^2$; (2)~that in such a series of +$2n + 1$~terms, the sum of the series is the number of terms +multiplied by the middle term; (3)~that in such a series of +$2n$~terms, the sum is half the number of terms multiplied by +the two middle terms.\Reference{6} + +For two centuries after the time of Hypsicles, arithmetic +%% -----File: 086.png---Folio 72------- +\index{Cube numbers}% +disappears from history. It is brought to light again about +100~\AD\ by \textbf{Nicomachus}, a Neo-Pythagorean, who inaugurated +\index{Nicomachus}% +the final era of Greek mathematics. From now on, arithmetic +was a favourite study, while geometry was neglected. Nicomachus +wrote a work entitled \textit{Introductio Arithmetica}, which +was very famous in its day. The great number of commentators +it has received vouch for its popularity. Boethius +\index{Boethius!ref.\ to}% +translated it into Latin. Lucian could pay no higher +compliment to a calculator than this: ``You reckon like +Nicomachus of Gerasa.'' The \textit{Introductio Arithmetica} was +the first exhaustive work in which arithmetic was treated +quite independently of geometry. Instead of drawing lines, +like Euclid, he illustrates things by real numbers. To be +\index{Euclid!ref.\ to}% +sure, in his book the old geometrical nomenclature is retained, +but the method is inductive instead of deductive. ``Its sole +business is classification, and all its classes are derived from, +and exhibited by, actual numbers.'' The work contains few +results that are really original. We mention one important +proposition which is probably the author's own. He states +that cubical numbers are always equal to the sum of successive +odd numbers. Thus, $8 = 2^{3} = 3 + 5$, $27 = 3^{3} = 7 + 9 + 11$, $64 += 4^{3} = 13 + 15 + 17 + 19$, and so on. This theorem was used +later for finding the sum of the cubical numbers themselves. +\textbf{Theon} of Smyrna is the author of a treatise on ``the mathematical +\index{Theon of Smyrna}% +rules necessary for the study of Plato.'' The work is +ill arranged and of little merit. Of interest is the theorem, +that every square number, or that number minus~$1$, is +divisible by $3$ or $4$ or both. A remarkable discovery is a +proposition given by \textbf{Iamblichus} in his treatise on Pythagorean +\index{Iamblichus}% +philosophy. It is founded on the observation that the +Pythagoreans called $1$, $10$, $100$, $1000$, units of the first, second, +third, fourth `course' respectively. The theorem is this: +If we add any three consecutive numbers, of which the highest +%% -----File: 087.png---Folio 73------- +\index{Arabic numerals and notation}% +is divisible by~$3$, then add the digits of that sum, then, again, +the digits of \emph{that} sum, and so on, the final sum will be~$6$. +Thus, $61 + 62 + 63 = 186$, $1 + 8 + 6 = 15$, $1 + 5 = 6$. This +discovery was the more remarkable, because the ordinary +Greek numerical symbolism was much less likely to suggest +any such property of numbers than our ``Arabic'' notation +\index{Notation!Arabic notation}% +would have been. + +The works of Nicomachus, Theon of Smyrna, Thymaridas, +\index{Thymaridas}% +and others contain at times investigations of subjects which +are really algebraic in their nature. Thymaridas in one place +uses the Greek word meaning ``unknown quantity'' in a way +which would lead one to believe that algebra was not far +\index{Algebra!early Greek}% +distant. Of interest in tracing the invention of algebra are +the arithmetical epigrams in the \textit{Palatine Anthology}, which contain +\index{Anthology, Palatine}% +\index{Palatine anthology}% +about fifty problems leading to linear equations. Before +the introduction of algebra these problems were propounded +as puzzles. A riddle attributed to Euclid and contained in +\index{Euclid!ref.\ to}% +the \textit{Anthology} is to this effect: A mule and a donkey were +walking along, laden with corn. The mule says to the +donkey, ``If you gave me one measure, I should carry twice +as much as you. If I gave you one, we should both carry +equal burdens. Tell me their burdens, O~most learned master +of geometry.''\Reference{6} + +It will be allowed, says Gow, that this problem, if authentic, +was not beyond Euclid, and the appeal to geometry smacks of +antiquity. A far more difficult puzzle was the famous `cattle-problem,' +\index{Cattle-problem}% +which Archimedes propounded to the Alexandrian +\index{Archimedes!ref.\ to}% +mathematicians. The problem is indeterminate, for from only +seven equations, eight unknown quantities in integral numbers +are to be found. It may be stated thus: The sun had a herd +of bulls and cows, of different colours. (1)~Of Bulls, the +white~($W$) were, in number, $(\frac{1}{2} + \frac{1}{3})$ of the blue~($B$) and yellow~($Y$): +the $B$ were $(\frac{1}{4} + \frac{1}{5})$ of the~$Y$ and piebald~($P$): the +%% -----File: 088.png---Folio 74------- +$P$ were $(\frac{1}{6} + \frac{1}{7})$ of the~$W$ and~$Y$. (2)~Of Cows, which had the +same colours ($w$, $b$, $y$, $p$), +\iffalse +\begin{align*}% [** PP: This commented code visually matches the original] +w=(\tfrac{1}{3} + \tfrac{1}{4})(B + b) + &: b = (\tfrac{1}{4} + \tfrac{1}{5})(P + p) : p = (\tfrac{1}{5} + \tfrac{1}{6})(Y + y) \\ + &: y = (\tfrac{1}{6} + \tfrac{1}{7})(W + w). +\end{align*} +\fi +%[** PP: Re-breaking and aligning on =, but otherwise retaining verbatim] +\begin{align*} +w = (\tfrac{1}{3} + \tfrac{1}{4})(B + b) : b + &= (\tfrac{1}{4} + \tfrac{1}{5})(P + p) : p \\ + &= (\tfrac{1}{5} + \tfrac{1}{6})(Y + y) : y + = (\tfrac{1}{6} + \tfrac{1}{7})(W + w). +\end{align*} +Find the number of bulls and cows.\Reference{6} Another problem in the +\textit{Anthology} is quite familiar to school-boys: ``Of four pipes, +one fills the cistern in one day, the next in two days, the third +in three days, the fourth in four days: if all run together, how +soon will they fill the cistern?'' A great many of these +problems, puzzling to an arithmetician, would have been solved +easily by an algebraist. They became very popular about the +time of Diophantus, and doubtless acted as a powerful stimulus +\index{Diophantus|(}% +on his mind. + +\textbf{Diophantus} was one of the last and most fertile mathematicians +of the second Alexandrian school. He died about +330~\AD. His age was eighty-four, as is known from an epitaph +to this effect: Diophantus passed $\frac{1}{6}$~of his life in childhood, $\frac{1}{12}$~in +youth, and $\frac{1}{7}$~more as a bachelor; five years after his marriage +was born a son who died four years before his father, at half +his father's age. The place of nativity and parentage of +Diophantus are unknown. If his works were not written in +Greek, no one would think for a moment that they were the +product of Greek mind. There is nothing in his works that +reminds us of the classic period of Greek mathematics. His +were almost entirely new ideas on a new subject. In the +circle of Greek mathematicians he stands alone in his specialty. +Except for him, we should be constrained to say that +among the Greeks \emph{algebra} was always an unknown science. +\index{Algebra!Diophantus|(}% + +Of his works we have lost the \textit{Porisms}, but possess a fragment +of \textit{Polygonal Numbers}, and seven books of his great work +on \textit{Arithmetica}, said to have been written in 13~books. + +If we except the Ahmes papyrus, which contains the first +\index{Ahmes!ref.\ to}% +%% -----File: 089.png---Folio 75------- +\index{Tentative assumption}% +suggestions of algebraic notation, and of the solution of +\index{Notation!in algebra}% +equations, then his \textit{Arithmetica} is the earliest treatise on +\index{Equations!theory of}% +algebra now extant. In this work is introduced the idea of +an algebraic equation expressed in algebraic symbols. His +treatment is purely analytical and completely divorced from +geometrical methods. He is, as far as we know, the first to +state that ``a negative number multiplied by a negative number +gives a positive number.'' This is applied to the multiplication +of differences, such as $(x - 1)(x - 2)$. It must be +remarked, however, that Diophantus had no notion whatever +of negative numbers standing by themselves. All he knew +were differences, such as $(2x - 10)$, in which $2x$ could not be +smaller than~$10$ without leading to an absurdity. He appears +to be the first who could perform such operations as $(x - 1) +× (x - 2)$ without reference to geometry. Such identities as +$(a + b)^2 = a^2 + 2ab + b^2$, which with Euclid appear in the elevated +rank of geometric theorems, are with Diophantus the +simplest consequences of the algebraic laws of operation. His +sign for subtraction was~\Glyph{subtract}, for equality~$\iota$. For unknown +quantities he had only one symbol,~$\varsigma$. He had no sign for +addition except juxtaposition. Diophantus used but few symbols, +and sometimes ignored even these by describing an operation +in words when the symbol would have answered just +as well. + +In the solution of simultaneous equations Diophantus adroitly +managed with only one symbol for the unknown quantities and +arrived at answers, most commonly, by the method of \emph{tentative +assumption}, which consists in assigning to some of the unknown +\index{Assumption, tentative}% +quantities preliminary values, that satisfy only one or two of +the conditions. These values lead to expressions palpably +wrong, but which generally suggest some stratagem by which +values can be secured satisfying all the conditions of the +problem. +%% -----File: 090.png---Folio 76------- +\index{Numbers!theory of numbers}% +\index{Theory of numbers}% + +Diophantus also solved determinate equations of the second +degree. We are ignorant of his method, for he nowhere goes +through with the whole process of solution, but merely states +the result. Thus, ``$84 x^2 + 7x = 7$, whence $x$ is found $= \frac{1}{4}$.'' +Notice he gives only one root. His failure to observe that a +quadratic equation has two roots, even when both roots are +positive, rather surprises us. It must be remembered, however, +that this same inability to perceive more than one out of +the several solutions to which a problem may point is common +to all Greek mathematicians. Another point to be observed +is that he never accepts as an answer a quantity which is +negative or irrational. + +Diophantus devotes only the first book of his \textit{Arithmetica} to +the solution of determinate equations. The remaining books +extant treat mainly of \emph{indeterminate quadratic equations} of the +\index{Quadratic equations}% +form $Ax^2 + Bx + C = y^2$, or of two simultaneous equations of the +same form. He considers several but not all the possible +cases which may arise in these equations. The opinion of +Nesselmann on the method of Diophantus, as stated by Gow, +\index{Nesselmann}% +is as follows: ``(1)~Indeterminate equations of the second +degree are treated completely only when the quadratic or +the absolute term is wanting: his solution of the equations +$Ax^{2} + C = y^{2}$ and $Ax^{2} + Bx + C = y^{2}$ is in many respects cramped. +(2)~For the `double equation' of the second degree he has a +definite rule only when the quadratic term is wanting in both +expressions: even then his solution is not general. More complicated +expressions occur only under specially favourable +circumstances.'' Thus, he solves $Bx + C^{2} = y^{2}$, $B_{1}x + C_1^2 = y_1^2$. + +The extraordinary ability of Diophantus lies rather in +another direction, namely, in his wonderful ingenuity to reduce +all sorts of equations to particular forms which he knows +how to solve. Very great is the variety of problems considered. +The 130~problems found in the great work of Diophantus contain +%% -----File: 091.png---Folio 77------- +\index{Arithmetic!Greek|)}% +\index{Greeks|)}% +over 50~different classes of problems, which are strung +together without any attempt at classification. But still more +multifarious than the problems are the solutions. General +methods are unknown to Diophantus. Each problem has its +own distinct method, which is often useless for the most +closely related problems. ``It is, therefore, difficult for a +modern, after studying 100~Diophantine solutions, to solve +the~101st.''\Reference{7} + +That which robs his work of much of its scientific value is +the fact that he always feels satisfied with one solution, though +his equation may admit of an indefinite number of values. +Another great defect is the absence of general methods. Modern +mathematicians, such as Euler, \DPtypo{La~Grange}{Lagrange}, Gauss, had to +\index{Euler!ref.\ to}% +\index{Gauss!ref.\ to}% +\index{Lagrange!ref.\ to}% +begin the study of indeterminate analysis anew and received +no direct aid from Diophantus in the formulation of methods. +In spite of these defects we cannot fail to admire the work +for the wonderful ingenuity exhibited therein in the solution +of particular equations. + +It is still an open question and one of great difficulty +whether Diophantus derived portions of his algebra from +\index{Algebra!Diophantus|)}% +\index{Diophantus|)}% +Hindoo sources or not. + +\Section{The Romans} +\index{Romans|(}% + +Nowhere is the contrast between the Greek and Roman +mind shown forth more distinctly than in their attitude toward +the mathematical science. The sway of the Greek was a +flowering time for mathematics, but that of the Roman a +period of sterility. In philosophy, poetry, and art the Roman +was an imitator. But in mathematics he did not even rise to +the desire for imitation. The mathematical fruits of Greek +genius lay before him untasted. In him a science which had +%% -----File: 092.png---Folio 78------- +no direct bearing on practical life could awake no interest. +As a consequence, not only the higher geometry of Archimedes +\index{Archimedes!ref.\ to}% +and Apollonius, but even the \textit{Elements} of Euclid, were entirely +\index{Apollonius!ref.\ to}% +\index{Euclid!ref.\ to}% +neglected. What little mathematics the Romans possessed +did not come from the Greeks, but from more ancient +sources. Exactly where and how it originated is a matter of +doubt. It seems most probable that the ``Roman notation,'' +\index{Notation!Roman}% +as well as the practical geometry of the Romans, came from +the old Etruscans, who, at the earliest period to which our +knowledge of them extends, inhabited the district between the +Arno and Tiber. + +Livy tells us that the Etruscans were in the habit of representing +the number of years elapsed, by driving yearly a nail +into the sanctuary of Minerva, and that the Romans continued +this practice. A less primitive mode of designating numbers, +presumably of Etruscan origin, was a notation resembling the +present ``Roman notation.'' This system is noteworthy from +the fact that a principle is involved in it which is not met +with in any other; namely, the principle of subtraction. If a +letter be placed before another of greater value, its value is +not to be added to, but subtracted from, that of the greater. +In the designation of large numbers a horizontal bar placed +over a letter was made to increase its value one thousand fold. +In fractions the Romans used the duodecimal system. +\index{Fractions!Roman}% + +Of arithmetical calculations, the Romans employed three +different kinds: Reckoning on the fingers, upon the abacus, +and by tables prepared for the purpose.\Reference{3} Finger-symbolism +was known as early as the time of King Numa, for he had +erected, says Pliny, a statue of the double-faced Janus, of +which the fingers indicated $365$ ($355$?), the number of days in +a year. Many other passages from Roman authors point out +the use of the fingers as aids to calculation. In fact, a finger-symbolism +of practically the same form was in use not only in +%% -----File: 093.png---Folio 79------- +\index{Canon paschalis}% +Rome, but also in Greece and throughout the East, certainly +as early as the beginning of the Christian era, and continued +to be used in Europe during the Middle Ages. We possess no +knowledge as to where or when it was invented. The second +mode of calculation, by the abacus, was a subject of elementary +\index{Abacus}% +\index{Calculation, origin of word}% +instruction in Rome. Passages in Roman writers indicate +that the kind of abacus most commonly used was covered with +dust and then divided into columns by drawing straight lines. +Each column was supplied with pebbles (calculi, whence `calculare' +and `calculate') which served for calculation. Additions +and subtractions could be performed on the abacus quite +easily, but in multiplication the abacus could be used only for +adding the particular products, and in division for performing +the subtractions occurring in the process. Doubtless at this +point recourse was made to mental operations and to the multiplication +table. Possibly finger-multiplication may also have +been used. But the multiplication of large numbers must, by +either method, have been beyond the power of the ordinary +arithmetician. To obviate this difficulty, the arithmetical +tables mentioned above were used, from which the desired +products could be copied at once. Tables of this kind were +prepared by \textit{Victorius} of Aquitania. His tables contain a +\index{Victorius}% +peculiar notation for fractions, which continued in use throughout +the Middle Ages. Victorius is best known for his \textit{canon +paschalis}, a rule for finding the correct date for Easter, which +he published in 457~\AD. + +Payments of interest and problems in interest were very old +among the Romans. The Roman laws of inheritance gave +rise to numerous arithmetical examples. Especially unique is +the following: A dying man wills that, if his wife, being with +child, gives birth to a son, the son shall receive~$\frac{2}{3}$ and she $\frac{1}{3}$~of +his estates; but if a daughter is born, she shall receive~$\frac{1}{3}$ and +his wife~$\frac{2}{3}$. It happens that twins are born, a boy and a girl. +%% -----File: 094.png---Folio 80------- +How shall the estates be divided so as to satisfy the will? +The celebrated Roman jurist, Salvianus Julianus, decided that +the estates shall be divided into seven equal parts, of which +the son receives four, the wife two, the daughter one. + +We next consider Roman geometry. He who expects to +\index{Geometry!Roman}% +find in Rome a science of geometry, with definitions, axioms, +theorems, and proofs arranged in logical order, will be disappointed. +The only geometry known was a \emph{practical} geometry, +which, like the old Egyptian, consisted only of empirical rules. +This practical geometry was employed in surveying. Treatises +thereon have come down to us, compiled by the Roman surveyors, +called \textit{agrimensores} or \textit{gromatici}. One would naturally +\index{Agrimensores}% +\index{Gromatici}% +expect rules to be clearly formulated. But no; they are left +to be abstracted by the reader from a mass of numerical examples. +``The total impression is as though the Roman gromatic +were thousands of years older than Greek geometry, and as +though a deluge were lying between the two.'' Some of their +rules were probably inherited from the Etruscans, but others +are identical with those of Heron. Among the latter is that +\index{Heron the Elder!ref.\ to}% +for finding the area of a triangle from its sides and the approximate +formula,~$\frac{13}{30}a^{2}$, for the area of equilateral triangles ($a$ +being one of the sides). But the latter area was also calculated +by the formulas $\frac{1}{2}(a^{2} + a)$ and $\frac{1}{2} a^{2}$, the first of which was +unknown to Heron. Probably the expression $\frac{1}{2}a^{2}$ was derived +from the Egyptian formula $\dfrac{a + b\Fstr}{2} · \dfrac{c + d}{2}$ for the determination +of the surface of a quadrilateral. This Egyptian formula +was used by the Romans for finding the area, not only of rectangles, +but of any quadrilaterals whatever. Indeed, the gromatici +considered it even sufficiently accurate to determine the +areas of cities, laid out irregularly, simply by measuring their +circumferences.\Reference{7} Whatever Egyptian geometry the Romans +possessed was transplanted across the Mediterranean at the +%% -----File: 095.png---Folio 81------- +\index{Julian calendar}% +time of \textit{Julius Cæsar}, who ordered a survey of the whole +\index{Caesar@{Cæsar}, Julius}% +empire to secure an equitable mode of taxation. Cæsar also +reformed the calendar, and, for that purpose, drew from +\index{Calendar}% +Egyptian learning. He secured the services of the Alexandrian +astronomer, \textit{Sosigenes}. +\index{Sosigenes}% + +In the fifth century, the Western Roman Empire was fast +falling to pieces. Three great branches---Spain, Gaul, and +the province of Africa---broke off from the decaying trunk. +In~476, the Western Empire passed away, and the Visigothic +chief, Odoacer, became king. Soon after, Italy was conquered +by the Ostrogoths under Theodoric. It is remarkable that +this very period of political humiliation should be the one +during which Greek science was studied in Italy most zealously. +School-books began to be compiled from the elements +of Greek authors. These compilations are very deficient, but +are of absorbing interest, from the fact that, down to the +twelfth century, they were the only sources of mathematical +knowledge in the Occident. Foremost among these writers is +\textbf{Boethius} (died~524). At first he was a great favourite of King +\index{Boethius}% +Theodoric, but later, being charged by envious courtiers with +treason, he was imprisoned, and at last decapitated. While +in prison he wrote \textit{On the Consolations of Philosophy}. As a +mathematician, Boethius was a Brobdingnagian among Roman +scholars, but a Liliputian by the side of Greek masters. He +wrote an \textit{Institutis Arithmetica}, which is essentially a translation +of the arithmetic of Nicomachus, and a \textit{Geometry} in +\index{Nicomachus!ref.\ to}% +several books. Some of the most beautiful results of Nicomachus +are omitted in Boethius' arithmetic. The first book +on geometry is an extract from Euclid's \textit{Elements}, which contains, +\index{Euclid!ref.\ to}% +in addition to definitions, postulates, and axioms, the +theorems in the first three books, without proofs. How can +this omission of proofs be accounted for? It has been argued +by some that Boethius possessed an incomplete Greek copy of +%% -----File: 096.png---Folio 82------- +\index{Apices of Boethius}% +\index{Gobar numerals}% +the \textit{Elements}; by others, that he had Theon's edition before +\index{Theon of Alexandria!ref.\ to}% +him, and believed that only the theorems came from Euclid, +while the proofs were supplied by Theon. The second book, +as also other books on geometry attributed to Boethius, +teaches, from numerical examples, the mensuration of plane +figures after the fashion of the agrimensores. + +A celebrated portion in the geometry of Boethius is that +pertaining to an abacus, which he attributes to the Pythagoreans. +\index{Abacus}% +A considerable improvement on the old abacus is +there introduced. Pebbles are discarded, and \textit{apices} (probably +small cones) are used. Upon each of these apices is drawn +a numeral giving it some value below~$10$. The names of +these numerals are pure Arabic, or nearly so, but are added, +apparently, by a later hand. These figures are obviously the +parents of our modern ``Arabic'' numerals. The~$0$~is not +mentioned by Boethius in the text. These numerals bear +striking resemblance to the Gubar-numerals of the West-Arabs, +\index{Gubar-numerals}% +which are admittedly of Indian origin. These facts +have given rise to an endless controversy. Some contended +that Pythagoras was in India, and from there brought the +\index{Pythagoras!ref.\ to}% +nine numerals to Greece, where the Pythagoreans used them +secretly. This hypothesis has been generally abandoned, for +it is not certain that Pythagoras or any disciple of his ever +was in India, nor is there any evidence in any Greek author, +that the apices were known to the Greeks, or that numeral +signs of any sort were used by them with the abacus. It is +improbable, moreover, that the Indian signs, from which the +apices are derived, are so old as the time of Pythagoras. +A second theory is that the \textit{Geometry} attributed to Boethius +is a forgery; that it is not older than the tenth, or possibly +the ninth, century, and that the apices are derived from the +Arabs. This theory is based on contradictions between passages +in the \textit{Arithmetica} and others in the \textit{Geometry}. But +%% -----File: 097.png---Folio 83------- +there is an Encyclopædia written by \textit{Cassiodorius} (died about +\index{Cassiodorius}% +570) in which both the arithmetic and geometry of Boethius +are mentioned. There appears to be no good reason for doubting +the trustworthiness of this passage in the Encyclopædia. +A third theory (Woepcke's) is that the Alexandrians either +\index{Woepcke}% +directly or indirectly obtained the nine numerals from the +Hindoos, about the second century~\AD, and gave them to +the Romans on the one hand, and to the Western Arabs +\index{Romans|)}% +on the other. This explanation is the most plausible. +%% -----File: 098.png---Folio 84------- + + +\Chapter{Middle Ages} + +\Section{The Hindoos} +\index{Hindoos|(}% + +\First{The} first people who distinguished themselves in mathematical +research, after the time of the ancient Greeks, belonged, +like them, to the Aryan race. It was, however, not a European, +but an Asiatic nation, and had its seat in far-off India. + +Unlike the Greek, Indian society was fixed into castes. The +only castes enjoying the privilege and leisure for advanced +study and thinking were the \emph{Brahmins}, whose prime business +was religion and philosophy, and the \emph{Kshatriyas}, who attended +to war and government. + +Of the development of Hindoo mathematics we know but +little. A few manuscripts bear testimony that the Indians +had climbed to a lofty height, but their path of ascent is no +longer traceable. It would seem that Greek mathematics grew +up under more favourable conditions than the Hindoo, for in +Greece it attained an independent existence, and was studied +for its own sake, while Hindoo mathematics always remained +merely a servant to astronomy. Furthermore, in Greece +mathematics was a science of the people, free to be cultivated +by all who had a liking for it; in India, as in Egypt, it was in +the hands chiefly of the priests. Again, the Indians were in +the habit of putting into verse all mathematical results they +obtained, and of clothing them in obscure and mystic language, +%% -----File: 099.png---Folio 85------- +which, though well adapted to aid the memory of him who +already understood the subject, was often unintelligible to the +uninitiated. Although the great Hindoo mathematicians +doubtless reasoned out most or all of their discoveries, yet +they were not in the habit of preserving the proofs, so that +the naked theorems and processes of operation are all that +have come down to our time. Very different in these respects +were the Greeks. Obscurity of language was generally +avoided, and proofs belonged to the stock of knowledge quite +as much as the theorems themselves. Very striking was the +difference in the bent of mind of the Hindoo and Greek; for, +while the Greek mind was pre-eminently \emph{geometrical}, the +Indian was first of all \emph{arithmetical}. The Hindoo dealt with +number, the Greek with form. Numerical symbolism, the +science of numbers, and algebra attained in India far greater +perfection than they had previously reached in Greece. On +the other hand, we believe that there was little or no geometry +in India of which the source may not be traced back to +Greece. Hindoo trigonometry might possibly be mentioned +as an exception, but it rested on arithmetic more than on +geometry. + +An interesting but difficult task is the tracing of the relation +between Hindoo and Greek mathematics. It is well +known that more or less trade was carried on between Greece +and India from early times. After Egypt had become a +Roman province, a more lively commercial intercourse sprang +up between Rome and India, by way of Alexandria. \textit{A~priori}, +it does not seem improbable, that with the traffic of merchandise +there should also be an interchange of ideas. That +communications of thought from the Hindoos to the Alexandrians +actually did take place, is evident from the fact that +certain philosophic and theologic teachings of the Manicheans, +Neo-Platonists, Gnostics, show unmistakable likeness to +%% -----File: 100.png---Folio 86------- +Indian tenets. Scientific facts passed also from Alexandria +to India. This is shown plainly by the Greek origin of some +of the technical terms used by the Hindoos. Hindoo astronomy +\index{Astronomy!Hindoo}% +was influenced by Greek astronomy. Most of the geometrical +knowledge which they possessed is traceable to +Alexandria, and to the writings of Heron in particular. In +algebra there was, probably, a mutual giving and receiving. +We suspect that Diophantus got the first glimpses of algebraic +\index{Diophantus!ref.\ to}% +knowledge from India. On the other hand, evidences have +been found of Greek algebra among the Brahmins. The +earliest knowledge of algebra in India may possibly have been +of Babylonian origin. When we consider that Hindoo scientists +looked upon arithmetic and algebra merely as tools +useful in astronomical research, there appears deep irony in +the fact that these secondary branches were after all the only +ones in which they won real distinction, while in their pet +science of astronomy they displayed an inaptitude to observe, +to collect facts, and to make inductive investigations. + +We shall now proceed to enumerate the names of the +leading Hindoo mathematicians, and then to review briefly +Indian mathematics. We shall consider the science only in +its complete state, for our data are not sufficient to trace the +history of the development of methods. Of the great Indian +mathematicians, or rather, astronomers,---for India had no +mathematicians proper,---\textbf{Aryabhatta} is the earliest. He was +\index{Aryabhatta}% +born 476~\AD, at Pataliputra, on the upper Ganges. His +celebrity rests on a work entitled \textit{Aryabhattiyam}, of which +the third chapter is devoted to mathematics. About one +hundred years later, mathematics in India reached the highest +mark. At that time flourished \textbf{Brahmagupta} (born 598). In +\index{Brahmagupta}% +628 he wrote his \textit{Brahma-sphuta-siddhanta} (``The Revised System +of Brahma''), of which the twelfth and eighteenth chapters +belong to mathematics. To the fourth or fifth century belongs +%% -----File: 101.png---Folio 87------- +\index{Arabic numerals and notation}% +\index{Numerals!Arabic}% +an anonymous astronomical work, called \textit{Surya-siddhanta} +(``Knowledge from the Sun''), which by native authorities +was ranked second only to the \textit{Brahma-siddhanta}, but is of interest +to us merely as furnishing evidence that Greek science +influenced Indian science even before the time of Aryabhatta. +The following centuries produced only two names of importance; +namely, \textbf{Cridhara}, who wrote a \textit{Ganita-sara} (``Quintessence +\index{Cridhara}% +of Calculation''), and \textbf{Padmanabha}, the author of an +\index{Padmanabha}% +algebra. The science seems to have made but little progress +at this time; for a work entitled \textit{Siddhantaciromani} (``Diadem +of an Astronomical System''), written by \textbf{Bhaskara Acarya} in +\index{Bhaskara}% +1150, stands little higher than that of Brahmagupta, written +over 500~years earlier. The two most important mathematical +chapters in this work are the \textit{Lilavati} ($=$~``the beautiful,'' \ie~the +noble science) and \textit{Viga-ganita} ($=$~``root-extraction''), devoted +to arithmetic and algebra. From now on, the Hindoos +in the Brahmin schools seemed to content themselves with +studying the masterpieces of their predecessors. Scientific +intelligence decreases continually, and in modern times a very +deficient Arabic work of the sixteenth century has been held +in great authority.\Reference{7} + +The mathematical chapters of the \textit{Brahma-siddhanta} and +\textit{Siddhantaciromani} were translated into English by H.~T. +Colebrooke, London, 1817. The \textit{Surya-siddhanta} was translated +\index{Colebrooke}% +by E.~Burgess, and annotated by W.~D. Whitney, New +\index{Whitney}% +Haven, Conn., 1860. + +The grandest achievement of the Hindoos and the one +which, of all mathematical inventions, has contributed most +to the general progress of intelligence, is the invention of +the principle of position in writing numbers. Generally we +speak of our notation as the ``Arabic'' notation, but it should +\index{Notation!Arabic notation}% +be called the ``Hindoo'' notation, for the Arabs borrowed it +from the Hindoos. That the invention of this notation was +%% -----File: 102.png---Folio 88------- +not so easy as we might suppose at first thought, may be +inferred from the fact that, of other nations, not even the +keen-minded Greeks possessed one like it. We inquire, who +invented this ideal symbolism, and when? But we know +neither the inventor nor the time of invention. That our +system of notation is of Indian origin is the only point of +which we are certain. From the evolution of ideas in general +we may safely infer that our notation did not spring into +existence a completely armed Minerva from the head of +Jupiter. The nine figures for writing the units are supposed +to have been introduced earliest, and the sign of zero and the +\index{Zero!(symbol for)}% +principle of position to be of later origin. This view receives +support from the fact that on the island of Ceylon a notation +resembling the Hindoo, but without the zero has been preserved. +We know that Buddhism and Indian culture were +transplanted to Ceylon about the third century after Christ, +and that this culture remained stationary there, while it made +progress on the continent. It seems highly probable, then, +that the numerals of Ceylon are the old, imperfect numerals +of India. In Ceylon, nine figures were used for the units, +nine others for the tens, one for~$100$, and also one for~$1000$. +These 20~characters enabled them to write all the numbers up +to~$9999$. Thus, $8725$ would have been written with six signs, +representing the following numbers: $8$,~$1000$, $7$,~$100$, $20$,~$5$. +These Singhalesian signs, like the old Hindoo numerals, are +supposed originally to have been the initial letters of the corresponding +numeral adjectives. There is a marked resemblance +between the notation of Ceylon and the one used by Aryabhatta +\index{Aryabhatta!ref.\ to}% +in the first chapter of his work, and there only. Although the +zero and the principle of position were unknown to the scholars +of Ceylon, they were probably known to Aryabhatta; for, in +the second chapter, he gives directions for extracting the square +and cube roots, which seem to indicate a knowledge of them. +%% -----File: 103.png---Folio 89------- +It would appear that the zero and the accompanying principle +of position were introduced about the time of Aryabhatta. +These are the inventions which give the Hindoo system its +great superiority, its admirable perfection. + +There appear to have been several notations in use in +different parts of India, which differed, not in principle, but +merely in the forms of the signs employed. Of interest is +also a \emph{symbolical system of position}, in which the figures +generally were not expressed by numerical adjectives, but by +objects suggesting the particular numbers in question. Thus, +for~$1$ were used the words \emph{moon}, \emph{Brahma}, \emph{Creator}, or \emph{form}; +for~$4$, the words \emph{Veda}, (because it is divided into four parts) +or \emph{ocean}, etc. The following example, taken from the \textit{Surya-siddhanta}, +illustrates the idea. The number $1,577,917,828$ is +expressed from right to left as follows: Vasu (a class of $8$~gods)% +${}+{}$ two${}+{}$eight${}+{}$mountains (the $7$~mountain-chains) +${}+{}$form${}+{}$digits (the $9$~digits)${}+{}$seven${}+{}$mountains${}+{}$lunar +days (half of which equal~$15$). The use of such notations +made it possible to represent a number in several different +ways. This greatly facilitated the framing of verses containing +arithmetical rules or scientific constants, which could +thus be more easily remembered. + +At an early period the Hindoos exhibited great skill in +calculating, even with large numbers. Thus, they tell us of +an examination to which Buddha, the reformer of the Indian +\index{Buddha}% +religion, had to submit, when a youth, in order to win the +maiden he loved. In arithmetic, after having astonished his +examiners by naming all the periods of numbers up to the~53d, +he was asked whether he could determine the number +of primary atoms which, when placed one against the other, +would form a line one mile in length. Buddha found the +required answer in this way: $7$~primary atoms make a very +minute grain of dust, $7$~of these make a minute grain of dust, +%% -----File: 104.png---Folio 90------- +\index{Arithmetic!Hindoo|(}% +$7$~of \emph{these} a grain of dust whirled up by the wind, and so on. +Thus he proceeded, step by step, until he finally reached the +length of a mile. The multiplication of all the factors gave +for the multitude of primary atoms in a mile a number consisting +of 15~digits. This problem reminds one of the `Sand-Counter' +\index{Sand-counter}% +of Archimedes. +\index{Archimedes!ref.\ to}% + +After the numerical symbolism had been perfected, figuring +was made much easier. Many of the Indian modes of +operation differ from ours. The Hindoos were generally +inclined to follow the motion from left to right, as in writing. +Thus, they \emph{added} the left-hand columns first, and made the +necessary corrections as they proceeded. For instance, they +would have added $254$ and $663$ thus: $2 + 6 = 8$, $5 + 6=11$, +which changes $8$ into~$9$, $4 + 3 = 7$. Hence the sum~$917$. In +\emph{subtraction} they had two methods. Thus in $821 - 348$ they +would say, $8\text{ from }11 = 3$, $4\text{ from }11 = 7$, $3\text{ from }7 = 4$. Or +they would say, $8\text{ from }11 = 3$, $5\text{ from }12 = 7$, $4\text{ from }8 = 4$. +In \emph{multiplication} of a number by another of only one digit, say +$569$ by~$5$, they generally said, $5·5 = 25$, $5·6 = 30$, which +changes $25$ into~$28$, $5·9 = 45$, hence the $0$ must be increased by~$4$. +The product is~$2845$. In the multiplication with each +other of many-figured numbers, they first multiplied, in the +manner just indicated, with the left-hand digit of the multiplier, +which was written above the multiplicand, and placed +the product above the multiplier. On multiplying with the +next digit of the multiplier, the product was not placed in +a new row, as with us, but the first product obtained was +corrected, as the process continued, by erasing, whenever +necessary, the old digits, and replacing them by new ones, +until finally the whole product was obtained. We who possess +the modern luxuries of pencil and paper, would not be likely +to fall in love with this Hindoo method. But the Indians +wrote ``with a cane-pen upon a small blackboard with a white, +%% -----File: 105.png---Folio 91------- +\index{Casting out the 9's}% +thinly liquid paint which made marks that could be easily +erased, or upon a white tablet, less than a foot square, strewn +with red flour, on which they wrote the figures with a small +stick, so that the figures appeared white on a red ground.''\Reference{7} +Since the digits had to be quite large to be distinctly legible, +and since the boards were small, it was desirable to have a +method which would not require much space. Such a one +was the above method of multiplication. Figures could be +easily erased and replaced by others without sacrificing neatness. +But the Hindoos had also other +ways of multiplying, of which we mention +the following: The %[** PP: Width-dependent line break] +%[Illustration] +\begin{wrapfigure}{r}{1.6in} +\hfill\Graphic{1.6in}{105a} +\end{wrapfigure} +tablet was +divided into squares like a chess-board. +Diagonals were also drawn, as seen in +the figure. The multiplication of $12 × +735 = 8820$ is exhibited in the adjoining diagram.\Reference{3} +The manuscripts +extant give no information of how \emph{divisions} were +executed. The correctness of their additions, subtractions, +and multiplications was tested ``by excess of $9$'s.'' In writing +fractions, the numerator was placed above the denominator, +but no line was drawn between them. + +We shall now proceed to the consideration of some arithmetical +problems and the Indian modes of solution. A favourite +method was that of \emph{inversion}. With laconic brevity, +Aryabhatta describes it thus: ``Multiplication becomes division, +\index{Aryabhatta!ref.\ to}% +division becomes multiplication; what was gain becomes +loss, what loss, gain; inversion.'' Quite different from this +quotation in style is the following problem from Aryabhatta, +which illustrates the method:\Reference[]{3} ``Beautiful maiden with beaming +eyes, tell me, as thou understandst the right method of +inversion, which is the number which multiplied by~$3$, then +increased by $\frac{3}{4}$ of the product, divided by~$7$, diminished by $\frac{1}{3}$ of +the quotient, multiplied by itself, diminished by~$52$, the square +%% -----File: 106.png---Folio 92------- +\index{Arithmetic!Hindoo|)}% +root extracted, addition of~$8$, and division by~$10$, gives the +number~$2$?'' The process consists in beginning with $2$ and +working backwards. Thus, $(2·10-8)^{2}+52 = 196$, $\sqrt {196}=14$, +and $14·\frac{3}{2}·7·\frac{4}{7} ÷ 3 = 28$, the answer. + +Here is another example taken from \textit{Lilavati}, a chapter in +Bhaskara's great work: ``The square root of half the number +\index{Bhaskara!ref.\ to|(}% +of bees in a swarm has flown out upon a jessamine-bush, $\frac{8}{9}$ of +the whole swarm has remained behind; one female bee flies +about a male that is buzzing within a lotus-flower into which +he was allured in the night by its sweet odour, but is now +imprisoned in it. Tell me the number of bees.'' Answer,~$72$. +The pleasing poetic garb in which all arithmetical problems +are clothed is due to the Indian practice of writing all school-books +in verse, and especially to the fact that these problems, +propounded as puzzles, were a favourite social amusement. +Says Brahmagupta: ``These problems are proposed simply for +\index{Brahmagupta!ref.\ to}% +pleasure; the wise man can invent a thousand others, or he +can solve the problems of others by the rules given here. As +the sun eclipses the stars by his brilliancy, so the man of +knowledge will eclipse the fame of others in assemblies of the +people if he proposes algebraic problems, and still more if he +solves them.'' + +The Hindoos solved problems in interest, discount, partnership, +alligation, summation of arithmetical and geometric +series, devised rules for determining the numbers of combinations +and permutations, and invented magic squares. It may +\index{Magic squares}% +here be added that chess, the profoundest of all games, had its +\index{Chess}% +origin in India. + +The Hindoos made frequent use of the ``rule of three,'' and +\index{Rule of three}% +also of the method of ``falsa positio,'' which is almost identical +\index{Falsa positio}% +with that of the ``tentative assumption'' of Diophantus. +\index{Assumption, tentative}% +\index{Tentative assumption}% +These and other rules were applied to a large number of +problems. +%% -----File: 107.png---Folio 93------- +\index{Notation!in algebra}% +\index{Quadratic equations}% + +Passing now to \emph{algebra}, we shall first take up the symbols +\index{Algebra!Hindoo|(}% +of operation. Addition was indicated simply by juxtaposition +as in Diophantine algebra; subtraction, by placing a dot over +the subtrahend; multiplication, by putting after the factors, +\emph{bha}, the abbreviation of the word \emph{bhavita}, ``the product''; +division, by placing the divisor beneath the dividend; square-root, +by writing \emph{ka}, from the word \emph{karana} (irrational), before +the quantity. The unknown quantity was called by Brahmagupta +\emph{yâvattâvat} (\emph{quantum tantum}). When several unknown +quantities occurred, he gave, unlike Diophantus, to each a +\index{Diophantus!ref.\ to}% +distinct name and symbol. The first unknown was designated +by the general term ``unknown quantity.'' The rest were +distinguished by names of colours, as the black, blue, yellow, +red, or green unknown. The initial syllable of each word +constituted the symbol for the respective unknown quantity. +Thus \emph{yâ} meant~$x$; \emph{kâ} (from $\text{\emph{kâlaka}} = \text{black}$) meant~$y$; \emph{yâ kâ +bha}, ``$x$~times~$y$''; \emph{ka}~$15$ \emph{ka}~$10$, ``$\sqrt{15} - \sqrt{10}$.'' + +The Indians were the first to recognise the existence of +absolutely negative quantities. They brought out the difference +\index{Negative quantities}% +between positive and negative quantities by attaching to +the one the idea of `possession,' to the other that of `debts.' +The conception also of opposite directions on a line, as an +interpretation of $+$~and~$-$ quantities, was not foreign to them. +They advanced beyond Diophantus in observing that a quadratic +has always two roots. Thus Bhaskara gives $x = 50$ and +$x = -5$ for the roots of $x^{2} - 45 x = 250$. ``But,'' says he, +``the second value is in this case not to be taken, for it is +inadequate; people do not approve of negative roots.'' Commentators +\index{Negative roots}% +speak of this as if negative roots were seen, but not +admitted. + +Another important generalisation, says Hankel, was this, +\index{Hankel!ref.\ to}% +that the Hindoos never confined their arithmetical operations +to rational numbers. For instance, Bhaskara showed how, +%% -----File: 108.png---Folio 94------- +\index{Irrationals}% +by the formula %[** PP: Displaying] +\[ +\sqrt{a + \sqrt{b}} + = \sqrt{\frac{a + \sqrt{a^{2} - b}}{2}} + + \sqrt{\frac{a - \sqrt{a^{2} - b}}{2}} +\] +the square root of the sum of rational and irrational numbers +\index{Square root}% +could be found. The Hindoos never discerned the dividing +line between numbers and magnitudes, set up by the Greeks, +which, though the product of a scientific spirit, greatly retarded +the progress of mathematics. They passed from magnitudes +to numbers and from numbers to magnitudes without +anticipating that gap which to a sharply discriminating mind +exists between the continuous and discontinuous. Yet by +doing so the Indians greatly aided the general progress of +mathematics. ``Indeed, if one understands by algebra the +application of arithmetical operations to complex magnitudes +of all sorts, whether rational or irrational numbers or space-magnitudes, +then the learned Brahmins of Hindostan are the +real inventors of algebra.''\Reference{7} + +Let us now examine more closely the Indian algebra. In +extracting the square and cube roots they used the formulas +$(a + b)^{2} = a^{2} + 2ab + b^{2}$ and $(a + b)^{3}= a^{3} + 3a^{2}b + 3ab^{2} + b^{3}$. +In this connection Aryabhatta speaks of dividing a number +into periods of two and three digits. From this we infer that +the principle of position and the zero in the numeral notation +were already known to him. In figuring with zeros, a statement +of Bhaskara is interesting. A fraction whose denominator +is zero, says he, admits of no alteration, though much be +added or subtracted. Indeed, in the same way, no change +takes place in the infinite and immutable Deity when worlds +are destroyed or created, even though numerous orders of beings +be taken up or brought forth. Though in this he apparently +evinces clear mathematical notions, yet in other places he +makes a complete failure in figuring with fractions of zero +\index{Fractions!Hindoo}% +denominator. + +In the Hindoo solutions of determinate equations, Cantor +%% -----File: 109.png---Folio 95------- +\index{Numbers!theory of numbers}% +\index{Theory of numbers}% +thinks he can see traces of Diophantine methods. Some +technical terms betray their Greek origin. Even if it be true +that the Indians borrowed from the Greeks, they deserve great +credit for improving and generalising the solutions of linear +and quadratic equations. Bhaskara advances far beyond the +\index{Bhaskara!ref.\ to|)}% +Greeks and even beyond Brahmagupta when he says that +\index{Brahmagupta!ref.\ to}% +``the square of a positive, as also of a negative number, +is positive; that the square root of a positive number is +twofold, positive and negative. There is no square root +of a negative number, for it is not a square.'' Of equations +of higher degrees, the Indians succeeded in solving +only some special cases in which both sides of the equation +could be made perfect powers by the addition of certain +terms to each. + +Incomparably greater progress than in the solution of determinate +equations was made by the Hindoos in the treatment +of \emph{indeterminate equations}. Indeterminate analysis was a +\index{Indeterminate analysis}% +\index{Indeterminate equations}% +subject to which the Hindoo mind showed a happy adaptation. +We have seen that this very subject was a favourite with Diophantus, +\index{Diophantus!ref.\ to}% +and that his ingenuity was almost inexhaustible in +devising solutions for particular cases. But the glory of +having invented \emph{general} methods in this most subtle branch +of mathematics belongs to the Indians. The Hindoo indeterminate +analysis differs from the Greek not only in method, +but also in aim. The object of the former was to find all +possible integral solutions. Greek analysis, on the other hand, +demanded not necessarily integral, but simply rational answers. +Diophantus was content with a single solution; the Hindoos +endeavoured to find all solutions possible. Aryabhatta gives +solutions in integers to linear equations of the form $ax ± by=c$, +where $a$,~$b$,~$c$ are integers. The rule employed is called the +\emph{pulveriser}. For this, as for most other rules, the Indians give +\index{Pulveriser}% +no proof. Their solution is essentially the same as the one of +%% -----File: 110.png---Folio 96------- +\index{Algebra!Hindoo|)}% +Euler. Euler's process of reducing $\dfrac{a}{b}$ to a continued fraction +\index{Euler!ref.\ to}% +amounts to the same as the Hindoo process of finding the +greatest common divisor of $a$~and~$b$ by division. This is frequently +called the Diophantine method. Hankel protests +\index{Hankel!ref.\ to}% +against this name, on the ground that Diophantus not only +\index{Diophantus!ref.\ to}% +never knew the method, but did not even aim at solutions +purely integral.\Reference{7} These equations probably grew out of problems +in astronomy. They were applied, for instance, to +determine the time when a certain constellation of the planets +would occur in the heavens. + +Passing by the subject of linear equations with more than +two unknown quantities, we come to indeterminate quadratic +equations. In the solution of $xy = ax + by + c$, they applied +the method re-invented later by Euler, of decomposing $(ab + c)$ +into the product of two integers $m·n$ and of placing $x = m + b$ +and $y = n + a$. + +Remarkable is the Hindoo solution of the quadratic equation +$cy^{2} = ax^{2} + b$. With great keenness of intellect they +recognised in the special case $y^{2} = ax^{2} + 1$ a fundamental +problem in indeterminate quadratics. They solved it by the +\emph{cyclic method}. ``It consists,'' says De~Morgan, ``in a rule for +\index{Cyclic method}% +\index{Demorgan@{De Morgan}!ref.\ to}% +finding an indefinite number of solutions of $y^{2} = ax^{2} +1$ ($a$~being +an integer which is not a square), by means of one solution +given or found, and of feeling for one solution by making a +solution of $y^2 = ax^{2} + b$ give a solution of $y^{2} = ax^{2} + b^{2}$. It +amounts to the following theorem: If $p$ and~$q$ be one set of +values of $x$ and~$y$ in $y^{2} = ax^{2} + b$ and $p'$ and~$q'$ the same or +another set, then $qp + pq'$ and $app' + qq'$ are values of $x$ and~$y$ +in $y^{2} = ax^{2} + b^{2}$. From this it is obvious that one solution of +$y^{2} = ax^{2} + 1$ may be made to give any number, and that if, +taking $b$ at pleasure, $y^{2} = ax^{2} + b^{2}$ can be solved so that $x$ and~$y$ +are divisible by~$b$, then one preliminary solution of $y^{2} = ax^{2} + 1$ +%% -----File: 111.png---Folio 97------- +\index{Pell's problem}% +\index{Pythagoras!ref.\ to}% +can be found. Another mode of trying for solutions is a +combination of the preceding with the \emph{cuttaca} (pulveriser).'' +These calculations were used in astronomy. + +Doubtless this ``cyclic method'' constitutes the greatest +\index{Cyclic method}% +invention in the theory of numbers before the time of Lagrange. +The perversity of fate has willed it, that the equation +$y^{2} = ax^{2} + 1$ should now be called \emph{Pell's} problem, while in +recognition of Brahmin scholarship it ought to be called the +``Hindoo problem.'' It is a problem that has exercised the +highest faculties of some of our greatest modern analysts. +By them the work of the Hindoos was done over again; for, +unfortunately, the Arabs transmitted to Europe only a small +part of Indian algebra and the original Hindoo manuscripts, +which we now possess, were unknown in the Occident. + +Hindoo \emph{geometry} is far inferior to the Greek. In it are +\index{Geometry!Hindoo}% +found no definitions, no postulates, no axioms, no logical +chain of reasoning or rigid form of demonstration, as with +Euclid. Each theorem stands by itself as an independent +\index{Euclid!ref.\ to}% +truth. Like the early Egyptian, it is empirical. Thus, in the +proof of the theorem of the right triangle, Bhaskara draws +\index{Bhaskara!ref.\ to}% +the %[** PP: Width-dependent line break] +% *[Illustration] +\begin{wrapfigure}[5]{r}{2.25in} +\hfill\Graphic{2.25in}{111a} +\end{wrapfigure} +right triangle four times in the square of the hypotenuse, +so that in the middle +there remains a square +whose side equals the +difference between the +two sides of the right +triangle. Arranging this square and the four triangles in a +different way, they are seen, together, to make up the sum +of the square of the two sides. ``Behold!'' says Bhaskara, +without adding another word of explanation. Bretschneider +\index{Bretschneider}% +conjectures that the Pythagorean proof was substantially the +same as this. In another place, Bhaskara gives a second +demonstration of this theorem by drawing from the vertex of +%% -----File: 112.png---Folio 98------- +\index{Ptolemy!ref.\ to}% +the right angle a perpendicular to the hypotenuse, and comparing +the two triangles thus obtained with the given triangle +to which they are similar. This proof was unknown in Europe +till Wallis \Change{rediscovered}{re-discovered} it. The Brahmins never inquired +\index{Wallis!ref.\ to}% +into the properties of figures. They considered only metrical +relations applicable in practical life. In the Greek sense, the +Brahmins never had a science of geometry. Of interest is +\index{Geometry!Hindoo}% +the formula given by Brahmagupta for the area of a triangle +\index{Brahmagupta!ref.\ to}% +in terms of its sides. In the great work attributed to Heron +\index{Heron the Elder!ref.\ to}% +the Elder this formula is first found. Whether the Indians +themselves invented it, or whether they borrowed it from +Heron, is a disputed question. Several theorems are given +by Brahmagupta on quadrilaterals which are true only of +those which can be inscribed on a circle---a limitation which +he omits to state. Among these is the proposition of Ptolemæus, +that the product of the diagonals is equal to the sum +of the products of the opposite sides. The Hindoos were +familiar with the calculation of the areas of circles and their +segments, of the length of chords and perimeters of regular +inscribed polygons. An old Indian tradition makes $\pi = 3$, +also $= \sqrt{10}$; but Aryabhatta gives the value $\frac{31416}{10000}$. Bhaskara +\index{Aryabhatta!ref.\ to}% +gives two values,---the `accurate,' $\frac{3927}{1250}$, and the `inaccurate,' +Archimedean value,~$\frac{22}{7}$. A commentator on \textit{Lilavati} says +that these values were calculated by beginning with a regular +inscribed hexagon, and applying repeatedly the formula +$AD = \sqrt{2 - \sqrt{\vphantom{\rule{0pt}{12pt}}4 - \smash[t]{\overline{AB}^{2}}}}$, wherein~$AB$ is the side of the given +polygon, and $AD$ that of one with double the number of +sides. In this way were obtained the perimeters of the +inscribed polygons of $12$, $24$, $48$, $96$, $192$, $384$~sides. Taking +the radius $= 100$, the perimeter of the last one gives the value +which Aryabhatta used for~$\pi$. +\index{p@{$\pi$}: values for!Hindoo}% + +Greater taste than for geometry was shown by the Hindoos +for \emph{trigonometry}. Like the Babylonians and Greeks, they +\index{Trigonometry|(}% +%% -----File: 113.png---Folio 99------- +divided the circle into quadrants, each quadrant into $90$~degrees +and $5400$ minutes. The whole circle was therefore +made up of $21,600$ equal parts. From Bhaskara's `accurate' +value for~$\pi$ it was found that the radius contained $3438$ of +these circular parts. This last step was not Grecian. The +Greeks might have had scruples about taking a part of a +\emph{curve} as the measure of a \emph{straight line}. Each quadrant was +divided into $24$~equal parts, so that each part embraced $225$ +units of the whole circumference, and corresponds to $3 \frac{3}{4}$ degrees. +Notable is the fact that the Indians never reckoned, like the +Greeks, with the whole chord of double the arc, but always +with the \emph{sine} (\emph{joa}) and \emph{versed sine}. Their mode of calculating +\index{Sine}% +\index{Versed sine}% +tables was theoretically very simple. The sine of~$90°$ was +equal to the radius, or~$3438$; the sine of~$30°$ was evidently +half that, or~$1719$. Applying the formula $\sin^{2}a + \cos^{2}a = r^{2}$, +they obtained $\sin 45° = \sqrt{\dfrac{r^2}{2}} = 2431$. Substituting for $\cos a$ +its equal $\sin(90 - a)$, and making $a = 60°$, they obtained +$\sin 60° = \dfrac{\sqrt{3r^2}}{2} = 2978$. With the sines of $90$, $60$, $45$, and~$30$ +as starting-points, they reckoned the sines of half the angles +by the formula $\versin 2a = 2 \sin^{2}a$, thus obtaining the sines +of $22°\ 30'$, $11°\ 15'$, $7°\ 30'$, $3°\ 45'$. They now figured out the +sines of the complements of these angles, namely, the sines +of $86°\ 15'$, $82°\ 30'$, $78°\ 45'$, $75°$, $67°\ 30'$; then they calculated +the sines of half these angles; then of their complements; +then, again, of half their complements; and so on. By this +very simple process they got the sines of angles at intervals of +$3°\ 45'$. In this table they discovered the unique law that if +$a$,~$b$,~$c$ be three successive arcs such that $a - b = b - c = 3°\ 45'$, +then $\sin a - \sin b = (\sin b - \sin c) - \dfrac{\sin b}{225}$. This formula was +afterwards used whenever a re-calculation of tables had to +be made. No Indian trigonometrical treatise on the triangle +%% -----File: 114.png---Folio 100------- +\index{Hindoos|)}% +\index{Trigonometry|)}% +is extant. In astronomy they solved plane and spherical +\index{Astronomy!Arabic}% +right triangles.\Reference{18} + +It is remarkable to what extent Indian mathematics enters +into the science of our time. Both the form and the spirit +of the arithmetic and algebra of modern times are essentially +Indian and not Grecian. Think of that most perfect of +mathematical symbolisms---the Hindoo notation, think of the +Indian arithmetical operations nearly as perfect as our own, +think of their elegant algebraical methods, and then judge +whether the Brahmins on the banks of the Ganges are not +entitled to some credit. Unfortunately, some of the most +brilliant of Hindoo discoveries in indeterminate analysis +reached Europe too late to exert the influence they would +have exerted, had they come two or three centuries earlier. + +\Section{The Arabs} +\index{Arabs|(}% + +After the flight of Mohammed from Mecca to Medina in +622~\AD, an obscure people of Semitic race began to play an +important part in the drama of history. Before the lapse of +ten years, the scattered tribes of the Arabian peninsula were +fused by the furnace blast of religious enthusiasm into a +powerful nation. With sword in hand the united Arabs +subdued Syria and Mesopotamia. Distant Persia and the +lands beyond, even unto India, were added to the dominions +of the Saracens. They conquered Northern Africa, and nearly +the whole Spanish peninsula, but were finally checked from +further progress in Western Europe by the firm hand of +Charles Martel (732~\AD). The Moslem dominion extended +now from India to Spain; but a war of succession to the +caliphate ensued, and in 755 the Mohammedan empire was +divided,---one caliph reigning at Bagdad, the other at Cordova +%% -----File: 115.png---Folio 101------- +\index{Indeterminate analysis}% +\index{Indeterminate equations}% +in Spain. Astounding as was the grand march of conquest by +the Arabs, still more so was the ease with which they put +aside their former nomadic life, adopted a higher civilisation, +and assumed the sovereignty over cultivated peoples. Arabic +was made the written language throughout the conquered +lands. With the rule of the Abbasides in the East began a +new period in the history of learning. The capital, Bagdad, +situated on the Euphrates, lay half-way between two old +centres of scientific thought,---India in the East, and Greece +in the West. The Arabs were destined to be the custodians +of the torch of Greek and Indian science, to keep it ablaze +during the period of confusion and chaos in the Occident, and +afterwards to pass it over to the Europeans. Thus science +passed from Aryan to Semitic races, and then back again +to the Aryan. The Mohammedans have added but little to +the knowledge in mathematics which they received. They +now and then explored a small region to which the path had +been previously pointed out, but they were quite incapable of +discovering new fields. Even the more elevated regions in +which the Hellenes and Hindoos delighted to wander---namely, +the Greek conic sections and the Indian indeterminate +\index{Conic sections!Arabs}% +analysis---were seldom entered upon by the Arabs. They +were less of a speculative, and more of a practical turn of +mind. + +The Abbasides at Bagdad encouraged the introduction of +the sciences by inviting able specialists to their court, irrespective +of nationality or religious belief. Medicine and +astronomy were their favourite sciences. Thus Haroun-al-Raschid, +\index{Astronomy!Arabic}% +the most distinguished Saracen ruler, drew Indian +physicians to Bagdad. In the year~772 there came to the +court of Caliph Almansur a Hindoo astronomer with astronomical +tables which were ordered to be translated into Arabic. +These tables, known by the Arabs as the \textit{Sindhind}, and +%% -----File: 116.png---Folio 102------- +\index{Sine}% +probably taken from the \textit{Brahma-sphuta-siddhanta} of Brahmagupta, +\index{Brahmagupta!ref.\ to}% +stood in great authority. They contained the important +Hindoo table of sines. + +Doubtless at this time, and along with these astronomical +tables, the Hindoo numerals, with the zero and the principle +\index{Numerals!Arabic}% +of position, were introduced among the Saracens. Before the +time of Mohammed the Arabs had no numerals. Numbers +were written out in words. Later, the numerous computations +connected with the financial administration over the conquered +lands made a short symbolism indispensable. In some localities, +the numerals of the more civilised conquered nations +were used for a time. Thus in Syria, the Greek notation was +\index{Notation!Arabic notation}% +retained; in Egypt, the Coptic. In some cases, the numeral +adjectives may have been abbreviated in writing. The \textit{Diwani-numerals}, +\index{Diwani-numerals}% +found in an Arabic-Persian dictionary, are supposed +to be such abbreviations. Gradually it became the practice to +employ the 28~Arabic letters of the alphabet for numerals, in +analogy to the Greek system. This notation was in turn +superseded by the Hindoo notation, which quite early was +adopted by merchants, and also by writers on arithmetic. Its +superiority was so universally recognised, that it had no rival, +except in astronomy, where the alphabetic notation continued +to be used. Here the alphabetic notation offered no great +disadvantage, since in the sexagesimal arithmetic, taken from +the \textit{Almagest}, numbers of generally only one or two places +had to be written.\Reference{7} + +As regards the form of the so-called Arabic numerals, the +\index{Arabic numerals and notation}% +statement of the Arabic writer \textit{Albiruni} (died~1039), who +\index{Albiruni!ref.\ to}% +spent many years in India, is of interest. He says that the +shape of the numerals, as also of the letters in India, differed +in different localities, and that the Arabs selected from the +various forms the most suitable. An Arabian astronomer +says there was among people much difference in the use of +%% -----File: 117.png---Folio 103------- +\index{Gobar numerals}% +symbols, especially of those for $5$, $6$, $7$, and~$8$. The symbols +used by the Arabs can be traced back to the tenth century. +We find material differences between those used by the +Saracens in the East and those used in the West. But +most surprising is the fact that the symbols of both the East +and of the West Arabs deviate so extraordinarily from the +Hindoo \textit{Devanagari} numerals ($=$~divine numerals) of to-day, +\index{Devanagari-numerals}% +\index{Numerals!Arabic}% +and that they resemble much more closely the apices of +\index{Apices of Boethius!ref.\ to}% +the Roman writer Boethius. This strange similarity on the +\index{Boethius!ref.\ to}% +one hand, and dissimilarity on the other, is difficult to explain. +The most plausible theory is the one of Woepcke: (1)~that +\index{Woepcke}% +about the second century after Christ, before the zero had +been invented, the Indian numerals were brought to Alexandria, +whence they spread to Rome and also to West Africa; +(2)~that in the eighth century, after the notation in India had +been already much modified and perfected by the invention of +the zero, the Arabs at Bagdad got it from the Hindoos; (3)~that +the Arabs of the West borrowed the Columbus-egg, the zero, +from those in the East, but retained the old forms of the nine +numerals, if for no other reason, simply to be contrary to their +political enemies of the East; (4)~that the old forms were +remembered by the West-Arabs to be of Indian origin, and +were hence called \textit{Gubar-numerals} ($=$~dust-numerals, in memory +\index{Gubar-numerals}% +of the Brahmin practice of reckoning on tablets strewn +with dust or sand; (5)~that, since the eighth century, the +numerals in India underwent further changes, and assumed +the greatly modified forms of the modern Devanagari-numerals.\Reference{3} +This is rather a bold theory, but, whether true or not, +it explains better than any other yet propounded, the relations +between the apices, the Gubar, the East-Arabic, and Devanagari +numerals. + +It has been mentioned that in 772 the Indian \textit{Siddhanta} was +brought to Bagdad and there translated into Arabic. There +%% -----File: 118.png---Folio 104------- +\index{Geometry!Arabic}% +\index{Ptolemy!ref.\ to}% +is no evidence that any intercourse existed between Arabic +and Indian astronomers either before or after this time, excepting +the travels of Albiruni. But we should be very slow +\index{Albiruni!ref.\ to}% +to deny the probability that more extended communications +actually did take place. + +Better informed are we regarding the way in which Greek +science, in successive waves, dashed upon and penetrated Arabic +soil. In Syria the sciences, especially philosophy and medicine, +were cultivated by Greek Christians. Celebrated were the +schools at Antioch and Emesa, and, first of all, the flourishing +Nestorian school at Edessa. From Syria, Greek physicians +and scholars were called to Bagdad. Translations of works +from the Greek began to be made. A large number of Greek +manuscripts were secured by Caliph \textit{Al Mamun} (813--833) from +the emperor in Constantinople and were turned over to Syria. +The successors of Al Mamun continued the work so auspiciously +begun, until, at the beginning of the tenth century, the +more important philosophic, medical, mathematical, and astronomical +works of the Greeks could all be read in the Arabic +tongue. The translations of mathematical works must have +been very deficient at first, as it was evidently difficult to +secure translators who were masters of both the Greek and +Arabic and at the same time proficient in mathematics. The +translations had to be revised again and again before they +were satisfactory. The first Greek authors made to speak in +Arabic were Euclid and Ptolemæus. This was accomplished +\index{Euclid!ref.\ to}% +during the reign of the famous Haroun-al-Raschid. A revised +\index{Haroun-al-Raschid}% +translation of Euclid's \textit{Elements} was ordered by Al Mamun. +\index{Elements (Euclid's)}% +As this revision still contained numerous errors, a new translation +was made, either by the learned Honein ben Ishak, or +\index{Honein ben Ishak}% +by his son, Ishak ben Honein. To the thirteen books of the +\index{Ishak ben Honein}% +\textit{Elements} were added the fourteenth, written by Hypsicles, +\index{Hypsicles!ref.\ to}% +and the fifteenth by Damascius. But it remained for Tabit +\index{Damascius!ref.\ to}% +%% -----File: 119.png---Folio 105------- +ben Korra to bring forth an Arabic Euclid satisfying every +\index{Tabit ben Korra!ref.\ to}% +need. Still greater difficulty was experienced in securing an +intelligible translation of the \textit{Almagest}. Among other important +\index{Almagest!ref.\ to}% +translations into Arabic were the works of Apollonius, +\index{Apollonius!ref.\ to}% +Archimedes, Heron, and Diophantus. Thus we see that in +\index{Archimedes!ref.\ to}% +\index{Diophantus!ref.\ to}% +\index{Heron the Elder!ref.\ to}% +the course of one century the Arabs gained access to the vast +treasures of Greek science. Having been little accustomed to +abstract thought, we need not marvel if, during the ninth century, +all their energy was exhausted merely in appropriating +the foreign material. No attempts were made at original +work in mathematics until the next century. + +In astronomy, on the other hand, great activity in original +\index{Astronomy!Arabic}% +research existed as early as the ninth century. The religious +observances demanded by Mohammedanism presented to astronomers +several practical problems. The Moslem dominions +being of such enormous extent, it remained in some localities +for the astronomer to determine which way the ``Believer'' +must turn during prayer that he may be facing Mecca. The +prayers and ablutions had to take place at definite hours during +the day and night. This led to more accurate determinations +of time. To fix the exact date for the Mohammedan +feasts it became necessary to observe more closely the motions +of the moon. In addition to all this, the old Oriental superstition +that extraordinary occurrences in the heavens in some +mysterious way affect the progress of human affairs added +increased interest to the prediction of eclipses.\Reference{7} + +For these reasons considerable progress was made. Astronomical +tables and instruments were perfected, observatories +erected, and a connected series of observations instituted. This +intense love for astronomy and astrology continued during the +whole Arabic scientific period. As in India, so here, we hardly +ever find a man exclusively devoted to pure mathematics. Most +of the so-called mathematicians were first of all astronomers. +%% -----File: 120.png---Folio 106------- +\index{Ptolemy!ref.\ to}% + +The first notable author of mathematical books was \textbf{Mohammed +ben Musa Al Hovarezmi}, who lived during the reign of Caliph +\index{Hovarezmi}% +\index{Mohammed ben Musa Hovarezmi}% +Al Mamun (813--833). He was engaged by the caliph in making +extracts from the \textit{Sindhind}, in revising the tablets of Ptolemæus, +in taking observations at Bagdad and Damascus, and in +measuring a degree of the earth's meridian. Important to us +is his work on algebra and arithmetic. The portion on arithmetic +\index{Arithmetic!Arabic}% +is not extant in the original, and it was not till 1857 +that a Latin translation of it was found. It begins thus: +``Spoken has Algoritmi. Let us give deserved praise to God, +our leader and defender.'' Here the name of the author, \textit{Al +Hovarezmi}, has passed into \textit{Algoritmi}, from which comes our +modern word, \emph{algorithm}, signifying the art of computing in +\index{Algorithm!origin of term}% +any particular way. The arithmetic of Hovarezmi, being +based on the principle of position and the Hindoo method of +calculation, ``excels,'' says an Arabic writer, ``all others in +brevity and easiness, and exhibits the Hindoo intellect and +sagacity in the grandest inventions.'' This book was followed +by a large number of arithmetics by later authors, which differed +from the earlier ones chiefly in the greater variety of +methods. Arabian arithmetics generally contained the four +operations with integers and fractions, modelled after the +Indian processes. They explained the operation of \emph{casting out +\index{Casting out the 9's}% +\index{Nines, casting out the}% +the~9's}, which was sometimes called the ``Hindoo proof.'' They +contained also the \textit{regula falsa} and the \textit{regula duorum falsorum}, +\index{Regula duorum falsorum}% +\index{Regula falsa}% +by which algebraical examples could be solved without algebra. +Both these methods were known to the Indians. The \textit{regula +falsa} or \textit{falsa positio} was the assigning of an assumed value to +the unknown quantity, which value, if wrong, was corrected +by some process like the ``rule of three.'' Diophantus used a +\index{Diophantus!ref.\ to}% +\index{Rule of three}% +method almost identical with this. The \textit{regula duorum falsorum} +was as follows:\Reference[]{7} To solve an equation $f(x) = V$, assume, +for the moment, two values for $x$; namely, $x = a$ and $x = b$. +%% -----File: 121.png---Folio 107------- +\index{Irrationals}% +Then form $f(a) = A$ and $f(b) = B$, and determine the errors +$V - A = E_a$ and $V - B = E_b$; then the required +$x = \dfrac{bE_a - aE_b}{E_a - E_b}$ +is generally a close approximation, but is absolutely accurate +whenever $f(x)$ is a linear function of~$x$. + +We now return to Hovarezmi, and consider the other part +\index{Hovarezmi!ref.\ to}% +\index{Mohammed ben Musa Hovarezmi!ref.\ to}% +of his work,---the \emph{algebra}. This is the first book known to +\index{Algebra!Arabic}% +\index{Algebra!origin of terms}% +contain this word itself as title. Really the title consists of +two words, \textit{aldshebr walmukabala}, the nearest English translation +of which is ``restoration'' and ``reduction.'' By ``restoration'' +was meant the transposing of negative terms to the +other side of the equation; by ``reduction,'' the uniting of +similar terms. Thus, $x^2 - 2x = 5x + 6$ passes by aldshebr into +$x^2 = 5x + 2x + 6$; and this, by walmukabala, into $x^2 = 7x + 6$. +The work on algebra, like the arithmetic, by the same author, +contains nothing original. It explains the elementary operations +and the solutions of linear and quadratic equations. +\index{Quadratic equations}% +From whom did the author borrow his knowledge of algebra? +That it came entirely from Indian sources is impossible, for +the Hindoos had no such rules like the ``restoration'' and +``reduction.'' They were, for instance, never in the habit of +making all terms in an equation positive, as is done by the +process of ``restoration.'' Diophantus gives two rules which +\index{Diophantus!ref.\ to}% +resemble somewhat those of our Arabic author, but the probability +that the Arab got all his algebra from Diophantus is +lessened by the considerations that he recognised both roots +of a quadratic, while Diophantus noticed only one; and that +the Greek algebraist, unlike the Arab, habitually rejected irrational +solutions. It would seem, therefore, that the algebra +of Hovarezmi was neither purely Indian nor purely Greek, +but was a hybrid of the two, with the Greek element predominating. + +The algebra of Hovarezmi contains also a few meagre +%% -----File: 122.png---Folio 108------- +fragments on \emph{geometry}. He gives the theorem of the right +\index{Geometry!Arabic}% +triangle, but proves it after Hindoo fashion and only for the +simplest case, when the right triangle is isosceles. He then +calculates the areas of the triangle, parallelogram, and circle. +For $\pi$ he uses the value~$3 \frac{1}{7}$, and also the two Indian, $\pi = \sqrt{10}$ +\index{p@{$\pi$}: values for!Arabic}% +and $\pi = \frac{62832}{20000}$. Strange to say, the last value was afterwards +forgotten by the Arabs, and replaced by others less +accurate. This bit of geometry doubtless came from India. +Later Arabic writers got their geometry almost entirely from +Greece. + +Next to be noticed are the three sons of \textbf{Musa ben Sakir}, +\index{Musa ben Sakir}% +who lived in Bagdad at the court of the Caliph Al Mamun. +They wrote several works, of which we mention a geometry +in which is also contained the well-known formula for the +area of a triangle expressed in terms of its sides. We are +told that one of the sons travelled to Greece, probably to +collect astronomical and mathematical manuscripts, and that +on his way back he made acquaintance with Tabit ben Korra. +\index{Tabit ben Korra}% +Recognising in him a talented and learned astronomer, Mohammed +procured for him a place among the astronomers at +the court in Bagdad. \textbf{Tabit ben Korra} (836--901) was born +at Harran in Mesopotamia. He was proficient not only in +astronomy and mathematics, but also in the Greek, Arabic, +and Syrian languages. His translations of Apollonius, Archimedes, +\index{Apollonius!ref.\ to}% +\index{Archimedes!ref.\ to}% +Euclid, Ptolemy, Theodosius, rank among the best. +\index{Euclid!ref.\ to}% +\index{Ptolemy!ref.\ to}% +\index{Theodosius!ref.\ to}% +His dissertation on \emph{amicable numbers} (of which each is the +\index{Numbers!amicable}% +\index{Numbers!theory of numbers}% +sum of the factors of the other) is the first known specimen +of original work in mathematics on Arabic soil. It shows +that he was familiar with the Pythagorean theory of numbers. +\index{Theory of numbers}% +Tabit invented the following rule for finding amicable numbers: +If $p = 3 · 2^{n} - 1$, $q = 3 · 2^{n-1} - 1$, $r = 9 · 2^{2n-1} - 1$ ($n$~being +a whole number) are three primes, then $a = 2^{n} pq$, $b = 2^{n} r$ are +a pair of amicable numbers. Thus, if $n = 2$, then $p = 11$, +%% -----File: 123.png---Folio 109------- +$q = 5$, $r = 71$, and $a = 220$, $b = 284$. Tabit also trisected an +angle. + +Foremost among the astronomers of the ninth century +ranked \textbf{Al Battani}, called \textit{Albategnius} by the Latins. Battan +\index{Albattani@{Al Battani}}% +in Syria was his birthplace. His observations were celebrated +for great precision. His work, \textit{De scientia stellarum}, was translated +into Latin by Plato Tiburtinus, in the twelfth century. +\index{Plato of Tivoli}% +Out of this translation sprang the word `sinus,' as the name +of a trigonometric function. The Arabic word for ``sine,'' +\index{Sine}% +\index{Sine!origin of term}% +\textit{dschiba}, was derived from the Sanscrit \textit{jiva}, and resembled +the Arabic word \textit{dschaib}, meaning an indentation or gulf. +Hence the Latin ``sinus.''\Reference{3} Al Battani was a close student +of Ptolemy, but did not follow him altogether. He took an +\index{Ptolemy!ref.\ to}% +important step for the better, when he introduced the Indian +``sine'' or \emph{half} the chord, in place of the \emph{whole} chord of +Ptolemy. Another improvement on Greek trigonometry made +\index{Trigonometry}% +by the Arabs points likewise to Indian influences. Propositions +and operations which were treated by the Greeks geometrically +are expressed by the Arabs algebraically. Thus, +\textit{Al Battani} at once gets from an equation +$\smash[b]{\dfrac{\sin \theta}{\cos \theta}} = D$, the value +of $\theta$ by means of +$\sin \theta = \dfrac{D\Fstr}{\sqrt{1 + D^2}}$,---a process unknown to +the ancients. He knows, of course, all the formulas for +spherical triangles given in the \textit{Almagest}, but goes further, +\index{Almagest!ref.\ to}% +and adds an important one of his own for oblique-angled +triangles; namely, $\cos a = \cos b \cos c + \sin b \sin c \cos A$. + +At the beginning of the tenth century political troubles +arose in the East, and as a result the house of the Abbasides +lost power. One province after another was taken, till, in~945, +all possessions were wrested from them. Fortunately, +the new rulers at Bagdad, the Persian Buyides, were as much +interested in astronomy as their predecessors. The progress +of the sciences was not only unchecked, but the conditions +%% -----File: 124.png---Folio 110------- +\index{Geometry!Arabic}% +\index{Regular solids}% +\index{Tycho Brahe}% +for it became even more favourable. The Emir \textit{Adud-ed-daula} +(978--983) gloried in having studied astronomy himself. His +son \textit{Saraf-ed-daula} erected an observatory in the garden of his +palace, and called thither a whole group of scholars.\Reference{7} Among +them were \textit{Abul Wefa}, \textit{Al Kuhi}, \textit{Al Sagani}. +\index{Abul Wefa}% + +\textbf{Abul Wefa} (940--998) was born at Buzshan in Chorassan, a +region among the Persian mountains, which has brought forth +many Arabic astronomers. He forms an important exception +to the unprogressive spirit of Arabian scientists by his brilliant +discovery of the \emph{variation} of the moon, an inequality +usually supposed to have been first discovered by Tycho +Brahe.\Reference{11} Abul Wefa translated Diophantus. He is one of the +\index{Brahe, Tycho}% +\index{Diophantus!ref.\ to}% +last Arabic translators and commentators of Greek authors. +The fact that he esteemed the algebra of Mohammed ben +Musa Hovarezmi worthy of his commentary indicates that +\index{Hovarezmi!ref.\ to}% +\index{Mohammed ben Musa Hovarezmi!ref.\ to}% +thus far algebra had made little or no progress on Arabic soil. +Abul Wefa invented a method for computing tables of sines +which gives the sine of half a degree correct to nine decimal +places. He did himself credit by introducing the \emph{tangent} into +trigonometry and by calculating a table of tangents. The +\index{Tangents!in trigonometry}% +\index{Trigonometry}% +first step toward this had been taken by Al Battani. Unfortunately, +\index{Albattani@{Al Battani}!ref.\ to}% +this innovation and the discovery of the moon's +variation excited apparently no notice among his contemporaries +and followers. ``We can hardly help looking upon this +circumstance as an evidence of a servility of intellect belonging +to the Arabian period.'' A treatise by Abul Wefa on +``geometric constructions'' indicates that efforts were being +made at that time to improve draughting. It contains a neat +construction of the corners of the regular \DPtypo{polyedrons}{polyhedrons} on the +circumscribed sphere. Here, for the first time, appears the +condition which afterwards became very famous in the Occident, +that the construction be effected with a single opening +of the compass. +%% -----File: 125.png---Folio 111------- +\index{Albiruni}% +\index{Alhogendi@{Al Hogendi}}% +\index{Cube numbers}% +\index{Indeterminate equations}% +\index{Quadratic equations}% + +\textbf{Al Kuhi}, the second astronomer at the observatory of the +\index{Alkuhi@{Al Kuhi}}% +emir at Bagdad, was a close student of Archimedes and +Apollonius. He solved the problem, to construct a segment +of a sphere equal in volume to a given segment and having +a curved surface equal in area to that of another given segment. +He, \textbf{Al Sagani}, and \textbf{Al Biruni} made a study of the +\index{Alsagani@{Al Sagani}}% +trisection of angles. \textbf{Abul Gud}, an able geometer, solved the +\index{Abul Gud}% +problem by the intersection of a parabola with an equilateral +hyperbola. + +The Arabs had already discovered the theorem that the +sum of two cubes can never be a cube. \textbf{Abu Mohammed Al +Hogendi} of Chorassan thought he had proved this, but we are +told that the demonstration was defective. Creditable work +in theory of numbers and algebra was done by \textbf{Al Karhi} of +\index{Algebra!Arabic}% +\index{Alkarhi@{Al Karhi}}% +\index{Fahri des Al Karhi}% +Bagdad, who lived at the beginning of the eleventh century. +His treatise on algebra is the greatest algebraic work of the +Arabs. In it he appears as a disciple of Diophantus. He +\index{Diophantus!ref.\ to}% +was the first to operate with higher roots and to solve equations +of the form $x^{2n} + ax^{n} = b$. For the solution of quadratic +equations he gives both arithmetical and geometric proofs. +He was the first Arabic author to give and prove the theorems +on the summation of the series:--- +\index{Series}% +\begin{align*} +1^2 + 2^2 + 3^2 + \dotsb + n^2 + &= (1 + 2 + \dotsb + n)\DPtypo{^}{\,}\frac{2n+1}{3}, \\ +1^3 + 2^3 + 3^3 + \dotsb + n^3 + &= (1 + 2 + \dotsb + n)^2. +\end{align*} + +Al Karhi also busied himself with indeterminate analysis. +\index{Indeterminate analysis}% +He showed skill in handling the methods of Diophantus, but +added nothing whatever to the stock of knowledge already +on hand. As a subject for original research, indeterminate +analysis was too subtle for even the most gifted of Arabian +minds. Rather surprising is the fact that Al Karhi's algebra +shows no traces whatever of Hindoo indeterminate analysis. +%% -----File: 126.png---Folio 112------- +\index{Alhayyami@{Al Hayyami}}% +\index{Arabic numerals and notation}% +\index{Cubic equations}% +\index{Notation!Arabic notation}% +But most astonishing it is, that an arithmetic by the same +author completely excludes the Hindoo numerals. It is constructed +\index{Numerals!Arabic}% +wholly after Greek pattern. Abul Wefa also, in the +\index{Abul Wefa!ref.\ to}% +second half of the tenth century, wrote an arithmetic in which +Hindoo numerals find no place. This practice is the very +opposite to that of other Arabian authors. The question, +why the Hindoo numerals were ignored by so eminent authors, +is certainly a puzzle. Cantor suggests that at one time there +\index{Cantor, M.}% +may have been rival schools, of which one followed almost +exclusively Greek mathematics, the other Indian. + +The Arabs were familiar with geometric solutions of quadratic +equations. Attempts were now made to solve cubic +equations geometrically. They were led to such solutions by +the study of questions like the Archimedean problem, demanding +the section of a sphere by a plane so that the two segments +shall be in a prescribed ratio. The first to state this +problem in form of a cubic equation was \textbf{Al Mahani} of Bagdad, +\index{Almahani@{Al Mahani}}% +while \textbf{Abu Gafar Al Hazin} was the first Arab to solve the +\index{Alhazin@{Al Hazin}}% +equation by conic sections. Solutions were given also by +\index{Conic sections!Arabs}% +Al Kuhi, Al Hasan ben Al Haitam, and others.\Reference{20} Another +\index{Alhaitam@{Al Haitam}!ref.\ to}% +\index{Alkuhi@{Al Kuhi}!ref.\ to}% +difficult problem, to determine the side of a regular heptagon, +required the construction of the side from the equation +$x^3 - x^2 - 2x + 1 = 0$. It was attempted by many and at last +solved by Abul Gud. + +The one who did most to elevate to a \emph{method} the solution +of algebraic equations by intersecting conics, was \textbf{Omar al +Hayyami} of Chorassan, about 1079~\AD\ He divides cubics into +two classes, the trinomial and quadrinomial, and each class +into families and species. Each species is treated separately +but according to a general plan. He believed that cubics +could not be solved by calculation, nor \Change{bi-quadratics}{biquadratics} by geometry. +He rejected negative roots and often failed to discover +\index{Negative roots}% +all the positive ones. Attempts at \Change{bi-quadratic}{biquadratic} equations +\index{Biquadratic equation}% +\index{Quadratic equations}% +%% -----File: 127.png---Folio 113------- +\index{Geometry!Arabic}% +were made by Abul Wefa,\Reference[]{20} who solved geometrically +\index{Abul Wefa!ref.\ to}% +$x^4 = a$ and $x^4 + ax^3 = b$. + +The solution of cubic equations by intersecting conics was +\index{Cubic equations}% +the greatest achievement of the Arabs in algebra. The foundation +to this work had been laid by the Greeks, for it was +Menæchmus who first constructed the roots of $x^3 - a = 0$ or +\index{Menæchmus!ref.\ to}% +$x^3 - 2 a^3 = 0$. It was not his aim to find the number corresponding +to~$x$, but simply to determine the side~$x$ of a cube +double another cube of side~$a$. The Arabs, on the other +hand, had another object in view: to find the roots of given +numerical equations. In the Occident, the Arabic solutions +of cubics remained unknown until quite recently. Descartes +\index{Descartes!ref.\ to}% +and Thomas Baker invented these constructions anew. The +\index{Baker, Th}% +works of Al Hayyami, Al Karhi, Abul Gud, show how the +\index{Abul Gud!ref.\ to}% +\index{Alhayyami@{Al Hayyami}!ref.\ to}% +\index{Alkarhi@{Al Karhi}}% +Arabs departed further and further from the Indian methods, +and placed themselves more immediately under Greek influences. +In this way they barred the road of progress against +themselves. The Greeks had advanced to a point where +material progress became difficult with their methods; but the +Hindoos furnished new ideas, many of which the Arabs now +rejected. + +With Al Karhi and Omar Al Hayyami, mathematics among +the Arabs of the East reached flood-mark, and now it begins to +ebb. Between 1100 and 1300~\AD\ come the crusades with +war and bloodshed, during which European Christians profited +much by their contact with Arabian culture, then far superior +to their own; but the Arabs got no science from the Christians +in return. The crusaders were not the only adversaries of the +Arabs. During the first half of the thirteenth century, they +had to encounter the wild Mongolian hordes, and, in~1256, were +conquered by them under the leadership of \textit{Hulagu}. The +caliphate at Bagdad now ceased to exist. At the close of the +fourteenth century still another empire was formed by Timur +%% -----File: 128.png---Folio 114------- +or \textit{Tamerlane}, the Tartar. During such sweeping turmoil, it +is not surprising that science declined. Indeed, it is a marvel +that it existed at all. During the supremacy of Hulagu, lived +\textbf{Nasir Eddin} (1201--1274), a man of broad culture and an able +\index{Nasir Eddin}% +\Addendlabel{114}% +astronomer. He persuaded Hulagu to build him and his associates +a large observatory at Maraga. Treatises on algebra, +geometry, arithmetic, and a translation of Euclid's \textit{Elements}, +\index{Elements (Euclid's)}% +\index{Euclid!ref.\ to}% +\index{Geometry!Arabic}% +were prepared by him. Even at the court of Tamerlane in +Samarkand, the sciences were by no means neglected. A +group of astronomers was drawn to this court. \textbf{Ulug Beg} +\index{Ulug Beg}% +(1393--1449), a grandson of Tamerlane, was himself an +astronomer. Most prominent at this time was \textbf{Al Kaschi}, the +\index{Alkaschi@{Al Kaschi}}% +author of an arithmetic. Thus, during intervals of peace, +science continued to be cultivated in the East for several +centuries. The last Oriental writer was \textit{Beha Eddin} (1547--1622). +\index{Beha Eddin}% +His \textit{Essence of Arithmetic} stands on about the same +level as the work of Mohammed ben Musa Hovarezmi, written +\index{Hovarezmi!ref.\ to}% +\index{Mohammed ben Musa Hovarezmi!ref.\ to}% +nearly 800 years before. + +``Wonderful is the expansive power of Oriental peoples, +with which upon the wings of the wind they conquer half +the world, but more wonderful the energy with which, in +less than two generations, they raise themselves from the +lowest stages of cultivation to scientific efforts.'' During +all these centuries, astronomy and mathematics in the Orient +greatly excel these sciences in the Occident. + +Thus far we have spoken only of the Arabs in the East. +Between the Arabs of the East and of the West, which were +under separate governments, there generally existed considerable +political animosity. In consequence of this, and of the +enormous distance between the two great centres of learning, +Bagdad and Cordova, there was less scientific intercourse +among them than might be expected to exist between peoples +having the same religion and written language. Thus the +%% -----File: 129.png---Folio 115------- +\index{Moors}% +course of science in Spain was quite independent of that in +Persia. While wending our way westward to Cordova, we +must stop in Egypt long enough to observe that there, too, +scientific activity was rekindled. Not Alexandria, but Cairo +with its library and observatory, was now the home of learning. +Foremost among her scientists ranked \textbf{Ben Junus} (died +\index{Ben Junus}% +1008), a contemporary of Abul Wefa. He solved some difficult +problems in spherical trigonometry. Another Egyptian astronomer +\index{Spherical trigonometry}% +\index{Trigonometry}% +\index{Trigonometry!spherical}% +was \textbf{Ibn Al Haitam} (died 1038), who wrote on geometric +\index{Alhaitam@{Al Haitam}}% +loci. Travelling westward, we meet in Morocco \textbf{Abul Hasan +\index{Abul Hasan}% +Ali}, whose treatise `on astronomical instruments' discloses a +thorough knowledge of the \textit{Conics} of Apollonius. Arriving +\index{Apollonius!ref.\ to}% +finally in Spain at the capital, Cordova, we are struck by the +magnificent splendour of her architecture. At this renowned +seat of learning, schools and libraries were founded during the +tenth century. + +Little is known of the progress of mathematics in Spain. +The earliest name that has come down to us is \textbf{Al Madshriti} +\index{Almadshriti@{Al Madshriti}}% +(died 1007), the author of a mystic paper on `amicable numbers.' +\index{Numbers!amicable}% +His pupils founded schools at Cordova, Dania, and +Granada. But the only great astronomer among the Saracens +in Spain is \textbf{Gabir ben Aflah} of Sevilla, frequently called \textit{Geber}. +\index{Gabir ben Aflah}% +He lived in the second half of the eleventh century. It was +formerly believed that he was the inventor of algebra, and that +\index{Algebra!Arabic}% +\index{Algebra!origin of terms}% +the word \emph{algebra} came from `Gabir' or `Geber.' He ranks +among the most eminent astronomers of this time, but, like so +many of his contemporaries, his writings contain a great deal +of mysticism. His chief work is an astronomy in nine books, of +\index{Astronomy!Arabic}% +which the first is devoted to trigonometry. In his treatment +of spherical trigonometry, he exercises great independence of +thought. He makes war against the time-honoured procedure +adopted by Ptolemy of applying ``the rule of six quantities,'' +\index{Ptolemy!ref.\ to}% +and gives a new way of his own, based on the `rule of four +%% -----File: 130.png---Folio 116------- +\index{Geber's theorem}% +quantities.' This is: If $PP_1$~and~$QQ_1$ be two arcs of great +circles intersecting in~$A$, and if $PQ$~and~$P_1Q_1$ be arcs of great +circles drawn perpendicular to~$QQ_1$, then we have the proportion +\[ +\sin AP : \sin PQ = \sin AP_1 : \sin P_1Q_1. +\] +From this he derives the formulas for spherical right triangles. +To the four fundamental formulas already given by Ptolemy, +he added a fifth, discovered by himself. If $a$,~$b$,~$c$, be the sides, +and $A$,~$B$,~$C$, the angles of a spherical triangle, right-angled at~$A$, +then $\cos B = \cos b \sin C$. This is frequently called ``Geber's +Theorem.'' Radical and bold as were his innovations in +\Addendlabel{116}% +spherical trigonometry, in plane trigonometry he followed +slavishly the old beaten path of the Greeks. Not even did he +adopt the Indian `sine' and `cosine,' but still used the Greek +\index{Sine}% +`chord of double the angle.' So painful was the departure +from old ideas, even to an independent Arab! After the time +of Gabir ben Aflah there was no mathematician among the +Spanish Saracens of any reputation. In the year in which +Columbus discovered America, the Moors lost their last foothold +\index{Moors}% +on Spanish soil. + +We have witnessed a laudable intellectual activity among +the Arabs. They had the good fortune to possess rulers +who, by their munificence, furthered scientific research. At +the courts of the caliphs, scientists were supplied with libraries +and observatories. A large number of astronomical and +mathematical works were written by Arabic authors. Yet +we fail to find a single important principle in mathematics +brought forth by the Arabic mind. Whatever discoveries +they made, were in fields previously traversed by the Greeks +or the Indians, and consisted of objects which the latter had +overlooked in their rapid march. The Arabic mind did not +possess that penetrative insight and invention by which mathematicians +in Europe afterwards revolutionised the science. +%% -----File: 131.png---Folio 117------- +The Arabs were learned, but not original. Their chief service +\index{Arabs|)}% +to science consists in this, that they adopted the learning of +Greece and India, and kept what they received with scrupulous +care. When the love for science began to grow in the +Occident, they transmitted to the Europeans the valuable +treasures of antiquity. Thus a Semitic race was, during the +Dark Ages, the custodian of the Aryan intellectual possessions. + + +\Section{Europe During the Middle Ages} +\index{Middle Ages|(}% + +With the third century after Christ begins an era of migration +of nations in Europe. The powerful Goths quit their +swamps and forests in the North and sweep onward in steady +southwestern current, dislodging the Vandals, Sueves, and +Burgundians, crossing the Roman territory, and stopping and +recoiling only when reaching the shores of the Mediterranean. +From the Ural Mountains wild hordes sweep down on the +Danube. The Roman Empire falls to pieces, and the Dark +Ages begin. But dark though they seem, they are the germinating +season of the institutions and nations of modern Europe. +The Teutonic element, partly pure, partly intermixed with the +Celtic and Latin, produces that strong and luxuriant growth, +the modern civilisation of Europe. Almost all the various +nations of Europe belong to the Aryan stock. As the Greeks +and the Hindoos---both Aryan races---were the great thinkers +of antiquity, so the nations north of the Alps became the great +intellectual leaders of modern times. + +\Subsection{Introduction of Roman Mathematics} +\index{Roman mathematics in Occident|(}% + +We shall now consider how these as yet barbaric nations of +the North gradually came in possession of the intellectual +%% -----File: 132.png---Folio 118------- +treasures of antiquity. With the spread of Christianity the +Latin language was introduced not only in ecclesiastical but +also in scientific and all important worldly transactions. Naturally +the science of the Middle Ages was drawn largely from +Latin sources. In fact, during the earlier of these ages Roman +authors were the only ones read in the Occident. Though +Greek was not wholly unknown, yet before the thirteenth +century not a single Greek scientific work had been read or +translated into Latin. Meagre indeed was the science which +could be gotten from Roman writers, and we must wait several +centuries before any substantial progress is made in mathematics. + +After the time of Boethius and Cassiodorius mathematical +\index{Boethius!ref.\ to}% +\index{Cassiodorius}% +activity in Italy died out. The first slender blossom of science +among tribes that came from the North was an encyclopædia +entitled \textit{Origines}, written by \textbf{Isidorus} (died 636 as bishop of +\index{Isidorus of Seville}% +Seville). This work is modelled after the Roman encyclopædias +of Martianus Capella of Carthage and of Cassiodorius. +Part of it is devoted to the quadrivium, arithmetic, music, +geometry, and astronomy. He gives definitions and grammatical +explications of technical terms, but does not describe the +modes of computation then in vogue. After Isidorus there +follows a century of darkness which is at last dissipated by +the appearance of \textbf{Bede the Venerable} (672--735), the most +\index{Bede, the Venerable}% +learned man of his time. He was a native of Ireland, then +the home of learning in the Occident. His works contain +treatises on the \textit{Computus}, or the computation of Easter-time, +\index{Computus}% +and on finger-reckoning. It appears that a finger-symbolism +\index{Finger-reckoning}% +was then widely used for calculation. The correct determination +of the time of Easter was a problem which in those days +greatly agitated the Church. It became desirable to have at +least one monk at each monastery who could determine the +day of religious festivals and could compute the calendar. +%% -----File: 133.png---Folio 119------- +Such determinations required some knowledge of arithmetic. +\index{Arithmetic!Middle Ages}% +Hence we find that the art of calculating always found some +little corner in the curriculum for the education of monks. + +The year in which Bede died is also the year in which +\textbf{Alcuin} (735--804) was born. Alcuin was educated in Ireland, +\index{Alcuin}% +and was called to the court of Charlemagne to direct the progress +of education in the great Frankish Empire. Charlemagne +was a great patron of learning and of learned men. In the +great sees and monasteries he founded schools in which were +taught the psalms, writing, singing, computation (\textit{computus}), +\index{Computus}% +and grammar. By \textit{computus} was here meant, probably, not +merely the determination of Easter-time, but the art of computation +in general. Exactly what modes of reckoning were +then employed we have no means of knowing. It is not likely +that Alcuin was familiar with the apices of Boethius or with +\index{Apices of Boethius!ref.\ to}% +the Roman method of reckoning on the abacus. He belongs +\index{Abacus}% +to that long list of scholars who dragged the theory of numbers +\index{Numbers!theory of numbers}% +\index{Theory of numbers}% +into theology. Thus the number of beings created by +God, who created all things well, is~$6$, because $6$~is a perfect +number (the sum of its divisors being $1 + 2 + 3 = 6$); $8$,~on +the other hand, is an imperfect number ($1 + 2 + 4 < 8$); hence +the second origin of mankind emanated from the number~$8$, +which is the number of souls said to have been in Noah's ark. + +There is a collection of ``Problems for Quickening the +Mind'' (\textit{propositiones ad acuendos iuvenes}), which are certainly +\index{Propositiones ad acuendos iuvenes}% +as old as 1000~\AD\ and possibly older. Cantor is of the opinion +that they were written much earlier and by Alcuin. The +following is a specimen of these ``Problems'': A dog chasing +a rabbit, which has a start of $150$~feet, jumps $9$~feet every time +the rabbit jumps~$7$. In order to determine in how many leaps +the dog overtakes the rabbit, $150$~is to be divided by~$2$. In +this collection of problems, the areas of triangular and quadrangular +pieces of land are found by the same formulas of +%% -----File: 134.png---Folio 120------- +\index{Anthology, Palatine}% +\index{Palatine anthology}% +\index{Propositiones ad acuendos iuvenes}% +\index{Sylvester II. (Gerbert)|(}% +approximation as those used by the Egyptians and given by +Boethius in his geometry. An old problem is the ``cistern-problem'' +(given the time in which several pipes can fill a +cistern singly, to find the time in which they fill it jointly), +which has been found previously in Heron, in the Greek \textit{Anthology}, +and in Hindoo works. Many of the problems show that +the collection was compiled chiefly from Roman sources. The +problem which, on account of its uniqueness, gives the most +positive testimony regarding the Roman origin is that on the +interpretation of a will in a case where twins are born. The +problem is identical with the Roman, except that different +ratios are chosen. Of the exercises for recreation, we mention +the one of the wolf, goat, and cabbage, to be rowed across a +river in a boat holding only one besides the ferry-man. Query: +How must he carry them across so that the goat shall not eat +the cabbage, nor the wolf the goat? The solutions of the +``problems for quickening the mind'' require no further knowledge +than the recollection of some few formulas used in +surveying, the ability to solve linear equations and to perform +the four fundamental operations with integers. Extraction +of roots was nowhere demanded; fractions hardly +\index{Fractions!Middle Ages}% +ever occur.\Reference{3} + +The great empire of Charlemagne tottered and fell almost +immediately after his death. War and confusion ensued. +Scientific pursuits were abandoned, not to be resumed until +the close of the tenth century, when under Saxon rule in Germany +and Capetian in France, more peaceful times began. +The thick gloom of ignorance commenced to disappear. The +zeal with which the study of mathematics was now taken up +by the monks is due principally to the energy and influence +of one man,---\textbf{Gerbert}. He was born in Aurillac in Auvergne. +\index{Gerbert|(}% +After receiving a monastic education, he engaged in study, +chiefly of mathematics, in Spain. On his return he taught +%% -----File: 135.png---Folio 121------- +school at Rheims for ten years and became distinguished for +his profound scholarship. By King Otto~I. and his successors +Gerbert was held in highest esteem. He was elected bishop +of Rheims, then of Ravenna, and finally was made Pope under +the name of Sylvester~II. by his former pupil Emperor Otho~III. +He died in~1003, after a life intricately involved in many +political and ecclesiastical quarrels. Such was the career of +the greatest mathematician of the tenth century in Europe. +By his contemporaries his mathematical knowledge was considered +wonderful. Many even accused him of criminal intercourse +with evil spirits. + +Gerbert enlarged the stock of his knowledge by procuring +copies of rare books. Thus in Mantua he found the geometry +\index{Geometry!Middle Ages}% +of Boethius. Though this is of small scientific value, yet it +\index{Boethius!ref.\ to}% +is of great importance in history. It was at that time the +only book from which European scholars could learn the elements +of geometry. Gerbert studied it with zeal, and is +generally believed himself to be the author of a geometry. +H.~Weissenborn denies his authorship, and claims that the +book in question consists of three parts which cannot come +from one and the same author.\Reference{21} This geometry contains +nothing more than the one of Boethius, but the fact that +occasional errors in the latter are herein corrected shows that +the author had mastered the subject. ``The first mathematical +paper of the Middle Ages which deserves this name,'' +says Hankel, ``is a letter of Gerbert to Adalbold, bishop of +Utrecht,'' in which is explained the reason why the area of a +triangle, obtained ``geometrically'' by taking the product of +the base by half its altitude, differs from the area calculated +``arithmetically,'' according to the formula +$\frac{1}{2} a (a + 1)$, used +by surveyors, where $a$ stands for a side of an equilateral triangle. +He gives the correct explanation that in the latter +formula all the small squares, in which the triangle is supposed +%% -----File: 136.png---Folio 122------- +to be divided, are counted in wholly, even though parts +of them project beyond it. + +Gerbert made a careful study of the arithmetical works of +Boethius. He himself published two works,---\textit{Rule of Computation +on the Abacus}, and \textit{A Small Book on the Division of +\index{Abacus}% +Numbers}. They give an insight into the methods of calculation +practised in Europe before the introduction of the +Hindoo numerals. Gerbert used the abacus, which was probably +unknown to Alcuin. \textbf{Bernelinus}, a pupil of Gerbert, +\index{Bernelinus}% +describes it as consisting of a smooth board upon which geometricians +were accustomed to strew blue sand, and then to draw +their diagrams. For arithmetical purposes the board was +divided into 30 columns, of which 3 were reserved for fractions, +while the remaining 27 were divided into groups with +3 columns in each. In every group the columns were marked +respectively by the letters C (\textit{centum}), D (\textit{decem}), and +S (\textit{singularis}) or M (\textit{monas}). Bernelinus gives the nine +numerals used, which are the apices of Boethius, and then +remarks that the Greek letters may be used in their place.\Reference{3} +By the use of these columns any number can be written +without introducing a zero, and all operations in arithmetic +\index{Arithmetic!Middle Ages}% +can be performed in the same way as we execute ours without +the columns, but with the symbol for zero. Indeed, the +methods of adding, subtracting, and multiplying in vogue +among the abacists agree substantially with those of to-day. +But in a division there is very great difference. The early rules +for division appear to have been framed to satisfy the following +three conditions: (1)~The use of the multiplication table shall +be restricted as far as possible; at least, it shall never be +required to multiply mentally a figure of two digits by another +of one digit. (2)~Subtractions shall be avoided as much as +possible and replaced by additions. (3)~The operation shall +proceed in a purely mechanical way, without requiring trials.\Reference{7} +%% -----File: 137.png---Folio 123------- +\index{Arithmetic!Middle Ages}% +That it should be necessary to make such conditions seems +strange to us; but it must be remembered that the monks of +the Middle Ages did not attend school during childhood and +learn the multiplication table while the memory was fresh. +Gerbert's rules for division are the oldest extant. They are +so brief as to be very obscure to the uninitiated. They were +probably intended simply to aid the memory by calling to +mind the successive steps in the work. In later manuscripts +they are stated more fully. In dividing any number by another +of one digit, say $668$~by~$6$, the divisor was first increased to~10 +by adding~4. The process is exhibited in the adjoining figure.\Reference{3} +As it continues, we must imagine the digits +which are crossed out, to be erased and then +replaced by the ones beneath. It is as follows: +$600 ÷ 10 = 60$, but, to rectify the error, $4 × 60$, +or~$240$, must be added; $200 ÷ 10 = 20$, but $4 × 20$, +or~$80$, must be added. We now write for +$60 + 40 + 80$, %[** PP: Width-dependent line break] +\begin{wrapfigure}[21]{r}{0.875in} +\setlength{\TmpLen}{1.75em}%[** PP: Sets width of \cancel{digit}] +\hfill\begin{tabular}{|*{3}{@{}c@{}|}} +\multicolumn{3}{@{}c@{}}{\smash[b]{\makebox[0pt][c]{\Graphic{3\TmpLen}{arc}}}} \\ +\multicolumn{2}{|@{}c@{}}{\smash{\Graphic{2\TmpLen}{arc}}} & \Graphic{\TmpLen}{arc} \\ + \sffamily C & \sffamily D & \sffamily S \\ + & & $6$ \\ + & & $4$ \\ +\hline + $6$ & $6$ & $8$ \\ +\hline + \cancel{6} & \cancel{6} & \cancel{8} \\ + \cancel{2} & \cancel{4} & \cancel{4} \\ + \cancel{1} & \cancel{8} & \cancel{8} \\ + \cancel{1} & \cancel{4} & \cancel{8} \\ + & \cancel{2} & $2$ \\ + & \cancel{4} & \\ + & \cancel{6} & \\ + & \cancel{2} & \\ + & \cancel{2} & \\ +\hline + & \cancel{6} & \cancel{6} \\ + & \cancel{2} & \cancel{2} \\ + & \cancel{1} & \cancel{2} \\ + & $1$ & $1$ +\end{tabular} +\end{wrapfigure} +its sum~$180$, and continue thus: +$100 ÷ 10 = 10$; the correction necessary is $4 × 10$, +or~$40$, which, added to~$80$, gives~$120$. Now +$100 ÷ 10 = 10$, and the correction $4 × 10$, together +with the~$20$, gives~$60$. Proceeding as +before, $60 ÷ 10 = 6$; the correction is $4 × 6 = 24$. +Now $20 ÷ 10 = 2$, the correction being $4 × 2 = 8$. +In the column of units we have now $8 + 4 + 8$, +or~$20$. As before, $20 ÷ 10 = 2$; the correction +is $2 × 4 = 8$, which is not divisible by~$10$, but +only by~$6$, giving the quotient~$1$ and the remainder~$2$. +All the partial quotients taken +together give $60 + 20 + 10 + 10 + 6 + 2 + 2 + 1 = 111$, and +the remainder~$2$. + +Similar but more complicated, is the process when the +divisor contains two or more digits. Were the divisor~27, +%% -----File: 138.png---Folio 124------- +\index{Roman mathematics in Occident|)}% +\index{Sylvester II. (Gerbert)|)}% +then the next higher multiple of~$10$, or~$30$, would be taken +for the divisor, but corrections would be required for the~$3$. +He who has the patience to carry such a division through +to the end, will understand why it has been said of Gerbert +that ``Regulas dedit, quæ a sudantibus abacistis vix intelliguntur.'' +He will also perceive why the Arabic method of +division, when first introduced, was called the \textit{divisio aurea}, +but the one on the abacus, the \textit{divisio ferrea}. + +In his book on the abacus, Bernelinus devotes a chapter to +fractions. These are, of course, the \textit{duodecimals}, first used +\index{Duodecimals}% +\index{Fractions!Middle Ages}% +\index{Fractions!duodecimal}% +by the Romans. For want of a suitable notation, calculation +with them was exceedingly difficult. It would be so even to +us, were we accustomed, like the early abacists, to express +them, not by a numerator or denominator, but by the application +of names, such as \textit{uncia} for~$\frac{1}{12}$, \textit{quincunx} for~$\frac{5}{12}$, \textit{dodrans} +for~$\frac{9}{12}$. + +In the tenth century, Gerbert was the central figure among +the learned. In his time the Occident came into secure possession +of all mathematical knowledge of the Romans. During +the eleventh century it was studied assiduously. Though +numerous works were written on arithmetic and geometry, +mathematical knowledge in the Occident was still very insignificant. +Scanty indeed were the mathematical treasures +obtained from Roman sources. + +\Subsection{Translation of Arabic Manuscripts} +\index{Arabic manuscripts|(}% + +By his great erudition and phenomenal activity, Gerbert +\index{Gerbert|)}% +infused new life into the study not only of mathematics, but +also of philosophy. Pupils from France, Germany, and Italy +gathered at Rheims to enjoy his instruction. When they +themselves became teachers, they taught of course not only +the use of the abacus and geometry, but also what they had +%% -----File: 139.png---Folio 125------- +\index{Geometry!Middle Ages}% +\index{Moors}% +\index{Sine}% +learned of the philosophy of Aristotle. His philosophy was +\index{Aristotle!ref.\ to}% +known, at first, only through the writings of Boethius. But +the growing enthusiasm for it created a demand for his complete +works. Greek texts were wanting. But the Latins +heard that the Arabs, too, were great admirers of Peripatetism, +and that they possessed translations of Aristotle's works and +commentaries thereon. This led them finally to search for +and translate Arabic manuscripts. During this search, mathematical +works also came to their notice, and were translated +into Latin. Though some few unimportant works may have +been translated earlier, yet the period of greatest activity +began about~1100. The zeal displayed in acquiring the +Mohammedan treasures of knowledge excelled even that of +the Arabs themselves, when, in the eighth century, they +plundered the rich coffers of Greek and Hindoo science. + +Among the earliest scholars engaged in translating manuscripts +into Latin was \textbf{Athelard of Bath}. The period of his +\index{Athelard of Bath}% +\Addendlabel{125}% +activity is the first quarter of the twelfth century. He +travelled extensively in Asia Minor, Egypt, and Spain, and +braved a thousand perils, that he might acquire the language +and science of the Mohammedans. He made the earliest +translations, from the Arabic, of Euclid's \textit{Elements} and of +\index{Elements (Euclid's)}% +\index{Euclid!ref.\ to}% +the astronomical tables of Mohammed ben Musa Hovarezmi. +\index{Hovarezmi!ref.\ to}% +\index{Mohammed ben Musa Hovarezmi!ref.\ to}% +In~1857, a manuscript was found in the library at Cambridge, +which proved to be the arithmetic by Mohammed ben Musa +in Latin. This translation also is very probably due to +Athelard. + +At about the same time flourished \textit{Plato of Tivoli} or \textit{Plato +\index{Plato of Tivoli}% +Tiburtinus}. He effected a translation of the astronomy of +Al Battani and of the \textit{Sphærica} of Theodosius. Through the +\index{Albattani@{Al Battani}!ref.\ to}% +\index{Theodosius!ref.\ to}% +former, the term \textit{sinus} was introduced into trigonometry. + +About the middle of the twelfth century there was a group +of Christian scholars busily at work at Toledo, under the +%% -----File: 140.png---Folio 126------- +\index{Algorithm!Middle Ages}% +\index{Arithmetic!Middle Ages}% +\index{Gerard of Cremona}% +\index{Sexagesimal system}% +leadership of Raymond, then archbishop of Toledo. Among +those who worked under his direction, \textbf{John of Seville} was +\index{John of Seville}% +most prominent. He translated works chiefly on Aristotelian +philosophy. Of importance to us is a \textit{liber algorismi}, compiled +by him from Arabic authors. On comparing works like +this with those of the abacists, we notice at once the most +\index{Abacists}% +striking difference, which shows that the two parties drew +from independent sources. It is argued by some that Gerbert +got his apices and his arithmetical knowledge, not from +\index{Apices of Boethius!ref.\ to}% +Boethius, but from the Arabs in Spain, and that part or the +whole of the geometry of Boethius is a forgery, dating from +the time of Gerbert. If this were the case, then the writings +of Gerbert would betray Arabic sources, as do those of John +of Seville. But no points of resemblance are found. Gerbert +could not have learned from the Arabs the use of the abacus, +\index{Abacus}% +because all evidence we have goes to show that they did not +employ it. Nor is it probable that he borrowed from the +Arabs the apices, because they were never used in Europe +except on the abacus. In illustrating an example in division, +mathematicians of the tenth and eleventh centuries state an +example in Roman numerals, then draw an abacus and insert +in it the necessary numbers with the apices. Hence it seems +probable that the abacus and apices were borrowed from the +same source. The contrast between authors like John of +Seville, drawing from Arabic works, and the abacists, consists +in this, that, unlike the latter, the former mention the Hindoos, +use the term \emph{algorism}, calculate with the zero, and do +not employ the abacus. The former teach the extraction of +roots, the abacists do not; they teach the sexagesimal fractions +\index{Fractions!sexagesimal}% +\index{Fractions!duodecimal}% +used by the Arabs, while the abacists employ the duodecimals +\index{Duodecimals}% +of the Romans.\Reference{3} + +A little later than John of Seville flourished \textbf{Gerard of +Cremona} in Lombardy. Being desirous to gain possession of +%% -----File: 141.png---Folio 127------- +\index{Alfonso's tables}% +\index{Alphonso's tables}% +\index{Arabic numerals and notation|(}% +\index{Elements (Euclid's)}% +\index{Geometry!Middle Ages}% +\index{Girard!ref.\ to}% +the \textit{Almagest}, he went to Toledo, and there, in~1175, translated +\index{Almagest!ref.\ to}% +this great work of Ptolemy. Inspired by the richness of +Mohammedan literature, he gave himself up to its study. He +translated into Latin over 70~Arabic works. Of mathematical +treatises, there were among these, besides the \textit{Almagest}, the +15~books of Euclid, the \textit{Sphærica} of Theodosius, a work of +\index{Euclid!ref.\ to}% +\index{Theodosius!ref.\ to}% +Menelaus, the algebra of Mohammed ben Musa Hovarezmi, the +\index{Hovarezmi!ref.\ to}% +\index{Mohammed ben Musa Hovarezmi!ref.\ to}% +astronomy of Dshabir ben Aflah, and others less important. +\index{Astronomy!Middle Ages}% +\index{Gabir ben Aflah!ref.\ to}% + +In the thirteenth century, the zeal for the acquisition of +Arabic learning continued. Foremost among the patrons of +science at this time ranked Emperor Frederick~II. of Hohenstaufen +(died 1250). Through frequent contact with Mohammedan +scholars, he became familiar with Arabic science. +He employed a number of scholars in translating Arabic +manuscripts, and it was through him that we came in possession +of a new translation of the \textit{Almagest}. Another royal +head deserving mention as a zealous promoter of Arabic +science was Alfonso~X. of Castile (died 1284). He gathered +around him a number of Jewish and Christian scholars, who +translated and compiled astronomical works from Arabic +sources. \textbf{Rabbi Zag} and \textbf{Iehuda ben Mose Cohen} were the +\index{Iehuda ben Mose Cohen}% +\index{Zag}% +most prominent among them. Astronomical tables prepared +by these two Jews spread rapidly in the Occident, and constituted +the basis of all astronomical calculation till the +sixteenth century.\Reference{7} The number of scholars who aided in +transplanting Arabic science upon Christian soil was large. +But we mention only one more. \textbf{Giovanni Campano} of Novara +\index{Giovanni Campano}% +(about 1260) brought out a new translation of Euclid, which +drove the earlier ones from the field, and which formed the +basis of the printed editions.\Reference{7} + +At the close of the twelfth century, the Occident was in +possession of the so-called Arabic notation. The Hindoo +\index{Notation!Arabic notation|(}% +methods of calculation began to supersede the cumbrous methods +%% -----File: 142.png---Folio 128------- +\index{Arabic manuscripts|)}% +\index{Elements (Euclid's)}% +inherited from Rome. Algebra, with its rules for solving +linear and quadratic equations, had been made accessible to +the Latins. The geometry of Euclid, the \textit{Sphærica} of Theodosius, +\index{Geometry!Middle Ages}% +the astronomy of Ptolemy, and other works were now +accessible in the Latin tongue. Thus a great amount of new +scientific material had come into the hands of the Christians. +The talent necessary to digest this heterogeneous mass of +knowledge was not wanting. The figure of Leonardo of Pisa +\index{Leonardo of Pisa}% +adorns the vestibule of the thirteenth century. + +It is important to notice that no work either on mathematics +or astronomy was translated directly from the Greek previous +to the fifteenth century. + +\Subsection{The First Awakening and its Sequel} + +Thus far, France and the British Isles have been the headquarters +of mathematics in Christian Europe. But at the +beginning of the thirteenth century the talent and activity +of one man was sufficient to assign the mathematical science +a new home in Italy. This man was not a monk, like Bede, +Alcuin, or Gerbert, but a merchant, who in the midst of +business pursuits found time for scientific study. \textbf{Leonardo +of Pisa} is the man to whom we owe the first renaissance of +mathematics on Christian soil. He is also called \textit{Fibonacci}, +\ie~son of Bonaccio. His father was secretary at one of the +numerous factories erected on the south and east coast of the +Mediterranean by the enterprising merchants of Pisa. He +made Leonardo, when a boy, learn the use of the abacus. The +boy acquired a strong taste for mathematics, and, in later years, +during his extensive business travels in Egypt, Syria, Greece, +and Sicily, collected from the various peoples all the knowledge +he could get on this subject. Of all the methods of +calculation, he found the Hindoo to be unquestionably the +%% -----File: 143.png---Folio 129------- +\index{Algorithm!Middle Ages}% +\index{Arabic numerals and notation|)}% +best. Returning to Pisa, he published, in~1202, his great +work, the \textit{Liber Abaci}. A revised edition of this appeared in~1228. +This work contains about all the knowledge the Arabs +possessed in arithmetic and algebra, and treats the subject in +a free and independent way. This, together with the other +books of Leonardo, shows that he was not merely a compiler, +or, like other writers of the Middle Ages, a slavish imitator +of the form in which the subject had been previously presented, +but that he was an original worker of exceptional +power. + +He was the first great mathematician to advocate the adoption +of the ``Arabic notation.'' The calculation with the zero +\index{Zero!origin of term}% +was the portion of Arabic mathematics earliest adopted by +the Christians. The minds of men had been prepared for the +reception of this by the use of the abacus and the apices. +\index{Abacus}% +\index{Apices of Boethius!ref.\ to}% +The reckoning with columns was gradually abandoned, and +the very word \emph{abacus} changed its meaning and became a +synonym for \emph{algorism}. For the zero, the Latins adopted +the name \emph{zephirum}, from the Arabic \textit{sifr} (\textit{sifra}=empty); +hence our English word \emph{cipher}. The new notation was +\index{Notation!Arabic notation|)}% +accepted readily by the enlightened masses, but, at first, +rejected by the learned circles. The merchants of Italy used +it as early as the thirteenth century, while the monks in the +monasteries adhered to the old forms. In~1299, nearly 100~years +after the publication of Leonardo's \textit{Liber Abaci}, the +Florentine merchants were forbidden the use of the Arabic +numerals in book-keeping, and ordered either to employ the +Roman numerals or to write the numeral adjectives out in +full. In the fifteenth century the abacus with its counters +ceased to be used in Spain and Italy. In France it was used +later, and it did not disappear in England and Germany before +the middle of the seventeenth century.\Reference{22} Thus, in the \textit{Winter's +Tale} (iv.~3), Shakespeare lets the clown be embarrassed by +%% -----File: 144.png---Folio 130------- +a problem which he could not do without counters. Iago +(in \textit{Othello}, i.~1) expresses his contempt for Michael Cassio, +``forsooth a great mathematician,'' by calling him a ``counter-caster.'' +So general, indeed, says Peacock, appears to have +\index{Peacock!ref.\ to, X}% +been the practice of this species of arithmetic, that its rules +\index{Arithmetic!Middle Ages}% +and principles form an essential part of the arithmetical +treatises of that day. The real fact seems to be that the old +methods were used long after the Hindoo numerals were in +common and general use. With such dogged persistency does +man cling to the old! + +The \textit{Liber Abaci} was, for centuries, the storehouse from +which authors got material for works on arithmetic and +algebra. In it are set forth the most perfect methods of +calculation with integers and fractions, known at that time; +the square and cube root are explained; equations of the first +and second degree leading to problems, either determinate +or indeterminate, are solved by the methods of `single' or +`double position,' and also by real algebra. The book contains +a large number of problems. The following was proposed +to Leonardo of Pisa by a magister in Constantinople, +as a difficult problem: If A gets from B $7$~denare, then A's +sum is five-fold B's; if B gets from A $5$~denare, then B's sum +is seven-fold A's. How much has each? The \textit{Liber Abaci} +contains another problem, which is of historical interest, +because it was given with some variations by Ahmes, 3000~years +\index{Ahmes!ref.\ to}% +earlier: $7$~old women go to Rome; each woman has +$7$~mules, each mule carries $7$~sacks, each sack contains $7$~loaves, +with each loaf are $7$~knives, each knife is put up in $7$~sheaths. +What is the sum total of all named? \textit{Ans.}~$137,256$.\Reference{3} + +In~1220, Leonardo of Pisa published his \textit{Practica Geometriæ}, +which contains all the knowledge of geometry and trigonometry +\index{Geometry!Middle Ages}% +transmitted to him. The writings of Euclid and of some +other Greek masters were known to him, either from Arabic +%% -----File: 145.png---Folio 131------- +\index{Theory of numbers}% +manuscripts directly or from the translations made by his +countrymen, Gerard of Cremona and Plato of Tivoli. Leonardo's +\textit{Geometry} contains an elegant geometrical demonstration +\index{Geometry!Middle Ages}% +of Heron's formula for the area of a triangle, as a +\index{Heron the Elder!ref.\ to}% +function of its three sides. Leonardo treats the rich material +before him with skill and Euclidean rigour. + +Of still greater interest than the preceding works are those +containing Fibonacci's original investigations. We must here +preface that after the publication of the \textit{Liber Abaci}, Leonardo +was presented by the astronomer Dominicus to Emperor +Frederick~II. of Hohenstaufen. On that occasion, John of +Palermo, an imperial notary, proposed several problems, which +Leonardo solved promptly. The first problem was to find a +number $x$, such that $x^2 + 5$ and $x^2 - 5$ are each square numbers. +\index{Numbers!theory of numbers}% +The answer is $x = 3\frac{5}{12}$; for $(3 \frac{5}{12})^2 + 5 = (4\frac{1}{12})^2$, $(3\frac{5}{12})^2 - 5 += (2\frac{7}{12})^2$. His masterly solution of this is given in his \textit{liber +quadratorum}, a copy of which work was sent by him to +Frederick~II\@. The problem was not original with John of +Palermo, since the Arabs had already solved similar ones. +Some parts of Leonardo's solution may have been borrowed +from the Arabs, but the method which he employed of building +squares by the summation of odd numbers is original with him. + +The second problem proposed to Leonardo at the famous +scientific tournament which accompanied the presentation of +this celebrated algebraist to that great patron of learning, +Emperor Frederick~II., was the solving of the equation +$x^3 + 2x^2 + 10x = 20$. As yet cubic equations had not been +solved algebraically. Instead of brooding stubbornly over +this knotty problem, and after many failures still entertaining +new hopes of success, he changed his method of inquiry and +showed by clear and rigorous demonstration that the roots +of this equation could not be represented by the Euclidean +irrational quantities, or, in other words, that they could not be +%% -----File: 146.png---Folio 132------- +constructed with the ruler and compass only. He contented +himself with finding a very close approximation to the +required root. His work on this cubic is found in the \textit{Flos}, +together with the solution of the following third problem +given him by John of Palermo: Three men possess in common +an unknown sum of money~$t$; the share of the first +is~$\dfrac{t}{2}$; that of the second,~$\dfrac{t}{3\Fstr}$; that of the third,~$\smash[t]{\dfrac{t}{6}}$. Desirous of +depositing the sum at a safer place, each takes at hazard a +certain amount; the first takes~$x$, but deposits only~$\dfrac{x}{2}$; the second +carries~$y$, but deposits only~$\smash{\dfrac{y}{3}}$; the third takes~$z$, and +deposits~$\smash[t]{\dfrac{z}{6}}$. Of the amount deposited each one must receive +exactly~$\frac{1}{3}$, in order to possess his share of the whole sum. +Find $x$, $y$,~$z$. Leonardo shows the problem to be indeterminate. +Assuming 7 for the sum drawn by each from the +deposit, he finds $t = 47$, $x = 33$, $y = 13$, $z = 1$. + +One would have thought that after so brilliant a beginning, +the sciences transplanted from Mohammedan to Christian soil +would have enjoyed a steady and vigorous development. But +this was not the case. During the fourteenth and fifteenth +centuries, the mathematical science was almost stationary. +Long wars absorbed the energies of the people and thereby +kept back the growth of the sciences. The death of Frederick~II. +in 1254 was followed by a period of confusion in Germany. +The German emperors and the popes were continually quarrelling, +and Italy was inevitably drawn into the struggles between +the Guelphs and the Ghibellines. France and England were +engaged in the Hundred Years' War (1338--1453). Then +followed in England the Wars of the Roses. The growth of +science was retarded not only by war, but also by the injurious +influence of scholastic philosophy. The intellectual +leaders of those times quarrelled over subtle subjects in metaphysics +%% -----File: 147.png---Folio 133------- +and theology. Frivolous questions, such as ``How +many angels can stand on the point of a needle?'' were discussed +with great interest. Indistinctness and confusion of +ideas characterised the reasoning during this period. Among +the mathematical productions of the Middle Ages, the works +of Leonardo of Pisa appear to us like jewels among quarry-rubbish. +\index{Leonardo of Pisa!ref.\ to}% +The writers on mathematics during this period were +not few in number, but their scientific efforts were vitiated +by the method of scholastic thinking. Though they possessed +the \textit{Elements} of Euclid, yet the true nature of a mathematical +\index{Elements (Euclid's)}% +proof was so little understood, that Hankel believes it no +exaggeration to say that ``since Fibonacci, not a single proof, +not borrowed from Euclid, can be found in the whole literature +of these ages, which fulfils all necessary conditions.'' + +The only noticeable advance is a simplification of numerical +operations and a more extended application of them. Among +the Italians are evidences of an early maturity of arithmetic. +\index{Arithmetic!Middle Ages}% +Peacock\Reference[]{22} says: The Tuscans generally, and the Florentines +\index{Peacock!ref.\ to, X}% +in particular, whose city was the cradle of the literature and +arts of the thirteenth and fourteenth centuries, were celebrated +for their knowledge of arithmetic and book-keeping, which +were so necessary for their extensive commerce; the Italians +were in familiar possession of commercial arithmetic long +before the other nations of Europe; to them we are indebted +for the formal introduction into books of arithmetic, under +distinct heads, of questions in the single and double rule of +three, loss and gain, fellowship, exchange, simple and compound +interest, discount, and so on. + +There was also a slow improvement in the algebraic notation. +\index{Notation!in algebra}% +The Hindoo algebra possessed a tolerable symbolic +\index{Algebra!Middle Ages}% +notation, which was, however, completely ignored by the Mohammedans. +In this respect, Arabic algebra approached +much more closely to that of Diophantus, which can scarcely +%% -----File: 148.png---Folio 134------- +\index{Albertus Magnus}% +\index{Exponents}% +\index{Stevin!ref.\ to}% +be said to employ symbols in a systematic way. Leonardo of +Pisa possessed no algebraic symbolism. Like the Arabs, he +expressed the relations of magnitudes to each other by lines +or in words. But in the mathematical writings of the monk +\textit{Luca Pacioli} (also called Lucas de Burgo sepulchri) symbols +\index{Pacioli!ref.\ to}% +began to appear. They consisted merely in abbreviations of +Italian words, such as \textit{p} for \textit{piu} (more), \textit{m} for \textit{meno} (less), \textit{co} +for \textit{cosa} (the thing or unknown quantity). ``Our present +notation has arisen by almost insensible degrees as convenience +\index{Notation!in algebra}% +suggested different marks of abbreviation to different +authors; and that perfect symbolic language which addresses +itself solely to the eye, and enables us to take in at a glance +the most complicated relations of quantity, is the result of a +large series of small improvements.''\Reference{23} + +We shall now mention a few authors who lived during the +thirteenth and fourteenth and the first half of the fifteenth +centuries. About the time of Leonardo of Pisa (1200~\AD), +lived the German monk \textbf{Jordanus Nemorarius}, who wrote a once +\index{Jordanus Nemorarius}% +famous work on the properties of numbers (1496), modelled +after the arithmetic of Boethius. The most trifling numeral +\index{Arithmetic!Middle Ages}% +\index{Boethius!ref.\ to}% +properties are treated with nauseating pedantry and prolixity. +A practical arithmetic based on the Hindoo notation was +also written by him. \textbf{John Halifax} (Sacro Bosco, died 1256) +\index{Halifax}% +taught in Paris and made an extract from the \textit{Almagest} containing +\index{Almagest!ref.\ to}% +only the most elementary parts of that work. This +extract was for nearly 400~years a work of great popularity +and standard authority. Other prominent writers are \textbf{Albertus +Magnus} and \textbf{George Purbach} in Germany, and \textbf{Roger Bacon} in +\index{Bacon, R.}% +\index{Purbach}% +England. It appears that here and there some of our modern +ideas were anticipated by writers of the Middle Ages. Thus, +\textbf{Nicole Oresme}, a bishop in Normandy (died 1382), first conceived +\index{Oresme}% +a notation of fractional powers, afterwards re-discovered +by Stevinus, and gave rules for operating with them. +%% -----File: 149.png---Folio 135------- +\index{Elements (Euclid's)}% +\index{Infinitesimals}% +\index{Infinity}% +His notation was totally different from ours. \textbf{Thomas Bradwardine}, +\index{Bradwardine}% +archbishop of Canterbury, studied star-polygons,---a +\index{Star-polygons}% +subject which has recently received renewed attention. The +first appearance of such polygons was with Pythagoras and +\index{Pythagoras!ref.\ to}% +his school. We next meet with such polygons in the geometry +of Boethius and also in the translation of Euclid from +\index{Boethius!ref.\ to}% +the Arabic by Athelard of Bath. Bradwardine's philosophic +\index{Athelard of Bath!ref.\ to}% +writings contain discussions on the infinite and the infinitesimal---subjects +never since lost sight of. To England +falls the honour of having produced the earliest European +writers on trigonometry. The writings of Bradwardine, of +\index{Trigonometry}% +Richard of Wallingford, and John Maudith, both professors +\index{Maudith}% +\index{Richard of Wallingford}% +at Oxford, and of Simon Bredon of Winchecombe, contain +\index{Bredon}% +trigonometry drawn from Arabic sources. + +The works of the Greek monk \textbf{Maximus Planudes}, who lived +\index{Planudes, M.}% +in the first half of the fourteenth century, are of interest only +as showing that the Hindoo numerals were then known in +Greece. A writer belonging, like Planudes, to the Byzantine +school, was \textbf{Moschopulus}, who lived in Constantinople in the +\index{Moschopulus}% +early part of the fifteenth century. To him appears to be +due the introduction into Europe of magic squares. He wrote +\index{Magic squares}% +a treatise on this subject. Magic squares were known to the +Arabs, and perhaps to the Hindoos. Mediæval astrologers +and physicians believed them to possess mystical properties +and to be a charm against plague, when engraved on silver +plate. + +In 1494 was \emph{printed} the \textit{Summa de Arithmetica, Geometria, +Proportione et Proportionalita}, written by the Tuscan monk +\textbf{Lucas Pacioli}, who, as we remarked, first introduced symbols +\index{Pacioli}% +in algebra. This contains all the knowledge of his day on +\index{Algebra!Middle Ages}% +arithmetic, algebra, and trigonometry, and is the first comprehensive +work which appeared after the \textit{Liber Abaci} of +Fibonacci. It contains little of importance which cannot be +%% -----File: 150.png---Folio 136------- +\index{Elements (Euclid's)}% +\index{Halifax!ref.\ to}% +found in Fibonacci's great work, published three centuries +earlier.\Reference{1} + +Perhaps the greatest result of the influx of Arabic learning +was the establishment of universities. What was their +\index{Universities of Cologne, Leipzig, Oxford, Paris, and Prague}% +attitude toward mathematics? The \textit{University of Paris}, so +famous at the beginning of the twelfth century under the +teachings of Abelard, paid but little attention to this science +during the Middle Ages. Geometry was neglected, and Aristotle's +logic was the favourite study. In~1336, a rule was +introduced that no student should take a degree without +attending lectures on mathematics, and from a commentary +on the first six books of Euclid, dated~1536, it appears that +\index{Euclid!ref.\ to}% +candidates for the degree of A.M. had to give an oath that +they had attended lectures on these books.\Reference{7} Examinations, +when held at all, probably did not extend beyond the first +book, as is shown by the nickname ``magister matheseos,'' +\index{Magister matheseos}% +applied to the Theorem of Pythagoras, the last in the first +book. More attention was paid to mathematics at the \textit{University +of Prague}, founded 1384. For the Baccalaureate degree, +students were required to take lectures on Sacro Bosco's +famous work on astronomy. Of candidates for the A.M. were +required not only the six books of Euclid, but an additional +knowledge of applied mathematics. Lectures were given on +the \textit{Almagest}. At the \textit{University of Leipzig}, the daughter of +\index{Almagest!ref.\ to}% +Prague, and at \textit{Cologne}, less work was required, and, as late +as the sixteenth century, the same requirements were made at +these as at Prague in the fourteenth. The universities of +Bologna, Padua, Pisa, occupied similar positions to the ones +in Germany, only that purely astrological lectures were given +in place of lectures on the \textit{Almagest}. At Oxford, in the +middle of the fifteenth century, the first two books of Euclid +were read.\Reference{6} + +Thus it will be seen that the study of mathematics was +%% -----File: 151.png---Folio 137------- +\index{Middle Ages|)}% +maintained at the universities only in a half-hearted manner. +No great mathematician and teacher appeared, to inspire the +students. The best energies of the schoolmen were expended +upon the stupid subtleties of their philosophy. The genius +of Leonardo of Pisa left no permanent impress upon the age, +\index{Leonardo of Pisa!ref.\ to}% +and another Renaissance of mathematics was wanted. +%% -----File: 152.png---Folio 138------- +\index{Elements (Euclid's)}% +\index{Geometry!Renaissance}% + + +\Chapter{Modern Europe} +\index{Modern Europe|etseq}% + +\First{We} find it convenient to choose the time of the capture of +Constantinople by the Turks as the date at which the Middle +Ages ended and Modern Times began. In~1453, the Turks +battered the walls of this celebrated metropolis with cannon, +and finally captured the city; the Byzantine Empire fell, to +rise no more. Calamitous as was this event to the East, it +acted favourably upon the progress of learning in the West. +A great number of learned Greeks fled into Italy, bringing +with them precious manuscripts of Greek literature. This +contributed vastly to the reviving of classic learning. Up +to this time, Greek masters were known only through the +often very corrupt Arabic manuscripts, but now they began +to be studied from original sources and in their own language. +The first English translation of Euclid was made in 1570 from +\index{Euclid!ref.\ to}% +the Greek by \textit{Sir Henry Billingsley}, assisted by \textit{John Dee}.\Reference{29} +\index{Billingsley}% +\index{Dee}% +About the middle of the fifteenth century, printing was invented; +books became cheap and plentiful; the printing-press +transformed Europe into an audience-room. Near the close of +the fifteenth century, America was discovered, and, soon after, +the earth was circumnavigated. The pulse and pace of the +world began to quicken. Men's minds became less servile; +they became clearer and stronger. The indistinctness of +thought, which was the characteristic feature of mediæval +learning, began to be remedied chiefly by the steady cultivation +%% -----File: 153.png---Folio 139------- +\index{Ptolemy!ref.\ to}% +of Pure Mathematics and Astronomy. Dogmatism was +attacked; there arose a long struggle with the authority of +the Church and the established schools of philosophy. The +Copernican System was set up in opposition to the time-honoured +\index{Copernican System}% +Ptolemaic System. The long and eager contest between +the two culminated in a crisis at the time of Galileo, and +\index{Galileo!ref.\ to}% +resulted in the victory of the new system. Thus, by slow +degrees, the minds of men were cut adrift from their old +scholastic moorings and sent forth on the wide sea of scientific +inquiry, to discover new islands and continents of truth. + +\Section{The Renaissance} +\index{Renaissance|(}% + +With the sixteenth century began a period of increased +intellectual activity. The human mind made a vast effort to +achieve its freedom. Attempts at its emancipation from +Church authority had been made before, but they were stifled +and rendered abortive. The first great and successful revolt +against ecclesiastical authority was made in Germany. The +new desire for judging freely and independently in matters +of religion was preceded and accompanied by a growing spirit +of scientific inquiry. Thus it was that, for a time, Germany +led the van in science. She produced \textit{Regiomontanus}, \textit{Copernicus, +\index{Copernicus}% +\index{Regiomontanus!ref.\ to}% +Rhæticus}, \textit{Kepler}, and \textit{Tycho Brahe}, at a period when +\index{Brahe, Tycho}% +\index{Kepler!ref.\ to}% +\index{Rhæticus!ref.\ to}% +\index{Tycho Brahe}% +France and England had, as yet, brought forth hardly any +great scientific thinkers. This remarkable scientific productiveness +was no doubt due, to a great extent, to the commercial +prosperity of Germany. Material prosperity is an essential +condition for the progress of knowledge. As long as every +individual is obliged to collect the necessaries for his subsistence, +there can be no leisure for higher pursuits. At this +time, Germany had accumulated considerable wealth. The +%% -----File: 154.png---Folio 140------- +Hanseatic League commanded the trade of the North. Close +commercial relations existed between Germany and Italy. +Italy, too, excelled in commercial activity and enterprise. +We need only mention Venice, whose glory began with the crusades, +and Florence, with her bankers and her manufacturers +of silk and wool. These two cities became great intellectual +centres. Thus, Italy, too, produced men in art, literature, and +science, who shone forth in fullest splendour. In fact, Italy +was the fatherland of what is termed the Renaissance. + +For the first great contributions to the mathematical sciences +we must, therefore, look to Italy and Germany. In Italy +brilliant accessions were made to algebra, in Germany to +\index{Algebra!Renaissance}% +astronomy and trigonometry. +\index{Trigonometry}% + +On the threshold of this new era we meet in Germany with +the figure of John Mueller, more generally called \textbf{Regiomontanus} +\index{Regiomontanus}% +(1436--1476). Chiefly to him we owe the revival of +trigonometry. He studied astronomy and trigonometry at +Vienna under the celebrated George Purbach. The latter +\index{Purbach!ref.\ to}% +perceived that the existing Latin translations of the \textit{Almagest} +\index{Almagest!ref.\ to}% +were full of errors, and that Arabic authors had not remained +true to the Greek original. Purbach therefore began to make +a translation directly from the Greek. But he did not live to +finish it. His work was continued by Regiomontanus, who +went beyond his master. Regiomontanus learned the Greek +language from Cardinal Bessarion, whom he followed to Italy, +where he remained eight years collecting manuscripts from +Greeks who had fled thither from the Turks. In addition to +the translation of and the commentary on the \textit{Almagest}, he +prepared translations of the \textit{Conics} of Apollonius, of Archimedes, +\index{Apollonius!ref.\ to}% +\index{Archimedes!ref.\ to}% +and of the mechanical works of Heron. Regiomontanus +\index{Heron the Elder!ref.\ to}% +and Purbach adopted the Hindoo \emph{sine} in place of the Greek +\index{Sine}% +\emph{chord of double the arc}. The Greeks and afterwards the Arabs +divided the radius into $60$~equal parts, and each of these again +%% -----File: 155.png---Folio 141------- +\index{Sine}% +into $60$~smaller ones. The Hindoos expressed the length of +the radius by parts of the circumference, saying that of the +$21,600$ equal divisions of the latter, it took $3438$ to measure +the radius. Regiomontanus, to secure greater precision, constructed +\index{Regiomontanus}% +one table of sines on a radius divided into $600,000$ +parts, and another on a radius divided decimally into $10,000,000$ +divisions. He emphasised the use of the \emph{tangent} in trigonometry. +\index{Trigonometry}% +Following out some ideas of his master, he calculated +a table of tangents. German mathematicians were not the +\index{Tangents!in trigonometry}% +first Europeans to use this function. In England it was known +a century earlier to Bradwardine, who speaks of tangent (\textit{umbra +\index{Bradwardine!ref.\ to}% +recta}) and cotangent (\textit{umbra versa}), and to John Maudith. +\index{Cotangent}% +\index{Maudith!ref.\ to}% +Regiomontanus was the author of an arithmetic and also of +a complete treatise on trigonometry, containing solutions of +both plane and spherical triangles. The form which he gave +to trigonometry has been retained, in its main features, to the +present day. + +Regiomontanus ranks among the greatest men that Germany +has ever produced. His complete mastery of astronomy and +mathematics, and his enthusiasm for them, were of far-reaching +influence throughout Germany. So great was his +reputation, that Pope Sixtus~IV. called him to Italy to +improve the calendar. Regiomontanus left his beloved city +\index{Calendar}% +of Nürnberg for Rome, where he died in the following year. + +After the time of Purbach and Regiomontanus, trigonometry +and especially the calculation of tables continued to occupy +German scholars. More refined astronomical instruments were +made, which gave observations of greater precision; but these +would have been useless without trigonometrical tables of corresponding +accuracy. Of the several tables calculated, that +by \textit{Georg Joachim} of Feldkirch in Tyrol, generally called +\textbf{Rhæticus}, deserves special mention. He calculated a table of +\index{Rhæticus}% +sines with the radius $= 10,000,000,000$ and from $10''$ to~$10''$; +%% -----File: 156.png---Folio 142------- +\index{Cubic equations|(}% +and, later on, another with the radius $= 1,000,000,000,000,000$, +and proceeding from $10''$ to~$10''$. He began also the construction +of tables of tangents and secants, to be carried to +\index{Secants}% +\index{Tangents!in trigonometry}% +the same degree of accuracy; but he died before finishing them. +For twelve years he had had in continual employment several +calculators. The work was completed by his pupil, \textbf{Valentine +Otho}, in~1596. This was indeed a gigantic work,---a monument +\index{Otho}% +of German diligence and indefatigable perseverance. +The tables were republished in 1613 by \textbf{Pitiscus}, who spared +\index{Pitiscus}% +no pains to free them of errors. Astronomical tables of +so great a degree of accuracy had never been dreamed of +by the Greeks, Hindoos, or Arabs. That Rhæticus was not a +\index{Rhæticus!ref.\ to}% +ready calculator only, is indicated by his views on \DPtypo{trignometrical}{trigonometrical} +lines. Up to his time, the trigonometric functions had +been considered always with relation to the arc; he was the +first to construct the right triangle and to make them depend +directly upon its angles. It was from the right triangle that +Rhæticus got his idea of calculating the hypotenuse; \ie~he +was the first to plan a table of secants. Good work in trigonometry +was done also by Vieta and Romanus. +\index{Romanus!ref.\ to}% +\index{Vieta!ref.\ to}% + +We shall now leave the subject of trigonometry to witness +the progress in the solution of algebraical equations. To do +so, we must quit Germany for Italy. The first comprehensive +algebra printed was that of Lucas Pacioli. He closes his +\index{Algebra!Renaissance|(}% +\index{Pacioli!ref.\ to}% +book by saying that the solution of the equations $x^{3} + mx = n$, +$x^{3} + n = mx$ is as impossible at the present state of science as +the quadrature of the circle. This remark doubtless stimulated +thought. The first step in the algebraic solution of +cubics was taken by \textbf{Scipio Ferro} (died 1526), a professor of +\index{Ferro, Scipio}% +mathematics at Bologna, who solved the equation $x^{3} + mx = n$. +Nothing more is known of his discovery than that he imparted +it to his pupil, \textit{Floridas}, in~1505. It was the practice in those +\index{Floridas}% +days and for two centuries afterwards to keep discoveries +%% -----File: 157.png---Folio 143------- +secret, in order to secure by that means an advantage over +rivals by proposing problems beyond their reach. This practice +gave rise to numberless disputes regarding the priority of +inventions. A second solution of cubics was given by \textit{Nicolo} +of Brescia (1506(?)--1557). When a boy of six, Nicolo was +so badly cut by a French soldier that he never again gained +the free use of his tongue. Hence he was called \textbf{Tartaglia}, +\index{Tartaglia|(}% +\ie~the stammerer. His widowed mother being too poor to +pay his tuition in school, he learned to read and picked up a +knowledge of Latin, Greek, and mathematics by himself. +Possessing a mind of extraordinary power, he was able to +appear as teacher of mathematics at an early age. In~1530, +one Colla proposed him several problems, one leading to the +\index{Colla}% +equation $x^3 + px^{2} = q$. Tartaglia found an imperfect method +for solving this, but kept it secret. He spoke about his secret +in public and thus caused Ferro's pupil, Floridas, to proclaim +his own knowledge of the form $x^{3} + mx = n$. Tartaglia, believing +him to be a mediocrist and braggart, challenged him to a +public discussion, to take place on the 22d of February, 1535. +Hearing, meanwhile, that his rival had gotten the method +from a deceased master, and fearing that he would be beaten +in the contest, Tartaglia put in all the zeal, industry, and +skill to find the rule for the equations, and he succeeded in it +ten days before the appointed date, as he himself modestly +says.\Reference{7} The most difficult step was, no doubt, the passing from +quadratic irrationals, used in operating from time of old, to +cubic irrationals. Placing $x = \sqrt[3]{t} - \sqrt[3]{u}$, Tartaglia perceived +that the irrationals disappeared from the equation $x^3 + mx = n$, +making $n = t - u$. But this last equality, together with +$(\frac{1}{3}m)^3 = tu$, gives at once +\[ +t = \sqrt{\left( \frac{n}{2} \right)^3 + \left( \frac{m}{3} \right)^3} + \frac{n}{2}, \quad +u = \sqrt{\left( \frac{n}{2} \right)^2 + \left( \frac{m}{2} \right)^3} - \frac{n}{2}. +\] +%% -----File: 158.png---Folio 144------- +This is Tartaglia's solution of $x^{3} + mx = n$. On the 13th~of +February, he found a similar solution for $x^{3} = mx + n$. The +contest began on the~22d. Each contestant proposed thirty +problems. The one who could solve the greatest number within +fifty days should be the victor. Tartaglia solved the thirty +problems proposed by Floridas in two hours; Floridas could +\index{Floridas}% +not solve any of Tartaglia's. From now on, Tartaglia studied +cubic equations with a will. In 1541 he discovered a general +solution for the cubic $x^{3} ± px^{2} = ± q$, by transforming it into +the form $x^{3} ± mx = ± n$. The news of Tartaglia's victory +spread all over Italy. Tartaglia was entreated to make known +his method, but he declined to do so, saying that after his +completion of the translation from the Greek of Euclid and +\index{Euclid!ref.\ to}% +Archimedes, he would publish a large algebra containing his +\index{Archimedes!ref.\ to}% +method. But a scholar from Milan, named \textbf{Hieronimo Cardano} +(1501--1576), after many solicitations, and after giving the +most solemn and sacred promises of secrecy, succeeded in +obtaining from Tartaglia a knowledge of his rules. + +At this time Cardan was writing his \textit{Ars Magna}, and he +\index{Cardan}% +knew no better way to crown his work than by inserting the +much sought for rules for solving cubics. Thus Cardan broke +his most solemn vows, and published in 1545 in his \textit{Ars Magna} +Tartaglia's solution of cubics. Tartaglia became desperate. +His most cherished hope, of giving to the world an immortal +work which should be the monument of his deep learning and +power for original research, was suddenly destroyed; for the +crown intended for his work had been snatched away. His +first step was to write a history of his invention; but, to completely +annihilate his enemies, he challenged Cardan and his +pupil Lodovico Ferrari to a contest: each party should propose +\index{Ferrari!ref.\ to}% +thirty-one questions to be solved by the other within fifteen +days. Tartaglia solved most questions in seven days, but the +other party did not send in their solution before the expiration +%% -----File: 159.png---Folio 145------- +of the fifth month; moreover, all their solutions except one +were wrong. A replication and a rejoinder followed. Endless +were the problems proposed and solved on both sides. The +dispute produced much chagrin and heart-burnings to the parties, +and to Tartaglia especially, who met with many other +disappointments. After having recovered himself again, Tartaglia +\index{Tartaglia|)}% +began, in 1556, the publication of the work which he +had had in his mind for so long; but he died before he reached +the consideration of cubic equations. Thus the fondest wish +\index{Cubic equations|)}% +of his life remained unfulfilled; the man to whom we owe the +greatest contribution to algebra made in the sixteenth century +was forgotten, and his method came to be regarded as the discovery +of Cardan and to be called Cardan's solution. + +Remarkable is the great interest that the solution of cubics +excited throughout Italy. It is but natural that after this +great conquest mathematicians should attack \Change{bi-quadratic}{biquadratic} equations. +As in the case of cubics, so here, the first impulse was +given by Colla, who, in 1540, proposed for solution the equation +\index{Colla}% +$x^{4} + 6x^{2} + 36 = 60x$. To be sure, Cardan had studied +particular cases as early as~1539. Thus he solved the equation +$13x^{2} = x^{4} + 2x^{3} + 2x + 1$ by a process similar to that employed +by Diophantus and the Hindoos; namely, by adding +to both sides $3x^{2}$ and thereby rendering both numbers +complete squares. But Cardan failed to find a general solution; +it remained for his pupil \textbf{Ferrari} to prop the reputation +\index{Ferrari}% +of his master by the brilliant discovery of the general +solution of \Change{bi-quadratic}{biquadratic} equations. Ferrari reduced Colla's +equation to the form $(x^{2}+ 6)^{2} = 60x + 6x^{2}$. In order to +give also the right member the form of a complete square +he added to both members the expression $2 (x^{2} + 6) y + y^{2}$, +containing a new unknown quantity~$y$. This gave him $(x^{2} + 6 ++ y)^{2} = (6 + 2y) x^{2} + 60x + (12y + y^{2})$. The condition that +the right member be a complete square is expressed by the +%% -----File: 160.png---Folio 146------- +\index{Biquadratic equation}% +cubic equation $(2y + 6) (12y + y^{2}) = 900$. Extracting the +square root of the \Change{bi-quadratic}{biquadratic}, he got $x^{2} + 6 + y = x \sqrt{2y + 6} ++ \dfrac{900\Fstr}{\sqrt{2y + 6}}$. Solving the cubic for~$y$ and substituting, it remained +only to determine~$x$ from the resulting quadratic. +Ferrari pursued a similar method with other numerical \Change{bi-quadratic}{biquadratic} +equations.\Reference{7} Cardan had the pleasure of publishing +this discovery in his \textit{Ars Magna} in~1545. Ferrari's solution +is sometimes ascribed to \textit{Bombelli}, but he is no more the discoverer +\index{Bombelli}% +of it than Cardan is of the solution called by his +name. + +To Cardan algebra is much indebted. In his \textit{Ars Magna} +he takes notice of negative roots of an equation, calling them +\index{Negative roots}% +\emph{fictitious}, while the positive roots are called \emph{real}. Imaginary +\index{Imaginary quantities}% +roots he does not consider; cases where they appear he calls +impossible. Cardan also observed the difficulty in the irreducible +case in the cubics, which, like the quadrature of the +circle, has since ``so much tormented the perverse ingenuity of +mathematicians.'' But he did not understand its nature. It remained +for \textbf{Raphael Bombelli} of Bologna, who published in 1572 +an algebra of great merit, to point out the reality of the apparently +imaginary expression which the root assumes, and thus +to lay the foundation of a more intimate knowledge of imaginary +quantities. + +After this brilliant success in solving equations of the third +and fourth degrees, there was probably no one who doubted, +that with aid of irrationals of higher degrees, the solution of +equations of any degree whatever could be found. But all +attempts at the algebraic solution of the quintic were fruitless, +and, finally, Abel demonstrated that all hopes of finding algebraic +\index{Abel!ref.\ to}% +solutions to equations of higher than the fourth degree +were purely Utopian. + +Since no solution by radicals of equations of higher degrees +%% -----File: 161.png---Folio 147------- +\index{Equations!numerical}% +\index{Falsa positio}% +could be found, there remained nothing else to be done than +the devising of rules by which at least the numerical values of +the roots could be ascertained. Cardan applied the Hindoo +rule of ``false position'' (called by him \textit{regula aurea}) to the +cubic, but this mode of approximating was exceedingly rough. +An incomparably better method was invented by \textbf{Franciscus +Vieta}, a French mathematician, whose transcendent genius +\index{Vieta}% +enriched mathematics with several important innovations. +Taking the equation $f(x) = Q$, wherein $f(x)$ is a polynomial +containing different powers of~$x$, with numerical coefficients, +and $Q$ is a given number, Vieta first substitutes in~$f(x)$ a +known approximate value of the root, and then shows that +another figure of the root can be obtained by division. A repetition +of the same process gives the next figure of the root, +and so on. Thus, in $x^2 + 14x = 7929$, taking $80$ for the approximate +root, and placing $x = 80 + b$, we get +\begin{DPalign*} +&(80 + b)^2 + 14 (80 + b) = 7929,\\ +\lintertext{or} &174b + b^2 = 409. +\end{DPalign*} +Since $174b$ is much greater than~$b^2$, we place $174b = 409$, and +obtain thereby $b = 2$. Hence the second approximation is~$82$. +Put $x = 82 + c$, then $(82 + c)^2 + 14(82 + c) = 7929$, or $178 c + c^2 += 57$. As before, place $178c = 57$, then $c = .3$, and the third +approximation gives $82.3$. Assuming $x = 82.3 + d$, and substituting, +gives $178.6 d + d^2 = 3.51$, and $178.6d = 3.51$, $\therefore d = .01$; +giving for the fourth approximation $82.31$. In the same way, +$e = .009$, and the value for the root of the given equation is +$82.319 \dots$. For this process, Vieta was greatly admired by his +contemporaries. It was employed by Harriot, Oughtred, Pell, +\index{Harriot!ref.\ to}% +\index{Oughtred!ref.\ to}% +\index{Pell}% +and others. Its principle is identical with the main principle +involved in the methods of approximation of Newton and +\index{Newton!ref.\ to}% +Horner. The only change lies in the arrangement of the +\index{Horner}% +%% -----File: 162.png---Folio 148------- +work. This alteration was made to afford facility and security +in the process of evolution of the root. + +We pause a moment to sketch the life of Vieta, the most +eminent French mathematician of the sixteenth century. He +was born in Poitou in 1540, and died in 1603 at Paris. He +was employed throughout life in the service of the state, +under Henry~III. and Henry~IV\@. He was, therefore, not a +mathematician by profession, but his love for the science was +so great that he remained in his chamber studying, sometimes +several days in succession, without eating and sleeping more +than was necessary to sustain himself. So great devotion to +abstract science is the more remarkable, because he lived at a +time of incessant political and religious turmoil. During the +war against Spain, Vieta rendered service to Henry~IV. by +deciphering intercepted letters written in a species of cipher, +and addressed by the Spanish Court to their governor of Netherlands. +The Spaniards attributed the discovery of the key +to magic. + +An ambassador from Netherlands once told Henry~IV. that +France did not possess a single geometer capable of solving a +problem propounded to geometers by a Belgian mathematician, +Adrianus Romanus. It was the solution of the equation of +\index{Romanus!ref.\ to}% +the forty-fifth degree:--- +\[ +45y - 3795y^3 + 95634y^5 - \dotsb + 945y^{41} - 45y^{43} + y^{45} = C. +\] +Henry~IV. called Vieta, who, having already pursued similar +investigations, saw at once that this awe-inspiring problem +was simply the equation by which $C=2\sin\phi$ was expressed +in terms of $y = 2 \sin \frac{1}{45}\phi$; that, since $45 = 3·3·5$, it was +necessary only to divide an angle once into 5~equal parts, and +then twice into~3,---a division which could be effected by corresponding +equations of the fifth and third degrees. Brilliant +was the discovery by Vieta of 23~roots to this equation, instead +%% -----File: 163.png---Folio 149------- +\index{Biquadratic equation}% +\index{Cubic equations}% +\index{Negative roots}% +\index{Notation!in algebra}% +of only one. The reason why he did not find 45~solutions, is +that the remaining ones involve negative sines, which were +unintelligible to him. Detailed investigations on the famous +old problem of the section of an angle into an odd number of +equal parts, led Vieta to the discovery of a trigonometrical +solution of Cardan's irreducible case in cubics. He applied +\index{Cardan!ref.\ to}% +the equation $\bigl(2 \cos \frac{1}{3} \phi\bigr)^3 - 3 \bigl(2 \cos \frac{1}{3} \phi\bigr) = 2 \cos \phi$ to the solution +of $x^3 - 3a^{2}x = a^{2}b$, when $a > \frac{1}{2} b$, by placing $x = 2a \cos \frac{1}{3}\phi$, and +determining~$\phi$ from $b = 2 a \cos \phi$. + +The main principle employed by him in the solution of +equations is that of \emph{reduction}. He solves the quadratic by +\index{Equations!solution of}% +making a suitable substitution which will remove the term +containing $x$ to the first degree. Like Cardan, he reduces the +general expression of the cubic to the form $x^{3}+ mx + n = 0$; +then, assuming $x = \bigl(\frac{1}{3}a - z^2\bigr) ÷ z$ and substituting, he gets +$z^6 - bz^{3} - \frac{1}{27} a^3 = 0$. Putting $z^3 = y$, he has a quadratic. In +the solution of \Change{bi-quadratics}{biquadratics}, Vieta still remains true to his +principle of reduction. This gives him the well-known cubic +resolvent. He thus adheres throughout to his favourite principle, +and thereby introduces into algebra a uniformity of +method which claims our lively admiration. In Vieta's algebra +we discover a partial knowledge of the relations existing +between the coefficients and the roots of an equation. He +shows that if the coefficient of the second term in an equation +of the second degree is minus the sum of two numbers whose +product is the third term, then the two numbers are roots of +the equation. Vieta rejected all except positive roots; hence +it was impossible for him to fully perceive the relations in +question. + +The most epoch-making innovation in algebra due to Vieta +is the denoting of general or indefinite quantities by letters +of the alphabet. To be sure, Regiomontanus and Stifel in +\index{Regiomontanus!ref.\ to}% +\index{Stifel!ref.\ to}% +Germany, and Cardan in Italy, used letters before him, but +%% -----File: 164.png---Folio 150------- +\index{Plus and minus, signs for}% +Vieta extended the idea and first made it an essential part of +algebra. The new algebra was called by him \textit{logistica speciosa} +in distinction to the old \textit{logistica numerosa}. Vieta's formalism +differed considerably from that of to-day. The equation +$a^3 + 3 a^2b + 3 ab^2 + b^3 = (a + b)^3$ was written by him ``$a \text{ cubus } + b +\text{ in } a \text{ quadr.\ } 3 + a \text{ in } b \text{ quadr.\ } 3 + b \text{ cubo æqualia } \overline{a + b} \text{ cubo}$.'' +In numerical equations the unknown quantity was denoted by~$N$, +its square by~$Q$, and its cube by~$C$. Thus the equation +$x^3 - 8x^2 +16x = 40$ was written $1C - 8Q + 16N \textit{ æqual.\ } 40$. +Observe that exponents and our symbol~($=$) for equality were +not yet in use; but that Vieta employed the Maltese cross~($+$) +as the short-hand symbol for addition, and the~($-$) for +subtraction. These two characters had not been in general +use before the time of Vieta. ``It is very singular,'' says +Hallam, ``that discoveries of the greatest convenience, and, +apparently, not above the ingenuity of a village schoolmaster, +should have been overlooked by men of extraordinary acuteness +like Tartaglia, Cardan, and Ferrari; and hardly less so +that, by dint of that acuteness, they dispensed with the aid of +these contrivances in which we suppose that so much of the +utility of algebraic expression consists.'' Even after improvements +in notation were once proposed, it was with extreme +\index{Notation!in algebra}% +slowness that they were admitted into general use. They +were made oftener by accident than design, and their authors +had little notion of the effect of the change which they were +making. The introduction of the $+$ and $-$ symbols seems +to be due to the Germans, who, although they did not enrich +algebra during the Renaissance with great inventions, as did +\index{Algebra!Renaissance|)}% +the Italians, still cultivated it with great zeal. The arithmetic +\index{Arithmetic!Renaissance}% +of \textbf{John Widmann}, printed \AD~1489 in Leipzig, is the +\index{Widmann}% +earliest book in which the $+$ and $-$ symbols have been found. +There are indications leading us to surmise that they were in +use first among merchants. They occur again in the arithmetic +%% -----File: 165.png---Folio 151------- +\index{Arithmetic!Renaissance}% +\index{Notation!in algebra}% +of Grammateus, a teacher at the University of Vienna. +\index{Grammateus}% +His pupil, Christoff Rudolff, the writer of the first text-book +\index{Rudolff}% +on algebra in the German language (printed in~1525), employs +these symbols also. So did Stifel, who brought out a second +\index{Stifel!ref.\ to}% +edition of Rudolff's \textit{Coss} in~1553. Thus, by slow degrees, +their adoption became universal. There is another short-hand +symbol of which we owe the origin to the Germans. In a +manuscript published sometime in the fifteenth century, a dot +placed before a number is made to signify the extraction of a +root of that number. This dot is the embryo of our present +symbol for the square root. Christoff Rudolff, in his algebra, +remarks that ``the radix quadrata is, for brevity, designated +in his algorithm with the character $\sqrt{\vphantom{4}}$, as~$\sqrt{4}$.'' Here the dot +has grown into a symbol much like our own. This same +symbol was used by \textit{Michael Stifel}. Our sign of equality is due +to \textbf{Robert Recorde} (1510--1558), the author of \textit{The Whetstone of +\index{Recorde}% +Witte} (1557), which is the first English treatise on algebra. +He selected this symbol because no two things could be more +equal than two parallel lines~$=$. The sign~$÷$ for division was +first used by \textit{Johann Heinrich Rahn}, a Swiss, in~1659, and +\index{Rahn}% +was introduced in England by \textit{John Pell} in~1668. +\index{Pell}% + +\textbf{Michael Stifel} (1486?--1567), the greatest German algebraist +\index{Stifel}% +of the sixteenth century, was born in Esslingen, and died in +Jena. He was educated in the monastery of his native place, +and afterwards became Protestant minister. The study of the +significance of mystic numbers in Revelation and in Daniel +drew him to mathematics. He studied German and Italian +works, and published in~1544, in Latin, a book entitled +\textit{Arithmetica integra}. Melanchthon wrote a preface to it. Its +three parts treat respectively of rational numbers, irrational +numbers, and algebra. Stifel gives a table containing the numerical +values of the binomial coefficients for powers below the~18th. +He observes an advantage in letting a geometric progression +%% -----File: 166.png---Folio 152------- +\index{Tartaglia!ref.\ to}% +correspond to an arithmetical progression, and arrives at +the designation of integral powers by numbers. Here are the +germs of the theory of exponents. In 1545 Stifel published +\index{Exponents}% +an arithmetic in German. His edition of Rudolff's \textit{Coss} contains +\index{Coss, term for algebra}% +rules for solving cubic equations, derived from the\DPnote{** PP: Removed duplicate `the'.} +\index{Cubic equations}% +writings of Cardan. +\index{Cardan!ref.\ to}% + +We remarked above that Vieta discarded negative roots of +\index{Negative roots}% +\index{Vieta!ref.\ to}% +equations. Indeed, we find few algebraists before and during +the Renaissance who understood the significance even of +negative quantities. Fibonacci seldom uses them. Pacioli +\index{Negative quantities}% +\index{Pacioli!ref.\ to}% +states the rule that ``minus times minus gives plus,'' but +applies it really only to the development of the product of +$(a - b) (c - d)$; purely negative quantities do not appear in +his work. The great German ``Cossist'' (algebraist), \textit{Michael +Stifel}, speaks as early as 1544 of numbers which are ``absurd'' +or ``fictitious below zero,'' and which arise when ``real numbers +above zero'' are subtracted from zero. Cardan, at last, speaks +of a ``pure minus''; ``but these ideas,'' says Hankel, ``remained +sparsely, and until the beginning of the seventeenth century, +mathematicians dealt exclusively with absolute positive quantities.'' +The first algebraist who occasionally places a purely +negative quantity by itself on one side of an equation, is +\textit{Harriot} in England. As regards the recognition of negative +\index{Harriot!ref.\ to}% +roots, Cardan and Bombelli were far in advance of all writers +\index{Bombelli!ref.\ to}% +of the Renaissance, including Vieta. Yet even they mentioned +these so-called false or fictitious roots only in passing, and +without grasping their real significance and importance. On +this subject Cardan and Bombelli had advanced to about the +same point as had the Hindoo Bhaskara, who saw negative roots, +\index{Bhaskara!ref.\ to}% +but did not approve of them. The generalisation of the conception +of quantity so as to include the negative, was an +exceedingly slow and difficult process in the development of +algebra. +\index{Algebra!Renaissance}% +%% -----File: 167.png---Folio 153------- +\index{Cubic equations}% +\index{Equations!solution of}% +\index{Trisection of angles}% + +We shall now consider the history of geometry during the +\index{Geometry!Renaissance}% +Renaissance. Unlike algebra, it made hardly any progress. +The greatest gain was a more intimate knowledge of Greek +geometry. No essential progress was made before the time of +Descartes. Regiomontanus, Xylander of Augsburg, Tartaglia, +\index{Regiomontanus!ref.\ to}% +\index{Tartaglia!ref.\ to}% +\index{Xylander}% +Commandinus of Urbino in Italy, Maurolycus, and others, +\index{Commandinus}% +\index{Maurolycus}% +made translations of geometrical works from the Greek. \textbf{John +Werner} of Nürnberg published in 1522 the first work on +\index{Werner}% +conics which appeared in Christian Europe. Unlike the +geometers of old, he studied the sections in relation with the +cone, and derived their properties directly from it. This mode +of studying the conics was followed by \textbf{Maurolycus} of Messina +(1494--1575). The latter is, doubtless, the greatest geometer +of the sixteenth century. From the notes of Pappus, he +\index{Pappus!ref.\ to}% +attempted to restore the missing fifth book of Apollonius +\index{Apollonius!ref.\ to}% +on \textit{maxima} and \textit{minima}. His chief work is his masterly +and original treatment of the conic sections, wherein he discusses +\index{Conic sections!Renaissance}% +tangents and asymptotes more fully than Apollonius +had done, and applies them to various physical and astronomical +problems. + +The foremost geometrician of Portugal was \textbf{Nonius}; of +\index{Nonius}% +France, before Vieta, was \textbf{Peter Ramus}, who perished in the +\index{Ramus}% +\index{Vieta!ref.\ to}% +massacre of St.~Bartholomew. \textit{Vieta} possessed great familiarity +with ancient geometry. The new form which he gave +to algebra, by representing general quantities by letters, enabled +him to point out more easily how the construction of +the roots of cubics depended upon the celebrated ancient problems +of the duplication of the cube and the trisection of an +\index{Duplication of the cube}% +angle. He reached the interesting conclusion that the former +problem includes the solutions of all cubics in which the radical +in Tartaglia's formula is real, but that the latter problem +includes only those leading to the irreducible case. + +The problem of the quadrature of the circle was revived in +%% -----File: 168.png---Folio 154------- +\index{Apollonius!ref.\ to}% +\index{Gregorian Calendar}% +this age, and was zealously studied even by men of eminence +and mathematical ability. The army of circle-squarers became +\index{Circle}% +most formidable during the seventeenth century. Among the +first to revive this problem was the German Cardinal \textbf{Nicolaus +Cusanus} (died 1464), who had the reputation of being a great +\index{Cusanus}% +logician. His fallacies were exposed to full view by Regiomontanus. +\index{Regiomontanus!ref.\ to}% +As in this case, so in others, every quadrator of +note raised up an opposing mathematician: Orontius was met +\index{Orontius}% +by Buteo and Nonius; Joseph Scaliger by Vieta, Adrianus +\index{Buteo}% +\index{Nonius!ref.\ to}% +\index{Scaliger}% +\index{Vieta!ref.\ to}% +Romanus, and Clavius; A. Quercu by Peter Metius. Two +\index{Clavius!ref.\ to}% +\index{Metius}% +\index{Quercu, a}% +\index{Romanus}% +\index{Romanus!ref.\ to}% +mathematicians of Netherlands, \textbf{Adrianus Romanus} and \textbf{Ludolph +\index{Ludolph}% +van Ceulen}, occupied themselves with approximating to the +ratio between the circumference and the diameter. The former +carried the value~$\pi$ to~$15$, the latter to~$35$, places. The +\index{p@{$\pi$}: values for!Ludolph's}% +value of~$\pi$ is therefore often named ``Ludolph's number.'' His +\index{Ludolph's number}% +performance was considered so extraordinary, that the numbers +were cut on his tomb-stone in St.~Peter's church-yard, at +Leyden. Romanus was the one who propounded for solution +that equation of the forty-fifth degree solved by Vieta. On +receiving Vieta's solution, he at once departed for Paris, to +make his acquaintance with so great a master. Vieta proposed +to him the Apollonian problem, to draw a circle touching +\index{Apollonian Problem}% +three given circles. ``Adrianus Romanus solved the problem +by the intersection of two hyperbolas; but this solution did not +possess the rigour of the ancient geometry. Vieta caused him +\index{Geometry!Renaissance}% +to see this, and then, in his turn, presented a solution which +had all the rigour desirable.''\Reference{25} Romanus did much toward +simplifying spherical trigonometry by reducing, by means of +\index{Trigonometry}% +certain projections, the $28$~cases in triangles then considered +to only six. + +Mention must here be made of the improvements of the +Julian calendar. The yearly determination of the movable +\index{Calendar}% +feasts had for a long time been connected with an untold +%% -----File: 169.png---Folio 155------- +amount of confusion. The rapid progress of astronomy led +to the consideration of this subject, and many new calendars +were proposed. Pope Gregory~XIII. convoked a large number +of mathematicians, astronomers, and prelates, who decided +upon the adoption of the calendar proposed by the Jesuit +\textbf{Lilius Clavius}. To rectify the errors of the Julian calendar +\index{Clavius}% +it was agreed to write in the new calendar the 15th~of +October immediately after the 4th~of October of the year +1582. The Gregorian calendar met with a great deal of opposition +both among scientists and among Protestants. Clavius, +who ranked high as a geometer, met the objections of the +former most ably and effectively; the prejudices of the latter +passed away with time. + +The passion for the study of mystical properties of numbers +descended from the ancients to the moderns. Much was +written on numerical mysticism even by such eminent men +as Pacioli and Stifel. The \textit{Numerorum Mysteria} of Peter +\index{Pacioli!ref.\ to}% +\index{Stifel!ref.\ to}% +Bungus covered $700$~quarto pages. He worked with great +\index{Bungus}% +industry and satisfaction on~$666$, which is the number of the +beast in Revelation (xiii.~18), the symbol of Antichrist. He +reduced the name of the `impious' Martin Luther to a form +which may express this formidable number. Placing $a = 1$, +$b = 2$,~etc, $k = 10$, $l = 20$,~etc., he finds, after misspelling the +name, that +$\text{M}_{(30)} \text{A}_{(1)} \text{R}_{(80)} \text{T}_{(100)} \text{I}_{(9)} \text{N}_{(40)}$ +$\text{L}_{(20)} \text{V}_{(200)} \text{T}_{(100)} \text{E}_{(5)} \text{R}_{(80)} \text{A}_{(1)}$ +constitutes the number required. These attacks on the great +reformer were not unprovoked, for his friend, Michael Stifel, +the most acute and original of the early mathematicians of +Germany, exercised an equal ingenuity in showing that the +above number referred to Pope Leo~X.,---a demonstration +which gave Stifel unspeakable comfort.\Reference{22} + +Astrology also was still a favourite study. It is well +\index{Astrology}% +known that Cardan, Maurolycus, Regiomontanus, and many +\index{Cardan!ref.\ to}% +\index{Maurolycus!ref.\ to}% +\index{Regiomontanus!ref.\ to}% +other eminent scientists who lived at a period even later than +%% -----File: 170.png---Folio 156------- +\index{Renaissance|)}% +this, engaged in deep astrological study; but it is not so generally +known that besides the occult sciences already named, +men engaged in the mystic study of star-polygons and magic +\index{Star-polygons}% +squares. ``The pentagramma gives you pain,'' says Faust to +Mephistopheles. It is of deep psychological interest to see +scientists, like the great Kepler, demonstrate on one page a +\index{Kepler!ref.\ to}% +theorem on star-polygons, with strict geometric rigour, while +on the next page, perhaps, he explains their use as amulets +or in conjurations.\Reference{1} Playfair, speaking of Cardan as an astrologer, +\index{Cardan!ref.\ to}% +\index{Playfair}% +calls him ``a melancholy proof that there is no folly +or weakness too great to be united to high intellectual attainments.''\Reference{26} +Let our judgment not be too harsh. The period +under consideration is too near the Middle Ages to admit of +complete emancipation from mysticism even among scientists. +Scholars like Kepler, Napier, Albrecht \Change{Duerer}{Dürer}, while in the +\index{Durer@{Dürer, A.}}% +\index{Napier, J.!ref.\ to}% +van of progress and planting one foot upon the firm ground +of truly scientific inquiry, were still resting with the other +foot upon the scholastic ideas of preceding ages. + + +\Section{Vieta to Descartes} + +The ecclesiastical power, which in the ignorant ages was an +unmixed benefit, in more enlightened ages became a serious +evil. Thus, in France, during the reigns preceding that of +Henry~IV., the theological spirit predominated. This is painfully +shown by the massacres of Vassy and of St.~Bartholomew. +Being engaged in religious disputes, people had no +leisure for science and for secular literature. Hence, down +to the time of Henry~IV., the French ``had not put forth a +single work, the destruction of which would now be a loss to +Europe.'' In England, on the other hand, no religious wars +were waged. The people were comparatively indifferent about +%% -----File: 171.png---Folio 157------- +\index{Menelaus!ref.\ to}% +religious strifes; they concentrated their ability upon secular +matters, and acquired, in the sixteenth century, a literature +which is immortalised by the genius of Shakespeare and +Spenser. This great literary age in England was followed +by a great scientific age. At the close of the sixteenth century, +the shackles of ecclesiastical authority were thrown off +by France. The ascension of Henry~IV. to the throne was +followed in 1598 by the Edict of Nantes, granting freedom +of worship to the Huguenots, and thereby terminating religious +wars. The genius of the French nation now began to blossom. +Cardinal Richelieu, during the reign of Louis~XIII., pursued +the broad policy of not favouring the opinions of any sect, but +of promoting the interests of the nation. His age was remarkable +for the progress of knowledge. It produced that +great secular literature, the counterpart of which was found +in England in the sixteenth century. The seventeenth century +was made illustrious also by the great French mathematicians, +Roberval, Descartes, Desargues, Fermat, and Pascal. + +More gloomy is the picture in Germany. The great changes +which revolutionised the world in the sixteenth century, and +which led England to national greatness, led Germany to +degradation. The first effects of the Reformation there were +salutary. At the close of the fifteenth and during the sixteenth +century, Germany had been conspicuous for her scientific +pursuits. She had been the leader in astronomy and +trigonometry. Algebra also, excepting for the discoveries in +cubic equations, was, before the time of Vieta, in a more +advanced state there than elsewhere. But at the beginning +of the seventeenth century, when the sun of science began to +rise in France, it set in Germany. Theologic disputes and +religious strife ensued. The Thirty Years' War (1618--1648) +proved ruinous. The German empire was shattered, and +became a mere lax confederation of petty despotisms. Commerce +%% -----File: 172.png---Folio 158------- +was destroyed; national feeling died out. Art disappeared, +and in literature there was only a slavish imitation +of French artificiality. Nor did Germany recover from this +low state for $200$~years; for in 1756 began another struggle, +the Seven Years' War, which turned Prussia into a wasted +land. Thus it followed that at the beginning of the seventeenth +century, the great Kepler was the only German mathematician +\index{Kepler!ref.\ to}% +of eminence, and that in the interval of $200$~years +between Kepler and Gauss, there arose no great mathematician +\index{Gauss!ref.\ to}% +in Germany excepting Leibniz. +\index{Leibniz!ref.\ to}% + +Up to the seventeenth century, mathematics was cultivated +but little in Great Britain. During the sixteenth century, she +brought forth no mathematician comparable with Vieta, Stifel, +or Tartaglia. But with the time of Recorde, the English +\index{Recorde!ref.\ to}% +became conspicuous for numerical skill. The first important +arithmetical work of English authorship was published in +Latin in 1522 by \textbf{Cuthbert Tonstall} (1474--1559). He had +\index{Tonstall}% +studied at Oxford, Cambridge, and Padua, and drew freely +from the works of Pacioli and Regiomontanus. Reprints of +\index{Pacioli!ref.\ to}% +\index{Regiomontanus!ref.\ to}% +his arithmetic appeared in England and France. After +\index{Arithmetic!Renaissance|(}% +Recorde the higher branches of mathematics began to be +studied. Later, Scotland brought forth Napier, the inventor +of logarithms. The instantaneous appreciation of their value +\index{Logarithms}% +is doubtless the result of superiority in calculation. In Italy, +and especially in France, geometry, which for a long time had +\index{Geometry!Renaissance}% +been an almost stationary science, began to be studied with +success. Galileo, Torricelli, Roberval, Fermat, Desargues, +Pascal, Descartes, and the English Wallis are the great revolutioners +of this science. Theoretical mechanics began to be +\index{Mechanics!Stevin and Galileo}% +studied. The foundations were laid by Fermat and Pascal +for the theory of numbers and the theory of probability. +\index{Probability}% + +We shall first consider the improvements made in the art +of calculating. The nations of antiquity experimented thousands +%% -----File: 173.png---Folio 159------- +\index{Arabic numerals and notation}% +of years upon numeral notations before they happened +to strike upon the so-called ``Arabic notation.'' In the simple +\index{Notation!Arabic notation}% +expedient of the cipher, which was introduced by the Hindoos +about the fifth or sixth century after Christ, mathematics received +one of the most powerful impulses. It would seem that +after the ``Arabic notation'' was once thoroughly understood, +decimal fractions would occur at once as an obvious extension +\index{Decimal fractions|(}% +\index{Fractions!decimal}% +\index{Fractions!continued}% +of it. But ``it is curious to think how much science had +attempted in physical research and how deeply numbers had +been pondered, before it was perceived that the all-powerful +simplicity of the `Arabic notation' was as valuable and as +manageable in an infinitely descending as in an infinitely +ascending progression.''\Reference{28} Simple as decimal fractions appear +to us, the invention of them is not the result of one mind or +even of one age. They came into use by almost imperceptible +degrees. The first mathematicians identified with their history +did not perceive their true nature and importance, and +failed to invent a suitable notation. The idea of decimal +fractions makes its first appearance in methods for approximating +to the square roots of numbers. Thus John of Seville, +\index{John of Seville}% +presumably in imitation of Hindoo rules, adds $2~n$~ciphers +to the number, then finds the square root, and takes this +\index{Square root}% +as the numerator of a fraction whose denominator is 1 followed +by $n$~ciphers. The same method was followed by +\textit{Cardan}, but it failed to be generally adopted even by his +\index{Cardan!ref.\ to}% +Italian contemporaries; for otherwise it would certainly have +been at least mentioned by \textit{Cataldi} (died 1626) in a work +\index{Cataldi}% +devoted exclusively to the extraction of roots. Cataldi finds +the square root by means of continued fractions---a method +\index{Continued fractions}% +ingenious and novel, but for practical purposes inferior to +Cardan's. \textbf{Orontius Finaeus} (died 1555) in France, and \textbf{William +\index{Finæus}% +Buckley} (died about 1550) in England extracted the +\index{Buckley}% +square root in the same way as Cardan and John of Seville. +%% -----File: 174.png---Folio 160------- +The invention of decimals is frequently attributed to Regiomontanus, +\index{Regiomontanus!ref.\ to}% +on the ground that instead of placing the sinus +totus, in trigonometry, equal to a multiple of~$60$, like the +\index{Trigonometry}% +Greeks, he put it $= 100,000$. But here the trigonometrical +lines were expressed in \emph{integers}, and not in fractions. Though +\index{Fractions!decimal}% +he adopted a decimal division of the radius, he and his successors +did not apply the idea outside of trigonometry and, +indeed, had no notion whatever of decimal \emph{fractions}. To +\textbf{Simon Stevin} of Bruges in Belgium (1548--1620), a man who +\index{Stevin}% +did a great deal of work in most diverse fields of science, we +owe the first systematic treatment of decimal fractions. In +his \textit{La Disme} (1585) he describes in very express terms the +advantages, not only of decimal fractions, but also of the +decimal division in systems of weights and measures. Stevin +applied the new fractions ``to all the operations of ordinary +arithmetic.''\Reference{25} What he lacked was a suitable notation. In +\index{Notation!in algebra}% +\index{Notation!decimal fractions}% +place of our decimal point, he used a cipher; to each place in +the fraction was attached the corresponding index. Thus, in +his notation, the number $5.912$ would be +$\overset{0}{5}\overset{1}{9}\overset{2}{1}\overset{3}{2}$ or +$5\CircleDigit{0}9\CircleDigit{1}1\CircleDigit{2}2\CircleDigit{3}$. +These indices, though cumbrous in practice, are of interest, +because they are the germ of an important innovation. To +Stevin belongs the honour of inventing our present mode of +designating powers and also of introducing fractional exponents +\index{Exponents}% +into algebra. Strictly speaking, this had been done much +earlier by \textit{Oresme}, but it remained wholly unnoticed. Not even +\index{Oresme!ref.\ to}% +Stevin's innovations were immediately appreciated or at once +accepted, but, unlike Oresme's, they remained a secure possession. +No improvement was made in the notation of decimals +till the beginning of the seventeenth century. After Stevin, +decimals were used by \textbf{Joost Bürgi}, a Swiss by birth, who prepared +\index{Burgi@{Bürgi}}% +a manuscript on arithmetic soon after 1592, and by +\textbf{Johann Hartmann Beyer}, who assumes the invention as his own. +\index{Beyer}% +In~1603, he published at Frankfurt on the Main a \textit{Logistica} +%% -----File: 175.png---Folio 161------- +\index{Arithmetic!Renaissance|)}% +\index{Trigonometry}% +\textit{Decimalis}. With Bürgi, a zero placed underneath the digit +in unit's place answers as sign of separation. Beyer's notation +resembles Stevin's. The decimal point, says Peacock, is +\index{Decimal point}% +\index{Peacock!ref.\ to, X}% +due to Napier, who in 1617 published his \textit{Rabdologia}, containing +\index{Napier, J.!ref.\ to}% +a treatise on decimals, wherein the decimal point is used +in one or two instances. In the English translation of Napier's +\textit{Mirifici logarithmorum canonis descriptio}, executed by Edward +Wright in 1616, and corrected by the author, the decimal +point occurs in the tables. There is no mention of decimals +in English arithmetics between 1619 and~1631. \textit{Oughtred} in +\index{Oughtred!ref.\ to}% +1631 designates the fraction $.56$ thus, $\olddecimal{0}{56}$. \textit{Albert Girard}, +\index{Girard!ref.\ to}% +a pupil of Stevin, in 1629 uses the point on one occasion. +John Wallis in 1657 writes \olddecimal{12}{345}, but afterwards in his +\index{Wallis!ref.\ to}% +algebra adopts the usual point. De Morgan says that ``to the +\index{Demorgan@{De Morgan}!ref.\ to}% +first quarter of the eighteenth century we must refer not only +the complete and final victory of the decimal point, but also +that of the now universal method of performing the operations +of division and extraction of the square root.\Reference{27} We have +dwelt at some length on the progress of the decimal notation, +because ``the history of language \ldots is of the highest order +of interest, as well as utility: its suggestions are the best +lesson for the future which a reflecting mind can have.''\Reference{27} + +The miraculous powers of modern calculation are due to +three inventions: the Arabic Notation, Decimal Fractions, and +\index{Decimal fractions|)}% +Logarithms. The invention of logarithms in the first quarter +\index{Logarithms|(}% +of the seventeenth century was admirably timed, for Kepler +\index{Kepler!ref.\ to}% +was then examining planetary orbits, and Galileo had just +\index{Galileo!ref.\ to}% +turned the telescope to the stars. During the Renaissance +German mathematicians had constructed trigonometrical +tables of great accuracy, but this greater precision enormously +increased the work of the calculator. It is no exaggeration +to say that the invention of logarithms ``by shortening the +labours doubled the life of the astronomer.'' Logarithms were +%% -----File: 176.png---Folio 162------- +\index{Euclid!ref.\ to}% +invented by \textbf{John Napier}, Baron of Merchiston, in Scotland +\index{Napier, J.}% +(1550--1617). It is one of the greatest curiosities of the +history of science that Napier constructed logarithms before +exponents were used. To be sure, Stifel and Stevin made +\index{Exponents}% +\index{Stevin!ref.\ to}% +\index{Stifel!ref.\ to}% +some attempts to denote powers by indices, but this notation +was not generally known,---not even to \textit{Harriot}, whose algebra +\index{Harriot!ref.\ to}% +appeared long after Napier's death. That logarithms flow +naturally from the exponential symbol was not observed until +much later. It was Euler who first considered logarithms as +being indices of powers. What, then, was Napier's line of +thought? + +Let $AB$ be a definite line, $DE$ a line extending from $D$ indefinitely. +Imagine two points starting at the same %[** PP: Width-dependent line break] +%[**Illustration: ] +\begin{wrapfigure}{l}{2.5in} +\Graphic{2.5in}{176a} +\end{wrapfigure} +moment; +the one moving from +$A$ toward~$B$, the other +from $D$ toward~$E$. Let +the velocity during the +first moment be the +same for both: let that of the point on line~$DE$ be uniform; +but the velocity of the point on $AB$ decreasing in such a way +that when it arrives at any point~$C$, its velocity is proportional +to the remaining distance~$BC$. While the first point moves +over a distance~$AC$, the second one moves over a distance~$DF$. +Napier calls $DF$ the logarithm of~$BC$. + +Napier's process is so unique and so different from all other +modes of presenting the subject that there cannot be the +shadow of a doubt that this invention is entirely his own; it +is the result of unaided, isolated speculation. He first sought +the logarithms only of sines; the line~$AB$ was the sine of~$90°$ +and was taken $= 10^7$; $BC$ was the sine of the arc, and $DF$ its +logarithm. We notice that as the motion proceeds, $BC$ decreases +in geometrical progression, while $DF$ increases in +arithmetical progression. Let $AB = a = 10^7$, let $x = DF$, +%% -----File: 177.png---Folio 163------- +$y = BC$, then $AC = a - y$. The velocity of the point~$C$ is +$\dfrac{d(a - y)}{dt} = y$; this gives $-\natlog y = t + c$. When $t = 0$, +then $y = a$ and $c = -\natlog a$. Again, let $\smash[t]{\dfrac{dx}{dt}} = a$ be the +velocity of the point~$F$, then $x = at$. Substituting for $t$~and~$c$ +their values and remembering that $a = 10^7$ and that by definition +$x = \Naplog y$, we get +\[ +\Naplog y = 10^7 \natlog \frac{10^7}{y}. +\] + +It is evident from this formula that Napier's logarithms are +\index{Napier, J.}% +not the same as the natural logarithms. Napier's logarithms +increase as the number itself decreases. He took the logarithm +of $\sin 90 = 0$; \ie~the logarithm of $10^7 = 0$. The logarithm +of $\sin \alpha$ increased from zero as $\alpha$ decreased from~$90°$. +Napier's genesis of logarithms from the conception of two +flowing points reminds us of Newton's doctrine of fluxions. +The relation between geometric and arithmetical progressions, +so skilfully utilised by Napier, had been observed by Archimedes, +Stifel, and others. Napier did not determine the base +to his system of logarithms. The notion of a ``base'' in fact +never suggested itself to him. The one demanded by his +reasoning is the reciprocal of that of the natural system, but +such a base would not reproduce accurately all of Napier's +figures, owing to slight inaccuracies in the calculation of the +tables. Napier's great invention was given to the world in +1614 in a work entitled \textit{Mirifici logarithmorum canonis descriptio}. +In it he explained the nature of his logarithms, and +gave a logarithmic table of the natural sines of a quadrant +from minute to minute. + +\textbf{Henry Briggs} (1556--1631), in Napier's time professor of +\index{Briggs}% +geometry at Gresham College, London, and afterwards +professor at Oxford, was so struck with admiration of +Napier's book, that he left his studies in London to do +%% -----File: 178.png---Folio 164------- +homage to the Scottish philosopher. Briggs was delayed in +his journey, and Napier complained to a common friend, ``Ah, +\index{Napier, J.!ref.\ to}% +John, Mr. Briggs will not come.'' At that very moment +knocks were heard at the gate, and Briggs was brought into +the lord's chamber. Almost one-quarter of an hour was +spent, each beholding the other without speaking a word. At +last Briggs began: ``My lord, I have undertaken this long +journey purposely to see your person, and to know by what +engine of wit or ingenuity you came first to think of this most +excellent help in astronomy, viz.~the logarithms; but, my +lord, being by you found out, I wonder nobody found it out +before, when now known it is so easy.''\Reference{28} Briggs suggested +to Napier the advantage that would result from retaining zero +for the logarithm of the whole sine, but choosing $10,000,000,000$ +for the logarithm of the $10$th part of that same sine, \ie~of +$5°\ 44' 22''$. Napier said that he had already thought of the +change, and he pointed out a slight improvement on Briggs' +idea; viz.~that zero should be the logarithm of~$1$, and +$10,000,000,000$ that of the whole sine, thereby making the +characteristic of numbers greater than unity positive and +not negative, as suggested by Briggs. Briggs admitted this +to be more convenient. The invention of ``Briggian logarithms'' +occurred, therefore, to Briggs and Napier independently. +The great practical advantage of the new system +was that its fundamental progression was accommodated to +the base, $10$, of our numerical scale. Briggs devoted all +his energies to the construction of tables upon the new plan. +Napier died in 1617, with the satisfaction of having found +in Briggs an able friend to bring to completion his unfinished +plans. In~1624 Briggs published his \textit{Arithmetica logarithmica}, +containing the logarithms to $14$~places of numbers, +from $1$ to~$20,000$ and from $90,000$ to~$100,000$. The gap from +$20,000$ to~$90,000$ was filled up by that illustrious successor of +%% -----File: 179.png---Folio 165------- +\index{Trigonometry}% +Napier and Briggs, \textbf{Adrian Vlacq} of Gouda in Holland. He +\index{Napier, J.!ref.\ to}% +\index{Vlacq}% +published in 1628 a table of logarithms from 1 to~$100,000$, of +which $70,000$ were calculated by himself. The first publication +of Briggian logarithms of trigonometric functions was made +in 1620 by \textbf{Gunter}, a colleague of Briggs, who found the logarithmic +\index{Gunter, E.}% +sines and tangents for every minute to seven places. +Gunter was the inventor of the words \emph{cosine} and \emph{cotangent}. +\index{Cosine}% +\index{Cotangent}% +Briggs devoted the last years of his life to calculating more +extensive Briggian logarithms of trigonometric functions, but +he died in~1631, leaving his work unfinished. It was carried +on by the English \textbf{Henry Gellibrand}, and then published by +\index{Gellibrand}% +Vlacq at his own expense. Briggs divided a degree into $100$~parts, +but owing to the publication by Vlacq of trigonometrical +tables constructed on the old sexagesimal division, Briggs' +innovation remained unrecognised. Briggs and Vlacq published +four fundamental works, the results of which ``have never been +superseded by any subsequent calculations.'' + +The first logarithms upon the \emph{natural} base $e$ were published +by \textbf{John Speidell} in his \textit{New Logarithmes} (London, 1619), which +\index{Speidell}% +contains the natural logarithms of sines, tangents, and secants. + +The only possible rival of John Napier in the invention of +logarithms was the Swiss \textbf{Justus Byrgius} (Joost Bürgi). He +\index{Burgi@{Bürgi}!ref.\ to}% +published a rude table of logarithms six years after the +appearance of the \textit{Canon Mirificus}, but it appears that he +conceived the idea and constructed that table as early, if not +earlier, than Napier did his. But he neglected to have the +results published until Napier's logarithms were known and +\index{Logarithms|)}% +admired throughout Europe. + +Among the various inventions of Napier to assist the +memory of the student or calculator, is ``Napier's rule of +circular parts'' for the solution of spherical right triangles. +\index{Napier's rule of circular parts}% +It is, perhaps, ``the happiest example of artificial memory +that is known.'' +%% -----File: 180.png---Folio 166------- + +The most brilliant conquest in algebra during the sixteenth +\index{Algebra!seventeenth century}% +century had been the solution of cubic and \Change{bi-quadratic}{biquadratic} equations. +\index{Equations!theory of}% +All attempts at solving algebraically equations of higher +degrees remaining fruitless, a new line of inquiry---the properties +of equations and their roots---was gradually opened up. +We have seen that Vieta had attained a partial knowledge of +the relations between roots and coefficients. \textbf{Peletarius}, a +\index{Peletarius}% +Frenchman, had observed as early as~1558, that the root of an +equation is a divisor of the last term. One who extended the +theory of equations somewhat further than Vieta, was \textbf{Albert +Girard} (1590--1634), a Flemish mathematician. Like Vieta, +\index{Girard}% +this ingenious author applied algebra to geometry, and was +the first who understood the use of negative roots in the +\index{Negative roots}% +solution of geometric problems. He spoke of imaginary quantities; +\index{Imaginary quantities}% +inferred by induction that every equation has as many +roots as there are units in the number expressing its degree; +and first showed how to express the sums of their powers in +terms of the coefficients. Another algebraist of considerable +power was the English \textbf{Thomas Harriot} (1560--1621). He +\index{Harriot}% +accompanied the first colony sent out by Sir Walter Raleigh +to Virginia. After having surveyed that country he returned +to England. As a mathematician, he was the boast of his +country. He brought the theory of equations under one +comprehensive point of view by grasping that truth in its +full extent to which Vieta and Girard only approximated; viz.~that +in an equation in its simplest form, the coefficient of +the second term with its sign changed is equal to the sum of +the roots; the coefficient of the third is equal to the sum +of the products of every two of the roots; etc. He was the +first to decompose equations into their simple factors; but, +since he failed to recognise imaginary and even negative roots, +he failed also to prove that every equation could be thus +decomposed. Harriot made some changes in algebraic notation, +%% -----File: 181.png---Folio 167------- +\index{Notation!in algebra}% +adopting small letters of the alphabet in place of the +capitals used by Vieta. The symbols of inequality $>$ and $<$ +\index{Vieta!ref.\ to}% +were introduced by him. Harriot's work, \textit{Artis Analyticæ +praxis}, was published in 1631, ten years after his death. +\textbf{William Oughtred} (1574--1660) contributed vastly to the propagation +\index{Oughtred}% +of mathematical knowledge in England by his treatises, +which were long used in the universities. He introduced $×$ +as symbol of multiplication, and $::$ as that of proportion. By +him ratio was expressed by only one dot. In the eighteenth +century \textit{Christian Wolf} secured the general adoption of the +\index{Wolf, C.!ref.\ to}% +dot as a symbol of multiplication, and the sign for ratio was +thereupon changed to two dots. Oughtred's ministerial duties +left him but little time for the pursuit of mathematics during +daytime, and evenings his economical wife denied him the +use of a light. + +Algebra was now in a state of sufficient perfection to enable +Descartes to take that important step which forms one of the +\index{Descartes!ref.\ to}% +grand epochs in the history of mathematics,---the application +of algebraic analysis to define the nature and investigate the +properties of algebraic curves. + +In geometry, the determination of the areas of curvilinear +\index{Geometry!Renaissance}% +figures was diligently studied at this period. \textbf{Paul Guldin} +\index{Guldin}% +(1577--1643), a Swiss mathematician of considerable note, +\Change{rediscovered}{re-discovered} the following theorem, published in his \textit{Centrobaryca}, +which has been named after him, though first found +in the \textit{Mathematical Collections} of Pappus: The volume of a +solid of revolution is equal to the area of the generating +figure, multiplied by the circumference described by the centre +of gravity. We shall see that this method excels that of +Kepler and Cavalieri in following a more exact and natural +\index{Cavalieri!ref.\ to}% +\index{Kepler!ref.\ to}% +course; but it has the disadvantage of necessitating the determination +of the centre of gravity, which in itself may be a +more difficult problem than the original one of finding the +%% -----File: 182.png---Folio 168------- +\index{Stereometry}% +volume. Guldin made some attempts to prove his theorem, +but Cavalieri pointed out the weakness of his demonstration. + +\textbf{Johannes Kepler} (1571--1630) was a native of Würtemberg +\index{Kepler|(}% +and imbibed Copernican principles while at the University of +Tübingen. His pursuit of science was repeatedly interrupted +by war, religious persecution, pecuniary embarrassments, frequent +changes of residence, and family troubles. In~1600 he +became for one year assistant to the Danish astronomer, Tycho +Brahe, in the observatory near Prague. The relation between +\index{Brahe, Tycho}% +the two great astronomers was not always of an agreeable +character. Kepler's publications are voluminous. His first +attempt to explain the solar system was made in~1596, when he +thought he had discovered a curious relation between the +five regular solids and the number and distance of the planets. +\index{Regular solids}% +The publication of this pseudo-discovery brought him much +fame. Maturer reflection and intercourse with Tycho Brahe +\index{Tycho Brahe}% +and Galileo led him to investigations and results more worthy of +\index{Galileo!ref.\ to}% +his genius---``Kepler's laws.'' He enriched pure mathematics +\index{Kepler's laws}% +as well as astronomy. It is not strange that he was interested +in the mathematical science which had done him so much +service; for ``if the Greeks had not cultivated conic sections, +\index{Conic sections!Kepler}% +Kepler could not have superseded Ptolemy.''\Reference{11} The Greeks +never dreamed that these curves would ever be of practical +use; Aristæus and Apollonius studied them merely to satisfy +their intellectual cravings after the ideal; yet the conic +sections assisted Kepler in tracing the march of the planets in +their elliptic orbits. Kepler made also extended use of logarithms +\index{Logarithms}% +and decimal fractions, and was enthusiastic in diffusing +a knowledge of them. At one time, while purchasing wine, he +was struck by the inaccuracy of the ordinary modes of determining +the contents of kegs. This led him to the study of +the volumes of solids of revolution and to the publication of +the \textit{Stereometria Doliorum} in~1615. In it he deals first with the +%% -----File: 183.png---Folio 169------- +\index{Definite integrals}% +\index{Infinitesimals}% +\index{Method of exhaustion!ref.\ to}% +\index{Rectification of curves}% +solids known to Archimedes and then takes up others. Kepler +\index{Archimedes!ref.\ to}% +introduced a new idea into geometry; namely, that of infinitely +great and infinitely small quantities. Greek mathematicians +always shunned this notion, but with it modern mathematicians +have completely revolutionised the science. In comparing +rectilinear figures, the method of superposition was employed +by the ancients, but in comparing rectilinear and curvilinear +figures with each other, this method failed because no addition +or subtraction of rectilinear figures could ever produce curvilinear +ones. To meet this case, they devised the Method of +Exhaustion, which was long and difficult; it was purely +\index{Exhaustion, method of}% +synthetical, and in general required that the conclusion should +be known at the outset. The new notion of infinity led +\index{Infinity}% +gradually to the invention of methods immeasurably more +powerful. Kepler conceived the circle to be composed of an +infinite number of triangles having their common vertices at +the centre, and their bases in the circumference; and the +sphere to consist of an infinite number of pyramids. He +applied conceptions of this kind to the determination of the +areas and volumes of figures generated by curves revolving +about any line as axis, but succeeded in solving only a few of +the simplest out of the 84~problems which he proposed for +investigation in his \textit{Stereometria}. + +Other points of mathematical interest in Kepler's works are +(1)~the statement of the earliest problem of inverse tangents; +\index{Inverse tangents (problem of)}% +\index{Tangents!inverse problem of}% +(2)~an investigation which amounts to the evaluation of +the definite integral $\displaystyle \int_{0}^{\phi} \sin \phi\, d\phi = 1 - \cos \phi$; (3)~the assertion +that the circumference of an ellipse, whose axes are $2a$ and +$2b$, is nearly $\pi (a + b)$; (4)~a passage from which it has been +inferred that Kepler knew the variation of a function near its +maximum value to disappear; (5)~the assumption of the principle +of continuity (which differentiates modern from ancient +\index{Continuity}% +geometry), when he shows that a parabola has a focus at +\index{Focus}% +%% -----File: 184.png---Folio 170------- +\index{Kepler|)}% +infinity, that lines radiating from this ``cæcus focus'' are +\index{Focus}% +parallel and have no other point at infinity. + +The \textit{Stereometria} led Cavalieri, an Italian Jesuit, to the +\index{Cavalieri}% +consideration of infinitely small quantities. \textbf{Bonaventura +Cavalieri} (1598--1647), a pupil of Galileo and professor at +\index{Galileo!ref.\ to}% +Bologna, is celebrated for his \textit{Geometria indivisibilibus continuorum +nova quadam ratione promota}, 1635. This work +expounds his method of Indivisibles, which occupies an intermediate +\index{Indivisibles|(}% +place between the method of exhaustion of the Greeks +and the methods of Newton and Leibniz. He considers lines +as composed of an infinite number of points, surfaces as composed +of an infinite number of lines, and solids of an infinite +number of planes. The relative magnitude of two solids or +surfaces could then be found simply by the summation of +series of planes or lines. For example, he finds the sum of the +squares of all lines making up a triangle equal to one-third +the sum of the squares of all lines of a parallelogram of equal +base and altitude; for if in a triangle, the first line at the apex +be~1, then the second is~2, the third is~3, and so on; and the +sum of their squares is +\[ +1^2 + 2^2 + 3^2 + \dotsb + n^2 = n(n + 1)(2n + 1) ÷ 6. +\] +In the parallelogram, each of the lines is $n$ and their number is +$n$; hence the total sum of their squares is $n^3$. The ratio +between the two sums is therefore +\[ +n(n + 1)(2n + 1) ÷ 6n^3 = \tfrac{1}{3}, +\] +since $n$ is infinite. From this he concludes that the pyramid +or cone is respectively $\frac{1}{3}$ of a prism or cylinder of equal base +and altitude, since the polygons or circles composing the former +decrease from the base to the apex in the same way as the +squares of the lines parallel to the base in a triangle decrease +from base to apex. By the Method of Indivisibles, Cavalieri +%% -----File: 185.png---Folio 171------- +solved the majority of the problems proposed by Kepler. +\index{Kepler!ref.\ to}% +Though expeditious and yielding correct results, Cavalieri's +method lacks a scientific foundation. If a line has absolutely +no width, then no number, however great, of lines can ever +make up an area; if a plane has no thickness whatever, then +even an infinite number of planes cannot form a solid. The +reason why this method led to correct conclusions is that one +area is to another area in the same ratio as the sum of the +series of lines in the one is to the sum of the series of lines in +the other. Though unscientific, Cavalieri's method was used +for fifty years as a sort of integral calculus. It yielded solutions +\index{Integral calculus}% +to some difficult problems. Guldin made a severe attack +\index{Guldin!ref.\ to}% +on Cavalieri and his method. The latter published in~1647, +after the death of Guldin, a treatise entitled \textit{Exercitationes +geometricæ sex}, in which he replied to the objections of his +opponent and attempted to give a clearer explanation of his +method. Guldin had never been able to demonstrate the +theorem named after him, except by metaphysical reasoning, +but Cavalieri proved it by the method of indivisibles. A +revised edition of the \textit{Geometry of Indivisibles} appeared in~1653. + +There is an important curve, not known to the ancients, +which now began to be studied with great zeal. Roberval +gave it the name of ``trochoid,'' Pascal the name of ``roulette,'' +\index{Roulette}% +\index{Trochoid}% +Galileo the name of ``cycloid.'' The invention of this curve +\index{Cycloid}% +\index{Galileo!ref.\ to}% +seems to be due to Galileo, who valued it for the graceful form +it would give to arches in architecture. He ascertained its +area by weighing paper figures of the cycloid against that of +the generating circle, and found thereby the first area to be +nearly but not exactly thrice the latter. A mathematical +determination was made by his pupil, \textbf{Evangelista Torricelli} +\index{Torricelli}% +(1608--1647), who is more widely known as a physicist than +as a mathematician. +%% -----File: 186.png---Folio 172------- + +By the Method of Indivisibles he demonstrated its area to +be triple that of the revolving circle, and published his +solution. This same quadrature had been effected a few years +earlier by Roberval in France, but his solution was not known +\index{Roberval}% +\index{Roberval!ref.\ to}% +to the Italians. Roberval, being a man of irritable and +violent disposition, unjustly accused the mild and amiable +Torricelli of stealing the proof. This accusation of plagiarism +created so much chagrin with Torricelli that it is considered +to have been the cause of his early death. \textbf{Vincenzo Viviani}, +\index{Viviani}% +another prominent pupil of Galileo, determined the tangent +to the cycloid. This was accomplished in France by Descartes +and Fermat. +\index{Fermat!ref.\ to}% + +In France, where geometry began to be cultivated with +greatest success, Roberval, Fermat, Pascal, employed the +Method of Indivisibles and made new improvements in it. +\textbf{Giles Persone de Roberval} (1602--1675), for forty years professor +of mathematics at the College of France in Paris, +claimed for himself the invention of the Method of Indivisibles. +Since his complete works were not published until +after his death, it is difficult to settle questions of priority. +Montucla and Chasles are of the opinion that he invented the +\index{Chasles!ref.\ to}% +\index{Montucla}% +method independent of and earlier than the Italian geometer, +though the work of the latter was published much earlier +than Roberval's. Marie finds it difficult to believe that the +\index{Marie, M.}% +Frenchman borrowed nothing whatever from the Italian, for +both could not have hit independently upon the word \textit{Indivisibles}, +which is applicable to infinitely small quantities, as +conceived by Cavalieri, but not as conceived by Roberval. +Roberval and Pascal improved the rational basis of the +Method of Indivisibles, by considering an area as made up +of an indefinite number of rectangles instead of lines, and +a solid as composed of indefinitely small solids instead of +surfaces. Roberval applied the method to the finding of +%% -----File: 187.png---Folio 173------- +\index{Indivisibles|)}% +areas, volumes, and centres of gravity. He effected the quadrature +of a parabola of any degree $y^m = a^{m-1} x$, and also of a +parabola $y^m = a^{m-n} x^n$. We have already mentioned his quadrature +of the cycloid. Roberval is best known for his method +\index{Cycloid}% +of drawing tangents. He was the first to apply motion to +\index{Tangents!in geometry}% +the resolution of this important problem. His method is +allied to Newton's principle of fluxions. Archimedes conceived +\index{Archimedes!ref.\ to}% +\index{Newton!ref.\ to}% +his spiral to be generated by a double motion. This +idea Roberval extended to all curves. Plane curves, as for +instance the conic sections, may be generated by a point +acted upon by two forces, and are the resultant of two +motions. If at any point of the curve the resultant be +resolved into its components, then the diagonal of the parallelogram +determined by them is the tangent to the curve +at that point. The greatest difficulty connected with this +ingenious method consisted in resolving the resultant into +components having the proper lengths and directions. Roberval +did not always succeed in doing this, yet his new idea was +a great step in advance. He broke off from the ancient +definition of a tangent as a straight line having only one point +in common with a curve,---a definition not valid for curves of +higher degrees, nor apt even in curves of the second degree to +bring out the properties of tangents and the parts they may +be made to play in the generation of the curves. The subject +of tangents received special attention also from Fermat, +\index{Fermat}% +\index{Fermat!ref.\ to}% +Descartes, and Barrow, and reached its highest development +\index{Barrow!ref.\ to}% +\index{Descartes!ref.\ to}% +after the invention of the differential calculus. Fermat and +Descartes defined tangents as secants whose two points of +intersection with the curve coincide; Barrow considered a +curve a polygon, and called one of its sides produced a tangent. + +A profound scholar in all branches of learning and a mathematician +of exceptional powers was \textbf{Pierre de Fermat} (1601--1665). +He studied law at Toulouse, and in 1631 was made +%% -----File: 188.png---Folio 174------- +\index{Midorge}% +councillor for the parliament of Toulouse. His leisure time +was mostly devoted to mathematics, which he studied with +irresistible passion. Unlike Descartes and Pascal, he led a +\index{Descartes!ref.\ to}% +quiet and unaggressive life. Fermat has left the impress of +his genius upon all branches of mathematics then known. A +great contribution to geometry was his \textit{De maximis et minimis}. +About twenty years earlier, Kepler had first observed that the +\index{Kepler!ref.\ to}% +increment of a variable, as, for instance, the ordinate of a +curve, is evanescent for values very near a maximum or a +minimum value of the variable. Developing this idea, Fermat +obtained his rule for maxima and minima. He substituted +\index{Maxima and minima}% +$x + e$ for $x$ in the given function of $x$ and then equated to each +other the two consecutive values of the function and divided +the equation by~$e$. If $e$ be taken~0, then the roots of this +equation are the values of~$x$, making the function a maximum +or a minimum. Fermat was in possession of this rule in~1629. +The main difference between it and the rule of the differential +\index{Differential calculus!alleged invention by Pascal}% +calculus is that it introduces the indefinite quantity~$e$ instead +of the infinitely small~$dx$. Fermat made it the basis for his +method of drawing tangents. + +Owing to a want of explicitness in statement, Fermat's +method of maxima and minima, and of tangents, was severely +attacked by his great contemporary, Descartes, who could +never be brought to render due justice to his merit. In the +ensuing dispute, Fermat found two zealous defenders in Roberval +and Pascal, the father; while Mydorge, Desargues, and +\index{Desargues!ref.\ to}% +Hardy supported Descartes. +\index{Hardy}% + +Since Fermat introduced the conception of infinitely small +differences between consecutive values of a function and arrived +at the principle for finding the maxima and minima, +it was maintained by Lagrange, Laplace, and Fourier, that +\index{Fourier!ref.\ to}% +\index{Lagrange!ref.\ to}% +\index{Laplace!ref.\ to}% +Fermat may be regarded as the first inventor of the differential +calculus. This point is not well taken, as will be seen +%% -----File: 189.png---Folio 175------- +from the words of Poisson, himself a Frenchman, who rightly +\index{Poisson!ref.\ to}% +says that the differential calculus ``consists in a system of +rules proper for finding the differentials of all functions, +rather than in the use which may be made of these infinitely +small variations in the solution of one or two isolated problems.'' + +A contemporary mathematician, whose genius excelled even +that of the great Fermat, was \textbf{Blaise Pascal} (1623--1662). He +\index{Pascal|(}% +was born at Clermont in Auvergne. In~1626 his father retired +to Paris, where he devoted himself to teaching his son, for he +would not trust his education to others. Blaise Pascal's genius +for geometry showed itself when he was but twelve years +old. His father was well skilled in mathematics, but did not +wish his son to study it until he was perfectly acquainted with +Latin and Greek. All mathematical books were hidden out of +his sight. The boy once asked his father what mathematics +treated of, and was answered, in general, ``that it was the +method of making figures with exactness, and of finding out +what proportions they relatively had to one another.'' He +was at the same time forbidden to talk any more about it, +or ever to think of it. But his genius could not submit to be +confined within these bounds. Starting with the bare fact +that mathematics taught the means of making figures infallibly +exact, he employed his thoughts about it and with a piece +of charcoal drew figures upon the tiles of the pavement, trying +the methods of drawing, for example, an exact circle or equilateral +triangle. He gave names of his own to these figures +and then formed axioms, and, in short, came to make perfect +demonstrations. In this way he arrived unaided at the theorem +that the sum of the three angles of a triangle is equal to +two right angles. His father caught him in the act of studying +this theorem, and was so astonished at the sublimity and +force of his genius as to weep for joy. The father now gave +%% -----File: 190.png---Folio 176------- +\index{Conic sections!more recent researches|(}% +him Euclid's \textit{Elements}, which he, without assistance, mastered +easily. His regular studies being languages, the boy employed +only his hours of amusement on the study of geometry, yet he +had so ready and lively a penetration that, at the age of sixteen, +he wrote a treatise upon conics, which passed for such a +surprising effort of genius, that it was said nothing equal to it +in strength had been produced since the time of Archimedes. +Descartes refused to believe that it was written by one so +young as Pascal. This treatise was never published, and is +now lost. Leibniz saw it in Paris and reported on a portion +\index{Leibniz!ref.\ to}% +of its contents. The precocious youth made vast progress in +all the sciences, but the constant application at so tender an +age greatly impaired his health. Yet he continued working, +and at nineteen invented his famous machine for performing +arithmetical operations mechanically. This continued strain +from overwork resulted in a permanent indisposition, and he +would sometimes say that from the time he was eighteen, +he never passed a day free from pain. At the age of +twenty-four he resolved to lay aside the study of the human +sciences and to consecrate his talents to religion. His Provincial +Letters against the Jesuits are celebrated. But at +times he returned to the favourite study of his youth. Being +kept awake one night by a toothache, some thoughts undesignedly +came into his head concerning the roulette or +cycloid; one idea followed another; and he thus discovered +\index{Cycloid}% +properties of this curve even to demonstration. A correspondence +between him and Fermat on certain problems was +the beginning of the theory of probability. Pascal's illness +increased, and he died at Paris at the early age of thirty-nine +years.\Reference{30} By him the answer to the objection to Cavalieri's +Method of Indivisibles was put in the clearest form. Like +\index{Indivisibles}% +Roberval, he explained ``the sum of right lines'' to mean ``the +sum of infinitely small rectangles.'' Pascal greatly advanced +%% -----File: 191.png---Folio 177------- +\index{Centre!of gravity}% +\index{Curves!quadrature of}% +\index{Involution of points}% +\index{Quadrature of curves}% +the knowledge of the cycloid. He determined the area of a +section produced by any line parallel to the base; the volume +generated by it revolving around its base or around the axis; +and, finally, the centres of gravity of these volumes, and also +of half these volumes cut by planes of symmetry. Before +publishing his results, he sent, in~1658, to all mathematicians +that famous challenge offering prizes for the first two solutions +of these problems. Only Wallis and A.~La~Louère competed +\index{Lalouere@{La Louère}}% +\index{Wallis!ref.\ to}% +for them. The latter was quite unequal to the task; +the former, being pressed for time, made numerous mistakes: +neither got a prize. Pascal then published his own solutions, +which produced a great sensation among scientific men. Wallis, +too, published his, with the errors corrected. Though not +competing for the prizes, Huygens, Wren, and Fermat solved +\index{Fermat!ref.\ to}% +\index{Huygens!ref.\ to}% +\index{Wren}% +some of the questions. The chief discoveries of \textbf{Christopher +Wren} (1632--1723), the celebrated architect of St.~Paul's +Cathedral in London, were the rectification of a cycloidal arc +\index{Rectification of curves}% +and the determination of its centre of gravity. Fermat +found the area generated by an arc of the cycloid. Huygens +invented the cycloidal pendulum. + +The beginning of the seventeenth century witnessed also +a revival of synthetic geometry. One who treated conics still +by ancient methods, but who succeeded in greatly simplifying +many prolix proofs of Apollonius, was \textbf{Claude Mydorge} in +\index{Mydorge}% +Paris (1585--1647), a friend of Descartes. But it remained +for \textbf{Girard Desargues} (1593--1662) of Lyons, and for Pascal, to +\index{Desargues}% +leave the beaten track and cut out fresh paths. They introduced +the important method of Perspective. All conics on +\index{Perspective}% +a cone with circular base appear circular to an eye at the apex. +Hence Desargues and Pascal conceived the treatment of the +\index{Pascal|)}% +conic sections as projections of circles. Two important and +beautiful theorems were given by Desargues: The one is on +the ``involution of the six points,'' in which a transversal +%% -----File: 192.png---Folio 178------- +\index{Brianchion}% +\index{Conic sections!more recent researches|)}% +\index{Hexagrammum mysticum}% +\index{Homological figures}% +meets a conic and an inscribed quadrangle; the other is that, +if the vertices of two triangles, situated either in space or in +a plane, lie on three lines meeting in a point, then their sides +meet in three points lying on a line; and conversely. This +last theorem has been employed in recent times by Brianchon, +Sturm, Gergonne, and Poncelet. Poncelet made it the basis +\index{Gergonne!ref.\ to}% +\index{Poncelet!ref.\ to}% +\index{Sturm, J.~C.~F.!ref.\ to}% +of his beautiful theory of homoligical figures. We owe to +Desargues the theory of involution and of transversals; also +the beautiful conception that the two extremities of a straight +line may be considered as meeting at infinity, and that parallels +\index{Infinity}% +differ from other pairs of lines only in having their points +of intersection at infinity. Pascal greatly admired Desargues' +\index{Pascal!ref.\ to}% +results, saying (in his \textit{Essais pour les Coniques}), ``I wish to +acknowledge that I owe the little that I have discovered on +this subject, to his writings.'' Pascal's and Desargues' writings +contained the fundamental ideas of modern synthetic +geometry. In Pascal's wonderful work on conics, written +at the age of sixteen and now lost, were given the theorem +on the anharmonic ratio, first found in Pappus, and also that +\index{Anharmonic ratio}% +\index{Pappus!ref.\ to}% +celebrated proposition on the mystic hexagon, known as +``Pascal's theorem,'' viz.~that the opposite sides of a hexagon +\index{Pascal's theorem}% +inscribed in a conic intersect in three points which are collinear. +This theorem formed the keystone to his theory. He +himself said that from this alone he deduced over 400~corollaries, +embracing the conics of Apollonius and many other +results. Thus the genius of Desargues and Pascal uncovered +several of the rich treasures of modern synthetic geometry; +but owing to the absorbing interest taken in the analytical +geometry of Descartes and later in the differential calculus, +the subject was almost entirely neglected until the present +century. + +In the theory of numbers no new results of scientific value +\index{Numbers!theory of numbers|(}% +\index{Theory of numbers|(}% +had been reached for over 1000~years, extending from the +%% -----File: 193.png---Folio 179------- +times of Diophantus and the Hindoos until the beginning of +\index{Diophantus!ref.\ to}% +the seventeenth century. But the illustrious period we are +now considering produced men who rescued this science from +the realm of mysticism and superstition, in which it had been +so long imprisoned; the properties of numbers began again +to be studied scientifically. Not being in possession of the +Hindoo indeterminate analysis, many beautiful results of the +Brahmins had to be re-discovered by the Europeans. Thus +a solution in integers of linear indeterminate equations was +re-discovered by the Frenchman \textbf{Bachet de Méziriac} (1581--1638), +\index{Meziriac@{Méziriac}}% +who was the earliest noteworthy European Diophantist. +In~1612 he published \textit{Problèmes plaisants et délectables qui se +font par les nombres}, and in 1621 a Greek edition of \textit{Diophantus} +with notes. The father of the modern theory of numbers is +\textbf{Fermat}. He was so uncommunicative in disposition, that he +\index{Fermat|(}% +generally concealed his methods and made known his results +only. In some cases later analysts have been greatly puzzled +in the attempt of supplying the proofs. Fermat owned a copy +of Bachet's \textit{Diophantus}, in which he entered numerous marginal +notes. In~1670 these notes were incorporated in a new +edition of \textit{Diophantus}, brought out by his son. Other theorems +on numbers, due to Fermat, were published in his \textit{Opera varia} +(edited by his son) and in Wallis's \textit{Commercium epistolicum} +\index{Wallis!ref.\ to}% +of~1658. Of the following theorems, the first seven are found +in the marginal notes:--- + +(1)~$x^n + y^n = z^n$ is impossible for integral values of $x$,~$y$, +and~$z$, when $n > 2$. Remark: ``I have found for this a truly +wonderful proof, but the margin is too small to hold it.'' +Repeatedly was this theorem made the prize question of +learned societies. It has given rise to investigations of +great interest and difficulty on the part of Euler, Lagrange, +\index{Euler!ref.\ to}% +\index{Lagrange!ref.\ to}% +Dirichlet, and Kummer. +\index{Dirichlet!ref.\ to}% +\index{Kummer!ref.\ to}% + +(2)~A prime of the form $4n + 1$ is only once the hypothenuse +\index{Prime numbers}% +%% -----File: 194.png---Folio 180------- +\index{Cube numbers}% +of a right triangle; its square is twice; its cube is three +times, etc. Example: $5^2 = 3^2 + 4^2$; $25^2 = 15^2 + 20^2 = 7^2 + 24^2$; +$125^2 = 75^2 + 100^2 = 35^2 +120^2 = 44^2 + 117^2$. + +(3)~A prime of the form $4n + 1$ can be expressed once, and +\index{Prime numbers}% +only once, as the sum of two squares. Proved by Euler. +\index{Euler!ref.\ to}% + +(4)~A number composed of two cubes can be resolved into +two other cubes in an infinite multiplicity of ways. + +(5)~Every number is either a triangular number or the sum +of two or three triangular numbers; either a square or the +\index{Numbers!triangular}% +sum of two, three, or four squares; either a pentagonal number +or the sum of two, three, four, or five pentagonal numbers; +similarly for polygonal numbers in general. The proof of this +and other theorems is promised by Fermat in a future work +which never appeared. This theorem is also given, with +others, in a letter of 1637(?) addressed to \textit{Pater Mersenne}. +\index{Mersenne}% + +(6)~As many numbers as you please may be found, such +that the square of each remains a square on the addition to +or subtraction from it of the sum of all the numbers. + +(7)~$x^4 + y^4 = z^2$ is impossible. + +(8)~In a letter of 1640 he gives the celebrated theorem +generally known as ``Fermat's theorem,'' which we state in +\index{Fermat's theorem}% +Gauss's notation: If $p$ is prime, and $a$ is prime to~$p$, then $a^{p-1} \equiv 1 \pmod{p}$. It was proved by Euler. + +(9)~Fermat died with the belief that he had found a long-sought-for +law of prime numbers in the formula $2^{2^n} + 1 =$ a +prime, but he admitted that he was unable to prove it rigorously. +The law is not true, as was pointed out by Euler in the +example $2^{2^5} + 1 = 4,294,967,297 = 6,700,417 \text{ times } 641$. The +American lightning calculator \textit{Zerah Colburn}, when a boy, +\index{Colburn, Z}% +readily found the factors, but was unable to explain the +method by which he made his marvellous mental computation. + +(10)~An odd prime number can be expressed as the difference +of two squares in one, and only one, way. This theorem, +%% -----File: 195.png---Folio 181------- +given in the \textit{Relation}, was used by Fermat for the decomposition +of large numbers into prime factors. + +(11)~If the integers $a$, $b$, $c$ represent the sides of a right +triangle, then its area cannot be a square number. This was +proved by Lagrange. + +(12)~Fermat's solution of $ax^2 + 1 = y^2$, where $a$ is integral +but not a square, has come down in only the broadest outline, +as given in the \textit{Relation}. He proposed the problem to the +Frenchman, \textit{Bernhard Frenicle de Bessy}, and in 1657 to all +\index{Bessy}% +living mathematicians. In England, Wallis and Lord Brounker +conjointly found a laborious solution, which was published in~1658, +and also in~1668, in an algebraical work brought out by +\textit{John Pell}. Though Pell had no other connection with the +\index{Pell}% +problem, it went by the name of ``Pell's problem.'' The first +\index{Pell's problem}% +solution was given by the Hindoos. + +We are not sure that Fermat subjected all his theorems to +rigorous proof. His methods of proof were entirely lost +until~1879, when a document was found buried among the +manuscripts of Huygens in the library of Leyden, entitled +\textit{Relation des découvertes en la science des nombres}. It appears +from it that he used an inductive method, called by him \textit{la +descente infinie ou indefinie}. He says that this was particularly +applicable in proving the impossibility of certain relations, as, +for instance, Theorem~11, given above, but that he succeeded +in using the method also in proving affirmative statements. +Thus he proved Theorem~3 by showing that if we suppose +there be a prime~$4n+1$ which does not possess this property, +then there will be a smaller prime of the form~$4n+1$ not +possessing it; and a third one smaller than the second, not +possessing it; and so on. Thus descending indefinitely, he +arrives at the number~$5$, which is the smallest prime factor +of the form $4n+1$. From the above supposition it would +follow that $5$ is not the sum of two squares---a conclusion +%% -----File: 196.png---Folio 182------- +\index{Mechanics!Stevin and Galileo}% +\index{Numbers!theory of numbers|)}% +\index{Theory of numbers|)}% +contrary to fact. Hence the supposition is false, and the +theorem is established. Fermat applied this method of descent +with success in a large number of theorems. By this +method Euler, Legendre, Dirichlet, proved several of his +enunciations and many other numerical propositions. + +A correspondence between \textit{Pascal} and \textit{Fermat} relating to a +\index{Pascal!ref.\ to}% +certain game of chance was the germ of the theory of probabilities, +which has since attained a vast growth. Chevalier de +Méré proposed to Pascal the fundamental problem, to determine +\index{Mere@{Méré}}% +the probability which each player has, at any given +\index{Probability}% +stage of the game, of winning the game. Pascal and Fermat +supposed that the players have equal chances of winning a +single point. + +The former communicated this problem to Fermat, who +\index{Fermat|)}% +studied it with lively interest and solved it by the theory of +combinations, a theory which was diligently studied both by +him and Pascal. The calculus of probabilities engaged the +attention also of Huygens. The most important theorem +\index{Huygens!ref.\ to}% +reached by him was that, if A has $p$~chances of winning a sum~$a$, +and $q$~chances of winning a sum~$b$, then he may expect to +win the sum $\dfrac{ap+bq}{p+q}$. The next great work on the theory of +probability was the \textit{Ars conjectandi} of Jakob Bernoulli. +\index{Bernoulli, James (born 1654)!ref.\ to}% + +Among the ancients, Archimedes was the only one who +\index{Archimedes!ref.\ to}% +attained clear and correct notions on theoretical statics. He +\index{Statics}% +had acquired firm possession of the idea of pressure, which +lies at the root of mechanical science. But his ideas slept +nearly twenty centuries, until the time of \textbf{Stevin} and \textbf{Galileo}. +\index{Galileo}% +\index{Stevin!ref.\ to}% +Stevin determined accurately the force necessary to sustain +a body on a plane inclined at any angle to the horizon. He +was in possession of a complete doctrine of equilibrium. While +Stevin investigated statics, Galileo pursued principally dynamics. +Galileo was the first to abandon the Aristotelian idea +that bodies descend more quickly in proportion as they are +%% -----File: 197.png---Folio 183------- +\index{Centrifugal force}% +\index{Laws of motion}% +heavier; he established the first law of motion; determined +\index{Motion, laws of}% +the laws of falling bodies; and, having obtained a clear notion +of acceleration and of the independence of different motions, +was able to prove that projectiles move in parabolic curves. +Up to his time it was believed that a cannon-ball moved +forward at first in a straight line and then suddenly fell +vertically to the ground. Galileo had an understanding of +\emph{centrifugal forces}, and gave a correct definition of \emph{momentum}. +Though he formulated the fundamental principle of statics, +known as the \emph{parallelogram of forces}, yet he did not fully +\index{Parallelogram of forces}% +recognise its scope. The principle of virtual velocities was +partly conceived by \textbf{Guido Ubaldo} (died 1607), and afterwards +\index{Ubaldo}% +more fully by Galileo. + +Galileo is the founder of the science of dynamics. Among +his contemporaries it was chiefly the novelties he detected in +the sky that made him celebrated, but Lagrange claims that +\index{Lagrange!ref.\ to}% +his astronomical discoveries required only a telescope and +perseverance, while it took an extraordinary genius to discover +laws from phenomena, which we see constantly and of +which the true explanation escaped all earlier philosophers. +The first contributor to the science of mechanics after Galileo +was Descartes. +\index{Descartes|(}% + +\Section{Descartes to Newton} + +Among the earliest thinkers of the seventeenth and eighteenth +centuries, who employed their mental powers toward the +destruction of old ideas and the up-building of new ones, ranks +\textbf{René Descartes} (1596--1650). Though he professed orthodoxy +in faith all his life, yet in science he was a profound sceptic. +He found that the world's brightest thinkers had been long +exercised in metaphysics, yet they had discovered nothing +%% -----File: 198.png---Folio 184------- +certain; nay, had even flatly contradicted each other. This +led him to the gigantic resolution of taking nothing whatever +on authority, but of subjecting everything to scrutinous examination, +according to new methods of inquiry. The certainty +of the conclusions in geometry and arithmetic brought out in +his mind the contrast between the true and false ways of +seeking the truth. He thereupon attempted to apply mathematical +reasoning to all sciences. ``Comparing the mysteries +of nature with the laws of mathematics, he dared to hope that +the secrets of both could be unlocked with the same key.'' +Thus he built up a system of philosophy called Cartesianism. + +Great as was Descartes' celebrity as a metaphysician, it +may be fairly questioned whether his claim to be remembered +by posterity as a mathematician is not greater. His philosophy +has long since been superseded by other systems, but the analytical +geometry of Descartes will remain a valuable possession +forever. At the age of twenty-one, Descartes enlisted in the +army of Prince Maurice of Orange. His years of soldiering +were years of leisure, in which he had time to pursue his +studies. At that time mathematics was his favourite science. +But in 1625 he ceased to devote himself to pure mathematics. +Sir William Hamilton is in error when he states that Descartes +\index{Hamilton, W.}% +considered mathematical studies absolutely pernicious as a +means of internal culture. In a letter to Mersenne, Descartes +says: ``M.~Desargues puts me under obligations on account of +\index{Desargues!ref.\ to}% +the pains that it has pleased him to have in me, in that he +shows that he is sorry that I do not wish to study more in geometry, +but I have resolved to quit only abstract geometry, that +is to say, the consideration of questions which \textit{serve only to +exercise the mind}, and this, in order to study another kind of +geometry, which has for its object the explanation of the +phenomena of nature\ldots. You know that all my physics is +nothing else than geometry.'' The years between 1629 and +%% -----File: 199.png---Folio 185------- +\index{Analytic geometry|(}% +\index{Co-ordinates}% +\index{Debaune@{De Baune}!ref.\ to}% +1649 were passed by him in Holland in the study, principally, +of physics and metaphysics. His residence in Holland was +during the most brilliant days of the Dutch state. In 1637 he +published his \textit{Discours de la Méthode}, containing among others +an essay of 106~pages on geometry. His \textit{Geometry} is not easy +reading. An edition appeared subsequently with notes by his +friend \textit{De Beaune}, which were intended to remove the difficulties. + +It is frequently stated that Descartes was the first to apply +algebra to geometry. This statement is inaccurate, for Vieta +and others had done this before him. Even the Arabs sometimes +used algebra in connection with geometry. The new +step that Descartes did take was the introduction into geometry +of an analytical method based on the notion of variables +and constants, which enabled him to represent curves by algebraic +equations. In the Greek geometry, the idea of motion +was wanting, but with Descartes it became a very fruitful +conception. By him a point on a plane was determined in +position by its distances from two fixed right lines or axes. +These distances varied with every change of position in +the point. This geometric idea of \emph{co-ordinate representation}, +together with the algebraic idea of \emph{two variables in one equation} +having an indefinite number of simultaneous values, furnished +a method for the study of loci, which is admirable for +the generality of its solutions. Thus the entire conic sections +of Apollonius is wrapped up and contained in a single equation +of the second degree. + +The Latin term for ``ordinate'' used by Descartes comes +from the expression \textit{lineæ ordinatæ}, employed by Roman surveyors +for parallel lines. The term \emph{abscissa} occurs for the +first time in a Latin work of 1659, written by \textit{Stefano degli +Angeli} (1623--1697), a professor of mathematics in Rome.\Reference{3} +\index{Angeli}% +Descartes' geometry was called ``analytical geometry,'' partly +%% -----File: 200.png---Folio 186------- +because, unlike the synthetic geometry of the ancients, it is +\index{Geometry!analytic|(}% +actually \emph{analytical}, in the sense that the word is used in +logic; and partly because the practice had then already +arisen, of designating by the term \emph{analysis} the calculus with +\index{Analysis!Descartes'}% +general quantities. + +The first important example solved by Descartes in his +geometry is the ``problem of Pappus''; viz.~``Given several +\index{Pappus!ref.\ to}% +straight lines in a plane, to find the locus of a point such that +the perpendiculars, or more generally, straight lines at given +angles, drawn from the point to the given lines, shall satisfy +the condition that the product of certain of them shall be in +a given ratio to the product of the rest.'' Of this celebrated +problem, the Greeks solved only the special case when the +number of given lines is four, in which case the locus of the +point turns out to be a conic section. By Descartes it was +solved completely, and it afforded an excellent example of +the use which can be made of his analytical method in the +study of loci. Another solution was given later by Newton +\index{Newton!ref.\ to}% +in the \textit{Principia}. + +The methods of drawing tangents invented by Roberval +\index{Tangents!in geometry}% +and Fermat were noticed earlier. Descartes gave a third +method. Of all the problems which he solved by his geometry, +none gave him as great pleasure as his mode of constructing +tangents. It is profound but operose, and, on that account, +inferior to Fermat's. His solution rests on the method of +\emph{Indeterminate Coefficients}, of which he bears the honour of +\index{Indeterminate coefficients}% +invention. Indeterminate coefficients were employed by him +also in solving \Change{bi-quadratic}{biquadratic} equations. +\index{Equations!solution of}% + +The essays of Descartes on dioptrics and geometry were +sharply criticised by Fermat, who wrote objections to the +former, and sent his own treatise on ``maxima and minima'' +\index{Maxima and minima}% +to show that there were omissions in the geometry. Descartes +thereupon made an attack on Fermat's method of tangents. +%% -----File: 201.png---Folio 187------- +Descartes was in the wrong in this attack, yet he continued +\index{Descartes!rule of signs}% +the controversy with obstinacy. He had a controversy also +with Roberval on the cycloid. This curve has been called +\index{Cycloid}% +\index{Roberval!ref.\ to}% +the ``Helen of geometers,'' on account of its beautiful properties +\index{Helen of geometers}% +and the controversies which their discovery occasioned. +Its quadrature by Roberval was generally considered a brilliant +achievement, but Descartes commented on it by saying +that any one moderately well versed in geometry might have +done this. He then sent a short demonstration of his own. +On Roberval's intimating that he had been assisted by a +knowledge of the solution, Descartes constructed the tangent +to the curve, and challenged Roberval and Fermat to do the +same. Fermat accomplished it, but Roberval never succeeded +in solving this problem, which had cost the genius of Descartes +but a moderate degree of attention. + +He studied some new curves, now called ``ovals of Descartes,'' +\index{Ovals of Descartes}% +which were intended by him to serve in the construction +of converging lenses, but which yielded no results +of practical value. + +The application of algebra to the doctrine of curved lines +\index{Algebra!seventeenth century}% +reacted favourably upon algebra. As an abstract science, +Descartes improved it by the systematic use of exponents and +\index{Exponents}% +by the full interpretation and construction of negative quantities. +\index{Negative quantities}% +Descartes also established some theorems on the theory of +equations. Celebrated is his ``rule of signs'' for determining +\index{Rule of signs}% +\index{Signs, rule of}% +the number of positive and negative roots; viz.~an equation +may have as many $+$~roots as there are variations of signs, and +as many $-$~roots as there are permanencies of signs. Descartes +was charged by Wallis with availing himself, without acknowledgment, +\index{Wallis!ref.\ to}% +of Harriot's theory of equations, particularly his mode +\index{Harriot!ref.\ to}% +of generating equations; but there seems to be no good ground +for the charge. Wallis also claimed that Descartes failed to +observe that the above rule of signs is not true whenever the +%% -----File: 202.png---Folio 188------- +\index{Apollonian Problem}% +equation has imaginary roots; but Descartes does not say that +the equation \emph{always has}, but that it \emph{may have} so many roots. +It is true that Descartes does not consider the case of imaginaries +directly, but further on in his \textit{Geometry} he gives +incontestable evidence of being able to handle this case +also. + +In mechanics, Descartes can hardly be said to have advanced +\index{Mechanics!Descartes, Wallis, Wren, Huygens, Newton}% +beyond Galileo. The latter had overthrown the ideas of +\index{Galileo!ref.\ to}% +Aristotle on this subject, and Descartes simply ``threw himself +upon the enemy'' that had already been ``put to the rout.'' +His statement of the first and second laws of motion was an +\index{Laws of motion}% +\index{Motion, laws of}% +improvement in form, but his third law is false in substance. +The motions of bodies in their direct impact was imperfectly +understood by Galileo, erroneously given by Descartes, and +first correctly stated by Wren, Wallis, and Huygens. +\index{Huygens!ref.\ to}% +\index{Wallis!ref.\ to}% +\index{Wren!ref.\ to}% + +One of the most devoted pupils of Descartes was the learned +\textit{Princess Elizabeth}, daughter of Frederick~V\@. She applied the +\index{Elizabeth, Princess}% +\index{Princess Elizabeth}% +new analytical geometry to the solution of the ``Apollonian +problem.'' His second royal follower was \textit{Queen Christina}, +the daughter of Gustavus Adolphus. She urged upon Descartes +to come to the Swedish court. After much hesitation +he accepted the invitation in~1649. He died at Stockholm one +year later. His life had been one long warfare against the +prejudices of men. + +It is most remarkable that the mathematics and philosophy +of Descartes should at first have been appreciated less by his +countrymen than by foreigners. The indiscreet temper of +Descartes alienated the great contemporary French mathematicians, +Roberval, Fermat, Pascal. They continued in investigations +of their own, and on some points strongly opposed +Descartes. The universities of France were under strict +ecclesiastical control and did nothing to introduce his mathematics +and philosophy. It was in the youthful universities of +%% -----File: 203.png---Folio 189------- +\index{Analytic geometry|)}% +\index{Debaune@{De Baune}}% +\index{Inverse tangents (problem of)}% +\index{Maxima and minima}% +Holland that the effect of Cartesian teachings was most +immediate and strongest. + +The only prominent Frenchman who immediately followed +in the footsteps of the great master was \textbf{De Beaune} (1601--1652). +He was one of the first to point out that the properties of a +curve can be deduced from the properties of its tangent. This +mode of inquiry has been called the \emph{inverse method of tangents}. +\index{Tangents!inverse problem of}% +He contributed to the theory of equations by considering for +\index{Equations!theory of}% +the first time the upper and lower limits of the roots of +numerical equations. + +In the Netherlands a large number of distinguished mathematicians +were at once struck with admiration for the Cartesian +geometry. Foremost among these are \textit{van Schooten}, \textit{John de +\index{Schooten, van}% +\index{Vanschooten@{Van Schooten}}% +Witt}, \textit{van Heuraet}, \textit{Sluze}, and \textit{Hudde}. \textbf{Van Schooten} (died 1660), +\index{Hudde}% +\index{Sluze}% +professor of mathematics at Leyden, brought out an edition +of Descartes' geometry, together with the notes thereon by +\index{Descartes!ref.\ to}% +De~Beaune. His chief work is his \textit{Exercitationes Mathematicæ}, +in which he applies the analytical geometry to the solution of +many interesting and difficult problems. The noble-hearted +\textbf{Johann de Witt}, grand-pensioner of Holland, celebrated as a +statesman and for his tragical end, was an ardent geometrician. +He conceived a new and ingenious way of generating conics, +which is essentially the same as that by projective pencils of +rays in modern synthetic geometry. He treated the subject +not synthetically, but with aid of the Cartesian analysis. +\textbf{René François de Sluze} (1622--1685) and \textbf{Johann Hudde} (1633--1704) +made some improvements on Descartes' and Fermat's +\index{Descartes|)}% +methods of drawing tangents, and on the theory of maxima and +minima. With Hudde, we find the first use of three variables +in analytical geometry. He is the author of an ingenious rule +\index{Geometry!analytic|)}% +for finding equal roots. We illustrate it by the equation +$x^3 - x^2 - 8x + 12 = 0$. Taking an arithmetical progression +3,~2, 1,~0, of which the highest term is equal to the degree of +%% -----File: 204.png---Folio 190------- +\index{Circle-squarers}% +\index{Quadrature of curves}% +the equation, and multiplying each term of the equation respectively +by the corresponding term of the progression, we get +$3x^3-2x^2-8x=0$, or $3x^2-2x-8=0$. This last equation +is by one degree lower than the original one. Find the G.C.D. +of the two equations. This is $x - 2$; hence 2~is one of the two +equal roots. Had there been no common divisor, then the +original equation would not have possessed equal roots. Hudde +gave a demonstration for this rule.\Reference{24} + +\textbf{Heinrich van Heuraet} must be mentioned as one of the earliest +\index{Heuraet}% +geometers who occupied themselves with success in the +rectification of curves. He observed in a general way that the +\index{Curves!quadrature of}% +\index{Rectification of curves}% +two problems of quadrature and of rectification are really +identical, and that the one can be reduced to the other. Thus +he carried the rectification of the hyperbola back to the +quadrature of the hyperbola. The semi-cubical parabola +\index{Parabola!semi-cubical}% +\index{Semi-cubical parabola}% +$y^3 = ax^2$ was the first curve that was ever rectified absolutely. +This appears to have been accomplished independently by Van +Heuraet in Holland and by \textbf{William Neil} (1637--1670) in England. +\index{Neil}% +According to Wallis the priority belongs to Neil. Soon +after, the cycloid was rectified by Wren and Fermat. +\index{Cycloid}% + +The prince of philosophers in Holland, and one of the +greatest scientists of the seventeenth century, was \textbf{Christian +Huygens} (1629--1695), a native of the\DPnote{** Uncapitalized in original} Hague. Eminent as a +\index{Huygens|(}% +physicist and astronomer, as well as mathematician, he was +a worthy predecessor of Sir Isaac Newton. He studied at +Leyden under the younger \textit{Van Schooten}. The perusal of +\index{Schooten, van!ref.\ to}% +\index{Vanschooten@{Van Schooten}!ref.\ to}% +some of his earliest theorems led Descartes to predict his +future greatness. In 1651 Huygens wrote a treatise in which +he pointed out the fallacies of Gregory St.~Vincent (1584--1667) +\index{Stvincent@{St.\ Vincent, Gregory}}% +\index{Vincent, Gregory St.}% +on the subject of quadratures. He himself gave a remarkably +close and convenient approximation to the length of a circular +arc. In 1660 and 1663 he went to Paris and to London. In +1666 he was appointed by Louis~XIV. member of the French +%% -----File: 205.png---Folio 191------- +\index{Analytic geometry}% +\index{Centre!of gravity}% +\index{Centre!of oscillation}% +\index{Mechanics!Descartes, Wallis, Wren, Huygens, Newton}% +Academy of Sciences. He was induced to remain in Paris +from that time until~1681, when he returned to his native +city, partly for consideration of his health and partly on +account of the revocation of the Edict of Nantes. + +The majority of his profound discoveries were made with +aid of the ancient geometry, though at times he used the +\index{Geometry!analytic}% +geometry of Descartes or of Cavalieri and Fermat. Thus, +\index{Descartes!ref.\ to}% +like his illustrious friend, Sir Isaac Newton, he always showed +\index{Newton!ref.\ to}% +partiality for the Greek geometry. Newton and Huygens +were kindred minds, and had the greatest admiration for each +other. Newton always speaks of him as the ``Summus +Hugenius.'' + +To the two curves (cubical parabola and cycloid) previously +\index{Cycloid}% +rectified he added a third,---the cissoid. He solved the +\index{Cissoid}% +problem of the catenary, determined the surface of the +\index{Catenary}% +parabolic and hyperbolic conoid, and discovered the properties +of the logarithmic curve and the solids generated by it. +Huygens' \textit{De horologio oscillatorio} (Paris, 1673) is a work that +ranks second only to the \textit{Principia} of Newton and constitutes +\index{Newton's Principia}% +\index{Principia (Newton's)}% +historically a necessary introduction to it.\Reference{13} The book opens +with a description of pendulum clocks, of which Huygens +\index{Pendulum}% +is the inventor. Then follows a treatment of accelerated +motion of bodies falling free, or sliding on inclined planes, or +on given curves,---culminating in the brilliant discovery that +the cycloid is the tautochronous curve. To the theory of +\index{Tautochronous curve}% +curves he added the important theory of ``evolutes.'' After +\index{Evolutes}% +explaining that the tangent of the evolute is normal to the +involute, he applied the theory to the cycloid, and showed by +simple reasoning that the evolute of this curve is an equal +cycloid. Then comes the complete general discussion of the +centre of oscillation. This subject had been proposed for +\index{Oscillation, centre of}% +investigation by Mersenne and discussed by Descartes and +\index{Mersenne}% +Roberval. In Huygens' assumption that the common centre +\index{Roberval!ref.\ to}% +%% -----File: 206.png---Folio 192------- +\index{Mechanics!Descartes, Wallis, Wren, Huygens, Newton}% +\index{Saturn's rings}% +\index{Undulatory theory of light}% +of gravity of a group of bodies, oscillating about a horizontal +axis, rises to its original height, but no higher, is expressed +for the first time one of the most beautiful principles of +dynamics, afterwards called the principle of the conservation +\index{Conservation!of \textit{vis viva}}% +of \textit{vis viva}.\Reference{32} The thirteen theorems at the close of the work +relate to the theory of centrifugal force in circular motion. +\index{Centrifugal force}% +This theory aided Newton in discovering the law of gravitation. +\index{Newton!ref.\ to}% + +Huygens wrote the first formal treatise on probability. He +\index{Probability}% +proposed the wave-theory of light and with great skill applied +geometry to its development. This theory was long neglected, +but was revived and successfully worked out by Young and +Fresnel a century later. Huygens and his brother improved +the telescope by devising a better way of grinding and polishing +lenses. With more efficient instruments he determined +the nature of Saturn's appendage and solved other astronomical +questions. Huygens' \textit{Opuscula posthuma} appeared +\index{Huygens|)}% +in~1703. + +Passing now from Holland to England, we meet there one +of the most original mathematicians of his day---\textbf{John Wallis} +\index{Wallis|(}% +(1616--1703). He was educated for the Church at Cambridge +and entered Holy Orders. But his genius was employed +chiefly in the study of mathematics. In 1649 he was appointed +Savilian professor of geometry at Oxford. He was one of +the original members of the Royal Society, which was founded +in~1663. Wallis thoroughly grasped the mathematical methods +both of Cavalieri and Descartes. His \textit{Conic Sections} is the +\index{Conic sections!more recent researches}% +\index{Descartes!ref.\ to}% +earliest work in which these curves are no longer considered +as sections of a cone, but as curves of the second degree, and +are treated analytically by the Cartesian method of co-ordinates. +In this work Wallis speaks of Descartes in the +highest terms, but in his \textit{Algebra} he, without good reason, +\index{Algebra!seventeenth century}% +accuses Descartes of plagiarising from Harriot. We have +\index{Harriot!ref.\ to}% +%% -----File: 207.png---Folio 193------- +\index{Analytic geometry}% +\index{Curves!quadrature of}% +\index{Descartes!rule of signs}% +\index{Equations!theory of}% +\index{Geometry!analytic}% +\index{Quadrature of curves}% +\index{Rule of signs}% +\index{Signs, rule of}% +already mentioned elsewhere Wallis's solution of the prize +questions on the cycloid, which were proposed by Pascal. + +The \textit{Arithmetic of Infinites}, published in 1655, is his greatest +work. By the application of analysis to the Method of Indivisibles, +\index{Indivisibles}% +he greatly increased the power of this instrument for +effecting quadratures. He advanced beyond Kepler by making +more extended use of the ``law of continuity'' and placing +\index{Continuity}% +full reliance in it. By this law he was led to regard the +denominators of fractions as powers with negative exponents. +Thus, the descending geometrical progression $x^3$,~$x^2$, $x^1$, $x^0$, if +continued, gives $x^{-1}$,~$x^{-2}$, $x^{-3}$, etc.; which is the same thing +as $\dfrac{1\Fstr}{x}$,~$\dfrac{1}{x^2}$,~$\dfrac{1}{x^3}$. The exponents of this geometric series are in +continued arithmetical progression, $3$,~$2$, $1$, $0$, $-1$, $-2$,~$-3$. +He also used fractional exponents, which, like the negative, +had been invented long before, but had failed to be generally +introduced. The symbol~$\infty$ for infinity is due to him. +\index{Infinity}% +\index{Infinity!symbol for}% + +Cavalieri and the French geometers had ascertained the +\index{Cavalieri!ref.\ to}% +formula for squaring the parabola of any degree, $y = x^m$, $m$~being +a positive integer. By the summation of the powers +of the terms of infinite arithmetical series, it was found that +the curve $y = x^m$ is to the area of the parallelogram having +the same base and altitude as $1$ is to $m + 1$. Aided by the law +of continuity, Wallis arrived at the result that this formula +holds true not only when $m$ is positive and integral, but also +when it is fractional or negative. Thus, in the parabola +$y = \sqrt{px\mathstrut}$, $m = \frac{1}{2}$; hence the area of the parabolic segment +is to that of the circumscribed rectangle as $1 : 1\frac{1}{2}$, or as $2 : 3$. +Again, suppose that in $y = x^m$, $m = -\frac{1}{2}$; then the curve is a +kind of hyperbola referred to its asymptotes, and the hyperbolic +space between the curve and its asymptotes is to the +corresponding parallelogram as $1 : \frac{1}{2}$. If $m = -1$, as in the +common equilateral hyperbola $y = x^{-1}$ or $xy = 1$, then this +ratio is $1 : -1 + 1$, or $1 : 0$, showing that its asymptotic space +%% -----File: 208.png---Folio 194------- +is infinite. But in the case when $m$~is greater than unity and +negative, Wallis was unable to interpret correctly his results. +For example, if $m = -3$, then the ratio becomes $1 : -2$, or +as unity to a negative number. What is the meaning of this? +Wallis reasoned thus: If the denominator is only zero, then +the area is already infinite; but if it is less than zero, then the +area must be more than infinite. It was pointed out later by +Varignon, that this space, supposed to exceed infinity, is really +finite, but taken negatively; that is, measured in a contrary +direction.\Reference{31} The method of Wallis was easily extended to +cases such as $y = ax^\frac{m}{n} + bx^\frac{p}{q}$ by performing the quadrature +for each term separately, and then adding the results. + +The manner in which Wallis studied the quadrature of the +circle and arrived at his expression for the value of $\pi$ is +\index{Circle}% +\index{p@{$\pi$}: values for!Wallis'}% +extraordinary. He found that the areas comprised between +the axes, the ordinate corresponding to $x$, and the curves +represented by the equations $y = (1 - x^2)^0$, $y = (1 - x^2)^1$, $y = +(1 - x^2)^2$, $y = (1 - x^2)^3$, etc., are expressed in functions of the +circumscribed rectangles having $x$ and $y$ for their sides, by the +quantities forming the series +\begin{align*} +&x,\\ +&x - \tfrac{1}{3}x^3,\\ +&x - \tfrac{2}{3}x^3 + \tfrac{1}{5}x^5,\\ +&x - \tfrac{3}{3}x^3 + \tfrac{3}{5}x^5 - \tfrac{1}{7}x^7, etc. +\end{align*} +When $x = 1$, these values become respectively $1$,~$\frac{2}{3}$, $\frac{8}{15}$, $\frac{48}{105}$, +etc. Now since the ordinate of the circle is $y = (1 - x^2)^\frac{1}{2}$, the +exponent of which is $\frac{1}{2}$ or the mean value between $0$ and $1$, the +question of this quadrature reduced itself to this: If $0$,~$1$, $2$, +$3$, etc., operated upon by a certain law, give $1$,~$\frac{2}{3}$, $\frac{8}{15}$, $\frac{48}{105}$, +what will $\frac{1}{2}$ give, when operated upon by the same law? He +attempted to solve this by \emph{interpolation}, a method first brought +\index{Interpolation}% +into prominence by him, and arrived by a highly complicated +%% -----File: 209.png---Folio 195------- +\index{Binomial formula}% +\index{p@{$\pi$}: values for!Wallis'}% +\index{Wallis|)}% +and difficult analysis at the following very remarkable expression: +\[ +\frac{\pi}{2} = \frac{2 · 2 · 4 · 4 · 6 · 6 · 8 · 8 \dotsm} + {1 · 3 · 3 · 5 · 5 · 7 · 7 · 9 \dotsm} +\] + +He did not succeed in making the interpolation itself, +because he did not employ literal or general exponents, and +could not conceive a series with more than one term and less +than two, which it seemed to him the interpolated series must +have. The consideration of this difficulty led Newton to the +\index{Newton!ref.\ to}% +discovery of the Binomial Theorem. This is the best place to +\index{Newton's discovery of binomial theorem}% +speak of that discovery. Newton virtually assumed that the +same conditions which underlie the general expressions for +the areas given above must also hold for the expression to be +interpolated. In the first place, he observed that in each +expression the first term is~$x$, that $x$~increases in odd powers, +that the signs alternate $+$~and~$-$, and that the second terms +$\frac{0}{3}x^3$,~$\frac{1}{3}x^3$, $\frac{2}{3}x^3$, $\frac{3}{3}x^3$, are in arithmetical progression. Hence the +first two terms of the interpolated series must be $x - \dfrac{\frac{1}{2}x^3}{3}$. +He next considered that the denominators $1$,~$3$, $5$, $7$, etc., are in +arithmetical progression, and that the coefficients in the numerators +in each expression are the digits of some power of +the number~$11$; namely, for the first expression, $11^0$ or~$1$; for +the second, $11^1$ or $1$,~$1$; for the third, $11^2$ or $1$, $2$,~$1$; for the +fourth, $11^3$ or $1$, $3$, $3$,~$1$; etc. He then discovered that, having +given the second digit (call it~$m$), the remaining digits can +be found by continual multiplication of the terms of the series +$\dfrac{m - 0}{1} · \dfrac{m - 1}{2} · \dfrac{m - 2}{3} · \dfrac{m - 3}{4} ·$ etc. Thus, if $m = 4$, then +$4 · \smash[b]{\dfrac{m - 1}{2}}$ gives~$6$; $6 · \smash[b]{\dfrac{m - 2}{3}}$ gives~$4$; $4 · \smash[b]{\dfrac{m - 3}{4}}$ gives~$1$. Applying +this rule to the required series, since the second term +is $\dfrac{\frac{1}{2}x^3\Fstr[12pt]}{3}$, we have $m = \frac{1}{2}$, and then get for the succeeding coefficients +%% -----File: 210.png---Folio 196------- +\index{Pacioli!ref.\ to}% +in the numerators respectively $-\frac{1}{8}\Fstr$, $+\frac{1}{16}$, $-\frac{5}{128}$, +etc.; hence the required area for the circular segment is +$x - \dfrac{\frac{1}{2}x^3\Fstr[12pt]}{3} - \dfrac{\frac{1}{8}x^5}{5} - \dfrac{\frac{1}{16}x^7}{7} -{}$ etc. Thus he found the interpolated +expression to be an infinite series, instead of one having more +than one term and less than two, as Wallis believed it must +be. This interpolation suggested to Newton a mode of expanding +$(1 - x^2)^\frac{1}{2}$, or, more generally, $(1-x^2)^m$, into a series. +He observed that he had only to omit from the expression +just found the denominators $1$, $3$, $5$, $7$, etc., and to lower +each power of $x$ by unity, and he had the desired expression. +In a letter to Oldenburg (June~13, 1676), Newton states the +theorem as follows: The extraction of roots is much shortened +by the theorem +\index{Binomial formula}% +\[ +(P + PQ)^\frac{m}{n} = P^\frac{m}{n} + \frac{m}{n} AQ + \frac{m-n}{2n} BQ + \frac{m-2n}{3n} CQ + \text{etc.}, +\] +where $A$ means the first term, $P^\frac{m}{n}$, $B$ the second term, $C$ the +third term, etc. He verified it by actual multiplication, but +gave no regular proof of it. He gave it for any exponent +whatever, but made no distinction between the case when the +exponent is positive and integral, and the others. + +It should here be mentioned that very rude beginnings of +the binomial theorem are found very early. The Hindoos and +\index{Newton's discovery of binomial theorem}% +Arabs used the expansions of $(a + b)^2$ and $(a + b)^3$ for extracting +roots; Vieta knew the expansion of $(a + b)^4$; but these +\index{Vieta!ref.\ to}% +were the results of simple multiplication without the discovery +of any law. The binomial coefficients for positive whole exponents +were known to some Arabic and European mathematicians. +Pascal derived the coefficients from the method of +\index{Pascal!ref.\ to}% +what is called the ``arithmetical triangle.'' Lucas de Burgo, +\index{Arithmetical triangle}% +Stifel, Stevinus, Briggs, and others, all possessed something +from which one would think the binomial theorem could have +%% -----File: 211.png---Folio 197------- +\index{Infinite series}% +been gotten with a little attention, ``if we did not know that +such simple relations were difficult to discover.'' + +Though Wallis had obtained an entirely new expression for~$\pi$, +\index{p@{$\pi$}: values for!Brouncker's}% +\index{Wallis!ref.\ to}% +he was not satisfied with it; for instead of a finite number +of terms yielding an absolute value, it contained merely an +infinite number, approaching nearer and nearer to that value. +He therefore induced his friend, \textbf{Lord Brouncker} (1620?-1684), +\index{Brouncker}% +the first president of the Royal Society, to investigate this +subject. Of course Lord Brouncker did not find what they +were after, but he obtained the following beautiful equality:--- +\[ +%[** PP: \cfrac leaves a bit too much vertical space] +\pi = \dfrac{4}{1 + + \dfrac{1\Fstr[10pt]}{2 + + \dfrac{9\Fstr[10pt]}{2 + + \dfrac{25\Fstr[10pt]}{2 + + \dfrac{49\Fstr[10pt]}{2 + \text{etc.}\Fstr[10pt]} + } + } + } +} +\] +Continued fractions, both ascending and descending, appear to +\index{Continued fractions}% +\index{Fractions!continued}% +have been known already to the Greeks and Hindoos, though +not in our present notation. Brouncker's expression gave +birth to the theory of continued fractions. + +Wallis' method of quadratures was diligently studied by +his disciples. Lord Brouncker obtained the first infinite +series for the area of an equilateral hyperbola between its +asymptotes. \textbf{Nicolaus Mercator} of Holstein, who had settled +\index{Mercator, N.}% +in England, gave, in his \textit{Logarithmotechnia} (London, 1668), a +similar series. He started with the grand property of the +equilateral hyperbola, discovered in 1647 by \textit{Gregory St.~Vincent}, +\index{Stvincent@{St.\ Vincent, Gregory}}% +\index{Vincent, Gregory St.}% +which connected the hyperbolic space between the +asymptotes with the natural logarithms and led to these +\index{Logarithms}% +logarithms being called hyperbolic. By it Mercator arrived +at the logarithmic series, which Wallis had attempted but +\index{Logarithmic series}% +failed to obtain. He showed how the construction of logarithmic +%% -----File: 212.png---Folio 198------- +\index{Prime and ultimate ratios}% +\index{Rectification of curves}% +tables could be reduced to the quadrature of hyperbolic +spaces. Following up some suggestions of Wallis, \textit{William +Neil} succeeded in rectifying the cubical parabola, and \textit{Wren} in +\index{Neil!ref.\ to}% +\index{Parabola}% +\index{Wren!ref.\ to}% +rectifying any cycloidal arc. + +A prominent English mathematician and contemporary of +Wallis was \textbf{Isaac Barrow} (1630-1677). He was professor of +\index{Barrow}% +mathematics in London, and then in Cambridge, but in 1669 +he resigned his chair to his illustrious pupil, Isaac Newton, +and renounced the study of mathematics for that of divinity. +As a mathematician, he is most celebrated for his method of +tangents. He simplified the method of Fermat by introducing +\index{Fermat!ref.\ to}% +\index{Tangents!direct problem of}% +two infinitesimals instead of one, and approximated to the +course of reasoning afterwards followed by Newton in his doctrine +on Ultimate Ratios. + +He considered the infinitesimal right triangle $ABB'$ having +%[** PP: Width-dependent line break] +%[Illustration] +\begin{wrapfigure}[8]{l}{2.125in} +\Graphic{2.125in}{212a} +\end{wrapfigure} +for its sides the difference between two successive ordinates, +the distance between them, and the portion of the curve intercepted +by them. This triangle is similar to $BPT$, formed by +the ordinate, the tangent, and the sub-tangent. Hence, if we +know the ratio of $B'A$ to $BA$, then we know the ratio of the +ordinate and the sub-tangent, and the tangent can be constructed +at once. For any curve, say $y^2 =px$, the ratio of $B'A$ +to $BA$ is determined from its +equation as follows: If $x$ receives +an infinitesimal increment +$PP'= e$, then $y$ receives +an increment $B'A = a$, and the +equation for the ordinate $B'P'$ +becomes $y^2+2ay+a^2=px+pe$. +Since $y^2 =px$, we get $2ay + a^2 = pe$; neglecting higher powers +of the infinitesimals, we have $2ay = pe$, which gives +\[ +a:e=p:2y=p:2\sqrt{px}. +\] +%% -----File: 213.png---Folio 199------- +But $a : e = \text{the ordinate} : \text{the sub-tangent}$; hence +\[ +p : 2\sqrt{px} = \sqrt{px} : \text{sub-tangent}, +\] +giving $2x$ for the value of the sub-tangent. This method differs +from that of the differential calculus only in notation.\Reference{31} + +\Section{Newton to Euler} + +It has been seen that in France prodigious scientific progress +was made during the beginning and middle of the seventeenth +century. The toleration which marked the reign of Henry IV. +and Louis~XIII. was accompanied by intense intellectual +activity. Extraordinary confidence came to be placed in the +power of the human mind. The bold intellectual conquests +of Descartes, Fermat, and Pascal enriched mathematics with +imperishable treasures. During the early part of the reign +of Louis~XIV. we behold the sunset splendour of this glorious +period. Then followed a night of mental effeminacy. This +lack of great scientific thinkers during the reign of Louis~XIV. +may be due to the simple fact that no great minds were born; +but, according to Buckle, it was due to the paternalism, to +the spirit of dependence and subordination, and to the lack +of toleration, which marked the policy of Louis~XIV\@. + +In the absence of great French thinkers, Louis~XIV. surrounded +himself by eminent foreigners. Römer from Denmark, +\index{Romer@{Römer}}% +Huygens from Holland, Dominic Cassini from Italy, +were the mathematicians and astronomers adorning his court. +They were in possession of a brilliant reputation before going +to Paris. Simply because they performed scientific work in +Paris, that work belongs no more to France than the discoveries +of Descartes belong to Holland, or those of Lagrange +to Germany, or those of Euler and Poncelet to Russia. We +%% -----File: 214.png---Folio 200------- +\index{Fluxions}% +must look to other countries than France for the great scientific +men of the latter part of the seventeenth century. + +About the time when Louis~XIV. assumed the direction +of the French government Charles II. became king of England. +At this time England was extending her commerce +and navigation, and advancing considerably in material prosperity. +A strong intellectual movement took place, which +was unwittingly supported by the king. The age of poetry +was soon followed by an age of science and philosophy. In +two successive centuries England produced Shakespeare and +Newton! +\index{Newton!ref.\ to}% + +Germany still continued in a state of national degradation. +The Thirty Years' War had dismembered the empire and +brutalised the people. Yet this darkest period of Germany's +history produced Leibniz, one of the greatest geniuses of +\index{Leibniz!ref.\ to}% +modern times. + +There are certain focal points in history toward which +the lines of past progress converge, and from which radiate +the advances of the future. Such was the age of Newton +and Leibniz in the history of mathematics. During fifty +years preceding this era several of the brightest and acutest +mathematicians bent the force of their genius in a direction +which finally led to the discovery of the infinitesimal calculus +by Newton and Leibniz. Cavalieri, Roberval, Fermat, Descartes, +Wallis, and others had each contributed to the new +geometry. So great was the advance made, and so near +was their approach toward the invention of the infinitesimal +analysis, that both Lagrange and Laplace pronounced their +countryman, Fermat, to be the true inventor of it. The differential +\index{Differential calculus}% +calculus, therefore, was not so much an individual +discovery as the grand result of a succession of discoveries +by different minds. Indeed, no great discovery ever flashed +upon the mind at once, and though those of Newton will +%% -----File: 215.png---Folio 201------- +influence mankind to the end of the world, yet it must be +admitted that Pope's lines are only a ``poetic fancy'':--- +\begin{center} +\footnotesize% +\settowidth{\TmpLen}{``\,God said, `Let Newton be,' and all was light.''}% +\parbox{\TmpLen}{% + ``\,Nature and Nature's laws lay hid in night; \\ + \settowidth{\TmpLen}{``\,}% + \makebox[\TmpLen][c]{}God said, `Let Newton be,' and all was light.''}% +\end{center} + +\textbf{Isaac Newton} (1642-1727) was born at Woolsthorpe, in +\index{Newton|(}% +Lincolnshire, the same year in which Galileo died. At his +birth he was so small and weak that his life was despaired of. +His mother sent him at an early age to a village school, and +in his twelfth year to the public school at Grantham. At +first he seems to have been very inattentive to his studies +and very low in the school; but when, one day, the little +Isaac received a severe kick upon his stomach from a boy +who was above him, he laboured hard till he ranked higher +in school than his antagonist. From that time he continued +to rise until he was the head boy.\Reference{33} At Grantham, Isaac +showed a decided taste for mechanical inventions. He constructed +a water-clock, a wind-mill, a carriage moved by the +person who sat in it, and other toys. When he had attained +his fifteenth year his mother took him home to assist her in +the management of the farm, but his great dislike for farm-work %[** PP: Only instance] +and his irresistible passion for study, induced her to +send him back to Grantham, where he remained till his +eighteenth year, when he entered Trinity College, Cambridge +(1660). Cambridge was the real birthplace of Newton's +genius. Some idea of his strong intuitive powers may be +drawn from the fact that he regarded the theorems of ancient +geometry as self-evident truths, and that, without any preliminary +study, he made himself master of Descartes' \textit{Geometry}. +He afterwards regarded this neglect of elementary geometry +a mistake in his mathematical studies, and he expressed to +Dr. Pemberton his regret that ``he had applied himself to the +\index{Pemberton}% +works of Descartes and other algebraic writers before he had +%% -----File: 216.png---Folio 202------- +\index{Binomial formula}% +\index{Exponents}% +considered the \textit{Elements} of Euclid with that attention which +so excellent a writer deserves.'' Besides Descartes' \textit{Geometry}, +he studied Oughtred's \textit{Clavis}, Kepler's \textit{Optics}, the works of +\index{Kepler!ref.\ to}% +\index{Oughtred!ref.\ to}% +Vieta, Schooten's \textit{Miscellanies}, Barrow's \textit{Lectures}, and the +\index{Barrow!ref.\ to}% +\index{Schooten, van!ref.\ to}% +\index{Vanschooten@{Van Schooten}!ref.\ to}% +\index{Vieta!ref.\ to}% +works of Wallis. He was particularly delighted with Wallis' +\index{Wallis!ref.\ to}% +\textit{Arithmetic of Infinites}, a treatise fraught with rich and varied +suggestions. Newton had the good fortune of having for +a teacher and fast friend the celebrated Dr. Barrow, who +had been elected professor of Greek in~1660, and was made +Lucasian professor of mathematics in~1663. The mathematics +of Barrow and of Wallis were the starting-points +from which Newton, with a higher power than his masters', +moved onward into wider fields. Wallis had effected the +quadrature of curves whose ordinates are expressed by any +\index{Curves!quadrature of}% +integral and positive power of $(1-x^2)$. We have seen how +Wallis attempted but failed to interpolate between the areas +thus calculated, the areas of other curves, such as that of +the circle; how Newton attacked the problem, effected the +interpolation, and discovered the Binomial Theorem, which +afforded a much easier and direct access to the quadrature +of curves than did the method of interpolation; for even +though the binomial expression for the ordinate be raised +to a fractional or negative power, the binomial could at once +be expanded into a series, and the quadrature of each separate +term of that series could be effected by the method of Wallis. +Newton introduced the system of literal indices. + +Newton's study of quadratures soon led him to another +and most profound invention. He himself says that in 1665 +and 1666 he conceived the method of fluxions and applied +\index{Fluxions|(}% +them to the quadrature of curves. Newton did not communicate +the invention to any of his friends till 1669, when +he placed in the hands of Barrow a tract, entitled \textit{De Analysi +per Æquationes Numero Terminorum Infinitas}, which was sent +%% -----File: 217.png---Folio 203------- +\index{Hudde!ref.\ to}% +by Barrow to Collins, who greatly admired it. In this treatise +\index{Barrow!ref.\ to}% +\index{Collins}% +the principle of fluxions, though distinctly pointed out, is only +partially developed and explained. Supposing the abscissa +to increase uniformly in proportion to the time, he looked +upon the area of a curve as a nascent quantity increasing +by continued fluxion in the proportion of the length of the +ordinate. The expression which was obtained for the fluxion +he expanded into a finite or infinite series of monomial terms, +\index{Infinite series}% +to which Wallis' rule was applicable. Barrow urged Newton +to publish this treatise; ``but the modesty of the author, of +which the excess, if not culpable, was certainly in the present +instance very unfortunate, prevented his compliance.''\Reference{26} Had +this tract been published then, instead of forty-two years +later, there would probably have been no occasion for that +long and deplorable controversy between Newton and Leibniz. + +For a long time Newton's method remained unknown, except +to his friends and their correspondents. In a letter +to Collins, dated December~10th, 1672, Newton states the fact +of his invention with one example, and then says: ``This +is one particular, or rather corollary, of a general method, +which extends itself, without any troublesome calculation, not +only to the drawing of tangents to any curve lines, whether +geometrical or mechanical, or anyhow respecting right lines +or other curves, but also to the resolving other abstruser +kinds of problems about the crookedness, areas, lengths, +centres of gravity of curves, etc.; nor is it (as Hudden's +method of Maximis and Minimis) limited to equations which +are free from surd quantities. This method I have interwoven +with that other of working in equations, by reducing +them to infinite series.'' + +These last words relate to a treatise he composed in the +year 1671, entitled \textit{Method of Fluxions}, in which he aimed +to represent his method as an independent calculus and as +%% -----File: 218.png---Folio 204------- +a complete system. This tract was intended as an introduction +to an edition of Kinckhuysen's \textit{Algebra}, which he had +\index{Kinckhuysen}% +undertaken to publish. ``But the fear of being involved in +disputes about this new discovery, or perhaps the wish to +render it more complete, or to have the sole advantage of +employing it in his physical researches, induced him to abandon +this design.''\Reference{33} + +Excepting two papers on optics, all of his works appear +to have been published only after the most pressing solicitations +of his friends and against his own wishes.\Reference{34} His researches +on light were severely criticised, and he wrote in~1675: +``I was so persecuted with discussions arising out of +my theory of light that I blamed my own imprudence for +parting with so substantial a blessing as my quiet to run +after a shadow.'' + +The \textit{Method of Fluxions}, translated by J. Colson from Newton's +\index{Colson}% +Latin, was first published in~1736, or sixty-five years +after it was written. In it he explains first the expansion +into series of fractional and irrational quantities,---a subject +which, in his first years of study, received the most careful +attention. He then proceeds to the solution of the two following +mechanical problems, which constitute the pillars, so +to speak, of the abstract calculus:--- + +``I\@. The length of the space described being continually +(\ie~at all times) given; to find the velocity of the motion at +any time proposed. + +``II\@. The velocity of the motion being continually given; to +find the length of the space described at any time proposed.'' + +Preparatory to the solution, Newton says: ``Thus, in the +equation $y = x^2$, if $y$ represents the length of the space at any +time described, which (time) another space~$x$, by increasing +with an uniform celerity~$\dot{x}$, measures and exhibits as described: +then $2x\dot{x}$ will represent the celerity by which the space~$y$, +%% -----File: 219.png---Folio 205------- +\index{Notation!differential calculus}% +at the same moment of time, proceeds to be described; and +contrarywise.'' + +``But whereas we need not consider the time here, any farther +than it is expounded and measured by an equable local +motion; and besides, whereas only quantities of the same +kind can be compared together, and also their velocities of +increase and decrease; therefore, in what follows I shall have +no regard to time formally considered, but I shall suppose +some one of the quantities proposed, being of the same kind, +to be increased by an equable fluxion, to which the rest may be +referred, as it were to time; and, therefore, by way of analogy, +it may not improperly receive the name of time.'' In this +statement of Newton there is contained a satisfactory answer +to the objection which has been raised against his method, +that it introduces into analysis the foreign idea of motion. A +quantity thus increasing by uniform fluxion, is what we now +call an independent variable. + +Newton continues: ``Now those quantities which I consider +as gradually and indefinitely increasing, I shall hereafter call +\emph{fluents}, or \emph{flowing quantities}, and shall represent them by the +\index{Fluents}% +final letters of the alphabet, $v$,~$x$, $y$, and~$z$; \dots and the velocities +by which every fluent is increased by its generating motion +(which I may call \emph{fluxions}, or simply velocities, or celerities), +I shall represent by the same letters pointed, thus, $\dot{v}$,~$\dot{x}$, $\dot{y}$,~$\dot{z}$. +That is, for the celerity of the quantity~$v$ I shall put~$\dot{v}$, and so +for the celerities of the other quantities $x$,~$y$, and~$z$, I shall put +$\dot{x}$,~$\dot{y}$, and~$\dot{z}$, respectively.'' It must here be observed that Newton +does not take the fluxions themselves infinitely small. +The ``moments of fluxions,'' a term introduced further on, are +\index{Moments in fluxionary calculus}% +infinitely small quantities. These ``moments,'' as defined and +used in the \textit{Method of Fluxions}, are substantially the differentials +of Leibniz. De Morgan points out that no small amount of +\index{Demorgan@{De Morgan}!ref.\ to}% +confusion has arisen from the use of the word \emph{fluxion} and the +%% -----File: 220.png---Folio 206------- +\index{Fluents}% +notation~$\dot{x}$ by all the English writers previous to~1704, excepting +Newton and Cheyne, in the sense of an infinitely small increment.\Reference{35} +\index{Cheyne}% +Strange to say, even in the \textit{Commercium Epistolicum} +\index{Commercium epistolicum}% +the words \emph{moment} and \emph{fluxion} appear to be used as synonymous. + +After showing by examples how to solve the first problem, +Newton proceeds to the demonstration of his solution:--- + +``The moments of flowing quantities (that is, their indefinitely +\index{Moments in fluxionary calculus}% +small parts, by the accession of which, in infinitely +small portions of time, they are continually increased) are as +the velocities of their flowing or increasing. + +``Wherefore, if the moment of any one (as~$x$) be represented +by the product of its celerity~$\dot{x}$ into an infinitely small quantity~$0$ +(\ie by~$\dot{x}0$), the moments of the others, $v$, $y$, $z$, will be represented +by $\dot{v}0$, $\dot{y}0$,~$\dot{z}0$; because $\dot{v}0$,~$\dot{x}0$, $\dot{y}0$, and~$\dot{z}0$ are to each other +as $\dot{v}$,~$\dot{x}$, $\dot{y}$, and~$\dot{z}$. + +``Now since the moments, as $\dot{x}0$ and~$\dot{y}0$, are the indefinitely +little accessions of the flowing quantities $x$~and~$y$, by which +those quantities are increased through the several indefinitely +little intervals of time, it follows that those quantities, $x$~and~$y$, +after any indefinitely small interval of time, become +$x + \dot{x}0$ and $y + \dot{y}0$, and therefore the equation, which at all +times indifferently expresses the relation of the flowing quantities, +will as well express the relation between $x + \dot{x}0$ and +$y + \dot{y}0$, as between $x$~and~$y$; so that $x + \dot{x}0$ and $y + \dot{y}0$ may +be substituted in the same equation for those quantities, instead +of $x$~and~$y$. Thus let any equation $x^3 - ax^2 + axy - y^3 = 0$ +be given, and substitute $x + \dot{x}0$ for~$x$, and $y + \dot{y}0$ for~$y$, and +there will arise +\[ +\left. +\begin{alignedat}{4} +&\phantom{a}x^3 &&+ 3x^2\dot{x}0 &&+ 3x\dot{x}0\dot{x}0 &&+ \dot{x}^3 0^3 \\ +-{}&ax^2 &&- 2ax\dot{x}0 &&- a\dot{x}0\dot{x}0 \\ ++{}&axy &&+ ay\dot{x}0 &&+ a\dot{x}0\dot{y}0 \\ +& &&+ ax\dot{y}0 \\ +-{}&y^3 &&- 3y^2\dot{y}0 &&- 3y\dot{y}0\dot{y}0 &&- \dot{y}^3 0^3 +\end{alignedat} +\right\} = 0. +\] +%% -----File: 221.png---Folio 207------- + +``Now, by supposition, $x^3-ax^2+axy-y^3=0$, which therefore, +being expunged and the remaining terms being divided +by 0, there will remain +\begin{align*} +3x^2\dot{x} &- 2ax\dot{x} + ay\dot{x} + ax\dot{y} - 3y^2\dot{y} + 3x\dot{x}\dot{x}0 - a\dot{x}\dot{x}0 + a\dot{x}\dot{y}0 \\ + &- 3y\dot{y}\dot{y}0 + \dot{x}^3 00 - \dot{y}^300 = 0. +\end{align*} +But whereas zero is supposed to be infinitely little, that it +may represent the moments of quantities, the terms that are +multiplied by it will be nothing in respect of the rest (\textit{termini +in eam ducti pro nihilo possunt haberi cum aliis collati}); therefore +I reject them, and there remains +\[ +3x^2\dot{x} - 2ax\dot{x} + ay\dot{x} + ax\dot{y} - 3y^2\dot{y} = 0, +\] +as above in Example~I.'' Newton here uses infinitesimals. +\index{Infinitesimals}% + +Much greater than in the first problem were the difficulties +encountered in the solution of the second problem, involving, +as it does, inverse operations which have been taxing the skill +of the best analysts since his time. Newton gives first a +special solution to the second problem in which he resorts +to a rule for which he has given no proof. + +In the general solution of his second problem, Newton +assumed homogeneity with respect to the fluxions and then +considered three cases: (1)~when the equation contains two +fluxions of quantities and but one of the fluents; (2)~when +the equation involves both the fluents as well as both the fluxions; +(3)~when the equation contains the fluents and the fluxions +of three or more quantities. The first case is the easiest +since it requires simply the integration of $\dfrac{dy}{dx}=f(x)$, to which +his ``special solution'' is applicable. The second case demanded +nothing less than the general solution of a differential +equation of the first order. Those who know what +efforts were afterwards needed for the complete exploration +of this field in analysis, will not depreciate Newton's work +%% -----File: 222.png---Folio 208------- +\index{Partial differential equations}% +even though he resorted to solutions in form of infinite series. +\index{Infinite series}% +Newton's third case comes now under the solution of partial +differential equations. He took the equation $2\dot{x} - \dot{z} + x\dot{y} = 0$ +and succeeded in finding a particular integral of it. + +The rest of the treatise is devoted to the determination of +maxima and minima, the radius of curvature of curves, and +\index{Maxima and minima}% +other geometrical applications of his fluxionary calculus. All +this was done previous to the year~1672. + +It must be observed that in the \textit{Method of Fluxions} (as well +as in his \textit{De Analysi} and all earlier papers) the method +employed by Newton is strictly infinitesimal, and in substance +like that of Leibniz. Thus, the original conception of the +\index{Leibniz!ref.\ to}% +calculus in England, as well as on the Continent, was based +on infinitesimals. The fundamental principles of the fluxionary +\index{Infinitesimals}% +calculus were first given to the world in the \textit{Principia}; but its +\index{Newton's Principia}% +\index{Principia (Newton's)}% +peculiar notation did not appear until published in the second +volume of Wallis' \textit{Algebra} in 1693. The exposition given in +the \textit{Algebra} was substantially a contribution of Newton; it +rests on infinitesimals. In the first edition of the \textit{Principia} +(1687) the description of fluxions is likewise founded on +infinitesimals, but in the second (1713) the foundation is +somewhat altered. In Book~II. Lemma~II. of the first edition +we read: ``Cave tamen intellexeris particulas finitas. \textit{Momenta +quam primum finitæ sunt magnitudinis, desinunt esse momenta. +Finiri enim repugnat aliquatenus perpetuo eorum incremento vel +decremento.} Intelligenda sunt principia jamjam nascentia +finitorum magnitudinum.'' In the second edition the two +sentences which we print in italics are replaced by the +following: ``Particulæ finitæ non sunt momenta sed quantitates +ipsæ ex momentis genitæ.'' Through the difficulty of +the phrases in both extracts, this much distinctly appears, +that in the first, moments are infinitely small quantities. +What else they are in the second is not clear.\Reference{35} In the +%% -----File: 223.png---Folio 209------- +\index{Fresnel's wave-surface}% +\textit{Quadrature of Curves} of 1704, the infinitely small quantity is +completely abandoned. It has been shown that in the \textit{Method +of Fluxions} Newton rejected terms involving the quantity 0, +because they are infinitely small compared with other terms. +This reasoning is evidently erroneous; for as long as 0 is a +quantity, though ever so small, this rejection cannot be made +without affecting the result. Newton seems to have felt this, +for in the \textit{Quadrature of Curves} he remarked that ``in mathematics +the minutest errors are not to be neglected" (errores +quam minimi in rebus mathematicis non sunt contemnendi). + +The early distinction between the system of Newton and +Leibniz lies in this, that Newton, holding to the conception +\index{Leibniz!ref.\ to}% +of velocity or fluxion, used the infinitely small increment as a +means of determining it, while with Leibniz the relation of the +infinitely small increments is itself the object of determination. +The difference between the two rests mainly upon a difference +in the mode of generating quantities.\Reference{35} + +We give Newton's statement of the method of fluxions or +rates, as given in the introduction to his \textit{Quadrature of Curves}. +``I consider mathematical quantities in this place not as +consisting of very small parts, but as described by a continued +motion. Lines are described, and thereby generated, not by +the apposition of parts, but by the continued motion of points; +superficies by the motion of lines; solids by the motion of +superficies; angles by the rotation of the sides; portions of +time by continual flux: and so on in other quantities. These +geneses really take place in the nature of things, and are +daily seen in the motion of bodies\dots. + +``Fluxions are, as near as we please (\textit{quam proxime}), as the +increments of fluents generated in times, equal and as small as +possible, and to speak accurately, they are in the prime ratio +of nascent increments; yet they can be expressed by any lines +whatever, which are proportional to them." +%% -----File: 224.png---Folio 210------- + +Newton exemplifies this last assertion by the problem of +tangency: Let $AB$ be the abscissa, $BC$~the ordinate, $VCH$~the +tangent, $Ec$~the increment of the ordinate, which produced +meets $VH$ at~$T$, and $Cc$~the increment of the curve. +The right line~$Cc$ being produced to~$K$, there are formed +three small triangles, the rectilinear~$CEc$, the mixtilinear~$CEc$, +and the rectilinear~$CET$. Of these, the first is evidently +the smallest, and the last the greatest. Now suppose +the ordinate~$bc$ to move into the place~$BC$, so that the +point~$c$ exactly coincides +with %[** PP: Width-dependent line break] +%[Illustration:] +\begin{wrapfigure}[8]{l}{2.75in} +\Graphic{2.75in}{224a} +\end{wrapfigure} +the +point $C$; $CK$, and +therefore the curve +$Cc$, is coincident +with the tangent~$CH$, +$Ec$ is absolutely +equal to $ET$, +and the mixtilinear +evanescent triangle $CEc$ is, in the last form, similar to the +triangle $CET$; and its evanescent sides $CE$, $Ec$, $Cc$, will be +proportional to $CE$, $ET$, and~$CT$, the sides of the triangle~$CET$. +Hence it follows that the fluxions of the lines $AB$, +$BC$, $AC$, being in the last ratio of their evanescent increments, +are proportional to the sides of the triangle~$CET$, or, which is +all one, of the triangle~$VBC$ similar thereunto. As long as +the points $C$~and~$c$ are distant from each other by an interval, +however small, the line~$CK$ will stand apart by a small angle +from the tangent~$CH$. But when $CK$ coincides with~$CH$, and +the lines $CE$, $Ec$, $cC$ reach their ultimate ratios, then the +points $C$~and~$c$ accurately coincide and are one and the same. +Newton then adds that ``in mathematics the minutest errors +are not to be neglected.'' This is plainly a rejection of the +postulates of Leibniz. The doctrine of infinitely small quantities +\index{Leibniz!ref.\ to}% +%% -----File: 225.png---Folio 211------- +is here renounced in a manner which would lead one +to suppose that Newton had never held it himself. Thus it +appears that Newton's doctrine was different in different +periods. Though, in the above reasoning, the Charybdis of +infinitesimals is safely avoided, the dangers of a Scylla stare +\index{Infinitesimals}% +us in the face. We are required to believe that a point may +be considered a triangle, or that a triangle can be inscribed +in a point; nay, that three dissimilar triangles become similar +and equal when they have reached their ultimate form in one +and the same point. + +In the introduction to the \textit{Quadrature of Curves} the fluxion +of $x^n$ is determined as follows:--- + +``In the same time that~$x$, by flowing, becomes~$x + 0$, the +power~$x^n$ becomes $(x+0)^n$, \ie~by the method of infinite series +\[ +x^n + n0x^{n-1} + \frac{n^2-n}{2} 0^{2} x^{n-2} + \text{etc.}, +\] +and the increments +\[ +0 \text{ and } n0 x^{n-1} + \frac{n^2-n}{2} 0^{2} x^{n-2} + \text{etc.}, +\] +are to one another as +\[ +1 \text{ to } nx^{n-1} + \frac{n^2-n}{2} 0 x^{n-2} + \text{etc.} +\] + +``Let now the increments vanish, and their last proportion +will be $1$~to~$nx^{n-1}$: hence the fluxion of the quantity~$x$ is to +the fluxion of the quantity~$x^n$ as~$1 : nx^{n-1}$. + +``The fluxion of lines, straight or curved, in all cases whatever, +as also the fluxions of superficies, angles, and other +quantities, can be obtained in the same manner by the method +of prime and ultimate ratios. But to establish in this way +the analysis of infinite quantities, and to investigate prime +and ultimate ratios of finite quantities, nascent or evanescent, +is in harmony with the geometry of the ancients; and I have +endeavoured to show that, in the method of fluxions, it is not +%% -----File: 226.png---Folio 212------- +\index{Astronomy!Newton|(}% +\index{Mechanics!Descartes, Wallis, Wren, Huygens, Newton|(}% +necessary to introduce into geometry infinitely small quantities.'' +This mode of differentiating does not remove all the +difficulties connected with the subject. When $0$~becomes +nothing, then we get the ratio $\dfrac{0}{0} = nx^{n-1}$, which needs further +elucidation. Indeed, the method of Newton, as delivered by +himself, is encumbered with difficulties and objections. Among +the ablest admirers of Newton, there have been obstinate disputes +respecting his explanation of his method of ``prime and +\index{Prime and ultimate ratios}% +ultimate ratios.'' + +The so-called ``method of limits'' is frequently attributed +\index{Limits, method of}% +to Newton, but the pure method of limits was never adopted +by him as his method of constructing the calculus. All he +did was to establish in his \textit{Principia} certain principles which +\index{Newton's Principia|(}% +\index{Principia (Newton's)|(}% +are applicable to that method, but which he used for a different +purpose. The first lemma of the first book has been made the +foundation of the method of limits:--- + +``Quantities and the ratios of quantities, which in any finite +time converge continually to equality, and before the end of +that time approach nearer the one to the other than by any +given difference, become ultimately equal.'' + +In this, as well as in the lemmas following this, there are +obscurities and difficulties. Newton appears to teach that a +variable quantity and its limit will ultimately coincide and be +equal. But it is now generally agreed that in the clearest +statements which have been made of the theory of limits, the +variable does not actually reach its limit, though the variable +may approach it as near as we please. + +The full title of Newton's \textit{Principia} is \textit{Philosophiæ Naturalis +Principia Mathematica}. It was printed in 1687 under +the direction, and at the expense, of Dr.~Edmund Halley. +A second edition was brought out in 1713 with many alterations +and improvements, and accompanied by a preface from +%% -----File: 227.png---Folio 213------- +\index{Fluxions|)}% +\index{Kepler's laws}% +\index{Laws of motion}% +\index{Motion, laws of}% +Mr.~Cotes. It was sold out in a few months, but a pirated +edition published in Amsterdam supplied the demand.\Reference{34} The +third and last edition which appeared in England during +Newton's lifetime was published in 1726 by Henry Pemberton. +The \textit{Principia} consists of three books, of which the first two, +constituting the great bulk of the work, treat of the mathematical +principles of natural philosophy, namely, the laws and +conditions of motions and forces. In the third book is drawn +up the constitution of the universe as deduced from the foregoing +principles. The great principle underlying this memorable +work is that of universal gravitation. The first book +\index{Gravitation, theory of}% +\index{Newton's discovery of universal gravitation}% +was completed on April~28, 1686. After the remarkably short +period of three months, the second book was finished. The +third book is the result of the next nine or ten months' +labours. It is only a sketch of a much more extended elaboration +of the subject which he had planned, but which was never +brought to completion. + +The law of gravitation is enunciated in the first book. Its +discovery envelops the name of Newton in a halo of perpetual +glory. The current version of the discovery is as follows: it +was conjectured by Hooke, Huygens, Halley, Wren, Newton, +\index{Halley}% +\index{Hooke}% +\index{Huygens!ref.\ to}% +\index{Wren!ref.\ to}% +and others, that, if Kepler's third law was true (its absolute +\index{Kepler!ref.\ to}% +accuracy was doubted at that time), then the attraction +between the earth and other members of the solar system +varied inversely as the square of the distance. But the proof +of the truth or falsity of the guess was wanting. In~1666 +Newton reasoned, in substance, that if $g$~represent the acceleration +of gravity on the surface of the earth, $r$~be the earth's +radius, $R$~the distance of the moon from the earth, $T$~the time +of lunar revolution, and $a$~a degree at the equator, then, if the +law is true, +\[ +g \frac{r^2}{R^2} = 4 \pi^2 \frac{R}{T^2}, \text{ or } g = \frac{4\pi}{T^2}\left(\frac{R}{r}\right)^3 · 180a. +\] +%% -----File: 228.png---Folio 214------- +\index{Centrifugal force}% +The data at Newton's command gave $R = 60.4r$, $T = 2,360,628$ +seconds, but $a$ only $60$ instead of $69\frac{1}{2}$ English miles. This +wrong value of~$a$ rendered the calculated value of~$g$ smaller +than its true value, as known from actual measurement. It +looked as though the law of inverse squares were not the true +law, and Newton laid the calculation aside. In~1684 he casually +ascertained at a meeting of the Royal Society that Jean +Picard had measured an arc of the meridian, and obtained a +\index{Picard, J.}% +more accurate value for the earth's radius. Taking the corrected +\index{Earth!size of}% +value for~$a$, he found a figure for~$g$ which corresponded +to the known value. Thus the law of inverse squares was +verified. In a scholium in the \textit{Principia}, Newton acknowledged +his indebtedness to Huygens for the laws on centrifugal +\index{Huygens!ref.\ to}% +force employed in his calculation. + +The perusal by the astronomer Adams of a great mass of +\index{Adams!ref.\ to}% +unpublished letters and manuscripts of Newton forming the +Portsmouth collection (which remained private property +until~1872, when its owner placed it in the hands of the +University of Cambridge) seems to indicate that the difficulties +encountered by Newton in the above calculation were of a +different nature. According to Adams, Newton's numerical +verification was fairly complete in~1666, but Newton had not +been able to determine what the attraction of a spherical shell +upon an external point would be. His letters to Halley show +\index{Halley}% +that he did not suppose the earth to attract as though all its +mass were concentrated into a point at the centre. He could +not have asserted, therefore, that the assumed law of gravity +was verified by the figures, though for long distances he might +have claimed that it yielded close approximations. When +Halley visited Newton in~1684, he requested Newton to determine +what the orbit of a planet would be if the law of attraction +were that of inverse squares. Newton had solved a +similar problem for Hooke in~1679, and replied at once that it +%% -----File: 229.png---Folio 215------- +\index{Ellipsoid!(attraction of)}% +\index{Solid of least resistance}% +was an ellipse. After Halley's visit, Newton, with Picard's +\index{Picard, J.}% +new value for the earth's radius, reviewed his early calculation, +\index{Earth!size of}% +and was able to show that if the distances between the +bodies in the solar system were so great that the bodies might +be considered as points, then their motions were in accordance +with the assumed law of gravitation. In 1685 he completed +his discovery by showing that a sphere whose density at any +point depends only on the distance from the centre attracts +an external point as though its whole mass were concentrated +at the centre.\Reference{34} + +Newton's unpublished manuscripts in the Portsmouth collection +show that he had worked out, by means of fluxions and +fluents, his lunar calculations to a higher degree of approximation +than that given in the \textit{Principia}, but that he was unable +to interpret his results geometrically. The papers in that collection +throw light upon the mode by which Newton arrived +at some of the results in the \textit{Principia}, as, for instance, the +famous construction in Book~II., Prop.~25, which is unproved +in the \textit{Principia}, but is demonstrated by him twice in a draft +of a letter to David Gregory, of Oxford.\Reference{34} +\index{Gregory, David F}% + +It is chiefly upon the \textit{Principia} that the fame of Newton +\index{Newton's Principia|)}% +\index{Principia (Newton's)|)}% +rests. Brewster calls it ``the brightest page in the records of +human reason.'' Let us listen, for a moment, to the comments +of Laplace, the foremost among those followers of Newton who +\index{Laplace!ref.\ to}% +grappled with the subtle problems of the motions of planets +under the influence of gravitation: ``Newton has well established +the existence of the principle which he had the merit +of discovering, but the development of its consequences and +advantages has been the work of the successors of this great +mathematician. The imperfection of the infinitesimal calculus, +when first discovered, did not allow him completely to +resolve the difficult problems which the theory of the universe +offers; and he was oftentimes forced to give mere hints, which +%% -----File: 230.png---Folio 216------- +\index{Astronomy!Newton|)}% +\index{Mechanics!Descartes, Wallis, Wren, Huygens, Newton|)}% +were always uncertain till confirmed by rigorous analysis. +Notwithstanding these unavoidable defects, the importance +and the generality of his discoveries respecting the system of +the universe, and the most interesting points of natural philosophy, +the great number of profound and original views, +which have been the origin of the most brilliant discoveries of +the mathematicians of the last century, which were all presented +with much elegance, will insure to the \textit{Principia} a lasting +pre-eminence over all other productions of the human +mind." + +Newton's \textit{Arithmetica Universalis}, consisting of algebraical +lectures delivered by him during the first nine years he was +professor at Cambridge, were published in~1707, or more than +thirty years after they were written. This work was published +by Mr.~Whiston. We are not accurately informed how +\index{Whiston}% +Mr.~Whiston came in possession of it, but according to some +authorities its publication was a breach of confidence on his +part. + +The \textit{Arithmetica Universalis} contains new and important +results on the theory of equations. His theorem on the +\index{Equations!theory of}% +sums of powers of roots is well known. Newton showed +that in equations with real coefficients, imaginary roots always +occur in pairs. His inventive genius is grandly displayed +in his rule for determining the inferior limit of the number +of imaginary roots, and the superior limits for the number +of positive and negative roots. Though less expeditious than +Descartes', Newton's rule always gives as close, and generally +\index{Descartes!ref.\ to}% +closer, limits to the number of positive and negative roots. +Newton did not prove his rule. It awaited demonstration +for a century and a half, until, at last, Sylvester established +\index{Sylvester~ref.\ to}% +a remarkable general theorem which includes Newton's rule +as a special case. + +The treatise on \textit{Method of Fluxions} contains Newton's method +%% -----File: 231.png---Folio 217------- +\index{Cubic curves}% +of approximating to the roots of numerical equations. This +is simply the method of Vieta improved. The same treatise +\index{Vieta!ref.\ to}% +contains ``Newton's parallelogram," which enabled him, in an +\index{Newton's parallelogram}% +equation, $f(x, y) = 0$, to find a series in powers of~$x$ equal to +the variable~$y$. The great utility of this rule lay in its determining +the \emph{form} of the series; for, as soon as the law was +known by which the exponents in the series vary, then the +expansion could be effected by the method of indeterminate +coefficients. The rule is still used in determining the infinite +branches to curves, or their figure at multiple points. Newton +gave no proof for it, nor any clue as to how he discovered it. +The proof was supplied half a century later, by Kaestner and +\index{Kaestner!ref.\ to}% +Cramer, independently.\Reference{37} +\index{Cramer}% + +In 1704 was published, as an appendix to the \textit{Opticks}, the +\textit{Enumeratio linearum tertii ordinis}, which contains theorems +on the theory of curves. Newton divides cubics into seventy-two +species, arranged in larger groups, for which his commentators +have supplied the names ``genera'' and ``classes,'' +recognising fourteen of the former and seven (or four) of the +latter. He overlooked six species demanded by his principles +of classification, and afterwards added by Stirling, Murdoch, +and Cramer. He enunciates the remarkable theorem that the +five species which he names ``divergent parabolas'' give by +\index{Divergent parabolas}% +their projection every cubic curve whatever. As a rule, the +tract contains no proofs. It has been the subject of frequent +conjecture how Newton deduced his results. Recently we have +gotten at the facts, since much of the analysis used by Newton +and a few additional theorems have been discovered among the +Portsmouth papers. An account of the four holograph manuscripts +on this subject has been published by W.~W. Rouse +Ball, in the \textit{Transactions of the London Mathematical Society} +\index{Ball, W.~W.~R.}% +(vol.~xx., pp.~104--143). It is interesting to observe how +Newton begins his research on the classification of cubic +%% -----File: 232.png---Folio 218------- +curves by the algebraic method, but, finding it laborious, +attacks the problem geometrically, and afterwards returns +again to analysis.\Reference{36} + +Space does not permit us to do more than merely mention +Newton's prolonged researches in other departments of science. +He conducted a long series of experiments in optics and is the +author of the corpuscular theory of light. The last of a +\index{Light, theory of}% +number of papers on optics, which he contributed to the Royal +Society, 1687, elaborates the theory of ``fits.'' He explained +the decomposition of light and the theory of the rainbow. By +him were invented the reflecting telescope and the sextant +\index{Sextant}% +(afterwards re-discovered by Thomas Godfrey of Philadelphia\Reference[]{2} +\index{Godfrey}% +and by John Hadley). He deduced a theoretical expression +\index{Hadley}% +for the velocity of sound in air, engaged in experiments on +chemistry, elasticity, magnetism, and the law of cooling, and +entered upon geological speculations. + +During the two years following the close of~1692, Newton +\index{Newton|)}% +suffered from insomnia and nervous irritability. Some thought +that he laboured under temporary mental aberration. Though +he recovered his tranquillity and strength of mind, the time +of great discoveries was over; he would study out questions +propounded to him, but no longer did he by his own accord +enter upon new fields of research. The most noted investigation +after his sickness was the testing of his lunar theory +by the observations of Flamsteed, the astronomer royal. In~1695 +\index{Flamsteed}% +he was appointed warden, and in~1699 master, of the +mint, which office he held until his death. His body was +interred in Westminster Abbey, where in~1731 a magnificent +monument was erected, bearing an inscription ending with, +``Sibi gratulentur mortales tale tantumque exstitisse humani +generis decus.'' It is not true that the Binomial Theorem is +also engraved on it. + +We pass to Leibniz, the second and independent inventor +%% -----File: 233.png---Folio 219------- +\index{Kuhn, J.}% +of the calculus. \textbf{Gottfried Wilhelm Leibniz} (1646--1716) was +\index{Leibniz|(}% +born in Leipzig. No period in the history of any civilised +nation could have been less favourable for literary and scientific +pursuits than the middle of the seventeenth century in Germany. +Yet circumstances seem to have happily combined +to bestow on the youthful genius an education hardly otherwise +obtainable during this darkest period of German history. +He was brought early in contact with the best of the culture +then existing. In his fifteenth year he entered the University +of Leipzig. Though law was his principal study, he applied +himself with great diligence to every branch of knowledge. +Instruction in German universities was then very low. The +higher mathematics was not taught at all. We are told that +a certain John Kuhn lectured on Euclid's \textit{Elements}, but that +his lectures were so obscure that none except Leibniz could +understand them. Later on, Leibniz attended, for a half-year, +at Jena, the lectures of Erhard Weigel, a philosopher and +\index{Weigel}% +mathematician of local reputation. In~1666 Leibniz published +a treatise, \textit{De Arte Combinatoria}, in which he does not pass +beyond the rudiments of mathematics. Other theses written +by him at this time were metaphysical and juristical in +character. A fortunate circumstance led Leibniz abroad. In~1672 +he was sent by Baron Boineburg on a political mission +to Paris. He there formed the acquaintance of the most +distinguished men of the age. Among these was Huygens, +\index{Huygens!ref.\ to}% +who presented a copy of his work on the oscillation of the +pendulum to Leibniz, and first led the gifted young German +to the study of higher mathematics. In~1673 Leibniz went +to London, and remained there from January till March. He +there became incidentally acquainted with the mathematician +Pell, to whom he explained a method he had found on the +\index{Pell}% +summation of series of numbers by their differences. Pell +told him that a similar formula had been published by Mouton +\index{Mouton}% +%% -----File: 234.png---Folio 220------- +\index{Calculating machines}% +\index{Inverse tangents (problem of)}% +\index{p@{$\pi$}: values for!Leibniz's}% +as early as~1670, and then called his attention to Mercator's +\index{Mercator, N.!ref.\ to}% +work on the rectification of the parabola. While in London, +Leibniz exhibited to the Royal Society his arithmetical machine, +\index{Arithmetical machine}% +\index{Machine, arithmetical}% +which was similar to Pascal's, but more efficient and +\index{Pascal!ref.\ to}% +perfect. After his return to Paris, he had the leisure to study +mathematics more systematically. With indomitable energy +he set about removing his ignorance of higher mathematics. +Huygens was his principal master. He studied the geometric +works of Descartes, Honorarius Fabri, Gregory St.~Vincent, +\index{Descartes!ref.\ to}% +and Pascal. A careful study of infinite series led him to the +\index{Infinite series}% +discovery of the following expression for the ratio of the +circumference to the diameter of the circle, previously discovered +by James Gregory:--- +\[ +\frac{\pi}{4} = 1 - \tfrac{1}{3} + \tfrac{1}{5} - \tfrac{1}{7} + \tfrac{1}{9} - \text{etc.} +\] +This elegant series was found in the same way as Mercator's +on the hyperbola. Huygens was highly pleased with it and +urged him on to new investigations. Leibniz entered into a +detailed study of the quadrature of curves and thereby became +\index{Curves!quadrature of}% +\index{Quadrature of curves}% +intimately acquainted with the higher mathematics. Among +the papers of Leibniz is still found a manuscript on quadratures, +written before he left Paris in~1676, but which was +never printed by him. The more important parts of it were +embodied in articles published later in the \textit{Acta Eruditorum}. + +In the study of Cartesian geometry the attention of Leibniz +was drawn early to the direct and inverse problems of tangents. +\index{Tangents!inverse problem of}% +The direct problem had been solved by Descartes for +the simplest curves only; while the inverse had completely +transcended the power of his analysis. Leibniz investigated +both problems for any curve; he constructed what he called +the \textit{triangulum characteristicum}---an infinitely small triangle +\index{Triangulum characteristicum}% +between the infinitely small part of the curve coinciding with +the tangent, and the differences of the ordinates and abscissas. +%% -----File: 235.png---Folio 221------- +\index{Differential calculus|(}% +A curve is here considered to be a polygon. The \textit{triangulum +characteristicum} is similar to the triangle formed by the tangent, +the ordinate of the point of contact, and the \DPtypo{subtangent}{sub-tangent}, +as well as to that between the ordinate, normal, +and sub-normal. It was first employed by Barrow in England, +\index{Barrow!ref.\ to}% +but appears to have been \DPtypo{reinvented}{re-invented} by Leibniz. From +it Leibniz observed the connection existing between the direct +and inverse problems of tangents. He saw also that the latter +could be carried back to the quadrature of curves. All these +results are contained in a manuscript of Leibniz, written in~1673. +One mode used by him in effecting quadratures was +as follows: The rectangle formed by a sub-tangent~$p$ and +an element~$a$ (\ie~infinitely small part of the abscissa) is +equal to the rectangle formed by the ordinate~$y$ and the element~$l$ +of that ordinate; or in symbols, $pa = yl$. But the +summation of these rectangles from zero on gives a right +triangle equal to half the square of the ordinate. Thus, +using Cavalieri's notation, he gets +\index{Cavalieri!ref.\ to}% +\index{Notation!differential calculus}% +\begin{align*} +&\omn pa = \omn yl = \frac{y^2}{2} + \quad \text{(\textit{omn.}\ meaning \textit{omnia}, all)}. +\intertext{But $y = \omn l$; hence} +&\overline{\omn \overline{\omn l}\, \frac{l}{a}} + = \overline{\frac{\omn l^2}{2a}}. +\end{align*} +This equation is especially interesting, since it is here that +Leibniz first introduces a new notation. He says: ``It will +be useful to write~$\int$ for \textit{omn.}, as $\int l$ for $\textit{omn.}\,l$, that is, the +sum of the~$l$'s''; he then writes the equation thus:--- +\[ +\frac{\int \overline{l^2}}{2a} + = \int \overline{{\textstyle\int \bar{l}} \frac{l}{a}}. +\] +From this he deduced the simplest integrals, such as +\[ +\int x = \frac{x^2}{2},\quad \int (x + y) = \int x + \int y. +\] +%% -----File: 236.png---Folio 222------- +\index{Inverse tangents (problem of)}% +\index{Quadrature of curves}% +Since the symbol of summation~$\int$ raises the dimensions, he +concluded that the opposite calculus, or that of differences~$d$, +would lower them. Thus, if $\int l = ya$, then $l = \dfrac{ya}{d}$. The +symbol~$d$ was at first placed by Leibniz in the denominator, +because the lowering of the power of a term was brought +about in ordinary calculation by division. The manuscript +giving the above is dated October~29th, 1675.\Reference{39} This, then, +was the memorable day on which the notation of the new +\index{Notation!differential calculus}% +calculus came to be,---a notation which contributed enormously +to the rapid growth and perfect development of the +calculus. + +Leibniz proceeded to apply his new calculus to the solution +of certain problems then grouped together under the name +of the Inverse Problems of Tangents. He found the cubical +\index{Tangents!inverse problem of}% +parabola to be the solution to the following: To find the +curve in which the sub-normal is reciprocally proportional +to the ordinate. The correctness of his solution was tested +by him by applying to the result Sluze's method of tangents +\index{Sluze!ref.\ to}% +and reasoning backwards to the original supposition. In the +solution of the third problem he changes his notation from +$\dfrac{x}{d}$ to the now usual notation~$dx$. It is worthy of remark +that in these investigations, Leibniz nowhere explains the +significance of $dx$ and~$dy$, except at one place in a marginal +note: ``Idem est $dx$~et~$\dfrac{x}{d}$, id est, differentia inter duas $x$~proximas.'' +Nor does he use the term \emph{differential}, but +always \emph{difference}. Not till ten years later, in the \textit{Acta +Eruditorum}, did he give further explanations of these symbols. +What he aimed at principally was to determine the +change an expression undergoes when the symbol $\int$~or~$d$ is +placed before it. It may be a consolation to students wrestling +with the elements of the differential calculus to know +that it required Leibniz considerable thought and attention\Reference[]{39} +%% -----File: 237.png---Folio 223------- +\index{Debaune@{De Baune}!ref.\ to}% +\index{Inverse tangents (problem of)}% +to determine whether $dx\,dy$ is the same as~$d(xy)$, and +$\dfrac{dx}{dy}$ the same as~$d \smash[t]{\dfrac{x}{y}}$. After considering these questions at +the close of one of his manuscripts, he concluded that the +expressions were not the same, though he could not give the +true value for each. Ten days later, in a manuscript dated +November~21, 1675, he found the equation $y\, d\overline{x} = d\,\overline{xy} - x\,d\overline{y}$, +giving an expression for~$d(xy)$, which he observed to be true +for all curves. He succeeded also in eliminating~$dx$ from +a differential equation, so that it contained only~$dy$, and +thereby led to the solution of the problem under consideration. +``Behold, a most elegant way by which the problems +of the inverse methods of tangents are solved, or at least +\index{Tangents!direct problem of}% +\index{Tangents!inverse problem of}% +are reduced to quadratures!'' Thus he saw clearly that the +inverse problems of tangents could be solved by quadratures, +or, in other words, by the integral calculus. In course of a +\index{Integral calculus}% +half-year he discovered that the direct problem of tangents, +too, yielded to the power of his new calculus, and that thereby +a more general solution than that of Descartes could be +\index{Descartes!ref.\ to}% +obtained. He succeeded in solving all the special problems +of this kind, which had been left unsolved by Descartes. +Of these we mention only the celebrated problem proposed +to Descartes by De~Beaune, viz.~to find the curve whose +ordinate is to its sub-tangent as a given line is to that part +of the ordinate which lies between the curve and a line drawn +from the vertex of the curve at a given inclination to the axis. + +Such was, in brief, the progress in the evolution of the new +calculus made by Leibniz during his stay in Paris. Before +his departure, in October, 1676, he found himself in possession +of the most elementary rules and formulæ of the infinitesimal +calculus. + +From Paris, Leibniz returned to Hanover by way of London +and Amsterdam. In London he met Collins, who showed him +\index{Collins}% +%% -----File: 238.png---Folio 224------- +\index{Tchirnhausen!ref.\ to}% +a part of his scientific correspondence. Of this we shall speak +later. In Amsterdam he discussed mathematics with Sluze, +\index{Sluze!ref.\ to}% +and became satisfied that his own method of constructing +tangents not only accomplished all that Sluze's did, but even +more, since it could be extended to three variables, by which +tangent planes to surfaces could be found; and especially, +since neither irrationals nor fractions prevented the immediate +application of his method. + +In a paper of July~11, 1677, Leibniz gave correct rules for +the differentiation of sums, products, quotients, powers, and +roots. He had given the differentials of a few negative and +fractional powers, as early as November, 1676, but had made +some mistakes. For $d\sqrt{x}$ he had given the erroneous value +$\smash[b]{\dfrac{1}{\sqrt{x}}}$, and in another place the value $-\frac{1}{2} x^{-\frac{1}{2}}$; for $d \dfrac{1}{\DPtypo{x^2}{x^3}}$\DPnote{** PP: Typo presumed on mathematical context.} occurs in +one place the wrong value, $-\smash{\dfrac{2}{x^2}}$, while a few lines lower is +given $-\dfrac{3}{x^4}$, its correct value. + +In 1682 was founded in Berlin the \textit{Acta Eruditorum}, a +journal usually known by the name of \textit{Leipzig Acts}. It +was a partial imitation of the French \textit{Journal des Savans} +(founded in 1665), and the literary and scientific review +published in Germany. Leibniz was a frequent contributor. +Tschirnhaus, who had studied mathematics in Paris with +Leibniz, and who was familiar with the new analysis of +Leibniz, published in the \textit{Acta \DPtypo{Eroditorum}{Eruditorum}} a paper on quadratures, +which consists principally of subject-matter communicated +by Leibniz to Tschirnhaus during a controversy +which they had had on this subject. Fearing that Tschirnhaus +might claim as his own and publish the notation and rules of +the differential calculus, Leibniz decided, at last, to make +public the fruits of his inventions. In~1684, or nine years +after the new calculus first dawned upon the mind of Leibniz, +and nineteen years after Newton first worked at fluxions, +%% -----File: 239.png---Folio 225------- +\index{Debaune@{De Baune}!ref.\ to}% +and three years before the publication of Newton's \textit{Principia}, +Leibniz published, in the \textit{Leipzig Acts}, his first paper on the +differential calculus. He was unwilling to give to the world +all his treasures, but chose those parts of his work which were +most abstruse and least perspicuous. This epoch-making paper +of only six pages bears the title: ``Nova methodus pro maximis +et minimis, itemque tangentibus, \Change{quae}{quæ} nec fractas nec irrationales +quantitates moratur, et singulare pro illis calculi +genus.'' The rules of calculation are briefly stated without +proof, and the meaning of $dx$~and~$dy$ is not made clear. It has +been inferred from this that Leibniz himself had no definite +and settled ideas on this subject. Are $dy$~and~$dx$ finite or +infinitesimal quantities? At first they appear, indeed, to have +been taken as finite, when he says: ``We now call any line +selected at random~$dx$, then we designate the line which is to~$dx$ +as $y$~is to the sub-tangent, by~$dy$, which is the difference of~$y$.'' +Leibniz then ascertains, by his calculus, in what way a +ray of light passing through two differently refracting media, +can travel easiest from one point to another; and then closes +his article by giving his solution, in a few words, of De +Beaune's problem. Two years later (1686) Leibniz published +in the \textit{Acta Eruditorum} a paper containing the rudiments of +the integral calculus. The quantities $dx$~and~$dy$ are there +treated as infinitely small. He showed that by the use of his +notation, the properties of curves could be fully expressed by +equations. Thus the equation +\[ +y = \sqrt{2x - x^2} + \int \frac{dx}{\sqrt{2x - x^2}} +\] +characterises the cycloid.\Reference{38} +\index{Cycloid}% + +The great invention of Leibniz, now made public by his +articles in the \textit{Leipzig Acts}, made little impression upon the +mass of mathematicians. In Germany no one comprehended +%% -----File: 240.png---Folio 226------- +\index{Analysis situs}% +\index{Singular solutions}% +\index{Tchirnhausen!ref.\ to}% +the new calculus except Tschirnhaus, who remained indifferent +to it. The author's statements were too short and succinct +to make the calculus generally understood. The first +to recognise its importance and to take up the study of it +were two foreigners,---the Scotchman \textit{John Craig}, and +\index{Craig, J.}% +the Swiss \textit{James Bernoulli}. The latter wrote Leibniz a +\index{Bernoulli, James (born 1654)!ref.\ to}% +\index{Bernoulli, John (born 1667)!ref.\ to}% +letter in~1687, wishing to be initiated into the mysteries +of the new analysis. Leibniz was then travelling abroad, +so that this letter remained unanswered till~\DPtypo{1790}{1690}. James +Bernoulli succeeded, meanwhile, by close application, in uncovering +the secrets of the differential calculus without assistance. +He and his brother John proved to be mathematicians +of exceptional power. They applied themselves to the new +science with a success and to an extent which made Leibniz +declare that it was as much theirs as his. Leibniz carried on +an extensive correspondence with them, as well as with other +mathematicians. In a letter to John Bernoulli he suggests, +among other things, that the integral calculus be improved by +reducing integrals back to certain fundamental irreducible +forms. The integration of logarithmic expressions was then +studied. The writings of Leibniz contain many innovations, +and anticipations of since prominent methods. Thus he made +use of variable parameters, laid the foundation of \textit{analysis in +situ}, introduced the first notion of determinants in his effort +\index{Determinants}% +to simplify the expression arising in the elimination of the +unknown quantities from a set of linear equations. He +resorted to the device of breaking up certain fractions into +the sum of other fractions for the purpose of easier integration; +he explicitly assumed the principle of continuity; he gave the +\index{Continuity}% +first instance of a ``singular solution,'' and laid the foundation +to the theory of envelopes in two papers, one of which contains +for the first time the terms \emph{co-ordinate} and \emph{axes of co-ordinates}. +\index{Co-ordinates!first use of term}% +He wrote on osculating curves, but his paper contained the +\index{Curves!osculating}% +\index{Curves!theory of}% +%% -----File: 241.png---Folio 227------- +\index{Fluxional controversy|(}% +error (pointed out by John Bernoulli, but not admitted by +him) that an osculating circle will necessarily cut a curve in +four consecutive points. Well known is his theorem on the +$n$th differential coefficient of the product of two functions of a +\index{Differential calculus|)}% +\index{Differential calculus!controversy between Newton and Leibniz|(}% +variable. Of his many papers on mechanics, some are valuable, +\index{Mechanics!Leibniz}% +while others contain grave errors. + +Before tracing the further development of the calculus we +shall sketch the history of that long and bitter controversy +between English and Continental mathematicians on the invention +of the calculus. The question was, did Leibniz invent it +independently of Newton, or was he a plagiarist? +\index{Newton, controversy with Leibniz|(}% + +We must begin with the early correspondence between the +parties appearing in this dispute. Newton had begun using +his notation of fluxions in 1666.\Reference{41} In~1669 Barrow sent Collins +\index{Barrow!ref.\ to}% +\index{Collins}% +Newton's tract, \textit{De Analysi per Equationes}, etc. + +The first visit of Leibniz to London extended from the 11th +of January until March,~1673. He was in the habit of committing +to writing important scientific communications received +from others. In~1890 Gerhardt discovered in the royal library +\index{Gerhardt}% +at Hanover a sheet of manuscript with notes taken by Leibniz +during this journey.\Reference{40} They are headed ``Observata Philosophica +in itinere Anglicano sub initium anni~1673.'' The sheet +is divided by horizontal lines into sections. The sections +given to Chymica, Mechanica, Magnetica, Botanica, Anatomica, +Medica, Miscellanea, contain extensive memoranda, while those +devoted to mathematics have very few notes. Under Geometrica +he says only this: ``Tangentes omnium figurarum. +Figurarum geometricarum explicatio per motum puncti in +moto lati.'' We suspect from this that Leibniz had read +Barrow's lectures. Newton is referred to only under Optica. +Evidently Leibniz did not obtain a knowledge of fluxions +during this visit to London, nor is it claimed that he did by +his opponents. +%% -----File: 242.png---Folio 228------- + +Various letters of Newton, Collins, and others, up to the +\index{Collins}% +beginning of~1676, state that Newton invented a method by +which tangents could be drawn without the necessity of freeing +their equations from irrational terms. Leibniz announced in +1674 to Oldenburg, then secretary of the Royal Society, that +\index{Oldenburg}% +he possessed very general analytical methods, by which he had +found theorems of great importance on the quadrature of the +circle by means of series. In answer, Oldenburg stated +Newton and James Gregory had also discovered methods of +\index{Gregory, James}% +quadratures, which extended to the circle. Leibniz desired to +have these methods communicated to him; and Newton, at +the request of Oldenburg and Collins, wrote to the former the +celebrated letters of June~13 and October~24, 1676. The first +contained the Binomial Theorem and a variety of other matters +relating to infinite series and quadratures; but nothing +directly on the method of fluxions. Leibniz in reply speaks +in the highest terms of what Newton had done, and requests +further explanation. Newton in his second letter just mentioned +explains the way in which he found the Binomial +Theorem, and also communicates his method of fluxions and +fluents in form of an anagram in which all the letters in the +sentence communicated were placed in alphabetical order. +Thus Newton says that his method of drawing tangents was +\[ +6\,\textit{a\,cc\,d\,æ }\ +13\,\textit{e\,ff }\ +7\,\textit{i }\ +3\,\textit{l }\ +9\,\textit{n }\ +4\,\textit{o }\ +4\,\textit{q\,rr }\ +4\,\textit{s }\ +9\,\textit{t }\ +12\,\textit{v\,x}. +\] +The sentence was, \Lat{``Data æquatione quotcunque fluentes +quantitates involvente fluxiones invenire, et vice versa.''} +(``Having any given equation involving never so many flowing +quantities, to find the fluxions, and vice versa.'') Surely this +anagram afforded no hint. Leibniz wrote a reply to Collins, +in which, without any desire of concealment, he explained the +principle, notation, and the use of the differential calculus. + +The death of Oldenburg brought this correspondence to a +%% -----File: 243.png---Folio 229------- +close. Nothing material happened till~1684, when Leibniz +published his first paper on the differential calculus in the +\textit{Leipzig Acts}, so that while Newton's claim to the priority of +invention must be admitted by all, it must also be granted +that Leibniz was the first to give the full benefit of the calculus +to the world. Thus, while Newton's invention remained a +secret, communicated only to a few friends, the calculus of +Leibniz was spreading over the Continent. No rivalry or +hostility existed, as yet, between the illustrious scientists. +Newton expressed a very favourable opinion of Leibniz's +inventions, known to him through the above correspondence +with Oldenburg, in the following celebrated scholium (\textit{Principia}, +\index{Newton's Principia}% +\index{Principia (Newton's)}% +first edition, 1687, Book~II., Prop.~7, scholium):--- + +``In letters which went between me and that most excellent +geometer, G.~G. Leibniz, ten years ago, when I signified that +I was in the knowledge of a method of determining maxima +and minima, of drawing tangents, and the like, and when I +concealed it in transposed letters involving this sentence (Data +æquatione, etc., above cited), that most distinguished man +wrote back that he had also fallen upon a method of the same +kind, and communicated his method, which hardly differed +from mine, except in his forms of words and symbols.'' + +As regards this passage, we shall see that Newton was afterwards +weak enough, as De Morgan says: ``First, to deny the +\index{Demorgan@{De Morgan}!ref.\ to}% +plain and obvious meaning, and secondly, to omit it entirely +from the third edition of the \textit{Principia}.'' On the Continent, +great progress was made in the calculus by Leibniz and his +coadjutors, the brothers James and John Bernoulli, and +\index{Bernoulli, James (born 1654)!ref.\ to}% +\index{Bernoulli, John (born 1667)!ref.\ to}% +Marquis de l'Hospital. In~1695 Wallis informed Newton by +\index{Lhospital@{L'Hospital}!ref.\ to}% +\index{Wallis!ref.\ to}% +letter that ``he had heard that his notions of fluxions passed +in Holland with great applause by the name of `Leibniz's +Calculus Differentialis.'\,'' Accordingly Wallis stated in the +preface to a volume of his works that the calculus differentialis +%% -----File: 244.png---Folio 230------- +was Newton's method of fluxions which had been +communicated to Leibniz in the Oldenburg letters. A review +of Wallis' works, in the \textit{Leipzig Acts} for~1696, reminded the +reader of Newton's own admission in the scholium above +cited. + +For fifteen years Leibniz had enjoyed unchallenged the +honour of being the inventor of his calculus. But in 1699 Fato +de Duillier, a Swiss, who had settled in England, stated in a +\index{Duillier}% +mathematical paper, presented to the Royal Society, his conviction +that Newton was the first inventor; adding that, +whether Leibniz, the second inventor, had borrowed anything +from the other, he would leave to the judgment of those who +had seen the letters and manuscripts of Newton. This was +the first distinct insinuation of plagiarism. It would seem that +the English mathematicians had for some time been cherishing +suspicions unfavourable to Leibniz. A feeling had doubtless +long prevailed that Leibniz, during his second visit to London +in~1676, had or might have seen among the papers of Collins, +\index{Collins}% +Newton's \textit{Analysis per æquationes}, etc., which contained applications +of the fluxionary method, but no systematic development +or explanation of it. Leibniz certainly did see at least +part of this tract. During the week spent in London, he took +note of whatever interested him among the letters and papers +of Collins. His memoranda discovered by Gerhardt in 1849 in +\index{Gerhardt}% +the Hanover library fill two sheets.\Reference{40} The one bearing on our +question is headed ``Excerpta ex tractatu Newtoni Msc.~de +Analysi per æquationes numero terminorum infinitas.'' The +notes are very brief, excepting those \textit{De Resolutione æquationum +affectarum}, of which there is an almost complete copy. +This part was evidently new to him. If he examined +Newton's entire tract, the other parts did not particularly +impress him. From it he seems to have gained nothing pertaining +to the infinitesimal calculus. By the previous introduction +%% -----File: 245.png---Folio 231------- +of his own algorithm he had made greater progress +than by what came to his knowledge in London. Nothing +mathematical that he had received engaged his thoughts in +the immediate future, for on his way back to Holland he composed +a lengthy dialogue on mechanical subjects. + +Duillier's insinuations lighted up a flame of discord which a +whole century was hardly sufficient to extinguish. Leibniz, +who had never contested the priority of Newton's discovery, +and who appeared to be quite satisfied with Newton's admission +in his scholium, now appears for the first time in the +controversy. He made an animated reply in the \textit{Leipzig Acts}, +and complained to the Royal Society of the injustice done him. + +Here the affair rested for some time. In the \textit{Quadrature of +Curves}, published~1704, for the first time, a formal exposition +of the method and notation of fluxions was made public. In~1705 +appeared an unfavourable review of this in the \textit{Leipzig +Acts}, stating that Newton uses and always has used fluxions +for the differences of Leibniz. This was considered by Newton's +friends an imputation of plagiarism on the part of their +chief, but this interpretation was always strenuously resisted +by Leibniz. Keill, professor of astronomy at Oxford, undertook +\index{Keill}% +with more zeal than judgment the defence of Newton. +In a paper inserted in the \textit{Philosophical Transactions} of~1708, +he claimed that Newton was the first inventor of fluxions and +``that the same calculus was afterward published by Leibniz, +the name and the mode of notation being changed.'' Leibniz +complained to the secretary of the Royal Society of bad treatment +and requested the interference of that body to induce +Keill to disavow the intention of imputing fraud. Keill was +not made to retract his accusation; on the contrary, was +authorised by Newton and the Royal Society to explain and +defend his statement. This he did in a long letter. Leibniz +thereupon complained that the charge was now more open than +%% -----File: 246.png---Folio 232------- +before, and appealed for justice to the Royal Society and to +Newton himself. The Royal Society, thus appealed to as a +judge, appointed a committee which collected and reported +upon a large mass of documents---mostly letters from and to +Newton, Leibniz, Wallis, Collins, etc. This report, called the +\index{Collins}% +\textit{Commercium Epistolicum}, appeared in the year~1712 and again +\index{Commercium epistolicum}% +in~1725, with a Recensio prefixed, and additional notes by Keill. +\index{Keill}% +The final conclusion in the \textit{Commercium Epistolicum} was +that Newton was the first inventor. But this was not to the +point. The question was not whether Newton was the first +inventor, but whether Leibniz had stolen the method. The +committee had not formally ventured to assert their belief +that Leibniz was a plagiarist. Yet there runs throughout the +document a desire of proving Leibniz guilty of more than +they meant positively to affirm. Leibniz protested only in +private letters against the proceeding of the Royal Society, +declaring that he would not answer an argument so weak. +John Bernoulli, in a letter to Leibniz, which was published +\index{Bernoulli, John (born 1667)!ref.\ to}% +later in an anonymous tract, is as decidedly unfair towards +Newton as the friends of the latter had been towards Leibniz. +Keill replied, and then Newton and Leibniz appear as mutual +accusers in several letters addressed to third parties. In a +letter to Conti, April~9, 1716, Leibniz again reminded Newton +of the admission he had made in the scholium, which he was +now desirous of disavowing; Leibniz also states that he +always believed Newton, but that, seeing him connive at +accusations which he must have known to be false, it was +natural that he (Leibniz) should begin to doubt. Newton +did not reply to this letter, but circulated some remarks among +his friends which he published immediately after hearing +of the death of Leibniz, November~14, 1716. This paper +of Newton gives the following explanation pertaining to the +scholium in question: ``He [Leibniz] pretends that in my +%% -----File: 247.png---Folio 233------- +\index{Differential calculus!controversy between Newton and Leibniz|)}% +\index{Fluxional controversy|)}% +book of principles I allowed him the invention of the calculus +differentialis, independently of my own; and that to attribute +this invention to myself is contrary to my knowledge there +avowed. But in the paragraph there referred unto I do not +find one word to this purpose.'' In the third edition of the +\textit{Principia}, 1726, Newton omitted the scholium and substituted +\index{Newton's Principia}% +\index{Principia (Newton's)}% +in its place another, in which the name of Leibniz does not +appear. + +National pride and party feeling long prevented the adoption +of impartial opinions in England, but now it is generally admitted +by nearly all familiar with the matter, that Leibniz +really was an independent inventor. Perhaps the most telling +evidence to show that Leibniz was an independent inventor +is found in the study of his mathematical papers (collected +and edited by C.~I. Gerhardt, in six volumes, Berlin, 1849--1860), +\index{Gerhardt}% +which point out a gradual and natural evolution of the +rules of the calculus in his own mind. ``There was throughout +the whole dispute,'' says De~Morgan, ``a confusion between +\index{Demorgan@{De Morgan}!ref.\ to}% +the knowledge of fluxions or differentials and that of a \emph{calculus} +of fluxions or differentials; that is, a digested method with +general rules.'' + +This controversy is to be regretted on account of the long +and bitter alienation which it produced between English and +Continental mathematicians. It stopped almost completely +all interchange of ideas on scientific subjects. The English +adhered closely to Newton's methods and, until about~1820, +\index{Newton, controversy with Leibniz|)}% +remained, in most cases, ignorant of the brilliant mathematical +discoveries that were being made on the Continent. The loss +in point of scientific advantage was almost entirely on the +side of Britain. The only way in which this dispute may be +said, in a small measure, to have furthered the progress of +mathematics, is through the challenge problems by which +each side attempted to annoy its adversaries. +%% -----File: 248.png---Folio 234------- +\index{Brachistochrone (line of swiftest descent)}% +\index{Curve of swiftest descent}% + +The recurring practice of issuing challenge problems was +inaugurated at this time by Leibniz. They were, at first, not +intended as defiances, but merely as exercises in the new calculus. +Such was the problem of the isochronous curve (to +\index{Isochronous curve}% +find the curve along which a body falls with uniform velocity), +proposed by him to the Cartesians in~1687, and solved by +James Bernoulli, himself, and John Bernoulli. James Bernoulli +\index{Bernoulli, John (born 1667)!ref.\ to}% +proposed in the Leipzig \textit{Journal} the question to find the +curve (the catenary) formed by a chain of uniform weight +\index{Catenary}% +suspended freely from its ends. It was resolved by Huygens, +\index{Huygens!ref.\ to}% +Leibniz, and himself. In~1697 John Bernoulli challenged the +best mathematicians in Europe to solve the difficult problem, +to find the curve (the cycloid) along which a body falls from +\index{Cycloid}% +one point to another in the shortest possible time. Leibniz +solved it the day he received it. Newton, de~l'Hospital, and +\index{Lhospital@{L'Hospital}!ref.\ to}% +the two Bernoullis gave solutions. Newton's appeared anonymously +in the \textit{Philosophical Transactions}, but John Bernoulli +recognised in it his powerful mind, ``anquam,'' he says, ``ex +ungue leonem.'' The problem of orthogonal trajectories (a +\index{Trajectories}% +system of curves described by a known law being given, to +describe a curve which shall cut them all at right angles) had +been long proposed in the \textit{Acta Eruditorum}, but failed at +first to receive much attention. It was again proposed in~1716 +by Leibniz, to feel the pulse of the English mathematicians. + +This may be considered as the first defiance problem professedly +aimed at the English. Newton solved it the same +evening on which it was delivered to him, although he was +much fatigued by the day's work at the mint. His solution, +as published, was a general plan of an investigation rather +than an actual solution, and was, on that account, criticised by +Bernoulli as being of no value. Brook Taylor undertook the +\index{Taylor, B.!ref.\ to}% +defence of it, but ended by using very reprehensible language. +%% -----File: 249.png---Folio 235------- +Bernoulli was not to be outdone in incivility, and made a +bitter reply. Not long afterwards Taylor sent an open defiance +to Continental mathematicians of a problem on the +integration of a fluxion of complicated form which was known +to very few geometers in England and supposed to be beyond +the power of their adversaries. The selection was injudicious, +for Bernoulli had long before explained the method of this +and similar integrations. It served only to display the skill +and augment the triumph of the followers of Leibniz. The +last and most unskilful challenge was by John Keill. The +\index{Keill}% +problem was to find the path of a projectile in a medium +which resists proportionally to the square of the velocity. +Without first making sure that he himself could solve it, +Keill boldly challenged Bernoulli to produce a solution. The +latter resolved the question in very short time, not only for a +resistance proportional to the square, but to any power of the +velocity. Suspecting the weakness of the adversary, he repeatedly +offered to send his solution to a confidential person +in London, provided Keill would do the same. Keill never +made a reply, and Bernoulli abused him and cruelly exulted +over him.\Reference{26} + +The explanations of the fundamental principles of the calculus, +as given by Newton and Leibniz, lacked clearness and +rigour. For that reason it met with opposition from several +quarters. In~1694 Bernard Nieuwentyt of Holland denied +\index{Nieuwentyt}% +the existence of differentials of higher orders and objected to +the practice of neglecting infinitely small quantities. These +objections Leibniz was not able to meet satisfactorily. In his +reply he said the value of $\dfrac{dy}{dx}$ in geometry could be expressed +as the ratio of finite quantities. In the interpretation of $dx$~and~$dy$ +Leibniz vacillated. At one time they appear in his +\index{Leibniz|)}% +writings as finite lines; then they are called infinitely small +%% -----File: 250.png---Folio 236------- +quantities, and again, \emph{quantitates inassignabiles}, which spring +from \emph{quantitates assignabiles} by the law of continuity. In this +last presentation Leibniz approached nearest to Newton. + +In England the principles of fluxions were boldly attacked +by Bishop Berkeley, the eminent metaphysician, who argued +\index{Berkeley}% +with great acuteness, contending, among other things, that +the fundamental idea of supposing a finite ratio to exist +between terms absolutely evanescent---``the ghosts of departed +quantities,'' as he called them---was absurd and unintelligible. +The reply made by Jurin failed to remove all the +\index{Jurin}% +objections. Berkeley was the first to point out what was +again shown later by Lazare Carnot, that correct answers were +\index{Carnot, Lazare!ref.\ to}% +reached by a ``compensation of errors.'' Berkeley's attack +was not devoid of good results, for it was the immediate cause +of the work on fluxions by Maclaurin. In France \textit{Michel +\index{Maclaurin!ref.\ to}% +Rolle} rejected the differential calculus and had a controversy +\index{Differential calculus|(}% +\index{Differential calculus!philosophy of}% +\index{Rolle!ref.\ to}% +with \textit{Varignon} on the subject. +\index{Varignon!ref.\ to}% + +Among the most vigorous promoters of the calculus on the +Continent were the Bernoullis. They and Euler made Basel +in Switzerland famous as the cradle of great mathematicians. +The family of Bernoullis furnished in course of a century +eight members who distinguished themselves in mathematics. +We subjoin the following genealogical table:--- +\begin{center} +\begin{minipage}{\linewidth} +\footnotesize +\noindent$\overbrace{ +\begin{tabular}{@{}lll@{}} +\textbf{Jacob}, 1654--1705 & \textit{Nicolaus} & \textbf{Johann}, 1667--1748 \\ +& \qquad$|$ & \qquad$|$ \\ +& \textit{Nicolaus}, 1687--1759 & \textit{Nicolaus}, 1695--1726 \\ +& & \textbf{Daniel}, 1700--1782 \\ +& & \textit{Johann}, 1710--1790 \\ +& \multicolumn{2}{c}{% +\qquad\makebox[0pt][c]{$\overbrace{ +\begin{tabular}{@{}ccc@{}} +\textit{Daniel} & \textit{Johann}, 1744--1807 & \textit{Jacob}, 1758--1789 +\end{tabular}}^{}$}} +\end{tabular}}^{\mbox{\small\textit{Nicolaus Bernoulli}, the Father}}$ +\end{minipage} +\end{center} +\index{Bernoullis, genealogical table of}% +Most celebrated were the two brothers Jacob (James) and +Johann (John), and Daniel, the son of John. James and +%% -----File: 251.png---Folio 237------- +\index{Integral calculus!origin of term}% +\index{Isoperimetrical figures}% +\index{Mechanics!Bernoullis}% +\index{Probability}% +John were staunch friends of Leibniz and worked hand in +\index{Leibniz!ref.\ to}% +hand with him. \textbf{James Bernoulli} (1654--1705) was born in +\index{Bernoulli, James (born 1654)}% +\index{Bernoulli, John (born 1667)!ref.\ to}% +Basel. Becoming interested in the calculus, he mastered it +without aid from a teacher. From 1687 until his death he +occupied the mathematical chair at the University of Basel. +He was the first to give a solution to Leibniz's problem of the +isochronous curve. In his solution, published in the \textit{Acta +Eruditorum}, 1690, we meet for the first time with the word +\emph{integral}. Leibniz had called the integral calculus \textit{calculus +summatorius}, but in 1696 the term \textit{calculus integralis} was +agreed upon between Leibniz and John Bernoulli. James +proposed the problem of the catenary, then proved the correctness +\index{Catenary}% +of Leibniz's construction of this curve, and solved the +more complicated problems, supposing the string to be (1)~of +variable density, (2)~extensible, (3)~acted upon at each point +by a force directed to a fixed centre. Of these problems he +published answers without explanations, while his brother +John gave in addition their theory. He determined the shape +of the ``elastic curve'' formed by an elastic plate or rod fixed +\index{Elastic curve}% +at one end and bent by a weight applied to the other end; of +the ``lintearia,'' a flexible rectangular plate with two sides +\index{Lintearia}% +fixed horizontally at the same height, filled with a liquid; of +the ``volaria,'' a rectangular sail filled with wind. He studied +\index{Volaria}% +the loxodromic and logarithmic spirals, in the last of which +\index{Spirals}% +he took particular delight from its remarkable property of +reproducing itself under a variety of conditions. Following +the example of Archimedes, he willed that the curve be engraved +upon his \Change{tombstone}{tomb-stone} with the inscription ``\textit{eadem mutata +resurgo}.'' In~1696 he proposed the famous problem of isoperimetrical +figures, and in 1701 published his own solution. He +wrote a work on \textit{Ars Conjectandi}, which is a development of +the calculus of probabilities and contains the investigation +now called ``Bernoulli's theorem'' and the so-called ``numbers +\index{Bernoulli's theorem}% +%% -----File: 252.png---Folio 238------- +\index{Brachistochrone (line of swiftest descent)}% +\index{Caustics}% +\index{Mechanics!Bernoullis}% +\index{Numbers of Bernoulli}% +of Bernoulli,'' which are in fact (though not so considered by +\index{Bernoulli, Daniel}% +\index{Bernoulli, Nicolaus (born 1695)}% +\index{Bernoulli, James (born 1654)}% +\index{Bernoulli, John (born 1667)}% +him) the coefficients of $\dfrac{x^n}{n!}$ in the expansion of $(e^x - 1)^{-1}$. Of +his collected works, in three volumes, one was printed in~1713, +the other two in~1744. + +\textbf{John Bernoulli} (1667--1748) was initiated into mathematics +by his brother. He afterwards visited France, where he met +Malebranche, Cassini, De Lahire, Varignon, and de l'Hospital. +For ten years he occupied the mathematical chair at Gröningen +and then succeeded his brother at Basel. He was one of the +most enthusiastic teachers and most successful original investigators +of his time. He was a member of almost every learned +society in Europe. His controversies were almost as numerous +as his discoveries. He was ardent in his friendships, but +unfair, mean, and violent toward all who incurred his dislike---even +his own brother and son. He had a bitter dispute +with James on the isoperimetrical problem. James convicted +him of several paralogisms. After his brother's death he +attempted to substitute a disguised solution of the former for +an incorrect one of his own. John admired the merits of +Leibniz and Euler, but was blind to those of Newton. He +\index{Newton!ref.\ to}% +immensely enriched the integral calculus by his labours. +Among his discoveries are the exponential calculus, the line +of swiftest descent, and its beautiful relation to the path +\index{Curve of swiftest descent}% +described by a ray passing through strata of variable density. +He treated trigonometry by the analytical method, studied +\index{Trigonometry}% +caustic curves and trajectories. Several times he was given +\index{Trajectories}% +prizes by the Academy of Science in Paris. + +Of his sons, \textbf{Nicholas} and \textbf{Daniel} were appointed professors +of mathematics at the same time in the Academy of St.~Petersburg. +The former soon died in the prime of life; the +latter returned to Basel in~1733, where he assumed the chair +of experimental philosophy. His first mathematical publication +%% -----File: 253.png---Folio 239------- +\index{Differential equations}% +was the solution of a differential equation proposed by +Riccati. He wrote a work on hydrodynamics. His investigations +\index{Hydrodynamics}% +\index{Riccati!ref.\ to}% +on probability are remarkable for their boldness and +\index{Probability}% +originality. He proposed the theory of \emph{moral expectation}, +\index{Moral expectation}% +which he thought would give results more in accordance with +our ordinary notions than the theory of \emph{mathematical probability}. +His ``moral expectation'' has become classic, but no +one ever makes use of it. He applies the theory of probability +to insurance; to determine the mortality caused by small-pox +\index{Insurance}% +at various stages of life; to determine the number of survivors +at a given age from a given number of births; to determine +how much inoculation lengthens the average duration of life. +He showed how the differential calculus could be used in the +theory of probability. He and Euler enjoyed the honour of +\index{Euler!ref.\ to}% +having gained or shared no less than ten prizes from the +Academy of Sciences in Paris. + +\textbf{Johann Bernoulli} (born 1710) succeeded his father in the +\index{Bernoulli, Nicolaus (born 1687)}% +\index{Bernoulli, James (born 1758)}% +\index{Bernoulli, John (born 1710)}% +\index{Bernoulli, John (born 1744)}% +professorship of mathematics at Basel. He captured three +prizes (on the capstan, the propagation of light, and the +magnet) from the Academy of Sciences at Paris. \textbf{Nicolaus +Bernoulli} (born 1687) held for a time the mathematical chair +at Padua which Galileo had once filled. \textbf{Johann Bernoulli} +(born 1744) at the age of nineteen was appointed astronomer +royal at Berlin, and afterwards director of the mathematical +department of the Academy. His brother \textit{Jacob} took upon +himself the duties of the chair of experimental physics at +Basel, previously performed by his uncle Jacob, and later +was appointed mathematical professor in the Academy at St.\ +Petersburg. + +Brief mention will now be made of some other mathematicians +belonging to the period of Newton, Leibniz, and the +elder Bernoullis. + +\textbf{Guillaume François Antoine l'Hospital} (1661--1704), a pupil +\index{Hospital,~l'}% +\index{Lhospital@{L'Hospital}}% +%% -----File: 254.png---Folio 240------- +\index{Analytic geometry}% +\index{Cycloid}% +\index{Equations!theory of}% +\index{Hospital,~l'}% +\index{Laplace!ref.\ to}% +\index{Lhospital@{L'Hospital}}% +of John Bernoulli, has already been mentioned as taking +part in the challenges issued by Leibniz and the Bernoullis. +He helped powerfully in making the calculus of Leibniz better +known to the mass of mathematicians by the publication of a +treatise thereon in 1696. This contains for the first time the +method of finding the limiting value of a fraction whose two +terms tend toward zero at the same time. + +Another zealous French advocate of the calculus was \textbf{Pierre +Varignon} (1654--1722). \textbf{Joseph Saurin} (1659--1737) solved the +\index{Saurin}% +\index{Varignon}% +delicate problem of how to determine the tangents at the +multiple points of algebraic curves. \textbf{François Nicole} (1683--1758) +\index{Curves!theory of}% +\index{Nicole}% +in 1717 issued the first systematic treatise on finite +differences, in which he finds the sums of a considerable +\Addendlabel{240}% +number of interesting series. He wrote also on roulettes, +particularly spherical epicycloids, and their rectification. Also +interested in finite differences was \textbf{Pierre Raymond de~Montmort} +\index{Finite differences}% +\index{Montmort, de}% +(1678--1719). His chief writings, on the theory of probability, +\index{Probability}% +served to stimulate his more distinguished successor, De~Moivre. +\index{Demoivre@{De Moivre}}% +\index{Moivre, de}% +\textbf{Jean Paul de~Gua} (1713--1785) gave the demonstration +\index{Gua, de}% +of Descartes' rule of signs, now given in books. This skilful +\index{Descartes!ref.\ to}% +geometer wrote in 1740 a work on analytical geometry, the +\index{Geometry!modern synthetic}% +object of which was to show that most investigations on curves +could be carried on with the analysis of Descartes quite as +easily as with the calculus. He shows how to find the tangents, +asymptotes, and various singular points of curves of all +degrees, and proved by perspective that several of these points +can be at infinity. A mathematician who clung to the methods +of the ancients was \textbf{Philippe de~Lahire} (1640--1718), a pupil of +\index{Lahire, de}% +Desargues. His work on conic sections is purely synthetic, +\index{Desargues!ref.\ to}% +but differs from ancient treatises in deducing the properties of +conics from those of the circle in the same manner as did +Desargues and Pascal. His innovations stand in close relation +\index{Pascal!ref.\ to}% +with modern synthetic geometry. He wrote on roulettes, on +%% -----File: 255.png---Folio 241------- +\index{Caustics}% +\index{Magic squares}% +\index{p@{$\pi$}: values for!Fagnano's}% +\index{Tchirnhausen}% +graphical methods, epicycloids, conchoids, and on magic +squares. \textbf{Michel Rolle} (1652--1719) is the author of a theorem +\index{Rolle}% +named after him. + +Of Italian mathematicians, Riccati and Fagnano must not +\index{Fagnano}% +\index{Riccati}% +remain unmentioned. \textbf{Jacopo Francesco, Count Riccati} (1676--1754) +is best known in connection with his problem, called +Riccati's equation, published in the \textit{Acta Eruditorum} in~1724. +He succeeded in integrating this differential equation for some +special cases. A geometrician of remarkable power was \textbf{Giulio +Carlo, Count de~Fagnano} (1682--1766). He discovered the following +formula, $\pi = 2i \log \dfrac{1 - i}{1 + i}$, in which he anticipated Euler +\index{Euler!ref.\ to}% +in the use of imaginary exponents and logarithms. His studies +\index{Exponents}% +\index{Imaginary quantities}% +on the rectification of the ellipse and hyperbola are the starting-points +of the theory of elliptic functions. He showed, for +\index{Elliptic functions}% +instance, that two arcs of an ellipse can be found in an indefinite +number of ways, whose difference is expressible by a +right line. + +In Germany the only noted contemporary of Leibniz is +\index{Leibniz!ref.\ to}% +\textbf{Ehrenfried Walter Tschirnhausen} (1651--1708), who discovered +the caustic of reflection, experimented on metallic reflectors +and large burning-glasses, and gave us a method of transforming +equations named after him. Believing that the most +\index{Equations!theory of}% +simple methods (like those of the ancients) are the most +correct, he concluded that in the researches relating to the +properties of curves the calculus might as well be dispensed +with. + +After the death of Leibniz there was in Germany not a +single mathematician of note. \textbf{Christian Wolf} (1679--1754), +\index{Wolf, C.}% +professor at Halle, was ambitious to figure as successor of +Leibniz, but he ``forced the ingenious ideas of Leibniz into a +pedantic scholasticism, and had the unenviable reputation of +having presented the elements of the arithmetic, algebra, and +%% -----File: 256.png---Folio 242------- +\index{Curves!theory of}% +\index{Differential calculus|)}% +\index{Strings, vibrating}% +\index{Vibrating strings}% +analysis developed since the time of the Renaissance in the +form of Euclid,---of course only in outward form, for into the +spirit of them he was quite unable to penetrate.''\Reference{16} + +The contemporaries and immediate successors of Newton in +Great Britain were men of no mean merit. We have reference +to Cotes, Taylor, Maclaurin, and De~Moivre. We are +\index{Cotes}% +\index{Demoivre@{De Moivre}}% +\index{Moivre, de}% +\index{Taylor, B.}% +told that at the death of \textbf{Roger Cotes} (1682--1716), Newton +exclaimed, ``If Cotes had lived, we might have known something.'' +It was at the request of Dr.~Bentley that Cotes +undertook the publication of the second edition of Newton's +\textit{Principia}. His mathematical papers were published after his +\index{Newton's Principia}% +\index{Principia (Newton's)}% +death by Robert Smith, his successor in the Plumbian professorship +\index{Smith, R.}% +at Trinity College. The title of the work, \textit{Harmonia +Mensurarum}, was suggested by the following theorem +contained in it: If on each radius vector, through a fixed point~$O$, +there be taken a point~$R$, such that the reciprocal of $OR$ be +the arithmetic mean of the reciprocals of $OR_1, OR_2, \ldots OR_n$, +then the locus of~$R$ will be a straight line. In this work +progress was made in the application of logarithms and the +\index{Logarithms}% +properties of the circle to the calculus of fluents. To Cotes +we owe a theorem in trigonometry which depends on the +\index{Trigonometry}% +forming of factors of~$x^n - 1$. Chief among the admirers of +Newton were Taylor and Maclaurin. The quarrel between +English and Continental mathematicians caused them to work +quite independently of their great contemporaries across the +Channel. + +\textbf{Brook Taylor} (1685--1731) was interested in many branches +of learning, and in the latter part of his life engaged mainly in +religious and philosophic speculations. His principal work, +\textit{Methodus incrementorum directa et inversa}, London, 1715--1717, +added a new branch to mathematics, now called ``finite differences.'' +\index{Finite differences}% +He made many important applications of it, particularly +to the study of the form of movement of vibrating +%% -----File: 257.png---Folio 243------- +\index{Centre!of oscillation}% +\index{Mechanics!Taylor}% +strings, first reduced to mechanical principles by him. This +work contains also ``Taylor's theorem,'' the importance of +\index{Taylor's theorem}% +which was not recognised by analysts for over fifty years, +until Lagrange pointed out its power. His proof of it does not +consider the question of convergency, and is quite worthless. +The first rigorous proof was given a century later by Cauchy. +\index{Cauchy!ref.\ to}% +Taylor's work contains the first correct explanation of astronomical +refraction. He wrote also a work on linear perspective, +a treatise which, like his other writings, suffers for +want of fulness and clearness of expression. At the age of +twenty-three he gave a remarkable solution of the problem of +the centre of oscillation, published in~1714. His claim to +\index{Oscillation, centre of}% +priority was unjustly disputed by John Bernoulli. +\index{Bernoulli, John (born 1667)!ref.\ to}% + +\textbf{Colin Maclaurin} (1698--1746) was elected professor of mathematics +\index{Maclaurin}% +at Aberdeen at the age of nineteen by competitive +examination, and in 1725 succeeded James Gregory at the University +\index{Gregory, James}% +of Edinburgh. He enjoyed the friendship of Newton, +\index{Newton!ref.\ to}% +and, inspired by Newton's discoveries, he published in 1719 his +\textit{Geometria Organica}, containing a new and remarkable mode +of generating conics, known by his name. A second tract, +\textit{De Linearum geometricarum Proprietatibus}, 1720, is remarkable +for the elegance of its demonstrations. It is based upon two +theorems: the first is the theorem of Cotes; the second is +\index{Cotes!ref.\ to}% +Maclaurin's: If through any point~$O$ a line be drawn meeting +the curve in $n$~points, and at these points tangents be drawn, +and if any other line through~$O$ cut the curve in $R_1$, $R_2$, etc., +and the system of $n$~tangents in $r_1$, $r_2$, etc., then $\sum \dfrac{1}{OR} = \sum \dfrac{1}{Or}$. +This and Cotes' theorem are generalisations of theorems of +Newton. Maclaurin uses these in his treatment of curves of +\index{Curves!theory of}% +the second and third degree, culminating in the remarkable +theorem that if a quadrangle has its vertices and the two +points of intersection of its opposite sides upon a curve of the +%% -----File: 258.png---Folio 244------- +third degree, then the tangents drawn at two opposite vertices +cut each other on the curve. He deduced independently +Pascal's theorem on the hexagram. The following is his extension +of this theorem (\textit{Phil.\ Trans.}, 1735): If a polygon +move so that each of its sides passes through a fixed point, +and if all its summits except one describe curves of the degrees +$m$,~$n$, $p$,~etc., respectively, then the free summit moves on a +curve of the degree $2 mnp\cdots$,\DPnote{** PP: Why dots?} which reduces to $mnp \cdots$ when +the fixed points all lie on a straight line. Maclaurin wrote on +\index{Maclaurin!ref.\ to}% +pedal curves. He is the author of an \textit{Algebra}. The object of +his treatise on \textit{Fluxions} was to found the doctrine of fluxions +on geometric demonstrations after the manner of the ancients, +and thus, by rigorous exposition, answer such attacks as Berkeley's +that the doctrine rested on false reasoning. The \textit{Fluxions} +contained for the first time the correct way of distinguishing +between maxima and minima, and explained their use in the +\index{Maxima and minima}% +theory of multiple points. ``Maclaurin's theorem'' was previously +given by James Stirling, and is but a particular case +\index{Stirling}% +of ``Taylor's theorem.'' Appended to the treatise on \textit{Fluxions} +is the solution of a number of beautiful geometric, mechanical, +and astronomical problems, in which he employs ancient +methods with such consummate skill as to induce Clairaut to +\index{Clairaut!ref.\ to}% +abandon analytic methods and to attack the problem of the +figure of the earth by pure geometry. His solutions commanded +the liveliest admiration of Lagrange. Maclaurin investigated +\index{Lagrange!ref.\ to}% +the attraction of the ellipsoid of revolution, and +showed that a homogeneous liquid mass revolving uniformly +around an axis under the action of gravity must assume the +form of an ellipsoid of revolution. Newton had given this +\index{Newton!ref.\ to}% +theorem without proof. Notwithstanding the genius of Maclaurin, +his influence on the progress of mathematics in Great +Britain was unfortunate; for, by his example, he induced his +countrymen to neglect analysis and to be indifferent to the +%% -----File: 259.png---Folio 245------- +wonderful progress in the higher analysis made on the Continent. + +It remains for us to speak of \textbf{Abraham de~Moivre} (1667--1754), +\index{Demoivre@{De Moivre}}% +\index{Moivre, de}% +who was of French descent, but was compelled to leave France +at the age of eighteen, on the Revocation of the Edict of Nantes. +He settled in London, where he gave lessons in mathematics. +He lived to the advanced age of eighty-seven and sank into a +state of almost total lethargy. His subsistence was latterly +dependent on the solution of questions on games of chance and +problems on probabilities, which he was in the habit of giving +at a tavern in St.~Martin's Lane. Shortly before his death he +declared that it was necessary for him to sleep ten or twenty +minutes longer every day. The day after he had reached the +total of over twenty-three hours, he slept exactly twenty-four +hours and then passed away in his sleep. De~Moivre enjoyed +the friendship of Newton and Halley. His power as a mathematician +lay in analytic rather than geometric investigation. +He revolutionised higher trigonometry by the discovery of the +\index{Trigonometry}% +theorem known by his name and by extending the theorems on +the multiplication and division of sectors from the circle to the +hyperbola. His work on the theory of probability surpasses +\index{Probability}% +anything done by any other mathematician except Laplace. +\index{Laplace!ref.\ to}% +His principal contributions are his investigations respecting +the Duration of Play, his Theory of Recurring Series, and +\index{Series}% +his extension of the value of Bernoulli's theorem by the aid +of Stirling's theorem.\Reference{42} His chief works are the \textit{Doctrine of +Chances}, 1716, the \textit{Miscellanea Analytica}, 1730, and his papers +in the \textit{Philosophical Transactions}. +%% -----File: 260.png---Folio 246------- + + +\Section{Euler, Lagrange, and Laplace} +\index{Euler!ref.\ to}% +\index{Lagrange!ref.\ to}% + +During the epoch of ninety years from 1730 to 1820 the French +and Swiss cultivated mathematics with most brilliant success. +No previous period had shown such an array of illustrious +names. At this time Switzerland had her Euler; France, her +Lagrange, Laplace, Legendre, and Monge. The mediocrity of +French mathematics which marked the time of Louis~XIV. +was now followed by one of the very brightest periods of all +history. England and Germany, on the other hand, which +during the unproductive period in France had their Newton +and Leibniz, could now boast of no great mathematician. +France now waved the mathematical sceptre. Mathematical +studies among the English and German people had sunk to +the lowest ebb. Among them the direction of original research +was ill-chosen. The former adhered with excessive partiality +to ancient geometrical methods; the latter produced the combinatorial +school, which brought forth nothing of value. + +The labours of Euler, Lagrange, and Laplace lay in higher +analysis, and this they developed to a wonderful degree. By +them analysis came to be completely severed from geometry. +During the preceding period the effort of mathematicians +not only in England, but, to some extent, even on the continent, +had been directed toward the solution of problems +clothed in geometric garb, and the results of calculation +were usually reduced to geometric form. A change now +took place. Euler brought about an emancipation of the +analytical calculus from geometry and established it as an +independent science. Lagrange and Laplace scrupulously +adhered to this separation. Building on the broad foundation +laid for higher analysis and mechanics by Newton +and Leibniz, Euler, with matchless fertility of mind, erected +%% -----File: 261.png---Folio 247------- +an elaborate structure. There are few great ideas pursued +by succeeding analysts which were not suggested by Euler, +or of which he did not share the honour of invention. +With, perhaps, less exuberance of invention, but with more +comprehensive genius and profounder reasoning, Lagrange +\index{Lagrange!ref.\ to}% +developed the infinitesimal calculus and put analytical +\index{Calculus!of variations}% +mechanics into the form in which we now know it. Laplace +applied the calculus and mechanics to the elaboration +of the theory of universal gravitation, and thus, largely extending +and supplementing the labours of Newton, gave a full +analytical discussion of the solar system. He also wrote an +epoch-marking work on Probability. Among the analytical +branches created during this period are the calculus of Variations +by Euler and Lagrange, Spherical Harmonics by Laplace +\index{Spherical Harmonics}% +and Legendre, and Elliptic Integrals by Legendre. +\index{Elliptic integrals}% +\index{Legendre!ref.\ to}% + +Comparing the growth of analysis at this time with the +growth during the time of Gauss, Cauchy, and recent mathematicians, +\index{Cauchy!ref.\ to}% +\index{Gauss!ref.\ to}% +we observe an important difference. During the +former period we witness mainly a development with reference +to \emph{form}. Placing almost implicit confidence in results of +calculation, mathematicians did not always pause to discover +rigorous proofs, and were thus led to general propositions, +some of which have since been found to be true in only special +cases. The Combinatorial School in Germany carried this +\index{Combinatorial School}% +tendency to the greatest extreme; they worshipped formalism +and paid no attention to the actual contents of formulæ. But +in recent times there has been added to the dexterity in the +formal treatment of problems, a much-needed rigour of demonstration. +A good example of this increased rigour is seen in +the present use of infinite series as compared to that of Euler, +\index{Infinite series}% +and of Lagrange in his earlier works. + +The ostracism of geometry, brought about by the master-minds %[** PP: Only instance] +of this period, could not last permanently. Indeed, a +%% -----File: 262.png---Folio 248------- +new geometric school sprang into existence in France before +the close of this period. Lagrange would not permit a single +\index{Lagrange!ref.\ to}% +diagram to appear in his \textit{Mécanique analytique}, but thirteen +years before his death, Monge published his epoch-making +\index{Monge!ref.\ to}% +\textit{Géometrie descriptive}. + +\textbf{Leonhard Euler} (1707--1783) was born in Basel. His father, +\index{Euler|(}% +a minister, gave him his first instruction in mathematics and +then sent him to the University of Basel, where he became a +favourite pupil of John Bernoulli. In his nineteenth year he +composed a dissertation on the masting of ships, which received +the second prize from the French Academy of Sciences. +When John Bernoulli's two sons, Daniel and Nicolaus, went to +Russia, they induced Catharine~I., in 1727, to invite their friend +Euler to St.~Petersburg, where Daniel, in 1733, was assigned to +the chair of mathematics. In 1735 the solving of an astronomical +problem, proposed by the Academy, for which several +eminent mathematicians had demanded some months' time, +was achieved in three days by Euler with aid of improved +methods of his own. But the effort threw him into a fever +and deprived him of the use of his right eye. With still +superior methods this same problem was solved later by the +illustrious Gauss in one hour!\Reference{47} The despotism of Anne~I. +\index{Gauss!ref.\ to}% +caused the gentle Euler to shrink from public affairs and to +devote all his time to science. After his call to Berlin by +Frederick the Great in 1747, the queen of Prussia, who +received him kindly, wondered how so distinguished a scholar +should be so timid and reticent. Euler naïvely replied, +``Madam, it is because I come from a country where, when one +speaks, one is hanged.'' In 1766 he with difficulty obtained +permission to depart from Berlin to accept a call by Catharine~II. +to St.~Petersburg. Soon after his return to Russia he +became blind, but this did not stop his wonderful literary +productiveness, which continued for seventeen years, until the +%% -----File: 263.png---Folio 249------- +\index{Beta function}% +\index{Gamma function}% +\index{Isoperimetrical figures}% +\index{Notation!trigonometry}% +day of his death.\Reference{45} He dictated to his servant his \textit{Anleitung +zur Algebra}, 1770, which, though purely elementary, is meritorious +as one of the earliest attempts to put the fundamental +processes on a sound basis. + +Euler wrote an immense number of works, chief of which +are the following: \textit{Introductio in analysin infinitorum}, 1748, +a work that caused a revolution in analytical mathematics, a +subject which had hitherto never been presented in so general +and systematic manner; \textit{Institutiones calculi differentialis}, 1755, +and \textit{Institutiones calculi integralis}, 1768--1770, which were the +most complete and accurate works on the calculus of that time, +\index{Calculus!of variations}% +and contained not only a full summary of everything then +known on this subject, but also the Beta and Gamma Functions +and other original investigations; \textit{Methodus inveniendi +lineas curvas maximi minimive proprietate gaudentes}, 1744, +which, displaying an amount of mathematical genius seldom +rivalled, contained his researches on the calculus of variations +(a subject afterwards improved by Lagrange), to the invention +of which Euler was led by the study of isoperimetrical curves, +the brachistochrone in a resisting medium, and the theory of +geodesics (subjects which had previously engaged the attention +\index{Geodesics}% +of the elder Bernoullis and others); the \textit{Theoria motuum planetarum +et cometarum}, 1744, \textit{Theoria motus lunæ}, 1753, \textit{Theoria +motuum lunæ}, 1772, are his chief works on astronomy; \textit{Ses +lettres à une princesse d'Allemagne sur quelques sujets de +Physique et de Philosophie}, 1770, was a work which enjoyed +great popularity. + +We proceed to mention the principal innovations and inventions +of Euler. He treated trigonometry as a branch of +\index{Trigonometry}% +analysis, introduced (simultaneously with Thomas Simpson in +\index{Simpson}% +England) the now current abbreviations for trigonometric +functions, and simplified formulæ by the simple expedient +of designating the angles of a triangle by $A$,~$B$,~$C$, and the +%% -----File: 264.png---Folio 250------- +\index{Bezout@{Bézout}!ref.\ to}% +opposite sides by $a$,~$b$,~$c$, respectively. He pointed out the +relation between trigonometric and exponential functions. In +a paper of 1737 we first meet the symbol~$\pi$ to denote $3.14159\ldots$.\Reference{21} +\index{p@{$\pi$}: values for!selection of letter~$\pi$}% +\Addendlabel{250}% +Euler laid down the rules for the transformation of co-ordinates +\index{Euler!ref.\ to}% +in space, gave a methodic analytic treatment of plane curves +and of surfaces of the second order. He was the first to +\index{Surfaces, theory of}% +discuss the equation of the second degree in three variables, +and to classify the surfaces represented by it. By criteria +analogous to those used in the classification of conics he +obtained five species. He devised a method of solving \Change{bi-quadratic}{biquadratic} +equations by assuming $x = \sqrt{\smash[b]{p}} + \sqrt{\smash[b]{q}} + \sqrt{r}$, with the +\index{Equations!solution of}% +\index{Equations!theory of}% +hope that it would lead him to a general solution of algebraic +equations. The method of elimination by solving a series of +\index{Elimination}% +linear equations (invented independently by \DPtypo{Bezout}{Bézout}) and the +method of elimination by symmetric functions, are due to him.\Reference{20} +\index{Symmetric functions}% +Far reaching are Euler's researches on logarithms. Leibniz +\index{Leibniz!ref.\ to}% +\index{Logarithms}% +and John Bernoulli once argued the question whether a +\index{Bernoulli, John (born 1667)!ref.\ to}% +negative number has a logarithm. Bernoulli claimed that +since $(-a)^2 = (+a)^2$, we have $\log(-a)^2= \log(+a)^2$ and +$2 \log(-a) = 2 \log(+a)$, and finally $\log(-a) = \log(+a)$. +Euler proved that $a$ has really an infinite number of logarithms, +all of which are imaginary when $a$ is negative, and all +except one when $a$ is positive. He then explained how +$\log(-a)^2$ might equal $\log(+a)^2$, and yet $\log(-a)$ not +equal $\log(+a)$. + +The subject of infinite series received new life from him. +\index{Infinite series}% +To his researches on series we owe the creation of the theory of +definite integrals by the development of the so-called \emph{Eulerian +integrals}. He warns his readers occasionally against the use +of divergent series, but is nevertheless very careless himself. +The rigid treatment to which infinite series are subjected now +was then undreamed of. No clear notions existed as to what +constitutes a convergent series. Neither Leibniz nor Jacob +%% -----File: 265.png---Folio 251------- +and John Bernoulli had entertained any serious doubt of the +\index{Bernoulli, Nicolaus (born 1687)}% +\index{Bernoulli, James (born 1654)!ref.\ to}% +\index{Bernoulli, John (born 1667)!ref.\ to}% +correctness of the expression $\frac{1}{2} = 1 - 1 + 1 - 1 +\dotsb$. Guido +Grandi went so far as to conclude from this that $\frac{1}{2} = 0 + 0 + +\index{Grandi}% +0 +\dotsb$. In the treatment of series Leibniz advanced a metaphysical +\index{Leibniz!ref.\ to}% +method of proof which held sway over the minds of +the elder Bernoullis, and even of Euler.\Reference{46} The tendency of +that reasoning was to justify results which seem to us now +highly absurd. The looseness of treatment can best be seen +from examples. The very paper in which Euler cautions +against divergent series contains the proof that +\begin{gather*} +\dotsb \frac{1}{n^2} + \frac{1}{n} + 1 + n + n^2 +\dotsb = 0 \text{ as follows:} \\ +n+n^2+\dotsb = \frac{n}{1-n},\quad 1+\frac{1}{n}+\frac{1}{n^2}+\dotsb=\frac{n}{n-1}; +\end{gather*} +these added give zero. Euler has no hesitation to write +$1-3+5-7+\dotsb=0$, and no one objected to such results +excepting Nicolaus Bernoulli, the nephew of John and Jacob. +Strange to say, Euler finally succeeded in converting Nicolaus +Bernoulli to his own erroneous views. At the present time +it is difficult to believe that Euler should have confidently +written $\sin\phi - 2 \sin 2\phi + 3 \sin 3\phi - 4 \sin 4\phi+\dotsb = 0$, but +such examples afford striking illustrations of the want of +scientific basis of certain parts of analysis at that time. +Euler's proof of the binomial formula for negative and +\index{Binomial formula}% +fractional exponents, which has been reproduced in elementary +text-books of even recent years, is faulty. A remarkable +development, due to Euler, is what he named the hypergeometric +series, the summation of which he observed to be +dependent upon the integration of a linear differential equation +of the second order, but it remained for Gauss to point +\index{Gauss!ref.\ to}% +out that for special values of its letters, this series represented +nearly all functions then known. + +Euler developed the calculus of finite differences in the first +\index{Finite differences}% +%% -----File: 266.png---Folio 252------- +\index{Numbers!theory of numbers}% +\index{Theory of numbers}% +chapters of his \textit{Institutiones calculi differentialis}, and then +deduced the differential calculus from it. He established a +theorem on homogeneous functions, known by his name, and +\index{Addition theorem of elliptic integrals}% +contributed largely to the theory of differential equations, a +\index{Differential equations}% +subject which had received the attention of Newton, Leibniz, +\index{Leibniz!ref.\ to}% +\index{Newton!ref.\ to}% +and the Bernoullis, but was still undeveloped. Clairaut, +\index{Clairaut!ref.\ to}% +Fontaine, and Euler about the same time observed criteria of +\index{Fontaine}% +integrability, but Euler in addition showed how to employ +them to determine integrating factors. The principles on +which the criteria rested involved some degree of obscurity. +The celebrated addition-theorem for elliptic integrals was first +\index{Elliptic integrals}% +established by Euler. He invented a new algorithm for +continued fractions, which he employed in the solution of +\index{Continued fractions}% +\index{Fractions!continued}% +the indeterminate equation $ax + by = c$. We now know that +substantially the same solution of this equation was given +$1000$ years earlier, by the Hindoos. By giving the factors of +the number $2^{2^n} + 1$ when $n = 5$, he pointed out that this expression +did not always represent primes, as was supposed by +Fermat. He first supplied the proof to ``Fermat's theorem,'' +\index{Fermat!ref.\ to}% +\index{Fermat's theorem}% +and to a second theorem of Fermat, which states that every +prime of the form $4n + 1$ is expressible as the sum of two +squares in one and only one way. A third theorem of Fermat, +that $x^n + y^n = z^n$, has no integral solution for values of $n$ +greater than~$2$, was proved by Euler to be correct when $n = 3$. +Euler discovered four theorems which taken together make +out the great law of quadratic reciprocity, a law independently +\index{Quadratic reciprocity}% +discovered by Legendre.\Reference{48} Euler enunciated and proved a +\index{Legendre!ref.\ to}% +well-known theorem, giving the relation between the number +of vertices, faces, and edges of certain polyhedra, which, +however, appears to have been known to Descartes. The +powers of Euler were directed also towards the fascinating +subject of the theory of probability, in which he solved some +\index{Probability}% +difficult problems. +%% -----File: 267.png---Folio 253------- +\index{Action, least}% + +Of no little importance are Euler's labours in analytical +mechanics. Says Whewell: ``The person who did most to +\index{Mechanics!Euler}% +\index{Whewell}% +give to analysis the generality and symmetry which are now +its pride, was also the person who made mechanics analytical; +I mean Euler.''\Reference{11} He worked out the theory of the rotation of +a body around a fixed point, established the general equations +of motion of a free body, and the general equation of hydrodynamics. +He solved an immense number and variety of mechanical +problems, which arose in his mind on all occasions. Thus, +on reading Virgil's lines, ``The anchor drops, the rushing keel +is staid,'' he could not help inquiring what would be the +ship's motion in such a case. About the same time as Daniel +Bernoulli he published the \textit{Principle of the Conservation of +\index{Conservation!of areas}% +Areas} and defended the principle of ``least action,'' advanced +\index{Areas, conservation of}% +\index{Least action}% +by Maupertius. He wrote also on tides and on sound. +\index{Maupertius}% + +Astronomy owes to Euler the method of the variation of +\index{Astronomy!more recent researches}% +arbitrary constants. By it he attacked the problem of perturbations, +explaining, in case of two planets, the secular variations +of eccentricities, nodes, etc. He was one of the first +to take up with success the theory of the moon's motion by +giving approximate solutions to the ``problem of three bodies.'' +\index{Problem of three bodies}% +\index{Three bodies, problem of}% +He laid a sound basis for the calculation of tables of the moon. +These researches on the moon's motion, which captured two +prizes, were carried on while he was blind, with the assistance +of his sons and two of his pupils. + +Most of his memoirs are contained in the transactions of +the Academy of Sciences at St.~Petersburg, and in those of +the Academy at Berlin. From 1728 to 1783 a large portion +of the Petropolitan transactions were filled by his writings. +He had engaged to furnish the Petersburg Academy with +memoirs in sufficient number to enrich its acts for twenty +years---a promise more than fulfilled, for down to 1818 the +volumes usually contained one or more papers of his. It has +%% -----File: 268.png---Folio 254------- +been said that an edition of Euler's complete works would fill +$16,000$ quarto pages. His mode of working was, first to concentrate +his powers upon a special problem, then to solve +separately all problems growing out of the first. No one +excelled him in dexterity of accommodating methods to special +problems. It is easy to see that mathematicians could not +long continue in Euler's habit of writing and publishing. The +material would soon grow to such enormous proportions as to +be unmanageable. We are not surprised to see almost the +opposite in Lagrange, his great successor. The great Frenchman +\index{Lagrange!ref.\ to}% +delighted in the general and abstract, rather than, like +Euler, in the special and concrete. His writings are condensed +and give in a nutshell what Euler narrates at great +\index{Euler|)}% +length. + +\textbf{Jean-le-Rond D'Alembert} (1717--1783) was exposed, when +\index{D'Alembert|(}% +\index{D'Alembert!ref.\ to}% +\index{D'Alembert's principle}% +an infant, by his mother in a market by the church of St.\ +Jean-le-Rond, near the Nôtre-Dame in Paris, from which he +derived his Christian name. He was brought up by the wife +of a poor glazier. It is said that when he began to show signs +of great talent, his mother sent for him, but received the +reply, ``You are only my step-mother; the glazier's wife is +my mother.'' His father provided him with a yearly income. +D'Alembert entered upon the study of law, but such was his +love for mathematics, that law was soon abandoned. At the +age of twenty-four his reputation as a mathematician secured +for him admission to the Academy of Sciences. In 1743 +appeared his \textit{Traité de dynamique}, founded upon the important +general principle bearing his name: The impressed forces are +equivalent to the effective forces. D'Alembert's principle +seems to have been recognised before him by Fontaine, and +\index{Fontaine}% +in some measure by John Bernoulli and Newton. D'Alembert +\index{Newton!ref.\ to}% +gave it a clear mathematical form and made numerous applications +of it. It enabled the laws of motion and the reasonings +%% -----File: 269.png---Folio 255------- +\index{Hydrodynamics}% +\index{Hydrostatics}% +\index{Infinite series}% +\index{Strings, vibrating}% +\index{Vibrating strings}% +depending on them to be represented in the most general +form, in analytical language. D'Alembert applied it in 1744 +in a treatise on the equilibrium and motion of fluids, in 1746 +to a treatise on the general causes of winds, which obtained +a prize from the Berlin Academy. In both these treatises, as +also in one of 1747, discussing the famous problem of vibrating +chords, he was led to partial differential equations. He was +\index{Partial differential equations}% +a leader among the pioneers in the study of such equations. +To the equation $\dfrac{\partial^2 y}{\partial t^2} = a^2 \dfrac{\partial^2 y}{\partial x^2}$, arising in the problem of vibrating +chords, he gave as the general solution, +\[ +y = f(x + at) + \phi(x - at), +\] +and showed that there is only one arbitrary function, if $y$ be +supposed to vanish for $x = 0$ and~$x=l$. Daniel Bernoulli, +\index{Bernoulli, Daniel!ref.\ to}% +starting with a particular integral given by Brook Taylor, +\index{Taylor, B.!ref.\ to}% +showed that this differential equation is satisfied by the +trigonometric series +\[ +y = \alpha \sin \frac{\pi x}{l} · \cos \frac{\pi t}{l} + + \beta \sin \frac{2 \pi x}{l}· \cos \frac{2 \pi t}{l} + \dotsb, +\] +and claimed this expression to be the most general solution. +Euler denied its generality, on the ground that, if true, the +doubtful conclusion would follow that the above series represents +any arbitrary function of a variable. These doubts were +dispelled by Fourier. Lagrange proceeded to find the sum +\index{Fourier!ref.\ to}% +\index{Lagrange!ref.\ to}% +of the above series, but D'Alembert rightly objected to his +process, on the ground that it involved divergent series.\Reference{46} +\index{Divergent series}% + +A most beautiful result reached by D'Alembert, with aid +of his principle, was the complete solution of the problem of +the precession of the equinoxes, which had baffled the talents +of the best minds. He sent to the French Academy in 1747, +on the same day with Clairaut, a solution of the problem of +\index{Clairaut!ref.\ to}% +three bodies. This had become a question of universal interest +%% -----File: 270.png---Folio 256------- +\index{Problem of three bodies}% +to mathematicians, in which each vied to outdo all others. +The problem of two bodies, requiring the determination of +their motion when they attract each other with forces inversely +proportional to the square of the distance between +them, had been completely solved by Newton. The ``problem +of three bodies'' asks for the motion of three bodies attracting +\index{Three bodies, problem of}% +each other according to the law of gravitation. Thus far, +the complete solution of this has transcended the power of +analysis. The general differential equations of motion were +\index{Differential calculus!philosophy of}% +stated by Laplace, but the difficulty arises in their integration. +\index{Laplace!ref.\ to}% +The ``solutions'' hitherto given are merely convenient methods +of approximation in special cases when one body is the sun, +disturbing the motion of the moon around the earth, or where +a planet moves under the influence of the sun and another +planet. + +In the discussion of the meaning of negative quantities, of +\index{Negative quantities}% +the fundamental processes of the calculus, and of the theory of +probability, D'Alembert paid some attention to the philosophy +of mathematics. His criticisms were not always happy. In +1754 he was made permanent secretary of the French Academy. +During the last years of his life he was mainly occupied with +the great French encyclopædia, which was begun by Diderot +and himself. D'Alembert declined, in 1762, an invitation of +\index{D'Alembert|)}% +Catharine~II. to undertake the education of her son. Frederick +the~Great pressed him to go to Berlin. He made a visit, but +declined a permanent residence there. + +\textbf{Alexis Claude Clairaut} (1713--1765) was a youthful prodigy. +\index{Clairaut|(}% +He read l'Hospital's works on the infinitesimal calculus and on +conic sections at the age of ten. In 1731 was published his +\textit{Recherches sur les courbes à double courbure}, which he had ready +for the press when he was sixteen. It was a work of remarkable +elegance and secured his admission to the Academy of +Sciences when still under legal age. In 1731 he gave a proof of +%% -----File: 271.png---Folio 257------- +\index{Astronomy!more recent researches}% +\index{Cubic curves}% +the theorem enunciated by Newton, that every cubic is a projection +\index{Newton!ref.\ to}% +of one of five divergent parabolas. Clairaut formed the +\index{Divergent parabolas}% +acquaintance of Maupertius, whom he accompanied on an expedition +\index{Maupertius}% +to Lapland to measure the length of a degree of the +meridian. At that time the shape of the earth was a subject +\index{Earth!figure of}% +of serious disagreement. Newton and Huygens had concluded +\index{Huygens!ref.\ to}% +from theory that the earth was flattened at the poles. About +1713 Dominico Cassini measured an arc extending from Dunkirk +\index{Cassini, D}% +to Perpignan and arrived at the startling result that the earth +is elongated at the poles. To decide between the conflicting +opinions, measurements were renewed. Maupertius earned by +his work in Lapland the title of ``earth flattener'' by disproving +the Cassinian tenet that the earth was elongated at the +poles, and showing that Newton was right. On his return, in +1743, Clairaut published a work, \textit{Théorie de la figure de la Terre}, +\index{Figure of the earth}% +which was based on the results of Maclaurin on homogeneous +ellipsoids. It contains a remarkable theorem, named after +Clairaut, that the sum of the fractions expressing the ellipticity +and the increase of gravity at the pole is equal to $2\frac{1}{2}$~times the +fraction expressing the centrifugal force at the equator, the +unit of force being represented by the force of gravity at the +equator. This theorem is independent of any hypothesis with +respect to the law of densities of the successive strata of the +earth. It embodies most of Clairaut's researches. Todhunter +says that ``in the figure of the earth no other person has +accomplished so much as Clairaut, and the subject remains at +present substantially as he left it, though the form is different. +The splendid analysis which Laplace supplied, adorned but did +not really alter the theory which started from the creative +hands of Clairaut.'' + +In 1752 he gained a prize of the St.~Petersburg Academy +for his paper on \textit{Théorie de la Lune}, in which for the first time +modern analysis is applied to lunar motion. This contained +%% -----File: 272.png---Folio 258------- +the explanation of the motion of the lunar apsides. This +motion, left unexplained by Newton, seemed to him at first +\index{Newton!ref.\ to}% +inexplicable by Newton's law, and he was on the point of +advancing a new hypothesis regarding gravitation, when, taking +\index{Gravitation, theory of}% +the precaution to carry his calculation to a higher degree +of approximation, he reached results agreeing with observation. +The motion of the moon was studied about the same +time by Euler and D'Alembert. Clairaut predicted that +\index{D'Alembert!ref.\ to}% +\index{Euler!ref.\ to}% +``Halley's Comet,'' then expected to return, would arrive at +\index{Halley's Comet}% +its nearest point to the sun on April~13, 1759, a date which +turned out to be one month too late. He was the first to +detect singular solutions in differential equations of the first +order but of higher degree than the first. + +In their scientific labours there was between Clairaut and +D'Alembert great rivalry, often far from friendly. The growing +ambition of Clairaut to shine in society, where he was a +\index{Clairaut|)}% +great favourite, hindered his scientific work in the latter part +of his life. + +\textbf{Johann Heinrich Lambert} (1728--1777), born at Mühl\-hausen +\index{Lambert|(}% +in Alsace, was the son of a poor tailor. While working at his +father's trade, he acquired through his own unaided efforts a +knowledge of elementary mathematics. At the age of thirty he +became tutor in a Swiss family and secured leisure to continue +his studies. In his travels with his pupils through Europe he +became acquainted with the leading mathematicians. In 1764 +he settled in Berlin, where he became member of the Academy, +and enjoyed the society of Euler and Lagrange. He received +a small pension, and later became editor of the Berlin \textit{Ephemeris}. +His many-sided scholarship reminds one of Leibniz. +In his \textit{Cosmological Letters} he made some remarkable prophecies +regarding the stellar system. In mathematics he made +several discoveries which were extended and overshadowed by +his great contemporaries. His first research on pure mathematics +%% -----File: 273.png---Folio 259------- +\index{Differential calculus!philosophy of}% +developed in an infinite series the root~$x$ of the equation +\index{Infinite series}% +$x^m + px = q$. Since each equation of the form $ax^r + bx^s = d$ +can be reduced to $x^m + px = q$ in two ways, one or the other of +the two resulting series was always found to be convergent, +and to give a value of~$x$. Lambert's results stimulated Euler, +\index{Euler!ref.\ to}% +who extended the method to an equation of four terms, and +particularly Lagrange, who found that a function of a root of +\index{Lagrange!ref.\ to}% +$a - x + \phi(x) = 0$ can be expressed by the series bearing his +name. In 1761 Lambert communicated to the Berlin Academy +a memoir, in which he proves that $\pi$ is irrational. This proof +\index{p@{$\pi$}: values for!proved to be irrational}% +is given in Note~IV. of Legendre's \textit{Géometrie}, where it is +\index{Legendre!ref.\ to}% +extended to~$\pi^2$. To the genius of Lambert we owe the introduction +\index{Lambert|)}% +into trigonometry of hyperbolic functions, which he +\index{Trigonometry}% +designated by $\textit{sinh}\,x$, $\textit{cosh}\,x$,~etc. His \textit{Freye Perspective}, 1759 +and 1773, contains researches on descriptive geometry, and +entitle him to the honour of being the forerunner of Monge. +\index{Monge!ref.\ to}% +In his effort to simplify the calculation of cometary orbits, he +was led geometrically to some remarkable theorems on conics, +for instance this: ``If in two ellipses having a common major +axis we take two such arcs that their chords are equal, and +that also the sums of the radii vectores, drawn respectively +from the foci to the extremities of these arcs, are equal to +each other, then the sectors formed in each ellipse by the arc +and the two radii vectores are to each other as the square +roots of the parameters of the ellipses.''\Reference{13} + +\textbf{John Landen} (1719--1790) was an English mathematician +\index{Landen}% +whose writings served as the starting-point of investigations +by Euler, Lagrange, and Legendre. Landen's capital discovery, +contained in a memoir of 1755, was that every arc of the +hyperbola is immediately rectified by means of two arcs of an +ellipse. In his ``residual analysis'' he attempted to obviate +the metaphysical difficulties of fluxions by adopting a purely +algebraic method. Lagrange's \textit{Calcul des Fonctions} is based +%% -----File: 274.png---Folio 260------- +\index{Witch of Agnesi}% +upon this idea. Landen showed how the algebraic expression +for the roots of a cubic equation could be derived by application +of the differential and integral calculus. Most of the +time of this suggestive writer was spent in the pursuits of +active life. + +\textbf{Étienne Bézout} (1730--1783) was a French writer of popular +\index{Bezout@{Bézout}}% +mathematical school-books. In his \textit{Théorie générale des Équations +Algébriques}, 1779, he gave the method of elimination by +\index{Bezout's@{Bézout's method of elimination}}% +linear equations (invented also by Euler). This method was +\index{Equations!solution of}% +first published by him in a memoir of 1764, in which he uses +determinants, without, however, entering upon their theory. +A beautiful theorem as to the degree of the resultant goes by +his name. + +\textbf{Louis Arbogaste} (1759--1803) of Alsace was professor of +\index{Arbogaste}% +mathematics at Strasburg. His chief work, the \textit{Calcul des +Dérivations}, 1800, gives the method known by his name, by +which the successive coefficients of a development are derived +from one another when the expression is complicated. De~Morgan +\index{Demorgan@{De Morgan}!ref.\ to}% +has pointed out that the true nature of derivation +is differentiation accompanied by integration. In this book +for the first time are the symbols of operation separated from +those of quantity. The notation $D_x y$ for $dy/dx$ is due to him. +\index{Notation!differential calculus}% + +\textbf{Maria Gaetana Agnesi} (1718--1799) of Milan, distinguished as +\index{Agnesi}% +a linguist, mathematician, and philosopher, filled the mathematical +chair at the University of Bologna during her father's +sickness. In 1748 she published her \textit{Instituzioni Analitiche}, +which was translated into English in 1801. The ``witch of +Agnesi'' or ``versiera'' is a plane curve containing a straight +line, $x = 0$, and a cubic $\left(\dfrac{y}{c}\right)^2 + 1 = \dfrac{c}{x}$. + +\textbf{Joseph Louis Lagrange} (1736--1813), one of the greatest +\index{Lagrange|(}% +mathematicians of all times, was born at Turin and died at +Paris. He was of French extraction. His father, who had +%% -----File: 275.png---Folio 261------- +\index{Isoperimetrical figures}% +charge of the Sardinian military chest, was once wealthy, but +lost all he had in speculation. Lagrange considered this loss +his good fortune, for otherwise he might not have made mathematics +the pursuit of his life. While at the college in Turin +his genius did not at once take its true bent. Cicero and Virgil +at first attracted him more than Archimedes and Newton. +He soon came to admire the geometry of the ancients, but the +perusal of a tract of Halley roused his enthusiasm for the +\index{Halley}% +analytical method, in the development of which he was destined +to reap undying glory. He now applied himself to +mathematics, and in his seventeenth year he became professor +of mathematics in the royal military academy at Turin. +Without assistance or guidance he entered upon a course of +study which in two years placed him on a level with the +greatest of his contemporaries. With aid of his pupils he +established a society which subsequently developed into the +Turin Academy. In the first five volumes of its transactions +appear most of his earlier papers. At the age of nineteen he +communicated to Euler a general method of dealing with +\index{Euler!ref.\ to}% +``isoperimetrical problems,'' known now as the Calculus of +\index{Calculus!of variations}% +Variations. This commanded Euler's lively admiration, and +he courteously withheld for a time from publication some +researches of his own on this subject, so that the youthful +Lagrange might complete \emph{his} investigations and claim the +invention. Lagrange did quite as much as Euler towards the +creation of the Calculus of Variations. As it came from Euler +it lacked an analytic foundation, and this Lagrange supplied. +He separated the principles of this calculus from geometric +considerations by which his predecessor had derived them. +Euler had assumed as fixed the limits of the integral, \ie~the +extremities of the curve to be determined, but Lagrange +removed this restriction and allowed all co-ordinates of the +curve to vary at the same time. Euler introduced in 1766 the +%% -----File: 276.png---Folio 262------- +\index{Acoustics}% +\index{Astronomy!more recent researches}% +name ``calculus of variations,'' and did much to improve this +science along the lines marked out by Lagrange. + +Another subject engaging the attention of Lagrange at +Turin was the propagation of sound. In his papers on this +subject in the \textit{Miscellanea Taurinensia}, the young mathematician +appears as the critic of Newton, and the arbiter between +\index{Newton!ref.\ to}% +Euler and D'Alembert. By considering only the particles +\index{D'Alembert!ref.\ to}% +\index{Euler!ref.\ to}% +which are in a straight line, he reduced the problem to the +same partial differential equation that represents the motions +of vibrating strings. The general integral of this was found by +\index{Strings, vibrating}% +\index{Vibrating strings}% +D'Alembert to contain two arbitrary functions, and the question +\index{Functions!arbitrary}% +now came to be discussed whether an arbitrary function +may be discontinuous. D'Alembert maintained the negative +against Euler, Daniel Bernoulli, and finally Lagrange,---arguing +\index{Bernoulli, Daniel!ref.\ to}% +that in order to determine the position of a point of the +chord at a time~$t$, the initial position of the chord must be +continuous. Lagrange settled the question in the affirmative. + +By constant application during nine years, Lagrange, at the +age of twenty-six, stood at the summit of European fame. +But his intense studies had seriously weakened a constitution +never robust, and though his physicians induced him to take +rest and exercise, his nervous system never fully recovered its +tone, and he was thenceforth subject to fits of melancholy. + +In 1764 the French Academy proposed as the subject of +a prize the theory of the libration of the moon. It demanded +an explanation, on the principle of universal gravitation, why +the moon always turns, with but slight variations, the same +\DPtypo{phase}{face} to the earth. Lagrange secured the prize. This success +encouraged the Academy to propose as a prize the theory +of the four satellites of Jupiter,---a problem of six bodies, +more difficult than the one of three bodies previously solved +by Clairaut, D'Alembert, and Euler. Lagrange overcame the +\index{Clairaut!ref.\ to}% +difficulties, but the shortness of time did not permit him to +%% -----File: 277.png---Folio 263------- +exhaust the subject. Twenty-four years afterwards it was +completed by Laplace. Later astronomical investigations of +\index{Laplace!ref.\ to}% +Lagrange are on cometary perturbations (1778 and 1783), on +Kepler's problem, and on a new method of solving the problem +\index{Kepler!ref.\ to}% +of three bodies. + +Being anxious to make the personal acquaintance of leading +mathematicians, Lagrange visited Paris, where he enjoyed the +stimulating delight of conversing with Clairaut, D'Alembert, +Condorcet, the Abbé Marie, and others. He had planned a +visit to London, but he fell dangerously ill after a dinner in +Paris, and was compelled to return to Turin. In 1766 Euler +left Berlin for St.~Petersburg, and he pointed out Lagrange as +the only man capable of filling the place. D'Alembert recommended +him at the same time. Frederick the Great thereupon +sent a message to Turin, expressing the wish of ``the +greatest king of Europe'' to have ``the greatest mathematician'' +at his court. Lagrange went to Berlin, and staid\DPnote{** PP: [sic], archaic form} there +twenty years. Finding all his colleagues married, and being +assured by their wives that the marital state alone is happy, +he married. The union was not a happy one. His wife +soon died. Frederick the Great held him in high esteem, +and frequently conversed with him on the advantages of perfect +regularity of life. This led Lagrange to cultivate regular +habits. He worked no longer each day than experience taught +him he could without breaking down. His papers were carefully +thought out before he began writing, and when he wrote +he did so without a single correction. + +During the twenty years in Berlin he crowded the transactions +of the Berlin Academy with memoirs, and wrote also +the epoch-making work called the \textit{Mécanique Analytique}. He +enriched algebra by researches on the solution of equations. +\index{Equations!solution of}% +There are two methods of solving directly algebraic equations,---that +of substitution and that of combination. The +%% -----File: 278.png---Folio 264------- +\index{Warring}% +former method was developed by Ferrari, Vieta, Tchirnhausen, +\index{Ferrari!ref.\ to}% +\index{Tchirnhausen!ref.\ to}% +\index{Vieta!ref.\ to}% +Euler, Bézout, and Lagrange; the latter by Vandermonde and +\index{Bezout@{Bézout}!ref.\ to}% +\index{Euler!ref.\ to}% +\index{Vandermonde!ref.\ to}% +Lagrange.\Reference{20} In the method of substitution the original forms +are so transformed that the determination of the roots is made +to depend upon simpler functions (resolvents). In the method +of combination auxiliary quantities are substituted for certain +simple combinations (``types'') of the unknown roots of the +equation, and auxiliary equations (resolvents) are obtained for +\index{Equations!numerical}% +these quantities with aid of the coefficients of the given equation. +Lagrange traced all known algebraic solutions of equations +to the uniform principle consisting in the formation and +solution of equations of lower degree whose roots are linear +functions of the required roots, and of the roots of unity. He +showed that the quintic cannot be reduced in this way, its +resolvent being of the sixth degree. His researches on the +theory of equations were continued after he left Berlin. In +the \textit{Résolution des équations numériques} (1798) he gave a +method of approximating to the real roots of numerical equations +by continued fractions. Among other things, it contains +also a proof that every equation must have a root,---a theorem +which appears before this to have been considered self-evident. +Other proofs of this were given by Argand, Gauss, and Cauchy. +\index{Argand!ref.\ to}% +\index{Cauchy!ref.\ to}% +\index{Gauss!ref.\ to}% +In a note to the above work Lagrange uses Fermat's theorem +\index{Fermat!ref.\ to}% +and certain suggestions of Gauss in effecting a complete algebraic +solution of any binomial equation. + +While in Berlin Lagrange published several papers on the +theory of numbers. In~1769 he gave a solution in integers of +\index{Numbers!theory of numbers}% +\index{Theory of numbers}% +indeterminate equations of the second degree, which resembles +the Hindoo cyclic method; he was the first to prove, in~1771, +``Wilson's theorem,'' enunciated by an Englishman, John +\index{Wilson}% +\index{Wilson's theorem}% +Wilson, and first published by Waring in his \textit{Meditationes +Algebraicæ}; he investigated in 1775 under what conditions +$±2$~and~$±5$ ($-1$~and~$±3$ having been discussed by Euler) +%% -----File: 279.png---Folio 265------- +are quadratic residues, or non-residues of odd prime numbers,~$q$; +he proved in 1770 Méziriac's theorem that every integer is +\index{Meziriac@{Méziriac}!ref.\ to}% +equal to the sum of four, or a less number, of squares. He +proved Fermat's theorem on $x^n + y^n = z^n$, for the case $n = 4$, +\index{Fermat!ref.\ to}% +also Fermat's theorem that, if $a^2 + b^2 = c^2$, then $ab$ is not a +square. + +In his memoir on Pyramids, 1773, Lagrange made considerable +use of determinants of the third order, and demonstrated +\index{Determinants}% +that the square of a determinant is itself a determinant. He +never, however, dealt explicitly and directly with determinants; +he simply obtained accidentally identities which are +now recognised as relations between determinants. + +Lagrange wrote much on differential equations. Though +\index{Differential equations}% +the subject of contemplation by the greatest mathematicians +(Euler, D'Alembert, Clairaut, Lagrange, Laplace), yet more +\index{D'Alembert!ref.\ to}% +\index{Euler!ref.\ to}% +than other branches of mathematics did they resist the systematic +application of fixed methods and principles. Lagrange +established criteria for singular solutions (\textit{Calcul des Fonctions}, +\index{Singular solutions}% +Lessons~14--17), which are, however, erroneous. He was the +first to point out the geometrical significance of such solutions. +He generalised Euler's researches on total differential equations +of two variables, and of the ninth order; he gave a solution +of partial differential equations of the first order (\textit{Berlin +Memoirs}, 1772 and 1774), and spoke of their singular solutions, +extending their solution in \textit{Memoirs} of 1779 and 1785 to equations +of any number of variables. The discussion on partial +differential equations of the second order, carried on by +D'Alembert, Euler, and Lagrange, has already been referred +to in our account of D'Alembert. + +While in Berlin, Lagrange wrote the ``\textit{Mécanique Analytique},'' +the greatest of his works (Paris, 1788). From the principle +of virtual velocities he deduced, with aid of the calculus of +\index{Calculus!of variations}% +\index{Virtual velocities}% +variations, the whole system of mechanics so elegantly and +%% -----File: 280.png---Folio 266------- +harmoniously that it may fitly be called, in Sir William +Rowan Hamilton's words, ``a kind of scientific poem.'' It is a +\index{Hamilton, W.~R.!ref.\ to}% +most consummate example of analytic generality. Geometrical +figures are nowhere allowed. ``On ne trouvera point de figures +dans cet ouvrage'' (Preface). The two divisions of mechanics---statics +\index{Mechanics!Lagrange}% +and dynamics---are in the first four sections of each +carried out analogously, and each is prefaced by a historic +sketch of principles. Lagrange formulated the principle of +least action. In their original form, the equations of motion +\index{Least action}% +involve the co-ordinates $x$,~$y$,~$z$, of the different particles $m$ or +$dm$ of the system. But $x$,~$y$,~$z$, are in general not independent, +and Lagrange introduced in place of them any variables $\xi$,~$\psi$,~$\phi$, +whatever, determining the position of the point at the time. +These may be taken to be independent. The equations of +motion may now assume the form +\[ +\frac{d}{dt}\, \frac{dT}{d\xi'} - \frac{dT}{d\xi} + \Xi = 0; +\] +or when $\Xi$,~$\Change{\psi}{\Psi}$, $\Change{\phi}{\Phi},\ldots$ are the partial differential coefficients +with respect to $\xi$,~$\psi$, $\phi,\ldots$ of one and the same function~$V$, +then the form +\[ +\frac{d}{dt}\, \frac{dT}{d\xi'} - \frac{dT}{d\xi} + \frac{dV}{d\xi} = 0. +\] +The latter is \textit{par excellence} the Lagrangian form of the equations +of motion. With Lagrange originated the remark that +mechanics may be regarded as a geometry of four dimensions. +To him falls the honour of the introduction of the potential +into dynamics.\Reference{49} Lagrange was anxious to have his \textit{Mécanique +Analytique} published in Paris. The work was ready for print +in 1786, but not till 1788 could he find a publisher, and then +only with the condition that after a few years he would purchase +all the unsold copies. The work was edited by +Legendre. +\index{Legendre!ref.\ to}% +%% -----File: 281.png---Folio 267------- + +After the death of Frederick the Great, men of science +were no longer respected in Germany, and Lagrange accepted +an invitation of Louis~XVI. to migrate to Paris. The French +queen treated him with regard, and lodging was procured for +him in the Louvre. But he was seized with a long attack of +melancholy which destroyed his taste for mathematics. For +two years his printed copy of the \textit{Mécanique}, fresh from the +press,---the work of a quarter of a century,---lay unopened on +his desk. Through Lavoisier he became interested in chemistry, +which he found ``as easy as algebra.'' The disastrous +\index{Algebra!Lagrange}% +crisis of the French Revolution aroused him again to activity. +About this time the young and accomplished daughter of the +astronomer Lemonnier took compassion on the sad, lonely +\index{Lemonnier}% +Lagrange, and insisted upon marrying him. Her devotion to +him constituted the one tie to life which at the approach of +death he found it hard to break. + +He was made one of the commissioners to establish weights +and measures having units founded on nature. Lagrange +strongly favoured the decimal subdivision, the general idea of +which was obtained from a work of Thomas Williams, London, +\index{Williams}% +1788. Such was the moderation of Lagrange's character, and +such the universal respect for him, that he was retained as president +of the commission on weights and measures even after it +had been \emph{purified} by the Jacobins by striking out the names +of Lavoisier, Laplace, and others. Lagrange took alarm at the +fate of Lavoisier, and planned to return to Berlin, but at the +establishment of the \textit{École Normale} in 1795 in Paris, he was +induced to accept a professorship. Scarcely had he time to +elucidate the foundations of arithmetic and algebra to young +pupils, when the school was closed. His additions to the +algebra of Euler were prepared at this time. In 1797 the +\index{Euler!ref.\ to}% +\textit{École Polytechnique} was founded, with Lagrange as one of +the professors. The earliest triumph of this institution was +%% -----File: 282.png---Folio 268------- +\index{Functions!theory of}% +\index{Theory of functions}% +the restoration of Lagrange to analysis. His mathematical +activity burst out anew. He brought forth the \textit{Théorie des +fonctions analytiques} (1797), \textit{Leçons sur le calcul des fonctions}, +a treatise on the same lines as the preceding (1801), and the +\textit{Résolution des équations \DPtypo{numeriques}{numériques}} (1798). In 1810 he +began a thorough revision of his \textit{Mécanique analytique}, but +he died before its completion. + +The \textit{Théorie des fonctions}, the germ of which is found in a +memoir of his of 1772, aimed to place the principles of the +calculus upon a sound foundation by relieving the mind of the +difficult conception of a limit or infinitesimal. John Landen's +\index{Landen!ref.\ to}% +residual calculus, professing a similar object, was unknown to +him. Lagrange attempted to prove Taylor's theorem (the +\index{Taylor's theorem}% +power of which he was the first to point out) by simple algebra, +and then to develop the entire calculus from that theorem. +The principles of the calculus were in his day involved in +philosophic difficulties of a serious nature. The infinitesimals +of Leibniz had no satisfactory metaphysical basis. In the +\index{Leibniz!ref.\ to}% +differential calculus of Euler they were treated as absolute +\index{Differential calculus!philosophy of}% +\index{Euler!ref.\ to}% +zeros. In Newton's limiting ratio, the magnitudes of which it +\index{Newton!ref.\ to}% +is the ratio cannot be found, for at the moment when they +should be caught and equated, there is neither arc nor chord. +The chord and arc were not taken by Newton as equal before +vanishing, nor after vanishing, but \emph{when} they vanish. ``That +method,'' said Lagrange, ``has the great inconvenience of considering +quantities in the state in which they cease, so to +speak, to be quantities; for though we can always well conceive +the ratios of two quantities, as long as they remain +finite, that ratio offers to the mind no clear and precise idea, +as soon as its terms become both nothing at the same time.'' +D'Alembert's method of limits was much the same as the +\index{D'Alembert!ref.\ to|(}% +\index{Limits, method of}% +method of prime and ultimate ratios. D'Alembert taught +\index{Prime and ultimate ratios}% +that a variable actually reached its limit. When Lagrange +%% -----File: 283.png---Folio 269------- +endeavoured to free the calculus of its metaphysical difficulties, +by resorting to common algebra, he avoided the whirlpool of +Charybdis only to suffer wreck against the rocks of Scylla. +The algebra of his day, as handed down to him by Euler, was +founded on a false view of infinity. No correct theory of +\index{Infinity}% +infinite series had then been established. Lagrange proposed +\index{Infinite series}% +to define the differential coefficient of $f(x)$ with respect to~$x$ +as the coefficient of~$h$ in the expansion of $f(x + h)$ by Taylor's +theorem, and thus to avoid all reference to limits. But he +used infinite series without ascertaining that they were convergent, +and his proof that $f(x + h)$ can always be expanded +in a series of ascending powers of~$h$, labours under serious +defects. Though Lagrange's method of developing the calculus +was at first greatly applauded, its defects were fatal, and to-day +his ``method of derivatives,'' as it was called, has been generally +\index{Derivatives, method of}% +abandoned. He introduced a notation of his own, but +\index{Notation!differential calculus}% +it was inconvenient, and was abandoned by him in the second +edition of his \textit{Mécanique}, in which he used infinitesimals. The +primary object of the \textit{Théorie des fonctions} was not attained, +but its secondary results were far-reaching. It was a purely +abstract mode of regarding functions, apart from geometrical +\index{Functions!theory of}% +or mechanical considerations. In the further development +of higher analysis a function became the leading idea, and +Lagrange's work may be regarded as the starting-point of the +theory of functions as developed by Cauchy, Riemann, Weierstrass, +\index{Theory of functions}% +and others. + +In the treatment of infinite series Lagrange displayed in +his earlier writings that laxity common to all mathematicians +of his time, excepting Nicolaus Bernoulli~II. and D'Alembert. +\index{Bernoulli, Nicolaus (born 1687)}% +But his later articles mark the beginning of a period of greater +rigour. Thus, in the \textit{Calcul de fonctions} he gives his theorem +on the limits of Taylor's theorem. Lagrange's mathematical +\index{Taylor's theorem}% +researches extended to subjects which have not been mentioned +%% -----File: 284.png---Folio 270------- +\index{Acoustics}% +\index{Probability}% +here---such as probabilities, finite differences, ascending +\index{Finite differences}% +continued fractions, elliptic integrals. Everywhere his +\index{Continued fractions}% +\index{Fractions!continued}% +wonderful powers of generalisation and abstraction are made +manifest. In that respect he stood without a peer, but +his great contemporary, Laplace, surpassed him in practical +\index{Laplace|(}% +sagacity. Lagrange was content to leave the application of +his general results to others, and some of the most important +researches of Laplace (particularly those on the velocity of +sound and on the secular acceleration of the moon) are implicitly +\index{Sound, velocity of}% +contained in Lagrange's works. + +Lagrange was an extremely modest man, eager to avoid +\index{Lagrange|)}% +controversy, and even timid in conversation. He spoke in +tones of doubt, and his first words generally were, ``Je ne +sais pas.'' He would never allow his portrait to be taken, +and the only ones that were secured were sketched without +his knowledge by persons attending the meetings of the +Institute. + +\textbf{Pierre Simon Laplace} (1749--1827) was born at Beau\-mont-en-Auge +in Normandy. Very little is known of his early +life. When at the height of his fame he was loath to speak +of his boyhood, spent in poverty. His father was a small +farmer. Some rich neighbours who recognised the boy's +talent assisted him in securing an education. As an extern +he attended the military school in Beaumont, where at an +early age he became teacher of mathematics. At eighteen +he went to Paris, armed with letters of recommendation to +D'Alembert, who was then at the height of his fame. The +letters remained unnoticed, but young Laplace, undaunted, +wrote the great geometer a letter on the principles of mechanics, +which brought the following enthusiastic response: +``You needed no introduction; you have recommended yourself; +my support is your due.'' D'Alembert secured him a +\index{D'Alembert!ref.\ to|)}% +position at the \textit{École Militaire} of Paris as professor of mathematics. +%% -----File: 285.png---Folio 271------- +\index{Circle!degrees of}% +His future was now assured, and he entered upon +those profound researches which brought him the title of ``the +Newton of France.'' With wonderful mastery of analysis, +Laplace attacked the pending problems in the application +of the law of gravitation to celestial motions. During the +\index{Gravitation, theory of}% +succeeding fifteen years appeared most of his original contributions +to astronomy. His career was one of almost uninterrupted +\index{Astronomy!more recent researches|(}% +prosperity. In 1784 he succeeded Bézout as examiner +to the royal artillery, and the following year he became member +of the Academy of Sciences. He was made president of +the Bureau of Longitude; he aided in the introduction of the +decimal system, and taught, with Lagrange, mathematics in +the \textit{École Normale}. When, during the Revolution, there arose +a cry for the reform of everything, even of the calendar, +\index{Calendar}% +Laplace suggested the adoption of an era beginning with the +year~1250, when, according to his calculation, the major axis +of the earth's orbit had been perpendicular to the equinoctial +line. The year was to begin with the vernal equinox, and the +zero meridian was to be located east of Paris by $185.30$~degrees +of the centesimal division of the quadrant, for by this meridian +\index{Division of the circle}% +the beginning of his proposed era fell at midnight. But the +revolutionists rejected this scheme, and made the start of the +new era coincide with the beginning of the glorious French +Republic.\Reference{50} + +Laplace was justly admired throughout Europe as a most +sagacious and profound scientist, but, unhappily for his reputation, +he strove not only after greatness in science, but also +after political honours. The political career of this eminent +scientist was stained by servility and suppleness. After the +18th of Brumaire, the day when Napoleon was made emperor, +Laplace's ardour for republican principles suddenly gave way +to a great devotion to the emperor. Napoleon rewarded this +devotion by giving him the post of minister of the interior, +%% -----File: 286.png---Folio 272------- +but dismissed him after six months for incapacity. Said +Napoleon, "Laplace ne saisissait aucune question sous son +véritable point de vue; il cherchait des subtilités partout, +n'avait que des idées problematiques, et portait enfin l'esprit +des infiniment petits jusque dans l'administration." Desirous +to retain his allegiance, Napoleon elevated him to the Senate +and bestowed various other honours upon him. Nevertheless, +he cheerfully gave his voice in 1814 to the dethronement of +his patron and hastened to tender his services to the Bourbons, +thereby earning the title of marquis. This pettiness of his +character is seen in his writings. The first edition of the +\textit{Système du monde} was dedicated to the Council of Five Hundred. +To the third volume of the \textit{Mécanique Céleste} is prefixed +a note that of all the truths contained in the book, that most +precious to the author was the declaration he thus made of +gratitude and devotion to the peace-maker of Europe. After +this outburst of affection, we are surprised to find in the editions +of the \textit{Théorie analytique des probabilités}, which appeared after +the Restoration, that the original dedication to the emperor is +suppressed. + +Though supple and servile in politics, it must be said that +in religion and science Laplace never misrepresented or concealed +his own convictions however distasteful they might be +to others. In mathematics and astronomy his genius shines +with a lustre excelled by few. Three great works did he give +to the scientific world,---the \textit{Mécanique Céleste}, the \textit{Exposition +du système du monde}, and the \textit{Théorie analytique des \DPtypo{probabilities}{probabilités}}. +Besides these he contributed important memoirs to the +French Academy. + +We first pass in brief review his astronomical researches. +In~1773 he brought out a paper in which he proved that the +mean motions or mean distances of planets are invariable or +merely subject to small periodic changes. This was the first +%% -----File: 287.png---Folio 273------- +and most important step in establishing the stability of the +solar system.\Reference{51} To Newton and also to Euler it had seemed +\index{Euler!ref.\ to}% +doubtful whether forces so numerous, so variable in position, +so different in intensity, as those in the solar system, could be +capable of maintaining permanently a condition of equilibrium. +Newton was of the opinion that a powerful hand must intervene +from time to time to repair the derangements occasioned +by the mutual action of the different bodies. This +paper was the beginning of a series of profound researches by +Lagrange and Laplace on the limits of variation of the various +\index{Lagrange!ref.\ to}% +elements of planetary orbits, in which the two great mathematicians +alternately surpassed and supplemented each other. +Laplace's first paper really grew out of researches on the +theory of Jupiter and Saturn. The behaviour of these planets +had been studied by Euler and Lagrange without receiving +satisfactory explanation. Observation revealed the existence +of a steady acceleration of the mean motions of our moon and +of Jupiter and an equally strange diminution of the mean motion +of Saturn. It looked as though Saturn might eventually +leave the planetary system, while Jupiter would fall into the +sun, and the moon upon the earth. Laplace finally succeeded +in showing, in a paper of 1784--1786, that these variations +(called the ``great inequality'') belonged to the class of ordinary +periodic perturbations, depending upon the law of attraction. +\index{Perturbations}% +The cause of so influential a perturbation was found in +the commensurability of the mean motion of the two planets. + +In the study of the Jovian system, Laplace was enabled to +determine the masses of the moons. He also discovered certain +very remarkable, simple relations between the movements +of those bodies, known as ``Laws of Laplace.'' His theory of +\index{Laws of Laplace}% +these bodies was completed in papers of 1788 and~1789. +These, as well as the other papers here mentioned, were published +in the \textit{\DPtypo{Mémoirs}{Mémoires} présentés par divers savans}. +The year +%% -----File: 288.png---Folio 274------- +\index{Nebular hypothesis}% +1787 was made memorable by Laplace's announcement that +the lunar acceleration depended upon the secular changes in +the eccentricity of the earth's orbit. This removed all doubt +then existing as to the stability of the solar system. The universal +validity of the law of gravitation to explain all motion +in the solar system was established. That system, as then +known, was at last found to be a complete machine. + +In 1796 Laplace published his \textit{Exposition du système du +monde}, a non-mathematical popular treatise on astronomy, +\index{Astronomy!more recent researches|)}% +ending with a sketch of the history of the science. In this +work he enunciates for the first time his celebrated nebular +hypothesis. A similar theory had been previously proposed +by Kant in~1755, and by Swedenborg; but Laplace does not +\index{Kant}% +\index{Swedenborg}% +appear to have been aware of this. + +Laplace conceived the idea of writing a work which should +contain a complete analytical solution of the mechanical problem +presented by the solar system, without deriving from +observation any but indispensable data. The result was the +\textit{Mécanique Céleste}, which is a systematic presentation embracing +all the discoveries of Newton, Clairaut, D'Alembert, Euler, +Lagrange, and of Laplace himself, on celestial mechanics. +\index{Mechanics!Laplace}% +The first and second volumes of this work were published in~1799; +the third appeared in~1802, the fourth in~1805. Of the +fifth volume, Books XI.~and~XII. were published in~1823; +Books XIII.,~XIV.,~XV. in~1824, and Book~XVI. in~1825. The +first two volumes contain the general theory of the motions +and figure of celestial bodies. The third and fourth volumes +give special theories of celestial motions,---treating particularly +of motions of comets, of our moon, and of other satellites. +The fifth volume opens with a brief history of celestial +mechanics, and then gives in appendices the results of the +author's later researches. The \textit{Mécanique Céleste} was such a +master-piece, and so complete, that Laplace's successors have +%% -----File: 289.png---Folio 275------- +been able to add comparatively little. The general part of +the work was translated into German by Joh.\ Karl Burkhardt, +\index{Burkhardt, J.~K.}% +and appeared in Berlin, 1800--1802. Nathaniel Bowditch +\index{Bowditch}% +brought out an edition in English, with an extensive commentary, +in Boston, 1829--1839. The \textit{Mécanique Céleste} is not +easy reading. The difficulties lie, as a rule, not so much in the +subject itself as in the want of verbal explanation. A complicated +chain of reasoning receives often no explanation whatever. +Biot, who assisted Laplace in revising the work for +\index{Biot}% +the press, tells that he once asked Laplace some explanation +of a passage in the book which had been written not long +before, and that Laplace spent an hour endeavouring to recover +the reasoning which had been carelessly suppressed with the +remark, ``Il est facile de voir.'' Notwithstanding the important +researches in the work, which are due to Laplace himself, +it naturally contains a great deal that is drawn from his predecessors. +It is, in fact, the organised result of a century of +patient toil. But Laplace frequently neglects to properly +acknowledge the source from which he draws, and lets the +reader infer that theorems and formulæ due to a predecessor +are really his own. + +We are told that when Laplace presented Napoleon with a +copy of the \textit{Mécanique Céleste}, the latter made the remark, +``M.~Laplace, they tell me you have written this large book on +the system of the universe, and have never even mentioned +its Creator.'' Laplace is said to have replied bluntly, ``Je +n'avais pas besoin de cette hypothèse-la.'' This assertion, +taken literally, is impious, but may it not have been intended +to convey a meaning somewhat different from its literal one? +Newton was not able to explain by his law of gravitation all +\index{Gravitation, theory of}% +questions arising in the mechanics of the heavens. Thus, +being unable to show that the solar system was stable, and +suspecting in fact that it was unstable, Newton expressed the +%% -----File: 290.png---Folio 276------- +opinion that the special intervention, from time to time, of a +powerful hand was necessary to preserve order. Now Laplace +was able to prove by the law of gravitation that the solar +system is stable, and in that sense may be said to have felt +no necessity for reference to the Almighty. + +We now proceed to researches which belong more properly +to pure mathematics. Of these the most conspicuous are on +the theory of probability. Laplace has done more towards +\index{Probability}% +advancing this subject than any one other investigator. He +published a series of papers, the main results of which were +collected in his \textit{Théorie analytique des probabilités}, 1812. The +third edition (1820) consists of an introduction and two books. +The introduction was published separately under the title, +\textit{Essai philosophique sur les probabilités}, and is an admirable +and masterly exposition without the aid of analytical formulæ +of the principles and applications of the science. The first +book contains the theory of generating functions, which are +applied, in the second book, to the theory of probability. +Laplace gives in his work on probability his method of +approximation to the values of definite integrals. The solution +of linear differential equations was reduced by him to +definite integrals. One of the most important parts of the +work is the application of probability to the method of least +squares, which is shown to give the most probable as well as +the most convenient results. + +The first printed statement of the principle of least squares +\index{Least squares}% +was made in 1806 by Legendre, without demonstration. Gauss +\index{Gauss!ref.\ to}% +\index{Legendre!ref.\ to}% +had used it still earlier, but did not publish it until 1809. +The first deduction of the law of probability of error that +appeared in print was given in 1808 by Robert Adrain in the +\index{Adrain}% +\textit{Analyst}, a journal published by himself in Philadelphia.\Reference{2} +Proofs of this law have since been given by Gauss, Ivory, +\index{Ivory!ref.\ to}% +Herschel, Hagen, and others; but all proofs contain some +\index{Hagen}% +\index{Herschel, J.~F.~W.!ref.\ to}% +%% -----File: 291.png---Folio 277------- +point of difficulty. Laplace's proof is perhaps the most satisfactory. + +Laplace's work on probability is very difficult reading, particularly +the part on the method of least squares. The +analytical processes are by no means clearly established or +free from error. ``No one was more sure of giving the +result of analytical processes correctly, and no one ever took +so little care to point out the various small considerations on +which correctness depends'' (De~Morgan). +\index{Demorgan@{De Morgan}!ref.\ to}% + +Of Laplace's papers on the attraction of ellipsoids, the most +\index{Attraction}% +important is the one published in 1785, and to a great extent +reprinted in the third volume of the \textit{Mécanique Céleste}. It +gives an exhaustive treatment of the general problem of +attraction of any ellipsoid upon a particle situated outside +\index{Ellipsoid!(attraction of)}% +or upon its surface. Spherical harmonics, or the so-called +``Laplace's coefficients,'' constitute a powerful analytic engine +\index{Laplace's coefficients}% +in the theory of attraction, in electricity, and magnetism. +The theory of spherical harmonics for two dimensions had +been previously given by Legendre. Laplace failed to make +due acknowledgment of this, and there existed, in consequence, +between the two great men, ``a feeling more +than coldness.'' The potential function,~$V$, is much used by +\index{Potential}% +Laplace, and is shown by him to satisfy the partial differential +equation $\dfrac{\partial^2V}{\partial x^2} + \dfrac{\partial^2V}{\partial y^2} + \dfrac{\partial^2V}{\partial z^2} = 0$. This is known as Laplace's +equation, and was first given by him in the more complicated +form which it assumes in polar co-ordinates. The notion +of potential was, however, not introduced into analysis by +Laplace. The honour of that achievement belongs to Lagrange.\Reference{49} +\index{Lagrange!ref.\ to}% + +Among the minor discoveries of Laplace are his method of +solving equations of the second, third, and fourth degrees, +\index{Equations!solution of}% +his memoir on singular solutions of differential equations, his +\index{Singular solutions}% +%% -----File: 292.png---Folio 278------- +\index{Acoustics}% +\index{Differential equations}% +researches in finite differences and in determinants, the establishment +\index{Determinants}% +\index{Finite differences}% +of the expansion theorem in determinants which had +been previously given by Vandermonde for a special case, the +\index{Vandermonde}% +\index{Vandermonde!ref.\ to}% +determination of the complete integral of the linear differential +equation of the second order. In the \textit{Mécanique Céleste} he +made a generalisation of Lagrange's theorem on the development +\index{Lagrange!ref.\ to}% +of functions in series known as Laplace's theorem. + +Laplace's investigations in physics were quite extensive. +We mention here his correction of Newton's formula on the +velocity of sound in gases by taking into account the changes +\index{Sound, velocity of}% +of elasticity due to the heat of compression and cold of rarefaction; +\index{Elasticity}% +his researches on the theory of tides; his mathematical +\index{Tides}% +theory of capillarity; his explanation of astronomical refraction; +\index{Capillarity}% +his formulæ for measuring heights by the barometer. + +Laplace's writings stand out in bold contrast to those of +Lagrange in their lack of elegance and symmetry. Laplace +\index{Laplace|)}% +looked upon mathematics as the tool for the solution of physical +problems. The true result being once reached, he spent +little time in explaining the various steps of his analysis, or +in polishing his work. The last years of his life were spent +mostly at Arcueil in peaceful retirement on a country-place, +where he pursued his studies with his usual vigour until his +death. He was a great admirer of Euler, and would often +\index{Euler!ref.\ to}% +say, ``Lisez Euler, lisez Euler, c'est notre maître à tous.'' + +\textbf{Abnit-Théophile Vandermonde} (1735--1796) studied music +during his youth in Paris and advocated the theory that all +art rested upon one general law, through which any one could +become a composer with the aid of mathematics. He was the +first to give a connected and logical exposition of the theory +of determinants, and may, therefore, almost be regarded as +the founder of that theory. He and Lagrange originated the +method of combinations in solving equations.\Reference{20} + +\textbf{Adrien Marie Legendre} (1752--1833) was educated at the +\index{Legendre|(}% +%% -----File: 293.png---Folio 279------- +\index{Elliptic functions}% +Collège Mazarin in Paris, where he began the study of mathematics +under Abbé Marie. His mathematical genius secured +\index{Marie, Abbé}% +for him the position of professor of mathematics at the military +school of Paris. While there he prepared an essay on +the curve described by projectiles thrown into resisting media +(ballistic curve), which captured a prize offered by the Royal +\index{Ballistic curve}% +Academy of Berlin. In 1780 he resigned his position in order +to reserve more time for the study of higher mathematics. +He was then made member of several public commissions. +In 1795 he was elected professor at the Normal School and +later was appointed to some minor government positions. +Owing to his timidity and to Laplace's unfriendliness toward +\index{Laplace!ref.\ to}% +him, but few important public offices commensurate with his +ability were tendered to him. + +As an analyst, second only to Laplace and Lagrange, Legendre +\index{Lagrange!ref.\ to}% +enriched mathematics by important contributions, mainly +on elliptic integrals, theory of numbers, attraction of ellipsoids, +and least squares. The most important of Legendre's +works is his \textit{Fonctions elliptiques}, issued in two volumes in +1825 and~1826. He took up the subject where Euler, Landen, +\index{Euler!ref.\ to}% +\index{Landen!ref.\ to}% +and Lagrange had left it, and for forty years was the only one +to cultivate this new branch of analysis, until at last Jacobi and +\index{Jacobi!ref.\ to}% +Abel stepped in with admirable new discoveries.\Reference{52} Legendre +\index{Abel!ref.\ to}% +imparted to the subject that connection and arrangement +which belongs to an independent science. Starting with an +integral depending upon the square root of a polynomial of +the fourth degree in~$x$, he showed that such integrals can be +brought back to three canonical forms, designated by $F(\phi)$, +$E(\phi)$, and~$\Pi(\phi)$, the radical being expressed in the form +$\Delta(\phi) = \sqrt{1 - k^2 \sin^2\phi}$. He also undertook the prodigious task +of calculating tables of arcs of the ellipse for different degrees +of amplitude and eccentricity, which supply the means of +integrating a large number of differentials. +%% -----File: 294.png---Folio 280------- +\index{Ellipsoid!(attraction of)}% +\index{Eulerian integrals}% +\index{Legendre's function}% +\index{Spherical trigonometry}% +\index{Trigonometry!spherical}% + +An earlier publication which contained part of his researches +on elliptic functions was his \textit{Calcul intégral} in three volumes +\index{Elliptic functions}% +(1811, 1816, 1817), in which he treats also at length of the +two classes of definite integrals named by him \emph{Eulerian}. He +tabulated the values of $\log \Gamma(p)$ for values of~$p$ between +$1$~and~$2$. + +One of the earliest subjects of research was the attraction +of spheroids, which suggested to Legendre the function~$P_n$, +named after him. His memoir was presented to the Academy +of Sciences in 1783. The researches of Maclaurin and Lagrange +\index{Lagrange!ref.\ to}% +\index{Maclaurin!ref.\ to}% +suppose the point attracted by a spheroid to be at the surface +or within the spheroid, but Legendre showed that in order to +determine the attraction of a spheroid on any external point +it suffices to cause the surface of another spheroid described +upon the same foci to pass through that point. Other memoirs +on ellipsoids appeared later. + +The two household gods to which Legendre sacrificed with +ever-renewed pleasure in the silence of his closet were the +elliptic functions and the theory of numbers. His researches +\index{Numbers!theory of numbers}% +\index{Theory of numbers}% +on the latter subject, together with the numerous scattered +fragments on the theory of numbers due to his predecessors +in this line, were arranged as far as possible into a systematic +whole, and published in two large quarto volumes, entitled +\textit{Théorie des nombres}, 1830. Before the publication of this +work Legendre had issued at divers times preliminary articles. +Its crowning pinnacle is the theorem of quadratic reciprocity, +\index{Quadratic reciprocity}% +previously indistinctly given by Euler without proof, but for the +\index{Euler!ref.\ to}% +first time clearly enunciated and partly proved by Legendre.\Reference{48} + +While acting as one of the commissioners to connect Greenwich +and Paris geodetically, Legendre calculated all the triangles +in France. This furnished the occasion of establishing +formulæ and theorems on geodesics, on the treatment of the +spherical triangle as if it were a plane triangle, by applying +%% -----File: 295.png---Folio 281------- +\index{Axioms (of geometry)}% +\index{Least squares}% +certain corrections to the angles, and on the method of least +squares, published for the first time by him without demonstration +in~1806. + +Legendre wrote an \textit{Éléments de Géométrie}, 1794, which +enjoyed great popularity, being generally adopted on the +Continent and in the United States as a substitute for Euclid. +\index{Euclid!ref.\ to}% +This great modern rival of Euclid passed through numerous +editions; the later ones containing the elements of trigonometry +and a proof of the irrationality of $\pi$ and~$\pi^2$. Much +\index{p@{$\pi$}: values for!proved to be irrational}% +attention was given by Legendre to the subject of parallel +lines. In the earlier editions of the \textit{Éléments}, he made direct +appeal to the senses for the correctness of the ``parallel-axiom.'' +He then attempted to demonstrate that ``axiom,'' but his +proofs did not satisfy even himself. In Vol.~XII. of the +Memoirs of the Institute is a paper by Legendre, containing +\index{Legendre|)}% +his last attempt at a solution of the problem. Assuming +space to be infinite, he proved satisfactorily that it is impossible +for the sum of the three angles of a triangle to exceed two +right angles; and that if there be any triangle the sum of +whose angles is two right angles, then the same must be true +of all triangles. But in the next step, to show that this sum +cannot be less than two right angles, his demonstration necessarily +failed. If it could be granted that the sum of the three +angles is always equal to two right angles, then the theory of +parallels could be strictly deduced. +\index{Parallels}% + +\textbf{Joseph Fourier} (1768--1830) was born at Auxerre, in central +\index{Fourier|(}% +France. He became an orphan in his eighth year. Through +the influence of friends he was admitted into the military +school in his native place, then conducted by the Benedictines +of the Convent of St.~Mark. He there prosecuted his studies, +particularly mathematics, with surprising success. He wished +to enter the artillery, but, being of low birth (the son of a +tailor), his application was answered thus: ``Fourier, not +%% -----File: 296.png---Folio 282------- +being noble, could not enter the artillery, although he were +a second Newton.''\Reference{53} He was soon appointed to the mathematical +\index{Newton!ref.\ to}% +chair in the military school. At the age of twenty-one +he went to Paris to read before the Academy of Sciences +a memoir on the resolution of numerical equations, which +\index{Equations!numerical}% +was an improvement on Newton's method of approximation. +This investigation of his early youth he never lost sight of. +He lectured upon it in the Polytechnic School; he developed +it on the banks of the Nile; it constituted a part of a work +entitled \textit{Analyse des equationes determines} (1831), which was +in press when death overtook him. This work contained +``Fourier's theorem'' on the number of real roots between +\index{Fourier's theorem}% +two chosen limits. Budan had published this result as early +\index{Budan}% +as 1807, but there is evidence to show that Fourier had established +it before Budan's publication. These brilliant results +were eclipsed by the theorem of Sturm, published in~1835. +\index{Sturm, J.~C.~F.!ref.\ to}% + +Fourier took a prominent part at his home in promoting +the Revolution. Under the French Revolution the arts and +sciences seemed for a time to flourish. The reformation of +the weights and measures was planned with grandeur of conception. +The Normal School was created in 1795, of which +Fourier became at first pupil, then lecturer. His brilliant +success secured him a chair in the Polytechnic School, the +duties of which he afterwards quitted, along with Monge and +\index{Monge!ref.\ to}% +Berthollet, to accompany Napoleon on his campaign to Egypt. +Napoleon founded the Institute of Egypt, of which Fourier +became secretary. In Egypt he engaged not only in scientific +work, but discharged important political functions. After +his return to France he held for fourteen years the prefecture +of Grenoble. During this period he carried on his elaborate +investigations on the propagation of heat in solid bodies, +published in 1822 in his work entitled \textit{La Theorie Analytique +de la Chaleur}. This work marks an epoch in the history of +%% -----File: 297.png---Folio 283------- +\index{Functions!arbitrary}% +\index{Infinite series}% +\index{Kelvin, Lord!ref.\ to}% +mathematical physics. ``Fourier's series'' constitutes its +\index{Fourier's series}% +gem. By this research a long controversy was brought to a +close, and the fact established that any arbitrary function +can be represented by a trigonometric series. The first +\index{Trigonometric series}% +announcement of this great discovery was made by Fourier +in 1807, before the French Academy. The trigonometric +series $\smash[b]{\sum\limits_{n=0}^{n=\infty} (a_n \sin nx + b_n \cos nx)}$ represents the function~$\phi(x)$ +for every value of~$x$, if the coefficients $a_n = \displaystyle\frac{1}{\pi} \int_{-\pi}^\pi \phi(x) \sin nx\, dx$, +and $b_n$ be equal to a similar integral. The weak point in +Fourier's analysis lies in his failure to prove generally that +the trigonometric series actually converges to the value of +the function. In 1827 Fourier succeeded Laplace as president +of the council of the Polytechnic School. + +Before proceeding to the origin of modern geometry we shall +speak briefly of the introduction of higher analysis into Great +Britain. This took place during the first quarter of this century. +The British began to deplore the very small progress +that science was making in England as compared with its +racing progress on the Continent. In 1813 the ``Analytical +\index{Analytical Society (in Cambridge)}% +Society'' was formed at Cambridge. This was a small club +established by George Peacock, John Herschel, Charles Babbage, +\index{Babbage}% +\index{Herschel, J.~F.~W.!ref.\ to}% +\index{Peacock!ref.\ to, X}% +and a few other Cambridge students, to promote, as it +was humorously expressed, the principles of pure ``$D$-ism,'' +that is, the Leibnizian notation in the calculus against those +\index{Notation!differential calculus}% +of ``dot-age,'' or of the Newtonian notation. This struggle +ended in the introduction into Cambridge of the notation +$\dfrac{dy}{dx}$, to the exclusion of the fluxional notation~$\dot{y}$. This +was a great step in advance, not on account of any great +superiority of the Leibnizian over the Newtonian notation, +but because the adoption of the former opened up to English +students the vast storehouses of continental discoveries. Sir +William Thomson, Tait, and some other modern writers find +\index{Tait}% +\index{Thomson, Sir William!ref.\ to}% +%% -----File: 298.png---Folio 284------- +\index{Arithmetical machine}% +\index{Calculating machines}% +\index{Fourier|)}% +\index{Machine, arithmetical}% +it frequently convenient to use both notations. Herschel, +\index{Herschel, J.~F.~W.!ref.\ to}% +Peacock, and Babbage translated, in 1816, from the French, +\index{Peacock}% +Lacroix's treatise on the differential and integral calculus, and +\index{Lacroix}% +added in 1820 two volumes of examples. Lacroix's was one +of the best and most extensive works on the calculus of that +time. Of the three founders of the ``Analytical Society,'' +Peacock afterwards did most work in pure mathematics. +Babbage became famous for his invention of a calculating +engine superior to Pascal's. It was never finished, owing +\index{Pascal!ref.\ to}% +to a misunderstanding with the government, and a consequent +failure to secure funds. John Herschel, the eminent +astronomer, displayed his mastery over higher analysis in +memoirs communicated to the Royal Society on new applications +of mathematical analysis, and in articles contributed +to cyclopædias on light, on meteorology, and on the history +of mathematics. + +\textbf{George Peacock} (1791--1858) was educated at Trinity College, +Cambridge, became Lowndean professor there, and later, dean +of Ely. His chief publications are his \textit{Algebra}, 1830 and~1842, +\index{Algebra!Peacock}% +and his \textit{Report on Recent Progress in Analysis}, which was the +first of several valuable summaries of scientific progress printed +in the volumes of the British Association. He was one of the +first to study seriously the fundamental principles of algebra, +and to fully recognise its purely symbolic character. He +advances, though somewhat imperfectly, the ``principle of the +permanence of equivalent forms.'' It assumes that the rules +applying to the symbols of arithmetical algebra apply also +in symbolical algebra. About this time D.~F. Gregory wrote +\index{Gregory, David F}% +a paper ``on the real nature of symbolical algebra,'' which +brought out clearly the commutative and distributive laws. +These laws had been noticed years before by the inventors +of symbolic methods in the calculus. It was Servois who +\index{Servois}% +introduced the names \emph{commutative} and \emph{distributive} in~1813. +%% -----File: 299.png---Folio 285------- +Peacock's investigations on the foundation of algebra were +considerably advanced by De~Morgan and Hankel. +\index{Demorgan@{De Morgan}!ref.\ to}% +\index{Hankel!ref.\ to}% + +\textbf{James Ivory} (1765--1842) was a Scotch mathematician who +\index{Ivory}% +for twelve years, beginning in 1804, held the mathematical +chair in the Royal Military College at Marlow (now at Sandhurst). +He was essentially a self-trained mathematician, and +almost the only one in Great Britain previous to the organisation +of the Analytical Society who was well versed in continental +mathematics. Of importance is his memoir (\textit{Phil.\ +Trans.}, 1809) in which the problem of the attraction of a +homogeneous ellipsoid upon an external point is reduced to +\index{Ellipsoid!(attraction of)}% +the simpler problem of the attraction of a related ellipsoid +upon a corresponding point interior to it. This is known as +``Ivory's theorem.'' He criticised with undue severity Laplace's +\index{Ivory's theorem}% +\index{Laplace!ref.\ to}% +solution of the method of least squares, and gave three proofs +\index{Least squares}% +of the principle without recourse to probability; but they are +\index{Probability}% +far from being satisfactory. + +\Subsection{The Origin of Modern Geometry} +\index{Geometry!modern synthetic|(}% + +By the researches of Descartes and the invention of the calculus, +the analytical treatment of geometry was brought into +great prominence for over a century. Notwithstanding the +efforts to revive synthetic methods made by Desargues, Pascal, +\index{Desargues!ref.\ to}% +\index{Pascal!ref.\ to}% +De~Lahire, Newton, and Maclaurin, the analytical method +\index{Delahire@{De Lahire}}% +\index{Maclaurin!ref.\ to}% +\index{Newton!ref.\ to}% +retained almost undisputed supremacy. It was reserved for +the genius of Monge to bring synthetic geometry in the +foreground, and to open up new avenues of progress. His +\textit{Géométrie descriptive} marks the beginning of a wonderful +development of modern geometry. + +Of the two leading problems of descriptive geometry, the +one---to represent by drawings geometrical magnitudes---was +brought to a high degree of perfection before the time of +%% -----File: 300.png---Folio 286------- +Monge; the other---to solve problems on figures in space +\index{Monge|(}% +by constructions in a plane---had received considerable attention +before his time. His most noteworthy predecessor +in descriptive geometry was the Frenchman Frézier (1682--1773). +\index{Descriptive geometry|(}% +\index{Frezier@{Frézier}}% +\index{Geometry!descriptive|(}% +But it remained for Monge to create descriptive +geometry as a \emph{distinct} branch of science by imparting to it +geometric generality and elegance. All problems previously +treated in a special and uncertain manner were referred +back to a few general principles. He introduced the line +of intersection of the horizontal and the vertical plane as +the axis of projection. By revolving one plane into the +other around this axis or ground-line, many advantages were +gained.\Reference{54} + +\textbf{Gaspard Monge} (1746--1818) was born at Beaune. The construction +of a plan of his native town brought the boy under +the notice of a colonel of engineers, who procured for him an +appointment in the college of engineers at Mézières. Being +of low birth, he could not receive a commission in the army, +but he was permitted to enter the annex of the school, where +surveying and drawing were taught. Observing that all the +operations connected with the construction of plans of fortification +were conducted by long arithmetical processes, he substituted +a geometrical method, which the commandant at first +refused even to look at, so short was the time in which it +could be practised; when once examined, it was received with +avidity. Monge developed these methods further and thus +created his descriptive geometry. Owing to the rivalry +between the French military schools of that time, he was not +permitted to divulge his new methods to any one outside of +this institution. In 1768 he was made professor of mathematics +at Mézières. In 1780, when conversing with two of his +pupils, S.~F. Lacroix and Gayvernon in Paris, he was obliged +\index{Lacroix}% +to say, ``All that I have here done by calculation, I could have +%% -----File: 301.png---Folio 287------- +\index{Imaginary quantities}% +done with the ruler and compass, but I am not allowed to +reveal these secrets to you.'' But Lacroix set himself to +\index{Lacroix}% +examine what the secret could be, discovered the processes, and +published them in 1795. The method was published by Monge +himself in the same year, first in the form in which the short-hand +writers took down his lessons given at the Normal School, +where he had been elected professor, and then again, in revised +form, in the \textit{Journal des écoles normales}. The next edition +occurred in 1798--1799. After an ephemeral existence of only +four months the Normal School was closed in~1795. In the +same year the Polytechnic School was opened, in the establishing +of which Monge took active part. He taught there +descriptive geometry until his departure from France to accompany +\index{Geometry!analytic}% +Napoleon on the Egyptian campaign. He was the first +president of the Institute of Egypt. Monge was a zealous +partisan of Napoleon and was, for that reason, deprived of all +his honours by Louis~XVIII\@. This and the destruction of the +Polytechnic School preyed heavily upon his mind. He did +not long survive this insult. + +Monge's numerous papers were by no means confined to descriptive +geometry. His analytical discoveries are hardly less +remarkable. He introduced into analytic geometry the methodic +\index{Analytic geometry}% +use of the equation of a line. He made important +contributions to surfaces of the second degree (previously +\index{Surfaces, theory of}% +studied by Wren and Euler) and discovered between the +\index{Euler!ref.\ to}% +\index{Wren!ref.\ to}% +theory of surfaces and the integration of partial differential +\index{Partial differential equations}% +equations, a hidden relation which threw new light upon both +subjects. He gave the differential of curves of curvature, +established a general theory of curvature, and applied it to the +ellipsoid. He found that the validity of solutions was not +impaired when imaginaries are involved among subsidiary +quantities. Monge published the following books: \textit{Statics}, +1786; \textit{Applications de l'algèbre à la géométrie}, 1805; \textit{Application +%% -----File: 302.png---Folio 288------- +\index{Brianchion}% +\index{Carnot, Lazare}% +de l'analyse à la géométrie}. The last two contain most of +his miscellaneous papers. + +Monge was an inspiring teacher, and he gathered around +him a large circle of pupils, among which were Dupin, Servois, +\index{Dupin}% +\index{Servois}% +Brianchon, Hachette, Biot, and Poncelet. +\index{Biot}% +\index{Hachette}% +\index{Poncelet!ref.\ to}% + +\textbf{Charles Dupin} (1784--1873), for many years professor of +mechanics in the Conservatoire des Arts et Métiers in Paris, +published in 1813 an important work on \textit{Développements de +géométrie}, in which is introduced the conception of conjugate +tangents of a point of a surface, and of the indicatrix.\Reference{53} It +contains also the theorem known as ``Dupin's theorem.'' +Surfaces of the second degree and descriptive geometry were +successfully studied by \textit{Jean Nicolas Pierre Hachette} (1769--1834), +who became professor of descriptive geometry at the +Polytechnic School after the departure of Monge for Rome and +Egypt. In 1822 he published his \textit{Traité de géométrie descriptive}. + +Descriptive geometry, which arose, as we have seen, in +technical schools in France, was transferred to Germany at +the foundation of technical schools there. G.~Schreiber, +\index{Schreiber}% +professor in Karlsruhe, was the first to spread Monge's +\index{Monge|)}% +geometry in Germany by the publication of a work thereon +in 1828--1829.\Reference{54} In the United States descriptive geometry was +\index{Descriptive geometry|)}% +\index{Geometry!descriptive|)}% +introduced in 1816 at the Military Academy in West Point +by Claude Crozet, once a pupil at the Polytechnic School in +\index{Crozet}% +Paris. Crozet wrote the first English work on the subject.\Reference{2} + +\textbf{Lazare Nicholas Marguerite Carnot} (1753--1823) was born at +Nolay in Burgundy, and educated in his native province. +He entered the army, but continued his mathematical studies, +and wrote in 1784 a work on machines, containing the earliest +proof that kinetic energy is lost in collisions of bodies. With +the advent of the Revolution he threw himself into politics, +and when coalesced Europe, in 1793, launched against France +a million soldiers, the gigantic task of organising fourteen +%% -----File: 303.png---Folio 289------- +\index{Brianchion}% +\index{Carnot, Lazare}% +\index{Differential calculus!philosophy of}% +\index{Dupin}% +armies to meet the enemy was achieved by him. He was +banished in 1796 for opposing Napoleon's \textit{coup d'état}. The +refugee went to Geneva, where he issued, in 1797, a work +still frequently quoted, entitled, \textit{Réflexions sur la Métaphysique +du Calcul Infinitésimal}. He declared himself as an ``irreconcilable +enemy of kings.'' After the Russian campaign he +offered to fight for France, though not for the empire. On +the restoration he was exiled. He died in Magdeburg. His +\textit{Géométrie de position}, 1803, and his \textit{Essay on Transversals}, +1806, are important contributions to modern geometry. While +Monge revelled mainly in three-dimensional geometry, Carnot +confined himself to that of two. By his effort to explain +the meaning of the negative sign in geometry he established +a ``geometry of position,'' which, however, is different from +the ``Geometrie der Lage'' of to-day. He invented a class +of general theorems on projective properties of figures, which +have since been pushed to great extent by Poncelet, Chasles, +\index{Chasles!ref.\ to}% +\index{Poncelet}% +and others. + +\textbf{Jean Victor Poncelet} (1788--1867), a native of Metz, took +part in the Russian campaign, was abandoned as dead on the +bloody field of Krasnoi, and taken prisoner to Saratoff. Deprived +there of all books, and reduced to the remembrance +of what he had learned at the Lyceum at Metz and the Polytechnic +School, where he had studied with predilection the +works of Monge, Carnot, and Brianchon, he began to study +mathematics from its elements. He entered upon original +researches which afterwards made him illustrious. While +in prison he did for mathematics what Bunyan did for +literature,---produced a much-read work, which has remained +of great value down to the present time. He returned to +France in 1814, and in 1822 published the work in question, +entitled, \textit{Traité des Propriétés projectives des figures}. In it +he investigated the properties of figures which remain unaltered +%% -----File: 304.png---Folio 290------- +by projection of the figures. The projection is not +effected here by parallel rays of prescribed direction, as with +Monge, but by central projection. Thus perspective projection, +used before him by Desargues, Pascal, Newton, and Lambert, +\index{Desargues!ref.\ to}% +\index{Lambert!ref.\ to}% +\index{Newton!ref.\ to}% +\index{Pascal!ref.\ to}% +was elevated by him into a fruitful geometric method. +In the same way he elaborated some ideas of De~Lahire, +\index{Delahire@{De Lahire}}% +Servois, and Gergonne into a regular method---the method +\index{Gergonne!ref.\ to}% +\index{Servois}% +of ``reciprocal polars.'' To him we owe the Law of Duality +\index{Duality}% +\index{Reciprocal polars}% +as a consequence of reciprocal polars. As an independent +principle it is due to Gergonne. Poncelet wrote much on +\index{Poncelet}% +applied mechanics. In 1838 the Faculty of Sciences was +\index{Mechanics!more recent work}% +enlarged by his election to the chair of mechanics. + +While in France the school of Monge was creating modern +geometry, efforts were made in England to revive Greek +geometry by \textbf{Robert Simson} (1687--1768) and \textbf{Matthew Stewart} +\index{Simson}% +\index{Stewart}% +(1717--1785). Stewart was a pupil of Simson and Maclaurin, +\index{Maclaurin!ref.\ to}% +and succeeded the latter in the chair at Edinburgh. During +the eighteenth century he and Maclaurin were the only prominent +mathematicians in Great Britain. His genius was ill-directed +by the fashion then prevalent in England to ignore +higher analysis. In his \textit{Four Tracts, Physical and Mathematical}, +1761, he applied geometry to the solution of difficult +\index{Geometry!modern synthetic|)}% +astronomical problems, which on the Continent were approached +analytically with greater success. He published, in +1746, \textit{General Theorems}, and in 1763, his \textit{Propositiones geometricæ +more veterum demonstratæ}. The former work contains +sixty-nine theorems, of which only five are accompanied +by demonstrations. It gives many interesting new results +on the circle and the straight line. Stewart extended some +theorems on transversals due to Giovanni Ceva (1648--1737), +\index{Ceva}% +an Italian, who published in 1678 at Mediolani a work containing +the theorem now known by his name. +%% -----File: 305.png---Folio 291------- +\index{Bolyai, Johann!ref.\ to}% +\index{Bolyai, Wolfgang!ref.\ to}% + + +\Chapter{Recent Times} + +\First{Never} more zealously and successfully has mathematics +been cultivated than in this century. Nor has progress, as +in previous periods, been confined to one or two countries. +While the French and Swiss, who alone during the preceding +epoch carried the torch of progress, have continued to develop +mathematics with great success, from other countries whole +armies of enthusiastic workers have wheeled into the front +rank. Germany awoke from her lethargy by bringing forward +Gauss, Jacobi, Dirichlet, and hosts of more recent men; +\index{Dirichlet!ref.\ to}% +\index{Gauss!ref.\ to}% +\index{Jacobi!ref.\ to}% +Great Britain produced her De~Morgan, Boole, Hamilton, +\index{Boole!ref.\ to}% +\index{Demorgan@{De Morgan}!ref.\ to}% +\index{Hamilton, W.~R.!ref.\ to}% +besides champions who are still living; Russia entered the +arena with her Lobatchewsky; Norway with Abel; Italy with +\index{Abel!ref.\ to}% +\index{Lobatchewsky!ref.\ to}% +Cremona; Hungary with her two Bolyais; the United States +\index{Cremona!ref.\ to}% +with Benjamin Peirce. +\index{Peirce, B.!ref.\ to}% + +The productiveness of modern writers has been enormous. +``It is difficult,'' says Professor Cayley,\Reference[]{56} ``to give an idea of +\index{Cayley!ref.\ to}% +the vast extent of modern mathematics. This word `extent' +is not the right one: I mean extent crowded with beautiful +detail,---not an extent of mere uniformity such as an objectless +plain, but of a tract of beautiful country seen at first in +the distance, but which will bear to be rambled through and +studied in every detail of hillside and valley, stream, rock, +wood, and flower.'' It is pleasant to the mathematician to +think that in his, as in no other science, the achievements of +%% -----File: 306.png---Folio 292------- +every age remain possessions forever; new discoveries seldom +disprove older tenets; seldom is anything lost or wasted. + +If it be asked wherein the utility of some modern extensions +of mathematics lies, it must be acknowledged that it is +at present difficult to see how they are ever to become applicable +to questions of common life or physical science. But +our inability to do this should not be urged as an argument +against the pursuit of such studies. In the first place, we +know neither the day nor the hour when these abstract +developments will find application in the mechanic arts, in +physical science, or in other branches of mathematics. For +example, the whole subject of graphical statics, so useful +\index{Graphical statics}% +to the practical engineer, was made to rest upon von Staudt's +\index{Von Staudt!ref.\ to}% +\textit{Geometrie der Lage}; Hamilton's ``principle of varying action'' +\index{Action!varying}% +\index{Hamilton, W.~R.!ref.\ to}% +\index{Varying action, principle of}% +has its use in astronomy; complex quantities, general integrals, +\index{Complex quantities}% +and general theorems in integration offer advantages in +the study of electricity and magnetism. ``The utility of such +researches,'' says Spottiswoode,\Reference[]{57} ``can in no case be discounted, +\index{Spottiswoode!ref.\ to}% +or even imagined beforehand. Who, for instance, would have +supposed that the calculus of forms or the theory of substitutions +\index{Calculus!of operations}% +\index{Substitutions, theory of}% +would have thrown much light upon ordinary equations; +or that Abelian functions and hyperelliptic transcendents +\index{Abelian functions}% +\index{Hyperelliptic functions}% +would have told us anything about the properties of curves; +\index{Curves!theory of}% +or that the calculus of operations would have helped us in +\index{Operations, calculus of}% +any way towards the figure of the earth?'' A second reason +\index{Earth!figure of}% +\index{Figure of the earth}% +in favour of the pursuit of advanced mathematics, even when +there is no promise of practical application, is this, that mathematics, +like poetry and music, deserves cultivation for its +own sake. + +The great characteristic of modern mathematics is its generalising +tendency. Nowadays little weight is given to isolated +theorems, ``except as affording hints of an unsuspected +new sphere of thought, like meteorites detached from some +%% -----File: 307.png---Folio 293------- +\index{Invariant}% +undiscovered planetary orb of speculation.'' In mathematics, +as in all true sciences, no subject is considered in itself alone, +but always as related to, or an outgrowth of, other things. +The development of the notion of continuity plays a leading +\index{Continuity}% +part in modern research. In geometry the principle of continuity, +\index{Geometry!modern synthetic|(}% +the idea of correspondence, and the theory of projection +\index{Correspondence, principle of}% +constitute the fundamental modern notions. Continuity +asserts itself in a most striking way in relation to the circular +points at infinity in a plane. In algebra the modern idea finds +\index{Infinity}% +expression in the theory of linear transformations and invariants, +and in the recognition of the value of homogeneity and +\index{Homogeneity}% +symmetry. + + +\Section{Synthetic Geometry} +\index{Synthetic geometry|(}% + +The conflict between geometry and analysis which arose +near the close of the last century and the beginning of the +present has now come to an end. Neither side has come +out victorious. The greatest strength is found to lie, not in +the suppression of either, but in the friendly rivalry between +the two, and in the stimulating influence of the one upon the +other. Lagrange prided himself that in his \textit{\DPtypo{Mecanique}{Mécanique} Analytique} +\index{Lagrange!ref.\ to}% +he had succeeded in avoiding all figures; but since his +time mechanics has received much help from geometry. + +Modern synthetic geometry was created by several investigators +about the same time. It seemed to be the outgrowth +of a desire for general methods which should serve as threads +of Ariadne to guide the student through the labyrinth of theorems, +corollaries, porisms, and problems. Synthetic geometry +was first cultivated by Monge, Carnot, and Poncelet in France; +\index{Carnot, Lazare!ref.\ to}% +\index{Monge!ref.\ to}% +\index{Poncelet!ref.\ to}% +it then bore rich fruits at the hands of Möbius and Steiner in +\index{Mobius@{Möbius}!ref.\ to}% +\index{Steiner!ref.\ to}% +Germany and Switzerland, and was finally developed to still +%% -----File: 308.png---Folio 294------- +\index{Anharmonic ratio}% +higher perfection by Chasles in France, von Staudt in Germany, +\index{Chasles!ref.\ to}% +\index{Von Staudt!ref.\ to}% +and Cremona in Italy. +\index{Cremona!ref.\ to|(}% + +\textbf{Augustus Ferdinand Möbius} (1790--1868) was a native of +\index{Mobius@{Möbius}}% +Schulpforta in Prussia. He studied at Göttingen under +Gauss, also at Leipzig and Halle. In Leipzig he became, in +\index{Gauss!ref.\ to}% +1815, privat-docent, the next year extraordinary professor of +astronomy, and in 1844 ordinary professor. This position he +held till his death. The most important of his researches +are on geometry. They appeared in \textit{Crelle's Journal}, and in +his celebrated work entitled \textit{Der Barycentrische Calcul}, Leipzig, +1827. As the name indicates, this calculus is based upon +properties of the centre of gravity.\Reference{58} Thus, that the point~$S$ +is the centre of gravity of weights $a$,~$b$, $c$,~$d$ placed at the +points $A$,~$B$, $C$,~$D$ respectively, is expressed by the equation +\[ +(a + b + c + d)S = aA + bB + cC + dD. +\] +His calculus is the beginning of a quadruple algebra, and contains +the germs of Grassmann's marvellous system. In designating +\index{Grassmann!ref.\ to}% +segments of lines we find throughout this work for the +first time consistency in the distinction of positive and negative +by the order of letters $AB$,~$BA$. Similarly for triangles +and tetrahedra. The remark that it is always possible to give +three points $A$,~$B$,~$C$ such weights $\alpha$,~$\beta$,~$\gamma$ that any fourth point~$M$ +in their plane will become a centre of mass, led Möbius +to a new system of co-ordinates in which the position of a +\index{Co-ordinates}% +point was indicated by an equation, and that of a line by +co-ordinates. By this algorithm he found by algebra many +geometric theorems expressing mainly invariantal properties,---for +example, the theorems on the anharmonic relation. +Möbius wrote also on statics and astronomy. He generalised +spherical trigonometry by letting the sides or angles of triangles +\index{Spherical trigonometry}% +\index{Trigonometry!spherical}% +exceed~$180°$. +%% -----File: 309.png---Folio 295------- +\index{Salmon}% + +\textbf{Jacob Steiner} (1796--1863), ``the greatest geometrician since +\index{Steiner}% +the time of Euclid,'' was born in Utzendorf in the Canton of +Bern. He did not learn to write till he was fourteen. At +eighteen he became a pupil of Pestalozzi. Later he studied +at Heidelberg and Berlin. When Crelle started, in 1826, the +celebrated mathematical journal bearing his name, Steiner and +Abel became leading contributors. In 1832 Steiner published +his \textit{Systematische Entwickelung der Abhängigkeit geometrischer +Gestalten von einander}, ``in which is uncovered the organism +by which the most diverse phenomena (\textit{Erscheinungen}) in +the world of space are united to each other.'' Through the +influence of Jacobi and others, the chair of geometry was +\index{Jacobi!ref.\ to}% +founded for him at Berlin in~1834. This position he occupied +until his death, which occurred after years of bad health. In +his \textit{Systematische Entwickelungen}, for the first time, is the +principle of duality introduced at the outset. This book and +von Staudt's lay the foundation on which synthetic geometry +\index{Von Staudt!ref.\ to}% +in its present form rests. Not only did he fairly complete the +theory of curves and surfaces of the second degree, but he +\index{Surfaces, theory of}% +made great advances in the theory of those of higher degrees. +In his hands synthetic geometry made prodigious progress. +New discoveries followed each other so rapidly that he often +did not take time to record their demonstrations. In an +article in \textit{Crelle's Journal} on \textit{Allgemeine Eigenschaften Algebraischer +\index{Crelle's Journal}% +Curven} he gives without proof theorems which were +declared by Hesse to be ``like Fermat's theorems, riddles to +\index{Hesse!ref.\ to}% +the present and future generations.'' Analytical proofs of +some of them have been given since by others, but Cremona +finally proved them all by a synthetic method. Steiner discovered +synthetically the two prominent properties of a surface +of the third order; viz.~that it contains twenty-seven +straight lines and a pentahedron which has the double points +for its vertices and the lines of the Hessian of the given surface +\index{Hessian}% +%% -----File: 310.png---Folio 296------- +for its edges.\Reference{55} The first property was discovered analytically +somewhat earlier in England by Cayley and Salmon, +\index{Cayley!ref.\ to}% +and the second by Sylvester. Steiner's work on this subject +\index{Steiner}% +\index{Sylvester!ref.\ to}% +was the starting-point of important researches by H.~Schröter, +\index{SchroterH@{Schröter, H.}!ref.\ to}% +F.~August, L.~Cremona, and R.~Sturm. Steiner made investigations +\index{August}% +\index{Cremona!ref.\ to|)}% +\index{Sturm, R.}% +by synthetic methods on maxima and minima, and +arrived at the solution of problems which at that time altogether +surpassed the analytic power of the calculus of variations. +\index{Calculus!of variations}% +He generalised the \textit{hexagrammum mysticum} and also +\index{Hexagrammum mysticum}% +Malfatti's problem.\Reference{59} Malfatti, in 1803, proposed the problem, +\index{Malfatti}% +\index{Malfatti's problem}% +to cut three cylindrical holes out of a three-sided prism in +such a way that the cylinders and the prism have the same +altitude and that the volume of the cylinders be a maximum. +This problem was reduced to another, now generally known +as Malfatti's problem: to inscribe three circles in a triangle +that each circle will be tangent to two sides of a triangle and +to the other two circles. Malfatti gave an analytical solution, +but Steiner gave without proof a construction, remarked that +there were thirty-two solutions, generalised the problem by +replacing the three lines by three circles, and solved the +analogous problem for three dimensions. This general problem +was solved analytically by C.~H. Schellbach (1809--1892) +\index{Schellbach}% +and Cayley, and by Clebsch with the aid of the addition +\index{Clebsch!ref.\ to}% +theorem of elliptic functions.\Reference{60} +\index{Elliptic functions}% + +Steiner's researches are confined to synthetic geometry. He +hated analysis as thoroughly as Lagrange disliked geometry. +\index{Lagrange!ref.\ to}% +Steiner's \textit{Gesammelte Werke} were published in Berlin in 1881 +and 1882. + +\textbf{Michel Chasles} (1793--1880) was born at Epernon, entered +\index{Chasles|(}% +the Polytechnic School of Paris in 1812, engaged afterwards +in business, which he later gave up that he might devote all +his time to scientific pursuits. In 1841 he became professor of +geodesy and mechanics at the Polytechnic School; later, +%% -----File: 311.png---Folio 297------- +\index{Cubic curves}% +``Professeur de Géométrie supérieure à la Faculté des Sciences +de Paris.'' He was a voluminous writer on geometrical subjects. +In 1837 he published his admirable \textit{Aperçu historique +sur l'origine et le développement des méthodes en géométrie}, containing +a history of geometry and, as an appendix, a treatise +``sur deux principes généraux de la Science.'' The \textit{Aperçu +historique} is still a standard historical work; the appendix +contains the general theory of Homography (Collineation) and +of duality (Reciprocity). The name \emph{duality} is due to Joseph +\index{Duality}% +Diaz Gergonne (1771--1859). Chasles introduced the term +\index{Gergonne}% +\emph{anharmonic ratio}, corresponding to the German \textit{Doppelverhältniss} +\index{Anharmonic ratio}% +and to Clifford's \emph{cross-ratio}. Chasles and Steiner +\index{Clifford!ref.\ to}% +\index{Steiner!ref.\ to}% +elaborated independently the modern synthetic or projective +geometry. Numerous original memoirs of Chasles were published +later in the \textit{Journal de l'École Polytechnique}. He gave +a reduction of cubics, different from Newton's in this, that the +\index{Newton!ref.\ to}% +five curves from which all others can be projected are symmetrical +with respect to a centre. In 1864 he began the publication, +in the \textit{Comptes rendus}, of articles in which he solves +by his ``method of characteristics'' and the ``principle of correspondence'' +\index{Characteristics, method of}% +\index{Correspondence, principle of}% +\index{Method of characteristics}% +an immense number of problems. He determined, +for instance, the number of intersections of two curves +in a plane. The method of characteristics contains the basis +of enumerative geometry. The application of the principle of +\index{Enumerative geometry}% +correspondence was extended by Cayley, A.~Brill, H.~G. Zeuthen, +\index{Brill, A.}% +\index{Cayley!ref.\ to}% +\index{Zeuthen!ref.\ to}% +H.~A. Schwarz, G.~H. Halphen (1844--1889), and others. +\index{Halphen!ref.\ to}% +\index{Schwarz!ref.\ to}% +The full value of these principles of Chasles was not brought +out until the appearance, in 1879, of the \textit{Kalkül der Abzählenden +Geometrie} by Hermann Schubert of Hamburg. This +\index{Schubert}% +work contains a masterly discussion of the problem of enumerative +geometry, viz.~to determine how many geometric figures +of given definition satisfy a sufficient number of conditions. +Schubert extended his enumerative geometry to $n$-dimensional +space.\Reference{55} +%% -----File: 312.png---Folio 298------- + +To Chasles we owe the introduction into projective geometry +of non-projective properties of figures by means of the infinitely +distant imaginary sphero-circle.\Reference{61} Remarkable is his +\index{Imaginary points, lines, etc}% +complete solution, in 1846, by synthetic geometry, of the +difficult question of the attraction of an ellipsoid on an external +\index{Ellipsoid!(attraction of)}% +point. This was accomplished analytically by Poisson in +\index{Poisson!ref.\ to}% +1835. The labours of Chasles and Steiner raised synthetic +\index{Steiner!ref.\ to}% +geometry to an honoured and respected position by the side +of analysis. + +\textbf{Karl Georg Christian von Staudt} (1798--1867) was born in +\index{Von Staudt}% +Rothenburg on the Tauber, and, at his death, was professor +in Erlangen. His great works are the \textit{Geometrie der Lage}, +Nürnberg, 1847, and his \textit{Beiträge zur Geometrie der Lage}, 1856--1860. +The author cut loose from algebraic formulæ and from +metrical relations, particularly the anharmonic ratio of Steiner +and Chasles, and then created a geometry of position, which +\index{Chasles|)}% +is a complete science in itself, independent of all measurements. +He shows that projective properties of figures have +no dependence whatever on measurements, and can be established +without any mention of them. In his theory of what +he calls ``Würfe,'' he even gives a geometrical definition of +a number in its relation to geometry as determining the position +of a point. The \textit{Beiträge} contains the first complete and +general theory of imaginary points, lines, and planes in projective +geometry. Representation of an imaginary point is +sought in the combination of an involution with a determinate +direction, both on the real line through the point. +While purely projective, von Staudt's method is intimately +related to the problem of representing by actual points and +lines the imaginaries of analytical geometry. This was systematically +undertaken by C.~F. Maximilien Marie, who worked, +\index{Marie, C. F. M.}% +however, on entirely different lines. An independent attempt +has been made recently (1893) by F.~H. Loud of Colorado +\index{Loud}% +%% -----File: 313.png---Folio 299------- +\index{Funicular polygons}% +College. Von~Staudt's geometry of position was for a long +\index{Von Staudt}% +time disregarded, mainly, no doubt, because his book is +extremely condensed. An impulse to the study of this subject +was given by Culmann, who rests his graphical statics upon +\index{Culmann}% +\index{Graphical statics}% +the work of von~Staudt. An interpreter of von~Staudt was +at last found in Theodor Reye of Strassburg, who wrote a +\index{Reye}% +\textit{Geometrie der Lage} in 1868. + +Synthetic geometry has been studied with much success by +\textbf{Luigi Cremona}, professor in the University of Rome. In +\index{Cremona}% +his \textit{Introduzione ad una teoria geometrica delle curve piane} +he developed by a uniform method many new results and +proved synthetically all important results reached before that +time by analysis. His writings have been translated into +German by M.~Curtze, professor at the gymnasium in Thorn. +\index{Curtze, M}% +The theory of the transformation of curves and of the correspondence +of points on curves was extended by him to three +dimensions. Ruled surfaces, surfaces of the second order, +\index{Surfaces, theory of}% +space-curves of the third order, and the general theory of +surfaces have received much attention at his hands. + +\textbf{Karl Culmann}, professor at the Polytechnicum in Zürich, +published an epoch-making work on \textit{Die graphische Statik}, +Zürich, 1864, which has rendered graphical statics a great +rival of analytical statics. Before Culmann, \textit{B.~E. Cousinery} +\index{Cousinery}% +had turned his attention to the graphical calculus, but he +made use of perspective, and not of modern geometry.\Reference{62} Culmann +is the first to undertake to present the graphical calculus +as a symmetrical whole, holding the same relation to the new +geometry that analytical mechanics does to higher analysis. +He makes use of the polar theory of reciprocal figures as +expressing the relation between the force and the funicular +polygons. He deduces this relation without leaving the plane +of the two figures. But if the polygons be regarded as projections +of lines in space, these lines may be treated as reciprocal +%% -----File: 314.png---Folio 300------- +\index{Culmann}% +\index{Non-Euclidean geometry|(}% +\index{Parallels}% +elements of a ``Nullsystem.'' This was done by \textit{Clerk +Maxwell} in 1864, and elaborated further by \textit{Cremona}.\Reference{63} The +\index{Cremona!ref.\ to}% +\index{Maxwell!ref.\ to}% +graphical calculus has been applied by \textit{O.~Mohr} of Dresden +\index{Mohr}% +to the elastic line for continuous spans. \textit{Henry~T. Eddy}, of +\index{Eddy}% +the Rose Polytechnic Institute, gives graphical solutions of +problems on the maximum stresses in bridges under concentrated +loads, with aid of what he calls ``reaction polygons.'' +\index{Reaction polygons}% +A standard work, \textit{La Statique graphique}, 1874, was issued by +Maurice Levy of Paris. +\index{Levy}% + +Descriptive geometry (reduced to a science by Monge in +\index{Descriptive geometry}% +\index{Geometry!descriptive}% +\index{Monge!ref.\ to}% +France, and elaborated further by his successors, \textit{Hachette}, +\index{Hachette}% +\textit{Dupin}, \textit{Olivier}, \textit{J.~de la Gournerie}) was soon studied also in +\index{Dupin!ref.\ to}% +\index{Gournerie}% +\index{Olivier}% +other countries. The French directed their attention mainly +to the theory of surfaces and their curvature; the Germans +and Swiss, through Schreiber, Pohlke, Schlessinger, and particularly +\index{Pohlke}% +\index{Schlessinger}% +\index{Schreiber}% +Fiedler, interwove projective and descriptive geometry. +\index{Fiedler}% +Bellavitis in Italy worked along the same line. The +\index{Bellavitis!ref.\ to}% +theory of shades and shadows was first investigated by the +French writers just quoted, and in Germany treated most +exhaustively by Burmester.\Reference{62} +\index{Burmester}% + +During the present century very remarkable generalisations +have been made, which reach to the very root of two of the +oldest branches of mathematics,---elementary algebra and +geometry. In algebra the laws of operation have been extended; +in geometry the axioms have been searched to the +\index{Axioms (of geometry)}% +bottom, and the conclusion has been reached that the space +defined by Euclid's axioms is not the only possible non-contradictory +space. Euclid proved (I.~27) that ``if a straight +line falling on two other straight lines make the alternate +angles equal to one another, the two straight lines shall be +parallel to one another.'' Being unable to prove that in every +other case the two lines are not parallel, he \emph{assumed} this to +be true in what is generally called the 12th ``axiom,'' by some +%% -----File: 315.png---Folio 301------- +the 11th ``axiom.'' But this so-called axiom is far from +axiomatic. After centuries of desperate but fruitless attempts +to prove Euclid's assumption, the bold idea dawned upon +the minds of several mathematicians that a geometry might +be built up without assuming the parallel-axiom. While +Legendre still endeavoured to establish the axiom by rigid +\index{Legendre!ref.\ to}% +proof, Lobatchewsky brought out a publication which assumed +\index{Lobatchewsky}% +the contradictory of that axiom, and which was the first of +a series of articles destined to clear up obscurities in the +fundamental concepts, and to greatly extend the field of +geometry. + +\textbf{Nicholaus Ivanovitch Lobatchewsky} (1793--1856) was born at +Makarief, in Nischni-Nowgorod, Russia, studied at Kasan, and +from 1827 to 1846 was professor and rector of the University +of Kasan. His views on the foundation of geometry were +first made public in a discourse before the physical and mathematical +faculty at Kasan, and first printed in the Kasan +\textit{Messenger} for 1829, and then in the \textit{Gelehrte Schriften der +Universität Kasan}, 1836--1838, under the title, ``New Elements +of Geometry, with a complete theory of Parallels.'' Being +\index{Parallels}% +in the Russian language, the work remained unknown to +foreigners, but even at home it attracted no notice. In 1840 +he published a brief statement of his researches in Berlin. +Lobatchewsky constructed an ``imaginary geometry,'' as he +\index{Imaginary geometry}% +called it, which has been described by Clifford as ``quite +simple, merely Euclid without the vicious assumption.'' A +remarkable part of this geometry is this, that through a +point an indefinite number of lines can be drawn in a plane, +none of which cut a given line in the same plane. A similar +system of geometry was deduced independently by the Bolyais +in Hungary, who called it ``absolute geometry.'' +\index{Absolute geometry}% + +\textbf{Wolfgang Bolyai de~Bolya} (1775--1856) was born in Szekler-Land, +\index{Bolyai, Wolfgang}% +Transylvania. After studying at Jena, he went to +%% -----File: 316.png---Folio 302------- +Göttingen, where he became intimate with Gauss, then nineteen +\index{Gauss!ref.\ to}% +years old. Gauss used to say that Bolyai was the only +\index{Bolyai, Johann}% +\index{Bolyai, Wolfgang}% +man who fully understood his views on the metaphysics of +mathematics. Bolyai became professor at the Reformed College +of Maros-Vásárhely, where for forty-seven years he had +for his pupils most of the present professors of Transylvania. +The first publications of this remarkable genius were dramas +and poetry. Clad in old-time planter's garb, he was truly +original in his private life as well as in his mode of thinking. +He was extremely modest. No monument, said he, should +stand over his grave, only an apple-tree, in memory of the +three apples; the two of Eve and Paris, which made hell out +of earth, and that of Newton, which elevated the earth again +\index{Newton!ref.\ to}% +into the circle of heavenly bodies.\Reference{64} His son, \textbf{Johann Bolyai} +(1802--1860), was educated for the army, and distinguished +himself as a profound mathematician, an impassioned violin-player, +and an expert fencer. He once accepted the challenge +of thirteen officers on condition that after each duel he might +play a piece on his violin, and he vanquished them all. + +The chief mathematical work of Wolfgang Bolyai appeared +in two volumes, 1832--1833, entitled \textit{Tentamen juventutem +studiosam in elementa matheseos puræ\ldots introducendi}. It +is followed by an appendix composed by his son Johann on +\textit{The Science Absolute of Space}. Its twenty-six pages make the +name of Johann Bolyai immortal. He published nothing else, +but he left behind one thousand pages of manuscript which +have never been read by a competent mathematician! His +father seems to have been the only person in Hungary who +really appreciated the merits of his son's work. For thirty-five +years this appendix, as also Lobatchewsky's researches, +remained in almost entire oblivion. Finally Richard Baltzer +\index{Baltzer, R.!ref.\ to}% +of the University of Giessen, in 1867, called attention to the +wonderful researches. Johann Bolyai's \textit{Science Absolute of} +%% -----File: 317.png---Folio 303------- +\textit{Space} and Lobatchewsky's \textit{Geometrical Researches on the +\index{Lobatchewsky!ref.\ to}% +Theory of Parallels} (1840) were rendered easily accessible to +\index{Parallels}% +American readers by translations into English made in 1891 +by George Bruce Halsted of the University of Texas. +\index{Halsted}% + +The Russian and Hungarian mathematicians were not the +only ones to whom pangeometry suggested itself. A copy of +the \textit{Tentamen} reached Gauss, the elder Bolyai's former room-mate %[** PP: Only instance] +\index{Gauss!ref.\ to}% +at Göttingen, and this Nestor of German mathematicians +was surprised to discover in it worked out what he himself +had begun long before, only to leave it after him in his +papers. As early as 1792 he had started on researches of that +character. His letters show that in 1799 he was trying to +prove \textit{a~priori} the reality of Euclid's system; but some time +\index{Euclid!ref.\ to}% +within the next thirty years he arrived at the conclusion +reached by Lobatchewsky and Bolyai. In 1829 he wrote to +Bessel, stating that his ``conviction that we cannot found +\index{Bessel!ref.\ to}% +geometry completely a~priori has become, if possible, still +firmer,'' and that ``if number is merely a product of our +mind, space has also a \emph{reality beyond} our mind of which we +cannot fully foreordain the laws \textit{a~priori}.'' The term \emph{non-Euclidean +geometry} is due to Gauss. It has recently been +brought to notice that \textit{Geronimo Saccheri}, a Jesuit father of +\index{Saccheri}% +Milan, in 1733 anticipated Lobatchewsky's doctrine of the +parallel angle. Moreover, G.~B. Halsted has pointed out +that in 1766 Lambert wrote a paper ``Zur Theorie der Parallellinien,'' +\index{Lambert!ref.\ to}% +published in the \textit{Leipziger Magazin für reine und +angewandte Mathematik}, 1786, in which: (1)~The failure of +the parallel-axiom in surface-spherics gives a geometry with +angle-sum $> 2$ right angles; (2)~In order to make intuitive +a geometry with angle-sum $< 2$ right angles we need the aid +of an ``imaginary sphere" (pseudo-sphere); (3)~In a space +with the angle-sum differing from 2 right angles, there is +an absolute measure (Bolyai's natural unit for length). +%% -----File: 318.png---Folio 304------- + +In 1854, nearly twenty years later, Gauss heard from his +\index{Gauss!ref.\ to}% +pupil, \textit{Riemann}, a marvellous dissertation carrying the discussion +\index{Riemann!ref.\ to}% +one step further by developing the notion of \nnobrkdash-ply +extended magnitude, and the measure-relations of which a +manifoldness of $n$ dimensions is capable, on the assumption +that every line may be measured by every other. Riemann +applied his ideas to space. He taught us to distinguish +between ``unboundedness'' and ``infinite extent.'' According +\index{Infinity}% +to him we have in our mind a more general notion of space, +\ie~a notion of non-Euclidean space; but we learn \emph{by experience} +that our physical space is, if not exactly, at least to high +degree of approximation, Euclidean space. Riemann's profound +dissertation was not published until 1867, when it +appeared in the \textit{Göttingen Abhandlungen}. Before this the +idea of \nnobrkdash-dimensions had suggested itself under various +\index{Hyperspace}% +aspects to Lagrange, Plücker, and H.~Grassmann. About the +\index{Grassmann!ref.\ to}% +\index{Lagrange!ref.\ to}% +\index{Plücker!ref.\ to}% +same time with Riemann's paper, others were published from +the pens of \textit{Helmholtz} and \textit{Beltrami}. These contributed powerfully +\index{Beltrami}% +\index{Helmholtz!ref.\ to}% +to the victory of logic over excessive empiricism. This +period marks the beginning of lively discussions upon this subject. +Some writers---Bellavitis, for example---were able to +\index{Bellavitis!ref.\ to}% +see in non-Euclidean geometry and \nnobrkdash-dimensional space nothing +but huge caricatures, or diseased outgrowths of mathematics. +Helmholtz's article was entitled \textit{Thatsachen, welche +der Geometrie zu Grunde liegen}, 1868, and contained many of +the ideas of Riemann. Helmholtz popularised the subject in +lectures, and in articles for various magazines. + +\textbf{Eugenio Beltrami}, born at Cremona, Italy, in 1835, and now +professor at Rome, wrote the classical paper \textit{Saggio di interpretazione +della geometria non-euclidea} (Giorn.\ di Matem.,~6), +which is analytical (and, like several other papers, should be +mentioned elsewhere were we to adhere to a strict separation +between synthesis and analysis). He reached the brilliant +%% -----File: 319.png---Folio 305------- +and surprising conclusion that the theorems of non-Euclidean +geometry find their realisation upon surfaces of constant negative +curvature. He studied, also, surfaces of constant positive +curvature, and ended with the interesting theorem that the +space of constant positive curvature is contained in the space +of constant negative curvature. These researches of Beltrami, +\index{Beltrami}% +Helmholtz, and Riemann culminated in the conclusion that +\index{Helmholtz!ref.\ to}% +\index{Riemann!ref.\ to}% +on surfaces of constant curvature we may have three geometries,---the +non-Euclidean on a surface of constant negative +curvature, the spherical on a surface of constant positive curvature, +and the Euclidean geometry on a surface of zero curvature. +The three geometries do not contradict each other, but +are members of a system,---a geometrical trinity. The ideas +of \DPtypo{hyper-space}{hyperspace} were brilliantly expounded and popularised in +\index{Hyperspace}% +England by Clifford. +\index{Clifford}% + +\textbf{William Kingdon Clifford} (1845--1879) was born at Exeter, +educated at Trinity College, Cambridge, and from 1871 until +his death professor of applied mathematics in University College, +London. His premature death left incomplete several +brilliant researches which he had entered upon. Among these +are his paper \textit{On Classification of Loci} and his \textit{Theory of +Graphs}. He wrote articles \textit{On the Canonical Form and +Dissection of a Riemann's Surface}, on \textit{Biquaternions}, and +an incomplete work on the \textit{Elements of Dynamic}. The +theory of polars of curves and surfaces was generalised by +him and by Reye. His classification of loci, 1878, being a +\index{Reye!ref.\ to}% +general study of curves, was an introduction to the study +of \nnobrkdash-dimensional space in a direction mainly projective. +This study has been continued since chiefly by G.~Veronese +\index{Veronese}% +of Padua, C.~Segre of Turin, E.~Bertini, F.~Aschieri, P.~Del~Pezzo +\index{Aschieri}% +\index{Bertini}% +\index{Delpezzo@{Del Pezzo}}% +\index{Segre}% +of Naples. + +Beltrami's researches on non-Euclidean geometry were followed, +in 1871, by important investigations of Felix Klein, +\index{Klein!ref.\ to}% +%% -----File: 320.png---Folio 306------- +\index{Clifford}% +\index{Parallels}% +resting upon Cayley's \textit{Sixth Memoir on Quantics}, 1859. The +\index{Cayley!ref.\ to}% +question whether it is not possible to so express the metrical +properties of figures that they will not vary by projection (or +linear transformation) had been solved for special projections +by Chasles, Poncelet, and E.~Laguerre (1834--1886) of Paris, +\index{Chasles!ref.\ to}% +\index{Laguerre}% +\index{Poncelet!ref.\ to}% +but it remained for Cayley to give a general solution by defining +the distance between two points as an arbitrary constant +multiplied by the logarithm of the anharmonic ratio in which +\index{Anharmonic ratio}% +the line joining the two points is divided by the fundamental +quadric. Enlarging upon this notion, Klein showed the independence +\index{Klein!ref.\ to}% +of projective geometry from the parallel-axiom, and +by properly choosing the law of the measurement of distance +deduced from projective geometry the spherical, Euclidean, +and pseudospherical geometries, named by him respectively +the elliptic, parabolic, and hyperbolic geometries. This suggestive +investigation was followed up by numerous writers, +particularly by G.~Battaglini of Naples, E.~d'Ovidio of Turin, +\index{Battaglini}% +\index{Dovidio@{D'Ovidio}}% +R.~de~Paolis of Pisa, F.~Aschieri, A.~Cayley, F.~Lindemann +\index{Aschieri}% +\index{Depaolis@{De Paolis}}% +\index{Lindemann!ref.\ to}% +of Munich, E.~Schering of Göttingen, W.~Story of Clark +\index{Schering}% +\index{Story}% +University, H.~Stahl of Tübingen, A.~Voss of Würzburg, +\index{Stahl}% +\index{Voss}% +Homersham Cox, A.~Buchheim.\Reference{55} The geometry of $n$ dimensions +\index{Buchheim!ref.\ to}% +\index{Cox}% +was studied along a line mainly metrical by a host of +writers, among whom may be mentioned Simon Newcomb of +\index{Newcomb!ref.\ to}% +the Johns Hopkins University, L.~Schläfli of Bern, W.~I. +\index{Schlafli@{Schläfli}}% +Stringham of the University of California, W.~Killing of +\index{Killing}% +\index{Stringham}% +Münster, T.~Craig of the Johns Hopkins, R.~Lipschitz of +\index{Craig, T.}% +\index{Lipschitz}% +Bonn. R.~S. Heath and Killing investigated the kinematics +\index{Heath}% +and mechanics of such a space. Regular solids in \nnobrkdash-dimensional +space were studied by Stringham, Ellery~W. Davis +\index{Davis, E.~W.}% +of the University of Nebraska, R.~Hoppe of Berlin, and +\index{Hoppe}% +others. Stringham gave pictures of projections upon our +space of regular solids in four dimensions, and Schlegel at +\index{Schlegel!ref.\ to, XII}% +Hagen constructed models of such projections. These are +%% -----File: 321.png---Folio 307------- +\index{Non-Euclidean geometry|)}% +among the most curious of a series of models published by +L.~Brill in Darmstadt. It has been pointed out that if a +\index{Brill, L.}% +fourth dimension existed, certain motions could take place +which we hold to be impossible. Thus Newcomb showed the +\index{Newcomb!ref.\ to}% +possibility of turning a closed material shell inside out by simple +flexure without either stretching or tearing; Klein pointed +\index{Klein!ref.\ to}% +out that knots could not be tied; Veronese showed that a +\index{Veronese!ref.\ to}% +body could be removed from a closed room without breaking +the walls; C.~S. Peirce proved that a body in four-fold space +\index{Peirce, C.~S.!ref.\ to}% +either rotates about two axes at once, or cannot rotate without +losing one of its dimensions. + +\Section{Analytic Geometry} +\index{Analytic geometry|(}% +\index{Geometry!analytic|(}% + +In the preceding chapter we endeavoured to give a flash-light %[** PP: Only instance] +view of the rapid advance of synthetic geometry. In +connection with hyperspace we also mentioned analytical +treatises. Modern synthetic and modern analytical geometry +have much in common, and may be grouped together +under the common name ``projective geometry.'' Each has +\index{Projective geometry}% +advantages over the other. The continual direct viewing of +figures as existing in space adds exceptional charm to the +study of the former, but the latter has the advantage in this, +that a well-established routine in a certain degree may outrun +thought itself, and thereby aid original research. While in +Germany Steiner and von Staudt developed synthetic geometry, +\index{Synthetic geometry|)}% +Plücker laid the foundation of modern analytic geometry. +\index{Geometry!modern synthetic|)}% +\index{Plücker|(}% + +\textbf{Julius Plücker} (1801--1868) was born at Elberfeld, in Prussia. +After studying at Bonn, Berlin, and Heidelberg, he spent +a short time in Paris attending lectures of Monge and his +pupils. Between 1826 and 1836 he held positions successively +at Bonn, Berlin, and Halle. He then became professor of +%% -----File: 322.png---Folio 308------- +physics at Bonn. Until 1846 his original researches were on +geometry. In 1828 and in 1831 he published his \textit{Analytisch-Geometrische +Entwicklungen} in two volumes. Therein he +adopted the abbreviated notation (used before him in a more +restricted way by Bobillier), and avoided the tedious process +\index{Bobillier}% +of algebraic elimination by a geometric consideration. In the +\index{Elimination}% +second volume the principle of duality is formulated analytically. +\index{Duality}% +With him duality and homogeneity found expression +\index{Homogeneity}% +already in his system of co-ordinates. The homogenous or +\index{Co-ordinates}% +tri-linear system used by him is much the same as the co-ordinates +of Möbius. In the identity of analytical operation +and geometric construction Plücker looked for the source of +\index{Plücker!ref.\ to}% +his proofs. The \textit{System der Analytischen Geometrie}, 1835, contains +a complete classification of plane curves of the third +order, based on the nature of the points at infinity. The +\index{Infinity}% +\textit{Theorie der Algebraischen Curven}, 1839, contains, besides an +enumeration of curves of the fourth order, the analytic relations +between the ordinary singularities of plane curves +known as ``Plücker's equations,'' by which he was able to +explain ``Poncelet's paradox.'' The discovery of these relations +\index{Poncelet!ref.\ to}% +\index{Poncelet's paradox}% +is, says Cayley, ``the most important one beyond all +\index{Cayley!ref.\ to}% +comparison in the entire subject of modern geometry.'' But +in Germany Plücker's researches met with no favour. His +method was declared to be unproductive as compared with +the synthetic method of Steiner and Poncelet! His relations +\index{Steiner!ref.\ to}% +with Jacobi were not altogether friendly. Steiner once +\index{Jacobi!ref.\ to}% +declared that he would stop writing for \textit{Crelle's Journal} if +Plücker continued to contribute to it.\Reference{66} The result was that +many of Plücker's researches were published in foreign journals, +and that his work came to be better known in France +and England than in his native country. The charge was +also brought against Plücker that, though occupying the chair +of physics, he was no physicist. This induced him to relinquish +%% -----File: 323.png---Folio 309------- +\index{Congruency of lines}% +mathematics, and for nearly twenty years to devote +his energies to physics. Important discoveries on Fresnel's +wave-surface, magnetism, spectrum-analysis were made by +him. But towards the close of his life he returned to his +first love,---mathematics,---and enriched it with new discoveries. +By considering space as made up of lines he created +a ``new geometry of space.'' Regarding a right line as a +curve involving four arbitrary parameters, one has the whole +system of lines in space. By connecting them by a single +relation, he got a ``complex'' of lines; by connecting them +\index{Complex of lines}% +with a twofold relation, he got a ``congruency'' of lines. His +first researches on this subject were laid before the Royal +Society in~1865. His further investigations thereon appeared +in 1868 in a posthumous work entitled \textit{Neue Geometrie des +Raumes gegründet auf die Betrachtung der geraden Linie als +Raumelement}, edited by Felix Klein. Plücker's analysis lacks +\index{Klein!ref.\ to}% +the elegance found in Lagrange, Jacobi, Hesse, and Clebsch. +\index{Clebsch!ref.\ to}% +\index{Hesse|(}% +\index{Hesse!ref.\ to}% +\index{Jacobi!ref.\ to}% +\index{Lagrange!ref.\ to}% +For many years he had not kept up with the progress of +geometry, so that many investigations in his last work had +already received more general treatment on the part of others. +The work contained, nevertheless, much that was fresh and +original. The theory of complexes of the second degree, left +unfinished by Plücker, was continued by Felix Klein, who +\index{Plücker|)}% +greatly extended and supplemented the ideas of his master. + +\textbf{Ludwig Otto Hesse} (1811--1874) was born at Königsberg, and +studied at the university of his native place under Bessel, +\index{Bessel!ref.\ to}% +Jacobi, Richelot, and F.~Neumann. Having taken the doctor's +\index{Neumann, C.!ref.\ to}% +\index{Neumann, F.~E.!ref.\ to}% +\index{Richelot!ref.\ to}% +degree in 1840, he became docent at Königsberg, and in 1845 +extraordinary professor there. Among his pupils at that time +were Durège, Carl Neumann, Clebsch, Kirchhoff. The Königsberg +\index{Durège!ref.\ to}% +\index{Kirchhoff!ref.\ to}% +period was one of great activity for Hesse. Every new +discovery increased his zeal for still greater achievement. +His earliest researches were on surfaces of the second order, +\index{Surfaces, theory of}% +%% -----File: 324.png---Folio 310------- +\index{Invariant}% +and were partly synthetic. He solved the problem to construct +any tenth point of such a surface when nine points are given. +The analogous problem for a conic had been solved by Pascal +\index{Pascal!ref.\ to}% +by means of the hexagram. A difficult problem confronting +mathematicians of this time was that of elimination. Plücker +\index{Elimination}% +had seen that the main advantage of his special method in +analytic geometry lay in the avoidance of algebraic elimination. +Hesse, however, showed how by determinants to make +algebraic elimination easy. In his earlier results he was +anticipated by Sylvester, who published his dialytic method +\index{Sylvester!ref.\ to}% +of elimination in 1840. These advances in algebra Hesse +applied to the analytic study of curves of the third order. By +linear substitutions, he reduced a form of the third degree in +three variables to one of only four terms, and was led to an +important determinant involving the second differential coefficient +of a form of the third degree, called the ``Hessian.'' +\index{Hessian}% +The ``Hessian'' plays a leading part in the theory of invariants, +a subject first studied by Cayley. Hesse showed that +his determinant gives for every curve another curve, such that +the double points of the first are points on the second, or +``Hessian." Similarly for surfaces (Crelle, 1844). Many of +\index{Surfaces, theory of}% +the most important theorems on curves of the third order are +due to Hesse. He determined the curve of the 14th order, +which passes through the 56~points of contact of the 28~bi-tangents +of a curve of the fourth order. His great memoir on +this subject (Crelle, 1855) was published at the same time as +was a paper by Steiner treating of the same subject. + +Hesse's income at Königsberg had not kept pace with his +growing reputation. Hardly was he able to support himself +and family. In 1855 he accepted a more lucrative position at +Halle, and in 1856 one at Heidelberg. Here he remained until +1868, when he accepted a position at a technic school in +Munich.\Reference{67} At Heidelberg he revised and enlarged upon his +%% -----File: 325.png---Folio 311------- +\index{Gournerie}% +\index{McCullagh}% +previous researches, and published in 1861 his \textit{Vorlesungen +über die Analytische Geometrie des Raumes, insbesondere über +Flächen 2.~Ordnung}. More elementary works soon followed. +While in Heidelberg he elaborated a principle, his ``Uebertragungsprincip." +According to this, there corresponds to every +point in a plane a pair of points in a line, and the projective +geometry of the plane can be carried back to the geometry of +points in a line. + +The researches of Plücker and Hesse were continued in England +\index{Hesse|)}% +by Cayley, Salmon, and Sylvester. It may be premised here +\index{Cayley!ref.\ to}% +\index{Salmon|(}% +\index{Sylvester!ref.\ to}% +that among the early writers on analytical geometry in England +was \textbf{James Booth} (1806--1878), whose chief results are embodied +\index{Booth}% +in his \textit{Treatise on Some New Geometrical Methods}; and \textbf{James +MacCullagh} (1809--1846), who was professor of natural philosophy +\index{MacCullagh}% +at Dublin, and made some valuable discoveries on the +theory of quadrics. The influence of these men on the +progress of geometry was insignificant, for the interchange of +scientific results between different nations was not so complete +at that time as might have been desired. In further illustration +of this, we mention that Chasles in France elaborated +\index{Chasles!ref.\ to}% +subjects which had previously been disposed of by Steiner in +\index{Steiner!ref.\ to}% +Germany, and Steiner published researches which had been +given by Cayley, Sylvester, and Salmon nearly five years +earlier. Cayley and Salmon in 1849 determined the straight +lines in a cubic surface, and studied its principal properties, +while Sylvester in 1851 discovered the pentahedron of such a +surface. Cayley extended Plücker's equations to curves of +higher singularities. Cayley's own investigations, and those +of M.~Nöther of Erlangen, G.~H. Halphen (1844--1889) of the +\index{Halphen}% +\index{Nother@{Nöther}}% +Polytechnic School in Paris, De~La~\DPtypo{Gournérie}{Gournerie} of Paris, A.~Brill +\index{Brill, A.}% +of Tübingen, lead to the conclusion that each higher singularity +of a curve is equivalent to a certain number of simple +singularities,---the node, the ordinary cusp, the double tangent, +%% -----File: 326.png---Folio 312------- +\index{Deficiency of curves}% +and the inflection. Sylvester studied the ``twisted Cartesian,'' +\index{Sylvester!ref.\ to}% +\index{Twisted Cartesian}% +a curve of the fourth order. Salmon helped powerfully +towards the spreading of a knowledge of the new algebraic and +geometric methods by the publication of an excellent series of +text-books (\textit{Conic Sections}, \textit{Modern Higher Algebra}, \textit{Higher +Plane Curves}, \textit{Geometry of Three Dimensions}), which have +been placed within easy reach of German readers by a free +translation, with additions, made by Wilhelm Fiedler of the +\index{Fiedler}% +Polytechnicum in Zürich. The next great worker in the field +of analytic geometry was Clebsch. +\index{Clebsch}% + +\textbf{Rudolf Friedrich Alfred Clebsch} (1833--1872) was born at +Königsberg in Prussia, studied at the university of that place +under Hesse, Richelot, F.~Neumann. From 1858 to 1863 he +\index{Hesse!ref.\ to}% +\index{Neumann, F.~E.!ref.\ to}% +\index{Richelot!ref.\ to}% +held the chair of theoretical mechanics at the Polytechnicum +in Carlsruhe. The study of Salmon's works led him into +algebra and geometry. In 1863 he accepted a position at the +University of \DPtypo{Giesen}{Giessen}, where he worked in conjunction with +Paul Gordan (now of Erlangen). In 1868 Clebsch went to +\index{Gordan}% +Göttingen, and remained there until his death. He worked +successively at the following subjects: Mathematical physics, +the calculus of variations and partial differential equations of +the first order, the general theory of curves and surfaces, +Abelian functions and their use in geometry, the theory of +\index{Abelian functions}% +invariants, and ``Flächenabbildung.''\Reference{68} He proved theorems +on the pentahedron enunciated by Sylvester and Steiner; he +\index{Steiner!ref.\ to}% +made systematic use of ``deficiency'' (\textit{Geschlecht}) as a fundamental +principle in the classification of algebraic curves. The +notion of deficiency was known before him to Abel and Riemann. +\index{Abel!ref.\ to}% +\index{Riemann!ref.\ to}% +At the beginning of his career, Clebsch had shown +how elliptic functions could be advantageously applied to +Malfatti's problem. The idea involved therein, viz.\ the use +\index{Malfatti's problem}% +of higher transcendentals in the study of geometry, led him +to his greatest discoveries. Not only did he apply Abelian +%% -----File: 327.png---Folio 313------- +\index{Armemante}% +\index{Ptolemy!ref.\ to}% +functions to geometry, but conversely, he drew geometry into +the service of Abelian functions. + +Clebsch made liberal use of determinants. His study of +\index{Clebsch}% +\index{Determinants}% +curves and surfaces began with the determination of the points +of contact of lines which meet a surface in four consecutive +points. Salmon had proved that these points lie on the intersection +\index{Salmon|)}% +of the surface with a derived surface of the degree +$11n - 24$, but his solution was given in inconvenient form. +Clebsch's investigation thereon is a most beautiful piece of +analysis. + +The representation of one surface upon another (\textit{Flächenabbildung}), +\index{Flächenabbildung}% +so that they have a $(1, 1)$ correspondence, was +thoroughly studied for the first time by Clebsch. The representation +of a sphere on a plane is an old problem which +drew the attention of Ptolemæus, Gerard Mercator, Lambert, +\index{Lambert!ref.\ to}% +\index{Mercator, G.}% +Gauss, Lagrange. Its importance in the construction of maps +\index{Gauss!ref.\ to}% +\index{Lagrange!ref.\ to}% +is obvious. Gauss was the first to represent a surface upon +another with a view of more easily arriving at its properties. +Plücker, Chasles, Cayley, thus represented on a plane the +\index{Cayley!ref.\ to}% +\index{Chasles!ref.\ to}% +\index{Plücker!ref.\ to}% +geometry of quadric surfaces; Clebsch and Cremona, that of +\index{Cremona!ref.\ to}% +cubic surfaces. Other surfaces have been studied in the same +way by recent writers, particularly M.~Nöther of Erlangen, +\index{Nother@{Nöther}}% +Armenante, Felix Klein, Korndörfer, Caporali, H.~G. Zeuthen +\index{Caporali}% +\index{Klein!ref.\ to}% +\index{Korndörfer}% +\index{Zeuthen}% +of Copenhagen. A fundamental question which has as yet +received only a partial answer is this: What surfaces can be +represented by a $(1, 1)$ correspondence upon a given surface? +This and the analogous question for curves was studied by +Clebsch. Higher correspondences between surfaces have been +investigated by Cayley and Nöther. The theory of surfaces +has been studied also by \textbf{Joseph Alfred Serret} (1819--1885), professor +\index{Serret}% +at the Sorbonne in Paris, \textbf{Jean Gaston Darboux} of Paris, +\index{Darboux}% +\textit{John Casey} of Dublin (died 1891), \textit{W.~R.~W. Roberts} of Dublin, +\index{Casey}% +\index{Roberts}% +\textit{H.~Schröter} (1829--1892) of Breslau. Surfaces of the +\index{SchroterH@{Schröter, H.}}% +%% -----File: 328.png---Folio 314------- +\index{Differential equations}% +\index{Fresnel's wave-surface}% +fourth order were investigated by Kummer, and Fresnel's +\index{Kummer!ref.\ to}% +wave-surface, studied by Hamilton, is a particular case of +\index{Hamilton, W.~R.!ref.\ to}% +Kummer's quartic surface, with sixteen canonical points and +sixteen singular tangent planes.\Reference{56} + +The infinitesimal calculus was first applied to the determination +of the measure of curvature of surfaces by Lagrange, +\index{Curvature, measure of}% +\index{Lagrange!ref.\ to}% +\index{Surfaces, theory of}% +Euler, and \DPtypo{Meunier}{Meusnier} (1754--1793) of Paris. Then followed the +\index{Euler!ref.\ to}% +\index{Meunier}% +researches of Monge and Dupin, but they were eclipsed by +\index{Dupin!ref.\ to}% +\index{Monge!ref.\ to}% +the work of Gauss, who disposed of this difficult subject in a +\index{Gauss!ref.\ to}% +way that opened new vistas to geometricians. His treatment +is embodied in the \textit{Disquisitiones generales circa superficies +curvas} (1827) and \textit{Untersuchungen über gegenstände\DPnote{** Non-capitalization seems to be a legitimate variant} der +höheren Geodäsie} of 1843 and~1846. He defined the measure +of curvature at a point to be the reciprocal of the product +of the two principal radii of curvature at that point. From +this flows the theorem of \textit{Johann August Grunert} (1797--1872; +\index{Grunert}% +professor in Greifswald), that the arithmetical mean of the +radii of curvature of all normal sections through a point is the +radius of a sphere which has the same measure of curvature +as has the surface at that point. Gauss's deduction of the +formula of curvature was simplified through the use of determinants +by \textit{Heinrich Richard Baltzer} (1818--1887) of Giessen.\Reference{69} +\index{Baltzer, R.}% +Gauss obtained an interesting theorem that if one surface be +developed (\textit{abgewickelt}) upon another, the measure of curvature +remains unaltered at each point. The question whether +two surfaces having the same curvature in corresponding +points can be unwound, one upon the other, was answered +by F.~Minding in the affirmative only when the curvature is +\index{Minding}% +constant. The case of variable curvature is difficult, and was +studied by Minding, J.~Liouville (1806--1882) of the Polytechnic +\index{Liouville!ref.\ to}% +School in Paris, Ossian Bonnet of Paris (died 1892). +\index{Bonnet, O.}% +Gauss's measure of curvature, expressed as a function of curvilinear +co-ordinates, gave an impetus to the study of differential-invariants, +\index{Co-ordinates}% +%% -----File: 329.png---Folio 315------- +\index{Analytic geometry|)}% +\index{Kelvin, Lord!ref.\ to}% +or differential-parameters, which have been +investigated by Jacobi, C.~Neumann, Sir James Cockle, +\index{Cockle}% +\index{Jacobi!ref.\ to}% +\index{Neumann, C.!ref.\ to}% +Halphen, and elaborated into a general theory by Beltrami, +\index{Beltrami!ref.\ to}% +\index{Halphen!ref.\ to}% +S.~Lie, and others. Beltrami showed also the connection +between the measure of curvature and the geometric axioms. +\index{Axioms (of geometry)}% + +Various researches have been brought under the head of +``analysis situs.'' The subject was first investigated by +\index{Analysis situs}% +Leibniz, and was later treated by Gauss, whose theory of +\index{Gauss!ref.\ to}% +\index{Leibniz!ref.\ to}% +knots (\textit{Verschlingungen}) has been employed recently by J.~B. +Listing, O.~Simony, F.~Dingeldey, and others in their ``topologic +\index{Dingeldey}% +\index{Listing}% +\index{Simony}% +studies.'' Tait was led to the study of knots by Sir +William Thomson's theory of vortex atoms. In the hands +\index{Thomson, Sir William!ref.\ to}% +of Riemann the analysis situs had for its object the determination +\index{Riemann!ref.\ to}% +of what remains unchanged under transformations +brought about by a combination of infinitesimal distortions. +In continuation of his work, Walter Dyck of Munich wrote on +\index{Dyck}% +the analysis situs of three-dimensional spaces. + +Of geometrical text-books not yet mentioned, reference +should be made to \textbf{Alfred Clebsch's} \textit{Vorlesungen über Geometrie}, +\index{Clebsch!ref.\ to}% +edited by Ferdinand Lindemann, now of Munich; \textbf{Frost's} +\index{Frost}% +\index{Lindemann}% +\textit{Solid Geometry}; \textbf{Durège's} \textit{Ebene Curven dritter Ordnung}. +\index{Durège!ref.\ to}% +\index{Geometry!analytic|)}% + + +\Section{Algebra} +\index{Algebra!recent|(}% + +The progress of algebra in recent times may be considered +under three principal heads: the study of fundamental laws +and the birth of new algebras, the growth of the theory of +equations, and the development of what is called modern +higher algebra. + +We have already spoken of George Peacock and D.~F. +\index{Peacock!ref.\ to, X}% +Gregory in connection with the fundamental laws of algebra. +\index{Gregory, David F}% +Much was done in this line by De~Morgan. +%% -----File: 330.png---Folio 316------- + +\textbf{Augustus De~Morgan} (1806--1871) was born at Madura (Madras), +\index{Demorgan@{De Morgan}}% +and educated at Trinity College, Cambridge. His scruples +about the doctrines of the established church prevented +him from proceeding to the M.A.~degree, and from sitting +for a fellowship. In 1828 he became professor at the newly +established University of London, and taught there until +1867, except for five years, from 1831--1835. De~Morgan was +a unique, manly character, and pre-eminent as a teacher. The +value of his original work lies not so much in increasing our +stock of mathematical knowledge as in putting it all upon a +thoroughly logical basis. He felt keenly the lack of close +reasoning in mathematics as he received it. He said once: +``We know that mathematicians care no more for logic than +\index{Logic}% +logicians for mathematics. The two eyes of exact science are +mathematics and logic: the mathematical sect puts out the +logical eye, the logical sect puts out the mathematical eye; +each believing that it can see better with one eye than with +two.'' De~Morgan saw with both eyes. He analysed logic +mathematically, and studied the logical analysis of the laws, +symbols, and operations of mathematics; he wrote a \textit{Formal +Logic} as well as a \textit{Double Algebra}, and corresponded both with +Sir William Hamilton, the metaphysician, and Sir William +\index{Hamilton, W.}% +\index{Hamilton, W.~R.!ref.\ to}% +Rowan Hamilton, the mathematician. Few contemporaries +were as profoundly read in the history of mathematics as +was De~Morgan. No subject was too insignificant to receive +his attention. The authorship of ``Cocker's Arithmetic'' and +the work of circle-squarers was investigated as minutely as was +\index{Circle-squarers}% +the history of the invention of the calculus. Numerous articles +of his lie scattered in the volumes of the \textit{Penny} and \textit{English +Cyclopædias}. His \textit{Differential Calculus}, 1842, is still a +standard work, and contains much that is original with the +author. For the \textit{Encyclopædia Metropolitana} he wrote on the +calculus of functions (giving principles of symbolic reasoning) +%% -----File: 331.png---Folio 317------- +and on the theory of probability. Celebrated is his \textit{Budget of +Paradoxes}, 1872. He published memoirs ``On the Foundation +of Algebra'' (\textit{Trans.\ of Cam.\ Phil.\ Soc.}, 1841, 1842, 1844, and~1847). + +In Germany symbolical algebra was studied by Martin Ohm, +\index{Ohm, M}% +who wrote a \textit{System der Mathematik} in~1822. The ideas of +Peacock and De~Morgan recognise the possibility of algebras +which differ from ordinary algebra. Such algebras were +indeed not slow in forthcoming, but, like non-Euclidean +geometry, some of them were slow in finding recognition. +This is true of Grassmann's, Bellavitis's, and Peirce's discoveries, +\index{Bellavitis!ref.\ to}% +\index{Grassmann!ref.\ to}% +\index{Peirce, B.!ref.\ to}% +but Hamilton's quaternions met with immediate +\index{Hamilton, W.~R.!ref.\ to}% +\index{Quaternions!ref.\ to}% +appreciation in England. These algebras offer a geometrical +interpretation of imaginaries. During the times of Descartes, +\index{Descartes!ref.\ to}% +Newton, and Euler, we have seen the negative and the imaginary, +\index{Euler!ref.\ to}% +\index{Newton!ref.\ to}% +$\sqrt{-1}$, accepted as numbers, but the latter was still +regarded as an algebraic fiction. The first to give it a geometric +picture, analogous to the geometric interpretation of the +negative, was \textit{H.~Kühn}, a teacher in Danzig, in a publication of +\index{Kuhn@{Kühn, H.}}% +1750--1751. He represented $a\sqrt{-1}$ by a line perpendicular +to the line~$a$, and equal to~$a$ in length, and construed $\sqrt{-1}$ as +the mean proportional between $+1$~and~$-1$. This same idea +was developed further, so as to give a geometric interpretation +of $a + \sqrt{-b}$, by \textit{Jean-Robert Argand} (1768--?) of Geneva, +\index{Argand}% +in a remarkable \textit{Essai} (1806).\Reference{70} The writings of Kühn and +Argand were little noticed, and it remained for Gauss to break +\index{Gauss!ref.\ to}% +down the last opposition to the imaginary. He introduced~$i$ as +an independent unit co-ordinate to~$1$, and $a + ib$ as a ``complex +\index{Complex quantities}% +number.'' The connection between complex numbers and +points on a plane, though artificial, constituted a powerful +aid in the further study of symbolic algebra. The mind +required a visual representation to aid it. The notion of +what we now call vectors was growing upon mathematicians, +%% -----File: 332.png---Folio 318------- +and the geometric addition of vectors in space was discovered +independently by Hamilton, Grassmann, and others, about the +\index{Grassmann!ref.\ to}% +\index{Hamilton, W.~R.}% +same time. + +\textbf{William Rowan Hamilton} (1805--1865) was born of Scotch +parents in Dublin. His early education, carried on at home, +was mainly in languages. At the age of thirteen he is said to +have been familiar with as many languages as he had lived +years. About this time he came across a copy of Newton's +\textit{Universal Arithmetic}. After reading that, he took up successively +analytical geometry, the calculus, Newton's \textit{Principia}, +Laplace's \textit{Mécanique Céleste}. At the age of eighteen he +published a paper correcting a mistake in Laplace's work. In +1824 he entered Trinity College, Dublin, and in 1827, while he +was still an undergraduate, he was appointed to the chair of +astronomy. His early papers were on optics. In 1832 he +predicted conical refraction, a discovery by aid of mathematics +which ranks with the discovery of Neptune by +Le~Verrier and Adams. Then followed papers on the \textit{Principle +of Varying Action} (1827) and a general method of +\index{Action!varying}% +\index{Varying action, principle of}% +dynamics (1834--1835). He wrote also on the solution of +\index{Dynamics}% +equations of the fifth degree, the hodograph, fluctuating +functions, the numerical solution of differential equations. +\index{Differential equations}% + +The capital discovery of Hamilton is his quaternions, in +\index{Quaternions}% +which his study of algebra culminated. In 1835 he published +in the \textit{Transactions of the Royal Irish Academy} his Theory of +Algebraic Couples. He regarded algebra ``as being no mere +art, nor language, nor primarily a science of quantity, but +rather as the science of order of progression.'' Time appeared +to him as the picture of such a progression. Hence his definition +of algebra as ``the science of pure time.'' It was the +subject of years' meditation for him to determine what he +should regard as the product of each pair of a system of perpendicular +directed lines. At last, on the 16th~of October, +%% -----File: 333.png---Folio 319------- +1843, while walking with his wife one evening, along the +Royal Canal in Dublin, the discovery of quaternions flashed +\index{Quaternions}% +upon him, and he then engraved with his knife on a stone in +Brougham Bridge the fundamental formula $i^2 = j^2 = k^2 = ijk = +-1$. At the general meeting of the Irish Academy, a month +later, he made the first communication on quaternions. An +account of the discovery was given the following year in the +\textit{Philosophical Magazine}. Hamilton displayed wonderful fertility +\index{Hamilton, W.~R.}% +in their development. His \textit{Lectures on Quaternions}, +delivered in Dublin, were printed in 1852. His \textit{Elements of +Quaternions} appeared in 1866. Quaternions were greatly +admired in England from the start, but on the Continent they +received less \DPtypo{attenttion}{attention}. P.~G. Tait's \textit{Elementary Treatise} +\index{Tait}% +helped powerfully to spread a knowledge of them in England. +Cayley, Clifford, and Tait advanced the subject somewhat by +\index{Cayley!ref.\ to}% +\index{Clifford!ref.\ to}% +original contributions. But there has been little progress in +recent years, except that made by Sylvester in the solution of +\index{Sylvester!ref.\ to}% +quaternion equations, nor has the application of quaternions +to physics been as extended as was predicted. The change +in notation made in France by Hoüel and by Laisant has been +\index{Houel@{Hoüel}}% +\index{Laisant}% +considered in England as a wrong step, but the true cause for +the lack of progress is perhaps more deep-seated. There is +indeed great doubt as to whether the quaternionic product can +claim a necessary and fundamental place in a system of vector +analysis. Physicists claim that there is a loss of naturalness +in taking the square of a vector to be negative. In order to +meet more adequately their wants, \textit{J.~W. Gibbs} of Yale University +\index{Gibbs!ref.\ to}% +and \textit{A.~Macfarlane} of the University of Texas, have +\index{Macfarlane}% +each suggested an algebra of vectors with a new notation. +Each gives a definition of his own for the product of two +vectors, but in such a way that the square of a vector is +positive. A third system of vector analysis has been used by +\textit{Oliver Heaviside} in his electrical researches. +\index{Heaviside}% +%% -----File: 334.png---Folio 320------- + +\textbf{Hermann Grassmann} (1809--1877) was born at Stettin, +\index{Grassmann|(}% +attended a gymnasium at his native place (where his father +was teacher of mathematics and physics), and studied theology +in Berlin for three years. In 1834 he succeeded Steiner as +\index{Steiner!ref.\ to}% +teacher of mathematics in an industrial school in Berlin, but +returned to Stettin in 1836 to assume the duties of teacher of +mathematics, the sciences, and of religion in a school there.\Reference{71} +Up to this time his knowledge of mathematics was pretty +much confined to what he had learned from his father, who +had written two books on ``Raumlehre'' and ``Grössenlehre.'' +But now he made his acquaintance with the works of Lacroix, +\index{Lacroix}% +Lagrange, and Laplace. He noticed that Laplace's results +\index{Laplace!ref.\ to}% +could be reached in a shorter way by some new ideas advanced +in his father's books, and he proceeded to elaborate this +abridged method, and to apply it in the study of tides. He +was thus led to a new geometric analysis. In 1840 he had +made considerable progress in its development, but a new +book of Schleiermacher drew him again to theology. In +1842 he resumed mathematical research, and becoming thoroughly +convinced of the importance of his new analysis, +decided to devote himself to it. It now became his ambition +to secure a mathematical chair at a university, but in this he +never succeeded. In 1844 appeared his great classical work, +the \textit{Lineale Ausdehnungslehre}, which was full of new and +\index{Ausdehnungslehre}% +strange matter, and so general, abstract, and out of fashion in +its mode of exposition, that it could hardly have had less +influence on European mathematics during its first twenty +years, had it been published in China.\DPnote{ouch} Gauss, Grunert, and +\index{Gauss!ref.\ to}% +\index{Grunert!ref.\ to}% +Möbius glanced over it, praised it, but complained of the +\index{Mobius@{Möbius}!ref.\ to}% +strange terminology and its ``philosophische Allgemeinheit.'' +Eight years afterwards, Bretschneider of Gotha was said to be +\index{Bretschneider}% +the only man who had read it through. An article in \textit{Crelle's +Journal}, in which Grassmann eclipsed the geometers of that +%% -----File: 335.png---Folio 321------- +\index{Curves!theory of}% +\index{Differential equations}% +\index{Matrices}% +time by constructing, with aid of his method, geometrically +any algebraic curve, remained again unnoticed. Need we marvel +if Grassmann turned his attention to other subjects,---to +Schleiermacher's philosophy, to politics, to philology? Still, +articles by him continued to appear in \textit{Crelle's Journal}, and in +1862 came out the second part of his \textit{Ausdehnungslehre}. It +\index{Ausdehnungslehre}% +was intended to show better than the first part the broad +scope of the Ausdehnungslehre, by considering not only geometric +applications, but by treating also of algebraic functions, +infinite series, and the differential and integral calculus. But +the second part was no more appreciated than the first. At the +age of fifty-three, this wonderful man, with heavy heart, gave +up mathematics, and directed his energies to the study of Sanskrit, +achieving in philology results which were better appreciated, +and which vie in splendour with those in mathematics. + +Common to the Ausdehnungslehre and to quaternions are geometric +addition, the function of two vectors represented in quaternions +by $S\alpha\beta$~and~$V\alpha\beta$, and the linear vector functions. The +quaternion is peculiar to Hamilton, while with Grassmann we +\index{Hamilton, W.~R.!ref.\ to}% +find in addition to the algebra of vectors a geometrical algebra +of wide application, and resembling Möbius's \textit{Barycentrische +\index{Mobius@{Möbius}!ref.\ to}% +Calcul}, in which the point is the fundamental element. Grassmann +developed the idea of the ``external product,'' the ``internal +product,'' and the ``open product.'' The last we now call +a matrix. His Ausdehnungslehre has very great extension, +having no limitation to any particular number of dimensions. +Only in recent years has the wonderful richness of +his discoveries begun to be appreciated. A second edition of +the \textit{Ausdehnungslehre} of 1844 was printed in 1877. C.~S. +Peirce gave a representation of Grassmann's system in the +\index{Peirce, C.~S.!ref.\ to}% +logical notation, and E.~W. Hyde of the University of Cincinnati +\index{Hyde}% +wrote the first text-book on Grassmann's calculus in +\index{Grassmann|)}% +the English language. +%% -----File: 336.png---Folio 322------- +\index{Aequipollences@{Æquipollences}}% + +Discoveries of less value, which in part covered those of +Grassmann and Hamilton, were made by \textit{Saint-Venant} (1797--1886), +\index{Saint-Venant!ref.\ to}% +who described the multiplication of vectors, and the +addition of vectors and oriented areas; by \textit{Cauchy}, whose +\index{Cauchy!ref.\ to}% +``clefs algébriques'' were units subject to combinatorial multiplication, +and were applied by the author to the theory of +elimination in the same way as had been done earlier by +Grassmann; by \textbf{Justus Bellavitis} (1803--1880), who published +\index{Bellavitis}% +in 1835 and 1837 in the \textit{Annali delle Scienze} his calculus of +æquipollences. Bellavitis, for many years professor at Padua, +was a self-taught mathematician of much power, who in his +thirty-eighth year laid down a city office in his native place, +Bassano, that he might give his time to science.\Reference{65} + +The first impression of Grassmann's ideas is marked in the +writings of \textbf{Hermann Hankel} (1839--1873), who published in +\index{Hankel}% +1867 his \textit{Vorlesungen über die Complexen Zahlen}. Hankel, +then docent in Leipzig, had been in correspondence with +Grassmann. The ``alternate numbers'' of Hankel are subject +\index{Alternate numbers}% +to his law of combinatorial multiplication. In considering +the foundations of algebra Hankel affirms the principle of +the permanence of formal laws previously enunciated incompletely +by Peacock. Hankel was a close student of mathematical +history, and left behind an unfinished work thereon. +Before his death he was professor at Tübingen. His \textit{\DPtypo{Complexe}{Complexen} +Zahlen} was at first little read, and we must turn to +\textbf{Victor Schlegel} of Hagen as the successful interpreter of Grassmann. +\index{Schlegel}% +Schlegel was at one time a young colleague of Grassmann +at the Marienstifts-Gymnasium in Stettin. Encouraged +by Clebsch, Schlegel wrote a \textit{System der Raumlehre} which +\index{Clebsch!ref.\ to}% +explained the essential conceptions and operations of the +Ausdehnungslehre. + +Multiple algebra was powerfully advanced by Peirce, whose +theory is not geometrical, as are those of Hamilton and Grassmann. +%% -----File: 337.png---Folio 323------- +\textbf{Benjamin Peirce} (1809--1880) was born at Salem, Mass., +\index{Peirce, B.}% +\index{Peirce, C.~S.}% +and graduated at Harvard College, having as undergraduate +carried the study of mathematics far beyond the limits of the +college course.\Reference{2} When Bowditch was preparing his translation +\index{Bowditch}% +and commentary of the \textit{Mécanique Céleste}, young Peirce +helped in reading the proof-sheets. He was made professor at +Harvard in 1833, a position which he retained until his death. +For some years he was in charge of the \textit{Nautical Almanac} +and superintendent of the United States Coast Survey. He +published a series of college text-books on mathematics, an +\textit{Analytical Mechanics}, 1855, and calculated, together with Sears~C. +Walker of Washington, the orbit of Neptune. Profound +\index{Walker}% +are his researches on \textit{Linear Associative Algebra}. The first of +\index{Linear associative algebra}% +several papers thereon was read at the first meeting of the +American Association for the Advancement of Science in 1864. +Lithographed copies of a memoir were distributed among friends +in 1870, but so small seemed to be the interest taken in this +subject that the memoir was not printed until 1881 (\textit{Am.\ Jour.\ +Math.}, Vol.~IV., No.~2). Peirce works out the multiplication +tables, first of \emph{single} algebras, then of \emph{double} algebras, and so +on up to sextuple, making in all 162~algebras, which he shows +to be possible on the consideration of symbols $A$,~$B$,~etc., +which are linear functions of a determinate number of letters +or units $i$,~$j$, $k$,~$l$,~etc., with coefficients which are ordinary analytical +magnitudes, real or imaginary,---the letters $i$,~$j$,~etc., +being such that every binary combination $i^2$,~$ij$, $ji$,~etc., is equal +to a linear function of the letters, but under the restriction of +satisfying the associative law.\Reference{56} \textit{Charles~S. Peirce}, a son of +Benjamin Peirce, and one of the foremost writers on mathematical +logic, showed that these algebras were all defective +\index{Logic}% +forms of quadrate algebras which he had previously discovered +by logical analysis, and for which he had devised a simple +notation. Of these quadrate algebras quaternions is a simple +%% -----File: 338.png---Folio 324------- +example; nonions is another. C.~S. Peirce showed that of all +linear associative algebras there are only three in which division +is unambiguous. These are ordinary single algebra, +ordinary double algebra, and quaternions, from which the +imaginary scalar is excluded. He showed that his father's +algebras are operational and matricular. Lectures on multiple +algebra were delivered by J.~J. Sylvester at the Johns Hopkins +\index{Sylvester!ref.\ to}% +University, and published in various journals. They treat +largely of the algebra of matrices. The theory of matrices +\index{Matrices}% +was developed as early as 1858 by Cayley in an important +\index{Cayley!ref.\ to}% +memoir which, in the opinion of Sylvester, ushered in the +reign of Algebra the Second. Clifford, Sylvester, H.~Taber, +\index{Clifford!ref.\ to}% +\index{Taber}% +C.~H. Chapman, carried the investigations much further. The +\index{Chapman}% +originator of matrices is really Hamilton, but his theory, published +\index{Hamilton, W.~R.!ref.\ to}% +in his \textit{Lectures on Quaternions}, is less general than that +of Cayley. The latter makes no reference to Hamilton. + +The theory of determinants\Reference[]{73} was studied by Hoëné Wronski +\index{Determinants}% +\index{Wronski}% +in Italy and J.~Binet in France; but they were forestalled by +\index{Binet}% +the great master of this subject, Cauchy. In a paper (\textit{Jour.\ +\index{Cauchy!ref.\ to}% +de l'ecole Polyt.}, IX.,~16) Cauchy developed several general +theorems. He introduced the name \emph{determinant}, a term +previously used by Gauss in the functions considered by him. +\index{Gauss!ref.\ to}% +In 1826 Jacobi began using this calculus, and he gave brilliant +\index{Jacobi!ref.\ to}% +proof of its power. In 1841 he wrote extended memoirs on +determinants in \textit{Crelle's Journal}, which rendered the theory +easily accessible. In England the study of linear transformations +of quantics gave a powerful impulse. Cayley developed +skew-determinants and Pfaffians, and introduced the use of +determinant brackets, or the familiar pair of upright lines. +More recent researches on determinants appertain to special +forms. ``Continuants'' are due to Sylvester; ``alternants,'' +originated by Cauchy, have been developed by Jacobi, N.~Trudi, +\index{Trudi}% +H.~Nägelbach, and G.~Garbieri; ``axisymmetric determinants,'' +\index{Garbieri}% +\index{Nagelbach@{Nägelbach}}% +%% -----File: 339.png---Folio 325------- +\index{Invariant}% +first used by Jacobi, have been studied by V.~A. Lebesgue, +\index{Lebesgue}% +Sylvester, and Hesse; ``circulants'' are due to E.~Catalan of +\index{Catalan, E.}% +\index{Hesse!ref.\ to}% +\index{Sylvester!ref.\ to}% +Liège, W.~Spottiswoode (1825--1883), J.~W.~L. Glaisher, and +\index{Glaisher!ref.\ to}% +\index{Spottiswoode}% +R.~F. Scott; for ``centro-symmetric determinants'' we are +\index{Determinants}% +\index{Scott}% +indebted to G.~Zehfuss. E.~B. Christoffel of Strassburg and +\index{Christoffel}% +\index{Zehfuss}% +G.~Frobenius discovered the properties of ``Wronskians,'' first +\index{Frobenius}% +used by Wronski. V.~Nachreiner and S.~Günther, both of +\index{Nachreiner}% +Munich, pointed out relations between determinants and continued +fractions; Scott uses Hankel's alternate numbers in his +\index{Hankel!ref.\ to}% +treatise. Text-books on determinants were written by Spottiswoode +(1851), Brioschi (1854), Baltzer (1857), Günther +\index{Baltzer, R.!ref.\ to}% +\index{Brioschi!ref.\ to}% +\index{Gunther@{Günther, S.}}% +(1875), Dostor (1877), Scott (1880), Muir (1882), Hanus +\index{Dostor}% +\index{Hanus}% +\index{Muir}% +(1886). + +Modern higher algebra is especially occupied with the +theory of linear transformations. Its development is mainly +the work of Cayley and Sylvester. +\index{Cayley}% + +\textbf{Arthur Cayley}, born at Richmond, in Surrey, in 1821, was +educated at Trinity College, Cambridge.\Reference{74} He came out Senior +Wrangler in 1842. He then devoted some years to the study +and practice of law. On the foundation of the Sadlerian professorship +at Cambridge, he accepted the offer of that chair, +thus giving up a profession promising wealth for a very +modest provision, but which would enable him to give all his +time to mathematics. Cayley began his mathematical publications +in the \textit{Cambridge Mathematical Journal} while he was +still an undergraduate. Some of his most brilliant discoveries +were made during the time of his legal practice. There is +hardly any subject in pure mathematics which the genius of +Cayley has not enriched, but most important is his creation +of a new branch of analysis by his theory of invariants. +Germs of the principle of invariants are found in the writings +of Lagrange, Gauss, and particularly of Boole, who +\index{Boole!ref.\ to}% +\index{Gauss!ref.\ to}% +\index{Lagrange!ref.\ to}% +showed, in 1841, that invariance is a property of discriminants +%% -----File: 340.png---Folio 326------- +generally, and who applied it to the theory of orthogonal +substitution. Cayley set himself the problem to determine +\index{Cayley}% +\textit{a~priori} what functions of the coefficients of a given equation +possess this property of invariance, and found, to begin with, +in 1845, that the so-called ``hyper-determinants'' possessed it. +Boole made a number of additional discoveries. Then Sylvester +\index{Sylvester}% +began his papers in the \textit{Cambridge and Dublin Mathematical +Journal} on the Calculus of Forms. After this, +discoveries followed in rapid succession. At that time Cayley +and Sylvester were both residents of London, and they +stimulated each other by frequent oral communications. It +has often been difficult to determine how much really belongs +to each. + +\textbf{James Joseph Sylvester} was born in London in 1814, and +educated at St.~Johns College, Cambridge. He came out +Second Wrangler in 1837. His Jewish origin incapacitated +him from taking a degree. In 1846 he became a student at +the Inner Temple, and was called to the bar in 1850. He +became professor of natural philosophy at University College, +London; then, successively, professor of mathematics at the +University of Virginia, at the Royal Military Academy in +Woolwich, at the Johns Hopkins University in Baltimore, +and is, since 1883, professor of geometry at Oxford. His +first printed paper was on Fresnel's optic theory, 1837. Then +followed his researches on invariants, the theory of equations, +theory of partitions, multiple algebra, the theory of numbers, +and other subjects mentioned elsewhere. About 1874 he took +part in the development of the geometrical theory of link-work %[** PP: Only instance] +movements, originated by the beautiful discovery of +A.~Peaucellier, Capitaine du Génie à Nice (published in +\index{Peaucellier}% +\textit{Nouvelles Annales}, 1864 and 1873), and made the subject +of close study by A.~B. Kempe. To Sylvester is ascribed the +\index{Kempe}% +general statement of the theory of contravariants, the discovery +\index{Contravariants}% +%% -----File: 341.png---Folio 327------- +\index{Macmahon}% +of the partial differential equations satisfied by the +invariants and covariants of binary quantics, and the subject +\index{Covariants}% +of mixed concomitants. In the \textit{American Journal of Mathematics} +are memoirs on binary and ternary quantics, elaborated +partly with aid of \textit{F.~Franklin}, now professor at the Johns +\index{Franklin}% +Hopkins University. At Oxford, Sylvester has opened up a +new subject, the theory of reciprocants, treating of the functions +\index{Reciprocants}% +of a dependent variable~$y$ and the functions of its differential +coefficients in regard to~$x$, which remain unaltered by +the interchange of $x$~and~$y$. This theory is more general than +one on differential invariants by Halphen (1878), and has +\index{Differential invariants}% +\index{Halphen!ref.\ to}% +been developed further by J.~Hammond of Oxford, McMahon +\index{Hammond, J}% +of Woolwich, A.~R. Forsyth of Cambridge, and others. Sylvester +\index{Forsyth}% +playfully lays claim to the appellation of the Mathematical +Adam, for the many names he has introduced into +mathematics. Thus the terms \emph{invariant}, \emph{discriminant}, \emph{Hessian}, +\index{Hessian}% +\emph{Jacobian}, are his. + +The great theory of invariants, developed in England mainly +by Cayley and Sylvester, came to be studied earnestly in Germany, +France, and Italy. One of the earliest in the field was +\textbf{Siegfried Heinrich Aronhold} (1819--1884), who demonstrated +\index{Aronhold}% +the existence of invariants, $S$~and~$T$, of the ternary cubic. +Hermite discovered evectants and the theorem of reciprocity +named after him. Paul Gordan showed, with the aid of +\index{Gordan}% +symbolic methods, that the number of distinct forms for a +binary quantic is finite. Clebsch proved this to be true for +\index{Clebsch!ref.\ to}% +quantics with any number of variables. A very much simpler +proof of this was given in 1891, by David Hilbert of Königsberg. +\index{Hilbert}% +In Italy, F.~Brioschi of Milan and \textit{Faà de Bruno} +\index{Brioschi}% +\index{Bruno, Faà de}% +(1825--1888) contributed to the theory of invariants, the +latter writing a text-book on binary forms, which ranks by +the side of Salmon's treatise and those of Clebsch and +Gordan. Among other writers on invariants are E.~B. Christoffel, +\index{Christoffel}% +%% -----File: 342.png---Folio 328------- +\index{Burkhardt, H.}% +\index{Invariant}% +\index{Sylvester!ref.\ to}% +\index{Tchirnhausen!ref.\ to}% +Wilhelm Fiedler, P.~A. McMahon, J.~W.~L. Glaisher of +\index{Fiedler}% +\index{Glaisher!ref.\ to}% +\index{McMahon}% +Cambridge, Emory McClintock of New York. McMahon discovered +\index{McClintock}% +that the theory of semi-invariants is a part of that of +\index{Semi-invariants}% +symmetric functions. The modern higher algebra has reached +\index{Symmetric functions}% +out and indissolubly connected itself with several other +branches of mathematics---geometry, calculus of variations, +\index{Calculus!of variations}% +mechanics. Clebsch extended the theory of binary forms to +\index{Clebsch!ref.\ to}% +\index{Mechanics!more recent work}% +ternary, and applied the results to geometry. Clebsch, Klein, +\index{Klein!ref.\ to}% +Weierstrass, Burckhardt, and Bianchi have used the theory of +\index{Bianchi}% +\index{Weierstrass!ref.\ to}% +invariants in hyperelliptic and Abelian functions. +\index{Abelian functions}% +\index{Hyperelliptic functions}% + +In the theory of equations Lagrange, Argand, and Gauss +\index{Equations!theory of|(}% +furnished proof to the important theorem that every algebraic +equation has a real or a complex root. Abel proved rigorously +\index{Abel!ref.\ to}% +that the general algebraic equation of the fifth or of higher +degrees cannot be solved by radicals (\textit{Crelle},~I., 1826). A modification +of Abel's proof was given by Wantzel. Before Abel, +\index{Wantzel}% +an Italian physician, \textit{Paolo Ruffini} (1765--1822), had printed +\index{Ruffini}% +proofs of the insolvability, which were criticised by his +countryman Malfatti. Though inconclusive, Ruffini's papers +\index{Malfatti}% +are remarkable as containing anticipations of Cauchy's theory +\index{Cauchy!ref.\ to}% +of groups.\Reference{76} A transcendental solution of the quintic involving +\index{Groups, theory of|(}% +elliptic integrals was given by Hermite (\textit{Compt.\ Rend.}, 1858, +\index{Elliptic integrals}% +\index{Hermite!ref.\ to}% +1865, 1866). After Hermite's first publication, Kronecker, in +\index{Kronecker!ref.\ to}% +1858, in a letter to Hermite, gave a second solution in which +was obtained a simple resolvent of the sixth degree. \textit{Jerrard}, in +\index{Jerrard}% +his \textit{Mathematical Researches} (1832--1835), reduced the quintic +to the trinomial form by an extension of the method of +Tschirnhausen. This important reduction had been effected +as early as 1786 by \textit{E.~S. Bring}, a Swede, and brought out +\index{Bring}% +in a publication of the University of Lund. Jerrard, like +Tschirnhausen, believed that his method furnished a general +algebraic solution of equations of any degree. In 1836 William~R. +Hamilton made a report on the validity of Jerrard's +\index{Hamilton, W.~R.!ref.\ to}% +%% -----File: 343.png---Folio 329------- +\index{Elliptic functions}% +\index{Hamilton's numbers}% +method, and showed that by his process the quintic could be +transformed to any one of the four trinomial forms. Hamilton +defined the limits of its applicability to higher equations. +Sylvester investigated this question, What is the lowest degree +an equation can have in order that it may admit of being +deprived of $i$~consecutive terms by aid of equations not higher +than $i$th~degree. He carried the investigation as far as $i = 8$, +and was led to a series of numbers which he named ``Hamilton's +numbers.'' A transformation of equal importance to Jerrard's +is that of Sylvester, who expressed the quintic as the sum of +three fifth-powers. The covariants and invariants of higher +equations have been studied much in recent years. + +Abel's proof that higher equations cannot always be solved +algebraically led to the inquiry as to what equations of a given +degree can be solved by radicals. Such equations are the +ones discussed by Gauss in considering the division of the +\index{Division of the circle}% +\index{Gauss!ref.\ to}% +circle. Abel advanced one step further by proving that an +\index{Circle!division of}% +irreducible equation can always be solved in radicals, if, of +two of its roots, the one can be expressed rationally in terms +of the other, provided that the degree of the equation is +prime; if it is not prime, then the solution depends upon +that of equations of lower degree. Through geometrical considerations, +Hesse came upon algebraically solvable equations +\index{Hesse!ref.\ to}% +of the ninth degree, not included in the previous groups. +The subject was powerfully advanced in Paris by the youthful +\textit{Evariste Galois} (born, 1811; killed in a duel, 1832), who +\index{Galois}% +introduced the notion of a group of substitutions. To him are +\index{Substitutions, theory of}% +due also some valuable results in relation to another set of +equations, presenting themselves in the theory of elliptic +functions, viz.\ the modular equations. Galois's labours gave +\index{Modular equations}% +birth to the important theory of substitutions, which has been +\index{Theory of substitutions}% +greatly advanced by \textit{C.~Jordan} of Paris, \textit{J.~A. Serret} (1819--1885) +\index{Jordan}% +of the Sorbonne in Paris, L.~Kronecker (1823--1891) of +\index{Kronecker}% +%% -----File: 344.png---Folio 330------- +\index{Sturm's theorem}% +\index{Warring}% +Berlin, Klein of Göttingen, M.~Nöther of Erlangen, C.~Hermite +\index{Hermite!ref.\ to}% +\index{Klein!ref.\ to}% +\index{Nother@{Nöther}}% +of Paris, A.~Capelli of Naples, L.~Sylow of Friedrichshald, +\index{Capelli}% +\index{Sylow}% +E.~Netto of Giessen. Netto's book, the \textit{Substitutionstheorie}, +\index{Netto}% +has been translated into English by F.~N. Cole of +\index{Cole}% +the University of Michigan, who contributed to the theory. +A simple group of 504~substitutions of nine letters, discovered +by Cole, has been shown by E.~H. Moore of the University of +\index{Moore}% +Chicago to belong to a doubly-infinite system of simple groups. +\index{Groups, theory of|)}% +The theory of substitutions has important applications in the +theory of differential equations. Kronecker published, in 1882, +\index{Kronecker!ref.\ to}% +his \textit{Grundzüge einer Arithmetischen Theorie der Algebraischen +Grössen}. + +Since Fourier and Budan, the solution of numerical equations +has been advanced by \textit{W.~G. Horner} of Bath, who gave +\index{Horner}% +an improved method of approximation (\textit{Philosophical Transactions}, +1819). \textbf{Jacques Charles François Sturm} (1803--1855), a +\index{Sturm, J.~C.~F.}% +native of Geneva, Switzerland, and the successor of Poisson +\index{Poisson!ref.\ to}% +in the chair of mechanics at the Sorbonne, published in 1829 +his celebrated theorem determining the number and situation +of roots of an equation comprised between given limits. Sturm +tells us that his theorem stared him in the face in the midst +of some mechanical investigations connected with the motion +of a compound pendulum.\Reference{77} This theorem, and Horner's +method, offer together sure and ready means of finding the +real roots of a numerical equation. + +The symmetric functions of the sums of powers of the roots +\index{Symmetric functions}% +of an equation, studied by Newton and Waring, was considered +\index{Newton!ref.\ to}% +more recently by Gauss, Cayley, Sylvester, Brioschi. Cayley +\index{Brioschi!ref.\ to}% +\index{Cayley!ref.\ to}% +\index{Gauss!ref.\ to}% +\index{Sylvester!ref.\ to}% +gives rules for the ``weight'' and ``order'' of symmetric functions. + +The theory of elimination was greatly advanced by Sylvester, +\index{Elimination}% +Cayley, Salmon, Jacobi, Hesse, Cauchy, Brioschi, and +\index{Cauchy!ref.\ to}% +\index{Hesse!ref.\ to}% +\index{Jacobi!ref.\ to}% +\index{Salmon}% +Gordan. Sylvester gave the dialytic method (\textit{Philosophical +\index{Dialytic method of elimination}% +\index{Gordan}% +%% -----File: 345.png---Folio 331------- +\index{Algebra!recent|)}% +Magazine}, 1840), and in 1852 established a theorem relating +to the expression of an eliminant as a determinant. Cayley +made a new statement of Bézout's method of elimination and +\index{Bezout's@{Bézout's method of elimination}}% +\index{Elimination}% +established a general theory of elimination (1852). + +\Section{Analysis} +\index{Analysis!modern|(}% + +Under this head we find it convenient to consider the subjects +of the differential and integral calculus, the calculus of +variations, infinite series, probability, and differential equations. +\index{Equations!theory of|)}% +Prominent in the development of these subjects was +Cauchy. +\index{Cauchy|(}% + +\textbf{Augustin-Louis Cauchy}\Reference[]{78} (1789--1857) was born in Paris, +and received his early education from his father. Lagrange +and Laplace, with whom the father came in frequent contact, +foretold the future greatness of the young boy. At the +École Centrale du Panthéon he excelled in ancient classical +studies. In 1805 he entered the Polytechnic School, and two +years later the École des Ponts et Chaussées. Cauchy left for +Cherbourg in 1810, in the capacity of engineer. Laplace's +\textit{Mécanique Céleste and} Lagrange's \textit{Fonctions Analytiques} were +among his book companions there. Considerations of health +induced him to return to Paris after three years. Yielding to +the persuasions of Lagrange and Laplace, he renounced engineering +in favour of pure science. We find him next holding +a professorship at the Polytechnic School. On the expulsion +of Charles~X., and the accession to the throne of Louis Philippe +in 1830, Cauchy, being exceedingly conscientious, found himself +unable to take the oath demanded of him. Being, in +consequence, deprived of his positions, he went into voluntary +exile. At Fribourg in Switzerland, Cauchy resumed his +studies, and in 1831 was induced by the king of Piedmont to +%% -----File: 346.png---Folio 332------- +accept the chair of mathematical physics, especially created +for him at the university of Turin. In 1833 he obeyed the +call of his exiled king, Charles~X., to undertake the education +of a grandson, the Duke of Bordeaux. This gave Cauchy an +opportunity to visit various parts of Europe, and to learn how +extensively his works were being read. Charles~X. bestowed +upon him the title of Baron. On his return to Paris in 1838, +a chair in the College de France was offered to him, but the +oath demanded of him prevented his acceptance. He was +nominated member of the Bureau of Longitude, but declared +ineligible by the ruling power. During the political events of +1848 the oath was suspended, and Cauchy at last became +professor at the Polytechnic School. On the establishment of +the second empire, the oath was re-instated, but Cauchy and +Arago were exempt from it. Cauchy was a man of great +\index{Arago}% +piety, and in two of his publications staunchly defended the +Jesuits. + +Cauchy was a prolific and profound mathematician. By a +prompt publication of his results, and the preparation of +standard text-books, he exercised a more immediate and +beneficial influence upon the great mass of mathematicians +than any contemporary writer. He was one of the leaders +in infusing rigour into analysis. His researches extended +over the field of series, of imaginaries, theory of numbers, +differential equations, theory of substitutions, theory of functions, +determinants, mathematical astronomy, light, elasticity, +etc.,---covering pretty much the whole realm of mathematics, +pure and applied. + +Encouraged by Laplace and Poisson, Cauchy published in +1821 his \textit{Cours d'Analyse de l'École Royale Polytechnique}, a +work of great merit. Had it been studied more diligently by +writers of text-books in England and the United States, many +a lax and loose method of analysis hardly as yet eradicated +%% -----File: 347.png---Folio 333------- +\index{Differential equations}% +from elementary text-books would have been discarded over +half a century ago. Cauchy was the first to publish a +rigorous proof of Taylor's theorem. He greatly improved +\index{Taylor's theorem}% +the exposition of fundamental principles of the differential +\index{Differential calculus!philosophy of}% +calculus by his mode of considering limits and his new theory +\index{Calculus!of variations|(}% +on the continuity of functions. The method of Cauchy and +\index{Continuity}% +Duhamel was accepted with favour by Hoüel and others. In +\index{Duhamel}% +England special attention to the clear exposition of fundamental +principles was given by De~Morgan. Recent American +\index{Demorgan@{De Morgan}!ref.\ to}% +treatises on the calculus introduce time as an independent variable, +and the allied notions of velocity and acceleration---thus +virtually returning to the method of fluxions. +\index{Fluxions}% + +Cauchy made some researches on the calculus of variations. +\index{Cauchy|)}% +This subject is now in its essential principles the same as +when it came from the hands of Lagrange. Recent studies +pertain to the variation of a double integral when the limits +are also variable, and to variations of multiple integrals in +general. Memoirs were published by Gauss in 1829, Poisson +\index{Gauss!ref.\ to}% +\index{Poisson!ref.\ to}% +in 1831, and Ostrogradsky of St.~Petersburg in 1834, without, +\index{Ostrogradsky}% +however, determining in a general manner the number and +form of the equations which must subsist at the limits in case +of a double or triple integral. In 1837 Jacobi published a +\index{Jacobi!ref.\ to}% +memoir, showing that the difficult integrations demanded by +the discussion of the second variation, by which the existence +of a maximum or minimum can be ascertained, are included +in the integrations of the first variation, and thus are superfluous. +This important theorem, presented with great brevity +by Jacobi, was elucidated and extended by V.~A. Lebesgue, +\index{Lebesgue}% +C.~E. Delaunay, Eisenlohr, S.~Spitzer, Hesse, and Clebsch. An +\index{Clebsch!ref.\ to}% +\index{Delaunay!ref.\ to}% +\index{Eisenlohr}% +\index{Hesse!ref.\ to}% +\index{Spitzer}% +important memoir by Sarrus on the question of determining +\index{Sarrus}% +the limiting equations which must be combined with the +indefinite equations in order to determine completely the +maxima and minima of multiple integrals, was awarded a +\index{Maxima and minima}% +%% -----File: 348.png---Folio 334------- +\index{Criteria of convergence|(}% +\index{Jellet}% +prize by the French Academy in 1845, honourable mention +being made of a paper by Delaunay. Sarrus's method was +\index{Delaunay!ref.\ to}% +simplified by Cauchy. In 1852 G.~Mainardi attempted to +\index{Mainardi}% +exhibit a new method of discriminating maxima and minima, +\index{Maxima and minima}% +and extended Jacobi's theorem to double integrals. Mainardi +and F.~Brioschi showed the value of determinants in exhibiting +\index{Brioschi!ref.\ to}% +\index{Determinants}% +the terms of the second variation. In 1861 \textit{Isaac Todhunter} +\index{Todhunter}% +(1820--1884) of St.~John's College, Cambridge, published +his valuable work on the \textit{History of the Progress of the Calculus +of Variations}, which contains researches of his own. In 1866 +he published a most important research, developing the theory +of discontinuous solutions (discussed in particular cases by +Legendre), and doing for this subject what Sarrus had done +for multiple integrals. + +The following are the more important authors of systematic +treatises on the calculus of variations, and the dates of publication: +\index{Calculus!of variations|)}% +Robert Woodhouse, Fellow of Caius College, Cambridge, +\index{Woodhouse}% +1810; Richard Abbatt in London, 1837; John Hewitt +\index{Abbatt}% +Jellett (1817--1888), once Provost of Trinity College, Dublin, +1850; G.~W. Strauch in \DPtypo{Zurich}{Zürich}, 1849; Moigno and Lindelöf, +\index{Lindelöf}% +\index{Moigno}% +\index{Strauch}% +1861; Lewis Buffett Carll of Flushing in New York, 1881. +\index{Carll}% + +The lectures on definite integrals, delivered by Dirichlet in +\index{Definite integrals}% +\index{Dirichlet!ref.\ to}% +1858, have been elaborated into a standard work by G.~F. +Meyer. The subject has been treated most exhaustively by +\index{Meyer, G.~F.}% +D.~Bierens de Haan of Leiden in his \textit{Exposé de la théorie des +\index{Haan}% +intégrals définies}, Amsterdam, 1862. + +The history of infinite series illustrates vividly the salient +\index{Infinite series|(}% +feature of the new era which analysis entered upon during the +\index{Analysis!modern|)}% +first quarter of this century. Newton and Leibniz felt the +\index{Leibniz!ref.\ to}% +\index{Newton!ref.\ to}% +necessity of inquiring into the convergence of infinite series, +\index{Convergence of series|(}% +but they had no proper criteria, excepting the test advanced +by Leibniz for alternating series. By Euler and his contemporaries +\index{Euler!ref.\ to}% +the \emph{formal} treatment of series was greatly extended, +%% -----File: 349.png---Folio 335------- +\index{Multiplication of series}% +while the necessity for determining the convergence was generally +lost sight of. Euler reached some very pretty results +on infinite series, now well known, and also some very +\index{Absolutely convergent series}% +absurd results, now quite forgotten. The faults of his time +found their culmination in the Combinatorial School in Germany, +\index{Combinatorial School}% +which has now passed into deserved oblivion. At the +beginning of the period now under consideration, the doubtful, +or plainly absurd, results obtained from infinite series stimulated +profounder inquiries into the validity of operations with +them. Their \emph{actual contents} came to be the primary, \emph{form} a +secondary, consideration. The first important and strictly +rigorous investigation of series was made by Gauss in connection +\index{Gauss!ref.\ to}% +with the hypergeometric series. The criterion developed +\index{Hypergeometric series}% +by him settles the question of convergence in every case +which it is intended to cover, and thus bears the stamp of +generality so characteristic of Gauss's writings. Owing to the +strangeness of treatment and unusual rigour, Gauss's paper +excited little interest among the mathematicians of that time. + +More fortunate in reaching the public was Cauchy, whose +\index{Cauchy!ref.\ to}% +\textit{Analyse Algébrique} of 1821 contains a rigorous treatment of +series. All series whose sum does not approach a fixed limit +as the number of terms increases indefinitely are called divergent. +Like Gauss, he institutes comparisons with geometric +series, and finds that series with positive terms are convergent +or not, according as the $n$th root of the $n$th term, or the ratio +of the $(n + 1)$th term and the $n$th term, is ultimately less +or greater than unity. To reach some of the cases where +these expressions become ultimately unity and fail, Cauchy +established two other tests. He showed that series with negative +\Addendlabel{335}% +terms converge when the absolute values of the terms +converge, and then deduces Leibniz's test for alternating +series. The product of two convergent series was not found +to be necessarily convergent. Cauchy's theorem that the +%% -----File: 350.png---Folio 336------- +product of two absolutely convergent series converges to the +product of the sums of the two series was shown half a century +later by F.~Mertens of Graz to be still true if, of the two +\index{Mertens}% +convergent series to be multiplied together, only one is absolutely +convergent. + +The most outspoken critic of the old methods in series +was Abel. His letter to his friend Holmboe (1826) contains +\index{Abel!ref.\ to}% +\index{Holmboe}% +severe criticisms. It is very interesting reading, even to +modern students. In his demonstration of the binomial theorem +he established the theorem that if two series and their +product series are all convergent, then the product series will +converge towards the product of the sums of the two given +series. This remarkable result would dispose of the whole +problem of multiplication of series if we had a universal +\index{Multiplication of series}% +practical criterion of convergency for semi-convergent series. +\index{Semi-convergent series}% +Since we do not possess such a criterion, theorems have been +recently established by A.~Pringsheim of Munich and A.~Voss +\index{Pringsheim|(}% +\index{Voss!ref.\ to}% +of Würzburg which remove in certain cases the necessity of +applying tests of convergency to the product series by the +application of tests to easier related expressions. Pringsheim +reaches the following interesting conclusions: The product +of two semi-convergent series can never converge absolutely, +but a semi-convergent series, or even a divergent series, multiplied +by an absolutely convergent series, \emph{may} yield an absolutely +convergent product. + +The researches of Abel and Cauchy caused a considerable +stir. We are told that after a scientific meeting in which +Cauchy had presented his first researches on series, Laplace +\index{Laplace!ref.\ to}% +hastened home and remained there in seclusion until he had +examined the series in his \textit{Mécanique Céleste}. Luckily, every +one was found to be convergent! We must not conclude, +however, that the new ideas at once displaced the old. On the +contrary, the new views were generally accepted only after a +%% -----File: 351.png---Folio 337------- +severe and long struggle. As late as 1844 De~Morgan began a +\index{Demorgan@{De Morgan}!ref.\ to}% +paper on ``divergent series'' in this style: ``I believe it will +\index{Absolutely convergent series}% +\index{Divergent series}% +be generally admitted that the heading of this paper describes +the only subject yet remaining, of an elementary character, on +which a serious schism exists among mathematicians as to the +absolute correctness or incorrectness of results.'' + +First in time in the evolution of more delicate criteria of +convergence and divergence come the researches of Josef Ludwig +Raabe (\textit{Crelle}, Vol.~IX.); then follow those of De~Morgan +\index{Raabe}% +as given in his calculus. De~Morgan established the logarithmic +criteria which were discovered in part independently +by J.~Bertrand. The forms of these criteria, as given by +\index{Bertrand}% +Bertrand and by Ossian Bonnet, are more convenient than +\index{Bonnet, O.!ref.\ to}% +De~Morgan's. It appears from Abel's posthumous papers +\index{Abel!ref.\ to}% +that he had anticipated the above-named writers in establishing +logarithmic criteria. It was the opinion of Bonnet +\index{Logarithmic criteria of convergence}% +that the logarithmic criteria never fail; but Du~Bois-Reymond +\index{Bois-Reymond, P.~du|(}% +and Pringsheim have each discovered series demonstrably +convergent in which these criteria fail to determine +the convergence. The criteria thus far alluded to have +been called by Pringsheim \emph{special} criteria, because they all +depend upon a comparison of the $n$th term of the series +with special functions $a^n$, $n^x$, $n(\log n)^x$, etc. Among the +first to suggest \emph{general} criteria, and to consider the subject +from a still wider point of view, culminating in a regular +mathematical theory, was Kummer. He established a theorem +\index{Kummer!ref.\ to}% +yielding a test consisting of two parts, the first part of +which was afterwards found to be superfluous. The study +of general criteria was continued by U.~Dini of Pisa, Paul +\index{Dini}% +Du~Bois-Reymond, G.~Kohn of Minden, and Pringsheim. +\index{Kohn}% +Du~Bois-Reymond divides criteria into two classes: criteria +of the \emph{first kind} and criteria of the \emph{second kind}, according as +the general $n$th term, or the ratio of the $(n + 1)$th term and +%% -----File: 352.png---Folio 338------- +the $n$th~term, is made the basis of research. Kummer's is a +\index{Kummer!ref.\ to}% +criterion of the second kind. A criterion of the first kind, +analogous to this, was invented by Pringsheim. From the +general criteria established by Du~Bois-Reymond and Pringsheim +respectively, all the special criteria can be derived. The +theory of Pringsheim is very complete, and offers, in addition +\index{Pringsheim|)}% +to the criteria of the first kind and second kind, entirely new +criteria of a \emph{third kind}, and also generalised criteria of the +second kind, which apply, however, only to series with never +\index{Absolutely convergent series}% +increasing terms. Those of the third kind rest mainly on the +consideration of the limit of the difference either of consecutive +terms or of their reciprocals. In the generalised criteria +of the second kind he does not consider the ratio of two consecutive +terms, but the ratio of any two terms however far +apart, and deduces, among others, two criteria previously given +by Kohn and Ermakoff respectively. + +Difficult questions arose in the study of Fourier's series.\Reference{79} +\index{Fourier's series}% +Cauchy was the first who felt the necessity of inquiring into +\index{Cauchy!ref.\ to}% +its convergence. But his mode of proceeding was found +by Dirichlet to be unsatisfactory. Dirichlet made the first +\index{Dirichlet!ref.\ to}% +thorough researches on this subject (\textit{Crelle}, Vol.~IV.). They +culminate in the result that whenever the function does not +become infinite, does not have an infinite number of discontinuities, +and does not possess an infinite number of +maxima and minima, then Fourier's series converges toward +the value of that function at all places, except points of +discontinuity, and there it converges toward the mean of +the two boundary values. Schläfli of Bern and Du~Bois-Reymond +\index{Schlafli@{Schläfli}!ref.\ to}% +expressed doubts as to the correctness of the mean +value, which were, however, not well founded. Dirichlet's +conditions are sufficient, but not necessary. Lipschitz, of +\index{Lipschitz!ref.\ to}% +Bonn, proved that Fourier's series still represents the function +when the number of discontinuities is infinite, and +%% -----File: 353.png---Folio 339------- +\index{Criteria of convergence|)}% +\index{Definite integrals}% +\index{Undulatory theory of light}% +established a condition on which it represents a function +having an infinite number of maxima and minima. Dirichlet's +\index{Dirichlet!ref.\ to}% +\index{Maxima and minima}% +belief that all continuous functions can be represented +by Fourier's series at all points was shared by Riemann and +\index{Fourier's series}% +\index{Riemann!ref.\ to}% +H.~Hankel, but was proved to be false by Du~Bois-Reymond +\index{Hankel!ref.\ to}% +and H.~A. Schwarz. +\index{Schwarz!ref.\ to}% + +Riemann inquired what properties a function must have, +so that there may be a trigonometric series which, whenever +\index{Trigonometric series}% +it is convergent, converges toward the value of the function. +He found necessary and sufficient conditions for this. They +do not decide, however, whether such a series actually represents +the function or not. Riemann rejected Cauchy's definition +\index{Cauchy!ref.\ to}% +of a definite integral on account of its arbitrariness, gave +a new definition, and then inquired when a function has an +integral. His researches brought to light the fact that continuous +functions need not always have a differential coefficient. +But this property, which was shown by Weierstrass to +\index{Weierstrass!ref.\ to}% +belong to large classes of functions, was not found necessarily +to exclude them from being represented by Fourier's series. +Doubts on some of the conclusions about Fourier's series were +thrown by the observation, made by Weierstrass, that the +integral of an infinite series can be shown to be equal to the +\index{Infinite series|)}% +sum of the integrals of the separate terms only when the series +converges \emph{uniformly} within the region in question. The subject +of uniform convergence was investigated by Philipp Ludwig +\index{Convergence of series|)}% +Seidel (1848) and G.~G. Stokes (1847), and has assumed +\index{Seidel}% +\index{Stokes!ref.\ to}% +great importance in Weierstrass' theory of functions. It +became necessary to prove that a trigonometric series representing +a continuous function converges uniformly. This was +done by Heinrich Eduard Heine (1821--1881), of Halle. Later +\index{Heine}% +researches on Fourier's series were made by G.~Cantor and +\index{Cantor, G.}% +Du~Bois-Reymond. +\index{Bois-Reymond, P.~du|)}% + +As compared with the vast development of other mathematical +%% -----File: 354.png---Folio 340------- +branches, the theory of probability has made very +\index{Probability}% +insignificant progress since the time of Laplace. Improvements +\index{Laplace!ref.\ to}% +and \DPtypo{simplications}{simplifications} in the mode of exposition have been +made by A.~De~Morgan, G.~Boole, A.~Meyer (edited by E.~Czuber), +\index{Boole!ref.\ to}% +\index{Czuber}% +\index{Demorgan@{De Morgan}!ref.\ to}% +\index{Meyer, A.}% +J.~Bertrand. Cournot's and Westergaard's treatment +\index{Bertrand}% +\index{Cournot}% +\index{Westergaard}% +of insurance and the theory of life-tables are classical. Applications +\index{Insurance}% +of the calculus to statistics have been made by L.~A.~J. +\index{Statistics}% +Quetelet (1796--1874), director of the observatory at Brussels; +\index{Quetelet}% +by Lexis; Harald Westergaard, of Copenhagen; and Düsing. +\index{Dusing@{Düsing}}% +\index{Lexis}% + +Worthy of note is the rejection of inverse probability by the +\index{Inverse probability}% +best authorities of our time. This branch of probability had +been worked out by Thomas Bayes (died 1761) and by Laplace +\index{Bayes}% +(Bk.~II., Ch.~VI. of his \textit{Théorie Analytique}). By it some +logicians have explained induction. For example, if a man, +\index{Induction}% +who has never heard of the tides, were to go to the shore of the +Atlantic Ocean and witness on $m$~successive days the rise of the +sea, then, says Quetelet, he would be entitled to conclude that +there was a probability equal to $\dfrac{m+1}{m+2}$ that the sea would rise +next day. Putting $m = 0$, it is seen that this view rests upon +the unwarrantable assumption that the probability of a totally +unknown event is~$\frac{1}{2}$, or that of all theories proposed for investigation +one-half are true. W.~S. Jevons in his \textit{Principles of +\index{Jevons}% +Science} founds induction upon the theory of inverse probability, +and F.~Y. Edgeworth also accepts it in his \textit{Mathematical +\index{Edgeworth}% +Psychics}. + +The only noteworthy recent addition to probability is the +subject of ``local probability,'' developed by several English +\index{Local probability}% +and a few American and French mathematicians. The earliest +problem on this subject dates back to the time of Buffon, the +\index{Buffon}% +naturalist, who proposed the problem, solved by himself and +Laplace, to determine the probability that a short needle, +thrown at random upon a floor ruled with equidistant parallel +%% -----File: 355.png---Folio 341------- +lines, will fall on one of the lines. Then came Sylvester's +\index{Sylvester!ref.\ to}% +four-point problem: to find the probability that four points, +\index{Four-point problem}% +\index{Probability}% +taken at random within a given boundary, shall form a re-entrant %[** PP: Only instance] +quadrilateral. Local probability has been studied in +England by A.~R. Clarke, H.~McColl, S.~Watson, J.~Wolstenholme, +\index{Clarke}% +\index{McColl}% +\index{Watson, S.}% +\index{Wolstenholme}% +but with greatest success by \textit{M.~W. Crofton} of the +\index{Crofton}% +military school at Woolwich. It was pursued in America by +E.~B. Seitz; in France by C.~Jordan, E.~Lemoine, E.~Barbier, +\index{Barbier}% +\index{Jordan!ref.\ to}% +\index{Lemoine}% +\index{Seitz}% +and others. Through considerations of local probability, +Crofton was led to the evaluation of certain definite integrals. +\index{Definite integrals}% + +The first full scientific treatment of differential equations +\index{Differential equations|(}% +was given by Lagrange and Laplace. This remark is especially +true of partial differential equations. The latter were investigated +\index{Partial differential equations|etseq}% +in more recent time by Monge, Pfaff, Jacobi, Émile Bour +\index{Bour}% +\index{Jacobi!ref.\ to}% +\index{Monge!ref.\ to}% +\index{Pfaff}% +(1831--1866) of Paris, A.~Weiler, Clebsch, A.~N. Korkine of St.~Petersburg, +\index{Clebsch!ref.\ to}% +\index{Korkine!ref.\ to}% +\index{Weiler}% +G.~Boole, A.~Meyer, Cauchy, Serret, Sophus Lie, +\index{Boole!ref.\ to}% +\index{Cauchy!ref.\ to}% +\index{Lie!ref.\ to}% +\index{Meyer, A.}% +\index{Serret!ref.\ to}% +and others. In 1873 their \DPtypo{reseaches}{researches}, on partial differential +equations of the first order, were presented in text-book form +by Paul Mansion, of the University of Gand. The keen +\index{Mansion}% +researches of \textbf{Johann Friedrich Pfaff} (1795--1825) marked a +decided advance. He was an intimate friend of young Gauss +at Göttingen. Afterwards he was with the astronomer Bode. +\index{Bode}% +Later he became professor at Helmstädt, then at Halle. By a +peculiar method, Pfaff found the general integration of partial +differential equations of the first order for any number +of variables. Starting from the theory of ordinary differential +equations of the first order in $n$~variables, he gives first their +general integration, and then considers the integration of the +partial differential equations as a particular case of the former, +assuming, however, as known, the general integration of differential +equations of any order between two variables. His +researches led Jacobi to introduce the name ``Pfaffian problem.'' +\index{Pfaffian problem}% +From the connection, observed by Hamilton, between +\index{Hamilton, W.~R.!ref.\ to}% +%% -----File: 356.png---Folio 342------- +\index{Maxima and minima}% +\index{Pfaffian problem}% +a system of ordinary differential equations (in analytical +mechanics) and a partial differential equation, Jacobi drew +the conclusion that, of the series of systems whose successive +integration Pfaff's method demanded, all but the first system +\index{Pfaff}% +were entirely superfluous. Clebsch considered Pfaff's problem +\index{Clebsch!ref.\ to}% +from a new point of view, and reduced it to systems of simultaneous +linear partial differential equations, which can be +established independently of each other without any integration. +Jacobi materially advanced the theory of differential +equations of the first order. The problem to determine unknown +functions in such a way that an integral containing +these functions and their differential coefficients, in a prescribed +manner, shall reach a maximum or minimum value, +demands, in the first place, the vanishing of the first variation +of the integral. This condition leads to differential equations, +the integration of which determines the functions. To ascertain +whether the value is a maximum or a minimum, the +second variation must be examined. This leads to new and +difficult differential equations, the integration of which, for +the simpler cases, was ingeniously deduced by Jacobi from +the integration of the differential equations of the first variation. +Jacobi's solution was perfected by Hesse, while Clebsch +\index{Hesse!ref.\ to}% +extended to the general case Jacobi's results on the second +variation. Cauchy gave a method of solving partial differential +\index{Cauchy!ref.\ to}% +equations of the first order having any number of +variables, which was corrected and extended by Serret, J.~Bertrand, +\index{Bertrand}% +\index{Serret!ref.\ to}% +O.~Bonnet in France, and Imschenetzky in Russia. +\index{Bonnet, O.!ref.\ to}% +\index{Imschenetzky}% +Fundamental is the proposition of Cauchy that every ordinary +differential equation admits in the vicinity of any non-singular +point of an integral, which is synectic within a certain circle +of convergence, and is developable by Taylor's theorem. +\index{Taylor's theorem}% +Allied to the point of view indicated by this theorem is that +of Riemann, who regards a function of a single variable as +\index{Riemann!ref.\ to}% +%% -----File: 357.png---Folio 343------- +\index{Logic}% +defined by the position and nature of its singularities, and +who has applied this conception to that linear differential +equation of the second order, which is satisfied by the hypergeometric +series. This equation was studied also by Gauss +\index{Gauss!ref.\ to}% +and Kummer. Its general theory, when no restriction is +\index{Kummer!ref.\ to}% +imposed upon the value of the variable, has been considered +by J.~Tannery, of Paris, who employed Fuchs' method of +\index{Fuchs}% +\index{Tannery}% +linear differential equations and found all of Kummer's +twenty-four integrals of this equation. This study has been +continued by Édouard Goursat of Paris. +\index{Goursat}% + +A standard text-book on \textit{Differential Equations}, including +original matter on integrating factors, singular solutions, and +especially on symbolical methods, was prepared in 1859 by +\textbf{George Boole} (1815--1864), at one time professor in Queen's +\index{Boole}% +University, Cork, Ireland. He was a native of Lincoln, and a +self-educated mathematician of great power. His treatise on +\textit{Finite Differences} (1860) and his \textit{Laws of Thought} (1854) are +\index{Finite differences}% +works of high merit. + +The fertility of the conceptions of Cauchy and Riemann +with regard to differential equations is attested by the +researches to which they have given rise on the part of +\textbf{Lazarus Fuchs} of Berlin (born 1835), \textbf{Felix Klein} of Göttingen +\index{Klein}% +(born 1849), \textbf{Henri Poincaré} of Paris (born 1854), and others. +\index{Poincaré}% +The study of linear differential equations entered a new +period with the publication of Fuchs' memoirs of 1866 +and~1868. Before this, linear equations with \emph{constant} coefficients +were almost the only ones for which general methods +of integration were known. While the general theory of +these equations has recently been presented in a new light +by Hermite, Darboux, and Jordan, Fuchs began the study +\index{Darboux}% +\index{Hermite!ref.\ to}% +\index{Jordan!ref.\ to}% +from the more general standpoint of the linear differential +equations whose coefficients are not constant. He directed +his attention mainly to those whose integrals are all regular. +%% -----File: 358.png---Folio 344------- +\index{Invariant}% +If the variable be made to describe all possible paths enclosing +one or more of the critical points of the equation, we +have a certain substitution corresponding to each of the +paths; the aggregate of all these substitutions being called +a \emph{group}. The forms of integrals of such equations were +examined by Fuchs and by G.~Frobenius by independent +\index{Frobenius}% +\index{Fuchs!ref.\ to}% +methods. Logarithms generally appear in the integrals of +a group, and Fuchs and Frobenius investigated the conditions +under which no logarithms shall appear. Through the study +of groups the reducibility or irreducibility of linear differential +\index{Groups, theory of|(}% +equations has been examined by Frobenius and Leo +Königsberger. The subject of linear differential equations, +\index{Konigsberger@{Königsberger}!ref.\ to}% +not all of whose integrals are regular, has been attacked by +G.~Frobenius of Berlin, W.~Thomé of Greifswald (born 1841), +\index{Thome@{Thomé}}% +and Poincaré, but the resulting theory of irregular integrals +\index{Irregular integrals}% +is as yet in very incomplete form. + +The theory of invariants associated with linear differential +equations has been developed by Halphen and by A.~R. Forsyth. +\index{Forsyth}% +\index{Halphen!ref.\ to}% + +The researches above referred to are closely connected with +the theory of functions and of groups. Endeavours have thus +\index{Theory of functions}% +been made to determine the nature of the function defined by +a differential equation from the differential equation itself, +and not from any analytical expression of the function, obtained +first by solving the differential equation. Instead of studying +the properties of the integrals of a differential equation for all +the values of the variable, investigators at first contented themselves +with the study of the properties in the vicinity of a +given point. The nature of the integrals at singular points +and at ordinary points is entirely different. \textit{Albert Briot} +\index{Briot}% +(1817--1882) and \textit{Jean Claude Bouquet} (1819--1885), both of +\index{Bouquet}% +Paris, studied the case when, near a singular point, the differential +equations take the form $(x-x_0) \dfrac{dy}{dx} = \displaystyle\int (xy)$. Fuchs +%% -----File: 359.png---Folio 345------- +\index{Kleinian groups}% +\index{Theory of functions}% +gave the development in series of the integrals for the particular +case of linear equations. Poincaré did the same for the +\index{Poincaré!ref.\ to}% +case when the equations are not linear, as also for partial +differential equations of the first order. The developments for +ordinary points were given by Cauchy and Madame Kowalevsky. +\index{Cauchy!ref.\ to}% +\index{Kowalevsky!ref.\ to}% + +The attempt to express the integrals by developments that +are always convergent and not limited to particular points in +a plane necessitates the introduction of new transcendents, for +the old functions permit the integration of only a small number +\index{Functions!theory of}% +of differential equations. Poincaré tried this plan with +linear equations, which were then the best known, having +been studied in the vicinity of given points by Fuchs, Thomé, +\index{Fuchs!ref.\ to}% +\index{Thome@{Thomé}!ref.\ to}% +Frobenius, Schwarz, Klein, and Halphen. Confining himself to +\index{Frobenius}% +\index{Halphen!ref.\ to}% +\index{Klein!ref.\ to}% +\index{Schwarz!ref.\ to}% +those with rational algebraical coefficients, Poincaré was able +to integrate them by the use of functions named by him \emph{Fuchsians}.\Reference{81} +He divided these equations into ``families.'' If the +integral of such an equation be subjected to a certain transformation, +the result will be the integral of an equation +belonging to the same family. The new transcendents have a +great analogy to elliptic functions; while the region of the +\index{Elliptic functions}% +latter may be divided into parallelograms, each representing a +group, the former may be divided into curvilinear polygons, +so that the knowledge of the function inside of one polygon +carries with it the knowledge of it inside the others. Thus +Poincaré arrives at what he calls \emph{Fuchsian groups}. He found, +\index{Fuchsian groups}% +moreover, that Fuchsian functions can be expressed as the +\index{Fuchsian functions}% +ratio of two transcendents (theta-fuchsians) in the same way +\index{Theta-fuchsians}% +that elliptic functions can be. If, instead of linear substitutions +with real coefficients, as employed in the above groups, +imaginary coefficients be used, then discontinuous groups are +obtained, which he called \emph{Kleinians}. The extension to non-linear %[** PP: Only instance] +equations of the method thus applied to linear equations +has been begun by Fuchs and Poincaré. +%% -----File: 360.png---Folio 346------- +\index{Abelian functions}% +\index{Algebraic functions}% +\index{Elliptic functions}% + +We have seen that among the earliest of the several kinds +of ``groups'' are the finite discontinuous groups (groups in +the theory of substitution), which since the time of Galois +have become the leading concept in the theory of algebraic +equations; that since 1876 Felix Klein, H.~Poincaré, and +\index{Klein!ref.\ to}% +\index{Poincaré!ref.\ to}% +others have applied the theory of finite and infinite discontinuous +groups to the theory of functions and of differential +\index{Theory of functions}% +equations. The finite continuous groups were first made +\index{Groups, theory of|)}% +the subject of general research in 1873 by Sophus Lie, now +\index{Lie}% +of Leipzig, and applied by him to the integration of ordinary +linear partial differential equations. + +Much interest attaches to the determination of those linear +differential equations which can be integrated by simpler +functions, such as algebraic, elliptic, or Abelian. This has +been studied by C.~Jordan, P.~Appel of Paris (born 1858), and +\index{Appel}% +\index{Jordan!ref.\ to}% +Poincaré. + +The mode of integration above referred to, which makes +known the properties of equations from the standpoint of the +theory of functions, does not suffice in the application of +differential equations to questions of mechanics. If we consider +\index{Mechanics!more recent work}% +the function as defining a plane curve, then the general +form of the curve does not appear from the above mode of +investigation. It is, however, often desirable to construct +the curves defined by differential equations. Studies having +this end in view have been carried on by Briot and Bouquet, +\index{Bouquet!ref.\ to}% +\index{Briot!ref.\ to}% +and by Poincaré.\Reference{81} + +The subject of singular solutions of differential equations +has been materially advanced since the time of Boole by G.~Darboux +\index{Boole!ref.\ to}% +\index{Darboux}% +and Cayley. The papers prepared by these mathematicians +\index{Cayley!ref.\ to}% +point out a difficulty as yet unsurmounted: whereas +a singular solution, from the point of view of the integrated +equation, ought to be a phenomenon of universal, or at least of +general occurrence, it is, on the other hand, a very special and +%% -----File: 361.png---Folio 347------- +exceptional phenomenon from the point of view of the differential +equation.\Reference{89} A geometrical theory of singular solutions +resembling the one used by Cayley was previously employed +by W.~W. Johnson of Annapolis. +\index{Johnson}% + +An advanced \textit{Treatise on Linear Differential Equations} +(1889) was brought out by Thomas Craig of the Johns Hopkins +\index{Craig, T.}% +University. He chose the algebraic method of presentation +followed by Hermite and Poincaré, instead of the geometric +\index{Hermite!ref.\ to}% +\index{Poincaré!ref.\ to}% +method preferred by Klein and Schwarz. A notable work, the +\index{Klein!ref.\ to}% +\index{Schwarz!ref.\ to}% +\textit{Traité d'Analyse}, is now being published by Émile Picard of +\index{Picard, E.}% +Paris, the interest of which is made to centre in the subject of +differential equations. +\index{Differential equations|)}% + + +\Section{Theory of Functions} +\index{Theory of functions|(}% + +We begin our sketch of the vast progress in the theory of +functions by considering the special class called elliptic functions. +\index{Elliptic functions|(}% +These were richly developed by Abel and Jacobi. +\index{Abel}% +\index{Jacobi!ref.\ to}% + +\textbf{Niels Henrick Abel} (1802--1829) was born at Findoë in Norway, +and was prepared for the university at the cathedral +school in Christiania. He exhibited no interest in mathematics +until 1818, when B.~Holmboe became lecturer there, +\index{Holmboe}% +and aroused Abel's interest by assigning original problems +to the class. Like Jacobi and many other young men who +became eminent mathematicians, Abel found the first exercise +of his talent in the attempt to solve by algebra the general +equation of the fifth degree. In 1821 he entered the University +in Christiania. The works of Euler, Lagrange, and +Legendre were closely studied by him. The idea of the inversion +of elliptic functions dates back to this time. His extraordinary +success in mathematical study led to the offer of a +stipend by the government, that he might continue his studies +%% -----File: 362.png---Folio 348------- +\index{Abelian functions}% +\index{Binomial formula}% +\index{Integral calculus}% +in Germany and France. Leaving Norway in 1825, Abel visited +\index{Abel}% +the astronomer, Schumacher, in Hamburg, and spent six +\index{Schumacher!ref.\ to}% +months in Berlin, where he became intimate with \textbf{August +Leopold Crelle} (1780--1855), and met Steiner. Encouraged by +\index{Crelle}% +\index{Steiner!ref.\ to}% +Abel and Steiner, Crelle started his journal in 1826. Abel +began to put some of his work in shape for print. His proof +of the impossibility of solving the general equation of the fifth +degree by radicals,---first printed in 1824 in a very concise +form, and difficult of apprehension,---was elaborated in greater +detail, and published in the first volume. He entered also +upon the subject of infinite series (particularly the binomial +\index{Infinite series}% +theorem, of which he gave in \textit{Crelle's Journal} a rigid general +investigation), the study of functions, and of the integral +calculus. The obscurities everywhere encountered by him +owing to the prevailing loose methods of analysis he endeavoured +to clear up. For a short time he left Berlin for Freiberg, +where he had fewer interruptions to work, and it was +there that he made researches on hyperelliptic and Abelian +\index{Hyperelliptic functions}% +functions. In July, 1826, Abel left Germany for Paris without +having met Gauss! Abel had sent to Gauss his proof of +\index{Gauss!ref.\ to}% +1824 of the impossibility of solving equations of the fifth +\index{Equations!solution of}% +degree, to which Gauss never paid any attention. This slight, +and a haughtiness of spirit which he associated with Gauss, +prevented the genial Abel from going to Göttingen. A similar +feeling was entertained by him later against Cauchy. Abel +\index{Cauchy!ref.\ to}% +remained ten months in Paris. He met there Dirichlet, +\index{Dirichlet!ref.\ to}% +Legendre, Cauchy, and others; but was little appreciated. +He had already published several important memoirs in +\textit{Crelle's Journal}, but by the French this new periodical was +as yet hardly known to exist, and Abel was too modest to +speak of his own work. Pecuniary embarrassments induced +him to return home after a second short stay in Berlin. At +Christiania he for some time gave private lessons, and served +%% -----File: 363.png---Folio 349------- +\index{Elliptic integrals}% +\index{Periodicity of functions}% +as docent. Crelle secured at last an appointment for him at +\index{Crelle!ref.\ to}% +Berlin; but the news of it did not reach Norway until after +the death of Abel at Froland.\Reference{82} + +At nearly the same time with Abel, Jacobi published articles +\index{Jacobi!ref.\ to}% +on elliptic functions. Legendre's favourite subject, so long +\index{Legendre!ref.\ to}% +neglected, was at last to be enriched by some extraordinary +discoveries. The advantage to be derived by inverting the +elliptic integral of the first kind and treating it as a function +of its amplitude (now called elliptic function) was recognised +by Abel, and a few months later also by Jacobi. A second +fruitful idea, also arrived at independently by both, is the +introduction of imaginaries leading to the observation that +the new functions simulated at once trigonometric and exponential +functions. For it was shown that while trigonometric +functions had only a real period, and exponential only an imaginary, +\index{Imaginary quantities}% +elliptic functions had both sorts of periods. These two +discoveries were the foundations upon which Abel and Jacobi, +each in his own way, erected beautiful new structures. Abel +developed the curious expressions representing elliptic functions +by infinite series or quotients of infinite products. +\index{Infinite products}% +\index{Infinite series}% +Great as were the achievements of Abel in elliptic functions, +they were eclipsed by his researches on what are now called +Abelian functions. Abel's theorem on these functions was +\index{Abelian functions}% +given by him in several forms, the most general of these +being that in his \textit{Mémoire sur une propriété générale d'une +classe très-étendue de fonctions transcendentes} (1826). The history +of this memoir is interesting. A few months after his +arrival in Paris, Abel submitted it to the French Academy. +Cauchy and Legendre were appointed to examine it; but said +\index{Cauchy!ref.\ to}% +nothing about it until after Abel's death. In a brief statement +of the discoveries in question, published by Abel in \textit{Crelle's +Journal}, 1829, reference is made to that memoir. This led +Jacobi to inquire of Legendre what had become of it. Legendre +%% -----File: 364.png---Folio 350------- +says that the manuscript was so badly written as to be +illegible, and that Abel was asked to hand in a better copy, +\index{Abel!ref.\ to}% +which he neglected to do. The memoir remained in Cauchy's +\index{Cauchy!ref.\ to}% +hands. It was not published until 1841. By a singular mishap, +the manuscript was lost before the proof-sheets were +read. + +In its form, the contents of the memoir belongs to the integral +calculus. Abelian integrals depend upon an irrational +\index{Abelian integrals}% +function $y$ which is connected with $x$ by an algebraic equation +$F(x, y) = 0$. Abel's theorem asserts that a sum of such +\index{Addition theorem of elliptic integrals}% +integrals can be expressed by a definite number $p$ of similar +integrals, where $p$ depends merely on the properties of the +equation $F(x, y) = 0$. It was shown later that $p$ is the deficiency +of the curve $F(x, y) = 0$. The addition theorems of +elliptic integrals are deducible from Abel's theorem. The +\index{Elliptic integrals}% +hyperelliptic integrals introduced by Abel, and proved by him +to possess multiple periodicity, are special cases of Abelian +\index{Periodicity of functions}% +integrals whenever $p= \text{or} > 3$. The reduction of Abelian to +elliptic integrals has been studied mainly by Jacobi, Hermite, +\index{Hermite!ref.\ to}% +\index{Jacobi!ref.\ to}% +Königsberger, Brioschi, Goursat, E.~Picard, and O.~Bolza of +\index{Bolza}% +\index{Brioschi!ref.\ to}% +\index{Goursat!ref.\ to}% +\index{Konigsberger@{Königsberger}!ref.\ to}% +\index{Picard, E.}% +the University of Chicago. + +Two editions of Abel's works have been published: the first +by Holmboe in 1839, and the second by Sylow and Lie in +\index{Holmboe}% +\index{Lie!ref.\ to}% +\index{Sylow!ref.\ to}% +1881. + +Abel's theorem was pronounced by Jacobi the greatest discovery +of our century on the integral calculus. The aged +\index{Integral calculus}% +Legendre, who greatly admired Abel's genius, called it ``\textit{monumentum +\index{Legendre!ref.\ to}% +aere perennius}.'' During the few years of work +allotted to the young Norwegian, he penetrated new fields of +research, the development of which has kept mathematicians +busy for over half a century. + +Some of the discoveries of Abel and Jacobi were anticipated +by Gauss. In the \textit{Disquisitiones Arithmeticæ} he observed +\index{Gauss!ref.\ to}% +%% -----File: 365.png---Folio 351------- +that the principles which he used in the division of the circle +were applicable to many other functions, besides the circular, +and particularly to the transcendents dependent on the integral +$\displaystyle\int \frac{dx}{\sqrt{1-x^4}}$. From this Jacobi\Reference[]{83} concluded that Gauss had +\index{Gauss!ref.\ to}% +\index{Jacobi|(}% +thirty years earlier considered the nature and properties of +elliptic functions and had discovered their double periodicity. +The papers in the collected works of Gauss confirm this conclusion. + +\textbf{Carl Gustav Jacob Jacobi}\Reference[]{84} (1804--1851) was born of Jewish +parents at Potsdam. Like many other mathematicians he was +initiated into mathematics by reading Euler. At the University +of Berlin, where he pursued his mathematical studies +independently of the lecture courses, he took the degree of +Ph.D. in 1825. After giving lectures in Berlin for two years, +he was elected extraordinary professor at Königsberg, and two +years later to the ordinary professorship there. After the +publication of his \textit{Fundamenta Nova} he spent some time in +travel, meeting Gauss in Göttingen, and Legendre, Fourier, +\index{Fourier!ref.\ to}% +\index{Legendre!ref.\ to}% +Poisson, in Paris. In 1842 he and his colleague, Bessel, attended +\index{Bessel!ref.\ to}% +\index{Poisson!ref.\ to}% +the meetings of the British Association, where they +made the acquaintance of English mathematicians. + +His early researches were on Gauss' approximation to the +value of definite integrals, partial differential equations, Legendre's +\index{Definite integrals}% +coefficients, and cubic residues. He read Legendre's +\textit{Exercises}, which give an account of elliptic integrals. When +he returned the book to the library, he was depressed in spirits +and said that important books generally excited in him new +ideas, but that this time he had not been led to a single original +thought. Though slow at first, his ideas flowed all the +richer afterwards. Many of his discoveries in elliptic functions +were made independently by Abel. Jacobi communicated +his first researches to \textit{Crelle's Journal}. In 1829, at the age +%% -----File: 366.png---Folio 352------- +\index{Kowalevsky!ref.\ to}% +of twenty-five, he published his \textit{Fundamenta Nova Theoriæ +Functionum Ellipticarum}, which contains in condensed form +the main results in elliptic functions. This work at once +secured for him a wide reputation. He then made a closer +study of theta-functions and lectured to his pupils on a new +\index{Theta-functions}% +theory of elliptic functions based on the theta-functions. He +developed a theory of transformation which led him to a multitude +of formulæ containing~$q$, a transcendental function of +the modulus, defined by the equation $q = e^{-\pi k'/k}$. He was also +led by it to consider the two new functions $\Eta$~and~$\Theta$, which +taken each separately with two different arguments are the +four (single) theta-functions designated by the $\Theta_1$, $\Theta_2$, $\Theta_3$, $\Theta_4$.\Reference{56} +In a short but very important memoir of 1832, he shows that +for the hyperelliptic integral of any class the direct functions +\index{Hyperelliptic integrals}% +to which Abel's theorem has reference are not functions of a +\index{Abel's theorem}% +single variable, such as the elliptic $\sn$,~$\cn$,~$\dn$, but functions of +$p$~variables.\Reference{56} Thus in the case $p = 2$, which Jacobi especially +considers, it is shown that Abel's theorem has reference to +two functions $\lambda(u, v)$, $\lambda_1(u, v)$, each of two variables, and +gives in effect an addition-theorem for the expression of the +functions $\lambda(u + u', v + v')$, $\lambda_1(u + u', v + v')$ algebraically in +terms of the functions $\lambda(u, v)$, $\lambda_1(u, v)$, $\lambda(u', v')$, $\lambda_1(u', v')$. By +the memoirs of Abel and Jacobi it may be considered that the +notion of the Abelian function of $p$~variables was established +and the addition-theorem for these functions given. Recent +studies touching Abelian functions have been made by Weierstrass, +\index{Abelian functions}% +\index{Weierstrass!ref.\ to}% +E.~Picard, Madame Kowalevski, and Poincaré. Jacobi's +\index{Poincaré!ref.\ to}% +work on differential equations, determinants, dynamics, and +the theory of numbers is mentioned elsewhere. + +In 1842 Jacobi visited Italy for a few months to recuperate +\index{Jacobi|)}% +his health. At this time the Prussian government gave him +a pension, and he moved to Berlin, where the last years of his +life were spent. +%% -----File: 367.png---Folio 353------- + +The researches on functions mentioned thus far have been +greatly extended. In 1858 \textbf{Charles Hermite} of Paris (born 1822), +\index{Hermite}% +introduced in place of the variable $q$ of Jacobi a new variable $\omega$ +\index{Jacobi!ref.\ to}% +connected with it by the equation $q = e^{i\pi\omega}$, so that $\omega = ik'/k$, and +was led to consider the functions $\phi(\omega)$, $\psi(\omega)$, $\chi(\omega)$.\Reference{56} Henry +Smith regarded a theta-function with the argument equal to +\index{Smith, H.!ref.\ to}% +zero, as a function of $\omega$. This he called an omega-function, +\index{Omega-function}% +while the three functions $\phi(\omega)$, $\psi(\omega)$, $\chi(\omega)$, are his modular +functions. Researches on theta-functions with respect to real +\index{Theta-functions}% +and imaginary arguments have been made by Meissel of Kiel, +\index{Meissel}% +J.~Thomae of Jena, Alfred Enneper of Göttingen (1830--1885). +\index{Enneper}% +\index{Thomae}% +A general formula for the product of two theta-functions was +given in 1854 by H.~Schröter of Breslau (1829--1892). These +\index{SchroterH@{Schröter, H.}!ref.\ to}% +functions have been studied also by Cauchy, Königsberger of +\index{Cauchy!ref.\ to}% +\index{Konigsberger@{Königsberger}}% +Heidelberg (born 1837), F.~S. Richelot of Königsberg (1808--1875), +\index{Richelot}% +Johann Georg Rosenhain of Königsberg (1816--1887), +\index{Rosenhain!ref.\ to}% +L.~Schläfli of Bern (born 1818).\Reference{85} +\index{Schlafli@{Schläfli}!ref.\ to}% + +Legendre's method of reducing an elliptic differential to its +\index{Legendre!ref.\ to}% +normal form has called forth many investigations, most important +of which are those of Richelot and of Weierstrass of +\index{Weierstrass!ref.\ to}% +Berlin. + +The algebraic transformations of elliptic functions involve +a relation between the old modulus and the new one which +Jacobi expressed by a differential equation of the third order, +and also by an algebraic equation, called by him ``modular +equation.'' The notion of modular equations was familiar to +\index{Modular equations}% +Abel, but the development of this subject devolved upon later +\index{Abel!ref.\ to}% +investigators. These equations have become of importance in +the theory of algebraic equations, and have been studied by +Sohnke, E.~Mathieu, L.~Königsberger, E.~Betti of Pisa (died +\index{Betti}% +\index{Mathieu!ref.\ to}% +\index{Sohnke}% +1892), C.~Hermite of Paris, Joubert of Angers, Francesco +\index{Joubert}% +Brioschi of Milan, Schläfli, H.~Schröter, M.~Gudermann of +\index{Brioschi!ref.\ to}% +\index{Gudermann}% +Cleve, Gützlaff. +\index{Gutzlaff@{Gützlaff}}% +%% -----File: 368.png---Folio 354------- +\index{Invariant}% +\index{Primary factors, Weierstrass' theory of}% + +Felix Klein of Göttingen has made an extensive study of +\index{Klein!ref.\ to}% +modular functions, dealing with a type of operations lying +\index{Modular functions}% +between the two extreme types, known as the theory of substitutions +\index{Theory of substitutions}% +and the theory of invariants and covariants. Klein's +\index{Covariants}% +theory has been presented in book-form by his pupil, Robert +Fricke. The bolder features of it were first published in his +\index{Fricke}% +\textit{Ikosaeder}, 1884. His researches embrace the theory of modular +functions as a specific class of elliptic functions, the +statement of a more general problem as based on the doctrine +of groups of operations, and the further development of the +subject in connection with a class of Riemann's surfaces. +\index{Riemann!ref.\ to}% + +The elliptic functions were expressed by Abel as quotients +of doubly infinite products. He did not, however, inquire +\index{Infinite products}% +rigorously into the convergency of the products. In 1845 +Cayley studied these products, and found for them a complete +\index{Cayley!ref.\ to}% +theory, based in part upon geometrical interpretation, which +he made the basis of the whole theory of elliptic functions. +Eisenstein discussed by purely analytical methods the general +\index{Eisenstein!ref.\ to}% +doubly infinite product, and arrived at results which have +been greatly simplified in form by the theory of primary +factors, due to Weierstrass. A certain function involving a +\index{Weierstrass!ref.\ to}% +doubly infinite product has been called by Weierstrass the +sigma-function, and is the basis of his beautiful theory of +\index{Sigma-function}% +elliptic functions. The first systematic presentation of Weierstrass' +theory of elliptic functions was published in 1886 by +G.~H. Halphen in his \textit{Théorie des fonctions elliptiques et des +\index{Halphen!ref.\ to}% +leurs applications}. Applications of these functions have been +given also by A.~G. Greenhill. Generalisations analogous to +\index{Greenhill}% +those of Weierstrass on elliptic functions have been made +by Felix Klein on hyperelliptic functions. +\index{Hyperelliptic functions}% + +Standard works on elliptic functions have been published by +\index{Elliptic functions|)}% +\textit{Briot and Bouquet} (1859), by \textit{Königsberger}, \textit{Cayley}, \textit{Heinrich +\index{Bouquet!ref.\ to}% +\index{Briot!ref.\ to}% +\index{Konigsberger@{Königsberger}!ref.\ to}% +Durège} of Prague (1821--1893), and others. +\index{Durège}% +%% -----File: 369.png---Folio 355------- + +Jacobi's work on Abelian and theta-functions was greatly +\index{Theta-functions}% +extended by \textbf{Adolph Göpel} (1812--1847), professor in a gymnasium +\index{Gopel@{Göpel}}% +near Potsdam, and \textbf{Johann Georg Rosenhain} of Königsberg +\index{Rosenhain}% +(1816--1887). Göpel in his \Lat{\textit{Theoriæ transcendentium primi +ordinis adumbratio levis}} (\textit{Crelle},~35, 1847) and Rosenhain in +several memoirs established each independently, on the analogy +of the single theta-functions, the functions of two variables, +called double theta-functions, and worked out in connection +with them the theory of the Abelian functions of two variables. +\index{Abelian functions|(}% +The theta-relations established by Göpel and Rosenhain received +for thirty years no further development, notwithstanding +the fact that the double theta series came to be of increasing +importance in analytical, geometrical, and mechanical problems, +and that Hermite and Königsberger had considered the +\index{Hermite!ref.\ to}% +\index{Konigsberger@{Königsberger}!ref.\ to}% +subject of transformation. Finally, the investigations of C.~W. +Borchardt of Berlin (1817--1880), treating of the representation +\index{Borchardt}% +of Kummer's surface by Göpel's biquadratic relation between +\index{Kummer!ref.\ to}% +four theta-functions of two variables, and researches of H.~H. +Weber of Marburg, F.~Prym of Würzburg, Adolf Krazer, and +\index{Krazer}% +\index{Prym}% +\index{Weber, H.~H.}% +Martin Krause of Dresden led to broader views. Researches +\index{Krause}% +on double theta-functions, made by Cayley, were extended to +\index{Cayley!ref.\ to}% +quadruple theta-functions by Thomas Craig of the Johns +\index{Craig, T.}% +Hopkins University. + +Starting with the integrals of the most general form and +considering the inverse functions corresponding to these integrals +(the Abelian functions of $p$~variables), \textit{Riemann} +\index{Riemann!ref.\ to}% +defined the theta-functions of $p$~variables as the sum of a +$p$\nobreakdash-tuply infinite series of exponentials, the general term depending +on $p$~variables. Riemann shows that the Abelian +functions are algebraically connected with theta-functions of +the proper arguments, and presents the theory in the broadest +form.\Reference{56} He rests the theory of the multiple theta-functions +upon the general principles of the theory of functions of a +complex variable. +%% -----File: 370.png---Folio 356------- +\index{Bernoulli, James (born 1758)}% +\index{Bernoulli, John (born 1667)!ref.\ to}% +\index{Riemann's surfaces!ref.\ to}% + +Through the researches of A.~Brill of Tübingen, M.~Nöther +\index{Brill, A.}% +\index{Nother@{Nöther}}% +of Erlangen, and Ferdinand Lindemann of Munich, made +\index{Lindemann!ref.\ to}% +in connection with Riemann-Roch's theorem and the theory +\index{Riemann!ref.\ to}% +of residuation, there has grown out of the theory of Abelian +functions a theory of algebraic functions and point-groups on +\index{Functions!definition of}% +\index{Functions!theory of|(}% +algebraic curves. + +Before proceeding to the general theory of functions, we +make mention of the ``calculus of functions,'' studied chiefly +\index{Calculus!of variations}% +by C.~Babbage, J.~F.~W. Herschel, and De~Morgan, which was +\index{Babbage}% +\index{Demorgan@{De Morgan}!ref.\ to}% +\index{Herschel, J.~F.~W.!ref.\ to}% +not so much a theory of functions as a theory of the solution +of functional equations by means of known functions or +symbols. + +The history of the general theory of functions begins with +the adoption of new definitions of a function. With the +Bernoullis and Leibniz, $y$ was called a function of~$x$, if there +\index{Leibniz!ref.\ to}% +existed an equation between these variables which made it +possible to calculate~$y$ for any given value of~$x$ lying anywhere +between $-\infty$~and~$+\infty$. The study of Fourier's theory +\index{Fourier!ref.\ to}% +of heat led Dirichlet to a new definition: $y$ is called a function +\index{Dirichlet!ref.\ to}% +of~$x$, if $y$ possess one or more definite values for each of certain +values that $x$ is assumed to take in an interval $x_0$~to~$x_1$. In +functions thus defined, there need be no analytical connection +between $y$~and~$x$, and it becomes necessary to look for possible +discontinuities. A great revolution in the ideas of a function +was brought about by Cauchy when, in a function as defined +\index{Cauchy!ref.\ to}% +by Dirichlet, he gave the variables imaginary values, and when +he extended the notion of a definite integral by letting the +variable pass from one limit to the other by a succession of +imaginary values along arbitrary paths. Cauchy established +several fundamental theorems, and gave the first great impulse +to the study of the general theory of functions. His researches +were continued in France by Puiseux and Liouville. But more +\index{Liouville!ref.\ to}% +\index{Puiseux}% +profound investigations were made in Germany by Riemann. +%% -----File: 371.png---Folio 357------- + +\textbf{Georg Friedrich Bernhard Riemann} (1826--1866) was born at +\index{Riemann|(}% +Breselenz in Hanover. His father wished him to study +theology, and he accordingly entered upon philological and +theological studies at Göttingen. He attended also some lectures +on mathematics. Such was his predilection for this +science that he abandoned theology. After studying for a +time under Gauss and Stern, he was drawn, in 1847, to Berlin +\index{Gauss!ref.\ to}% +\index{Stern}% +by a galaxy of mathematicians, in which shone Dirichlet, +\index{Dirichlet!ref.\ to}% +Jacobi, Steiner, and Eisenstein. Returning to Göttingen in +\index{Eisenstein!ref.\ to}% +\index{Jacobi!ref.\ to}% +\index{Steiner!ref.\ to}% +1850, he studied physics under Weber, and obtained the +\index{Weber, W.~E.!ref.\ to}% +doctorate the following year. The thesis presented on that +occasion, \textit{Grundlagen für eine allgemeine Theorie der Funktionen +einer veränderlichen complexen Grösse}, excited the admiration of +Gauss to a very unusual degree, as did also Riemann's trial +lecture, \textit{Ueber die Hypothesen welche der Geometrie zu Grunde +liegen}. Riemann's Habilitationsschrift was on the Representation +of a Function by means of a Trigonometric Series, in +\index{Trigonometric series}% +which he advanced materially beyond the position of Dirichlet. +Our hearts are drawn to this extraordinarily gifted but +shy genius when we read of the timidity and nervousness +displayed when he began to lecture at Göttingen, and of his +jubilation over the unexpectedly large audience of eight students +at his first lecture on differential equations. + +Later he lectured on Abelian functions to a class of three +\index{Abelian functions|)}% +only,---Schering, Bjerknes, and Dedekind. Gauss died in 1855, +\index{Bjerknes, C. A.}% +\index{Dedekind!ref.\ to}% +\index{Schering!ref.\ to}% +and was succeeded by Dirichlet. On the death of the latter, +in 1859, Riemann was made ordinary professor. In 1860 he +visited Paris, where he made the acquaintance of French +mathematicians. The delicate state of his health induced +him to go to Italy three times. He died on his last trip at +Selasca, and was buried at Biganzolo. + +Like all of Riemann's researches, those on functions were +profound and far-reaching. He laid the foundation for a +%% -----File: 372.png---Folio 358------- +\index{Kelvin, Lord!ref.\ to}% +general theory of functions of a complex variable. The theory +of potential, which up to that time had been used only in +\index{Potential}% +mathematical physics, was applied by him in pure mathematics. +He accordingly based his theory of functions on the +partial differential equation, $\dfrac{\partial^2u}{\partial x^2} + \dfrac{\partial^2 u}{\partial y^2} = \Delta u = 0$, which must +hold for the analytical function $w = u + iv$ of $z = x + iy$. It +had been proved by Dirichlet that (for a plane) there is always +one, and only one, function of $x$~and~$y$, which satisfies $\Delta u = 0$, +and which, together with its differential quotients of the first +two orders, is for all values of $x$~and~$y$ within a given area +one-valued and continuous, and which has for points on the +boundary of the area arbitrarily given values.\Reference{86} Riemann +called this ``Dirichlet's principle,'' but the same theorem was +stated by Green and proved analytically by Sir William +\index{Green!ref.\ to}% +Thomson. It follows then that $w$ is uniquely determined for +\index{Thomson, Sir William!ref.\ to}% +all points within a closed surface, if $u$ is arbitrarily given +for all points on the curve, whilst $v$ is given for one point +within the curve. In order to treat the more complicated +case where $w$ has $n$~values for one value of~$z$, and to observe +the conditions about continuity, Riemann invented the celebrated +surfaces, known as ``Riemann's surfaces,'' consisting +\index{Riemann's surfaces}% +of $n$~coincident planes or sheets, such that the passage from +one sheet to another is made at the branch-points, and that the +$n$~sheets form together a multiply-connected surface, which +can be dissected by cross-cuts into a singly-connected surface. +The $n$-valued function~$w$ becomes thus a one-valued function. +Aided by researches of J.~Lüroth of Freiburg and of Clebsch, +\index{Clebsch!ref.\ to}% +\index{Luroth@{Lüroth}}% +W.~K. Clifford brought Riemann's surface for algebraic functions +\index{Clifford!ref.\ to}% +\Addendlabel{358}% +to a canonical form, in which only the two last of the $n$~leaves +are multiply-connected, and then transformed the surface into +the surface of a solid with $p$~holes. A.~Hurwitz of Zürich +\index{Hurwitz}% +discussed the question, how far a Riemann's surface is determinate +%% -----File: 373.png---Folio 359------- +\index{Thomson's theorem}% +by the assignment of its number of sheets, its branch-points +and branch-lines.\Reference{62} + +Riemann's theory ascertains the criteria which will determine +an analytical function by aid of its discontinuities and +boundary conditions, and thus defines a function independently +of a mathematical expression. In order to show that +two different expressions are identical, it is not necessary to +transform one into the other, but it is sufficient to prove the +agreement to a far less extent, merely in certain critical points. + +Riemann's theory, as based on Dirichlet's principle (Thomson's +\index{Dirichlet!ref.\ to}% +theorem), is not free from objections. It has become +evident that the existence of a derived function is not a consequence +of continuity, and that a function may be integrable +\index{Continuity}% +without being differentiable. It is not known how far the +methods of the infinitesimal calculus and the calculus of +variations (by which Dirichlet's principle is established) can +be applied to an unknown analytical function in its generality. +Hence the use of these methods will endow the functions with +properties which themselves require proof. Objections of this +kind to Riemann's theory have been raised by Kronecker, +\index{Kronecker!ref.\ to}% +Weierstrass, and others, and it has become doubtful whether +\index{Weierstrass}% +\index{Weierstrass!ref.\ to}% +his most important theorems are actually proved. In consequence +of this, attempts have been made to graft Riemann's +speculations on the more strongly rooted methods of Weierstrass. +The latter developed a theory of functions by starting, +not with the theory of potential, but with analytical +expressions and operations. Both applied their theories to +Abelian functions, but there Riemann's work is more general.\Reference{86} +\index{Abelian functions}% + +The theory of functions of one complex variable has been +studied since Riemann's time mainly by \textbf{Karl Weierstrass} of +\index{Riemann|)}% +Berlin (born 1815), \textbf{Gustaf Mittag-Leffler} of Stockholm (born +\index{Mittag-Leffler}% +1846), and Poincaré of Paris. Of the three classes of such +\index{Poincaré!ref.\ to}% +%% -----File: 374.png---Folio 360------- +\index{Fuchsian functions}% +\index{Hyperelliptic functions}% +functions (viz.\ functions uniform throughout, functions uniform +only in lacunary spaces, and non-uniform functions) +Weierstrass showed that those functions of the first class +which can be developed according to ascending powers of~$x$ +into converging series, can be decomposed into a product of an +infinite number of primary factors. A primary factor of the +\index{Primary factors, Weierstrass' theory of}% +species~$n$ is the product $\left(1- \dfrac{x}{a}\right)e^{\Change{P_{(x)}}{P(x)}}$, $\Change{P_{(x)}}{P(x)}$ being an entire polynomial +of the $n$th~degree. A function of the species~$n$ is one, +all the primary factors of which are of species~$n$. This classification +gave rise to many interesting problems studied also +by Poincaré. + +The first of the three classes of functions of a complex +variable embraces, among others, functions having an infinite +number of singular points, but no singular lines, and at the +same time no isolated singular points. These are Fuchsian +functions, existing throughout the whole extent. Poincaré +first gave an example of such a function. + +Uniform functions of two variables, unaltered by certain +linear substitutions, called hyperfuchsian functions, have been +studied by E.~Picard of Paris, and by Poincaré.\Reference{81} +\index{Picard, E.}% + +Functions of the second class, uniform only in lacunary +spaces, were first pointed out by Weierstrass. The Fuchsian +and the Kleinian functions do not generally exist, except in +\index{Kleinian functions}% +the interior of a circle or of a domain otherwise bounded, and +are therefore examples of functions of the second class. +Poincaré has shown how to generate functions of this class, +and has studied them along the lines marked out by Weierstrass. +Important is his proof that there is no way of +generalising them so as to get rid of the lacunæ. + +Non-uniform functions are much less developed than the +preceding classes, even though their properties in the vicinity +of a given point have been diligently studied, and though +%% -----File: 375.png---Folio 361------- +\index{Conform representation of surfaces}% +\index{Infinite series}% +much light has been thrown on them by the use of Riemann's +surfaces. With the view of reducing their study to that of +uniform transcendents, Poincaré proved that if $y$ is any +analytical non-uniform function of~$x$, one can always find a +variable~$z$, such that $x$~and~$y$ are uniform functions of~$z$. + +Weierstrass and Darboux have each given examples of continuous +\index{Darboux}% +\index{Weierstrass!ref.\ to}% +functions having no derivatives. Formerly it had +been generally assumed that every function had a derivative. +\Addendlabel{361}% +Ampère was the first who attempted to prove analytically +\index{Ampère!ref.\ to}% +(1806) the existence of a derivative, but the demonstration +is not valid. In treating of discontinuous functions, Darboux +established rigorously the necessary and sufficient condition +that a continuous or discontinuous function be susceptible of +integration. He gave fresh evidence of the care that must +be exercised in the use of series by giving an example of a +series always convergent and continuous, such that the series +formed by the integrals of the terms is always convergent, and +yet does not represent the integral of the first series.\Reference{87} + +The general theory of functions of two variables has been +investigated to some extent by Weierstrass and Poincaré. + +\textbf{H.~A. Schwarz} of Berlin (born 1845), a pupil of Weierstrass, +\index{Schwarz}% +has given the conform\DPnote{** PP: [sic]} representation (\emph{Abbildung}) of various +surfaces on a circle. In transforming by aid of certain +substitutions a polygon bounded by circular arcs into another +also bounded by circular arcs, he was led to a remarkable +differential equation $\psi(u', t) = \psi(u, t)$, where $\psi(u, t)$ is the +expression which Cayley calls the ``Schwarzian derivative,'' +\index{Schwarzian derivative}% +and which led Sylvester to the theory of reciprocants. +\index{Reciprocants}% +\index{Sylvester!ref.\ to}% +Schwarz's developments on minimum surfaces, his work on +hypergeometric series, his inquiries on the existence of solutions +\index{Hypergeometric series}% +to important partial differential equations under prescribed +conditions, have secured a prominent place in mathematical +literature. +%% -----File: 376.png---Folio 362------- +\index{Irrationals}% + +The modern theory of functions of one real variable was +first worked out by H.~Hankel, Dedekind, G.~Cantor, Dini, and +\index{Cantor, G.}% +\index{Dedekind!ref.\ to}% +\index{Dini!ref.\ to}% +\index{Hankel!ref.\ to}% +Heine, and then carried further, principally, by Weierstrass, +\index{Heine!ref.\ to}% +\index{Weierstrass!ref.\ to}% +Schwarz, Du~Bois-Reymond, Thomae, and Darboux. Hankel +\index{Bois-Reymond, P.~du}% +\index{Darboux}% +\index{Schwarz!ref.\ to}% +\index{Thomae}% +established the principle of the condensation of singularities; +\index{Condensation of singularities}% +Dedekind and Cantor gave definitions for irrational numbers; +\index{Numbers!theory of numbers|(}% +definite integrals were studied by Thomae, Du~Bois-Reymond, +\index{Definite integrals}% +and Darboux along the lines indicated by the definitions of +such integrals given by Cauchy, Dirichlet, and Riemann. Dini +\index{Cauchy!ref.\ to}% +\index{Dirichlet!ref.\ to}% +\index{Riemann!ref.\ to}% +wrote a text-book on functions of a real variable (1878), which +was translated into German, with additions, by J.~Lüroth and +\index{Luroth@{Lüroth}!ref.\ to}% +A.~Schepp. Important works on the theory of functions are +\index{Schepp}% +the \textit{Cours de M.~Hermite}, \textit{Tannery's Théorie des Fonctions +\index{Hermite!ref.\ to}% +\index{Tannery!ref.\ to}% +d'une variable seule}, \textit{A Treatise on the Theory of Functions} by +\textit{James Harkness} and \textit{Frank Morley}, and \textit{Theory of Functions of +\index{Functions!theory of|)}% +\index{Harkness}% +\index{Morley}% +\index{Theory of functions|)}% +a Complex Variable} by \textit{A.~R. Forsyth}. +\index{Forsyth}% + +\Section{Theory of Numbers} +\index{Theory of numbers|(}% + +``Mathematics, the queen of the sciences, and arithmetic, +the queen of mathematics.'' Such was the dictum of Gauss, +\index{Gauss!ref.\ to}% +who was destined to revolutionise the theory of numbers. +When asked who was the greatest mathematician in Germany, +Laplace answered, Pfaff. When the questioner said +\index{Laplace!ref.\ to}% +\index{Pfaff!ref.\ to}% +he should have thought Gauss was, Laplace replied, ``Pfaff +is by far the greatest mathematician in Germany; but Gauss +is the greatest in all Europe.''\Reference{83} Gauss is one of the three +greatest masters of modern analysis,---Lagrange, Laplace, +\index{Lagrange!ref.\ to}% +Gauss. Of these three contemporaries he was the youngest. +While the first two belong to the period in mathematical history +preceding the one now under consideration, Gauss is the +one whose writings may truly be said to mark the beginning +%% -----File: 377.png---Folio 363------- +\index{Imaginary quantities}% +\index{Infinite series}% +\index{Kaestner}% +of our own epoch. In him that abundant fertility of invention, +displayed by mathematicians of the preceding period, +is combined with an absolute rigorousness in demonstration +which is too often wanting in their writings, and which the +ancient Greeks might have envied. Unlike Laplace, Gauss +\index{Gauss|(}% +\index{Laplace!ref.\ to}% +strove in his writings after perfection of form. He rivals +Lagrange in elegance, and surpasses this great Frenchman in +\index{Lagrange!ref.\ to}% +rigour. Wonderful was his richness of ideas; one thought followed +another so quickly that he had hardly time to write +down even the most meagre outline. At the age of twenty +Gauss had overturned old theories and old methods in all +branches of higher mathematics; but little pains did he take +to publish his results, and thereby to establish his priority. +He was the first to observe rigour in the treatment of infinite +series, the first to fully recognise and emphasise the importance, +and to make systematic use of determinants and of +\index{Determinants}% +imaginaries, the first to arrive at the method of least squares, +\index{Least squares}% +the first to observe the double periodicity of elliptic functions. +\index{Elliptic functions}% +He invented the heliotrope and, together with Weber, the +\index{Heliotrope}% +\index{Weber, W.~E.!ref.\ to}% +bifilar magnetometer and the declination instrument. He +reconstructed the whole of magnetic science. + +\textbf{Carl Friedrich Gauss}\Reference[]{47} (1777--1855), the son of a bricklayer, +was born at Brunswick. He used to say, jokingly, that he +could reckon before he could talk. The marvellous aptitude +for calculation of the young boy attracted the attention of +Bartels, afterwards professor of mathematics at Dorpat, who +brought him under the notice of Charles William, Duke of +Brunswick. The duke undertook to educate the boy, and sent +him to the Collegium Carolinum. His progress in languages +there was quite equal to that in mathematics. In 1795 he +went to Göttingen, as yet undecided whether to pursue philology +or mathematics. Abraham Gotthelf Kästner, then professor +of mathematics there, and now chiefly remembered for +%% -----File: 378.png---Folio 364------- +his \textit{Geschichte der Mathematik} (1796), was not an inspiring +teacher. At the age of nineteen Gauss discovered a method of +inscribing in a circle a regular polygon of seventeen sides, and +this success encouraged him to pursue mathematics. He worked +quite independently of his teachers, and while a student at +Göttingen made several of his greatest discoveries. Higher +arithmetic was his favourite study. Among his small circle +of intimate friends was Wolfgang Bolyai. After completing +\index{Bolyai, Wolfgang!ref.\ to}% +his course he returned to Brunswick. In 1798 and 1799 he +repaired to the university at \Change{Helmstadt}{Helmstädt} to consult the library, +and there made the acquaintance of Pfaff, a mathematician of +much power. In 1807 the Emperor of Russia offered Gauss a +chair in the Academy at St.~Petersburg, but by the advice of +the astronomer Olbers, who desired to secure him as director +\index{Olbers}% +of a proposed new observatory at Göttingen, he declined the +offer, and accepted the place at Göttingen. Gauss had a +marked objection to a mathematical chair, and preferred the +post of astronomer, that he might give all his time to science. +He spent his life in Göttingen in the midst of continuous +work. In 1828 he went to Berlin to attend a meeting of +scientists, but after this he never again left Göttingen, except +in 1854, when a railroad was opened between Göttingen and +Hanover. He had a strong will, and his character showed +a curious mixture of self-conscious dignity and child-like +simplicity. He was little communicative, and at times +morose. + +A new epoch in the theory of numbers dates from the publication +of his \textit{Disquisitiones Arithmeticæ}, Leipzig, 1801. The +beginning of this work dates back as far as 1795. Some of its +results had been previously given by Lagrange and Euler, but +\index{Euler!ref.\ to}% +\index{Lagrange!ref.\ to}% +were reached independently by Gauss, who had gone deeply +into the subject before he became acquainted with the writings +of his great predecessors. The \textit{Disquisitiones Arithmeticæ} +%% -----File: 379.png---Folio 365------- +\index{Congruencies, theory of}% +was already in print when Legendre's \textit{Théorie des Nombres} +\index{Legendre!ref.\ to}% +appeared. The great law of quadratic reciprocity, given in +\index{Quadratic reciprocity}% +the fourth section of Gauss' work, a law which involves the +whole theory of quadratic residues, was discovered by him by +induction before he was eighteen, and was proved by him one +year later. Afterwards he learned that Euler had imperfectly +\index{Euler!ref.\ to}% +enunciated that theorem, and that Legendre had attempted +to prove it, but met with apparently insuperable difficulties. +In the fifth section Gauss gave a second proof of this ``gem'' +of higher arithmetic. In 1808 followed a third and fourth +demonstration; in 1817, a fifth and sixth. No wonder that +he felt a personal attachment to this theorem. Proofs were +given also by Jacobi, Eisenstein, Liouville, Lebesgue, A.~Genocchi, +\index{Eisenstein!ref.\ to}% +\index{Genocchi}% +\index{Jacobi!ref.\ to}% +\index{Lebesgue}% +\index{Liouville!ref.\ to}% +Kummer, M.~A. Stern, Chr.\ Zeller, Kronecker, +\index{Kronecker!ref.\ to}% +\index{Kummer!ref.\ to}% +\index{Stern}% +\index{Zeller}% +Bouniakowsky, E.~Schering, J.~Petersen, Voigt, E.~Busche, +\index{Bouniakowsky}% +\index{Busche}% +\index{Petersen}% +\index{Schering!ref.\ to}% +\index{Voigt}% +and Th.\ Pepin.\Reference{48} The solution of the problem of the representation +\index{Pepin}% +of numbers by binary quadratic forms is one of the +great achievements of Gauss. He created a new algorithm by +introducing the theory of congruences. The fourth section +of the \textit{Disquisitiones Arithmeticæ}, treating of congruences of +the second degree, and the fifth section, treating of quadratic +forms, were, until the time of Jacobi, passed over with universal +neglect, but they have since been the starting-point of a long +series of important researches. The seventh or last section, +developing the theory of the division of the circle, was received +\index{Circle!division of}% +\index{Division of the circle}% +from the start with deserved enthusiasm, and has since been +repeatedly elaborated for students. A standard work on +\textit{Kreistheilung} was published in 1872 by Paul Bachmann, then +\index{Bachmann!ref.\ to}% +of Breslau. Gauss had planned an eighth section, which was +omitted to lessen the expense of publication. His papers on +the theory of numbers were not all included in his great treatise. +Some of them were published for the first time after his death +in his collected works (1863--1871). He wrote two memoirs on +%% -----File: 380.png---Folio 366------- +\index{Action, least}% +\index{Capillarity}% +\index{Ellipsoid!(attraction of)}% +\index{Geodesy}% +the theory of biquadratic residues (1825 and 1831), the second +\index{Biquadratic residues}% +of which contains a theorem of biquadratic reciprocity. + +Gauss was led to astronomy by the discovery of the planet +\index{Astronomy!more recent researches}% +Ceres at Palermo in 1801. His determination of the elements +of its orbit with sufficient accuracy to enable Olbers to \Change{rediscover}{re-discover} +it, made the name of Gauss generally known. In 1809 +he published the \textit{Theoria motus corporum coelestium}, which +contains a discussion of the problems arising in the determination +of the movements of planets and comets from +observations made on them under any circumstances. In it +are found four formulæ in spherical trigonometry, now usually +called ``Gauss' Analogies,'' but which were published somewhat +\index{Gauss' Analogies}% +earlier by Karl Brandon Mollweide of Leipzig (1774--1825), +\index{Mollweide}% +and earlier still by Jean Baptiste Joseph Delambre +\index{Delambre}% +(1749--1822).\Reference{44} Many years of hard work were spent in the +astronomical and magnetic observatory. He founded the +German Magnetic Union, with the object of securing continuous +\index{German Magnetic Union}% +observations at fixed times. He took part in geodetic +observations, and in 1843 and 1846 wrote two memoirs, \textit{Ueber +Gegenstände der höheren Geodesie}. He wrote on the attraction +of homogeneous ellipsoids, 1813. In a memoir on capillary +attraction, 1833, he solves a problem in the calculus of +\index{Calculus!of variations}% +variations involving the variation of a certain double integral, +the limits of integration being also variable; it is the earliest +example of the solution of such a problem. He discussed the +problem of rays of light passing through a system of lenses. + +Among Gauss' pupils were Christian Heinrich Schumacher, +\index{Schumacher}% +Christian Gerling, Friedrich Nicolai, August Ferdinand +\index{Gerling}% +\index{Nicolai}% +Möbius, Georg Wilhelm Struve,\DPnote{** PP: [sic] no `and'} Johann Frantz Encke. +\index{Encke}% +\index{Mobius@{Möbius}!ref.\ to}% +\index{Struve}% + +Gauss' researches on the theory of numbers were the starting-point +for a school of writers, among the earliest of whom +was Jacobi. The latter contributed to \textit{Crelle's Journal} an article +on cubic residues, giving theorems without proofs. After the +\index{Cubic residues}% +%% -----File: 381.png---Folio 367------- +\index{Elliptic functions}% +publication of Gauss' paper on biquadratic residues, giving +the law of biquadratic reciprocity, and his treatment of complex +numbers, Jacobi found a similar law for cubic residues. +\index{Jacobi!ref.\ to}% +By the theory of elliptical functions, he was led to beautiful +theorems on the representation of numbers by $2$,~$4$,~$6$, and~$8$ +squares. Next come the researches of Dirichlet, the expounder +\index{Dirichlet|(}% +of Gauss, and a contributor of rich results of his own. + +\textbf{Peter Gustav Lejeune Dirichlet}\Reference[]{88} (1805--1859) was born in +Düren, attended the gymnasium in Bonn, and then the +Jesuit gymnasium in Cologne. In 1822 he was attracted to +Paris by the names of Laplace, Legendre, Fourier, Poisson, +\index{Fourier!ref.\ to}% +\index{Legendre!ref.\ to}% +Cauchy. The facilities for a mathematical education there +were far better than in Germany, where Gauss was the only +great figure. He read in Paris Gauss' \textit{Disquisitiones Arithmeticæ}, +a work which he never ceased to admire and study. +Much in it was simplified by Dirichlet, and thereby placed +within easier reach of mathematicians. His first memoir on +the impossibility of certain indeterminate equations of the +fifth degree was presented to the French Academy in 1825. +He showed that Fermat's equation, $x^n + y^n = z^n$, cannot exist +\index{Fermat!ref.\ to}% +when $n = 5$. Some parts of the analysis are, however, +Legendre's. Euler and Lagrange had proved this when $n$~is~$3$ +\index{Euler!ref.\ to}% +\index{Lagrange!ref.\ to}% +and~$4$, and Lamé proved it when $n = 7$. Dirichlet's acquaintance +\index{Lame@{Lamé}!ref.\ to}% +with Fourier led him to investigate Fourier's series. He +\index{Fourier's series}% +became docent in Breslau in 1827. In 1828 he accepted a +position in Berlin, and finally succeeded Gauss at Göttingen +in 1855. The general principles on which depends the average +number of classes of binary quadratic forms of positive +and negative determinant (a subject first investigated by +Gauss) were given by Dirichlet in a memoir, \textit{Ueber die Bestimmung +\index{Gauss|)}% +der mittleren Werthe in der Zahlentheorie}, 1849. More +recently F.~Mertens of Graz has determined the asymptotic +\index{Mertens}% +values of several numerical functions. Dirichlet gave some +%% -----File: 382.png---Folio 368------- +attention to prime numbers. Gauss and Legendre had given +\index{Prime numbers}% +expressions denoting approximately the asymptotic value of +the number of primes inferior to a given limit, but it remained +for Riemann in his memoir, \textit{Ueber die Anzahl der Primzahlen +\index{Riemann!ref.\ to}% +unter einer gegebenen Grösse}, 1859, to give an investigation of +the asymptotic frequency of primes which is rigorous. Approaching +the problem from a different direction, \textit{Patnutij +Tchebycheff}, formerly professor in the University of St.~Petersburg +\index{Tchebycheff}% +(born 1821), established, in a celebrated memoir, \textit{Sur les +Nombres Premiers}, 1850, the existence of limits within which +the sum of the logarithms of the primes~$P$, inferior to a given +number~$x$, must be comprised.\Reference{89} This paper depends on very +elementary considerations, and, in that respect, contrasts +strongly with Riemann's, which involves abstruse theorems +of the integral calculus. Poincaré's papers, Sylvester's contraction +\index{Integral calculus}% +\index{Poincaré!ref.\ to}% +\index{Sylvester!ref.\ to}% +of Tchebycheff's limits, with reference to the distribution +of primes, and researches of J.~Hadamard (awarded the +\index{Hadamard}% +\textit{Grand prix} of 1892), are among the latest researches in this +line. The enumeration of prime numbers has been undertaken +at different times by various mathematicians. In 1877 the +British Association began the preparation of factor-tables, +\index{Factor-tables}% +under the direction of J.~W.~L. Glaisher. The printing, by +\index{Glaisher!ref.\ to}% +the Association, of tables for the sixth million marked the +completion of tables, to the preparation of which Germany, +France, and England contributed, and which enable us to +resolve into prime factors every composite number less than +$9,000,000$. + +Miscellaneous contributions to the theory of numbers were +made by \textit{Cauchy}. He showed, for instance, how to find all +\index{Cauchy!ref.\ to}% +the infinite solutions of a homogeneous indeterminate equation +of the second degree in three variables when one solution is +given. He established the theorem that if two congruences, +which have the same modulus, admit of a common solution, +%% -----File: 383.png---Folio 369------- +\index{Covariants}% +\index{Dirichlet|)}% +the modulus is a divisor of their resultant. \textbf{Joseph Liouville} +\index{Liouville}% +(1809--1882), professor at the \DPtypo{Collége}{Collège} de France, investigated +mainly questions on the theory of quadratic forms of two, and +of a greater number of variables. Profound researches were +instituted by \textbf{Ferdinand Gotthold Eisenstein} (1823--1852), of +\index{Eisenstein}% +Berlin. Ternary quadratic forms had been studied somewhat +by Gauss, but the extension from two to three indeterminates +was the work of Eisenstein who, in his memoir, \textit{Neue Theoreme +der höheren Arithmetik}, defined the ordinal and generic +characters of ternary quadratic forms of uneven determinant; +and, in case of definite forms, assigned the weight of any order +or genus. But he did not publish demonstrations of his results. +In inspecting the theory of binary cubic forms, he was +led to the discovery of the first covariant ever considered in +analysis. He showed that the series of theorems, relating to +the presentation of numbers by sums of squares, ceases when +the number of squares surpasses eight. Many of the proofs +omitted by Eisenstein were supplied by Henry Smith, who +\index{Smith, H.}% +was one of the few Englishmen who devoted themselves to the +study of higher arithmetic. + +\textbf{Henry John Stephen Smith}\Reference[]{90} (1826--1883) was born in London, +and educated at Rugby and at Balliol College, Oxford. +Before 1847 he travelled much in Europe for his health, +and at one time attended lectures of Arago in Paris, but +after that year he was never absent from Oxford for a +single term. In 1861 he was elected Savilian professor of +geometry. His first paper on the theory of numbers appeared +in 1855. The results of ten years' study of everything published +on the theory of numbers are contained in his Reports +which appeared in the British Association volumes from 1859 +to 1865. These reports are a model of clear and precise +exposition and perfection of form. They contain much original +matter, but the chief results of his own discoveries were +%% -----File: 384.png---Folio 370------- +printed in the \textit{Philosophical Transactions} for 1861 and 1867. +They treat of linear indeterminate equations and congruences, +and of the orders and genera of ternary quadratic forms. He +established the principles on which the extension to the general +case of $n$ indeterminates of quadratic forms depends. +He contributed also two memoirs to the \textit{Proceedings of the +Royal Society} of 1864 and 1868, in the second of which he +remarks that the theorems of Jacobi, Eisenstein, and Liouville, +\index{Eisenstein!ref.\ to}% +\index{Jacobi!ref.\ to}% +\index{Liouville!ref.\ to}% +relating to the representation of numbers by $4$,~$6$,~$8$ +squares, and other simple quadratic forms are deducible by a +uniform method from the principles indicated in his paper. +Theorems relating to the case of $5$~squares were given by +Eisenstein, but Smith completed the enunciation of them, and +\index{Smith, H.}% +added the corresponding theorems for $7$~squares. The solution +of the cases of $2$,~$4$,~$6$ squares may be obtained by elliptic +functions, but when the number of squares is odd, it involves +processes peculiar to the theory of numbers. This class of +theorems is limited to $8$~squares, and Smith completed the +group. In ignorance of Smith's investigations, the French +Academy offered a prize for the demonstration and completion +of Eisenstein's theorems for $5$~squares. This Smith had +accomplished fifteen years earlier. He sent in a dissertation +in 1882, and next year, a month after his death, the prize was +awarded to him, another prize being also awarded to H.~Minkowsky +\index{Minkowsky}% +of Bonn. The theory of numbers led Smith to the +study of elliptic functions. He wrote also on modern geometry. +\index{Elliptic functions}% +His successor at Oxford was J.~J. Sylvester. +\index{Sylvester!ref.\ to}% + +\textbf{Ernst Eduard Kummer} (1810--1893), professor in the University +\index{Kummer}% +of Berlin, is closely identified with the theory of numbers. +Dirichlet's work on complex numbers of the form $a + ib$, +introduced by Gauss, was extended by him, by Eisenstein, +and Dedekind. Instead of the equation $x^4 - 1 = 0$, the roots +of which yield Gauss' units, Eisenstein used the equation +%% -----File: 385.png---Folio 371------- +$x^3 - 1 = 0$ and complex numbers $a + b\rho$ ($\rho$~being a cube root +of unity), the theory of which resembles that of Gauss' numbers. +Kummer passed to the general case $x^n - 1 = 0$ and got +\index{Kummer}% +complex numbers of the form $\alpha = a_1A_1 + a_2A_2 + a_3A_3 + \dotsb$, +where $a_i$~are whole real numbers, and $A_i$~roots of the above +equation.\Reference{59} Euclid's theory of the greatest common divisor +is not applicable to such complex numbers, and their prime +factors cannot be defined in the same way as prime factors of +common integers are defined. In the effort to overcome this +difficulty, Kummer was led to introduce the conception of +``ideal numbers.'' These ideal numbers have been applied by +\index{Ideal numbers}% +G.~Zolotareff of St.~Petersburg to the solution of a problem +\index{Zolotareff}% +of the integral calculus, left unfinished by Abel (\textit{Liouville's +\index{Abel!ref.\ to}% +\index{Integral calculus}% +Journal}, Second Series, 1864, Vol.~IX.). \textbf{Julius Wilhelm Richard +Dedekind} of Braunschweig (born 1831) has given in the second +\index{Dedekind}% +edition of Dirichlet's \textit{Vorlesungen über Zahlentheorie} a new +\index{Dirichlet!ref.\ to}% +theory of complex numbers, in which he to some extent +deviates from the course of Kummer, and avoids the use of +ideal numbers. Dedekind has taken the roots of any irreducible +equation with integral coefficients as the units for his complex +numbers. Attracted by Kummer's investigations, his +pupil, \textbf{Leopold Kronecker} (1823--1891) made researches which +he applied to algebraic equations. + +On the other hand, efforts have been made to utilise in the +theory of numbers the results of the modern higher algebra. +Following up researches of Hermite, \textbf{Paul Bachmann} of Münster +\index{Bachmann}% +investigated the arithmetical formula which gives the automorphics +of a ternary quadratic form.\Reference{89} The problem of the +equivalence of two positive or definite ternary quadratic forms +was solved by L.~Seeber; and that of the arithmetical automorphics +\index{Seeber}% +of such forms, by Eisenstein. The more difficult problem +\index{Eisenstein!ref.\ to}% +of the equivalence for indefinite ternary forms has been +investigated by Edward Selling of Würzburg. On quadratic +\index{Selling}% +%% -----File: 386.png---Folio 372------- +\index{Negative quantities}% +\index{Partition of numbers}% +\index{Theory of numbers|)}% +forms of four or more indeterminates little has yet been done. +Hermite showed that the number of non-equivalent classes of +\index{Hermite!ref.\ to}% +quadratic forms having integral coefficients and a given discriminant +is finite, while Zolotareff and A.~N. Korkine, both +\index{Korkine}% +\index{Zolotareff!ref.\ to}% +of St.~Petersburg, investigated the minima of positive quadratic +forms. In connection with binary quadratic forms, Smith +\index{Smith, H.!ref.\ to}% +established the theorem that if the joint invariant of two +properly primitive forms vanishes, the determinant of either +of them is represented primitively by the duplicate of the +other. + +The interchange of theorems between arithmetic and algebra +is displayed in the recent researches of J.~W.~L. Glaisher +\index{Glaisher}% +of Trinity College (born 1848) and Sylvester. Sylvester gave +a Constructive Theory of Partitions, which received additions +from his pupils, F.~Franklin and G.~S. Ely. +\index{Ely}% +\index{Franklin}% + +The conception of ``number'' has been much extended in +our time. With the Greeks it included only the ordinary +positive whole numbers; Diophantus added rational fractions +\index{Diophantus!ref.\ to}% +\index{Numbers!definitions of numbers}% +to the domain of numbers. Later negative numbers and +imaginaries came gradually to be recognised. Descartes fully +grasped the notion of the negative; Gauss, that of the imaginary. +\index{Imaginary quantities}% +With Euclid, a ratio, whether rational or irrational, was +not a number. The recognition of ratios and irrationals as +\index{Irrationals}% +\index{Ratios}% +numbers took place in the sixteenth century, and found expression +with Newton. By the ratio method, the continuity of the +\index{Continuity}% +\index{Newton!ref.\ to}% +real number system has been based on the continuity of space, +but in recent time three theories of irrationals have been +advanced by Weierstrass, J.~W.~R. Dedekind, G.~Cantor, and +\index{Cantor, G.}% +\index{Dedekind!ref.\ to}% +\index{Weierstrass!ref.\ to}% +Heine, which prove the continuity of numbers without borrowing +\index{Heine!ref.\ to}% +it from space. They are based on the definition of numbers +\index{Numbers!theory of numbers|)}% +by regular sequences, the use of series and limits, and some +new mathematical conceptions. +%% -----File: 387.png---Folio 373------- +\index{Astronomy!more recent researches|(}% + + +\Section{Applied Mathematics} +\index{Applied mathematics|(}% + +Notwithstanding the beautiful developments of celestial +mechanics reached by Laplace at the close of the eighteenth +\index{Laplace!ref.\ to}% +century, there was made a discovery on the first day of the +present century which presented a problem seemingly beyond +the power of that analysis. We refer to the discovery of Ceres +by Piazzi in Italy, which became known in Germany just after +\index{Piazzi}% +the philosopher Hegel had published a dissertation proving \textit{a~priori} +\index{Hegel}% +that such a discovery could not be made. From the +positions of the planet observed by Piazzi its orbit could not +be satisfactorily calculated by the old methods, and it remained +for the genius of Gauss to devise a method of calculating +\index{Gauss!ref.\ to}% +elliptic orbits which was free from the assumption of a small +eccentricity and inclination. Gauss' method was developed +further in his \textit{Theoria Motus}. The new planet was re-discovered +with aid of Gauss' data by Olbers, an astronomer +\index{Olbers}% +who promoted science not only by his own astronomical +studies, but also by discerning and directing towards astronomical +pursuits the genius of Bessel. +\index{Bessel|(}% + +\textbf{Friedrich Wilhelm Bessel}\Reference[]{91} (1784--1846) was a native of +Minden in Westphalia. Fondness for figures, and a distaste +for Latin grammar led him to the choice of a mercantile +career. In his fifteenth year he became an apprenticed clerk +in Bremen, and for nearly seven years he devoted his days to +mastering the details of his business, and part of his nights to +study. Hoping some day to become a supercargo on trading +expeditions, he became interested in observations at sea. With +a sextant constructed by him and an ordinary clock he determined +the latitude of Bremen. His success in this inspired +him for astronomical study. One work after another was +mastered by him, unaided, during the hours snatched from +%% -----File: 388.png---Folio 374------- +sleep. From old observations he calculated the orbit of +Halley's comet. Bessel introduced himself to Olbers, and +\index{Halley's Comet}% +submitted to him the calculation, which Olbers immediately +sent for publication. Encouraged by Olbers, Bessel turned +his back to the prospect of affluence, chose poverty and the +stars, and became assistant in J.~H. Schröter's observatory at +\index{SchroterJH@{Schröter, J.~H.}}% +Lilienthal. Four years later he was chosen to superintend +the construction of the new observatory at Königsberg.\Reference{92} In +the absence of an adequate mathematical teaching force, Bessel +was obliged to lecture on mathematics to prepare students for +astronomy. He was relieved of this work in 1825 by the +arrival of Jacobi. We shall not recount the labours by which +\index{Jacobi!ref.\ to}% +Bessel earned the title of founder of modern practical astronomy +and geodesy. As an observer he towered far above +Gauss, but as a mathematician he reverently bowed before the +genius of his great contemporary. Of Bessel's papers, the one +of greatest mathematical interest is an ``\textit{Untersuchung des +Theils der planetarischen Störungen, welcher aus der Bewegung +der Sonne ensteht}'' (1824), in which he introduces a class of +transcendental functions, $J_n(x)$, much used in applied mathematics, +and known as ``Bessel's functions.'' He gave their +\index{Bessel's functions}% +principal properties, and constructed tables for their evaluation. +Recently it has been observed that Bessel's functions +appear much earlier in mathematical literature.\Reference{98} +Such functions of the zero order occur in papers of Daniel +Bernoulli (1732) and Euler on vibration of heavy strings suspended +\index{Euler!ref.\ to}% +from one end. All of Bessel's functions of the first +kind and of integral orders occur in a paper by Euler (1764) on +the vibration of a stretched elastic membrane. In 1878 Lord +Rayleigh proved that Bessel's functions are merely particular +\index{Rayleigh, Lord!ref.\ to}% +cases of Laplace's functions. J.~W.~L. Glaisher illustrates +\index{Glaisher!ref.\ to}% +by Bessel's functions his assertion that mathematical branches +growing out of physical inquiries as a rule ``lack the easy flow +%% -----File: 389.png---Folio 375------- +\index{Bessel|)}% +or homogeneity of form which is characteristic of a mathematical +theory properly so called.'' These functions have been +studied by C.~Th.\ Anger of Danzig, O.~Schlömilch of Dresden, +\index{Anger}% +\index{Schlomilch@{Schlömilch}}% +R.~Lipschitz of Bonn (born 1832), Carl Neumann of Leipzig +\index{Lipschitz!ref.\ to}% +\index{Neumann, C.}% +(born 1832), Eugen Lommel of Leipzig, I.~Todhunter of St.~John's +\index{Lommel}% +\index{Todhunter!ref.\ to}% +College, Cambridge. + +Prominent among the successors of Laplace are the following: +\index{Laplace!ref.\ to}% +\textit{Siméon Denis Poisson} (1781--1840), who wrote in 1808 +\index{Poisson!ref.\ to}% +a classic \textit{Mémoire sur les inégalités séculaires des moyens mouvements +des planètes}. \textit{Giovanni Antonio Amadeo Plana} (1781--1864) +\index{Plana}% +of Turin, a nephew of Lagrange, who published in 1811 +a \Ita{\textit{Memoria sulla teoria dell'\DPnote{** PP: [sic] space} attrazione degli sferoidi ellitici}}, and +contributed to the theory of the moon. \textbf{Peter Andreas Hansen} +\index{Hansen}% +(1795--1874) of Gotha, at one time a clockmaker in Tondern, +then Schumacher's assistant at Altona, and finally director of +the observatory at Gotha, wrote on various astronomical subjects, +but mainly on the lunar theory, which he elaborated in +his work \textit{Fundamenta nova investigationes orbitæ veræ quam +Luna perlustrat} (1838), and in subsequent investigations +embracing extensive lunar tables. \textbf{George Biddel Airy} (1801--1892), +\index{Airy}% +royal astronomer at Greenwich, published in 1826 his +\textit{Mathematical Tracts on the Lunar and Planetary Theories}. +These researches have since been greatly extended by him. +\textbf{August Ferdinand Möbius} (1790--1868) of Leipzig wrote, in 1842, +\index{Mobius@{Möbius}!ref.\ to}% +\textit{Elemente der Mechanik des Himmels}. \textbf{Urbain Jean Joseph Le +Verrier} (1811--1877) of Paris wrote\DPtypo{,}{} the \textit{Recherches Astronomiques}, +\index{Leverrier@{Le Verrier}}% +constituting in part a new elaboration of celestial +mechanics, and is famous for his theoretical discovery of +Neptune. \textbf{John Couch Adams} (1819--1892) of Cambridge +\index{Adams}% +\index{Neptune, discovery of}% +divided with Le~Verrier the honour of the mathematical discovery +of Neptune, and pointed out in 1853 that Laplace's +explanation of the secular acceleration of the moon's mean +motion accounted for only half the observed acceleration. +%% -----File: 390.png---Folio 376------- +\textbf{Charles Eugène Delaunay} (born 1816, and drowned off Cherbourg +\index{Delaunay}% +in 1872), professor of mechanics at the Sorbonne in +Paris, explained most of the remaining acceleration of the +moon, unaccounted for by Laplace's theory as corrected by +\index{Laplace!ref.\ to}% +Adams, by tracing the effect of tidal friction, a theory +previously suggested independently by Kant, Robert Mayer, +\index{Kant}% +\index{Mayer!ref.\ to}% +and William Ferrel of Kentucky. \textbf{George Howard Darwin} of +\index{Darwin}% +\index{Ferrel!ref.\ to}% +Cambridge (born 1845) made some very remarkable investigations +in 1879 on tidal friction, which trace with great +certainty the history of the moon from its origin. He has +since studied also the effects of tidal friction upon other +bodies in the solar system. Criticisms on some parts of his +researches have been made by James Nolan of Victoria. \textbf{Simon +\index{Nolan}% +Newcomb} (born 1835), superintendent of the \textit{Nautical Almanac} +\index{Nautical almanac, United States}% +\index{Newcomb}% +at Washington, and professor of mathematics at the Johns +Hopkins University, investigated the errors in Hansen's tables +of the moon. For the last twelve years the main work of the +\textit{U.~S. Nautical Almanac} office has been to collect and discuss +data for new tables of the planets which will supplant the +tables of Le~Verrier. \textit{G.~W. Hill} of that office has contributed +\index{Hill}% +\index{Leverrier@{Le Verrier}!ref.\ to}% +an elegant paper on certain possible abbreviations in the computation +of the long-period of the moon's motion due to the +direct action of the planets, and has made the most elaborate +determination yet undertaken of the inequalities of the moon's +motion due to the figure of the earth. He has also computed +certain lunar inequalities due to the action of Jupiter. + +The mathematical discussion of Saturn's rings was taken up +\index{Saturn's rings}% +first by Laplace, who demonstrated that a homogeneous solid +ring could not be in equilibrium, and in 1851 by B.~Peirce, +\index{Peirce, B.!ref.\ to}% +who proved their non-solidity by showing that even an irregular +solid ring could not be in equilibrium about Saturn. The +mechanism of these rings was investigated by James Clerk +Maxwell in an essay to which the Adams prize was awarded. +\index{Maxwell!ref.\ to}% +%% -----File: 391.png---Folio 377------- +\index{Algebraic functions!integrals}% +\index{Mechanics!more recent work|(}% +He concluded that they consisted of an aggregate of unconnected +particles. + +The problem of three bodies has been treated in various +\index{Problem of three bodies}% +\index{Three bodies, problem of}% +ways since the time of Lagrange, but no decided advance +towards a more complete algebraic solution has been made, +and the problem stands substantially where it was left by him. +He had made a reduction in the differential equations to the +seventh order. This was elegantly accomplished in a different +way by Jacobi in 1843. \textit{R.~Radau} (\textit{Comptes Rendus}, LXVII., +\index{Jacobi!ref.\ to}% +\index{Radau}% +1868, p.~841) and \textit{Allégret} (\textit{Journal de Mathématiques}, 1875, +\index{Allégret}% +p.~277) showed that the reduction can be performed on the +equations in their original form. Noteworthy transformations +and discussions of the problem have been given by J.~L.~F. +Bertrand, by Émile Bour (1831--1866) of the Polytechnic School +\index{Bertrand}% +\index{Bour}% +in Paris, by Mathieu, Hesse, J.~A. Serret. H.~Bruns of Leipzig +\index{Bruns}% +\index{Hesse!ref.\ to}% +\index{Mathieu!ref.\ to}% +\index{Serret!ref.\ to}% +has shown that no advance in the problem of three or of $n$ +bodies may be expected by algebraic integrals, and that we +must look to the modern theory of functions for a complete +solution (\textit{Acta Math.}, XI., p.~43).\Reference{93} + +Among valuable text-books on mathematical astronomy rank +the following works: \textit{Manual of Spherical and Practical Astronomy} +by \textit{Chauvenet} (1863), \textit{Practical and Spherical Astronomy} +\index{Chauvenet}% +by \textit{Robert Main} of Cambridge, \textit{Theoretical Astronomy} by \textit{James~C. +\index{Astronomy!more recent researches|)}% +\index{Main}% +Watson} of Ann Arbor (1868), \textit{Traité élémentaire de Mécanique +\index{Watson, J.~C.}% +Céleste} of \textit{H.~Resal} of the Polytechnic School in Paris, +\index{Resal}% +\textit{Cours d'Astronomie de l'École Polytechnique} by \textit{Faye}, \textit{Traité +\index{Faye}% +de Mécanique Céleste} by \textit{Tisserand}, \textit{Lehrbuch der Bahnbestimmung} +\index{Tisserand}% +by \textit{T.~Oppolzer}, \textit{Mathematische Theorien der Planetenbewegung} +\index{Oppolzer}% +by \textit{O.~Dziobek}, translated into English by M.~W. +\index{Dziobek}% +Harrington and W.~J. Hussey. +\index{Harrington}% +\index{Hussey}% + +During the present century we have come to recognise the +advantages frequently arising from a geometrical treatment of +mechanical problems. To Poinsot, Chasles, and Möbius we +\index{Chasles!ref.\ to}% +\index{Mobius@{Möbius}!ref.\ to}% +\index{Poinsot!ref.\ to}% +%% -----File: 392.png---Folio 378------- +\index{Dynamics|(}% +\index{Kowalevsky!ref.\ to}% +\index{Variation of arbitrary consonants}% +owe the most important developments made in geometrical +mechanics. \textbf{Louis Poinsot} (1777--1859), a graduate of the +\index{Poinsot}% +Polytechnic School in Paris, and for many years member of +the superior council of public instruction, published in 1804 +his \textit{Éléments de Statique}. This work is remarkable not only +as being the earliest introduction to synthetic mechanics, but +also as containing for the first time the idea of couples, which +was applied by Poinsot in a publication of 1834 to the theory +of rotation. A clear conception of the nature of rotary +motion was conveyed by Poinsot's elegant geometrical representation +by means of an ellipsoid rolling on a certain fixed +\index{Ellipsoid!(attraction of)}% +plane. This construction was extended by Sylvester so as +\index{Sylvester!ref.\ to}% +to measure the rate of rotation of the ellipsoid on the plane. + +A particular class of dynamical problems has recently been +treated geometrically by \textbf{Sir Robert Stawell Ball}, formerly +\index{Ball, Sir~R.~S.}% +astronomer royal of Ireland, now Lowndean Professor of +Astronomy and Geometry at Cambridge. His method is given +in a work entitled \textit{Theory of Screws}, Dublin, 1876, and in +\index{Screws, theory of}% +subsequent articles. Modern geometry is here drawn upon, +as was done also by Clifford in the related subject of Biquaternions. +\index{Biquaternions}% +\index{Clifford!ref.\ to}% +Arthur Buchheim of Manchester (1859--1888), +\index{Buchheim}% +showed that Grassmann's Ausdehnungslehre supplies all the +\index{Ausdehnungslehre}% +\index{Grassmann!ref.\ to}% +necessary materials for a simple calculus of screws in elliptic +space. Horace Lamb applied the theory of screws to the question +\index{Lamb}% +of the steady motion of any solid in a fluid. + +Advances in theoretical mechanics, bearing on the integration +and the alteration in form of dynamical equations, +were made since Lagrange by Poisson, William Rowan Hamilton, +\index{Hamilton, W.~R.!ref.\ to}% +\index{Lagrange!ref.\ to}% +\index{Poisson!ref.\ to}% +Jacobi, Madame Kowalevski, and others. Lagrange had +\index{Jacobi!ref.\ to}% +established the ``Lagrangian form'' of the equations of +motion. He had given a theory of the variation of the +arbitrary constants which, however, turned out to be less +fruitful in results than a theory advanced by Poisson.\Reference{99} Poisson's +%% -----File: 393.png---Folio 379------- +\index{Geodesics}% +\index{Ultimate multiplier, theory of}% +theory of the variation of the arbitrary constants and +the method of integration thereby afforded marked the first +onward step since Lagrange. Then came the researches of +Sir William Rowan Hamilton. His discovery that the integration +\index{Hamilton, W.~R.!ref.\ to}% +of the dynamic differential equations is connected with +the integration of a certain partial differential equation of the +\index{Partial differential equations}% +first order and second degree, grew out of an attempt to deduce, +by the undulatory theory, results in geometrical optics previously +\index{Undulatory theory of light}% +based on the conceptions of the emission theory. The +\textit{Philosophical Transactions} of 1833 and 1834 contain Hamilton's +papers, in which appear the first applications to mechanics +of the principle of varying action and the characteristic +\index{Action!varying}% +\index{Varying action, principle of}% +function, established by him some years previously. The +object which Hamilton proposed to himself is indicated by +the title of his first paper, viz.\ the discovery of a function +by means of which all integral equations can be actually +represented. The new form obtained by him for the equation +of motion is a result of no less importance than that which +was the professed object of the memoir. Hamilton's method +of integration was freed by Jacobi of an unnecessary complication, +\index{Jacobi!ref.\ to}% +and was then applied by him to the determination of a +geodetic line on the general ellipsoid. With aid of elliptic \DPtypo{coordinates}{co-ordinates} +\index{Co-ordinates}% +\index{Elliptic co-ordinates}% +\index{Ellipsoid!(attraction of)}% +Jacobi integrated the partial differential equation +and expressed the equation of the geodetic in form of a +relation between two Abelian integrals. Jacobi applied to +\index{Abelian integrals}% +differential equations of dynamics the theory of the ultimate +multiplier. The differential equations of dynamics are only +one of the classes of differential equations considered by +Jacobi. Dynamic investigations along the lines of Lagrange, +Hamilton, and Jacobi were made by Liouville, A.~Desboves, +\index{Desboves}% +\index{Liouville!ref.\ to}% +Serret, J.~C.~F. Sturm, Ostrogradsky, J.~Bertrand, Donkin, +\index{Bertrand}% +\index{Donkin}% +\index{Ostrogradsky}% +\index{Serret!ref.\ to}% +\index{Sturm, J.~C.~F.!ref.\ to}% +Brioschi, leading up to the development of the theory of a +\index{Brioschi!ref.\ to}% +system of canonical integrals. +%% -----File: 394.png---Folio 380------- +\index{Hydrodynamics}% +\index{Ignoration of co-ordinates}% +\index{Kowalevsky}% + +An important addition to the theory of the motion of a solid +body about a fixed point was made by Madame \textbf{Sophie de +Kowalevski}\Reference[]{96} (1853--1891), who discovered a new case in which +the differential equations of motion can be integrated. By +the use of theta-functions of two independent variables she +\index{Theta-functions}% +furnished a remarkable example of how the modern theory of +functions may become useful in mechanical problems. She +was a native of Moscow, studied under Weierstrass, obtained +the doctor's degree at Göttingen, and from 1884 until her +death was professor of higher mathematics at the University +of Stockholm. The research above mentioned received the +Bordin prize of the French Academy in 1888, which was +doubled on account of the exceptional merit of the paper. + +There are in vogue three forms for the expression of the +kinetic energy of a dynamical system: the Lagrangian, the +Hamiltonian, and a modified form of Lagrange's equations in +which certain velocities are omitted. The kinetic energy +is expressed in the first form as a homogeneous quadratic +function of the velocities, which are the time-variations of the +co-ordinates of the system; in the second form, as a homogeneous +quadratic function of the momenta of the system; +the third form, elaborated recently by Edward John Routh +\index{Routh}% +of Cambridge, in connection with his theory of ``ignoration of +co-ordinates,'' and by A.~B. Basset, is of importance in hydrodynamical +\index{Basset}% +problems relating to the motion of perforated solids +in a liquid, and in other branches of physics. + +In recent time great practical importance has come to be +attached to the principle of mechanical similitude. By it one +\index{Similitude (mechanical)}% +can determine from the performance of a model the action of +the machine constructed on a larger scale. The principle was +first enunciated by Newton (\textit{Principia}, Bk.~II., Sec.~VIII., +\index{Newton!ref.\ to}% +Prop.~32), and was derived by Bertrand from the principle +\index{Bertrand}% +of virtual velocities. A corollary to it, applied in ship-building, +%% -----File: 395.png---Folio 381------- +\index{Hydrodynamics|(}% +goes by the name of William Froude's law, but was enunciated +\index{Froude}% +also by Reech. +\index{Reech}% + +The present problems of dynamics differ materially from +those of the last century. The explanation of the orbital and +axial motions of the heavenly bodies by the law of universal +gravitation was the great problem solved by Clairaut, Euler, +D'Alembert, Lagrange, and Laplace. It did not involve the +consideration of frictional resistances. In the present time +the aid of dynamics has been invoked by the physical +sciences. The problems there arising are often complicated +by the presence of friction. Unlike astronomical problems of +a century ago, they refer to phenomena of matter and motion +that are usually concealed from direct observation. The great +pioneer in such problems is Lord Kelvin. While yet an +\index{Kelvin, Lord!ref.\ to}% +undergraduate at Cambridge, during holidays spent at the +seaside, he entered upon researches of this kind by working +out the theory of spinning tops, which previously had been +only partially explained by Jellet in his \textit{Treatise on the Theory +\index{Jellet!ref.\ to}% +of Friction} (1872), and by Archibald Smith. +\index{Smith, A.}% + +Among standard works on mechanics are \textbf{Jacobi's} \textit{Vorlesungen +\index{Jacobi!ref.\ to}% +über Dynamik}, edited by Clebsch, 1866; \textbf{Kirchhoff's} \textit{Vorlesungen +\index{Clebsch!ref.\ to}% +\index{Kirchhoff!ref.\ to}% +über mathematische Physik}, 1876; \textbf{Benjamin Peirce's} +\index{Peirce, B.!ref.\ to}% +\textit{Analytic Mechanics}, 1855; \textbf{Somoff's} \textit{Theoretische Mechanik}, +\index{Mechanics!more recent work|)}% +\index{Somoff}% +1879; \textbf{Tait and Steele's} \textit{Dynamics of a Particle}, 1856; \textbf{Minchin's} +\index{Minchin}% +\index{Steele}% +\index{Tait}% +\textit{Treatise on Statics}; \textbf{Routh's} \textit{Dynamics of a System of Rigid +\index{Dynamics|)}% +\index{Routh!ref.\ to}% +Bodies}; \textbf{Sturm's} \textit{Cours de Mécanique de l'École Polytechnique}. +\index{Sturm, J.~C.~F.!ref.\ to}% + +The equations which constitute the foundation of the theory +of fluid motion were fully laid down at the time of Lagrange, +but the solutions actually worked out were few and mainly +of the irrotational type. A powerful method of attacking +problems in fluid motion is that of images, introduced in 1843 +\index{Images, theory of}% +by George Gabriel Stokes of Pembroke College, Cambridge. +\index{Stokes}% +It received little attention until Sir William Thomson's discovery +\index{Thomson, Sir William!ref.\ to}% +%% -----File: 396.png---Folio 382------- +\index{Kelvin, Lord!ref.\ to}% +\index{Vortex rings}% +of electrical images, whereupon the theory was extended +by Stokes, Hicks, and Lewis. In 1849, Thomson gave the +\index{Hicks}% +\index{Lewis}% +\index{Stokes!ref.\ to}% +\index{Thomson, J.~J.}% +\index{Thomson, Sir William!ref.\ to}% +maximum and minimum theorem peculiar to hydrodynamics, +which was afterwards extended to dynamical problems in +general. + +A new epoch in the progress of hydrodynamics was created, +in 1856, by Helmholtz, who worked out remarkable properties +\index{Helmholtz!ref.\ to}% +of rotational motion in a homogeneous, incompressible fluid, +devoid of viscosity. He showed that the vortex filaments in +such a medium may possess any number of knottings and twistings, +but are either endless or the ends are in the free surface +of the medium; they are indivisible. These results suggested +to Sir William Thomson the possibility of founding on them a +new form of the atomic theory, according to which every atom +\index{Atomic theory}% +is a vortex ring in a non-frictional ether, and as such must be +absolutely permanent in substance and duration. The vortex-atom +theory is discussed by J.~J. Thomson of Cambridge +(born 1856) in his classical treatise on the \textit{Motion of Vortex +Rings}, to which the Adams Prize was awarded in 1882. +Papers on vortex motion have been published also by Horace +\index{Vortex motion}% +Lamb, Thomas Craig, Henry~A. Rowland, and Charles Chree. +\index{Chree}% +\index{Craig, T.}% +\index{Lamb}% +\index{Rowland}% + +The subject of jets was investigated by Helmholtz, Kirchhoff, +\index{Jets}% +\index{Kirchhoff!ref.\ to}% +Plateau, and Rayleigh; the motion of fluids in a fluid by +\index{Plateau}% +Stokes, Sir W.~Thomson, Köpcke, Greenhill, and Lamb; the +\index{Greenhill}% +\index{Kopcke@{Köpcke}}% +theory of viscous fluids by Navier, Poisson, Saint-Venant, +\index{Navier!ref.\ to}% +\index{Poisson!ref.\ to}% +\index{Saint-Venant!ref.\ to}% +Stokes, O.~E. Meyer, Stefano, Maxwell, Lipschitz, Craig, +\index{Lipschitz!ref.\ to}% +\index{Maxwell!ref.\ to}% +\index{Meyer, O.~E.}% +\index{Stefano}% +Helmholtz, and A.~B. Basset. Viscous fluids present great +\index{Basset}% +difficulties, because the equations of motion have not the same +degree of certainty as in perfect fluids, on account of a deficient +theory of friction, and of the difficulty of connecting +\index{Friction, theory of}% +oblique pressures on a small area with the differentials of the +velocities. + +Waves in liquids have been a favourite subject with English +\index{Waves|(}% +%% -----File: 397.png---Folio 383------- +\index{Kelvin, Lord!ref.\ to}% +\index{Solitary wave}% +mathematicians. The early inquiries of Poisson and +\index{Poisson!ref.\ to}% +Cauchy were directed to the investigation of waves produced +\index{Cauchy!ref.\ to}% +by disturbing causes acting arbitrarily on a small portion +of the fluid. The velocity of the long wave was given +\index{Long wave}% +approximately by Lagrange in 1786 in case of a channel of +\index{Lagrange!ref.\ to}% +rectangular cross-section, by Green in 1839 for a channel of +\index{Green!ref.\ to}% +triangular section, and by P.~Kelland for a channel of any +\index{Kelland}% +uniform section. Sir George B. Airy, in his treatise on \textit{Tides +\index{Airy!ref.\ to}% +\index{Tides}% +and Waves}, discarded mere approximations, and gave the exact +equation on which the theory of the long wave in a channel of +uniform rectangular section depends. But he gave no general +solutions. J.~McCowan of University College at Dundee +\index{McCowan}% +discusses this topic more fully, and arrives at exact and +complete solutions for certain cases. The most important +application of the theory of the long wave is to the explanation +of tidal phenomena in rivers and estuaries. + +The mathematical treatment of solitary waves was first +taken up by S.~Earnshaw in 1845, then by Stokes; but the first +\index{Earnshaw}% +\index{Stokes!ref.\ to}% +sound approximate theory was given by J.~Boussinesq in 1871, +\index{Boussinesq}% +who obtained an equation for their form, and a value for the +velocity in agreement with experiment. Other methods of +approximation were given by Rayleigh and J.~McCowan. In +\index{Rayleigh, Lord!ref.\ to}% +connection with deep-water waves, Osborne Reynolds gave in +\index{Reynolds}% +1877 the dynamical explanation for the fact that a group +of such waves advances with only half the rapidity of the +individual waves. + +The solution of the problem of the general motion of an +ellipsoid in a fluid is due to the successive labours of Green +\index{Ellipsoid!motion of}% +(1833), Clebsch (1856), and Bjerknes (1873). The free +\index{Bjerknes, C. A.}% +\index{Clebsch!ref.\ to}% +motion of a solid in a liquid has been investigated by W.~Thomson, +\index{Thomson, Sir William!ref.\ to}% +Kirchhoff, and Horace Lamb. By these labours, the +\index{Kirchhoff!ref.\ to}% +\index{Lamb}% +motion of a single solid in a fluid has come to be pretty well +understood, but the case of two solids in a fluid is not developed +%% -----File: 398.png---Folio 384------- +\index{Hydrodynamics|)}% +\index{Spheroid (liquid)}% +so fully. The problem has been attacked by W.~M. +Hicks. +\index{Hicks}% + +The determination of the period of oscillation of a rotating +liquid spheroid has important bearings on the question of the +origin of the moon. G.~H. Darwin's investigations thereon, +\index{Darwin!ref.\ to}% +viewed in the light of Riemann's and Poincaré's researches, +\index{Poincaré!ref.\ to}% +\index{Riemann!ref.\ to}% +seem to disprove Laplace's hypothesis that the moon separated +\index{Laplace!ref.\ to}% +from the earth as a ring, because the angular velocity was too +great for stability; Darwin finds no instability. + +The explanation of the contracted vein has been a point of +\index{Contracted vein}% +much controversy, but has been put in a much better light by +the application of the principle of momentum, originated by +Froude and Rayleigh. Rayleigh considered also the reflection +\index{Froude}% +\index{Rayleigh, Lord!ref.\ to}% +of waves, not at the surface of separation of two uniform +media, where the transition is abrupt, but at the confines of +two media between which the transition is gradual. + +The first serious study of the circulation of winds on the +\index{Winds|(}% +earth's surface was instituted at the beginning of the second +quarter of this century by \textit{H.~W. Dové}, \textit{William~C. Redfield}, and +\index{Dové}% +\index{Redfield}% +\textit{James~P. Espy}, followed by researches of \textit{W.~Reid}, \textit{Piddington}, +\index{Espy}% +\index{Piddington}% +\index{Reid}% +and \textit{Elias Loomis}. But the deepest insight into the wonderful +\index{Loomis}% +correlations that exist among the varied motions of the +atmosphere was obtained by \textbf{William Ferrel} (1817--1891). He +\index{Ferrel}% +was born in Fulton County, Pa., and brought up on a farm. +Though in unfavourable surroundings, a burning thirst for +knowledge spurred the boy to the mastery of one branch after +another. He attended Marshall College, Pa., and graduated +in 1844 from Bethany College. While teaching school he +became interested in meteorology and in the subject of tides. +\index{Meteorology|(}% +In 1856 he wrote an article on ``the winds and currents of the +ocean.'' The following year he became connected with the +\textit{Nautical Almanac}. A mathematical paper followed in 1858 +on ``the motion of fluids and solids relative to the earth's +%% -----File: 399.png---Folio 385------- +surface.'' The subject was extended afterwards so as to +embrace the mathematical theory of cyclones, tornadoes, +water-spouts, etc. In 1885 appeared his \textit{Recent Advances in +Meteorology}. In the opinion of a leading European meteorologist +(\textit{Julius Hann} of Vienna), Ferrel has ``contributed more +\index{Hann}% +to the advance of the physics of the atmosphere than any +other living physicist or meteorologist.'' + +Ferrel teaches that the air flows in great spirals toward the +poles, both in the upper strata of the atmosphere and on the +earth's surface beyond the 30th~degree of latitude; while +the return current blows at nearly right angles to the above +spirals, in the middle strata as well as on the earth's surface, +in a zone comprised between the parallels $30°$~N. and $30°$~S. The +idea of three superposed currents blowing spirals was first +advanced by James Thomson, but was published in very +\index{Thomson, J.}% +meagre abstract. + +Ferrel's views have given a strong impulse to theoretical +research in America, Austria, and Germany. Several objections +raised against his argument have been abandoned, or +have been answered by W.~M. Davis of Harvard. The mathematical +\index{Davis, W.~M.}% +analysis of F.~Waldo of Washington, and of others, +\index{Waldo}% +has further confirmed the accuracy of the theory. The transport +of Krakatoa dust and observations made on clouds point +toward the existence of an upper east current on the equator, +and Pernter has mathematically deduced from Ferrel's theory +\index{Pernter, J.~M.}% +the existence of such a current. + +Another theory of the general circulation of the atmosphere +was propounded by Werner Siemens of Berlin, in which an +\index{Siemens}% +attempt is made to apply thermodynamics to aërial currents. +\index{Thermodynamics}% +Important new points of view have been introduced recently +by Helmholtz, who concludes that when two air currents blow +\index{Helmholtz!ref.\ to}% +one above the other in different directions, a system of air +waves must arise in the same way as waves are formed on the +\index{Waves|)}% +%% -----File: 400.png---Folio 386------- +\index{Meteorology|)}% +\index{Winds|)}% +sea. He and A.~Oberbeck showed that when the waves on the +\index{Oberbeck}% +sea attain lengths of from $16$ to $33$~feet, the air waves must +attain lengths of from $10$ to $20$~miles, and proportional depths. +Superposed strata would thus mix more thoroughly, and their +energy would be partly dissipated. From hydrodynamical +equations of rotation Helmholtz established the reason why +\index{Helmholtz!ref.\ to}% +the observed velocity from equatorial regions is much less in +a latitude of, say, $20°$ or~$30°$, than it would be were the movements +unchecked. + +About 1860 acoustics began to be studied with renewed +\index{Acoustics}% +zeal. The mathematical theory of pipes and vibrating strings +had been elaborated in the eighteenth century by Daniel Bernoulli, +\index{Bernoulli, Daniel!ref.\ to}% +D'Alembert, Euler, and Lagrange. In the first part of +\index{D'Alembert!ref.\ to}% +\index{Euler!ref.\ to}% +\index{Lagrange!ref.\ to}% +the present century Laplace corrected Newton's theory on the +\index{Laplace!ref.\ to}% +\index{Newton!ref.\ to}% +velocity of sound in gases, Poisson gave a mathematical discussion +\index{Poisson!ref.\ to}% +of torsional vibrations; Poisson, Sophie Germain, and +\index{Germain, Sophie!ref.\ to}% +Wheatstone studied Chladni's figures; Thomas Young and the +\index{Chladni's figures}% +\index{Wheatstone}% +\index{Young!ref.\ to}% +brothers Weber developed the wave-theory of sound. \textbf{Sir J.~F.~W. +Herschel} wrote on the mathematical theory of sound for +\index{Herschel, J.~F.~W.}% +the \textit{Encyclopædia Metropolitana}, 1845. Epoch-making were +Helmholtz's experimental and mathematical researches. In +his hands and Rayleigh's, Fourier's series received due +\index{Fourier's series}% +\index{Rayleigh, Lord}% +attention. Helmholtz gave the mathematical theory of beats, +difference tones, and summation tones. \textbf{Lord Rayleigh} (John +William Strutt) of Cambridge (born 1842) made extensive +\index{Strutt, J.~W.}% +mathematical researches in acoustics as a part of the theory of +vibration in general. Particular mention may be made of his +discussion of the disturbance produced by a spherical obstacle +on the waves of sound, and of phenomena, such as sensitive +flames, connected with the instability of jets of fluid. In 1877 +\index{Jets}% +and 1878 he published in two volumes a treatise on \textit{The Theory +of Sound}. Other mathematical researches on this subject have +been made in England by Donkin and Stokes. +\index{Stokes!ref.\ to}% +%% -----File: 401.png---Folio 387------- + +The theory of elasticity\Reference[]{42} belongs to this century. Before +\index{Elasticity|(}% +1800 no attempt had been made to form general equations for +the motion or equilibrium of an elastic solid. Particular problems +had been solved by special hypotheses. Thus, James +Bernoulli considered elastic laminæ; Daniel Bernoulli and +\index{Bernoulli, James (born 1758)}% +Euler investigated vibrating rods; Lagrange and Euler, the +\index{Euler!ref.\ to}% +\index{Lagrange!ref.\ to}% +\index{Vibrating rods}% +equilibrium of springs and columns. The earliest investigations +of this century, by Thomas Young (``Young's modulus of +\index{Young!ref.\ to}% +elasticity'') in England, J.~Binet in France, and G.~A.~A. Plana +\index{Binet}% +\index{Plana}% +in Italy, were chiefly occupied in extending and correcting the +earlier labours. Between 1830 and 1840 the broad outline of the +modern theory of elasticity was established. This was accomplished +almost exclusively by French writers,---Louis-Marie-Henri +Navier (1785--1836), Poisson, Cauchy, Mademoiselle +\index{Cauchy!ref.\ to}% +\index{Navier}% +\index{Poisson}% +\index{Poisson!ref.\ to}% +Sophie Germain (1776--1831), Félix Savart (1791--1841). +\index{Germain, Sophie}% +\index{Savart}% + +\textbf{Siméon Denis Poisson}\Reference[]{94} (1781--1840) was born at Pithiviers. +The boy was put out to a nurse, and he used to tell that when +his father (a common soldier) came to see him one day, the +nurse had gone out and left him suspended by a thin cord to a +nail in the wall in order to protect him from perishing under +the teeth of the carnivorous and unclean animals that roamed +on the floor. Poisson used to add that his gymnastic efforts +when thus suspended caused him to swing back and forth, and +thus to gain an early familiarity with the pendulum, the study +of which occupied him much in his maturer life. His father +destined him for the medical profession, but so repugnant was +this to him that he was permitted to enter the Polytechnic +School at the age of seventeen. His talents excited the interest +of Lagrange and Laplace. At eighteen he wrote a memoir +on finite differences which was printed on the recommendation +of Legendre. He soon became a lecturer at the school, and +continued through life to hold various government scientific +posts and professorships. He prepared some 400 publications, +%% -----File: 402.png---Folio 388------- +\index{Capillarity}% +\index{Kelvin, Lord!ref.\ to}% +mainly on applied mathematics. His \textit{Traité de Mécanique}, +2~vols., 1811 and 1833, was long a standard work. He wrote +on the mathematical theory of heat, capillary action, probability +of judgment, the mathematical theory of electricity and +magnetism, physical astronomy, the attraction of ellipsoids, +definite integrals, series, and the theory of elasticity. He was +considered one of the leading analysts of his time. + +His work on elasticity is hardly excelled by that of Cauchy, +\index{Cauchy!ref.\ to}% +and second only to that of Saint-Venant. There is hardly a +problem in elasticity to which he has not contributed, while +many of his inquiries were new. The equilibrium and motion +of a circular plate was first successfully treated by him. +Instead of the definite integrals of earlier writers, he used +preferably finite summations. Poisson's contour conditions +for elastic plates were objected to by Gustav Kirchhoff of +\index{Kirchhoff!ref.\ to}% +Berlin, who established new conditions. But Thomson and +\index{Thomson, Sir William!ref.\ to}% +Tait in their \textit{Treatise on Natural Philosophy} have explained +\index{Tait}% +the discrepancy between Poisson's and Kirchhoff's boundary +conditions, and established a reconciliation between them. + +Important contributions to the theory of elasticity were +made by Cauchy. To him we owe the origin of the theory +of stress, and the transition from the consideration of the +force upon a molecule exerted by its neighbours to the consideration +of the stress upon a small plane at a point. He +anticipated Green and Stokes in giving the equations of isotropic +\index{Green!ref.\ to}% +\index{Stokes!ref.\ to}% +elasticity with two constants. The theory of elasticity +was presented by Gabrio Piola of Italy according to the principles +\index{Piola}% +of Lagrange's \textit{Mécanique Analytique}, but the superiority +of this method over that of Poisson and Cauchy is far from +evident. The influence of temperature on stress was first +investigated experimentally by Wilhelm Weber of Göttingen, +\index{Weber, W.~E.!ref.\ to}% +and afterwards mathematically by Duhamel, who, assuming +\index{Duhamel}% +Poisson's theory of elasticity, examined the alterations of +%% -----File: 403.png---Folio 389------- +form which the formulæ undergo when we allow for changes +of temperature. Weber was also the first to experiment +on elastic after-strain. Other important experiments were +made by different scientists, which disclosed a wider range +of phenomena, and demanded a more comprehensive theory. +Set was investigated by Gerstner (1756--1832) and Eaton +\index{Gerstner}% +Hodgkinson, while the latter physicist in England and Vicat +\index{Hodgkinson}% +\index{Vicat}% +(1786--1861) in France experimented extensively on absolute +strength. Vicat boldly attacked the mathematical theories of +flexure because they failed to consider shear and the time-element. +\index{Flexure, theory of}% +As a result, a truer theory of flexure was soon propounded +by Saint-Venant. Poncelet advanced the theories of +\index{Poncelet!ref.\ to}% +\index{Saint-Venant!ref.\ to}% +resilience and cohesion. + +\textbf{Gabriel Lamé}\Reference[]{94} (1795--1870) was born at Tours, and graduated +\index{Lame@{Lamé}}% +\index{Lame@{Lamé}!ref.\ to}% +at the Polytechnic School. He was called to Russia +with Clapeyron and others to superintend the construction of +bridges and roads. On his return, in 1832, he was elected +professor of physics at the Polytechnic School. Subsequently +he held various engineering posts and professorships in Paris. +As engineer he took an active part in the construction of the +first railroads in France. Lamé devoted his fine mathematical +talents mainly to mathematical physics. In four works: +\textit{Leçons sur les fonctions inverses des transcendantes et les surfaces +isothermes}; \textit{Sur les coordonnées curvilignes et leurs diverses +applications}; \textit{Sur la théorie analytique de la chaleur}; \textit{Sur la +théorie mathématique de l'élasticité des corps solides} (1852), and +in various memoirs he displays fine analytical powers; but a +certain want of physical touch sometimes reduces the value of +his contributions to elasticity and other physical subjects. In +considering the temperature in the interior of an ellipsoid +under certain conditions, he employed functions analogous to +Laplace's functions, and known by the name of ``Lamé's functions.'' +\index{Lame's@{Lamé's functions}}% +A problem in elasticity called by Lamé's name, viz.\ +%% -----File: 404.png---Folio 390------- +\index{Jellet!ref.\ to}% +to investigate the conditions for equilibrium of a spherical +elastic envelope subject to a given distribution of load on +the bounding spherical surfaces, and the determination of the +resulting shifts is the only completely general problem on +elasticity which can be said to be completely solved. He +deserves much credit for his derivation and transformation +of the general elastic equations, and for his application of +them to double refraction. Rectangular and triangular membranes +were shown by him to be connected with questions in +the theory of numbers. The field of photo-elasticity was +entered upon by Lamé, F.~E. Neumann, Clerk Maxwell. +\index{Maxwell!ref.\ to}% +\index{Neumann, F.~E.!ref.\ to}% +Stokes, Wertheim, R.~Clausius, Jellett, threw new light upon +\index{Clausius!ref.\ to}% +\index{Light, theory of}% +\index{Stokes!ref.\ to}% +\index{Wertheim}% +the subject of ``rari-constancy'' and ``multi-constancy,'' which +\index{Multi-constancy}% +\index{Rari-constancy}% +has long divided elasticians into two opposing factions. The +uni-constant isotropy of Navier and Poisson had been questioned +\index{Navier!ref.\ to}% +\index{Poisson!ref.\ to}% +by Cauchy, and was now severely criticised by Green +\index{Cauchy!ref.\ to}% +\index{Green!ref.\ to}% +and Stokes. + +\textbf{Barré de Saint-Venant} (1797--1886), ingénieur des ponts et +\index{Saint-Venant}% +chaussées, made it his life-work to render the theory of +elasticity of practical value. The charge brought by practical +engineers, like Vicat, against the theorists led Saint-Venant to +\index{Vicat!ref.\ to}% +place the theory in its true place as a guide to the practical +man. Numerous errors committed by his predecessors were +removed. He corrected the theory of flexure by the consideration +of slide, the theory of elastic rods of double curvature by +the introduction of the third moment, and the theory of torsion +by the discovery of the distortion of the primitively +plane section. His results on torsion abound in beautiful +graphic illustrations. In case of a rod, upon the side surfaces +of which no forces act, he showed that the problems of flexure +and torsion can be solved, if the end-forces are distributed +over the end-surfaces by a definite law. Clebsch, in his +\index{Clebsch!ref.\ to|(}% +\textit{Lehrbuch der Elasticität}, 1862, showed that this problem is +%% -----File: 405.png---Folio 391------- +reversible to the case of side-forces without end-forces. +Clebsch\Reference[]{68} extended the research to very thin rods and to very +thin plates. Saint-Venant considered problems arising in the +scientific design of built-up artillery, and his solution of them +differs considerably from Lamé's solution, which was popularised +by Rankine, and much used by gun-designers. In Saint-Venant's +translation into French of Clebsch's \textit{Elasticität}, he +develops extensively a double-suffix notation for strain and +stresses. Though often advantageous, this notation is cumbrous, +and has not been generally adopted. \textit{Karl Pearson}, +\index{Pearson}% +professor in University College, London, has recently examined +mathematically the permissible limits of the application +of the ordinary theory of flexure of a beam. + +The mathematical theory of elasticity is still in an unsettled +condition. Not only are scientists still divided into two +schools of ``rari-constancy'' and ``multi-constancy,'' but difference +\index{Multi-constancy}% +of opinion exists on other vital questions. Among the +numerous modern writers on elasticity may be mentioned +Émile Mathieu (1835--1891), professor at Besançon, Maurice +\index{Mathieu}% +Levy of Paris, Charles Chree, superintendent of the Kew Observatory, +\index{Chree}% +\index{Levy}% +A.~B. Basset, Sir William Thomson (Lord Kelvin) +\index{Kelvin, Lord!ref.\ to}% +\index{Thomson, Sir William!ref.\ to}% +of Glasgow, J.~Boussinesq of Paris, and others. Sir William +\index{Boussinesq}% +Thomson applied the laws of elasticity of solids to the investigation +of the earth's elasticity, which is an important element +\index{Earth!rigidity of}% +in the theory of ocean-tides. If the earth is a solid, then its +elasticity co-operates with gravity in opposing deformation +due to the attraction of the sun and moon. Laplace had +shown how the earth would behave if it resisted deformation +only by gravity. Lamé had investigated how a solid sphere +would change if its elasticity only came into play. Sir +William Thomson combined the two results, and compared +them with the actual deformation. Thomson, and afterwards +G.~H. Darwin, computed that the resistance of the earth to +\index{Darwin!ref.\ to}% +%% -----File: 406.png---Folio 392------- +tidal deformation is nearly as great as though it were of steel. +This conclusion has been confirmed recently by Simon Newcomb, +\index{Newcomb!ref.\ to}% +from the study of the observed periodic changes in +latitude. For an ideally rigid earth the period would be $360$~days, +\index{Latitude, periodic changes in}% +but if as rigid as steel, it would be~$441$, the observed +period being $430$~days. + +Among text-books on elasticity may be mentioned the works +\index{Elasticity|)}% +of Lamé, Clebsch, Winkler, Beer, Mathieu, W.~J. Ibbetson, and +\index{Beer}% +\index{Clebsch!ref.\ to|)}% +\index{Ibbetson}% +\index{Lame@{Lamé}!ref.\ to}% +\index{Mathieu!ref.\ to}% +\index{Winkler}% +F.~Neumann, edited by O.~E. Meyer. +\index{Meyer, O.~E.}% +\index{Neumann, F.~E.!ref.\ to}% + +Riemann's opinion that a \emph{science} of physics only exists since +\index{Riemann!ref.\ to}% +the invention of differential equations finds corroboration even +in this brief and fragmentary outline of the progress of mathematical +physics. The undulatory theory of light, first advanced +\index{Undulatory theory of light|(}% +by Huygens, owes much to the power of mathematics: +\index{Huygens!ref.\ to}% +by mathematical analysis its assumptions were worked out +to their last consequences. \textbf{Thomas Young}\Reference[]{95} (1773--1829) was +\index{Young}% +the first to explain the principle of interference, both of +light and sound, and the first to bring forward the idea +of transverse vibrations in light waves. Young's explanations, +not being verified by him by extensive numerical calculations, +attracted little notice, and it was not until \textbf{Augustin +Fresnel} (1788--1827) applied mathematical analysis to a much +\index{Fresnel}% +greater extent than Young had done, that the undulatory +theory began to carry conviction. Some of Fresnel's mathematical +assumptions were not satisfactory; hence Laplace, +\index{Laplace!ref.\ to}% +Poisson, and others belonging to the strictly mathematical +\index{Poisson!ref.\ to}% +school, at first disdained to consider the theory. By their +opposition Fresnel was spurred to greater exertion. Arago +\index{Arago}% +was the first great convert made by Fresnel. When polarisation +and double refraction were explained by Young and +Fresnel, then Laplace was at last won over. Poisson drew +from Fresnel's formulæ the seemingly paradoxical deduction +that a small circular disc, illuminated by a luminous point, +%% -----File: 407.png---Folio 393------- +\index{Kelvin, Lord!ref.\ to}% +\index{MacCullagh!ref.\ to}% +must cast a shadow with a bright spot in the centre. But +this was found to be in accordance with fact. The theory +was taken up by another great mathematician, Hamilton, who +\index{Hamilton, W.~R.!ref.\ to}% +from his formulæ predicted conical refraction, verified experimentally +by Lloyd. These predictions do not prove, however, +\index{Lloyd}% +that Fresnel's formulæ are correct, for these prophecies might +have been made by other forms of the wave-theory. The +theory was placed on a sounder dynamical basis by the writings +of Cauchy, Biot, Green, C.~Neumann, Kirchhoff, McCullagh, +\index{Biot}% +\index{Cauchy!ref.\ to}% +\index{Green!ref.\ to}% +\index{Kirchhoff!ref.\ to}% +\index{McCullagh}% +\index{Neumann, C.!ref.\ to}% +Stokes, Saint-Venant, Sarrau, Lorenz, and Sir William Thomson. +\index{Lorenz}% +\index{Saint-Venant!ref.\ to}% +\index{Sarrau}% +\index{Stokes!ref.\ to}% +\index{Thomson, Sir William!ref.\ to}% +In the wave-theory, as taught by Green and others, the +luminiferous ether was an incompressible elastic solid, for +\index{Ether, luminiferous}% +the reason that fluids could not propagate transverse vibrations. +But, according to Green, such an elastic solid would +transmit a longitudinal disturbance with infinite velocity. +Stokes remarked, however, that the ether might act like +a fluid in case of finite disturbances, and like an elastic solid +in case of the infinitesimal disturbances in light propagation. + +Fresnel postulated the density of ether to be different in +different media, but the elasticity the same, while C.~Neumann +and McCullagh assume the density uniform and the elasticity +different in all substances. On the latter assumption the +direction of vibration lies in the plane of polarisation, and not +perpendicular to it, as in the theory of Fresnel. + +While the above writers endeavoured to explain all optical +properties of a medium on the supposition that they arise +entirely from difference in rigidity or density of the ether in +the medium, there is another school advancing theories in +which the mutual action between the molecules of the body +and the ether is considered the main cause of refraction and +dispersion.\Reference{100} The chief workers in this field are J.~Boussinesq, +\index{Boussinesq}% +W.~Sellmeyer, Helmholtz, E.~Lommel, E.~Ketteler, W.~Voigt, +\index{Helmholtz!ref.\ to}% +\index{Ketteler}% +\index{Lommel}% +\index{Sellmeyer}% +\index{Voigt}% +and Sir William Thomson in his lectures delivered at the +%% -----File: 408.png---Folio 394------- +\index{Kelvin, Lord!ref.\ to}% +\index{Undulatory theory of light|)}% +Johns Hopkins University in 1884. Neither this nor the +first-named school succeeded in explaining all the phenomena. +A third school was founded by Maxwell. He proposed the +\index{Maxwell!ref.\ to}% +electro-magnetic theory, which has received extensive development +\index{Electro-magnetic theory of light}% +recently. It will be mentioned again later. According +to Maxwell's theory, the direction of vibration does not lie +exclusively in the plane of polarisation, nor in a plane perpendicular +to it, but something occurs in both planes---a magnetic +vibration in one, and an electric in the other. Fitzgerald and +\index{Fitzgerald}% +Trouton in Dublin verified this conclusion of Maxwell by +\index{Trouton}% +experiments on electro-magnetic waves. + +Of recent mathematical and experimental contributions to +optics, mention must be made of H.~A. Rowland's theory of +\index{Rowland}% +concave gratings, and of A.~A. Michelson's work on interference, +\index{Michelson}% +and his application of interference methods to astronomical +measurements. + +In electricity the mathematical theory and the measurements +\index{Electricity|(}% +of \textbf{Henry Cavendish} (1731--1810), and in magnetism +\index{Cavendish}% +the measurements of \textbf{Charles Augustin Coulomb} (1736--1806), +\index{Coulomb}% +became the foundations for a system of measurement. For +electro-magnetism the same thing was done by \textbf{Andrè Marie +Ampère} (1775--1836). The first complete method of measurement +\index{Ampère}% +was the system of absolute measurements of terrestrial +magnetism introduced by \textit{Gauss} and \textbf{Wilhelm Weber} (1804--1891) +\index{Weber, W.~E.}% +and afterwards extended by Wilhelm Weber and F.~Kohlrausch +\index{Kohlrausch}% +to electro-magnetism and electro-statics. In 1861 +the British Association and the Royal Society appointed a +special commission with Sir William Thomson at the head, to +\index{Thomson, Sir William!ref.\ to}% +consider the unit of electrical resistance. The commission +recommended a unit in principle like W. Weber's, but greater +than Weber's by a factor of~$10^7$.\Reference{101} The discussions and labours +on this subject continued for twenty years, until in 1881 a +general agreement was reached at an electrical congress in Paris. +%% -----File: 409.png---Folio 395------- +\index{Kelvin, Lord|(}% +\index{Kelvin, Lord!ref.\ to}% + +A function of fundamental importance in the mathematical +theories of electricity and magnetism is the ``potential.'' It +\index{Potential}% +was first used by Lagrange in the determination of gravitational +attractions in 1773. Soon after, Laplace gave the +\index{Laplace!ref.\ to}% +celebrated differential equation, +\[ +\frac{\partial^2V}{\DPtypo{d}{\partial}x^2} + +\frac{\partial^2V}{\DPtypo{d}{\partial}y^2} + +\frac{\partial^2V}{\DPtypo{d}{\partial}z^2} = 0, +\] +which was extended by Poisson by writing $-4\pi k$ in place of +\index{Poisson!ref.\ to}% +zero in the right-hand member of the equation, so that it +applies not only to a point external to the attracting mass, +but to any point whatever. The first to apply the potential +function to other than gravitation problems was \textbf{George Green} +\index{Green}% +\index{Green!ref.\ to}% +(1793--1841). He introduced it into the mathematical theory +of electricity and magnetism. Green was a self-educated man +who started out as a baker, and at his death was fellow of +Caius College, Cambridge. In 1828 he published by subscription +at Nottingham a paper entitled \textit{Essay on the application +of mathematical analysis to the theory of electricity and magnetism}. +It escaped the notice even of English mathematicians +until 1846, when Sir William Thomson had it reprinted in +\index{Thomson, Sir William}% +\index{Thomson, Sir William!ref.\ to}% +\textit{Crelle's Journal}, vols.~xliv.\ and xlv. It contained what is now +known as ``Green's theorem'' for the treatment of potential. +Meanwhile all of Green's general theorems had been re-discovered +by Sir William Thomson, Chasles, Sturm, and Gauss. +The term \emph{potential function} is due to Green. Hamilton used +the word \emph{force-function}, while Gauss, who about 1840 secured +\index{Force-function}% +the general adoption of the function, called it simply \emph{potential}. + +Large contributions to electricity and magnetism have been +made by \textbf{William Thomson}. He was born in 1824 at Belfast, +Ireland, but is of Scotch descent. He and his brother James +studied in Glasgow. From there he entered Cambridge, and +was graduated as Second Wrangler in 1845. William Thomson, +%% -----File: 410.png---Folio 396------- +\index{Addition theorem of elliptic integrals}% +Sylvester, Maxwell, Clifford, and J.~J. Thomson are a group +\index{Clifford!ref.\ to}% +\index{Maxwell!ref.\ to}% +\index{Sylvester!ref.\ to}% +\index{Thomson, J.~J.!ref.\ to}% +\index{Thomson, Sir William}% +of great men who were Second Wranglers at Cambridge. At +the age of twenty-two W.~Thomson was elected professor of +natural philosophy in the University of Glasgow, a position +which he has held ever since. For his brilliant mathematical +and physical achievements he was knighted, and in 1892 was +made Lord Kelvin. His researches on the theory of potential +\index{Kelvin, Lord|)}% +are epoch-making. What is called ``Dirichlet's principle'' +\index{Dirichlet!ref.\ to}% +was discovered by him in 1848, somewhat earlier than by +Dirichlet. We owe to Sir William Thomson new synthetical +methods of great elegance, viz.\ the theory of electric images +and the method of electric inversion founded thereon. By +them he determined the distribution of electricity on a bowl, +a problem previously considered insolvable. The distribution +of static electricity on conductors had been studied before this +mainly by Poisson and Plana. In 1845 F.~E. Neumann of +\index{Neumann, F.~E.!ref.\ to}% +\index{Plana}% +\index{Poisson!ref.\ to}% +Königsberg developed from the experimental laws of Lenz the +mathematical theory of magneto-electric induction. In 1855 +W.~Thomson predicted by mathematical analysis that the discharge +of a Leyden jar through a linear conductor would in +\index{Leyden jar}% +certain cases consist of a series of decaying oscillations. This +was first established experimentally by Joseph Henry of +\index{Henry}% +Washington. William Thomson worked out the electro-static +induction in submarine cables. The subject of the screening +effect against induction, due to sheets of different metals, was +worked out mathematically by Horace Lamb and also by +\index{Lamb}% +Charles Niven. W.~Weber's chief researches were on electro-dynamics. +\index{Niven}% +\index{Weber, W.~E.!ref.\ to}% +Helmholtz in 1851 gave the mathematical theory +\index{Helmholtz!ref.\ to}% +of the course of induced currents in various cases. \textbf{Gustav +Robert Kirchhoff}\Reference[]{97} (1824--1887) investigated the distribution of +\index{Kirchhoff}% +\index{Kirchhoff!ref.\ to}% +a current over a flat conductor, and also the strength of current +in each branch of a network of linear conductors. + +The entire subject of electro-magnetism was revolutionised +%% -----File: 411.png---Folio 397------- +\index{Conservation!of energy}% +\index{Energy, conservation of}% +by \textbf{James Clerk Maxwell} (1831--1879). He was born near +\index{Maxwell}% +Edinburgh, entered the University of Edinburgh, and became +a pupil of Kelland and Forbes. In 1850 he went to Trinity +\index{Forbes}% +\index{Kelland}% +College, Cambridge, and came out Second Wrangler, E.~Routh +\index{Routh!ref.\ to}% +being Senior Wrangler. Maxwell then became lecturer at +Cambridge, in 1856 professor at Aberdeen, and in 1860 +professor at King's College, London. In 1865 he retired to +private life until 1871, when he became professor of physics +at Cambridge. Maxwell not only translated into mathematical +language the experimental results of Faraday, but established +the electro-magnetic theory of light, since verified experimentally +by Hertz. His first researches thereon were published +in 1864. In 1871 appeared his great \textit{Treatise on Electricity +and Magnetism}. He constructed the electro-magnetic theory +from general equations, which are established upon purely +dynamical principles, and which determine the state of the +electric field. It is a mathematical discussion of the stresses +and strains in a dielectric medium subjected to electro-magnetic +forces. The electro-magnetic theory has received developments +from Lord Rayleigh, J.~J. Thomson, H.~A. Rowland, R.~T. +\index{Rayleigh, Lord!ref.\ to}% +\index{Rowland}% +\index{Thomson, J.~J.!ref.\ to}% +Glazebrook, H.~Helmholtz, L.~Boltzmann, O.~Heaviside, J.~H. +\index{Boltzmann}% +\index{Glazebrook}% +\index{Heaviside}% +\index{Helmholtz}% +Poynting, and others. \textbf{Hermann von~Helmholtz} turned his +\index{Poynting}% +attention to this part of the subject in 1871. He was born +in 1821 at Potsdam, studied at the University of Berlin, and +published in 1847 his pamphlet \textit{Ueber die Erhaltung der Kraft}. +He became teacher of anatomy in the Academy of Art in +Berlin. He was elected professor of physiology at Königsberg +in 1849, at Bonn in 1855, at Heidelberg in 1858. It was +at Heidelberg that he produced his work on \textit{Tonempfindung}. +In 1871 he accepted the chair of physics at the University of +Berlin. From this time on he has been engaged chiefly on +inquiries in electricity and hydrodynamics. Helmholtz aimed +to determine in what direction experiments should be made to +%% -----File: 412.png---Folio 398------- +\index{Electricity|)}% +\index{Helmholtz}% +\index{Thermodynamics|(}% +decide between the theories of W.~Weber, F.~E. Neumann, +\index{Neumann, F.~E.}% +\index{Weber, W.~E.!ref.\ to}% +Riemann, and Clausius, who had attempted to explain electro-dynamic +\index{Clausius!ref.\ to}% +\index{Riemann!ref.\ to}% +phenomena by the assumption of forces acting at a distance +between two portions of the hypothetical electrical fluid,---the +intensity being dependent not only on the distance, but also +on the velocity and acceleration,---and the theory of Faraday +\index{Faraday}% +and Maxwell, which discarded action at a distance and assumed +\index{Maxwell!ref.\ to}% +stresses and strains in the dielectric. His experiments favoured +the British theory. He wrote on abnormal dispersion, and +created analogies between electro-dynamics and hydrodynamics. +Lord Rayleigh compared electro-magnetic problems +\index{Rayleigh, Lord!ref.\ to}% +with their mechanical analogues, gave a dynamical theory of +diffraction, and applied Laplace's coefficients to the theory of +\index{Laplace!ref.\ to}% +radiation. Rowland made some emendations on Stokes' paper +\index{Rowland}% +\index{Stokes!ref.\ to}% +on diffraction and considered the propagation of an arbitrary +electro-magnetic disturbance and spherical waves of light. +Electro-magnetic induction has been investigated mathematically +by Oliver Heaviside, and he showed that in a cable it is +\index{Heaviside}% +an actual benefit. Heaviside and Poynting have reached +\index{Poynting}% +remarkable mathematical results in their interpretation and +development of Maxwell's theory. Most of Heaviside's papers +have been published since 1882; they cover a wide field. + +One part of the theory of capillary attraction, left defective +by Laplace, namely, the action of a solid upon a liquid, and +the mutual action between two liquids, was made dynamically +perfect by Gauss. He stated the rule for angles of contact +\index{Gauss!ref.\ to}% +between liquids and solids. A similar rule for liquids was +established by Ernst Franz Neumann. Chief among recent +workers on the mathematical theory of capillarity are Lord +\index{Capillarity}% +Rayleigh and E.~Mathieu. +\index{Mathieu!ref.\ to}% + +The great principle of the conservation of energy was +\index{Conservation!of energy}% +\index{Energy, conservation of}% +established by \textbf{Robert Mayer} (1814--1878), a physician in +\index{Mayer}% +Heilbronn, and again independently by Colding of Copenhagen, +\index{Colding}% +%% -----File: 413.png---Folio 399------- +\index{Kelvin, Lord!ref.\ to}% +Joule, and Helmholtz. \textbf{James Prescott Joule} (1818--1889) +\index{Joule}% +determined experimentally the mechanical equivalent +of heat. Helmholtz in 1847 applied the conceptions of the +\index{Heat, theory of|(}% +transformation and conservation of energy to the various +branches of physics, and thereby linked together many well-known +phenomena. These labours led to the abandonment +of the corpuscular theory of heat. The mathematical treatment +of thermic problems was demanded by practical considerations. +Thermodynamics grew out of the attempt to +determine mathematically how much work can be gotten out +of a steam engine. \textbf{Sadi-Carnot}, an adherent of the corpuscular +\index{Carnot, Sadi}% +theory, gave the first impulse to this. The principle known +by his name was published in 1824. Though the importance +of his work was emphasised by \textit{B.~P.~E. Clapeyron}, it did not +\index{Clapeyron}% +meet with general recognition until it was brought forward +by William Thomson. The latter pointed out the necessity +\index{Thomson, Sir William!ref.\ to}% +of modifying Carnot's reasoning so as to bring it into accord +with the new theory of heat. William Thomson showed in +1848 that Carnot's principle led to the conception of an +absolute scale of temperature. In 1849 he published ``an +account of Carnot's theory of the motive power of heat, with +numerical results deduced from Regnault's experiments.'' In +February, 1850, \textbf{Rudolph Clausius} (1822--1888), then in Zürich +\index{Clausius}% +(afterwards professor in Bonn), communicated to the Berlin +Academy a paper on the same subject which contains the +Protean second law of thermodynamics. In the same month +\textbf{William John~M. Rankine} (1820--1872), professor of engineering +\index{Rankine}% +and mechanics at Glasgow, read before the Royal Society +of Edinburgh a paper in which he declares the nature of +heat to consist in the rotational motion of molecules, and +arrives at some of the results reached previously by Clausius. +He does not mention the second law of thermodynamics, but +in a subsequent paper he declares that it could be derived +%% -----File: 414.png---Folio 400------- +\index{Kelvin, Lord!ref.\ to}% +from equations contained in his first paper. His proof of +the second law is not free from objections. In March, 1851, +appeared a paper of William Thomson which contained a +\index{Thomson, Sir William!ref.\ to}% +perfectly rigorous proof of the second law. He obtained it +before he had seen the researches of Clausius. The statement +\index{Clausius!ref.\ to|(}% +of this law, as given by Clausius, has been much +criticised, particularly by Rankine, Theodor Wand, P.~G. +\index{Rankine!ref.\ to}% +\index{Wand}% +Tait, and Tolver Preston. Repeated efforts to deduce it from +\index{Preston}% +\index{Tait}% +general mechanical principles have remained fruitless. The +science of thermodynamics was developed with great success +by Thomson, Clausius, and Rankine. As early as 1852 +Thomson discovered the law of the dissipation of energy, +\index{Dissipation of energy}% +deduced at a later period also by Clausius. The latter designated +the non-transformable energy by the name \emph{entropy}, +\index{Entropy}% +and then stated that the entropy of the universe tends +toward a maximum. For entropy Rankine used the term +\emph{thermodynamic function}. Thermodynamic investigations have +been carried on also by G.~Ad.\ Hirn of Colmar, and Helmholtz +\index{Helmholtz!ref.\ to}% +\index{Hirn}% +(monocyclic and polycyclic systems). Valuable graphic +methods for the study of thermodynamic relations were devised +in 1873--1878 by \textbf{J.~Willard Gibbs} of Yale College. +\index{Gibbs}% +Gibbs first gives an account of the advantages of using +various pairs of the five fundamental thermodynamic quantities +for graphical representation, then discusses the entropy-temperature +and entropy-volume diagrams, and the volume-energy-entropy +surface (described in Maxwell's \textit{Theory of +\index{Maxwell!ref.\ to}% +Heat}). Gibbs formulated the energy-entropy criterion of +equilibrium and stability, and expressed it in a form applicable +to complicated problems of dissociation. Important +works on thermodynamics have been prepared by Clausius +in 1875, by R.~Rühlmann in 1875, and by Poincaré in 1892. +\index{Poincaré!ref.\ to}% +\index{Ruhlmann@{Rühlmann}}% + +In the study of the law of dissipation of energy and the +principle of least action, mathematics and metaphysics met on +%% -----File: 415.png---Folio 401------- +\index{Action, least}% +\index{Thermodynamics|)}% +common ground. The doctrine of least action was first propounded +\index{Least action}% +by Maupertius in 1744. Two years later he proclaimed +\index{Maupertius}% +it to be a universal law of nature, and the first +scientific proof of the existence of God. It was weakly supported +by him, violently attacked by König of Leipzig, and +\index{Konig@{König}}% +keenly defended by Euler. Lagrange's conception of the principle +\index{Euler!ref.\ to}% +\index{Lagrange!ref.\ to}% +of least action became the mother of analytic mechanics, +\index{Mechanics!more recent work}% +but his statement of it was inaccurate, as has been remarked +by Josef Bertrand in the third edition of the \textit{Mécanique Analytique}. +\index{Bertrand}% +The form of the principle of least action, as it now +exists, was given by Hamilton, and was extended to electro-dynamics +\index{Hamilton, W.~R.!ref.\ to}% +by F.~E. Neumann, Clausius, Maxwell, and Helmholtz. +\index{Helmholtz!ref.\ to}% +\index{Maxwell!ref.\ to}% +\index{Neumann, F.~E.!ref.\ to}% +To subordinate the principle to all reversible processes, +Helmholtz introduced into it the conception of the ``kinetic +potential.'' In this form the principle has universal validity. + +An offshoot of the mechanical theory of heat is the modern +kinetic theory of gases, developed mathematically by \textit{Clausius}, +\index{Gases!Kinetic theory of|(}% +\index{Kinetic theory of gases|(}% +\textit{Maxwell}, \textit{Ludwig Boltzmann} of Munich, and others. The first +suggestions of a kinetic theory of matter go back as far as the +time of the Greeks. The earliest work to be mentioned here is +that of Daniel Bernoulli, 1738. He attributed to gas-molecules +\index{Bernoulli, Daniel!ref.\ to}% +great velocity, explained the pressure of a gas by molecular +bombardment, and deduced Boyle's law as a consequence of +\index{Boyle's law}% +his assumptions. Over a century later his ideas were taken +up by Joule (in~1846), A.~K. Krönig (in~1856), and Clausius +\index{Joule!ref.\ to}% +\index{Krönig}% +(in~1857). Joule dropped his speculations on this subject +when he began his experimental work on heat. Krönig +\index{Heat, theory of|)}% +explained by the kinetic theory the fact determined experimentally +by Joule that the internal energy of a gas is not +altered by expansion when no external work is done. Clausius +took an important step in supposing that molecules may have +rotary motion, and that atoms in a molecule may move relatively +to each other. He assumed that the force acting +%% -----File: 416.png---Folio 402------- +between molecules is a function of their distances, that temperature +depends solely upon the kinetic energy of molecular +motions, and that the number of molecules which at any +moment are so near to each other that they perceptibly influence +each other is comparatively so small that it may be +neglected. He calculated the average velocities of molecules, +and explained evaporation. Objections to his theory, raised +by Buy's-Ballot and by Jochmann, were satisfactorily answered +\index{Buy's-Ballot}% +\index{Jochmann}% +by Clausius and Maxwell, except in one case where an additional +\index{Maxwell!ref.\ to}% +hypothesis had to be made. Maxwell proposed to himself +the problem to determine the average number of molecules, +the velocities of which lie between given limits. His expression +therefor constitutes the important law of distribution of +velocities named after him. By this law the distribution of +molecules according to their velocities is determined by the +same formula (given in the theory of probability) as the distribution +of empirical observations according to the magnitude +of their errors. The average molecular velocity as deduced +by Maxwell differs from that of Clausius by a constant factor. +\index{Clausius!ref.\ to|)}% +Maxwell's first deduction of this average from his law of distribution +was not rigorous. A sound derivation was given by +O.~E. Meyer in 1866. Maxwell predicted that so long as +\index{Meyer, O.~E.}% +Boyle's law is true, the coefficient of viscosity and the coefficient +of thermal conductivity remain independent of the pressure. +His deduction that the coefficient of viscosity should +be proportional to the square root of the absolute temperature +appeared to be at variance with results obtained from pendulum +experiments. This induced him to alter the very foundation +of his kinetic theory of gases by assuming between +the molecules a repelling force varying inversely as the fifth +power of their distances. The founders of the kinetic theory +had assumed the molecules of a gas to be hard elastic spheres; +but Maxwell, in his second presentation of the theory in 1866, +%% -----File: 417.png---Folio 403------- +\index{Applied mathematics|)}% +\index{Thomson, Sir William!ref.\ to}% +went on the assumption that the molecules behave like centres +of forces. He demonstrated anew the law of distribution +of velocities; but the proof had a flaw in argument, pointed +out by Boltzmann, and recognised by Maxwell, who adopted +\index{Boltzmann}% +\index{Maxwell!ref.\ to}% +a somewhat different form of the distributive function in a +paper of 1879, intended to explain mathematically the effects +observed in Crookes' radiometer. Boltzmann gave a rigorous +\index{Radiometer}% +general proof of Maxwell's law of the distribution of velocities. + +None of the fundamental assumptions in the kinetic theory +of gases leads by the laws of probability to results in very +close agreement with observation. Boltzmann tried to establish +kinetic theories of gases by assuming the forces between +molecules to act according to different laws from those previously +assumed. Clausius, Maxwell, and their predecessors +took the mutual action of molecules in collision as repulsive, +but Boltzmann assumed that they may be attractive. \DPtypo{Experiment}{Experiments} +of Joule and Lord Kelvin seem to support the latter +\index{Joule!ref.\ to}% +\index{Kelvin, Lord!ref.\ to}% +assumption. + +Among the latest researches on the kinetic theory is Lord +\index{Gases!Kinetic theory of|)}% +\index{Kinetic theory of gases|)}% +Kelvin's disproof of a general theorem of Maxwell and Boltzmann, +asserting that the average kinetic energy of two given +portions of a system must be in the ratio of the number of +degrees of freedom of those portions. +%% -----File: 418.png---Folio 404------- + +\clearpage +\phantomsection +\pdfbookmark[-1]{Back Matter}{Back Matter} + +\NChapter[1]{Addenda} + +\footnotesize +\Addendref{14}. The new \textit{Akhmim papyrus}, written in Greek, is probably the +copy of an older papyrus, antedating Heron's works, and is the oldest +extant text-book on practical Greek arithmetic. It contains, besides +arithmetical examples, a table for finding ``unit-fractions,'' identical in +scope with that of Ahmes, and, like Ahmes's, without a clue as to its mode +of construction. See \textsc{Biblioth.\ Math.}, 1893, p.~79--89. The papyrus is +edited by J.~Baillet (\textit{Mémoires publiés par les membres de la mission +archéologique française au Caire}, T.~IX., 1\textsuperscript{r}~\textit{fascicule}, Paris, 1892, p.~1--88). + +\Addendref{39}. Chasles's or Simson's definition of a Porism is preferable to +Proclus's, given in the text. See Gow, p.~217--221. + +\Addendref{114}. Nasir Eddin for the first time elaborated trigonometry independently +of astronomy and to such great perfection that, had his work +been known, Europeans of the 15th~century might have spared their +labours. See \textsc{Biblioth.\ Math.}, 1893, p.~6. + +\Addendref{116}. This law of sines was probably known before Gabir ben +Aflah to Tabit ben Korra and others. See \textsc{Biblioth.\ Math.}, 1893, p.~7. + +\Addendref{125}. Athelard was probably not the first to translate Euclid's +\textit{Elements} from the Arabic. See M.~Cantor's \textsc{Vorlesungen}, Vol.~II., +p.~91,~92. + +\Addendref{240}. G.~Eneström argues that Taylor and not Nicole is the real +inventor of finite differences. See \textsc{Biblioth.\ Math.}, 1893, p.~91. + +\Addendref{250}. An earlier publication in which $3.14159\ldots$ is designated by~$\pi$, +is W.~Jones's \textit{Synopsis palmariorum matheseos}, London, 1706, p.~243, +263~\textit{et~seq.} See \textsc{Biblioth.\ Math.}, 1894, p.~106. + +\Addendref{335}. Before Gauss a theorem on convergence, usually attributed +to Cauchy, was given by Maclaurin (\textit{Fluxions}, §~350). A rule of convergence +was deduced also by Stirling. See \textit{Bull.\ N.~Y. Math.\ Soc.}, Vol.~III., +p.~186. + +\Addendref{358}. The surface of a solid with $p$~holes was considered before +Clifford by Tonelli, and was probably used by Riemann himself. See +\textsc{Math.\ Annalen}, Vol.~45, p.~142. + +\Addendref{361}. As early as 1835, Lobachevsky showed in a memoir the +necessity of distinguishing between continuity and differentiability. See +G.~B. Halsted's transl.\ of A.~Vasiliev's \textit{Address} on Lobachevsky, p.~23. + +\textit{Recent deaths.} Johann Rudolf Wolf, Dec.~6, 1893; Heinrich Hertz, +Jan.~1, 1894; Eugène Catalan, Feb.~14, 1894; Hermann von~Helmholtz, +Sept.~8, 1894; Arthur Cayley, Jan.~26, 1895. +%% -----File: 419.png---Folio 405------- + +\normalsize +\cleardoublepage + +\IndexBookmark +\fancyhead[C]{\textsc{INDEX}} +\printindex + + +% **** Need to replace index text + +%% -----File: 437.png---Folio 423------- + +\Catalog + +\CatalogSection{ALGEBRA.} + +\Entry{ALDIS: A Text-book of Algebra.} By W.~S.~\Au{Aldis}. 12mo. \$\Pr{1.90}. + +\Entry{{BALL} \Rm{(W.~W.~R.):} Elementary Algebra.} 16mo. \$\Pr{1.25}. + +\Entry{CHRYSTAL: Algebra.} An Elementary Text-book for the Higher Classes + of Secondary Schools and Colleges. 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Paper. \Pr{75}~cents each. + +\Entry{MURDOCK: Notes on Electricity and Magnetism.} Designed as a companion +to Silvanus~P. Thompson's ``Elementary Lessons in Electricity +and Magnetism.'' \Au{By J.~B. Murdock}, Lieut.\ U.S.N\@. 18mo. New +Edition. \Pr{60}~cents. + +\Entry{POOLE: The Practical Telephone Handbook.} By \Au{Joseph Poole}. With +227~Illustrations. Small crown 8vo. \$\Pr{1.00}. +%% -----File: 457.png---Folio 443------- + +\Entry{PREECE \Rm{and} STUBBS: A Manual of Telephony.} By \Au{Wm.~Henry +Preece} and \Au{Arthur~J. Stubbs}. \$\Pr{4.50}. + +\Entry{RUSSELL: Electric Light Cables and the Distribution of Electricity.} +By \Au{Stuart~A. Russell}, \Degr{A.M.}, \Degr{I.C.E\@.} With over 100~Illustrations. +12mo. \$2.25. + +\Entry{STEWART \Rm{and} GEE: Practical Physics for Schools and Junior Students +of Colleges.} By \Au{Balfour Stewart}, \Degr{M.A.}, \Degr{LL.D.}, \Degr{F.R.S.}, and +\Au{W.~W. Haldane Gee}, \Degr{B.Sc.} + +\Subentry\Ti{Vol.~I\@.} Electricity and Magnetism. 16mo. \Pr{60}~cents. + +\Subentry\textbf{Lessons in Elementary Practical Physics.} By \Au{Balfour Stewart}, +\Degr{M.A.}, \Degr{LL.D.}, \Degr{F.R.S.}, and \Au{W.~W. Haldane Gee}, \Degr{B.Sc.} + +\Subentry\Ti{Vol.~II\@.} Electricity and Magnetism. 12mo. \$\Pr{2.25}. + +\Entry{THOMSON: Notes on Recent Researches in Electricity and Magnetism.} +Intended as a Sequel to Professor \Au{Clerk Maxwell}'s ``Treatise on +Electricity and Magnetism.'' By \Au{J.~J. Thomson}. 8vo. \$\Pr{4.50}. + +\Entry{THOMSON: Reprints of Papers of Electrostatics and Magnetism.} By +Sir \Au{William Thomson}, \Degr{D.C.L.}, \Degr{LL.D.}, \Degr{F.R.S.}, \Degr{F.R.S.E\@.} 8vo. \$\Pr{5.00}. + +\Entry{THOMPSON: Elementary Lessons in Electricity and Magnetism.} By +\Au{Silvanus~P. Thompson}, \Degr{D.Sc.}, \Degr{B.A.}, \Degr{F.R.A.S\@.} New Edition. With +Illustrations. 16mo. \$\Pr{1.25}. + +\Subentry Notes to the same, by \Au{J.~B. Murdock}. \Pr{60}~cents. + +\Entry{WALKER: How to Light a Colliery by Electricity.} 4to. Limp. \Pr{75}~cents. + +\Subentry\textbf{Town Lighting by Electricity.} (\emph{In the Press.}) + +\Entry{WATSON \Rm{and} BURBURY: The Mathematical Theory of Electricity +and Magnetism.} By \Au{H.~W. Watson}, \Degr{D.Sc.}, \Degr{F.R.S.}, and \Au{S.~H. Burbury}, +\Degr{M.A\@.} + +\Subentry\Ti{Vol.~I\@.} Electrostatics. 8vo. \$\Pr{2.75}. + +\settowidth{\TmpLen}{\Ti{Vol.~I\@.}} +\Subentry\makebox[\TmpLen][r]{\Ti{II\@.}} Magnetism and Electrodynamics. 8vo. \$\Pr{2.60}. + + +\CatalogSection{HISTORICAL.} + +\Entry{BALL:} Works by \Au{Walter~W. Rouse Ball}. + +\Subentry\textbf{A Short Account of the History of Mathematics.} 12mo. \$\Pr{2.60}. + +\Subentry\textbf{History of the Study of Mathematics at Cambridge.} 12mo. \$\Pr{1.90}. + +\Entry{BARROW: Mathematical Works.} Edited by \Au{W.~Whewell}. \$\Pr{2.25}. + +\Entry{CAYLEY \Rm{(Arthur):} The Collected Mathematical Papers of.} To be +completed in ten volumes. Vols.~I.--VI. published. \$\Pr{5.50}~each. + +\Entry{GOW: A Short History of Greek Mathematics.} By \Au{J.~Gow}. 8vo. \$\Pr{3.00}. + +\Entry{HEATH: Diophantos of Alexandria.} A Study in the History of Greek +Algebra. By \Au{T.~L. Heath}, \Degr{B.A\@.} 8vo. \$\Pr{2.25}. + +\Entry{KLEIN: Lectures on Mathematics.} \textit{The Evanston Colloquium.} Reported +by \Au{Alexander Ziwet}. \$\Pr{1.50}. + +\Entry{SMITH \Rm{(Henry J. S.):} Mathematical Papers.} With Portrait and Memoir. +2~vols. (\emph{In the Press.}) + +\Entry{WOOLWICH: Mathematical Papers.} Edited by \Au{E.~J. Brooksmith}, \Degr{B.A\@.} +\$\Pr{1.75}. +%% -----File: 458.png---Folio 444------- +%[** PP: Page of special ads; adding hrule] + +\setlength{\Titleskip}{2pt} +\begin{center}\rule{1.5in}{0.5pt}\end{center} + +\begin{center} +{\SizeB \bfseries A SHORT ACCOUNT}\\[\Titleskip] +{\SizeD OF}\\[\Titleskip] +{\SizeA \bfseries\sffamily THE\quad HISTORY\quad OF\quad MATHEMATICS.}\\[\Titleskip] +{\SizeC \Au{By WALTER W. ROUSE BALL},}\\[\Titleskip] +{\SizeE \textsc{Fellow and Tutor of Trinity College, Cambridge.}}\\[\Titleskip] +{\SizeD Second Edition, Revised.\quad \$\Pr{3.25}.} +\end{center} + +\SizeE% +``While technical and exact enough to be of value to the specialist in mathematics as a +handy book of reference, it is so clearly and familiarly written, that it is the best work on this +subject for the general reader that we know of\ldots. From this history, or historical sketch, +the intelligent reader can gain a very complete view of the progress of mathematical science +from its beginnings until its contemporary differentiation into numerous specialties,---each +of them important and difficult enough to detain for a lifetime a brilliant mind,---all of which +are fruitful in their applications to the various phases of modern science and modern +industry.''---\textit{Science}. + +\normalsize +\ifthenelse{\boolean{ForPrinting}}{% + \vfil\clearpage +}{% +\begin{center}\rule{1.5in}{0.5pt}\end{center} +} + +\begin{center} +{\SizeB \bfseries A HISTORY}\\[\Titleskip] +{\SizeD OF}\\[\Titleskip] +\ifthenelse{\boolean{ForPrinting}}{% + \makebox[0pt][c]{\SizeA \bfseries\sffamily THE STUDY OF MATHEMATICS AT CAMBRIDGE.}\\[\Titleskip] +}{% + \makebox[0pt][c]{\SizeB \bfseries\sffamily THE STUDY OF MATHEMATICS AT CAMBRIDGE.}\\[\Titleskip] +} +{\SizeC \Au{By WALTER W. ROUSE BALL},}\\[\Titleskip] +{\SizeD 12mo.\quad \$\Pr{1.90}.} +\end{center} + +\begin{center}\rule{1.5in}{0.5pt}\end{center} + +\begin{center} +{\SizeB \bfseries A SHORT HISTORY}\\[\Titleskip] +{\SizeD OF}\\[\Titleskip] +{\SizeA \bfseries\sffamily GREEK\quad MATHEMATICS.}\\[\Titleskip] +{\SizeC \Au{By JAMES GOW},}\\[\Titleskip] +{\SizeD 8vo.\quad \$\Pr{3.00}.} +\end{center} + +\SizeE +``\ldots Evidently the production of a scholar, and the result of years of laborious research. +Mr.~Gow divides his history into three parts. The first treats of the decimal scale and Egyptian +arithmetic; the second and third parts are concerned with Greek arithmetic and geometry\ldots. +The largest part of Mr.~Gow's history, and that which will probably be the most interesting +to the general mathematical reader, is justly devoted to geometry; for it is in this +department of mathematics that the acuteness of the Greek mind is most conspicuously seen, +and that the continuity of mathematical discovery can be more fully traced\ldots. The interesting +character of the notes is quite a feature of the book, which is in this respect distinguished +from almost all histories of mathematics\ldots. It must be to all students of +mathematics a most welcome and instructive volume.''---\Au{J.~S. Mackay}, in \textit{The Academy}. + +\normalsize +\begin{center}\rule{1.5in}{0.5pt}\end{center} + +\begin{center} +{\SizeA \bfseries\sffamily DIOPHANTOS\quad OF\quad ALEXANDRIA:}\\[\Titleskip] +{\SizeD A STUDY IN THE HISTORY OF}\\[\Titleskip] +{\SizeA \bfseries\sffamily GREEK\quad ALGEBRA.}\\[\Titleskip] +{\SizeC \Au{By T.~S. HEATH}, \Degr{B.A.},}\\[\Titleskip] +{\SizeE \textsc{Scholar of Trinity College, Cambridge.}}\\[\Titleskip] +{\SizeD 8vo.\quad \$\Pr{2.00}.} +\end{center} + +\begin{center}\rule{1.5in}{0.5pt}\end{center} + +\begin{center} +{\Large THE\quad MACMILLAN\quad COMPANY,}\\[\Titleskip] +{\large 66 \ FIFTH \ AVENUE, \ NEW \ YORK.} +\end{center} + + +%%%%%%%%%%%%%%%%%%%%%%%%% GUTENBERG LICENSE %%%%%%%%%%%%%%%%%%%%%%%%%% + +\Licensing + +\begin{PGtext} +End of Project Gutenberg's A History of Mathematics, by Florian Cajori + +*** END OF THIS PROJECT GUTENBERG EBOOK A HISTORY OF MATHEMATICS *** + +***** This file should be named 31061-pdf.pdf or 31061-pdf.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/3/1/0/6/31061/ + +Produced by Andrew D. 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Anyone seeking to utilize +this eBook outside of the United States should confirm copyright +status under the laws that apply to them. diff --git a/README.md b/README.md new file mode 100644 index 0000000..bac96da --- /dev/null +++ b/README.md @@ -0,0 +1,2 @@ +Project Gutenberg (https://www.gutenberg.org) public repository for +eBook #31061 (https://www.gutenberg.org/ebooks/31061) |