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diff --git a/26839-t/26839-t.tex b/26839-t/26839-t.tex new file mode 100644 index 0000000..60c2efa --- /dev/null +++ b/26839-t/26839-t.tex @@ -0,0 +1,20066 @@ +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % +% % +% The Project Gutenberg eBook of Mathematical Recreations and Essays, % +% by W. W. Rouse Ball % +% % +% This eBook is for the use of anyone anywhere in the United States and % +% most other parts of the world 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. If you are not located in the United States, you % +% will have to check the laws of the country where you are located before % +% using this eBook. % +% % +% % +% Title: Mathematical Recreations and Essays % +% % +% Author: W. W. Rouse Ball % +% % +% Release Date: October 8, 2008 [eBook #26839] % +% [Most recently updated: October 14, 2021] % +% % +% Language: English % +% % +% Character set encoding: UTF-8 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK MATHEMATICAL RECREATIONS *** % +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{26839} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% memoir: Advanced book class. Required. %% +%% inputenc: read latin-1 input code. Required. %% +%% amsmath: AMS mathematics enhancements. Required. %% +%% amssymb: extra AMS mathematics symbols. Required. %% +%% hyperref: Hypertext embellishments for pdf output. Required. %% +%% Driver option needs to be set explicitly. %% +%% perpage: Resets footnote markers every page. Required. %% +%% multirow: Allows vertical spans in tables. Required. %% +%% longtable: Allows tables to span pagebreaks. Required. %% +%% psibycus: For authentic polytonic Greek. Strongly recommended. %% +%% If absent, Greek will be faked using math fonts. %% +%% graphicx: Standard interface for graphics inclusion. Required. %% +%% Driver option needs to be set explicitly. %% +%% wrapfig: Allows placement of graphics inside text cutouts. Required. %% +%% flafter: Stops graphics floating backwards. Required. %% +%% varioref: Allows text references to "above", "below", "opposite" etc %% +%% to be automatically adapted to different paginations and %% +%% float locations. Required. %% +%% indentfirst: Alters default indentation of initial paragraph in a %% +%% section. Optional. If omitted, initial paragraphs will %% +%% not be indented, unlike the original book. %% +%% afterpage: Allows insertion of something (eg a longtable, which is %% +%% too big to float) on the next clear page. Required. %% +%% multicol: automatically balance index columns. Recommended. %% +%% %% +%% %% +%% Producer's Comments: %% +%% %% +%% A mathematically straightforward text with lots of stylistic %% +%% challenges. %% +%% %% +%% %% +%% Things to Check: %% +%% %% +%% hyperref and graphicx driver option matches workflow: OK %% +%% color driver option matches workflow (color package is called %% +%% by hyperref, so may rely on color.cfg): OK %% +%% Spellcheck: OK %% +%% Smoothreading pool: Yes %% +%% LaCheck: OK %% +%% Lprep/gutcheck: OK %% +%% PDF pages: 377 %% +%% PDF page size: 499 x 709pt (b5) in print format %% +%% 648 x 432pt in screen format %% +%% PDF bookmarks: created but closed by default %% +%% PDF document info: filled in %% +%% PDF Reader displays document title in window title bar %% +%% ToC page numbers: should have a small "PAGE" at the top of each page %% +%% Images: One EPS/pdf publisher's logo %% +%% One EPS/pdf for blackletter on title page %% +%% Two EPS/pdf (for blackletter part titles) used twice each in %% +%% print format; coloured versions ending with -s are used only %% +%% in screen format %% +%% 34 EPS/pdf illustrations %% +%% Some illustrations draw outside their bounding boxes, so if %% +%% using automated conversion from eps to some other format %% +%% check that they aren't cropped. For example, check that %% +%% illus066 (p91 in screen format, p43 in print) shows %% +%% label B at the lower right. %% +%% Warnings: There should be no warnings in either format. %% +%% Underfull/overfull boxes: For print format there is one overfull hbox %% +%% caused by an alignment being a litle over 1pt too wide. %% +%% There are also three overfull vboxes (each by 1pt) from the %% +%% multirows used on p97. %% +%% For screen format there is one overfull hbox with a comma %% +%% protruding into the margin on p362, and no overfull vboxes. %% +%% For print format there is a multitude of underfull vboxes, %% +%% but no underfull hboxes. The 38 underfull vboxes are due to %% +%% there being insufficient breakable material between %% +%% unbreakable stuff that cannot be floated, combined with %% +%% fairly rigid spacing settings to try to keep consistent text %% +%% blocks on facing pages. Local looser settings could be used %% +%% to make the underfulls go away, but the output would look %% +%% much the same so there's not a lot of point. %% +%% For screen format there should be no underfull boxes at all %% +%% (the screen format has more flexible parameter settings than %% +%% the print format). %% +%% Fonts: latin1 characters, eg "3×3n=9n" about halfway down p4 %% +%% "money less than £12" near bottom of p8. %% +%% Fonts: psibycus for polytonic greek (if ibycus package used) %% +%% Index: Index references in footnotes are fragile--see comments in %% +%% in preamble code below for more detail. Check that (for %% +%% example) the index contains only one entry for Bachet's %% +%% Problèmes, Oughtred's Recreations and Ozanam's Récréations. %% +%% %% +%% Command block: %% +%% pdflatex x7 %% +%% makeindex -r %% +%% pdflatex x3 %% +%% %% +%% Compile History: %% +%% %% +%% Oct 08: dcwilson. %% +%% Compiled with pdfLaTeX SEVEN times, followed by makeindex -r %% +%% and another THREE times through pdfLaTeX. %% +%% MiKTeX 2.7, Windows XP Pro %% +%% Compiled with LaTeX SEVEN times, followed by makeindex -r %% +%% and another THREE times through LaTeX. %% +%% MiKTeX 2.7, Windows XP Pro %% +%% DVIPS 5.96dev used to create the PostScript file. %% +%% Acrobat Distiller 8.1.2 used to generate PDF output. %% +%% %% +%% %% +%% October 2008: pglatex. %% +%% Compile this project with: %% +%% pdflatex 26839-t.tex ..... SEVEN times %% +%% makeindex -r 26839-t.idx %% +%% pdflatex 26839-t.tex ..... THREE times %% +%% %% +%% pdfeTeX, Version 3.141592-1.30.5-2.2 (Web2C 7.5.5) %% +%% %% +%% October 2021: okrick. %% +%% MiKTeX Console 4.3, Windows 10 Home %% +%% TeXworks 0.6.6 used to generate PDF output. %% +%% %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\listfiles +% +% Compilation instructions +% +% This file has been written to produce output in either print-oriented +% or screen-oriented format. The default is print-oriented; change +% \Papertrue to \Paperfalse below to get screen-oriented output. +% +% This file needs to be run through LaTeX more that the "usual" three +% times. This is because an iterative method is used to to get the +% table of contents (ToC) to format. The number of iterations required +% depends on the length of the table and the state of the .aux file when +% you start (it's probably best to delete any old .aux files before +% beginning this compilation process). Once the ToC shows the word "PAGE" +% just above the first page reference on each page of the ToC the process +% has converged. (The file also uses longtables, but these should +% definitely have stabilised by the time the ToC is OK.) +% Then you need to generate the index (using makeindex -r; the -r +% is to disable implicit ranges because all ranges have been explicitly +% coded already), and do another three runs through LaTeX. +% For the print-oriented format you probably need 5 LaTeX runs +% followed by makeindex and another 3 LaTeX runs. For the +% screen-oriented format you probably need 7 initial LaTeX runs +% followed by makeindex and another 3 LaTeX runs. +% +% The illustrations and diagrams are in Encapsulated PostScript, but +% compiled PDF versions are also supplied. If you are using an application +% to prepare the final output that doesn't cope with .eps or .pdf directly, +% check the graphics carefully because some draw outside their bounding boxes. +% It may be important to specify the appropriate "paper" size (b5 for +% the print-oriented and 648x432pt for screen-oriented: see below). +% For example, dvips -t b5 ... +% +% We use the memoir documentclass. This can be obtained from CTAN +% if it's not already part of your TeX distribution. +% +% We use the hyperref package: make sure your configuration's default +% driver is appropriate, or add an explicit driver option to the +% invocation of hyperref below. +% +% We use the amsmath and amssymb packages. These can be obtained from +% CTAN or from the American Mathematical Society if they're not already +% part of your TeX distribution. +% +% We use the multirow package. This can be obtained from CTAN +% if it's not already part of your TeX distribution. +% +% We use the perpage package. This can be obtained from CTAN +% if it's not already part of your TeX distribution. +% +% We use the wrapfig package. This can be obtained from CTAN +% if it's not already part of your TeX distribution. +% +% We use the ibycus package for a tiny bit of polytonic classical +% Greek. This package and the associated fonts can also be obtained +% from CTAN, and we strongly recommend installing it. However, if +% the package isn't installed the code will approximate the Greek +% using standard mathematical symbols. +% +% Other packages used are part of the LaTeX base, graphics or tools +% bundles, and so should be already present in your TeX distribution. +% +\makeatletter +% +% Formatting for screen or paper is different +% +\newif\ifPaper +\Papertrue +% Change the line above to \Paperfalse +% if you want to get screen-oriented output +% +% NB we use "openany" because the original has chapters +% beginning on both recto and verso pages +\ifPaper + \documentclass[b5paper,12pt,twoside,openany,onecolumn]{memoir}[2005/09/25] + \setlrmarginsandblock{2.3cm}{2.6cm}{*} + \setulmarginsandblock{3.1cm}{2.2cm}{*} + \setlength{\headsep}{1cm} +\else + \documentclass[ebook,landscape,14pt,oneside,openany,onecolumn]{memoir}[2005/09/25] + \setlrmarginsandblock{2cm}{2cm}{*} + \setulmarginsandblock{1.5cm}{1cm}{*} + \setlength{\headsep}{0.7cm} +\fi +\setlength{\footskip}{0.6cm} +\fixthelayout +\typeoutlayout +% +% font and accent stuff +% this is the section most likely to require modification +% +% need a 30pt bold in screen format: perhaps \usepackage{type1cm} would be easier +\ifPaper\else + \DeclareFontShape{OT1}{cmr}{bx}{n} + {<5><6><7><8><9>gen*cmbx% + <10><10.95>cmbx10% + <12><14.4><17.28><20.74><24.88><29.86><35.83>cmbx12}{} +\fi +% Courier, for the PG licence stuff +\DeclareRobustCommand\ttfamily % Courier, for the PG licence stuff + {\not@math@alphabet\ttfamily\mathtt + \fontfamily{pcr}\fontencoding{T1}\selectfont} +% blackletter, for part headings and "London" on title page (see below) +% we resort to a graphics file +\newdimen\PartHeadHeight +\newcommand\PartOneText{\texorpdfstring{\protect\UsePartImage1}{Mathematical Recreations}.} +\newcommand\PartTwoText{\texorpdfstring{\protect\UsePartImage2}{Miscellaneous Essays and Problems}.} +\def\UsePartImage#1{\includegraphics[height=\PartHeadHeight]{./images/part#1head\if@mainmatter\else\ifPaper\else-s\fi\fi}\@gobble} + +% +% There are a couple of bits of classical Greek: +% [Greek: hoi polloi] +% [Greek: Kuklou metresis] +% We use the ibycus package if available, but if not we +% have a fallback to use math greek and fake the accents +\GenericInfo{*** }{***\MessageBreak + Important Note: this document contains\MessageBreak + a small amount of classical Greek; see comments in TeX source\@gobble} +\IfFileExists{psibycus.sty} +{% use ibycus greek with Type 1 fonts + \GenericInfo{*** }{*** Attempting to use ibycus polytonic Greek\MessageBreak\expandafter\@gobble\@gobble} + \usepackage{psibycus}[2004/10/18] + \def\hoipolloi{{\greek{oi( polloi'}}} + \def\Kukloumetresis{{\greek{Ku'klou me'trhsis}}} +}{% else fake with math greek + \def\RoughBreathing{\mathaccent"012C } + \def\hoipolloi{$o\RoughBreathing{\vphantom{(}\iota}$ + $\pi{o}\lambda\lambda{o}\acute\iota$} + \def\Kukloumetresis{$K\!\acute\upsilon\kappa\lambda{o}\upsilon$ + $\mu\acute\epsilon\tau\!\rho\eta\sigma\iota\varsigma$} + \GenericInfo{*** }{*** Faking breathed Greek using math\MessageBreak\expandafter\@gobble\@gobble} +} + +\usepackage[utf8]{inputenc}[2004/02/05] %changed latin1 to utf8 +% \DeclareInputText{176}{\ensuremath{{^\circ}}} % suppress warnings about \textdegree in math +\DeclareUnicodeCharacter{00E3}{\^a} % added ã +\DeclareUnicodeCharacter{00EA}{\^e} % added ê + +% mathematics +\usepackage[reqno]{amsmath}[2000/07/18] +\usepackage[psamsfonts]{amssymb}[2002/01/22] +% we only want inline equations to break where we explicitly allow +\binoppenalty=\@M +\relpenalty=\@M +\def\dotsc{\allowbreak\ldots} +\let\dotm\cdot +\let\epsilon\varepsilon +\let\phi\varphi +\def\Therefore{\therefore\;} +\DeclareMathOperator{\cosec}{cosec} +\def\maketag@@@#1{\hbox to2cm{\m@th\normalfont\dotfill#1}} +\def\tagform@#1{\maketag@@@{(\ignorespaces#1\unskip\@@italiccorr),}} +\def\Tagform@#1{\maketag@@@{(\ignorespaces#1\unskip\@@italiccorr).}} +\def\Tag{\let\tagform@\Tagform@\tag} +% for aligned equations with commentary +% should be replaced with amsmath equivalent if I can find one! +% left text & LHS & RHS & right text \cr +\newskip\@Centering \@Centering=0pt plus 1000pt minus 1000pt +\newenvironment{LRalign}{\[\let\\\cr\LR@lign}{\]\aftergroup\ignorespaces} +\def\LR@lign#1\end{\displ@y \tabskip=\displaywidth + \halign to\displaywidth{\kern-\displaywidth + \rlap{\@lign##}\tabskip=\@Centering + &\hfil$\@lign\displaystyle{##}$\tabskip=\z@ + &$\@lign\displaystyle{{}##}$\hfil\tabskip=\@Centering + &\llap{\@lign##}\tabskip=\z@\crcr + #1\crcr}\end} + +% to avoid over/underfull boxes without using explicit linebreaks +% when a paragraph contains lots of difficult-to-hyphenate stuff +\def\stretchyspace{\spaceskip0.5em plus 0.5em minus 0.25em} + +% footnotes +\setlength{\footmarksep}{\z@} +\setlength{\footmarkwidth}{1.3em} +\newfootnoteseries{T} % for transcriber's notes +\renewcommand{\m@make@footnotetext}[1]{% fixing buglet in recent versions of memoir + \@namelongdef{@footnotetext#1}##1{% + \insert\@nameuse{footins#1}{% +% \def\baselinestretch{\m@m@singlespace}% + \reset@font\@nameuse{foottextfont#1}% + \@preamfntext + \hsize\columnwidth + \protected@edef\@currentlabel{% + \csname p@footnote#1\endcsname\@nameuse{@thefnmark#1}% + }% + \color@begingroup + \@nameuse{@makefntext#1}{% + \rule\z@\footnotesep{\@nameuse{foottextfont#1}\ignorespaces ##1}% bug was here + \@finalstrut\strutbox}% + \color@endgroup}\m@mmf@prepare}} +\plainfootstyle{T} % default +\renewcommand{\thefootnote}{\BringhurstX{footnote}} +\footmarkstyle{#1\hfill} +\footmarkstyleT{#1.\hfill} +\renewcommand{\foottextfont}{\footnotesize\normalfont} +\let\foottextfontT\foottextfont +\usepackage{perpage}[2002/12/20] +\MakePerPage{footnote} +\MakePerPage{footnoteT} +\newif\ifEditorials +% there are two kinds of transcriber's notes: +% those to do with the text (typos etc), and "editorial" +% comments/elaborations +% PG requires \Editorialsfalse, but if you're not a purist +% change to \Editorialstrue +\Editorialsfalse +\ifEditorials + \let\Editorial\footnoteT\GenericInfo{\@spaces\@spaces\@spaces + }{NB: Including transcriber's editorialisations} +\else + \let\Editorial\@gobble\GenericInfo{\@spaces\@spaces\@spaces + }{NB: Suppressing transcriber's editorialisations} +\fi +\def\BringhurstX#1{\expandafter\@BringhurstX\csname c@#1\endcsname} +\def\@BringhurstX#1{\ifcase#1\or*\or\dag\or\ddag\or\S\or$\|$\or\P + \or**\or\dag\dag\or\ddag\ddag\or\S\S\or$\|\|$\or\P\P\else?\fi} +% Sometimes the author has multiple references on a page to the same footnote +% so we have to go to some lengths to discover if there's an intervening +% pagebreak or not since our pagination is unlikely to match the original. +% If there's a pagebreak we need a second copy of the footnote, +% but if not we need to duplicate the footnotemark. +% We need to insert labels for both footnotemark locations; +% the label names are passed as #1 and #2 +% Syntax is \multifootnote{label 1}{label 2}{footnote text} +% Not sure if this is totally stable... +% Perhaps we could do this by accessing the perpage +% machinery somehow? +\def\multifootnote#1#2{% + \expandafter\ifx\csname r@#1\endcsname\relax + % labels undefined: don't do anything fancy this time + \let\next\footnote + \else + \@tempcnta\expandafter\expandafter\expandafter + \@secondoffive\csname r@#1\endcsname\relax + \@tempcntb\expandafter\expandafter\expandafter + \@secondoffive\csname r@#2\endcsname\relax + \ifnum\@tempcnta=\@tempcntb + % footnotes are on same page; duplicate footnotemark + \addtocounter{footnote}{-1}\footnotemark\let\next\@gobble + \else + % pagebreak intervenes; duplicate entire footnote + \let\next\footnote + \fi\fi\next} +% The conventions regarding "ibidem" are that you can only use it +% when the citation is the same as that immediately preceding it +% *on the same spread* +% Since this may change with the pagination, we can't simply +% reproduce the author's original "ibid"s (unless they're inside +% a single footnote), but must check they are still on the +% same spread as what is being ibided +% #1 is case flag, #2 is label for other reference, +% #3 is alternative text for when ibid isn't allowed +\def\DP@ibid#1#2#3{\global\advance\c@vrcnt\tw@ + \label{ib@d:\the\c@vrcnt}\expandafter + \ifx\csname r@#2\endcsname\relax + % label not defined so no fancy stuff this time + \csname #1bid\endcsname + \else\expandafter + \ifx\csname r@ib@d:\the\c@vrcnt\endcsname\relax + % companion label not defined so avoid fancy stuff this time + \csname #1bid\endcsname + \else % check for matching spread + \@tempcnta\expandafter\expandafter\expandafter + \@secondoffive\csname r@#2\endcsname\relax + \@tempcntb\expandafter\expandafter\expandafter + \@secondoffive\csname r@ib@d:\the\c@vrcnt\endcsname\relax + \ifnum\@tempcnta=\@tempcntb % same page: use ibid + \csname #1bid\endcsname + \else % not same page, so should have a < b + \if@twoside % spreads make sense + \ifodd\@tempcntb + % should be on same spread, unless b=a+3 (highly unlikely) + \csname #1bid\endcsname + \else #3\fi + \else #3\fi + \fi + \fi\fi} +\def\ibidref{\DP@ibid i} +\def\Ibidref{\DP@ibid I} + % We want the footnotes to be out of the way at the bottom, but memoir + % sets \raggedbottom in screen mode, and in print mode there is sometimes + % insufficient stretch on a page. +\renewcommand*{\footnoterule}{\kern-3\p@\ifPaper\vglue0\p@ plus6\p@\else\vfill\fi + \hrule width 0.4\columnwidth \kern 2.6\p@} + +% We set up the version of the verbatim package embedded +% in the memoir class to wrap nicely +\setlength{\verbatimindent}{.25in} +\wrappingon +\addto@hook\afterevery@verbatim{\parindent\z@\relax} +\setverbatimfont{\normalfont\ttfamily\ifPaper\tiny\else\Small\fi} % 8pt for B5, 11pt for screen +\addto@hook\every@verbatim{\PGhook} +\let\verbatimbreakchar\empty +\let\PGhook\empty +{\catcode`\L\active +\gdef\PGlicencelink{\catcode`\L\active\letL\PGlinklicence}} +\def\PGlinklicence{\@ifnextchar i{\PG@lli}{L}} +\def\PG@lli#1{\@ifnextchar c{\PG@llii}{Li}} +\def\PG@llii#1{\@ifnextchar e{\PG@lliii}{Lic}} +\def\PG@lliii#1{\@ifnextchar n{\PG@lliv}{Lice}} +\def\PG@lliv#1{\@ifnextchar s{\PG@llv}{Licen}} +\def\PG@llv#1{\@ifnextchar e{\PG@llvi}{Licens}} +\def\PG@llvi#1{\hyperlink{PGlicence}{License}} +\def\PGheaderhook{\catcode`\L\active} + +% half-title, title and copyright pages +\aliaspagestyle{title}{empty} +\setlength{\droptitle}{-2em} +\pretitle{\begin{center}\bfseries\ifPaper\LARGE + \else\fontsize{30}{36}\selectfont\fi} +\renewcommand{\maketitlehookb}{\vspace{\z@\@plus.5fill}%\ifPaper\vspace{24pt}\else\medskip\fi + \begin{center}\tiny\bfseries BY\end{center}} +\posttitle{\par\end{center}} +\preauthor{\begin{center}\Large} +\postauthor{\par\end{center}} +\def\affiliation#1{\def\theAffiliation{#1}} +\def\Edition#1{\def\theEdition{#1}} +\def\Publisher#1{\def\thePublisher{#1}} +\renewcommand{\maketitlehookc}{\begin{center}\tiny + \textsc{\theAffiliation}\par\end{center}} +\predate{\vspace{\z@\@plus.5fill}\begin{center}\Small\textit{\theEdition}\end{center} + \vspace{\z@\@plus.5fill}\begin{center}\large\textsc{\thePublisher}\par\normalsize\oldstylenums} +\postdate{\par\bigskip\SMALL[\textit{\theRights}]\end{center}\vspace{-1.5em}} +\def\Rights#1{\def\theRights{#1}} +\let\transcribersnotes\@empty +\let\transcribersNotes\@empty +\newcommand{\transcribersnote}[1]{% + \@ifnotempty{#1}{\g@addto@macro\transcribersnotes{#1\par}% + \@xp\@ifempty\@xp{\transcribersNotes}% + {\renewcommand{\transcribersNotes}{note}} + {\renewcommand{\transcribersNotes}{notes}}}} + +\def\makecopyrightpage{% production credits and transcriber's notes + \begingroup\pagestyle{empty} + \ifPaper + \null\vfil + \transcribersnote{This document is designed for two-sided printing. Consequently, the + many hyperlinked cross-references are not visually distinguished. + The document can be recompiled for more comfortable on-screen viewing: + see comments in source \LaTeX\ code. } + \else + \transcribersnote{This document is designed for on-screen viewing. + It can be recompiled for two-sided printing: see comments in source \LaTeX\ code. + Alternatively, print this on-screen version 2-up.} + \fi + \begin{center} +Produced by Joshua Hutchinson, David Starner, David Wilson +and the Online Distributed Proofreading Team at +http://www.pgdp.net + \end{center} + \vfil\vfil + \vfil + \vbox{\Small\hsize=.75\textwidth\parindent=\z@\parskip=.75em + \textit{Transcriber's \transcribersNotes}\par\medskip\raggedright + \transcribersnotes\par} + \ifPaper\newpage\else\eject\fi\endgroup} + +% parts, chapters and sections +% (for part i; we redefine sections for part ii later) +\usepackage{indentfirst}[1995/11/23] +\renewcommand{\printpartname}{\partnamefont\MakeUppercase{\partname}} +\renewcommand{\printpartnum}{\partnumfont \thepart.} +\renewcommand{\midpartskip}{\par\vskip3pc} +\renewcommand{\afterpartskip}{\vskip6pc\begin{adjustwidth}{5pc}{5pc}\small + \itshape~\par\DP@PartQuote\end{adjustwidth}\global\let\DP@PartQuote\empty + \vfil\newpage} +\newcommand{\PartQuote}[1]{\long\def\DP@PartQuote{#1}} +\let\DP@PartQuote\empty +\renewcommand{\parttitlefont}{\PartHeadHeight=\ifPaper17pt\else39pt\fi} +\makechapterstyle{rouseball}{% + \ifPaper\setlength{\beforechapskip}{4pc} + \else\setlength{\beforechapskip}{2pc}\fi + \renewcommand{\chapnumfont}{\normalfont\huge\bfseries} + \renewcommand{\printchapternum}{\chapnumfont\Roman{chapter}.} + \renewcommand{\printchaptername}{\begin{center}\chapnumfont + \MakeUppercase{\chaptername}} + \renewcommand{\printchapternonum}{\begin{center}} + \setlength{\midchapskip}{3pc} + \renewcommand{\chaptitlefont}{\normalfont\Large\bfseries} + \renewcommand{\printchaptertitle}[1]{\chaptitlefont\MakeUppercase{##1}} + \setlength{\afterchapskip}{\@ne pc} + \renewcommand{\afterchaptertitle}{\end{center}\nobreak\vskip\afterchapskip} + } +\chapterstyle{rouseball} +\makechapterstyle{advert}{% + \renewcommand{\printchapternum}{\chapnumfont\Roman{chapter}} + \renewcommand{\printchaptername}{\begin{center}} + \let\printchapternonum\printchaptername + \setlength{\beforechapskip}{\z@} + \setlength{\afterchapskip}{0.5em} + \renewcommand{\chaptitlefont}{\normalfont\Large\bfseries} + \renewcommand{\afterchaptertitle}{\end{center}\nobreak\vskip\afterchapskip} + } +% kill section numbers so we can use unstarred versions to get bookmarks easily +\setsecnumformat{} +\setsecnumdepth{section} % we want bookmarks down to this level +\maxsecnumdepth{subsection} +\maxtocdepth{subsection} +\setlength{\parindent}{1.8em} +\setlength{\leftmargini}{1.8em} +\setsecheadstyle{\normalfont\normalsize\scshape} +\setsecindent{\parindent} +\setaftersecskip{-1.5em} +% because we are using run-in heads we need to force the paragraph to start +% otherwise the hyperlinks/contents page number can be on the previous page +% This means we can't have any blank lines between the \section and the +% beginning of the section text and we need to be careful about line endings +\def\@xsect #1{\@tempskipa #1\relax + \ifdim \@tempskipa >\z@ \par \nobreak \vskip \@tempskipa \@afterheading + \else \@nobreakfalse \global \@noskipsectrue \everypar {\if@noskipsec + \global \@noskipsecfalse {\setbox \z@ \lastbox }\clubpenalty \@M + \begingroup \@svsechd \endgroup \unskip \@tempskipa #1\relax + \hskip -\@tempskipa \else \clubpenalty \@clubpenalty \everypar {}\fi + }\leavevmode + \fi \ignorespaces} +% we need this because the \phantomsection inserted by hyperref blocks the +% \ignorespaces at the end of section header processing +% and if we have a \phantomsection in vertical mode a page break +% can happen between the link and the "section's" text +\AtBeginDocument{\let\PGph@ntomsecti@n\phantomsection + \def\phantomsection{\leavevmode\PGph@ntomsecti@n\ignorespaces}} +\setsubsecheadstyle{\normalfont\normalsize\itshape} +\setsubsecindent{\parindent} +\setbeforesubsecskip{\z@} +\setaftersubsecskip{-1em} +% section/subsection headings get an added period, +% because we don't want the period in the ToC +\def\Sectionformat#1#2{#1.} +\def\chapindex{\specialindex{\jobname}{chapter}} +\newcommand{\ssection}{% Like a section but without the preceding big skip + \sechook% + \@startsection{section}{1}% level 1 + {\secindent}% heading indent + {-0.5ex \@plus -0.1ex \@minus -.2ex}% skip before the heading + {\aftersecskip}% skip after the heading + {\normalfont\secheadstyle}} % font +% In part ii the author uses slightly different heading styles +% so we set up switches for these here +% For Chapter VIII the section headings are italic +% and centered rather than small caps and run-in as in part i. +\def\UseChapterVIIIHeadings{% + \setsecheadstyle{\normalfont\normalsize\centering\itshape} + \setsecindent{0pt} + \setaftersecskip{2.3ex plus.2ex}} +% For Chapter XIV we need to restore the section headings to the style used in part i +% although now subsections have a small space above +% and the sections etc are numbered +\def\UseChapterXIVHeadings{% + \setsecheadstyle{\normalfont\normalsize\scshape} + \setsecindent{\parindent} + \setaftersecskip{-1.5em} + \setbeforesubsecskip{-1ex plus -.2ex minus -.1ex} + \defaultsecnum + \renewcommand{\thesection}{\Roman{section}.} + \renewcommand{\thesubsection}{{\upshape(\roman{subsection})}}} + + +% table of contents +\renewcommand{\contentsname}{TABLE OF CONTENTS.} +\setpnumwidth{2.75em} +\setrmarg{3.5em} +\setlength{\cftpartnumwidth}{\z@} +\renewcommand{\cftpartfillnum}[1]{\par} +\def\partnumberline#1#2.{\vbox{% + \centerline{PART #1.}\ifPaper\kern4ex\else\kern2ex\fi + \centerline{\PartHeadHeight=17pt#2.}\kern3ex\vfil}% + \aftergroup\setpartlinkheight} +\def\setpartlinkheight{\baselineskip=\ifPaper5\else4\fi\baselineskip} +\setlength{\cftchapternumwidth}{\z@} +\renewcommand{\cftchapterfillnum}[1]{\par} +\renewcommand{\cftchapterfont}{\scshape} +\def\chapternumberline#1#2.{\vbox{% + \centerline{\if!#1!\else Chapter \@Roman{#1}.\fi\qquad#2.}\par\kern2ex}% + \aftergroup\setchaplinkheight} +\def\setchaplinkheight{\baselineskip=2.5\baselineskip} +\cftsetindents{section}{\z@}{2.3em} +\cftsetindents{subsection}{1.5em}{3.2em} +\let\numberline\@gobble % suppresses section etc numbers +% the following machinations are to (a) encourage pagebreaks in the ToC +% at chapter divisions, (b) to discourage separating subsections from their +% section, and (c) to write "PAGE" just above the first section entry on any +% page [NB if ToCPAGE has any vertical size the pagination frequently fails +% to stabilise; since it's always at the top of a page or just below a +% chapter line it can never overlap anything] +\newcount\DPtoc +\DPtoc\m@ne +\def\ToCPAGE{\hrule height\z@ depth\z@ + \setbox\z@=\vbox{\rightline{\SMALL PAGE}}\ht\z@=\z@\dp\z@=\z@\vskip-1ex + \box\z@\vskip1ex} +\def\firstfilbreak{\let\secfilbreak\filbreak} +\AtBeginDocument{% so we don't get clobbered by hyperref + \let\DPaddcontentsline\addcontentsline + \def\addcontentsline#1#2#3{\hbox to\z@{\hss % this is because somewhere space is leaking so we neutralise it + \expandafter\ifx\csname l@#2\endcsname\l@chapter + \addtocontents{#1}{\protect\bigskip\protect\filbreak\protect\let + \protect\secfilbreak\protect\firstfilbreak}% + \DPaddcontentsline{#1}{#2}{#3}% + \else\expandafter\ifx\csname l@#2\endcsname\l@subsection + \addtocontents{#1}{\protect\nobreak}% + \DPaddcontentsline{#1}{#2}{#3\protect\tocsecspacer}% + \else\expandafter\ifx\csname l@#2\endcsname\l@section + \addtocontents{#1}{\protect\edef\protect\DP@tmp{{\protect\expandafter + \protect\ifx\protect\csname r@toc:subsec:\number\DPtoc + \protect\endcsname\protect\relax??\protect\else + \protect\expandafter\protect\expandafter\protect\expandafter + \protect\@secondoffive\protect\csname r@toc:subsec:\number\DPtoc + \protect\endcsname\protect\fi}}}% + \global\advance\DPtoc\@ne\relax + \addtocontents{#1}{\protect\edef\protect\DP@tmp{\protect\DP@tmp + {\protect\expandafter\protect\ifx\protect + \csname r@toc:subsec:\number\DPtoc\protect\endcsname\protect + \relax??\protect\else\protect\expandafter\protect\expandafter + \protect\expandafter\protect\@secondoffive\protect + \csname r@toc:subsec:\number\DPtoc\protect\endcsname\protect\fi}}}% + \addtocontents{#1}{\protect\expandafter\protect\nametest\protect\DP@tmp}% + \addtocontents{#1}{\protect\secfilbreak\protect\ifsamename\protect\else + \protect\ToCPAGE\protect\fi}% + \DPaddcontentsline{#1}{#2}{#3\protect\tocsecspacer\protect + \label{toc:subsec:\number\DPtoc}}% + \else + \DPaddcontentsline{#1}{#2}{#3}% + \fi\fi\fi\hss}\ignorespaces}% + } +\def\tocsecbox#1{\ifPaper\vbox{\leftskip\z@\expandafter\@tempdima\@pnumwidth + \advance\hsize-\@tempdima\raggedright\hyphenpenalty=10000 + \emergencystretch.5\hsize + \noindent\vrule height12pt width\z@ depth\z@ % to give the illusion of +% evenly-spaced lines, together with the vphantom inserted above + \hangindent1.5em\hangafter1\parfillskip\z@#1\cftsectionleader\null + \par}\aftergroup\settocsecboxlinkheight\else#1\fi} +\ifPaper\def\settocsecboxlinkheight{\baselineskip=2\baselineskip}\fi +\def\tocsecspacer{\vphantom{\normalfont Pp}} + +% headers and footers +\copypagestyle{frontstuff}{headings} + \makeevenhead{frontstuff}{\normalfont\SMALL\thepage}{\normalfont + \SMALL\MakeUppercase{\leftmark}}{} + \makeoddhead{frontstuff}{}{\normalfont\SMALL\MakeUppercase{\rightmark}}% + {\normalfont\SMALL\thepage} +\makepsmarks{frontstuff}{% + \let\@mkboth\markboth + \def\chaptermark##1{% + \markboth{\MakeUppercase{##1}}{\MakeUppercase{##1}}}% + \def\tocmark{% + \markboth{\MakeUppercase{\contentsname}}{\MakeUppercase{\contentsname}}}% + } +\makeevenfoot{frontstuff}{}{\SMALL\wmc}{} +\makeoddfoot{frontstuff}{}{\SMALL\wmc}{} +\pagestyle{frontstuff} +\copypagestyle{part}{plain} +\copypagestyle{chapter}{plain} +\ifPaper + % because we are using a 2-page spread format we make a bit of fuss about page-turning hyphenation + % the large \brokenpenalty makes it hard to end the next (odd/recto) page with a hyphen + \makeevenfoot{part}{}{\normalfont\SMALL\wmc\thepage}{\global\brokenpenalty10000} + % the small \brokenpenalty makes it less hard to end the next (even/verso) page with a hyphen + % (because the other end of the hyphen will still be visible on the same spread) + \makeoddfoot{part}{}{\normalfont\SMALL\wmc\thepage}{\global\brokenpenalty150} + \makeevenfoot{chapter}{}{\normalfont\SMALL\wmc\thepage}{\global\brokenpenalty10000} + \makeoddfoot{chapter}{}{\normalfont\SMALL\wmc\thepage}{\global\brokenpenalty150} +\else + \makeevenfoot{part}{}{\normalfont\SMALL\wmc\thepage}{} + \makeoddfoot{part}{}{\normalfont\SMALL\wmc\thepage}{} + \makeevenfoot{chapter}{}{\normalfont\SMALL\wmc\thepage}{} + \makeoddfoot{chapter}{}{\normalfont\SMALL\wmc\thepage}{} +\fi +\copypagestyle{mainstuff}{headings} +\makepsmarks{mainstuff}{% + \let\@mkboth\markboth + \def\chaptermark##1{% + \markboth{\MakeUppercase{##1}}{\MakeUppercase{##1}}}% + \def\sectionmark##1{% + \markright{\MakeUppercase{##1}.}}% note the added period + \let\subsectionmark\sectionmark + \def\indexmark{\markboth{\MakeUppercase{\indexname}.}% + {\MakeUppercase{\indexname}.}}% + } +\def\mainstuffChapNumOdd{CH. \Roman{chapter}]} +\def\mainstuffChapNumEven{[CH. \Roman{chapter}} +\ifPaper + \makeevenhead{mainstuff}{\normalfont\SMALL\thepage}{\normalfont + \SMALL\MakeUppercase{\leftmark}}{\normalfont\SMALL\mainstuffChapNumEven} + \makeoddhead{mainstuff}{\normalfont\SMALL\mainstuffChapNumOdd}{\normalfont + \SMALL\MakeUppercase{\rightmark}}{\normalfont\SMALL\thepage} + \makeevenfoot{mainstuff}{}{\SMALL\wmc}{\global\brokenpenalty10000} + \makeoddfoot{mainstuff}{}{\SMALL\wmc}{\global\brokenpenalty150} +\else + \makeoddhead{mainstuff}{\normalfont\SMALL\mainstuffChapNumOdd}{\normalfont + \SMALL\MakeUppercase{\ifodd\count\z@\rightmark\else\leftmark + \fi}}{\normalfont\SMALL\thepage} + \makeoddfoot{mainstuff}{}{\SMALL\wmc}{} + +\fi + +% make cross-page hyphenation difficult by default +\brokenpenalty10000 + +\copypagestyle{adverts}{headings} +\makeevenhead{adverts}{\normalfont\SMALL\thepage}{}{} +\makeoddhead{adverts}{}{}{\normalfont\SMALL\thepage} +\makeevenfoot{adverts}{}{\SMALL\wmc}{} +\makeoddfoot{adverts}{}{\SMALL\wmc}{} +\copypagestyle{licence}{headings} +\makeevenhead + {licence}{\normalfont\SMALL\thepage}{\normalfont\SMALL LICENSING.}{} +\makeoddhead + {licence}{}{\normalfont\SMALL LICENSING.}{\normalfont\SMALL\thepage} +\makeevenfoot{licence}{}{\SMALL\wmc}{} +\makeoddfoot{licence}{}{\SMALL\wmc}{} + +% for quotations, which in this book were printed full-width in a smaller +% font but we will use small font *and* slightly narrower text block +\AtBeginDocument{\let\QuoteFont\Small} +\newenvironment{Quotation}{\QuoteFont\partopsep\z@\quotation}{\par\medskip + \global\advance\@listdepth\m@ne} +% for exam questions in Chapter vii: smaller font but full-width +\newenvironment{ExamQuestions}{\QuoteFont\medskip\def\Q[##1]{\par\indent + \hbox to\parindent{##1\hss}\ignorespaces}}{\medskip} + +% illustrations +% (external files are provided as both .eps and .pdf; +% a few are bitmaps lifted from page scans, most from hand-coded PostScript) +\usepackage{flafter}[2000/07/23] +\ifPaper % defaults are not stretchy enough + \setlength\textfloatsep{20\p@ \@plus 6\p@ \@minus 4\p@} + \setlength\intextsep {14\p@ \@plus 8\p@ \@minus 4\p@} +\fi +% driver should be specified in graphics.cfg; +% if not, add explicit option to graphicx call +\usepackage[final]{graphicx}[1999/02/16] +\GenericWarning{*** }{***\MessageBreak + Important Note: this document comes with a full set of\MessageBreak + graphics in EPS (Encapsulated PostScript, not MetaPost) format\MessageBreak + and an equivalent full set compiled as PDF. If your workflow\MessageBreak + requires another conversion make sure you know what you're doing!\MessageBreak\expandafter\@gobble\@gobble} +% we limit file extensions because different TeX systems search in different orders +% and the default ordering of extensions may lead to the wrong file being located +\ifpdf\DeclareGraphicsExtensions{.pdf}\else\DeclareGraphicsExtensions{.eps}\fi + +% to refer to an uncaptioned illustration depending on +% where LaTeX has actually floated it to +% although it doesn't help with deciding above/below on +% a particular page +\usepackage{varioref}[2004/02/27] +% add hook to enable uppercasing first character +\newif\ifVariorefUC +\VariorefUCfalse +\def\Vpageref{\VariorefUCtrue\vpageref} +\def\Varioif#1#2{\ifVariorefUC#1\else#2\fi\VariorefUCfalse} +% add links to references which don't call \pageref +\def\reftextfaceafter {\Varioif Oon the \Acrobatmenu + {NextPage}{\reftextvario{facing}{next} page}} +\def\reftextfacebefore{\Varioif Oon the \Acrobatmenu + {PrevPage}{\reftextvario{facing}{preceding} page}} +\def\reftextafter {\Varioif Oon the \Acrobatmenu + {NextPage}{\reftextvario{following}{next} page}} +\def\reftextbefore {\Varioif Oon the \Acrobatmenu + {PrevPage}{\reftextvario{preceding}{previous} page}} +\def\reftextcurrent{\Varioif Oon \reftextvario{this}{the current} page} +\def\reftextfaraway#1{\Varioif Oon page~\pageref{#1}} +\def\reftextpagerange#1#2{\Varioif Oon pages~\pageref{#1}--\pageref{#2}} +% suppress a link if it goes to the same page +% requires destination to have been created by \DPlabel +\def\vhyperlink{\begingroup\@ifstar + {\vhyperl@nkstar}{\vhyperl@nk}} +\def\vhyperl@nkstar#1#2{% + \def\reftextfaceafter{\unskip#2}\let\reftextfacebefore\reftextfaceafter + \def\reftextafter{\hyperlink{#1}{#2}}\let\reftextbefore\reftextafter + \let\reftextcurrent\reftextfaceafter + \def\reftextfaraway##1{\hyperlink{#1}{#2}}\let\vref@space\relax + \vp@geref{#1}{}\endgroup} +\def\vhyperl@nk#1#2{% + \def\reftextfaceafter{#2}\let\reftextfacebefore\reftextfaceafter + \def\reftextafter{\hyperlink{#1}{#2}}\let\reftextbefore\reftextafter + \let\reftextcurrent\reftextfaceafter + \def\reftextfaraway##1{\hyperlink{#1}{#2}}\let\vref@space\space + \vp@geref{#1}{}\endgroup} +% captions look like \textit{Figure} \lowercaseroman{n}. +% but since they aren't always numbered consecutively we use +% \legend (with hard-coded numbers if necessary) rather than \caption +\captionstyle{\centering} +\captiontitlefont{\normalfont\Small\itshape} +\def\Uproman#1{\upshape\@roman{#1}.} +% For magic squares we use a picture environment rather than tabular/array +% We first redefine the horizontal/vertical lines for the picture environment +% to give them squarecap ends a la PostScript: this makes the corners of +% frames join up neatly +\def\@hline{\advance\@linelen\@wholewidth + \ifnum \@xarg <\z@ \hskip -\@linelen \hskip\@halfwidth + \else\hskip-\@halfwidth\fi + \vrule \@height \@halfwidth \@depth \@halfwidth \@width \@linelen + \ifnum \@xarg <\z@ \hskip -\@linelen \hskip\@halfwidth + \else\hskip-\@halfwidth\fi} +\def\@upline{% + \hb@xt@\z@{\advance\@linelen\@halfwidth\hskip -\@halfwidth + \vrule \@width \@wholewidth \@height \@linelen \@depth \@halfwidth\hss}} +\def\@downline{% + \hb@xt@\z@{\advance\@linelen\@halfwidth\hskip -\@halfwidth + \vrule \@width \@wholewidth \@height \@halfwidth \@depth \@linelen \hss}} +% Now we set up a MagicSquare environment, which is a bit like a tabular +% except that the entries must be enclosed in braces if they have > 1 digit +% \begin{MagicSquare}{horiz order}[optional vertical order, defaults to square] +% {entry} & {entry} & ... & {entry}\\ +% \end{MagicSquare} +\def\Sq@r#1{\vbox to\SqHt{\vss\hbox to\SqWd{\smaller\hss$\vphantom + {\SqHtDefault}#1$\hss}\vss}} +\def\SqHtDefault{1} +\def\Cell(#1,#2;#3){\put(#1,#2){\framebox{\Sq@r{#3}}}} +\def\MagicSquare#1{\catcode`\&=\active + \@ifnextchar[{\M@gicSqu@re#1}{\M@gicSqu@re#1[#1]}} +\long\def\M@gicSqu@re#1[#2]#3\end{% + \let\\\MSqN@xtC@ll + \begin{picture}(#1,#2) + \@tempcnta=#2 + \MSqN@xtC@ll#3 + \put(0,0){\line(0,1){#2}} + \put(0,0){\line(1,0){#1}} + \put(#1,0){\line(0,1){#2}} + \put(0,#2){\line(1,0){#1}} + \end{picture}\end} +\let\endMagicSquare\empty +\let\MSqHorizAdvance\@ne +\let\MSqVertAdvance\m@ne +\long\def\MSqN@xtCell#1{\advance\@tempcntb\MSqHorizAdvance + \Cell(\@tempcntb,\@tempcnta;#1)} +\long\def\MSqN@xtC@ll{\ifnum\@tempcnta=\z@\let\next\relax\else + \advance\@tempcnta\MSqVertAdvance\@tempcntb=-\MSqHorizAdvance + \let\next\MSqN@xtCell\fi\next} +{\catcode`\&=\active +\global\let&=\MSqN@xtCell} +\unitlength=1.5em +\linethickness{0.14em} +\fboxsep\z@ +\def\SqHt{1.5em} +\def\SqWd{1.5em} +\RequirePackage{wrapfig}[2003/01/31] +% for the diagrams explaining Chinese rings +\def\RingDiagram#1#2#3#4#5#6#7{% +\begin{picture}(42,8) +\put(0,0){\line(0,1){8}} +\put(0,4){\line(1,0){42}} +\put(4,\ifcase#1 1\or7\fi){\circle{3}} +\put(10,\ifcase#2 1\or7\fi){\circle{3}} +\put(16,\ifcase#3 1\or7\fi){\circle{3}} +\put(22,\ifcase#4 1\or7\fi){\circle{3}} +\put(28,\ifcase#5 1\or7\fi){\circle{3}} +\put(34,\ifcase#6 1\or7\fi){\circle{3}} +\put(40,\ifcase#7 1\or7\fi){\circle{3}} +\end{picture}\\} +\def\RingDiag#1#2#3#4#5{% +\begin{picture}(42,10) +\put(0,0){\line(0,1){8}} +\put(0,4){\line(1,0){42}} +\put(4,\ifcase#1 1\or7\fi){\circle{3}} +\put(13,\ifcase#2 1\or7\fi){\circle{3}} +\put(22,\ifcase#3 1\or7\fi){\circle{3}} +\put(31,\ifcase#4 1\or7\fi){\circle{3}} +\put(40,\ifcase#5 1\or7\fi){\circle{3}} +\end{picture}} +% note the multirow package does not supply a date in \ProvidesFile format +% we used multirow.sty V1.6 version (5-May-2004) +\usepackage{multirow} + +% to deal with the scanned page breaks +% add a "draft" option to the documentclass invocation +% to see the scan numbers +\ifdraftdoc +\def\PG#1 #2.png#3 +{\marginpar{\noindent\null\hfill\Small #2.png}} +\def\PGx#1 #2.png#3 +{} +\ifPaper\else\advance\marginparwidth15pt\fi +\else +\def\PG#1 #2.png#3 +{} +\let\PGx\PG +\fi + +% for thought breaks +% if it won't fit on the page, we squeeze in a rule instead +\newcommand\ThoughtBreakDP{\noindent + \hbox to\textwidth{\hglue\z@\@plus\@ne fil*\hglue\z@\@plus0.2fil*\hglue\z@ + \@plus0.2fil*\hglue\z@\@plus0.2fil*\hglue\z@\@plus0.2fil*\hglue\z@ + \@plus\@ne fil}} +\newcommand\ThoughtBreakRule{\hbox to\textwidth{\hfil + \vrule height\z@ depth.4pt width.3\textwidth\relax\hfil}} +\let\Th@ughtBre@k\ThoughtBreakRule +\newcommand\ThoughtBreakSpace{\vskip1.5em plus .5em\relax} +\newcommand\ThoughtBreak{\par\hrule height\z@ depth\z@ + \nobreak\begingroup + \setbox\@tempboxa=\vbox{\ThoughtBreakSpace\Th@ughtBre@k}% + \dimen@\pagegoal\advance\dimen@-\pagetotal % space left on page + \ifdim\ht\@tempboxa>\dimen@ % not enough space left + \ifdim\dimen@>\z@ % but there is some space to fill + \vskip-\pagedepth\@plus\@ne fil + \fi + \setbox\@tempboxa=\vbox{\ThoughtBreakRule}% without any space before + \ht\@tempboxa\z@\dp\@tempboxa\z@\box\@tempboxa + \break + \else % the normal thought break should fit + \box\@tempboxa + \ThoughtBreakSpace + \fi\endgroup} + +% PDF stuff: links, document info, etc +% Originally coded with a PostScript workflow in mind; +% if default driver given in hyperref.cfg is not suitable, +% add appropriate explicit option to hyperref call +\usepackage[final,colorlinks]{hyperref}[2003/11/30] +% we check if the driver is the useless (for pdf) "hypertex", +% and if so we force dvips instead and issue a warning +\usepackage{memhfixc}[2004/05/13] +\providecommand{\ebook}{2xydw} % real ebook number supplied during WW +\hypersetup{pdftitle=The Project Gutenberg eBook \#\ebook: Mathematical Recreations and Essays, + pdfsubject=Mathematical Recreations and Essays, + pdfauthor=W. W. Rouse Ball, + pdfkeywords={David Starner, Joshua Hutchinson, David Wilson, + Project Gutenberg Online Distributed Proofreading Team}, + pdfstartview=Fit, + pdfstartpage=1, + pdfpagemode=UseNone, + pdfdisplaydoctitle, + bookmarksopen, + bookmarksopenlevel=1, + linktocpage=false} +\ifPaper + \hypersetup{pdfpagescrop=0 0 499 709, b5paper, % b5 176x250mm + pdfpagelayout=TwoPageRight, + plainpages=false, linkcolor=\ifdraftdoc blue\else black\fi, + menucolor=\ifdraftdoc blue\else black\fi, + urlcolor=\ifdraftdoc magenta\else black\fi} +\else + \hypersetup{pdfpagescrop=0 0 648 432, % ebook 9x6" + pdfpagelayout=SinglePage, linkcolor=blue, menucolor=blue, + urlcolor=magenta} + \ifpdf\else % here we assume dvips or dvipsone is being used + \AtBeginDocument{\special{! <</PageSize [648 432]>> setpagedevice}} + \fi +\fi +% pdf annotations +\ifpdf % using native pdftex + \def\wm{\noindent\kern.5\textwidth\pdfannot width\textwidth height12pt {\wmGuts}} + \def\gext{pdf} +\else\expandafter\ifx\csname pdfmark\endcsname\relax\else % pdfmarks: we're using dvips or dvipsone + \def\wm{\kern-12pt\noindent\kern.5\textwidth\rlap{\pdfmark[\phantom{\vrule width\textwidth + height12pt}]{pdfmark=/ANN, Raw={\wmGuts}}}} + \def\gext{eps} +\fi\fi +\def\wmGuts{/Subtype /FreeText + /Contents (\string\200 Project \string\200 Gutenberg \string\200 \#\ebook\ \string\200) + /DA ([0.6875 0.6875 0.6875] r + /CoBo 8 Tf) /BS << /W 0 >> /F 37 /Q 1 /Title (PG) + /DS(font: bold Courier,monospace 8.0pt; 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#2; #3, #4, #5, #6){($#1$;~$#2$; $#3$,~$#4$, $#5$,~$#6$)} +% common abbreviations (can be modified if desired) +\def\IE{\textit{i.e.}} +\def\Eg{\textit{ex.~gr.}} +\def\EG{\textit{Ex.~gr.}} +\def\eg{\textit{e.g.}} +\def\ibid{\textit{ibid.}} +\def\Ibid{\textit{Ibid.}} +\def\etseq{\textit{et seq.}} +\def\sic{\textit{sic}} +% The standard summation symbol seems too overwhelming when used inline, +% especially without limits +% We could use \Sigma, but it is subtly different from \sum, hence... +\def\textsum{\raise.4ex\hbox{$\m@th\mathsmaller{\sum}$}} + +% a spacing kludge for one array +\def\DParraykludge{\setbox0=\copy\@arstrutbox\@tempdima=\ht0 + \advance\@tempdima by3pt\ht0=\@tempdima\box0 } + +% details for printing out the index +% +% Important note: because of the way memoir double-handles certain commands, +% whenever there's (for example) \textit{foo} inside an \index inside a footnote, +% it ends up in the .idx as "\textit {foo}" and if there are other \index commands +% (not in footnotes) for the same index entry, then these need to write the \textit +% explicitly with the two spaces, whereas inside the footnote it should be \textit{foo} +% with no spaces. Otherwise the \index commands from inside the footnotes won't get +% included with those from outside, and there will be two sets of entries in the index. +% This is important when maintining this file, because the apparently extraneous double +% spaces in various \index commands are intentional. See for example the various index +% entries for Bachet's Problèmes, Oughtred's Recreations and Ozanam's Recreations. +\noindexintoc +% to get balanced columns without doing it manually + + +\usepackage{multicol}[2006/05/18] +\onecolindextrue +\def\preindexhook{\phantomsection + \addtocontents{toc}{\protect\bigskip} + \addtocontents{toc}{\protect\let\protect\l@chapter\protect\l@section} + \DPaddcontentsline{toc}{chapter}{\protect\textsc{\indexname}} + \setlength{\columnseprule}{\z@}% + \setlength{\columnsep}{\indexcolsep}% + \multicols{2}\def\endtheindex + {\def\@currenvir{multicols}\endmulticols}} +\def\hyperspindexpage(#1)#2{\hyperlink{page.#1}{chap. \textsc{\@roman{#2}}}} +\def\PrintTheIndex{\begingroup +\small +% although the original starts the index recto (p379) there's no reason +% to suppose this is essential (p378 is not blank) since some chapters +% start verso in the original +\clearpage +\let\mainstuffChapNumOdd\empty +\let\mainstuffChapNumEven\empty +\setlength\indexcolsep{15pt} +\ifPaper + \renewcommand{\indexspace}{\par\penalty-3000 \vskip 10pt plus5pt minus3pt\relax} +\else + \renewcommand{\indexspace}{\par\penalty-3000 \vskip 12pt plus6pt minus4pt\relax} +\fi +\raggedright\hyphenpenalty=10000\emergencystretch.5\hsize +\printindex +\endgroup} +\def\Printer#1{\vspace*{\z@\@plus1fill}{\centering + \miniscule\hrule\smallskip#1\par}\eject} + +% hyphenation hints +\hyphenation{Win-ches-ter se-cond theo-ry} + +% avoid really short final lines of paragraphs if possible +\parfillskip=\z@\@plus0.85\textwidth + +% hard-code various document dimensions to try to minimise +% effects of changes to class/package defaults +% (these values mostly based on mempatch 4.9a) +% +% tiny amount of stretch in the parskip improves appearance of slightly-short pages +\parskip\z@\@plus\p@\relax +% these values are quite different from earlier versions of mempatch (eg 4.5) +% but seem to work quite well +\renewcommand*{\defaultlists}{% + \setlength{\partopsep}{0.2\onelineskip \@plus 0.1\onelineskip + \@minus 0.1\onelineskip}% + \parsepi = 0.3333\onelineskip \@plus 0.1667\onelineskip \@minus \p@ + \itemsepi = \parsepi + \topsepi = 0.6667\onelineskip \@plus 0.3333\onelineskip + \@minus 0.2\onelineskip + \parsepii = 0.1667\onelineskip \@plus \p@ \@minus \p@ + \topsepii = \parsepi + \topsepiii = \parsepii} +\defaultlists +\@listi + +\makeatother + +\makeindex + +\begin{document} +\PGx---File: 001.png-------------------------------------------------- +% MacMillan logo +\begingroup +\pagestyle{empty} +\ifPaper\pagenumbering{Alph}\fi % to ensure unique hyperref page anchors +\let\PGhook\PGlicencelink +\begin{verbatim} +The Project Gutenberg eBook of Mathematical Recreations and Essays, by W. W. Rouse Ball + +This eBook is for the use of anyone anywhere in the United States and +most other parts of the world 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. If you are not located in the United States, you +will have to check the laws of the country where you are located before +using this eBook. + +Title: Mathematical Recreations and Essays + +Author: W. W. Rouse Ball + +Release Date: October 8, 2008 [eBook #26839] +[Most recently updated: October 14, 2021] + +Language: English + +Character set encoding: UTF-8 + +*** START OF THIS PROJECT GUTENBERG EBOOK MATHEMATICAL RECREATIONS *** +\end{verbatim} +\clearpage +\null\vfil +\begin{center} +\includegraphics[height=1cm]{./images/macmillan2} +\vfil +\textit{\SMALL First Edition, Feb. $1892$. Reprinted, May, $1892$.} + +\textit{\SMALL Second Edition, $1896$. Reprinted, $1905$.} + +%% Note: the 1917 edition shows +%% First Edition, February, 1892. +%% Second Edition, May, 1892. +%% Third Edition, 1896. +%% Fourth Edition, 1905. +%% Fifth Edition, 1911. +%% Sixth Edition, 1914. +%% Seventh Edition, 1917. +%% which provides a timeline more consistent with the "Note to the fourth edition" below! + +\end{center} +\ifPaper\cleartorecto\else\newpage\fi +\endgroup +\ifPaper + \frontmatter +\else + \frontmatter* +\fi +\title{MATHEMATICAL\\ +RECREATIONS AND ESSAYS} + +\author{W.W. ROUSE BALL} +\affiliation{Fellow and Tutor of Trinity College, Cambridge.} +\Edition{FOURTH EDITION} + +\Publisher{\includegraphics[width=49bp]{./images/london}\\ % blackletter text in image +MACMILLAN AND CO., Limited\\ +\tiny NEW YORK: THE MACMILLAN COMPANY} +\date{1905} +\Rights{All rights reserved.} + +\maketitle + +\newpage + +\ifEditorials\else + \transcribersnote{Most of the open questions discussed by the author were +settled during the twentieth century.} +\fi + +\transcribersnote{The author's footnotes are labelled using printer's + marks\footnotemark; footnotes showing where corrections to the text + have been made are labelled numerically\ifEditorials, + as are explanatory notes\fi\footnotemarkT.} + +\transcribersnote{\SMALL Minor typographical corrections are documented in the \LaTeX\ source.} + + +\makecopyrightpage +\ifPaper\cleartorecto\fi +\PGx---File: 002.png-------------------------------------------------- +% MATHEMATICAL +% RECREATIONS AND ESSAYS +% +% BY +% W.W. ROUSE BALL +% FELLOW AND TUTOR OF TRINITY COLLEGE, CAMBRIDGE. +% +% \textit{FOURTH EDITION} +% +% London: +% MACMILLAN AND CO., LIMITED +% NEW YORK: THE MACMILLAN COMPANY +% 1905 +% [\textit{All rights reserved.}] +% +\PGx---File: 003.png-------------------------------------------------- +% \textit{First Edition, Feb. 1892. Reprinted, May, 1892 +% Second Edition, 1896. Reprinted, 1905}. +\PGx---File: 004.png------------------------------------------------------ + +\chapter*[Preface]{PREFACE TO THE FIRST EDITION.} + +{\advance\baselineskip1ex +\DPpdfbookmark[0]{Preface to the First Edition}{Preface*1} +\textsc{The} following pages contain an account of certain mathematical +recreations, problems, and speculations of past +and present times. I hasten to add that the conclusions are +of no practical use, and most of the results are not new. If +therefore the reader proceeds further he is at least forewarned. + +At the same time I think I may assert that many of the +diversions---particularly those in the latter half of the book---are +interesting, not a few are associated with the names of +distinguished mathematicians, while hitherto several of the +memoirs quoted have not been easily accessible to English +readers. + +The book is divided into two parts, but in both parts I have +included questions which involve advanced mathematics. + +The \emph{\hyperlink{part.1}{first part}} consists of seven chapters, in which are +included various problems and amusements of the kind usually +called \emph{mathematical recreations}. The questions discussed in +the first of these chapters are connected with \hyperlink{chapter.1}{arithmetic}; those +in the second with \hyperlink{chapter.2}{geometry}; and those in the third relate to +\hyperlink{chapter.3}{mechanics}. The \hyperlink{chapter.4}{fourth chapter} contains an account of some +miscellaneous problems which involve both number and situation; +the \hyperlink{chapter.5}{fifth chapter} contains a concise account of magic +squares; and the \hyperlink{chapter.6}{sixth and seventh} chapters deal with some +\PG----File: 005.png----------------------------------------------------- +unicursal problems. Several of the questions mentioned in +the first three chapters are of a somewhat trivial character, +and had they been treated in any standard English work to +which I could have referred the reader, I should have pointed +them out. In the absence of such a work, I thought it best +to insert them and trust to the judicious reader to omit them +altogether or to skim them as he feels inclined. + +The \emph{\hyperlink{part.2}{second part}} consists of five chapters, which are mostly +\emph{historical}. They deal respectively with \hyperlink{chapter.8}{three classical problems} +in geometry---namely, the duplication of the cube, the trisection +of an angle, and the quadrature of the circle---\hyperlink{chapter.10}{astrology}, +the hypotheses as to the nature of \hyperlink{chapter.12}{space} and \hyperlink{chapter.14}{mass}, and a +means of \hyperlink{chapter.13}{measuring time}. + +I have inserted detailed references, as far as I know, as +to the sources of the various questions and solutions given; +also, wherever I have given only the result of a theorem, I have +tried to indicate authorities where a proof may be found. In +general, unless it is stated otherwise, I have taken the references +direct from the original works; but, in spite of considerable +time spent in verifying them, I dare not suppose that they are +free from all errors or misprints. + +I shall be grateful for notices of additions or corrections +which may occur to any of my readers. + +}\bigskip +\vbox{\rightline{W.W. ROUSE BALL} + \bigskip + \setbox0=\hbox{\small\textsc{Trinity College, Cambridge.}} + \setbox1=\hbox to\wd0{\small\hfil\textit{February}, 1892.\hfil} + \indent\box0\par + \indent\box1 +} + +\PG----File: 006.png------------------------------------------------------- + +{\ifPaper\else\setlength{\beforechapskip}{0.75pc}\fi +\chapter*{NOTE TO THE FOURTH EDITION.} + +\advance\baselineskip1ex +\DPpdfbookmark[0]{Note to the Fourth Edition}{Preface*2} +\textsc{In} this edition I have inserted in the earlier chapters +descriptions of several additional Recreations involving elementary +mathematics, and I have added in the second part +chapters on the \emph{History of the \hyperlink{chapter.7}{Mathematical Tripos} at Cambridge}, +\emph{\hyperlink{chapter.9}{Mersenne's} \hyperlink{chapter.9}{Numbers}}, and \emph{\hyperlink{chapter.11}{Cryptography and Ciphers}}. + +It is with some hesitation that I include in the book the +chapters on \emph{\hyperlink{chapter.10}{Astrology}} and \emph{\hyperlink{chapter.11}{Ciphers}}, for these subjects are only +remotely connected with Mathematics, but to afford myself +some latitude I have altered the title of the \hyperlink{part.2}{second part} to +\emph{Miscellaneous Essays and Problems}. + +}\ifPaper\bigskip\else\smallskip\fi +\vbox{\rightline{W.W.R.B.} + \ifPaper\bigskip\else\smallskip\fi + \setbox0=\hbox{\small\textsc{Trinity College, Cambridge.}} + \setbox1=\hbox to\wd0{\small\hfil13 \textit{May}, 1905.\hfil} + \indent\box0\par + \indent\box1 +} + +\PGx---File: 007.png---------------------------------------------------- +\PGx---File: 008.png------------------------------------------------------ +\PGx---File: 009.png------------------------------------------------------ +\PGx---File: 010.png------------------------------------------------------ +\PGx---File: 011.png------------------------------------------------------ +\PGx---File: 012.png------------------------------------------------------- +\PGx---File: 013.png------------------------------------------------------ +\PGx---File: 014.png------------------------------------------------------- +\PGx---File: 015.png------------------------------------------------------ +\PG----File: 016.png------------------------------------------------------- +%ERRATA. +% +%Page 36, line 9. \emph{After} all \emph{insert} even. % [** fixed] +% +%Page 58, line 16. \emph{For} 13 \emph{read} 15. % [** fixed] +% +%Page 229, line 3. \emph{For} 1850 \emph{read} 1851. % [** fixed] +% +%Page 232, line 4. \emph{Before} 1854 \emph{insert} 1853 and. % [** fixed] +% +%Page 363, footnote \dag, and Page 364, footnote *. \emph{These footnotes +%should be interchanged.} % [** fixed] +% +\PG----File: 017.png------------------------------------------------------ + +\clearpage +\DPpdfbookmark[0]{Table of Contents}{ToC*1} +\tableofcontents* + +% now disable formatting commands used inside ToC, so bookmarks come out OK +% (could also use \pdfstringdefDisableCommands, but this is less work) +\let\tocsecbox\empty +\let\tocsecspacer\empty + +\PG----File: 018.png------------------------------------------------------ +\ifPaper + \mainmatter +\else + \mainmatter* +\fi +\pagestyle{mainstuff} + +%PART I. +\PartQuote{``Les hommes ne sont jamais plus ingénieux que dans l'invention +des jeux; l'esprit s'y trouve à son aise\textellipsis. Après les jeux +qui dépendent uniquement des nombres viennent les jeux où +entre la situation\textellipsis. Après les jeux où n'entrent +que le nombre et la situation viendraient les jeux où entre +le mouvement\textellipsis. Enfin +il serait a souhaiter qu'on eût un cours entier des jeux, traités +mathématiquement.'' \hfill\qquad\penalty-1000\null\hfill +\hbox{\emph{(Leibnitz\index{Leibnitz on Games}: letter to De~Montmort\index +{DeMontmort@De Montmort}\index{Montmort, De}, July~29, 1715.)}}} +\part{\PartOneText} + +\PGx---File: 019.png------------------------------------------------------- + + + + +%CHAPTER I. +\chapter[Some Arithmetical Questions.][Arithmetical Recreations.]% +{Some Arithmetical Questions.} + +\textsc{The} interest excited by statements of the relations +between\chapindex{Arithmetical Recreations@\nobreak--- \textsc{Recreations}} +numbers of certain forms has been often remarked. The +majority of works on mathematical recreations include several +such problems, which are obvious to any one acquainted with +the elements of algebra, but which to many who are ignorant +of that subject possess the same kind of charm that some +mathematicians find in the more recondite propositions of +higher arithmetic. I shall devote the bulk of this chapter to +these elementary problems, but I append a few remarks on +one or two questions in the theory of numbers. + +Before entering on the subject of the chapter, I may add +that a large proportion of the elementary questions mentioned +here and in the following two chapters are taken from one +of two sources. The first of these is the classical \textit{Problèmes +plaisans et délectables}, by C.G.~Bachet\index +{Bachet@Bachet's \textit {Problèmes}|(}, sieur de Méziriac, of +which the first edition was published in 1612 and the second +in 1624: it is to the edition of 1624 that the references hereafter +given apply. Several of Bachet's problems are taken +from the writings of Alcuin\index{Alcuin}, Pacioli di~Burgo\index +{Lucas di Burgo}\index{Pacioli di Burgo}, Tartaglia\index{Tartaglia}, or +Cardan\index{Cardan}, and possibly some of them are of oriental origin, +but I have made no attempt to add such references. The +other source to which I alluded above is Ozanam's \textit{Récréations +mathématiques et physiques}\index +{Ozanam@Ozanam's \textit {Récréations}}. +The greater portion of the original +edition, published in two volumes at Paris in 1694, was a +compilation from the works of Bachet, Leurechon\index{Leurechon}, +Mydorge\index{Mydorge}, +\PG----File: 020.png------------------------------------------------------- +van Etten\index{Etten, van}\index{VanEtten@Van Etten}, and Oughtred\index +{Oughtreds@Oughtred's \textit {Recreations}}: this part is excellent, +but the +same cannot be said of the additions due to Ozanam\index +{Ozanam@Ozanam's \textit {Récréations}}. In the +\textit{Biographie Universelle} allusion is made to subsequent editions +issued in 1720, 1735, 1741, 1778, and 1790; doubtless these +references are correct, but the following editions, all of which +I have seen, are the only ones of which I have any knowledge. +In 1696 an edition was issued at Amsterdam. In +1723---six years after the death of Ozanam---one was issued +in three volumes, with a supplementary fourth volume, containing +(among other things) an appendix on puzzles: I +believe that it would be difficult to find in any of the books +current in England on mathematical amusements as many as +a dozen puzzles which are not contained in one of these four +volumes. Fresh editions were issued in 1741, 1750 (the +second volume of which bears the date 1749), 1770, and +1790. The edition of 1750 is said to have been corrected +by Montucla\index{Montucla} on condition that his name should not be +associated with it; but the edition of 1790 is the earliest one +in which reference is made to these corrections, though the +editor is referred to only as Monsieur M***. Montucla\index{Montucla} +expunged most of what was actually incorrect in the older +editions, and added several historical notes, but unfortunately +his scruples prevented him from striking out the accounts of +numerous trivial experiments and truisms which overload the +work. An English translation of the original edition appeared +in 1708, and I believe ran through four editions, the last of +them being published in Dublin in 1790. Montucla's\index{Montucla} revision +of 1790 was translated by C.~Hutton\index{Hutton, C.}, and editions of this +were issued in 1803, in 1814, and (in one volume) in 1840: +my references are to the editions of 1803 and 1840. + +\ThoughtBreakSpace + +I proceed now to enumerate some of the elementary questions +connected with numbers which for nearly three centuries +have formed a large part of most compilations of mathematical +amusements. They are given here mainly for their historical---not +for their arithmetical---interest; and perhaps a mathematician +\PG----File: 021.png----------------------------------------------------- +may well omit them, and pass at once to the latter +part of this chapter. + +These questions are of the nature of tricks or puzzles +and I follow the usual course and present them in that form. +I may note however that most of them are not worth proposing, +even as tricks, unless either the \textit{modus operandi} is +disguised or the result arrived at is different from that +expected; but, as I am not writing on conjuring, I refrain +from alluding to the means of disguising the operations +indicated, and give merely a bare enumeration of the steps +essential to the success of the method used, though I may +recall the fundamental rule that no trick, however good, will +bear immediate repetition, and that, if it is necessary to +appear to repeat it, a different method of obtaining the result +should be used. + +\ssection[Elementary Questions on Numbers (Miscellaneous)]% +[Elementary Tricks and Problems]{To find a number selected by some one} +There are\index{Arithmetical Puzzles@\textsc{Arithmetical Puzzles}|(}% +\index{NumbersPuzzle@\nobreak--- \textsc{Puzzles with}|(}% +\index{PuzzlesArith@\textsc{Puzzles}, Arithmetical|(}% +\index{Tricks@\textsc{Tricks with Numbers}|(} +innumerable ways of finding a number chosen by some one, +provided the result of certain operations on it is known. I +confine myself to methods typical of those commonly used. +Any one acquainted with algebra will find no difficulty in +modifying the rules here given or framing new ones of an +analogous nature. + +\subsection*{First Method\/\protect\footnote +{Bachet, \textit{Problèmes plaisans}, Lyons, 1624, problem~\textsc{i}, +p.~53.}} (i)~Ask the person who has chosen the +number to treble it. (ii)~Enquire if the product is even or +odd: if it is even, request him to take half of it; if it is odd, +request him to add unity to it and then to take half of it. +(iii)~Tell him to multiply the result of the second step by $3$. +(iv)~Ask how many integral times $9$ divides into the latter +product: suppose the answer to be $n$. (v)~Then the number +thought of was $2n$ or $2n + 1$, according as the result of step (i) +was even or odd. + +The demonstration is obvious. Every even number is of +the form $2n$, and the successive operations applied to this +give (i)~$6n$, which is even; (ii)~$\frac{1}{2}6n = 3n$; (iii)~$3 \times + 3n = 9n$; +(iv)~$\frac{1}{9}9n = n$; (v)~$2n$. Every odd number is of the form +\PG----File: 022.png------------------------------------------------------ +$2n + 1$, and the successive operations applied to this give +(i)~$6n + 3$, which is odd; (ii)~$\tfrac{1}{2}(6n + 3 + 1) = 3n + 2$; +(iii)~$3 (3n + 2) = 9n + 6$; (iv)~$\tfrac{1}{9} (9n + 6) = n + \text +{a remainder}$; +(v)~$2n+1$. These results lead to the rule given above. + +\subsection*{Second Method\/\protect\footnote +{A similar rule was given by Bachet, problem~\textsc{iv}, p.~74. +}} Ask the person who has chosen the +number to perform in succession the following operations. +(i)~To multiply the number by $5$. (ii)~To add $6$ to the product. +(iii)~To multiply the sum by $4$. (iv)~To add $9$ to the +product. (v)~To multiply the sum by $5$. Ask to be told the +result of the last operation: if from this product $165$ is subtracted, +and then the remainder is divided by $100$, the quotient +will be the number thought of originally. + +For let $n$ be the number selected. Then the successive +operations applied to it give (i)~$5n$; (ii)~$5n + 6$; (iii)~$20n + 24$; +(iv)~$20n+33$; (v)~$100n+ 165$. Hence the rule. + +\subsection*{Third Method\/\protect\footnote +{Bachet, problem~\textsc{v}, p.~80.}} Request the person who has thought of +the number to perform the following operations. (i)~To +multiply it by any number you like, say,~$a$. (ii)~To divide the +product by any number, say,~$b$. (iii)~To multiply the quotient +by~$c$. (iv)~To divide this result by~$d$. (v)~To divide the +final result by the number selected originally. (vi)~To add +to the result of operation (v) the number thought of at first. +Ask for the sum so found: then, if $ac/bd$ is subtracted from +this sum, the remainder will be the number chosen originally. + +For, if $n$ was the number selected, the result of the first four +operations is to form $nac/bd$; operation (v) gives $ac/bd$; and +(vi) gives $n + (ac/bd)$, which number is mentioned. But $ac/bd$ +is known; hence, subtracting it from the number mentioned, +$n$ is found. Of course $a$, $b$, $c$, $d$ may have any numerical values +it is liked to assign to them. For example, if $a =12$, $b = 4$, +$c = 7$, $d = 3$ it is sufficient to subtract $7$ from the final result +in order to obtain the number originally selected. + +\subsection*{Fourth Method\/\protect\footnote +{\textit{Educational Times}, London, May~1, 1895, vol.~\textsc{xlviii}, +p.~234.}} Ask some one to select a number less +\PG----File: 023.png----------------------------------------------------- +than $90$. (i)~Request him to multiply it by $10$, and to add +any number he pleases, $a$, which is less than $10$. (ii)~Request +him to divide the result of step (i) by $3$, and to mention the +remainder, say, $b$. (iii)~Request him to multiply the quotient +obtained in step (ii) by $10$, and to add any number he pleases, +$c$, which is less than $10$. (iv)~Request him to divide the result +of step (iii) by $3$, and to mention the remainder, say $d$, and +the third digit (from the right) of the quotient; suppose +this digit is $e$. Then, if the numbers $a$, $b$, $c$, $d$, $e$ are known, +the original number can be at once determined. In fact, if +the number is $9x + y$, where $x \ngtr 9$ and $y \ngtr 8$, and if $r$ is +the remainder when $a-b + 3 (c-d)$ is divided by $9$, we have +$x = e$, $y=9-r$. + +The demonstration is not difficult. For if the selected +number is $9x + y$, step (i) gives $90x + 10y + a$; (ii)~let +$y + a = 3n + b$, then the quotient obtained in step (ii) is +$30x + 3y + n$; step (iii) gives $300x + 30y + 10n + c$; (iv)~let +$n + c = 3m + d$, then the quotient obtained in step (iv) is +$100x+ 10y+ 3n + m$, which I will denote by $Q$. Now the +third digit in $Q$ must be $x$, because, since $y\ngtr 8$ and $a \ngtr 9$, +we have $n \ngtr 5$; and since $n \ngtr 5$ and $c \ngtr 9$, we have +$m \ngtr 4$; +therefore $10y + 3n + m \ngtr 99$. Hence the third or hundreds +digit in $Q$ is $x$. + +Again, from the relations $y + a = 3n + b$ and $n + c = 3m + d$, +we have $9m-y = a-b + 3(c-d)$: hence, if $r$ is the remainder +when $a-b +3(c-d)$ is divided by $9$, we have $y=9-r$. [This +is always true, if we make $r$ positive; but if $a-b + 3 (c-d)$ +is negative, it is simpler to take $y$ as equal to its numerical +value; or we may prevent the occurrence of this case by +assigning proper values to $a$ and $c$.] Thus $x$ and $y$ are both +known, and therefore the number selected, namely $9x + y$, is +known. + +\subsection*{Fifth Method\/\protect\footnote +{Bachet, problem~\textsc{vi}, p.~84: Bachet added, on p.~87, a note on the +previous history of the problem. +}} Ask any one to select a number less +than $60$. (i)~Request him to divide it by $3$ and mention the +\PG----File: 024.png------------------------------------------------------ +remainder; suppose it to be $a$. (ii)~Request him to divide it +by $4$, and mention the remainder; suppose it to be $b$. (iii)~Request +him to divide it by $5$, and mention the remainder; +suppose it to be $c$. Then the number selected is the remainder +obtained by dividing $40a + 45b + 36c$ by $60$. + +This method can be generalized and then will apply to any +number chosen. Let $a', b', c', \ldots$ be a series of numbers prime +to one another, and let $p$ be their product. Let $n$ be any +number less than $p$, and let $a, b, c, \ldots$ be the remainders +when $n$ is divided by $a', b', c', \ldots$ respectively. Find a number +$A$ which is a multiple of the product $b'c'd'\dotsm$ and which +exceeds by unity a multiple of $a'$. Find a number $B$ which +is a multiple of $a'c'd'\dotsm$ and which exceeds by unity a multiple +of $b'$; and similarly find analogous numbers $C, D, \dots$. Rules +for the calculation of $A, B, C, \ldots$ are given in the theory of +numbers, but in general, if the numbers $a', b', c', \ldots$ are small, +the corresponding numbers, $A, B, C, \ldots$ can be found by inspection. +I proceed to show that $n$ is equal to the remainder +when $Aa + Bb + Cc + \dotsb$ is divided by $p$. + +Let $N = Aa + Bb + Cc + \dotsb$, and let $M(x)$ stand for a +multiple of $x$. + +Now $A = M(a') + 1$, therefore $Aa = M(a') + a$. Hence, if +the first term in $N$, that is $Aa$, is divided by $a'$, the remainder +is $a$. Again, $B$ is a multiple of $a'c'd'\dotsm$. Therefore $Bb$ is +exactly divisible by $a'$. Similarly $Cc, Dd, \ldots$ are each exactly +divisible by $a'$. Thus every term in $N$, except the first, is +exactly divisible by $a'$. Hence, if $N$ is divided by $a'$, the +remainder is $a$. But if $n$ is divided by $a'$, the remainder is $a$. +\begin{LRalign} +Therefore & N-n &= M(a')\,. \\ +Similarly & N-n &= M(b')\,, \\ + & N-n &= M(c')\,, \\ + & \multispan{2}{\dotfill} \\ +\indent But $a', b', c', \ldots$ are prime to one another.\\ +&\Therefore N-n &= M(a'b'c' \dotsm)\rlap{${} = M(p)\,,$} \\ +that is,& N &= M(p) + n\,.\\ +\end{LRalign} +\PG----File: 025.png------------------------------------------------------- +Now $n$ is less than $p$, hence if $N$ is divided by $p$, the +remainder is $n$. + +The rule given by Bachet corresponds to the case of $a' = 3$, +$b'= 4$, $c' = 5$, $p = 60$, $A = 40$, $B = 45$, $C = 36$. If the number +chosen is less than 420, we may take $a' = 3$, $b' = 4$, $c' = 5$, $d'= 7$, +$p = 420$, $A = 280$, $B = 105$, $C = 336$, $D = 120$. + + +\ssection*{To find the result of a series of operations performed +on any number (\emph{unknown to the questioner}) without asking +any questions} All rules for solving such problems ultimately +depend on so arranging the operations that the number disappears +from the final result. Four examples will suffice. + +\subsection*{First Example\/\protect\footnote +{Bachet, problem~\textsc{viii}, p.~102. +}} Request some one to think of a number. +Suppose it to be $n$. Ask him (i)~to multiply it by any number +you please (say) $a$; (ii)~then to add (say) $b$; (iii)~then to divide +the sum by (say) $c$. (iv)~Next, tell him to take $a/c$ of the +number originally chosen; and (v)~to subtract this from the +result of the third operation. The result of the first three +operations is $(na + b)/c$, and the result of operation (iv) is +$na/c$: the difference between these is $b/c$, and therefore is +known to you. For example, if $a = 6$, $b = 12$, $c = 4$, and +$a/c = 1\frac{1}{2}$, then the final result is $3$. + +\subsection*{Second Example\/\protect\footnote +{Bachet, problem~\textsc{xiii}, p.~123: Bachet presented the above trick in +a somewhat more general form, but one which is less effective in practice. +}} Ask $A$ to take any number of counters +that he pleases: suppose that he takes $n$ counters. (i)~Ask +some one else, say $B$, to take $p$ times as many, where $p$ is +any number you like to choose. (ii)~Request $A$ to give $q$ of +his counters to $B$, where $q$ is any number you like to select. +(iii)~Next, ask $B$ to transfer to $A$ a number of counters equal +to $p$ times as many counters as $A$ has in his possession. Then +there will remain in $B$'s hands $q (p + 1)$ counters: this number +is known to you; and the trick can be finished either by +mentioning it or in any other way you like. + +The reason is as follows. The result of operation (ii) is +that $B$ has $pn + q$ counters, and $A$ has $n-q$ counters. The +\PG----File: 026.png----------------------------------------------------- +result of (iii) is that $B$ transfers $p(n-q)$ counters to $A$: hence +he has left in his possession $(pn + q)-p (n-q)$ counters, that +is, he has $q (p + 1)$. + +For example, if originally $A$ took any number of counters, +then (if you chose $p$ equal to $2$), first you would ask $B$ to take +twice as many counters as $A$ had done; next (if you chose $q$ +equal to $3$) you would ask $A$ to give $3$ counters to $B$; and +then you would ask $B$ to give to $A$ a number of counters equal +to twice the number then in $A$'s possession; after this was +done you would know that $B$ had $3 (2 + 1)$, that is, $9$ left. + +This trick (as also some of the following problems) may be +performed equally well with one person, in which case $A$ may +stand for his right hand and $B$ for his left hand. + +\subsection*{Third Example} Ask some one to perform in succession +the following operations. (i)~Take any number of three +digits. (ii)~Form a new number by reversing the order of +the digits. (iii)~Find the difference of these two numbers. +(iv)~Form another number by reversing the order of the digits +in this difference. (v)~Add together the results of (iii) and (iv). +Then the sum obtained as the result of this last operation will +be $1089$. + +An illustration and the explanation of the rule are given +below. +\[ +\begin{array}{crr@{}c@{}r@{}c@{}l} +\text{(i)} &237 &100a &{}+{}&10b &{}+{}&c\\ +\addlinespace +\text{(ii)} &732 &100c &{}+{}&10b &{}+{}&a\\ +\cmidrule(lr){2-2}\cmidrule(lr){3-7} +\text{(iii)} &495 &\multicolumn{1}{l@{}}{100(a-c-1)} &+&90 &+&(10+c-a)\\ +\addlinespace +\text{(iv)} &594 &\multicolumn{1}{l@{}}{100(10+c-a)} &+&90 &+&(a-c-1)\\ +\cmidrule(lr){2-2}\cmidrule(lr){3-7} +\text{(v)} &1089 &\multicolumn{1}{c}{900} &+&180 &+&9\\ +\cmidrule(lr){2-2}\cmidrule(lr){3-7} +\specialrule{0pt}{-1pt}{-1pt} % yes I know double rules are frowned upon! +\cmidrule(lr){2-2}\cmidrule(lr){3-7} +\end{array} +\] + +\subsection*{Fourth Example\/\protect\footnote +{\textit{Educational Times Reprints}, 1890, vol.~\textsc{liii}, p.~78.}} +The following trick\index{Money, Question on|(} depends on the same +principle. Ask some one to perform in succession the following +operations. (i)~To write down any sum of money less than +\pounds12; the number of pounds not being the same as the number +of pence. (ii)~To \emph{reverse} this sum, that is, to write down a +\PG----File: 027.png------------------------------------------------------ +sum of money obtained from it by interchanging the numbers +of pounds and pence. (iii)~To find the difference between the +results of (i) and (ii). (iv)~To reverse this difference. (v)~To +add together the results of (iii) and (iv). Then this sum will +be \pounds12.~18\textit{s}.~11\textit{d}. + +For instance, take the sum \pounds10.~17\textit{s}.~5\textit{d}.; we have +\[ +\begin{tabular}{rlrrr} +& & \pounds. & \textit{s}.& \textit{d}. \\ +(i) & \hbox to2cm{\dotfill} & 10 & 17 & 5 \\ +\addlinespace +(ii) & \dotfill & 5 & 17 & 10 \\ +\cmidrule(lr){3-5} +(iii) & \dotfill & 4 & 19 & 7 \\ +\addlinespace +(iv) & \dotfill & 7 & 19 & 4 \\ +\cmidrule(lr){3-5} +(v) & \dotfill & 12 & 18 & 11 \\ +\cmidrule(lr){3-5} +\specialrule{0pt}{-1pt}{-1pt} % yes I know double rules are frowned upon! +\cmidrule(lr){3-5} +\end{tabular} +\] + +The following work explains the rule, and shows that the +final result is independent of the sum written down initially. +\[ +\begin{tabular}{rlccc} +& & \pounds. & \textit{s}. & \textit{d}. \\ +(i) & \hbox to2cm{\dotfill} & $a$ & $b$ & $c$ \\ +\addlinespace +(ii) & \dotfill & $c$ & $b$ & $a$ \\ +\cmidrule(lr){3-5} +(iii) & \dotfill & $a-c-\phantom{1}1$ & $\quad 19 \quad$ & $c-a+12$ \\ +\addlinespace +(iv) & \dotfill & $c-a + 12$ & $19$ & $a-c-\phantom{1}1$ \\ +\cmidrule(lr){3-5} +(v) & \dotfill & $11$ & $38$ & $11$ \\ +\cmidrule(lr){3-5} +\specialrule{0pt}{-1pt}{-1pt} % yes I know double rules are frowned upon! +\cmidrule(lr){3-5} +\end{tabular} +\] + +The rule can be generalized to cover any system of monetary +units\index{Money, Question on|)}. + +\ssection*{Problems involving Two Numbers} I proceed next to +give a couple of examples of a class of problems which involve +two numbers. + +\subsection*{First Example\/\protect\footnote +{Bachet, problem~\textsc{ix}, p.~107. +}} Suppose that there are two numbers, one +even and the other odd, and that a person $A$ is asked to select +one of them, and that another person $B$ takes the other. It is +desired to know whether $A$ selected the even or the odd number. +Ask $A$ to multiply his number by 2 (or any even number) and +$B$ to multiply his by $3$ (or any odd number). Request them +\PG----File: 028.png----------------------------------------------------- +to add the two products together and tell you the sum. If it +is even, then originally $A$ selected the odd number, but if it is +odd, then originally $A$ selected the even number. The reason +is obvious. + +\subsection*{Second Example\/\protect\footnote +{Bachet, problem~\textsc{xi}, p.~113.}} +The above rule was extended by Bachet +to any two numbers, provided they were prime to one another +and one of them was not itself a prime. Let the numbers be +$m$ and $n$, and suppose that $n$ is exactly divisible by $p$. Ask $A$ +to select one of these numbers, and $B$ to take the other. Choose +a number prime to $p$, say $q$. Ask $A$ to multiply his number by +$q$, and $B$ to multiply his number by $p$. Request them to add +the products together and state the sum. Then $A$ originally +selected $m$ or $n$, according as this result is not or is divisible +by $p$. For example, $m = 7$, $n= 15$, $p = 3$, $q=2$. + +\ssection*{Problems depending on the Scale of Notation} Many% +\index{Denary scale of notation}% +\index{Notation, Denary scale of}% +\index{Scale of Notation, Denary}% +\index{Scale Puzzles@\nobreak--- Puzzles dependent on|(} +of the rules for finding two or more numbers depend on the +fact that in arithmetic an integral number is denoted by +a succession of digits, where each digit represents the product +of that digit and a power of ten, and the number is equal +to the sum of these products. For example, $2017$ signifies +$(2 \times 10^3) + (0 \times 10^2) + (1 \times 10)+ 7$; that is, the $2$ +represents +$2$ thousands, \IE\ the product of $2$ and $10^3$, the $0$ represents +$0$ hundreds, \IE\ the product of $0$ and $10^2$; the $1$ represents +$1$ ten, \IE\ the product of $1$ and $10$, and the $7$ represents +$7$ units. Thus every digit has a local value. + +The application to tricks connected with numbers will be +understood readily from three illustrative examples. + +\subsection*{First Example\/\protect\footnote +{Some similar questions were given by Bachet in problem~\textsc{xii}, p.~117; +by Oughtred\index{Oughtreds@Oughtred's \textit{Recreations}} in his \textit +{Mathematicall Recreations} (translated from or founded +on van Etten's\index{Etten, van}\index{VanEtten@Van Etten} work of 1633), +London, 1653, problem \textsc{xxxiv}; and by +Ozanam\index{Ozanam@Ozanam's \textit{Récréations}}, part~\textsc{i}, +chapter~\textsc{x}.}} +A common conjuring trick is to ask a boy +among the audience to throw two dice, or to select at random +from a box a domino on each half of which is a number. The +boy is then told to recollect the two numbers thus obtained, to +\PG----File: 029.png---------------------------------------------------- +choose either of them, to multiply it by $5$, to add $7$ to the +result, to double this result, and lastly to add to this the other +number. From the number thus obtained, the conjurer subtracts +$14$, and obtains a number of two digits which are the +two numbers chosen originally. + +For suppose that the boy selected the numbers $a$ and $b$. +Each of these is less than ten---dice or dominoes ensuring this. +The successive operations give (i)~$5a$; (ii)~$5a + 7$; (iii)~$10a + 14$; +(iv)~$10a + 14 + b$. Hence, if 14 is subtracted from the final +result, there will be left a number of two digits, and these +digits are the numbers selected originally. An analogous +trick might be performed in other scales of notation if it was +thought necessary to disguise the process further. + +\subsection*{Second Example\protect\footnote +{Bachet gave some similar questions in problem~\textsc{xii}, p.~117.}} +Similarly, if three numbers, say, $a$, $b$, $c$, +are chosen, then, if each of them is less than ten, they +can be found by the following rule. (i)~Take one of the +numbers, say, $a$, and multiply it by $2$. (ii)~Add $3$ to the +product; the result is $2a + 3$. (iii)~Multiply this by $5$, and +add $7$ to the product; the result is $10a + 22$. (iv)~To +this sum add the second number. (v)~Multiply the result +by $2$. (vi)~Add $3$ to the product. (vii)~Multiply by $5$, and +add the third number to the product. The result is +$100a+ 10b + c + 235$. Hence, if the final result is known, it +is sufficient to subtract $235$ from it, and the remainder will +be a number of three digits. These digits are the numbers +chosen originally. + +I have seen a similar rule applied to determine the birthday +and age of some one in the audience. The result is a number +of six digits, of which the first two digits give the day of +the month, the middle two digits the number of the month, +and the last two digits the present age. + +\subsection*{Third Example\protect\footnote +{A similar question was given by Laisant\index{Laisant, C.A.} and +Perrin\index{Perrin} in their \textit{Algèbre}, +Paris, 1892; and in \textit{L'Illustration} for July~13, 1895.}} +The following rule for finding a man's +age is of the same kind. Take the tens digit of the year of +\PG----File: 030.png---------------------------------------------------- +birth; (i)~multiply it by $5$; (ii)~to the product add $2$; +(iii)~multiply the result by $2$; (iv)~to this product add the +units digit of the birth-year; (v)~subtract the sum from $110$. +The result is the man's age in 1906. + +The algebraic proof of the rule is obvious. Let $a$ and $b$ be +the tens and units digits of the birth-year. The successive +operations give (i)~$5a$; (ii)~$5a + 2$; (iii)~$10a + 4$ (iv)~$10a + 4 + b$; +(v)~$106 -(10a + b)$, which is his age in 1906. The rule can be +easily adapted to give the age in any specified year. + +\ssection*{Other Problems with numbers in the denary scale} +I may mention here two or three other slight problems\index +{Denary scale of notation} +dependent on the common scale of notation, which, as far as I +am aware, are unknown to most compilers of books of puzzles. + +\subsection*{First Problem} The first of them is as follows. Take any +number of three digits: reverse the order of the digits: subtract +the number so formed from the original number: then, if +the last digit of the difference is mentioned, all the digits in +the difference are known. + +For let $a$ be the hundreds digit of the number chosen, $b$ be +the tens digit, and $c$ be the units digit. Therefore the number +is $100a + 10b +c$. The number obtained by reversing the digits +is $100c + 10b + a$. The difference of these numbers is equal to +$(100a + c)-(100c + a)$, that is, to $99 (a-c)$. But $a-c$ is not +greater than $9$, and therefore the remainder can only be $99$, +$198$, $297$, $396$, $495$, $594$, $693$, $792$, or $891$---in each case the +middle digit being $9$ and the digit before it (if any) being equal +to the difference between $9$ and the last digit. Hence, if the +last digit is known, so is the whole of the remainder. + +\subsection*{Second Problem} The second problem is somewhat similar +and is as follows. (i)~Take any number; (ii)~reverse the +digits; (iii)~find the difference between the number formed in +(ii) and the given number; (iv)~multiply this difference by +any number you like to name; (v)~cross out any digit except +a nought; (vi)~read the remainder. Then the sum of the +digits in the remainder subtracted from the next highest multiple +of nine will give the figure struck out. +\PG----File: 031.png---------------------------------------------------- + +This follows at once from the fact that the result of operation +(iii)---and therefore also of operation (iv)---is necessarily a +multiple of nine, and it is known that the sum of the digits of +every multiple of nine is itself a multiple of nine\index +{Scale Puzzles@\nobreak--- Puzzles dependent on|)}. + +\subsection*{Miscellaneous Questions} Besides these problems, properly +so called, there are numerous questions on numbers which can +be solved empirically, but which are of no special mathematical +interest. + +As an instance I may quote a question which attracted +some attention in London in 1893, and may be enunciated +as follows. With the seven digits $9$, $8$, $7$, $6$, $5$, $4$, $0$ express +three numbers whose sum is $82$: each digit, being used +only once, and the use of the usual notations for fractions +being allowed. One solution is $80.6\dot{9} + .7\dot{4} + .\dot{5}$. Similar +questions are with the ten digits, $9$, $8$, $7$, $6$, $5$, $4$, $3$, $2$, +$1$, $0$, to +express numbers whose sum is unity; a solution is $35/70$ and +$148/296$. If the sum were $100$, a solution would be $50$, $49$, $1/2$, +% silently making $\frac{1}{2}$ look like the other fractions here +and $38/76$. A less straightforward question would be, with +the nine digits, $9$, $8$, $7$, $6$, $5$, $4$, $3$, $2$, $1$, to express four +numbers +whose sum is $100$; a solution is $78$, $15$, $\sqrt[2]{9}$, and +$\sqrt[3]{64}$. +% [*Note: the explicit "2" in the square root is necessary for the trick] + +\ssection*{Problems with a series of things which are numbered} +Any collection of things which can be distinguished one from +the other---especially if numbered consecutively---afford easy +concrete illustrations of questions depending on these elementary +properties of numbers. As examples I proceed to +enumerate a few familiar tricks. The first two of these are +commonly shown by the use of a \emph{watch}, the last three are +best exemplified by the use of a \emph{pack of playing cards}, +which readily lend themselves to such illustrations, and I +present them in these forms. + +\subsection*{First Example\protect\footnote +{Bachet, problem~\textsc{xx}, p.~155; Oughtred\index +{Oughtreds@Oughtred's \textit{Recreations}}, +\textit{Mathematicall Recreations}, +London, 1653, p.~28.}} +The first of these examples\index{Watch Problem|(} is connected +with the hours marked on the face of a watch. In this +puzzle some one is asked to think of some hour, say, $m$, and +then to touch a number that marks another hour, say, $n$. +\PG----File: 032.png------------------------------------------------------ +Then if, beginning with the number touched, he taps each +successive hour marked on the face of the watch, going in the +opposite direction to that in which the hands of the watch +move, and reckoning to himself the taps as $m$, $(m + 1)$,~\&c., +the $(n +12)$th tap will be on the hour he thought of. For +example, if he thinks of \textsc{v} and touches \textsc{ix}, then, if he taps +successively \textsc{ix}, \textsc{viii}, \textsc{vii}, \textsc{vi},~\ldots, +going backwards and reckoning +them respectively as $5$, $6$, $7$, $8$,~\ldots, the tap which he +reckons as $21$ will be on the \textsc{v}. + +The reason of the rule is obvious, for he arrives finally at +the $(n + 12-m)$th hour from which he started. Now, since he +goes in the opposite direction to that in which the hands of +the watch move, he has to go over $(n-m)$ hours to reach the +hour $m$: also it will make no difference if in addition he goes +over $12$ hours, since the only effect of this is to take him +once completely round the circle. Now $(n + 12-m)$ is always +positive, since $m<12$, and therefore if we make him pass over +$(n+12-m)$ hours we can give the rule in a form which is +equally valid whether $m$ is greater or less than $n$. + +\subsection*{Second Example} The following is another well-known +way of indicating on a watch-dial an hour selected by some +one. I do not know who first invented it. If the hour is +tapped by a pencil beginning at \textsc{vii} and proceeding backwards +round the dial to \textsc{vi}, \textsc{v},~\&c., and if the person who +selected the number counts the taps, reckoning from the hour +selected (thus, if he selected \textsc{x}, he would reckon the first tap +as the $11$th), then the $20$th tap as reckoned by him will be on +the hour chosen. + +For suppose he selected the $n$th hour. Then the $8$th tap +is on \textsc{xii} and is reckoned by him as the $(n + 8)$th. The tap +which he reckons as $(n + 9)$th is on \textsc{xi}, and generally the tap +which he reckons as $(n + p)$th is on the hour $(20-p)$. Hence, +putting $p-20-n$, the tap which he reckons as $20$th is on +the hour $n$. Of course the hours indicated by the first seven +taps are immaterial\index{Watch Problem|)}. + +\subsection*{Extension} It is obvious that the same trick can be +\PG----File: 033.png----------------------------------------------------- +performed with any collection of $m$ things, such as cards or +dominoes, which are distinguishable one from the other, +provided $m < 20$. For suppose the $m$ things are arranged on +a table in some numerical order, and the $n$th thing is selected +by a spectator. Then the first $(19-m)$ taps are immaterial, +the $(20-m)$th tap must be on the $m$th thing and be reckoned +by the spectator as the $(n + 20-m)$th, the $(20-m + 1)$th tap +must be on the $(m-1)$th thing and be reckoned as the +$(n + 20-m + 1)$th, and finally the $(20-n)$th tap will be on +the $n$th thing and is reckoned as the $20$th tap. + +\subsection*{Third Example} The following example rests on an\index +{Cards, Problems with|(} \hypertarget{chapter:1:cards} +{extension} of the method used in the last question; it is +very simple, but I have never seen it previously described +in print. Suppose that a pack of $n$ cards is given to some +one who is asked to select one out of the first $m$ cards and to +remember (but not to mention) what is its number from the +top of the pack (say it is actually the $x$th card in the pack). +Then take the pack, reverse the order of the top $m$ cards +(which can be easily effected by shuffling), and transfer $y$ cards +(where $y < n-m$) from the bottom to the top of the pack. +The effect of this is that the card originally chosen is now the +$(y + m-x + 1)$th from the top. Return to the spectator the +pack so rearranged, and ask that the top card be counted as +the $(x + 1)$th, the next as the $(x+2)$th, and so on, in which +case the card originally chosen will be the $(y + m + 1)$th. +Now $y$ and $m$ can be chosen as we please, and may be varied +every time the trick is performed; thus any one unskilled in +arithmetic will not readily detect the \textit{modus operandi}. + +\subsection*{Fourth Example\protect\footnote +{A particular case of this problem was +given by Bachet, problem~\textsc{xvii}, p.~138.}} +Place a card on the table, and on it +place as many other cards from the pack as with the number +of pips on the card will make a total of twelve. For example, +if the card placed first on the table is the five of clubs, then +seven additional cards must be placed on it. The court cards +may have any values assigned to them, but usually they are +\PG----File: 034.png---------------------------------------------------- +reckoned as tens. This is done again with another card, and +thus another pile is formed. The operation may be repeated +either only three or four times or as often as the pack will +permit of such piles being formed. If finally there are $p$ such +piles, and if the number of cards left over is $r$, then the sum +of the number of pips on the bottom cards of all the piles will +be $13 (p - 4) + r$. + +For, if $x$ is the number of pips on the bottom card of a +pile, the number of cards in that pile will be $13 - x$. A similar +argument holds for each pile. Also there are $52$ cards in the +pack; and this must be equal to the sum of the cards in the +$p$ piles and the $r$ cards left over. +\begin{align*} +\Therefore (13 - x_1) + (13 - x_2) + \dotsb + (13 - x_p) + r &= 52\,,\\ +\Therefore 13p - (x_1 + x_2 + \dotsb + x_p) + r &= 52\,,\\ +\Therefore x_1 + x_2 + \dotsb + x_p &= 13p - 52 + r\\ + &= 13 (p - 4) + r\,. +\end{align*} + +More generally, if a pack of $n$ cards is taken, and if in each +pile the sum of the pips on the bottom card and the number of +cards put on it is equal to $m$, then the sum of the pips on the +bottom cards of the piles will be $(m + 1) p + r -n$. In an écarté +pack $n = 32$, and it is convenient to take $m = 15$. + +\subsection*{Fifth Example} It may be noticed that cutting a pack\index +{Cutting Cards, Problems on} +of cards never alters the relative position of the cards provided +that, if necessary, we regard the top card as following immediately +after the bottom card in the pack. This is used in +the following trick\footnote +{Bachet, problem~\textsc{xix}, p.~152.}. Take a pack, and deal the cards face +upwards on the table, calling them one, two, three,~\&c.\ as you +put them down, and noting in your own mind the card first +dealt. Ask some one to select a card and recollect its number. +Turn the pack over, and let it be cut (not shuffled) as often as +you like. Enquire what was the number of the card chosen. +Then, if you deal, and as soon as you come to the original first +card begin (silently) to count, reckoning this as one, the +selected card will appear at the number mentioned. Of course, +\PG----File: 035.png---------------------------------------------------- +if all the cards are dealt before reaching this number, you +must turn the cards over and go on counting continuously. + +Another similar trick is performed by handing the pack +face upwards to some one, and asking him to select a card +and state its number, reckoning from the top; suppose it to +be the $n$th. Next, ask him to choose a number at which it +shall appear in the pack; suppose he selects the $m$th. Take +the pack and secretly move $m - n$ cards from the bottom to +the top (or if $n$ is greater than $m$, then $n - m$ from the top to +the bottom) and of course the card will be in the required +position\index{Cards, Problems with|)}. + +\markright{Medieval Problems in Arithmetic.} +\section*{Medieval Problems in Arithmetic} Before leaving the\index +{Medieval Problems|(} +subject of these elementary questions, I may mention a few +problems which for centuries have appeared in nearly every +collection of mathematical recreations, and therefore may claim +what is almost a prescriptive right to a place here. + +\subsection*{First Example\protect\footnote +{Some similar problems were given by Bachet, appendix, problem \textsc{iii}, +p.~206; problem \textsc{ix}, p.~233; by Oughtred\index +{Oughtreds@Oughtred's \textit{Recreations}} in his \textit{Recreations}, +p.~22: and by Ozanam\index{Ozanam@Ozanam's \textit{Récréations}}, 1803 +edition, vol.~\textsc{i}, p.~174; 1840 edition, p.~79. Earlier +instances occur in Tartaglia's\index{Tartaglia} writings.}} +The following is a sample of one class +of these puzzles. Three men robbed a gentleman of a vase\index +{Vase Problem|(}, containing $24$ ounces of balsam. Whilst running away they +met in a wood with a glass-seller, of whom in a great hurry +they purchased three vessels. On reaching a place of safety +they wished to divide the booty, but they found that their +vessels contained $5$, $11$, and $13$ ounces respectively. How could +they divide the balsam into equal portions? + +Problems like this can be worked out only by trial: there +are several solutions, of which one is as follows. +\[ +\begin{tabularx}{\ifPaper.9\else.7\fi\textwidth}{l >{\centering}X + >{\centering}X r @{}>{~}r @{ } l r @{ } l r @{ } l r @{ } l} +\multicolumn{4}{l@{}}{The vessels can contain\dotfill} & 24 & oz. & + 13 & oz. & 11 & oz. & 5 & oz. \\ +\multicolumn{4}{l@{}}{Their contents originally are~} & 24 & \ldots & 0 & + \ldots & 0 & \ldots & 0 & \ldots \\ +\multicolumn{4}{l@{}}{First, make their contents\dotfill} & 0 & \ldots & + 8 & \ldots & 11 & \ldots & 5 & \ldots \\ +Second, & '' & '' & \ldots & 16 & \ldots & 8 & + \ldots & 0 & \ldots & 0 & \ldots \\ +Third, & '' & '' & \ldots & 16 & \ldots & 0 & + \ldots & 8 & \ldots & 0 & \ldots \\ +\PGx---File: 036.png---------------------------------------------------- +Fourth, & '' & '' & \ldots & 3 & \ldots & 13 & + \ldots & 8 & \ldots & 0 & \ldots \\ +Fifth, & '' & '' & \ldots & 3 & \ldots & 8 & + \ldots & 8 & \ldots & 5 & \ldots \\ +Sixth, & '' & '' & \ldots & 8 & \ldots & 8 & + \ldots & 8 & \ldots & 0 & \ldots \\ +\end{tabularx}\index{Vase Problem|)} +\] + +\subsection*{Second Example\protect\footnote +{Bachet, problem~\textsc{xxii}, p.~170.}} The next of these is a not uncommon +game, played by two people, say $A$ and $B$. $A$ begins by +mentioning some number not greater than (say) six, $B$ may +add to that any number not greater than six, $A$ may add +to that again any number not greater than six, and so on. +He wins who is the first to reach (say) $50$. Obviously, if $A$ +calls $43$, then whatever $B$ adds to that, $A$ can win next time. +Similarly, if $A$ calls $36$, $B$ cannot prevent $A$'s calling $43$ the +next time. In this way it is clear that the key numbers are +those forming the arithmetical progression $43$, $36$, $29$, $22$, $15$, +$8$, $1$; and whoever plays first ought to win. + +Similarly, if no number greater than $m$ may be added at +any one time, and $n$ is the number to be called by the victor, +then the key numbers will be those forming the arithmetical +progression whose common difference is $m + 1$ and whose +smallest term is the remainder obtained by dividing $n$ by +$m + 1$. + +The same game may be played in another form by placing +$n$ coins, matches, or other objects on a table, and directing +each player in turn to take away not more than $m$ of them. +Whoever takes away the last coin wins. Obviously the key +numbers are multiples of $m + 1$, and the first player who is +able to leave an exact multiple of $(m + 1)$ coins can win. +Perhaps a better form of the game is to make that player lose +who takes away the last coin, in which case each of the key +numbers exceeds by unity a multiple of $m + 1$. + +Mr~Loyd\index{Loyd, S.} has also suggested\footnote +{\textit{Tit-Bits}, London, July~17, Aug.~7, 1897.} a modification which is +equivalent to placing $n$ counters in the form of a circle, and +allowing each player in succession to take away not more than +$m$ of them which are in unbroken sequence: $m$ being less than +\PG----File: 037.png--------------------------------------------------- +$n$ and greater than unity. In this case the second of the two +players can always win. + +\subsection*{Recent Extension of this Problem} The games last described +are very simple, but if we impose on the original problem the +additional restriction that each player may not add the same +number more than three times, the analysis becomes by no +means easy. It is difficult in this case to say whether it is an +advantage to begin or not. I have never seen this extension +described in print, and I will therefore enunciate it at length. + +Suppose that each player is given eighteen cards, three of +them marked $6$, three marked $5$, three marked $4$, three +marked $3$, three marked $2$, and three marked $1$. They play +alternately; $A$ begins by playing one of his cards; then +$B$ plays one of his, and so on. He wins who first plays a +card which makes the sum of the points or numbers on all the +cards played exactly equal to $50$, but he loses if he plays +a card which makes this sum exceed $50$. The game can be +played mentally or by noting the numbers on a piece of paper, +and in practice it is unnecessary to use cards. + +Thus, if they play as follows $A$,~$4$; $B$,~$3$; $A$,~$1$; $B$,~$6$; $A$,~$3$; +$B$,~$4$; $A$,~$4$; $B$,~$5$; $A$,~$4$; $B$,~$4$; $A$,~$5$; the game +stands at $43$. +$B$ can now win, for he may safely play $3$, since $A$ has not +another $4$ wherewith to follow it; and if $A$ plays less than $4$, +$B$ will win the next time. Again, if they play thus, $A$,~$6$; +$B$,~$3$; $A$,~$1$; $B$,~$6$; $A$,~$3$; $B$,~$4$; $A$,~$2$; $B$,~$5$; +$A$,~$1$; $B$,~$5$; +$A$,~$2$; $B$,~$5$; $A$,~$2$; $B$,~$3$; $A$ is now forced to play $1$, and +$B$ wins by playing~$1$. + +The game can be also played if each player is given only +two cards of each kind. + +\subsection*{Third Example} The following medieval problem\index +{Three-Things Problem|(} is somewhat +more elaborate. Suppose that three people, $P$, $Q$, $R$, +select three things, which we may denote by $a$, $e$, $i$, respectively, +and that it is desired to find by whom each object was +selected\footnote{Bachet, problem~\textsc{xxv}, p.~187.}. + +Place $24$ counters on a table. Ask $P$ to take one counter, +\PG----File: 038.png------------------------------------------------------- +$Q$ to take two counters, and $R$ to take three counters. Next, +ask the person who selected $a$ to take as many counters as he +has already, whoever selected $e$ to take twice as many counters +as he has already, and whoever selected $i$ to take four times as +many counters as he has already. Note how many counters +remain on the table. There are only six ways of distributing +the three things among $P$, $Q$, and $R$; and the number of +counters remaining on the table is different for each way. +The remainders may be $1$, $2$, $3$, $5$, $6$, or $7$. + +Bachet summed up the results in the mnemonic line +\emph{Par fer} ($1$) \emph{César} ($2$) \emph{jadis} ($3$) +\emph{devint} ($5$) \emph{si grand} ($6$) \emph{prince} ($7$). +Corresponding to any remainder is a word or words containing +two syllables: for instance, to the remainder $5$ corresponds +the word \emph{devint}. The vowel in the first syllable indicates the +thing selected by $P$, the vowel in the second syllable indicates +the thing selected by $Q$, and of course $R$ selected the remaining +thing. \emph{Salve certa animae semita vita quies} was suggested by +Oughtred\index{Oughtreds@Oughtred's \textit {Recreations}}\footnote +{\textit{Mathematicall Recreations}, London, 1653, p.~20. +} as an alternative mnemonic line. + +\subsection*{Extension} M.~Bourlet\index{Bourlet}, in the course of a very +kindly notice\footnote +{\textit{Bulletin des sciences mathématiques}, Paris, 1893, +vol.~\textsc{xvii}, pp.~105--107.} of the second edition of this work, +has given a much +neater solution of the above question, and has extended the +problem to the case of $n$ people, $P_0, P_1, P_2, \dotsc, P_{n-1}$, each of +whom selects one object, out of a collection of $n$ objects, such +as dominoes or cards. It is required to know which domino +or card was selected by each person. + +Let us suppose the dominoes to be denoted or marked by +the numbers $0, 1, \dotsc, n-1$, instead of by vowels. Give one +counter to $P_1$, two counters to $P_2$, and generally $k$ counters to +$P_k$. Note the number of counters left on the table. Next +ask the person who had chosen the domino $0$ to take as many +counters as he had already, and generally whoever had chosen +the domino $h$ to take $n^h$ times as many dominoes as he had +already: thus if $P_k$ had chosen the domino numbered $h$, he +\PG----File: 039.png------------------------------------------------------- +would take $n^h k$ counters. Note the total number of counters +taken, \IE~$\textsum n^hk$. Divide it by $n$, then the remainder will be +the number on the domino selected by $P_0$; divide the quotient +by $n$, and the remainder will be the number on the domino +selected by $P_1$; divide this quotient by $n$, and the remainder +will be the number on the domino selected by $P_2$; and so on. +In other words, if the number of counters taken is expressed +in the scale of notation whose radix is $n$, then the $(h + 1)$th +digit from the right will give the number on the domino +selected by $P_h$. + +Thus in Bachet's problem with $3$ people and $3$ dominoes, +we should first give one counter to $Q$, and two counters to $R$, +while $P$ would have no counters; then we should ask the +person who selected the domino marked $0$ or $a$ to take as +many counters as he had already, whoever selected the +domino marked $1$ or $e$ to take three times as many counters +as he had already, and whoever selected the domino marked +$2$ or $i$ to take nine times as many counters as he had already. +By noticing the original number of counters, and observing +that $3$ of these had been given to $Q$ and $R$, we should know +the total number taken by $P$, $Q$, and $R$. If this number +were divided by $3$, the remainder would be the number of the +domino chosen by $P$; if the quotient were divided by $3$ the +remainder would be the number of the domino chosen by $Q$; +and the final quotient would be the number of the domino +chosen by $R$\index{Three-Things Problem|)}. + +I may add that Bachet also discussed the case when $n = 4$, +which had been previously considered by Diego Palomino\index{Diego Palomino}% +\index{Palomino} in +1599, but as M.~Bourlet's\index{Bourlet} method is general, it is unnecessary +to discuss further particular cases. + +\subsection*{Decimation} The last of these antique problems to which\index +{Decimation|(}\label{page:DecimationStart} +I referred consists in placing men round a circle so that if +every $n$th man is killed the remainder shall be certain specified +individuals. When decimation was a not uncommon +punishment a knowledge of this kind may have had practical +interest. +\PG----File: 040.png------------------------------------------------------- + +Hegesippus\index{Hegesippus on Decimation}\footnote +{\textit{De Bello Judaico}, bk.~\textsc{iii}, chaps.~16--18. +} says that Josephus\index{Josephus on Decimation} saved his life by such a +device. According to his account, after the Romans had +captured Jotopat, Josephus and forty other Jews took refuge +in a cave. Josephus, much to his disgust, found that all +except himself and one other man were resolved to kill themselves, +so as not to fall into the hands of their conquerors. +Fearing to show his opposition too openly he consented, but +declared that the operation must be carried out in an orderly +way, and suggested that they should arrange themselves round +a circle and that every third person should be killed until +but one man was left, who must then commit suicide. It is +alleged that he placed himself and the other man in the $31$st +and $16$th place respectively, with a result which will be easily +foreseen. + +The question is usually presented in the following form. A +ship, carrying as passengers fifteen Turks and fifteen Christians, +encountered a storm, and the pilot declared that, in order to +save the ship and crew, one-half of the passengers must be +thrown into the sea. To choose the victims the passengers +were placed round a circle, and it was agreed that every ninth +man should be cast overboard, reckoning from a certain point. +It is desired to find an arrangement by which all the Christians +should be saved.\footnote +{Bachet, problem~\textsc{xxiii}, p.~174. The same problem had been +previously enunciated by Tartaglia\index{Tartaglia}.} + +Problems like this can be easily solved by counting, but it +is impossible to give a general rule. In this case, the Christians, +reckoning from the man first counted, must occupy the +places $1$, $2$, $3$, $4$, $10$, $11$, $13$, $14$, $15$, $17$, $20$, $21$, +$25$, $28$, $29$. +This arrangement can be recollected by the positions of the +vowels in the following doggerel rhyme, +\begin{quote} +\emph{From numbers' aid and art, never will fame depart,} +\end{quote} +where $a$ stands for $1$, $e$ for $2$, $i$ for $3$, $o$ for $4$, and $u$ for +$5$. +Hence (looking only at the vowels in the verse) the order is +\PG----File: 041.png------------------------------------------------------- +$4$ Christians, $5$ Turks, $2$ Christians, $1$ Turk, $3$ Christians, +$1$ Turk, $1$ Christian, $2$ Turks, $2$ Christians, $3$ Turks, $1$ Christian, +$2$ Turks, $2$ Christians, $1$ Turk. Other similar mnemonic lines +in French and in Latin were given by Bachet\index +{Bachet@Bachet's \textit {Problèmes}|)} and by Ozanam\index +{Ozanam@Ozanam's \textit {Récréations}} +respectively\index{Decimation|)}\index{Medieval Problems|)}\index +{NumbersPuzzle@\nobreak--- \textsc{Puzzles with}|)}\label{page:DecimationEnd}. + +\bigbreak +\section{Arithmetical Fallacies} I insert next some instances\index +{Arithmetical Fallacies|(}% +\index{FallaciesArith@\textsc{Fallacies, Arithmetical}|(} +of demonstrations\footnote +{Of the fallacies given in the text, the first, second, and third, are +well known; the fourth is not new, but the earliest work in which I +recollect seeing it is my \textit{Algebra}, Cambridge, 1890, p.~430; the fifth +is given in G.C.~Chrystal's \textit{Algebra}, Edinburgh, 1889, +vol.~\textsc{ii}, p.~159; the +eighth is due to G.T.~Walker\index{Walker, G.T.}, and, as far as I know, has +not appeared in +any other book; the ninth is due to D'Alembert\index{DAlembert@D'Alembert}; +and the tenth to +F.~Galton\index{Galton}. A mechanical demonstration that $1 = 2$ was given by +R.~Chartres\index{Chartres, R.} in \textit{Knowledge}, July, 1891. +J.L.F.~Bertrand\index{Bertrand, J.L.F.} pointed out +that a demonstration that $1=-1$ can be also obtained from the proposition +in the Integral Calculus that, if the limits are constant, the order +of integration is indifferent; hence the integral to $x$ +(from $x = 0$ to $x = 1$) +of the integral to $y$ (from $y = 0$ to $y =1$) of a function $\phi$ should +be equal +to the integral to $y$ (from $y = 0$ to $y = 1$) of the integral to $x$ (from +$x=0$ to $x = 1$) of $\phi$, but if $\phi=(x^2-y^2)/(x^2 + y^2)^2$, this gives +$\frac{1}{4}\pi =-\frac{1}{4}\pi$.} leading to arithmetical results which are +obviously impossible. I include algebraical proofs as well as +arithmetical ones. The fallacies are so patent that in preparing +the first and second editions I did not think such +questions worth printing, but, as some correspondents have +expressed a contrary opinion, I give them for what they are +worth. + +\subsection*{First fallacy} One of the oldest of these---and not a very +interesting specimen---is as follows. Suppose that $a = b$, then +\begin{align*} +ab & = a^2\,. \\ +\Therefore ab-b^2 & = a^2 - b^2\,. \\ +\Therefore b(a-b) & = (a+b)(a-b)\,. \\ +\Therefore b & = a + b\,. \\ +\Therefore b & = 2b\,. \\ +\Therefore 1 & = 2\,. +\end{align*} +\PG----File: 042.png----------------------------------------------------- + +\subsection*{Second Fallacy} Another instance, almost as puerile, is as +follows. Let $a$ and $b$ be two unequal numbers, and let $c$ be +their arithmetic mean, hence +\begin{align*} +a + b &= 2c\,.\\ +\Therefore (a + b)(a - b) &= 2c(a - b)\,.\\ +\Therefore a^2 - 2ac &= b^2 - 2bc\,.\\ +\Therefore a^2 - 2ac + c^2 &= b^2 - 2bc + c^2\,.\\ +\Therefore (a - c)^2 &= (b - c)^2\,.\\ +\Therefore a &= b\,. +\end{align*} + +\subsection*{Third Fallacy} Another example, the idea of which is +due to John Bernoulli\index{Bernoulli, John}, may be stated as follows. +\begin{LRalign} +\indent We have & (-1)^2 &= 1\,. \\ +Take logarithms, & \Therefore 2 \log(-1) &= \log 1 = 0\,. \\ + & \Therefore \log(-1) &= 0\,. \\ + & \Therefore -1 &= e^0\,. \\ + & \Therefore -1 &= 1\,.\\ +\end{LRalign} + +The same argument may be expressed thus. Let $x$ be a +quantity which satisfies the equation +\begin{LRalign} +&e^x &= -1\,.\\ +Square both sides,&\Therefore e^{2x} &= 1\,. \\ +&\Therefore 2x &= 0\,.\\ +&\Therefore x &= 0\,.\\ +&\Therefore e^x &= e^0\,.\\ +But $e^x = -1$ and $e^0 = 1$,& \Therefore -1 &= 1\,.\\ +\end{LRalign} + +\subsection*{Fourth Fallacy} As yet another instance, we know that +\[ + \log(1 + x) = x - \tfrac{1}{2}x^2 + \tfrac{1}{3}x^3 - \dotsb\,. +\] +If $x = 1$, the resulting series is convergent; hence we have +\begin{align*} + \log 2 &= 1 - \tfrac{1}{2} + \tfrac{1}{3} - \tfrac{1}{4} ++ \tfrac{1}{5} - \tfrac{1}{6} + \tfrac{1}{7} - \tfrac{1}{8} ++ \tfrac{1}{9} - \dotsb\,. \\ +\Therefore 2 \log 2 &= 2 - 1 + \tfrac{2}{3} - \tfrac{1}{2} ++ \tfrac{2}{5} - \tfrac{1}{3} + \tfrac{2}{7} - \tfrac{1}{4} ++ \tfrac{2}{9} - \dotsb\,. +\end{align*} +\PG----File: 043.png----------------------------------------------------- +Taking those terms together which have a common denominator, +we obtain +\begin{LRalign} +& 2 \log 2 & = 1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} + + \frac{1}{7} - \frac{1}{4} + \frac{1}{9}-\dotsb \\ % inserted final - +& & = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + + \frac{1}{5}-\dotsb \\ +& & = \log 2\,. \\ +Hence&2 &= 1\,.\\ +\end{LRalign} + +\subsection*{Fifth Fallacy} This fallacy is very similar to that last +given. We have +\begin{align*} + \log 2 & = \textstyle 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + + \frac{1}{5} - \frac{1}{6}+\dotsb \\ + & = \textstyle \left( 1 + \frac{1}{3} + \frac{1}{5} + \dotsb\right) - + \left( \frac{1}{2} + \frac{1}{4} + \frac{1}{6}+\dotsb\right) \\ + & = \textstyle \left \{ \left( 1 + \frac{1}{3} + \frac{1}{5} + + \dotsb \right) + + \left( \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \dotsb\right) + \right\} - 2 \left( \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + + \dotsb \right) \\ + & = \textstyle \left\{ 1 + \frac{1}{2} + \frac{1}{3} + \dotsb\right\} + - \left( 1 + \frac{1}{2} + \frac{1}{3} + \dotsb\right) \\ + & = 0\,. +\end{align*} + +The error in each of the foregoing examples is obvious, but +the fallacies in the next examples are concealed somewhat +better. + +\medskip +\subsection*{Sixth Fallacy} We can write the identity $\sqrt{-1} = \sqrt{-1}$ +in the form +% +\begin{LRalign} + & \sqrt{\frac{-1}{1}} & = \sqrt{\frac{1}{-1}}\, , \\ +hence& \frac{\sqrt{-1}}{\sqrt{1}} & = \frac{\sqrt{1}}{\sqrt{-1}} \, , \\ +therefore& (\sqrt{-1})^2 & = (\sqrt{1})^2 \, , \\ +that is,& -1 & = 1 \, . +\end{LRalign} + +\subsection*{Seventh Fallacy} Again, we have +\begin{LRalign} + &\sqrt{a} \times \sqrt{b} & = \sqrt{ab} \, . \\ +\indent Hence& \sqrt{-1} \times \sqrt{-1} & = \sqrt{(-1) (-1)} \, , \\ +therefore& (\sqrt{-1})^2 & =\sqrt{1} \, , \\ +that is,& -1 & = 1 \, . +\end{LRalign} + +\PG----File: 044.png----------------------------------------------------- +\subsection*{Eighth Fallacy} The following demonstration depends on +the fact that an algebraical identity is true whatever be the +symbols used in it, and it will appeal only to those who are +familiar with this fact. + +We have, as an identity, +\[ + \sqrt{x-y} = i \sqrt{y-x} +\tag{i} +\] +where $i$ stands either for $+ \sqrt{-1}$ or for $- \sqrt{-1}$. Now an +\emph{identity} in $x$ and $y$ is necessarily true whatever numbers $x$ +and $y$ may represent. First put $x = a$ and $y = b$, +\[ + \Therefore \sqrt{a - b} = i \sqrt{b - a} + \Tag{ii} +\] + Next put $x = b$ and $y = a$, +\[ + \Therefore \sqrt{b - a} = i \sqrt{a - b} + \Tag{iii} +\] +Also since (i) is an identity, it follows that in +(ii) and (iii) the symbol $i$ must be the same, that +is, it represents $+ \sqrt{-1}$ or $- \sqrt{-1}$ in both cases. Hence, +from (ii) and (iii), we have +\begin{LRalign} + & \sqrt{a-b}\; \sqrt{b - a} & = i^2 \sqrt{b-a}\; \sqrt{a - b}\,, \\ + & \Therefore 1 & = i^2\,, \\ +that is& 1 &= -1 \, . +\end{LRalign} + +\subsection*{Ninth Fallacy} The following fallacy is due to +D'Alembert\index{DAlembert@D'Alembert}\footnote +{\textit{Opuscules mathématiques}, Paris, 1761, vol.~\textsc{i}, +p.~201.}. We know that if the product of two numbers is equal to the +product of two other numbers, the numbers will be in proportion, and +from the definition of a proportion it follows that if the first term +is greater than the second, then the third term will be greater than +the fourth: thus, if $ad=bc$, then $a:b = c:d$, and if in this +proportion $a > b$, then $c > d$. Now if we put $a = d =1$ and $b = c += -1$ we have four numbers which satisfy the relation $ad = bc$ and such +that $a>b$; hence, by the proposition, $c > d$, that is, $-1 > 1$, which is +absurd. + +\subsection*{Tenth Fallacy} The mathematical theory of probability\index +{Probabilities, Fallacies in} leads +to various paradoxes: of these one specimen\footnote{See \textit{Nature}, +Feb.~15, March~1, 1894, vol.~\textsc{xlix}, pp.~365--366, 413.} will suffice. +\PG----File: 045.png------------------------------------------------------- +Suppose three coins to be thrown up and the fact whether each +comes down head or tail to be noticed. The probability that +all three coins come down head is clearly $(\frac{1}{2})^3$, that is, +is $\frac{1}{8}$; +similarly the probability that all three come down tail is $\frac{1}{8}$: +hence the probability that all the coins come down alike +(\IE\ either all of them heads or all of them tails) is $\frac{1}{4}$. But, +of three coins thus thrown up, at least two must come down alike; +now the probability that the third coin comes down head is $\frac{1}{2}$ +and the probability that it comes down tail is $\frac{1}{2}$, thus the +probability that it comes down the same as the other two coins +is $\frac{1}{2}$: hence the probability that all the coins come down alike +is $\frac{1}{2}$. I leave to my readers to say whether either of these +conflicting conclusions is right and if so, which\index +{Arithmetical Fallacies|)}% +\index{FallaciesArith@\textsc{Fallacies, Arithmetical}|)}. + +\subsection*{Arithmetical Problems} To the above examples I may add +the following questions, which I have often propounded in past +years: though not fallacies, they may serve to illustrate the +fact that the answer to an arithmetical question is frequently +different to what a hasty reader might suppose. + +The first of these questions is as follows. Two clerks +are engaged, one at a salary commencing at the rate of (say) +\pounds100 a year with a rise of \pounds20 every year, the other at a +salary commencing at the same rate (\pounds100 a year) with a +rise of \pounds5 every half-year, in each case payments being made +half-yearly: which has the larger income? The answer is the +latter; for in the first year the first clerk receives \pounds100, but +the second clerk receives \pounds50 and \pounds55 as his two half-yearly +payments and thus receives in all \pounds105. In the second year +the first clerk receives \pounds120, but the second clerk receives +\pounds60 and \pounds65 as his two half-yearly payments and thus receives +in all \pounds125. In fact the second clerk will always receive +\pounds5 a year more than the first clerk. + +As another question take the following. A man bets $1/n$th +of his money on an even chance (say tossing heads or tails +with a penny): he repeats this again and again, each time +betting $1/n$th of all the money then in his possession. If, +finally, the number of times he has won is equal to the number +\PG----File: 046.png----------------------------------------------------- +of times he has lost, has he gained or lost by the transaction? +He has, in fact, lost. + +Here is another simple question to which not unfrequently +I have received incorrect answers. One tumbler is half-full +of wine, another is half-full of water: from the first tumbler +a teaspoonful of wine is taken out and poured into the +tumbler containing the water: a teaspoonful of the mixture +in the second tumbler is then transferred to the first tumbler. +As the result of this double transaction, is the quantity of +wine removed from the first tumbler greater or less than the +quantity of water removed from the second tumbler? Nineteen +people out of twenty will say it is greater, but this is not +the case\index{Tricks@\textsc{Tricks with Numbers}|)}. + + +\subsection*{Routes on a Chess-Board} A not uncommon problem\index +{Routes on a Chess-board} can +be generalised as\index{Chess-board, problems@\nobreak--- problems} +follows\footnote +{The substance of the problem was given in a scholarship paper set +at Cambridge about 30 years ago, and possibly was not new then.}. +Construct a rectangular board of +$mn$ cells (or small squares) by ruling $m+1$ vertical lines and +$n+1$ horizontal lines. It is required to know how many routes +can be taken from the top left-hand corner to the bottom right-hand +corner, the motion being along the ruled lines and its +direction being always either vertically downwards or horizontally +from left to right. The answer is $(m + n) ! / m! n !$: thus +on a square board containing $16$ cells (\IE\ one-quarter of a chess-board), +where $m=n = 4$, there are $70$ such routes; while on a +common chess-board, where $m=n=8$, there are no less than +12870 such routes. A similar theorem can be enunciated for +a parallelopiped. + +Another problem of a somewhat similar type is the determination +of the number of closed routes through $mn$ points +arranged in $m$ rows and $n$ columns, following the lines of the +quadrilateral net-work, and passing once and only once through +each point\footnote +{See C.F.~Sainte-Marie in \textit{L'Intermédiaire des mathématiciens}, +Paris, vol.~\textsc{xi}, March, 1904, pp.~86--88.}. + +\markright{Permutation Problems.} +\subsection*{Permutation Problems} As other simple illustrations of\index +{Permutation Problems} +\PG----File: 047.png----------------------------------------------------- +the very large number of ways in which combinations of even +a few things can be arranged, I may note that as many as +$19,958400$ distinct skeleton cubes\index{Skeleton Cubes}\index +{Cubes, Skeleton} can be formed with twelve +differently coloured rods of equal +length\footnote{\textit{Mathematical Tripos}, Cambridge, Part~I, +1894.}; again there are +$3,979614,965760$ ways of arranging a set of twenty-eight +dominoes (\IE\ a set from double zero to double six) in a line, +with like numbers in contact\footnote +{Reiss\index{Reiss} in \textit{Annali di matematica}, +Milan, November, 1871, vol.~\textsc{v}, pp.~63--120.}; while there are no less +than $53644,737765,488792,839237,440000$ possible different distributions +of hands at whist\index{Whist, Number of Hands at}\index +{Cards, Problems with} with a pack of fifty-two +cards\footnote{That is $(52!)/(13!)^4$.}. + +\subsection*{Voting Problems} Here is a simple example on combinations +dealing with the cumulative vote\index{Cumulative Vote}\index +{Voting, Question on} as affecting the +representation of a minority. If there are $p$ electors each +having $r$ votes of which not more than $s$ may be given to one +candidate, and $n$ men are to be elected, then the least number +of supporters who can secure the election of a candidate must +exceed $pr/(ns + r)$. + +\subsection*{Exploration Problems} Another common question is concerned\index +{Exploration Problems} +with the maximum distance into a desert which could +be reached from a frontier settlement by the aid of a party of +$n$ explorers, each capable of carrying provisions that would last +one man for $a$ days. The answer is that the man who reaches +the greatest distance will occupy $na/(n + 1)$ days before he +returns to his starting point. If in the course of their +journey they may make depôts, the longest possible journey +will occupy $\frac{1}{2}a (1 + \frac{1}{2} + \frac{1}{3}+ \dotsb + 1/n)$ days. +Further extensions by the use of horses and cycles will suggest themselves. + +\ThoughtBreakSpace +Here I conclude my account of such of these easy problems\index +{Arithmetical Puzzles@\textsc{Arithmetical Puzzles}|)} +on numbers or elementary algebra as seemed worth reproducing. +It will be noticed that the majority of them either are due to +Bachet or were collected by him in his classical \textit{Problèmes}\index +{Bachet@Bachet's \textit {Problèmes}|(}; but +it should be added that besides the questions I have mentioned +\PG----File: 048.png----------------------------------------------------- +he enunciated, even if he did not always solve, some other +problems of greater interest. One instance will suffice. + +\ssection[Bachet's Weights Problem]{Bachet's Weights Problem\protect\footnote +{Bachet, Appendix, +problem~\textsc{v}, p.~215.}} Among the more difficult\index +{Weights@\textsc{Weights Problem, the}|(} +problems proposed by Bachet was the determination of the +least number of weights which would serve to weigh any +integral number of pounds from $1$~lb.\ to $40$~lbs.\ inclusive. +Bachet gave two solutions: namely, (i)~the series of weights +of $1$, $2$, $4$, $8$, $16$, and $32$~lbs.; (ii)~the series of weights of +$1$, $3$, $9$, and $27$~lbs. + +If the weights may be placed in only one of the scale-pans, +the first series gives a solution, as had been pointed out in +1556 by Tartaglia\index{Tartaglia}\footnote +{\textit{Trattato de' numeri e misure}, Venice, +1556, vol.~\textsc{ii}, bk.~\textsc{i}, chap.~\textsc{xvi}, art.~32.}. + +Bachet, however, assumed that any weight might be +placed in either of the scale-pans. In this case the second +series gives the least possible number of weights required. His +reasoning is as follows. To weigh $1$~lb.\ we must have a $1$~lb.\ +weight. To weigh $2$~lbs.\ we must have in addition either a +$2$~lb.\ weight or a $3$~lb.\ weight; but, if we are confined to only +one new weight (in addition to the $1$~lb.\ we have got already), +then with no weight greater than $3$~lbs.\ could we weigh $2$~lbs.: +if we use a $2$~lb.\ weight we then can weigh $1$~lb., $2$~lbs., and +$3$~lbs., but if we use a $3$~lb.\ weight we then can weigh $1$~lb., +$(3-1)$~lbs., $3$~lbs., and $(3 + 1)$~lbs.; hence a $3$~lb.\ weight is +preferable. Similarly, to enable us to weigh $5$~lbs.\ we must have +another weight not greater than $9$~lbs., and a weight of $9$~lbs.\ +enables us to weigh every weight from $1$~lb.\ to $13$~lbs.; hence +it is the best to choose. The next weight required will be +$2(1 + 3 + 9) + 1$~lb., that is, will be $27$~lbs.; and this enables +us to weigh from $1$~lb.\ to $40$~lbs.\ Thus only four weights are +required, namely, $1$~lb., $3$~lbs., $3^2$~lbs., and $3^3$~lbs. + +We can show similarly that the series of weights of $1$, $3$, +$3^2$, $\dots$, $3^{n-1}$~lbs.\ will enable us to weigh any integral number +of pounds from $1$~lb.\ to $(1+3 + 3^2+ \dotsb 3^{n-1})$~lbs., that is, to +\PG----File: 049.png----------------------------------------------------- +$\frac{1}{2}(3^n-1)$~lbs. This is the least number with which the problem +can be effected. + +To determine the arrangement of the weights to weigh any +given mass we have only to express the number of pounds in +it as a number in the ternary scale of notation, except that in +finding the successive digits we must make every remainder +either $0$, $1$, or $-1$: to effect this a remainder $2$ must be written +as $3-1$, that is, the quotient must be increased by unity, in +which case the remainder is $-1$. This is explained in most +text-books on algebra. + +Bachet's argument does not prove that his result is unique +or that it gives the least possible number of weights required. +These omissions have been supplied by Major MacMahon\index{MacMahon|(}, +who has discussed the far more difficult problem (of which +Bachet's is a particular case) of the determination of all possible +sets of weights, not necessarily unequal, which enable us to +weigh any integral number of pounds from $1$ to $n$ inclusive, +(i)~when the weights may be placed in only one scale-pan, and +(ii)~when any weight may be placed in either scale-pan. He +has investigated also the modifications of the results which are +necessary when we impose either or both of the further conditions +(\textit{a})~that no other weighings are to be possible, and + (\textit{b})~that +each weighing is to be possible in only one way, that is, is to +be unique\footnote +{See his article in the \textit{Quarterly Journal of Mathematics}, 1886, +vol.~\textsc{xxi}, pp.~367--373. An account of the method is given in +\textit{Nature}, Dec.~4, 1890, vol.~\textsc{xlii}, pp.~113--114.}. + +The method for case (i) consists in resolving $1 + x + x^2 + \dotsb + x^n$ +into factors, each factor being of the form $1 + x^a+ x^{2a} + \dotsb +x^{ma}$; +the number of solutions depends on the composite character of +$n + 1$. The method for case (ii) consists in resolving the expression +$x^{-n} + x^{-n+1}+ \dotsb\allowbreak + x^{-1} + 1 + x + \dotsb\allowbreak + + x^{n-1} + x^n$ into factors, +each factor being of the form $x^{-ma} + \dotsb\allowbreak + + x^{-a} + 1 + x^a+ \dotsb + x^{ma}$; +the number of solutions depends on the composite character of +$2n+1$. + +Bachet's problem falls under case (ii), $n = 40$. MacMahon's +\PG----File: 050.png----------------------------------------------------- +analysis shows that there are eight such ways of factorizing +$x^{-40} + x^{-39} +\allowbreak\dotsb\allowbreak+ 1 + x^{39} + x^{40}$. +First, there is the +expression itself in which $a=1$, $m= 40$. Second, the expression is +equal to $(1-x^{81})/x^{40} (1-x)$, which can be resolved into the +product of $(1-x^3)/x (1-x)$ and $(1-x^{81})/x^{39} (1-x^3)$; hence it +can be resolved into two factors of the form given above, in +one of which $a=1$, $m=1$, and in the other, $a =3$, $m=13$. +Third, similarly, it can be resolved into two such factors, in +one of which $a=1$, $m = 4$, and in the other $a = 9$, $m = 4$. +Fourth, it can be resolved into three such factors, in one of +which $a = 1$, $m= 1$, in another $a = 3$, $m = 1$, and in the other, +$a = 9$, $m = 4$. Fifth, it can be resolved into two such factors, +in one of which $a = 1$, $m = 13$, and in the other $a = 27$, $m = 1$. +Sixth, it can be resolved into three such factors, in one of +which $a=1$, $m = 1$, in another $a = 3$, $m = 4$, and in the other +$a = 27$, $m = 1$. Seventh, it can be resolved into three such +factors, in one of which $a = 1$, $m= 4$, in another $a = 9$, $m=1$, +and in the other $a= 27$, $m=1$. Eighth, it can be resolved +into four such factors, in one of which $a= 1$, $m= 1$, in another +$a = 3$, $m = 1$, in another $a = 9$, $m = 1$, and in the other $a = 27$, +$m= 1$. + +These results show that there are eight possible sets of +weights with which any integral number of pounds from $1$ to +$40$ can be weighed subject to the conditions (ii), (\textit{a}), and +(\textit{b}). +If we denote $p$ weights each equal to $w$ by $w^p$, these eight +solutions are $1^{40}$; $1$, $3^{13}$; $1^4$, $9^4$; $1$, $3$, $9^4$; +$1^{13}$, $27$; $1$, $3^4$, $27$; +$1^4$, $9$, $27$; $1$, $3$, $9$, $27$. +The last of these is Bachet's\index{Bachet@Bachet's \textit {Problèmes}|)} +solution: not only is it that in which the least number of weights are +employed, but it is also the only unique one in which all the +weights are unequal% +\index{MacMahon|)}% +\index{PuzzlesArith@\textsc{Puzzles}, Arithmetical|)}% +\index{Weights@\textsc{Weights Problem, the}|)}. + +\section{Problems in Higher Arithmetic} At the commencement% +\index{Arithmetic, Higher@\textsc{Arithmetic, Higher}|(}% +\index{Higher@\textsc{Higher Arithmetic}|(}% +\index{NumbersTheory@\nobreak--- \textsc{Theory of}|(} +of this chapter I alluded to the special interest which +many mathematicians find in the theorems of higher arithmetic: +such, for example, as that every prime of the form +$4n+1$ and every power of it is expressible as the sum of two +\PG----File: 051.png------------------------------------------------------ +squares\footnote{Fermat's\index{Fermat, P.} \textit{Diophantus}, Toulouse, +1670, bk.~\textsc{iii}, prop.~22, p.~127; or +Brassinne's \textit{Précis}, Paris, 1853, p.~65.}, +and the first and second powers can be expressed +thus in only one way. For instance, $13 = 3^2 + 2^2$, +$13^2 = 12^2 + 5^2$, +$13^3 = 46^2 + 9^2$, and so on. Similarly $41 = 5^2 + 4^2$, +$41^2 = 40^2 + 9^2$, +$41^3 = 236^2+115^2$, and so on. + +Propositions such as the one just quoted may be found in +text-books on the theory of numbers and therefore lie outside +the limits of this work, but there are one or two questions in +higher arithmetic involving points not yet quite cleared up +which may find a place here. + +\ssection*{Primes} The first of these is concerned with the possibility +of determining readily whether a given number is prime\index +{Primes@\textsc{Primes}} or +not. Euler\index{Euler} and Gauss\index{Gauss} attached great importance to +this problem, but failed to establish any conclusive test. It would +seem, however, that Fermat\index{Fermat, P.} possessed some means of finding +from its form whether a given number (at any rate if one of +certain known forms) was prime or not. Thus, in answer to +Mersenne\index{Mersenne on Primes} who asked if he could tell without much +trouble whether the number $100895,598169$ was a prime, Fermat wrote +on April~7, 1643, that it was the product of $898423$ and $112303$, +both of which were primes. I have indicated elsewhere one +way by which this result can be found, and Mr~F.W.~Laurence +has indicated another which may have been that used by +Fermat in this particular case. + +\markright{Mersenne's Numbers.} +\ssection*{Mersenne's Numbers\protect\footnote +{For references, see \protect\hyperlink{chapter.9}{chapter~ix} below.}} +Another illustration, confirmatory\index +{MersenneNos@\textsc{Mersenne's Numbers}|(}\DPlabel{Mersenne:I} +of the opinion that Fermat or some of his contemporaries had +a test by which it was possible to find out whether certain +numbers were prime, may be drawn from Mersenne's \textit{Cogitata +Physico-Mathematica} which was published in 1644. In the +preface to that work it is asserted that in order that $2^p-1$ +may be prime, the only values of $p$, not greater than $257$, +which are possible are $1$, $2$, $3$, $5$, $7$, $13$, $17$, $19$, $31$, $67$, +$127$, and $257$: I conjecture that the number $67$ is a misprint +for $61$. With this correction the statement appears to be +\PG----File: 052.png------------------------------------------------------ +true\Editorial +{It's not: see \href{http://www.mersenne.org}{www.mersenne.org}}, +and it has been verified for all except nineteen values +of $p$: namely, $71$, $101$, $103$, $107$, $109$, $137$, $139$, $149$, $157$, +$163$, $167$, $173$, $181$, $193$, $199$, $227$, $229$, $241$, and $257$. Of +these values, Mersenne asserted that $p = 257$ makes $2^p-1$ +a prime, and that the other values make $2^p-1$ a composite +number. The demonstrations for the cases when $p = 89$, $127$ +have not been published; nor have the actual factors of $2^p-1$ +when $p = 89$ been as yet determined: the discovery of these +factors may be commended to those interested in the theory +of numbers. + +Mersenne's result could not be obtained empirically, and it +is impossible to suppose that it was worked out for every case; +hence it would seem that whoever first enunciated it was +acquainted with certain theorems in higher arithmetic which +have not been re-discovered\index{MersenneNos@\textsc{Mersenne's Numbers}|)}. + +\markright{Perfect Numbers.} +\ssection*{Perfect Numbers\protect\footnote +{On the theory of perfect numbers, see bibliographical references by +H.~Brocard, \textit{L'Intermédiaire des mathématiciens}, Paris, 1895, +vol.~\textsc{ii}, pp.~52--54; and 1905, vol.~\textsc{xii}, p.~19.}} +The theory of \emph{perfect numbers}\index{Perfect@\textsc{Perfect Numbers}} +depends directly on that of Mersenne's Numbers. A number is +said to be perfect if it is equal to the sum of all its integral +subdivisors. Thus the subdivisors of $6$ are $1$, $2$, and $3$; the +sum of these is equal to $6$; hence $6$ is a perfect number. + +It is probable that all perfect numbers are included in the +formula $2^{p-1} (2^p-1)$, where $2^p-1$ is a prime. Euclid\index{Euclid} +proved that any number of this form is perfect; Euler\index{Euler} showed +that the formula includes all even perfect numbers; and there is +reason to believe---though a rigid demonstration is wanting---that +an odd number cannot be perfect. If we assume that the +last of these statements is true, then every perfect number is +of the above form. It is easy to establish that every number +included in this formula (except when $p = 2$) is congruent to +unity to the modulus $9$, that is, when divided by $9$ leaves a +remainder $1$; also that either the last digit is a $6$ or the last +two digits are $28$. + +Thus, if $p = 2$, $3$, $5$, $7$, $13$, $17$, $19$, $31$, $61$, then by + Mersenne's rule +\PG----File: 053.png------------------------------------------------------ +the corresponding values of $2^p-1$ are prime; they are $3$, $7$, $31$, +$127$, $8191$, $131071$, $524287$, $2147483647$, $2305843009213693951$; +and the corresponding perfect numbers are $6$, $28$, $496$, $8128$, +$33550336$, $8589869056$, $137438691328$, $2305843008139952128$, +and \hfil\allowbreak\hfilneg$2658455991569831744654692615953842176$. + +\markright{Goldbach's Theorem.} +\ssection*{Goldbach's Theorem} Another interesting problem\index +{Goldbach's Theorem} in +higher arithmetic is the question whether there are any even +integers which cannot be expressed as a sum of two primes. +Probably there are none. The expression of all even\footnoteT +{`even' inserted as per errata sheet} integers not +greater than $5000$ in the form of a sum of two primes has +been effected\footnote{\textit{Transactions of the Halle Academy + (Naturforschung)}, vol.~\textsc{lxxii}, +Halle, 1897, pp.~5--214: see also \textit{L'Intermédiaire des +mathématiciens}, 1903, vol.~\textsc{x}, and 1904, vol.~\textsc{xi}.}, +but a general demonstration that all even +integers can be so expressed is wanting. + +\markright{Lagrange's Theorem.} +\ssection*{Lagrange's Theorem\protect\footnote +{\textit{Nouveaux Mémoires de l'Académie Royale des Sciences}, Berlin, +1775, p.~356.}} +Another theorem in higher arithmetic\index{Lagrange's Theorem} +which, as far as I know, is still unsolved, is to the effect +that every prime of the form $4n - 1$ is the sum of a prime of +the form $4n + 1$ and of double a prime of the form $4n + 1$; +for example, $23 = 13 + 2 \times 5$. Lagrange, however, added that +it was only by induction that he arrived at the result. + +\ssection*{Fermat's Theorem on Binary Powers}% +\addcontentsline{toc}{subsection}{Fermat's \protect\textit{Theorem on + Binary Powers}}% +\markright{Fermat's Theorem on Binary Powers.}% +Fermat\label{page:Fermat}% +\index{Binary@\textsc{Binary Powers}, Fermat on|(}% +\index{Fermat, P.|(}% +\index{Fermat on Binary Powers|(} enriched +mathematics with a multitude of new propositions. +With two exceptions all these have been proved subsequently +to be true. The first of these exceptions is his \emph{theorem on +binary powers}, in which he asserted that all numbers of the +form $2^m + 1$, where $m = 2^n$, are primes\footnote +{Letter of Oct.~18, 1640, \textit{Opera}, Toulouse, 1679, p.~162: or +Brassinne's \textit{Précis}, p.~143.}, +but he added that, +though he was convinced of the truth of this proposition, he +could not obtain a valid demonstration. + +It may be shown that $2^m +1$ is composite if $m$ is not a +power of $2$, but of course it does not follow that $2^m + 1$ is a +prime if $m$ is a power of $2$. As a matter of fact the theorem +\PG----File: 054.png----------------------------------------------------- +is not true. In 1732 +Euler\index{Euler}\footnote +{\textit{Commentarii Academiae Scientiarum Petropolitanae}, St Petersburg, +1738, vol.~\textsc{vi}, p.~104; see also \textit +{Novi Comm. Acad. Sci. Petrop.}, St Petersburg, +1764, vol.~\textsc{ix}, p.~101: or \textit +{Commentationes Arithmeticae Collectae}, +St Petersburg, 1849, vol.~\textsc{i}, pp.~2, 357.} +showed that if $n = 5$ the formula +gives $4294,967297$, which is equal to $641 \times 6,700417$: curiously +enough, these factors can be deduced at once from Fermat's\index +{Binary@\textsc{Binary Powers}, Fermat on|)}\index{Fermat on Binary Powers|)} +remark on the possible factors of numbers of the form $2^m \pm 1$, +from which it may be shown that the prime factors (if any) +of $2^{32} + 1$ must be primes of the form $64n + 1$. + +During the last thirty years it has been +shown\footnote{For the factors and bibliographical references, see the memoir +by A.J.C.~Cunningham\index{Cunningham, A.J.C.} and A.E.~Western\index +{Western on Binary Powers}, \textit{Transactions of the London +Mathematical Society}, May~14, 1903, series~2, vol.~\textsc{i}, p.~175.} +that the +resulting numbers are composite when $n = 6$, $9$, $11$, $12$, $18$, $23$, +$36$, and $38$: the two last numbers contain many thousands of +millions of digits. I believe that Eisenstein\index{Eisenstein} asserted that +the number of primes of the form $2^m + 1$, where $m = 2^n$, is infinite: +the proof has not been published, but perhaps it might throw +some light on the general theory. + +\ssection*{Fermat's Last Theorem}% +\addcontentsline{toc}{subsection}{Fermat's \protect\textit{Last Theorem}}% +\markright{Fermat's Last Theorem.}% +I pass now to the only other +assertion made by Fermat which has not been proved hitherto\Editorial +{Andrew Wiles' proof appeared in 1995: \textit{Annals of Mathematics}, +vol.~\textsc{cxli}, pp.~443--551}. +This, which is sometimes known as \textit{Fermat's Last Theorem}\index +{FermatsLast@\textsc{Fermat's Last Theorem}|(}, is +to the effect\footnote +{Fermat's enunciation will be found in his edition of \textit{Diophantus}, +Toulouse, 1670, bk.~\textsc{ii}, qu.~8, p.~61; or Brassinne's +\textit{Précis}, Paris, 1853, +p.~53. For bibliographical references, see \textit{L'Intermédiaire des +mathématiciens}, 1905, vol.~\textsc{xii}, pp.~11, 12.} +that no integral values of $x$, $y$, $z$ can be found +to satisfy the equation $x^n + y^n = z^n$, if $n$ is an integer greater +than $2$. This proposition has acquired extraordinary celebrity +from the fact that no general demonstration of it has been +given, but there is no reason to doubt that it is true. + +Fermat seems to have discovered its truth first\footnote +{See a letter from Fermat quoted in my \textit{History of Mathematics}, +London, chapter~\textsc{xv}.} +for the case +$n = 3$, and then for the case $n = 4$. His proof for the former +of these cases is lost, but that for the latter is +extant\footnote{Fermat's \textit{Diophantus}, note on p.~339; +or Brassinne's \textit{Précis}, p.~127.}, and a +\PG----File: 055.png----------------------------------------------------- +similar proof for the case of $n=3$ was given by +Euler\index{Euler}\footnote{Euler's Algebra (English trans. 1797), +vol.~\textsc{ii}, chap.~\textsc{xv}, p.~247.}. These +proofs depend upon showing that, if three integral values of +$x$, $y$, $z$ can be found which satisfy the equation, then it will be +possible to find three other and smaller integers which also +satisfy it: in this way finally we show that the equation must +be satisfied by three values which obviously do not satisfy it. +Thus no integral solution is possible. It would seem that this +method is inapplicable except when $n = 3$ and $n = 4$. + +Fermat's discovery of the general theorem was made later. +An easy demonstration can be given on the assumption that +every number can be resolved into prime (complex) factors in +one and only one way. That assumption has been made by +some writers, but it is not universally true. It is possible that +Fermat made some such supposition, though it is perhaps more +probable that he discovered a rigorous demonstration. At any +rate he asserts definitely that he had a valid proof---demonstratio +mirabilis sane---and the fact that every other theorem +on the subject which he stated he had proved has been subsequently +verified must weigh strongly in his favour; especially +as in making the one statement in his writings which is not +correct he was scrupulously careful to add that he could not +obtain a satisfactory demonstration of it. + +It must be remembered that Fermat was a mathematician +of quite the first rank who had made a special study of the +theory of numbers. That subject is in itself one of peculiar +interest and elegance, but its conclusions have little practical +importance, and since his time it has been discussed by only +a few mathematicians, while even of them not many have made +it their chief study. This is the explanation of the fact that +it took more than a century before some of the simpler results +which Fermat had enunciated were proved, and thus it is not +surprising that a proof of the theorem which he succeeded in +establishing only towards the close of his life should involve +great difficulties. + +\PG----File: 056.png----------------------------------------------------- +In 1823 Legendre\index{Legendre}\footnote +{Reprinted in his \textit{Théorie des Nombres}, Paris, 1830, +vol.~\textsc{ii}, pp.~361--368: see also pp.~5, 6.} +obtained a proof for the case of $n = 5$; +in 1832 Lejeune Dirichlet\index{Dirichlet on Fermat's Theorem}\index +{Lejeune Dirichlet on Fermat}\footnote +{\textit{Crelle's Journal}, 1832, vol.~\textsc{ix}, pp.~390--393.} +gave one for $n=14$, and in 1840 +Lamé\index{Lame@Lamé} and Lebesgue\index +{Lebesgue on Fermat's Theorem}\footnote +{\textit{Liouville's Journal}, 1841, vol.~\textsc{v}, pp.~195--215, +276--9, 348--9.} gave proofs for $n=7$. + +The proposition appears to be true universally, and in 1849 +Kummer\index{Kummer on Fermat's Theorem}\footnote +{References to Kummer's Memoirs are given in Smith's\index +{Smith, H@\protect\nobreak--- Hen., on Numbers} Report to +the British Association on the Theory of Numbers, London, 1860.}, +by means of ideal primes, proved it to be so for all +numbers except those (if any) which satisfy three conditions. +It is not known whether any number can be found to satisfy +these conditions, but it seems unlikely, and it has been shown +that there is no number less than $100$ which does so. The +proof is complicated and difficult, and there can be little +doubt is based on considerations unknown to Fermat. I +may add that to prove the truth of the proposition when $n$ is +greater than $4$, it obviously is sufficient to confine ourselves to +cases where $n$ is a prime, and the first step in Kummer's +demonstration is to show that in such cases one of the numbers +$x$, $y$, $z$ must be divisible by $n$. + +Naturally there has been much speculation as to how Fermat +arrived at the result. The modern treatment of higher +arithmetic is founded on the special notation and processes +introduced by Gauss\index{Gauss}, who pointed out that the theory of +discrete magnitude is essentially different from that of continuous +magnitude, but until the end of the last century the +theory of numbers was treated as a branch of algebra, and such +proofs by Fermat as are extant involve nothing more than +elementary geometry and algebra, and indeed some of his +arguments do not involve any symbols. This has led some +writers to think that Fermat used none but elementary +algebraic methods. This may be so, but the following remark, +which I believe is not generally known, rather points to the +opposite conclusion. He had proposed, as a problem to the +\PG----File: 057.png----------------------------------------------------- +English mathematicians, to show that there was only one +integral solution of the equation $x^2 + 2 = y^3$: the solution +evidently being $x = 5, y = 3$. On this he has a +note\footnote{Fermat's \textit{Diophantus}, bk.~\textsc{vi}, prop.~19, +p.~320; or Brassinne's \textit{Précis}, p.~122.} to the +effect that there was no difficulty in finding a solution in +rational fractions, but that he had discovered an entirely new +method---sane pulcherrima et subtilissima---which enabled him +to solve such questions in integers. It was his intention to +write a work\footnote{Fermat's \textit{Diophantus}, bk.~\textsc{iv}, +prop.~31, p.~181; or Brassinne's \textit{Précis}, p.~82.} +on his researches in the theory of numbers, but +it was never completed, and we know but little of his methods +of analysis. I venture however to add my private suspicion +that continued fractions played a not unimportant part in his +researches, and as strengthening this conjecture I may note +that some of his more recondite results---such as the theorem +that a prime of the form $4n + 1$ is expressible as the sum of +two squares---may be established with comparative ease by +properties of such fractions% +\index{Arithmetic, Higher@\textsc{Arithmetic, Higher}|)}% +\index{Fermat, P.|)}% +\index{FermatsLast@\textsc{Fermat's Last Theorem}|)}% +\index{Higher@\textsc{Higher Arithmetic}|)}% +\index{NumbersTheory@\nobreak--- \textsc{Theory of}|)}. + +\PG----File: 058.png----------------------------------------------------- + + +%CHAPTER II. + +\chapter{Some Geometrical Questions.} + +\textsc{In} this chapter I propose to enumerate certain geometrical\chapindex +{Geometrical Recreations@\textsc{Geometrical Recreations}} +questions the discussion of which will not involve necessarily +any considerable use of algebra or arithmetic. Unluckily no +writer like Bachet has collected and classified problems of this +kind, and I take the following instances from my note-books +with the feeling that they represent the subject but imperfectly. + +The first part of the chapter is devoted to questions which +are of the nature of formal propositions: the last part contains +a description of various trivial puzzles and games, which the +older writers would have termed geometrical, but which the +reader of to-day may omit without loss. + +In accordance with the rule I laid down for myself in the +preface, I exclude the detailed discussion of theorems which +involve advanced mathematics. Moreover (with one possible +exception) I exclude also any mention of the numerous geometrical +paradoxes which depend merely on the inability of the +eye to compare correctly the dimensions of figures when their +relative position is changed. This apparent deception does +not involve the conscious reasoning powers, but rests on the +inaccurate interpretation by the mind of the sensations derived +through the eyes, and I do not consider such paradoxes as +coming within the domain of mathematics. + +\PG----File: 059.png----------------------------------------------------- +\section{Geometrical Fallacies} Most educated Englishmen are% +\index{FallaciesGeom@\nobreak--- \textsc{Geometrical}|(}% +\index{Geometrical Fallacies@\textsc{Geometrical Fallacies}|(} +acquainted with the series of logical propositions in geometry +associated with the name of Euclid\index{Euclid}, but it is not known so +generally that these propositions were supplemented originally +by certain exercises. Of such exercises Euclid issued three +series: two containing easy theorems or problems, and the +third consisting of geometrical fallacies, the errors in which +the student was required to find. + +The collection of fallacies prepared by Euclid\index{Euclid} is lost, and +tradition has not preserved any record as to the nature of the +erroneous reasoning or conclusions; but, as an illustration of +such questions, I append two or three demonstrations, leading +to obviously impossible results, which perhaps may amuse any +one to whom they are new. I leave the discovery of the errors +to the ingenuity of my readers. + +\subsection*{First Fallacy} \emph{To prove that a right angle is equal to an +angle which is greater than a right angle.} Let $ABCD$ be a +rectangle. From $A$ draw a line $AE$ outside the rectangle, +equal to $AB$ or $DC$ and making an acute angle with $AB$, as +\begin{figure*}[!hbt] +\centerline{\ifpdf\includegraphics[height=6.5cm,viewport=0 0 415 350]{./images/illus059.pdf} % size graphic using BoundingBox + \else\includegraphics[height=6.5cm]{./images/illus059.eps}\fi} +\end{figure*} +indicated in the diagram. Bisect $CB$ in $H$, and through $H$ +draw $HO$ at right angles to $CB$. Bisect $CE$ in $K$, and through +$K$ draw $KO$ at right angles to $CE$. Since $CB$ and $CE$ are not +\PG----File: 060.png----------------------------------------------------- +parallel the lines $HO$ and $KO$ will meet (say) at $O$. Join $OA$, +$OE$, $OC$, and $OD$. + +The triangles $ODC$ and $OAE$ are equal in all respects. +For, since $KO$ bisects $CE$ and is perpendicular to it, we have +$OC= OE$. Similarly, since $HO$ bisects $CB$ and $DA$ and is perpendicular +to them, we have $OD = OA$. Also, by construction, +$DC = AE$. Therefore the three sides of the triangle $ODC$ are +equal respectively to the three sides of the triangle $OAE$. +Hence, by Euc.~\textsc{i}.~8, the triangles are equal; and therefore the +angle $ODC$ is equal to the angle $OAE$. + +Again, since $HO$ bisects $DA$ and is perpendicular to it, we +have the angle $ODA$ equal to the angle $OAD$. + +Hence the angle $ADC$ (which is the difference of $ODC$ and +$ODA$) is equal to the angle $DAE$ (which is the difference of +$OAE$ and $OAD$). But $ADC$ is a right angle, and $DAE$ is +necessarily greater than a right angle. Thus the result is +impossible. + +\subsection*{Second Fallacy\protect\footnote +{See a note by M.~Coccoz\index{Coccoz} in \textit{L'Illustration}, Paris, + Jan.~12, 1895.}} +\emph{To prove that a part of a line is equal to +the whole line.} Let $ABC$ be a triangle; and, to fix our ideas, +let us suppose that the triangle is scalene, that the angle $B$ is +\begin{figure*}[!hbt] +\centerline{\includegraphics[height=3.3cm]{./images/illus060}} +\end{figure*} +acute, and that the angle $A$ is greater than the angle $C$. From +$A$ draw $AD$ making the angle $BAD$ equal to the angle $C$, and +cutting $BC$ in $D$. From $A$ draw $AE$ perpendicular to $BC$. + +The triangles $ABC$, $ABD$ are equiangular; hence, by Euc.~\textsc{vi}.~19, +% slight alteration: using displayed equation +\[ +\triangle ABC : \triangle ABD = AC^2 : AD^2\,. +\] +\PG----File: 061.png------------------------------------------------------ +Also the triangles $ABC$, $ABD$ are of equal altitude: hence, by +Euc.~\textsc{vi}.~1, +\begin{align*} + \triangle ABC : \triangle ABD &= BC : BD\,, \\ + \Therefore AC^2 : AD^2 &= BC : BD\,. \\ + \Therefore \frac{AC^2}{BC} &= \frac{AD^2}{BD}\,. \\ +\intertext{Hence, by Euc.~\textsc{ii}.~13, } + \frac{AB^2 + BC^2 - 2BC \dotm BE}{BC} +&=\frac{AB^2 + BD^2 - 2BD \dotm BE}{BD}\,. \\ + \Therefore \frac{AB^2}{BC} +BC -2BE &= \frac{AB^2}{BD} +BD -2BE\,. \\ + \Therefore \frac{AB^2}{BC} - BD &= \frac{AB^2}{BD} - BC\,. \\ + \Therefore \frac{AB^2 - BC \dotm BD}{BC} +&= \frac{AB^2 - BC \dotm BD}{BD}\,. \\ + \Therefore BC &= BD\,, +\end{align*} +a result which is impossible. + +\subsection*{Third Fallacy} \emph{To prove that every triangle is isosceles.} +Let $ABC$ be any triangle. Bisect $BC$ in $D$, and through $D$ +draw $DO$ perpendicular to $BC$. Bisect the angle $BAC$ by $AO$. + +First. If $DO$ and $AO$ do not meet, then they are parallel. +Therefore $AO$ is at right angles to $BC$. Therefore $AB= AC$. + +Second. If $DO$ and $AO$ meet, let them meet in $O$. Draw +$OE$ perpendicular to $AC$. Draw $OF$ +perpendicular to $AB$. Join $OB$, $OC$. + +\begin{wrapfigure}{o}{6cm} +\ifpdf\includegraphics[width=6cm,viewport=0 5 360 245]{./images/illus061.pdf} % size graphic using BoundingBox +\else\includegraphics[width=6cm]{./images/illus061.eps}\fi +\end{wrapfigure} + +Let us begin by taking the case +where $O$ is inside the triangle, in +which case $E$ falls on $AC$ and $F$ on +$AB$. %Corrected BC to AB to fix typo in original + +The triangles $AOF$ and $AOE$ are +equal, since the side $AO$ is common, +angle $OAF = \text{angle}$ $OAE$, and angle $OFA= \text{angle}$ $OEA$. Hence +$AF=AE$. Also, the triangles $BOF$ and $COE$ are equal. For +\PG----File: 062.png----------------------------------------------------- +since $OD$ bisects $BC$ at right angles, we have $OB=OC$; also, +since the triangles $AOF$ and $AOE$ are equal, we have +$OF=OE$; lastly, the angles at $F$ and $E$ are right angles. +Therefore, by Euc.~\textsc{i}.~47 and \textsc{i}.~8, the triangles $BOF$ +and $COE$ are equal. Hence $FB=EC$. + +Therefore $AF+FB=AE+EC$, that is, $AB=AC$. + +The same demonstration will cover the case where $DO$ and +$AO$ meet at $D$, as also the case where they meet outside $BC$ +but so near it that $E$ and $F$ fall on $AC$ and $AB$ and not on +$AC$ and $AB$ produced. + +Next take the case where $DO$ and $AO$ meet outside the +triangle, and $E$ and $F$ fall on $AC$ and $AB$ produced. Draw +$OE$ perpendicular to $AC$ produced. Draw $OF$ perpendicular +to $AB$ produced. Join $OB$, $OC$. + +\begin{figure*}[!hbt] +\centerline{\includegraphics[width=\ifPaper.6\else.5\fi\textwidth]{./images/illus062}} +\end{figure*} + +Following the same argument as before, from the equality +of the triangles $AOF$ and $AOE$, we obtain $AF=AE$; and, +from the equality of the triangles $BOF$ and $COE$, we obtain +$FB=EC$. Therefore $AF-FB=AE-EC$, that is, $AB=AC$. + +Thus in all cases, whether or not $DO$ and $AO$ meet, and +whether they meet inside or outside the triangle, we have +$AB = AC$: and therefore every triangle is isosceles, a result +which is impossible. +% The explanation of the fallacy is that in fact there is a case not covered: +% F falls between A and B but E falls outside A and C (or vice versa), so +% AB = AF + FB but AC = AE - CE = AF - FB + +\subsection*{Fourth Fallacy} I am indebted to Captain Turton\index +{Turton, W.H.} for the following ingenious fallacy; it appeared for the first +time in the third edition of this work. + +On the hypothenuse, $BC$, of an isosceles right-angled +\PG----File: 063.png----------------------------------------------------- +triangle, $DBC$, describe an equilateral triangle $ABC$, the +vertex $A$ being on the same side of the base as $D$ is. On $CA$ +take a point $H$ so that $CH = CD$. Bisect $BD$ in $K$. Join $HK$ +and let it cut $CB$ (produced) in $L$. Join $DL$. Bisect $DL$ at +$M$, and through $M$ draw $MO$ perpendicular to $DL$. Bisect +$HL$ at $N$, and through $N$ draw $NO$ perpendicular to $HL$. +Since $DL$ and $HL$ intersect, therefore $MO$ and $NO$ will also +intersect; moreover, since $BDC$ is a right angle, $MO$ and $NO$ +both slope away from $DC$ and therefore they will meet on the +side of $DL$ remote from $A$. Join $OC$, $OD$, $OH$, $OL$. + +The triangles $OMD$ and $OML$ are equal, hence $OD = OL$. +Similarly the triangles $ONL$ and $ONH$ are equal, hence +$OL = OH$. Therefore $OD = OH$. Now in the triangles $OCD$ +and $OCH$, we have $OD= OH$, $CD = CH$ (by construction), and +$OC$ common, hence (by Euc.~\textsc{i}.~8) the angle $OCD$ is equal to +the angle $OCH$, which is absurd. + +\subsection*{Fifth Fallacy\protect\footnote +{\textit{Mathesis}, October, 1893, series~2, vol.~\textsc{iii}, p.~224.}} +\emph{To prove that, if two opposite sides of a +quadrilateral are equal, the other two sides must be parallel.} +Let $ABCD$ be a quadrilateral such that $AB$ is equal to $DC$. +Bisect $AD$ in $M$, and through $M$ draw $MO$ at right angles to +$AD$. Bisect $BC$ in $N$, and draw $NO$ at right angles to $BC$. + +If $MO$ and $NO$ are parallel, then $AD$ and $BC$ (which are +at right angles to them) are also parallel. + +If $MO$ and $NO$ are not parallel, let them meet in $O$; then +$O$ must be either inside the quadrilateral as in the left-hand +\begin{figure*}[!hbt] +\centering +\ifPaper\else\hspace*{\fill}\fi +\begin{minipage}{\ifPaper0.45\textwidth\else0.33\textwidth\fi} +\centerline{\includegraphics[width=\textwidth]{./images/illus063a}} +\end{minipage} +\hfill +\begin{minipage}{\ifPaper0.45\textwidth\else0.33\textwidth\fi} +\centerline{\includegraphics[width=\textwidth]{./images/illus063b}} +\end{minipage} +\ifPaper\else\hspace*{\fill}\fi +\end{figure*} +diagram or outside the quadrilateral as in the right-hand +diagram. Join $OA$, $OB$, $OC$, $OD$. + +\PG----File: 064.png------------------------------------------------------- +Since $OM$ bisects $AD$ and is perpendicular to it, we have +$OA =\allowbreak OD$, and the angle $OAM$ equal to the angle $ODM$. +Similarly $OB = OC$, and the angle $OBN$ equal to the angle +$OCN$. Also by hypothesis $AB = DC$, hence, by Euc.~\textsc{i}.~8, the +triangles $OAB$ and $ODC$ are equal in all respects, and therefore +the angle $AOB$ is equal to the angle $DOC$. + +Hence in the left-hand diagram the sum of the angles +$AOM$, $AOB$ is equal to the sum of the angles $DOM$, $DOC$; +and in the right-hand diagram the difference of the angles +$AOM$, $AOB$ is equal to the difference of the angles $DOM$, $DOC$; +and therefore in both cases the angle $MOB$ is equal to the +angle $MOC$, \IE\ $OM$ (or $OM$ produced) bisects the angle $BOC$. +But the angle $NOB$ is equal to the angle $NOC$, \IE\ $ON$ bisects +the angle $BOC$; hence $OM$ and $ON$ coincide in direction. +Therefore $AD$ and $BC$, which are perpendicular to this direction, +must be parallel. This result is not universally true, +and the above demonstration contains a flaw. + +\subsection*{Sixth Fallacy} The following argument is taken from a +text-book on electricity, published in 1889 by two distinguished +mathematicians, in which it was presented as valid. A given +vector $OP$ of length $l$ can be resolved in an infinite number +of ways into two vectors $OM$, $MP$, of lengths $l'$, $l''$, and we +can make $l'/l''$ have any value we please from nothing to +infinity. Suppose that the system is referred to rectangular +axes $Ox$, $Oy$; and that $OP$, $OM$, $MP$ make respectively angles +$\theta$, $\theta'$, $\theta''$ with $Ox$. Hence, by projection on $Oy$ +and on $Ox$, we have +\begin{LRalign} +&l\sin\theta &= l'\sin\theta' + l''\sin\theta''\,,\\ +&l\cos\theta &= l'\cos\theta' + l''\cos\theta''\,.\\ +Therefore +&\tan\theta &= \frac{n\sin\theta' + \sin\theta''}{n\cos\theta' + + \cos\theta''}\,,\\ +\end{LRalign} +where $n = l'/l''$. This result is true whatever be the value of $n$. +But $n$ may have any value (\Eg~$n = \infty$, or $n = 0$), hence +$\tan\theta = \tan\theta' = \tan\theta''$, which obviously is impossible. + +\subsection*{Seventh Fallacy} Here is a fallacious investigation, to +\PG----File: 065.png------------------------------------------------------ +which Mr~Chartres\index{Chartres, R.} first called my attention, of the value +of $\pi$: it is founded on well-known quadratures. The area of the +semi-ellipse bounded by the minor axis is (in the usual notation) +equal to $\frac{1}{2}\pi ab$. If the centre is moved off to an +indefinitely great distance along the major axis, the ellipse +degenerates into a parabola, and therefore in this particular +limiting position the area is equal to two-thirds of the circumscribing +rectangle. But the first result is true whatever be +the dimensions of the curve. +\begin{align*} +\Therefore \tfrac{1}{2}\pi ab & = \tfrac{2}{3}a \times 2b,\\ +\Therefore \pi & = 8/3, +\end{align*} +a result which is obviously untrue% +\index{FallaciesGeom@\nobreak--- \textsc{Geometrical}|)}% +\index{Geometrical Fallacies@\textsc{Geometrical Fallacies}|)}. + +\section{Geometrical Paradoxes} To the above examples I may +add the following questions, which, though not exactly +fallacious, lead to results which at a hasty glance appear +impossible. + +\subsection*{First Paradox} The first is a problem, sent to me by +Mr~Renton\index{Renton}, to rotate a plane lamina (say, for instance, a +sheet of paper) through four right angles so that the effect is +equivalent to turning it through only one right angle. + +If it is desired that the effect shall be equivalent to turning +it through a right angle about a point $O$, the solution is as +follows. Describe on the lamina a square $OABC$. Rotate +the lamina successively through two right angles about the +diagonal $OB$ as axis and through two right angles about the +side $OA$ as axis, and the required result will be attained. + +\subsection*{Second Paradox} As in arithmetic, so in geometry, the +theory of probability\index{Probabilities, Fallacies in} lends itself to +numerous paradoxes. +Here is a very simple illustration. A stick is broken at +random into three pieces. It is possible to put them together +into the shape of a triangle provided the length of the +longest piece is less than the sum of the other two pieces +(\textit{cf.} Euc.~\textsc{i}.~20), that is, provided the length of the +longest piece is less than half the length of the stick. But the +probability that a fragment of a stick shall be half the +\PG----File: 066.png--------------------------------------------------------- +original length of the stick is $\frac{1}{2}$. Hence the probability that +a triangle can be constructed out of the three pieces into +which the stick is broken would appear to be $\frac{1}{2}$. This is not +true, for actually the probability is $\frac{1}{4}$. + +\subsection*{Third Paradox} The following example illustrates how +easily the eye may be deceived in demonstrations obtained by +actually dissecting\index{Dissection, Proofs by|(} the figures and +re-arranging the parts. In +fact proofs by superposition should be regarded with considerable +distrust unless they are supplemented by mathematical +reasoning. The well-known proofs of the propositions +Euclid~\textsc{i}.~32\index{Eucy@Euclid \textsc{i}. 32} +and Euclid~\textsc{i}.~47\index{Eucz@Euclid \textsc{i}. 47} can be so +supplemented and +are valid. On the other hand, as an illustration of how +deceptive a non-mathematical proof may be, I here mention +the familiar paradox that a square of paper, subdivided like +a chessboard into $64$ small squares, can be cut into four pieces +\begin{figure*}[!hbt] +\centerline{\ifpdf\includegraphics[width=.9\textwidth,viewport=0 0 450 170]{./images/illus066.pdf} % size graphic using BoundingBox +\else\includegraphics[width=.9\textwidth]{./images/illus066.eps}\fi\DPlabel{illus066}} +\end{figure*} +which being put together form a figure containing $65$\index +{Sixty-five Puzzle} such small squares\footnote +{I do not know who discovered this paradox. It is given in various +modern books, but I cannot find an earlier reference to it than one by +Prof.\ G.H.~Darwin\index{Darwin, G.H.}, \textit{Messenger of Mathematics}, +1877, vol.~\textsc{vi}, p.~87.}. +This is effected by cutting the original square +into four pieces in the manner indicated by the thick lines in +the \vhyperlink{illus066}{first figure}. If these four pieces are put +together in the shape of a rectangle in the way shown in the +\vhyperlink{illus066}{second figure} +it will appear as if this rectangle contains $65$ of the small +squares. + +\PG----File: 067.png------------------------------------------------------ +This phenomenon, which in my experience non-mathematicians find +perplexing, is due to the fact that the edges +of the four pieces of paper, which in the second figure lie along +the diagonal $AB$, do not coincide exactly in direction. In +reality they include a small lozenge or diamond-shaped figure, +whose area is equal to that of one of the $64$ small squares in +the original square, but whose length $AB$ is much greater than +its breadth. The diagrams show that the angle between the +two sides of this lozenge which meet at $A$ is +$\tan^{-1}\frac{2}{5} - \tan^{-1}\frac{3}{8}$, +that is, is $\tan^{-1}\frac{1}{46}$, which is less than +$1\frac{1}{4}^{\circ}$. To enable the +eye to distinguish so small an angle as this the dividing lines +in the first figure would have to be cut with extreme accuracy +and the pieces placed together with great care. + +The paradox depends upon the relation $5 \times 13 - 8^2 = 1$. Similar +results can be obtained from the formulae $13 \times 34 - 21^2 = 1$, +$34 \times 89 - 55^2 = 1$,\textellipsis; or from the formulae +$5^2 - 3 \times 8 = 1$, +$13^2 - 8 \times 21 = 1$, $34^2 - 21 \times 55 = 1$,\textellipsis. +These numbers are obtained by finding convergents to the continued fraction +\[ +1 + \frac{1}{1} \genfrac{}{}{0pt}{}{}{+} % \underset{+}{} gives a too-small + + \frac{1}{1} \genfrac{}{}{0pt}{}{}{+} + \frac{1}{1} \genfrac{}{}{0pt}{}{}{+} \dotsb\, . +\] + +A similar paradox for a square of $17$ cells, by which it was +shown that $289$ was equal to $288$, was alluded to by Ozanam\footnote +{Ozanam\index{Ozanam@Ozanam's \textit{Récréations}}, 1803 edition, +vol.~\textsc{i}, p.~299.} +who gave also the diagram for dividing a rectangle of $11$ by +$3$ into two rectangles whose dimensions appear to be $5$ by $4$ +and $7$ by $2$. + +\subsection*{Turton's Seventy-Seven Puzzle} A far better dissection +puzzle was invented by Captain Turton\index{Turton, W.H.}\index +{Seventy-seven Puzzle}. In this a piece of +cardboard, $11$ inches by $7$ inches, subdivided into $77$ small +equal squares, each $1$ inch by $1$ inch, can be cut up and +re-arranged so as to give $78$ such equal squares, each $1$ inch +by $1$ inch, of which $77$ are arranged in a rectangle of the same +dimensions as the original rectangle from one side of which +projects a small additional square. The construction is ingenious, +\PG----File: 068.png------------------------------------------------------ +but cannot be described without the use of a model. +The trick consists in utilizing the fact that cardboard has a +sensible thickness. Hence the edges of the cuts can be +bevelled, but in the model the bevelling is so slight as to be +imperceptible save on a very close scrutiny. The play thus +given in fitting the pieces together permits the apparent +production of an additional square\index{Dissection, Proofs by|)}. + +\section[Colouring Maps][The Four-Colour Theorem.]{Colouring Maps} +I proceed next to mention the% +\index{Colouring@\textsc{Colouring Maps}|(}% +\index{Four-Colour Theorem|(}% +\index{Map@\textsc{Map Colour Theorem}|(} +geometrical proposition that \emph{not more than four colours are +necessary in order to colour a map of a country (divided into +districts) in such a way that no two contiguous districts shall +be of the same colour}. By contiguous districts are meant +districts having a common \emph{line} as part of their boundaries: +districts which touch only at points are not contiguous in this +sense. + +The problem was mentioned by A.F.~Möbius\footnote +{\textit{Leipzig Transactions} (\textit{Math.-phys. Classe}), 1885, +vol.~\textsc{xxxvii}, pp.~1--6.} in his +Lectures in 1840, but it was not until Francis Guthrie\index +{Guthrie on colouring maps}\footnote +{\textit{Proceedings of the Royal Society of Edinburgh}, July~19, 1880, +vol.~\textsc{x}, p.~728.} +communicated it to De~Morgan\index{DeMorgan@De Morgan, A.} about 1850 that +attention was +generally called to it: it is said that the fact had been +familiar to practical map-makers for a long time previously. +Through De~Morgan the proposition became generally known; +and in 1878 Cayley\index{Cayley}\footnote +{\textit{Proceedings of the London Mathematical Society}, 1878, +vol.~\textsc{ix}, +p.~148, and \textit{Proceedings of the Royal Geographical Society}, 1879, +N.S., vol.~\textsc{i}, p.~259.} recalled attention to it by stating that +he did not know of any rigorous proof of it. + +Probably the following argument, though not a formal +demonstration, will satisfy the reader that the result is +true. + +Let $A$, $B$, $C$ be three contiguous districts, and let $X$ be any +other district contiguous with all of them. Then $X$ must +\PG----File: 069.png------------------------------------------------------ +lie either wholly outside the external boundary of the area +$ABC$ or wholly inside the internal boundary, that is, it must +occupy a position either like $X$ or like $X'$. In either case +every remaining occupied area in the figure is enclosed by +the boundaries of not more than three districts: hence there is +no possible way of drawing another area $Y$ which shall be +contiguous with $A$, $B$, $C$, and $X$. In other words, it is possible +to draw on a plane four areas which are contiguous, but it is +not possible to draw five such areas. + +\begin{figure*}[!hbt] +\[\includegraphics +[height=\ifPaper8cm\else.7\textheight\fi]{./images/illus069}\label{illus:069}\] +\end{figure*} + +If $A$, $B$, $C$ are not contiguous, each with the other, or if $X$ +is not contiguous with $A$, $B$, and $C$, it is not necessary to +colour them all differently, and thus the most unfavourable +case is that already treated. Moreover any of the above areas +may diminish to a point and finally disappear without affecting +the argument. + +That we may require at least four colours is obvious from +the diagram \vpageref{illus:069}, %[*Note: originally "above diagram"] +since in that case the areas $A$, $B$, $C$, and $X$ +would have to be coloured differently. + +A proof of the proposition involves difficulties of a high +order, which as yet have baffled all attempts to surmount +them. + +\PG----File: 070.png------------------------------------------------------ +The argument by which the truth of the proposition +was formerly supposed to be demonstrated was given by +A.B.~Kempe\index{Kempe on Colouring Maps}\footnote +{He sent his first demonstration across the Atlantic to the \textit{American +Journal of Mathematics}, 1879, vol.~\textsc{ii}, pp.~193--200; but +subsequently he communicated it in simplified forms to the London Mathematical +Society, \textit{Transactions}, 1879, vol.~\textsc{x}, pp.~229--231, and to +\textit{Nature}, Feb.~26, +1880, vol.~\textsc{xxi}, pp.~399--400.} in 1879, but there is a flaw\footnote +{See articles by P.J.~Heawood\index{Heawood on Colouring Maps} in the \textit +{Quarterly Journal of Mathematics}, +London, 1890, vol.~\textsc{xxiv}, pp.~332--338; and 1897, vol.~\textsc{xxxi}, +pp.~270--285.} +in it. + +In 1880, Tait\index{Tait} published a solution\footnote +{\textit{Proceedings of the Royal Society of Edinburgh}, July~19, 1880, +vol.~\textsc{x}, p.~729; and \textsc{Philosophical Magazine}, January, 1884, +series~5, vol.~\textsc{xvii}, p.~41.} depending on the +theorem that if a closed network of lines joining an even +number of points is such that three and only three lines meet +at each point then three colours are sufficient to colour the +lines in such a way that no two lines meeting at a point are of +the same colour; a closed network being supposed to exclude +the case where the lines can be divided into two groups +between which there is but one connecting line. His deduction +therefrom that four colours will suffice for a map was +given in the last edition of this work. The demonstration +appeared so straightforward that at first it was generally +accepted, but it would seem that it too involves a fallacy\footnote +{See J.~Peterson\index{Peterson on maps} of Copenhagen, \textit +{L'Intermédiaire des mathématiciens}, +vol.~\textsc{v}, 1898, pp.~225--227; and vol.~\textsc{vi}, 1899, pp.~36--38.}. +The proof however leads to the interesting corollary that four +colours may not suffice for a map drawn on a multiply-connected +surface such as an anchor ring. + +Although a proof of the theorem is still wanting\Editorial +{Appel and Haken's controversial computer-assisted proof +appeared in 1977: \textit{Illinois Journal of Mathematics}, +vol.~\textsc{xxi}, pp.~429--490 and 491--567.}, no one +has succeeded in constructing a plane map which requires +more than four tints to colour it, and there is no reason to +doubt the correctness of the statement that it is not necessary +to have more than four colours for any plane map. The +number of ways which such a map can be coloured with four +\PG----File: 071.png------------------------------------------------------ +tints has been also considered\footnote +{See A.C.~Dixon\index{Dixon, A.C.}, \textit{Messenger of Mathematics}, +Cambridge, 1902--3, vol.~\textsc{xxxii}, pp.~81--83.}, +but the results are not +sufficiently interesting to require mention here% +\index{Colouring@\textsc{Colouring Maps}|)}% +\index{Four-Colour Theorem|)}% +\index{Map@\textsc{Map Colour Theorem}|)}. + +\section[Physical Geography][Hills and Dales.]% +{Physical Configuration of a Country} As I have been +alluding to maps, I may here mention that the theory of% +\index{Geography@\textsc{Geography, Physical}|(}% +\index{Hills@\textsc{Hills and Dales}|(}% +\index{Physical@\textsc{Physical Geography}|(} +the representation of the physical configuration of a country +by means of lines drawn on a map was discussed, by Cayley\index{Cayley} +and Clerk Maxwell\index{Maxwell, J. Clerk}\footnote +{Cayley on `\textit{Contour and Slope Lines},' \textit{Philosophical + Magazine}, +London, October, 1859, series~4, vol.~\textsc{xviii}, pp.~264--268; + \textit{Collected +Works}, vol.~\textsc{iv}, pp.~108--111. J.~Clerk Maxwell on `\textit{Hills + and Dales},' +\textit{Philosophical Magazine}, December, 1870, series~4, vol.~\textsc{xl}, + pp.~421--427; +\textit{Collected Works}, vol.~\textsc{ii}, pp.~233--240.}. +They showed that a certain relation +exists between the number of hills, dales, passes,~\&c.\ which +can co-exist on the earth or on an island. I proceed to give a +summary of their nomenclature and conclusions. + +All places whose heights above the mean sea level are +equal are on the same level. The locus of such points on a +map is indicated by a \emph{contour-line}\index{Contour-lines}. Roughly +speaking, an island is bounded by a contour-line. It is usual to draw the +successive contour-lines on a map so that the difference between +the heights of any two successive lines is the same, and thus +the closer the contour-lines the steeper is the slope, but the +heights are measured dynamically by the amount of work to +be done to go from one level to the other and not by linear +distances. + +A contour-line in general will be a closed curve. This +curve may enclose a region of elevation: if two such regions +meet at a point, that point will be a crunode (\IE\ a real double +point) on the contour-line through it, and such a point is +called a \emph{pass}. The contour-line may enclose a region of depression: +if two such regions meet at a point, that point +will be a crunode on the contour-line through it, and such +a point is called a \emph{fork} or bar. As the heights of the corresponding +level surfaces become greater, the areas of the regions +\PG----File: 072.png------------------------------------------------------ +of elevation become smaller, and at last become reduced to +points: these points are the \emph{summits} of the corresponding +mountains. Similarly as the level surface sinks the regions of +depression contract, and at last are reduced to points: these +points are the \emph{bottoms} (or immits) of the corresponding valleys. + +Lines drawn so as to be everywhere at right angles to +the contour-lines are called \emph{lines of slope}\index{Lines of Slope}. If +we go up a line of slope generally we shall reach a summit, and if we go +down such a line generally we shall reach a bottom: we may +come however in particular cases either to a pass or to a fork. +Districts whose lines of slope run to the same summit are +\emph{hills}. Those whose lines of slope run to the same bottom are +\emph{dales}. A \emph{watershed} is the line of slope from a summit to a +pass or a fork, and it separates two dales. A \emph{watercourse} is +the line of slope from a pass or a fork to a bottom, and it +separates two hills. + +If $n + 1$ regions of elevation or of depression meet at a +point, the point is a multiple point on the contour-line drawn +through it; such a point is called a pass or a fork of the +$n$th order, and must be counted as $n$ separate passes (or forks). +If one region of depression meets another in several places at +once, one of these must be taken as a fork and the rest as passes. + +Having now a definite geographical terminology we can +apply geometrical propositions to the subject. Let $h$ be the +number of hills on the earth (or an island), then there will be +also $h$ summits; let $d$ be the number of dales, then there will +be also $d$ bottoms; let $p$ be the whole number of passes, $p_1$ that +of single passes, $p_2$ of double passes, and so on; let $f$ be the +whole number of forks, $f_1$ that of single forks, $f_2$ of double +forks, and so on; let $w$ be the number of watercourses, then +there will be also $w$ watersheds\index{Watersheds and Watercourses}. Hence, +by the theorems of Cauchy\index{Cauchy} and Euler\index{Euler}, +\begin{LRalign} +& h &=1 + p_1 + 2p_2 + \dotsb\,, \\ +& d &=1 + f_1 + 2f_2 + \dotsb\,, \\ +and & w &=2(p_1 + f_1) + 3(p_2 + f_2)) + \dotsb\,.\\ +\end{LRalign} + +\PG----File: 073.png--------------------------------------------------------- +The above results can be extended to the case of a multiply-connected +closed surface% +\index{Geography@\textsc{Geography, Physical}|)}% +\index{Hills@\textsc{Hills and Dales}|)}% +\index{Physical@\textsc{Physical Geography}|)}. + +\section*{Games} Leaving now the question of formal geometrical +propositions, I proceed to enumerate a few games or puzzles\index +{PuzzlesGeom@\nobreak--- Geometrical|(} +which depend mainly on the relative position of things, but +I postpone to \hyperlink{chapter.4}{chapter~\textsc{iv}} the discussion +of such amusements +of this kind as necessitate any considerable use of arithmetic +or algebra. Some writers regard draughts, solitaire, chess +and such like games as subjects for geometrical treatment +in the same way as they treat dominoes\index{Dominoes}, backgammon, and +games with dice in connection with arithmetic: but these +discussions require too many artificial assumptions to correspond +with the games as actually played or to be interesting. + +The amusements to which I refer are of a more trivial +description, and it is possible that a mathematician may like +to omit the remainder of the chapter. In some cases it is +difficult to say whether they should be classified as mainly +arithmetical or geometrical, but the point is of no importance. + +\ssection{Statical Games of Position} Of the innumerable statical% +\index{Counters, Games with|(}% +\index{GamesStatic@\nobreak--- Statical|(}% +\index{Statical@\textsc{Statical Games}|(} +games involving geometry of position I shall mention only +three or four. + +\subsection[Three-in-a-row. \texorpdfstring{\protect\quad Extension to $p$-in-a-row}{ Extension to p-in-a-row}]% +[Three-in-a-row.]{Three-in-a-row} +First, I may mention the game of three-in-a-row\index{Row, Counters in a|(}% +\index{Three-in-a-row@\textsc{Three-in-a-row}|(}, +of which noughts and crosses\index{Noughts and Crosses}, one form of merrilees, +and go-bang are well-known examples. These games are +played on a board---generally in the form of a square containing +$n^2$ small squares or cells. The common practice is for +one player to place a white counter or piece or to make a cross +on each small square or cell which he occupies: his opponent +similarly uses black counters or pieces or makes a nought on +each square which he occupies. Whoever first gets three (or +any other assigned number) of his pieces in three adjacent cells +and in a straight line wins. The mathematical theory for a +board of $9$ cells has been worked out completely, and there is +no difficulty in extending it to one of $16$ cells: but the analysis +is lengthy and not particularly interesting. Most of these +\PG----File: 074.png--------------------------------------------------------- +games were known to the ancients\footnote +{\ifPaper\stretchyspace\fi +Becq de~Fouquières\index{DeFouqui@De Fouquières}% +\index{Fouqu@Fouquières on Ancient Games}, \textit{Les jeux des anciens}, +second edition, Paris, 1873, chap.~\textsc{xviii}.}, and it is for that +reason I mention them here. + +\subsection*{Three-in-a-row. Extension} I may, however, add an +elegant but difficult extension which has not previously found +its way, so far as I am aware, into any book of mathematical +recreations. The problem is to place $n$ counters on a plane +so as to form as many rows as possible, each of which shall +contain three and only three counters\footnote +{\textit{Educational Times Reprints}, 1868, vol.~\textsc{viii}, p.~106; +\Ibid\ 1886, vol.~\textsc{xlv}, pp.~127--128.}. + +It is easy to arrange the counters in a number of rows +equal to the integral part of $\frac{1}{8}(n-1)^2$. This can be effected by +the following construction. Let $P$ be any point on a cubic. +Let the tangent at $P$ cut the curve again in $Q$. Let the tangent +at $Q$ cut the curve in $A$. Let $PA$ cut the curve in $B$, $QB$ cut +it in $C$, $PC$ cut it in $D$, $QD$ cut it in $E$, and so on. Then the +counters must be placed at the points $P, Q, A, B,\dots$. Thus $9$ +counters can be placed in 8 such rows; 10 counters in 10 rows; +15 counters in 24 rows; 81 counters in 800 rows; and so on. + +If however the point $P$ is a pluperfect point of the $n$th order +on the cubic, then Sylvester\index{Sylvester} proved that the above +construction gives a number of rows equal to the integral part of +$\frac{1}{6}(n-1)(n-2)$. Thus 9 counters can be arranged in 9 rows, +which is a well-known and easy puzzle; 10 counters in 12 rows; +15 counters in 30 rows; and so on. + +Even this however is an inferior limit and may be exceeded---for +instance, Sylvester stated that 9 counters can be +placed in 10 rows, each containing three counters; I do not +know how he placed them, but one way of so arranging them +is by putting them at points whose coordinates are $(2, 0)$, $(2, 2)$, +$(2, 4)$, $(4, 0)$, $(4, 2)$, $(4, 4)$, $(0, 0)$, $(3, 2)$, $(6, 4)$; another +way is by putting them at the points $(0, 0)$, $(0, 2)$, $(0, 4)$, $(2, 1)$, +$(2, 2)$, $(2, 3)$, $(4, 0)$, $(4, 2)$, $(4, 4)$; more generally, the angular +points of a regular hexagon and the three points of intersection +\PG----File: 075.png--------------------------------------------------------- +of opposite sides form such a group, and therefore any +projection of that figure will give a solution. + +Thus at present it is not possible to say what is the maximum +number of rows of three which can be formed from $n$ +counters placed on a plane. + +\markright{$p$-in-a-row} +\subsection*{Extension to $p$-in-a-row} The problem mentioned above at +once suggests the extension of placing $n$ counters so as to form +as many rows as possible, each of which shall contain $p$ and +only $p$ counters. Such problems can be often solved immediately +by placing at infinity the points of intersection of +some of the lines, and (if it is so desired) subsequently projecting +the diagram thus formed so as to bring these points to +a finite distance. One instance of such a solution is given +above. + +As easy examples I may give the arrangement of $16$ +counters in $15$ rows\footnoteT{`13' corrected to '15' as per errata sheet}, each +containing $4$ counters; and the +arrangement of $19$ counters in $10$ rows, each containing $5$ +counters. A solution of the second of these problems can be +obtained by placing counters at the $19$ points of intersection of +the $10$ lines $x=\pm a$, $x=\pm b$, $y=\pm a$, $y=\pm b$, $y=\pm x$: of +these points two are at infinity. The first problem I leave to the +ingenuity of my readers\index{Counters, Games with|)}% +\index{Row, Counters in a|)}% +\index{Three-in-a-row@\textsc{Three-in-a-row}|)}. + +\subsection[Tesselation. \texorpdfstring{\protect\quad}{} Cross-Fours]% +[Tesselation. Cross-Fours.]{Tesselation} +Another of these statical recreations is known +as tesselation\index{Tesselation|(} and consists in the formation of +geometrical +designs or mosaics\index{Mosaic Pavements} by means of tesselated tiles. + +To those who have never looked into the matter it may be +surprising that patterns formed by the use of square tiles (of +which one-half bounded by a diagonal is white and the other +half black) should be subject to mathematical analysis. In +view of the discussion of this subject by Montucla\index{Montucla}\footnote +{See Ozanam\index{Ozanam@Ozanam's \textit{Récréations}}, 1803 edition, + vol.~\textsc{i}, p.~100; 1840 edition, p.~46.}, Lucas\index +{Lucas, E.}\footnote +{Lucas, \textit{Récréations Mathématiques}, Paris, 1882--3, + vol.~\textsc{ii}, part~4: +hereafter I shall refer to this work by the name of the author.}, +and other writers it would be hard to refuse to call the +formation of such patterns a mathematical amusement, but +the treatment is (perhaps necessarily) somewhat empirical, +\PG----File: 076.png--------------------------------------------------------- +and though there are some interesting puzzles of this kind, I +do not propose to describe them here. + +Sylvester\index{Sylvester}\footnote +{\EG\ see the \textit{Educational Times Reprints}, London, 1868, + vol.~\textsc{x}, +pp.~74--76, 112: see also vol.~\textsc{xlv}, p.~127; vol.~\textsc{lvi}, + pp.~97--99.} +proposed a modified tesselation problem which +consists in forming anallagmatic squares\index{Anallagmatic Squares}, that is, +squares such that in every row and every column the number of changes of +colour or the number of permanences is constant, the tiles used +being square white tiles and square black tiles. + +If more than two colours are used, the problems become +increasingly difficult. As a simple instance of this class of +problems I may mention one, sent to me by a correspondent +who termed it \emph{Cross-Fours}\index{Cross-fours}, wherein sixteen square +counters are used, the upper half of each being yellow, red, pink, +or blue, and the lower half being gold, green, black, or white, +no two counters being coloured alike. Such counters can be +arranged in the form of a square so that in each vertical, +horizontal, and diagonal line there shall be 8 colours and no +more: they can be also arranged so that in each of these ten +lines there shall be 6 colours and no more, or 5 colours and no +more, or 4 colours and no more. Puzzles of this kind are but +little known; they are however not uninstructive\index{Tesselation|)}. + +\subsection{Colour-Cube Problem} As an example of a recreation% +\index{Colour-cube@\textsc{Colour-cube Problem}|(} +\index{Cubes, Coloured|(} +analogous to tesselation I will mention the colour-cube problem; +I select this partly because it is one of the most difficult +of such puzzles, but chiefly because it has been subjected\footnote +{By Major MacMahon\index{MacMahon}; an abstract of his paper, read before the +London Mathematical Society on Feb.~9, 1893, was given in \textit{Nature}, +Feb.~23, 1893, vol.~\textsc{xlvii}, p.~406.} to +mathematical analysis. + +Stripped of mathematical technicalities the problem may +be enunciated as follows. A cube has six faces, and if six +colours are chosen we can paint each face with a different +colour. By permuting the order of the colours we can obtain +thirty such cubes, no two of which are coloured alike. Take +any one of these cubes, $K$, then it is desired to select eight +\PG----File: 077.png-------------------------------------------------- +out of the remaining twenty-nine cubes, such that they can be +arranged in the form of a cube (whose linear dimensions are +double those of any of the separate cubes) coloured like the +cube $K$, and placed so that where any two cubes touch each +other the faces in contact are coloured alike. + +Only one collection of eight cubes can be found to satisfy +these conditions. To pick out these eight cubes empirically +would be out of the question, but the mathematical analysis +enables us to select them by the following rule. Take any +face of the cube $K$: it has four angles, and at each angle +three colours meet. By permuting the colours cyclically we +can obtain from each angle two other cubes, and the eight +cubes so obtained are those required. + +For instance suppose that the six colours are indicated +by the letters $a$, $b$, $c$, $d$, $e$, $f$. Let the cube $K$ be put on a +table, and to fix our ideas suppose that the face coloured $f$ is +at the bottom, the face coloured $a$ is at the top, and the faces +coloured $b$, $c$, $d$, and $e$ front respectively the east, north, west, +and south points of the compass. I may denote such an +arrangement by \Cube(f; a; b, c, d, e). One cyclical permutation +of the colours which meet at the north-east corner of the +top face gives the cube \Cube(f; c; a, b, d, e), and a second cyclical +permutation gives the cube \Cube(f; b; c, a, d, e). Similarly +cyclical permutations of the colours which meet at the north-west +corner of the top face of $K$ give the cubes \Cube(f; d; b, a, c, e) +and \Cube(f; c; b, d, a, e). Similarly from the top south-west +corner of $K$ we get the cubes \Cube(f; e; b, c, a, d) and +\Cube(f; d; b, c, e, a): and from the top south-east corner we get % note colon is clear in scan +the cubes \Cube(f; e; a, c, d, b) and \Cube(f; b; e, c, d, a). + +The eight cubes being thus determined it is not difficult to +arrange them in the form of a cube coloured similarly to $K$, +and subject to the condition that faces in contact are coloured +alike; in fact they can be arranged in two ways to satisfy +these conditions. One such way, taking the cubes in the +numerical order given above, is to put the cubes $3$, $6$, $8$, and $2$ +at the SE, NE, NW, and SW corners of the bottom face; of +\PG----File: 078.png-------------------------------------------------- +course each placed with the colour $f$ at the bottom, while $3$ +and $6$ have the colour $b$ to the east, and $2$ and $8$ have the +colour $d$ to the west: the cubes $7$, $1$, $4$, and $5$ will then form +the SE, NE, NW, and SW corners of the top face; of course +each placed with the colour $a$ at the top, while $7$ and $1$ have +the colour $b$ to the east, and $5$ and $4$ have the colour $d$ to the +west. If however $K$ is not given, then, without the aid of +mathematical analysis, it is a difficult puzzle to arrange the +eight cubes in the form of a cube coloured similarly to one of +the other twenty-two cubes and subject to the condition that +faces in contact are coloured alike% +\index{Colour-cube@\textsc{Colour-cube Problem}|)}% +\index{Cubes, Coloured|)}% +\index{GamesStatic@\nobreak--- Statical|)}% +\index{Statical@\textsc{Statical Games}|)}. + +It is easy to make similar puzzles in two dimensions which +are fairly difficult; it is somewhat surprising that none are to +be bought, but I have never seen any except those that I have +made myself. + +\ssection[Dynamical Games of Position][Dynamical Geometrical Games.]% +{Dynamical Games of Position} Games which are played +by moving pieces on boards of various shapes% +\index{Dynamical@\textsc{Dynamical Games}|(}% +\index{Games@\textsc{Games}, Dynamical|(}---such as merrilees, +fox and geese, solitaire, backgammon, draughts, and +chess---present more interest. In general, however, they permit +of so many movements of the pieces that any mathematical +analysis of them becomes too intricate to follow out completely. +Probably this is obvious, but it may emphasize the impossibility +of discussing such games effectively if I add that it has +been shown that in a game of chess there may be as many +as $197299$ ways of playing the first four moves, and nearly +$72000$ different positions at the end of the first four moves +(two on each side), of which $16556$ arise when the players +move pawns only\footnote +{\textit{L'Intermédiaire des mathématiciens}, Paris, December, 1903, + vol.~\textsc{x}, +pp.~305--308: also \textit{Royal Engineers Journal}, London, August--November, +1889; or \textit{British Association Transactions}, 1890, p.~745.}. + +Games in which the possible movements are very limited +may be susceptible of mathematical treatment. One or two +of these are given in the next chapter: here I shall confine +myself mainly to puzzles and simple amusements. + +\subsection{Shunting Problems} The first I will mention is a little +\PG----File: 079.png-------------------------------------------------- +puzzle which I bought some years ago and which was described +as the ``Great Northern puzzle% +\index{Railway Puzzles (shunting)|(}% +\index{Shunting@\textsc{Shunting Problems}|(}.'' It is typical of a good many +problems connected with the shunting of trains, and though it +rests on a most improbable hypothesis, I give it as a specimen +of its kind. + +\begin{figure*}[!hbt] +\centerline{\includegraphics +[width=\ifPaper.8\else.6\fi\textwidth]{./images/illus079}} +\end{figure*} + +The puzzle shows a railway, $DEF$, with two sidings, $DBA$ +and $FCA$, connected at $A$. The portion of the rails at $A$ +which is common to the two sidings is long enough to permit +of a single wagon, like $P$ or $Q$, running in or out of it; but +is too short to contain the whole of an engine, like $R$. Hence, +if an engine runs up one siding, such as $DBA$, it must come +back the same way. + +Initially a small block of wood, $P$, coloured to represent a +wagon, is placed at $B$; a similar block, $Q$, is placed at $C$; and +a longer block of wood, $R$, representing an engine, is placed at +$E$. The problem is to use the engine $R$ to interchange the +wagons $P$ and $Q$, without allowing any flying shunts. + +This is effected thus. (i)~$R$ pushes $P$ into $A$. (ii)~$R$ returns, +pushes $Q$ up to $P$ in $A$, couples $Q$ to $P$, draws them both out +to $F$, and then pushes them to $E$. (iii)~$P$ is now uncoupled, +$R$ takes $Q$ back to $A$, and leaves it there. (iv)~$R$ returns to $P$, +pulls $P$ back to $C$, and leaves it there. (v)~$R$ running successively +through $F$, $D$, $B$ comes to $A$, draws $Q$ out, and leaves +it at $B$. + +A somewhat similar puzzle, on sale in the streets in 1905, +is made as follows. A loop-line $BGE$ connects two points $B$ +and $E$ on a railway track $AF$, which is supposed blocked at +both ends, as shown in the diagram. In the model, the track +\PG----File: 080.png-------------------------------------------------- +$AF$ is 9 inches long, $AB = EF = 1\frac{5}{6}$ inches, and $AH = FK = BC += DE = \frac{1}{4}$ inch. On the track and loop are eight wagons, +\begin{figure*}[!hbt] +\centerline{\includegraphics +[width=\ifPaper.9\else.8\fi\textwidth]{./images/illus080}} +\end{figure*} +numbered successively $1$ to $8$, each one inch long and one-quarter +of an inch broad, and an engine of the same dimensions. +Originally the wagons are on the track from $A$ to $F$ +and in the order $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, and the engine is on +the loop. The construction and the initial arrangement ensure that +at any one time there cannot be more than eight vehicles on +the track. Also if eight vehicles are on it only the penultimate +vehicle at either end can be moved on to the loop, but if less than +eight are on the track then the last two vehicles at either end +can be moved on to the loop. If the points at each end of the loop-line +are clear, it will hold four, but not more than four, vehicles. +The object is to reverse the order of the wagons on the track, +so that from $A$ to $F$ they will be numbered successively $8$ to $1$; +and to do this by means which will involve as few transferences +of the engine or a wagon to or from the loop as is possible. + +Other shunting problems are not uncommon, but these two +examples will suffice% +\index{Railway Puzzles (shunting)|)}% +\index{Shunting@\textsc{Shunting Problems}|)}. + +\subsection{Ferry-Boat Problems} Everybody is familiar with the story\index +{Ferry@\textsc{Ferry-boat Problems}|(} +of the showman who was travelling with a wolf, a goat, and a +basket of cabbages; and for obvious reasons was unable to leave +the wolf alone with the goat, or the goat alone with the cabbages. +The only means of transporting them across a river was a boat +so small that he could take in it only one of them at a time. +The problem is to show how the passage could be effected\footnote +{Ozanam\index{Ozanam@Ozanam's \textit{Récréations}}, 1803 edition, + vol.~\textsc{i}, p.~171; 1840 edition, p.~77.}. + +A similar problem, given by Alcuin\index{Alcuin}, Tartaglia\index +{Tartaglia}, and others, is as follows\footnote +{Bachet\index{Bachet@Bachet's \textit{Problèmes}}, Appendix, + problem~\textsc{iv}, p.~212.}. +Three beautiful ladies have for husbands three +men, who are as jealous as they are young, handsome, and +\PG----File: 081.png-------------------------------------------------- +gallant. The party are travelling, and find on the bank of +a river, over which they have to pass, a small boat which can +hold no more than two persons. How can they pass, it being +agreed that, in order to avoid scandal, no woman shall be left +in the society of a man unless her husband is present? + +The method of transportation to be used in the above cases +is obvious, and can be illustrated practically by using six court +cards\index{Cards, Problems with} out of a pack. Another problem similar to + the one last +mentioned is the case of $n$ married couples who have to cross +a river by means of a boat which can be rowed by one person +and will carry $n - 1$ people, but not more, with the condition +that no woman is to be in the society of a man unless her +husband is present. Alcuin's problem is the case of $n = 3$. +Let $y$ denote the number of passages from one bank to the +other which will be necessary. Then it has been shown that +if $n = 3$, $y = 11$; if $n = 4$, $y = 9$; and if $n > 4$, $y = 7$; +the demonstration presents no difficulty. + +The following analogous problem is due to the late Prof.\ +Lucas\index{Lucas, E.}\footnote +{Lucas, vol.~\textsc{i}, pp.~15--18, 237--238.}. +To find the smallest number $x$ of persons that a boat +must be able to carry in order that $n$ married couples may by +its aid cross a river in such a manner that no woman shall +remain in the company of any man unless her husband is +present; it being assumed that the boat can be rowed by one +person only. Also to find the least number of passages, say $y$, +from one bank to the other which will be required. M.~Delannoy\index{Delannoy} +has shown that if $n = 2$, then $x = 2$, and $y = 5$. If $n = 3$, then +$x = 2$, and $y = 11$. If $n = 4$, then $x = 3$, and $y = 9$. If $n = 5$, then +$x = 3$, and $y = 11$. And finally if $n > 5$, then $x = 4$, and $y = 2n - 1$. + +M.~De~Fonteney\index{DeFont@De Fonteney on Ferry Problem}\index +{Fonteney on Ferry Problem} has remarked that, if there was an island +in the middle of the river, the passage might be always effected +by the aid of a boat which could carry only two persons. If there +are only two or only three couples the island is unnecessary, +and the case is covered by the preceding method. If $n > 3$ +then the least number of passages from land to land which will +be required is $8 (n - 1)$. + +\PG----File: 082.png-------------------------------------------------- +His solution is as follows. The first nine passages will +be the same, no matter how many couples there may be: +the result is to transfer one couple to the island and one +couple to the second bank. The result of the next eight +passages is to transfer one couple from the first bank to the +second bank: this series of eight operations must be repeated +as often as necessary until there is left only one couple on the +first bank, only one couple on the island, and all the rest on +the second bank. The result of the last seven passages is to +transfer all the couples to the second bank. + +The solution for the case when there are four couples may +be represented as follows. Let $A$ and $a$, $B$ and $b$, $C$ and $c$, $D$ +and $d$, be the four couples. The letters in the successive lines +indicate the positions of the men and their respective wives +after different passages of the boat. + +\label{LT:row:1}% +\begin{longtable}{c@{}rcc@{}c@{}c@{}cc@{}c@{}c@{}cc@{}c@{}c@{}cc@{}c@{} + c@{}cc@{}c@{}c@{}cc@{}c@{}c@{}c} +& & & \multicolumn{8}{c}{\Small First Bank} + & \multicolumn{8}{c}{\Small Island} + & \multicolumn{8}{c}{\Small Second Bank} \\ +\multicolumn{3}{l}{Initially}&$A$&$B$&$C$&$D$&$a$&$b$&$c$&$d$ + &.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.\\ +After\kern-.5em&1st&\kern-.5em passage&$A$&$B$&$C$&$D$&.&.&$c$&$d$ + &.&.&.&.&$a$&$b$&.&.&.&.&.&.&.&.&.&.\endfirsthead +& & & \multicolumn{8}{c}{\Small First Bank} + & \multicolumn{8}{c}{\Small Island} + & \multicolumn{8}{c}{\Small Second Bank}\endhead +\Ditto{1}{2}{After}&2nd&\Ditto{1}{2}{\kern-.5em passage}&$A$&$B$&$C$&$D$ +&.&$b$&$c$&$d$&.&.&.&.&$a$&.&.&.&.&.&.&.&.&.&.&.\label{LT:row:2}\\ +\Ditto{2}{3}{After}&3rd&\Ditto{2}{3}{\kern-.5em passage}&$A$&$B$&$C$&$D$ +&.&.&.&$d$&.&.&.&.&$a$&$b$&$c$&.&.&.&.&.&.&.&.&.\label{LT:row:3}\\ +\Ditto{3}{4}{After}&4th&\Ditto{3}{4}{\kern-.5em passage}&$A$&$B$&$C$&$D$ +&.&.&$c$&$d$&.&.&.&.&$a$&$b$&.&.&.&.&.&.&.&.&.&.\label{LT:row:4}\\ +\Ditto{4}{5}{After}&5th&\Ditto{4}{5}{\kern-.5em passage}&.&.&$C$&$D$&.&. +&$c$&$d$&$A$&$B$&.&.&$a$&$b$&.&.&.&.&.&.&.&.&.&.\label{LT:row:5}\\ +\Ditto{5}{6}{After}&6th&\Ditto{5}{6}{\kern-.5em passage}&.&.&$C$&$D$&.&. +&$c$&$d$&$A$&$B$&.&.&.&.&.&.&.&.&.&.&$a$&$b$&.&.\label{LT:row:6}\\ +\Ditto{6}{7}{After}&7th&\Ditto{6}{7}{\kern-.5em passage}&.&.&$C$&$D$&.&. +&$c$&$d$&$A$&$B$&.&.&.&$b$&.&.&.&.&.&.&$a$&.&.&.\label{LT:row:7}\\ +\Ditto{7}{8}{After}&8th&\Ditto{7}{8}{\kern-.5em passage}&.&.&$C$&$D$&.&. +&$c$&$d$&.&.&.&.&.&$b$&.&.&$A$&$B$&.&.&$a$&.&.&.\label{LT:row:8}\\ +\Ditto{8}{9}{After}&9th&\Ditto{8}{9}{\kern-.5em passage}&.&.&$C$&$D$&.&. +&$c$&$d$&.&$B$&.&.&.&$b$&.&.&$A$&.&.&.&$a$&.&.&.\label{LT:row:9}\\ +\Ditto{9}{10}{After}&10th&\Ditto{9}{10}{\kern-.5em passage}&.&$B$&$C$&$D$ +&.&.&$c$&$d$&.&.&.&.&.&$b$&.&.&$A$&.&.&.&$a$&.&.&.\label{LT:row:10}\\ +\Ditto{10}{11}{After}&11th&\Ditto{10}{11}{\kern-.5em passage}&.&$B$&$C$&$D$ +&.&.&.&.&.&.&.&.&.&$b$&$c$&$d$&$A$&.&.&.&$a$&.&.&.\label{LT:row:11}\\ +\Ditto{11}{12}{After}&12th&\Ditto{11}{12}{\kern-.5em passage}&.&$B$&$C$&$D$ +&.&.&.&$d$&.&.&.&.&.&$b$&$c$&.&$A$&.&.&.&$a$&.&.&.\label{LT:row:12}\\ +\Ditto{12}{13}{After}&13th&\Ditto{12}{13}{\kern-.5em passage}&.&.&.&$D$&.&. +&.&$d$&.&$B$&$C$&.&.&$b$&$c$&.&$A$&.&.&.&$a$&.&.&.\label{LT:row:13}\\ +\Ditto{13}{14}{After}&14th&\Ditto{13}{14}{\kern-.5em passage}&.&.&.&$D$&.&. +&.&$d$&.&.&.&.&.&$b$&$c$&.&$A$&$B$&$C$&.&$a$&.&.&.\label{LT:row:14}\\ +\Ditto{14}{15}{After}&15th&\Ditto{14}{15}{\kern-.5em passage}&.&.&.&$D$&.&. +&.&$d$&.&.&.&.&$a$&$b$&$c$&.&$A$&$B$&$C$&.&.&.&.&.\label{LT:row:15}\\ +\Ditto{15}{16}{After}&16th&\Ditto{15}{16}{\kern-.5em passage}&.&.&.&$D$&.&. +&.&$d$&.&.&.&.&.&$b$&.&.&$A$&$B$&$C$&.&$a$&.&$c$&.\label{LT:row:16}\\ +\Ditto{16}{17}{After}&17th&\Ditto{16}{17}{\kern-.5em passage}&.&.&.&$D$&.&. +&.&$d$&.&$B$&.&.&.&$b$&.&.&$A$&.&$C$&.&$a$&.&$c$&.\label{LT:row:17}\\ +\Ditto{17}{18}{After}&18th&\Ditto{17}{18}{\kern-.5em passage}&.&$B$&.&$D$&. +&.&.&$d$&.&.&.&.&.&$b$&.&.&$A$&.&$C$&.&$a$&.&$c$&.\label{LT:row:18}\\ +\Ditto{18}{19}{After}&19th&\Ditto{18}{19}{\kern-.5em passage}&.&.&.&.&.&.&. +&$d$&.&$B$&.&$D$&.&$b$&.&.&$A$&.&$C$&.&$a$&.&$c$&.\label{LT:row:19}\\ +\Ditto{19}{20}{After}&20th&\Ditto{19}{20}{\kern-.5em passage}&.&.&.&.&.&.&. +&$d$&.&.&.&.&.&$b$&.&.&$A$&$B$&$C$&$D$&$a$&.&$c$&.\label{LT:row:20}\\ +\Ditto{20}{21}{After}&21st&\Ditto{20}{21}{\kern-.5em passage}&.&.&.&.&.&.&. +&$d$&.&.&.&.&.&$b$&$c$&.&$A$&$B$&$C$&$D$&$a$&.&.&.\label{LT:row:21}\\ +\Ditto{21}{22}{After}&22nd&\Ditto{21}{22}{\kern-.5em passage}&.&.&.&.&.&.&. +&$d$&.&.&.&.&.&.&.&.&$A$&$B$&$C$&$D$&$a$&$b$&$c$&.\label{LT:row:22}\\ +\Ditto{22}{23}{After}&23rd&\Ditto{22}{23}{\kern-.5em passage}&.&.&.&.&.&.&$c$ +&$d$&.&.&.&.&.&.&.&.&$A$&$B$&$C$&$D$&$a$&$b$&.&.\label{LT:row:23}\\ +\Ditto{23}{24}{After}&24th&\Ditto{23}{24}{\kern-.5em passage}&.&.&.&.&.&.&. +&.&.&.&.&.&.&.&.&.&$A$&$B$&$C$&$D$&$a$&$b$&$c$&$d$\label{LT:row:24}\\ +\end{longtable} +\PGx---File: 083.png------------------------------------------------------ + +Prof.\ G.~Tarry\index{Tarry} has suggested an extension of the problem, +which still further complicates its solution. He supposes that +each husband travels with a harem of $m$ wives or concubines; +moreover, as Mohammedan women are brought up in seclusion, +it is reasonable to suppose that they would be unable to row a +boat by themselves without the aid of a man. But perhaps the +difficulties attendant on the travels of one wife may be deemed +sufficient for Christians, and I content myself with merely +mentioning the increased anxieties experienced by Mohammedans +in similar circumstances\index{Ferry@\textsc{Ferry-boat Problems}|)}. + +{\ifPaper\stretchyspace\fi +\subsection[Geodesic Problems][Geodesics.]{Geodesics} +Geometrical problems connected with finding +the shortest routes from one point to another on a curved +surface are often difficult, but geodesics\index +{Geodesic@\textsc{Geodesic Problems}|(} on a flat surface or +flat surfaces are in general readily determinable. + +}I append an instance\footnote +{I heard a similar question propounded at Cambridge in 1903, but +the only place where I have seen it in print is the \textit{Daily Mail}, +London, February~1, 1905.}, but I should have hesitated to do so +had not experience shown that some readers do not readily see +the solution. It is as follows: A room is $30$ feet long, $12$ feet +wide, and $12$ feet high. On the middle line of one of the +smaller side walls and one foot from the ceiling is a wasp. +On the middle line of the opposite wall and $11$ feet from the +ceiling is a fly. The wasp catches the fly by crawling all the +way to it: the fly, paralysed by fear, remaining still. The +problem is to find the shortest route that the wasp can follow. + +To obtain a solution we observe that we can cut a sheet of +paper so that, when folded properly, it will make a model to +\PG----File: 084.png------------------------------------------------------ +scale of the room. This can be done in several ways. If, +when the paper is again spread out flat, we can join the points +representing the wasp and the fly by a straight line lying +wholly on the paper we shall obtain a geodesic route between +them. Thus the problem is reduced to finding the way of +cutting out the paper which gives the shortest route of the +kind. + +\begin{figure*}[!hbt] +\centerline{\includegraphics +[height=\ifPaper8cm\else.7\textheight\fi]{./images/illus084}\DPlabel{illus084}} +\end{figure*} + +\vhyperlink{illus084}{Here} is the diagram corresponding to a solution of the +above question, where $A$ represents the floor, $B$ and $D$ the +longer side-walls, $C$ the ceiling, and $W$ and $F$ the positions on +the two smaller side-walls occupied initially by the wasp and +fly. In the diagram the square of the distance between $W$ +and $F$ is $(32)^2 + (24)^2$; hence the distance is $40$ feet\index +{Geodesic@\textsc{Geodesic Problems}|)}. + +\subsection[Problems with Counters placed in a row][Problems with Counters.]% +{Problems with Counters placed in a row} +Numerous dynamical% +\index{Counters, Games with|(}% +\index{GamesWith@\nobreak--- with Counters|(}% +\index{Pawns@\textsc{Pawns, Games with}|(}% +\index{Row, Counters in a|(} +problems and puzzles may be illustrated with a box of +counters, especially if there are counters of two colours. Of +course coins or pawns or cards will serve equally well. I proceed +to enumerate a few of these played with counters placed +in a row. + +\subsection*{First Problem with Counters} The following problem must +be familiar to many of my readers. Ten counters (or coins) are +placed in a row. Any counter may be moved over two of +\PG----File: 085.png-------------------------------------------------- +those adjacent to it on the counter next beyond them. It is +required to move the counters according to the above rule so +that they shall be arranged in five equidistant couples. + +If we denote the counters in their initial positions by the +numbers $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, we proceed as + follows. Put +$7$ on $10$, then $5$ on $2$, then $3$ on $8$, then $1$ on $4$, and lastly + $9$ on +$6$. Thus they are arranged in pairs on the places originally +occupied by the counters $2$, $4$, $6$, $8$, $10$. + +Similarly by putting $4$ on $1$, then $6$ on $9$, then $8$ on $3$, then +$10$ on $7$, and lastly $2$ on $5$, they are arranged in pairs on the +places originally occupied by the counters $1$, $3$, $5$, $7$, $9$. + +If two superposed counters are reckoned as only one, +solutions analogous to those given above will be obtained by +putting $7$ on $10$, then $5$ on $2$, then $3$ on $8$, then $1$ on $6$, and +lastly $9$ on $4$; or by putting $4$ on $1$, then $6$ on $9$, then $8$ on $3$, +then $10$ on $5$, and lastly $2$ on $7$\footnote +{Note by J.~Fitzpatrick\index{Fitzpatrick, J.} to a French translation of the +third edition of this work, Paris, 1898.}. + +There is a somewhat similar game played with eight counters, +but in this case the four couples finally formed are not equidistant. +Here the transformation will be effected if we move +$5$ on $2$, then $3$ on $7$, then $4$ on $1$, and lastly $6$ on $8$. This form +of the game is applicable equally to $(8 + 2n)$ counters, for if we +move $4$ on $1$ we have left on one side of this couple a row of +$(8 + 2n - 2)$ counters. This again can be reduced to one of +$(8 + 2n - 4)$ counters, and in this way finally we have left $8$ +counters which can be moved in the way explained above. + +A more complete generalization would be the case of $n$ +counters, where each counter might be moved over the $m$ +counters adjacent to it on to the one beyond them. + +\subsection*{Second Problem with Counters} Another problem of a +somewhat similar kind is due to Tait\index{Tait}\footnote +{\textit{Philosophical Magazine}, London, January, 1884, series~5, +vol.~\textsc{xvii}, p.~39.}. Place four florins +(or white counters) and four halfpence (or black counters) +alternately in a line in contact with one another. It is required +\PG----File: 086.png-------------------------------------------------- +in four moves, each of a pair of two contiguous pieces, +without altering the relative position of the pair, to form +a continuous line of four halfpence followed by four florins. + +His solution is as follows. Let a florin be denoted by $a$ +and a halfpenny by $b$, and let $\times \times$ denote two contiguous +blank spaces. Then the successive positions of the pieces may +be represented thus: +\[ +\def\arraycolsep{4pt} +\begin{array}{>{$}l<{\dotfillatleast3$}cccccccccc@{.}} +Initially&\times&\times&a&b&a&b&a&b&a&b\\ +After the first move&b&a&a&b&a&b&a&\times&\times&b\\ +After the second move&b&a&a&b&\times&\times&a&a&b&b\\ +After the third move&b&\times&\times&b&a&a&a&a&b&b\\ +After the fourth move&b&b&b&b&a&a&a&a&\times&\times\\ +\end{array} +\] + +The operation is conducted according to the following rule. +Suppose the pieces to be arranged originally in circular order, +with two contiguous blank spaces, then we always move to +the blank space for the time being that pair of coins which +occupies the places next but one and next but two to the +blank space on one assigned side of it. + +A similar problem with $2n$ counters---$n$ of them being +white and $n$ black---will at once suggest itself, and, if $n$ is +greater than $4$, it can be solved in $n$ moves. I have however +failed to find a simple rule which covers all cases alike, but +solutions, due to M.~Delannoy\index{Delannoy}, have been given\footnote +{\textit{La Nature}, June, 1887, p.~10.} for the four +cases where $n$ is of the form $4m$, $4m + 2$, $4m + 1$, or $4m + 3$; in +the first two cases the first $\frac{1}{2}n$ moves are of pairs of dissimilar +counters and the last $\frac{1}{2}n$ moves are of pairs of similar counters; +in the last two cases, the first move is similar to that given % +% [**Note: original wording 'on the last page' silently altered because +% (a) the reference isn't % accurate (the reference is to the array above) and +% (b) it's easier than trying to decide if there's an intervening page break +% in the LaTeX version +above, namely, of the penultimate and antepenultimate +counters to the beginning of the row, the next $\frac{1}{2} (n - 1)$ moves +are of pairs of dissimilar counters, and the final $\frac{1}{2} (n - 1)$ +moves are of similar counters. + +The problem is also capable of solution if we substitute +the restriction that at each move the pair of counters taken up +\PG----File: 087.png-------------------------------------------------- +must be moved to one of the two ends of the row instead of +the condition that the final arrangement is to be continuous. + +Tait\index{Tait} suggested a variation of the problem by making it a +condition that the two coins to be moved shall also be made to +interchange places; in this form it would seem that $5$ moves are +required; or, in the general case, $n + 1$ moves are required. + +\subsection[Problems on a Chess-board with Counters or Pawns]% +[Problems with Counters or Pawns]% +{Problems on a Chess-board with Counters or Pawns} The +following three problems require the use of a chess-board as well% +\index{Chess-board, Games@\textsc{Chess-board, Games on}|(}% +\index{Chess-board, problems@\nobreak--- problems|(} +as of counters or pieces of two colours. It is more convenient +to move a pawn than a counter, and if therefore I describe them +as played with pawns it is only as a matter of convenience and +not that they have any connection with chess. The first is +characterized by the fact that in every position not more than +two moves are possible; in the second and third problems not +more than four moves are possible in any position. With these +limitations, analysis is possible. I shall not discuss the similar +problems in which more moves are possible. + +\subsection*{First Problem with Pawns\protect\footnote +{Lucas\index{Lucas, E.}, vol.~\textsc{ii}, part~5, pp.~141-143.}} +On a row of seven squares +on a chess-board $3$ white pawns (or counters), denoted in the +diagram by ``$a$''s, are placed on the $3$ squares at one end, and +$3$ black pawns (or counters), denoted by ``$b$''s, are placed on +the $3$ squares at the other end---the middle square being left +vacant. Each piece can move only in one direction; the ``$a$'' +pieces can move from left to right, and the ``$b$'' pieces from +right to left. If the square next to a piece is unoccupied, it +\begin{figure*}[!h] +\centering +\begin{picture}(7,1) +\Cell(0,0;a) +\Cell(1,0;a) +\Cell(2,0;a) +\Cell(3,0; ) +\Cell(4,0;b) +\Cell(5,0;b) +\Cell(6,0;b) +\end{picture} +\end{figure*} +can move on to that; or if the square next to it is occupied by +a piece of the opposite colour and the square beyond that is +unoccupied, then it can, like a queen in draughts, leap over +that piece on to the unoccupied square beyond it. The object +is to get all the white pawns in the places occupied initially +by the black pawns and vice versa. + +The solution requires $15$ moves. It may be effected by +moving first a white pawn, then successively two black pawns +\PG----File: 088.png-------------------------------------------------- +then three white pawns, then three black pawns, then three +white pawns, then two black pawns, and then one white pawn. +We can express this solution by saying that if we number the +cells (a term used to describe each of the small squares on a +chess-board) consecutively, then initially the vacant space +occupies the cell $4$ and in the successive moves it will occupy +the cells $3$, $5$, $6$, $4$, $2$, $1$, $3$, $5$, $7$, $6$, $4$, $2$, $3$, +$5$, $4$. Of these moves, six are simple and nine are leaps. + +Similarly if we have $n$ white pawns at one end of a row +of $2n + 1$ cells, and $n$ black pawns at the other end, they can +be interchanged in $n (n + 2)$ moves, by moving in succession +$1$ pawn, $2$ pawns, $3$ pawns,~\ldots, $n - 1$ pawns, $n$ pawns, $n$ pawns, +$n$ pawns, $n - 1$ pawns,~\ldots, $2$ pawns, and $1$ pawn---all the pawns +in each group being of the same colour and different from +that of the pawns in the group preceding it. Of these moves +$2n$ are simple and $n^2$ are leaps\index{Row, Counters in a|)}. + +\subsection*{Second Problem with Pawns\protect\footnote +{Lucas\index{Lucas, E.}, vol.~\textsc{ii}, part~5, p.~144.}} +A similar game may be +played on a rectangular or square board. The case of a square +board containing $49$ cells, or small squares, will illustrate this +sufficiently: in this case the initial position is shown in the +annexed diagram where the ``$a$''s denote the pawns or pieces +\begin{figure*}[!hbt] +\centering +\begin{MagicSquare}{7} + a & a & a & a & b & b & b \\ + a & a & a & a & b & b & b \\ + a & a & a & a & b & b & b \\ + a & a & a & {} & b & b & b \\ + a & a & a & b & b & b & b \\ + a & a & a & b & b & b & b \\ + a & a & a & b & b & b & b +\end{MagicSquare} +\end{figure*} +of one colour, and the ``$b$''s those of the other colour. The ``$a$'' +pieces can move horizontally from left to right or vertically +down, and the ``$b$'' pieces can move horizontally from right to +left or vertically up, according to the same rules as before. + +\PG----File: 089.png----------------------------------------------------- +The solution reduces to the preceding case. The pieces +in the middle column can be interchanged in $15$ moves. In +the course of these moves every one of the seven cells in that +column is at some time or other vacant, and whenever that +is the case the pieces in the row containing the vacant cell +can be interchanged. To interchange the pieces in each of +the seven rows will require 15 moves. Hence to interchange all +the pieces will require $15 + (7 \times 15)$ moves, that is, $120$ moves. + +If we place $2n(n + 1)$ white pawns and $2n(n + 1)$ black +pawns in a similar way on a square board of $(2n + 1)^2$ cells, +we can transpose them in $2n (n + 1)(n + 2)$ moves: of these +$4n(n + 1)$ are simple and $2n^2 (n + 1)$ are leaps. + +\subsection*{Third Problem with Pawns} The following analogous, +though somewhat more complicated, game was I believe +originally published in the first edition of this work: but I find +\begin{figure*}[!hbt] +\centering +\begin{picture}(5,5) +\Cell(0,4;a)\Cell(1,4;b)\Cell(2,4;c)\Cell(3,4; )\Cell(4,4; ) +\Cell(0,3;d)\Cell(1,3;e)\Cell(2,3;f)\Cell(3,3; )\Cell(4,3; ) +\Cell(0,2;g)\Cell(1,2;h)\Cell(2,2;*)\Cell(3,2;H)\Cell(4,2;G) +\Cell(0,1; )\Cell(1,1; )\Cell(2,1;F)\Cell(3,1;E)\Cell(4,1;D) +\Cell(0,0; )\Cell(1,0; )\Cell(2,0;C)\Cell(3,0;B)\Cell(4,0;A) +\put(0,2){\line(0,1){3}} +\put(0,2){\line(1,0){2}} +\put(0,5){\line(1,0){3}} +\put(2,0){\line(0,1){2}} +\put(2,0){\line(1,0){3}} +\put(5,0){\line(0,1){3}} +\put(3,3){\line(1,0){2}} +\put(3,3){\line(0,1){2}} +\end{picture} +\end{figure*} +that it has been since widely distributed in connexion with an +advertisement and probably now is well-known. On a square +board of 25 cells, place eight white pawns or counters on the +cells denoted by small letters in the annexed diagram, and +eight black pawns or counters on the cells denoted by capital +letters: the cell marked with an asterisk ($*$) being left blank. +Each pawn can move according to the laws already explained---the +white pawns being able to move only horizontally from +left to right or vertically downwards, and the black pawns being +able to move only horizontally from right to left or vertically +upwards. The object is to get all the white pawns in the +places initially occupied by the black pawns and vice versa. +No moves outside the dark line are permitted. + +\PG----File: 090.png------------------------------------------------------ +Since there is only one cell on the board which is unoccupied, +and since no diagonal moves and no backward moves are +permitted, it follows that at each move not more than two +pieces of either colour are capable of moving. There are however +a very large number of solutions. The following empirical +solution in forty-eight moves is one way of effecting the transfer---the +letters indicating the cells \emph{from} which the pieces are +successively moved: +\[\def\arraycolsep{2pt} +\begin{array}{ccccccccccccccccccccccccc} +h&H&*&f&F&E&H&G&*&c&b&h&g&d&f&F&C&*&h&H&B&A&C&*\\ +c&a&b&h&H&*&c&f&F&D&G&H&B&C&*&g&h&e&f&F&*&h&H&*\rlap{$\;$.} +\end{array} +\] +It will be noticed that the first twenty-four moves lead to a +symmetrical position, and that the next twenty-three moves +can be at once obtained by writing the first twenty-three +moves in reverse order and interchanging small and capital +letters\index{Counters, Games with|)}% +\index{GamesWith@\nobreak--- with Counters|)}% +\index{Pawns@\textsc{Pawns, Games with}|)}. + +Probably, were it worth the trouble, the mathematical +theory of games such as that just described might be worked +out by the use of Vandermonde's\index{Vandermonde} notation, described later +in \hyperlink{section*.140}{chapter~\textsc{vi}}, or by the analogous method +employed in the theory of the game of solitaire\footnote +{On the theory of the solitaire, see Reiss\index{Reiss}, `\textit{Beiträge +zur Theorie des Solitär-Spiels},' \textit{Crelle's Journal}, Berlin, 1858, +vol.~\textsc{liv}, pp.~344--379; and +Lucas\index{Lucas, E.}, vol.~\textsc{i}, part~\textsc{v}, pp.~89--141. +}. I believe that this has not been +done, and I do not think it would repay the labour involved. + +\markright{Problems with Chess-pieces} +\subsection*{Problems on a Chess-board with Chess-pieces} There are +several mathematical recreations with chess-pieces, other than +pawns, somewhat similar to those given above. One of these, +on the determination of the ways in which eight queens can be +placed on a board so that no queen can take any other, is given +later in \hyperlink{section.4.4}{chapter~\textsc{iv}}. +Another, on the path to be followed by a +knight which is moved on a chess-board so that it shall occupy +every cell once and only once, is given in \hyperlink +{section.6.5}{chapter~\textsc{vi}}. Here +I will mention one of the simplest of such problems, which is +interesting from the fact that it is given in Guarini's manuscript +\PG----File: 091.png------------------------------------------------------ +written in 1512; it was quoted by Lucas\index{Lucas, E.}, but so far as +I know has not been otherwise published. + +\subsection*{Guarini's Problem}% +\addcontentsline{toc}{subsection}{Guarini's Problem} +On a board of nine cells\index{Guarini's Problem}, such as that\index +{Chess-board, knights@\nobreak--- knights' moves on} +drawn below, the two white knights are placed on the two top +\begin{figure*}[!hbt] +\centering +\begin{MagicSquare}{3} +a & C & d\\ +D & {} & B\\ +b & A & c +\end{MagicSquare} +\end{figure*} +corner cells ($a$,~$d$), and the two black knights on the two +bottom corner cells ($b$,~$c$): the other cells are left vacant. It +is required to move the knights so that the white knights shall +occupy the cells $b$ and $c$, while the black shall occupy the cells +$a$ and $d$. + +The solution is tolerably obvious. First, move the pieces +from $a$ to $A$, from $b$ to $B$, from $c$ to $C$, and from $d$ to $D$. Next, +move the pieces from $A$ to $d$, from $B$ to $a$, from $C$ to $b$, and +from $D$ to $c$. The effect of these eight moves is the same as if +the original square had been rotated through one right angle. +Repeat the above process, that is, move the pieces successively +from $a$ to $A$, from $b$ to $B$, from $c$ to $C$, from $d$ to $D$; from $A$ +to $d$, from $B$ to $a$, from $C$ to $b$, and from $D$ to $c$. The required +result is then attained% +\index{Chess-board, Games@\textsc{Chess-board, Games on}|)}% +\index{Chess-board, problems@\nobreak--- problems|)}% +\index{Dynamical@\textsc{Dynamical Games}|)}% +\index{Games@\textsc{Games}, Dynamical|)}. + +\section[Geometrical Puzzles (rods, strings, \protect\&c.)]% +[Geometrical Puzzles]{Geometrical Puzzles with Rods, etc} Another species +of geometrical puzzles, to which here I will do no more than +allude, are made of steel rods, or of wire, or of wire and string. +Numbers of these are often sold in the streets of London +for a penny each, and some of them afford ingenious problems +in the geometry of position. Most of them could hardly +be discussed without the aid of diagrams, but they are +inexpensive to construct, and in fact innumerable puzzles +on geometry of position can be made with a couple of +stout sticks and a ball of string, or even with only a box +of matches: several examples are given in the appendix to +the fourth volume of the 1723 edition of Ozanam's\index +{Ozanam@Ozanam's \textit {Récréations}} work. +\PG----File: 092.png----------------------------------------------------- +I will mention, as an easy example, analogous to one group +of the string puzzles, that any one can take off his waistcoat +(which may be unbuttoned) without taking off his coat\index +{Coat and Waistcoat Trick}, and +without pulling the waistcoat over the head like a jersey. + +This last feat may serve to show the difficulty of mentally +realizing the effect of geometrical alterations in a figure unless +they are of the simplest character. + +\section{Paradromic Rings} The fact just stated is illustrated +by the familiar experiment of making \emph{paradromic rings}\index +{Paradromic@\textsc{Paradromic Rings}|(} by +cutting a paper ring prepared in the following manner. + +Take a strip of paper or piece of tape, say, for convenience, +an inch or two wide and at least nine or ten inches long, +rule a line in the middle down the length $AB$ of the strip, +gum one end over the other end $B$, and we get a ring like +a section of a cylinder. If this ring is cut by a pair of scissors +along the ruled line we obtain two rings exactly like the first, +except that they are only half the width. Next suppose that +the end $A$ is twisted through two right angles before it is +gummed to $B$ (the result of which is that the back of the +strip at $A$ is gummed over the front of the strip at $B$), then a +cut along the line will produce only one ring. Next suppose +that the end $A$ is twisted once completely round (\IE\ through +four right angles) before it is gummed to $B$, then a similar cut +produces two interlaced rings. If any of my readers think +that these results could be predicted off-hand, it may be +interesting to them to see if they can predict correctly the +effect of again cutting the rings formed in the second and +third experiments down their middle lines in a manner similar +to that above described. + +The theory is due to J.B.~Listing\index +{Listing@Listing's \textit{Topologie}}\footnote +{\textit{Vorstudien zur Topologie, Die Studien}, Göttingen, 1847, +part~\textsc{x}.} who discussed the case +when the end $A$ receives $m$ half-twists, that is, is twisted +through $m\pi$, before it is gummed to $B$. + +If $m$ is even we obtain a surface which has two sides and +\PG----File: 093.png---------------------------------------------------- +two edges, which are termed paradromic. If the ring is cut +along a line midway between the edges, we obtain two rings, +each of which has $m$ half-twists, and which are linked together +$\frac{1}{2}m$ times. + +If $m$ is odd we obtain a surface having only one side and +one edge. If this ring is cut along its mid-line, we obtain +only one ring, but it has $2m$ half-twists, and if $m$ is greater +than unity it is knotted\index{Paradromic@\textsc{Paradromic Rings}|)}% +\index{PuzzlesGeom@\nobreak--- Geometrical|)}. + +\PG----File: 094.png---------------------------------------------------- + + + +%CHAPTER III. + +\chapter[Some Mechanical Questions.][Mechanical Recreations.]% +{Some Mechanical Questions.} + +\textsc{I proceed} now to enumerate a few questions connected\chapindex +{Mechanical Recreations@\textsc{Mechanical Recreations}} +with mechanics which lead to results that seem to me interesting +from a historical point of view or paradoxical. Problems +in mechanics generally involve more difficulties than +problems in arithmetic, algebra, or geometry, and the explanations +of some phenomena---such as those connected with the +flight of birds---are still incomplete, while the explanations of +many others of an interesting character are too difficult to +find a place in a non-technical work. Here, however, I shall +confine myself to questions which, like those treated in the +two preceding chapters, are of an elementary, not to say +trivial, character; and the conclusions are well-known to +mathematicians. + +I assume that the reader is acquainted with the fundamental +ideas of kinematics and dynamics, and is familiar with +the three Newtonian laws\index{MotionLaw@Motion, Laws of}\index +{Newtonian Laws of Motion|(}; namely, first that a body will +continue in its state of rest or of uniform motion in a straight +line unless compelled to change that state by some external +force: second, that the change of momentum per unit of time +is proportional to the external force and takes place in the +direction of it: and third, that the action of one body on +another is equal in magnitude but opposite in direction to +the reaction of the second body on the first. The first and +second laws state the principles required for solving any +question on the motion of a particle under the action of given +\PG----File: 095.png------------------------------------------------------ +forces. The third law supplies the additional principle required +for the solution of problems in which two or more particles +influence one another. + +\section[Paradoxes on Motion][Zeno's Paradoxes.]{Motion} +The difficulties connected with the idea of \emph{motion} +have been for a long time a favourite subject for paradoxes% +\index{MotionParadox@\nobreak--- Paradoxes on|(}% +\index{PuzzlesMech@\nobreak--- Mechanical|(}, +some of which bring us into the realm of the philosophy of +mathematics. + +\subsection*{Zeno's Paradoxes on Motion} One of the earliest of these\index +{Zeno on Motion|(}\index{FallaciesMech@\nobreak--- \textsc{Mechanical}|(} +is the remark of Zeno to the effect that since an arrow cannot +move where it is not, and since also it cannot move where it +is (\IE\ in the space it exactly fills), it follows that it cannot +move at all. The answer that the very idea of the motion of +the arrow implies the passage from where it is to where it is +not was rejected by Zeno, who seems to have thought that the +appearance of motion of a body was a phenomenon caused by +the successive appearances of the body at rest but in different +positions. + +Zeno also asserted that the idea of motion was itself +inconceivable, for what moves must reach the middle of its +course before it reaches the end. Hence the assumption of +motion presupposes another motion, and that in turn another, +and so ad infinitum. His objection was in fact analogous to +the biological difficulty expressed by Swift\index{Swift}:--- +\begin{verse}\small +\leavevmode\llap{``}% +So naturalists observe, a flea hath smaller fleas that on him prey.\\ +And these have smaller fleas to bite 'em. And so proceed ad infinitum.'' +\end{verse} + +\bigskip +\noindent Or as De~Morgan\index{DeMorgan@De Morgan, A.} preferred to put it +\begin{verse}\small +\leavevmode\llap{``}% +Great fleas have little fleas upon their backs to bite 'em,\\ +And little fleas have lesser fleas, and so ad infinitum.\\ +And the great fleas themselves, in turn, have greater fleas to go on;\\ +While these have greater still, and greater still, and so on.'' +\end{verse} + +\ThoughtBreakSpace +\subsection*{Achilles and the Tortoise} Zeno's paradox about Achilles\index +{Achilles and the Tortoise} +and the tortoise is known even more widely. The assertion +was that if Achilles ran ten times as fast as a tortoise, yet +if the tortoise had (say) $1000$ yards start it could never be +\PG----File: 096.png------------------------------------------------------- +overtaken. To establish this, Zeno argued that when Achilles +had gone the $1000$ yards, the tortoise would still be $100$ yards +in front of him; by the time he had covered these $100$ yards, +it would still be $10$ yards in front of him; and so on for ever. +Thus Achilles would get nearer and nearer to the tortoise +but would never overtake it. Zeno regarded this as confirming +his view that the popular idea of motion is self-contradictory. + +\subsection*{Zeno's Paradox on Time} The fallacy of Achilles and +the Tortoise is usually explained by saying that though the +time required to overtake the tortoise can be divided into an +infinite number of intervals, as stated in the argument, yet +these intervals get smaller and smaller in geometrical progression, +and the sum of them all is a finite time: after the lapse of +that time Achilles would be in front of the tortoise. Probably +Zeno would have replied that this explanation rests on the +assumption that space and time are infinitely divisible, +propositions which he would not admit. He seems further +to have contended that while, to an accurate thinker, the +notion of the infinite divisibility of time was impossible, it +was equally impossible to think of a minimum measure of +time. For suppose, he argued, that $\tau$ is the smallest conceivable +interval, and suppose that three horizontal lines composed +of three consecutive spans $abc$, $a'b'c'$, $a''b''c''$ are placed so +that $aa'a''$, $bb'b''$, $cc'c''$ are vertically over one another. Imagine +the second line moved as a whole one span to the right in the +time $\tau$, and simultaneously the third line moved as a whole +one span to the left. Then $b$, $a'$, $c''$ will be vertically over +one another. And in this duration $\tau$ (which by hypothesis is +indivisible) $c'$ must have passed vertically over $a''$. Hence the +duration is divisible, contrary to the hypothesis\index{Zeno on Motion|)}. + +\markright{The Paradox of Tristram Shandy.} +\subsection*{The Paradox of Tristram Shandy} Mr~Russell\index +{Russell, B.A.W.} has enunciated\footnote +{B.A.W.~Russell, \textit{Principles of Mathematics}, Cambridge, 1903, +vol.~\textsc{i}, p.~358. +} a paradox somewhat similar to that of Achilles and +the Tortoise, save that the intervals of time considered get +\PG----File: 097.png----------------------------------------------------- +longer and longer during the course of events. Tristram +Shandy, as we know, took two years writing the history of +the first two days of his life, and lamented that, at this rate, +material would accumulate faster than he could deal with it, +so that he could never come to an end, however long he lived. +But had he lived long enough, and not wearied of his task, +then, even if his life had continued as eventfully as it began, +no part of his biography would have remained unwritten. +For if he wrote the events of the first day in the first year, he +would write the events of the $n$th day in the $n$th year, hence +in time the events of any assigned day would be written, and +therefore no part of his biography would remain unwritten. +This argument might be put in the form of a demonstration +that the part of a magnitude may be equal to the whole of it. + +Questions, such as those given above, which are concerned +with the continuity and extent of space and time involve +difficulties of a high order. + +\markright{Angular Motion.} +\subsection*{Angular Motion} A non-mathematician finds additional\index +{Angular Motion} +difficulties in the idea of angular motion. For instance, here +is a well-known proposition on motion in an equiangular spiral +(of which the result is true on the ordinary conventions of +mathematics) which shows that a body, moving with uniform +velocity and as slowly as we please, may in a finite time whirl +round a fixed point an infinite number of times. + +The equiangular spiral is the trace of a point $P$, which +moves along a line $OP$, the line $OP$ turning round a fixed +point $O$ with uniform angular velocity while the distance of +$P$ from $O$ decreases with the time in geometrical progression. +If the radius vector rotates through four right angles we have +one convolution of the curve. All convolutions are similar, +and the length of each convolution is a constant fraction, say +$1/n$th, that of the convolution immediately outside it. Inside +any given convolution, there are an infinite number of convolutions +which get smaller and smaller as we get nearer the +pole. Now suppose a point $Q$ to move uniformly along the +spiral from any point towards the pole. If it covers the first +\PG----File: 098.png----------------------------------------------------- +convolution in $a$ seconds, it will cover the next in $a/n$ seconds, +the next in $a/n^2$ seconds, and so on, and will finally reach the +pole in $(a + a/n + a/n^2 + a/n^3 + \dotsb)$ seconds, that is, in +$an/(n-1)$ seconds. The velocity is uniform, and yet in a finite +time, $Q$ will have traversed an infinite number of convolutions +and therefore have circled round the pole an infinite number +of times\footnote +{The proposition is put in this form in J.~Richard's\index +{Richard, J.} \textit{Philosophie des +math\-é\-mat\-iques}, Paris, 1903, pp.~119--120.}\index +{MotionParadox@\nobreak--- Paradoxes on|)}. + +\markright{Simple Relative Motion.} +\subsection*{Simple Relative Motion} Even if the philosophical difficulties +suggested by Zeno are settled or evaded, the mere idea +of relative motion\index +{Relative Motion} has been often found to present difficulties, +and Zeno himself failed to explain a simple phenomenon +involving the principle. As one of the easiest examples of +this kind, I may quote the common question of how many +trains going from $B$ to $A$ a passenger from $A$ to $B$ would +meet and pass on his way, assuming that the journey either +way takes $4\frac{1}{2}$ hours and that the trains start from each end +every hour. The answer is $9$. Or again this: Take two +pennies, face upwards on a table and edges in contact. +Suppose that one is fixed and that the other rolls on it without +slipping, making one complete revolution round it and +returning to its initial position. How many revolutions round +its own centre has the rolling coin made? The answer is $2$\index +{FallaciesMech@\nobreak--- \textsc{Mechanical}|)}. + +\subsection*{Laws of Motion}% +\addcontentsline{toc}{section}{Force, Inertia, Centrifugal Force}% +\markright{Laws of Motion.}% +I proceed next to make a few remarks +on points connected with the laws of motion\index +{MotionLaw@Motion, Laws of|(}. + +The first law of motion is often said to define \emph{force}\index +{Force, Definition of}, but +it is in only a qualified sense that this is true. Probably +the meaning of the law is best expressed in Clifford's\index{Clifford} phrase, +that force is ``the description of a certain kind of motion''---in +other words it is not an entity but merely a convenient +way of stating, without circumlocution, that a certain kind of +motion is observed. + +It is not difficult to show that any other interpretation +lands us in difficulties. Thus some authors use the law to +justify a definition that force is that which moves a body or +\PG----File: 099.png------------------------------------------------------- +changes its motion; yet the same writers speak of a steam-engine +moving a train. It would seem then that, according +to them, a steam-engine is a force. That such statements are +current may be fairly reckoned among mechanical paradoxes. + +The idea of force is difficult to grasp. How many people, +for instance, could predict correctly what would happen in a +question as simple as the following? A rope (whose weight +may be neglected) hangs over a smooth pulley; it has one end +fastened to a weight of $10$ stone, and the other end to a sailor +of weight $10$ stone, the sailor and the weight hanging in the +air. The sailor begins to climb up the rope; will the weight +move at all; and, if so, will it rise or fall? + +It will be noted that in the first law of motion it is asserted +that, unless acted on by an external force, a body in motion +continues to move (i)~with uniform velocity, and (ii)~in a +straight line. + +The tendency of a body to continue in its state of rest +or of uniform motion is called its \emph{inertia}\index{Inertia}. +This tendency +may be used to explain various common phenomena and +experiments. Thus, if a number of dominoes or draughts are +arranged in a vertical pile, a sharp horizontal blow on one of +those near the bottom will send it out of the pile, and those +above will merely drop down to take its place---in fact they +have not time to change their relative positions before there +is sufficient space for them to drop vertically as if they were a +solid body. + +This also is the principle on which depends the successful +playing of ``Aunt Sally,'' and the performance of numerous +tricks, described in collections of mathematical puzzles\footnote +{See \textit{Les récréations scientifiques} by G.~Tissandier\index +{Tissandier}, where several +ingenious illustrations of inertia are given.}. + +The statement about inertia\index{Inertia} in the first law may be taken +to imply that a body set in rotation about a principal axis +passing through its centre of mass will continue to move with +a uniform angular velocity and to keep its axis of rotation fixed +in direction. The former of these statements is the assumption +\PG----File: 100.png----------------------------------------------------- +on which our measurement of time is based as mentioned below +in \hyperlink{chapter.13}{chapter~\textsc{xiii}}. +The latter assists us to explain the motion of +a projectile in a resisting fluid. It affords the explanation of +why the barrel of a rifle is grooved; and why, similarly, anyone +who has to throw a flat body of irregular shape (such as a card) +in a given direction usually gives it a rapid rotatory motion +about a principal axis. Elegant illustrations of the fact just +mentioned are afforded by a good many of the tricks of acrobats, +though the full explanation of most of them also introduces +other considerations. Thus when some few years ago the +Japanese village at Knightsbridge was one of the shows of +London, there were some acrobats there who tossed on to the +top surface of an umbrella a penny so that it alighted on its +edge, and then, by turning round the stick of the umbrella +rapidly, the coin was caused to rotate, but as the umbrella +moved away underneath it the coin remained apparently +stationary and standing upright, while by diminishing or increasing +the angular velocity of the umbrella the penny was +caused to run forwards or backwards. This is not a difficult +trick to execute. + +The tendency of a body in motion to continue to move +in a straight line is sometimes called its \textit{centrifugal force}\index +{Centrifugal Force|(}. +Thus, if a train is running round a curve, it tends to move in +a straight line, and is constrained only by the pressure of the +rails to move in the required direction. Hence it presses on +the outer rail of the curve. This pressure can be diminished +to some extent both by raising the outer rail, and by putting +a guard rail, parallel and close to the inner rail, against which +the wheels on that side also will press. + +An illustration of this fact occurred in a little known incident +of the American civil war\footnote +{\textit{Capturing a Locomotive} by W.~Pittenger\index +{Pittenger}, London, 1882, p.~104.}. In the spring of 1862 a +party of volunteers from the North made their way to the +rear of the Southern armies and seized a train, intending to +destroy, as they passed along it, the railway which was the +main line of communication between various confederate corps +\PG----File: 101.png------------------------------------------------- +and their base of operations. They were however detected +and pursued. To save themselves, they stopped on a sharp +curve and tore up some rails so as to throw the engine which +was following them off the line. Unluckily for themselves +they were ignorant of dynamics and tore up the inner rails of +the curve, an operation which did not incommode their +pursuers\index{Centrifugal Force|)}. + +The second law gives us the means of measuring mass, +force, and therefore \emph{work}\index{Work|(}. +A given agent in a given time can +do only a definite amount of work. This is illustrated by the +fact that although, by means of a rigid lever and a fixed +fulcrum, any force however small may be caused to move any +mass however large, yet what is gained in power is lost in +speed---as the popular phrase runs. + +Montucla\index{Montucla}\footnote +{Ozanam\index{Ozanam@Ozanam's \textit{Récréations}}, 1803 edition, +vol.~\textsc{ii}, p.~18; 1840 edition, p.~202.} +inserted a striking illustration of this principle +founded on the well-known story of Archimedes\index{Archimedes} who is said +to have declared to Hiero\index +{Hiero of Syracuse} that, were he but given a fixed +fulcrum, he could move the world. Montucla\index{Montucla} calculated the +mass of the earth and, assuming that a man could work incessantly +at the rate of $116$ foot-lbs.\ per second, which is a very +high estimate, he found that it would take over three billion +centuries, \IE\ $3 \times 10^{14}$ years, before a mass equal to that of the +earth was moved as much as one inch against gravity at the +surface of the earth: to move it one inch along a horizonal % [** this isn't a typo!] +plane would take about $74000$ centuries. + +\subsection*{Stability of Equilibrium}% +\addcontentsline{toc}{section}{Work, Stability of Equilibrium, \protect\&c.} +It is known to all those who\index +{Equilibrium, Puzzles on|(}\index{Stability of Equilibrium|(} +have read the elements of mechanics that the centre of gravity +of a body, which is resting in equilibrium under its own weight, +must be vertically above its base: also, speaking generally, +we may say that, if every small displacement has the effect of +raising the centre of gravity, then the equilibrium is stable, +that is, the body when left to itself will return to its original +position; but, if a displacement has the effect of lowering the +centre of gravity, then for that displacement the equilibrium +is unstable; while, if every displacement does not alter the +\PG----File: 102.png----------------------------------------------------- +height above some fixed plane of the centre of gravity, then +the equilibrium is neutral. In other words, if in order to cause +a displacement work has to be done against the forces acting +on the body, then for that displacement the equilibrium is +stable, while if the forces do work the equilibrium is unstable. + +A good many of the simpler mechanical toys and tricks +afford illustrations of this principle. + +\markright{Magic Bottles.} +\subsection*{Magic Bottles\protect\footnote +{Ozanam\index{Ozanam@Ozanam's \textit{Récréations}}, +1803 edition, vol.~\textsc{ii}, p.~15; 1840 edition, p.~201.}} +Among the most common of such toys are +the small bottles\index{Magic Bottles}---trays of which may be seen +any day in the +streets of London---which keep always upright, and cannot +be upset until their owner orders them to lie down. Such a +bottle is made of thin glass or varnished paper fixed to the +plane surface of a solid hemisphere or smaller segment of +a sphere. Now the distance of the centre of gravity of a +homogeneous hemisphere from the centre of the sphere is +three-eighths of the radius, and the mass of the glass or +varnished paper is so small compared with the mass of the +lead base that the centre of gravity of the whole bottle is +still within the hemisphere. Let us denote the centre of the +hemisphere by $C$, and the centre of gravity of the bottle by $G$. + +If such a bottle is placed with the hemisphere resting on a +horizontal plane and $GC$ vertical, any small displacement on the +plane will tend to raise $G$, and thus the equilibrium is stable. +This may be seen also from the fact that when slightly displaced +there is brought into play a couple, of which one force +is the reaction of the table passing through $C$ and acting +vertically upward, and the other the weight of the bottle +acting vertically downward at $G$. If $G$ is below $C$, this couple +tends to restore the bottle to its original position. + +If there is dropped into the bottle a shot or nail so heavy +as to raise the centre of gravity of the whole above $C$, then +the equilibrium is unstable, and, if any small displacement is +given, the bottle falls over on to its side. + +Montucla\index{Montucla} says that in his time it was not uncommon to +see boxes of tin soldiers mounted on lead hemispheres, and +\PG----File: 103.png----------------------------------------------------- +when the lid of the box was taken off the whole regiment +sprang to attention. + +In a similar way we may explain how to balance a pencil +in a vertical position, with its point resting on the top of one's +finger, an experiment which is described in nearly every book +of puzzles\footnote +{\EG\ Oughtred\index{Oughtreds@Oughtred's \textit{Recreations}}, +\textit{Mathematical Recreations}, p.~24; Ozanam\index +{Ozanam@Ozanam's \textit{Récréations}}, 1803 +edition, vol.~\textsc{ii}, p.~14; 1840 edition, p.~200.}. +This is effected by taking a penknife, of which +one blade is opened through an angle of (say) $120^{\circ}$, and sticking +the blade in the pencil so that the handle of the penknife is +below the finger. The centre of gravity is thus brought below +the point of support, and a small displacement given to the +pencil will raise the centre of gravity of the whole: thus the +equilibrium is stable. + +Other similar tricks are the suspension of a bucket over +the edge of a table by a couple of sticks, and the balancing of +a coin on the edge of a wine-glass by the aid of a couple of +forks\footnote +{Oughtred\index{Oughtreds@Oughtred's \textit{Recreations}}, p.~30; +Ozanam\index{Ozanam@Ozanam's \textit{Récréations}}, 1803 edition, +vol.~\textsc{ii}, p.~12; 1840 edition, +p.~199.}---the +sticks or forks being so placed that the centre of +gravity of the whole is vertically below the point of support +and its depth below it a maximum. + +The toy representing a horseman, whose motion continually +brings him over the edge of a table into a position which seems +to ensure immediate destruction, is constructed in somewhat +the same way. A wire has one end fixed to the feet of the +rider; the wire is curved downwards and backwards, and at +the other end is fixed a weight. When the horse is placed so +that his hind legs are near the edge of the table and his forefeet +over the edge, the weight is under his hind feet. Thus +the whole toy forms a pendulum with a curved instead of a +straight rod. Hence the farther it swings over the table, the +higher is the centre of gravity raised, and thus the toy tends +to return to its original position of equilibrium. + +An elegant modification of the prancing horse was brought +out at Paris in 1890 in the shape of a toy made of tin and in +\PG----File: 104.png----------------------------------------------------- +the figure of a man\footnote +{\textit{La Nature}, Paris, March, 1891.}. The legs are pivoted +so as to be movable +about the thighs, but with a wire check to prevent too long +a step, and the hands are fastened to the top of a $\bigcap$-shaped +wire weighted at its ends. If the figure is placed on a narrow +sloping plank or strip of wood passing between the legs of the +$\bigcap$, then owing to the $\bigcap$-shaped wire any lateral displacement +of the figure will raise its centre of gravity, and thus for any +such displacement the equilibrium is stable. Hence, if a slight +lateral disturbance is given, the figure will oscillate and will +rest alternately on each foot: when it is supported by one foot +the other foot under its own weight moves forwards, and thus +the figure will walk down the plank though with a slight +reeling motion. Shortly after the publication of the third +edition of this book an improved form of this toy, in the +shape of a walking elephant made in heavy metal, was issued +in England, and probably in that form it is now familiar to +all who are interested in noticing street toys\index +{Equilibrium, Puzzles on|)}\index{Stability of Equilibrium|)}. + +\markright{Columbus's Egg.} +\subsection*{Columbus's Egg} The toy known as Columbus's egg\index +{Columbus's Egg Puzzle} depends +on the same principle as the magic bottle, though it leads to +the converse result. The shell of the egg is made of tin and +cannot be opened. Inside it and fastened to its base is a +hollow truncated tin cone, and there is also a loose marble +inside the shell. If the egg is held properly, the marble runs +inside the cone and the egg will stand on its base, but so long +as the marble is outside the cone, the egg cannot be made to +stand on its base. + +\markright{Cones running up hill.} +\subsection*{Cones running up hill\protect\footnote +{Ozanam\index{Ozanam@Ozanam's \textit{Récréations}}, 1803 edition, +vol.~\textsc{ii}, p.~49; 1840 edition, p.~216.}} The +experiment to make a double +cone run up hill\index{Cones moving uphill} depends on the same principle +as the toys +above described; namely, on the tendency of a body to take a +position so that its centre of gravity is as low as possible. In +this case it produces the optical effect of a body moving by +itself up a hill. + +Usually the experiment is performed as follows. Arrange +two sticks in the shape of a $\bigvee$, with the apex on a table and +\PG----File: 105.png----------------------------------------------------- +the two upper ends resting on the top edge of a book placed +on the table. Take two equal cones fixed base to base, and place +them with the curved surfaces resting on the sticks near the +apex of the $\bigvee$, the common axis of the cones being horizontal +and parallel to the edge of the book. Then, if properly +arranged, the cones will run up the plane formed by the +sticks. + +The explanation is obvious. The centre of gravity of the +cones moves in the vertical plane midway between the two +sticks and it occupies a lower position as the points of contact +on the sticks get farther apart. Hence as the cone rolls up +the sticks its centre of gravity descends% +\index{MotionLaw@Motion, Laws of|)}% +\index{Newtonian Laws of Motion|)}% +\index{PuzzlesMech@\nobreak--- Mechanical|)}% +\index{Work|)}. + +\section{Perpetual Motion} The idea of making a machine which% +\index{FallaciesMech@\nobreak--- \textsc{Mechanical}|(}% +\index{MotionPerp@\nobreak--- Perpetual|(}% +\index{Perpetual@\textsc{Perpetual Motion}|(} +once set going would continue to go for ever by itself has been +the ignis fatuus of self-taught mechanicians in much the same +way as the quadrature of the circle has been of self-taught +geometricians. + +Now the obvious meaning of the third law of motion is +that a force is only one aspect of a stress, and that whenever a +force is caused another equal and opposite one is brought also +into existence---though it may act upon a different body, and +thus be immaterial for the particular problem considered. The +law however is capable of another +interpretation\footnote{Newton's\index{Newton} \textit{Principia}, last +paragraph of the Scholium to the Laws of Motion.}, namely, +that the rate at which an agent does work (that is, its action) +is equal to the rate at which work is done against it (that is, +its reaction). If it is allowable to include in the reaction the +rate at which kinetic energy is being produced, and if work is +taken to include that done against molecular forces, then it +follows from this interpretation that the work done by an +agent on a system is equivalent to the total increase of energy, +that is, the power of doing work. Hence in an isolated system +the total amount of energy is constant. If this is granted, +then since friction and some molecular dissipation of energy +\PG----File: 106.png----------------------------------------------------- +cannot be wholly prevented, it must be impossible to construct +in an isolated system a machine capable of perpetual motion. + +I do not propose to describe in detail the various machines +for producing perpetual motion which have been suggested, but +I may add that a number of them are equivalent essentially to +the one of which a section is represented in the accompanying +figure. + +\begin{figure*}[!hbt] +\centerline{\includegraphics[width=6cm]{./images/illus106}} +\end{figure*} + +It consists of two concentric vertical wheels in the same +plane, and mounted on a horizontal axle through their centre, $C$. +The space between the wheels is divided into compartments by +spokes inclined at a constant angle to the radii to the points +whence they are drawn, and each compartment contains a +heavy bullet. Apart from these bullets, the wheels would be +in equilibrium. Each bullet tends to turn the wheels round +their axle, and the moment which measures this tendency is +the product of the weight of the bullet and its distance from +the vertical through $C$. + +The idea of the constructors of such machines was that, as +the bullet in any compartment would roll under gravity to the +lowest point of the compartment, the bullets on the right-hand +side of the diagram would be farther from the vertical through +$C$ than those on the left. Hence the sum of the moments of +the weights of the bullets on the right would be greater than +the sum of the moments of those on the left. Thus the wheels +\PG----File: 107.png----------------------------------------------------- +would turn continually in the same direction as the hands of a +watch. The fallacy in the argument is obvious. + +Another large group of machines for producing perpetual +motion depended on the use of a magnet to raise a mass which +was then allowed to fall under gravity. Thus, if the bob of a +simple pendulum was made of iron, it was thought that magnets +fixed near the highest points which were reached by the bob in +the swing of the pendulum would draw the bob up to the same +height in each swing and thus give perpetual motion. + +Of course it is only in isolated systems that the total amount +of energy is constant, and, if a source of external energy can be +obtained from which energy is continually introduced into the +system, perpetual motion is, in a sense, possible; though even +here materials would ultimately wear out. The solar heat and +the tides are among the most obvious of such sources. + +There was at Paris in the latter half of the eighteenth\label +{page:PerpClockStart} +century a clock\index{Clocks} which was an ingenious illustration of such +perpetual motion\footnote +{Ozanam\index{Ozanam@Ozanam's \textit{Récréations}}, 1803 edition, +vol.~\textsc{ii}, p.~105; 1840 edition, p.~238.}. The energy which was +stored up in it to +maintain the motion of the pendulum was provided by the +expansion of a silver rod. This expansion was caused by the +daily rise of temperature, and by means of a train of levers it +wound up the clock. There was a disconnecting apparatus, so +that the contraction due to a fall of temperature produced no +effect, and there was a similar arrangement to prevent overwinding. +I believe that a rise of eight or nine degrees +Fahrenheit was sufficient to wind up the clock for twenty-four +hours. + +I have in my possession a watch\index{Watches}, which produces the +same effect by somewhat different means. Inside the case +is a steel weight, and if the watch is carried in a pocket this +weight rises and falls at every step one takes, somewhat after +the manner of a pedometer. The weight is raised by the +action of the person who has it in his pocket in taking a +step, and in falling it winds up the spring of the watch. +On the face is a small dial showing the number of hours for +\PG----File: 108.png----------------------------------------------------- +which the watch is wound up. As soon as the hand of this +dial points to fifty-six hours, the train of levers which winds +up the watch disconnects automatically, so as to prevent overwinding +the spring, and it reconnects again as soon as the +watch has run down eight hours. The watch\index{Watches} is an excellent +time-keeper, and a walk of about a couple of miles is sufficient +to wind it up for twenty-four hours% +\index{MotionPerp@\nobreak--- Perpetual|)}% +\index{Perpetual@\textsc{Perpetual Motion}|)}% +\label{page:PerpClockEnd}. + +\section{Models} I may add here the observation, which is well\index +{Models@\textsc{Models}|(} +known to mathematicians, but is a perpetual source of disappointment +to ignorant inventors, that it frequently happens +that an accurate model of a machine will work satisfactorily +while the machine itself will not do so. + +One reason for this is as follows. If all the parts of a model +are magnified in the same proportion, say $m$, and if thereby a +line in it is increased in the ratio $m:1$, then the areas and +volumes in it will be increased respectively in the ratios $m^2:1$ +and $m^3:1$. For example, if the side of a cube is doubled then +a face of it will be increased in the ratio $4: 1$ and its volume +will be increased in the ratio $8:1$. + +Now if all the linear dimensions are increased $m$ times, +then some of the forces that act on a machine (such, for +example, as the weight of part of it) will be increased $m^3$ times, +while others which depend on area (such as the sustaining +power of a beam) will be increased only $m^2$ times. Hence the +forces that act on the machine and are brought into play by +the various parts may be altered in different proportions, and +thus the machine may be incapable of producing results similar +to those which can be produced by the model. + +The same argument has been adduced in the case of animal +life to explain why very large specimens of any particular breed +or species are usually weak. For example, if the linear dimensions +of a bird were increased $n$ times, the work necessary to +give the power of flight would have to be increased no less +than $n^7$ times\footnote +{Helmholtz\index{Helmholtz}\index{Von Helmholz}, \textit{Gesammelte +Abhandlungen}, Leipzig, 1881, vol.~\textsc{i}, p.~165.}. Again, if the +linear dimensions of a man +\PG----File: 109.png------------------------------------------------------ +of height $5$~ft.\ $10$~in.\ were increased by one-seventh his height +would become $6$~ft.\ $8$~in., but his weight would be increased +in the ratio $512: 343$ (\IE\ about half as much again), while +the cross sections of his legs, which would have to bear this +weight, would be increased only in the ratio $64:49$; thus +in some respects he would be less efficient than before. Of +course the increased dimensions, length of limb, or size of +muscle might be of greater advantage than the relative loss +of strength; hence the problem of what are the most efficient +proportions is not simple, but the above argument will serve +to illustrate the fact that the working of a machine may not +be similar to that of a model of it\index +{FallaciesMech@\nobreak--- \textsc{Mechanical}|)}\index{Models@\textsc{Models}|)}. + +\ThoughtBreakSpace +Leaving now these elementary considerations I pass on to +some other mechanical questions. + +\ssection{Sailing quicker than the Wind} As a kinematical +paradox I may allude to the possibility of \emph{sailing quicker than +the wind blows}\index{Sailing@\textsc{Sailing}, Theory of|(}, +a fact which strikes many people as curious. + +The explanation\footnote +{Ozanam\index{Ozanam@Ozanam's \textit{Récréations}}, 1803 edition, +vol.~\textsc{iii}, pp.~359, 367; 1840 edition, pp.~540, +543.} depends on the consideration of the +velocity of the wind relative to the boat. Perhaps, however, +a non-mathematician will find the solution simplified if I consider +first the effect of the wind-pressure on the back of the +sail which drives the boat forward, and second the resistance +to motion caused by the sail being forced through the air. + +When the wind is blowing against a plane sail the resultant +pressure of the wind on the sail may be resolved into two +components, one perpendicular to the sail (but which in general +is not a function only of the component velocity in that direction, +though it vanishes when that component vanishes) and +the other parallel to its plane. The latter of these has no +effect on the motion of the ship. The component perpendicular +to the sail tends to move the ship in that direction. +This pressure, normal to the sail, may be resolved again into +two components, one in the direction of the keel of the boat, +\PG----File: 110.png------------------------------------------------------ +the other in the direction of the beam of the boat. The +former component drives the boat forward, the latter to leeward. +It is the object of a boat-builder to construct the boat +on lines so that the resistance of the water to motion forward +shall be as small as possible, and the resistance to motion in +a perpendicular direction (\IE\ to leeward) shall be as large as +possible; and I will assume for the moment that the former +of these resistances may be neglected, and that the latter is +so large as to render motion in that direction impossible. + +Now, as the boat moves forward, the pressure of the air +on the front of the sail will tend to stop the motion. As +long as its component normal to the sail is less than the +pressure of the wind behind the sail and normal to it, the +resultant of the two will be a force behind the sail and normal +\begin{figure*}[!hbt] +\centerline{\includegraphics[width=9cm]{./images/illus110}} +\end{figure*} +to it which tends to drive the boat forwards. But as the +velocity of the boat increases, a time will arrive when the +pressure of the wind is only just able to balance the resisting +force which is caused by the sail moving through the air. The +velocity of the boat will not increase beyond this, and the +motion will be then what mathematicians describe as ``steady.'' + +In the accompanying figure, let $BAR$ represent the keel +of a boat, $B$ being the bow, and let $SAL$ represent the sail. +Suppose that the wind is blowing in the direction $W\!A$ with +a velocity $u$; and that this direction makes an angle $\theta$ with +\PG----File: 111.png----------------------------------------------------- +the keel, \IE\ angle $W\!AR = \theta$. Suppose that the sail is set +so as to make an angle $\alpha$ with the keel, \IE\ angle $BAS = \alpha$, and +therefore angle $W\!AL=\theta + \alpha$. Suppose finally that $v$ is the +velocity of the boat in the direction $AB$. + +I have already shown that the solution of the problem +depends on the relative directions and velocities of the +wind and the boat; hence to find the result reduce the boat to +rest by impressing on it a velocity $v$ in the direction $BA$. +The resultant velocity of $v$ parallel to $BA$ and of $u$ parallel +to $W\!A$ will be parallel to $SL$, if +$v \sin \alpha = u \sin (\theta + \alpha)$; and in +this case the resultant pressure perpendicular to the sail +vanishes. + +Thus, for steady motion we have $v \sin \alpha=u \sin (\theta + \alpha)$. +Hence, whenever $\sin (\theta + \alpha)>\allowbreak\sin \alpha$, we have +$v>u$. Suppose, +to take one instance, the sail to be fixed, that is, suppose $\alpha$ to +be a constant. Then $v$ is a maximum if $\theta + \alpha = \frac{1}{2}\pi$, +that is, if +$\theta$ is equal to the complement of $\alpha$. In this case we have +$v = u \cosec \alpha$, and therefore $v$ is greater than $u$. Hence, if the +wind makes the same angle $\alpha$ abaft the beam that the sail +makes with the keel, the velocity of the boat will be greater +than the velocity of the wind. + +Next, suppose that the boat is running close to the wind, +so that the wind is before the beam (see figure +\vhyperlink{Figure:112}{below}), +% [**Note: original wording "on next page"] +then in the same way as before we have +$v \sin \alpha = u \sin (\theta + \alpha)$, +or $v \sin \alpha = u \sin \phi$, where +$\phi = \text{angle } W\!AS = \pi - \theta - \alpha$. Hence +$v = u \sin \phi \cosec \alpha$. + +Let $w$ be the component velocity of the boat in the teeth +of the wind, that is, in the direction $AW$. Then we have +$w =\allowbreak v \cos BAW =\allowbreak v \cos (\alpha + \phi) = +\allowbreak u \sin \phi \cosec \alpha \cos (\alpha + \phi)$. If $\alpha$ +is constant, this is a maximum when +$\phi = \frac{1}{4}\pi - \frac{1}{2}\alpha$; and, if $\phi$ +has this value, then $w = \frac{1}{2}u (\cosec\alpha - 1)$. This formula shows +that $w$ is greater than $u$, if $\sin\alpha< \frac{1}{3}$. Thus, if the sails +can be set so that $\alpha$ is less than $\sin^{-1}\frac{1}{3}$, that is, +rather less +than $19^{\circ} 29'$, and if the wind has the direction above assigned, +then the component velocity of the boat in the face of the +wind is greater than the velocity of the wind. + +\PG----File: 112.png------------------------------------------------------ +The above theory is curious, but it must be remembered +that in practice considerable allowance has to be made for +the fact that no boat for use on water can be constructed in +\begin{figure*}[!hbt] +\centerline{\includegraphics[width=9cm]{./images/illus112}\DPlabel{Figure:112}} +\end{figure*} +which the resistance to motion in the direction of the keel +can be wholly neglected, or which would not drift slightly to +leeward if the wind was not dead astern. Still this makes less +difference than might be thought by a landsman. In the case +of boats sailing on smooth ice the assumptions made are substantially +correct, and the practical results are said to agree +closely with the theory. + +\section[Boat moved by a rope inside the boat][Boat moved by a Rope.]% +{Boat moved by a Rope} There is a form of boat-racing\index +{Boat-racing with a rope|(}, +occasionally used at regattas, which affords a somewhat curious +illustration of certain mechanical principles. The only thing +supplied to the crew is a coil of rope, and they have, without +leaving the boat, to propel it from one point to another as +rapidly as possible. The motion is given by tying one end of +the rope to the after thwart, and giving the other end a series +of violent jerks in a direction parallel to the keel. I am told +that in still water a pace of two or three miles an hour can +be thus attained. + +\PG----File: 113.png------------------------------------------------------ +The chief cause for this result seems to be that the friction +between the boat and the water retards all relative motion, +but is not great enough to materially affect motion caused by +a sufficiently big impulse. Hence the usual movements of the +crew in the boat do not sensibly move the centre of gravity of +themselves and the boat, but this does not apply to an impulsive +movement, and if the crew in making a jerk move their +centre of gravity towards the bow $n$ times more rapidly than +it returns after the jerk, then the boat is impelled forwards +at least $n$ times more than backwards: hence on the whole +the motion is forwards\index +{Boat-racing with a rope|)}\index{Sailing@\textsc{Sailing}, Theory of|)}. + +\section[Results dependent on Hauksbee's Law][Motion of Fluids.]% +{Motion of Fluids and Motion in Fluids} The theories +of \emph{motion of fluids}\index{Fluid Motion|(} and +\emph{motion in fluids}\index{Motion in Fluids|(} involve considerable +difficulties. Here I will mention only one or two instances---mainly +illustrations of Hauksbee's Law. + +\subsection*{Hauksbee's Law} When a fluid is in rapid motion\index +{Hauksbee@\textsc{Hauksbee's Law}|(} the +pressure is less than when it is at rest\footnote +{See Besant\index{Besant on Hauksbee's Law}, \textit{Hydromechanics}, +Cambridge, 1867, art.~149, where however +it is assumed that the pressure is proportional to the density. +Hauksbee was the earliest writer who called attention to the problem, +but I do not know who first explained the phenomenon; some references +to it are given by Willis\index{Willis on Hauksbee's law}, +\textit{Cambridge Philosophical Transactions}, 1830, +vol.~\textsc{iii}, pp.~129--140.}. Thus, if a current +of air is moving in a tube, the pressure on the sides of the +tube is less than when the air is at rest---and the quicker the +air moves the smaller is the pressure. This fact was noticed +by Hauksbee nearly two centuries ago. In an elastic perfect +fluid in which the pressure is proportional to the density, the +law connecting the pressure, $p$, and the steady velocity, $v$, is +$p = \Pi\alpha^{-v^2}$ where $\Pi$ and $\alpha$ are constants: the +establishment of +corresponding formula for gases where the pressure is proportional +to a power of the density presents no difficulty. + +This principle is illustrated by a twopenny toy, on sale in +most toy-shops, called the \emph{pneumatic mystery}. It consists of a +tube, with a cup-shaped end in which rests a wooden ball. If +the tube is held in a vertical position, with the mouthpiece at +\PG----File: 114.png------------------------------------------------------- +the upper end and the cup at the lower end, then, if anyone +blows hard through the tube and places the ball against the +cup, the ball will remain suspended there. The explanation is +that the pressure of the air below the ball is so much greater +than the pressure of the air in the cup that the ball is held up. + +The same effect may be produced by fastening to one end +of a tube a piece of cardboard having a small hole in it. If +a piece of paper is placed over the hole and the experimenter +blows through the tube, the paper will not be detached from +the card but will bend so as to allow the egress of the air. + +An exactly similar experiment, described in many text-books +on hydromechanics, is made as follows. To one end of a +straight tube a plane disc is fitted which is capable of sliding +on wires projecting from the end of the tube. If the disc is +placed at a small distance from the end, and anyone blows +steadily into the tube, the disc will be drawn towards the tube +instead of being blown off the wires, and will oscillate about a +position near the end of the tube. + +In the same way we may make a tube by placing two books +on a table with their backs parallel and an inch or so apart and +laying a sheet of newspaper over them. If anyone blows +steadily through the tube so formed, the paper will be sucked in +instead of being blown out. + +The following experiment is explicable by the same argument. +On the top of a vertical axis balance a thin horizontal +rod. At each end of this rod fasten a small vertical square or +sail of thin cardboard---the two sails being in the same plane. +If anyone blows close to one of these squares and in a direction +parallel to its plane, the square will move towards the side on +which one is blowing, and the rod with the two sails will +rotate about the axis. + +The experiments above described can be performed so as +to illustrate Hauksbee's Law; but unless care is taken other +causes will be also introduced which affect the phenomena: +it is however unnecessary for my purpose to go into these +details. +\PG----File: 115.png------------------------------------------------------- + +\subsection[Cut on a tennis-ball. \texorpdfstring{\protect\quad}{} Spin on a cricket-ball]% +[Spin on Tennis and Cricket Balls.]{Cut on a Tennis-Ball} +Racquet and tennis players know +that if a strong cut is given to a ball it can be made to% +\index{Cut on a Tennis-ball|(}% +\index{Racquet-ball, Cut on|(}% +\index{Tennis-ball, Cut on|(} +rebound off a vertical wall and then (without striking the +floor or any other wall) return and hit the wall again. + +This affords another illustration of Hauksbee's Law. The +explanation\footnote +{See Magnus\index{Magnus on Hauksbee's Law} on +`\textit{Die Abweichung der Geschosse}' in the \textit{Abhandlungen +der Akademie der Wissenschaften}, Berlin, 1852, pp.~1--23; +Lord Rayleigh\index{Rayleigh}, +`\textit{On the irregular flight of a tennis ball},' +\textit{Messenger of Mathematics}, +Cambridge, 1878, vol.~\textsc{vii}, pp.~14--16. +} is that the cut causes the ball to rotate rapidly +about an axis through its centre of figure, and the friction of the +surface of the ball on the air produces a sort of whirlpool. This +rotation is in addition to its motion of translation. Suppose the +ball to be spherical and rotating about an axis through its centre +perpendicular to the plane of the paper in the direction of the +arrow-head, and at the same time moving through still air from +left to right parallel to $PQ$. Any motion of the ball perpendicular +to $PQ$ will be produced by the pressure of the air on the +surface of the ball, and this pressure will, by Hauksbee's Law, +be greatest where the velocity of the air relative to the ball is +least, and vice versa. To find the velocity of the air relative +to the ball we may reduce the centre of the ball to rest, and +suppose a stream of air to impinge on the surface of the ball +moving with a velocity equal and opposite to that of the +centre of the ball. The air is not frictionless, and therefore +the air in contact with the surface of the ball will be set in +motion, by the rotation of the ball and will form a sort of +whirlpool rotating in the direction of the arrow-head in the +figure. To find the actual velocity of this air relative to the +ball we must consider how the motion due to the whirlpool is +affected by the motion of the stream of air parallel to $QP$. +The air at $A$ in the whirlpool is moving against the stream of +air there, and therefore its velocity is retarded: the air at $B$ +in the whirlpool is moving in the same direction as the stream +of air there, and therefore its velocity is increased. Hence the +relative velocity of the air at $A$ is less than that at $B$, and +\PG----File: 116.png------------------------------------------------------- +since the pressure of the air is greatest where the velocity is +least, the pressure of the air on the surface of the ball at $A$ is +greater than on that at $B$, Hence the ball is forced by this +\begin{figure*}[!hbt] +\centerline{\includegraphics[height=\ifPaper3.5cm\else5cm\fi]{./images/illus116}} +\end{figure*} +pressure in the direction from the line $PQ$, which we may +suppose to represent the section of the vertical wall in a +racquet-court. In other words, the ball tends to move at right +angles to the line in which its centre is moving and in the +direction in which the surface of the front of the ball is being +carried by the rotation\index{Hauksbee@\textsc{Hauksbee's Law}|)}. + +In the case of a lawn tennis-ball, the shape of the ball is +altered by a strong cut, and this introduces additional complications% +\index{Cut on a Tennis-ball|)}% +\index{Racquet-ball, Cut on|)}% +\index{Tennis-ball, Cut on|)}. + +\subsection*{Spin on a Cricket-Ball} The curl of a cricket-ball% +\index{Spin on Cricket-ball}\index{Cricket-Ball, Spin on} in its +flight through the air, caused by a spin given by the bowler in +delivering the ball, is explained by the same reasoning. + +Thus suppose the ball is delivered in a direction lying in a +vertical plane containing the two middle stumps of the wickets. +A spin round a horizontal axis parallel to the crease in a +direction which the bowler's umpire would describe as positive, +namely, counter clock-wise, will, in consequence of the friction +of the air, cause it to drop, and therefore decrease the length +of the pitch. A spin in the opposite direction will cause it to +rise, and therefore lengthen the pitch. A spin round a vertical +axis in the positive direction, as viewed from above, will make +it curl sideways in the air to the left, that is, from leg to off. +A spin in the opposite direction will make it curl to the right. +A spin given to the ball round the direction of motion of the +\PG----File: 117.png-------------------------------------------------- +centre of the ball will not sensibly affect the motion through +the air, though it would cause the ball, on hitting the ground, +to break. Of course these various kinds of spin can be +combined. + +\ThoughtBreakSpace +The questions involving the application of Hauksbee's Law +are easy as compared with many of the problems in fluid +motion. The analysis required to attack most of these problems +is beyond the scope of this book, but one of them may +be worth mentioning even though no explanation is given. + +\subsection*{The Theory of the Flight of Birds}% +\addcontentsline{toc}{section}{Flight of Birds}% +\markright{Flight of Birds.}% +A mechanical problem\index{Birds, Flight of|(} +of great interest is the explanation of the means by which +birds are enabled to fly for considerable distances with no +(perceptible) motion of the wings. Albatrosses, to take an +instance of special difficulty, have been known to follow for +some days ships running at the rate of nine or ten knots, and +sometimes for considerable periods there is no motion of the +wings or body which can be detected, while even if the bird +moved its wings it is not easy to understand how it has the +muscular energy to propel itself so rapidly and for such a length +of time. Of this phenomenon various explanations\footnote +{See G.H.~Bryan\index{Bryan on Bird Flight} in the +\textit{Transactions of the British Association} for +1896, vol.~\textsc{lxvi}, pp.~726-728.} have +been suggested. Notable among these are Mr~Maxim's\index +{Maxim on Bird Flight} of +upward air-currents, Lord Rayleigh's\index{Rayleigh} of variations of the +wind velocity at different heights above the ground, Dr~S.P.~Langley's\index +{Langley on Bird Flight} +of the incessant occurrence of gusts of wind separated by lulls, +and Dr~Bryan's of vortices in the atmosphere. + +It now seems reasonably certain that the second and third of +these sources of energy account for at least a portion of the +observed phenomena. The effect of the third cause may be +partially explained by noting that the centre of gravity of the +bird with extended wings is slightly below the aeroplane or wing +surface, so that the animal forms a sort of parachute. The effect +of a sudden gust of wind upon such a body is that the aeroplane +is set in motion more rapidly than the suspended mass, causing +\PG----File: 118.png------------------------------------------------------ +the structure to heel over so as to receive the wind on the under +surface of the aeroplane, and this lifts the suspended mass +giving it an upward velocity. When the wind falls the greater +inertia of the mass carries it on upwards causing the aeroplane +to again present its under side to the air; and if while the +parachute is in this position the wind is still blowing from the +side, the suspended mass is again lifted. Thus the more the +bird is blown about, the more it rises in the air; actually birds +in flight are carried up by a sudden side gust of wind as we +should expect from this theory. + +The fact that the bird is in motion tends also to keep it up, +for it has been recently shown that a horizontal plane under +the action of gravity falls to the ground more slowly if it is +travelling through the air with horizontal velocity than it +would do if allowed to fall vertically, hence the bird's forward +motion causes it to fall through a smaller height between +successive gusts of wind than it would do if it were at rest, +Moreover it has been proved experimentally that the horsepower +required to support a body in horizontal flight by means +of an aeroplane is less for high than for low speeds: hence +when a side-wind (that is, a wind at right angles to the bird's +course) strikes the bird, the lift is increased in consequence +of the bird's forward velocity\index{Birds, Flight of|)}\index +{Fluid Motion|)}\index{Motion in Fluids|)}. + +\section{Curiosa Physica} When I was writing the first edition\index +{Curiosa Physica|(} +of these ``Recreations,'' I put together a chapter, following +this one, on ``Some Physical Questions,'' dealing with problems +such as, in the Theory of Sound\index +{Sound, Problem in}, the explanation of the +fact that in some of Captain Parry's\index +{Parry on Sound} experiments the report +of a cannon\index{Gun, Report of}, when fired, travelled so much more rapidly +than the sound of the human voice that observers heard the +report of the cannon when fired before that of the order to +fire it\footnote +{The fact is well authenticated. Mr~Earnshaw\index{Earnshaw, S.} +(\textit{Philosophical Transactions}, London, 1860, pp.~133--148) +explained it by the acceleration of +a wave caused by the formation of a kind of bore, a view accepted +by Clerk Maxwell and most physicists, but Sir George Airy\index +{Airy, Sir Geo.} thought that the +explanation was to be found in physiology; see Airy's \textit{Sound}, second +edition, London, 1871, pp.~141, 142. +}: in the Kinetic Theory of Gases, the complications in +\PG----File: 119.png------------------------------------------------------ +our universe that might be produced by ``Maxwell's demon''\index +{Maxwell's Demon}\footnote +{See \textit{Theory of Heat}, by J.~Clerk Maxwell\index{Maxwell, J. Clerk}, +second edition, London, 1872, p.~308.}: +in the Theory of Optics, the explanation of the Japanese\index +{Japanese Magic Mirrors}\index{Magic Mirrors}\index{Mirrors, Magic} +``magic mirrors,''\footnote +{See a memoir by W.E.~Ayrton\index{Ayrton on Magic Mirrors} and J.~Perry\index +{Perry on Magic Mirrors}, \textit{Proceedings of the +Royal Society of London}, part~\textsc{i}, 1879, vol.~\textsc{xxviii}, +pp.~127--148.} +which reflect the pattern on the back of +the mirror (on which the light does not fall): to which I +might add the theory of the ``spectrum top\index{Spectrum Top},'' by means +of which a white surface, on which some black lines are +drawn, can be moved so as to give the impression\footnote +{See letters from Mr~C.E.~Benham\index{Benham on Spectrum Top} and others in +\textit{Nature}, 1894--5; +and a memoir by Prof.\ Liveing\index{Liveing on the Spectrum Top}, +Cambridge Philosophical Society, November~26, 1894.} that the +lines are coloured (red, green, blue, slate, or drab), and the +curious fact that the colours change with the direction of +rotation: it has also been recently shown that if two trains +of waves, whose lengths are in the ratio $m-1: m+1$, be +superposed\index{Waves, Superposition of}, then every $m$th wave in the +system will be big---thus +the current opinion that every ninth wave in the open +sea is bigger than the other waves may receive scientific +confirmation. There is no lack of interesting and curious +phenomena in physics, and in some branches, notably in +electricity and magnetism, the difficulty is rather one of selection, +but I felt that the connection with mathematics was in +general either too remote or too technical to justify the insertion +of such a collection in a work on elementary mathematical +recreations, and therefore I struck out the chapter. I +mention the fact now partly to express the hope that some +physicist will one day give us a collection of the kind, partly +to suggest these questions to those who are interested in such +matters\index{Curiosa Physica|)}. + +\PG----File: 120.png------------------------------------------------------ +% CHAPTER IV. + +\chapter[Some Miscellaneous Questions.]% +[Miscellaneous Mathematical Recreations.]{Some Miscellaneous Questions.} + +\textsc{I propose} to discuss in this chapter the mathematical theory +of certain of the more common mathematical amusements and +games. Some of these might have been treated in the first +two chapters, but, since most of them involve mixed geometry +and algebra, it is rather more convenient to deal with +them apart from the problems and puzzles which have been +described already. This division, however, is by no means well +defined, and the arrangement is based on convenience rather +than on any logical distinction. + +The majority of the questions here enumerated have no +connection one with another, and I jot them down almost at +random. + +I shall discuss in succession the \emph{Fifteen Puzzle}, the \emph{Tower +of Hanoï}, \emph{Chinese Rings}, the \emph{Eight Queens Problem}, the +\emph{Fifteen School-Girls Problem}, and some miscellaneous \emph +{Problems connected with a pack of cards}. + +\ssection[The Fifteen Puzzle]{The Fifteen Puzzle\protect\footnote +{There are two articles on the subject in the \textit{American Journal of +Mathematics} 1879, vol.~\textsc{ii}, by Professors Woolsey Johnson\index +{Johnson on Fifteen Puzzle} and Story\index{Story on the Fifteen Puzzle}; +but the whole theory is deducible immediately from the proposition I +give above in the text.}} +Some years ago the so-called +\emph{fifteen puzzle}\index{Fifteen@\textsc{Fifteen Puzzle}|(} was on sale +in all toy-shops. It consists of a +shallow wooden box---one side being marked as the top---in the +form of a square, and contains fifteen square blocks or counters +\PG----File: 121.png------------------------------------------------------- +numbered $1$, $2$, $3 \ldots$ up to $15$. The box will hold just sixteen +such counters, and, as it contains only fifteen, they can be +moved about in the box relatively to one another. Initially +they are put in the box in any order, but leaving the sixteenth +cell or small square empty; the puzzle is to move them so +that finally they occupy the position shown in the first of the +annexed figures. + +\begin{figure*}[!hbt] +\centering +\ifPaper\else\hspace*{\fill}\fi +\begin{minipage}{\ifPaper0.45\textwidth\else0.33\textwidth\fi} +\centerline{\includegraphics[width=\textwidth]{./images/illus121a}} +\end{minipage} +\hfill +\begin{minipage}{\ifPaper0.45\textwidth\else0.33\textwidth\fi} +\centerline{\includegraphics[width=\textwidth]{./images/illus121b}} +\end{minipage} +\ifPaper\else\hspace*{\fill}\fi +\label{illus:121} +\end{figure*} + +We may represent the various stages in the game by supposing +that the blank space, occupying the sixteenth cell, is +moved over the board, ending finally where it started. + +The route pursued by the blank space may consist partly of +tracks followed and again retraced, which have no effect on the +arrangement, and partly of closed paths travelled round, which +necessarily are cyclical permutations of an odd number of +counters. No other motion is possible. + +Now a cyclical permutation of $n$ letters is equivalent to +$n-1$ simple interchanges; accordingly an odd cyclical permutation +is equivalent to an even number of simple interchanges. +Hence, if we move the counters so as to bring the blank space +back into the sixteenth cell, the new order must differ from +the initial order by an even number of simple interchanges. If +\PG----File: 122.png------------------------------------------------------- +therefore the order we want to get can be obtained from this +initial order only by an odd number of interchanges, the +problem is incapable of solution; if it can be obtained by an +even number, the problem is possible. + +Thus the order in the second of the diagrams given +\vpageref{illus:121} +%[**Note: original wording "on the opposite page" +is deducible from that in the first diagram by six +interchanges; namely, by interchanging the counters $1$ and $2$, $3$ +and $4$, $5$ and $6$, $7$ and $8$, $9$ and $10$, $11$ and $12$. Hence the one +can be deduced from the other by moving the counters about in the box. + +If however in the second diagram the order of the last +three counters had been $13$, $15$, $14$, then it would have required +seven interchanges of counters to bring them into the order +given in the first diagram. Hence in this case the problem +would be insoluble. + +{\renewcommand\tabcolsep{4pt} % for several tabulars +The easiest way of finding the number of simple interchanges +necessary in order to obtain one given arrangement +from another is to make the transformation by a series of +cycles. For example, suppose that we take the counters in +the box in any definite order, such as taking the successive +rows from left to right, and suppose the original order and the +final order to be respectively +\[ +\begin{tabularx}{\textwidth}{@{}X@{}rrrrrrrrrrrrrrr@{}X@{}} +& $1$, & $13$, & $2$, & $3$, & $5$, & $7$, & $12$, & $8$, & + $15$, & $6$, & $9$, & $4$, & $11$, & $10$, & $14$,&\\ +and& $11$, & $2$, & $3$, & $4$, & $5$, & $6$, & $7$, & + $1$, & $9$, &$10$, & $13$, & $12$, & $8$, & $14$, & $15$.& +\end{tabularx} +\] +We can deduce the second order from the first by $12$ simple +interchanges. The simplest way of seeing this is to arrange the +process in three separate cycles as follows:--- +\[ +\begin{tabular} +{rrr@{\hglue8pt}|@{\hglue8pt}rrrrrrrrrrr@{\hglue8pt}|@{\hglue8pt}r} +$1$, & $11$, & $8\;$; & $13$, & $2$, & $3$, & $4$, & $12$, & $7$, & + $6$, & $10$, & $14$, & $15$, & $9\;$; & $5$. \\ +$11$, & $8$, & $1\;$; & $2$, & $3$, & $4$, & $12$, & $7$, & $6$, & + $10$, & $14$, & $15$, & $9$, & $13\;$; & $5$. +\end{tabular} +\] +Thus, if in the first row of figures $11$ is substituted for $1$, then +$8$ for $11$, then $1$ for $8$, we have made a cyclical interchange of +$3$ numbers, which is equivalent to $2$ simple interchanges (namely, +interchanging $1$ and $11$, and then $1$ and $8$). Thus the whole +process is equivalent to one cyclical interchange of $3$ numbers, +another of $11$ numbers, and another of $1$ number. Hence it is +\PG----File: 123.png------------------------------------------------------- +equivalent to $(2 + 10 + 0)$ simple interchanges. This is an even +number, and thus one of these orders can be deduced from the +other by moving the counters about in the box. + +It is obvious that, if the initial order is the same as the +required order except that the last three counters are in the +order $15$, $14$, $13$, it would require one interchange to put them +in the order $13$, $14$, $15$; hence the problem is insoluble. + +If however the box is turned through a right angle, so as +to make $AD$ the top, this rotation will be equivalent to $13$ +simple interchanges. For, if we keep the sixteenth square +always blank, then such a rotation would change any order +such as +\[ +\begin{tabularx}{\textwidth}{@{}X@{}rrrrrrrrrrrrrrr@{}X@{}} + & $1$, & $2$, & $3$, & $4$, & $5$, & $6$, & $7$, & $8$, & $9$, & + $10$, & $11$, & $12$, & $13$, & $14$, & $15$,&\\ +to & $13$, & $9$, & $5$, & $1$, & $14$, & $10$, & $6$, & $2$, & $15$, & + $11$, & $7$, & $3$, & $12$, & $8$, & $4$,& +\end{tabularx} +\] +which is equivalent to $13$ simple interchanges. Hence it will +change the arrangement from one in which a solution is impossible +to one where it is possible, and vice versa. + +Again, even if the initial order is one which makes a solution +impossible, yet if the first cell and not the last is left +blank it will be possible to arrange the fifteen counters in +their natural order. For, if we represent the blank cell by $b$, +this will be equivalent to changing the order +\[ +\begin{tabular}{rrrrrrrrrrrrrrrr} +$1$, & $2$, & $3$, & $4$, & $5$, & $6$, & $7$, & $8$, & $9$, & $10$, & + $11$, & $12$, & $13$, & $14$, & $15$, & $b$, \\ +$b$, &$1$, & $2$, & $3$, & $4$, & $5$, & $6$, & $7$, & $8$, & $9$, & + $10$, & $11$, & $12$, & $13$, & $14$, & $15$\phantom{,}\rlap{$\;$:} +\end{tabular} +\] +this is a cyclical interchange of $16$ things and therefore is +equivalent to $15$ simple interchanges. Hence it will change +the arrangement from one in which a solution is impossible to +one where it is possible, and vice versa.} + +It is evident that the above principles are applicable +equally to a rectangular box containing $mn$ cells or spaces and +$mn-1$ counters which are numbered. Of course $m$ may be +equal to $n$. If such a box is turned through a right angle, and +$m$ and $n$ are both even, it will be equivalent to $mn-3$ simple +interchanges---and thus will change an impossible position to +a possible one, and vice versa---but unless both $m$ and $n$ are +\PG----File: 124.png--------------------------------------------------------- +even the rotation is equivalent to only an even number of +interchanges. Similarly, if either $m$ or $n$ is even, and it is +impossible to solve the problem when the last cell is left blank, +then it will be possible to solve it by leaving the first cell +blank. + +The problem may be made more difficult by limiting the +possible movements by fixing bars inside the box which will +prevent the movement of a counter transverse to their directions. +We can conceive also of a similar cubical puzzle, but +we could not work it practically except by sections\index +{Fifteen@\textsc{Fifteen Puzzle}|)}. + + +\section{The Tower of Hanoï} I may mention next the ingenious% +\index{Hanoi@\textsc{Hanoï, Tower of}|(}% +\index{Tower@\textsc{Tower of Hanoï}|(} +puzzle known as the \emph{Tower of Hanoï}. It was brought out in +1883 by M.~Claus\index{Claus} (Lucas\index{Lucas, E.}). + +It consists of three pegs fastened to a stand, and of +eight circular discs of wood or cardboard each of which has +a hole in the middle so that a peg can be put through it. +These discs are of different radii, and initially they are placed +all on one peg, so that the biggest is at the bottom, and the +radii of the successive discs decrease as we ascend: thus the +smallest disc is at the top. This arrangement is called the +\emph{Tower}. The problem is to shift the discs from one peg to +another in such a way that a disc shall never rest on one +smaller than itself, and finally to transfer the tower (\IE\ all +the discs in their proper order) from the peg on which they +initially rested to one of the other pegs. + +The method of effecting this is as follows. (i)~If initially +there are $n$ discs on the peg $A$, the first operation is to transfer +gradually the top $n-1$ discs from the peg $A$ to the peg $B$, +leaving the peg $C$ vacant: suppose that this requires $x$ separate +transfers. (ii)~Next, move the bottom disc to the peg $C$. +(iii)~Then, reversing the first process, transfer gradually the +$n-1$ discs from $B$ to $C$, which will necessitate $x$ transfers. +Hence, if it requires $x$ transfers of simple discs to move a +tower of $n-1$ discs, then it will require $2x +1$ separate +transfers of single discs to move a tower of $n$ discs. Now +\PG----File: 125.png--------------------------------------------------------- +with $2$ discs it requires $3$ transfers, \IE\ $2^2-1$ transfers; +hence with $3$ discs the number of transfers required will be +$2 (2^2-1) + 1$, that is, $2^3-1$. Proceeding in this way we see +that with a tower of $n$ discs it will require $2^n-1$ transfers +of single discs to effect the complete transfer. Thus the eight +discs of the puzzle will require $255$ single transfers. The result +can be also obtained by the theory of finite differences. It +will be noticed that every alternate move consists of a transfer +of the smallest disc from one peg to another, the pegs being +taken in cyclical order. + +M.~De~Parville\index{DeParville@De Parville on Tower of Hanoï}\index +{Parville@Parville on Tower of Hanoï} gives an account of the origin +of the toy +which is a sufficiently pretty conceit to deserve repetition\footnote +{\textit{La Nature}, Paris, 1884, part~\textsc{i}, pp.~285--286.}. +In the great temple at Benares, says he, beneath the dome +which marks the centre of the world, rests a brass-plate in +which are fixed three diamond needles, each a cubit high +and as thick as the body of a bee. On one of these needles, +at the creation, God placed sixty-four discs of pure gold, the +largest disc resting on the brass plate, and the others getting +smaller and smaller up to the top one. This is the Tower of +Bramah. Day and night unceasingly the priests transfer the +discs from one diamond needle to another according to the +fixed and immutable laws of Bramah, which require that the +priest must not move more than one disc at a time and that +he must place this disc on a needle so that there is no smaller +disc below it. When the sixty-four discs shall have been thus +transferred from the needle on which at the creation God +placed them to one of the other needles, tower, temple, and +Brahmins alike will crumble into dust, and with a thunder-clap +the world will vanish. Would that English writers were +in the habit of inventing equally interesting origins for the +puzzles they produce! + +The number of separate transfers of single discs which the +Brahmins must make to effect the transfer of the tower is +$2^{64}-1$, that is, is $18,446744,073709,551615$: a number which, +\PG----File: 126.png--------------------------------------------------------- +even if the priests never made a mistake, would require many +thousands of millions of years to carry out% +\index{Hanoi@\textsc{Hanoï, Tower of}|)}% +\index{Tower@\textsc{Tower of Hanoï}|)}. + +\section[Chinese Rings]{Chinese Rings\protect\footnote +{It was described by Cardan\index{Cardan} in 1550 in his \textit +{De Subtilitate}, bk.~\textsc{xv}, +paragr.~2, ed.\ Sponius, vol.~\textsc{iii}, p.~587; by Wallis\index +{Wallis, J.} in his \textit{Algebra}, second +edition, 1693, \textit{Opera}, vol.~\textsc{ii}, chap.~111, pp.~472--478; +and allusion is +made to it also in Ozanam's\index{Ozanam@Ozanam's \textit{Récréations}} +\textit{Récréations}, 1723 edition, vol.~\textsc{iv}, p.~439. +}} A somewhat more elaborate toy, known\index +{Chinese rings@\textsc{Chinese rings}|(} +as \emph{Chinese Rings}, which is on sale in most English toy-shops, +is represented in the accompanying figure. It consists of a +\begin{figure*}[!hbt] +\centerline{\includegraphics[width=11.5cm]{./images/illus126}} +\end{figure*} +number of rings hung upon a bar in such a manner that the +ring at one end (say $A$) can be taken off or put on the bar +at pleasure; but any other ring can be taken off or put on +only when the one next to it towards $A$ is on, and all the +rest towards $A$ are off the bar. The order of the rings cannot +be changed. + +Only one ring can be taken off or put on at a time. [In +the toy, as usually sold, the first two rings form an exception +to the rule. Both these can be taken off or put on together. +To simplify the discussion I shall assume at first that only +one ring is taken off or put on at a time.] I proceed to show +that, if there are $n$ rings, then in order to disconnect them +\PG----File: 127.png--------------------------------------------------------- +from the bar, it will be necessary to take a ring off or to +put a ring on either $\frac{1}{3}(2^{n+1}-1)$ times or +$\frac{1}{3}(2^{n+1}-2)$ times according as $n$ is odd or even. + +Let the taking a ring off the bar or putting a ring on the +bar be called a \emph{step}. It is usual to number the rings from the +free end $A$. Let us suppose that we commence with the first +$m$ rings off the bar and all the rest on the bar; and suppose +that then it requires $x-1$ steps to take off the next ring, +that is, it requires $x-1$ additional steps to arrange the rings +so that the first $m + 1$ of them are off the bar and all the +rest are on it. Before taking these steps we can take off the +$(m + 2)$th ring and thus it will require $x$ steps from our initial +position to remove the $(m + 1)$th and $(m + 2)$th rings. + +Suppose that these $x$ steps have been made and that thus +the first $m + 2$ rings are off the bar and the rest on it, and +let us find how many additional steps are now necessary to +take off the $(m + 3)$th and $(m + 4)$th rings. To take these +off we begin by taking off the $(m + 4)$th ring: this requires +$1$ step. Before we can take off the $(m + 3)$th we must arrange +the rings so that the $(m + 2)$th is on and the first $m + 1$ rings +are off: to effect this, (i)~we must get the $(m + 1)$th ring on +and the first $m$ rings off, which requires $x-1$ steps, (ii)~then +we must put on the $(m + 2)$th ring, which requires $1$ step, +(iii)~and lastly we must take the $(m + 1)$th ring off, which +requires $x-1$ steps: thus this series of movements requires in +all $\{2 (x-1) + 1\}$ steps. Next we can take the $(m + 3)$th ring +off, which requires $1$ step; this leaves us with the first $m + 1$ +rings off, the $(m + 2)$th on, the $(m + 3)$th off and all the rest on. +Finally to take off the $(m + 2)$th ring, (i)~we get the $(m + 1)$th +ring on and the first $m$ rings off, which requires $x-1$ steps, +(ii)~we take off the $(m + 2)$th ring, which requires $1$ step, +(iii)~we take $(m+1)$th ring off, which requires $x-1$ steps: +thus this series of movements requires $\{2 (x-1) + 1\}$ steps. + +Therefore, if when the first $m$ rings are off it requires $x$ +steps to take off the $(m + 1)$th and $(m + 2)$th rings, then the +number of additional steps required to take off the $(m + 3)$th +\PG----File: 128.png------------------------------------------------------ +and $(m + 4)$th rings is $1 + \{2(x-1) + 1\} + 1 + \{2(x-1) + 1\}$, +that is, is $4x$. + +To find the whole number of steps necessary to take off +an odd number of rings we proceed as follows. + +To take off the first ring requires 1 step;\\ +{\renewcommand\tabcolsep{4pt} +\begin{tabular}{@{}p{\parindent}@{}llllc@{}l} +$\Therefore$&to take off the first&$3$&rings requires &$4$ +&additional steps&;\\ +$\Therefore$&\hfil"\hfil"\hfil&$5$&\hfil"\hfil"\hfil&$4^2$&"\hfil"\hfil&. +\end{tabular}\\ +In this way we see that the number of steps required to take +off the first $2n+ 1$ rings is $1 + 4 + 4^2 + \dotsb + 4^n$, which is equal +to $\tfrac{1}{3}(2^{2n+2} - 1)$. + +To find the number of steps necessary to take off an even +number of rings we proceed in a similar manner. + +\noindent +\begin{tabular}{@{}p{\parindent}@{}llllc@{}l} +&To take off the first&$2$&rings requires&\multicolumn{2}{l}{$2$ steps;}\\ +$\Therefore$&to take off the first&$4$&rings requires&$2\times4$ +&additional steps&;\\ +$\Therefore$&\hfil"\hfil"\hfil&$6$&\hfil"\hfil"\hfil&$2\times4^2$ +&"\hfil"\hfil&. +\end{tabular}}\\ +In this way we see that the number of steps required to take +off the first $2n$ rings is +$2 + (2 \times 4) + (2 \times 4^2) + \dotsb + (2 \times 4^{n-1})$, +which is equal to $\tfrac{1}{3}(2^{2n+1}-2)$. + +If we take off or put on the first two rings in one step +instead of two separate steps, these results become respectively +$2^{2n}$ and $2^{2n-1}-1$. + +I give the above analysis because it is the direct solution +of a problem attacked by Cardan\index{Cardan} in 1550 and by Wallis +\index{Wallis, J.} in +1693---in both cases unsuccessfully---and which at one time +attracted some attention. I proceed next to give another +solution, more elegant though rather artificial. + +This solution, which is due to M.~Gros\index{Gros on Chinese Rings}\footnote +{\textit{Théorie du Baguenodier}, by L.~Gros, Lyons, 1872. I take the +account of this from Lucas\index{Lucas, E.}, vol.~\textsc{i}, part~7.}, +depends on a +convention by which any position of the rings is denoted by +a certain number expressed in the binary scale of notation +in such a way that a step is indicated by the addition or +subtraction of unity. + +Let the rings be indicated by circles: if a ring is on the +bar, it is represented by a circle drawn above the bar; if the +ring is off the bar, it is represented by a circle below the bar. +\PG----File: 129.png--------------------------------------------------------- +Thus figure~i below represents a set of seven rings of which +the first two are off the bar, the next three are on it, the sixth +is off it, and the seventh is on it. + +Denote the rings which are on the bar by the digits $1$ or $0$ +alternately, reckoning from left to right, and denote a ring +which is off the bar by the digit assigned to that ring +on the bar which is nearest to it on the left of it, or by +a $0$ if there is no ring to the left of it. + +Thus the three positions indicated below are denoted respectively +by the numbers written below them. The position +represented in figure~ii is obtained from that in figure~i by +putting the first ring on to the bar, while the position represented +\begin{figure*}[!hbt] +\ifPaper\vspace*{1.25cm}\fi % better to pad the diagram than have big gaps in the text +\unitlength=0.175em +\centering +\begin{minipage}{0.3\textwidth} +\centering +\RingDiagram1011100 +$1101000$ +\legend{Figure \Uproman{1}} +\end{minipage} +\hfill +\begin{minipage}{0.3\textwidth} +\centering +\RingDiagram1011101 +$1101001$ +\legend{Figure \Uproman{2}} +\end{minipage} +\hfill +\begin{minipage}{0.3\textwidth} +\centering +\RingDiagram1010100 +$1100111$ +\legend{Figure \Uproman{3}} +\end{minipage} +\end{figure*} +in figure~iii is obtained from that in figure~i by taking +the fourth ring off the bar. + +It follows that every position of the rings is denoted by a +number expressed in the binary scale: moreover, since in going +from left to right every ring on the bar gives a variation (that +is, $1$ to $0$ or $0$ to $1$) and every ring off the bar gives a continuation, +the effect of a step by which a ring is taken off or +put on the bar is either to subtract unity from this number or +to add unity to it. For example, the number denoting the +position of the rings in figure~ii is obtained from the number +denoting that in figure~i by adding unity to it. Similarly the +number denoting the position of the rings in figure~iii is obtained +from the number denoting that in figure~i by subtracting unity +from it. + +The position when all the seven rings are off the bar is +denoted by the number $0000000$: when all of them are on +the bar by the number $1010101$. Hence to change from one +\PG----File: 130.png-------------------------------------------------------- +position to the other requires a number of steps equal to the +difference between these two numbers in the binary scale. The +first of these numbers is $0$: the second is equal to $2^6 + 2^4 + 2^2 + 1$, +that is, to $85$. Therefore $85$ steps are required. In a similar +way we may show that to put on a set of $2n + 1$ rings requires +$(1+2^1 + 2^2 + \ldots + 2^{2n})$ steps, that is, +$\frac{1}{3} (2^{2n+2}-1)$ steps; and to put +on a set of $2n$ rings requires $(2 + 2^3 + \ldots + 2^{2n-1})$ steps, +that is, $\frac{1}{3}(2^{n+1}-2)$ steps. + +I append a table indicating the steps necessary to take off +the first four rings from a set of five rings. The diagrams +in the middle column show the successive position of the rings +after each step. The number following each diagram indicates +\begin{figure*}[!hbt] +\ifPaper\vspace*{0.5cm}\fi % better to pad the diagram than have big gaps in the text +\unitlength=0.15em +\def\tabcolsep{12pt} +\centering +\begin{tabular}{l@{}l@{}cc@{\hglue5pt}c@{\hglue3pt}l} +Initial~\null&\multicolumn{2}{@{}l}{position}&\RingDiag11111&$10101$&\\ +After&1st~\null&step&\RingDiag11101&$10110$&\multirow{2}*[-2pt]{\Huge\}}\\ +\null\quad"&2nd& "&\RingDiag11100&$10111$\\ +\null\quad"&3rd& "&\RingDiag10100&$11000$\\ +\null\quad"&4th& "&\RingDiag10101&$11001$&\multirow{2}*[-2pt]{\Huge\}}\\ +\null\quad"&5th& "&\RingDiag10111&$11010$\\ +\null\quad"&6th& "&\RingDiag10110&$11011$\\ +\null\quad"&7th& "&\RingDiag10010&$11100$\\ +\null\quad"&8th& "&\RingDiag10011&$11101$\\ +\null\quad"&9th& "&\RingDiag10001&$11110$&\multirow{2}*[-2pt]{\Huge\}}\\ +\null\quad"&\llap{1}0th& "&\RingDiag10000&$11111$ +\end{tabular} +\ifPaper\vspace*{0.5cm}\fi +\end{figure*} +that position, each number being obtained from the one above +it by the addition of unity. The steps which are bracketed together +can be made in one movement, and, if thus effected, the +whole process is completed in $7$ movements instead of $10$ steps: +this is in accordance with the formula given above. + +\PG----File: 131.png-------------------------------------------------------- +M.~Gros\index{Gros on Chinese Rings} asserted that it is possible to take +from $64$ to +$80$ steps a minute, which in my experience is a rather high +estimate. If we accept the lower of these numbers, it would +be possible to take off $10$ rings in less than $8$ minutes; to take +off $25$ rings would require more than $582$ days, each of ten +hours work; and to take off $60$ rings would necessitate no less +than $768614,336404,564650$ steps, and would require nearly +$55000,000000$ years work---assuming of course that no mistakes +were made\index{Chinese rings@\textsc{Chinese rings}|)}. + +\section[The Eight Queens Problem]{The Eight Queens Problem\protect\footnote +{On the history of this problem see W.~Ahrens\index{Ahrens}, \textit +{Mathematische Unterhaltungen +und Spiele}, Leipzig, 1901, chap.~\textsc{ix}---a work issued subsequent +to the third edition of this book.}} The determination of the +number of ways in which eight queens% +\index{Eight Queens@\textsc{Eight Queens Problem}|(}% +\index{Queens@\textsc{Queens Problem, Eight}|(}% +\index{Queens, Problems with|(} can be placed on a +chess-board---or more generally, in which $n$ queens can be +placed on a board of $n^2$ cells---so that no queen can take any +other was proposed originally by Nauck\index{Nauck} in 1850. + +In 1874 Dr~S.~Günther\index{Gunther@Günther, S.}\footnote +{Grunert's \textit{Archiv der Mathematik und Physik}, 1874, vol.~\textsc{lvi} +pp.~281--292.} suggested a method of solution +by means of determinants. For, if each symbol represents +the corresponding cell of the board, the possible solutions for +a board of $n^2$ cells are given by those terms, if any, of the +determinant +\[\left|\; +\begin{matrix} +a_1 & b_2 & c_3 & d_4 & \hdotsfor{2} \\ +\beta_2 & a_3 & b_4 & c_5 & \hdotsfor{2} \\ +\gamma_3 & \beta_4 & a_5 & b_6 & \hdotsfor{2} \\ +\delta_4 & \gamma_5 & \beta_6 & a_7 & \hdotsfor{2} \\ +\hdotsfor{6} \\ +\hdotsfor{4} & a_{2n-3} & b_{2n-2} \\ +\hdotsfor{4} & \beta_{2n-2} & a_{2n-1} +\end{matrix}\; +\right| +\] +in which no letter and no suffix appears more than once. + +The reason is obvious. Every term in a determinant +contains one and only one element out of every row and out +\PG----File: 132.png---------------------------------------------------- +of every column: hence any term will indicate a position on +the board in which the queens cannot take one another by +moves rook-wise. Again in the above determinant the letters +and suffixes are so arranged that all the same letters and +all the same suffixes lie along bishop's paths: hence, if we +retain only those terms in each of which all the letters and all +the suffixes are different, they will denote positions in which +the queens cannot take one another by moves bishop-wise. +It is clear that the signs of the terms are immaterial. + +In the case of an ordinary chess-board the determinant +is of the $8$th order, and therefore contains $8!$, that is, $40320$ +terms, so that it would be out of the question to use this +method for the usual chess-board of $64$ cells or for a board of +larger size unless some way of picking out the required terms +could be discovered. + +A way of effecting this was suggested by Dr~J.W.L.~Glaisher\index +{Glaisher, J.W.L.}\footnote +{\textit{Philosophical Magazine}, London, December, 1874, series~4, +vol.~\textsc{xlviii}, pp.~457--467.} +in 1874, and as far as I am aware the theory remains as he left +it. He showed that if all the solutions of $n$ queens on a board +of $n^2$ cells were known, then all the solutions of a certain type +for $n+1$ queens on a board of $(n+1)^2$ cells could be deduced, +and that all the other solutions of $n+1$ queens on a board of +$(n+1)^2$ cells could be obtained without difficulty. The method +will be sufficiently illustrated by one instance of its application. + +It is easily seen that there are no solutions when $n=2$ and +$n=3$. If $n=4$ there are two terms in the determinant which +give solutions, namely, $b_2c_5\gamma_3\beta_6$ and $c_3\beta_2b_6\gamma_5$. +To find the +solutions when $n=5$, Glaisher\index{Glaisher, J.W.L.} proceeded thus. +In this case +Günther's\index{Gunther@Günther, S.} determinant is +\[ +\left\vert +\begin{matrix} +a_1 & b_2 & c_3 & d_4 & e_5 \\ +\beta_2 & a_3 & b_4 & c_5 & d_6 \\ +\gamma_3 & \beta_4 & a_5 & b_6 & c_7 \\ +\delta_4 & \gamma_5 & \beta_6 & a_7 & b_8 \\ +\epsilon_5 & \delta_6 & \gamma_7 & \beta_8 & a_9 +\end{matrix} +\right\vert +\] +\PG----File: 133.png------------------------------------------------------ +To obtain those solutions (if any) which involve $a_9$ it is sufficient +to append $a_9$ to such of the solutions for a board of $16$ +cells as do not involve $a$. As neither of those given above +involves an $a$ we thus get two solutions, namely, +$b_2 c_5 \gamma_3 \beta_6 a_9$ and +$c_3 \beta_2 b_6 \gamma_5 a_9$. The solutions which involve $a_1$, $e_5$ +and $\epsilon_5$ can be +written down by symmetry. The eight solutions thus obtained +are all distinct; we may call them of the first type. + +The above are the only solutions which can involve elements +in the corner squares of the determinant. Hence the remaining solutions are +obtainable from the determinant +\[ +\begin{vmatrix} + 0 & b_2 & c_3 & d_4 & 0 \\ + \beta_2 & a_3 & b_4 & c_5 & d_6 \\ + \gamma_3 & \beta_4 & a_5 & b_6 & c_7 \\ + \delta_4 & \gamma_5 & \beta_6 & a_7 & b_8 \\ + 0 & \delta_6 & \gamma_7 & \beta_8 & 0 +\end{vmatrix} +\] +If, in this, we take the minor of $b_2$ and in it replace by zero +every term involving the letter $b$ or the suffix $2$ we shall get +all solutions involving $b_2$. But in this case the minor at +once reduces to $d_6 a_5 \delta_4 \beta_8$. We thus get one solution, namely, +$b_2 d_6 a5 \delta_4 \beta_8$. The solutions which involve +$\beta_2$, $\delta_4$, $\delta_6$, $\beta_8$, $b_8$, $d_6$, and +$d_4$ can be obtained by symmetry. Of these eight solutions it +is easily seen that only two are distinct: these may be called +solutions of the second type. + +Similarly the remaining solutions must be obtained from +the determinant +\[ +\begin{vmatrix} + 0 & 0 & c_3 & 0 & 0 \\ + 0 & a_3 & b_4 & c_5 & 0 \\ + \gamma_3 & \beta_4 & a_5 & b_6 & c_7 \\ + 0 & \gamma_5 & \beta_6 & a_7 & 0 \\ + 0 & 0 & \gamma_7 & 0 & 0 +\end{vmatrix} +\] + +If, in this, we take the minor of $c_3$, and in it replace by +zero every term involving the letter $c$ or the suffix $3$, we shall +get all the solutions which involve $c_3$. But in this case the +\PG----File: 134.png---------------------------------------------------- +minor vanishes. Hence there is no solution involving $c_3$, and +therefore by symmetry no solutions which involve $\gamma_3$, $\gamma_7$, +or $c_3$. +Had there been any solutions involving the third element in +the first or last row or column of the determinant we should +have described them as of the third type. + +Thus in all there are ten and only ten solutions, namely, +eight of the first type, two of the second type, and none of +the third type. + +Similarly, if $n = 6$, we obtain no solutions of the first type, +four solutions of the second type, and no solutions of the +third type; that is, four solutions in all. If $n = 7$, we obtain +sixteen solutions of the first type, twenty-four solutions of the +second type, no solutions of the third type, and no solutions +of the fourth type; that is, forty solutions in all. If $n = 8$, +we obtain sixteen solutions of the first type, fifty-six solutions +of the second type, and twenty solutions of the third type, +that is, ninety-two solutions in all. + +It will be noticed that all the solutions of one type are not +always distinct. In general, from any solution seven others +can be obtained at once. Of these eight solutions, four consist +of the initial or fundamental solution and the three similar +ones obtained by turning the board through one, two, or three +right angles; the other four are the reflexions of these in a +mirror: but in any particular case it may happen that the +reflexions reproduce the originals, or that a rotation through +one or two right angles makes no difference. Thus on boards +of $4^2$, $5^2$, $6^2$, $7^2$, $8^2$, $9^2$, $10^2$ cells there are +respectively $1$, $2$, $1$, $6$, +$12$, $46$, $92$ fundamental solutions; while altogether there are +respectively $2$, $10$, $4$, $40$, $92$, $352$, $724$ solutions. + +The following collection of fundamental solutions may +interest the reader. The positions on the board of the queens +are indicated by digits: the first digit represents the number +of the cell occupied by the queen in the first column reckoned +from one end of the column, the second digit the number in the +second column, and so on. Thus on a board of $4^2$ cells the +solution $3142$ means that one queen is on the $3$rd square of +\PG----File: 135.png------------------------------------------------------ +the first column, one on the $1$st square of the second column, +one on the $4$th square of the third column, and one on the +$2$nd square of the fourth column. If a fundamental solution +gives rise to only four solutions the number which indicates +it is placed in curved brackets,~($\;$); if it gives rise to only +two solutions the number which indicates it is placed in +square brackets,~[$\;$]; the other fundamental solutions give rise +to eight solutions each. + +On a board of $4^2$ cells there is $1$ fundamental solution: +namely, [$3142$]. + +On a board of $5^2$ cells there are $2$ fundamental solutions: +namely, $14253$, [$25314$]. + +On a board of $6^2$ cells there is $1$ fundamental solution: +namely, ($246135$). + +On a board of $7^2$ cells there are $6$ fundamental solutions: +namely, $1357246$, $3572461$, ($5724613$), $4613572$, $3162574$, +($2574136$). + +{\ifPaper\stretchyspace\fi +On a board of $8^2$ cells there are $12$ fundamental solutions: +namely, $25713864$, $57138642$, $71386425$, $82417536$, $68241753$, +$36824175$, +$64713528$, $36814752$, $36815724$, $72418536$, $26831475$, +($64718253$).} The arrangement in this order is due to Mr~Oram\index +{Oram on Eight Queens|(}. +It will be noticed that the $10$th, $11$th, and $12$th solutions +somewhat resemble the $4$th, $6$th, and $7$th respectively. The +$6$th solution is the only one in which no three queens are in +a straight line. + +On a board of $9^2$ cells there are $46$ fundamental solutions; +one of them is $248396157$. On a board of $10^2$ cells there are +$92$ fundamental solutions; these were given by Dr A.~Pein\index +{Pein on Ten Queens}\footnote +{\textit{Aufstellung von $n$ Königinnen auf einem Schachbrett von $n^2$ +Feldern}, Leipzig, 1889.}; +one of them is $2468t13579$, where $t$ stands for ten. On a board +of $11^2$ cells there are $341$ fundamental solutions; these have +been given by Dr T.B.~Sprague\index{Sprague on Eleven Queens}\footnote +{\textit{Proceedings of the Edinburgh Mathematical Society}, +vol.~\textsc{xvii}, 1898--9, pp.~43--68.}: +one of them is $15926t37e48$. +I may add that for a board of $n^2$ cells there is always a +\PG----File: 136.png------------------------------------------------------ +symmetrical solution of the form $246 \ldots n135\ldots (n-1)$, when +$n= 6m$ or $n = 6m + 4$, Also Mr~Oram has shown that for a +board of $n^2$ cells, when $n$ is a prime, cyclical arrangements +of the $n$ natural numbers, other than in their natural order, +will give solutions; see, for instance, the solution quoted +above\index{Oram on Eight Queens|)}. + +The puzzle in the form of a board of $36$ squares is sold +in the streets of London for a penny, a small wooden board +being ruled in the manner shown in the diagram and having +holes drilled in it at the points marked by dots. The object +is to put six pins into the holes so that no two are connected +by a straight line.\ifPaper\par\fi +\[ +\includegraphics[width=5cm]{./images/illus136}\label{illus:136} +\] + +\subsection*{Other Problems with Queens}% +\addcontentsline{toc}{section}{Other Problems with Queens and Chess-pieces}% +\markright{Problems with Queens.}% +Captain Turton\index{Turton, W.H.} called my +attention to two other problems of a somewhat analogous +character, neither of which, as far as I know, has been +published elsewhere, or solved otherwise than empirically. + +The first of these is to place eight queens on a chess-board +so as to command the fewest possible squares. Thus, if queens +are placed on cells $1$ and $2$ of the second column, on cell 2 of +the sixth column, on cells $1$, $3$, and $7$ of the seventh column, +and on cells $2$ and $7$ of the eighth column, eleven cells on the +board will not be in check; the same number can be obtained +by other arrangements. Is it possible to place the eight +queens so as to leave more than eleven cells out of check? +I have never succeeded in doing so, nor in showing that it is +impossible to do it. + +\PG----File: 137.png--------------------------------------------------------- +The other problem is to place $m$ queens ($m$ being less than +$5$) on a chess-board so as to command as many cells as possible. +For instance, four queens can be placed in several ways on +the board so as to command $58$ cells besides those on which +the queens stand, thus leaving only $2$ cells which are not +commanded: \Eg\ this is effected if the queens are placed +on cell $5$ of the third column, cell $1$ of the fourth column, cell +$6$ of the seventh column, and cell $2$ of the eighth column; or +on cell $1$ of the first column, cell $7$ of the third column, cell +$3$ of the fifth column, and cell $5$ of the seventh column. A +similar problem is to determine the minimum number and +the position of queens which can be placed on a board of $n^2$ +cells so as to occupy or command every cell. It would seem +that, even with the additional restriction that no queen shall +be able to take any other queen, there are no less than ninety-one +typical solutions in which five queens can be placed on a +chess-board so as to command every cell\footnote +{\textit{L'Intermédiaire des mathématiciens}, Paris, 1901, +vol.~\textsc{viii}, p.~88.\label{ibid:1}}% +\index{Eight Queens@\textsc{Eight Queens Problem}|)}% +\index{Queens@\textsc{Queens Problem, Eight}|)}% +\index{Queens, Problems with|)}. + +\subsection*{Extension to other Chess Pieces} Analogous problems may% +\index{Chess-board, Games@\textsc{Chess-board, Games on}}% +\index{Chess-board, problems@\nobreak--- problems} +be proposed with other chess-pieces. For instance, questions +as to the maximum number of knights which can be placed +on a board of $n^2$ cells so that no knight can take any other, +and the minimum number of knights which can be placed +on it so as to occupy or command every cell have been +propounded\footnote +{\Ibidref{ibid:1}{\textit{L'Intermédiaire des mathématiciens}, Paris}, +March, 1896, vol.~\textsc{iii}, p.~58; 1897, vol.~\textsc{iv}, +p.~15, 254; and 1898, vol.~\textsc{v}, p.~87.}. + +Similar problems have also been proposed for $k$ kings +placed on a chess-board of $n^2$ cells\footnote +{\Ibidref{ibid:1}{\textit{L'Intermédiaire des mathématiciens}, Paris}, +June, 1901, p.~140.}. It has been asserted +that, if $k = 2$, the number of ways in which two kings can be +placed on a board so that they may not occupy adjacent +squares is $\frac{1}{2}(n-1)(n-2) (n^2 + 3n-2)$. Similarly, if $k=3$, the +number of ways in which three kings can be placed on a board +so that no two of them occupy adjacent squares is said to be +$\frac{1}{6}(n-1) (n-2) (n^4+ 3n^3-20n^2-30n + 132)$. +\PG----File: 138.png--------------------------------------------------------- + +{\ifPaper\stretchyspace\fi +\section{The Fifteen School-Girls Problem} This problem% +\index{Fifteen School@\textsc{Fifteen School-girls}|(}% +\index{Kirkman's Prob@\textsc{Kirkman's Problem}|(}% +\index{School@\textsc{School-girls, Fifteen}|(}---which +was first enunciated by Mr~T.P.~Kirkman\index{Kirkman@Kirkman, T.P.}, and is +sometimes known} as \emph{Kirkman's problem}\footnote +{It was published first in the \textit{Lady's and Gentleman's Diary} for 1850, +p.~48, and has been the subject of numerous memoirs. Among these +I may single out the papers by A.~Cayley\index{Cayley} in the \textit +{Philosophical Magazine}, +July, 1850, series~3, vol.~\textsc{xxxvii}, pp.~50--53; by T.P.~Kirkman in the +\textit{Cambridge and Dublin Mathematical Journal}, 1850, vol.~\textsc{v}, +p.~260; by +R.R.~Anstice\index{Anstice}, \Ibid, 1852, vol.~\textsc{vii}, pp.~279--292; +by B.~Pierce\index{Pierce on Kirkman's Problem}, \textit{Gould's +Journal}, Cambridge, U.S., 1860, vol.~\textsc{vi}, pp.~169--174; by +T.P.~Kirkman, +\textit{Philosophical Magazine}, March, 1862, series~4, vol.~\textsc{xxiii}, +pp.~198--204; +by W.S.B.~Woolhouse\index{Woolhouse, Kirkman's Problem} in the \textit +{Lady's Diary} for 1862, pp.~84--88, and for +1863, pp.~79--90, and in the \textit{Educational Times Reprints}, 1867, +vol.~\textsc{viii}, +pp.~76--83; by J.~Power\index{Power, Kirkman's Problem} in the \textit +{Quarterly Journal of Mathematics}, 1867, +vol.~\textsc{viii}, pp.~236--251; by A.H.~Frost\index{Frost, A.H.}, \Ibid, +1871, vol.~\textsc{xi}, pp.~26--37; +by E.~Carpmael\index{Carpmael, Kirkman's Problem} in the \textit +{Proceedings of the London Mathematical Society}, +1881, vol.~\textsc{xii}, pp.~148--156; by Lucas\index{Lucas, E.} in his +\textit{Récréations}, vol.~\textsc{ii}, +part~vi; by A.C.~Dixon\index{Dixon, A.C.} in the \textit{Messenger of +Mathematics}, Cambridge, +October, 1893, vol.~\textsc{xxiii}, pp.~88--89; and by W.~Burnside\index +{Burnside, Kirkman's Problem}, \Ibid, 1894, +vol.~\textsc{xxiii}, pp.~137--143. It has also, since the issue of my third +edition, +been discussed by W.~Ahrens\index{Ahrens} in his \textit{Mathematische +Unterhaltungen und +Spiele}, Leipzig, 1901, chapter~xiv.}---consists in arranging +fifteen things in different sets of triplets. It is usually +presented in the form that a school-mistress was in the habit +of taking her girls for a daily walk. The girls were fifteen +in number, and were arranged in five rows of three each so +that each girl might have two companions. The problem is +to dispose them so that for seven consecutive days no girl will +walk with any of her school-fellows more than once. More +generally we may require to arrange $3m$ girls in triplets to +walk out for $\frac{1}{2}(3m-1)$ days, so that no girl will walk with +any of her school-fellows more than once. + +The theory of the formation of all such possible triplets in +the case of fifteen girls is not difficult, but the extension to +$3m$ girls is, as yet, unsolved\Editorial +{In 1971 Ray-Chaudhuri and Wilson proved that Kirkman triple systems of +order $\nu$ exist if and only if $\nu\equiv3\pmod 6$}. +I proceed to describe three +methods of solution: these methods are analytical, but I may +add that the problem can be also attacked by geometrical +methods. + +\PG----File: 139.png------------------------------------------------------ +\subsection*{Frost's Method} +The first of these solutions is due to Mr~Frost\index{Frost, A.H.|(}. +A full exposition of it would occupy a good deal of +space, but I hope that the following sketch will make the +process intelligible. + +Denote one of the girls by $k$. Her companions on each +day are different: suppose that on Sunday they are $a_1$ and $a_2$, +on Monday $b_1$ and $b_2$, and so on, and finally on Saturday $g_1$ +and $g_2$. Hence for each day we have one triplet, and we have +to find four others, but in each of the latter no two like letters +can occur together, that is, the three letters in any of them +must be all different. + +Let $a$ stand for $a_1$, or $a_2$, $b$ for $b_1$ or $b_2$, and so on. The +suffixes $1$ and $2$ are called complementary. Then, since the +three letters in each of the triplets we are trying to find must +be different, we must make some arrangement such as putting +$a$ with $bc$, $de$, and $fg$; and, if so, $b$ may be associated with $df$ +and $eg$; and $c$ with $dg$ and $ef$. Thus there are seven possible +triads, such as $abc$, $ade$, $afg$, $bdf$, $beg$, $cdg$, and $cef$. Moreover +each of these may stand for any one of four triplets: for +instance, the triad $bdf$ may stand for any of the triplets +$b_1 d_1 f_1$, $b_1 d_2 f_2$, $b_2 d_1 f_2$, $b_2 d_2 f_1$. + +The four triads which do not involve $a$ must be placed in the +Sunday column, the four which do not involve $b$ in the Monday +column, and so on. Thus each triad will occur four times. + +It only remains to insert the proper suffixes. This is done +as follows. Take one triad, such as $bdf$, and insert a different +set of suffixes each time that it occurs; for instance, the four +sets given above. Next, the other like letters ($b$, $d$, or $f$ as the +case may be) in these four columns must have the complementary +suffixes attached. + +After this is done, the next triplet in the Sunday column +will be $b_2 eg$. The triad $beg$ occurs in four columns and includes +four possible triplets, such as +$b_2 e_1 g_1$, $b_2 e_2 g_2$, $b_1 e_1 g_2$, $b_1 e_2 g_1$. Insert +these, and then give the complementary suffixes to the other +like letters in these four columns. + +In this way the arrangement is constructed gradually, by +\PG----File: 140.png------------------------------------------------------ +taking one triad at a time, inserting the proper suffixes to the +four triplets included in it, and then the complementary +suffixes in the other like letter in the same columns. + +One final arrangement, thus obtained, is as follows:{\ifPaper\smaller\fi +\[ +\begin{array}{|c|c|c|c|c|c|c|} +\hline +\text{\smaller Sunday}\DParraykludge + & \text{\smaller Monday} & \text{\smaller Tuesday} + & \text{\smaller Wednesday} & \text{\smaller Thursday} + & \text{\smaller Friday} & \text{\smaller Saturday}\\[3pt] +\hline + k a_1 a_2 \DParraykludge +& k b_1 b_2 & k c_1 c_2 & k d_1 d_2 +& k e_1 e_2 & k f_1 f_2 & k g_1 g_2 +\\ + b_1 d_1 f_1 & a_1 d_2 e_2 & a_1 d_1 e_1 & a_2 b_2 c_2 +& a_2 b_1 c_1 & a_1 b_2 c_1 & a_1 b_1 c_2 +\\ + b_2 e_1 g_1 & a_2 f_2 g_2 & a_2 f_1 g_1 & a_1 f_2 g_1 +& a_1 f_1 g_2 & a_2 d_2 e_1 & a_2 d_1 e_2 +\\ + c_1 d_2 g_2 & c_1 d_1 g_1 & b_1 d_2 f_2 & b_1 e_1 g_2 +& b_2 d_1 f_2 & b_1 e_2 g_1 & b_2 d_2 f_1 +\\ + c_2 e_2 f_2 & c_2 e_1 f_1 & b_2 e_2 g_2 & c_1 e_2 f_1 +& c_2 d_2 g_1 & c_2 d_1 g_2 & c_1 e_1 f_2 +\\[6pt] +\hline +\end{array} +\] +}We might obtain other solutions by selecting other seven +triads or by choosing other arrangements of the suffixes in +each triad (or by merely interchanging letters and suffixes in +the above order). By these means Mr~Power\index{Power, Kirkman's Problem} +showed that +there are no less than $15567,552000$ different solutions; but, +since the total number of ways in which the school can walk +out for a week in triplets is $(455)^7$, the probability that any +chance way satisfies Kirkman's condition is very small. + +Frost's method is applicable to the case of $2^{2n}-1$ girls +walking out for $2^{2n-1}-1$ days in triplets. The detailed +solution for $63$ girls walking out for $31$ days, which corresponds +to $n = 3$, have been given\index{Frost, A.H.|)}. + +\subsection*{Anstice's Method} Another method of attacking\index{Anstice|(} +the problem is due to Mr Anstice; it is illustrated by the following +elegant solution, by which from the order on Sunday we can +obtain the order on the following six days by a cyclical permutation. +Let the girls be denoted respectively by the letters +$k$, $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$, $a_7$, +$b_1$, $b_2$, $b_3$, $b_4$, $b_5$, $b_6$, $b_7$; and suppose +the order on Sunday to be +\[ + k a_1 b_1,\; a_2 a_3 a_5,\; a_4 b_3 b_6,\; a_6 b_2 b_7,\; a_7 b_4 b_5\,. +\] +Then, if the suffixes are permuted cyclically, we obtain six +other arrangements which satisfy the conditions of the problem: +the reason being that in the above arrangement the difference +\PG----File: 141.png---------------------------------------------------- +of the suffixes of every pair of like letters---such as either the +``$a$''s or the ``$b$''s---in a triplet is different for each triplet, as +also is the difference of the suffixes of every pair of unlike +letters which are in a triplet. + +Two other arrangements for Sunday, from which those for +the remaining days are obtainable by cyclical permutations +can be formed. These are $ka_1b_1$, $a_2a_3a_5$, $a_4b_5b_7$, $a_6b_3b_4$, +$a_7b_2b_6$; +and $ka_1b_1$, $a_2a_3a_5$, $a_4b_2b_6$, $a_6b_5b_7$, $a_7b_3b_4$. + +Anstice's method is applicable to the case of $2p+1$ girls +walking out for $p$ days in triplets so that no pair may walk +together more than once, provided $p$ is a prime of the form +$12m+7$. In such a case he showed how to construct a fundamental +arrangement for one day from which the arrangements +for the remaining $p-1$ days can be obtained by cyclical +permutations of suffixes. The number of such fundamental +arrangements is $3 (2m+1)(3m+1)$. + +The problem of $15$ girls corresponds to $m = 0$, and the +three fundamental Anstician arrangements are given above. +If $m=1$ we have the problem of $39$ girls. One Anstician +arrangement in this case is as follows: $ka_1b_1$, $a_2a_8a_{12}$, +$a_5a_7a_{10}$, +$a_6a_{17}a_{18}$, $a_3b_{10}b_{15}$, $a_4b_3b_5$, $a_9b_{18}b_{19}$, +$a_{11}b_8b_{14}$, $a_{13}b_9b_{17}$, $a_{14}b_{12}b_{16}$, +$a_{15}b_4b_7$, $a_{16}b_2b_{11}$, $a_{19}b_6b_{13}$. If $m=2$ we have the +problem of $63$ +girls, of which Frost has given one solution; and so on.\index{Anstice|)} + +\subsection*{Gill's Method} Another method of attacking the problem +has been suggested to me by Mr~T.H.~Gill\index{Gill, Kirkman's Problem|(}. +Representing +the girls by $a_1, a_2, a_3, \dotsc, a_{3m}$ he (i)~forms one triplet of the +type $a_1a_{m+1}a_{2m+1}$, from which, by cyclical permutation of the +suffixes $1, 2, \dotsc, 3m$ he obtains $m$ triplets which constitute an +arrangement for one day, and (ii)~forms $\frac{1}{2}(m-1)$ other triplets +such that the three differences of the suffixes are different, +from which, by cyclical permutations of the suffixes, the +arrangements for the remaining $\frac{3}{2}(m-1)$ other days can be +obtained. Thus in the case of $15$ girls, the triplet $a_1a_6a_{11}$ +gives, by cyclical permutations of the suffixes, an arrangement +for the first day and two triplets such as $a_1a_2a_5$, $a_1a_3a_9$ enable +us to form $30$ triplets from which an arrangement for the +\PG----File: 142.png---------------------------------------------------- +other six days can be found. Here is a solution thus +determined. +\[\text{\relsize{-2} +\begin{tabular}{l r@.r@.r r@.r@.r r@.r@.r r@.r@.r r@.r@.r} + First Day: & 1& 6&11;& 2& 7&12;& 3& 8&13;& 4& 9&14;& 5&10&15\,.\\ + Second Day: & 1& 2& 5;& 3& 4& 7;& 8& 9&12;&10&11&14;&13&15& 6\,.\\ + Third Day: & 2& 3& 6;& 4& 5& 8;& 9&10&13;&11&12&15;&14& 1& 7\,.\\ + Fourth Day: & 5& 6& 9;& 7& 8&11;&12&13& 1;&14&15& 3;& 2& 4&10\,.\\ + Fifth Day: & 7& 9&15;& 8&10& 1;& 3& 5&11;& 4& 6&12;&13&14& 2\,.\\ + Sixth Day: & 9&11& 2;&10&12& 3;& 5& 7&13;& 6& 8&14;&15& 1& 4\,.\\ + Seventh Day:& 11&13& 4;&12&14& 5;&15& 2& 8;& 1& 3& 9;& 6& 7&10\,.\\ +\end{tabular}}\] +But, although this method gives triplets with which the +problem can be solved, the final arrangement is empirical\index +{Gill, Kirkman's Problem|)}. + +A solution of the problem of $21$ girls for $10$ days can be +got by the same method: $a_1a_8a_{15}$ giving $7$ triplets which +constitute an arrangement for one day; and $a_1a_2a_6$, $a_1a_3a_{11}$, +$a_1a_4a_{10}$ giving $63$ triplets from which an arrangement for the +other nine days can be formed. Here is the solution thus +determined. +\[\hss\ifPaper\def\tabcolsep{4pt}\fi +\text{\relsize{-2}\begin{tabular} +{l r@.r@.r r@.r@.r r@.r@.r r@.r@.r r@.r@.r r@.r@.r r@.r@.r} +First Day:& 1&8&15;& 2&9&16;& 3&10&17;& 4&11&18;& 5&12&19;& + 6&13&20;& 7&14&21\,.\\ +Second Day:& 1&2&6;& 4&5&9;& 7&8&12;& 10&11&15;& 13&14&18;& + 16&17&21;& 19&20&3\,.\\ +Third Day:& 7&10&16;& 8&11&17;& 12&15&21;& 18&19&2;& 20&1&9;& + 3&5&13;& 4&6&14\,.\\ +Fourth Day:& 13&16&1;& 14&17&2;& 18&21&6;& 3&4&8;& 5&7&15;& + 9&11&19;& 10&12&20\,.\\ +Fifth Day:& 4&7&13;& 5&8&14;& 9&12&18;& 15&16&20;& 17&19&6;& + 21&2&10;& 1&3&11\,.\\ +Sixth Day;& 1&4&10;& 2&5&11;& 6&9&15;& 12&13&17;& 14&16&3;& + 18&20&7;& 19&21&8\,.\\ +Seventh Day:& 2&3&7;& 5&6&10;& 8&9&13;& 11&12&16;& 14&15&19;& + 17&18&1;& 20&21&4\,.\\ +Eighth Day:& 10&13&19;& 11&14&20;& 15&18&3;& 21&1&5;& 2&4&12;& + 6&8&16;& 7&9&17\,.\\ +Ninth Day:& 16&19&4;& 17&20&5;& 21&3&9;& 6&7&11;& 8&10&18;& + 12&14&1;& 13&15&2\,.\\ +Tenth Day:& 19&1&7;& 20&2&8;& 3&6&12;& 9&10&14;& 11&13&21;& + 15&17&4;& 16&18&5\,.\\ +\end{tabular}}\hss\] + +I should be interested if any of my readers could give me +a similar solution of the analogous arrangement of $33$ girls for +$16$ days formed from typical triplet suffixes like $1$, $12$, $23$; $1$, +$2$, $10$; $1$, $3$, $16$; $1$, $4$, $18$; $1$, $5$, $11$; $1$, $6$, $13$; +or from other +sets of triplets formed in a similar way so that (except in the +first triplet) the differences of the suffixes are all different. + +{\ifPaper\stretchyspace\fi +\subsection*{Walecki's Theorem} Lastly, Walecki\index +{Walecki}---quoted by Lucas\index{Lucas, E.}---has +shown that}, if a solution for the case of $n$ girls walking +\PG----File: 143.png---------------------------------------------------- +out in triplets for $\frac{1}{2}(n-1)$ days is known, then a solution for +$3n$ girls walking out for $\frac{1}{2}(3n-1)$ days can be deduced. + +For if an arrangement of the $n$ girls, $a_1, a_2, \dotsc, a_n$ for +$\frac{1}{2}(n-1)$ days is known; and also one of the $n$ girls, +$b_1, b_2,\dotsc, b_n$; +and also one of the $n$ girls $c_1, c_2, \dotsc, c_n$; then an arrangement +of these $3n$ girls for $\frac{1}{2}(n-1)$ days is known. A set of $n$ +triplets for another day will be given by $a_mb_{m+k}c_{m + 2k}$ where $m$ +is put equal to $1, 2, \dotsc, n$ successively. Here $k$ may have any +of the $n$ values, $0, 1, 2, \dotsc, (n-1)$; but, wherever a suffix is +greater than $n$, it is to be divided by $n$ and only the remainder +retained. Hence altogether we have an arrangement for +${n + \frac{1}{2} (n-1)}$ days, \IE\ for $\frac{1}{2}(3n-1)$ days. + +The arrangement of $3$ girls for one day is obvious. Hence, +by Walecki's\index{Walecki} theorem, we can deduce at once an arrangement +of $3^m$ girls for $\frac{1}{2}(3^m-1)$ days. And, generally, as I have +given solutions of the problem in the case of $3n$ girls when +$n= 1$, $3$, $5$, $7$, $9$, $13$, $15$, it follows that for the same values +of $n$, +we can solve the analogous arrangements of $3^m \times n$ girls. + +To the original theorem J.J.~Sylvester\index{Sylvester}\footnote +{\textit{Philosophical Magazine}, July, 1850, series~3, vol.~\textsc{xxxvii}, +p.~52; +a solution by Sylvester is given in the \textit{Philosophical Magazine}, May, +1861, series~4, vol.~\textsc{xxi}, p.~371.} added the +corollary that the school of $15$ girls could walk out in +triplets on $13\times 7$ days until every possible triplet had walked +abreast once. + +The generalized problem of finding the greatest number of +ways in which $x$ girls walking in rows of $a$ abreast can be +arranged so that every possible combination of $b$ of them may +walk abreast once and only once has been solved for various +cases. Suppose that this greatest number of ways is $y$. It +is obvious that, if all the $x$ girls are to walk out each day in +rows of $a$ abreast, then $x$ must be an exact multiple of $a$ and +the number of rows formed each day is $x/a$. If such an +arrangement can be made for $z$ days, then we have a solution +of the problem to arrange $x$ girls to walk out in rows of $a$ +abreast for $z$ days so that they all go out each day and so that +\PG----File: 144.png---------------------------------------------------- +every possible combination of $b$ girls may walk together once, +and only once. In the corresponding generalization of Kirkman's +problem no companionship of girls which has occurred +once may occur again, but it does not follow necessarily that +every possible companionship must occur once. + +An example where the solution is obvious is if $x=2n$, $a=2$, +$b=2$, in which case $y=n(2n-1)$, $z=2n-1$. + +If we take the case $x=15$, $a=3$, $b=2$, we find $y=35$; and +it happens that these $35$ rows can be divided into $7$ sets, +each of which contains all the symbols; hence $z=7$. More +generally, if $x=5 \times 3^m$, $a=3$, $b=2$, we find $y=\frac{3}{2}(x-1)/x$, +$z=\frac{1}{2}(x-1)$. It will be noticed that in the solutions of the +original fifteen school-girls problem and of Walecki's extension +of it given above every possible pair of girls walk together +once; hence we might infer that in these cases we could +determine $z$ as well as $y$. + +The results of the last paragraph were given by Kirkman\index +{Kirkman@Kirkman, T.P.}\footnote +{\textit{Cambridge and Dublin Mathematical Journal}, 1850, vol.~\textsc{v}, +pp.~255--262.\label{ibid:2}} +in 1850. In the same memoir he also proved that, if $p$ is a +prime, and if $x=p^m$, $a=p$, $b=2$, then $y=(p^m-1)/(p-1)$; +if $x=(p^2+p+1)(p+1)$ where $p^2+p+1$ has no divisor less +than $p+1$, $a=p+1$, $b=2$, then $y=x(x-1)/p(p+1)$; if +$x=p^3+p+1$, $a=p+1$, $b=2$, then $y=x$; and Sylvester's\index{Sylvester} +result that if $x=15$, $a=3$, $b=3$, $y=455$, $z=91$. Three years +later Kirkman\footnote +{\Ibidref{ibid:2}{\textit{Cambridge and Dublin Mathematical Journal}}, +1853, vol.~\textsc{viii}, pp.~38--42.} +solved the problem when $x=2^n$, $a=4$, +$b=3$. Lastly, in 1893, Sylvester\footnote +{\textit{Messenger of Mathematics}, February, 1893, vol.~\textsc{xxii}, +pp.~159--160.} published the solution when +$x=9$, $a=3$, $b=3$, in which case $y=84$, $z=28$; and stated that +a similar method was applicable when $x=3^m$, $a=3$, $b=3$: thus +$9$ girls can be arranged to walk out $28$ times (say $4$ times a +day for a week) so that in any day the same pair never are +together more than once and so that at the end of the week +each girl has been associated with every possible pair of her +schoolfellows. + +\PG----File: 145.png---------------------------------------------------- +In 1867 Mr~S.~Bills\index{Bills on Kirkman's Problem}\footnote +{\textit{Educational Times Reprints}, London, 1867, vol.~\textsc{viii}, +pp.~32--33.\label{ibid:3}} showed that if $x=7$, $a=3$, $b=2$, +then $y=7$: if $x=15$, $a=3$, $b=2$, then $y=35$: if $x=31$, $a=3$, +$b=2$, then $y=155$: and the method by which these results +are proved will give the value of $y$, if $x=2^n-1$, $a=3$, $b=2$. +Shortly afterwards Mr~W.~Lea\index{Lea on Kirkman's Problem}\footnote +{\Ibidref{ibid:3}{\textit{Educational Times Reprints}, London}, +1868, vol.~\textsc{ix}, pp.~35--36; and 1874, vol.~\textsc{xxii}, +pp.~74--76; see also the volume for 1869, vol.~\textsc{xi}, p.~97.} showed +that if $x=11$, +$a=5$, $b=4$, then $y=66$; also that if $x=16$, $a=4$, $b=3$, then +$y=140$; the latter result is a particular case of Kirkman's +theorems. It will be noticed that these writers did not confine +their discussion to cases where $x$ is an exact multiple of $a$\index +{Fifteen School@\textsc{Fifteen School-girls}|)}% +\index{Kirkman's Prob@\textsc{Kirkman's Problem}|)}% +\index{School@\textsc{School-girls, Fifteen}|)}. + +\section[Problems connected with a pack of cards]% +{Problems connected with a Pack of Cards} I mentioned +in \hyperlink{chapter:1:cards}{chapter~\textsc{i}} that an ordinary pack +of playing cards\index{Cards, Problems with|(} +could be used to illustrate many tricks depending on simple +properties of numbers. Most of these involve the relative +position of the cards. The principle of solution generally +consists in re-arranging the pack in a particular manner so as +to bring the card into some definite position. Any such rearrangement +is a species of shuffling. + +I shall treat in succession of problems connected with\index +{Shuffling@\textsc{Shuffling Cards}|(} +\emph{shuffling a pack}, \emph{arrangements by rows and columns}, the +\emph{determination of a pair out of $\frac{1}{2}n(n+1)$ pairs}, +\emph{Gergonne's pile +problem}, and the game known as \emph{the mouse trap}. + +\section*{Shuffling a Pack}% +\addcontentsline{toc}{subsection}{Monge on shuffling a pack of cards} +Any system of \emph{shuffling a pack}\index{Monge on Shuffling Cards|(} of +cards, if carried out consistently, leads to an arrangement +which can be calculated; but tricks that depend on it generally +require considerable technical skill. + +Suppose for instance that a pack of $n$ cards is shuffled, as +is not unusual, by placing the second card on the first, the +third below these, the fourth above them, and so on. The +theory of this system of shuffling is due to Monge\footnote +{Monge's investigations are printed in the \textit{Mémoires de l'Académie +des Sciences}, Paris, 1773, pp.~390--412. Among those who have studied +the subject afresh I may in particular mention V.~Bouniakowski\index +{Bouniakowski, V., on shuffling}, \textit{Bulletin +physico-mathématique de St Pétersbourg}, 1857, vol.~\textsc{xv}, +pp.~202--205, +summarised in the \textit{Nouvelles annales de mathématiques}, 1858, +pp.~66--67; T.~de St~Laurent\index{DeSaint@De St Laurent}\index +{StL@St Laurent on cards}, \textit{Mémoires de l'Académie de Gard}, 1865; +L.~Tanner\index{Tanner on Shuffling Cards}, \textit{Educational Times +Reprints}, 1880, vol.~\textsc{xxxiii}, pp.~73--75; +and M.J.~Bourget\index{Bourget, M.J., on shuffling}, \textit{Liouville's +Journal}, 1882, pp.~413--434. The solutions +given by Prof.\ Tanner are simple and concise.}. The +\PG----File: 146.png---------------------------------------------------- +following are some of the results and are not difficult to prove +directly. + +If the pack contains $6p+4$ cards, the $(2p + 2)$th card will +occupy the same position in the shuffled pack. For instance, +if a complete pack of $52$ cards is shuffled as described above, +the $18$th card will remain the $18$th card. + +Again, if a pack of $10p + 2$ cards is shuffled in this way, +the $(2p + 1)$th and the $(6p \times 2)$th cards will interchange places. +For instance, if an écarté pack of $32$ cards is shuffled as described +above, the $7$th and the $20$th cards will change places. + +More generally, one shuffle of a pack of $2p$ cards will +move the card which was in the $x_0$th place to the $x_1$th place, +where $x_1=\frac{1}{2}(2p+x_0+1)$ if $x_0$ is odd, and +$x_1=\frac{1}{2}(2p-x_0 + 2)$ +if $x_0$ is even, from which the above results can be deduced. +By repeated applications of the above formulae we can +show that the effect of $m$ such shuffles is to move the card +which was initially in the $x_0$th place to the $x_m$th place where +$2^{m+1}x_m=(4p+1)(2^{m-1} \pm 2^{m-2} \pm \dotsb \pm 2 \pm 1) +\pm 2x_0 + 2^m \pm 1$, the +sign $\pm$ representing an ambiguity of sign. + +Again, in any pack of $n$ cards after a certain number of +shufflings, not greater than $n$, the cards will return to their +primitive order. This will always be the case as soon as the +original top card occupies that position again. To determine +the number of shuffles required for a pack of $2p$ cards, it is +sufficient to put $x_m=x_0$ and find the smallest value of $m$ +which satisfies the resulting equation for all values of $x_0$ +from $1$ to $2p$. It follows that, if $m$ is the least number +which makes $4^m-1$ divisible by $4p+1$, then $m$ shuffles will be +required if either $2^m+1$ or $2^m-1$ is divisible by $4p+1$, otherwise +$2m$ shuffles will be required. The number for a pack of +\PG----File: 147.png---------------------------------------------------- +$2p+1$ cards is the same as that for a pack of $2p$ cards. With +an écarté pack of $32$ cards, six shuffles are sufficient; with +a pack of $2^n$ cards, $n+1$ shuffles are sufficient; with a full +pack of $52$ cards, twelve shuffles are sufficient; with a pack +of $13$ cards ten shuffles are sufficient; while with a pack of +$50$ cards fifty shuffles are required; and so on\index +{Monge on Shuffling Cards|)}. + +Mr W.H.H.~Hudson\index{Hudson, W.H.H., on cards}\footnote +{\textit{Educational Times Reprints}, London, 1865, vol.~\textsc{ii}, p.~105.} +has also shown that, whatever is +the law of shuffling, yet if it is repeated again and again on +a pack of $n$ cards, the cards will ultimately fall into their +initial positions after a number of shufflings not greater than +the greatest possible \textsc{l.c.m.} of all numbers whose sum is $n$. + +For suppose that any particular position is occupied after +the $1$st, $2$nd, $\dots$, $p$th shuffles by the cards +$A_1, A_2, \dotsc, A_p$ respectively, +and that initially the position is occupied by the +card $A_0$. Suppose further that after the $p$th shuffle $A_0$ returns +to its initial position, therefore $A_0=A_p$. Then at the second +shuffling $A_2$ succeeds $A_1$ by the same law by which $A_1$ succeeded +$A_0$ at the first; hence it follows that previous to the +second shuffling $A_2$ must have been in the place occupied by +$A_1$ previous to the first. Thus the cards which after the +successive shuffles take the place initially occupied by $A_1$ are +$A_2, A_3, \dotsc, A_p, A_1$; that is, after the $p$th shuffle $A_1$ +has returned +to the place initially occupied by it: and so for all the +other cards $A_2, A_3, \dotsc, A_{p-1}$. + +Hence the cards $A_1, A_2, \dotsc, A_p$ form a cycle of $p$ cards, one +or other of which is always in one or other of $p$ positions in the +pack, and which go through all their changes in $p$ shufflings. +Let the number $n$ of the pack be divided into $p, q, r, \ldots$ such +cycles, whose sum is $n$; then the \textsc{l.c.m.} of $p, q, r, \ldots$ +is the +utmost number of shufflings necessary before all the cards will +be brought back to their original places. + +In the case of a pack of $52$ cards, the greatest \textsc{l.c.m.} of +numbers whose sum is $52$ will be found by trial to be +$180180$\index{Shuffling@\textsc{Shuffling Cards}|)}. + +\PG----File: 148.png---------------------------------------------------- +\section*{Arrangements by Rows and Columns}% +\addcontentsline{toc}{subsection}{Arrangement by rows and columns} +A not uncommon +trick, which rests on a species of shuffling, depends on the +obvious fact that if $n^2$ cards are arranged in the form of a +square of $n$ rows, each containing $n$ cards, then any card +will be defined if the row and the column in which it lies are +mentioned. + +This information is generally elicited by first asking in +which row the selected card lies, and noting the extreme left-hand +card of that row. The cards in each column are then +taken up, face upwards, one at a time beginning with the +lowest card of each column and taking the columns in their +order from right to left--each card taken up being placed on +the top of those previously taken up. The cards are then +dealt out again in rows, from left to right, beginning with the +top left-hand corner, and a question is put as to which row +contains the card. The selected card will be that card in the +row mentioned which is in the same vertical column as the +card which was originally noted. + +The above is the form in which the trick is usually +presented, but it is greatly improved by allowing the pack to +be cut as often as is liked before the cards are re-dealt, and +then giving one cut at the end so as to make the top card in +the pack one of those originally in the top row. + +The explanation is obvious. For, if $16$ cards are taken, +\begin{figure*}[!hbt] +\def\SqHt{3em} +\def\SqWd{2em} +\unitlength=1em +\centering +\hspace*{\fill} +\begin{minipage}{0.4\textwidth} +\centering +\begin{picture}(11,15) +\Cell(0,12;1)\Cell(3,12;2)\Cell(6,12;3)\Cell(9,12;4) +\Cell(0,8;5)\Cell(3,8;6)\Cell(6,8;7)\Cell(9,8;8) +\Cell(0,4;9)\Cell(3,4;10)\Cell(6,4;11)\Cell(9,4;12) +\Cell(0,0;13)\Cell(3,0;14)\Cell(6,0;15)\Cell(9,0;16) +\end{picture} +\legend{Figure \Uproman{1}} +\end{minipage} +\hfill +\begin{minipage}{0.4\textwidth} +\centering +\begin{picture}(11,15) +\Cell(0,12;1)\Cell(3,12;5)\Cell(6,12;9)\Cell(9,12;13) +\Cell(0,8;2)\Cell(3,8;6)\Cell(6,8;10)\Cell(9,8;14) +\Cell(0,4;3)\Cell(3,4;7)\Cell(6,4;11)\Cell(9,4;15) +\Cell(0,0;4)\Cell(3,0;8)\Cell(6,0;12)\Cell(9,0;16) +\end{picture} +\legend{Figure \Uproman{2}} +\end{minipage} +\hspace*{\fill} +\end{figure*} +the first and second arrangements may be represented by +\PG----File: 149.png---------------------------------------------------- +figures~i and ii. For example, if we are told that in figure~i +the card is in the third row, it must be either $9$, $10$, $11$, or $12$: +hence, if we know in which row of figure~ii it lies, it is determined. +If we allow the pack to be cut between the deals, +we must secure somehow that the top card is either $1$, $2$, $3$, +or $4$, since that will leave the cards in each row of figure~ii +unaltered though the positions of the rows will be changed. + +\section*{Determination of a selected pair of cards out of +$\protect\frac{1}{2}n(n+1)$ given pairs\protect\footnote +{Bachet\index{Bachet@Bachet's \textit{Problèmes}}, problem~\textsc{xvii}, +avertissement, p.~146 \etseq}}% +\addcontentsline{toc}{subsection}{Determination of one out of +\texorpdfstring{$\protect\frac12n(n+1)$}{\textonehalf n(n+1)} given couples} +Another common trick is to throw +twenty cards on to a table in ten couples, and ask someone +to select one couple\index{Pairs of Cards Trick|(}. +The cards are then taken up, and dealt +out in a certain manner into four rows each containing five +cards. If the rows which contain the given cards are indicated, +the cards selected are known at once. + +This depends on the fact that the number of homogeneous +products of two dimensions which can be formed out of four +things is $10$. Hence the homogeneous products of two dimensions +formed out of four things can be used to define ten things. + +Suppose that ten pairs of cards are placed on a table and +someone selects one couple. Take up the cards in their +\begin{figure*}[!hbt] +\centering +\def\SqHt{3em} +\def\SqWd{2em} +\unitlength=1em +\begin{picture}(14,15) +\Cell(0,12;1)\Cell(3,12;2)\Cell(6,12;3)\Cell(9,12;5)\Cell(12,12;7) +\Cell(0,8;4)\Cell(3,8;9)\Cell(6,8;10)\Cell(9,8;11)\Cell(12,8;13) +\Cell(0,4;6)\Cell(3,4;12)\Cell(6,4;15)\Cell(9,4;16)\Cell(12,4;17) +\Cell(0,0;8)\Cell(3,0;14)\Cell(6,0;18)\Cell(9,0;19)\Cell(12,0;20) +\end{picture} +\end{figure*} +couples. Then the first two cards form the first couple, the +next two the second couple, and so on. Deal them out in +four rows each containing five cards according to the scheme +shown in the diagram. % [**Note: originally "above"] + +\PG----File: 150.png---------------------------------------------------- +The first couple ($1$ and $2$) are in the first row. Of the next +couple ($3$ and $4$), put one in the first row and one in the +second. Of the next couple ($5$ and $6$), put one in the first +row and one in the third, and so on, as indicated in the +diagram. After filling up the first row proceed similarly with +the second row, and so on. + +Enquire in which rows the two selected cards appear. If +only one line, the $m$th, is mentioned as containing the cards +then the required pair of cards are the $m$th and $(m+1)$th +cards in that line. These occupy the clue squares of that +line. Next, if two lines are mentioned, then proceed as +follows. Let the two lines be the $p$th and the $q$th and suppose +$q>p$. Then that one of the required cards which is in the +$q$th line will be the $(q-p)$th card which is below the first +of the clue squares in the $p$th line. The other of the required +cards is in the $p$th line and is the $(q-p)$th card to the right of +the second of the clue squares. + +Bachet's rule, in the form in which I have given it, is +applicable to a pack of $n(n+1)$ cards divided into couples, +and dealt in $n$ rows each containing $n+1$ cards; for there are +$\frac{1}{2}n(n+1)$ such couples, also there are $\frac{1}{2}n(n+1)$ +homogeneous +products of two dimensions which can be formed out of $n$ +things. Bachet gave the diagrams for the cases of $20$, $30$, and +$42$ cards: these the reader will have no difficulty in constructing +for himself, and I have enunciated the rule for $20$ cards in +a form which covers all the cases. + +I have seen the same trick performed by means of a sentence +and not by numbers. If we take the case of ten couples, then +after collecting the pairs the cards must be dealt in four rows +each containing five cards, in the order indicated by the +sentence \emph{Matas dedit nomen Cocis}. This sentence must be +imagined as written on the table, each word forming one line, +The first card is dealt on the $M$. The next card (which is the +pair of the first) is placed on the second $m$ in the sentence, +that is, third in the third row. The third card is placed on +the $a$. The fourth card (which is the pair of the third) is +\PG----File: 151.png--------------------------------------------------------- +placed on the second $a$, that is, fourth in the first row. Each +of the next two cards is placed on a $t$, and so on. Enquire +in which rows the two selected cards appear. If two rows +are mentioned, the two cards are on the letters common to +the words that make these rows. If only one row is mentioned, +the cards are on the two letters common to that row. + +The reason is obvious: let us denote each of the first +pair by an $a$, and similarly each of any of the other pairs +by an $e$, $i$, $o$, $c$, $d$, $m$, $n$, $s$, or $t$ respectively. +Now the sentence +\emph{Matas dedit nomen Cocis} contains four words each of five +letters; ten letters are used, and each letter is repeated only +twice. Hence, if two of the words are mentioned, they will +have one letter in common, or, if one word is mentioned, it will +have two like letters. + +To perform the same trick with any other number of cards +we should require a different sentence. + +The number of homogeneous products of three dimensions +which can be formed out of four things is $20$, and of these the +number consisting of products in which three things are alike +and those in which three things are different is $8$. This leads +to a trick with $8$ trios of things, which is similar to that last +given--the cards being arranged in the order indicated by the +sentence \emph{Lanata levete livini novoto}. + +I believe that these arrangements by sentences are +known, but I am not aware who invented them\index{Pairs of Cards Trick|)}. + +\section*{Gergonne's Pile Problem}% +\addcontentsline{toc}{subsection}{Gergonne's Pile Problem} +Before discussing Gergonne's% +\index{Gergonne@\textsc{Gergonne's Problem}|(}% +\index{Pile@\textsc{Pile Problems}|(} +theorem I will describe the familiar three pile problem, the +theory of which is included in his results. + +\subsection*{The Three Pile Problem\protect\footnote +{The trick is mentioned by Bachet\index{Bachet@Bachet's \textit{Problèmes}}, +problem \textsc{xviii}, p.~143, but his +analysis of it is insufficient.}} This trick\index +{Three-pile@\textsc{Three-pile Problem}|(} is usually performed +as follows. Take $27$ cards and deal them into three piles, +face upwards. By ``dealing'' is to be understood that the top +card is placed as the bottom card of the first pile, the second +card in the pack as the bottom card of the second pile, the +\PG----File: 152.png---------------------------------------------------- +third card as the bottom card of the third pile, the fourth card +on the top of the first one, and so on: moreover I assume +that throughout the problem the cards are held in the hand +face upwards. The result can be modified to cover any other +way of dealing. + +Request a spectator to note a card, and remember in which +pile it is. After finishing the deal, ask in which pile the card +is. Take up the three piles, placing that pile between the +other two. Deal again as before, and repeat the question as +to which pile contains the given card. Take up the three +packs again, placing the pile which now contains the selected +card between the other two. Deal again as before, but in +dealing note the middle card of each pile. Ask again for the +third time in which pile the card lies, and you will know that +the card was the one which you noted as being the middle +card of that pile. The trick can be finished then in any way +that you like. The usual method\label{page:152}---but a very clumsy one---is +to take up the three piles once more, placing the named +pile between the other two as before, when the selected card +will be the middle one in the pack, that is, if $27$ cards are +used it will be the fourteenth. + +The trick is often performed with $15$ cards or with $21$ cards, +in either of which cases the same rule holds. + +\subsection*{Gergonne's Generalization} The general theory for a pack +of $m^m$ cards was given by M.~Gergonne\footnote +{Gergonne's \textit{Annales de Mathématiques}, Nismes, 1813--4, +vol.~\textsc{iv}, pp.~276--283.}. Suppose the pack +is arranged in $m$ piles, each containing $m^{m-1}$ cards, and that, +after the first deal, the pile indicated as containing the selected +card is taken up $a$th; after the second deal, is taken up $b$th; +and so on, and finally after the $m$th deal, the pile containing +the card is taken up $k$th. Then when the cards are collected +after the $m$th deal the selected card will be $n$th from the top +where +\[ +\begin{array}{ll} +\text{if $m$ is even,}& n=km^{m-1}-jm^{m-2} + \dotsb + bm - a + 1\,,\\ +\text{if $m$ is odd,}& n=km^{m-1}-jm^{m-2} + \dotsb - bm + a\,. +\end{array} +\] + +\PG----File: 153.png--------------------------------------------------------- +For example, if a pack of $256$ cards (\IE\ $m = 4$) was given, +and anyone selected a card out of it, the card could be determined +by making four successive deals into four piles of +$64$ cards each, and after each deal asking in which pile the +selected card lay. The reason is that after the first deal you +know it is one of sixty-four cards. In the next deal these +sixty-four cards are distributed equally over the four piles, +and therefore, if you know in which pile it is, you will know +that it is one of sixteen cards. After the third deal you know +it is one of four cards. After the fourth deal you know which +card it is. + +Moreover, if the pack of $256$ cards is used, it is immaterial +in what order the pile containing the selected card +is taken up after a deal. For, if after the first deal it is taken +up $a$th, after the second $b$th, after the third $c$th, and after +the fourth $d$th, the card will be the $(64d-16c + 4b-a + 1)$th +from the top of the pack, and thus will be known. We need +not take up the cards after the fourth deal, for the same +argument will show that it is the $(64-16c + 4b-a + 1)$th in +the pile then indicated as containing it. Thus if $a = 3$, $b = 4$, +$c = 1$, $d = 2$, it will be the $62$nd card in the pile indicated after +the fourth deal as containing it and will be the $126$th card in +the pack as then collected. + +In exactly the same way a pack of twenty-seven cards may +be used, and three successive deals, each into three piles of +nine cards, will suffice to determine the card. If after the +deals the pile indicated as containing the given card is taken +up $a$th, $b$th, and $c$th respectively, then the card will be the +$(9c-3b + a)$th in the pack or will be the $(9-3b + a)$th card +in the pile indicated after the third deal as containing it. + +The method of proof will be illustrated sufficiently by +considering the usual case of a pack of twenty-seven cards, for +which $m = 3$, which are dealt into three piles each of nine cards. + +Suppose that, after the first deal, the pile containing the +selected card is taken up $a$th: then (i)~at the top of the pack +there are $a-1$ piles each containing nine cards; (ii)~next +\PG----File: 154.png----------------------------------------------------- +there are $9$ cards, of which one is the selected card; and +(iii)~lastly there are the remaining cards of the pack. The +cards are dealt out now for the second time: in each pile the +bottom $3(a-1)$ cards will be taken from (i), the next $3$ +cards from (ii), and the remaining $9-3a$ cards from (iii). + +Suppose that the pile now indicated as containing the +selected card is taken up $b$th: then (i)~at the top of the +pack are $9(b-1)$ cards; (ii)~next are $9-3a$ cards; (iii)~next +are $3$ cards, of which one is the selected card; and (iv)~lastly +are the remaining cards of the pack. The cards are dealt out +now for the third time: in each pile the bottom $3 (b-1)$ +cards will be taken from (i), the next $3-a$ cards will be +taken from (ii), the next card will be one of the three cards in +(iii), and the remaining $8-3b+a$ cards are from (iv). + +Hence, after this deal, as soon as the pile is indicated, it +is known that the card is the $(9-3b + a)$th from the top +of that pile. If the process is continued by taking up this +pile as $c$th, then the selected card will come out in the place +$9(c-1) + (8-3b + a) + 1$ from the top, that is, will come out +as the $(9c-3b + a)$th card. + +Since, after the third deal, the position of the card in the +pile then indicated is known, it is easy to notice the card, in +which case the trick can be finished in some way more effective +than dealing again. + +If we put the pile indicated always in the middle of the +pack we have $a=2$, $b = 2$, $c = 2$, hence $n = 9c-3b + a= 14$, +which is the form in which the trick is usually presented, as +was explained above on page \pageref{page:152}. + +I have shown that if $a$, $b$, $c$ are known, then $n$ is determined. +We may modify the rule so as to make the selected +card come out in any assigned position, say the $n$th. In this +case we have to find values of $a$, $b$, $c$ which will satisfy the +equation $n = 9c-3b + a$, where $a$, $b$, $c$ can have only the values +$1$, $2$, or $3$. + +Hence, if we divide $n$ by $3$ and the remainder is $1$ or $2$, this +remainder will be $a$; but, if the remainder is $0$, we must decrease +\PG----File: 155.png------------------------------------------------------ +the quotient by unity so that the remainder is $3$, and this +remainder will be $a$. In other words $a$ is the smallest positive +number (exclusive of zero) which must be subtracted from $n$ +to make the difference a multiple of $3$. + +Next let $p$ be this multiple, \IE~$p$ is the next lowest integer +to $n/3$: then $3p = 9c-3b$, therefore $p = 3c-b$. Hence $b$ is +the smallest positive number (exclusive of zero) which must +be added to $p$ to make the sum a multiple of $3$, and $c$ is that +multiple. + +A couple of illustrations will make this clear. Suppose +we wish the card to come out $22$nd from the top, therefore +$22 = 9c-3b + a$. The smallest number which must be subtracted +from $22$ to leave a multiple of $3$ is $1$, therefore $a = 1$. +Hence $22 = 9c-3b + 1$, therefore $7 = 3c-b$. The smallest +number which must be added to $7$ to make a multiple of $3$ +is $2$, therefore $b = 2$. Hence $7 = 3c-2$, therefore $c = 3$. Thus +$a = 1$, $b=2$, $c = 3$. + +Again, suppose the card is to come out $21$st. Hence +$21 = 9c-3b+a$. Therefore $a$ is the smallest number which +subtracted from $21$ makes a multiple of $3$, therefore $a= 3$. +Hence $6 = 3c-b$. Therefore $b$ is the smallest number which +added to $6$ makes a multiple of $3$, therefore $b = 3$. Hence +$9 = 3c$, therefore $c = 3$. Thus $a = 3$, $b = 3$, $c = 3$. + +If any difficulty is experienced in this work, we can proceed +thus. Let $a = x+1$, $b=3-y$, $c=z+1$; then $x$, $y$, $z$ +may have only the values $0$, $1$, or $2$. In this case Gergonne's +equation takes the form $9z + 3y + x = n-1$. Hence, if $n-1$ +is expressed in the ternary scale of notation, $x$, $y$, $z$ will be +determined, and therefore $a$, $b$, $c$ will be known. + +The rule in the case of a pack of $m^m$ cards is exactly +similar. We want to make the card come out in a given +place. Hence, in Gergonne's formula, we are given $n$ and +we have to find $a, b, \dotsc, k$. We can effect this by dividing $n$ +continually by $m$, with the convention that the remainder are +to be alternately positive and negative and that their numerical +values are to be not greater than $m$ or less than unity\index +{Gergonne@\textsc{Gergonne's Problem}|)}. + +\PG----File: 156.png----------------------------------------------------- +An analogous theorem with a pack of $lm$ cards can be constructed. +C.T.~Hudson\index{Hudson, C.T., on cards} and L.E.~Dickson\index +{Dickson, L.E.}\footnote +{\textit{Educational Times Reprints}, 1868, vol.~\textsc{ix}, pp.~89--91; +and \textit{Bulletin} +of the American Mathematical Society, New York, April, 1895, vol.~\textsc{i}, +pp.~184--186.} have discussed +the general case where such a pack is dealt $n$ times, each time +into $l$ piles of $m$ cards; and they have shown how the piles +must be taken up in order that after the $n$th deal the selected +card may be $r$th from the top. + +The principle will be sufficiently illustrated by one example +treated in a manner analogous to the cases already +discussed. For instance, suppose that an écarté pack of $32$ +cards is dealt into $4$ piles each of $8$ cards, and that the pile +which contains some selected card is picked up $a$th. Suppose +that on dealing again into four piles, one pile is indicated as +containing the selected card, the selected card cannot be one +of the bottom $2 (a-1)$ cards, or of the top $8-2a$ cards, but +must be one of the intermediate 2 cards, and the trick can be +finished in any way, as for instance by the common conjuring +ambiguity of asking someone to choose one of them, leaving it +doubtful whether the one he takes is to be rejected or retained% +\index{Three-pile@\textsc{Three-pile Problem}|)}% +\index{Pile@\textsc{Pile Problems}|)}. + +\section*{The Mouse Trap}% +\addcontentsline{toc}{subsection}{The Mouse Trap. \texorpdfstring{\protect\quad}{} Treize} +I will conclude this chapter with the\index +{Mousetrap@\textsc{Mousetrap, Game of}|(} +bare mention of another game of cards, known as the \emph{mouse +trap}, the discussion of which involves some rather difficult +algebraic analysis. + +It is played as follows. A set of cards, marked with the +numbers $1, 2, 3, \dotsc, n$, is dealt in any order, face upwards, +in the form of a circle. The player begins at any card and +counts round the circle always in the same direction. If the +$k$th card has the number $k$ on it---which event is called a +\emph{hit}---the player takes up the card and begins counting afresh. +According to Cayley\index{Cayley}, the player wins if he thus takes up all +the cards, and the cards win if at any time the player counts +up to $n$ without being able to take up a card. + +\PG----File: 157.png------------------------------------------------------- +For example, if a pack of only four cards is used and these +cards come in the order, $3214$, then the player would obtain +the second card $2$ as a hit, next he would obtain $1$ as a hit, +but if he went on for ever he would not obtain another hit. +On the other hand, if the cards in the pack were initially in +the order $1423$, the player would obtain successively all four +cards in the order $1$, $2$, $3$, $4$. + +The problem may be stated as the determination of what +hits and how many hits can be made with a given number of +cards; and what permutations will give a certain number of +hits in a certain order. + +Cayley\footnote +{\textit{Quarterly Journal of Mathematics}, 1878, vol.~\textsc{xv}, +pp.~8--10.\label{ibid:4}} showed that there are $9$ arrangements of a pack +of four cards in which no hit will be made, $7$ arrangements in +which only one hit will be made, $3$ arrangements in which +only two hits will be made, and $5$ arrangements in which four +hits will be made. + +Prof.\ Steen\index{Steen on the Mousetrap}\footnote +{\Ibidref{ibid:4}{\textit{Quarterly Journal of Mathematics}}, +vol.~\textsc{xv}, pp.~230--241.} has investigated the general +theory for a pack +of $n$ cards. He has shown how to determine the number of +arrangements in which $x$ is the first hit [Arts.~3--5]; the +number of arrangements in which $1$ is the first hit and $x$ +is the second hit [Art.~6]; and the number of arrangements +in which $2$ is the first hit and $x$ the second hit [Arts.~7--8]; +but beyond this point the theory has not been carried. It +is obvious that, if there are $n-1$ hits, the $n$th hit will +necessarily follow\index{Mousetrap@\textsc{Mousetrap, Game of}|)}. + +\ssection*{Treize} The French game of \emph{treize}\index +{Treize, Game of} is very similar. It is +played with a full pack of fifty-two cards (knave, queen, and +king counting as $11$, $12$, and $13$ respectively). The dealer +calls out $1, 2, 3, \dotsc, 13$, as he deals the $1$st, $2$nd, $3$rd, +$\dots$, $13$th +cards respectively. At the beginning of a deal the dealer +offers to lay or take certain odds that he will make a hit in +the thirteen cards next dealt. + +One of the innumerable forms of \emph{patience} is played in a +similar way\index{Cards, Problems with|)}. +\PG----File: 158.png--------------------------------------------------------- + + +%CHAPTER V. + +\chapter{Magic Squares.} + +A \emph{magic square} consists of a number of integers arranged\chapindex +{Magic Square@\textsc{Magic Squares}} +in the form of a square, so that the sum of the numbers in +every row, in every column, and in each diagonal is the same. +If the integers are the consecutive numbers from $1$ to $n^2$ the +square is said to be of the $n$th order, and it is easily seen +that in this case the sum of the numbers in any row, column, +or diagonal is equal to $\frac{1}{2}n(n^2 +1)$: this number may be +denoted by $N$. I confine my account to such magic squares, +that is, to squares formed with consecutive integers, from +$1$ upwards. + +Thus the first $16$ integers, arranged in either of the forms +given in figures~i and ii below, form a magic square of the +\begin{figure*}[!hbt] +\centering +\hspace*{\fill} +\begin{minipage}{0.3\textwidth} +\centering +\begin{MagicSquare}{4} +1 & {15} & {14} & 4 \\ +{12} & 6 & 7 & 9 \\ +8 & {10} & {11} & 5 \\ +{13} & 3 & 2 & {16} +\end{MagicSquare} +\legend{Figure \Uproman1} +\end{minipage} +\hfill +\begin{minipage}{0.3\textwidth} +\centering +\begin{MagicSquare}{4} +{15} & {10} & 3 & 6 \\ +4 & 5 & {16} & 9 \\ +{14} & {11} & 2 & 7 \\ +1 & 8 & {13} & {12} \\ +\end{MagicSquare} +\legend{Figure \Uproman2} +\end{minipage} +\hspace*{\fill} +\label{figure:i} +\end{figure*} +fourth order, the sum of the numbers in any row, column, +or diagonal being $34$. Similarly figures~iii and v on page~\pageref +{figure:iii}, % [page 144] +figure~viii on page~\pageref{figure:vi}, % [page 147] +and figures~xii and xiii on page~\pageref{figure:xii}, % [page 159] +show magic squares of the fifth order; and figure~xi on +page \pageref{figure:ix} % [page 155] +shows a magic square of the sixth order; and figures~xiv +and xv on pages~\pageref{figure:xiv}, \pageref{figure:xv}, % [pages 160, 161] +show magic squares of the eighth order. + +\phantomsection +\addcontentsline{toc}{section}{Notes on the History of Magic Squares} +The formation of these squares is an old amusement, and +\PG----File: 159.png----------------------------------------------------- +in times when mystical philosophical ideas were associated +with particular numbers it was natural that such arrangements +should be deemed to possess magical properties. Magic squares +of an odd order were constructed in India before the Christian +era according to a law of formation which is explained hereafter. +Their introduction into Europe appears to have been +due to Moschopulus\index{Moschopulus}, who lived at Constantinople in the +early part of the fifteenth century, and enunciated two methods +for making such squares. The majority of the medieval +astrologers and physicians were much impressed by such arrangements. +In particular the famous Cornelius Agrippa\index{Agrippa, Cornelius} +(1486--1535) constructed magic squares of the orders $3$, $4$, $5$, +$6$, $7$, $8$, $9$, which were associated respectively with the seven +astrological ``planets''\index{Astrological Planets}\index +{PlanetsA@Planets (astrological)}: namely, Saturn, Jupiter, Mars, the +Sun, Venus, Mercury, and the Moon. He taught that a +square of one cell, in which unity was inserted, represented +the unity and eternity of God; while the fact that a square +of the second order could not be constructed illustrated the +imperfection of the four elements, air, earth, fire, and water; +and later writers added that it was symbolic of original sin. +A magic square engraved on a silver plate was sometimes +prescribed as a charm against the plague, and one, namely, +that represented in figure~i on page~\pageref{figure:i}, +is drawn in the % [*Note: originally "the last page"] +picture of Melancholy, painted about the year 1500 by Albert +Dürer\index{Durer@Dürer, A.}. Such charms are still worn in the East. + +The development of the theory has been due mainly to +French mathematicians. Bachet\index{Bachet@Bachet's \textit {Problèmes}|(} +gave a rule for the construction +of any square of an odd order in a form substantially +equivalent to one of the rules given by Moschopulus\index{Moschopulus}. The +formation of magic squares, especially of even squares, was +considered by Frénicle\index{Frenicle@Frénicle, Magic Squares} and +Fermat. The theory was continued +by Poignard\index{Poignard, Magic Squares}, De la~Hire\index +{DelaHire@De la Hire on Magic Squares}\index{LaHire@La Hire}, +Sauveur\index{Sauveur, Magic Squares}, D'Ons-en-bray\index +{Donsenbray@D'Ons-en-bray, Magic Squares}\index +{Onsenbray@Ons-en-bray on Magic Squares}, +and Des~Ourmes\index{DesOurmes@Des Ourmes on Magic Squares}\index +{Ourmes on Magic Squares}. Ozanam\index +{Ozanam@Ozanam's \textit {Récréations}} included in his work an essay on +magic squares which was amplified by Montucla\index{Montucla}. From this +and from De la Hire's memoirs the larger part of the materials +for this chapter are derived. Like most algebraical problems, +\PG----File: 160.png---------------------------------------------------- +the construction of magic squares attracted the attention of +Euler\index{Euler}, but he did not advance the general theory. In 1837 +an elaborate work on the subject was compiled by B.~Violle\index +{Violle, Magic Squares}, +which is useful as containing numerous illustrations. I give +the references in a footnote\footnote +{Bachet\index{Bachet@Bachet's \textit{Problèmes}|)}, \textit{Problèmes +plaisans}, Lyons, 1624, problem~\textsc{xxi}, p.~161; +Frénicle\index{Frenicle@Frénicle, Magic Squares}, \textit{Divers +Ouvrages de Mathématique par Messieurs de l'Académie +des Sciences}, Paris, 1693, pp.~423--483; with an appendix (pp.~484--507), +containing diagrams of all the possible magic squares of the fourth order, +$880$ in number: Fermat\index{Fermat, P.}, \textit{Opera Mathematica}, +Toulouse, 1679, pp.~173--178; +or Brassinne's \textit{Précis}, Paris, 1853, pp.~146--149: Poignard\index +{Poignard, Magic Squares}, \textit{Traité +des Quarrés Sublimes}, Brussels, 1704: De la~Hire\index +{DelaHire@De la Hire on Magic Squares}\index{LaHire@La Hire}, \textit +{Mémoires de l'Académie +des Sciences} for 1705, Paris, 1706, part~\textsc{i}, pp.~127--171; +part~\textsc{ii}, pp.~364--382: +Sauveur\index{Sauveur, Magic Squares}, \textit{Construction des Quarrés +Magiques}, Paris, 1710: D'Ons-en-bray, +\textit{Mémoires de l'Académie des Sciences} for 1750, Paris, 1754, +pp.~241--271: +Des~Ourmes, \textit{Mémoires de Mathématique et de Physique} (French +Academy), Paris, 1763, vol.~\textsc{iv}, pp.~196--241: Ozanam\index +{Ozanam@Ozanam's \textit{Récréations}} and Montucla\index{Montucla}, +\textit{Récréations}, part~\textsc{i}, chapter~\textsc{xii}: +Euler\index{Euler}, \textit{Commentationes Arithmeticae +Collectae}, St Petersburg, 1849, vol.~\textsc{ii}, pp.~593--602: +Violle, \textit{Traité +Complet des Carrés Magiques}, 3~vols, Paris, 1837--8. A sketch of the +history of the subject is given in chap.~iv of S.~Günther's\index +{Gunther@Günther, S.} \textit{Geschichte der +mathematischen Wissenschaften}, Leipzig, 1876. See also +W.~Ahrens\index{Ahrens}, \textit{Mathematische Unterhaltungen und Spiele}, +Leipzig, 1901, chapter~xii.}. + +I shall confine myself to establishing rules for the construction +of squares subject to no conditions beyond those +given in the definition. Rules sufficient for this purpose are +contained in the works to which I have just referred and on +which I have based this sketch; some extensions and developments +will be found in the memoirs mentioned below\footnote +{In England the subject has been studied by R.~Moon\index{Moon, R.}, +\textit{Cambridge +Mathematical Journal}, 1845, vol.~\textsc{iv}, pp.~209--214; +H.~Holditch\index{Holditch on Magic Squares}, \textit{Quarterly +Journal of Mathematics}, London, 1864, vol.~\textsc{vi}, pp.~181--189; +W.H.~Thompson\index{Thompson on Magic Squares}, +\Ibid, 1870, vol.~\textsc{x}, pp.~186--202; J.~Horner\index +{Horner on Magic Squares}, \Ibid, 1871, +vol.~\textsc{xi}, pp.~57--65, 123--132, 213--224; S.M.~Drach\index +{Drach on Magic Squares}, \textit{Messenger of +Mathematics}, Cambridge, 1873, vol.~\textsc{ii}, pp.~169--174, 187; +A.H.~Frost\index{Frost, A.H.}\label{footnote:frost:160}, +\textit{Quarterly Journal of Mathematics}, London, 1878, vol.~\textsc{xv}, +pp.~34--49, +93--123, 366--368, in which the results of previous memoirs are included: +there are also some pamphlets and articles on it of a more popular character. +Of recent Continental works on the subject I have no complete bibliography, +and probably it is better to omit all rather than give an imperfect list.}. I +\PG----File: 161.png----------------------------------------------------- +shall commence by giving rules for the construction of a square +of an odd order, and then shall proceed to similar rules for +one of an even order. + +It will be convenient to use the following terms. The +spaces or small squares occupied by the numbers are called +\emph{cells}. The diagonal from the top left-hand cell to the bottom +right-hand cell is called the \emph{leading diagonal} or \emph{left diagonal}. +The diagonal from the top right-hand cell to the bottom left-hand +cell is called the \emph{right diagonal}. + +\ssection[Construction of Odd Magic Squares]% +[Magic Squares of an odd order.]{Magic Squares of an odd order} +I proceed to give +methods for constructing \emph{odd magic squares}, but for simplicity +I shall apply them to the formation of squares of the fifth +order, though exactly similar proofs will apply equally to any +odd square. + +\subsection*{De la Loubère's Method\protect\footnote +{De la~Loubère, \textit{Du Royaume de Siam} (Eng. Trans.), London, 1693, +vol.~\textsc{ii}, pp.~227--247. De la~Loubère was the envoy of +Louis~XIV\index{Louis XIV of France} to +Siam in 1687--8, and there learnt this method.}}% +\addcontentsline{toc}{subsection}{Method of De la Loubère} +If the reader% +\index{DelaLoub@De la Loubère on Magic Squares|(}% +\index{LaLoub@La Loubère|(}% +\index{Loubere@Loubère, de la|(} +will look at \vhyperlink{figure:iii}{figure~iii} +he will see one way in which such a square containing $25$ cells +can be constructed. The middle cell in the top row is occupied +by $1$. The successive numbers are placed in their natural +\begin{figure*}[!hbt] +\centering +\ifPaper\small\unitlength=1.5em\hspace*{-\textwidth}\fi +\begin{minipage}{0.25\textwidth} +\centering +\begin{MagicSquare}{5} + {17}& {24}& 1& 8& {15}\\ + {23}& 5& 7& {14}& {16}\\ + 4& 6& {13}& {20}& {22}\\ + {10}& {12}& {19}& {21}& 3\\ + {11}& {18}& {25}& 2& 9 +\end{MagicSquare} +\legend{\hbox to.25\textwidth{\hss De la Loubère's Method.\hss}\break +Figure \Uproman3} +\end{minipage} +\ifPaper\else\hfill\fi +\begin{minipage}{\ifPaper.55\else0.45\fi\textwidth} +\centering +\def\SqWd{3.375em} +\unitlength=.375em +\def\MSqHorizAdvance{9} +\def\MSqVertAdvance{-4} +\begin{MagicSquare}{45}[20] +{15+2} & {20+4} & {\phantom{0}0+1} & {\phantom{0}5+3} & {10+5} \\ +{20+3} & {\phantom{0}0+5} & {\phantom{0}5+2} & {10+4} & {15+1} \\ +{\phantom{0}0+4} & {\phantom{0}5+1} & {10+3} & {15+5} & {20+2} \\ +{\phantom{0}5+5} & {10+2} & {15+4} & {20+1} & {\phantom{0}0+3} \\ +{10+1} & {15+3} & {20+5} & {\phantom{0}0+2} & {\phantom{0}5+4} +\end{MagicSquare} +\legend{De la Loubère's Method.\break +Figure \Uproman4} +\end{minipage} +\ifPaper\else\hfill\fi +\begin{minipage}{0.25\textwidth} +\centering +\begin{MagicSquare}{5} +{23}& 6& {19}& 2& {15} \\ +{10}& {18}& 1& {14}& {22} \\ +{17}& 5& {13}& {21}& 9 \\ + 4& {12}& {25}& 8& {16} \\ +{11}& {24}& 7& {20}& 3 +\end{MagicSquare} +\legend{Bachet's Method.\break +Figure \Uproman5} +\end{minipage} +\ifPaper\hspace*{-\textwidth}\fi +\DPlabel{figure:iii} +\end{figure*} +order in a diagonal line which slopes upwards to the right, +except that (i)~when the top row is reached the next number +is written in the bottom row as if it came immediately above +the top row; (ii)~when the right-hand column is reached, the +next number is written in the left-hand column, as if it +\PG----File: 162.png------------------------------------------------------ +immediately succeeded the right-hand column; and (iii)~when +a cell which has been filled up already, or when the top +right-hand square is reached, the path of the series drops to +the row vertically below it and then continues to mount again. +Probably a glance at the diagram in \vhyperlink{figure:iii}{figure~iii} +will make this clear. + +The reason why such a square is magic can be explained +best by expressing the numbers in the scale of notation whose +radix is $5$ (or $n$, if the magic square is of the order $n$), except +that $5$ is allowed to appear as a unit-digit and $0$ is not allowed +to appear as a unit-digit. The result is shown in +\vhyperlink{figure:iii}{figure~iv}. +From that figure it will be seen that the method of construction +ensures that every row and every column shall contain +one and only one of each of the unit-digits $1$, $2$, $3$, $4$, $5$, the sum +of which is $15$; and also one and only one of each of the radix-digits +$0$, $5$, $10$, $15$, $20$, the sum of which is $50$. Hence, as +far as rows and columns are concerned, the square is magic. +Moreover if the square is odd, each of the diagonals will +contain one and only one of each of the unit-digits $1$, $2$, $3$, $4$, $5$. +Also the leading diagonal will contain one and only one of the +radix-digits $0$, $5$, $10$, $15$, $20$, the sum of which is $50$; and +if, as is the case in the square drawn above, the number $10$ +is the radix-digit to be added to the unit-digits in the right +diagonal, then the sum of the radix-digits in that diagonal +is also $50$. Hence the two diagonals also possess the magical +property. + +And generally if a magic square of an odd order $n$ is +constructed by De la~Loubère's method, every row and every +column must contain one and only one of each of the unit-digits +$1, 2, 3 \dotsc, n$; and also one and only one of each of the +radix-digits $0, n, 2n, \dotsc, n(n-1)$. Hence, as far as rows and +columns are concerned, the square is magic. Moreover each +diagonal will either contain one and only one of the unit-digits +or will contain $n$ unit-digits each equal to $\frac{1}{2}(n+1)$. It will +also either contain one and only one of the radix-digits or will +contain $n$ radix-digits each equal to $\frac{1}{2}n(n-1)$. Hence the +\PG----File: 163.png------------------------------------------------------ +two diagonals will also possess the magical property. Thus +the square will be magic. + +I may notice here that, if we place $1$ in any cell and fill +up the square by De la~Loubère's rule, we shall obtain a +square that is magic in rows and in columns, but it will not +in general be magic in its diagonals. + +It is evident that other squares can be derived from De la~Loubère's +square by permuting the symbols properly. For +instance, in \vhyperlink{figure:iii}{figure~iv}, +we may permute the symbols $1$, $2$, $3$, $4$, $5$ +in $5!$ ways, and we may permute the symbols $0$, $5$, $15$, $20$ in +% [Note: factorial notation silently updated +$4!$ ways. Any one of these $5!$ arrangements combined with +any one of these $4!$ arrangements will give a magic square. +Hence we can obtain $2880$ magic squares of the fifth order of +this kind, though only $720$ of them are really distinct. Other +squares can however be deduced, for it may be noted that +from any magic square, whether even or odd, other magic +squares of the same order can be formed by the mere interchange +of the row and the column which intersect in a cell +on a diagonal with the row and the column which intersect +in the complementary cell of the same diagonal% +\index{DelaLoub@De la Loubère on Magic Squares|)}% +\index{LaLoub@La Loubère|)}% +\index{Loubere@Loubère, de la|)}. + +\subsection*{Bachet's Method\protect\footnote +{Bachet, Problem~\textsc{xxi}, p.~161. +}}\addcontentsline{toc}{subsection}{Method of Bachet} +Another method, very similar to that\index +{Bachet@Bachet's \textit {Problèmes}} +of De la~Loubère, for constructing an odd magic square is as +follows. We begin by placing $1$ in the cell above the middle +one (that is, in a square of the fifth order in the cell occupied +by the number $7$ in \vhyperlink{figure:iii}{figure~iii}), +and then we write the successive +numbers in a diagonal line sloping upwards to the right, subject +to the condition that when the cases (i) and (ii) mentioned +under De la~Loubère's method occur the rules there given are +followed, but when the case (iii) occurs the path of the series +rises \emph{two} rows, \IE\ it is continued from one cell to the cell next +but one vertically above it, if this cell is above the top row the +path continues from the corresponding cell in one of the bottom +two rows following the analogy of rule~(i) in De la~Loubère's +method. Such a square is delineated in figure~v on page~\pageref{figure:iii}. +Bachet's method leads ultimately to this arrangement; except +\PG----File: 164.png------------------------------------------------------ +that the rules are altered so as to make the line slope downwards. +This method also gives $720$ magic squares of the fifth +order. + +\subsection*{De la Hire's Method\protect +\footnote{\textit{Mémoires de l'Académie des Sciences} for 1705, +part~\textsc{i}, pp.~127--171.}}% +\addcontentsline{toc}{subsection}{Method of De la Hire} +I shall now give another rule% +\index{DelaHire@De la Hire on Magic Squares|(}% +\index{LaHire@La Hire|(} for +the formation of odd magic squares. To form an odd magic +square of the order $n$ by this method, we begin by constructing +two subsidiary squares, one of the unit-digits, $1, 2, \dotsc, n$, and +the other of multiples of the radix, namely, $0, n, 2n, \dotsc, (n-1)n$. +We then form the magic square by adding together the numbers +in the corresponding cells in the two subsidiary squares. + +De la~Hire gave several ways of constructing such subsidiary +squares. I select the following method (props.~x and +xiv of his memoir) as being the simplest, but I shall apply it +to form a square of only the fifth order. It leads to the same +results as the second of the two rules given by Moschopulus\index +{Moschopulus}. + +The first of the subsidiary squares (figure~vi, \vpageref[below]{figure:vi}), +is constructed thus. First, $3$ is put in the top left-hand corner, +and then the numbers $1$, $2$, $4$, $5$ are written in the other cells +of the top line (in any order). Next, the number in each cell +\begin{figure*}[!hbt] +\centering +\begin{minipage}{.3\textwidth} +\centering +\begin{MagicSquare}{5} +3&4&1&5&2\\ +2&3&4&1&5\\ +5&2&3&4&1\\ +1&5&2&3&4\\ +4&1&5&2&3 +\end{MagicSquare} +\legend{{\smaller First Subsidiary Square}\break +Figure \Uproman{6}} +\end{minipage} +\hfill +\begin{minipage}{.3\textwidth} +\centering +\begin{MagicSquare}{5} +{15}&0&{20}&5&{10}\\ +0&{20}&5&{10}&{15}\\ +{20}&5&{10}&{15}&0\\ +5&{10}&{15}&0&{20}\\ +{10}&{15}&0&{20}&5 +\end{MagicSquare} +\legend{{\smaller Second Subsidiary Square}\break +Figure \Uproman{7}} +\end{minipage} +\hfill +\begin{minipage}{.3\textwidth} +\centering +\begin{MagicSquare}{5} +{18}&4&{21}&{10}&{12}\\ +2&{23}&9&{11}&{20}\\ +{25}&7&{13}&{19}&1\\ +6&{15}&{17}&3&{24}\\ +{14}&{16}&5&{22}&8 +\end{MagicSquare} +\legend{{\smaller Resulting Magic Square}\break +Figure \Uproman{8}} +\end{minipage} +\DPlabel{figure:vi} +\end{figure*} +of the top line is repeated in all the cells which lie in a diagonal +line sloping downwards to the right (see \vhyperlink{figure:vi}{figure~vi}) +according to the rule~(ii) in De la~Loubère's method. The cells filled +by the same number form a \emph{broken diagonal}. It follows that every row +and every column contains one and only one $1$, one and only +one $2$, and so on. Hence the sum of the numbers in every row +\PG----File: 165.png------------------------------------------------------ +and in every column is equal to $15$; also, since we placed $3$, +which is the average of these numbers, in the top left-hand +corner, the sum of the numbers in the left diagonal is $15$; +and, since the right diagonal contains one and only one of +each of the numbers $1$, $2$, $3$, $4$, and $5$, the sum of the numbers +in that diagonal also is $15$. + +The second of the subsidiary squares (\vhyperlink*{figure:vi}{figure~vii}) +is constructed in a similar way with the numbers $0$, $5$, $10$, $15$, and +$20$, except that the mean number $10$ is placed in the top right-hand +corner; and the broken diagonals formed of the same +numbers all slope downwards to the left. It follows that every +row and every column in \vhyperlink{figure:vi}{figure~vii} contains one and +only one $0$, one and only one $5$, and so on; hence the sum of the +numbers in every row and every column is equal to $50$. Also +the sum of the numbers in each diagonal is equal to $50$. + +If now we add together the numbers in the corresponding +cells of these two squares, we shall obtain $25$ numbers such +that the sum of the numbers in every row, every column, +and each diagonal is equal to $15 + 50$, that is, to $65$. This is +represented in figure viii. Moreover, no two cells in that figure +contain the same number. For instance, the numbers $21$ to $25$ +can occur only in those five cells which in +\vhyperlink{figure:vi}{figure~vii} are occupied +by the number $20$, but the corresponding cells in +\vhyperlink{figure:vi}{figure~vi} +contain respectively the numbers $1$, $2$, $3$, $4$, and $5$; and thus +in \vhyperlink{figure:vi}{figure~viii} +each of the numbers from $21$ to $25$ occurs once +and only once. De la~Hire preferred to have the cells in the +subsidiary squares which are filled by the same number +connected by a knight's move and not by a bishop's move; +and usually his rule is enunciated in that form. + +By permuting the numbers $1$, $2$, $4$, $5$ in +\vhyperlink{figure:vi}{figure~vi} we get +$4!$ other arrangements, each of which combined with that in +\vhyperlink{figure:vi}{figure~vii} would give a magic square. +Similarly by permuting the numbers $0$, $5$, $15$, $20$ in +\vhyperlink{figure:vi}{figure~vii} we obtain $4!$ other +squares, each of which might be combined with any of the +$4!$ arrangements deduced from +\vhyperlink{figure:vi}{figure~vi}. Hence altogether +% [*Note: again silently altering factorial notation] +\PG----File: 166.png------------------------------------------------------ +we can obtain in this way $576$ magic squares of the fifth +order\index{DelaHire@De la Hire on Magic Squares|)}% +\index{LaHire@La Hire|)}. + +There is yet another method of constructing odd squares +which is due to Poignard\index{Poignard, Magic Squares}, and was improved by +De la~Hire in the memoir already cited. I shall not discuss it, because, +though for certain assigned values of $n$ it is simpler than the +methods which I have given, it depends on the form of $n$, +and particularly on the number of prime factors of $n$. In the +case of a square of the fifth order, this gives an even larger +number of magic squares than the methods of De la~Loubère, +Bachet, and De la~Hire. I may also add that it has been +shown that magic squares whose order is a prime number can +be constructed by a rule similar to De la~Loubère's, except +that we begin by placing $1$ in the bottom left-hand cell, and +the subsequent consecutive numbers fill cells forming a knight's +path on the square and not a bishop's path. A square of +the fifth order of this kind is given in figure~xiii on page +\pageref{figure:xii}. +There are $2880$ magic squares of the fifth order of this kind. + +De la~Hire\index{DelaHire@De la Hire on Magic Squares}\index +{LaHire@La Hire} showed that, apart from mere inversions, there +were $57600$ magic squares of the fifth order which could be +formed by the methods he enumerated. Taking account of +other methods, it would seem that the total number of magic +squares of the fifth order is very large, perhaps exceeding +$500000$. + +\ssection[Construction of Even Magic Squares]% +[Magic squares of an even order.]{Magic squares of an even order} +The above methods +are inapplicable to squares of an even order. I proceed to +give two methods for constructing any \emph{even magic square} of +an order higher than two. + +It will be convenient to use the following terms. Two +rows which are equidistant, the one from the top, the other +from the bottom, are said to be \emph{complementary}. Two columns +which are equidistant, the one from the left-hand side, the +other from the right-hand side, are said to be \emph{complementary}. +Two cells in the same row, but in complementary columns, are +said to be \emph{horizontally related}. Two cells in the same column, +but in complementary rows, are said to be \emph{vertically related}. +\PG----File: 167.png------------------------------------------------------ +Two cells in complementary rows and columns are said to be +\emph{skewly related}; thus, if the cell $b$ is horizontally related to the +cell $a$, and the cell $d$ is vertically related to the cell $a$, then the +cells $b$ and $d$ are skewly related; in such a case if the cell $c$ +is vertically related to the cell $b$, it will be horizontally related +to the cell $d$, and the cells $a$ and $c$ are skewly related: the +cells $a$, $b$, $c$, $d$ constitute an \emph{associated group}, and if the +square is divided into four equal quarters, one cell of an associated +group is in each quarter. + +A \emph{horizontal interchange} consists in the interchange of the +numbers in two horizontally related cells. A \emph{vertical interchange} +consists in the interchange of the numbers in two +vertically related cells. A \emph{skew interchange} consists in the +interchange of the numbers in two skewly related cells. A +\emph{cross interchange} consists in the change of the numbers in any +cell and in its horizontally related cell with the numbers in +the cells skewly related to them; hence, it is equivalent to two +vertical interchanges and two horizontal interchanges. + +\subsection*{First Method\protect\footnote +{See an article in the \textit{Messenger of Mathematics}, Cambridge, +September, 1893, vol.~\textsc{xxiii}, pp.~65--69. +}}\addcontentsline{toc}{subsection}{First Method} +This method is the simplest with which I +am acquainted, and I believe, at any rate as far as concerns +singly-even squares, was published for the first time in 1893. + +Begin by filling the cells of the square with the numbers +$1, 2, \dotsc, n^2$ in their natural order commencing (say) with the +top left-hand corner, writing the numbers in each row from +left to right, and taking the rows in succession from the top. +I will commence by proving that a certain number of horizontal +and vertical interchanges in such a square must make it magic, +and will then give a rule by which the cells whose numbers +are to be interchanged can be at once picked out. + +First, we may notice that the sum of the numbers in each +diagonal is equal to $N$, where $N = \frac{1}{2}n (n^2 + 1)$; hence the +diagonals are already magic, and will remain so if the numbers +therein are not altered. + +\PG----File: 168.png--------------------------------------------------- +Next, consider the rows. The sum of the numbers in the +$x$th row from the top is $N-\frac{1}{2}n^2(n-2x+1)$. The sum of +the numbers in the complementary row, that is, the $x$th row +from the bottom, is $N + \frac{1}{2}n^2(n-2x + 1)$. Also the number +in any cell in the $x$th row is less than the number in the +cell vertically related to it by $n(n-2x + 1)$. Hence, if in +these two rows we make $\frac{1}{2}n$ interchanges of the numbers +which are situated in vertically related cells, then we +increase the sum of the numbers in the $x$th row by +$\frac{1}{2}n \times n(n-2x + 1)$, and therefore make that row magic; +while we decrease the sum of the numbers in the +complementary row by the same number, and therefore make that +row magic. Hence, if in every pair of complementary rows +we make $\frac{1}{2}n$ interchanges of the numbers situated in vertically +related cells, the square will be made magic in rows. But, +in order that the diagonals may remain magic, either we must +leave both the diagonal numbers in any row unaltered, or we +must change both of them with those in the cells vertically +related to them. + +The square is now magic in diagonals and in rows, and it +remains to make it magic in columns. Taking the original +arrangement of the numbers (in their natural order) we might +have made the square magic in columns in a similar way to +that in which we made it magic in rows. The sum of the +numbers originally in the $y$th column from the left-hand side +is $N-\frac{1}{2}n (n-2y + 1)$. The sum of the numbers originally in +the complementary column, that is, the $y$th column from the +right-hand side, is $N + \frac{1}{2}n(n-2y +1)$. Also the number +originally in any cell in the $y$th column was less than the +number in the cell horizontally related to it by $n-2y+1$. +Hence, if in these two columns we had made $\frac{1}{2}n$ interchanges +of the numbers situated in horizontally related cells, we should +have made the sum of the numbers in each column equal to $N$. +If we had done this in succession for every pair of complementary +columns, we should have made the square magic in +columns. But, as before, in order that the diagonals might +\PG----File: 169.png------------------------------------------------------ +remain magic, either we must have left both the diagonal +numbers in any column unaltered, or we must have changed +both of them with those in the cells horizontally related to +them. + +It remains to show that the vertical and horizontal interchanges, +which have been considered in the last two paragraphs, +can be made independently, that is, that we can make these +interchanges of the numbers in complementary columns in such +a manner as will not affect the numbers already interchanged +in complementary rows. This will require that in every column +there shall be exactly $\frac{1}{2}n$ interchanges of the numbers in +vertically related cells, and that in every row there shall be +exactly $\frac{1}{2}n$ interchanges of the numbers in horizontally related +cells. I proceed to show how we can always ensure this, if $n$ +is greater than $2$. I continue to suppose that the cells are +initially filled with the numbers $1, 2, \dotsc, n^2$ in their natural +order, and that we work from that arrangement. + +A \emph{doubly-even square} is one where $n$ is of the form $4m$. +If the square is divided into four equal quarters, the first +quarter will contain $2m$ columns and $2m$ rows. In each of +these columns take $m$ cells so arranged that there are also +$m$ cells in each row, and change the numbers in these $2m^2$ +cells and the $6m^2$ cells associated with them by a cross interchange. +The result is equivalent to $2m$ interchanges in every +row and in every column, and therefore renders the square +magic. + +One way of selecting the $2m^2$ cells in the first quarter is to +divide the whole square into sixteen subsidiary squares each +containing $m^2$ cells, which we may represent by the diagram +% [*Note: moved diagram down three lines; originally "above" +below, and then we may take either the cells in the $a$ squares +\begin{figure*}[!hbt] +\centering +\begin{MagicSquare}{4} +a&b&b&a\\ +b&a&a&b\\ +b&a&a&b\\ +a&b&b&a +\end{MagicSquare} +\end{figure*} +\PG----File: 170.png---------------------------------------------------- +or those in the $b$ squares; thus, if every number in the eight +$a$ squares is interchanged with the number skewly related to +it the resulting square is magic. A magic square of the eighth +order, constructed in this way, is shown in figure~xv on page +\pageref{figure:xv}. + +Another way of selecting the $2m^2$ cells in the first quarter +would be to take the first $m$ cells in the first column, the cells +$2$ to $m + 1$ in the second column, and so on, the cells $m + 1$ to +$2m$ in the $(m + 1)$th column, the cells $m + 2$ to $2m$ and the +first cell in the $(m + 2)$th column, and so on, and finally the +$2m$th cell and the cells $1$ to $m - 1$ in the $2m$th column. + +A \emph{singly-even square} is one where $n$ is of the form +$2(2m + 1)$. If the square is divided into four equal quarters, +the first quarter will contain $2m + 1$ columns and $2m + 1$ rows. +In each of these columns take $m$ cells so arranged that there +are also $m$ cells in each row: as, for instance, by taking the +first $m$ cells in the first column, the cells $2$ to $m + 1$ in the +second column, and so on, the cells $m + 2$ to $2m + 1$ in the +$(m + 2)$th column, the cells $m + 3$ to $2m + 1$ and the first cell +in the $(m + 3)$th column, and so on, and finally the $(2m + 1)$th +cell and the cells $1$ to $m - 1$ in the $(2m + 1)$th column. Next +change the numbers in these $m (2m + 1)$ cells and the $3m (2m + 1)$ +cells associated with them by cross interchanges. The result +is equivalent to $2m$ interchanges in every row and in every +column. In order to make the square magic we must have +$\frac{1}{2}n$, that is, $2m + 1$ such interchanges in every row and in +every column, that is, we must have one more interchange in +every row and in every column. This presents no difficulty, +for instance, in the arrangement indicated above the numbers +in the $(2m + 1)$th cell of the first column, in the first cell of +the second column, in the second cell of the third column, and +so on, to the $2m$th cell in the $(2m + 1)$th column may be interchanged +with the numbers in their vertically related cells; +this will make all the rows magic. Next, the numbers in the +$2m$th cell of the first column, in the $(2m + 1)$th cell of the +second column, in the first cell of the third column, in the +second cell of the fourth column, and so on, to the $(2m - 1)$th +\PG----File: 171.png---------------------------------------------------- +cell of the $(2m+1)$th column may be interchanged with those +in the cells horizontally related to them; and this will make +the columns magic without affecting the magical properties of +the rows. + +It will be observed that we have implicitly assumed that $m$ +is not zero, that is, that $n$ is greater than 2; also it would seem +that, if $m=1$ and therefore $n=6$, then the numbers in the +diagonal cells must be included in those to which the cross +interchange is applied, but, if $n>6$, this is not necessary, +though it may be convenient. + +The construction of odd magic squares and of doubly-even +magic squares is very easy. But though the rule given above +for singly-even squares is not difficult, it is tedious of application. +It is unfortunate that no more obvious rule---such, for +instance, as one for bordering a doubly-even square---can be +suggested for writing down instantly and without thought +singly-even magic squares. + +\subsection*{De la Hire's Method\protect +\footnote{The rule is due to De la~Hire (part~2 of his memoir) and is given +by Montucla\index{Montucla} in his edition of Ozanam's\index +{Ozanam@Ozanam's \textit{Récréations}} work: I have used the modified +enunciation of it inserted in Labosne's edition of Bachet's\index +{Bachet@Bachet's \textit{Problèmes}} \textit{Problèmes}, as +it saves the introduction of a third subsidiary square. I do not know to +whom the modification is due.}}% +\addcontentsline{toc}{subsection}{Method of De la Hire and Labosne} +I now proceed to give another way due to De la~Hire% +\index{DelaHire@De la Hire on Magic Squares|(}% +\index{LaHire@La Hire|(}% +\index{Labosne on Magic Squares}, of constructing any even magic square of +an order higher than two. + +In the same manner as in his rule for making odd magic +squares, we begin by constructing two subsidiary squares, one +of the unit-digits, $1,2,3,\dotsc,n$, and the other of the radix-digits +$0,n,2n,\dotsc,(n-1)n$. We then form the magic square +by adding together the numbers in the corresponding cells in +the two subsidiary squares. Following the analogy of the +notation used above, two numbers which are equidistant from +the ends of the series $1,2,3,\dotsc,n$ are said to be \emph{complementary}. +Similarly numbers which are equidistant from the +ends of the series $0,n,2n,\dotsc,(n-1)n$ are said to be \emph{complementary}. + +\PG----File: 172.png---------------------------------------------------- +For simplicity I shall apply this method to construct a +magic square of only the sixth order, though an exactly similar +method will apply to any even square of an order higher than +the second. + +The first of the subsidiary squares ({figure~ix}) is constructed +as follows. First, the cells in the leading diagonal are filled with +\begin{figure*}[!hbt] +\centering +\ifPaper\small\unitlength=1.5em\fi +\begin{minipage}{.3\textwidth} +\centering +\begin{MagicSquare}{6} +1 & 5 & 4 & 3 & 2 & 6 \\ +6 & 2 & 4 & 3 & 5 & 1 \\ +6 & 5 & 3 & 4 & 2 & 1 \\ +1 & 5 & 3 & 4 & 2 & 6 \\ +6 & 2 & 3 & 4 & 5 & 1 \\ +1 & 2 & 4 & 3 & 5 & 6 +\end{MagicSquare} +\legend{{\smaller First Subsidiary Square}\break +Figure \Uproman{9}} +\end{minipage} +\hfill +\begin{minipage}{.3\textwidth} +\centering +\begin{MagicSquare}{6} + 0 & {30} & {30} & 0 & {30} & 0 \\ +{24} & 6 & {24} & {24} & 6 & 6 \\ +{18} & {18} & {12} & {12} & {12} & {18} \\ +{12} & {12} & {18} & {18} & {18} & {12} \\ + 6 & {24} & 6 & 6 & {24} & {24} \\ +{30} & 0 & 0 & {30} & 0 & {30} +\end{MagicSquare} +\legend{{\smaller Second Subsidiary Square}\break +Figure \Uproman{10}} +\end{minipage} +\hfill +\begin{minipage}{.3\textwidth} +\centering +\begin{MagicSquare}{6} + 1 & {35} & {34} & 3 & {32} & 6 \\ +{30} & 8 & {28} & {27} & {11} & 7 \\ +{24} & {23} & {15} & {16} & {14} & {19} \\ +{13} & {17} & {21} & {22} & {20} & {18} \\ +{12} & {26} & 9 & {10} & {29} & {25} \\ +{31} & 2 & 4 & {33} & 5 & {36} +\end{MagicSquare} +\legend{{\smaller Resulting Magic Square}\break +Figure \Uproman{11}} +\end{minipage} +\DPlabel{figure:ix} +\end{figure*} +the numbers $1$, $2$, $3$, $4$, $5$, $6$ placed in any order whatever that +puts complementary numbers in complementary positions (\Eg\ in +the order $2$, $6$, $3$, $4$, $1$, $5$, or in their natural order +$1$, $2$, $3$, $4$, $5$, $6$). +Second, the cells vertically related to these are filled respectively +with the same numbers. Third, each of the remaining +cells in the first vertical column is filled either with the same +number as that already in two of them or with the complementary +number (\Eg\ in \vhyperlink{figure:ix}{figure~ix} with a ``1'' or a ``6'') +in any way, provided that there are an equal number of each +of these numbers in the column, and subject also to the +provisoes mentioned in the next paragraph but one. Fourth, +the cells horizontally related to those in the first column +are filled with the complementary numbers. Fifth, the remaining +cells in the second and third columns are filled in +an analogous way to that in which those in the first column +were filled: and then the cells horizontally related to them +are filled with the complementary numbers. The square so +formed is necessarily magic in rows, columns, and diagonals. + +The second of the subsidiary squares (\vhyperlink*{figure:ix}{figure~x}) +is constructed as follows. First, the cells in the leading diagonal +\PG----File: 173.png------------------------------------------------- +are filled with the numbers $0$, $6$, $12$, $18$, $24$, $30$ placed in +any order whatever that puts complementary numbers in +complementary positions. Second, the cells horizontally related +to them are filled respectively with the same numbers. Third, +each of the remaining cells in the first horizontal row is filled +either with the same number as that already in two of them +or with the complementary number (\Eg\ in \vhyperlink{figure:ix}{figure~x} +with a ``0'' or a ``30'') in any way, provided (i)~that there are an +equal number of each of these numbers in the row, and (ii)~that +if any cell in the first row of \vhyperlink{figure:ix}{figure~ix} and its +vertically related cell are filled with complementary numbers, then the +corresponding cell in the first row of \vhyperlink{figure:ix}{figure~x} and +its horizontally related cell must be occupied by the same number\footnote +{The insertion of this step evades the necessity of constructing (as +Montucla\index{Montucla} did) a third subsidiary square.}. +Fourth, the cells vertically related to those in the first row +are filled with the complementary numbers. Fifth, the remaining +cells in the second and the third rows are filled +in an analogous way to that in which those in the first +row were filled: and then the cells vertically related to them +are filled with the complementary numbers. The square so +formed is necessarily magic in rows, columns, and diagonals. + +It remains to show that proviso~(ii) in the third step described +in the last paragraph can be satisfied always. In a +doubly even square, that is, one in which $n$ is divisible by $4$, +we need not have any complementary numbers in vertically +related cells in the first subsidiary square unless we please, but +even if we like to insert them they will not interfere with +the satisfaction of this proviso. In the case of a singly even +square, that is, one in which $n$ is divisible by $2$, but not +by $4$, we cannot satisfy the proviso if any horizontal row +in the first square has all its vertically related squares, other +than the two squares in the diagonals, filled with complementary +numbers. Thus in the case of a singly even square +it will be necessary in constructing the first square to take +care in the third step that in every row at least one cell which +\PG----File: 174.png------------------------------------------------- +is not in a diagonal shall have its vertically related cell filled +with the same number as itself: this is always possible if $n$ is +greater than $2$. + +The required magic square will be constructed if in each +cell we place the sum of the numbers in the corresponding cells +of the subsidiary squares, figures~ix and x. The result of this +is given in \vhyperlink{figure:ix}{figure~xi}. +The square is evidently magic. Also +every number from $1$ to $36$ occurs once and only once, for +the numbers from $1$ to $6$ and from $31$ to $36$ can occur only +in the top or the bottom rows, and the method of construction +ensures that the same number cannot occur twice. Similarly +the numbers from $7$ to $12$ and from $25$ to $30$ occupy two +other rows, and no number can occur twice; and so on. The +square in figure~i on page \pageref{figure:i} may be constructed by the +above rules; and the reader will have no difficulty in applying +them to any other even square% +\index{DelaHire@De la Hire on Magic Squares|)}% +\index{LaHire@La Hire|)}. + +\ssection[Composite Magic Squares][Composite and Bordered Squares.]% +{Other Methods for Constructing any Magic Square} +The above methods appear to me to be the simplest which +have been proposed. There are however \emph{two other methods}\index +{Composite Magic Squares}, of +less generality, to which I will allude briefly in passing. +Both depend on the principle that, if every number in a +magic square is multiplied by some constant, and a constant +is added to the product, the square will remain magic. + +The \emph{first method} applies only to such squares as can be +divided into smaller magic squares of some order higher +than two. It depends on the fact that, if we know how to +construct magic squares of the $m$th and $n$th orders, we can +construct one of the $mn$th order. For example, a square of +$81$ cells may be considered as composed of $9$ smaller squares +each containing $9$ cells, and by filling the cells in each of these +small squares in the same relative order and taking the small +squares themselves in the same order, the square can be constructed +easily. Such squares are called \emph{composite magic +squares}. + +\phantomsection +\addcontentsline{toc}{section}{Bordered Magic Squares} +The \emph{second method}, which was introduced by Frénicle\index +{Frenicle@Frénicle, Magic Squares}, +consists in surrounding a magic square with a \emph{border}\index +{Bordered Magic Squares|(}. Thus +\PG----File: 175.png------------------------------------------------------- +in figure~xii \vpageref{figure:xii} the inner square is magic, and it is +surrounded with a border in such a way that the whole +square is also magic. In this manner from the magic square +of the $3$rd order we can build up successively squares of the +orders $5$, $7$, $9$,~\&c., that is, any odd magic square. Similarly +from the magic square of the $4$th order we can build up +successively any higher even magic square. + +If we construct a magic square of the first $n^2$ numbers by bordering +a magic square of $(n-2)^2$ numbers, the usual process is to +reserve for the $4 (n-1)$ numbers in the border the first $2 (n-1)$ +natural numbers and the last $2(n-1)$ numbers. Now the sum +of the numbers in each line of a square of the order $(n-2)$ is +$\frac{1}{2}(n-2)\{(n-2)^2+1\}$, and the average is +$\frac{1}{2}\{(n-2)^2+1\}$. +Similarly the average number in a square of the $n$th order is +$\frac{1}{2}(n^2+1)$. The difference of these is $2(n-1)$. We begin +then by taking any magic square of the order $(n-2)$, and we +add to every number in it $2(n-1)$; this makes the average +number $\frac{1}{2}(n^2 + 1)$. + +The numbers reserved for the border occur in pairs, $n^2$ and +$1$, $n^2-1$ and $2$, $n^2-2$ and $3$,~\&c., such that the average of each +pair is $\frac{1}{2}(n^2 + 1)$, and they must be bordered on the square +so that these numbers are opposite to one another. Thus the bordered +square will be necessarily magic, provided that the sum of the +numbers in two adjacent sides of the external border is correct. +The arrangement of the numbers in the borders will be +somewhat facilitated if the number $n^2 + 1-p$ (which has to be +placed opposite to the number $p$) is denoted by $\overline{p}$, +but it is not worth while going into further details here. + +It will illustrate sufficiently the general method if I explain +how the square in \vhyperlink{figure:xii}{figure~xii} is constructed. +A magic square of +the third order is formed by De la~Loubère's rule, and to every +number in it $8$ is added: the result is the inner square in +\vhyperlink{figure:xii}{figure~xii}. +The numbers not used are $25$ and $1$, $24$ and $2$, +$23$ and $3$, $22$ and $4$, $21$ and $5$, $20$ and $6$, $19$ and $7$, +$18$ and $8$. The sum of each pair is $26$, and obviously they must be +placed at opposite ends of any line. + +\PG----File: 176.png------------------------------------------------------ +I believe that with a little patience a magic square of any +order can be thus built up, and of course it will have the +property that, if each border is successively stripped off, the +square will still remain magic. Some examples are given by +Violle\index{Violle, Magic Squares}. This is the method of construction +commonly adopted +\begin{figure*}[!hbt] +\centering +\hspace*{\fill} +\begin{minipage}{.4\textwidth} +\centering +\begin{MagicSquare}{5} +1 & 2 & {19} & {20} & {23} \\ +{18} & {16} & 9 & {14} & 8 \\ +{21} & {11} & {13} & {15} & 5 \\ +{22} & {12} & {17} & {10} & 4 \\ +3 & {24} & 7 & 6 & {25} \\ +\put(1,1){\line(0,1){3}} +\put(1,1){\line(1,0){3}} +\put(4,1){\line(0,1){3}} +\put(1,4){\line(1,0){3}} +\end{MagicSquare} +\legend{Bordered Magic Square.\break +Figure \Uproman{12}} +\end{minipage} +\hfill +\begin{minipage}{.4\textwidth} +\centering +\begin{MagicSquare}{5} +7 & {20} & 3 & {11} & {24} \\ +{13} & {21} & 9 & {17} & 5 \\ +{19} & 2 & {15} & {23} & 6 \\ +{25} & 8 & {16} & 4 & {12} \\ +1 & {14} & {22} & {10} & {18} +\end{MagicSquare} +\legend{Nasik Magic Square.\break +Figure \Uproman{13}} +\end{minipage} +\hspace*{\fill} +\DPlabel{figure:xii} +\end{figure*} +by self-taught mathematicians, many of whom seem to think +that the empirical formation of such squares is a valuable +discovery\index{Bordered Magic Squares|)}. + +There are magic circles, rectangles, crosses, diamonds, stars, +and other figures: also magic cubes, cylinders, and spheres. +The theory of the construction of such figures is of no value, +and I cannot spare the space to describe rules for forming +them. + +\section{Hyper-Magic Squares} In recent times attention has been\index +{Hyper-magic Squares|(} +mainly concentrated on the formation of magic squares with +the imposition of additional conditions; some of the resulting +problems involve mathematical difficulties of a high order. + +\subsection[Pan-diagonal or Nasik Squares][Hyper-Magic Squares.]% +{Nasik Squares} In one species of hyper-magic squares\index +{Nasik Squares|(} the +squares are formed so that the sums of the numbers along +all the rows and columns, both diagonals, and all the broken +diagonals are the same. In England these are called \emph{nasik +squares} or \emph{pan-diagonal magic squares}: in France \emph{carrés +diaboliques}\index{Diabolic Squares|(} or \emph{carrés magiquement +magiques}. These squares +were mentioned by De la~Hire, Sauveur\index{Sauveur, Magic Squares}, +and Euler\index{Euler}; but the +theory is mainly due to Mr~A.H.~Frost\index{Frost, A.H.}, who has expounded +it in the memoirs mentioned in the footnote on page~\pageref +{footnote:frost:160}, and +\PG----File: 177.png------------------------------------------------------ +to M.~Frolow\index{Frolow on Magic Squares}, who treated it in two memoirs, +St Petersburg, +1884, and Paris, 1886. Of course a nasik square can be +divided by a vertical or horizontal cut and the pieces interchanged +without affecting the magical property. By one +vertical and one horizontal transposition of this kind any +number can be moved to any specified cell. + +A nasik square of the fourth order is represented in +figure~ii on page \pageref{figure:i}, and one of the fifth order is +represented +in figure~xiii \vpageref{figure:xii}. Nasik squares of the order $6n \pm 1$ +can be constructed by rules analogous to those given by +De la~Loubère\index{DelaLoub@De la Loubère on Magic Squares}, +except that a knight's and not a bishop's move must +be used in connecting cells filled by consecutive numbers and +that for orders higher than five special rules for going from +the cell occupied by the number $kn$ to that occupied by the +number $kn + 1$ have to be laid down\index +{Diabolic Squares|)}\index{Nasik Squares|)}. + +\subsection*{Doubly-Magic Squares}% +\addcontentsline{toc}{subsection}{Doubly Magic Squares} +In another species of hyper-magic\index{Doubly Magic Squares|(} +squares the problem is to construct a magic square of the $n$th +order in such a way that if the number in each cell is replaced +\begin{figure*}[!hbt] +\centering +\begin{MagicSquare}{8} +5 & {31} & {35} & {60} & {57} & {34} & 8 & {30} \\ +{19} & 9 & {53} & {46} & {47} & {56} & {18} & {12} \\ +{16} & {22} & {42} & {39} & {52} & {61} & {27} & 1 \\ +{63} & {37} & {25} & {24} & 3 & {14} & {44} & {50} \\ +{26} & 4 & {64} & {49} & {38} & {43} & {13} & {23} \\ +{41} & {51} & {15} & 2 & {21} & {28} & {62} & {40} \\ +{54} & {48} & {20} & {11} & {10} & {17} & {55} & {45} \\ +{36} & {58} & 6 & {29} & {32} & 7 & {33} & {59} +\end{MagicSquare} +\legend{A Doubly-Magic Square.\break +Figure \Uproman{14}} +\DPlabel{figure:xiv} +\end{figure*} +by its $m$th power the resulting square shall also be magic. +Here for example (see \vhyperlink{figure:xiv}{figure~xiv}) +is a magic square\footnote +{See M.~Coccoz\index{Coccoz} in \textit{L'Illustration}, May~29, 1897.} +of the eighth order, the sum of the numbers in each line being equal +to $260$, so constructed that if the number in each cell is +\PG----File: 178.png------------------------------------------------------ +replaced by its square the resulting square is also magic (the +sum of the numbers in each line being equal to $11180$)% +\index{Doubly Magic Squares|)}% +\index{Hyper-magic Squares|)}. + +\section{Magic Pencils} Hitherto I have concerned myself with% +\index{Magic Pencils@\textsc{Magic Pencils}|(}% +\index{Pencils@\textsc{Pencils, Magic}|(} +numbers arranged in lines. By reciprocating the figures composed +of the points on which the numbers are placed we obtain +a collection of lines forming pencils, and, if these lines be +numbered to correspond with the points, the pencils will be +magic\footnote +{See \textit{Magic-Reciprocals} by G.~Frankenstein\index +{Frankenstein on Magic Pencils}, Cincinnati, 1875.}. +Thus, in a magic square of the $n$th order, we arrange +$n^2$ consecutive numbers to form $2n + 2$ lines, each containing +$n$ numbers so that the sum of the numbers in each line is the +same. Reciprocally we can arrange $n^2$ lines, numbered consecutively +to form $2n + 2$ pencils, each containing $n$ lines, +so that in each pencil the sum of the numbers designating the +lines is the same. + +For instance, \vhyperlink{figure:xv}{figure~xv} represents a magic +square of $64$ +\begin{figure*}[!hbt] +\centering +\begin{MagicSquare}{8} + 1 & 2 & {62} & {61} & {60} & {59} & 7 & 8 \\ + 9 & {10} & {54} & {53} & {52} & {51} & {15} & {16} \\ +{48} & {47} & {19} & {20} & {21} & {22} & {42} & {41} \\ +{40} & {39} & {27} & {28} & {29} & {30} & {34} & {33} \\ +{32} & {31} & {35} & {36} & {37} & {38} & {26} & {25} \\ +{24} & {23} & {43} & {44} & {45} & {46} & {18} & {17} \\ +{49} & {50} & {14} & {13} & {12} & {11} & {55} & {56} \\ +{57} & {58} & 6 & 5 & 4 & 3 & {63} & {64} \\ +\end{MagicSquare} +\legend{Figure \Uproman{15}} +\DPlabel{figure:xv} +\end{figure*} +consecutive numbers arranged to form 18 lines, each of 8 +numbers. Reciprocally, \vhyperlink{figure:xvi}{figure~xvi} represents 64 lines +arranged to form 18 pencils, each of 8 lines. The method of construction +is fairly obvious. The eight-rayed pencil, vertex $O$, is +cut by two parallels perpendicular to the axis of the pencil, +\PG----File: 179.png------------------------------------------------------- +and all the points of intersection are joined cross-wise. This +gives 8 pencils, with vertices $A, B, \dotsc, H$; 8 pencils, with +vertices $A',\ldots H'$; one pencil with its vertex at $O$; and +one pencil with its vertex on the axis of the last-named +pencil. + +\begin{figure*}[!hbt] +\centerline{\includegraphics[height=.7\textheight]{./images/illus179}} +\legend{Figure \Uproman{16}} +\DPlabel{figure:xvi} +\end{figure*} + +\PG----File: 180.png------------------------------------------------------ +The sum of the numbers in each of the 18 lines in +\vhyperlink{figure:xv}{figure~xv} is the same. +To make \vhyperlink{figure:xvi}{figure~xvi} correspond to this we must +number the lines in the pencil $A$ from left to right, $1, 9, \dotsc, 57$, +following the order of the numbers in the first column of the +square: the lines in pencil $B$ must be numbered similarly to +correspond to the numbers in the second column of the square, +and so on. To prevent confusion in the figure I have not +inserted the numbers, but it will be seen that the method of +construction ensures that the sum of the 8 numbers which +designate the lines in each of these 18 pencils is the same. + +We can proceed a step further, if the resulting figure is +cut by two other parallel lines perpendicular to the axis, and +if the points of their intersection with the cross-joins be +joined cross-wise, these new cross-joins will intersect on the +axis of the original pencil or on lines perpendicular to it. +The whole figure will now give $8^3$ lines, arranged in $244$ pencils +each of $8$ rays, and will be the reciprocal of a magic cube of +the $8$th order. If we reciprocate back again we obtain a +representation in a plane of a magic cube% +\index{Magic Pencils@\textsc{Magic Pencils}|)}% +\index{Pencils@\textsc{Pencils, Magic}|)}. + +\section[Magic Puzzles][Magic Puzzles.]{Magic Square Puzzles} +Many empirical problems, closely +related to magic squares, will suggest themselves; but most of\index +{Magic Square Puzzles|(} +them are more correctly described as ingenious puzzles than +as mathematical recreations. The following will serve as +specimens. + +\subsection*{Magic Card Square\protect\footnote +{Ozanam\index{Ozanam@Ozanam's \textit{Récréations}}, +1723 edition, vol.~\textsc{iv}, p.~434.}}% +\addcontentsline{toc}{subsection}{Card Square} +The first of these is the familiar +problem of placing the sixteen court cards (taken out of a +pack) in the form of a square so that no row, no column, and +neither of the diagonals shall contain more than one card of +each suit and one card of each rank. The solution presents no +difficulty, and is indicated in figure~xviii \vpageref[below]{figure:xviii}. + +\subsection*{Euler's Officers Problem\protect\footnote +{Euler's \textit{Commentationes Arithmeticae}, St~Petersburg, 1849, +vol~\textsc{ii}, +pp.~302--361. See also a paper by G.~Tarry\index{Tarry} in the \textit +{Comptes rendus} +of the French Association for the Advancement of Science, Paris, 1900, +vol.~\textsc{ii}, pp.~170--203; and various notes in \textit +{L'Intermédiaire des mathématiciens}, +Paris, vol.~\textsc{iii}, 1896, pp.~17, 90; vol.~\textsc{v}, 1898, +pp.~83, 176, 252, % last two digits illegible in scan; these taken from TIA scan of 7th edition +vol.~\textsc{vi}, 1899, p.~251; vol.~\textsc{vii}, 1900, pp.~14, 311.}}% +\addcontentsline{toc}{subsection}{Euler's Officers Problem} +A similar problem, proposed +\PG----File: 181.png------------------------------------------------------ +by Euler\index{Euler|(} in 1779, consists in arranging, if it be possible, +thirty-six officers taken from six regiments---the officers being +in six groups, each consisting of six officers of equal rank, one +drawn from each regiment; say officers of rank $a$, $b$, $c$, $d$, $e$, $f$, +drawn from the $1$st, $2$nd, $3$rd, $4$th, $5$th, and $6$th regiments---in +a solid square formation of six by six, so that each row and +each file shall contain one and only one officer of each rank +and one and only one officer from each regiment. The problem +is insoluble. + +\subsection*{Extension of Euler's Problem} +More generally\index{Euler|)} we may +investigate the arrangement on a chess-board, containing $n^2$ +cells, of $n^2$ counters (the counters being divided into $n$ groups, +each group consisting of $n$ counters of the same colour +numbered consecutively $1, 2, \dotsc, n$) so that each row and each +% [*Note: silently added commas] +column shall contain no two counters of the same colour or +marked with the same number. + +For instance, if $n=3$, with three red counters $a_1$, $a_2$, $a_3$, +three white counters $b_1,$ $b_2$, $b_3$, and three black counters +$c_1$, $c_2$, $c_3$, +we can satisfy the conditions by arranging them as in figure~xvii +\vpageref[below]{figure:xviii}. +If $n = 4$, then with counters $a_1$, $a_2$, $a_3$, $a_4$; $b_1$, $b_2$, +$b_3$, $b_4$; $c_1$, $c_2$, $c_3$, $c_4$; $d_1$, $d_2$, $d_3$, $d_4$, +we can arrange them as in +figure~xviii \vpageref[below]{figure:xviii}. A solution when $n = 5$ +is indicated in figure~xix. + +\begin{figure*}[!hbt] +\centering +\ifPaper\else\hspace*{\fill}\fi +\begin{minipage}[b]{5.5em} +\centering +\begin{MagicSquare}{3} +{a_1} & {b_2} & {c_3} \\ +{b_2} & {c_1} & {a_2} \\ +{c_3} & {a_3} & {b_1} +\end{MagicSquare} +\legend{Figure \Uproman{17}} +\end{minipage} +\hfill +\begin{minipage}[b]{7em} +\centering +\begin{MagicSquare}{4} +{a_1} & {b_2} & {c_3} & {d_4} \\ +{c_4} & {d_3} & {a_2} & {b_1} \\ +{d_2} & {c_1} & {b_4} & {a_3} \\ +{b_3} & {a_4} & {d_1} & {c_2} +\end{MagicSquare} +\legend{Figure \Uproman{18}} +\end{minipage} +\hfill +\begin{minipage}[b]{8.5em} +\centering +\begin{MagicSquare}{5} +{a_1} & {b_2} & {c_3} & {d_4} & {e_5} \\ +{b_5} & {c_1} & {d_2} & {e_3} & {a_4} \\ +{c_4} & {d_5} & {e_1} & {a_2} & {b_3} \\ +{d_3} & {e_4} & {a_5} & {b_1} & {c_2} \\ +{e_2} & {a_3} & {b_4} & {c_5} & {d_1} +\end{MagicSquare} +\legend{Figure \Uproman{19}} +\end{minipage} +\ifPaper\else\hspace*{\fill}\fi +\label{figure:xviii} +\end{figure*} + +The problem is soluble if $n$ is odd or if $n$ is of the form +$4m$. If solutions when $n=a$ and when $n=b$ are known, a +\PG----File: 182.png---------------------------------------------------- +solution when $n=ab$ can be written down at once. The +theory is closely connected with that of magic squares and +need not be here discussed further. + +\subsection*{Magic Domino Squares}% +\addcontentsline{toc}{subsection}{Domino Squares} +Analogous problems can be +made with dominoes\index{Dominoes}. An ordinary set of dominoes, ranging +from double zero to double six, contains $28$ dominoes. Each +domino is a rectangle formed by fixing two small square +blocks together side by side: of these $56$ blocks, eight are +\begin{figure*}[!hbt] +\centering +\includegraphics[height=\ifPaper.8\textwidth\else.6\textheight\fi]{./images/illus182} +\legend{Magic Domino Square.} +\label{illus:182} +\end{figure*} +blank, on each of eight of them is one pip, on each of another +eight of them are two pips, and so on. It is required to +arrange the dominoes so that the $56$ blocks form a square of +$7$ by $7$ bordered by one line of $7$ blank squares and so that +the sum of the pips in each row, each column, and in the two +diagonals of the square is equal to $24$. A solution\footnote +{See \textit{L'Illustration}, July~10, 1897.} is given +\vpageref[above]{illus:182}. + +Similarly, a set of dominoes, ranging from double zero +to double $n$, contains $\frac{1}{2}(n+1)(n+2)$ dominoes and therefore +\PG----File: 183.png---------------------------------------------------- +$(n+1)(n+2)$ blocks. Can these dominoes be arranged in the +form of a square of $(n+1)^2$ cells, bordered by a row of blanks, +so that the sum of the pips in each row, each column, and in +the two diagonals of the square is equal to $\frac{1}{2}n(n+2)$? + +\subsection*{Magic Coin Squares\protect\footnote +{See \textit{The Strand Magazine}, London, December, 1896, pp.~720, 721. +}}\addcontentsline{toc}{subsection}{Coin Squares} +There are somewhat similar questions +concerned with coins. Here is one applicable to a +square of the third order divided into nine cells, as in figure~xvii +\ifPaper\vpageref[above]{figure:xviii}\else\vpageref{figure:xviii}\fi. % [*Note: originally "above"] +% In print mode, vpageref gets into an infinite loop here (there is always a "labels have changed warning") +If a five-shilling piece is placed in the middle +cell $c_1$ and a florin in the cell below it, namely, in $a_3$ it is +required to place the fewest possible current English coins in +the remaining seven cells so that in each cell there is at least +one coin, so that the total value of the coins in every cell is +different, and so that the sum of the values of the coins in +each row, column, and diagonal is fifteen shillings: it will be +found that thirteen additional coins will suffice. A similar +problem is to place ten current English postage stamps, all but +two being different, in the nine cells so that the sum of the +values of the stamps in each row, column, and diagonal is +ninepence.\index{Magic Square Puzzles|)} + + +\PG----File: 184.png---------------------------------------------------- +%CHAPTER VI. + +\chapter{Unicursal Problems.} + +\textsc{I propose} to consider in this chapter some problems which\chapindex +{Unicursal Problems@\textsc{Unicursal Problems}} +arise out of the theory of unicursal curves. I shall commence +with \emph{Euler's Problem and Theorems}, and shall apply the +results briefly to the theories of \emph{Mazes} and \emph{Geometrical Trees}. +The reciprocal unicursal problems of the \emph{Hamilton Game} and +the \emph{Knight's Path on a Chess-board} will be discussed in the +latter half of the chapter. + +\section{Euler's Problem} Euler's problem% +\index{Euler'sUni@\textsc{Euler's Unicursal Problem}|(}% +\index{Konigsberg@Königsberg Problem|(} +has its origin in a memoir\footnote +{\textit{Solutio problematis ad Geometriam situs pertinentis}, +\textit{Commentarii +Academiae Scientiarum Petropolitanae} for 1736, St~Petersburg, 1741, +vol.~\textsc{viii}, pp.~128--140. This has been translated into French +by M.~Ch.~Henry\index{Henry on Unicursal Problems}; +see Lucas\index{Lucas, E.}, vol.~\textsc{i}, part~2, pp.~21--33. +} presented by him in 1736 to the St~Petersburg +Academy, in which he solved a question then under discussion +as to whether it was possible to take a walk in the town of +Königsberg in such a way as to cross every bridge in it once +and only once. + +The town is built near the mouth of the river Pregel, +which there takes the form indicated \vpageref[below]{illus:185a} +and includes the +%[*Note: Illustration moved from 185.png to improve screen version] +\begin{figure*}[!hbt] +\centerline{\includegraphics[width=.8\textwidth]{./images/illus185a}} +\label{illus:185a} +\end{figure*} +island of Kneiphof. In 1759 there were (and according to +Baedeker there are still) seven bridges in the positions shown +in the diagram, and it is easily seen that with such an +arrangement the problem is insoluble. Euler however did not +\PG----File: 185.png---------------------------------------------------- +confine himself to the case of Königsberg, but discussed the +general problem of any number of islands connected in any +way by bridges. It is evident that the question will not be +affected if we suppose the islands to diminish to points and +the bridges to lengthen out. In this way we ultimately obtain +a geometrical figure or network. In the Königsberg problem +this figure is of the shape indicated \vpageref[below]{illus:185b}, +the areas being +represented by the points $A$, $B$, $C$, $D$, and the bridges being +represented by the lines $l$, $m$, $n$, $p$, $q$, $r$, $s$. + +\begin{figure*}[!hbt] +\centerline{\includegraphics[width=.6\textwidth]{./images/illus185b}} +\label{illus:185b} +\end{figure*} + +Euler's problem consists therefore in finding whether a +given geometrical figure can be described by a point moving +so as to traverse every line in it once and only once. A more +general question is to determine how many strokes are necessary +to describe such a figure so that no line is traversed +twice: this is covered by the rules hereafter given. The +figure may be either in three or in two dimensions, and +it may be represented by lines, straight, curved, or tortuous, +\PG----File: 186.png---------------------------------------------------- +joining a number of given points, or a model may be +constructed by taking a number of rods or pieces of string +furnished at each end with a hook so as to allow of any +number of them being connected together at one point. + +The theory of such figures is included as a particular case +in the propositions proved by Listing\index +{Listing@Listing's \textit{Topologie}} in his \textit{Topologie}\footnote +{\textit{Die Studien}, Göttingen, 1847, part~x. See also Tait\index +{Tait} on \textit{Listing's Topologie}, +\textit{Philosophical Magazine}, London, January, 1884, series~5, +vol.~\textsc{xvii}, pp.~30--46.}. I +shall, however, adopt here the methods of Euler, and I shall +begin by giving some definitions, as it will enable me to put +the argument in a more concise form. + +\phantomsection +\addcontentsline{toc}{subsection}{Definitions} +A \emph{node} (or isle) is a point to or from which lines are +drawn. A \emph{branch} (or bridge or path) is a line connecting +two consecutive nodes. An \emph{end} (or hook) is the point at +each termination of a branch. The \emph{order} of a node is the +number of branches which meet at it. A node to which only +one branch is drawn is a \emph{free} node or a free end. A node +at which an even number of branches meet is an \emph{even} node: +evidently the presence of a node of the second order is immaterial. +A node at which an odd number of branches meet +is an \emph{odd} node. A figure is closed if it has no free end: +such a figure is often called a closed network. + +A \emph{route} consists of a number of branches taken in consecutive +order and so that no branch is traversed twice. A +closed % [*Note: silently correcting obvious typo "close"] +route terminates at the point from which it started. +A figure is described \emph{unicursally} when the whole of it is +traversed in one route. + +\phantomsection +\addcontentsline{toc}{subsection}{Euler's Theorems} +The following are Euler's results. (i)~In a closed network +the number of odd nodes is even. (ii)~A figure which +has no odd node can be described unicursally, in a re-entrant +route, by a moving point which starts from any point on it. +(iii)~A figure which has two and only two odd notes can be +described unicursally by a moving point which starts from one +of the odd nodes and finishes at the other. (iv)~A figure +which has more than two odd nodes cannot be described +\PG----File: 187.png------------------------------------------------------ +completely in one route; to which Listing\index +{Listing@Listing's \textit{Topologie}} added the corollary +that a figure which has $2n$ odd nodes, and no more, can +be described completely in $n$ separate routes. I now proceed +to prove these theorems. + +\subsection*{First} \emph{The number of odd nodes in a closed network is +even.} + +Suppose the number of branches to be $b$. Therefore the +number of hooks is $2b$. Let $k_n$ be the number of nodes of +the $n$th order. Since a node of the $n$th order is one at +which $n$ branches meet, there are $n$ hooks there. Also since +the figure is closed, $n$ cannot be less than $2$. +\[\def\tabcolsep{0pt} +\begin{tabularx}{\textwidth}{XrrrlX} +& $\Therefore 2k_2 +{}$& $3k_3 +{}$& $4k_4 + \dotsb $&${} + nk_n ++ \dotsb = 2b\,.$& \\ + Hence & & $ 3k_3 +{}$&$ 5k_5 + \dotsb $& \quad is even. \\ +& $\Therefore$ & $ k_3+{}$&$ k_5 + \dotsb $& \quad is even. +\end{tabularx} +\] + +\subsection*{Second} \emph{A figure which has no odd node can be described +unicursally in a re-entrant route.} + +Since the route is to be re-entrant it will make no difference +where it commences. Suppose that we start from a node +$A$. Every time our route takes us through a node we use +up one hook in entering it and one in leaving it. There +are no odd nodes, therefore the number of hooks at every +node is even: hence, if we reach any node except $A$, we +shall always find a hook which will take us into a branch +previously untraversed. Hence the route will take us finally +to the node $A$ from which we started. If there are more than +two hooks at $A$, we can continue the route over one of the +branches from $A$ previously untraversed, but in the same way +as before we shall finally come back to $A$. + +It remains to show that we can arrange our route so as +to make it cover all the branches. Suppose each branch of +the network to be represented by a string with a hook at each +end, and that at each node all the hooks there are fastened +together. The number of hooks at each node is even, and if +\PG----File: 188.png---------------------------------------------------- +they are unfastened they can be re-coupled together in pairs, +the arrangement of the pairs being immaterial. The whole +network will then form one or more closed curves, since now +each node consists merely of two ends hooked together. + +If this random coupling gives us one single curve then the +proposition is proved; for starting at any point we shall go +along every branch and come back to the initial point. But if +this random coupling produces anywhere an isolated loop, $L$, +then where it touches some other loop, $M$, say at the node $P$, +unfasten the four hooks there (viz.\ two of the loop $L$ and two +of the loop $M$) and re-couple them in any other order: then +the loop $L$ will become a part of the loop $M$. In this way, by +altering the couplings, we can transform gradually all the +separate loops into parts of only one loop. + +For example, take the case of three isles, $A$, $B$, $C$, each +connected with both the others by two bridges. The most +unfavourable way of re-coupling the ends at $A$, $B$, $C$ would be +to make $ABA$, $ACA$, and $BCB$ separate loops. The loops +$ABA$ and $ACA$ are separate and touch at $A$; hence we should +re-couple the hooks at $A$ so as to combine $ABA$ and $ACA$ into +\begin{figure*}[!hbt] +\centerline{\includegraphics[width=7cm]{./images/illus188}} +\end{figure*} +one loop $ABACA$. Similarly, by re-arranging the couplings of +the four hooks at $B$, we can combine the loop $BCB$ with +$ABACA$ and thus make only one loop. + +I infer from Euler's language that he had attempted to +solve the problem of giving a practical rule which would enable +one to describe such a figure unicursally without knowledge of +\PG----File: 189.png---------------------------------------------------- +its form, but that in this he was unsuccessful. He however +added that any geometrical figure can be described completely +in a single route provided each part of it is described twice and +only twice, for, if we suppose that every branch is duplicated, +there will be no odd nodes and the figure is unicursal. In +this case any figure can be described completely without knowing +its form: rules to effect this are given below. + +\subsection*{Third} \emph{A figure which has two and only two odd nodes can +be described unicursally by a point which starts from one of +the odd nodes and finishes at the other odd node.} + +This at once reduces to the second theorem. Let $A$ and $Z$ +be the two odd nodes. First, suppose that $Z$ is not a free +end. We can, of course, take a route from $A$ to $Z$; if we +imagine the branches in this route to be eliminated, it will +remove one hook from $A$ and make it even, will remove two +hooks from every node intermediate between $A$ and $Z$ and +therefore leave each of them even, and will remove one hook +from $Z$ and therefore will make it even. All the remaining +network is now even: hence, by Euler's second proposition, +it can be described unicursally, and, if the route begins at +$Z$, it will end at $Z$. Hence, if these two routes are taken +in succession, the whole figure will be described unicursally, +beginning at $A$ and ending at $Z$. Second, if $Z$ is a free +end, then we must travel from $Z$ to some node, $Y$, at which +more than two branches meet. Then a route from $A$ to $Y$ +which covers the whole figure exclusive of the path from $Y$ +to $Z$ can be determined as before and must be finished by +travelling from $Y$ to $Z$. + +\subsection*{Fourth} \emph{A figure having $2n$ odd nodes, and no more, can +be described completely in $n$ separate routes.} + +If any route starts at an odd node, and if it is continued +until it reaches a node where no fresh path is open to it, this +latter node must be an odd one. For every time we enter +an even node there is necessarily a way out of it; and similarly +every time we go through an odd node we use up one hook in +entering and one hook in leaving, but whenever we reach it as +\PG----File: 190.png---------------------------------------------------- +the end of our route we use only one hook. If this route is +suppressed there will remain a figure with $2n-2$ odd nodes. +Hence $n$ such routes will leave one or more networks with +only even nodes. But each of these must have some node +common to one of the routes already taken and therefore can +be described as a part of that route. Hence the complete +passage will require n and not more than $n$ routes. It follows, +as stated by Euler, that, if there are more than two odd +nodes, the figure cannot be traversed completely in one +route. + +\phantomsection +\addcontentsline{toc}{subsection}{Examples} +The Königsberg bridges lead to a network with four odd +nodes; hence, by Euler's fourth proposition, it cannot be +described unicursally in a single journey, though it can be +traversed completely in two separate routes. + +The first and second diagrams figured \vpageref[below]{illus:190} contain +only even nodes, and therefore each of them can be described unicursally. +\begin{figure*}[!hbt] +\centerline{\ifpdf\includegraphics[width=\ifPaper\else.7\fi\textwidth,viewport=0 0 200 70]{./images/illus190.pdf} % size graphic using BoundingBox +\else\includegraphics[width=\ifPaper\else.7\fi\textwidth]{./images/illus190.eps}\fi} +\label{illus:190} +\end{figure*} +The first of these---a re-entrant pentagon---was one +of the Pythagorean symbols\index{Pythagorean Symbol}. +The other is the so-called sign-manual of Mohammed\index +{Mohammed's sign-manual}, said to have been originally traced +in the sand by the point of his scimetar without taking the +scimetar off the ground or retracing any part of the figure---which, +as it contains only even nodes, is possible. The third +diagram is taken from Tait's\index{Tait} article: it contains only two +odd nodes, and therefore can be described unicursally if we +start from one of them and finish at the other. + +As other examples I may note that the geometrical figure +\PG----File: 191.png---------------------------------------------------- +formed by taking a $(2n+1)$gon and joining every angular +point with every other angular point is unicursal. On the +other hand a chess-board, divided as usual by straight lines +into $64$ cells, has $28$ odd nodes and $53$ even nodes: hence it +would require $14$ separate pen-strokes to trace out all the +boundaries without going over any more than once. Again, +the diagram on page \pageref{illus:136} has $20$ odd nodes and therefore +would require $10$ separate pen-strokes to trace it out. + +It is well known that a curve which has as many nodes as +is consistent with its degree is unicursal. + +\section[Mazes][Mazes and Labyrinths.]{Mazes} +Everyone has read of the labyrinth\index +{Labyrinths|(} of Minos\index{Minos} in +Crete and of Rosamund's Bower. A few modern mazes\index{Mazes|(} exist +here and there---notably one, which is a very poor specimen +of its kind, at Hampton Court\index{Hampton Court, Maze at}---and +in one of these, or at any +rate on a drawing of one, most of us have threaded our way +to the interior. I proceed now to consider the manner in +which any such construction may be completely traversed even +by one who is ignorant of its plan. + +The theory of the description of mazes is included in +Euler's theorems given above. The paths in the maze are +what previously we have termed branches, and the places +where two or more paths meet are nodes. The entrance to +the maze, the end of a blind alley, and the centre of the maze +are free ends and therefore odd nodes. + +If the only odd nodes are the entrance to the maze and the +centre of it--which will necessitate the absence of all blind +alleys--the maze can be described unicursally. This follows +from Euler's third proposition. Again, no matter how many +odd nodes there may be in a maze, we can always find a +route which will take us from the entrance to the centre +without retracing our steps, though such a route will take us +through only a part of the maze. But in neither of the cases +mentioned in this paragraph can the route be determined +without a plan of the maze. + +A plan is not necessary, however, if we make use of +\PG----File: 192.png---------------------------------------------------- +Euler's suggestion, and suppose that every path in the maze +is duplicated. In this case we can give definite rules for the +complete description of the whole of any maze, even if we are +entirely ignorant of its plan. Of course to walk twice over +every path in a labyrinth is not the shortest way of arriving +at the centre, but, if it is performed correctly, the whole maze +is traversed, the arrival at the centre at some point in the +course of the route is certain, and it is impossible to lose one's +way\index{Euler'sUni@\textsc{Euler's Unicursal Problem}|)}% +\index{Konigsberg@Königsberg Problem|)}. + +\phantomsection +\addcontentsline{toc}{subsection}{Rules for completely traversing a Maze} +I need hardly explain why the complete description of such +a duplicated maze is possible, for now every node is even, +and hence, by Euler's second proposition, if we begin at the +entrance we can traverse the whole maze; in so doing we +shall at some point arrive at the centre, and finally shall +emerge at the point from which we started. This description +will require us to go over every path in the maze twice, and as +a matter of fact the two passages along any path will be always +made in opposite directions. + +If a maze is traced on paper, the way to the centre is +generally obvious, but in an actual labyrinth it is not so easy +to find the correct route unless the plan is known. In order +to make sure of describing a maze without knowing its plan it +is necessary to have some means of marking the paths which +we traverse and the direction in which we have traversed them---for +example, by drawing an arrow at the entrance and end of +every path traversed, or better perhaps by marking the wall on +the right-hand side, in which case a path may not be entered +when there is a mark on each side of it. If we can do this, +and if when a node is reached, we take, if it be possible, some +path not previously used, or, if no other path is available, we +enter on a path already traversed once only, we shall completely +traverse any maze in two dimensions\footnote +{See \textit{Le problème des labyrinthes} by G.~Tarry\index{Tarry}, +\textit{Nouvelles Annales de +math\-é\-mat\-iques}, May, 1895, series~3, vol.~\textsc{xiv}. +}. Of course a +path must not be traversed twice in the same direction, a +\PG----File: 193.png---------------------------------------------------- +path already traversed twice (namely, once in each direction) +must not be entered, and at the end of a blind alley it is +necessary to turn back along the path by which it was +reached. + +\phantomsection +\addcontentsline{toc}{subsection}{Notes on the History of Mazes} +I think most people would understand by a maze a series of +interlacing paths through which some route can be obtained +leading to a space or building at the centre of the maze. I +believe that few, if any, mazes of this type existed in classical +or medieval times. + +One class of what the ancients called mazes or labyrinths +seems to have comprised any complicated building with numerous +vaults and passages\footnote +{For instance, see the descriptions of the labyrinth at Lake Moeris +given by Herodotus\index{Herodotus on Lake Moeris}, bk.~ii, c.~148; +Strabo\index{Strabo on Lake Moeris}, bk.~xvii, c.~1, art.~37; +Diodorus\index{Diodorus on Lake Moeris}, bk.~i, cc.~61, 66; and +Pliny\index{Pliny}, \textit{Hist. Nat.}, bk.~xxxvi, c.~13, arts.~84--89. +On these and other references see A.~Wiedemann\index +{Wiedemann on Lake Moeris}, \textit{Herodots +zweites Buch}, Leipzig, 1890, p.~522 \etseq\ See also Virgil\index{Virgil}, +\textit{Aeneid}, bk.~v, +c.~v, 588; Ovid\index{Ovid}, \textit{Met.}, bk.~viii, c.~5, 159; % silently correcting viii. to viii, +Strabo, bk.~viii, c.~6.}. Such a building might be termed a +labyrinth, but it is not what is usually understood by the word. +The above rules would enable anyone to traverse the whole of +any structure of this kind. I do not know if there are any +accounts or descriptions of Rosamund's Bower\index{Rosamund's Bower} +other than those +by Drayton\index{Drayton}, Bromton\index{Bromton}, and Knyghton\index +{Knyghton}: in the opinion of some, +these imply that the bower was merely a house, the passages +in which were confusing and ill-arranged. + +Another class of ancient mazes consisted of a tortuous path +confined to a small area of ground and leading to a place or +shrine in the centre\footnote +{On ancient and medieval labyrinths---particularly of this kind---see +an article by Mr~E.~Trollope\index{Trollope on Mazes} in \textit +{The Archaeological Journal}, 1858, vol.~\textsc{xv}, +pp.~216--235, from which much of the historical information given above +is derived}. This is a maze in which there is no +chance of taking a wrong turning; but, as the whole area +can be occupied by the windings of one path, the distance +to be traversed from the entrance to the centre may be +considerable, even though the piece of ground covered by the +maze is but small. + +\PG----File: 194.png---------------------------------------------------- +The traditional form of the labyrinth\index +{Cretan Labyrinth}\index{Daedalus, Labyrinth of} constructed for +the Minotaur\index{Minotaur} is a specimen of this class. It was delineated +on the reverses of the coins of Cnossus\index{Cnossus, Coins of}, +specimens of which +are not uncommon; one form of it is indicated in the +\vpageref[accompanying diagram][diagram ]{illus:194} +(figure~i). The design really is the +same as that drawn in figure~ii, as can be easily seen by +bending round a circle the rectangular figure there given. + +Mr~Inwards\index{Inwards on the Cretan Maze} has suggested\footnote +{\textit{Knowledge}, London, October, 1892.} that this design on the coins +of Cnossus may be a survival from that on a token given by +\begin{figure*}[!hbt] +\centering +\begin{minipage}{.3\textwidth} +\centerline{\includegraphics[width=\textwidth]{./images/illus194a}} +\legend{Figure \Uproman{1}} +\end{minipage} +\hfill +\begin{minipage}{.6\textwidth} +\centerline{\includegraphics[width=\textwidth]{./images/illus194b}} +\legend{Figure \Uproman{2}} +\end{minipage} +\label{illus:194} +\end{figure*} +the priests as a clue to the right path in the labyrinth there. +Taking the circular form of the design shown above he supposed +each circular wall to be replaced by two equidistant +walls separated by a path, and thus obtained a maze to which +the original design would serve as the key. The route thus +indicated may be at once obtained by noticing that when a +node is reached (\IE\ a point where there is a choice of paths) +the path to be taken is that which is next but one to that by +which the node was approached. This maze may be also +threaded by the simple rule of always following the wall on +the right-hand side or always that on the left-hand side. +The labyrinth may be somewhat improved by erecting a few +additional barriers, without affecting the applicability of the +above rules, but it cannot be made really difficult. This +makes a pretty toy, but though the conjecture on which it is +founded is ingenious it must be regarded as exceedingly +\PG----File: 195.png--------------------------------------------------- +improbable. Another suggestion is that the curved line on +the reverse of the coins indicated the form of the rope held +by those taking part in some rhythmic dance; while others +consider that the form was gradually evolved from the widely +prevalent svastika\index{Svastika}. + +Copies of the maze of Cnossus were frequently engraved +on Greek and Roman gems; similar but more elaborate +designs are found in numerous Roman mosaic pavements\index +{Mosaic Pavements}\footnote +{See \Eg\ Breton's\index{Breton on Mosaics} \textit{Pompeia}, p.~303.}. +A copy of the Cretan labyrinth\index{Cretan Labyrinth} was embroidered +on many of +the state robes of the later Emperors, and, apparently thence, +was copied on to the walls and floors of various churches\footnote +{Ozanam\index{Ozanam, A.F., on Labyrinths}, \textit +{Graphia aureae urbis Romae}, pp.~92, 178.}. +At a later time in Italy and in France these mural and pavement +decorations were developed into scrolls of great complexity, +but consisting, as far as I know, always of a single +line. Some of the best specimens now extant are on the walls +of the cathedrals at Lucca\index{Lucca, Labyrinth at}, Aix\index +{Aix, Labyrinth at} in Provence, and Poitiers\index{Poitiers, Labyrinth at}; +and on the floors of the churches of Santa Maria in Trastevere\index +{Trastevere, Labyrinth at} +at Rome\index{Rome, Labyrinth at}, San~Vitale at Ravenna\index +{Ravenna, Labyrinth at}, Notre Dame at St~Omer\index +{StOmer@St Omer, Labyrinth at}, +and the cathedral at Chartres\index{Chartres, Labyrinth at}. +It is possible that they were +used to represent the journey through life as a kind of pilgrim's +progress. + +In England these mazes were usually, perhaps always, cut +in the turf adjacent to some religious house or hermitage: and +there are some slight reasons for thinking that, when traversed +as a religious exercise, a \emph{pater} or \emph{ave} had to be repeated at +every turning. After the Renaissance, such labyrinths were +frequently termed Troy-towns\index{Troy-towns} or Julian's bowers\index +{Julian's Bowers}. Some +of the best specimens, which are still extant, are those +at Rockliff Marshes\index{Rockliff Marshes, Labyrinth at}, Cumberland; +Asenby\index{Asenby, Labyrinth at}, Yorkshire; +Alkborough\index{Alkborough, Labyrinth at}, Lincolnshire; +Wing\index{Wing, Labyrinth at}, Rutlandshire; +Boughton-Green\index{Boughton Green, Labyrinth at}, Northamptonshire; +Comberton\index{Comberton, Labyrinth at}, Cambridgeshire; +Saffron Walden\index{Saffron Walden, Labyrinth at}, Essex; +and Chilcombe\index{Chilcombe, Labyrinth at}, near Winchester. + +The modern maze seems to have been introduced---probably +from Italy---during the Renaissance, and many of the +\PG----File: 196.png----------------------------------------------------- +palaces and large houses built in England during the Tudor +and the Stuart periods had labyrinths attached to them. +Those adjoining the royal palaces at Southwark\index +{Southwark, Labyrinth at}, Greenwich\index{Greenwich, Labyrinth at}, +and Hampton Court\index{Hampton Court, Maze at} were particularly well +known from their +vicinity to the capital. The last of these was designed by +London and Wise\index{London and Wise} in 1690, for William~III\index +{William III of England}, who had a fancy +for such conceits: a plan of it is given in various guide-books. +\begin{figure*}[!hbt] +\centerline{\includegraphics[width=\ifPaper.8\else.6\fi\textwidth]{./images/illus196}} +\legend{\textsc{Maze at Hampton Court.}} +\end{figure*} +For the majority of the sight-seers who enter, it is sufficiently +elaborate; but it is an indifferent construction, for it can be +described completely by always following the hedge on one +side (either the right hand or the left hand), and no node is +of an order higher than three. + +Unless at some point the route to the centre forks and +subsequently the two forks reunite, forming a loop in which +the centre of the maze is situated, the centre can be reached +by the rule just given, namely, by following the wall on one +side---either on the right hand or on the left hand. No +labyrinth is worthy of the name of a puzzle which can be +threaded in this way. Assuming that the path forks as +described above, the more numerous the nodes and the higher +their order the more difficult will be the maze, and the +difficulty might be increased considerably by using bridges and +tunnels so as to construct a labyrinth in three dimensions. +In an ordinary garden and on a small piece of ground, often +\PG----File: 197.png------------------------------------------------------ +of an inconvenient shape, it is not easy to make a maze which +fulfils these conditions. +\vpageref[Here][Here ]{illus:197} is a plan of one which I put up +\begin{figure*}[!hbt] +\centerline{\includegraphics +[height=\ifPaper.6\textwidth\else.7\textheight\fi]{./images/illus197}} +\label{illus:197} +\end{figure*} +in my own garden on a plot of ground which would not allow +of more than $36$ by $23$ paths, but it will be noticed that none +of the nodes are of a high order\index{Labyrinths|)}\index{Mazes|)}. + +\section{Geometrical Trees} Euler's original investigations\index +{Trees@\textsc{Trees, Geometrical}|(} were +confined to a closed network. In the problem of the maze it +was assumed that there might be any number of blind alleys +in it, the ends of which formed free nodes. We may now +progress one step farther, and suppose that the network or +closed part of the figure diminishes to a point. This last +arrangement is known as a \emph{tree}. The number of unicursal +descriptions necessary to completely describe a tree is called +the \emph{base} of the ramification\index{Ramification}. + +We can illustrate the possible form of these trees by rods, +having a hook at each end. Starting with one such rod, we +can attach at either end one or more similar rods. Again, +on any free hook we can attach one or more similar rods, +and so on. Every free hook, and also every point where two +\PG----File: 198.png------------------------------------------------------ +or more rods meet, are what hitherto we have called nodes. +The rods are what hitherto we have termed branches or paths. + +The theory of trees---which already plays a somewhat +important part in certain branches of modern analysis, and +possibly may contain the key to certain chemical and biological +theories---originated in a memoir by Cayley\index{Cayley}\footnote +{\textit{Philosophical Magazine}, March, 1857, series~4, vol.~\textsc{xiii}, +pp.~172--176; or \textit{Collected Works}, Cambridge, 1890, +vol.~\textsc{iii}, no.~203, pp.~242--346: +see also the paper on double partitions, \textit{Philosophical Magazine}, +November, 1860, series~4, vol.~\textsc{xx}, pp.~337--341. On the number of +trees with a given number of nodes, see the \textit +{Quarterly Journal of Mathematics}, +London, 1889, vol.~\textsc{xxiii}, pp.~376--378. The connection with +chemistry was first pointed out in Cayley's paper on isomers, \textit +{Philosophical Magazine}, +June, 1874, series~4, vol.~\textsc{xlvii}, pp.~444--447, and +was treated more fully in his report on trees to the British Association in +1875, \textit{Reports}, pp.~257--305.}, written in +1856. The discussion of the theory has been analytical rather +than geometrical. I content myself with noting the following results. + +The number of trees with $n$ given nodes is $n^{n-2}$. If $A_n$ is +the number of trees with $n$ branches, and $B_n$ the number of +trees with $n$ free branches which are bifurcations at least, +then +\begin{align*} +(1-x)^{-1} (1-x^2)^{-A_1} (1-x^3)^{-A_2} \dotsm + & = 1 + A_1 x + A_2 x^2 + A_3 x^3 + \dotsb\, ,\\ +(1-x)^{-1} (1-x^2)^{-B_2} (1-x^3)^{-B_3} \dotsm + & = 1 + x + 2B_2 x^2 + 2B_3 x^3 + \dotsb\,. +\end{align*} +Using these formulae we can find successively the values of +$A_1,A_2,\dots$, and $B_1,B_2,\dots$. The values of $A_n$ when $n = 2$, $3$, +$4$, $5$, $6$, $7$, are $2$, $4$, $9$, $20$, $48$, $115$; and of $B_n$ are +$1$, $2$, $5$, $12$, +$33$, $90$\index{Trees@\textsc{Trees, Geometrical}|)}\index{Ramification}. + +\ThoughtBreakSpace +I turn next to consider some problems where it is desired +to find a route which will pass once and only once through +each node of a given geometrical figure. This is the reciprocal +of the problem treated in the first part of this chapter, and is +a far more difficult question. I am not aware that the general +theory has been considered by mathematicians, though two +\PG----File: 199.png----------------------------------------------------- +special cases---namely, the \emph{Hamiltonian} (or Icosian) \emph{Game} +and the \emph{Knight's Path on a Chess-Board}---have been treated in +some detail; and I confine myself to a discussion of these. + +\section{The Hamiltonian Game} The Hamiltonian Game% +\index{Dodecahedron@\textsc{Dodecahedron Game}|(}% +\index{Hamilton, Sir Wm.|(}% +\index{Hamiltonian@\textsc{Hamiltonian Game}|(}% +\index{Icosian@\textsc{Icosian Game}|(} consists +in the determination of a route along the edges of a regular +dodecahedron which will pass once and only once through +every angular point. Sir William Hamilton\footnote +{See \textit{Quarterly Journal of Mathematics}, London, 1862, +vol.~\textsc{v}, p.~305; +or \textit{Philosophical Magazine}, January, 1884, series~5, +vol.~\textsc{xvii}, p.~42; also +Lucas\index{Lucas, E.}, vol.~\textsc{ii}, part~vii.}, who invented +this game---if game is the right term for it---denoted the +twenty angular points on the solid by letters which stand for +various towns. The thirty edges constitute the only possible +paths. The inconvenience of using a solid is considerable, +and the dodecahedron may be represented conveniently in +perspective by a flat board marked as shown in the first of +the annexed diagrams. The second and third diagrams will +answer our purpose equally well and are easier to draw. + +\begin{figure*}[!hbt] +\centerline{\includegraphics[width=\textwidth]{./images/illus199}} +\label{illus:199} +\end{figure*} + +The first problem is go ``all round the world,'' that is, +starting from any town, to go to every other town once and +only once and to return to the initial town; the order of the $n$ +towns to be first visited being assigned, where $n$ is not greater +than five. + +Hamilton's rule for effecting this was given at the meeting +in 1857 of the British Association at Dublin. At each +angular point there are three and only three edges. Hence, +\PG----File: 200.png----------------------------------------------------- +if we approach a point by one edge, the only routes open to +us are one to the right, denoted by $r$, and one to the left, +denoted by $l$. It will be found that the operations indicated +on opposite sides of the following equalities are equivalent, +\[ +lr^2l = rlr,\quad rl^2r = lrl,\quad lr^3l=r^2,\quad rl^3r=l^2\,. +\] +Also the operation $l^5$ or $r^5$ brings us back to the initial point: +we may represent this by the equations +\[ +l^5=1,\quad r^5=1\,. +\] + +To solve the problem for a figure having twenty angular +points we must deduce a relation involving twenty successive +operations, the total effect of which is equal to unity. By +repeated use of the relation $l^2 = rl^3r$ we see that +\begin{LRalign} +&1=l^5=l^2l^3=(rl^3r)l^3&=\{rl^3\}^2=\{r(rl^3r)l\}^2\\ +&&=\{r^2l^3rl\}^2=\{r^2(rl^3r)lrl\}^2=\{r^3l^3rlrl\}^2\,.\\ +Therefore && \{r^3l^3(rl)^2\}^2=1&\hbox to2cm{\dotfill(i),}\\ +and similarly& & \{l^3r^3(lr)^2\}^2=1&\hbox to2cm{\dotfill(ii).}\\ +\end{LRalign} +Hence on a dodecahedron either of the operations +\begin{gather*} +r~r~r~l~l~l~r~l~r~l~r~r~r~l~l~l~r~l~r~l\tag{i}\\ +l~l~l~r~r~r~l~r~l~r~l~l~l~r~r~r~l~r~l~r\tag{ii} +\end{gather*} +indicates a route which takes the traveller through every town, +The arrangement is cyclical, and the route can be commenced +at any point in the series of operations by transferring the +proper number of letters from one end to the other. The +point at which we begin is determined by the order of certain +towns which is given initially. + +Thus, suppose that we are told that we start from $F$ and +then successively go to $B$, $A$, $U$, and $T$, and we want to find +a route from $T$ through all the remaining towns which will +end at $F$. If we think of ourselves as coming into $F$ from +$G$, the path $FB$ would be indicated by $l$, but if we think of +ourselves as coming into $F$ from $E$, the path $FB$ would be +indicated by $r$. The path from $B$ to $A$ is indicated by $l$, +and so on. Hence our first paths are indicated either by $l\;l\;l\;r$ +or by $r\;l\;l\;r$. The latter operation does not occur either in (i) +\PG----File: 201.png----------------------------------------------------- +or in (ii), and therefore does not fall within our solutions. The +former operation may be regarded either as the $1$st, $2$nd, $3$rd, +and $4$th steps of (ii), or as the $4$th, $5$th, $6$th, and $7$th steps +of (i). Each of these leads to a route which satisfies the +problem. These routes are +\begin{LRalign} +&F~B~A~U~T~P~O~N~C~D~E~J~K~L~M~Q~R~S~H~G~F\,,&&\\ +and&F~B~A~U~T~S~R~K~L~M~Q~P~O~N~C~D~E~J~H~G~F\,.&&\\ +\end{LRalign} + +It is convenient to make a mark or to put down a counter +at each corner as soon as it is reached, and this will prevent +our passing through the same town twice. + +A similar game may be played with other solids provided +that at each angular point three and only three edges meet. Of +such solids a tetrahedron and a cube are the simplest instances, +but the reader can make for himself any number of plane +figures representing such solids similar to those drawn +\vpageref{illus:199}. % [*Note: originally "on the last page" +Some of these were indicated by Hamilton. In all +such cases we must obtain from the formulae analogous to +those given above cyclical relations like (i) or (ii) there given. +The solution will then follow the lines indicated above. This +method may be used to form a rule for describing any maze in +which no node is of an order higher than three. + +For solids having angular points where more than three +edges meet---such as the octahedron where at each angular +point four edges meet, or the icosahedron where at each +angular point five edges meet---we should at each point have +more than two routes open to us; hence (unless we suppress +some of the edges) the symbolical notation would have to be +extended before it could be applied to these solids. I offer +the suggestion to anyone who is desirous of inventing a new +game. + +Another and a very elegant solution of the Hamiltonian +dodecahedron problem has been given by M.~Hermary\index{Hermary}. It +consists in unfolding the dodecahedron into its twelve pentagons, +each of which is attached to the preceding one by only +one of its sides; but the solution is geometrical, and not +directly applicable to more complicated solids. + +\PG----File: 202.png----------------------------------------------------- +Hamilton suggested as another problem to start from any +town, to go to certain specified towns in an assigned order, +then to go to every other town once and only once, and to end +the journey at some given town. He also suggested the consideration +of the way in which a certain number of towns +should be blocked so that there was no passage through them, +in order to produce certain effects. These problems have not, +so far as I know, been subjected to mathematical analysis% +\index{Dodecahedron@\textsc{Dodecahedron Game}|)}% +\index{Hamilton, Sir Wm.|)}% +\index{Hamiltonian@\textsc{Hamiltonian Game}|)}% +\index{Icosian@\textsc{Icosian Game}|)}. + +\section{Knight's Path on a Chess-Board} Another geometrical% +\index{Chess-board, Games@\textsc{Chess-board, Games on}|(}% +\index{Chess-board, knights@\nobreak--- knights' moves on|(}% +\index{Chess-board, problems@\nobreak--- problems|(}% +\index{Knight@\textsc{Knight's Path Problem}|(}% +\index{Routes on a Chess-board} +problem on which a great deal of ingenuity has been expended, +and of a kind somewhat similar to the Hamiltonian game, +consists in moving a knight on a chess-board in such a manner +that it shall move successively on to every cell\footnote +{The $64$ small squares into which a chess-board is divided are termed +\emph{cells}\index{Cells of a Chess-board}.} once and only +once. The literature on this subject is so extensive\footnote +{For a bibliography see A.~van der Linde\index{Linde on Knight's Path}, +\textit{Geschichte und Literatur +des Schachspiels}, Berlin, 1874, vol.~\textsc{ii}, pp.~101--111. +On the problem +see a memoir by P.~Volpicelli\index{Volpicelli on Knight's Path} +in \textit{Atti della Reale Accademia dei Lincei}, +Rome, 1872, vol.~\textsc{xxv}, pp.~87--162: also \textit +{Applications de l'Analyse +Mathématique au Jeu des échecs}, by C.F.~de~Jaenisch\index +{Jaenisch}, 3~vols., St~Petersburg, +1862--3; and General Parmentier\index{Parmentier on Knight's Path}, +\textit{Association Française pour +l'avancement des Sciences}, 1891, 1892, 1894.} that I +make no pretence to give a full account of the various methods +for solving the problem, and I shall content myself by putting +together a few notes on some of the solutions I have come +across, particularly on those due to De~Moivre, Euler, Vandermonde\index +{Vandermonde}, Warnsdorff, and Roget. + +On a board containing an even number of cells the path +may or may not be re-entrant, but on a board containing an +odd number of cells it cannot be re-entrant. For, if a knight +begins on a white cell, its first move must take it to a black +cell, the next to a white cell, and so on. Hence, if its path +passes through all the cells, then on a board of an odd number +of cells the last move must leave it on a cell of the same colour +\PG----File: 203.png----------------------------------------------------- +as that on which it started, and therefore these cells cannot be +connected by one move. + +\phantomsection +\addcontentsline{toc}{subsection}{Method of De Montmort and De Moivre} +The earliest solutions of which I have any knowledge are +those given about the end of the seventeenth century by +De~Montmort\index{DeMontmort@De Montmort}\index{Montmort, De} and +De~Moivre\index{DeMoivre@De Moivre, on Knight's Move}\index +{Moivre, A. De}\footnote +{I do not know where they were published originally; they were +quoted by Ozanam\index{Ozanam@Ozanam's \textit{Récréations}} +and Montucla\index{Montucla}, see Ozanam, 1803 edition, vol.~\textsc{i}, +p.~178; 1840 edition, p.~80.}. They apply to the ordinary +chess-board of $64$ cells, and depend on dividing (mentally) the +board into an inner square containing sixteen cells surrounded +by an outer ring of cells two deep. If initially the knight is +placed on a cell in the outer ring, it moves round that ring +always in the same direction so as to fill it up completely---only +going into the inner square when absolutely necessary. +When the outer ring is filled up the order of the moves +required for filling the remaining cells presents but little difficulty. +If initially the knight is placed on the inner square +the process must be reversed. The method can be applied to +square and rectangular boards of all sizes. It is illustrated +sufficiently by De Moivre's solution which is given +\vpageref[below]{DeMoivre}, +\begin{figure*}[!hbt] +\centering +\hspace*{\fill} +\begin{minipage}[b]{.4\textwidth} +\centering +\begin{MagicSquare}{8} +{34} & {49} & {22} & {11} & {36} & {39} & {24} & 1 \\ +{21} & {10} & {35} & {50} & {23} & {12} & {37} & {40} \\ +{48} & {33} & {62} & {57} & {38} & {25} & 2 & {13} \\ +9 & {20} & {51} & {54} & {63} & {60} & {41} & {26} \\ +{32} & {47} & {58} & {61} & {56} & {53} & {14} & 3 \\ +{19} & 8 & {55} & {52} & {59} & {64} & {27} & {42} \\ +{46} & {31} & 6 & {17} & {44} & {29} & 4 & {15} \\ +7 & {18} & {45} & {30} & 5 & {16} & {43} & {28} +\end{MagicSquare} +\legend{De Moivre's Solution.} +\end{minipage} +\hfill +\begin{minipage}[b]{.4\textwidth} +\centering +\begin{MagicSquare}{6} +{30} & {21} & 6 & {15} & {28} & {19} \\ +7 & {16} & {29} & {20} & 5 & {14} \\ +{22} & {31} & 8 & {35} & {18} & {27} \\ +9 & {36} & {17} & {26} & {13} & 4 \\ +{32} & {23} & 2 & {11} & {34} & {25} \\ +1 & {10} & {33} & {24} & 3 & {12} +\end{MagicSquare} +\legend{Euler's Thirty-six Cell Solution.} +\end{minipage} +\hspace*{\fill} +\label{DeMoivre} +\end{figure*} +where the numbers indicate the order in which the cells +are occupied successively. I place by its side a somewhat +similar re-entrant solution, due to Euler\index{Euler}, for a board of +\PG----File: 204.png----------------------------------------------------- +$36$ cells. If a chess-board is used it is convenient to place +a counter on each cell as the knight leaves it. + +\phantomsection +\addcontentsline{toc}{subsection}{Method of Euler} +The next serious attempt to deal with the subject was +made by Euler\index{Euler|(}\footnote +{\textit{Mémoires de Berlin} for 1759, Berlin, 1766, pp.~310--337; or +\textit{Commentationes +Arithmeticae Collectae}, St~Petersburg, 1849, vol.~\textsc{i}, pp.~337--355.} +in 1759: it was due to a suggestion made by +L.~Bertrand\index{Bertrand,~L. (of Geneva)} of Geneva, who subsequently +(in 1778) issued an +account of it. This method is applicable to boards of any +shape and size, but in general the solutions to which it +leads are not symmetrical and their mutual connexion is not +apparent. + +Euler commenced by moving the knight at random over +the board until it has no move open to it. With care this +will leave only a few cells not traversed: denote them by +$a, b, \dots$. His method consists in establishing certain rules by +which these vacant cells can be interpolated into various parts +of the circuit, and also by which the circuit can be made +re-entrant. + +The following example, mentioned by Legendre\index{Legendre} as one of +exceptional difficulty, illustrates the method. Suppose that we +\begin{figure*}[!hbt] +\centering +\ifPaper\else\hspace*{\fill}\fi +\begin{minipage}{.4\textwidth} +\centering +\begin{MagicSquare}{8} +{55} & {58} & {29} & {40} & {27} & {44} & {19} & {22} \\ +{60} & {39} & {56} & {43} & {30} & {21} & {26} & {45} \\ +{57} & {54} & {59} & {28} & {41} & {18} & {23} & {20} \\ +{38} & {51} & {42} & {31} & 8 & {25} & {46} & {17} \\ +{53} & {32} & {37} & {a} & {47} & {16} & 9 & {24} \\ +{50} & 3 & {52} & {33} & {36} & 7 & {12} & {15} \\ +1 & {34} & 5 & {48} & {b} & {14} & {c} & {10} \\ +4 & {49} & 2 & {35} & 6 & {11} & {d} & {13} +\end{MagicSquare} +\legend{Figure \Uproman{1}} +\end{minipage} +\hfill +\begin{minipage}{.4\textwidth} +\centering +\begin{MagicSquare}{8} +{22} & {25} & {50} & {39} & {52} & {35} & {60} & {57} \\ +{27} & {40} & {23} & {36} & {49} & {58} & {53} & {34} \\ +{24} & {21} & {26} & {51} & {38} & {61} & {56} & {59} \\ +{41} & {28} & {37} & {48} & 3 & {54} & {33} & {62} \\ +{20} & {47} & {42} & {13} & {32} & {63} & 4 & {55} \\ +{29} & {16} & {19} & {46} & {43} & 2 & 7 & {10} \\ +{18} & {45} & {14} & {31} & {12} & 9 & {64} & 5 \\ +{15} & {30} & {17} & {44} & 1 & 6 & {11} & 8 +\end{MagicSquare} +\legend{Figure \Uproman{2}} +\end{minipage} +\ifPaper\else\hspace*{\fill}\fi +\legend{Example of Euler's Method.} +\DPlabel{Euler:i} +\end{figure*} +have formed the route given in \vhyperlink{Euler:i}{figure~i}; namely, $1$, +%[*Note: originally "figure i above", but in screen the diagram came out below +% and \vpageref got stuck in a pagebreak loop :-(] +$2$, $3$, $\dots$, $59$, $60$; and that there are four cells left +untraversed, namely, $a$, $b$, $c$, $d$. + +We begin by making the path $1$ to $60$ re-entrant. The +\PG----File: 205.png----------------------------------------------------- +cell $1$ commands a cell $p$, where $p$ is $32$, $52$, or $2$. The cell $60$ +commands a cell $q$, where $q$ is $29$, $59$, or $51$. Then, if any of +these values of $p$ and $q$ differ by unity, we can make the route +re-entrant. This is the case here if $p=52$, $q=51$. Thus the +cells $1, 2, 3, \dotsc, 51$; $60, 59, \dotsc, 52$ form a re-entrant route of +$60$ moves. Hence, if we replace the numbers $60, 59, \dotsc, 52$ by +$52, 53, \dotsc, 60$, the steps will be numbered consecutively. I +recommend the reader who wishes to follow the subsequent +details of Euler's argument to construct this square on a piece +of paper before proceeding further. + +Next, we add the cells $a$, $b$, $d$ to this route. In the new +diagram of $60$ cells formed as above the cell $a$ commands the +cells there numbered $51$, $53$, $41$, $25$, $7$, $5$, and $3$. It is +indifferent which of these we select: suppose we take $51$. Then +we must make $51$ the last cell of the route of $60$ cells, so that +we can continue with $a$, $b$, $d$. Hence, if the reader will add $9$ +to every number on the diagram he has constructed, and then +replace $61, 62, \dotsc, 69$ by $1, 2, \dotsc, 9$, he will have a route which +starts from the cell occupied originally by $60$, the $60$th move +is on to the cell occupied originally by $51$, and the $61$st, $62$nd, +$63$rd moves will be on the cells $a$, $b$, $d$ respectively. + +It remains to introduce the cell $c$. Since $c$ commands the +cell now numbered $25$, and $63$ commands the cell now numbered +$24$, this can be effected in the same way as the first +route was made re-entrant. In fact the cells numbered $1, +2, \dotsc, 24$; $63, 62, \dotsc, 25, c$ form a knight's path. Hence we +must replace $63, 62, \dotsc, 25$ by the numbers $25, 26, \dotsc, 63$, and +then we can fill up $c$ with $64$. We have now a route which +covers the whole board. + +Lastly, it remains to make this route re-entrant. First, we +must get the cells $1$ and $64$ near one another. This can be +effected thus. Take one of the cells commanded by $1$, such as +$28$, then $28$ commands $1$ and $27$. Hence the cells $64, 63, \dotsc, 28$; +$1, 2, \dotsc, 27$ form a route; and this will be represented in the +diagram if we replace the cells numbered $1, 2, \dotsc, 27$ by $27, +26, \dotsc, 1$. + +\PG----File: 206.png-------------------------------------------------- +The cell now occupied by $1$ commands the cells $26$, $38$, $54$, +$12$, $14$, $16$, $28$; and the cell occupied by $64$ commands the +cells $13$, $43$, $63$, $55$. The cells $13$ and $14$ are consecutive, and +therefore the cells $64, 63, \dotsc, 14$; $1, 2, \dotsc, 13$ form a route. +Hence we must replace the numbers $1, 2, \dotsc, 13$ by $13, 12, \dotsc, 1$, +and we obtain a re-entrant route covering the whole board, +which is represented in the second of the diagrams given +\vpageref{Euler:i}. %[*Note: originally "above"] +Euler showed how seven other re-entrant routes can +be deduced from any given re-entrant route. + +It is not difficult to apply the method so as to form a route +which begins on one given cell and ends on any other given +cell. + +Euler next investigated how his method could be modified +so as to allow of the imposition of additional restrictions. + +An interesting example of this kind is where the first $32$ +moves are confined to one half of the board. One solution +of this is delineated \vpageref[below]{figure:Roget}. +The order of the first $32$ moves +\begin{figure*}[!hbt] +\centering +\ifPaper\else\hspace*{\fill}\fi +\begin{minipage}{.4\textwidth} +\centering +\begin{MagicSquare}{8} +{58} & {43} & {60} & {37} & {52} & {41} & {62} & {35} \\ +{49} & {46} & {57} & {42} & {61} & {36} & {53} & {40} \\ +{44} & {59} & {48} & {51} & {38} & {55} & {34} & {63} \\ +{47} & {50} & {45} & {56} & {33} & {64} & {39} & {54} \\ +{22} & 7 & {32} & 1 & {24} & {13} & {18} & {15} \\ +{31} & 2 & {23} & 6 & {19} & {16} & {27} & {12} \\ + 8 & {21} & 4 & {29} & {10} & {25} & {14} & {17} \\ + 3 & {30} & 9 & {20} & 5 & {28} & {11} & {26}\\ +\put(0,4){\line(1,0){8}} +\end{MagicSquare} +\legend{Euler's Half-board Solution.} +\end{minipage} +\hfill +\begin{minipage}{.4\textwidth} +\centering +\begin{MagicSquare}{8} +{50} & {45} & {62} & {41} & {60} & {39} & {54} & {35} \\ +{63} & {42} & {51} & {48} & {53} & {36} & {57} & {38} \\ +{46} & {49} & {44} & {61} & {40} & {59} & {34} & {55} \\ +{43} & {64} & {47} & {52} & {33} & {56} & {37} & {58} \\ +{26} & 5 & {24} & 1 & {20} & {15} & {32} & {11} \\ +{23} & 2 & {27} & 8 & {29} & {12} & {17} & {14} \\ + 6 & {25} & 4 & {21} & {16} & {19} & {10} & {31} \\ + 3 & {22} & 7 & {28} & 9 & {30} & {13} & {18}\\ +\put(0,4){\line(1,0){8}} +\end{MagicSquare} +\legend{Roget's Half-board Solution.} +\end{minipage} +\ifPaper\else\hspace*{\fill}\fi +\label{figure:Roget} +\end{figure*} +can be determined by Euler's method. It is obvious that, if +to the number of each such move we add $32$, we shall have +a corresponding set of moves from $33$ to $64$ which would cover +the other half of the board; but in general the cell numbered +$33$ will not be a knight's move from that numbered $32$, +nor will $64$ be a knight's move from $1$. + +\PG----File: 207.png----------------------------------------------------- +Euler however proceeded to show how the first $32$ moves +might be determined so that, if the half of the board containing +the corresponding moves from $33$ to $64$ was twisted +through two right angles, the two routes would become +united and re-entrant. If $x$ and $y$ are the numbers of a +cell reckoned from two consecutive sides of the board, we +may call the cell whose distances are respectively $x$ and $y$ +from the opposite sides a complementary cell. Thus the cells +$(x, y)$ and $(9-x, 9-y)$ are complementary, where $x$ and $y$ +denote respectively the column and row occupied by the cell. +Then in Euler's solution the numbers in complementary cells +differ by $32$: for instance, the cell $(3, 7)$ is complementary to +the cell $(6, 2)$, the one is occupied by $57$, the other by $25$. + +Roget's method\index{Roget, P.M.}, which is described later, can be also +applied to give half-board solutions. The result is indicated +\vpageref{figure:Roget}. % [*Note: originally "above"] +The close of Euler's memoir is devoted to showing how +the method could be applied to crosses and other rectangular +figures. I may note in particular his elegant re-entrant symmetrical +solution for a square of $100$ cells\index{Euler|)}. + +\phantomsection +\addcontentsline{toc}{subsection}{Method of Vandermonde} +The next attempt of any special interest is due to Vandermonde\index +{Vandermonde}\footnote +{\textit{L'Histoire de l'Académie des Sciences} for 1771, Paris, +1774, pp. 566-574.}, +who reduced the problem to arithmetic. His idea was +to cover the board by two or more independent routes taken at +random, and then to connect the routes. He defined the position +of a cell by a fraction $x/y$, whose numerator $x$ is the +number of the cell from one side of the board, and whose denominator +$y$ is its number from the adjacent side of the board; +this is equivalent to saying that $x$ and $y$ are the co-ordinates of +a cell. In a series of fractions denoting a knight's path, the +differences between the numerators of two consecutive fractions +can be only one or two, while the corresponding difference +between their denominators must be two or one respectively. +Also $x$ and $y$ cannot be less than $1$ or greater than $8$. The +notation is convenient, but Vandermonde applied it merely +to obtain a particular solution of the problem for a board of +\PG----File: 208.png---------------------------------------------------- +$64$ cells: the method by which he effected this is analogous to +that established by Euler, but it is applicable only to squares +of an even order. The route that he arrives at is defined in +his notation by the following fractions. + +\noindent{\baselineskip=1.5\baselineskip +$\frac{5}{5}$, $\frac{4}{3}$, $\frac{2}{4}$, $\frac{4}{5}$, $\frac{5}{3}$, +$\frac{7}{4}$, $\frac{8}{2}$, $\frac{6}{1}$, +$\frac{7}{3}$, $\frac{8}{1}$, $\frac{6}{2}$, +$\frac{8}{3}$, $\frac{7}{1}$, $\frac{5}{2}$, +$\frac{6}{4}$, $\frac{8}{5}$, $\frac{7}{7}$, +$\frac{5}{8}$, $\frac{6}{6}$, $\frac{5}{4}$, +$\frac{4}{6}$, $\frac{2}{5}$, $\frac{1}{7}$, +$\frac{3}{8}$, +$\frac{2}{6}$, $\frac{1}{8}$, $\frac{3}{7}$, $\frac{1}{6}$, $\frac{2}{8}$, +$\frac{4}{7}$, $\frac{3}{5}$, $\frac{1}{4}$, +$\frac{2}{2}$, $\frac{4}{1}$, $\frac{3}{3}$, +$\frac{1}{2}$, $\frac{3}{1}$, $\frac{2}{3}$, +$\frac{1}{1}$, $\frac{3}{2}$, $\frac{1}{3}$, +$\frac{2}{1}$, $\frac{4}{2}$, $\frac{3}{4}$, +$\frac{1}{5}$, $\frac{2}{7}$, $\frac{4}{8}$, +$\frac{3}{6}$, +$\frac{4}{4}$, $\frac{5}{6}$, $\frac{7}{5}$, $\frac{8}{7}$, $\frac{6}{8}$, +$\frac{7}{6}$, $\frac{8}{8}$, $\frac{6}{7}$, +$\frac{8}{6}$, $\frac{7}{8}$, $\frac{5}{7}$, +$\frac{6}{5}$, $\frac{8}{4}$, $\frac{7}{2}$, +$\frac{5}{1}$, $\frac{6}{3}$. + +}\medskip +The path is re-entrant but unsymmetrical. Had he transferred +the first three fractions to the end of this series he +would have obtained two symmetrical circuits of thirty-two +moves joined unsymmetrically, and might have been enabled +to advance further in the problem. Vandermonde\index{Vandermonde} +also considered the case of a route in a cube. + +In 1773 Collini\index{Collini on Chess}\footnote +{\textit{Solution du Problème du Cavalier au Jeu des échecs}, Mannheim, +1773.} proposed the exclusive use of symmetrical +routes arranged without reference to the initial cell, but connected +in such a manner as to permit of our starting from +it. This is the foundation of the modern manner of attacking +the problem. The method was re-invented in 1825 by +Pratt\index{Pratt on Knight's Path}\footnote +{\textit{Studies of Chess}, sixth edition, London, 1825.}, +and in 1840 by Roget\index{Roget, P.M.}, and has been subsequently +employed by various writers. Neither Collini nor Pratt showed +skill in using this method. The rule given by Roget is described +later. + +\phantomsection +\addcontentsline{toc}{subsection}{Method of Warnsdorff} +One of the most ingenious of the solutions of the knight's +path is that given in 1823 by Warnsdorff\index +{Warnsdorff, Knight's Path|(}\footnote +{\textit{Des Rösselsprunges einfachste und allgemeinste Lösung}, +Schmalkalden, 1823: see Jaenisch\index{Jaenisch}, +vol.~\textsc{ii}, pp.~56--61, 273--289.}. His rule is that +the knight must be moved always to one of the cells from +which it will command the fewest squares not already traversed. +The solution is not symmetrical and not re-entrant; +moreover it is difficult to trace practically. The rule has not +been proved to be true, but no exception to it is known: +apparently it applies also to all rectangular boards which can +\PG----File: 209.png----------------------------------------------------- +be covered completely by a knight. It is somewhat curious +that in most cases a single false step, except in the last three +or four moves, will not affect the result. + +Warnsdorff added that when, by the rule, two or more cells +are open to the knight, it may be moved to either or any of +them indifferently. This is not so, and with great ingenuity +two or three cases of failure have been constructed, but it +would require exceptionally bad luck to happen accidentally +on such a route\index{Warnsdorff, Knight's Path|)}. + +The above methods have been applied to boards of various +shapes, especially to boards in the form of rectangles, crosses, +and circles\footnote +{See \Eg\ T.~Ciccolini's\index{Ciccolini on Chess} work \textit +{Del Cavallo degli Scacchi}, Paris, 1836.}. + +All the more recent investigations impose additional restrictions: +such as to require that the route shall be re-entrant, or +more generally that it shall begin and terminate on given +cells. + +\phantomsection +\addcontentsline{toc}{subsection}{Method of Roget} +The best complete solution with which I am acquainted---and +one which I believe is not generally known---is due to +Roget\index{Roget, P.M.|(}\footnote +{\textit{Philosophical Magazine}, April, 1840, series~3, vol.~\textsc{xvi}, +pp.~305--309; see also the \textit{Quarterly Journal of Mathematics} for 1877, +vol.~\textsc{xiv}, pp.~354--359. Some solutions, founded on Roget's method, +are given in the \textit{Leisure Hour}, Sept.~13, 1873, pp. 587--590; +see also \Ibid, Dec.~20, 1873, pp.~813--815. +}. It divides the whole route into four circuits, which +can be combined so as to enable us to begin on any cell +and terminate on any other cell of a different colour. Hence, +if we like to select this last cell at a knight's move from the +initial cell, we obtain a re-entrant route. On the other hand, +the rule is applicable only to square boards containing $(4n)^2$ +cells: for example, it could not be used on the board of the +French \emph{jeu des dames}, which contains $100$ cells. + +Roget began by dividing the board of $64$ cells into four +quarters. Each quarter contains $16$ cells, and these $16$ cells +can be arranged in $4$ groups, each group consisting of $4$ cells +\PG----File: 210.png------------------------------------------------------- +which form a closed knight's path. All the cells in each such +path are denoted by the same letter $l$, $e$, $a$, or $p$, as the case +nay be. The path of $4$ cells indicated by the consonants $l$ and +the path indicated by the consonants $p$ are diamond-shaped: +the paths indicated respectively by the vowels $e$ and $a$ are +square-shaped, as may be seen by looking at one of the four +quarters in figure~i \vpageref[below]{Roget:i}. + +\begin{figure*}[!hbt] +\centering +\ifPaper\else\hspace*{\fill}\fi +\begin{minipage}{.4\textwidth} +\centering +\def\SqHtDefault{lp} +\begin{MagicSquare}{8} +l & e & a & p & l & e & a & p \\ +a & p & l & e & a & p & l & e \\ +e & l & p & a & e & l & p & a \\ +p & a & e & l & p & a & e & l \\ +l & e & a & p & l & e & a & p \\ +a & p & l & e & a & p & l & e \\ +e & l & p & a & e & l & p & a \\ +p & a & e & l & p & a & e & l \\ +\put(0,4){\line(1,0){8}} +\put(4,0){\line(0,1){8}} +\end{MagicSquare} +\legend{Roget's Solution \upshape(i).} +\end{minipage} +\hfill +\begin{minipage}{.4\textwidth} +\centering +\begin{MagicSquare}{8} +{34} & {51} & {32} & {15} & {38} & {53} & {18} & 3 \\ +{31} & {14} & {35} & {52} & {17} & 2 & {39} & {54} \\ +{50} & {33} & {16} & {29} & {56} & {37} & 4 & {19} \\ +{13} & {30} & {49} & {36} & 1 & {20} & {55} & {40} \\ +{48} & {63} & {28} & 9 & {44} & {57} & {22} & 5 \\ +{27} & {12} & {45} & {64} & {21} & 8 & {41} & {58} \\ +{62} & {47} & {10} & {25} & {60} & {43} & 6 & {23} \\ +{11} & {26} & {61} & {46} & 7 & {24} & {59} & {42}\\ +\put(0,4){\line(1,0){8}} +\put(4,0){\line(0,1){8}} +\end{MagicSquare} +\legend{Roget's Solution \upshape(ii).} +\end{minipage} +\ifPaper\else\hspace*{\fill}\fi +\label{Roget:i} +\end{figure*} + +Now all the $16$ cells on a complete chess-board which are +marked with the same letter can be combined into one circuit, +and wherever the circuit begins we can make it end on any +other cell in the circuit, provided it is of a different colour +to the initial cell. If it is indifferent on what cell the +circuit terminates we may make the circuit re-entrant, and +in this case we can make the direction of motion round each +group (of $4$ cells) the same. For example, all the cells +marked $p$ can be arranged in the circuit indicated by the +successive numbers $1$ to $16$ in figure~ii \vpageref[above]{Roget:i}. +Similarly all +the cells marked $a$ can be combined into the circuit indicated +by the numbers $17$ to $23$; all the $l$ cells into the circuit $33$ to +$48$; and all the $e$ cells into the circuit $49$ to $64$. Each of the +circuits indicated above is symmetrical and re-entrant. The +consonant and the vowel circuits are said to be of opposite +kinds. + +\PG----File: 211.png------------------------------------------------------- +The general problem will be solved if we can combine the +four circuits into a route which will start from any given cell, +and terminate on the $64$th move on any other given cell of a +different colour. To effect this Roget gave the two following +rules. + +First.\quad If the initial cell and the final cell are denoted the +one by a consonant and the other by a vowel, take alternately +circuits indicated by consonants and vowels, beginning with +the circuit of $16$ cells indicated by the letter of the initial +cell and concluding with the circuit indicated by the letter of +the final cell. + +Second.\quad If the initial cell and the final cell are denoted +both by consonants or both by vowels, first select a cell, $Y$, in +the same circuit as the final cell, $Z$, and one move from it, +next select a cell, $X$, belonging to one of the opposite circuits +and one move from $Y$. This is always possible. Then, leaving +out the cells $Z$ and $Y$, it always will be possible, by the rule +already given, to travel from the initial cell to the cell $X$ in +$62$ moves, and thence to move to the final cell on the $64$th +move. + +In both cases however it must be noticed that the cells in +each of the first three circuits will have to be taken in such an +order that the circuit does not terminate on a corner, and it +may be desirable also that it should not terminate on any of +the border cells. This will necessitate some caution. As far +as is consistent with these restrictions it is convenient to +make these circuits re-entrant, and to take them and every +group in them in the same direction of rotation. + +As an example, suppose that we are to begin on the cell +numbered $1$ in figure~ii \vpageref{Roget:i}, which is one of those in a +$p$ circuit, and to terminate on the cell numbered $64$, which is +one of those in an $e$ circuit. This falls under the first rule: +hence first we take the $16$ cells marked $p$, next the $16$ cells +marked $a$, then the $16$ cells marked $l$, and lastly the $16$ cells +marked $e$. One way of effecting this is shown in the diagram. +Since the cell $64$ is a knight's move from the initial cell the +\PG----File: 212.png------------------------------------------------------- +route is re-entrant. Also each of the four circuits in the +diagram is symmetrical, re-entrant, and taken in the same +direction, and the only point where there is any apparent +breach in the uniformity of the movement is in the passage +from the cell numbered $32$ to that numbered $33$. + +A rule for re-entrant routes, similar to that of Roget, has +been given by various subsequent writers, especially by +De~Polignac\index{DePolignac@De Polignac on Knight's Move}\index +{Polignac on Knight's Path}\footnote +{\textit{Comptes Rendus}, April, 1861; and \textit{Bulletin de la +Société Mathématique de France}, 1881, vol.~\textsc{ix}, pp. 17--24.} +and by Laquière\index{Laquiere@Laquière on Knight's Path}\footnote +{\textit{Bulletin de la Société Mathématique de France}, 1880, +vol.~\textsc{viii}, pp.~82--102, 132--158.}, who have stated it at much +greater length. Neither of these authors seems to have been +aware of Roget's theorems. De~Polignac, like Roget, illustrates +the rule by assigning letters to the various squares in +the way explained above, and asserts that a similar rule is +applicable to all even squares. + +Roget's method can be also applied to two half-boards, as +indicated in the figure given above on page~\pageref{figure:Roget}\index +{Roget, P.M.|)}. + +\phantomsection +\addcontentsline{toc}{subsection}{Method of Moon} +Another way of dividing the board into closed circuits +which can be connected was given in $1843$ by Moon\index{Moon, R.}\footnote +{\textit{Cambridge Mathematical Journal}, 1843, vol.~\textsc{iii}, +pp.~233--236.}. He +\begin{figure*}[!hbt] +\centering +\ifPaper\else\hspace*{\fill}\fi +\begin{minipage}{.4\textwidth} +\centering +\begin{MagicSquare}{8} +a & b & c & d & a & b & c & d \\ +c & d & a & b & c & d & a & b \\ +b & a & A & B & C & D & d & c \\ +d & c & C & D & A & B & b & a \\ +a & b & B & A & D & C & c & d \\ +c & d & D & C & B & A & a & b \\ +b & a & d & c & b & a & d & c \\ +d & c & b & a & d & c & b & a \\ +\put(2,2){\line(1,0){4}} +\put(2,2){\line(0,1){4}} +\put(2,6){\line(1,0){4}} +\put(6,2){\line(0,1){4}} +\end{MagicSquare} +\legend{Moon's Solution.} +\end{minipage} +\hfill +\begin{minipage}{.4\textwidth} +\centering +\begin{MagicSquare}{8} +{63} & {22} & {15} & {40} & 1 & {42} & {59} & {18} \\ +{14} & {39} & {64} & {21} & {60} & {17} & 2 & {43} \\ +{37} & {62} & {23} & {16} & {41} & 4 & {19} & {58} \\ +{24} & {13} & {38} & {61} & {20} & {57} & {44} & 3 \\ +{11} & {36} & {25} & {52} & {29} & {46} & 5 & {56} \\ +{26} & {51} & {12} & {33} & 8 & {55} & {30} & {45} \\ +{35} & {10} & {49} & {28} & {53} & {32} & {47} & 6 \\ +{50} & {27} & {34} & 9 & {48} & 7 & {54} & {31} +\end{MagicSquare} +\legend{Jaenisch's Solution\index{Jaenisch}.} +\end{minipage} +\ifPaper\else\hspace*{\fill}\fi +\label{Jaenisch} +\end{figure*} +divided the board into a central square containing $4^2$ cells and +a surrounding annulus (see figure \vpageref{Jaenisch}). The annulus may be +\PG----File: 213.png------------------------------------------------------ +divided into four closed circuits, each containing $12$ cells: these +are marked respectively with the letters $a$, $b$, $c$, $d$. The central +square may be divided similarly into four closed circuits, each +containing $4$ cells, denoted by the letters $A$, $B$, $C$, $D$. We can +connect these routes as follows. If we are moving outwards +from the central square to the annulus we can go from a cell $A$ +either to $b$ or to $c$ or to $d$ (but not to $a$) and similarly for the +other letters. If we are moving inwards from the annulus to +the central square we must go from $a$ to $D$, or $d$ to $A$, or $b$ to +$C$, or $c$ to $B$, as the case may be. Thus if the initial cell is +on $a$, we might take either of the cycles +$a\ D\ b\ C\ d\ A\ c\ B$, or +$a\ D\ c\ B\ d\ A\ b\ C$. By following these rules we always can +connect the routes into one path, but in general it will not +be re-entrant. It is convenient to take the cells in each circuit +in one and the same direction, but a circuit in the outer annulus +must not end in a corner cell, and to avoid this we may +have to alter the direction in which a circuit is taken. + +Moon's\index{Moon, R.} rule can be modified to cover the case of any doubly +even square board, and the path can be made to begin and end +on any two given squares, but I do not propose to go further +into details. + +\phantomsection +\addcontentsline{toc}{subsection}{Method of Jaenisch} +The method which Jaenisch\index{Jaenisch} gives as the most fundamental +is not very different from that of Roget. It leads to eight +forms, similar to that in the diagram printed \vpageref{Jaenisch}, in which +the sum of the numbers in every column and every row +is $260$; but although symmetrical it is not in my opinion so +easy to reproduce as that given by Roget. + +\phantomsection +\addcontentsline{toc}{subsection}{Number of possible routes} +It is as yet impossible to say how many solutions of +the problem exist. Legendre\index{Legendre}\footnote +{\textit{Théorie des Nombres}, Paris, 2nd edition, 1830, vol.~\textsc{ii}, +p.~165.} mentioned the question, but +Minding\index{Minding on Knight's Path}\footnote +{\textit{Cambridge and Dublin Mathematical Journal}, 1852, vol.~\textsc{vii}, +pp.~147--156; and \textit{Crelle's Journal}, 1853, vol.~\textsc{xliv}, +pp.~73--82.} was the earliest writer to attempt to answer it. +More recent investigations have shown that on the one hand +the number of possible routes is less\footnote +{Jaenisch\index{Jaenisch}, vol.~\textsc{ii}, p.~268.} than the number of +\PG----File: 214.png------------------------------------------------------- +combinations of $168$ things taken $63$ at a time, and on the +other hand is greater than $31,054144$---since this latter number +is the number of re-entrant paths of a particular type\footnote +{\textit{Bulletin de la Société Mathématique de France}, 1881, +vol.~\textsc{ix}, pp.~1--17.}. + +\phantomsection +\addcontentsline{toc}{section}{Paths of other Chess-Pieces} +There are many similar problems in which it is required to +determine routes by which a piece moving according to certain +laws (\Eg\ a chess-piece such as a king, knight,~\&c.) can +travel from a given cell over a board so as to occupy +successively all the cells, or certain specified cells, once and +only once, and terminate its route in a given cell. + +Euler's method can be applied to find routes of this kind: +for instance, he applied it to find a re-entrant route by which +a piece that moved two cells forward like a castle and then +one cell like a bishop would occupy in succession all the black +cells on the board. As another instance, a castle, placed on a +chess-board of $n^2$ cells, can always be moved in such a manner +that it shall move successively on to every cell once and only +once; moreover, starting on any cell, its path can be made to +terminate, if $n$ be even, on any other cell of a different colour, +and, if $n$ be odd, on any other cell of the same colour\footnote +{\textit{L'Intermédiaire des mathématiciens}, Paris, July, 1901, +p.~153.}. But it will suffice to have discussed the classical problem of the +determination of a knight's path on an ordinary chess-board, +and I need not enter on the discussion of other similar +problems\index{Chess-board, Games@\textsc{Chess-board, Games on}|)}% +\index{Chess-board, knights@\nobreak--- knights' moves on|)}% +\index{Chess-board, problems@\nobreak--- problems|)}% +\index{Knight@\textsc{Knight's Path Problem}|)}. + +\PG----File: 215.png----------------------------------------------------- +% the original is verso (p198), so no need for \cleartorecto +\PartQuote{``No man of science should think it a waste of time to learn +something of the history of his own subject; nor is the investigation +of laborious methods now fallen into disuse, or of errors +once commonly accepted the least valuable of mental disciplines.'' + +\bigskip +``The most worthless book of a bygone day is a record +worthy of preservation. Like a telescopic star, its obscurity +may render it unavailable for most purposes; but it serves, +in hands which know how to use it, to determine the places of +more important bodies.'' \hfill\qquad\ifPaper\penalty-100\fi +\null\hfill\textsc{(De Morgan\index{DeMorgan@De Morgan, A.}.)}} + +\addtocontents{toc}{\string\newpage} +\part{\PartTwoText} + +\PGx---File: 216.png----------------------------------------------------- +% Note that in this chapter vol. numbers are lowercase, not small caps, roman +\chapter{The Mathematical Tripos.} % VII + +\textsc{The} Mathematical Tripos has played so prominent a part% + \chapindex{Acts or Disputations}% + \chapindex{Cambridge, Mathematics@\textsc{Cambridge, Mathematics}}% + \chapindex{Cambridge, Studies@\nobreak--- \textsc{Studies at}}% + \chapindex{Mathematics, Cambridge@\textsc{Mathematics, Cambridge}}% + \chapindex{Moderators}% + \chapindex{Optimes}% + \chapindex{Senate-House@\textsc{Senate-House Examination}}% + \chapindex{Tripos, Math@\textsc{Tripos, Mathematical}}% + \chapindex{Wranglers} +in the history of education at Cambridge and of mathematics\chapindex +{Mathematics, Cambridge@\textsc{Mathematics, Cambridge}} +in England, that a sketch of its development\footnote +{The following pages are mostly summarised from my\index{Ball} +\textit{History of the Study of Mathematics at Cambridge}, Cambridge 1889. +The subject is also treated in Whewell's\index{Whewell, W.} \textit +{Liberal Education}, Cambridge, three parts, 1845, 1850, 1853; +Wordsworth's\index{Wordsworth, C.} \textit{Scholae Academicae}, Cambridge, +1877; my own\index{Ball} \textit{Origin and History of the Mathematical +Tripos}, Cambridge, 1880; and Dr~Glaisher's\index{Glaisher, J.W.L.} +Presidential Address to the London Mathematical Society, \textit +{Transactions}, vol~\textsc{xviii}, 1886, pp.~4--38. +} may be interesting to general readers. + +\phantomsection +\addcontentsline{toc}{section}{Medieval Course of Studies: Acts} +So far as mathematics is concerned the history of the +University before Newton may be summed up very briefly. +The University was founded towards the end of the twelfth +century. Throughout the middle ages the studies were +organised on lines similar to those at Paris and Oxford. To +qualify for a degree it was necessary to perform various +exercises, and especially to keep a number of \emph{acts} or to oppose +acts kept by other students. An act consisted in effect of a +debate in Latin, thrown, at any rate in later times, into +syllogistic form. It was commenced by one student, the +\emph{respondent}, stating some proposition, often propounded in the +form of a thesis, which was attacked by one or more \emph{opponents}, +the discussion being controlled by a graduate. The +teaching was largely in the hands of young graduates---every +master of arts being compelled to reside and teach for at least +one year---though no doubt Colleges and private hostels supplemented +this instruction in the case of their own students. + +\PG----File: 217.png------------------------------------------------------ +\phantomsection +\addcontentsline{toc}{section}{The Renaissance at Cambridge} +\phantomsection +\addcontentsline{toc}{subsection}{Rise of a Mathematical School} +The Reformation in England was mainly the work of +Cambridge divines, and in the University the Renaissance was +warmly welcomed. In spite of the disorder and confusion of +the Tudor period, new studies and a system of professional +instruction were introduced. Probably the science (as distinct +from the art) of mathematics, save so far as involved in the +quadrivium, was still an exotic study, but it was not wholly +neglected. Tonstall\index{Tonstall, C.}, subsequently the most eminent +English arithmetician of his time, migrated, perhaps about 1495, from +Balliol College, Oxford, to King's Hall, Cambridge, and in +1530 the University appointed a mathematical lecturer in the +person of Paynell\index{Paynell, N.} of Pembroke Hall. Most of the subsequent +English mathematicians of the Tudor period seem to have +been educated at Cambridge; of these I may mention Record\index{Record, R.}, +who migrated, probably about 1535, from Oxford, Dee\index{Dee, J.}, +Digges\index{Digges, T.}, +Blundeville\index{Blundeville, T.}, +Buckley\index{Buckley, W.}, +Billingsley\index{Billingsley, H.}, +Hill\index{Hill, T.}, +Bedwell\index{Bedwell, T.}, +Hood\index{Hood, T.}, +Richard and John Harvey\index{Harvey, J.}\index{Harvey, R.}, +Edward Wright\index{Wright, E.}, +Briggs\index{Briggs, H.}, +and Oughtred\index{Oughtred, W.}. +The Elizabethan statutes restricted liberty of +thought and action in many ways, but, in spite of the civil +and religious disturbances of the early half of the 17th century +the mathematical school continued to grow. Horrox\index{Horrox, J.}, +Seth Ward\index{Ward, S.}, +Foster\index{Foster, S.}, +Rooke\index{Rooke, L.}, +Gilbert Clerke\index{Clerke, G.}, +Pell, Wallis\index{Wallis, J.}, +Barrow\index{Barrow, I.}, +Dacres\index{Dacres, A.}, +and Morland\index{Morland, S.} may be cited as prominent Cambridge +mathematicians of the time. + +Newton's\index{Newton} mathematical career dates from 1665; his +reputation, abilities, and influence attracted general attention to +the subject. He created a school of mathematics and mathematical +physics, among the earliest members of which I note +the names of Laughton\index{Laughton, R.}, +Samuel Clarke\index{Clarke, S.}, +Craig\index{Craig, J.}, +Flamsteed\index{Flamsteed, J.}, +Whiston\index{Whiston, W.}, +Saunderson\index{Saunderson, N.}, +Jurin\index{Jurin, J.}, +Taylor\index{Taylor, B.}, +Cotes\index{Cotes, R.}, and Robert Smith\index{Smith, R@\nobreak--- R.}. +Since then Cambridge has been regarded, as in a special sense, +the home of English mathematicians, and from 1706 onwards +we have fairly complete accounts of the course of reading and +work of mathematical students there. + +Until less than a century ago the form of the method of +qualifying for a degree remained substantially unaltered, but +\PG----File: 218.png---------------------------------------------------- +the subject-matter of the discussions varied from time to time +with the prevalent studies of the place. + +\phantomsection +\addcontentsline{toc}{section}{Subject-Matter of Acts at different periods} +After the Renaissance some of the statutable exercises were +``huddled,'' that is, were reduced to a mere form. To huddle\index{Huddling} +an act, the proctor generally asked some question such as \emph{Quid +est nomen} to which the answer usually expected was \emph{Nescio}. +In these exercises considerable license was allowed, particularly +if there were any play on the words involved. For example, +T.~Brasse, of Trinity, was accosted with the question, \emph{Quid +est aes?} to which he answered, \emph{Nescio nisi finis examinationis.} +It should be added that retorts such as these were only allowed +in the pretence exercises, and a candidate who in the actual +examination was asked to give a definition of happiness and +replied an exemption from Payne---that being the name of the +moderator then presiding---was plucked for want of discrimination +in time and place. In earlier years even the farce of +huddling\index{Huddling} seems to have been unnecessary, for it was said +in 1675 that it was not uncommon for the proctors to take ``cautions +for the performance of the statutable exercises, and accept +the forfeit of the money so deposited in lieu of their performance.'' + +In medieval times acts had been usually kept on some +scholastic question or on a proposition taken from the \emph{Sentences}. +About the end of the fifteenth century religious questions, such +as the interpretation of Biblical texts, began to be introduced, +some fifty or sixty years later the favourite subjects were +drawn either from dogmatic theology or from philosophy. +In the seventeenth century the questions were usually philosophical, +but in the eighteenth century, under the influence of +the Newtonian school, a large proportion of them were +mathematical. + +Further details about these exercises and specimens of +acts kept in the 18th century are given in my\index{Ball} \textit +{History of Mathematics at Cambridge.} Here I will only say that they +provided an admirable training in the art of presenting an +argument, and in dialectical skill in attack and defence. The +\PG----File: 219.png------------------------------------------------------ +mental strain in a contested act was severe. De~Morgan\index +{DeMorgan@De Morgan, A.}, describing his act kept in 1826, wrote\footnote +{\textit{Budget of Paradoxes}, by A.~De~Morgan, London, 1872, p.~305.}, + +\begin{Quotation} +I was badgered for two hours with arguments given and answered in +Latin,---or what we call Latin---against Newton's first section, Lagrange's +derived functions, and Locke on innate principles. And though I took +off everything, and was pronounced by the moderator to have disputed +\textit{magno honore}, I never had such a strain of thought in my life. +For the inferior opponents were made as sharp as their betters by their +tutors, who kept lists of queer objections drawn from all quarters. +\end{Quotation} + +Had the language of the discussions been changed to English, +as was repeatedly urged from 1774 onwards, these exercises +might have been retained with advantage, but the barbarous +Latin and the syllogistic form in which they were carried on +prejudiced their retention. + +About 1830 a custom grew up for the respondent and +opponents to meet previously and arrange their arguments +together. The discussions then became an elaborate farce, +and were a mere public performance of what had been +already rehearsed. Accordingly the moderators of 1839 took +the responsibility of abandoning them. This action was +singularly high-handed, since a report of May~30, 1838, had +recommended that they should be continued, and there was +no reason why they should not have been reformed and +retained as a useful feature in the scheme of study. + +\phantomsection +\addcontentsline{toc}{section}{Degree Lists} +On the result of the acts a list of those qualified to +receive degrees was drawn up. This list was not arranged +strictly in order of merit, because the proctors could insert +names anywhere in it, but by the beginning of the 18th +century this power had become restricted to the right reserved +to the vice-chancellor, the senior regent, and each +proctor to place in the list one candidate anywhere he +liked---a right which continued to exist till 1828, though it was +not exercised after 1797. Subject to the granting of these +honorary degrees\index{Honorary Optimes}, this final list was arranged in +order of merit into wranglers\index{Wranglers|(} and senior optimes\index +{Optimes|(}\index{Senior Optimes|(}, junior optimes\index{Junior Optimes|(}, +and +\PG----File: 220.png------------------------------------------------------ +poll-men\index{Poll-Men}. The bachelors on receiving their degrees took +seniority according to their order on this list. The title +\emph{wrangler} is derived from these contentious discussions\index +{Wranglers|)}; the +title \emph{optime} from the customary compliment given by the +moderator to a successful disputant, \textit{Domine\textellipsis, optime +disputasti}, or even \textit{optime quidem disputasti}\index +{Optimes|)}\index{Senior Optimes|)}\index{Junior Optimes|)}, +and the title of \emph{poll-man} +from the description of this class as \hoipolloi. + +The final exercises for the B.A. degree were never huddled, +and until 1839 were carried out strictly. University officials +were responsible for approving the subject-matter of these +acts. Stupid men offered some irrefutable truism, but the +ambitious student courted reputation by affirming some +paradox. Probably all honour men kept acts, but poll-men +were deemed to comply with the regulations by keeping +opponencies. The proctors were responsible for presiding at +these acts, or seeing that competent graduates did so. In and +after 1649 two examiners were specially appointed for this +purpose. In 1680\footnote{See Grace of October 25, 1680.} +these examiners were appointed by the +Senate with the title of moderator, and with the joint stipend +of four shillings for everyone graduating as B.A. during their +year of office. In 1688 the joint stipend of the moderators +was fixed at \pounds40 a year. The moderators, like the proctors, +were nominated by the Colleges in rotation. + +\phantomsection +\addcontentsline{toc}{section}{Oral Examinations always possible} +From the earliest times the proctors had the power of +questioning a candidate at the end of a disputation, and +probably all candidates for a degree attended the public +schools on certain days to give an opportunity to the +proctors, or any master that liked, to examine them\footnote +{\EG\ see De~la~Pryme's\index{DelaPryme@De la Pryme} account of his +graduation in 1694, +\textit{Surtees Society}, vol.~\textsc{liv}, 1870, p.~32.}, though +the opportunity was not always used. Different candidates +attended on different days. Probably such examinations were +conducted in Latin. But soon after 1710\footnote +{W.~Reneu\index{Reneu, W.}, in his letters of 1708--1710 describing the +course for the +B.A.~degree, makes no mention of the Senate-House examination, and I +think it is a reasonable inference that it had not then been established.} +the moderators +\PG----File: 221.png------------------------------------------------------ +or proctors began the custom of summoning on one day in +January all candidates whom they proposed to question. +The examination was held in public, and from it the Senate-House +Examination arose. The examination at this time did +not last more than one day, and was, there can be no doubt, +partly on philosophy and partly on mathematics. It is +believed that it was always conducted in English, and it is +likely that its rapid development was largely due to this. + +\phantomsection +\addcontentsline{toc}{section}{Public Oral Examinations become +customary, 1710--30} +This introduction of a regular oral examination seems to +have been largely due to the fact that when, in 1710, George~I\index +{George I of England} +gave the Ely library to the University, it was decided to assign +for its reception the old Senate-House---now the Catalogue +Room in the Library---and to build a new room for the +meetings of the Senate. Pending the building of the new +Senate-House the books were stored in the Schools. As the +Schools were thus rendered unavailable for keeping acts, +considerable difficulty was found in arranging for all the +candidates to keep the full number of statutable exercises, +and thus obtaining opportunities to compare them one with +another: hence the introduction of a supplementary oral +examination. The advantages of this examination as providing +a ready means of testing the knowledge and abilities +of the candidates were so patent that it was retained when +the necessity for some system of the kind had passed away, +and finally it became systematized into an organized test to +which all questionists were subjected. + +\phantomsection +\addcontentsline{toc}{subsection}{Additional work thrown on Moderators. +Stipends raised} +In 1731 the University raised the joint stipend of the +moderators to \pounds60 ``in consideration of their additional +trouble in the Lent Term.'' This would seem to indicate +that the Senate-House Examination had then taken formal +shape, and perhaps that a definite scheme for its conduct had +become customary. + +\phantomsection +\addcontentsline{toc}{subsection}{Facilitates order of merit} +As long as the order of the list of those approved for +degrees was settled on the result of impressions derived from +acts kept by the different candidates at different times and on +different subjects, it was impossible to arrange the men in +\PG----File: 222.png---------------------------------------------------- +strict order of merit, nor was much importance attached to +the order. But, with the introduction of an examination of +all the candidates on one day, much closer attention was paid +to securing a strict order of merit, and more confidence was +felt in the published order. It seems to have been consequent +on this that in and after 1747 the final lists were freely circulated, +and it was further arranged that the names of the +honorary optimes\index{Honorary Optimes} should be indicated. +In the lists given in +the Calendars issued subsequent to 1799 these names are +struck out. It is only in exceptional cases that we are +acquainted with the true order for the earlier tripos lists, +but in a few cases the relative positions of the candidates +are known; for example, in 1680 Bentley\index{Bentley, R.} came out as third +though he was put down as sixth in the list of wranglers. + +\phantomsection +\addcontentsline{toc}{section}{Scheme of Examination in 1750} +\phantomsection +\addcontentsline{toc}{subsection}{Right of M.A.s to take part in it} +Of the detailed history of the examination until the +middle of the eighteenth century we know nothing. From +1750 onwards, however, we have more definite accounts of +it. At this time, it would seem that all the men from each +College were taken together as a class, and questions passed +down by the proctors or moderators till they were answered: +but the examination remained entirely oral, and technically +was regarded as subsidiary to the discussions which had been +previously held in the schools. As each class contained men +of very different abilities a custom grew up by which every +candidate was liable to be taken aside to be questioned by +any M.A. who wished to do so, and this was regarded as an +important part of the examination. The subjects were mathematics +and philosophy. The examination now continued for +two days and a half. At the conclusion of the second day +the moderators received the reports of those masters of arts +who had voluntarily taken part in the examination, and +provisionally settled the final list; while the last half-day was +used in revising and re-arranging the order of merit. + +Richard Cumberland\index{Cumberland, R.|(} has left an account of +the tests to +which he was subjected when he took his B.A. degree in +1751. Clearly the disputations still played an important +\PG----File: 223.png------------------------------------------------------ +part, and it is difficult to say what weight was attached to +the subsequent Senate-House examination; his reference to it +is only of a general character. After saying that he kept +two acts and two opponencies he continues\footnote +{\textit{Memoirs of Richard Cumberland}, London, 1806, pp.~78, 79.}: + +\begin{Quotation} +The last time I was called upon to keep an act in the schools I sent +in three questions to the Moderator, which he withstood as being all +mathematical, and required me to conform to the usage of proposing one +metaphysical question in the place of that, which I should think fit to +withdraw. This was ground I never liked to take, and I appealed +against his requisition: the act was accordingly put by till the matter +of right should be ascertained by the statutes of the university, and in +the result of that enquiry it was given for me, and my question +stood\textellipsis. +I yielded now to advice, and paid attention to my health, till we were +cited to the senate house to be examined for our Bachelor's degree. It +was hardly ever my lot during that examination to enjoy any respite. +I seemed an object singled out as every man's mark, and was kept +perpetually at the table under the process of question and answer.\index +{Cumberland, R.|)} +\end{Quotation} + +It was found possible by means of the new examination to +differentiate the better men more accurately than before; and +accordingly, in 1753, the first class was subdivided into two, +called respectively wranglers and senior optimes\index{Optimes}\index +{Senior Optimes}, a division which is still maintained. + +The semi-official examination by M.A.s was regarded as +the more important part of the test, and the most eminent +residents in the University took part in it. Thus John Fenn\index{Fenn, J.}, +of Caius, 5th wrangler in 1761, writes\footnote +{Quoted by C.~Wordsworth\index{Wordsworth, C.}, \textit{Scholae Academicae}, +Cambridge, 1877, pp.~30--31.}: + +\begin{Quotation} +On the following Monday, Tuesday, and Wednesday, we sat in the +Senate-house for public examination; during this time I was officially +examined by the Proctors and Moderators, and had the honor of being +taken out for examination by Mr~Abbot\index{Abbot, W.}, the celebrated +mathematical +tutor of St~John's College, by the eminent professor of mathematics +Mr~Waring\index{Waring, E.}, of Magdalene, and by Mr~Jebb\index{Jebb, J.} +of Peterhouse, a man +thoroughly versed in the academical studies. +\end{Quotation} + +\noindent This irregular examination by any master who chose to take +part in it constantly gave rise to accusations of partiality. + +\PG----File: 224.png------------------------------------------------------ +\phantomsection +\addcontentsline{toc}{section}{Scheme of Examination in 1763} +In 1763 the traditional rules for the conduct of the +examination took more definite shape. Henceforth the +examiners used the disputations only as a means of classifying +the men roughly. On the result of their ``acts,'' and +probably partly also of their general reputation, the candidates +were divided into eight classes, each arranged in alphabetical +order. The subsequent position of the men in the class was +determined solely by the Senate-House examination. The +first two classes comprised all who were expected to be +wranglers, the next four classes included the other candidates +for honours, and the last two classes consisted of poll-men +only. Practically anyone placed in either of the first two +classes was allowed, if he wished, to take an aegrotat senior +optime\index{Senior Optimes}, and thus escape all further examination: +this was called gulphing it. All the men from one College were no +longer taken together, but each class was examined separately +and \textit{vivâ voce}; and hence, since all the students comprised in +each class were of about equal attainments, it was possible to +make the examination more effective. Richard Watson\index{Watson, R.}, of +Trinity, claimed that this change was made by him when +acting as moderator in 1763. He says\footnote +{\textit{Anecdotes of the Life of Richard Watson by Himself}, London, 1817, +pp.~18, 19.}: + +\begin{Quotation} +There was more room for partiality\textellipsis then [\IE\ in 1759] than % NB tight ellipsis matches original +there is +now; and I attribute the change, in a great degree, to an alteration +which I introduced the first year I was moderator [\IE\ in 1763], and +which has been persevered in ever since. At the time of taking their +Bachelor of Arts' degree, the young men are examined in classes, and +the classes are now formed according to the abilities shown by individuals +in the schools. By this arrangement, persons of nearly equal merits are +examined in the presence of each other, and flagrant acts of partiality +cannot take place. Before I made this alteration, they were examined in +classes, but the classes consisted of members of the same College, and +the best and worst were often examined together. +\end{Quotation} + +\noindent It is probable that before the examination in the Senate-House +began a candidate, if manifestly placed in too low a class, was +\PG----File: 225.png------------------------------------------------------ +allowed the privilege of challenging the class to which he was +assigned. Perhaps this began as a matter of favour, and was +only granted in exceptional cases, but a few years later it +became a right which every candidate could exercise; and I +think that it is partly to its development that the ultimate +predominance of the tripos over the other exercises for the +degree is due. + +In the same year, 1763, it was decided that the relative +position of the senior and second wranglers, namely, Paley\index +{Paley, W.}, of +Christ's, and Frere\index{Frere, J.}, of Caius, was to be decided by the +Senate-House examination and not by the disputations. Henceforward +distinction in the Senate-House examination was +regarded as the most important honour open to undergraduates. + +\phantomsection +\addcontentsline{toc}{section}{Foundations of Smith's Prizes, 1768} +In 1768 Dr~Smith\index{Smith, R@\nobreak--- R.}, of Trinity College, +founded prizes for +mathematics and natural philosophy open to two commencing +bachelors. The examination followed immediately after the +Senate-House examination, and the distinction, being much +coveted, tended to emphasize the mathematical side of the +normal University education of the best men. Since 1883 the +prizes have been awarded on the result of dissertations\footnote +{See Grace of October 25, 1885; and the \textit{Cambridge University +Reporter}, October 23 and 30, 1883.}. + +\phantomsection +\addcontentsline{toc}{section}{Introduction of a Written Examination, +circ.\ 1770} +Until now the Senate-House examination had been oral, +but about this time, \textit{circ}.\ 1770, it began to be the custom to +dictate some or all of the questions and to require answers to +be written. Only one question was dictated at a time, and a +fresh one was not given out until some student had solved +that previously read: a custom which by causing perpetual +interruptions to take down new questions must have proved +very harassing. We are perhaps apt to think that an examination +conducted by written papers is so natural that the +custom is of long continuance, but I know no record of any +in Europe earlier than the eighteenth century. Until 1830 +the questions for the Smith's Prize were dictated. + +\PG----File: 226.png------------------------------------------------------ +\phantomsection +\addcontentsline{toc}{section}{Description of the Examination in 1772} +The following description of the Senate-House examination +as it existed in 1772 is given by Jebb\index{Jebb, J.|(}\footnote +{\textit{The Works of J.~Jebb}, London, 1787, vol.~ii, pp.~290--297.}. + +\begin{Quotation} +The moderators, some days before the arrival of the time prescribed +by the vice-chancellor, meet for the purpose of forming the students into +divisions of six, eight, or ten, according to their performance in the +schools, with a view to the ensuing examination. + +Upon the first of the appointed days, at eight o'clock in the morning, +the students enter the senate-house, preceded by a master of arts from +each college, who\textellipsis is called the ``father'' of the % NB tight ellipsis matches original +college\textellipsis + +After the proctors have called over the names, each of the moderators +sends for a division of the students: they sit with him round a table, +with pens, ink, and paper, before them: he enters upon his task of +examination, and does not dismiss the set till the hour is expired. This +examination has now for some years been held in the english language. + +The examination is varied according to the abilities of the students. +The moderator generally begins with proposing some questions from +the six books of Euclid, plain trigonometry, and the first rules of +algebra. If any person fails in an answer, the question goes to the next. +From the elements of mathematics, a transition is made to the four +branches of philosophy, viz.\ mechanics, hydrostatics, apparent astronomy, +and optics, as explained in the works of Maclaurin, Cotes, Helsham, +Hamilton, Rutherforth, Keill, Long, Ferguson, and Smith. If the +moderator finds the set of questionists, under examination, capable of +answering him, he proceeds to the eleventh and twelfth books of Euclid, +conic sections, spherical trigonometry, the higher parts of algebra, and +sir Isaac Newton's Principia; more particularly those sections, which +treat of the motion of bodies in eccentric and revolving orbits; the +mutual action of spheres, composed of particles attracting each other +according to various laws; the theory of pulses, propagated through +elastic mediums; and the stupendous fabric of the world. Having +closed the philosophical examination, he sometimes asks a few questions +in Locke's Essay on the human understanding, Butler's Analogy, or +Clarke's Attributes. But as the highest academical distinctions are +invariably given to the best proficients in mathematics and natural +philosophy, a very superficial knowledge in morality and metaphysics +will suffice. + +When the division under examination is one of the highest classes, +problems are also proposed, with which the student retires to a distant +part of the senate-house; and returns, with his solution upon paper, to +the moderator, who, at his leisure, compares it with the solutions of +other students, to whom the same problems have been proposed. + +\PG----File: 227.png------------------------------------------------------ +The extraction of roots, the arithmetic of surds, the invention of +divisers, the resolution of quadratic, cubic, and biquadratic equations; +together with the doctrine of fluxions\index{Fluxions}, and its application +to the solution +of questions ``de maximis et minimis,'' to the finding of areas, to the +rectification of curves, the investigation of the centers of gravity and +oscillation, and to the circumstances of bodies, agitated, according to +various laws, by centripetal forces, as unfolded, and exemplified, in the +fluxional treatises of Lyons, Saunderson, Simpson, Emerson, Maclaurin, +and Newton, generally form the subject matter of these problems. + +When the clock strikes nine, the questionists are dismissed to breakfast: +they return at half past nine, and stay till eleven; they go in again +at half past one, and stay till three; and, lastly, they return at half-past +three, and stay till five. + +The hours of attendance are the same upon the subsequent day. + +On the third day they are finally dismissed at eleven. + +During the hours of attendance, every division is twice examined in +form, once by each of the moderators, who are engaged for the whole +time in this employment. + +As the questionists are examined in divisions of only six or eight at a +time, but a small portion of the whole number is engaged, at any +particular hour, with the moderators; and, therefore, if there were no +further examination, much time would remain unemployed. + +But the moderator's inquiry into the merits of the candidates forms +the least material part of the examination. + +The ``fathers'' of the respective colleges, zealous for the credit of the +societies, of which they are the guardians, are incessantly employed in +examining those students, who appear most likely to contest the palm of +glory with their sons. + +This part of the process is as follows: + +The father of a college takes a student of a different college aside, +and, sometimes for an hour and an half together, strictly examines him +in every part of mathematics and philosophy, which he professes to have +read. + +After he hath, from this examination, formed an accurate idea of the +student's abilities and acquired knowledge, he makes a report of his +absolute or comparative merit to the moderators, and to every other +father who shall ask him the question. + +Besides the fathers, all masters of arts, and doctors, of whatever +faculty they be, have the liberty of examining whom they please; and +they also report the event of each trial, to every person who shall make +the inquiry. + +The moderators and fathers meet at breakfast, and at dinner. From +the variety of reports, taken in connection with their own examination, +\PG----File: 228.png------------------------------------------------------ +the former are enabled, about the close of the second day, so far to settle +the comparative merits of the candidates, as to agree upon the names of +four-and-twenty, who to them appear most deserving of being distinguished +by marks of academical approbation. + +These four-and-twenty [wranglers and senior optimes] are recommended +to the proctors for their private examination; and, if approved +by them, and no reason appears against such placing of them from any +subsequent inquiry, their names are set down in two divisions, according +to that order, in which they deserve to stand; are afterwards printed; +and read over upon a solemn day, in the presence of the vice-chancellor, +and of the assembled university. + +The names of the twelve [junior optimes\index{Junior Optimes}], who, +in the course of the +examination, appear next in desert, are also printed, and are read over, +in the presence of the vice-chancellor, and of the assembled university, +upon a day subsequent to the former\textellipsis + +The students, who appear to have merited neither praise nor censure, +[the poll-men], pass unnoticed: while those, who have taken no pains +to prepare themselves for the examination, and have appeared with +discredit in the schools, are distinguished by particular tokens of disgrace. +\end{Quotation} + +\noindent Jebb's statement about the number of wranglers and senior +optimes is only approximate. + +It may be added that it was now frankly recognized +that the examination was competitive\footnote +{``Emulation, which is the principle upon which the plan is +constructed.'' \textit{The Works of J.~Jebb}, London, 1787, +vol.~iii, p.~261.}. Also that though +it was open to any member of the Senate to take part +in it, yet the determination of the relative merit of the +students was entirely in the hands of the moderators\footnote +{\textit{The Works of J.~Jebb}\index{Jebb, J.|)}, London, 1787, +vol.~iii, p.~272.}. +Although the examination did not occupy more than three +days it must have been a severe physical trial to anyone who +was delicate. It was held in winter and in the Senate-House. +That building was then noted for its draughts, and was not +warmed in any way: and according to tradition, on one +occasion the candidates on entering in the morning found the +ink in the pots on their desks frozen. + +\phantomsection +\addcontentsline{toc}{section}{Scheme of Examination in 1779} +The University was not altogether satisfied\footnote +{See Graces of July~5, 1773, and of February~17, 1774.} with the +\PG----File: 229.png--------------------------------------------------- +scheme in force, and in 1779\footnote +{See Graces of March~19, 20, 1779.} the scheme of examination was +amended in various respects. In particular the examination +was extended to four days, a third day being given up entirely +to natural religion, moral philosophy, and Locke\index{Locke, J.}. It was +further announced\footnote +{Notice issued by the Vice-Chancellor, dated May~19, 1779.} +that a candidate would not receive credit +for advanced subjects unless he had satisfied the examiners +in Euclid and elementary Natural Philosophy. + +\phantomsection +\addcontentsline{toc}{subsection}{System of Brackets} +A system of brackets\index{Brackets, in Tripos} or ``classes quam minimae'' +was now introduced. Under this system the examiners issued on +the morning of the fourth day a provisional list of men who +had obtained honours, with the names of those of about equal +merit bracketed, and that day was devoted to arranging the +names in each bracket in order of merit: the examiners being +given explicit authority to invite the assistance of others in +this work. Whether at this time a candidate could request +to be re-examined with the view of being moved from one +bracket to another is uncertain, but later this also was +allowed. + +Under the scheme of 1779 also the number of examiners +was increased to four, the moderators of one year becoming, +as a matter of course, the examiners of the next. Thus of +the four examiners in each year, two had taken part in +the examination of the previous year, and the continuity of +the system of examination was maintained. The names of the +moderators appear on the tripos lists, but the names of the +examiners were not printed on the lists till some years later. + +The right of any M.A. to take part in the examination +was not affected, though henceforth it was exercised more +sparingly, and I believe was not insisted on after 1785. But +it became a regular custom for the moderators to invite +particular M.A.s to examine and compare specified candidates. +Milner\index{Milner, I.}, of Queens', was constantly asked to assist +in this way. + +It was not long before it became an established custom +that a candidate, who was dissatisfied with the class in which +\PG----File: 230.png------------------------------------------------------ +he had been placed as the result of his disputations, might +challenge it before the examination began. This power seems +to have been used but rarely; it was, however, a recognition +of the fact that a place in the tripos list was to be determined +by the Senate-House examination alone, and the examiners +soon acquired the habit of settling the preliminary classes +without exclusive reference to the previous disputations. + +\phantomsection +\addcontentsline{toc}{section}{Problem Papers in 1785 and 1786} +The earliest papers actually set in the Senate-House, and +now extant, are two problem papers\index{Problem Papers} set in 1785 and 1786 +by W.~Hodson\index{Hodson, W.}, of Trinity, then a proctor. The autograph +copies from which he gave out the questions were luckily +preserved, and are in the library\footnote +{The \textit{Challis Manuscripts}\index{Challis MSS},~\textsc{iii}, 61.} +of Trinity College. They +must be almost the last problem papers which were dictated, +instead of being printed and given as a whole to the +candidates. + +The problem paper\index{Problem Papers|(} in 1786 was as follows: + +\begin{ExamQuestions} +\Q[1.] To determine the velocity with which a Body must be thrown, in +a direction parallel to the Horizon, so as to become a secondary planet to +the Earth; as also to describe a parabola, and never return. + +\Q[2.] To demonstrate, supposing the force to vary as $\dfrac{1}{D^2}$, +how far a +body must fall both within and without the Circle to acquire the Velocity +with which a body revolves in a Circle. + +\Q[3.] Suppose a body to be turned (\sic) upwards with the Velocity with +which it revolves in an Ellipse, how high will it ascend? The same is +asked supposing it to move in a parabola. + +\Q[4.] Suppose a force varying first as $\dfrac{1}{D^3}$, secondly in a +greater ratio +than $\dfrac{1}{D^2}$ but less than $\dfrac{1}{D^3}$, and thirdly in a +less ratio than $\dfrac{1}{D^2}$, in each +of these Cases to determine whether at all, and where the body parting +from the higher Apsid will come to the lower. + +\Q[5.] To determine in what situation of the moon's Apsid they go most +forwards, and in what situation of her Nodes the Nodes go most backwards, +and why? + +\Q[6.] In the cubic equation $x^3 + qx + r = 0$ which wants the second term; +supposing $x = a + b$ and $3ab=-q$, to determine the value of $x$. + +\Q[7.] To find the fluxion\index{Fluxions} of $x^r \times (y^n + z^m)^{1/q}$. +% silently altering notation for fractional exponent + +\PG----File: 231.png------------------------------------------------------ +\Q[8.] To find the fluent of $\dfrac{a \dot{x}}{a+x}$. + +\Q[9.] To find the fluxion\index{Fluxions} of the $m^{\text{th}}$ power +of the Logarithm of $x$. + +\Q[10.] Of right-angled Triangles containing a given Area to find that +whereof the sum of the two legs $AB + BC$ shall be the least possible. +[This and the two following questions are illustrated by diagrams. The +angle at $B$ is the right angle.] + +\Q[11.] To find the Surface of the Cone $ABC$. [The cone is a right one +on a circular base.] + +\Q[12.] To rectify the arc $DB$ of the semicircle $DBV$\index +{Problem Papers|)}. +\end{ExamQuestions} + +In cases of equality in the Senate-House examination the +acts were still taken into account in settling the tripos order: +and in 1786 when the second, third, and fourth wranglers +came out equal in the examination a memorandum was published +that the second place was given to that candidate who +\textit{dialectis magis est versatus}, and the third place to that one +who \textit{in scholis sophistarum melius disputavit}. + +There seem to have been considerable intervals in the +examination by the moderators, and the examinations by the +extraneous examiners took place in these intervals. Those +candidates who at any time were not being examined occupied +themselves with amusements, provided they were not too +boisterous and obvious: probably dice and cards played a +large part in them. Gunning\index{Gunning, H.} in an amusing account of his +examination in 1788 talks of games with a teetotum\footnote +{H.~Gunning, \textit{Reminiscences}, second edition, London, 1855, vol.~i, +p.~82.} in +which he took part on the Wednesday (when Locke\index{Locke, J.} and +Paley\index{Paley, W.} formed the subjects of examination), but ``which was +carried on with great spirit\textellipsis by considerable numbers during % NB tight ellipsis matches original +the whole of the examination.'' + +About this time, 1790, the custom of printing\index{Examination, Printed} +the problem papers\index{Problem Papers} +was introduced, but until 1828 the other papers continued +to be dictated. Since 1827 all the papers have been printed. + +I insert here the following letter\footnote +{C.~Wordsworth\index{Wordsworth, C.}, \textit{Scholae Academicae}, +Cambridge, 1877, pp.~322--23.} from William Gooch\index{Gooch, W.}, of +\PG----File: 232.png------------------------------------------------------ +Caius, in which he describes his examination in the Senate-House +in 1791. It must be remembered that it is the letter +of an undergraduate addressed to his father and mother, and +was not intended either for preservation or publication: a fact +which certainly does not detract from its value. + +\begin{Quotation} +\phantomsection +\addcontentsline{toc}{section}{Description of the Examination in 1791} +\emph{Monday} $\frac{1}{4}$ aft. 12.\hfil\break\indent +We have been examin'd this Morning in pure Mathematics \& I've +hitherto kept just about even with Peacock which is much more than I +expected. We are going at 1~o'clock to be examin'd till 3 in Philosophy. + +From 1 till 7 I did more than Peacock; But who did most at Moderator's +Rooms this Evening from 7 till 9, I don't know yet;---but I did +above three times as much as the Sen\textsuperscript{r} Wrangler last year, +yet I'm afraid not so much as Peacock. + +Between One \& three o'Clock I wrote up 9 sheets of Scribbling Paper +so you may suppose I was pretty fully employ'd. + +\emph{Tuesday Night.}\hfil\break\indent +I've been shamefully us'd by Lax\index{Lax, W.} to-day;---Tho' his anxiety for +Peacock must (of course) be very great, I never suspected that his Partially +(\sic) w\textsuperscript{d} get the better of his Justice. I had entertain'd +too high an +opinion of him to suppose it.---he gave Peacock a long private Examination +\& then came to me (I hop'd) on the same subject, but 'twas only to +\emph{Bully} me as much as he could,---whatever I said (tho' right) he tried +to convert into Nonsense by seeming to misunderstand me. However I +don't entirely dispair of being first, tho' you see Lax\index{Lax, W.} seems +determin'd that I shall not.---I had no Idea (before I went into the +Senate-House) of being able to contend at all with Peacock. + +\emph{Wednesday evening.}\hfil\break\indent +Peacock \& I are still in perfect Equilibrio \& the Examiners themselves +can give no guess yet who is likely to be first;---a New Examiner +(Wood\index{Wood, J.} of St.~John's, who is reckon'd the first Mathematician +in the University, +for Waring\index{Waring, E.} doesn't reside) was call'd solely to examine +Peacock +\& me only.---but by this new Plan nothing is yet determin'd.---So Wood +is to examine us again to-morrow morning. + +\emph{Thursday evening.}\hfil\break\indent +Peacock is declar'd first \& I second,---Smith of this Coll.\ is either +8\textsuperscript{th} or 9\textsuperscript{th} \& Lucas is either +10\textsuperscript{th} or 11\textsuperscript{th}.---Poor Quiz Carver is +one of the +\hoipolloi;---I'm perfectly \emph{satisfied} that the Senior Wranglership +is Peacock's due, but \textit{certainly} not so very indisputably as +Lax\index{Lax, W.} pleases to represent it---I +understand that \emph{he} asserts 'twas 5 to 4 in Peacock's favor. Now +Peacock \& I have explain'd to each other how we went on, \& can \emph{prove +indisputably} that it wasn't 20 to 19 in his favor;---I \emph{cannot} +therefore be +\PG----File: 233.png------------------------------------------------------ +displeas'd for being plac'd second, tho' I'm provov'd (\sic) with +Lax\index{Lax, W.} for +his false report (so much beneath the Character of a Gentleman.)--- + +N.B. it is my very \emph{particular Request} that you don't mention +Lax's\index{Lax, W.} behaviour to me to any one. +\end{Quotation} + +Such was the form ultimately taken by the Senate-House +examination, a form which it substantially retained without +alteration for nearly half-a-century. It soon became the sole +test by which candidates were judged. The University was +not obliged to grant a degree to anyone who performed the +statutable exercises, and it was open to the University to +refuse to pass a supplicat for the B.A. degree unless the % NB "supplicat" = "a formal petition for a degree or for incorporation" (OED) +% or http://www.admin.cam.ac.uk/offices/students/praelectors/supplicat.pdf +candidate had presented himself for the Senate-House examination. +In 1790 James Blackburn\index{Blackburn, J.}, of Trinity, a questionist +of exceptional abilities, was informed that in spite of his good +disputations he would not be allowed a degree unless he also +satisfied the examiners in the tripos. He accordingly solved +one ``very hard problem,'' though in consequence of a dispute +with the authorities he refused to attempt any more.\footnote +{Gunning\index{Gunning, H.}, \textit{Reminiscences}, second edition, +London, 1855, vol.~i, p.~182.} + +\phantomsection +\addcontentsline{toc}{subsection}{The Poll Part of the Examination} +It will be recollected that the examination was now compulsory +on all candidates pursuing the normal course for the +B.A. degree. In 1791 the University laid down rules\footnote +{See Grace of April~8, 1791.} for +its conduct, so far as it concerned poll-men, decreeing that +those who passed were to be classified in four divisions or +classes, the names in each class to be arranged alphabetically, +but not to be printed on the official tripos lists. The classes in +the final lists must be distinguished from the eight preliminary +classes issued before the commencement of the examination. +The men in the first six preliminary classes were expected to +take honours; those in the seventh and eighth preliminary +classes were \emph{primâ facie} poll-men. + +\phantomsection +\addcontentsline{toc}{section}{A Pass Standard introduced} +In 1799 the moderators announced\footnote +{Communicated by the moderators to fathers of Colleges on +January~18, 1799, and agreed to by the latter.} +that for the future +they would require every candidate to show a competent +\PG----File: 234.png------------------------------------------------------ +knowledge of the first book of Euclid, arithmetic, vulgar and +decimal fractions, simple and quadratic equations, and Locke +and Paley\index{Paley, W.}. Paley's works seem to be held in esteem by +modern divines, and his \textit{Evidences}, though not his \textit{Philosophy}, +still remains (1905) one of the subjects of the Previous +Examination, but his contemporaries thought less highly of +his writings, or at any rate of his Philosophy. Thus Best is +quoted by Wordsworth\index{Wordsworth, C.}\footnote +{C.~Wordsworth, \textit{Scholae Academicae}, Cambridge, 1877, p.~123.} +as saying of Paley's\index{Paley, W.} \textit{Philosophy}, +``The tutors of Cambridge no doubt neutralize by their +judicious remarks, when they read it to their pupils, all that +is pernicious in its principles'': so also Richard Watson\index{Watson, R.}, +Bishop of Llandaff, in his anecdotal autobiography\footnote +{\textit{Anecdotes of the Life of R.~Watson}, London, 1817, p.~19.}, says, in +describing the Senate-House examination in which Paley was +senior wrangler, that Paley was afterwards known to the +world by many excellent productions, ``though there are some\textellipsis % NB tight ellipsis matches original +principles in his philosophy which I by no means approve.'' + +In 1800 the moderators extended to all men in the first +four preliminary classes the privilege of being allowed to +attempt the problem papers: hitherto this privilege had been +confined to candidates placed in the first two classes. Until +1828 the problem papers were set in the evenings, and in the +rooms of the moderator. + +\phantomsection +\addcontentsline{toc}{section}{Problem Papers from 1802 onwards} +The \emph{University Calendars}\index{Calendars, University} date from 1796, +and from 1802 to 1882 inclusive contain the printed tripos papers\index +{Problem Papers} of the previous January. The papers from 1801 to 1820 +and from 1838 to 1849 inclusive were also published in separate volumes, +which are to be found in most public libraries. No problems +were ever set to the men in the seventh and eighth preliminary +classes, which contained the poll-men. None of the bookwork +papers of this time are now extant, but it is believed that they +contained but few riders. Many of the so-called problems +were really pieces of bookwork or easy riders: it must however +\PG----File: 235.png------------------------------------------------------ +be remembered that the text-books then in circulation were +inferior and incomplete as compared with modern ones. + +\phantomsection +\addcontentsline{toc}{section}{Description of the Examination in 1802} +The \emph{Calendar} of 1802 contains a diffuse account of the +examination. It commences as follows: + +\begin{Quotation} +On the Monday morning, a little before eight o'clock, the students, +generally about a hundred, enter the Senate-House, preceded by a master +of arts, who on this occasion is styled the father of the College to which +he belongs. On two pillars at the entrance of the Senate-House are hung +the classes and a paper denoting the hours of examination of those who +are thought most competent to contend for honours. Immediately after +the University clock has struck eight, the names are called over, and the +absentees, being marked, are subject to certain fines. The classes to be +examined are called out, and proceed to their appointed tables, where +they find pens, ink, and paper provided in great abundance. In this +manner, with the utmost order and regularity, two-thirds of the young +men are set to work within less than five minutes after the clock has +struck eight. There are three chief tables, at which six examiners preside. +At the first, the senior moderator of the present year and the junior +moderator of the preceding year. At the second, the junior moderator +of the present and the senior moderator of the preceding year. At the +third, two moderators of the year previous to the two last, or two examiners +appointed by the Senate. The two first tables are chiefly allotted +to the six first classes; the third, or largest, to the \hoipolloi. + +The young men hear the propositions or questions delivered by the +examiners; they instantly apply themselves; demonstrate, prove, work +out and write down, fairly and legibly (otherwise their labour is of little +avail) the answers required. All is silence; nothing heard save the voice +of the examiners; or the gentle request of some one, who may wish a +repetition of the enunciation. It requires every person to use the utmost +dispatch; for as soon as ever the examiners perceive anyone to have +finished his paper and subscribed his name to it another question is +immediately given\textellipsis + +The examiners are not seated, but keep moving round the tables, both +to judge how matters proceed and to deliver their questions at proper +intervals. The examination, which embraces arithmetic, algebra, +fluxions\index{Fluxions}, the doctrine of infinitesimals and increments, +geometry, trigonometry, +mechanics, hydrostatics, optics, and astronomy, in all their +various gradations, is varied according to circumstances: no one can +anticipate a question, for in the course of five minutes he may be dragged +from Euclid to Newton, from the humble arithmetic of Bonnycastle to +the abstruse analytics of Waring\index{Waring, E.}. While this examination +is proceeding +at the three tables between the hours of eight and nine, printed problems +\PG----File: 236.png---------------------------------------------------- +are delivered to each person of the first and second classes; these he takes +with him to any window he pleases, where there are pens, ink, and paper +prepared for his operations. +\end{Quotation} + +The examination began at eight. At nine o'clock the +papers had to be given up, and half-an-hour was allowed for +breakfast. At half-past nine the candidates came back, and +were examined in the way described above till eleven, when +the Senate-House was again cleared. An interval of two +hours then took place. At one o'clock all returned to be again +examined. At three the Senate-House was cleared for half-an-hour, +and, on the return of the candidates, the examination +was continued till five. At seven in the evening the first four +classes went to the senior moderator's rooms to solve problems. +They were finally dismissed for the day at nine, after eight +hours of examination. The work of Tuesday was similar to +that of Monday: Wednesday was partly devoted to logic and +moral philosophy. At eight o'clock on Thursday morning a +first list was published with all candidates of about equal +merits bracketed\index{Brackets, in Tripos}. Until nine o'clock a +candidate had the +right to challenge anyone above him to an examination to +see which was the better. At nine a second list came out, +and a candidate's right of challenge was then confined to the +bracket immediately above his own. If he proved himself the +equal of the man so challenged his name was transferred to +the upper bracket. To challenge and then to fail to substantiate +the claim to removal to a higher bracket was considered +rather ridiculous. Revised lists were published at 11~a.m., +3~p.m., and 5~p.m., according to the results of the examination +during that day. At five the whole examination ended. The +proctors, moderators, and examiners then retired to a room +under the Public Library to prepare the list of honours, which +was sometimes settled without much difficulty in a few hours, +but sometimes not before 2~a.m.\ or 3~a.m.\ the next morning. +The name of the senior wrangler was generally announced at +midnight, and the rest of the list the next morning. In 1802 +there were eighty-six candidates for honours, and they were +\PG----File: 237.png------------------------------------------------------ +divided into fifteen brackets, the first and second brackets containing +each one name only, and the third bracket four names. + +It is clear from the above account that the competition\index +{Competition, in Tripos} +fostered by the examination had developed so much as to +threaten to impair its usefulness as guiding the studies of the +men. On the other hand, there can be no doubt that the +carefully devised arrangements for obtaining an accurate order +of merit stimulated the best men to throw all their energies +into the work for the examination. It is easy to point out +the usual double-edged result of a strict order of merit. The +problem before the University was to retain its advantages +while checking any abuses to which it might lead. + +It was the privilege of the moderators to entertain the +proctors and some of the leading resident mathematicians the +night before the issue of the final list, and to communicate +that list in confidence to their guests. This pleasant custom +survived till 1884. I revived the practice in 1890 when +acting as senior moderator, but it seems to have now ceased. + +\phantomsection +\addcontentsline{toc}{section}{Scheme of Reading in 1806} +In 1806 Sir Frederick Pollock\index{Pollock, F.|(} was senior wrangler, +and in 1869 in answer to an appeal from De~Morgan\index +{DeMorgan@De Morgan, A.} for an account +of the mathematical study of men at the beginning of the +century he wrote a letter\footnote +{\textit{Memoir of A.~de~Morgan}, London, 1882, pp.~387--392.} +which is sufficiently interesting to bear reproduction: + +\begin{Quotation} +I shall write in answer to your inquiry, \emph{all} about my books, my +studies, and my degree, and leave you to settle all about the proprieties +which my letter may give rise to, as to egotism, modesty,~\&c. The only +books I read the first year were Wood's\index{Wood, J.} \textit{Algebra} +(as far as quadratic equations), Bonnycastle's\index{Bonnycastle, J.} ditto, +and \textit{Euclid} (Simpson's)\index{Simpson's Euclid}. In the second +year I read Wood (beyond quadratic equations), and Wood and Vince\index +{Vince, S.}, for what they called the \emph{branches}. In the third year +I read the \emph{Jesuit's} Newton\index{Newton} and Vince's \textit +{Fluxions}\index{Fluxions}; these were all the \emph{books}, but there were +certain \textsc{mss}.\ floating about which I copied---which belonged to +Dealtry\index{Dealtry, W.}, +second wrangler in Kempthorne's year. I have no doubt that I had read +less and seen fewer books than any senior wrangler of about my time, or +any period since; but what I knew I knew thoroughly, and it was completely +at my fingers' ends. I consider that I was the last \emph{geometrical} +and \emph{fluxional} senior wrangler; I was not up to the \emph{differential} +calculus, +\PG----File: 238.png------------------------------------------------------ +and never acquired it. I went up to college with a knowledge of Euclid +and algebra to quadratic equations, nothing more; and I never read any +second year's lore during my first year, nor any third year's lore during +my second; my \emph{forte} was, that what I \emph{did} know I \emph +{could produce at any moment with \textsc{perfect} accuracy}. +I could repeat the first book of Euclid +word by word and letter by letter. During my first year I was not a +`\emph{reading}' man (so called); I had no expectation of honours or +a fellowship, +and I attended all the lectures on all subjects---Harwood's anatomical, +Woollaston's chemical, and Farish's mechanical lectures---but the examination +at the end of the first year revealed to me my powers. I was not +only in the first class, but it was generally understood I was +\emph{first} in the +first class; neither I nor any one for me expected I should get in at all. +Now, as I had taken no pains to prepare (taking, however, marvellous +pains while the examination was going on), I knew better than any one +else the value of my \emph{examination qualities} (great rapidity and perfect +accuracy); and I said to myself, `If you're not an ass, you'll be senior +wrangler;' and \emph{I took to `reading' accordingly}. A curious circumstance +occurred when the Brackets came out in the Senate-house declaring the +result of the examination: I saw at the top the name of Walter +\emph{bracketed alone} (as he was); +in the bracket below were \emph{Fiott}, \emph{Hustler}\index{Hustler, J.D.}, +\emph{Jephson}. I +looked down and could not find my own name till I got to Bolland, when +my pride took fire, and I said, `I must have beaten \emph{that man}, so I will +look up again;' and on looking up carefully I found the nail had been +passed through my name, and I was at the top bracketed \emph{alone}, even +above Walter. You may judge what my feelings were at this discovery; +it is the only instance of two such brackets, and it made my fortune---that +is, made me independent, and gave me an immense college reputation. +It was said I was more than half of the examination before any +one else. The two moderators were Hornbuckle\index{Hornbuckle, T.W.}, +of St~John's, and Brown (Saint Brown)\index{Brown, J. (Saint)}, of Trinity. +The Johnian congratulated me. I said +perhaps I might be challenged; he said, `Well, if you are you're quite +safe---you may sit down and do nothing, and no one would get up to you +in a whole day.'\textellipsis + +Latterly the Cambridge examinations seem to turn upon very different +matters from what prevailed in my time. I think a Cambridge education +has for its object to make good members of society---not to extend science +and make profound mathematicians. The tripos questions in the Senate-house +ought not to go beyond certain limits, and geometry ought to be +cultivated and encouraged much more than it is. +\end{Quotation} + +To this De~Morgan replied\index{DeMorgan@De Morgan, A.}: + +\begin{Quotation} +Your letter suggests much, because it gives possibility of answer. +The \emph{branches} of algebra of course mainly refer to the second part of +\PG----File: 239.png------------------------------------------------------ +Wood\index{Wood, J.}, now called the theory of equations. Waring\index +{Waring, E.} was his guide. +Turner---whom you must remember as head of Pembroke, senior wrangler +of 1767---told a young man in the hearing of my informant to be sure +and attend to quadratic equations. `It was a quadratic,' said he, `made +me senior wrangler.' It seems to me that the Cambridge \emph{revivers} were +Waring, Paley\index{Paley, W.}, Vince\index{Vince, S.}, +Milner\index{Milner, I.}. + +You had Dealtry's\index{Dealtry, W.} \textsc{mss}. He afterwards published +a very good book on fluxions\index{Fluxions}. +He merged his mathematical fame in that of a Claphamite +Christian. It is something to know that the tutor's \textsc{ms}.\ was +in vogue in 1800--1806. + +Now---how did you get your conic sections? How much of Newton\index{Newton} +did you read? From Newton direct, or from tutor's manuscript? + +Surely Fiott was our old friend Dr~Lee. I missed being a pupil of +Hustler\index{Hustler, J.D.} by a few weeks. He retired just before I went +up in February 1823. The echo of Hornbuckle's\index{Hornbuckle, T.W.} answer +to you about the challenge has lighted on Whewell\index{Whewell, W.}, +who, it is said, wanted to challenge Jacob\index{Jacob, E.}, and +was answered that he could not beat [him] if he were to write the +whole day and the other wrote nothing. I do not believe that Whewell +would have listened to any such dissuasion. + +I doubt your being the last fluxional senior wrangler. So far as I +know, Gipps, Langdale, Alderson, Dicey, Neale, may contest this point +with you. +\end{Quotation} + +The answer of Sir Frederick Pollock to these questions is +dated August~7, 1869, and is as follows. + +\begin{Quotation} +You have put together as \emph{revivers} five very different men. +Woodhouse\index{Woodhouse, R.} +was better than Waring\index{Waring, E.}, who could not prove +Wilson's\index{Wilson's Theorem} (Judge of C.P.) +guess about the property of prime numbers; but Woodhouse (I think) +did prove it, and a beautiful proof it is. Vince\index{Vince, S.} was a +bungler, and I think utterly insensible of mathematical beauty. + +Now for your questions. I did not get my conic sections from Vince. +I copied a \textsc{ms}.\ of Dealtry\index{Dealtry, W.}. I fell in love +with the cone and its sections, +and everything about it. I have never forsaken my favourite pursuit; +I delighted in such problems as two spheres touching each other and also +the inside of a hollow cone,~\&c. As to Newton\index{Newton}, I read a +good deal (men \emph{now} read nothing), but I read much of the notes. +I detected a blunder which nobody seemed to be aware of. +Tavel\index{Tavel, G.F.}, tutor of Trinity, was not; +and he argued very favourably of me in consequence. The application +of the Principia I got from \textsc{mss}. The blunder was this: in +calculating the resistance of a globe at the end of a cylinder oscillating +in a resisting medium they had forgotten to notice that there is a difference +between the resistance to a globe and a circle of the same diameter. + +\PG----File: 240.png------------------------------------------------------ +The story of Whewell\index{Whewell, W.} and Jacob cannot be true. Whewell +was a very, \emph{very} considerable man, I think not a \emph{great} man. +I have no doubt Jacob\index{Jacob, E.} +beat him in accuracy, but the supposed answer \emph{cannot} be true; it is a +mere echo of what actually passed between me and Hornbuckle\index +{Hornbuckle, T.W.} on the +day the Tripos came out---for the truth of which I vouch. I think the +examiners are taking too \emph{practical} a turn; it is a waste of time +to calculate \emph{actually} a longitude by the help of logarithmic tables +and lunar observations. It would be a fault not to know \emph{how}, +but a greater to be handy at it\index{Pollock, F.|)}. +\end{Quotation} + +A few minor changes in the Senate-House examinations +were made in 1808\footnote{See Graces, December 15, 1808.}. +A fifth day was added to the examination. +Of the five days thus given up to it three were devoted +to mathematics, one to logic, philosophy, and religion, and +one to the arrangement of the brackets. Apart from the +evening paper the examination on each of the first three +days lasted six hours. Of these eighteen hours, eleven were +assigned to book-work and seven to problems. The problem +papers were set from 6 to 10 in the evening. + +A letter from Whewell\index{Whewell, W.} dated January~19, 1816, describes +his examination in the Senate-House\index{Competition, in Tripos}\footnote +{S.~Douglas\index{Douglas, S.}, \textit{Life of W.~Whewell}, London, +1881, p.~20.}. + +\begin{Quotation} +Jacob\index{Jacob, E.}. Whewell. Such is the order in which we are fixed +after a week's examination\textellipsis I had before been given to understand % NB tight ellipsis matches original +that a great +deal depended upon being able to write the greatest possible quantity in +the smallest time, but of the rapidity which was actually necessary I had +formed the most distant idea. I am upon no occasion a quick writer, +and upon subjects where I could not go on without sometimes thinking a +little I soon found myself considerably behind. I was therefore surprised, +and even astonished, to find myself bracketed off, as it is called, in the +second place; that is, on the day when a new division of the classes is +made for the purpose of having a closer examination of the respective +merits of men who come pretty near to each other, I was not classed +with anybody, but placed alone in the second bracket. The man who is +at the head of the list is of Caius College, and was always expected to be +very high, though I do not know that anybody expected to see him so +decidedly superior as to be bracketed off by himself. +\end{Quotation} + +\noindent The tendency to cultivate mechanical rapidity was a grave +evil, and lasted long after Whewell's time. According to +\PG----File: 241.png------------------------------------------------------ +rumour the highest honours in 1845 were obtained, to the +general regret of the University, by assiduous practice in +writing\index{Competition, in Tripos}\footnote +{For a contemporary account of this see C.A.~Bristed\index{Bristed, C.A.}, +\textit{Five Years in an English University}, New York, 1852, pp.~233--239.}. + +\phantomsection +\addcontentsline{toc}{section}{Introduction of modern analytical notation} +The devotion of the Cambridge school to geometrical and +fluxional\index{Fluxions} methods has led to its isolation from contemporary +continental mathematicians. Early in the nineteenth century +the evil consequence of this began to be recognized; and it +was felt to be little less than a scandal that the researches +of Lagrange\index{Lagrange}, Laplace\index{Laplace}, and Legendre\index +{Legendre} were unknown to many +Cambridge mathematicians save by repute. An attempt to +explain the notation and methods of the calculus as used on +the Continent was made by R.~Woodhouse\index{Woodhouse, R.}, who stands out +as the apostle of the new movement. It is doubtful if he +could have brought analytical methods into vogue by himself; +but his views were enthusiastically adopted by three students, +Peacock, Babbage\index{Babbage, C.}, and Herschel\index{Herschel, Sir John}, +who succeeded in carrying +out the reforms he had suggested. They created an Analytical +Society which Babbage explained was formed to advocate +``the principles of pure $d$-ism as opposed to the \emph{dot}-age of the +University.'' The character of the instruction in mathematics +at the University has at all times largely depended on the +text-books then in use, and the importance of good books +of this class was emphasized by a traditional rule that +questions should not be set on a new subject in the tripos +unless it had been discussed in some treatise suitable and +available for Cambridge students\footnote +{See \Eg, the Grace of November~14, 1827, referred to below.}. +Hence the importance +attached to the publication of the work on analytical trigonometry +by Woodhouse\index{Woodhouse, R.} in 1809, and of the works on the +differential calculus issued by members of the Analytical +Society in 1816 and 1820. + +In 1817 Peacock, who was moderator, introduced the +symbols for differentiation into the papers set in the Senate-House +examination. But his colleague continued to use the +\PG----File: 242.png----------------------------------------------------- +fluxional\index{Fluxions} notation. Peacock himself wrote on March~17 of +1817 (\IE\ shortly after the examination) on the subject as +follows\footnote +{\textit{Proceedings of the Royal Society}, London, 1859, vol.~ix, +pp.~538--9.}: + +\begin{Quotation} +I assure you\textellipsis that I shall never cease to exert myself to the % NB tight ellipsis matches original +utmost in the cause of reform, and that I will never decline any office which +may increase my power to effect it. I am nearly certain of being nominated +to the office of Moderator in the year 1818--19, and as I am an examiner +in virtue of my office, for the next year I shall pursue a course even +more decided than hitherto, since I shall feel that men have been +prepared for the change, and will then be enabled to have acquired a +better system by the publication of improved elementary books. I have +considerable influence as a lecturer, and I will not neglect it. It is by +silent perseverance only that we can hope to reduce the many-headed +monster of prejudice, and make the University answer her character as +the loving mother of good learning and science. +\end{Quotation} + +In 1818 all candidates for honours, that is, all men in +the first six preliminary classes, were allowed to attempt the +problems: this change was made by the moderators. + +In 1819 G.~Peacock, who was again moderator, induced +his colleague to adopt the new notation. It was employed +in the next year by Whewell, and in the following year +by Peacock again. Henceforth the calculus in its modern +language and analytical methods were freely used, new subjects +were introduced, and for many years the examination +provided a mathematical training fairly abreast of the times. + +By this time the disputations had ceased to have any +immediate effect on a man's place in the tripos. Thus +Whewell\index{Whewell, W.}\footnote +{\textit{Whewell's Writings and Correspondence}, ed.\ Todhunter, London, +1876, vol.~ii, p.~36.}, writing about his duties as moderator in 1820, +said: + +\begin{Quotation} +You would get very exaggerated ideas of the importance attached to it +[an Act] if you were to trust Cumberland; I believe it was formerly more +thought of than it is now. It does not, at least immediately, produce +any effect on a man's place in the tripos, and is therefore considerably +less attended to than used to be the case, and in most years is not very +interesting after the five or six best men: so that I look for a considerable +\PG----File: 243.png----------------------------------------------------- +exercise of, or rather demand for, patience on my part. The other part +of my duty in the Senate House consists in manufacturing wranglers, +senior optimes, etc.\ and is, while it lasts, very laborious. +\end{Quotation} + +Of the examination itself in this year he wrote as follows\footnote +{S.~Douglas\index{Douglas, S.}, \textit{Life of Whewell}, London, 1881, +p.~56.}: + +\begin{Quotation} +The examination in the Senate House begins to-morrow, and is rather +close work while it lasts. We are employed from seven in the morning +till five in the evening in giving out questions and receiving written +answers to them; and when that is over, we have to read over all the +papers which we have received in the course of the day, to determine who +have done best, which is a business that in numerous years has often +kept the examiners up the half of every night; but this year is not +particularly numerous. In addition to all this, the examination is conducted +in a building which happens to be a very beautiful one, with a marble +floor and a highly ornamented ceiling; and as it is on the model of a +Grecian temple, and as temples had no chimneys, and as a stove or a fire +of any kind might disfigure the building, we are obliged to take the +weather as it happens to be, and when it is cold we have the full benefit +of it---which is likely to be the case this year. However, it is only a few +days, and we have done with it. +\end{Quotation} + +\noindent A sketch of the examination in the previous year from the +point of view of an examinee was given by J.M.F.~Wright\index +{Wright, J.M.F.}\footnote +{\textit{Alma Mater}, London, 1827, vol.~ii, pp.~58--98.}, +but there is nothing of special interest in it. + +Sir George Airy\index{Airy, Sir Geo.}\footnote +{See \textit{Nature}, vol.~35, Feb.~24, 1887, pp.~397--399.} +gave the following sketch of his recollections +of the reading and studies of undergraduates of his +time and of the tripos of 1823, in which he had been senior +wrangler: + +\begin{Quotation} +At length arrived the Monday morning on which the examination +for the B.A. degree was to begin\textellipsis. We were all marched in +a body to the +Senate-House and placed in the hands of the Moderators. How the +``candidates for honours'' were separated from the \hoipolloi\ I do not +know, I presume that the Acts and the Opponencies had something to do +with it. The honour candidates were divided into six groups: and of +these Nos.~1 and 2 (united), Nos.~3 and 4 (united), and Nos.~5 and 6 +(united), received the questions of one Moderator. No.~1, Nos.~2 and 3 +(united), Nos.~4 and 5 (united), and No.~6, received those of the other +Moderator. The Moderators were reversed on alternate days. There +\PG----File: 244.png----------------------------------------------------- +were no printed question-papers: each examiner had his bound manuscript +of questions, and he read out his first question; each of the +examinees who thought himself able proceeded to write out his answer, +and then orally called out ``Done.'' The Moderator, as soon as he +thought proper, proceeded with another question. I think there was +only one course of questions on each day (terminating before 3~o'clock, +for the Hall dinner). The examination continued to Friday mid-day. +On Saturday morning, about 8~o'clock, the list of honours (manuscript) +was nailed on the door of the Senate-House. +\end{Quotation} + +\phantomsection +\addcontentsline{toc}{section}{Alterations in Schemes of Study, 1824} +It must be remembered that for students pursuing the +normal course the Senate-House examination still provided +the only avenue to a degree. That examination involved a +knowledge of the elements of moral philosophy and theology, +an acquaintance with the rules of formal logic, and the power +of reading and writing scholastic Latin, but mathematics was +the predominant subject, and this led to a certain one-sidedness +in education. The evil of this was generally recognized, and +in 1822 various reforms were introduced in the University +curriculum; in particular the Previous Examination was +established for students in their second year, the subjects +being prescribed Greek and Latin works, a Gospel, and Paley's\index{Paley, W.} +\textit{Evidences}. Set classical books were introduced in the final +examination of poll-men; and another honour or tripos +examination was established for classical students. These +alterations came into effect in 1824; and henceforth the +Senate-House examination, so far as it related to mathematical +students, was known as the Mathematical Tripos. + +\phantomsection +\addcontentsline{toc}{section}{Scheme of Examination in 1827} +In 1827 the scheme of examination in the Mathematical +Tripos was revised. By regulations\footnote +{See the Grace, November~14, 1827.} which came into operation +in January, 1828, another day was added, so that the +examination extended over four days, exclusive of the day +of arranging the brackets; the number of hours of examination +was twenty-three, of which seven were assigned to +problems. On the first two days all the candidates had the +same questions proposed to them, inclusive of the evening +\PG----File: 245.png----------------------------------------------------- +problems, and the examination on those days excluded the +higher and more difficult parts of mathematics, in order, in +the words of the report, ``that the candidates for honours +may not be induced to pursue the more abstruse and profound +mathematics, to the neglect of more elementary knowledge.'' +Accordingly, only such questions as could be solved without +the aid of the differential calculus were set on the first day, +and those set on the second day involved only its elementary +applications. The classes were reduced to four, determined +as before by the exercises in the schools. The regulations +of 1827 definitely prescribed that all the papers should be +printed\index{Examination, Printed}. +They are also noticeable as being the last which +gave the examiners power to ask \emph{vivâ voce} questions, though +such questions were restricted to ``propositions contained in +the mathematical works commonly in use in the University, +or examples and explanations of such propositions.'' It was +further recommended that no paper should contain more +questions than well-prepared students could be expected to +answer within the time allowed for it, but that if any +candidate, before the end of the time, had answered all +the questions in the paper, the examiners might propose +additional questions \emph{vivâ voce}. The power of granting +honorary optime\index{Optimes}\index{Honorary Optimes} degrees now +ceased; it had already fallen +into abeyance. Henceforth the examination was conducted +under definite rules, and I no longer concern myself with +the traditions of the examination. + +In the same year as these changes became effective the +examination for the poll degree\index{Poll Examinations} was separated +from the tripos with different sets of papers and a different schedule +of subjects\footnote +{See Grace, May~21, 1828, confirming a Report of March~27, 1828.}. +It was, however, still nominally considered +as forming part of the Senate-House examination, and until +1858 those who obtained a poll degree were arranged in +four classes, described as fourth, fifth, sixth, and seventh, +as if in continuation of the junior optimes or third class +of the tripos. The year 1828 therefore shews us the +\PG----File: 246.png----------------------------------------------------- +Senate-House examination dividing into two distinct parts; +one known as the mathematical tripos, the other as the +poll examination\index{Poll Examinations}. In 1851\footnoteT +{`1850' corrected to `1851' as per errata sheet} the classical tripos\index +{Classical Tripos} was made +independent of the mathematical tripos, and thus provided a +separate avenue to a degree. Historically, the examination +usually known as ``the General'' represents the old Senate-House +examination for the poll-men, but gradually it has +been moved to an earlier period in the normal course taken +by the men. In 1852 another set of examinations, at first +called ``the professor's examinations,'' and now somewhat +modified and known as ``the Specials,'' was instituted for all +poll-men to take before they could qualify for a degree. In 1858 +the fiction that the poll-examinations\index{Poll Examinations} were part of +the Senate-House examination was abandoned, and subsequently they +have been treated as providing an independent method of +obtaining the degree: thus now the mathematical tripos is the +sole representative of the old Senate-House examination. Since +1858 numerous other ways of obtaining the degree have been +established, and it is now possible to get it by shewing +proficiency in very special, or even technical subjects. + +\phantomsection +\addcontentsline{toc}{section}{Scheme of Examination in 1833} +Further changes in the mathematical tripos were introduced +in 1833\footnote{See the Grace of April~6, 1832.}. +The duration of the examination, before the issue +of the brackets\index{Brackets, in Tripos}, was extended to five days, +and the number of +hours of examination on each day was fixed at five and a-half. +Seven and a-half hours were assigned to problems. The examination +on the first day was confined to subjects that did +not require the differential calculus, and only the simplest +applications of the calculus were permitted on the second and +third days. During the first four days of the examination +the same papers were set to all the candidates alike, but on the +fifth day the examination was conducted according to classes. +No reference was made to \emph{vivâ voce} questions, and the +preliminary classification of the brackets only survived in a +permission to re-examine candidates if it were found necessary. +This permissive rule remained in force till 1848, but I believe +\PG----File: 247.png----------------------------------------------------- +that in fact it was never used. In December, 1834, a few +unimportant details were amended. + +\phantomsection +\addcontentsline{toc}{subsection}{All the papers marked} +Mr Earnshaw\index{Earnshaw, S.}, the senior moderator in 1836, informed me +that he believed that the tripos of that year was the earliest +one in which all the papers were marked, and that in previous +years the examiners had partly relied on their impression of +the answers given. + +\phantomsection +\addcontentsline{toc}{section}{Scheme of Examination in 1839} +New regulations came into force\footnote +{See Grace of May~30, 1838.} in 1839. The examination +now lasted for six days, and continued as before for five +hours and a-half each day. Eight and a-half hours were +assigned to problems. Throughout the whole examination the +same papers were set to all candidates, and no reference was +made to any preliminary classes. It was no doubt in accordance +with the spirit of these changes that the acts in the +schools should be abolished, but they were discontinued by +the moderators of 1839 without the authority of the Senate. +The examination was for the future confined\footnote +{Under a badly-worded grace passed on May~11, 1842, on the +recommendation of a syndicate on theological studies, candidates for +mathematical honours were, after 1846, required to attend the poll +examination on Paley's Moral Philosophy, the New Testament and +Ecclesiastical History. This had not been the intention of the Senate, +and on March~14, 1855, a grace was passed making this clear.} +to mathematics. + +In the same year in which the new scheme came into force +a proposal to again reopen the subject was rejected (March~6). + +The difficulty of bringing professorial lectures into relation +with the needs of students has more than once been before +the University. The desirability of it was emphasized by a +Syndicate in February, 1843, which recommended conferences +at stated intervals between the mathematical professors and +examiners. This report foreshadowed the creation of a Mathematical +Board\index{Board, Mathematical}, but it was rejected by the Senate +on March~31. + +\phantomsection +\addcontentsline{toc}{section}{Scheme of Examination in 1848} +A few years later the scheme of the examination was again +reconstructed by regulations\footnote +{See Grace of May~13, 1846, confirming a report of March~23, 1846.} +which came into effect in 1848. +The duration of the examination was extended to eight days. +\PG----File: 248.png------------------------------------------------------ +The examination lasted in all forty-four and a-half hours, +twelve of which were devoted to problems. The first three +days were assigned to specified elementary subjects; in the +papers set on these days riders were to be set as well as bookwork, +but the methods of analytical geometry and the calculus +were excluded. After the first three days there was a +short interval, at the end of which the examiners issued a list +of those who had so acquitted themselves as to deserve mathematical +honours. Only those whose names were contained in +this list were admitted to the last five days of the examination, +which was devoted to the higher parts of mathematics. After +the conclusion of the examination the examiners, taking into +account the whole eight days, brought out the list arranged +in order of merit. No provision was made for any rearrangement +of this list corresponding to the examination of the +brackets\index{Brackets, in Tripos}. The arrangements of 1848 +remained in force till 1873. + +\phantomsection +\addcontentsline{toc}{section}{Creation of a Board of Mathematical Studies} +In the same year as these regulations came into force, a +Board of Mathematical Studies (consisting of the mathematical +professors, and the moderators and examiners for the current +year and the two preceding years) was constituted\footnote +{See Grace of October~31, 1848.} by the +Senate. From that time forward their minutes supply a permanent +record of the changes gradually introduced into the +tripos. I do not allude to subsequent changes which only +concern unimportant details of the examination. + +In May, 1849, the Board\index{Board, Mathematical} issued a report in which, +after giving a review of the past and existing state of the mathematical +studies in the University, they recommended that the +mathematical theories of electricity, magnetism, and heat +should not be admitted as subjects of examination. In the +following year they issued a second report, in which they +recommended the omission of elliptical integrals, Laplace's +coefficients, capillary attraction, and the figure of the earth +considered as heterogeneous, as well as a definite limitation +of the questions in lunar and planetary theory. In making +\PG----File: 249.png----------------------------------------------------- +these recommendations the Board were only giving expression +to what had become the practice in the examination. + +I may, in passing, mention a curious attempt which was +made in 1853 and\footnoteT{`1853 and' inserted as per errata sheet} 1854 +to assist candidates in judging of the relative +difficulty of the questions asked. This was effected by giving +to the candidates, at the same time as the examination paper, +a slip of paper on which the marks assigned for the bookwork +and rider for each question were printed. I mention +the fact merely because these things are rapidly forgotten and +not because it is of any intrinsic value. I possess a complete +set of slips which came to me from Dr~Todhunter\index{Todhunter, J.}. + +In 1856 there was an amusing difference of opinion +between the Vice-Chancellor and the moderators. The Vice-Chancellor +issued a notice to say that for the convenience of +the University he had directed the tripos lists to be published +at 8.0~a.m.\ as well as at 9.0~a.m., but when the University +arrived at 8.0 the moderators said that they should not read +the list until 9.0. + +\phantomsection +\addcontentsline{toc}{section}{Scheme of Examination in 1873} +Considerable changes in the scheme of examination were +introduced in 1873. On December~5, 1865, the Board had +recommended the addition of Laplace's coefficients and the +figure of the earth considered as heterogeneous as subjects of +the examination; the report does not seem to have been +brought before the Senate, but attention was called to the +fact that certain departments of mathematics and mathematical +physics found no place in the tripos schedules, and +were neglected by most students. Accordingly a syndicate +was appointed on June~6, 1867, to consider the matter, and +a scheme drawn up by them was approved in 1868\footnote +{See Grace of June~2, 1868. It was carried by a majority of only five +in a house of 75.} and +came into effect in 1873. The new scheme of examination +was framed on the same lines as that of 1848. The +subjects in the first three days were left unchanged, but +an extra day was added, devoted to the elements of +\PG----File: 250.png----------------------------------------------------- +mathematical physics. The essence of the modification was the +greatly extended range of subjects introduced into the schedule +of subjects for the last five days, and their arrangement in +divisions, the marks awarded to the five divisions being approximately +those awarded to the three days in proportion +to $2$, $1$, $1$, $1$, $2/3$ to $1$ respectively. Under the new regulations +the number of examiners was increased from four to five. + +The assignment of marks to groups of subjects was made +under the impression that the best candidates would concentrate +their abilities on a selection of subjects from the various +divisions. But it was found that, unless the questions were +made extremely difficult, more marks could be obtained by +reading superficially all the subjects in the five divisions than +by attaining real proficiency in a few of the higher ones: +while the wide range of subjects rendered it practically impossible +to thoroughly cover all the ground in the time allowed. +The failure was so pronounced that in 1877 another syndicate +was appointed to consider the mathematical studies and +examinations of the University. They presented an elaborate +scheme, but on May~13, 1878, some of the most important +parts of it were rejected and their subsequent proposals, +accepted on November~21, 1878 (by 62 to 49), represented a +compromise which pleased few members of the Senate\footnote +{See Graces of May~17, 1877; May~29, 1878; and November~21, 1878: +and the \textit{Cambridge University Reporter}, April~2, May~14, June~4, +October~29, November~12, and November~26, 1878.}. + +\phantomsection +\addcontentsline{toc}{section}{Scheme of Examination in 1882} +Under the new scheme which came into force in 1882 +the tripos was divided into two portions: the first portion +was taken at the end of the third year of residence, +the range of subjects being practically the same as in the +regulations of 1848, and the result brought out in the +customary order of merit. The second portion was held in +the following January, and was open only to those who had +been wranglers in the preceding June. This portion was confined +to higher mathematics and appealed chiefly to specialists. +The result was brought out in three classes, each arranged in +\PG----File: 251.png----------------------------------------------------- +alphabetical order. The moderators and examiners conducted +the whole examination without any extraneous aid. + +In the next year or two further amendments were made\footnote +{See the Graces of December~13, 1883; June~12, 1884; February~10, +1885; October~29, 1885; and June~1, 1886.}, +moving the second part to the June of the fourth year, +throwing it open to all men who had graduated in the tripos +of the previous June, and transferring the conduct of the +examination in Part~2 to four examiners nominated by the +Board: this put it largely under the control of the professors. +The range of subjects of Part~2 was also greatly extended, +and candidates were encouraged to select only a few of them. +It was further arranged that Part~1 might be taken at the +end of a man's second year of residence, though in that case +it would not qualify for a degree. A student who availed +himself of this leave could take Part~2 at the end either of his +third or of his fourth year as he pleased. The tripos is +still (1905) carried on under the scheme of 1886. + +\phantomsection +\addcontentsline{toc}{subsection}{Fall in number of students +reading mathematics} +The general effect of these changes was to destroy the +homogeneity of the tripos. Objections to the new scheme +were soon raised. Especially, it was said---whether rightly +or wrongly---that Part~1 contained too many technical subjects +to serve as a general educational training for any save mathematicians; +that the distinction of a high place in the historic +list produced on its results tended to prevent the best men +taking it in their second year, though by this time they had +read sufficiently to be able to do so; and that Part~2 was so +constructed as to appeal only to professional mathematicians, +and that thus the higher branches of mathematics were neglected +by all save a few specialists. + +Whatever value be attached to these opinions, the number +of students studying mathematics fell rapidly under the scheme +of 1886. In 1899 the Board proposed\footnote +{See Reports dated November~7, 1899, and January~20, 1900.} further changes. +These seemed to some members of the Senate to be likely to still +further decrease the number of men who took up the subject +\PG----File: 252.png----------------------------------------------------- +as one of general education. At any rate the two main proposals +were rejected (February~15, 1900) by votes of 151 to +130 and 161 to 129. + +\phantomsection +\addcontentsline{toc}{section}{Origin of term Tripos} +The curious origin of the term tripos\index{Tripos, Origin of term|(} +has been repeatedly +told, and an account of it may fitly close this chapter. +Formerly there were three principal occasions on which questionists +were admitted to the title or degree of bachelor. The +first of these was the comitia priora, held on Ash-Wednesday, +for the best men in the year. The next was the comitia +posteriora, which was held a few weeks later, and at which +any student who had distinguished himself in the quadragesimal +exercises subsequent to Ash-Wednesday had his +seniority reserved to him. Lastly, there was the comitia +minora, for students who had in no special way distinguished +themselves. In the fifteenth century an important part in +the ceremony on each of these occasions was taken by a +certain ``ould bachilour,'' who sat upon a three-legged stool +or tripos before the proctors and tested the abilities of the +would-be graduates by arguing some question with the ``eldest +son,'' who was selected from them as their representative. +To assist the latter in what was often an unequal contest his +``father,'' that is, the officer of his college who was to present +him for his degree, was allowed to come to his assistance. + +Originally the ceremony was a serious one, and had a +certain religious character. It took place in Great St~Mary's +Church, and marked the admission of the student to a position +with new responsibilities, while the season of Lent was chosen +with a view to bring this into prominence. The Puritan party +objected to the observance of such ecclesiastical ceremonies, +and in the course of the sixteenth century they introduced +much license and buffoonery into the proceedings. The part +played by the questionist became purely formal. A serious +debate still sometimes took place between the father of the +senior questionist and a regent master who represented the +University; but the discussion was prefaced by a speech by +the bachelor, who came to be called Mr~Tripos just as we +\PG----File: 253.png----------------------------------------------------- +speak of a judge as the bench, or of a rower as an oar. +Ultimately public opinion permitted Mr~Tripos to say pretty +much what he pleased, so long as it was not dull and was +scandalous. The speeches he delivered or the verses he +recited were generally preserved by the Registrary, and were +known as the tripos verses: originally they referred to the +subjects of the disputations then propounded. The earliest +copies now extant are those for 1575. + +\phantomsection +\addcontentsline{toc}{subsection}{Tripos Verses} +The University officials, to whom the personal criticisms +in which the tripos indulged were by no means pleasing, +repeatedly exhorted him to remember ``while exercising his +privilege of humour, to be modest withal.'' In 1740, says +Mr~Mullinger\index{Mullinger, J.B.}\footnote +{J.B.~Mullinger, \textit{The University of Cambridge}, Cambridge, vol.~i, +1873, pp.~175, 176.}, ``the authorities after condemning the excessive +license of the tripos announced that the comitia at +Lent would in future be conducted in the Senate-House; and +all members of the University, of whatever order or degree, +were forbidden to assail or mock the disputants with scurrilous +jokes or unseemly witticisms. About the year 1747--8, the +moderators initiated the practice of printing the honour lists +on the back of the sheets containing the tripos verses, and +after the year 1755 this became the invariable practice. By +virtue of this purely arbitrary connection these lists themselves +became known as the tripos; and eventually the examination +itself, of which they represented the results, also became +known by the same designation.'' + +The tripos ceased to deliver his speech about 1750, but +the issue of tripos verses continued for nearly 150 years +longer. During the latter part of this time they consisted of +four sets of verses, usually in Latin, but occasionally in Greek, +in which current topics in the University were treated lightly +or seriously as the writer thought fit. They were written +for the proctors and moderators by undergraduates or commencing +bachelors, who were supposed each to receive a pair +\PG----File: 254.png----------------------------------------------------- +of white kid gloves in recognition of their labours. Thus +gradually the word tripos changed its meaning ``from a thing +of wood to a man, from a man to a speech, from a speech to +sets of verses, from verses to a sheet of coarse foolscap paper, +from a paper to a list of names, and from a list of names to a +system of examination\footnote +{Wordsworth\index{Wordsworth, C.}, \textit{Scholae Academicae}, Cambridge, +1877, p.~21.}.'' + +In 1895 the proctors and moderators, without consulting +the Senate, sent in no verses, and thus, in spite of widespread +regret, an interesting custom of many centuries standing was +destroyed. No doubt it may be argued that the custom had +never been embodied in statute or ordinance, and thus was +not obligatory. Also it may be said that its continuance was +not of material benefit to anybody. I do not think that such +arguments are conclusive, and personally I regret the disappearance +of historic ties unless it can be shown that they +cause inconvenience, which of course in this case could not be +asserted\index{Tripos, Origin of term|)}. +\PG----File: 255.png------------------------------------------------------ + +% CHAPTER VIII. +\UseChapterVIIIHeadings + +\chapter{Three Geometrical Problems.} + + +\phantomsection +\addcontentsline{toc}{section}{The Three Problems} +\textsc{Among} the more interesting geometrical problems of +antiquity are three questions which attracted the special +attention of the early Greek mathematicians. Our knowledge +of geometry is derived from Greek sources, and thus these +questions have attained a classical position in the history of +the subject. The three questions to which I refer are (i)~the +duplication of a cube, that is, the determination of the side +of a cube whose volume is double that of a given cube; (ii)~the +trisection of an angle; and (iii)~the squaring of a circle, that +is, the determination of a square whose area is equal to that +of a given circle---each problem to be solved by a geometrical +construction involving the use of straight lines and circles only, +that is, by Euclidean geometry. + +With the restriction last mentioned all three problems are +insoluble\footnote +{F.~Klein\index{Klein}, \textit{Vorträge über ausgewählte Fragen +der Elementargeometrie}, Leipzig, 1895.}. +To duplicate a cube the length of whose side is $a$, +we have to find a line of length $x$, such that $x^3 = 2a^3$. Again, +to trisect a given angle, we may proceed to find the sine of the +angle, say $a$, then, if $x$ is the sine of an angle equal to one-third +of the given angle, we have $4x^3=3x-a$. Thus the first and second +problems, when considered analytically, require the solution of +a cubic equation; and since a construction by means of circles +(whose equations are of the form $x^2 + y^2 + ax + by + c = 0$) and +straight lines (whose equations are of the form +$\alpha x + \beta y +\gamma = 0$) +\PG----File: 256.png------------------------------------------------------ +cannot be equivalent to the solution of a cubic equation, it is +inferred that the problems are insoluble if in our constructions +we are restricted to the use of circles and right lines. If the +use of the conic sections is permitted, both of these questions +can be solved in many ways. The third problem is different in +character, but under the same restrictions it also is insoluble. + +I propose to give some of the constructions which have +been proposed for solving the first two of these problems. To +save space, I shall not draw the necessary diagrams, and in +most cases I shall not add the proofs: the latter present but +little difficulty. I shall conclude with some historical notes on +approximate solutions of the quadrature of the circle. + +\section[The Duplication of the Cube]% +{The Duplication of the Cube\protect\footnote +{See \textit{Historia Problematis de Cubi Duplicatione} by N.T.~Reimer\index +{Reimer on Delian Problem}, +Göttingen, 1798; and \textit{Historia Problematis Cubi Duplicandi} +by C.H.~Biering\index{Biering on Delian Problem}, Copenhagen, 1844: also +\textit{Das Delische Problem}, by A.~Sturm\index{Sturm, A.}, +Linz, 1895--7. Some notes on the subject are given in my\index{Ball} +\textit{History of Mathematics}.}} + +\phantomsection +\addcontentsline{toc}{subsection}{Legendary origin of the problem} +The problem of the duplication of the cube\index +{Cube, Dup@\textsc{Cube, Duplication of}|(}% +\index{Duplication@\textsc{Duplication of Cube}|(} was known in +ancient times as the Delian problem\index{Delian@\textsc{Delian Problem}|(}, +in consequence of a legend that the Delians had consulted Plato\index +{Plato on Delian Problem} on the subject. +In one form of the story, which is related by Philoponus\index +{Philoponus on Delian Problem}\footnote +{\textit{Philoponus ad Aristotelis Analytica Posteriora}, bk.~\textsc{i}, +chap.~vii.}, it +is asserted that the Athenians in 430~\textsc{b.c.}, when suffering from +the plague of eruptive typhoid fever, consulted the oracle at +Delos as to how they could stop it. Apollo replied that they +must double the size of his altar which was in the form of a +cube. To the unlearned suppliants nothing seemed more easy, +and a new altar was constructed either having each of its edges +double that of the old one (from which it followed that the +volume was increased eight-fold) or by placing a similar cubic +altar next to the old one. Whereupon, according to the +legend, the indignant god made the pestilence worse than +before, and informed a fresh deputation that it was useless to +\PG----File: 257.png------------------------------------------------------ +trifle with him, as his new altar must be a cube and have a +volume exactly double that of his old one. Suspecting a +mystery the Athenians applied to Plato\index{Plato on Delian Problem}, +who referred them to +the geometricians. The insertion of Plato's name is an obvious +anachronism. Eratosthenes\index{Eratosthenes}\footnote +{\textit{Archimedis Opera cum Eutocii Commentariis}, ed.\ Torelli, Oxford, +1792, p.~144; ed.\ Heiberg, Leipzig, 1880--1, vol.~\textsc{iii}, pp.~104--107. +} relates a somewhat similar +story, but with Minos\index{Minos} as the propounder of the problem. + +In an Arab work, the Greek legend was distorted +into the following extraordinarily impossible piece of history, +which I cite as a curiosity of its kind. ``Now in the days +of Plato,'' says the writer, ``a plague broke out among the +children of Israel. Then came a voice from heaven to one +of their prophets, saying, `Let the size of the cubic altar be +doubled, and the plague will cease'; so the people made another +altar like unto the former, and laid the same by its side. +Nevertheless the pestilence continued to increase. And again +the voice spake unto the prophet, saying, `They have made a +second altar like unto the former, and laid it by its side, but +that does not produce the duplication of the cube.' Then +applied they to Plato\index{Plato on Delian Problem}, the Grecian sage, +who spake to them, +saying, `Ye have been neglectful of the science of geometry, +and therefore hath God chastised you, since geometry is the +most sublime of all the sciences.' Now, the duplication of +a cube depends on a rare problem in geometry, namely\textellipsis''. +And then follows the solution of Apollonius\index{Apollonius}, which is given +later. + +If $a$ is the length of the side of the given cube and $x$ that +of the required cube, we have $x^3= 2a^3$, that is, $x: a = \sqrt[3]{2}: 1$. +It is probable that the Greeks were aware that the latter ratio +is incommensurable, in other words, that no two integers +can be found whose ratio is the same as that of $\sqrt[3]{2}: 1$, +but it did not therefore follow that they could not find the +ratio by geometry: in fact, the side and diagonal of a square +are instances of lines whose numerical measures are incommensurable. + +\PG----File: 258.png------------------------------------------------------ +I proceed now to give some of the geometrical constructions +which have been proposed for the duplication of the cube\footnote +{On the application to this problem of the traditional Greek methods +of analysis by Hero\index{Hero@Hero of Alexandria on $\pi$} and +Philo\index{Philo} (leading to the solution by the use of +Apollonius's circle), by Nicomedes\index{Nicomedes} (leading to +the solution by the use of the conchoid\index{Conchoid, the}), +and by Pappus\index{Pappus} (leading to the solution by the use of the +cissoid\index{Cissoid, the}), see \textit{Geometrical Analysis} by +J.~Leslie\index{Leslie, J.}, Edinburgh, second edition, +1811, pp.~247--250, 453.}. +With one exception, I confine myself to those which can be +effected by the aid of the conic sections. + +\phantomsection +\addcontentsline{toc}{section}{Lemma of Hippocrates} +Hippocrates\index{Hippocrates of Chios}\footnote +{Proclus, ed.\ Friedlein, pp.~212, 213.} +(circ.\ 420~\textsc{b.c.}) was perhaps the earliest mathematician +who made any progress towards solving the problem. +He did not give a geometrical construction, but he reduced +the question to that of finding two means between one +straight line ($a$), and another twice as long ($2a$). If these +means are $x$ and $y$, we have $a: x = x: y = y: 2a$, from which it +follows that $x^3 = 2a^3$. It is in this form that the problem is +always presented now. Formerly any process of solution by +finding these means was called a mesolabum\index{Mesolabum}. + +\phantomsection +\addcontentsline{toc}{section}{\protect\tocsecbox +{Solutions of Archytas, Plato, Menaechmus, Apollonius, and Sporus}} +One of the first solutions of the problem was that given +by Archytas\index{Archytas on Delian Problem}\footnote +{\textit{Archimedis Opera}, ed.\ Torelli, p.~143; ed.\ Heiberg, +vol.~\textsc{iii}, pp.~98--103.\label{ibid:5}} +in or about the year 400~\textsc{b.c.} His construction +is equivalent to the following. On the diameter $OA$ of the +base of a right circular cylinder describe a semicircle whose +plane is perpendicular to the base of the cylinder. Let the +plane containing this semicircle rotate round the generator +through $O$, then the surface traced out by the semicircle will +cut the cylinder in a tortuous curve. This curve will itself be +cut by a right cone, whose axis is $OA$ and semi-vertical angle +is (say) $60^{\circ}$, in a point $P$, such that the projection of $OP$ on +the base of the cylinder will be to the radius of the cylinder in +the ratio of the side of the required cube to that of the given +cube. Of course the proof given by Archytas is geometrical; +and it is interesting to note that in it he shows himself familiar +with the results of the propositions Euc.~\textsc{iii},~18, +\textsc{iii},~35, and +\PG----File: 259.png---------------------------------------------------- +\textsc{xi},~19. To show analytically that the construction is correct, +take $OA$ as the axis of $x$, and the generator of the cylinder +drawn through $O$ as axis of $z$, then with the usual notation, +in polar coordinates, if $a$ is the radius of the cylinder, we +have for the equation of the surface described by the semicircle +$r = 2a \sin\theta$; for that of the cylinder $r \sin\theta = 2a \cos\phi$; +and for that of the cone $\sin\theta \cos\phi = \frac{1}{2}$. +These three surfaces +cut in a point such that $\sin^3\theta = \frac{1}{2}$, and therefore +$(r\sin\theta)^3=2a^3$. +Hence the volume of the cube whose side is $r \sin\theta$ is twice +that of the cube whose side is $a$. + +The construction attributed to Plato\index{Plato on Delian Problem}\footnote +{\Ibidref{ibid:5}{\textit{Archimedis Opera}}, +ed.\ Torelli, p.~135; ed.\ Heiberg, +vol.~\textsc{iii}, pp.~66--71.\label{ibid:6}} (circ.\ 360~\textsc{b.c.}) +depends on the theorem that, if $CAB$ and $DAB$ are two right-angled +triangles, having one side, $AB$, common, their other +sides, $AD$ and $BC$, parallel, and their hypothenuses, $AC$ and +$BD$, at right angles, then if these hypothenuses cut in $P$, we +have $PC : PB = PB : PA = PA : PD$. Hence, if such a figure +can be constructed having $PD = 2PC$, the problem will be +solved. It is easy to make an instrument by which the figure +can be drawn. + +The next writer whose name is connected with the problem +is Menaechmus\index{Menaechmus}\footnote +{\Ibidref{ibid:6}{\textit{Archimedis Opera}}, +ed.\ Torelli, pp.~141--143; ed.\ Heiberg, vol.~\textsc{iii}, +pp.~92--99.\label{ibid:7}}, who in or about 340~\textsc{b.c.} gave two +solutions of it. + +In the first of these he pointed out that two parabolas +having a common vertex, axes at right angles, and such that +the latus rectum of the one is double that of the other will +intersect in another point whose abscissa (or ordinate) will +give a solution. If we use analysis this is obvious; for, if +the equations of the parabolas are $y^2 = 2ax$ and $x^2=ay$, they +intersect in a point whose abscissa is given by $x^3 = 2a^3$. It is +probable that this method was suggested by the form in which +Hippocrates had cast the problem: namely, to find $x$ and $y$ so +that $a : x = x : y = y : 2a$, whence we have $x^2 = ay$ and $y^2 = 2ax$. + +The second solution given by Menaechmus was as follows. +\PG----File: 260.png---------------------------------------------------- +Describe a parabola of latus rectum $l$. Next describe a rectangular +hyperbola, the length of whose real axis is $4l$, and +having for its asymptotes the tangent at the vertex of the parabola +and the axis of the parabola. Then the ordinate and the +abscissa of the point of intersection of these curves are the +mean proportionals between $l$ and $2l$. This is at once obvious +by analysis. The curves are $x^2 = ly$ and $xy = 2l^2$. These +cut in a point determined by $x^3 = 2l^3$ and $y^3 = 4l^3$. Hence +$l : x = x : y = y : 2l$. + +The solution of Apollonius\index{Apollonius}\footnote +{\Ibidref{ibid:7}{\textit{Archimedis Opera}}, +ed.\ Torelli, p.~137; ed.\ Heiberg, +vol.~\textsc{iii}, pp.~76--79. The solution is given in my\index{Ball} +\textit{History of Mathematics}, London, 1901, p.~84.\label{ibid:8}}, +which was given about +220~\textsc{b.c.}, was as follows. The problem is to find two mean +proportionals between two given lines. Construct a rectangle +$OADB$, of which the adjacent sides $OA$ and $OB$ are respectively +equal to the two given lines. Bisect $AB$ in $C$. With $C$ +as centre describe a circle cutting $OA$ produced in $a$ and +cutting $OB$ produced in $b$, so that $aDb$ shall be a straight +line. If this circle can be so described, it will follow that +$OA : Bb = Bb : Aa = Aa : OB$, that is, $Bb$ and $Aa$ are the two +mean proportionals between $OA$ and $OB$. It is impossible +to construct the circle by Euclidean geometry, but Apollonius +gave a mechanical way of describing it. + +The only other construction of antiquity to which I will +refer is that given by Diocles\index{Diocles on Delian Problem} +and Sporus\index{Sporus on Delian Problem}\footnote +{\Ibidref{ibid:8}{\textit{Archimedis Opera}}, +ed.\ Torelli, pp.~138, 139, 141; ed.\ Heiberg, vol.~\textsc{iii}, +pp.~78--84, 90--93.}. It is as follows. +Take two sides of a rectangle $OA$, $OB$, equal to the two lines +between which the means are sought. Suppose $OA$ to be the +greater. With centre $O$ and radius $OA$ describe a circle. Let +$OB$ produced cut the circumference in $C$ and let $AO$ produced +cut it in $D$. Find a point $E$ on $BC$ so that if $DE$ cuts $AB$ +produced in $F$ and cuts the circumference in $G$, then $FE=EG$. +If $E$ can be found, then $OE$ is the first of the means between +$OA$ and $OB$. Diocles invented the cissoid\index{Cissoid, the} +in order to determine +\PG----File: 261.png------------------------------------------------------ +$E$, but it can be found equally conveniently by the aid of +conics. + +\phantomsection +\addcontentsline{toc}{section}{\protect\tocsecbox +{Solutions of Vieta, Descartes, Gregory of St~Vincent, and Newton}} +In more modern times several other solutions have been +suggested. I may allude in passing to three given by Huygens\index +{Huygens}\footnote{\textit{Opera Varia}, Leyden, 1724, pp.~393--396.}, +but I will enunciate only those proposed respectively by Vieta\index{Vieta}, +Descartes\index{Descartes}, Gregory of St Vincent\index +{Gregoryof@Gregory of St Vincent}\index{StVin@St Vincent, Gregory of}, +and Newton\index{Newton}. + +Vieta's\index{Vieta} construction is as follows\footnote +{\textit{Opera Mathematica}, ed.\ Schooten, Leyden, 1646, prop,~\textsc{v}, +pp.~242--243.}. Describe a circle, centre +$O$, whose radius is equal to half the length of the larger of the +two given lines. In it draw a chord $AB$ equal to the smaller +of the two given lines. Produce $AB$ to $E$ so that $BE = AB$. +Through $A$ draw a line $AF$ parallel to $OE$. Through $O$ draw a +line $DOCFG$, cutting the circumference in $D$ and $C$, cutting $AF$ +in $F$, and cutting $BA$ produced in $G$, so that $GF = OA$. If +this line can be drawn then $AB : GC = GC : GA = GA : CD$. + +Descartes\index{Descartes} pointed out\footnote +{\textit{Geometria}, bk.~\textsc{iii}, ed.\ Schooten, Amsterdam, 1659, p.~91.} +that the curves $x^2 = ay$ and +$x^2 + y^2 = ay + bx$ cut in a point $(x, y)$ such that +$a : x = x : y = y : b$. +Of course this is equivalent to the first solution given by +Menaechmus, but Descartes preferred to use a circle rather +than a second conic. + +Gregory's construction was given in the form of the following +theorem\footnote +{Gregory of St Vincent, \textit{Opus Geometricum Quadraturae Circuli}, +Antwerp, 1647, bk.~\textsc{vi}, prop.~138, p.~602.}. +The hyperbola drawn through the point of intersection +of two sides of a rectangle so as to have the two other +sides for its asymptotes meets the circle circumscribing the +rectangle in a point whose distances from the asymptotes are +the mean proportionals between two adjacent sides of the rectangle. +This is the geometrical expression of the proposition +that the curves $xy = ab$ and $x^2 + y^2 = ay + bx$ cut in a point +$(x, y)$ such that $a : x = x : y = y : b$. + +One of the constructions proposed by Newton\index{Newton} +is as follows\footnote +{\textit{Arithmetica Universalis}, Ralphson's (second) edition, +1728, p.~242; see also pp.~243, 245.}. +\PG----File: 262.png------------------------------------------------------ +Let $OA$ be the greater of two given lines. Bisect $OA$ in $B$. +With centre $O$ and radius $OB$ describe a circle. Take a point +$C$ on the circumference so that $BC$ is equal to the other of the +two given lines. From $O$ draw $ODE$ cutting $AC$ produced in +$D$, and $BC$ produced in $E$, so that the intercept $DE=OB$. +Then $BC : OD = OD : CE = CE : OA$. Hence $OD$ and $CE$ are +two mean proportionals between any two lines $BC$ and $OA$% +\index{Cube, Dup@\textsc{Cube, Duplication of}|)}% +\index{Delian@\textsc{Delian Problem}|)}% +\index{Duplication@\textsc{Duplication of Cube}|)}. + +\section[The Trisection of an Angle]{The Trisection of an Angle\footnote +{On the bibliography of the subject see the supplements to \textit +{L'Intermédiaire des Mathématiciens}, Paris, May and June, 1904.}} + +The trisection of an angle\index{Trisection@\textsc{Trisection of Angle}|(} +is the second of these classical +problems, but tradition has not enshrined its origin in romance. +The following two constructions are among the oldest and best +known of those which have been suggested; they are quoted +by Pappus\index{Pappus}\footnote +{Pappus, \textit{Mathematicae Collectiones}, bk.~\textsc{iv}, props.~32, 33 +(ed. Commandino, Bonn, 1670, pp.~97--99). On the application to this +problem of the traditional Greek methods of analysis see \textit{Geometrical +Analysis}, by J.~Leslie\index{Leslie, J.}, Edinburgh, second edition, 1811, +pp.~245--247.}, but I do not know to whom they were due originally. + +\phantomsection +\addcontentsline{toc}{section}{Solutions quoted by Pappus (three)} +The first of them is as follows. Let $AOB$ be the given +angle. From any point $P$ in $OB$ draw $PM$ perpendicular to +$OA$. Through $P$ draw $PR$ parallel to $OA$. On $MP$ take a point +$Q$ so that if $OQ$ is produced to cut $PR$ in $R$ then +$QR = 2\dotm OP$. +If this construction can be made, then $AOR=\frac{1}{3}AOB$. The +solution depends on determining the position of $R$. This +was effected by a construction which may be expressed analytically +thus. Let the given angle be $\tan^{-1} (b/a)$. Construct +the hyperbola $xy=ab$, and the circle $(x-a)^2 + (y-b)^2 = 4(a^2 + b^2)$. +Of the points where they cut, let $x$ be the abscissa which is +greatest, then $PR = x-a$, and $\tan^{-1} (b/x)=\frac{1}{3}\tan^{-1}(b/a)$. + +The second construction is as follows. Let $AOB$ be the +given angle. Take $OB=OA$, and with centre $O$ and radius $OA$ +describe a circle. Produce $AO$ indefinitely and take a point $C$ +on it external to the circle so that if $CB$ cuts the circumference +\PG----File: 263.png------------------------------------------------------ +in $D$ then $CD$ shall be equal to $OA$. Draw $OE$ parallel to $CDB$. +Then, if this construction can be made, $AOE = \frac{1}{3} AOB$. The +ancients determined the position of the point $C$ by the aid of +the conchoid\index{Conchoid, the}: it could be also found by the use of +the conic sections. + +I proceed to give a few other solutions; confining myself +to those effected by the aid of conics. + +Among the other constructions given by Pappus\index{Pappus}\footnote +{Pappus, bk.~\textsc{iv}, prop.~34, pp.~99--104.} I may +quote the following. Describe a hyperbola whose eccentricity +is two. Let its centre be $C$ and its vertices $A$ and $A'$. Produce +$CA'$ to $S$ so that $A'S = CA'$. On $AS$ describe a segment +of a circle to contain the given angle. Let the orthogonal +bisector of $AS$ cut this segment in $O$. With centre $O$ and +radius $OA$ or $OS$ describe a circle. Let this circle cut the +branch of the hyperbola through $A'$ in $P$. Then $SOP = \frac{1}{3}SOA$. + +\phantomsection +\addcontentsline{toc}{section}{Solutions of Descartes, Newton, +Clairaut, and Chasles} +In modern times one of the earliest of the solutions by a +direct use of conics was suggested by Descartes\index{Descartes}, who +effected it by the intersection of a circle and a parabola. His +construction\footnote +{\textit{Geometria}, bk.~\textsc{iii}, ed.\ Schooten, Amsterdam, 1659, p.~91.} +is equivalent to finding the points of intersection +other than the origin, of the parabola $y^2 = \frac{1}{4}x$ and the circle +$x^2 + y^2 - \frac{13}{4}x + 4ay = 0$. The ordinates of these points are +given by the equation $4y^3 = 3y - a$. The smaller positive root +is the sine of one-third of the angle whose sine is $a$. The +demonstration is ingenious. + +One of the solutions proposed by Newton\index{Newton} is practically +equivalent to the third one which is quoted above from Pappus\index{Pappus}. +It is as follows\footnote +{\textit{Arithmetica Universalis}, problem \textsc{xlii}, Ralphson's (second) +edition, London, 1728, p.~148; see also pp.~243--245.}. +Let $A$ be the vertex of one branch of a +hyperbola whose eccentricity is two, and let $S$ be the focus of +the other branch. On $AS$ describe the segment of a circle +containing an angle equal to the supplement of the given +angle. Let this circle cut the $S$ branch of the hyperbola in +$P$. Then $PAS$ will be equal to one-third of the given angle. + +\PG----File: 264.png------------------------------------------------------ +The following elegant solution is due to Clairaut\index +{Clairaut on Trisection of Angle}\footnote +{I believe that this was first given by Clairaut, but I have mislaid +my reference. The construction occurs as an example in the \textit{Geometry +of Conics}, by C.~Taylor\index{Taylor, Ch., on Trisection}, Cambridge, 1881, +No.~308, p.~126.}. Let +$AOB$ be the given angle. Take $OA = OB$, and with centre $O$ +and radius $OA$ describe a circle. Join $AB$, and trisect it in +$H$, $K$, so that $AH= HK= KB$. Bisect the angle $AOB$ by $OC$ +cutting $AB$ in $L$. Then $AH = 2 \dotm HL$. With focus $A$, vertex +$H$, and directrix $OC$, describe a hyperbola. Let the branch of +this hyperbola which passes through $H$ cut the circle in $P$. +Draw $PM$ perpendicular to $OC$ and produce it to cut the circle +in $Q$. Then by the focus and directrix property we have +$AP : PM = AH : HL = 2 : 1$, $\Therefore AP = 2 \dotm PM = PQ$. Hence, by +symmetry, $AP = PQ = QR$. $\Therefore AOP = POQ = QOR$. + +I may conclude by giving the solution which Chasles\index +{Chasles on Trisection of Angle}\footnote +{\textit{Traité des sections coniques}, Paris, 1865, art.~37, p.~36.} +regards as the most fundamental. It is equivalent to the +following proposition. If $OA$ and $OB$ are the bounding radii +of a circular arc $AB$, then a rectangular hyperbola having $OA$ +for a diameter and passing through the point of intersection of +$OB$ with the tangent to the circle at $A$ will pass through one +of the two points of trisection of the arc\index +{Trisection@\textsc{Trisection of Angle}|)}. + +% in screen format there's a seriously bad interaction between footnotes +% and pagination resulting in a widowed section heading, hence the hard pagebreak +\ifPaper\else\clearpage\fi +\section[The Quadrature of the Circle]{The Quadrature of the Circle\footnote +{See Montucla's\index{Montucla} \textit{Histoire des Recherches sur la +Quadrature du Cercle}, edited by P.L.~Lacroix\index{Lacroix}, Paris, 1831; +also various articles by A.~De~Morgan\index{DeMorgan@De Morgan, A.}, +and especially his \textit{Budget of Paradoxes}, London, 1872. A popular +sketch of the subject has been compiled by H.~Schubert\index +{Schubert@Schubert on $\pi$}, \textit{Die Quadratur +des Zirkels}, Hamburg, 1889; and since the publication of the earlier +editions of these \textit{Recreations} Prof.\ F.~Rudio\index +{Rudio@Rudio on $\pi$} of Zurich has given an +analysis of the arguments of Archimedes\index{Archimedes}, +Huygens\index{Huygens}, Lambert\index{Lambert@Lambert on $\pi$}, and +Legendre\index{Legendre} on the subject, with an introduction on the history +of the problem, Leipzig, 1892.}} + +The object of the third of the classical problems was the +\index{Circle@\textsc{Circle, Quadrature of}|(}% +\index{Squaring@\textsc{Squaring the Circle}|(}% +\index{Quadrature@\textsc{Quadrature of Circle}|(} +determination of a side of a square whose area should be equal +to that of a given circle. + +\PG----File: 265.png---------------------------------------------------- +The investigation, previous to the last two hundred years, +of this question was fruitful in discoveries of allied theorems, +but in more recent times it has been abandoned by those who +are able to realize what is required. The history of this subject +has been treated by competent writers in such detail that I +shall content myself with a very brief allusion to it. + +Archimedes\index{Archimedes} showed\footnote +{\textit{Archimedis Opera}, \Kukloumetresis, +prop.~\textsc{i}, ed.\ Torelli, pp.~203--205; +ed.\ Heiberg, vol.~\textsc{i}, pp.~258--261, vol.~\textsc{iii}, pp.~269--277.} +(what possibly was known before) that +the problem is equivalent to finding the area of a right-angled +triangle whose sides are equal respectively to the perimeter of +the circle and the radius of the circle. Half the ratio of these +lines is a number, usually denoted by $\pi$\index{P@$\pi$|(}. + +\phantomsection +\addcontentsline{toc}{section}{Incommensurability of \texorpdfstring +{$\pi$}{pi}} +That this number is incommensurable had been long +suspected, and has been now demonstrated. The earliest +analytical proof of it was given by Lambert\index +{Lambert@Lambert on $\pi$}\footnote +{\textit{Mémoires de l'Académie de Berlin} for 1761, Berlin, 1768, +pp.~265--322.} in 1761; in +1803 Legendre\index{Legendre}\footnote +{Legendre's \textit{Geometry}, Brewster's translation, Edinburgh, 1824, +pp.~239--245.} extended the proof to show that $\pi^2$ was also +incommensurable; and recently Lindemann\index +{Lindemann@Lindemann on $\pi$}\footnote +{Ueber die Zahl $\pi$, \textit{Mathematische Annalen}, Leipzig, 1882, +vol.~\textsc{xx}, pp.~213--225. The proof leads to the conclusion that, +if $x$ is a root of a rational integral algebraical equation, then $e^x$ +cannot be rational: hence, if $\pi i$ was the root of such an equation, +$e^{\pi i}$ could not be rational; but $e^{\pi i}$ is equal to $-1$, and +therefore is rational; hence $\pi i$ cannot be the root of +such an algebraical equation, and therefore neither can $\pi$.} +has shown that $\pi$ cannot be the root of a rational algebraical equation. + +An earlier attempt by James Gregory\index{GregoryJ@Gregory, Jas.} to give a +geometrical demonstration of this is worthy of notice. Gregory proved\footnote +{\textit{Vera Circuli et Hyperbolae Quadratura}, Padua, 1668: this is +reprinted in Huygens's\index{Huygens} \textit{Opera Varia}, Leyden, +1724, pp. 405--462.} +that the ratio of the area of any arbitrary sector to that of +the inscribed or circumscribed polygons is not expressible by +a finite number of algebraical terms. Hence he inferred that +the quadrature was impossible. This was accepted by Montucla\index{Montucla}, +\PG----File: 266.png---------------------------------------------------- +but it is not conclusive, for it is conceivable that some particular +sector might be squared, and this particular sector +might be the whole circle. + +In connection with Gregory's proposition above cited, +I may add that Newton\index{Newton}\footnote +{\textit{Principia}, bk.~\textsc{i}, section~\textsc{vi}, +lemma~\textsc{xxviii}.} proved that in any closed oval an +arbitrary sector bounded by the curve and two radii cannot +be expressed in terms of the co-ordinates of the extremities +of the arc by a finite number of algebraical terms. The argument +is condensed and difficult to follow: the same reasoning +would show that a closed oval curve cannot be represented by +an algebraical equation in polar co-ordinates. From this +proposition no conclusion as to the quadrature of the circle +is to be drawn, nor did Newton draw any. In the earlier +editions of this work I expressed an opinion that the result +presupposed a particular definition of the word oval, but on +more careful reflection I think that the conclusion is valid +without restriction. + +With the aid of the quadratrix, or the conchoid\index{Conchoid, the}, or the +cissoid\index{Cissoid, the}, the quadrature of the circle is easy, but the +construction of those curves assumes a knowledge of the value of $\pi$, +and thus the question is begged. + +\markright{Approximations to the value of $\pi$.} +\phantomsection +\addcontentsline{toc}{section}{Definitions of \texorpdfstring{$\pi$}{pi}} +I need hardly add that, if $\pi$ represented merely the ratio of +the circumference of a circle to its diameter, the determination +of its numerical value would have but slight interest. It is however +a mere accident that $\pi$ is defined usually in that way, and +it really represents a certain number which would enter into +analysis from whatever side the subject was approached. + +I recollect a distinguished professor explaining how different +would be the ordinary life of a race of beings born, as easily +they might be, so that the fundamental processes of arithmetic, +algebra and geometry were different to those which seem to +us so evident, but, he added, it is impossible to conceive of a +universe in which $e$ and $\pi$ should not exist. + +I have quoted elsewhere an anecdote, which perhaps will +bear repetition, that illustrates how little the usual definition +\PG----File: 267.png------------------------------------------------------ +of $\pi$ suggests its properties. De~Morgan\index +{DeMorgan@De Morgan, A.} was explaining to an +actuary what was the chance that a certain proportion of some +group of people would at the end of a given time be alive; and +quoted the actuarial formula, involving $\pi$, which, in answer to +a question, he explained stood for the ratio of the circumference +of a circle to its diameter. His acquaintance, who had so far +listened to the explanation with interest, interrupted him and +exclaimed, ``My dear friend, that must be a delusion, what can +a circle have to do with the number of people alive at the end +of a given time?'' In reality the fact that the ratio of the +length of the circumference of a circle to its diameter is the +number denoted by $\pi$ does not afford the best analytical +definition of $\pi$, and is only one of its properties. + +\phantomsection +\addcontentsline{toc}{section}{Origin of symbol \texorpdfstring{$\pi$}{pi}} +The use of a single symbol to denote this number $3.14159\ldots$ +seems to have been introduced about the beginning of the +eighteenth century. W.~Jones\index{Jones@Jones on $\pi$}\footnote +{\textit{Synopsis Palmariorum Matheseos}, London, 1706, pp.~243, 263 \etseq} +in 1706 represented it by $\pi$; +a few years later\footnote +{See notes by G.~Eneström\index{Enestrom@Eneström on $\pi$} in the +\textit{Bibliotheca Mathematica}, Stockholm, +1889, vol.~\textsc{iii}, p.~28; \Ibid, 1890, vol.~\textsc{iv}, p.~22.} +John Bernoulli denoted it by $c$; Euler\index{Euler} in +1734 used $p$, and in 1736 used $c$; Chr. Goldback in 1742 used +$\pi$; and after the publication of Euler's \textit{Analysis} the symbol +$\pi$ was generally employed. + +\phantomsection +\addcontentsline{toc}{section}{Methods of approximating to the +numerical value of \texorpdfstring{$\pi$}{pi}} +The numerical value of $\pi$ can be determined by either of +two methods with as close an approximation to the truth as is +desired. + +The first of these methods is geometrical. It consists in +calculating the perimeters of polygons inscribed in and circumscribed +about a circle, and assuming that the circumference +of the circle is intermediate between these perimeters\footnote +{The history of this method has been written by K.E.I.~Selander\index +{Selander@Selander on $\pi$}, +\textit{Historik öfver Ludolphska Talet}, Upsala, 1868.}. The +approximation would be closer if the areas and not the +perimeters were employed. The second and modern method +rests on the determination of converging infinite series for $\pi$. + +We may say that the $\pi$-calculators who used the first +\PG----File: 268.png------------------------------------------------------ +method regarded $\pi$ as equivalent to a geometrical ratio, but +those who adopted the modern method treated it as the symbol +for a certain number which enters into numerous branches +of mathematical analysis. + +\phantomsection +\addcontentsline{toc}{section}{Geometrical methods of approximation} +It may be interesting if I add here a list of some of the +approximations to the value of $\pi$ given by various writers\footnote +{For the methods used in classical times and the results obtained, +see the notices of their authors in M.~Cantor's\index{Cantor@Cantor on $\pi$} +\textit{Geschichte der Mathematik}, +Leipzig, vol.~\textsc{i}, 1880. For medieval and modern approximations, see +the article by A.~De~Morgan\index{DeMorgan@De Morgan, A.} on the Quadrature +of the Circle in vol.~\textsc{xix} of +the \textit{Penny Cyclopaedia}, London, 1841; with the additions given by +B.~de~Haan\index{DeHaan@De Haan on $\pi$}\index{Haan@Haan, de, on $\pi$} +in the \textit{Verhandelingen} of Amsterdam, 1858, vol.~\textsc{iv}, p.~22: +the conclusions were tabulated, corrected, and extended by +Dr~J.W.L.~Glaisher\index{Glaisher, J.W.L.} +in the \textit{Messenger of Mathematics}, Cambridge, 1873, vol.~\textsc{ii}, +pp.~119--128; and \Ibid, 1874, vol.~\textsc{iii}, pp.~27--46.}. +This will indicate incidentally those who have studied the subject +to the best advantage. + +\phantomsection +\addcontentsline{toc}{subsection}{Results of Egyptians, Babylonians, Jews} +The ancient Egyptians\footnote +{\textit{Ein mathematisches Handbuch der alten Aegypter} (\IE\ the Rhind +papyrus\index{Rhind Papyrus}), by A.~Eisenlohr\index{Eisenlohr on Ahmes}, +Leipzig, 1877, arts. 100--109, 117, 124.} +took $256/81$ as the value of $\pi$, this +is equal to $3.1605\dots$; but the rougher approximation of $3$ was +used by the Babylonians\footnote +{Oppert\index{Oppert@Oppert on $\pi$}, \textit{Journal Asiatique}, +August, 1872, and October, 1874.} and by the Jews\footnote +{1~Kings, ch.~7, ver.~23; 2~Chronicles, ch.~4, ver.~2.}. It is not unlikely +that these numbers were obtained empirically. + +We come next to a long roll of Greek mathematicians who +attacked the problem. Whether the researches of the members +of the Ionian School, the Pythagoreans, Anaxagoras\index{Anaxagoras}, +Hippias\index{Hippias}, Antipho\index{Antipho}, +and Bryso\index{Bryso} led to numerical approximations for the +value of $\pi$ is doubtful, and their investigations need not +detain us. The quadrature of certain lunes by Hippocrates\index +{Hippocrates of Chios} of Chios is ingenious and correct, +but a value of $\pi$ cannot be +thence deduced; and it seems likely that the later members +of the Athenian School concentrated their efforts on other +questions. + +It is probable that Euclid\index{Euclid}\footnote +{These results can be deduced from Euc.~\textsc{iv}, 15, and \textsc{iv}, 8: +see also book~\textsc{xii}, prop.~16.}, the illustrious founder of the +\PG----File: 269.png------------------------------------------------------ +Alex\-and\-rian School, was aware that $\pi$ was greater than $3$ and +less than $4$, but he did not state the result explicitly. + +\phantomsection +\addcontentsline{toc}{subsection}{Results of Archimedes and other +Greek writers} +The mathematical treatment of the subject began with +Ar\-chi\-medes\index{Archimedes}, who proved that $\pi$ is less than +$3\frac{1}{7}$ and greater +than $3\frac{10}{71}$, that is, it lies between $3.1428\ldots$ and +$3.1408\dots$. He established\footnote +{\textit{Archimedis Opera}, \Kukloumetresis, prop.~\textsc{iii}, ed.\ Torelli, +Oxford, 1792, pp.~205--216; ed.\ Heiberg, Leipzig, 1880, vol.~\textsc{i}, +pp.~263-271.} this by inscribing in a circle and circumscribing +about it regular polygons of $96$ sides, then determining by +geometry the perimeters of these polygons, and finally assuming +that the circumference of the circle was intermediate between +these perimeters: this leads to the result +$6336/2017\frac14 <\pi<14688/4673\frac12$,\footnoteT{Inserted $14688/$} +% http://itech.fgcu.edu/faculty/clindsey/mhf4404/archimedes/archimedes.html +from which he deduced the limits given above. This method +is equivalent to using the proposition $\sin\theta < \theta < \tan\theta$, +where $\theta= \pi/96$: the values of $\sin\theta$ and $\tan\theta$ were +deduced by +Archimedes from those of $\sin\frac13 \pi$ and $\tan\frac13\pi$ by repeated +bisections of the angle. With a polygon of $n$ sides this process +gives a value of $\pi$ correct to at least the integral part +of $(2\log n - 1.19)$ places of decimals. The result given by +Archimedes is correct to $2$ places of decimals. His analysis +leads to the conclusion that the perimeters of these polygons +for a circle whose diameter is $4970$ feet would lie between +$15610$ feet and $15620$ feet---actually it is about $15613$ feet +$9$ inches. + +Apollonius\index{Apollonius} discussed these results, but his criticisms have +been lost. + +Hero of Alexandria\index{Hero@Hero of Alexandria on $\pi $} gave\footnote +% the space after the \pi is required because the other index reference to +% Hero is inside a footnote and somehow garners a space on the way through +{\textit{Mensurae}, ed.\ Hultsch, Berlin, 1864, p.~188.} +the value $3$, but he quoted\footnote +{\textit{Geometria}, ed.\ Hultsch, Berlin, 1864, pp.~115, 136.} +the +result $22/7$: possibly the former number was intended only +for rough approximations. + +The only other Greek approximation that I need mention +is that given by Ptolemy\index{Ptolemy}\footnote +{\textit{Almagest}, bk.~\textsc{vi}, chap.~7; ed.\ Halma, vol.~\textsc{i}, +p.~421.}, who asserted that $\pi = 3^{\circ} 8' 30''$. +This is equivalent to taking +$\pi = 3 + \frac8{60} + \frac{30}{3600} =\allowbreak +3\frac{17}{120} =\allowbreak 3.141\dot6$. + +\PG----File: 270.png---------------------------------------------------- +\phantomsection +\addcontentsline{toc}{subsection}{Results of Roman surveyors and Gerbert} +The Roman surveyors seem to have used $3$, or sometimes $4$, +for rough calculations. For closer approximations they often +employed $3\frac{1}{8}$ instead of $3\frac{1}{7}$, since the fractions then +introduced are more convenient in duodecimal arithmetic. On the other +hand Gerbert\index{Gerbert}\footnote +{\textit{\OE{}uvres de Gerbert}, ed.\ Olleris, Clermont, 1867, p. 453.} +recommended the use of $22/7$. + +Before coming to the medieval and modern European +mathematicians it may be convenient to note the results arrived +at in India and the East. + +\phantomsection +\addcontentsline{toc}{subsection}{Results of Indian and Eastern writers} +Baudhayana\index{Baudhayana on $\pi$}\footnote +{The \textit{Sulvasutras} by G.~Thibaut\index{Thibaut on Baudhayana}, +\textit{Asiatic Society of Bengal}, 1875, +arts.~26--28.} took $49/16$ as the value of $\pi$. + +Arya-Bhata\index{Arya@Arya Bhata on $\pi$}\footnote +{\textit{Leçons de calcul d'Aryabhata}, by L.~Rodet\index +{Rodet on Arya-Bhata} in the \textit{Journal Asiatique}, +1879, series~7, vol.~\textsc{xiii}, pp.~10, 21.}, circ.\ 530, gave +$62832/20000$, which is equal +to $3.1416$. He showed that, if $a$ is the side of a regular +polygon of $n$ sides inscribed in a circle of unit diameter, and if +$b$ is the side of a regular inscribed polygon of $2n$ sides, then +$b^2=\frac{1}{2}-\frac{1}{2}(1-a^2)^{\frac{1}{2}}$. From the side of an +inscribed hexagon, he found +successively the sides of polygons of $12$, $24$, $48$, $96$, $192$, and +$384$ sides. The perimeter of the last is given as equal to $\sqrt{9.8694}$, +from which his result was obtained by approximation. + +Brahmagupta\index{Brahmagupta on $\pi$}\footnote +{\textit{Algebra\textellipsis from Brahmegupta and Bhascara}, trans. by % NB tight ellipsis matches original +H.T.~Colebrooke\index{Colebrooke on Indian Algebra}, +London, 1817, chap.~\textsc{xii}, art.~40, p.~308.\label{ibid:9}}, +circ.\ 650, gave $\sqrt{10}$, which is equal to +$3.1622\dots$. He is said to have obtained this value by inscribing +in a circle of unit diameter regular polygons of $12$, $24$, $48$, and +$96$ sides, and calculating successively their perimeters, which +he found to be $\sqrt{9.65}$, $\sqrt{9.81}$, $\sqrt{9.86}$, $\sqrt{9.87}$ +respectively; and +to have assumed that as the number of sides is increased +indefinitely the perimeter would approximate to $\sqrt{10}$. + +Bhaskara\index{Bhaskara@Bhaskara on $\pi$}, circ.\ 1150, gave two +approximations. One\footnote +{\Ibidref{ibid:9}{\textit{Algebra\textellipsis Bhascara}}, +p.~87.\label{ibid:10}}---possibly +copied from Arya-Bhata, but said to have been calculated +afresh by Archimedes's method from the perimeters of regular +polygons of $384$ sides---is $3927/1250$, which is equal to $3.1416$: +\PG----File: 271.png---------------------------------------------------- +the other\footnote +{\Ibidref{ibid:10}{\textit{Algebra\textellipsis Bhascara}}, +p.~95.} is $\frac{754}{240}$, +which is equal to $3.141\dot6$, but it is uncertain +whether this was not given only as an approximate value. + +Among the Arabs the values $22/7$, $\sqrt{10}$, and $62832/20000$ +were given by Alkarisimi\index{Alkarisimi@Alkarisimi on $\pi$}\footnote +{\textit{The Algebra of Mohammed ben Musa}, ed.\ by F.~Rosen\index +{Rosen@Rosen on Arab values of $\pi$}, London, 1831, +pp.~71--72.}, circ.\ 830; and no doubt were derived +from Indian sources. He described the first as an approximate +value, the second as used by geometricians, and the third as +used by astronomers. + +In Chinese\index{Chinese@Chinese on $\pi$} works the values $3$, +$\frac{22}{7}$, $\frac{157}{50}$ are said to +occur: probably the last two results were copied from the +Arabs. + +\phantomsection +\addcontentsline{toc}{subsection}{Results of European writers, 1200--1630} +Returning to European mathematicians, we have the +following successive approximations to the value of $\pi$: many of +those prior to the eighteenth century having been calculated +originally with the view of demonstrating the incorrectness of +some alleged quadrature. + +Leonardo of Pisa\index{Leonardo@Leonardo of Pisa on $\pi$}\footnote +{Boncompagni's \textit{Scritti di Leonardo}, vol.~\textsc{ii} (\textit +{Practica Geometriae}), Rome, 1862, p.~90.}, in the thirteenth century, +gave for $\pi$ the value $1440/458\frac{1}{3}$ +which is equal to $3.1418\dots$. In the +fifteenth century, Purbach\index{Purbach@Purbach on $\pi$}\footnote +{Appendix to the \textit{De Triangulis} of Regiomontanus\index +{Muller@Müller (Regiomontanus)}\index{Regiom@Regiomontanus on $\pi$}, +Basle, 1541, p.~131.} gave or quoted the value +$62832/20000$, which is equal to $3.141\dot6$; Cusa\index{Cusa on $\pi$} +believed that the accurate value was $\frac{3}{4}(\sqrt{3} + \sqrt{6})$ which +is equal to $3.1423\dots$; and, in 1464, Regiomontanus\footnote +{In his correspondence with Cardinal Cusa, \textit{De Quadratura Circuli}, +Nuremberg, 1533, wherein he proved that Cusa's result was wrong. +I cannot quote the exact reference, but the figures are given by competent +writers and I have no doubt are correct.} is said to have given +a value equal to $3.14243$. + +Vieta\index{Vieta}\footnote +{\textit{Canon Mathematicus seu ad Triangula}, Paris, 1579, pp.~56, 66: +probably this work was printed for private circulation only, +it is very rare.}, in 1579, showed that $\pi$ was greater than +\PG----File: 272.png------------------------------------------------------ +$31415926535/10^{10}$, and less than $31415926537/10^{10}$. This +was deduced from the perimeters of the inscribed and circumscribed +polygons of $6 \times 2^{16}$ sides, obtained by repeated use of +the formula $2 \sin^2\frac{1}{2}\theta = 1-\cos \theta$. +He also gave\footnote +{\textit{Vietae Opera}\index{Vieta}, ed.\ Schooten, Leyden, 1646, p.~400. +} a result equivalent to the formula +\[ +\frac{2}{\pi} = +\frac{\surd 2}{2} +\frac{\surd (2+\surd 2)}{2} +\frac{\surd \{2+\surd (2+\surd 2)\}}{2} +\dotsm\;. +\] + +The father of Adrian Metius\index{Metius@Metius on $\pi$}\footnote +{\textit{Arithmeticae libri duo et Geometriae}, by A.~Metius, Leyden, 1626, +pp.~88, 89. [Probably issued originally in 1611.]}, in 1585, gave $355/113$, +which is equal to $3.14159292\dots$, and is correct to $6$ places +of decimals. This was a curious and lucky guess, for all that +he proved was that $\pi$ was intermediate between $377/120$ and +$333/106$, whereon he jumped to the conclusion that he should +obtain the true fractional value by taking the mean of the +numerators and the mean of the denominators of these fractions. + +In 1593 Adrian Romanus\index{Romanus@Romanus on $\pi$}\footnote +{\textit{Ideae Mathematicae}, Antwerp, 1593: a rare work, which I have +never been able to consult.} calculated the perimeter of the +inscribed regular polygon of $1073,741824$ (\IE~$2^{30}$) sides, from +which he determined the value of $\pi$ correct to $15$ places of +decimals. + +L.~van Ceulen\index{Ceulen@Ceulen, van, on $\pi$}\index +{VanCeulen@Van Ceulen on $\pi$} devoted no inconsiderable part of his life to +the subject. In 1596\footnote +{\textit{Vanden Circkel}, Delf, 1596, fol.~14, p.~1; or +\textit{De Circulo}, Leyden, 1619, p.~3.} +he gave the result to $20$ places of decimals: +this was calculated by finding the perimeters of the +inscribed and circumscribed regular polygons of $60 \times 2^{33}$ sides, +obtained by the repeated use of a theorem of his discovery +equivalent to the formula $1-\cos A = 2 \sin^2\frac{1}{2}A$. I possess a +finely executed engraving of him of this date, with the result +printed round a circle which is below his portrait. He died in +1610, and by his directions the result to $35$ places of decimals +\PG----File: 273.png------------------------------------------------------ +(which was as far as he had then calculated it) was engraved on +his tombstone\footnote +{The inscription is quoted by Prof.\ de~Haan\index{DeHaan@De Haan on $\pi$}% +\index{Haan@Haan, de, on $\pi$} in the \textit{Messenger of +Mathematics}, 1874, vol.~\textsc{iii}, p.~25.} +in St Peter's Church, Leyden. His posthumous arithmetic\footnote +{\textit{De Arithmetische en Geometrische Fondamenten}, Leyden, 1615, p.~163; +or p.~144 of the Latin translation by W.~Snell, published at Leyden in +1615 under the title \textit{Fundamenta Arithmetica et Geometrica}. This was +reissued, together with a Latin translation of the \textit{Vanden Circkel}, +in 1619, under the title \textit{De Circulo}; in which see pp.~3, 29--32, 92.} +contains the result to $32$ places; this was obtained +by calculating the perimeter of a polygon, the number of whose +sides is $2^{62}$, \IE\ $4,611686,018427,387904$. Van~Ceulen\index +{Ceulen@Ceulen, van, on $\pi$}\index{VanCeulen@Van Ceulen on $\pi$} also +compiled a table of the perimeters of various regular polygons. + +Willebrord Snell\index{Snell@Snell on $\pi$}\footnote +{\textit{Cyclometricus}, Leyden, 1621, p.~55.}, +in 1621, obtained from a polygon of $2^{30}$ +sides an approximation to $34$ places of decimals. This is less +than the numbers given by van Ceulen, but Snell's method +was so superior that he obtained his $34$ places by the use of a +polygon from which van Ceulen had obtained only $14$ (or +perhaps $16$) places. Similarly, Snell obtained from a hexagon +an approximation as correct as that for which Archimedes\index{Archimedes} +had required a polygon of $96$ sides, while from a polygon of $96$ sides +he determined the value of $\pi$ correct to seven decimal places +instead of the two places obtained by Archimedes. The reason +is that Archimedes, having calculated the lengths of the sides +of inscribed and circumscribed regular polygons of $n$ sides, +assumed that the length of $1/n$th of the perimeter of the circle +was intermediate between them; whereas Snell constructed +from the sides of these polygons two other lines which gave +closer limits for the corresponding arc. His method depends +on the theorem +$3 \sin\theta /(2 + \cos\theta) < \theta +< (2 \sin\frac{1}{3}\theta + \tan\frac{1}{3}\theta)$, by +the aid of which a polygon of $n$ sides gives a value of $\pi$ correct +to at least the integral part of $(4 \log n - .2305)$ places of +decimals, which is more than twice the number given by the +older rule. Snell's proof of his theorem is incorrect, though +the result is true. + +\PG----File: 274.png---------------------------------------------------- +Snell also added a table\footnote +{It is quoted by Montucla, ed.\ 1831, p.~70.} of the perimeters of all regular +inscribed and circumscribed polygons, the number of whose +sides is $10\times 2^n$ where $n$ is not greater than $19$ and not less than +$3$. Most of these were quoted from van Ceulen, but some were +recalculated. This list has proved useful in refuting circle-squarers. +A similar list was given by James Gregory\index +{GregoryJ@Gregory, Jas.}\footnote +{\textit{Vera Circuli et Hyperbolae Quadratura}, prop.~29, quoted by +Huygens\index{Huygens}, \textit{Opera Varia}, Leyden, 1724, p.~447.}. + +In 1630 Grienberger\index{Grienberger@Grienberger on $\pi$}\footnote +{\textit{Elementa Trigonometrica}, Rome, 1630, end of preface.}, +by the aid of Snell's theorem, carried +the approximation to $39$ places of decimals. He was the last +mathematician who adopted the classical method of finding the +perimeters of inscribed and circumscribed polygons. Closer +approximations serve no useful purpose. Proofs of the theorems +used by Snell\index{Snell@Snell on $\pi$} and other calculators +in applying this +method were given by Huygens\index{Huygens} in a work\footnote +{\textit{De Circula Magnitudine Inventa}, 1654; \textit{Opera Varia}, +pp.~351--387. The proofs are given in G.~Pirie's\index{Pirie@Pirie on $\pi$} +\textit{Geometrical Methods of Approximating +the Value of $\pi$}, London, 1877, pp.~21--23.} which may be +taken as closing the history of this method. + +\phantomsection +\addcontentsline{toc}{section}{Theorems of Wallis and Brouncker} +In 1656 Wallis\index{Wallis, J.}\footnote +{\textit{Arithmetica Infinitorum}, Oxford, 1656, prop.~191. An analysis of +the investigation by Wallis was given by Cayley\index{Cayley}, +\textit{Quarterly Journal of +mathematics}, 1889, vol.~\textsc{xxiii}, pp.~165--169.} proved that +\[ +\frac{\pi}{2}=\frac{2\dotm 2\dotm 4\dotm 4\dotm 6\dotm 6\dotsm} +{1\dotm 3\dotm 3\dotm 5\dotm 5\dotm 7\dotm 7\dotsm}\;, +\] +and quoted a proposition given a few years earlier by Viscount +Brouncker\index{Brouncker on $\pi$} to the effect that +\[ +\frac{\pi}{4}=1+\frac{1^2}{2} \genfrac{}{}{0pt}{}{}{+} +\frac{3^2}{2} \genfrac{}{}{0pt}{}{}{+} \frac{5^2}{2} +\genfrac{}{}{0pt}{}{}{+}\ldots\;, +\] +but neither of these theorems was used to any large extent for +calculation. + +Subsequent calculators have relied on converging infinite +series, a method that was hardly practicable prior to the +\PG----File: 275.png------------------------------------------------------ +invention of the calculus, though Descartes\index{Descartes}\footnote +{See Euler's\index{Euler} paper in the \textit{Novi Commentarii Academiae +Scientiarum}, St Petersburg, 1763, vol.~\textsc{viii}, pp.~157--168.} +had indicated a geometrical process which was equivalent to the use of +such a series. The employment of infinite series was proposed +by James Gregory\footnote +{See the letter to Collins\index{Collins, Letter from J.~Gregory}, dated +Feb.~15, 1671, printed in the \textit{Commercium +Epistolicum}, London, 1712, p.~25, and in the Macclesfield Collection, +\textit{Correspondence of Scientific Men of the Seventeenth Century}, +Oxford, 1841, vol.~\textsc{ii}, p.~216.}, who established the theorem that +$\theta = \tan\theta - \frac{1}{3}\tan^3\theta + \frac{1}{5}\tan^5\theta + - \dotsb$, the result being true only +if $\theta$ lies between $-\frac{1}{4}\pi$ and $\frac{1}{4}\pi$. + +\phantomsection +\addcontentsline{toc}{section}{Analytical methods of approximation. +Gregory's series} +\phantomsection +\addcontentsline{toc}{subsection}{Results of European writers, 1699--1873} +The first mathematician to make use of Gregory's series\index +{GregorysS@Gregory's Series} +for obtaining an approximation to the value of $\pi$ was Abraham +Sharp\index{Sharp@Sharp on $\pi$}\footnote +{See \textit{Life of A.~Sharp} by W.~Cudworth\index{Cudworth on Sharp}, +London, 1889, p.~170. +Sharp's work is given in one of the preliminary discourses (p.~53 \etseq) +prefixed to H.~Sherwin's \textit{Mathematical Tables}\index +{Sherwin's Tables}. The tables were issued at +London in 1705: probably the discourses were issued at the same time, +though the earliest copies I have seen were printed in 1717.}, who, in 1699, +on the suggestion of Halley\index{Halley@Halley on $\pi$}, determined +it to $72$ places of decimals ($71$ correct). He obtained this +value by putting $\theta = \frac{1}{6}\pi$ in Gregory's series. + +Machin\index{Machin@Machin's series for $\pi$}\footnote +{W.~Jones's\index{Jones@Jones on $\pi$} \textit{Synopsis Palmariorum}, +London, 1706, p.~243; and Maseres, \textit{Scriptores Logarithmici}, +London, 1796, vol.~\textsc{iii}, pp.~vii--ix, 155--164.}, earlier than 1706, +gave the result to $100$ places (all correct). +He calculated it by the formula +\[ +\tfrac{1}{4}\pi = 4 \tan^{-1}\tfrac{1}{5} -\tan^{-1}\tfrac{1}{239}\;. +\] + +De~Lagny\index{DeLagny@De Lagny on $\pi$}\index +{Lagny@Lagny on $\pi$}\footnote +{\textit{Histoire de l'Académie} for 1719, Paris, 1721, p.~144.}, in 1719, +gave the result to $127$ places of decimals ($112$ correct), calculating +it by putting $\theta = \frac{1}{6}\pi$ in Gregory's series. + +Hutton\index{Hutton, C.}\footnote +{\textit{Philosophical Transactions}, 1776, vol.~\textsc{lxvi}, +pp.~476--492.}, in 1776, and Euler\index{Euler}\footnote +{\textit{Nova Acta Academiae Scientiarum Petropolitanae} for 1793, +St~Petersburg, 1798, vol.~\textsc{xi}, pp.~133--149: the memoir was read +in 1779.}, in 1779, suggested the use of the formulae +$\frac{1}{4}\pi =\allowbreak \tan^{-1}\frac{1}{2} +\allowbreak +\tan^{-1}\frac{1}{3}$ or +$\frac{1}{4}\pi = 5 \tan^{-1}\frac{1}{7} + 2\tan^{-1}\frac{3}{79}$, +\PG----File: 276.png---------------------------------------------------- +but neither carried the approximation as far as had been done +previously. + +Vega\index{Vega@Vega on $\pi$}, in 1789\footnote +{\textit{Nova Acta Academiae Scientiarum Petropolitanae} for 1790, +St~Petersburg, 1795, vol.~\textsc{ix}, p.~41.}, gave the value of +$\pi$ to $143$ places of decimals ($126$ correct); and, in 1794\footnote +{\textit{Thesaurus Logarithmorum} (\textit{logarithmisch-trigonometrischer +Tafeln}), Leipzig, 1794, p.~633.}, to $140$ places ($136$ correct). + +Towards the end of the last century Baron Zach\index{Zach on $\pi$} saw in +the Radcliffe Library, Oxford, a manuscript by an unknown +author which gives the value of $\pi$ to $154$ places of decimals +($152$ correct). + +In 1841 Rutherford\index{Rutherford@Rutherford on $\pi$}\footnote +{\textit{Philosophical Transactions}, 1841, p.~283.} calculated it to +$208$ places of decimals ($152$ correct), using the formula +$\frac{1}{4}\pi=4\tan^{-1}\frac{1}{5} - \tan^{-1}\frac{1}{70} + + \tan^{-1}\frac{1}{99}$. + +In 1844 Dase\index{Dase@Dase on $\pi$}\footnote +{\textit{Crelle's Journal}, 1844, vol.~\textsc{xxvii}, p.~198.} calculated +it to $205$ places of decimals ($200$ correct), using the formula +$\frac{1}{4}\pi= \tan^{-1}\frac{1}{2} +\tan^{-1}\frac{1}{5} + + \tan^{-1}\frac{1}{8}$. + +In 1847 Clausen\index{Clausen@Clausen on $\pi$}\footnote +{Schumacher\index{Schumacher}, \textit{Astronomische Nachrichten}, +vol.~\textsc{xxv}, col.~207.} carried the approximation to $250$ places +of decimals ($248$ correct), calculating it independently by the +formulae $\frac{1}{4}\pi = 2\tan^{-1}\frac{1}{3} + \tan^{-1}\frac{1}{7}$ +and $\frac{1}{4}\pi = 4\tan^{-1}\frac{1}{5}-\tan^{-1}\frac{1}{239}$. + +In 1853 Rutherford\footnote +{\textit{Proceedings of the Royal Society}, Jan.~20, 1853, vol.~\textsc{vi}, +pp.~273-275.} carried his former approximation +to $440$ places of decimals (all correct), and William Shanks\index +{Shanks@Shanks on $\pi$} +prolonged the approximation to $530$ places. In the same year +Shanks published an approximation to $607$ places\footnote +{\textit{Contributions to Mathematics}, W.~Shanks, London, 1853, +pp.~86, 87.}: and in +1873 he carried the approximation to $707$ places of decimals\footnote +{\textit{Proceedings of the Royal Society}, 1872--3, vol.~\textsc{xxi}, +p.~318; 1873--4, vol.~\textsc{xxii}, p.~45.}. +These were calculated from Machin's\index{Machin@Machin's series for $\pi$} +formula. + +In 1853 Richter\index{Richter@Richter on $\pi$}, presumably in ignorance of +what had been done in England, found the value of $\pi$ to $333$ +places\footnote +{\textit{Grunert's Archiv}, vol.~\textsc{xxi}, p.~119.\label{ibid:11}} of +\PG----File: 277.png------------------------------------------------------ +decimals ($330$ correct); in 1854 he carried the approximation +to $400$ places\footnote +{\Ibidref{ibid:11}{\textit{Grunert's Archiv}}, +vol.~\textsc{xxiii}, p.~476: the approximation given in vol.~\textsc{xxii}, +p.~473, is correct only to $330$ places.\label{ibid:12}}; +and in 1855 carried it to $500$ places\footnote +{\Ibidref{ibid:12}{\textit{Grunert's Archiv}}, +vol.~\textsc{xxv}, p.~472; and \textit{Elbingen Anzeigen}, No.~85.}. + +Of the series and formulae by which these approximations +have been calculated, those used by Machin and Dase are +perhaps the easiest to employ. Other series which converge +rapidly are the following, +\[ +\frac{\pi}{6} = \frac{1}{2} + \frac{1}{2}\dotm \frac{1}{3\dotm 2^3} + + \frac{1\dotm 3}{2\dotm 4} \dotm \frac{1}{5\dotm 2^5} + \dotsb\;, +\] +and +\[ +\frac{\pi}{4} = 2 + 22\tan^{-1}\frac{1}{28} + + \tan^{-1}\frac{1}{443} - 5\tan^{-1}\frac{1}{1393} - + 10\tan^{-1}\frac{1}{11018}\;, +\] +the latter of these is due to Mr~Escott\footnote +{\textit{L'Intermédiaire des mathématiciens}, Paris, Dec.~1896, +vol.~\textsc{iii}, p.~276.}. + +As to those writers who believe that they have squared the +circle their number is legion and, in most cases, their ignorance +profound, but their attempts are not worth discussing here. +``Only prove to me that it is impossible,'' said one of them, +``and I will set about it immediately''; and doubtless the +statement that the problem is insoluble has attracted much +attention to it. + +\phantomsection +\addcontentsline{toc}{section}{Geometrical approximations} +Among the geometrical ways of approximating to the truth +the following is one of the simplest. Inscribe in the given +circle a square, and to three times the diameter of the circle +add a fifth of a side of the square, the result will differ from +the circumference of the circle by less than one-seventeen-thousandth +part of it. + +\phantomsection +\addcontentsline{toc}{section}{Approximations by the theory of probability} +An approximate value of $\pi$ has been obtained experimentally +by the theory of probability\index{Probabilities@Probabilities and $\pi$}. +On a plane a number +of equidistant parallel straight lines, distance apart $a$, are ruled; +and a stick of length $l$, which is less than $a$, is dropped on to +the plane. The probability that it will fall so as to lie across +one of the lines is $2l/\pi a$. If the experiment is repeated many +\PG----File: 278.png------------------------------------------------------ +hundreds of times, the ratio of the number of favourable cases +to the whole number of experiments will be very nearly equal +to this fraction: hence the value of $\pi$ can be found. In 1855 +Mr A.~Smith\index{Smith, A@Smith, A., on $\pi$}\label +{Smith:footnote:278}\footnote +{A.~De~Morgan\index{DeMorgan@De Morgan, A.}, \textit{Budget of Paradoxes}, +London, 1872, pp.~171, 172 [quoted from an article by De~Morgan published +in 1861].} of Aberdeen made $3204$ trials, and deduced +$\pi= 3.1553$. A pupil of Prof.\ De~Morgan\label +{DeMorgan:footnote:278}\multifootnote +{Smith:footnote:278}{DeMorgan:footnote:278}{A.~De~Morgan\index +{DeMorgan@De Morgan, A.}, \textit{Budget of Paradoxes}, London, 1872, +pp.~171, 172 [quoted from an article by De~Morgan published in 1861].}, +from $600$ trials, +deduced $\pi = 3.137$. In 1864 Captain Fox\index{Fox@Fox on $\pi$}\footnote +{\textit{Messenger of Mathematics}, Cambridge, 1873, vol.~\textsc{ii}, +pp.~113, 114.} made $1120$ trials +with some additional precautions, and obtained as the mean +value $\pi = 3.1419$. + +Other similar methods of approximating to the value of $\pi$ +have been indicated. For instance, it is known that if two +numbers are written down at random, the probability that +they will be prime to each other is $6/\pi^2$. Thus, in one case\footnote +{Note on $\pi$ by R.~Chartres\index{Chartres, R.}. \textit +{Philosophical Magazine}, London, series~6, +vol.~\textsc{xxxix}, March, 1904, p.~315.} where each of $50$ students wrote +down $5$ pairs of numbers +at random, $154$ of the pairs were found to consist of numbers +prime to each other. This gives $6/\pi^2 = 154/250$, from which +we get $\pi=3.12$\index{Circle@\textsc{Circle, Quadrature of}|)}% +\index{Squaring@\textsc{Squaring the Circle}|)}% +\index{Quadrature@\textsc{Quadrature of Circle}|)}% +\index{P@$\pi$|)}. + +\PG----File: 279.png------------------------------------------------------ +% CHAPTER IX. + +\chapter{Mersenne's Numbers.} + + +\textsc{One} of the unsolved riddles of higher arithmetic, to which% +\chapindex{MersenneNos@\textsc{Mersenne's Numbers}}% +\chapindex{Theory@\textsc{Theory of Numbers}} +I have alluded in \hyperlink{Mersenne:I}{Chapter~I}, +is the discovery of the method +by which Mersenne or his contemporaries determined values +of $p$ which make a number of the form $2^p-1$ a prime. It is +convenient to describe such primes as \emph{Mersenne's Numbers}\index +{MersenneNos@\textsc{Mersenne's Numbers}}. +In this chapter, for shortness, I use $N$ to denote a number of +the form $2^p-1$. In a memoir in the \textit{Messenger of Mathematics} +in 1891 I\index{Ball} gave a brief sketch of the history of the problem. +I here repeat the facts in somewhat more detail, and add a +sketch of methods used in attacking the problem. + +\phantomsection +\addcontentsline{toc}{section}{Mersenne's Enunciation of the Theorem} +Mersenne's enunciation of the results associated with his +name is in the preface to his \textit{Cogitata}\footnote +{\textit{Cogitata Physico-Mathematica}, Paris, 1644, praefatio generalis, +article~19.}. The passage is as follows: + +\begin{Quotation} +``Vbi fuerit operae pretium aduertere \textsc{xxviii} numeros a Petro Bungo +pro perfectis exhibitos, capite \textsc{xxviii}, libri de Numeris, +non esse omnes Perfectos, quippe $20$ sunt imperfecti, adeovt [adeunt?] +solos octo perfectos habeat\textellipsis qui sunt è regione tabulae Bungi, % NB tight ellipsis matches original +$1$, $2$, $3$, $4$, $8$, $10$, $12$, et $29$: +quique soli perfecti sunt, vt qui Bungum habuerint, errori medicinam +faciant. + +Porrò numeri perfecti adeo rari sunt, vt vndecim dumtaxat potuerint +hactenus inueniri: hoc est, alii tres a Bongianis differentes: neque +enim vllus est alius perfectus ab illis octo, nisi superes exponentem +numerum $62$, progressionis duplae ab $1$ incipientis. Nonus enim perfectus +est potestas exponentis $68$ minus $1$. Decimus, potestas exponentis +\PG----File: 280.png------------------------------------------------------ +$128$, minus $1$. Vndecimus denique, potestas $258$, minus $1$, hoc est +potestas $257$, vnitate decurtata, multiplicata per potestatem $256$. + +Qui vndecim alios repererit, nouerit se analysim omnem, quae fuerit +hactenus, superasse: memineritque interea nullum esse perfectum à +$17000$ potestate ad $32000$; \&~nullum potestatum interuallum tantum +assignari posse, quin detur illud absque perfectis. Verbi gratia, si +fuerit exponens $1050000$, nullus erit numerus progressionis duplae vsque +ad $2090000$, qui perfectis numeris\index +{NumbersPerfect@\textsc{Numbers, Perfect}}\index +{Perfect@\textsc{Perfect Numbers}} seruiat, hoc est qui minor vnitate, +primus existat. + +Vnde clarum est quà m rari sint perfecti numeri, \&~quà m merito viris +perfectis comparentur; esseque vnam ex maximis totius Matheseos +difficultatibus, praescriptam numerorum perfectorum multitudinum +exhibere; quemadmodum \&~agnoscere num dati numeri $15$, aut $20$ +caracteribus constantes, sint primi necne, cùm nequidem saeculum +integrum huic examini, quocumque modo hactenus cognito, sufficiat. +\end{Quotation} % [NB no closing " in original] + +It is evident that, if $p$ is not a prime, then $N$ is composite, +and two or more of its factors can be written down by inspection. +Hence we may confine ourselves to prime values of $p$. +Mersenne\index{MersenneNos@\textsc{Mersenne's Numbers}}, in effect, +asserted that the only values of $p$, not +greater than $257$, which make $N$ a prime, are $1$, $2$, $3$, $5$, $7$, $13$, +$17$, $19$, $31$, $67$, $127$, $257$: I assume that the number $67$ is a +misprint for $61$. With this correction we have no reason to +doubt the truth of the statement, but it has not been definitely +established. + +\phantomsection +\addcontentsline{toc}{section}{List of known results} +There are $56$ primes not greater than $257$. The determination +of the prime or composite character of $N$ for the +9 cases when $p$ is less than $20$ presents no difficulty: in only +one of them is $N$ composite. For 2 of the remaining 47 cases +(namely, when $p = 23$ and $37$) the decomposition of $N$ had +been given by Fermat\index{Fermat, P.}. For 9 of them (namely, when $p = 29$, +$43$, $73$, $83$, $131$, $179$, $191$, $239$, $251$) the factors of $N$ were +given by Euler\index{Euler}. He also proved that $N$ was prime when +$p = 31$. Plana\index{Plana, G.A.A.} gave the factors of $N$ when $p = 41$. +Landry\index{Landry} and Le~Lasseur\index{LeLass@Le Lasseur} +discovered the factors in 10 cases (namely, +when $p=47$, $53$, $59$, $79$, $97$, $113$, $151$, $211$, $223$, and $233$), +but their analysis has not been published. Seelhoff\index{Seelhoff} +showed that $N$ was prime when $p = 61$, Cunningham\index{Cunningham, A.J.C.} +gave the factors when $p = 197$, and Cole\index{Cole, F.N.} +the factors when $p = 67$. Statements +\PG----File: 281.png------------------------------------------------------ +have been made that the composite character of $N$ when +$p = 89$, and its prime character when $p=127$ have been +proved, but the proofs have not been published or verified. + +\phantomsection +\addcontentsline{toc}{section}{Cases awaiting verification} +Thus there are $21$ values of $p$ for which Mersenne's statement +still awaits verification. These are $71$, $89$, $101$, $103$, $107$, $109$, +$127$, $137$, $139$, $149$, $157$, $163$, $167$, $173$, $181$, $193$, +$199$, $227$, $229$, $241$, $257$. For these values $N$ is (according to +Mersenne\index{MersenneNos@\textsc{Mersenne's Numbers}}) prime when +$p= 127$, and $257$, and is composite for +the other values, but as explained above it is probable that +the character of $N$ is known when $p = 89$ and $127$. + +To put the matter in another way. According to Mersenne's +statement (corrected by the substitution of $61$ for $67$), $44$ +of the $56$ primes less than $258$ make $N$ composite and the +remaining $12$ primes make $N$ prime. In $25$ out of the $44$ cases +in which $N$ is said to be composite we know its factors, and +in $19$ cases the statement is still unverified. In $10$ out of the +$12$ cases in which he said that $N$ was prime his statement has +been verified, and in $2$ cases it is still unverified. + +From the wording of the last clause in the above quotation +it has been conjectured that the result had been communicated +to Mersenne, and that he published it without being aware of +how it was proved. In itself this seems probable. He was +a good mathematician, but not an exceptional genius. It +would be strange if he established a proposition which has +baffled Euler, Lagrange, Legendre, Gauss, Jacobi, and other +mathematicians of the first rank; but if the proposition is +due to Fermat\index{Fermat, P.}, with whom Mersenne was in constant +correspondence, the case is altered, and not only is the absence of +a demonstration explained, but we cannot be sure that we +have attacked the problem on the best lines. + +The known results as to the prime or composite character +of $N$, and in the latter case its smallest factor, are given in the +table \vpageref{table:Mersenne}. The cases that remain as yet +% [* originally "on the opposite page"] +unverified are marked with an asterisk. + +\PG----File: 282.png------------------------------------------------------ +\afterpage{\clearpage{\small\label{table:Mersenne} +\begin{longtable}{r|c|c|l} +\multicolumn{1}{c|}{$p$} & \emph{Value of $N = 2^p - 1$} & & \\ +\hline +$ 1$ & $ 1 $ & prime & \endfirsthead +\multicolumn{1}{c|}{$p$} & \emph{Value of $N = 2^p - 1$} & & \\ +\hline\endhead +$ 2$ & $ 3 $ & prime & \\ +$ 3$ & $ 7 $ & prime & \\ +$ 5$ & $ 31 $ & prime & \\ +$ 7$ & $ 127 $ & prime & \\ +$ 11$ & $ 2047 = 23 \times 89 $ & composite & \\ +$ 13$ & $ 8191 $ & prime & \\ +$ 17$ & $ 131071 $ & prime & \\ +$ 19$ & $ 524287 $ & prime & \\ +$ 23$ & $ 8388607 = 47 \times 178481 $ & composite + & Fermat\index{Fermat, P.} \\ +$ 29$ & $ 536870911 = 233 \times 1103 \times 2089 $ & composite + & Euler\index{Euler} \\ +$ 31$ & $ 2147483647 $ & prime & Euler\index{Euler} \\ +$ 37$ & $ 137438953471 = 223 \times 616318177 $ & composite + & Fermat\index{Fermat, P.} \\ +$ 41$ & $ 2199023255551 = 13367 \times 164511353 $ & composite + & Plana\index{Plana, G.A.A.} \\ +$ 43$ & $ 8796093022207 = 431 \times 9719 \times 2099863 $ & composite + & Euler\index{Euler} \\ +$ 47$ & $ 2351 \times 4513 \times 13264529 $ & composite + & Landry\index{Landry} \\ +$ 53$ & $ 6361 \times 69431 \times 20394401 $ & composite + & Landry\index{Landry} \\ +$ 59$ & $ 179951 \times 3203431780337 $ & composite + & Landry\index{Landry} \\ +$ 61$ & $ 2305843009213693951 $ & prime & Seelhoff\index{Seelhoff} \\ +$ 67$ & \rlap{$\equiv 0~(193707721) $}\kern4em & composite + & Cole\index{Cole, F.N.} \\ +$ 71$ & $ 2361183241434822606847 $ & $*$ & \\ +$ 73$ & \rlap{$\equiv 0~(439) $}\kern4em & composite + & Euler\index{Euler} \\ +$ 79$ & \rlap{$\equiv 0~(2687) $}\kern4em & composite + & Le Lasseur\index{LeLass@Le Lasseur} \\ +$ 83$ & \rlap{$\equiv 0~(167) $}\kern4em & composite + & Euler\index{Euler} \\ +$ 89$ & $ 618970019642690137449562111 $ & $*$ & \\ +$ 97$ & \rlap{$\equiv 0~(11447) $}\kern4em & composite + & Le Lasseur\index{LeLass@Le Lasseur} \\ +$101$ & $ 2535301200456458802993406410751 $ & $*$ & \\ +$103$ & $ 10141204801825835211973625643007 $ & $*$ & \\ +$107$ & $ 162259276829213363391578010288127 $ & $*$ & \\ +$109$ & $ 649037107316853453566312041152511 $ & $*$ & \\ +$113$ & \rlap{$\equiv 0~(3391) $}\kern4em & composite + & Le Lasseur\index{LeLass@Le Lasseur} \\ +$127$ & \ifPaper\smaller\fi$ 170141183460469231731687303715884105727 $ + & $*$ & \\ +$131$ & \rlap{$\equiv 0~(263) $}\kern4em & composite + & Euler\index{Euler} \\ +$137$ & $ \quad\dotfill\quad $ & $*$ & \\ +$139$ & $ \quad\dotfill\quad $ & $*$ & \\ +$149$ & $ \quad\dotfill\quad $ & $*$ & \\ +$151$ & \rlap{$\equiv 0~(18121) $}\kern4em & composite + & Le Lasseur\index{LeLass@Le Lasseur} \\ +$157$ & $ \quad\dotfill\quad $ & $*$ & \\ +$163$ & $ \quad\dotfill\quad $ & $*$ & \\ +$167$ & $ \quad\dotfill\quad $ & $*$ & \\ +$173$ & $ \quad\dotfill\quad $ & $*$ & \\ +$179$ & \rlap{$\equiv 0~(359) $}\kern4em & composite + & Euler\index{Euler} \\ +$181$ & $ \quad\dotfill\quad $ & $*$ & \\ +$191$ & \rlap{$\equiv 0~(383) $}\kern4em & composite + & Euler\index{Euler} \\ +$193$ & $ \quad\dotfill\quad $ & $*$ & \\ +$197$ & \rlap{$\equiv 0~(7487) $}\kern4em & composite + & \ifPaper\rlap{\Small Cunningham}\else Cunningham\fi\index{Cunningham, A.J.C.} \\ +$199$ & $ \quad\dotfill\quad $ & $*$ & \\ +$211$ & \rlap{$\equiv 0~(15193) $}\kern4em & composite + & Le Lasseur\index{LeLass@Le Lasseur} \\ +$223$ & \rlap{$\equiv 0~(18287) $}\kern4em & composite + & Le Lasseur\index{LeLass@Le Lasseur} \\ +$227$ & $ \quad\dotfill\quad $ & $*$ & \\ +$229$ & $ \quad\dotfill\quad $ & $*$ & \\ +$233$ & \rlap{$\equiv 0~(1399) $}\kern4em & composite + & Le Lasseur\index{LeLass@Le Lasseur} \\ +$239$ & \rlap{$\equiv 0~(479) $}\kern4em & composite + & Euler\index{Euler} \\ +$241$ & $ \quad\dotfill\quad $ & $*$ & \\ +$251$ & \rlap{$\equiv 0~(503) $}\kern4em & composite + & Euler\index{Euler} \\ +$257$ & $ \quad\dotfill\quad $ & $*$ & +\end{longtable}}} + +\PG----File: 283.png----------------------------------------------------- +Before describing the methods used for attacking the +problem it will be convenient to state in more detail when +and by whom these results were established. + +The factors (if any) of such values of $N$ as are less than +a million can be verified easily: they have been known for a +long time, and I need not allude to them in detail. + +\phantomsection +\addcontentsline{toc}{section}{History of Investigations} +The factors of $N$ when $p=11$, $23$, and $37$ had been indicated +by Fermat\index{Fermat, P.}\footnote +{\textit{Oeuvres de Fermat}, Paris, vol.~\textsc{ii}, 1894, p.~210; +or \textit{Opera Mathematica}, Toulouse, 1679, p.~164; or Brassinne's +\textit{Précis}, Paris, 1853, p.~144.}, +some four years prior to the publication of +Mersenne's work, in a letter dated October~18, 1640. The +passage is as follows: + +\begin{Quotation} +En la progression double, si d'un nombre quarré, généralement +parlant, vous ôtez $2$ ou $8$ ou $32$~\&c., les nombres premiers moindres +de l'unite qu'un multiple du quaternaire, qui mesureront le reste, feront +l'effet requis. Comme de $25$, qui est un quarré, ôtez $2$; le reste $23$ +mesurera la 11\textsuperscript{e} puissance $-1$; ôtez $2$ de $49$, +le reste $47$ mesurera la 23\textsuperscript{e} puissance +$-1$. Ôtez $2$ de $225$, le reste $223$ mesurera la +37\textsuperscript{e} puissance $-1$,~\&c. +\end{Quotation} + +The factors of $N$ when $p=29$, $43$, and $73$ were given by +Euler\index{Euler}\footnote +{\textit{Commentarii Academiae Scientiarum Petropolitanae}, 1738, +vol.~\textsc{vi}, p.~103; or \textit{Commentationes Arithmeticae Collectae}, +vol.~\textsc{i}, p.~2.} in 1732. The fact that $N$ is composite for the +values $p = 83$, $131$, $179$, $191$, $239$, and $251$ follows from a +proposition enunciated, at the same time, by Euler\index{Euler} to the +effect that, if $4n + 3$ and $8n + 7$ are primes, then $2^{4n+3}-1 \equiv 0 +\pmod{8n + 7}$. This was proved by Lagrange\index{Lagrange}\footnote +{\textit{Nouveaux Mémoires de l'Académie des Sciences de Berlin}, 1775, +pp.~323--356.} in his classical +memoir of 1775. The proposition also covers the cases of +$p=11$ and $p = 23$. This is the only general theorem on the +subject which is yet established. + +The fact that $N$ is prime when $p = 31$ was proved by +Euler\index{Euler}\footnote +{\textit{Histoire de l'Académie des Sciences} for 1772, Berlin, 1774, p.~36. +See also Legendre\index{Legendre}, \textit{Théorie des Nombres}, +third edition, Paris, 1830, vol.~\textsc{i}, pp.~222--229.} in 1771. +Fermat\index{Fermat, P.} had asserted, in the letter mentioned +\PG----File: 284.png---------------------------------------------------- +above, that the only possible prime factors of $2^p\pm1$, when +$p$ was prime, were of the form $np + 1$, where $n$ is an +integer. This was proved by Euler\index{Euler}\footnote +{\textit{Novi Commentarii Academiae Scientiarum Petropolitanae}, +vol.~\textsc{i}, p.~20; or \textit{Commentationes Arithmeticae Collectae}, +St~Petersburg, 1849, vol.~\textsc{i}, pp.~55, 56.} in 1748, who added +that, since $2^p\pm1$ is odd, every factor of it must be odd, +and therefore if $p$ is odd $n$ must be even. But if $p$ is a given +number we can define $n$ much more closely, and Euler\index{Euler} showed +that the prime factors (if any) of $2^{31}-1$ were necessarily +primes of the form $248n + 1$ or $248n + 63$; also they must be +less than $\sqrt{2^{31}-1}$, that is, than $46339$. Hence it is necessary +to try only forty divisors to see if $2^{31}-1$ is a prime or +composite. + +The factors of $N$ when $p = 41$ were given by Plana\index +{Plana, G.A.A.}\footnote{G.A.A.~Plana, \textit{Memorie della Reale +Accademia delle Scienze di Torino}, Series~2, vol.~\textsc{xx}, 1863, +p.~130.\label{ibid:13}} in 1859. He showed that the prime factors (if any) +are primes of the form $328n + 1$ or $328n + 247$, and lie between +$1231$ and $\sqrt{2^{41}-1}$, that is, $1048573$. Hence it is necessary +to try only $513$ divisors to see if $2^{41}-1$ is composite: the +seventeenth of these divisors gives the required factors. This +is the same method of attacking the problem which was used +by Euler in 1771, but it would be very laborious to employ it +for values of $p$ greater than $41$. Plana\index +{Plana, G.A.A.}\footnote +{\Ibidref{ibid:13}{Plana}, p.~137.} added the forms of +the prime divisors of $N$, if $p$ is not greater than $101$. + +That $N$ is prime when $p = 127$ seems to have been verified +by Lucas\index{Lucas, E.}\footnote +{\textit{Sur la Théorie des Nombres Premiers}, Turin, 1876, p.~11; and +\textit{Recherches sur les Ouvrages de Léonard de Pise}, Rome, 1877, p.~26, +quoted by Lieut.-Colonel A.J.C.~Cunningham\index{Cunningham, A.J.C.}, +\textit{Proceedings of the London Mathematical Society}, Nov.~14, 1895, +vol.~\textsc{xxvii}, p.~54.} in 1876 and 1877. The demonstration has not +been published. + +The discovery of the factors of $N$ for the values $p = 47$, +$53$, and $59$ is due apparently to the late F.~Landry\index{Landry}, who +\PG----File: 285.png------------------------------------------------------ +established theorems on the factors (if any) of numbers of +certain forms. Instead of publishing his results he issued a +challenge to all mathematicians to solve the general problem. +This is contained in a rare pamphlet published at Paris in +1867, of which I possess a copy, in which the factors of +certain numbers are given, and on page~8 of which it is +implied that he had obtained the factors of $2^p-1$ when $p = 47$, +$53$, and $59$. He seems to have communicated his results to +Lucas\index{Lucas, E.}, who quoted them in the memoir cited +below\label{Lucas:footnote1:285}\footnote +{\textit{American Journal of Mathematics}, 1878, vol.~\textsc{i}, p.~236.}. + +The factors of $N$ when $p = 79$ and $113$ were given first +by Le~Lasseur\index{LeLass@Le Lasseur}, and were quoted by Lucas in the same +memoir\label{Lucas:footnote2:285}\multifootnote +{Lucas:footnote1:285}{Lucas:footnote2:285}{\textit +{American Journal of Mathematics}, 1878, vol.~\textsc{i}, p.~236.}. + +A factor of $N$ when $p = 233$ was discovered later by Le~Lasseur\index +{LeLass@Le Lasseur}, +and was quoted by Lucas\index{Lucas, E.} in 1882\footnote +{\textit{Récréations}, 1882--3, vol.~\textsc{i}, p.~241.\label{ibid:14}}. + +The factors of $N$ when $p = 97$, $151$, $211$, and $223$ were +determined subsequently by Le~Lasseur\index{LeLass@Le Lasseur}, and were +quoted by Lucas\index{Lucas, E.}\footnote +{\Ibidref{ibid:14}{\textit{Récréations}}, +vol.~\textsc{ii}, p.~230.} in 1883. + +That $N$ is prime when $p=61$ had been conjectured by +Landry\index{Landry} and in 1886 a demonstration was offered by +Seelhoff\index{Seelhoff}\footnote +{P.H.H.~Seelhoff, \textit{Zeitschrift für Mathematik und Physik}, 1886, +vol.~\textsc{xxxi}, p.~178.}. +His demonstration is open to criticism, but the fact has been +verified by others\footnote +{See Weber-Wellstein\index{Weber-Wellstein}, \textit{Encyclopaedie der +Elementar-Mathematik}, p.~48; and F.N.~Cole\index{Cole, F.N.}, +\textit{Bulletin of the American Mathematical Society}, +December, 1903, p.~136.}, and may be accepted as proved. + +That $N$ is composite when $p = 89$ seems to have been verified +by Lucas\index{Lucas, E.}\footnote +{\textit{Théorie des Nombres}, Paris, 1891, p.~376.} in 1891, +but the demonstration has not been published, +nor have the actual factors been discovered. + +That $7487$ is a factor of $N$ when $p=197$ was shown by +A.J.C.~Cunningham\index{Cunningham, A.J.C.} in 1895\footnote +{\textit{Proceedings of the London Mathematical Society}, March~14, 1895, +vol.~\textsc{xxvi}, p.~261.}. + +\PG----File: 286.png------------------------------------------------------ +That $N$ is not prime when $p = 67$ seems to have been verified +by Lucas\index{Lucas, E.}\footnote +{\textit{Sur la Théorie des Nombres Premiers}, Turin, 1876, p.~11, +quoted by Lieut-Colonel A.J.C.~Cunningham\index{Cunningham, A.J.C.}, +\textit{Proceedings of the London Mathematical Society}, Nov.~14, 1895, +vol.~\textsc{xxvii}, p.~54, and \textit{Recherches sur +les Ouvrages de Léonard de Pise}, Rome, 1877, p.~26.} +in 1876 and 1877. The composite nature of\footnoteT{Corrected: originally $N$ $2^p=1$} +$N=2^p - 1$ when $p = 67$ was confirmed by E.~Fauquembergue\index +{Fauquembergue, E.}\footnote +{\textit{L'Intermédiaire des mathématiciens}, Paris, Sept.~1894, +vol.~\textsc{i}, p.~148.}, +and was also implied by Lucas\index{Lucas, E.}\footnote +{\textit{Théorie des Nombres}, Paris, 1891, p.~376.} +in 1891. The factors were given by F.N.~Cole\index{Cole, F.N.}\footnote +{\textit{On the Factoring of Large Numbers}, \textit{Bulletin of the American +Mathematical Society}, December, 1903, pp.~134--137.} in 1903. + +Bickmore\index{Bickmore, C.E.} in the memoir\footnote +{C.E.~Bickmore, \textit{Messenger of Mathematics}, Cambridge, 1895, +vol.~\textsc{xxv}, p.~19.} cited below showed that $1433$ +is another factor of $N$ if $p = 179$; and that $1913$ and $5737$ +are other factors of $N$ if $p = 239$. + +\ThoughtBreakSpace +\phantomsection +\addcontentsline{toc}{section}{Methods used in attacking the problem} +I turn next to consider the methods by which these results +can be obtained. It is impossible to believe that the statement +made by Mersenne rested on an empirical conjecture, +but the riddle as to how it was discovered is still, after nearly +250 years, unsolved. + +I cannot offer any solution of the riddle. But it may be +interesting to indicate some ways which have been used in +attacking the problem. The object is to find a prime divisor +$q$ (other than $N$ and $1$) of a number $N$ when $N$ is of the form +$2^p-1$ and $p$ is a prime. It can be easily shown that $q$ must +be of the form $2pt + 1$. Also $q$ must be of one of the forms +$8i \pm 1$: for $N$ is of the form $2A^2 - B^2$, where $A$ is even and +$B$ odd, hence\footnote +{Legendre\index{Legendre}, \textit{Théorie des Nombres}, third edition, +Paris, 1830, vol.~\textsc{i}, +§~143. In the case of Mersenne's numbers, $B = b = 1$.} +any factor of it must be of the form $2a^2 - b^2$, +that is, of the form $8i \pm 1$, and 2 must be a quadratic residue +of $q$. The theory of residues is, however, of but little use in +finding factors of the cases that still await solution, though +the possibility some day of finding a complete series of solutions +\PG----File: 287.png--------------------------------------------------- +by properties of residues must not be neglected\footnote +{For methods of finding the residue indices of $2$ see Bickmore\index +{Bickmore, C.E.}, \textit{Messenger of Mathematics}, May, 1895, +vol.~\textsc{xxv}, pp.~15--21; see also +Lieut-Colonel A.J.C.~Cunningham\index{Cunningham, A.J.C.} on $2$ as a $16$-ic +residue, \textit{Proceedings of the London Mathematical Society}, 1895--6, +vol.~\textsc{xxvii}, pp.~85--122.}. Our present +knowledge of the means of factorizing $N$ has been summed +up in the statement\footnote +{\textit{Transactions of the British Association for the Advancement of +Science} (Ipswich Meeting), 1895, p.~614.} that a prime factor of the form +$2pt + 1$ can be found directly by rules due to Legendre\index{Legendre}, +Gauss\index{Gauss}, and +Jacobi\index{Jacobi}, when $t = 1$, $3$, $4$, $8$, or $12$; and that a factor +of the form $2ptt' + 1$ can be found indirectly by a method due to +Bickmore\index{Bickmore, C.E.} when $t = 1$, $3$, $4$, $8$, or $12$, and $t'$ +is an odd integer +greater than $3$. But this only indicates how little has yet +been done towards finding a general solution of the problem. + +\subsection[By trial of divisors of known forms][Mersenne's Numbers]% +{First} There is the simple but crude method of trying all +possible prime divisors $q$ which are of the form $2pt + 1$ as well +as of one of the forms $8i\pm 1$. + +The chief known results for the smaller factors may be +summarized by saying that a prime of this form will divide $N$ +when $t=1$, if $p = 11$, $23$, $83$, $131$, $179$, $191$, $239$, or $251$; +when $t = 3$, if $p = 37$, $73$, or $233$; when $t = 5$, if $p = 43$; +when $t= 15$, if $p = 113$; when $t = 17$, if $p = 79$; when $t = 19$, +if $p = 29$, or $197$; when $t=25$, if $p = 47$; when $t = 41$, if $p = 223$; +when $t = 59$, if $p = 97$; when $t= 163$, if $p = 41$; when $t= 1525$, if +$p = 59$; when $t = 4$, if $p=11$, $29$, $179$, or $239$; when $t = 8$, +if $p = 11$; when $t = 12$, if $p = 239$; when $t = 36$, if $p = 29$, or +$211$; when $t = 60$, if $p = 53$, or $151$; and when $t= 1445580$, +if $p = 67$. + +Of the 25 cases in which we know that Mersenne's statement +of the composite character of $N$ is correct all save 3 can +be easily verified by trial in this way. For neglecting all +values of $t$ not exceeding, say, $60$ which make $q$ either +composite or not of one of the forms $8i\pm 1$ we have in each +case only some $20$ or so divisors to try. Of the 3 other cases +in which Mersenne's statement of the composite character of +\PG----File: 288.png------------------------------------------------------ +$N$ has been verified, one verification ($p = 41$) is due to Plana\index +{Plana, G.A.A.}, who frankly confessed that the result was reached ``par un +heureux hasard''; a second is due to Landry\index{Landry} ($p = 59$), who did +not explain how he obtained the factors; and the third is due +to Cole\index{Cole, F.N.} ($p = 67$), who established it by the use of +quadratic residues of $N$, involving laborious numerical work. + +Of the 10 cases in which we know that Mersenne's statement +of the prime character of $N$ is correct all save one may +be verified by trial in this way, for the number of possible +factors is not large. The exception is the case where $p$ equals +$61$, which Seelhoff\index{Seelhoff} and others have shown to be prime. + +Thus practically we may say that simple empirical trials +would at once lead us to all the conclusions known except in +the case of $p = 41$ due to Plana\index{Plana, G.A.A.}, of $p = 59$ +due to Landry\index{Landry}, of +$p = 61$ due to Seelhoff, and of $p = 67$ due to Cole\index{Cole, F.N.}. +In fact, save for these four results the conclusions of all mathematicians +to date could be obtained by anyone by a few hours' +arithmetical work. + +As $p$ increases the number of factors to be tried increases +so fast that, if $p$ is large, it would be practically impossible +to apply the test to obtain large factors. This is an important +point, for it has been asserted that in the cases still awaiting +verification there are no factors less than $50,000$. Hence, +we may take it as reasonably certain that this cannot have +been the method by which the result was originally obtained; +nor, as here enunciated, is it likely to give many factors not +yet known. Of course it is possible there may be ways by +which the number of possible values of $t$ might be further +limited, and if we could find them we might thus diminish +the number of possible factors to be tried, but it will be +observed that the values of $N$ which still await verification +are very large, for instance, when $p = 257$, $N$ contains no less +than $78$ digits. + +It is hardly necessary to add that if $q$ is known and is +not very large we can determine whether or not it is a factor +of $N$ without the labour of performing the division. + +\PG----File: 289.png------------------------------------------------------ +For instance, if we want to verify that $q = 13367$ is a +factor of $N$ when $p = 41$, we proceed thus. Take the power +of $2$ nearest to $q$ or to its square-root. We have to the +modulus $q$ +\[ +\begin{array}{rl@{}l} + & 2^{14} & {}= 16384\equiv 3017\equiv 7\times 431\,,\\ + \Therefore\quad & 2^{28} & {}\equiv 49(-1377) \equiv -638\,, \\ + \Therefore\quad & 2^{27} & {}\equiv -319\,, \\ + \Therefore\quad & 2^{14+27} & {}\equiv (3017) (-319) \equiv 1\,, \\ + \Therefore\quad & 2^{41} & {}\equiv 1\;. +\end{array} +\] + +\subsection[By indeterminate equations][Mersenne's Numbers]% +{Second} We can proceed by reducing the problem to the +solution of an indeterminate equation. + +It is clear that we can obtain a factor of $N$ if we can +express it as the difference of the squares, or more generally +of the $n$th powers, of two integers $u$ and $v$. Further, if we +can express a multiple of $N$, say $mN$, in this form, we can +find a factor of $mN$ and (with certain obvious limitations as to +the value of $m$) this will lead to a factor of $N$. It may be +also added that if $m$ can be found so that $N/m$ is expressible +as a continued fraction of a certain form, a certain continuant\footnote +{See J.G.~Birch\index{Birch, J.G.} in the \textit{Messenger of Mathematics}, +August, 1902, vol.~\textsc{xxii}, pp.~52--55.} +defined by the form of the continued fraction is a factor of $N$. + +Since $N$ can always be expressed as the difference of two +squares, this method seems a natural one by which to attack +the problem. If we put +\[ + N = n^2 + a = (n + b)^2 - (b^2 + 2bn - a), +\] +we can make use of the known forms of $u$ and $v$, and thus +obtain an indeterminate equation between two variables $x$ +and $y$ of the form +\[ + x^2 = (2py + H)^2 - 4(K - y) +\] +where $H$ and $K$ are numbers which can be easily calculated. +\PG----File: 290.png------------------------------------------------------ +Integral values of $x$ and $y$ where $y < K$ will determine values +of $u$ and $v$, and thus give factors of $N$. + +We can also attack the problem by indeterminate equations +in another way. For the factors must be of the form $2pt + 1$ +and $8ps + 1$, hence +\begin{LRalign} +&(2pt + 1) (8ps + 1) &= N\,,\\ +&&= 2^p - 1\,,\\ +&&= 2(2^{p - 1} - 1) + 1\,,\\ +&\Therefore 4s + t + 8pst &= (2^{p - 1} - 1)/p\,,\\ +&&= \text{(say)} \alpha + 8p\beta\,. \\ +\indent Hence& 4s + t = \alpha + 8px, & \text{ and } st = \beta - x\,, +\end{LRalign} +where $x \ngtr \beta$ and $t$ is odd. These results again lead to an +indeterminate equation. + +But, in both cases, unless $p$ is small, the resulting equations +are intractable. + +\subsection[By properties of quadratic forms][Mersenne's Numbers]% +{Third} A not uncommon method of attacking problems +such as this, dealing with the factorization of large numbers, +is through the theory of quadratic forms\footnote +{For a sketch of this see G.B.~Mathews\index{Mathews, G.B.}, +\textit{Theory of Numbers}, part~1, Cambridge, 1891, pp.~261-271. +See also F.N.~Cole's\index{Cole, F.N.} paper, On the Factoring of +Large Numbers, \textit{Bulletin of the American Mathematical +Society}, December, 1903, pp.~134-137; and \textit{Quadratic Partitions} by +A.J.C.~Cunningham\index{Cunningham, A.J.C.}, London, 1904.}. At best this is +a difficult method to use, and we have no reason to think that +it would have been employed by a mathematician of the +seventeenth century. I here content myself with alluding to it. + +\subsection[By the use of a \textit{Canon Arithmeticus}][Mersenne's Numbers]% +{Fourth} There is yet another way in which the problem +might be attacked. The problem will be solved if we can find +an odd prime $q$ so that to it as modulus $2^{p+y} \equiv z$, +and $2^y \equiv z$, +where $y$ and $z$ may have any values we like to choose. If +such values of $q$, $y$, and $z$ can be found, we have +$2^y (2^p - 1) \equiv 0$. +Therefore $2^p = 1$, that is, $q$ is a divisor of $N$. + +\PG----File: 291.png--------------------------------------------------- +For example, to the modulus $23$, we have +\begin{LRalign} +& 2^8 &\equiv 3\,,\\ +& 2^{16} &\equiv 3^2\,.\\ +Also & 2^5 &\equiv 3^2\,.\\ +Therefore & 2^{16} - 2^5 &\equiv 0\,,\\ +& \Therefore 2^{11} - 1 &\equiv 0\,.\\ +\end{LRalign} +Without going further into the matter we may say that the +\emph{à priori} determination of the values $q$, $y$, and $z$ introduces +us to an almost untrodden field. It is just possible (though I +should suppose unlikely) that the key to the riddle is to be +found on methods of finding $q$, $y$, $z$, to satisfy the above +conditions. For instance, if we could say what was the +remainder when $2^x$ was divided by a prime $q$ of the form +$2pt + 1$, and if the remainders were the same when $x = u$ and +$x = v$, then to the modulus $q$ we should have, $2^u \equiv 2^v$, and +therefore $2^{u-v} \equiv 1$. + +It should however be noted that Jacobi's\index{Jacobi} \textit +{Canon Arithmeticus} +and the similar canon drawn up by Cunningham\index{Cunningham, A.J.C.} +would, if carried far enough, enable us to solve the problem +by this method. Cunningham's \textit{Canon} gives the solution of +the congruence $2^x \equiv R$ for all prime moduli less than $1000$, +but it is of no use in determining factors of $N$ larger than +$1000$. It is however possible that if $R$ or $q$ have certain +forms such a canon might be constructed, and thus lead to +a solution of the problem. + +\subsection[By properties of binary powers][Mersenne's Numbers]% +{Fifth} It is noteworthy that the odd values of $p$ +specified by Mersenne are primes of one of the four forms +$2^q \pm 1$ or $2^q \pm 3$, but it is not the case that all primes of these +forms make $N$ a prime, for instance, $N$ is composite if +$p = 2^3 + 3 = 11$ or if $p = 2^5 - 3 = 29$. + +This fact has suggested to more than one mathematician +the possibility that some test as to the prime or composite +character of $N$ when $p$ is of one of these forms may be discoverable. +Of course this is merely a conjecture. There is however +\PG----File: 292.png------------------------------------------------------- +this to say for it, that we know that Fermat\index{Fermat, P.}\footnote +{\EG, see above, page \pageref{page:Fermat}.} had paid +attention to numbers of this form. + +\subsection[By the use of the binary scale][Mersenne's Numbers]% +{Sixth} The number $N$ when expressed in the binary +scale, consists of $1$ repeated $p$ times. This has suggested +whether the work connected with the determination of factors +of $N$ might not with advantage be expressed in the binary +scale. A method based on the use of properties of this +scale has been indicated by G.~de~Longchamps\index +{DeLongchamps@De Longchamps, G.}\footnote +{\textit{Comptes rendus de~l'Académie des Sciences}, Paris, Nov. 1877, +vol.~\textsc{lxxxv}, pp.~950--952.}, but as there +given it would be unlikely to lead to the discovery of large +divisors. I am, however, inclined to think that greater advantages +would be gained by working in a scale whose radix +was $4p$ or may-be $8p$---the resulting numbers being then +expressed by a reasonably small number of digits. In fact +when expressed in the latter scale in only one out of the 25 +cases in which the factors of $N$ are known does the smallest +factor contain more than two digits. + +\subsection[By the use of Fermat's Theorem][Mersenne's Numbers]% +{Seventh} I have reserved to the last the description of +the method which seems to me to be the most hopeful. + +We know by Fermat's Theorem that if $x + 1$ is a prime +then $2^x-1$ is divisible by $x + 1$. Hence if $2pt + 1$ is a +prime we have, to the modulus $2pt + 1$ +\begin{gather*} +2^{2pt}-1 \equiv 0\,,\\ +\Therefore (2^p-1)(1+2^p+2^{2p}+\dotsb+2^{(2t-1)p}) \equiv 0\,. +\end{gather*} +Hence, a divisor of $2^p-1$ will be known, if we can find a +value of $t$ such that $2pt + 1$ is prime and the second factor +is prime to it. + +This method may be used to establish Euler's theorem +of 1732. For if we put $t=1$, and if $2p+1$ is a prime, we +have, to the modulus $2p + 1$ +\[ +(2^p-1)(2^p + 1)\equiv 0\,. +\] +Hence $2^p\equiv1$ if $2^p + 1$ is prime to $2p+1$. This is the +\PG----File: 293.png------------------------------------------------------ +case if $p = 4m + 3$. Hence $2p + 1$ is a factor of $N$ if $p = 11$, +$23$, $83$, $131$, $179$, $191$, $239$, and $251$, for in these cases +$2p + 1$ is prime. + +The problem of Mersenne's Numbers is a particular case +of the determination of the factors of $a^n - 1$. This has been +the subject of investigations by many mathematicians: an +outline of their conclusions has been given by Bickmore\index +{Bickmore, C.E.}\footnote +{\textit{Messenger of Mathematics}, Cambridge, 1895--6, vol.~\textsc{xxv}, +pp.~1--44; also 1896--7, vol.~\textsc{xxvi}, pp.~1--38; see also a note by +Mr E.B.~Escott\index{Escott, E.B.} in the \textit{Messenger}, 1903--4, +vol.~\textsc{xxxiii}, p.~49.\label{ibid:15}}. +I ought also to add a reference to the general method suggested +by F.W.~Lawrence\index{Lawrence, F.W.}\footnote +{\Ibidref{ibid:15}{\textit{Messenger of Mathematics}, Cambridge}, +1894--5, vol.~\textsc{xxiv}, +pp.~100--109; \textit{Quarterly Journal of Mathematics}, 1896, +vol.~\textsc{xxviii}, pp.~285--311; +and \textit{Transactions of the London Mathematical Society}, May~13, 1897, +vol.~\textsc{xxviii}, pp.~465--475.} for the factorization of any high +number: it is possible that Fermat used some method +analogous to this. + +\phantomsection +\addcontentsline{toc}{section}{Mechanical methods of Factorizing Numbers} +Finally, I should add that machines\footnote +{F.W.~Lawrence\index{Lawrence, F.W.}, \textit +{Quarterly Journal of Mathematics}, 1896, already quoted, pp.~310--311; +see also C.A.~Laisant\index{Laisant, C.A.}, \textit{Comptes Rendus +Association Français pour l'avancement des sciences}, 1891 (Marseilles), +vol.~\textsc{xx}, pp.~165--8.} have been devised +for investigating whether a number is prime, but I do not +know that any have been constructed suitable for numbers as +large as those involved in the numbers in question\Editorial{The primality +or otherwise of all Mersenne numbers $2^p-1$ up to $p=6972593$ has +been decided: thirty-eight are prime (see +\href{http://www.mersenne.org}{www.mersenne.org}). +Of the asterisked entries in the author's table on page~\pageref +{table:Mersenne}, $2^{89}-1$, $2^{107}-1$ and $2^{127}-1$ are prime and +the rest (including $2^{257}-1)$ are composite.}. + +\PG----File: 294.png------------------------------------------------------ + +% CHAPTER X. + +\chapter{Astrology.} + +\textsc{Astrologers} professed to be able to foretell the future% +\chapindex{Astrology@\textsc{Astrology}}% +\chapindex{Horoscopes@\textsc{Horoscopes}}% +\chapindex{Raphael on Astrology}, +and within certain limits to control it. I propose to give in +this chapter a concise account of the rules they used for this +purpose\footnote +{I have relied mainly on the \textit{Manual of Astrology} by Raphael---whose +real name was R.C.~Smith---London, 1828, to which the references to +Raphael hereafter given apply; and on Cardan's\index{Cardan} writings, +especially his commentary on Ptolemy's\index{Ptolemy} work and his +\textit{Geniturarum Exempla}. I am indebted +also for various references and gossip to Whewell's\index{Whewell, W.} +\textit{History of the Inductive Sciences}; to various works by Raphael, +published in London between 1825 and 1832; and to a pamphlet by +M.~Uhlemann\index{Uhlemann on Astrology}, entitled +\textit{Grundzüge der Astronomie und Astrologie}, Leipzig, 1857.}. + +I have not attempted to discuss the astrology of periods +earlier than the middle ages, for the technical laws of the +ancient astrology are not known with accuracy. At the same +time there is no doubt that, as far back as we have any definite +historical information, the art was practised in the East; that +thence it was transplanted to Egypt, Greece, and Rome; and +that the medieval astrology was founded on it. It is probable +that the rules did not differ materially from those described in +this chapter\footnote +{On the influences attributed to the planets, see \textit{The Dialogue of +Bardesan on Fate}\index{Bardesan on Fate}, translated by +W.~Cureton\index{Cureton on Syriac Astrology} in the +\textit{Spicilegium Syriacum}, London, 1855.}, +and it may be added that the more intelligent +thinkers of the old world recognised that the art had no valid +\PG----File: 295.png------------------------------------------------------ +pretences to accuracy. I may note also that the history of +the development of the art ceases with the general acceptance +of the Copernican theory, after which the practice of astrology +rapidly became a mere cloak for imposture. + +All the rules of the medieval astrology---to which I confine +myself---are based on the Ptolemaic astronomy, and originate +in the \textit{Tetrabiblos}\footnote +{There is an English translation by J.~Wilson\index{Wilson on Ptolemy}, +London [\emph{n.d.}]; and +a French translation is given in Halma's edition of Ptolemy's works.} +which is said, it may be falsely, to have +been written by Ptolemy\index{Ptolemy} himself. The art was developed by +numerous subsequent writers, especially by Albohazen\index +{Albohazen on Astrology}\footnote +{\textit{De judiciis astrorum}, ed.\ Liechtenstein, Basle, 1571.}, and +Firmicus\index{Firmicus on Astrology}. The last of these collected the +works of most of his predecessors in a volume\footnote +{\textit{Astronomicorum}, eight books, Venice, 1499.}, which remained a +standard authority until the close of the sixteenth century. + +I may begin by reminding the reader that though there +was a fairly general agreement as to the methods of procedure +and interpretation---which alone I attempt to describe---yet +there was no such thing as a fixed code of rules or a standard +text-book. It is therefore difficult to reduce the rules to any +precise and definite form, and almost impossible, within the +limits of a chapter, to give detailed references. At the same +time the practice of the elements of the art was tolerably well +established and uniform, and I feel no doubt that my account, +as far as it goes, is substantially correct. + +\phantomsection +\addcontentsline{toc}{section}{Astrology. Two branches: +natal and horary astrology} +There were two distinct problems with which astrologers +concerned themselves. One was the determination in general +outline of the life and fortunes of an enquirer: this was known +as \emph{natal astrology}\index{Natal Astrology}, and was effected by +the erection of a \emph{scheme +of nativity}. The other was the means of answering any specific +question about the individual: this was known as \emph{horary +astrology}\index{Horary Astrology}. Both depended on the casting or erecting +of a \emph{horoscope}. The person for whom it was erected was known as +the \emph{native}. + +\PG----File: 296.png---------------------------------------------------- +\phantomsection +\addcontentsline{toc}{section}{Rules for casting and reading a horoscope} +\phantomsection +\addcontentsline{toc}{subsection}{Houses and their significations} +A horoscope was cast according to the following rules\index +{HoroscopesRulesCast@\nobreak--- Rules to cast}\footnote{Raphael, pp.~91--109.}. The +space between two concentric and similarly situated squares +was divided into twelve spaces, as shown in the annexed +diagram. These twelve spaces were known technically as +\emph{houses}\index{Houses, Astrological}; they were numbered consecutively +$1, 2, \dotsc, 12$ (see +figure); and were described as the first house, the second +\begin{figure*}[!hbt] +\centerline{\includegraphics{./images/illus296}} +\end{figure*} +house, and so on. The dividing lines were termed \emph{cusps}\index +{Cusps, Astrological}: the +line between the houses $12$ and $1$ was called the cusp of +the first house, the line between the houses $1$ and $2$ was called +the cusp of the second house, and so on, finally the line +between the houses $11$ and $12$ was called the cusp of the +twelfth house. Each house had also a name of its own---thus +the first house was called the ascendant house, the eighth +house was called the house of death, and so on---but as +these names are immaterial for my purpose I shall not define +them. + +Next, the positions which the various astrological signs +and planets had occupied at some definite time and place (for +\PG----File: 297.png------------------------------------------------------ +instance, the time and place of birth of the native, if his nativity +was being cast) were marked on the celestial sphere. This +sphere was divided into twelve equal spaces by great circles +drawn through the zenith, the angle between any two consecutive +circles being $30^{\circ}$. The first circle was drawn through the +East point, and the space between it and the next circle towards +the North corresponded to the first house, and sometimes +was called the first house. The next space, proceeding from East +to North, corresponded to the second house, and so on. Each +of the twelve spaces between these circles corresponded to +one of the twelve houses, and each of the circles to one of +the cusps. + +In delineating\index{HoroscopeRulesCast@\nobreak--- Rules to cast}\footnote +{Raphael, pp.~118--131.} a horoscope, it was usual to begin by +inserting the zodiacal signs. A zodiacal sign\index +{Zodiac, Signs in Astrology} extends over $30^{\circ}$, +and was marked on the cusp which passed through it: by its +side was written a number indicating the distance to which its +influence extended in the earlier of the two houses divided by +the cusp. Next the position of the planets in these signs were +calculated, and each planet was marked in its proper house +and near the cusp belonging to the zodiacal sign in which +the planet was then situated: it was followed by a number +indicating its right ascension measured from the beginning of +the sign. The name of the native and the date for which the +horoscope was cast were inserted usually in the central square. +The \hyperlink{illus:307}{diagram} near the end of this chapter is a +facsimile of the horoscope of Edward~VI\index{Edward VI} as cast by Cardan +and will serve as an illustration of the above remarks. + +We are now in a position to explain how a horoscope was +\emph{read}\index{HoroscopesRulesRead@\nobreak--- Rules to read|(} or interpreted. +Each house was associated with certain +definite questions and subjects, and the presence or absence in +that house of the various signs and planets gave the answer to +these questions or information on these subjects. + +These questions cover nearly every point on which information +would be likely to be sought. They may be classified +roughly as follows. For the answer, so far as it concerns the +\PG----File: 298.png------------------------------------------------------ +native, to all questions connected with his life and health, look +in house~1; for questions connected with his wealth, refer to +house~2; for his kindred and communications to him, refer to +3; for his parents and inheritances, refer to 4; for his children +and amusements, refer to 5; for his servants and illnesses, +refer to 6; for his marriage and amours, refer to 7; for his +death, refer to 8; for his learning, religion and travels, refer +to 9; for his trade and reputation, refer to 10; for his friends, +refer to 11; and finally for questions connected with his +enemies, refer to house 12. + +\phantomsection +\addcontentsline{toc}{subsection}{Planets and their significations} +I proceed to describe briefly the influences of the planets% +\index{PlanetsA@Planets (astrological)}% +\index{PlanetsS@\nobreak--- Signification of|(}, +and shall then mention those of the zodiacal signs; I should +note however that in practice the signs were in many respects +more influential than the planets. + +The astrological ``planets''\index{Astrological Planets} were seven in +number, and included the Sun and the Moon. They were Saturn or the +Great Infortune, Jupiter or the Great Fortune, Mars or the +Lesser Infortune, the Sun, Venus or the Lesser Fortune, +Mercury, and the Moon: the above order being that of their +apparent times of rotation round the earth. + +Each of them had a double signification. In the first place +it impressed certain characteristics, such as good fortune, +feebleness,~\&c., on the dealings of the native with the subjects +connected with the house in which it was located; and in the +second place it imported certain objects into the house which +would affect the dealings of the native with the subjects of +that house. + +To describe the exact influence of each planet in each +house would involve a long explanation, but the general effect +of their presence may be indicated roughly as follows\footnote +{Raphael, pp.~70--90; pp.~204--209.}. The +presence of Saturn is malignant: that of Jupiter is propitious: +that of Mars is on the whole injurious: that of the Sun +indicates respectability and moderate success: that of Venus +is rather favourable: that of Mercury implies rapid practical +action: and lastly the presence of the Moon merely faintly +\PG----File: 299.png------------------------------------------------------ +reflects the influence of the planet nearest her, and suggests +rapid changes and fickleness. Besides the planets, the Moon's +nodes and some of the more prominent fixed stars\footnote +{Raphael, pp.~129--131,} also had certain influences. + +These vague terms may be illustrated by taking a few +simple cases. + +For example, in casting a nativity, the life, health, and +general career of the native were determined by the first or +ascendant house, whence comes the expression that a man's +fortune is in the ascendant. Now the most favourable planet +was Jupiter. Therefore, if at the instant of birth Jupiter was +in the first house, the native might expect a long, happy, +healthy life; and being born under Jupiter he would have a +``jovial'' disposition. On the other hand, Saturn was the most +unlucky of all the planets, and was as potent as malignant. If +at the instant of birth he was in the first house, his potency +might give the native a long life, but it would be associated +with an angry and unhappy temper, a spirit covetous, revengeful, +stern, and unloveable, though constant in friendship +no less than in hate, which was what astrologers meant by a +``saturnine'' character. Similarly a native born under Mercury, +that is, with Mercury in the first house, would be of a mercurial +nature, while anyone born under Mars would have a martial +bent. + +Moreover it was the prevalent opinion that a jovial person +would have his horoscope affected by Jupiter, even if that +planet had not been in the ascendant at the time of birth. +Thus the horoscope of an adult depended to some extent on his +character and previous life. It is hardly necessary to point +out how easily this doctrine enabled an astrologer to make the +prediction of the heavens agree with facts that were known +or probable. + +In the same way the other houses are affected. For instance, +no astrologer, who believed in the art, would have +wished to start on a long journey when Saturn was in the +\PG----File: 300.png----------------------------------------------------- +ninth house or house of travels; and, if at the instant of birth +Saturn was in that house, the native always would incur +considerable risk on his journeys. + +Moreover every planet was affected to some extent by its +aspect (conjunction, opposition, or quadrature) to every other +planet according to elaborate rules\footnote +{Raphael, pp. 132--170.} which depended on their +positions and directions of motion: in particular the angular +distance between the Sun and the Moon---sometimes known +as the ``part of fortune''---was regarded as specially important, +and this distance affected the whole horoscope. In general, +conjunction was favourable, quadrature unfavourable, and +opposition ambiguous. + +Each planet not only influenced the subjects in the house in +which it was situated, but also imported certain objects into +the house. Thus Saturn was associated with grandparents, +paupers, beggars, labourers, sextons, and gravediggers. If, for +example, he was present in the fourth house, the native might +look for a legacy from some such person; if he was present +in the twelfth house, the native must be careful of the consequences +of the enmity of any such person; and so on. + +Similarly Jupiter was associated generally with lawyers, +priests, scholars, and clothiers; but, if he was conjoined with a +malignant planet, he represented knaves, cheats, and drunkards. +Mars indicated soldiers (or, if in a watery sign, sailors on ships +of war), masons, doctors, smiths, carpenters, cooks, and tailors; +but, if afflicted with Mercury or the Moon, he denoted the +presence of thieves. The Sun implied the action of kings, +goldsmiths, and coiners; but, if afflicted by a malignant planet, +he denoted false pretenders. Venus imported musicians, embroiderers, +and purveyors of all luxuries; but, if afflicted, +prostitutes and bullies. Mercury imported astrologers, philosophers, +mathematicians, statesmen, merchants, travellers, men +of intellect, and cultured workmen; but, if afflicted, he signified +the presence of pettifoggers, attorneys, thieves, messengers, +\PG----File: 301.png----------------------------------------------------- +footmen, and servants. Lastly, the presence of the Moon +introduced sailors and those engaged in inferior offices\index +{PlanetsS@\nobreak--- Signification of|)}. + +\phantomsection +\addcontentsline{toc}{subsection}{Zodiacal signs and their significations} +I come now to the influence and position of the zodiacal\index +{Zodiac, Signs in Astrology} +signs. So far as the first house was concerned, the sign of the +zodiac which was there present was even more important than +the planet or planets, for it was one of the most important +indications of the durations of life. + +Each sign was connected with certain parts of the body---\Eg\ +Aries influenced the head, neck and shoulders---and that +part of the body was affected according to the house in which +the sign was. Further each sign was associated with certain +countries and connected the subjects of the house in which the +sign was situated with those countries: \Eg\ Aries was +associated especially with events in England, France, Syria, +Verona, Naples,~\&c. + +The sign in the first house determined also the character +and appearance of the native\footnote +{Raphael, pp. 61--69.}. Thus the character of a native +born under Aries (\textit{m}) was passionate; under Taurus (\textit{f}) was +dull and cruel; under Gemini (\textit{m}) was active and ingenious; +under Cancer (\textit{f}) was weak and yielding; under Leo (\textit{m}) was +generous, resolute, and ambitious; under Virgo (\textit{f}) was sordid +and mean; under Libra (\textit{m}) was amorous and pleasant; under +Scorpio (\textit{f}) was cold and reserved; under Sagittarius (\textit{m}) +was generous, active, and jolly; under Capricorn (\textit{f}) was weak and +narrow; under Aquarius (\textit{m}) was honest and steady; and under +Pisces (\textit{f}) was phlegmatic and effeminate. + +Moreover the signs were regarded as alternately masculine +and feminine, as indicated above by the letters \textit{m} or \textit{f} +placed after each sign. A masculine sign is fortunate, and all planets +situated in the same house have their good influence rendered +thereby more potent and their unfavourable influence mitigated. +But all feminine signs are unfortunate, their direct effect is evil, +and they tend to nullify all the good influence of any planet +which they afflict (\IE\ with which they are connected), and to +increase all its evil influences, while they also import an element +\PG----File: 302.png------------------------------------------------------ +of fickleness into the house and often turn good influences into +malignant ones. The precise effect of each sign was different +on every planet\index{HoroscopesRulesRead@\nobreak--- Rules to read|)}. + +\phantomsection +\addcontentsline{toc}{section}{Knowledge that rules were worthless} +I think the above account is sufficient to enable the reader +to form a general idea of the manner in which a horoscope was +cast and interpreted, and I do not propose to enter into further +details. This is the less necessary as the rules---especially as +to the relative importance to be assigned to various planets +when their influence was conflicting---were so vague that astrologers +had little difficulty in finding in the horoscope of a client +any fact about his life of which they had information or any +trait of character which they suspected him to possess. + +That this vagueness was utilized by quacks is notorious, +but no doubt many an astrologer in all honesty availed himself +of it, whether consciously or unconsciously. It must be remembered +also that the rules were laid down at a time when men +were unacquainted with any exact science, with the possible +exception of mathematics, and further that, if astrology had +been reduced to a series of inelastic rules applicable to all horoscopes, +the number of failures to predict the future correctly +would have rapidly led to a recognition of the folly of the art. +As it was, the failures were frequent and conspicuous enough +to shake the faith of most thoughtful men. Moreover it was a +matter of common remark that astrologers showed no greater +foresight in meeting the difficulties of life than their neighbours, +while they were neither richer, wiser, nor happier for +their supposed knowledge. But though such observations were +justified by reason they were often forgotten in times of difficulty +and danger. A prediction of the future and the promise +of definite advice as to the best course of action, revealed by the +heavenly bodies themselves, appealed to the strongest desires of +all men, and it was with reluctance that the futility of the +advice was gradually recognized. + +The objections to the scheme had been stated clearly by +several classical writers. Cicero\index{Cicero on Astrology}\footnote +{Cicero, \textit{De Divinatione}, \textsc{ii}, 42.} pointed out that +not one of +\PG----File: 303.png------------------------------------------------------ +the futures foretold for Pompey\index{Pompey}, Crassus\index{Crassus}, +and Caesar\index{Caesar, Julius}\index{Julius Caesar} had been +verified by their subsequent lives, and added that the planets, +being almost infinitely distant, cannot be supposed to affect +us. He also alluded to the fact, which was especially pressed by +Pliny\index{Pliny}\footnote +{Pliny, \textit{Historia Naturalis}, \textsc{vii}, 49; \textsc{xxix}, 1.}, +that the horoscopes of twins are practically identical +though their careers are often very different, or as Pliny put it, +every hour in every part of the world are born lords and slaves, +kings and beggars. + +In answer to the latter obvious criticism astrologers replied +by quoting the anecdote of Publius Nigidius Figulus\index +{Figulus on Astrology}\index{Nigidius on Astrology}, a +celebrated Roman astrologer of the time of Julius Caesar. It +is said that when an opponent of the art urged as an objection +the different fates of persons born in two successive instants, +Nigidius bade him make two contiguous marks on a potter's +wheel, which was revolving rapidly near them. On stopping +the wheel, the two marks were found to be far removed from +each other. Nigidius received the name of Figulus, the +potter, in remembrance of this story, but his argument, says +St~Augustine\index{Augustine on Astrology}\footnote +{St~Augustine, \textit{De Civitate Dei}, bk.~\textsc{v}, chap.~iii; +\textit{Opera omnia}, ed.\ Migne, vol.~\textsc{vii}, p.~143.}, who +gives us the narrative, was as fragile as +the ware which the wheel manufactured. + +On the other hand Seneca\index{Seneca on Astrology} and Tacitus\index +{Tacitus on Astrology} may be cited as +being on the whole favourable to the claims of astrology, +though both recognized that it was mixed up with knavery and +fraud. An instance of successful prediction which is given +by the latter of these writers\footnote +{\textit{Annales}, \textsc{vi}, 22: quoted by Whewell\index{Whewell, W.}, +\textit{History of the Inductive Sciences}, vol.~\textsc{i}, p.~313.} +may be used more correctly as +an illustration of how the ordinary professors of the art varied +their predictions to suit their clients and themselves. The +story deals with the first introduction of the astrologer Thrasyllus\index +{Thrasyllus on Astrology} to the emperor Tiberius\index +{Tiberius on Astrology}. Those who were brought to +Tiberius on any important matter were admitted to an interview +in an apartment situated on a lofty cliff in the island +\PG----File: 304.png------------------------------------------------------ +of Capreae. They reached this place by a narrow path overhanging +the sea, accompanied by a single freedman of great +bodily strength; and on their return, if the emperor had conceived +any doubts of their trustworthiness, a single blow buried +the secret and its victim in the ocean. After Thrasyllus had, +in this retreat, stated the results of his art as they concerned +the emperor, the latter asked the astrologer whether he had +calculated how long he himself had to live. The astrologer +examined the aspect of the stars, and while he did this showed, +as the narrative states, hesitation, alarm, increasing terror, and +at last declared that the present hour was for him critical, +perhaps fatal. Tiberius\index{Tiberius on Astrology} embraced him, and told +him he was right in supposing he had been in danger but that he should +escape it; and made him thenceforth a confidential counsellor. +But Thrasyllus\index{Thrasyllus on Astrology} would have been but a sorry +astrologer had he not foreseen such a question and prepared an answer which +he thought fitted to the character of his patron. + +A somewhat similar story is told\footnote +{\textit{Personal Characteristics from French History}, by Baron +F.~Rothschild\index{Rothschild, F.}, +London, 1896, p.~10. The story was introduced by Sir Walter +Scott\index{Scott, Sir Walter} in Quentin Durward (chap.~\textsc{xv}).} +of Louis~XI\index{Louis XI of France} of France. +He sent for a famous astrologer whose death he was meditating, +and asked him to show his skill by foretelling his own future. +The astrologer replied that his fate was uncertain, but it was +so inseparably interwoven with that of his questioner that the +latter would survive him but by a few hours, whereon the +superstitious monarch not only dismissed him uninjured, but +took steps to secure his subsequent safety. The same anecdote +is also related of a Scotch student who, being captured by +Algerian pirates, predicted to the Sultan that their fates +were so involved that he should predecease the Sultan by +only a few weeks. This may have been good enough for a +barbarian, but with a civilized monarch it probably would in +most cases be less effectual, as certainly it is less artistic, +than the answer of Thrasyllus. + +\PG----File: 305.png------------------------------------------------------ +\medskip +\phantomsection +\addcontentsline{toc}{section}{Notable instances of horoscopy} +I may conclude by mentioning a few notable cases of horoscopy. + +\phantomsection +\addcontentsline{toc}{subsection}{Lilly's prediction of the Great Fire +and Plague} +Among the most successful instances of horoscopy enumerated +by Raphael\footnote +{\textit{Manual of Astrology}, p.~37.} is one by W.~Lilly\index +{Lilly on Astrology}, given in his \textit{Monarchy +or No Monarchy}, published in 1651, in which he predicted a +plague in London so terrible that the number of deaths should +exceed the number of coffins and graves, to be followed by ``an +exorbitant fire.'' The prediction was amply verified in 1665 +and 1666. In fact Lilly's success was embarrassing, for the +Committee of the House of Commons, which sat to investigate +the causes of the fire and ultimately attributed it to the papists, +thought that he must have known more about it than he +chose to declare, and on Oct.~25, 1666, summoned him before +them. I may add that Lilly proved himself a match for his +questioners. + +\phantomsection +\addcontentsline{toc}{subsection}{Flamsteed's guess} +An even more curious instance of a lucky hit is told of +Flamsteed\index{Flamsteed on Astrology}\footnote +{The story, though in a slightly different setting, is given in \textit{The +London Chronicle}, Dec.~3, 1771, and it is there stated that Flamsteed +attributed the result to the direct action of the devil.}, +the first astronomer royal. It is said that an +old lady who had lost some property wearied Flamsteed by +her perpetual requests that he would use his observatory to +discover her property for her. At last, tired out with her importunities, +he determined to show her the folly of her demand +by making a prediction, and, after she had found it false, to +explain again to her that nothing else could be expected. +Accordingly he drew circles and squares round a point that +represented her house and filled them with all sorts of mystical +symbols. Suddenly striking his stick into the ground he said, +``Dig there and you will find it.'' The old lady dug in the spot +thus indicated, and found her property; and it may be conjectured +that she believed in astrology for the rest of her life. + +Perhaps the belief that the royal observatory was built for +such purposes may be still held, for De~Morgan\index +{DeMorgan@De Morgan, A.}, writing in +1850, says that ``persons still send to Greenwich to have their +\PG----File: 306.png------------------------------------------------------- +fortunes told, and in one case a young gentleman wrote to +know who his wife was to be, and what fee he was to remit.'' + +It is easier to give instances of success in horoscopy than of +failure. Not only are all ambiguous predictions esteemed to +be successful, but it is notorious that prophecies which have +been verified by the subsequent course of events are remembered +and quoted, while the far more numerous instances in which the +prophecies have been falsified are forgotten or passed over in +silence. + +\phantomsection +\addcontentsline{toc}{subsection}{Cardan's horoscope of Edward VI} +As exceptionally well-authenticated instances of failures +I may mention the twelve cases collected by Cardan\index{Cardan|(} in his +\textit{Geniturarum Exempla}. These are good examples because +Cardan was not only the most eminent astrologer of his time, +but was a man of science, and perhaps it is not too much to say +was accustomed to accurate habits of thought; moreover, as far +as I can judge, he was perfectly honest in his belief in astrology. +To English readers the most interesting of these is the horoscope +of Edward~VI\index{Edward VI|(} of England, the more so as Cardan has left +a full account of the affair, and has entered into the reasons of +his failure to predict Edward's death. + +To show how Cardan came to be mixed up in the transaction +I should explain that in 1552 Cardan went to Scotland to +prescribe for John Hamilton\index{Hamilton, Archbishop}, the archbishop of +St~Andrews, who was ill with asthma and dropsy and about whose treatment +the physicians had disagreed\footnote +{Luckily they left voluminous reports on the case and the proper +treatment for it. The only point on which there was a general agreement +was that the phlegm, instead of being expectorated, collected in his +Grace's brains, and that thereby the operations of the intellect were +impeded. Cardan was celebrated for his success in lung diseases, and his +remedies were fairly successful in curing the asthma. His fee was $500$ +crowns for travelling expenses from Pavia, $10$ crowns a day, and the +right to see other patients; the archbishop actually gave him $2300$ crowns +in money and numerous presents in kind; his fees from other persons +during the same time must have amounted to about an equal sum (see +Cardan's \textit{De Libris Propriis}, ed.\ 1557, pp.~159--175; \textit +{Consilia Medica}, \textit{Opera}, vol.~\textsc{ix}, pp.~124--148; +\textit{De Vita Propria}, ed.\ 1557, pp.~138, 193 \etseq).}. +On his return through +\PG----File: 307.png------------------------------------------------------- +London, Cardan stopped with Sir John Cheke\index{Cheke, Sir John}, the +Professor of Greek at Cambridge, who was tutor to the young king. Six +months previously, Edward had been attacked by measles and +small-pox which had made his health even weaker than before. +The king's guardians were especially anxious to know how long +he would live, and they asked Cardan to cast Edward's nativity +with particular reference to that point. + +The Italian was granted an audience in October, of which +he wrote a full account in his diary, quoted in the \textit{Geniturarum +Exempla}. The king, says he\footnote +{I quote from Morley's\index{Morley on Cardan} translation, vol.~\textsc{ii}, +p.~135 \etseq}, was ``of a stature somewhat +below the middle height, pale faced, with grey eyes, a grave +aspect, decorous, and handsome. He was rather of a bad habit +of body than a sufferer from fixed diseases, and had a somewhat +projecting shoulder-blade.'' But, he continues, he was a +boy of most extraordinary wit and promise. He was then +but fifteen years old and he was already skilled in music and +dialectics, a master of Latin, English, French, and fairly proficient +\begin{figure*}[htb] +\centerline{\hypertarget{illus:307}{\includegraphics +[height=\ifPaper9cm\else.7\textheight\fi]{./images/illus307}\index +{EdwardHoro@\protect\nobreak--- Horoscope of}\index +{HoroscopesExample@\protect\nobreak--- Example of}}}\label{illus:307} +\end{figure*} +\PG----File: 308.png------------------------------------------------------ +in Greek, Italian, and Spanish. He ``filled with the +highest expectation every good and learned man, on account of +his ingenuity and suavity of manners\textellipsis. When a royal gravity +was called for, you would think that it was an old man you saw, +but he was bland and companionable as became his years. He +played upon the lyre, took concern for public affairs, was liberal +of mind, and in these respects emulated his father\index +{Henry VIII of England}, who, while he +studied to be [too] good, managed to seem bad.'' And in another +place\footnote{\textit{De Rerum Varietate}, p.~285.} +he describes him as ``that boy of wondrous hopes.'' At +the close of the interview Cardan begged leave to dedicate to +Edward a work on which he was then engaged. Asked the +subject of the work, Cardan replied that he began by showing +the cause of comets. The subsequent conversation, if it is +reported correctly, shows good sense and considerable logical +skill on the part of the young king. + +I have reproduced \vpageref{illus:307} a facsimile of Cardan's original +drawing of Edward's horoscope. The horoscope was cast and +read with unusual care. I need not quote the minute details +given about Edward's character and subsequent career, but +obviously the predictions were founded on the impressions derived +from the above-mentioned interview. The conclusion +about his length of life was that he would certainly live past +middle age, though after the age of 55 years 3 months and +17 days various diseases would fall to his lot\footnote +{\textit{Geniturarum Exempla}, p.~19.\label{ibid:16}}. + +In the following July the king died\index{Edward VI|)}, and Cardan felt it +necessary for his reputation to explain the cause of his error. +The title of his dissertation is \emph{Quae post consideravi de +eodem}\footnote{\Ibidref{ibid:16}{\textit{Geniturarum Exempla}}, p.~23.}. +In effect his explanation is that a weak nativity can never +be predicted from a single horoscope, and that to have ensured +success he must have cast the nativity of every one with whom +Edward had come intimately into contact; and, failing the +necessary information to do so, the horoscope could be regarded +only as a probable prediction\index{Cardan|)}. + +\PG----File: 309.png------------------------------------------------------ +This was the argument usually offered to account for non-success. +A better defence would have been the one urged +by Raphael\footnote{\textit{The Familiar Astrologer}, London, 1832, p.~248.} +and by Southey\index{Southey on Astrology}\footnote +{\textit{The Doctor}, chap.~92.} that there might be other +planets unknown to the astrologer which had influenced +the horoscope, but I do not think that medieval astrologers +assigned this reason for failure. + +I have not alluded to the various adjuncts of the art, but +astrologers so frequently claimed the power to be able to raise +spirits\index{Spirits, Raising} that perhaps I may be pardoned for remarking +that I believe some of the more important and elaborate of these +deceptions were effected not infrequently by means of a magic +lantern, the pictures being sometimes thrown on to a mirror, +and at other times on to a thick cloud of smoke which caused +the images to move and finally disappear in a fantastic way +capable of many explanations\footnote +{See \Eg\ the life of Cellini\index{Cellini}, chap.~\textsc{xiii}, +Roscoe's translation, pp.~144-146. See also Sir David Brewster's\index +{Brewster, Sir David} \textit{Letters on Natural Magic}.}. + +I would conclude by repeating again that though the +practice of astrology was so often connected with impudent +quackery, yet one ought not to forget that nearly every +physician and man of science in medieval Europe was an +astrologer. These observers did not consider that its rules +were definitely established, and they laboriously collected much +of the astronomical evidence that was to crush their art. Thus, +though there never was a time when astrology was not practised +by knaves, there was a period of intellectual development when +it was honestly accepted as a difficult but a real science. + +\PG----File: 310.png------------------------------------------------------- + + + + +% CHAPTER XI + +\chapter{Cryptographs and Ciphers.} + + +\textsc{The} art of constructing cryptographs or +ciphers---intelligible\chapindex{Ciphers@\textsc{Ciphers}} +\chapindex{Cryptography@\textsc{Cryptography}}% +\chapindex{Secret@\textsc{Secret Communications}}% +to those who know the key and unintelligible to others---has +been studied for centuries. Their usefulness on certain +occasions, especially in time of war, is obvious, while it may +be a matter of great importance to those from whom the key +is concealed to discover it. But the romance connected with +the subject, the not uncommon desire to discover a secret, and +the implied challenge to the ingenuity of all from whom the +key is hidden, have attracted to the subject the attention of +many to whom its utility is a matter of indifference. + +The leading authorities on the subject, few of which are +less than a century old, are enumerated in an article by J.E.~Bailey\index +{Bailey, J.E.} in the ninth edition of the \textit{Encyclopaedia Britannica}, +and references to various historic ciphers are there given. +My knowledge of the subject, however, is limited to ciphers +which I have met with in the course of casual reading, and I +prefer to discuss the subject as it has presented itself to me, +with no attempt to make it historically complete and no +reference to other authorities. In fact the theory of the +subject is not sufficiently important to make it worth while +to try to deal with it historically or exhaustively. + +Most writers use the words cryptograph and cipher as +synonymous. I employ them, however, with different meanings, +which I proceed to define. + +\PG----File: 311.png------------------------------------------------------- +\phantomsection +\addcontentsline{toc}{section}{A Cryptograph. Definition. Illustration} +A cryptograph may be defined\index +{Cryptographs, Def@\textsc{Cryptographs}, Definition of} +as a manner of writing in +which the letters or symbols employed are used in their +normal sense, but are so arranged that the communication is +intelligible only to those possessing the key. The word is +sometimes used to denote the communication made. A simple +example is a communication in which every word is spelt +backwards. Thus: +\[ +\text{\emph{ymene deveileb ot eb gniriter troper noitisop no ssorc daor.}} +\] + +\phantomsection +\addcontentsline{toc}{section}{A Cipher. Definition. Illustration} +A cipher may be defined\index{Ciphers, Definition@\nobreak--- Definition of} +as a manner of writing by +characters arbitrarily invented or by an arbitrary use of +letters, words, or characters in other than their ordinary +sense, intelligible only to those possessing the key. The +word is sometimes used to denote the communication made. +A simple example is when each letter is replaced by the +one that immediately follows it in the natural order of the +alphabet, \emph{a} being replaced by \emph{b}, \emph{b} by \emph{c}, +and so on, and finally +\emph{z} by \emph{a}. In this cipher the above message would read: +\[ +\text{\emph{fofnz cfmjfwfe up cf sfujsjoh sfqpsu qptjujpo po dsptt spbe.}} +\] + +\phantomsection +\addcontentsline{toc}{section}{Essential Features of Cryptographs +and Ciphers} +In both cryptographs and ciphers the essential feature is +that the communication may be freely given to all the world +though it is unintelligible save to those who possess the key. +The key must not be accessible to anyone, and if possible it +should be known only to those using the cryptograph or +cipher. The art of constructing a cryptograph lies in the +concealment of the proper order of the essential letters or +words: the art of constructing a cipher lies in concealing +what letters or words are represented by the symbols used. +In an actual communication cipher symbols may be arranged +cryptographically, and thus further hinder a reading of the +message. Thus the message given above would read in a +cryptographic cipher as +\[ +\text{\emph{znfof efwfjmfc pu fc hojsjufs uspqfs opjujtpq op ttpsd ebps.}} +\] +If the message were sent in Latin or some foreign language it +would further diminish the chance of it being read by a +\PG----File: 312.png------------------------------------------------------ +stranger through whose hands it passed. But I may confine +myself to messages in English, and for the present to simple +cryptographs and ciphers. + +A communication in cryptograph or cipher must be in +writing or in some permanent form. Thus to make small +muscular movements---such, \Eg, as talking on the fingers, +or breathing long and short in the Morse dot and dash system, +or making use of pre-arranged signs by a fan or stick, or +flashing signals by light---do not here concern us. + +Again, the mere fact that the message is concealed or +conveyed secretly does not make it a cryptograph or cipher. +The majority of stories dealing with secret communications +are concerned with the artfulness with which the message +is concealed or conveyed and have nothing to do with +cryptographs or ciphers. Many of the ancient instances of +secret communication are of this type\footnote +{A long list of classical authorities for different devices used in +ancient times for concealing messages is given in \textit{Mercury} by +J.~Wilkins\index{Wilkins on Ciphers}, London, 1641, pp.~27--36.}. +Illustrations are to +be found in messages conveyed by pigeons, or wrapped round +arrows shot over the head of a foe, or written on the paper +wrapping of a cigarette, or the use of ink which becomes +visible only when the recipient treats the paper on which it +is written by some chemical or physical process. + +Again, a communication in a foreign language or in any +recognized notation like shorthand is not an instance of a +cipher. A letter in Chinese or Polish or Russian might be +often used for conveying a secret message from one part of +England to another, but it fails to fulfil our test that if +published to all the world it would be concealed from everyone, +unless submitted to some special investigation. On the other +hand, in practice, foreign languages or systems of shorthand +which are but little known may serve to conceal a communication +better than an easy cipher, for in the last case +the key may be found with but little trouble, while in the +other cases, though the key may be accessible, it is probable +\PG----File: 313.png------------------------------------------------------- +that there are only a few who know where to look for it. +An illustration of this is afforded by the system used by +Pepys in writing his Diary which is further alluded to below. + +\phantomsection +\addcontentsline{toc}{section}{Cryptographs of Three Types. +Illustrations}\markright{Cryptographs} +I proceed to enumerate some of the better known types of +cryptographs. There are at least three distinct types. The\index +{Cryptographs, Three@\nobreak--- Three types of|(} +first type comprises those in which the order of the letters +is changed in some pre-arranged manner. The second type +comprises those in which the concealment is due to the introduction +of non-significant letters. The third type comprises +those in which the letters used are written in fragments. +The types are not exclusive, and any particular cryptograph +may comprise the distinctive feature of two or all the types. + +\phantomsection +\addcontentsline{toc}{subsection}{Order of letters re-arranged} +A cryptograph of the first type is one in which the +successive letters of the message are re-arranged in some +pre-determined manner. + +One of the most obvious cryptographs of this type is to +write each word or the message itself backwards. He would, +however, be a careless reader who could be deceived by this. +Here is an instance in which the whole message is written +backwards: +\[ +\text{\emph{tsop yb tnes tnemeerga fo seniltuo smret ruo tpecca yeht.}} +\] +In such a case it is unnecessary to indicate the division into +words by leaving spaces between them, and we might divide +the letters artificially, as thus: +\[ +\text{\emph{Ts opybtne stne meer gafos eniltu osmret ruot peccaye ht.}} +\] + +Systems of this kind which depend on altering the places +of letters or lines in some pre-arranged manner have always +been common. I quote a couple of instances\footnote +{\textit{Mercury, or the Secret and Swift Messenger}, by +\hypertarget{footnote:wilkins}{J.~Wilkins}, London, +1641, pp.~50--52.} from Wilkins's\index{Wilkins on Ciphers} +book to which I have already referred---it was a work which +seems to have been studied diligently by many of those who +took part in the civil disturbances of the 17th century, and +gives an excellent account of some of the easier systems of +cryptographs and ciphers. + +\PG----File: 314.png------------------------------------------------------ +The first example I take from him is where the letters +which make up the communication are written vertically up +or down. Thus the message: \emph{The pestilence continues to +increase} might be written thus: +\[\def\arraycolsep{1pt} +\begin{matrix} +\emph{e}&\emph{i}&\emph{o}&\emph{t}&\emph{n}&\emph{l}&\emph{i}&\emph{t}\\ +\emph{s}&\emph{n}&\emph{t}&\emph{i}&\emph{o}&\emph{e}&\emph{t}&\emph{h}\\ +\emph{a}&\emph{c}&\emph{s}&\emph{n}&\emph{c}&\emph{n}&\emph{s}&\emph{e}\\ +\emph{e}&\emph{r}&\emph{e}&\emph{u}&\emph{e}&\emph{c}&\emph{e}&\emph{p} +\end{matrix} +\] + +Again, Wilkins\index{Wilkins on Ciphers} says that the cryptograph may be +yet further obscured by placing the letters which make up the +message in any pre-arranged but discontinuate order. For +instance if the message runs to four lines we may put the +first letter at the beginning of the first line, the second at +the beginning of the fourth line, the third at the end of the +first line, the fourth at the end of the fourth line, the fifth at +the beginning of the second line, the sixth at the beginning of +the third line, the seventh at the end of the second line, the +eighth at the end of the third line, and so on. Thus the +message: \emph{Wee shall make an Irruption upon the Enemie, from +the North, at ten of the clock this night} would read thus: +\[ +\begin{array}{l} +\emph{Wm rpeta hhs cteinpke}\\ +\emph{haih fonoih kftoe nil}\\ +\emph{anoerr ocgt tthmnu rl}\\ +\emph{eauo mhtei nlen ettes}, +\end{array} +\] +where, to obscure the message further, it is divided arbitrarily +into what appear to be words. + +Another instance of a cryptograph of this type may be +constructed thus. First, by writing the message in lines +of some arranged length, say, for instance, each containing +seventeen letters---the letters in successive lines being +arranged vertically under those corresponding to them in +the upper line---and either leaving no spaces between the +words or inserting some pre-arranged letter or letters or digits +between them, such as $j$, $q$, $z$. The message can be then sent +as a cryptograph by writing the letters in order in successive +\PG----File: 315.png--------------------------------------------------- +vertical lines. This only comes to saying that we write +successively the 1st, 18th, 35th letters of the original message, +and then the 2nd, 19th, 36th letters, and so on. To confuse +the decipherer the final reading may be arbitrarily put into +what might represent words. If, however, we know the clue +number, say $c$, it is easy enough to read the communication. +For if it divides into the number of letters $n$ times with a +remainder $r$ it suffices to re-write the message in lines putting +$n + 1$ letters in each of the first $r$ lines, and $n$ letters in each +of the last $c - r$ lines, and then the communication can be +read by reading the columns downwards. For instance, if +the following communication, containing $270$ letters, were +received: +{\CryptoSetup +\emph{A\.h\.t\.z\.e\.i\.p\.q\.h\.g\.e\.s\.o\.a\.e\.o\.u\.a\.z\.s\.e\.s\.e\.% +w\.a\.e\.q\.t\.m\.u\.s\.f\.d\.t\.b\.e\.n\.z\.c\.e\.s\.j\.t\.e\.o\.t\.t\.q\.% +i\.z\.y\.c\.z\.% +h\.t\.z\.j\.i\.o\.a\.r\.h\.q\.e\.t\.t\.j\.r\.f\.e\.s\.f\.t\.n\.z\.m\.r\.o\.% +o\.m\.o\.h\.y\.e\.a\.r\.z\.i\.a\.q\.n\.e\.o\.r\.n\.b\.r\.e\.o\.t\.l\.e\.n\.% +n\.k\.a\.e\.r\.w\.i\.z\.e\.s\.j\.u\.% +a\.s\.j\.o\.d\.e\.z\.w\.j\.z\.z\.s\.z\.j\.b\.r\.r\.i\.t\.t\.j\.n\.f\.j\.l\.% +w\.e\.u\.z\.r\.o\.q\.y\.f\.o\.h\.t\.q\.a\.y\.e\.i\.z\.s\.l\.e\.o\.p\.j\.i\.% +d\.i\.h\.a\.l\.o\.a\.l\.h\.p\.e\.p\.k\.r\.% +h\.e\.a\.n\.a\.z\.s\.r\.v\.l\.i\.i\.m\.o\.s\.i\.a\.d\.y\.g\.t\.p\.e\.k\.i\.% +j\.s\.c\.e\.r\.q\.v\.v\.j\.q\.j\.q\.a\.j\.q\.n\.y\.j\.i\.n\.t\.k\.a\.e\.h\.% +s\.b\.h\.s\.n\.b\.g\.o\.a\.o\.t\.q\.% +e\.t\.q\.e\.u\.u\.e\.s\.a\.y\.q\.u\.r\.n\.t\.p\.e\.b\.q\.s\.t\.z\.a\.m\.z\.% +t\.q\.r\.j}}, and the clue number were \emph{17} we +should put \emph{16} letters in each of the first \emph{15} lines and +\emph{15} letters +in each of the last \emph{2} lines. The communication could then be +discovered by reading the columns downwards: the letters +\emph{j}, \emph{q}, and \emph{z} marking the ends of words. + +Another cryptograph of this type may be constructed by +arranging the letters cyclically, and agreeing that the communication +is to be made by selected letters, as, for instance, +every seventh, second, seventh, second, and so on. Thus if the +communication were \emph{Ammunition too low to allow of a sortie}, +which consists of $32$ letters, the successive significant letters +would come in the order $7$, $9$, $16$, $18$, $25$, $27$, $2$, $4$, $13$, +$15$, $24$, $28$, $5$, $8$, $20$, $22$, $1$, $6$, $21$, $26$, $11$, $14$, +$32$, $10$, $31$, $12$, $17$, $23$, $3$, +$29$, $30$, $19$---the numbers being selected as in the decimation +problem given above on pages \pageref{page:DecimationStart}--\pageref +{page:DecimationEnd}, and being struck out +from the $32$ cycle as soon as they are determined. The +above communication would then read +{\CryptoSetup +\emph{T\.t\.r\.i\.o\.o\.a\.l\.m\.o\.l\.a\.o\.o\.n\.m\.s\.u\.e\.o\.a\.% +w\.o\.t\.n\.l\.i\.o\.t\.i\.f\.w}}. This is a good cryptograph, but it is +troublesome to construct, especially if the message is long, and for that +reason is not to be recommended. + +\PG----File: 316.png------------------------------------------------------- +\phantomsection +\addcontentsline{toc}{subsection}{Use of non-significant symbols. The Grille} +A cryptograph of the second type is one in which the +message is expressed in ordinary writing, but in it are +introduced a number of dummies or non-significant letters or +digits thus concealing which of the letters are relevant. + +One way of picking out those letters which are relevant +is by the use of a perforated card of the shape of (say) a +sheet of note-paper, which when put over such a sheet permits +only such letters as are on certain portions of it to be visible. +Such a card is known as a \emph{grille}\index{Grille, The}. +An example of a grille +with four openings is figured \vpageref[below]{Grille}. A communication made +\begin{figure*}[!hbt] +\centering +\def\MSqHorizAdvance{3} +\def\SqWd{4.5em} +\begin{MagicSquare}{18}[6] +{} & {} & {} & {} & {} & {} \\ +{} & {} & {} & {} & {} & {} \\ +{} & {} & {} & {} & {} & {} \\ +{} & {} & {} & {} & {} & {} \\ +{} & {} & {} & {} & {} & {} \\ +{} & {} & {} & {} & {} & {}\\ +\linethickness{0.24em} +\put(3,4){\line(1,0){3}} +\put(3,4){\line(0,1){1}} +\put(3,5){\line(1,0){3}} +\put(6,4){\line(0,1){1}} +\put(9,2){\line(1,0){3}} +\put(9,2){\line(0,1){1}} +\put(9,3){\line(1,0){3}} +\put(12,2){\line(0,1){1}} +\put(15,1){\line(1,0){3}} +\put(15,1){\line(0,1){1}} +\put(15,2){\line(1,0){3}} +\put(15,5){\line(1,0){3}} +\put(15,5){\line(0,1){1}} +\sffamily\bfseries +\put(0,5.5){\llap{A }} +\put(18,5.5){\rlap{ B}} +\put(18,0){\rlap{ C}} +\put(0,0){\llap{D }} +\end{MagicSquare} +\label{Grille} +\end{figure*} +in this way may be easily concealed from anyone who does +not possess a card of the same pattern. If the recipient +possesses such a card he has only to apply it in order to read +the message. + +The use of the grille may be rendered less easy to detect +if it be used successively in different positions, for instance, +with the edges $AB$ and $CD$ successively put along the top of +the paper containing the message. \Vpageref[Below]{Grillex}, for instance, +is a message which, with the aid of the grille figured +\vpageref[above]{Grille}, is at +once intelligible. On applying the grille to it with the line +$AB$ along the top $HK$ we get the first half of the communication, +namely, \emph{1000 rifles se}. On applying the grille with the +line $CD$ along the top $HK$ we get the rest of the message, +namely, \emph{nt to L to-day}. The other spaces in the paper are +filled with non-significant letters or numerals in any way we +please. Of course any one using such a grille would not divide +\PG----File: 317.png------------------------------------------------------- +the sheet of paper on which the communication was written +into cells, but in the figure I have done so in order to render +the illustration clearer. + +\begin{figure*}[!hbt] +\centering +\def\MSqHorizAdvance{3} +\def\SqWd{4.5em} +\begin{MagicSquare}{18}[6] +{981} & {264} & {070} & {523} & {479} & {100} \\ +{NTT} & {ORI} & {SON} & {SON} & {AHY} & {DTC} \\ +{BFS} & {PUM} & {OLT} & {KFE} & {LJO} & {EGX} \\ +{AEU} & {QJT} & {EGO} & {FLE} & {HVE} & {WLA} \\ +{FML} & {AES} & {REM} & {REM} & {ODA} & {SSE} \\ +{YZZ} & {EPD} & {QJC} & {EKS} & {TIM} & {OEF} \\ +\sffamily\bfseries +\put(0,5.5){\llap{$H$ }} +\put(18,5.5){\rlap{ $K$}} +\end{MagicSquare} +\label{Grillex} +\end{figure*} + +We can avoid the awkward expedient of having to use a +perforated card, which may fall into undesired hands, by +introducing a certain pre-arranged number of dummies or +non-significant letters or symbols between those which make +up the message. Thus, to take an extreme case, we might +arrange that only every $101$st letter should form our communication, +and the intervening $100$ letters should be written +at random. But such a communication would be $101$ times +longer than the message, a nearly fatal objection if it had to +be written in a hurry or telegraphed. A better method, and +one which is not easily discovered by a stranger, is to arrange +that (say) only every alternate second and third letter shall +be relevant. Thus the first, third, sixth, eighth, eleventh, etc., +letters are those that make up the message. Such a communication +would be only two and a half times as long as +the message, but even that might be a great disadvantage +if time in sending the message was of importance. For a +message written at leisure this need not matter much, and +in such a code the introduction of a sufficient number of +unnecessary letters in some pre-arranged manner gives an +effectual means of conveying a message in secret. + +We can also avoid the use of a perforated card if it be +arranged that every $n$\textsuperscript{th} word shall give the message, +the other +\PG----File: 318.png------------------------------------------------------- +words being non-significant, though of course inserted as far as +possible so as to make the complete communication run as a +whole. But the difficulty of composing a document of this +kind and its great length render it unsuitable for any purpose +except an occasional communication composed at leisure and +sent in writing. This method is said to have been used by +the Earl of Argyle\index{Argyle} when plotting against +James~II\index{James II of England}. + +Similarly any system that rests on picking out certain +letters in a document, which letters form a communication in +ordinary writing, is a cryptograph. Thus a communication +conveyed by a newspaper, in which the letters making up the +message are indicated by pen dots or pin pricks or in some +other agreed way, is a cryptograph. + +\phantomsection +\addcontentsline{toc}{subsection}{Use of broken symbols. The Scytale} +A kind of secret writing which may perhaps be considered +to constitute a third type of cryptograph is a communication +on paper which is legible only when the paper is folded in a +particular way. An example is a message written across the +edges of a strip of paper wrapped spiral-wise round a stick +called a \emph{scytale}\index{Scytale, The}. When the paper is unwound +and taken off the stick the letters appear broken, and may seem to consist +of arbitrary signs, but by wrapping the paper round a similar +stick the message can be again read. This system is said to +have been used by the Lacedemonians\footnote +{For references, see Wilkins\index{Wilkins on Ciphers}, +\hyperlink{footnote:wilkins}{\textit{supra}}, p.~38.}. The concealment +can never have been effectual against an intelligent reader +who got possession of the paper. + +The defect of the method is that the broken letters at +once attract attention and suggest the system used. If the +fact can be concealed that the visible symbols are parts of +letters the cryptograph would be much improved. As an +illustration take the +\vpageref[appended communication][communication ]{illus:319} which is said to +have been given to the Young Pretender\index{Pretender, The Young} +during his wanderings after Culloden. +\begin{figure*}[!hbt] +\centerline{\includegraphics[width=\textwidth]{./images/illus319}} +\label{illus:319} +\end{figure*} +If it be creased along the lines $BB$ and +$CC$ ($CC$ being along the second line of the second score), and +then folded over, with $B$ inside, so that the crease $C$ lies over +the line $A$ (which is the second line of the first score) thus +\PG----File: 319.png------------------------------------------------------ +leaving only the top and bottom of the piece of paper +visible, it will be found to read \emph{conceal yourself, your foes +look for you}. I have seen what +purports to be the original, but of +the truth of the anecdote I know +nothing, and the desirability of concealing +himself must have been so +patent that it was hardly necessary +to communicate it by a cryptograph\index +{Cryptographs, Three@\nobreak--- Three types of|)}. + +\phantomsection +\addcontentsline{toc}{section}{Ciphers. + Use of arbitrary symbols unnecessary}\markright{Ciphers} +I proceed next to some of the +more common types of ciphers\index{Ciphers, Four@\nobreak--- Four types of|(}. It +is immaterial whether we invent +characters to denote the various +letters; or whether we employ special +symbols to represent them, such as +the symbol \hbox{(} for \emph{a}, the symbol \hbox{:} for \emph{b}, +and so on; or whether we use the +letters in a non-natural sense, such +as the letter \emph{z} for \emph{a}, the letter \emph{y} +for \emph{b}, and so on. The rules for +reading the cipher will be the same +in each case. + +In early times it was a common +practice to invent arbitrary symbols +to represent the letters. If the +symbols are invented for the purpose they provoke attention, +hence it would seem that preferably we should use symbols +which are not likely to attract special notice. For instance, +the symbols may be musical notes, in which case the message +would appear as a piece of music. Geometrical figures have +also been used for the same purpose. It is not even necessary +to employ written signs. Natural objects have often been +used, as in a necklace of beads, or a bouquet of flowers, +where the different shaped or coloured beads or different +flowers stand for different letters or words. An even more +subtle form of disguising the cipher is to make the different +\PG----File: 320.png---------------------------------------------------- +distances between consecutive knots or beads indicate the +different letters. + +Of all such systems we may say that a careful scrutiny +shows that different symbols are being used, and as soon as +the various symbols are distinguished one from the other +no additional complication is introduced, while for practical +purposes they are more trouble to send or receive than those +written in symbols in current use. Accordingly I confine +myself to ciphers written by the use of the current letters +and numerals. + +\phantomsection +\addcontentsline{toc}{section}{Ciphers of Four Types} +It is convenient to divide ciphers into four classes. The +first class comprises ciphers in which the same letter or word +is always represented by the same symbol, and this symbol +always represents the same letter or word. The second class +comprises ciphers in which the same letter or word is, in some +or all cases, represented by more than one symbol, and this +symbol always represents the same letter or word. The third +class comprises ciphers in which the same symbol represents +sometimes one letter or word and sometimes another. The +fourth class comprises ciphers in which each letter or word is +always represented by the same symbol, but more than one +letter or word may be represented by the same symbol. + +\phantomsection +\addcontentsline{toc}{subsection}{Ciphers of the First Type. Illustrations} +A cipher of the first type then is one in which the same +letter or word is always represented by the same symbol, and +this symbol always represents the same letter or word. + +Perhaps the simplest illustration of a cipher of this type +is to employ one language, but written as far as practical in +the alphabet of another language. It is said that during the +Indian Mutiny messages in English, but written in Greek +characters, were used freely, and successfully baffled the +ingenuity of the enemy, into whose hands they fell. If this +is true, the intelligence of the Hindoos must have been much +less than that with which they are usually credited. The +device, however, is an old one, for we are told\footnote +{Sir John Hayward\index{Hayward, J.}, \textit{Life of Edward~VI.}, +edition of 1636, p.~20.} that +Edward~VI\index{Edward VI} was accustomed to make notes in cipher ``with +\PG----File: 321.png------------------------------------------------------ +Greek characters, to the end that they who waited on him +should not read them.'' + +A common cipher of this type is made by using the actual +letters of the alphabet, but in a non-natural sense as indicating +other letters. Thus we may use each letter to represent +the one immediately following it in the natural order of the +alphabet---the letters being supposed to be cyclically arranged---\emph{a} +standing for \emph{b} wherever it occurs, \emph{b} standing for \emph{c}, and +so on, and finally \emph{z} standing for \emph{a}. Or more generally we +may write the letters of the alphabet in a line, and under them +re-write the letters in any order we like. For instance +\[ +\vbox{\itshape\def\tabcolsep{.25em}\centering +\begin{tabular}{ccccc ccccc ccccc ccccc ccccc c} +a & b & c & d & e & f & g & h & i & j & k & l & m & n & o & p & q & r & + s & t & u & v & w & x & y & z \\ +o & l & k & m & a & z & s & q & x & e & u & f & y & r & t & h & c & w & + b & v & n & i & d & g & j & p +\end{tabular}} +\] +In such a scheme, we must in our communication replace \emph{a} +by \emph{o}, \emph{b} by \emph{l}, etc. The recipient will prepare a key by +rearranging the letters in the second line in their natural order +and placing under them the corresponding letter in the first +line. Then wherever \emph{a} comes in the message he receives he +will replace it by \emph{e}; similarly he will replace \emph{b} by \emph{s}, +and so on. + +A cipher of this kind is not uncommonly used in military +signalling, the order of the letters being given by the use of +a key-word. If, for instance, \emph{Pretoria} is chosen as the key-word, +we write the letters in this order, striking out any +which occur more than once, and continue with the unused +letters of the alphabet in their natural order, writing the +whole in two lines thus: +\[ +\vbox{\itshape\def\tabcolsep{.25em}\centering +\begin{tabular}{ccccc ccccc ccc} +p & r & e & t & o & i & a & b & c & d & f & g & h \\ +z & y & x & w & v & u & s & q & n & m & l & k & j +\end{tabular}} +\] +Then in using the cipher \emph{p} is replaced by \emph{z} and +\textit{vice versâ}, +\emph{r} by \emph{y}, and so on. A long message in such a cipher would be +easily discoverable, but it is rapidly composed by the sender +and read by the receiver, and for some purposes may be +useful, especially if the discovery of the purport of the +message is, after a few hours, immaterial. + +\PG----File: 322.png------------------------------------------------------ +A summary of the usual rules for reading ciphers of this +type, whether written in English, French, German, Italian, +Dutch, Latin, or Greek, was given by D.A.~Conradus\index{Conradus, D.A.} in +1742\footnote +{\textit{Gentleman's Magazine}, 1742, vol.~\textsc{xii}, pp.~133--135, +185--186, 241--242, 473--475. See also the \textit{Collected Works of +E.A.~Poe\index{Poe, E.A.}} in 4 volumes, vol.~\textsc{i}, p.~30 \etseq}; +and similar rules have been given by various later +writers. In English the letter which occurs most frequently +is \emph{e}. The next most common letters are said to be \emph{t}, \emph{a}, +\emph{o}, and \emph{i}; then \emph{n}; then \emph{r}, \emph{s}, and \emph{h}; +then \emph{d} and \emph{l}; then \emph{c}, \emph{w}, \emph{u}, +and \emph{m}; then \emph{f}, \emph{y}, \emph{g}, \emph{p}, and \emph{b}; +then \emph{v} and \emph{k}; and then \emph{x}, \emph{q}, \emph{j}, +and \emph{z}. The most common double letters are \emph{ee}, \emph{ll}, +\emph{oo}, and \emph{ss}; +while in more than half the cases of a double letter at the end +of a word, the letter is either \emph{l} or \emph{s}. Also, \emph{t} and +\emph{h} form a +common conjunction. I need not, however, go here into further +details of this kind. + +Assuming that the division into words is given, that non-significant +symbols are not introduced, and that the problem +is not complicated by the avoidance of the use of common +words, a communication of any considerable length can usually +be read with but little difficulty. The hints given by Conradus +will at once suggest certain hypotheses as to which letters +stand for which. Taking one of these hypotheses we write +the message, replacing the symbols by the letters we conjecture +that they represent and replace the others by dots. +If the hypothesis is tenable, the arrangement will probably +suggest some of the missing letters. If, for example, we find +two words \hbox{emph{s-all}} and \hbox{emph{t-e}} where the missing letter is represented +by the same symbol, the first word shows us that the +missing letter is \emph{h}, \emph{m}, or \emph{t}, and the second word +shews us +that it must be \emph{e}, \emph{h}, \emph{i}, or \emph{o}, hence it must be +\emph{h}. Every fresh +letter so determined makes the hypothesis more probable and +renders it easier to guess what the remaining symbols represent. +The chief difficulty is to get a working hypothesis for +the first few letters---if it is the true solution, probably the +\PG----File: 323.png------------------------------------------------------ +puzzle will be readily solved---but to make up a working +hypothesis for even a few letters requires patience. + +Ciphers of this class in which the division between the +words is given are to be avoided. If we leave a space +between such words a would-be decipherer is given an immense +help. He will naturally try if a word denoted by a single +symbol can be an \emph{i} or an \emph{a}, while the words of two or three +letters will often stand revealed and so provide a definite +groundwork on which he can construct the key. A long +word may also betray the secret. For instance, if the +decipherer has reason to suspect that the message related to +something connected with Birmingham, and he found that a +particular word of ten letters had its second and fifth letters +alike, as also its fourth and tenth letters, he would naturally see +how the key would work if the word represented Birmingham, +and on this hypothesis would at once know the letters represented +by eight symbols. With reasonable luck this should +suffice to enable him to tell if the hypothesis was tenable. +The effect of this can be avoided by leaving no spaces between +the words, but this might lead to confusion and is not to +be recommended. We can also use letters which occur but +rarely, such as \emph{j}, \emph{q}, \emph{x}, \emph{z}, to separate words, +and probably this is the best method. + +Ciphers of this type suggest themselves naturally to those +approaching the subject for the first time, and are commonly +made by merely shifting the letters a certain number of places +forward. If this is done we may decrease the risk of detection +by altering the amount of shifting at short (and preferably +irregular) intervals. Thus it may be agreed that if initially +we shift every letter one place forward then whenever we +come to the letter (say) \emph{n} we shall shift every letter one more +place forward. In this way the cipher changes continually, +and is essentially changed to one of the third class; but even +with this improvement it is probable that an expert would +decode a fairly long message without much difficulty. + +We can have ciphers for numerals as well as for letters: +\PG----File: 324.png------------------------------------------------------ +such ciphers are common in many shops. Any word or sentence +containing ten different letters will answer the purpose. Thus, +an old tradesman of my acquaintance used the excellent precept +\emph{Be just O Man}---the first letter representing 1, the second $2$, +and so on. In this cipher the price $10$/$6$ would be marked +\emph{bn}/\emph{t}. This is an instance of a cipher of the first type. + +\phantomsection +\addcontentsline{toc}{subsection}{Ciphers of the Second Type. Illustrations} +A cipher of the second type is one in which the same +letter or word is, in some or all cases, represented by more +than one symbol, and this symbol always represents the same +letter or word. Such ciphers were uncommon before the +Renaissance, but the fact that to those who held the key they +were not more difficult to write or read than ciphers of the +first type, while the key was not so easily discovered, led +to their common adoption in the seventeenth century. + +A simple instance of such a cipher is given by the use of +numerals to denote the letters of the alphabet. Thus \emph{a} may +be represented by $11$ or by $37$ or by $63$, \emph{b} by $12$ or by $38$ or +by $64$, and so on, and finally \emph{z} by $36$ or by $62$ or by $88$, while +we can use $89$ or $90$ to signify the end of a word and the +numbers $91$ to $99$ to denote words or sentences which constantly +occur. Of course in practice no one would employ +the numbers in an order like this, which suggests their +meaning, but it will serve to illustrate the principle. I have +deliberately used numbers of only two digits, as the recipient +can then point off the symbols used in twos, and will know +that each pair of symbols represents a letter, word, or sentence +in the message. A disadvantage of this cipher is that +since each letter is denoted by two symbols the length of the +message is doubled by putting it in cipher. + +The cipher can be improved by introducing after every +(say) eleventh digit a non-significant digit. If this is done +the recipient of the message must erase every twelfth digit +before he begins to read the message. With this addition the +difficulty of discovering the key is considerably increased. + +The same principle is sometimes applied with letters instead +of numbers. For instance, if we take a word (say) of $n$ letters, +\PG----File: 325.png----------------------------------------------------- +preferably all different, and construct a table as shown below +of $n^2$ cells, each cell is defined by two letters of the key-word. +Thus, if we choose the word \emph{smoking-cap} we shall have $100$ +\begin{figure*}[!hbt] +\centering +\makeatletter +\def\Sq@r#1{\vbox to\SqHt{\vss\hbox to\SqWd{\smaller\hss + \vphantom{yl}#1\hss}\vss}} +\begin{MagicSquare}{11} +{}& {\emph{S}} & {\emph{M}} & {\emph{O}} & {\emph{K}} & {\emph{I}} + & {\emph{N}} & {\emph{G}} & {\emph{C}} & {\emph{A}} & {\emph{P}} \\ +{\emph{S}} & a & b & c & d & e & f & g & h & i & j \\ +{\emph{M}} & k & l & m & n & o & p & q & r & s & t \\ +{\emph{O}} & u & v & w & x & y & z & a & b & c & d \\ +{\emph{K}} & e & f & g & h & i & j & k & l & m & n \\ +{\emph{I}} & o & p & q & r & s & t & u & v & w & x \\ +{\emph{N}} & y & z & a & b & c & d & e & f & g & h \\ +{\emph{G}} & i & j & k & l & m & n & o & p & q & r \\ +{\emph{C}} & s & t & u & v & w & x & y & z & {}& {}\\ +{\emph{A}} & {}& {}& {}& {}& {}& {}& {}& {}& {}& {}\\ +{\emph{P}} & {}& {}& {}& {}& {}& {}& {}& {}& {}& {}\\ +\put(1,0){\line(0,1){11}} +\put(0,10){\line(1,0){11}} +\end{MagicSquare} +\end{figure*} +cells, and each cell is determined uniquely by the two letters +denoting its row and column. If we fill these cells in order +with the letters of the alphabet we shall have a system similar +to that explained above, where \emph{a} will be denoted by \emph{ss} +or \emph{og} +or \emph{no}, and so for the other letters. The last $22$ cells may be +used to denote the first $22$ letters of the alphabet, or better, +three or four of them may be used after the end of a word to +show that it is ended, and the rest may be used to denote +words or sentences which are likely to occur frequently. + +Like the similar cipher with numbers this can be improved +by introducing after every $m$th letter any single letter which +it is agreed shall be non-significant. To decipher a communication +so written it is necessary to know the clue-word +and the clue-number. + +Here for instance is a communication written in the above +cipher with the clue-word \emph{smoking-cap}, and with $7$ as the +clue-number: +{\CryptoSetup +\emph{n\.g\.m\.k\.s\.i\.g\.r\.i\.o\.i\.c\.p\.s\.s\.a\.m\.c\.k\.s\.c\.a\.k% +\.q\.i\.g\.n\.a\.s\.s\.n\.x\.m\.i\.g\.p\.o\.a\.s\.u\.i\.a\.m\.n\.o\.c\.m% +\.p\.a\.m\.i\.n\.s\.c\.n\.o\.g\.c\.p\.n\.c\.i\.s\.y\.i\.k\.s\.k\.a\.m\.s% +\.s\.s\.g\.n\.n%\.n removed to allow deciphering +\.c\.a\.e\.k\.k\.n\.o\.o\.m\.k\.h\.s\.c\.p\.c\.m\.s\.c% +\.b\.g\.p\.n\.g\.s\.i\.a\.% +\PG----File: 326.png------------------------------------------------------ +w\.s\.s\.g\.i\.g\.g\.n\.d\.i\.i\.c\.a}}\footnoteT +{The original text read \textellipsis\emph{sssgnnn}\textellipsis, which +leads to gobbledegook in the deciphered message.}. +% Enemy are in force at the ford and have three guns +In this sentence the letters denoting the 79th, +80th, 81st, and 82nd cells have been used to denote the end of +a word, and no use has been made of the last 18 cells. + +Another cipher of this type is made as follows\footnote +{The method is well known. It is mentioned by E.A.~Poe\index{Poe, E.A.}, +\textit{Collected Works}, vol.~\textsc{iii}, pp.~338--9, but is much older.}. +The sender and recipient of the message furnish themselves with +identical copies of some book. In the cipher only numerals are +used, and these numerals indicate the locality of the letters in +the book. For example, the first letter in the communication +might be indicated by 79--8--5, meaning that it is the 5th letter +in the 8th line of the 79th page. But though secrecy might +be secured, it would be very tedious to prepare or decode a +message, and the method is not as safe as some of those +described below. + +Another cipher of this type is for the sender and receiver +to agree on some common book of reference and to agree +further on a number which, if desired, may be communicated +as part of the message. To employ this cipher the page of +the book indicated by the given number must be used. The +first letter in it is taken to signify \emph{a}, the next \emph{b}, and so +on--any letter which occurs a second time or more frequently +being neglected. It may be also arranged that after $n$ letters +of the message have been ciphered, the next $n$ letters shall be +written in a similar cipher taken from the $p$th following page +of the book, and so on. Thus the possession of the code-book +would be of little use to anyone who did not also know the +numbers employed. It is so easy to conceal the clue number +that with ordinary prudence it would be almost impossible for +an unauthorized person to discover a message sent in this +cipher. The clue number may be communicated indirectly in +many ways. For instance, it may be arranged that the +number to be used shall be the number sent, plus (say) $q$, or +that the number to be used shall be an agreed multiple of the +number actually sent. + +\PG----File: 327.png------------------------------------------------------ +\phantomsection +\addcontentsline{toc}{subsection}{Ciphers of the Third Type. Illustrations} +A cipher of the third type is one in which the same +symbol represents sometimes one letter or word and sometimes +another. Usually such ciphers are easily made or read +by those who have the key, but are difficult to discover by +those who do not possess it. + +A simple example is the employment of pre-arranged +numbers in shifting forward the letters that make the communication. +For instance, if we agree on the clue number +(say) $4276$, then the first letter in the communication is +replaced by the fourth letter which follows it in the natural +order of the alphabet: for instance, if it were an \emph{a} it would +be replaced by \emph{e}. The next letter is replaced by the second +letter which follows it in the natural order of the alphabet: +for instance, if it were an \emph{a} it would be replaced by \emph{c}. +The next letter is replaced by the seventh after it. The next by the +sixth after it. The next by the fourth, and so on to the end +of the message. Of course to read the message the recipient +would reverse the process. If the letters of the alphabet are +written at uniform intervals along a ruler, and another ruler +similarly marked with the digits can slide along it, the letter +corresponding to the shifting of any given number of places +can be read at once. + +It would be undesirable to allow the division into words +to appear in the message, and either the words must be run +on continuously, or preferably the less common letters \emph{j, q, z} +may be used to mark the division of words. It is also well +to conceal the number of digits in the clue-number. This +can be done and the cipher much improved by inserting after +every (say) $m$th letter a non-significant letter. + +Here for instance is a communication written in this +cipher with the clue-numbers $4276$ and $7$: +{\CryptoSetup +\emph{a\.t\.p\.z\.n\.h\.v\.a\.x\.u\.x\.h\.i\.e\.p\.x\.% +a\.f\.w\.g\.h\.z\.n\.i\.y\.p\.r\.p\.s\.i\.k\.b\.d\.k\.z\.y\.y\.g\.k\.q\.% +p\.r\.g\.e\.z\.u\.y\.t\.l\.k\.o\.b\.l\.d\.i\.f\.e\.b\.z\.m\.x\.l\.p\.o\.% +g\.q\.u\.y\.i\.t\.c\.m\.g\.x\.% +k\.c\.k\.u\.e\.x\.v\.s\.q\.k\.a\.z\.i\.a\.g\.g\.s\.i\.g\.a\.y\.t\.n\.v\.% +v\.s\.s\.t\.y\.v\.u\.a\.s\.l\.y\.w\.g\.j\.u\.z\.m\.c\.s\.f\.c\.t\.q\.b\.% +p\.w\.j\.v\.a\.e\.p\.f\.x\.h\.i\.% +b\.w\.p\.x\.i\.u\.l\.t\.x\.l\.a\.v\.v\.t\.q\.z\.o\.x\.w\.k\.v\.t\.u\.v\.% +v\.f\.h\.e\.q\.b\.x\.n\.p\.v\.i\.s\.m\.p\.h\.z\.m\.q\.t\.u\.w\.x\.j\.y\.% +k\.e\.e\.v\.l\.t\.i\.f}}. +% Writ for vacant seat will be issued next week Executive Conservative +% committee meets tomorrow Essential you should be present Opposition +% to your nomination threatened +The recipient would begin by striking out every eighth letter. +He would then shift back every letter 4, 2, 7, 6, 4, 2,~\&c., +\PG----File: 328.png------------------------------------------------------ +places respectively, and in reading it would leave out the +letters \emph{j}, \emph{q}, and \emph{z} as only marking the ends of words. + +This is an excellent cipher, and it has the additional merit +of not materially lengthening the message. It can be rendered +still more difficult by arranging that either or both the clue-numbers +shall be changed according to some definite scheme, +and it may be further agreed that they shall change automatically +every day or week. + +A somewhat similar system was proposed by Wilkins\index +{Wilkins on Ciphers}\label{page:Wilkins}\footnote +{\textit{Mercury}, by J.~Wilkins, London, 1641, pp.~59, 60.}. +He took a key-word, such as \emph{prudentia}, and constructed as +many alphabets as there were letters in it, each alphabet +being arranged cyclically and beginning respectively with the +letters $p$, $r$, $u$, $d$, $e$, $n$, $t$, $i$, and $a$. +He thus got a table like +the following, giving nine possible letters which might stand +for any letter of the alphabet\Editorial +{Except \emph{j}; perhaps the cipher was intended for use with Latin.}. +Using this we may vary the +\begin{figure*}[!hbt] +\centering +\small\unitlength=1.4em +\makeatletter +\def\Sq@r#1{\vbox to1.4em{\vss\hbox to1.4em{\smaller + \hss\vphantom{yl}#1\hss}\vss}} +\hspace*{-\textwidth} +\begin{MagicSquare}{25}[10] + a & b & c & d & e & f & g & h & i & k & l & m & n + & o & p & q & r & s & t & u & v & w & x & y & z \\ + p & q & r & s & t & u & v & w & x & y & z & a & b + & c & d & e & f & g & h & i & k & l & m & n & o \\ + r & s & t & u & v & w & x & y & z & a & b & c & d + & e & f & g & h & i & k & l & m & n & o & p & q \\ + u & v & w & x & y & z & a & b & c & d & e & f & g + & h & i & k & l & m & n & o & p & q & r & s & t \\ + d & e & f & g & h & i & k & l & m & n & o & p & q + & r & s & t & u & v & w & x & y & z & a & b & c \\ + e & f & g & h & i & k & l & m & n & o & p & q & r + & s & t & u & v & w & x & y & z & a & b & c & d \\ + n & o & p & q & r & s & t & u & v & w & x & y & z + & a & b & c & d & e & f & g & h & i & k & l & m \\ + t & u & v & w & x & y & z & a & b & c & d & e & f + & g & h & i & k & l & m & n & o & P & q & r & s \\ + i & k & l & m & n & o & p & q & r & s & t & u & v + & w & x & y & z & a & b & c & d & e & f & g & h \\ + a & b & c & d & e & f & g & h & i & k & l & m & n + & o & p & q & r & s & t & u & v & w & x & y & z \\ +\put(0,9){\line(1,0){25}} +\end{MagicSquare} +\hspace*{-\textwidth} +\end{figure*} +cipher in successive words or letters of the communication. +Thus the message \emph{The prisoners have mutinied and seized the +railway station} would, according as the cipher changes in successive +words or letters, read as \emph{Hwt fhziedvhi bupy pxwmqmhg +erh ervmrq max zirteig station} or as \emph{Hyy svvlwnthm lehx +\PG----File: 329.png---------------------------------------------------- +uukzgmiq tvd gvcciq mqe frcoanr atpkcrr}. I have taken +Wilkins's key-word, but it is obvious that it would be desirable +to omit \emph{a} wherever it appears in it, since otherwise, if the +cipher changes in successive words, some of the words may +appear unaltered in the cipher, as is shown in the first of the +examples given above. + +\phantomsection +\addcontentsline{toc}{subsection}{Ciphers of the Fourth Type. Illustrations} +A cipher of the fourth type is one in which each letter is +always represented by the same symbol, but more than one +letter may be represented by the same symbol. Such ciphers +were not uncommon at the beginning of the nineteenth century, +and were usually framed by means of a key-sentence containing +about as many letters as there are letters in the alphabet. + +Thus if the key-phrase is \emph{The fox jumped over the garden +gate}, we write under it the letters of the alphabet in their +usual sequence as shown below: +\[ +\vbox{\itshape\def\tabcolsep{.15em}\centering +\begin{tabular}{cccccccccccccccccccccccccccccccccccc} +T&h&e&~&f&o&x&~&j&u&m&p&e&d&~&o&v&e&r&~&t&h&e&~&g&a&r&d&e&n&~&g&a&t&e&. \\ +a&b&c& &d&e&f& &g&h&i&j&k&l& &m&n&o&p& &q&r&s& &t&u&v&w&x&y& &z&a&b&c&. \\ +\end{tabular}} +\] +Then we write the message replacing \textit{a} by \textit{t} or \textit{a}, +\textit{b} by \textit{h} +or \textit{t}, \textit{c} by \textit{e}, \textit{d} by \textit{f}, and so on. +Here is such a message. +\textit{M foemho nea ge eoo jmdhohg avf teg ev ume afrmeo}. +% I desire you to see Gilbert and act on his advice +But it will +be observed that in the cipher \textit{a} may represent \textit{a} or +\textit{u}, \textit{d} may +represent \textit{l} or \textit{w}, \textit{e} may represent \textit{c} or +\textit{k} or \textit{o} or \textit{s} or \textit{x}, \textit{g} may +represent \textit{t} or \textit{z}, \textit{h} may represent \textit{b} or +\textit{r}, \textit{o} may represent \textit{e} or +\textit{m}, \textit{r} may represent \textit{p} or \textit{v}, and +\textit{t} may represent \textit{a} or \textit{b} or \textit{q}. +And the recipient, in deciphering it, must judge as best he can +which is the right meaning to be assigned to these letters +when they appear. + +An instance of a cipher of the fourth type is afforded by a +note sent by the Duchess de~Berri\index{Berri, de}\index +{DeBerri@De Berri} to her adherents in Paris, +in which she employed the key phrase +\[ +\vbox{\itshape\def\tabcolsep{.15em}\centering +\begin{tabular}{cccccccccccccccccccccccccc} +l&e&~&g&o&u&v&e&r&n&e&m&e&n&t&~&p&r&o&v&i&s&o&i&r&e.\\ +a&b& &c&d&e&f&g&h&i&j&k&l&m&n& &o&p&q&r&s&t&u&v&x&y. +\end{tabular}} +\] +Hence in putting her message into cipher she replaced \textit{a} by +\textit{l}, +\textit{b} by \textit{e}, \textit{c} by \textit{g}, and so on. She forgot +however to supply the +\PG----File: 330.png------------------------------------------------------- +key to the recipients of the message, but her friend Berryer +had little difficulty in reading it by the aid of the rules +I have indicated, and thence deduced the key-phrase she +had employed\index{Ciphers, Four@\nobreak--- Four types of|)}. + +\phantomsection +\addcontentsline{toc}{section}{Requisites in a good Cipher} +Having considered various classes of cryptographs and +ciphers I may now consider what features we should regard +as important in choosing a cipher intended for practical +use\index{Ciphers, Requisites@\nobreak--- Requisites for good|(}. + +In the first place, it is obvious that the means employed +should be such as not to excite suspicion if the communication +falls into unauthorized hands. But this is a counsel of +perfection, and almost impossible to attain. + +In the second place, we may say that, under modern +conditions in war, finance, or diplomacy, a cipher may be +useless unless it can be telegraphed or telephoned. If this is +deemed important, it will practically restrict us to the use of +the 26 letters of the alphabet, the 10 numerical symbols for +the digits, to which if we like we may add a few additional +marks such as punctuation stops, brackets,~\&c. The same +condition will require that the message should not be +unduly lengthened by being turned into cipher. Hence +any considerable use of non-significant symbols is to be +deprecated. + +In the third place, the key to the cipher should be such +that it can be easily reproduced from memory. For, if the +key is so elaborate that those who use it are obliged to preserve +it in some tangible and accessible form, unauthorized +persons may obtain the power of reading messages. Hence +the key should be reproducible at will. Further, it is desirable +that the key should be of such a character that it (or a change +of it) can be telegraphed or otherwise communicated without +the probability of exciting suspicion. + +In the fourth place, a cipher should be capable of change +at short intervals. So that if the reading of one message in +it be discovered subsequent messages may be undecipherable +even though the system used is unaltered. + +\PG----File: 331.png------------------------------------------------------ +Lastly, no ambiguity should be possible in deciphering +the communication. This will exclude ciphers of the fourth +type. + +Accordingly in choosing a good cipher we should seek for +one in which only current letters, symbols, or words are +employed; such that its use does not unduly lengthen the +message; such that the key to it can be reproduced at will +and need not be kept in a form which might betray the secret +to an unauthorized person; such that the key to it changes +or can be changed at short intervals; and such that it is not +ambiguous. Many ciphers of the second and third types +fulfil these conditions, but it is generally desirable to avoid +ciphers of the first type unless circumstances permit of the +free use of a code-book\index{Ciphers, Requisites@\nobreak--- Requisites for good|)}. + +\phantomsection +\addcontentsline{toc}{section}{Cipher Machines} +The use of instruments giving a cipher, which is or can be +varied constantly and automatically, has been often recommended. +Several have been constructed on the lines of the +well-known letter-locks\footnote +{See, for instance, the descriptions of those devised by Sir Charles +Wheatstone\index{Wheatstone on Ciphers}, given in his \textit +{Scientific Papers}, London, 1879, pp.~342--347; +and by Capt.\ Bazeries in \textit{Comptes Rendus, Association Français +pour l'avancement des sciences}, vol.~\textsc{xx} (Marseilles), 1891, +p.~160, \etseq}. The possession of the key of the +instrument as well as a knowledge of the clue-word is +necessary to enable anyone to read a message, but the risk of +some instrument, when set, falling into unauthorized hands +must be taken into account. Since equally good ciphers can +be constructed without the use of mechanical devices I do not +think their employment can be recommended. + +\phantomsection +\addcontentsline{toc}{section}{Historical Ciphers} +This chapter has already run to such a length that I +cannot find space to describe more than one or two ciphers +that appear in history or fiction\index +{Ciphers, Historical@\nobreak--- Historical|(}, but, we may say that +until recently most of the historical ciphers are not difficult +to read. + +\phantomsection +\addcontentsline{toc}{subsection}{Julius Caesar, Augustus} +It is said that Julius Caesar\index{Caesar, Julius}\index{Julius Caesar} +in making secret memoranda +was accustomed to move every letter four places forward, +\PG----File: 332.png------------------------------------------------------ +writing \emph{d} for \emph{a}, \emph{e} for \emph{b},~\&c. This would be a +very easy +instance of a cipher of the first type, but it may have been +effective at that time. His nephew Augustus\index{Augustus} sometimes used +a similar cipher, in which each letter was moved forward one +place\footnote +{Of some of Caesar's correspondence, Suetonius\index{Suetonius} says +(cap.~56) \emph{si +quis investigare et persequi velit, quartam elementorum literam, id est, +d pro a, et perinde reliquas commutet.} And of Augustus he says (cap.~88) +\emph{quoties autem per notas scribit, b pro a, c pro b, ac deinceps eadem +ratione, sequentes literas ponit; pro x autem duplex a.}}. + +\phantomsection +\addcontentsline{toc}{subsection}{Bacon} +Bacon\index{Bacon, Francis} proposed a cipher in which each letter was +denoted +by a group of five letters consisting of \emph{A} and \emph{B} only. Since +there are 32 such groups, he had 6 symbols to spare, which +he could use to separate words or to which he could assign +special meanings. A message in this cipher would be five +times as long as the original message. This may be compared +with the far superior system of the five (or four) digit codebook +system in use at the present time. + +\phantomsection +\addcontentsline{toc}{subsection}{Charles I} +Charles~I\index{Charles I|(} used ciphers freely in important +correspondence---the +majority being of the second type. He was foolish +enough to take a cabinet containing many confidential letters +in cipher, to some of which their readings were appended, +on the field of Naseby, where they fell into the hands of +Fairfax\footnote +{\textit{First Report of the Royal Commission on Historical Manuscripts}, +1870, pp.~2, 4.}. The House of Commons sent them to a committee +presided over by a Mr~Tate\index{Tate}. It is commonly believed that the +Committee referred the papers to J.~Wallis\index{Wallis, J.}\footnote +{See his letters on Cryptography, \textit{Opera}, vol.~\textsc{iii}, +pp.~659--672.}, then Fellow +of Queens' College, Cambridge, and subsequently Savilian +Professor at Oxford, who discovered the key to them. At +any rate the letters were read. + +In these ciphers each letter was represented by a number. +The clues to some of the ciphers were provided by the King +who had written over the number the letter which it represented, +as shown in the following quotation: +\PG----File: 333.png---------------------------------------------------- +\[ +\vbox{\itshape\def\tabcolsep{.3em}\centering +\begin{tabular}{cccccccccccccccccc} +&c&a&t&o&l&i&c&k&s&i&n&F\\ +&$11$&$18$&$45$&$35$&$23$&$27$&$11$&$25$&$47$&$28$&$40$&$148$& + \multicolumn{4}{l}{\upshape haue layed}\\[6pt] +t&h&i&r&p&u&r&s&e&s&t&o&g&e&t&h&e&r\\ +$45$&$31$&$27$&$51$&$33$&$62$&$50$&$47$&$7$&$48$&$45$&$35$&$21$& + $7$&$46$&$32$&$7$&$51$\\[6pt] +f&o&r&&s&u&p&l&y&of&a&r&m&e&s.&--&--&--\\ +$15$&$35$&$50$&a&$47$&$62$&$33$&$23$&$74$&k$1$&$17$&$51$&$42$&$7$& + $47$.&--&--&-- +\end{tabular}} +\] + +The published letters show that the King used different +ciphers at different times, though perhaps he used the same +one in all correspondence with any particular person, but the +general character of those he employed is the same. The +sentence quoted above is taken from a letter from Queen +Henrietta Maria\index{Henrietta Maria} of January~26, 1643. In this and +another +letter a few months later \textit{a} is represented by 17 or 18, +\textit{b} by 13, +\textit{c} by 11 or 12, \textit{d} by 5, \textit{e} by 7 or 8 or 9 or 10, +\textit{f} by 15 or 16, +\textit{g} by 21, \textit{h} by 31 or 32, \textit{i} by 27 or 28, +\textit{k} by 25, \textit{l} by 23 or 24, +\textit{m} by 42 or 44, \textit{n} by 39 or 40 or 41, +\textit{o} by 35 or 36 or 37 or 38, +\textit{p} by 33 or 34, \textit{r} by 50 or 51 or 52, +\textit{s} by 47 or 48, \textit{t} by 45 or +46, \textit{u} by 62 or 63, \textit{w} by 58, and \textit{y} by 74 or 77. +Numbers of +three digits were used to represent particular people or places. +Thus 148 stood for \emph{France}, 189 for the \emph{King}, 260 for the +\emph{Queen}, 354 for \emph{Prince Rupert}, and so on. Further, there +were a few special symbols, thus \textit{k}$1$ stood for \emph{but}\Editorial +{Or maybe \emph{of}, as in the preceding example.}, +\textit{n}$1$ for \emph{to}, and +\textit{f}$1$ for \emph{is}. The numbers 2 to 4 and 65 to 72 were +non-significant, +and were to be struck out or neglected by the recipient +of the message. Each symbol is separated from that which +follows it by a full-stop. + +The Queen seems to have found writing in cipher a great +trouble. In the letter from which I have already quoted a +sentence she says \textellipsis\ \emph{que je suis extrement tourmantee du +mal de teete qui fait que je mesteray en syfre par un autre se qui +jovois fait moy mesme}, and she uses the cipher only for the +particular words it was desired to conceal. Thus she writes +\emph{Mr Capell nous a fait voir que sy} 27, 23,~\&c.,~\&c. If by this +she saved herself trouble, she did it at the cost of rendering +the cipher much easier to read. + +\PG----File: 334.png------------------------------------------------------ +The system used by Charles was in considerable repute +during the seventeenth century, but even without extraneous +help it is possible for a diligent student to discover the key +if the message is fairly long. An excellent illustration of this +fact is to be found in the writings of the late Sir Charles +Wheatstone\index{Wheatstone on Ciphers}. A paper in cipher, every page +of which was +initialled by Charles~I, and countersigned by Lord Digby\index{Digby, Lord}, +was purchased some years ago by the British Museum. It +was believed to be a state paper of importance. It consists +of a series of numbers (about 150 different symbols being used) +without any clue to their meaning, or any indication of a +division between the words employed. The task of reading +it was rendered the more difficult by the supposition, which +proved incorrect, that the document was in English; but +notwithstanding this, Sir Charles Wheatstone\index{Wheatstone on Ciphers} +discovered the key\footnote +{The document, its translation, and the key used are given in +Wheatstone's \textit{Scientific Papers}, London, 1879, pp. 321--341.}. +In this cipher \emph{a} was represented by any of the numbers +$12$, $13$, $14$, $15$, $16$, or $17$, \emph{b} by $18$ or $19$, and so on, +while some +$65$ special words were represented by particular numbers. + +I may note in passing that Charles also used a species of +shorthand, in which the letters were represented by four +strokes varying in length and position. Essentially the system +is simple, though it is troublesome to read or write\index{Charles I|)}. + +\phantomsection +\addcontentsline{toc}{subsection}{Pepys} +The famous diary of Samuel Pepys\index{Pepys, S.|(} is commonly said to +have been written in cipher, but in reality it is written in +shorthand according to a system invented by T.~Shelton\index +{Shelton, T.}\footnote +{\textit{Tachy-graphy} by T.~Shelton. The earliest edition I have seen is +dated 1641. A somewhat similar system by W.~Cartwright\index{Cartwright, W.} +was issued by J.~Rich\index{Rich, J.} under the title \textit{Semographie}, +London, 1644.}. +It is however somewhat difficult to read, for the vowels are +usually omitted, and Pepys used some arbitrary signs for +terminations, particles, and certain words---so far turning it +into a cipher. Further, in certain places, when the matter is +such that it can hardly be expressed with decency, he changed +\PG----File: 335.png------------------------------------------------------ +from English to a foreign language, or inserted non-significant +letters. Shelton's\index{Shelton, T.} system had been forgotten when +attention was first attracted to the diary. Accordingly we may say +that, to those who first tried to read it, it was written in +cipher, but Pepys's contemporaries would have properly described +it as being written in shorthand, though with a few +modifications of his own invention\index{Pepys, S.|)}. + +A system of shorthand specially invented for the purpose +is a true cipher. One such system in which each letter is +represented either by a dot or by a line of constant length +was used by the Earl of Glamorgan\index{Glamorgan, Earl of}, better known +by his subsequent title as Marquis of Worcester\index{Worcester, Marquis of}, +in 1645, as also by +Charles~I\index{Charles I}. in some of his private correspondence. It is a +cipher of the first type and has the defects inherent in almost +every cipher of this kind: in fact Glamorgan's letter was +deciphered, and the system discovered by Mr~Dircks\index +{Dircks, H.}\footnote +{\textit{Life of the Marquis of Worcester} by H.~Dircks, London, 1865. +Worcester's system of shorthand was described by him in his \textit +{Century of Inventions}, London, 1663, sections 3, 4, 5.}. +Obsolete systems of shorthand\footnote +{Various systems, including those used in classical and medieval +times, are described in the \textit{History of Shorthand} by +T.~Anderson\index{Anderson, T.}, London, 1882.} might be thus used to form +an effective cipher. + +\phantomsection +\addcontentsline{toc}{subsection}{De Rohan} +It is always difficult to read a very short message in +cipher, since necessarily the clues are few in number. When +the Chevalier de~Rohan\index{DeRohan@De Rohan} was sent to the Bastille, +on suspicion +of treason, there was no evidence against him except what +might be extracted from Monsieur Latruaumont\index{Latruaumont}. The latter +died without making any admission. De~Rohan's friends had +arranged with him to communicate the result of Latruaumont's +examination, and accordingly in sending him some fresh body +linen they wrote on one of the shirts \emph{Mg dulhxcclgu ghj yxuj, +lm ct ulgc alj}. +% Le prisonnier est mort, il na rien dit +For twenty-four hours de~Rohan pored over the +message, but, failing to read it, he admitted his guilt, and was +executed November~27, 1674. + +\PG----File: 336.png------------------------------------------------------ +The cipher is a very simple one of the first type, but the +communication is so short that unless the key were known it +would not be easy to read it. Had de~Rohan suspected that +the second word was \emph{prisonnier}, it would have given him 7 +out of the 12 letters used, and as the first and third words +suggest the symbols used for \emph{l} and \emph{t}, he could hardly have +failed to read the message. + +\phantomsection +\addcontentsline{toc}{subsection}{Marie Antoinette} +The \vpageref[following cipher][cipher ]{Antoinette} is said to have been +employed by Marie Antoinette\index{Marie Antoinette}\footnote +{The key is given, but without explanation, in \textit{Juniper Hall}, by +C.~Hill\index{Hill, C.}, London, 1904, p.~13.}. +I take it that it was used in the method +\ifPaper\begin{figure*}[!hbt] +\else\begin{figure*}[p]\relsize{-2}\fi +\itshape\centering +\ifPaper + \def\SqHt{3em} + \def\SqWd{1.5em} + \unitlength=1.5em +\else % we have to squeeze a bit tighter + \def\SqHt{2.8em} + \def\SqWd{1.4em} + \unitlength=1.4em\fi +\def\BigSqr#1{\vbox to\SqHt{\vss\setbox0=\hbox to\SqHt + {\relsize{2}\hss\vphantom{1}#1\/\hss}\dp0=0pt\box0\vss}} +\def\BigCell(#1,#2;#3){\put(#1,#2){\framebox{\BigSqr{#3}}}} +\def\Sqr#1#2{\vbox to\SqHt{\vss\vss\setbox0=\hbox to\SqWd + {\smaller\hss\vphantom{1}#1\/\hss}\dp0=0pt\box0% + \setbox0=\hbox to\SqWd{\smaller\hss\vphantom{1}#2\/\hss + }\dp0=0pt\box0\vss\vss}} +\def\Cell(#1,#2;#3#4){\put(#1,#2){\framebox{\Sqr{#3}{#4}}}} +\begin{picture}(13,22) +\BigCell(0,20;AB)\Cell(2,20;AO)\Cell(3,20;BP)\Cell(4,20;CQ) + \Cell(5,20;DR)\Cell(6,20;ES)\Cell(7,20;FT)\Cell(8,20;GV) + \Cell(9,20;HX)\Cell(10,20;IY)\Cell(11,20;LZ)\Cell(12,20;MN) +\BigCell(0,18;CD)\Cell(2,18;MZ)\Cell(3,18;AN)\Cell(4,18;BO) + \Cell(5,18;CP)\Cell(6,18;DQ)\Cell(7,18;ER)\Cell(8,18;FS) + \Cell(9,18;GT)\Cell(10,18;HV)\Cell(11,18;IX)\Cell(12,18;LY) +\BigCell(0,16;EF)\Cell(2,16;LN)\Cell(3,16;MO)\Cell(4,16;AP) + \Cell(5,16;BQ)\Cell(6,16;CR)\Cell(7,16;DS)\Cell(8,16;ET) + \Cell(9,16;FV)\Cell(10,16;GX)\Cell(11,16;HY)\Cell(12,16;IZ) +\BigCell(0,14;GH)\Cell(2,14;IN)\Cell(3,14;LO)\Cell(4,14;MP) + \Cell(5,14;AQ)\Cell(6,14;BR)\Cell(7,14;CS)\Cell(8,14;DT) + \Cell(9,14;EV)\Cell(10,14;FX)\Cell(11,14;GY)\Cell(12,14;HZ) +\BigCell(0,12;IL)\Cell(2,12;HN)\Cell(3,12;IO)\Cell(4,12;LP) + \Cell(5,12;MQ)\Cell(6,12;AR)\Cell(7,12;BS)\Cell(8,12;CT) + \Cell(9,12;DV)\Cell(10,12;EX)\Cell(11,12;FY)\Cell(12,12;GZ) +\BigCell(0,10;MN)\Cell(2,10;GN)\Cell(3,10;HO)\Cell(4,10;IP) + \Cell(5,10;LQ)\Cell(6,10;MR)\Cell(7,10;AS)\Cell(8,10;BT) + \Cell(9,10;CV)\Cell(10,10;DX)\Cell(11,10;EY)\Cell(12,10;FZ) +\BigCell(0,8;OP)\Cell(2,8;FN)\Cell(3,8;GO)\Cell(4,8;HP) + \Cell(5,8;IQ)\Cell(6,8;LR)\Cell(7,8;MS)\Cell(8,8;AT) + \Cell(9,8;BV)\Cell(10,8;CX)\Cell(11,8;DY)\Cell(12,8;EZ) +\BigCell(0,6;QR)\Cell(2,6;EN)\Cell(3,6;FO)\Cell(4,6;GP) + \Cell(5,6;HQ)\Cell(6,6;IR)\Cell(7,6;LS)\Cell(8,6;MT) + \Cell(9,6;AV)\Cell(10,6;BX)\Cell(11,6;CY)\Cell(12,6;DZ) +\BigCell(0,4;ST)\Cell(2,4;DN)\Cell(3,4;EO)\Cell(4,4;FP) + \Cell(5,4;GQ)\Cell(6,4;HR)\Cell(7,4;IS)\Cell(8,4;LT) + \Cell(9,4;MV)\Cell(10,4;AX)\Cell(11,4;BY)\Cell(12,4;CZ) +\BigCell(0,2;VX)\Cell(2,2;CN)\Cell(3,2;DO)\Cell(4,2;EP) + \Cell(5,2;FQ)\Cell(6,2;GR)\Cell(7,2;HS)\Cell(8,2;IT) + \Cell(9,2;LV)\Cell(10,2;MX)\Cell(11,2;AY)\Cell(12,2;BZ) +\BigCell(0,0;YZ)\Cell(2,0;BN)\Cell(3,0;CO)\Cell(4,0;DP) + \Cell(5,0;EQ)\Cell(6,0;FR)\Cell(7,0;GS)\Cell(8,0;HT) + \Cell(9,0;IV)\Cell(10,0;LX)\Cell(11,0;MY)\Cell(12,0;AZ) +\put(0,0){\line(0,1){22}} +\put(0,0){\line(1,0){13}} +\put(13,0){\line(0,1){22}} +\put(0,22){\line(1,0){13}} +\end{picture} +\label{Antoinette} +\end{figure*} +\PG----File: 337.png------------------------------------------------------ +indicated on page \pageref{page:Wilkins} above. If so, the first word in the +communication would be rewritten according to the scheme +given in the first line, \emph{a} being replaced by \emph{o}, and +\emph{vice versâ}, +\emph{b} by \emph{p}, and so on. The second word would be rewritten +according to the scheme in the second line, and so on. + +\phantomsection +\addcontentsline{toc}{subsection}{The Code Dictionary} +One of the modern systems is the five digit code-book +cipher\index{Code-Book Ciphers}, to which I have already alluded. +According to the +general belief, it is frequently employed in certain official +communications at the present day. A code dictionary is +prepared in which every word likely to be used is printed, +and the words are numbered consecutively $00000$, $00001, \ldots$ +up, if necessary, to $99999$. Thus each word is represented +% removed obvious misprint comma after "if" +by a number of five digits, and there are $10^5$ such +numbers available. The message is first written down in +words. Below that it is written in numbers, each word being +replaced by the number corresponding to it. To each of these +numbers is added some definite prearranged clue-number---the +words in the dictionary being assumed to be arranged +cyclically, so that if the resulting number exceeds $10^5$ it is +denoted only by the excess above $10^5$. The resulting numbers +are sent as a message. On receipt of a message it is divided +into consecutive groups of five numbers, each group representing +a word. From each number is subtracted the prearranged +clue-number, and then the message can be read off by the +code dictionary. When a code message is published by the +Government receiving it, the construction of the sentences is +usually altered before publication, so that the key may not be +discoverable by anyone in possession of the code-book or who +has seen the cipher message. This is a rule applicable to all +cryptographs and ciphers. + +This is a cipher with $10^5$ symbols, and as each symbol +consists of five digits, a message of $n$ words is denoted by $5n$ +digits, and probably is not longer than the message when +written in the ordinary way. Since however the number of +words required is less than $10^5$, the spare numbers may be +used to represent collocations of words which constantly occur, +and if so the cipher message may be slightly shortened. + +\PG----File: 338.png------------------------------------------------------ +If the clue number is the same all through the message +it would be possible by not more than $10^5$ trials to discover +the message. This is not a serious risk, but, slight though it +is, it can be avoided if the clue number is varied; the clue +number might, for instance, be $781$ for the first three words, +$791$ for the next five words, $801$ for the next seven words, +and so on. Further it may be arranged that the clue numbers +shall be changed every day; thus on the seventh day of the +month they might be $781$, $791$,~\&c., and on the eighth day +$881$, $891$,~\&c., and so on. + +This cipher can however be further improved by inserting +at some step, say after each $m$th digit, an unmeaning digit. +For example, if, in the original message written in numbers, we +insert a $9$ after every seven digits we shall get a collection of +words (each represented by five digits), most of which would +have no connection with the original message, and probably +the number of digits used in the message itself would no +longer be a multiple of $5$. Of course the receiver has only to +reverse the process in order to read the message. + +It is however unnecessary to use five symbols for each +word. For if we make a similar code with the twenty-six +letters of the alphabet instead of the ten digits, four letters +for each word or phrase would give us $26^4$, that is, $456976$ +possible variations. Thus the message would be shorter and +the power of the code increased. Further, if we like to use +the ten digits and the twenty-six letters of the alphabet---all +of which are easily telegraphed---we could, by only using +three symbols, obtain $36^3$, that is, $46656$ possible words, which +would be sufficient for all practical purposes. + +This code, at any rate with these modifications, is undecipherable +by strangers, but it has the disadvantages that those +who use it must always have the code dictionary available, +and that it takes a considerable time to code or decode +a communication. For practical purposes its use would be +confined to communications which could be deciphered at +leisure in an office, It is especially suitable in the case of +\PG----File: 339.png------------------------------------------------------ +communications between officials, each supplied with a competent +staff of secretaries or clerks---as from an ambassador +to his chief, or a commander in the field to his war office. +It is an excellent example of a cipher of the first type, but it +is not clear that it possesses any superiority over some of the +simple ciphers of the third type. + +\phantomsection +\addcontentsline{toc}{subsection}{Poe's Writings} +One of the best known writers on the subject of cryptographs +and ciphers is E.A.~Poe\index{Poe, E.A.}, indeed probably a good +many readers have made their first acquaintance with a +cipher in his story of \textit{The Gold Bug}, the interest of which +turns on reading a simple cipher of the first type. In earlier +times J.~Tritheim\index{Tritheim, J.} of Spanheim, +G.~Porta\index{Porta, G.} of Naples, Cardan\index{Cardan}, +Niceron\index{Niceron}, and J.~Wilkins\index{Wilkins on Ciphers} +occupied much the same position, +while whenever ciphers were freely used skilful decipherers +seem to have arisen.\ifPaper\enlargethispage{12pt}\fi + +Poe\index{Poe, E.A.} wrote an essay on cryptography in which he said +that it may be roundly asserted that human ingenuity cannot +concoct a cipher which human ingenuity cannot resolve---a +conclusion which is hardly justified by the known facts. In +an earlier article he once made a similar remark so far as +ciphers of the first class are concerned, with the implied +limitation that only $26$ symbols may be used. In this sense +the observation is correct. His assertion excited some attention, +and numerous communications in cipher were sent to +him. More than one of his correspondents did not play the +game fairly, not only employing foreign languages, but using +several different ciphers in the same communication. Nevertheless +he resolved all except one; and he proved that this +last was a fraud, being merely a jargon of random characters, +having no meaning whatever\index{Ciphers, Historical@\nobreak--- Historical|)}. + +\PG----File: 340.png------------------------------------------------------ +% CHAPTER XII +\chapter[Hyper-space.]{Hyper-space\protect\footnotemark.} + +\textsc{I propose} to devote the remaining pages to the +consideration\chapindex{Hyper-space@\textsc{Hyper-space}}% +\chapindex{Geometry, Non-Euclidean@\textsc{Geometry, Non-Euclidean}}% +\chapindex{Space@\textsc{Space}, Properties of}, +from the point of view of a mathematician, of certain +properties of space, time, and matter, and to a sketch of some +hypotheses as to their nature. I shall not discuss the metaphysical\footnotetext +{% we put this below the first few lines to force +% some text on the page under the heading! +On the possibility of the existence of space of more than three +dimensions see C.H.~Hinton\index{Hinton on Space}, \textit +{Scientific Romances}, London, 1886, a most +interesting work, from which I have derived much assistance in compiling +the earlier part of this chapter; his later work, \textit +{The Fourth Dimension}, +London, 1904, may be also consulted. See also G.F.~Rodwell\index +{Rodwell on Hyper-space}, \textit{Nature}, +May~1, 1873, vol.~\textsc{viii}, pp.~8, 9; and E.A.~Abbott\index +{Abbott, E.A.}, \textit{Flatland}, London, 1884. +\endgraf +The theory of Non-Euclidean geometry is due primarily to +Lobatschewsky\index{Lobatschewsky, N.I.}, +\textit{Geometrische Untersuchungen zur Theorie der Parallellinien}, +Berlin, 1840 (originally given in a lecture in 1826); to Gauss\index{Gauss} +(\Eg\ letters to Schumacher, May~17, 1831, July~12, 1831, and Nov.~28, 1846, +printed in Gauss's collected works); and to J.~Bolyai, Appendix to the +first volume of his father's \textit{Tentamen}, Maros-Vásárkely, 1832; +though the subject had been discussed by J.~Saccheri\index{Saccheri, J.} +as long ago as 1733: its development was mainly the work of +G.F.B.~Riemann\index{Riemann, G.F.B.}, \textit{Ueber die Hypothesen +welche der Geometrie zu Grunde liegen}, written in 1854, \textit +{Göttinger Abhandlungen}, 1866--7, vol.~\textsc{xiii}, pp.~131--152 +(translated in \textit{Nature}, May~1 and 8, 1873, vol.~\textsc{viii}, +pp.~14--17, 36--37); H.L.F.~von Helmholtz\index{Helmholtz}\index +{Von Helmholz}, \textit{Göttinger Nachrichten}, June~3, 1868, +pp.~193--221; and E.~Beltrami\index{Beltrami on Space}, \textit{Saggio di +Interpretazione della Geometria non-Euclidea}, Naples, 1868, and the +\textit{Annali di Matematica}, series~2, vol.~\textsc{ii}, pp.~232--255: +see an article by von Helmholtz in the \textit{Academy}, Feb.~12, 1870, +vol.~\textsc{i}, pp.~128--131. Within the last +twenty-five years the theory has been treated by several mathematicians. +\endgraf +A bibliography of hyper-space, compiled by G.B.~Halsted\index +{Halsted on Hyper-space}, appeared in the \textit{American Journal of +Mathematics}, vol.~\textsc{i} (1878), pp.~261--276, +384--385; and vol.~\textsc{ii} (1879), pp.~65--70.} +\PG----File: 341.png--------------------------------------------------------- +theories that profess to account for the origin of our +conceptions of them, for these theories lead to no practical +result and rest on assertions which are incapable of definite +proof---a foundation which does not commend itself to a scientific +student. Space, time, and matter cannot be defined; but +the means of measuring them and the investigation of their +properties fall within the domain of mathematics. + +I devote this chapter to considerations connected with +space, leaving the subjects of time and mass to the following +two chapters. + +\phantomsection +\addcontentsline{toc}{section}{Two subjects of speculation on Hyper-space} +I shall confine my remarks on the properties of space to +two speculations which recently have attracted considerable +attention. These are (i)~the possibility of the existence of +space of more than three dimensions, and (ii)~the possibility +of kinds of geometry, especially of two dimensions, other than +those which are treated in the usual text-books. These problems +are related. The term hyper-space was used originally +of space of more than three dimensions, but now it is often +employed to denote also any Non-Euclidean space. I attach +the wider meaning to it, and it is in that sense that this +chapter is on the subject of hyper-space. + +\phantomsection +\addcontentsline{toc}{section}{Space of two dimensions and of one dimension} +In regard to the first of these questions, the conception of +a world of more than three dimensions is facilitated by the fact +that there is no difficulty in imagining a world confined to only +two dimensions---which we may take for simplicity to be a +plane, though equally well it might be a spherical or other surface. +We may picture the inhabitants of flatland\index{Flat-land|(} as moving +either on the surface of a plane or between two parallel and +adjacent planes. They could move in any direction along the +plane, but they could not move perpendicularly to it, and +would have no consciousness that such a motion was possible. +We may suppose them to have no thickness, in which case +they would be mere geometrical abstractions: or, preferably, +we may think of them as having a small but uniform thickness, +in which case they would be realities. + +\PG----File: 342.png--------------------------------------------------------- +Several writers have amused themselves by expounding and +illustrating the conditions of life in such a world. To take a +very simple instance, in flatland---or any even dimensional +space---a knot is impossible, a simple alteration which alone +would make some difference in the experience of the inhabitants +as compared with our own. + +If an inhabitant of flatland was able to move in three +dimensions, he would be credited with supernatural powers by +those who were unable so to move; for he could appear or +disappear at will, could (so far as they could tell) create matter +or destroy it, and would be free from so many constraints to +which the other inhabitants were subject that his actions +would be inexplicable by them. + +We may go one step lower, and conceive of a world of one +dimension---like a long tube---in which the inhabitants could +move only forwards and backwards. In such a universe there +would be lines of varying lengths, but there could be no +geometrical figures. To those who are familiar with space of +higher dimensions, life in line-land\index{Line-land} would seem somewhat +dull. It is commonly said that an inhabitant could know only two +other individuals; namely, his neighbours, one on each side. +If the tube in which he lived was itself of only one dimension, +this is true; but we can conceive an arrangement of tubes in +two or three dimensions, where an occupant would be conscious +of motion in only one dimension, and yet which would permit +of more variety in the number of his acquaintances and conditions +of existence. + +\phantomsection +\addcontentsline{toc}{section}{Space of four dimensions} +Our conscious life is in three dimensions, and naturally the +idea occurs whether there may not be a fourth dimension. No +inhabitant of flatland could realize what life in three dimensions +would mean, though, if he evolved an analytical geometry +applicable to the world in which he lived, he might be able to +extend it so as to obtain results true of that world in three +dimensions which would be to him unknown and inconceivable. +Similarly we cannot realize what life in four dimensions +is like, though we can use analytical geometry to obtain results +\PG----File: 343.png--------------------------------------------------------- +true of that world, or even of worlds of higher dimensions. +Moreover the analogy of our position to the inhabitants of flatland +enables us to form some idea of how inhabitants of space +of four dimensions would regard us. + +\phantomsection +\addcontentsline{toc}{subsection}{Existence in such a world} +Just as the inhabitants of flatland might be conceived as +being either mere geometrical abstractions, or real and of a +uniform thickness in the third dimension, so, if there is a fourth +dimension, we may be regarded either as having no thickness +in that dimension, in which event we are mere (geometrical) +abstractions---as indeed idealist philosophers have asserted to +be the case---or as having a uniform thickness in that dimension, +in which event we are living in four dimensions although +we are not conscious of it. In the latter case it is reasonable +to suppose that the thickness in the fourth dimension of bodies +in our world is small and possibly constant; it has been conjectured +also that it is comparable with the other dimensions +of the molecules of matter, and if so it is possible that the constitution +of matter and its fundamental properties may supply +experimental data which will give a physical basis for proving +or disproving the existence of this fourth dimension. + +If we could look down on the inhabitants of flatland we +could see their anatomy and what was happening inside them. +Similarly an inhabitant of four-dimensional space could see +inside us. + +An inhabitant of flatland could get out of a room, such as a +rectangle, only through some opening, but, if for a moment he +could step into three dimensions, he could reappear on the other +side of any boundaries placed to retain him. Similarly, if we +came across persons who could move out of a closed prison-cell +without going through any of the openings in it, there might +be some reason for thinking that they did it by passing first +in the direction of the fourth dimension and then back again +into our space. This however is unknown. + +Again, if a finite solid was passed slowly through flatland, +the inhabitants would be conscious only of that part of it +which was in their plane. Thus they would see the shape of +\PG----File: 344.png--------------------------------------------------------- +the object gradually change and ultimately vanish. In the +same way, if a body of four dimensions was passed through our +space, we should be conscious of it only as a solid body, namely, +the section of the body by our space, whose form and appearance +gradually changed and perhaps ultimately vanished. It +has been suggested that the birth, growth, life, and death of +animals may be explained thus as the passage of finite four-dimensional +bodies through our three-dimensional space. I +believe that this idea is due to Mr~Hinton\index{Hinton on Space}. + +The same argument is applicable to all material bodies. +The impenetrability and inertia of matter are necessary consequences; +the conservation of energy follows, provided that the +velocity with which the bodies move in the fourth dimension +is properly chosen: but the indestructibility of matter rests on +the assumption that the body does not pass completely through +our space. I omit the details connected with change of density +as the size of the section by our space varies. + +\phantomsection +\addcontentsline{toc}{subsection}{Arguments in favour of the existence +of such a world} +We cannot prove the existence of space of four dimensions, +but it is interesting to enquire whether it is probable that +such space actually exists. To discuss this, first let us consider +how an inhabitant of flatland might find arguments to support +the view that space of three dimensions existed, and then +let us see whether analogous arguments apply to our world. +I commence with considerations based on geometry and then +proceed to those founded on physics. + +Inhabitants of flatland would find that they could have two +triangles of which the elements were equal, element to element, +and yet which could not be superposed. We know that the +explanation of this fact is that, in order to superpose them, one +of the triangles would have to be turned over so that its undersurface +came on to the upper side, but of course such a movement +would be to them inconceivable. Possibly however they +might have suspected it by noticing that inhabitants of one-dimensional +space might experience a similar difficulty in +comparing the equality of two lines, $ABC$ and $CB'A'$, each +defined by a set of three points. We may suppose that the +\PG----File: 345.png--------------------------------------------------------- +lines are equal and such that corresponding points in them +could be superposed by rotation round $C$---a movement +inconceivable to the inhabitants---but an inhabitant of such +a world in moving along from $A$ to $A'$ would not arrive at the +corresponding points in the two lines in the same relative +order, and thus might hesitate to believe that they were +equal. Hence inhabitants of flatland might infer by analogy +that by turning one of the triangles over through three-dimensional +space they could make them coincide. + +We have a somewhat similar difficulty in our geometry. +We can construct triangles in three dimensions---such as two +spherical triangles---whose elements are equal respectively one +to the other, but which cannot be superposed. Similarly we +may have two spirals whose elements are equal respectively, +one having a right-handed twist and the other a left-handed +twist, but it is impossible to make one fill exactly the same +parts of space as the other does. Again, we may conceive of +two solids, such as a right hand and a left hand, which are +exactly similar and equal but of which one cannot be made to +occupy exactly the same position in space as the other does. +Those are difficulties similar to those which would be experienced +by the inhabitants of flatland in comparing triangles; +and it may be conjectured that in the same way as such +difficulties in the geometry of an inhabitant in space of one +dimension are explicable by temporarily moving the figure +into space of two dimensions by means of a rotation round a +point, and as such difficulties in the geometry of flatland are +explicable by temporarily moving the figure into space of +three dimensions by means of a rotation round a line, so +such difficulties in our geometry would disappear if we could +temporarily move our figures into space of four dimensions by +means of a rotation round a plane---a movement which of +course is inconceivable to us. + +Next we may enquire whether the hypothesis of our existence +in a space of four dimensions affords an explanation of +any difficulties or apparent inconsistencies in our physical +\PG----File: 346.png--------------------------------------------------------- +science\footnote +{See a note by myself\index{Ball} in the \textit{Messenger of Mathematics}, +Cambridge, 1891, vol.~\textsc{xxi}, pp.~20--24, from which the above argument +is extracted. The question has been treated by Mr~Hinton\index +{Hinton on Space} on similar lines.}. +The current conception of the luminiferous ether, +the explanation of gravity, and the fact that there are only a +finite number of kinds of matter, all the atoms of each kind +being similar, present such difficulties and inconsistencies. +To see whether the hypothesis of a four-dimensional space +gives any aid to their elucidation, we shall do best to consider +first the analogous problems in two dimensions. + +We live on a solid body, which is nearly spherical, and +which moves round the sun under an attraction directed to it. +To realize a corresponding life in flatland we must suppose +that the inhabitants live on the rim of a (planetary) disc +which rotates round another (solar) disc under an attraction +directed towards it. We may suppose that the planetary +world thus formed rests on a smooth plane, or other surface of +constant curvature; but the pressure on this plane and even +its existence would be unknown to the inhabitants, though +they would be conscious of their attraction to the centre of the +disc on which they lived. Of course they would be also aware +of the bodies, solid, liquid, or gaseous, which were on its rim, +or on such points of its interior as they could reach. + +Every particle of matter in such a world would rest on this +plane medium. Hence, if any particle was set vibrating, it +would give up a part of its motion to the supporting plane. +The vibrations thus caused in the plane would spread out in +all directions, and the plane would communicate vibrations to +any other particles resting on it. Thus any form of energy +caused by vibrations, such as light, radiant heat, electricity, +and possibly attraction, could be transmitted from one point +to another without the presence of any intervening medium +which the inhabitants could detect. + +If the particles were supported on a uniform elastic plane +film, the intensity of the disturbance at any other point would +vary inversely as the distance of the point from the source of +\PG----File: 347.png--------------------------------------------------------- +disturbance; if on a uniform elastic solid medium, it would +vary inversely as the square of that distance. But, if the +supporting medium was vibrating, then, wherever a particle +rested on it, some of the energy in the plane would be given +up to that particle, and thus the vibrations of the intervening +medium would be hindered when it was associated with +matter. + +If the inhabitants of this two-dimensional world were +sufficiently intelligent to reason about the manner in which +energy was transmitted they would be landed in a difficulty. +Possibly they might be unable to explain gravitation between +two particles---and therefore between the solar disc and their +disc---except by supposing vibrations in a rigid medium between +the two particles or discs. Again, they might be able to detect +that radiant light and heat, such as the solar light and heat, +were transmitted by vibrations transverse to the direction from +which they came, though they could realize only such vibrations +as were in their plane, and they might determine experimentally +that in order to transmit such vibrations a medium +of great rigidity (which we may call ether) was necessary. Yet +in both the above cases they would have also distinct evidence +that there was no medium capable of resisting motion in the +space around them, or between their disc and the solar disc. +The explanation of these conflicting results lies in the fact that +their universe was supported by a plane, of which they were +necessarily unconscious, and that this rigid elastic plane was +the ether which transmitted the vibrations\index{Flat-land|)}. + +Now suppose that the bodies in our universe have a uniform +thickness in the fourth dimension, and that in that direction +our universe rests on a homogeneous elastic body whose thickness +in that direction is small and constant. The transmission +of force and radiant energy, without the intervention of an +intervening medium, may be explained by the vibrations of +the supporting space, even though the vibrations are not themselves +in the fourth dimension. Also we should find, as in +fact we do, that the vibrations of the luminiferous ether are +\PG----File: 348.png----------------------------------------------------- +hindered when it is associated with matter. I have assumed +that the thickness of the supporting space is small and uniform, +because then the intensity of the energy transmitted from a +source to any point would vary inversely as the square of the +distance, as is the case; whereas if the supporting space was a +body of four dimensions, the law would be that of the inverse +cube of the distance. + +The application of this hypothesis to the third difficulty +mentioned above---namely, to show why there are in our universe +only a finite number of kinds of atoms, all the atoms of +each kind having in common a number of sharply defined +properties---will be given later\footnote +{See below, p.~\pageref{page:373} (3).}. + +Thus the assumption of the existence of a four-dimensional +homogeneous elastic body on which our three-dimensional +universe rests, affords an explanation of some difficulties in +our physical science. + +It may be thought that it is hopeless to try to realize a +figure in four dimensions. Nevertheless attempts have been +made to see what the sections of such a figure would look +like. + +If the boundary of a solid is $\phi(x, y, z) = 0$, we can obtain +some idea of its form by taking a series of plane sections by +planes parallel to $z=0$, and mentally superposing them. In four +dimensions the boundary of a body would be $\phi(x, y, z, \omega) = 0$, +and attempts have been made to realize the form of such a +body by making models of a series of solids in three dimensions +formed by sections parallel to $\omega = 0$. Again, we can represent +a solid in perspective by taking sections by three co-ordinate +planes. In the case of a four-dimensional body the section +by each of the four co-ordinate solids will be a solid, and +attempts have been made by drawing these to get an idea of +the form of the body. Of course a four-dimensional body will +be bounded by solids. + +The possible forms of regular bodies in four dimensions, +\PG----File: 349.png----------------------------------------------------- +analogous to polyhedrons in space of three dimensions, have +been discussed by Mr~Stringham\index{Stringham on Hyper-space}\footnote +{\textit{American Journal of Mathematics}, 1880, vol.~\textsc{iii}, +pp.~1--14.}. + +\ThoughtBreakSpace + +\phantomsection +\addcontentsline{toc}{section}{Non-Euclidean Geometries} +I now turn to the second of the two problems mentioned +at the beginning of the chapter: namely, the possibility of +there being kinds of geometry other than those which are +treated in the usual elementary text-books\index +{NonEuclid@\textsc{Non-Euclidean Geometry}|(}. This subject is so +technical that in a book of this nature I can do little more +than give a sketch of the argument on which the idea is based. + +\phantomsection +\addcontentsline{toc}{section}{Euclid's axioms and postulates. +The parallel postulate} +The Euclidean system of geometry\index +{Euclidean Geometry|(}\index{Euclidean Space|(}, with which alone most +people are acquainted, rests on a number of independent +axioms and postulates. Those which are necessary for Euclid's +geometry have, within recent years, been investigated and +scheduled. They include not only those explicitly given by +him, but some others which he unconsciously used. If these +are varied, or other axioms are assumed, we get a different +series of propositions, and any consistent body of such propositions +constitutes a system of geometry. Hence there is +no limit to the number of possible Non-Euclidean geometries +that can be constructed. + +Among Euclid's axioms and postulates\index{EuclidAx@Euclid's Axioms \&c.} +is one on parallel lines\index +{EuclidAxPar@\nobreak--- Parallel Postulate}\index{Parallels, Theory of}, +which is usually stated in the form that if a straight +line meets two straight lines, so as to make the two interior +angles on the same side of it taken together less than two +right angles, then these straight lines being continually produced +will at length meet upon that side on which are the +angles which are less than two right angles. Expressed in +this form the axiom is far from obvious, and from early times +numerous attempts have been made to prove it\footnote +{Some of the more interesting and plausible attempts have been +collected by J.~Richard\index{Richard, J.} in his \textit +{Philosophie de Mathématiques}, Paris, 1903.}. All such +attempts failed, and it is now known that the axiom cannot +be deduced from the other axioms assumed by Euclid. It +can be replaced by other statements about parallel lines, such +as that the distance between two parallel lines is always the +\PG----File: 350.png----------------------------------------------------- +same, but such alternative statements, though perhaps \textit{primâ facie} +more axiomatic, are not to be preferred to Euclid's form, +since his statement brings out prominently a characteristic +feature of the space with which he is concerned. + +\phantomsection +\addcontentsline{toc}{section}{Hyperbolic Geometry of two dimensions} +The earliest conception of a body of Non-Euclidean geometry +was due to the discovery, made independently by +Saccheri\index{Saccheri, J.}, Lobatschewsky\index{Lobatschewsky, N.I.}, +and John Bolyai\index{Bolyai, J.}, that a consistent +system of geometry of two dimensions can be produced on +the assumption that the axiom on parallels is not true, and +that through a point a number of straight (that is, geodetic) +lines can be drawn parallel to a given straight line. The +resulting geometry is called \emph{hyperbolic}\index{Hyperbolic Geometry|(}. + +\phantomsection +\addcontentsline{toc}{section}{Elliptic Geometry of two dimensions} +Riemann\index{Riemann, G.F.B.} later distinguished between boundlessness of +space and its infinity, and showed that another consistent +system of geometry of two dimensions can be constructed in +which all straight lines are of a finite length, so that a particle +moving along a straight line will return to its original position. +This leads to a geometry of two dimensions, called +\emph{elliptic geometry}\index{Elliptic Geometry|(}, +analogous to the hyperbolic geometry, but characterized +by the fact that through a point no straight line can be +drawn which, if produced far enough, will not meet any other +given straight line. This can be compared with the geometry +of figures drawn on the surface of a sphere. + +\phantomsection +\addcontentsline{toc}{subsection}{Elliptic, Parabolic and Hyperbolic +Geometries compared} +Thus according as no straight line, or only one straight +line, or a pencil of straight lines can be drawn through a +point parallel to a given straight line, we have three systems +of geometry of two dimensions known respectively as elliptic, +parabolic\index{Parabolic Geometry|(} or homaloidal\index +{Homaloidal Geometry} or Euclidean, and hyperbolic. + +In the parabolic and hyperbolic systems straight lines are +infinitely long. In the elliptic they are finite. In the hyperbolic +system there are no similar figures of unequal size; the +area of a triangle can be deduced from the sum of its angles, +which is always less than two right angles; and there is a finite +maximum to the area of a triangle. In the elliptic system all +straight lines are of the same finite length; any two lines +intersect; and the sum of the angles of a triangle is greater +than two right angles. In the elliptic system it is possible +\PG----File: 351.png----------------------------------------------------- +to get from one point to a point on the other side of a plane +without passing through the plane, namely, by going the other +way round the straight line joining the two points; thus a +watch-dial moving face upwards continuously forward in a +plane in a straight line in the direction from the mark \textsc{vi} to +the mark \textsc{xii} will finally appear to a stationary observer +with its face downwards; and if originally the mark \textsc{iii} was +to the right of the observer it will finally be on his left hand% +\index{Elliptic Geometry|)}% +\index{Hyperbolic Geometry|)}. + +In spite of these and other peculiarities of hyperbolic and +elliptical geometries, it is impossible to prove by observation +that one of them is not true of the space in which we live. +For in measurements in each of these geometries we must +have a unit of distance; and if we live in a space whose +properties are those of either of these geometries, and such +that the greatest distances with which we are acquainted +(\Eg\ the distances of the fixed stars) are immensely smaller +than any unit, natural to the system, then it may be impossible +for our observations to detect the discrepancies between the +three geometries. It might indeed be possible by observations +of the parallaxes of stars to prove that the parabolic system and +either the hyperbolic or elliptic system were false, but never +can it be proved by measurements that Euclidean geometry +is true. Similar difficulties might arise in connection with +excessively minute quantities. In short, though the results of +Euclidean geometry are more exact than present experiments +can verify for finite things, such as those with which we have +to deal, yet for much larger things or much smaller things or +for parts of space at present inaccessible to us they may not +be true\index{Parabolic Geometry|)}. + +If however we go a step further and ask what is meant by +saying that a geometry is true or false, I can only quote the +remark of Poincaré\index{Poincare@Poincaré, H.}, +that the selection of a geometry is really +a matter of convenience, and that that geometry is the best +which enables us to state the physical laws in the simplest +form. This opinion has been strongly controverted, but at +any rate it expresses one view of the question. + +\phantomsection +\addcontentsline{toc}{section}{Non-Euclidean Geometries of three or +more dimensions} +The above refers only to hyper-space of two dimensions. +\PG----File: 352.png----------------------------------------------------- +Naturally there arises the question whether there are different +kinds of Non-Euclidean space of three or more dimensions. +Riemann\index{Riemann, G.F.B.} showed that there are three kinds of +Non-Euclidean space of three dimensions having properties analogous to the +three kinds of Non-Euclidean space of two dimensions already +discussed. These are differentiated by the test whether at +every point no geodetical surface, or one geodetical surface, or +a fasciculus of geodetical surfaces can be drawn parallel to a +given surface: a geodetical surface being defined as such that +every geodetic line joining two points on it lies wholly on the +surface. It may be added that each of the three systems of +geometry of two dimensions described above may be deduced +as properties of a surface in each of these three kinds of +Non-Euclidean space of three dimensions. + +It is evident that the properties of Non-Euclidean space +of three dimensions are deducible only by the aid of mathematics, +and cannot be illustrated materially, for in order to +realize or construct surfaces in Non-Euclidean space of two +dimensions we think of or use models in space of three dimensions; +similarly the only way in which we could construct +models illustrating Non-Euclidean space of three dimensions +would be by utilizing space of four dimensions. + +We may proceed yet further and conceive of Non-Euclidean +geometries of more than three dimensions, but this remains, as +yet, an unworked field. + +Returning to the former question of Non-Euclidean +geometries, I wish again to emphasize the fact that, if the +axioms enunciated by Euclid are replaced by others, it is +possible to construct other consistent systems of geometry. +Some of these are interesting, but those which have been +mentioned above have a special importance, from the somewhat +sensational fact that they lead to no results necessarily +inconsistent with the properties, as far as we can observe +them, of the space in which we live; we are not at present +acquainted with any other systems which are consistent with +our experience\index{Euclidean Geometry|)}\index +{Euclidean Space|)}\index{NonEuclid@\textsc{Non-Euclidean Geometry}|)}. +\PG----File: 353.png----------------------------------------------------- +% CHAPTER XIII + +\chapter{Time and its Measurement.} + +\textsc{The} problems connected with time are totally different\chapindex +{Time@\textsc{Time}} +in character from those concerning space which I discussed +in the last chapter. I there stated that the life of people +living in space of one dimension would be uninteresting, +and that probably they would find it impossible to realize +life in space of higher dimensions. In questions connected +with time we find ourselves in a somewhat similar position. +Mentally, we can realize a past and a future---thus going backwards +and forwards---actually we go only forwards. Hence +time is analogous to space of one dimension. Were our time +of two dimensions, the conditions of our life would be infinitely +varied, but we can form no conception of what such a phrase +means, and I do not think that any attempts have been made +to work it out. + +I shall concern myself here mainly with questions concerning +the measurement of time\index{Time, Measure@\nobreak--- Measurement of|(}, +and shall treat them rather +from a historical than from a philosophical point of view. + +In order to measure anything we must have an unalterable +unit\index{Time, Units@\nobreak--- Units of|(} of the same kind, +and we must be able to determine +how often that unit is contained in the quantity to be +measured. Hence only those things can be measured which +are capable of addition to things of the same kind. + +Thus to measure a length we may take a foot-rule, and +by applying it to the given length as often as is necessary, +we shall find how many feet the length contains. But in +\PG----File: 354.png----------------------------------------------------- +comparing lengths we assume as the result of experience that +the length of the foot-rule is constant, or rather that any +alteration in it can be determined; and, if this assumption +was denied, we could not prove it, though, if numerous repetitions +of the experiment under varying conditions always +gave the same result, probably we should feel no doubt as +to the correctness of our method. + +It is evident that the measurement of time is a more +difficult matter. We cannot keep a unit by us in the same +way as we can keep a foot-rule; nor can we repeat the measurement +over and over again, for time once passed is gone for +ever. Hence we cannot appeal directly to our sensations to +justify our measurement. Thus, if we say that a certain +duration is four hours, it is only by a process of reasoning +that we can show that each of the hours is of the same +duration. + +\phantomsection +\addcontentsline{toc}{section}{Units for measuring durations +(days, weeks, months, years)} +The establishment of a scientific unit for measuring durations +has been a long and slow affair. The process seems to +have been as follows. Originally man observed that certain +natural phenomena recurred after the interval of a day, say +from sunrise to sunrise. Experience---for example, the amount +of work that could be done in it---showed that the length of +every day was about the same, and, assuming that this was +accurately so, man had a unit by which he could measure +durations. The present subdivision of a day into hours, +minutes, and seconds is artificial, and apparently is derived +from the Babylonians. + +Similarly a month and a year are natural units of time +though it is not easy to determine precisely their beginnings +and endings. + +So long as men were concerned merely with durations +which were exact multiples of these units or which needed +only a rough estimate, this did very well; but as soon as they +tried to compare the different units or to estimate durations +measured by part of a unit they found difficulties. In +particular it cannot have been long before it was noticed that +\PG----File: 355.png----------------------------------------------------- +the duration of the same day differed in different places, and +that even at the same place different days differed in duration +at different times of the year, and thus that the duration of a +day was not an invariable unit. + +The question then arises as to whether we can find a fixed +unit by which a duration can be measured, and whether we +have any assurance that the seconds and minutes used to-day +for that purpose are all of equal duration. To answer +this we must see how a mathematician would define a unit +of time\index{Day@Day, Definition of}. Probably he would say +that experience leads us to +believe that, if a rigid body is set moving in a straight line +without any external force acting on it, it will go on moving +in that line; and those times are taken to be equal in which +it passes over equal spaces: similarly, if it is set rotating about +a principal axis passing through its centre of mass, those times +are taken to be equal in which it turns through equal angles. +Our experiences are consistent with this statement, and that +is as high an authority as a mathematician hopes to get. + +The spaces and the angles can be measured, and thus durations +can be compared. Now the earth may be taken roughly +as a rigid body rotating about a principal axis passing through +its centre of mass, and subject to no external forces affecting +its rotation: hence the time it takes to turn through four +right angles, \IE\ through $360^{\circ}$, is always the same; this is +called a sidereal day\index{Sidereal Time}: the time to turn through +one twenty-fourth part of $360^{\circ}$, \IE\ through $15^{\circ}$, +is an hour\index{Hours, definition of}: the time +to turn through one-sixtieth part of $15^{\circ}$, \IE\ through $15'$, is a +minute\index{Minutes, def.\ of}\index{Seconds, def.\ of}: and so on. + +If, by the progress of astronomical research, we find that +there are external forces affecting the rotation of the earth, +mathematics would have to be invoked to find what the +time of rotation would be if those forces ceased to act, and +this would give us a correction to be applied to the unit +chosen. In the same way we may say that although an increase +of temperature affects the length of a foot-rule, yet its +change of length can be determined, and thus applied as a +\PG----File: 356.png----------------------------------------------------- +correction to the foot-rule when it is used as the unit of +length. As a matter of fact there is reason to think that +the earth takes about one sixty-sixth of a second longer to +turn through four right angles now than it did 2500 years +ago, and thus the duration of a second is just a trifle longer +to-day than was the case when the Romans were laying the +foundations of the power of their city. + +The sidereal day can be determined only by refined astronomical +observations and is not a unit suitable for ordinary +purposes. The relations of civil life depend mainly on the +sun, and he is our natural time keeper. The true solar day\index +{Day, Sidereal@\nobreak--- Sidereal and Solar}\index{Solar Time} +is the time occupied by the earth in making one revolution +on its axis relative to the sun; it is true noon when the +sun is on the meridian. Owing to the motion of the sun +relative to the earth, the true solar day is about four minutes +longer than a sidereal day. + +The true solar day is not however always of the same +duration. This is inconvenient if we measure time by clocks +(as now for nearly two centuries has been usual in Western +Europe) and not by sun-dials, and therefore we take the +average\index{Mean Time} +duration of the true solar day as the measure of a day: this +is called the mean solar day. Moreover to define the noon of +a mean solar day we suppose a point to move uniformly round +the ecliptic coinciding with the sun at each apse, and further +we suppose a fictitious sun, called the mean sun\index +{Sun-the@Sun, the Mean}, to move +in the celestial equator so that its distance from the first +point of Aries is the same as that of this point: it is mean +noon when this mean sun is on the meridian. The mean solar +day is divided into hours, minutes\index{Minutes, def.\ of}, +and seconds\index{Seconds, def.\ of}; and these +are the usual units of time in civil life\index +{Time, Measure@\nobreak--- Measurement of|)}. + +The time indicated by our clocks and watches\index{Watches} is mean solar +time; that marked on ordinary sun-dials is true solar time. +The difference between them is the equation of time\index +{Time, equation@\nobreak--- Equation of}: this may +amount at some periods of the year to a little more than a +quarter of an hour. In England we take the Greenwich +meridian as our origin for longitudes, and instead of local +\PG----File: 357.png----------------------------------------------------- +mean solar time we take Greenwich mean solar time as the +civil standard. + +Of course mean time\index{Mean Time} is a comparatively recent invention. +The French were the last civilized nation to abandon the use +of true time: this was in 1816. + +Formerly there was no common agreement as to when the +day began\index{Day, Commencement@\nobreak--- Commencement of}. +In parts of ancient Greece and in Japan the +interval from sunrise to sunset was divided into 12 hours\index +{Hours, definition of}, +and that from sunset to sunrise into 12 hours. The Jews, +Chinese, Athenians, and, for a long time, the Italians, +divided their day into 24 hours\index{Hours, definition of}, +beginning at the hour of +sunset, which of course varies every day: this method is said +to be still used in certain villages near Naples, except that +the day begins half-an-hour after sunset---the clocks being +re-set once a week. Similarly the Babylonians, Assyrians, +Persians, and until recently the modern Greeks and the inhabitants +of the Balearic Islands counted the twenty-four +hours of the day from sunrise. Until the middle of last +century, the inhabitants of Basle reckoned the twenty-four +hours from our 11.0~p.m. The ancient Egyptians and Ptolemy\index{Ptolemy} +counted the twenty-four hours from noon: this is the practice +of modern astronomers. In Western Europe the day is taken +to begin at midnight---as was first suggested by Hipparchus\index +{Hipparchus on hours of day}---and is divided into two equal periods +of twelve hours each. + +The week of seven days is an artificial unit of time. It +had its origin in the East, and was introduced into the West +by the Roman emperors, and, except during the French +Revolution, has been subsequently in general use among +civilized races. The names of the days\index +{Days of Week, Names of|(}\index{Week, Names of Days|(} are derived from the +seven astrological planets\index{Astrological Planets}\index +{PlanetsA@Planets (astrological)}, arranged, as was customary, in the +order of their apparent times of rotation round the earth, +namely, Saturn, Jupiter, Mars, the Sun, Venus, Mercury, and +the Moon. The twenty-four hours of the day were dedicated +successively to these planets: and the day was consecrated to +the planet of the first hour. + +Thus if the first hour was dedicated to Saturn, the second +\PG----File: 358.png----------------------------------------------------- +would be dedicated to Jupiter, and so on; but the day would +be Saturn's day. The twenty-fourth hour of Saturn's day +would be dedicated to Mars, thus the first hour of the next day +would belong to the Sun; and the day would be Sun's day. +Similarly the next day would be Moon's day; the next, Mars's +day; the next, Mercury's day: the next, Jupiter's day; and +the next, Venus's day\index +{Days of Week, Names of|)}\index{Week, Names of Days|)}. + +\phantomsection +\addcontentsline{toc}{section}{The Civil Calendar (Julian, Gregorian, \&c.)} +The astronomical month\index{Months} is a natural unit of time depending +on the motion of the moon, and containing about $29\frac{1}{2}$ days. +The months of the calendar\index +{Calendar, the Civil@\textsc{Calendar}, the Civil|(}\index{Year, Civil|(} +have been evolved gradually as +convenient divisions of time, and their history is given in +numerous astronomies. In the original Julian arrangement\index +{Calendar, the Julian@\nobreak--- the Julian|(}\index{Julian Calendar|(} +the months in a leap year\index{Leap-year|(} contained alternately +31 and 30 days, +while in other years February had 29 days. This was altered +by Augustus\index{Augustus} in order that his month should not be +inferior to one named after his uncle. + +The solar tropical year is another natural unit of time. +According to a recent determination, it contains 365.242216 +days, that is, 365\textsuperscript{d.} 5\textsuperscript{h.} +48\textsuperscript{m.} 47\textsuperscript{s.}.4624. + +The Egyptians knew that it contained between 365 and 366 +days, but the Romans did not profit by this information, for +Numa\index{Numa on the Year} is said to have reckoned 355 days +as constituting +a year---extra months being occasionally intercalated, so that +the seasons might recur at about the same period of the year\index +{Time, Units@\nobreak--- Units of|)}. + +In 46~\textsc{b.c.} Julius Caesar\index{Caesar, Julius} decreed that +thenceforth the year +should contain 365 days, except that in every fourth or leap +year one additional day should be introduced. He ordered +this rule to come into force on January~1, 45~\textsc{b.c.} The change +was made on the advice of Sosigenes of Alexandria\index +{Sosigenes on Calendar}. + +It must be remembered that the year 1~\textsc{a.d.} follows immediately +1~\textsc{b.c.}, that is, there is no year 0, and thus 45~\textsc{b.c.} +would be a leap year. All historical dates are given now as if the +Julian calendar was reckoned backwards as well as forwards +from that year\footnote +{Herschel\index{Herschel, Sir John}, \textit{Astronomy}, London, 11th~ed. +1871, arts.~916--919.}. As a matter of fact, owing to a mistake in +the original decree, the Romans, during the first $36$ years after +\PG----File: 359.png---------------------------------------------------- +45~\textsc{b.c.}, intercalated the extra day every third year, thus +producing an error of 3 days. This was remedied by Augustus\index{Augustus}, +who directed that no intercalation of an extra day should be +made in any of the twelve years \textsc{a.u.c.}~746 to 757 inclusive, +but that the intercalation should be again made in the year +\textsc{a.u.c.}~761 (that is, 8~\textsc{a.d.}) and every succeeding +fourth year. + +The Julian calendar made the year, on an average, contain +365.25 days. The actual value is, very approximately, +365.242216 days. Hence the Julian year is too long by about +$11\frac{1}{4}$ minutes: this produces an error of nearly one day in 128 +years. If the extra day in every thirty-second leap year had +been omitted---as was suggested by some unknown Persian +astronomer---the error would have been less than one day in +100,000 years. It may be added that Sosigenes\index{Sosigenes on Calendar} +was aware that his rule made the year slightly too long. + +The error in the Julian calendar of rather more than eleven +minutes a year gradually accumulated, until in the sixteenth +century the seasons arrived some ten days earlier than they +should have done. In 1582 Gregory~XIII\index{Gregory@Gregory XIII|(} +corrected this by +omitting ten days from that year, which therefore contained +only 355 days. At the same time he decreed that thenceforth +every year which was a multiple of a century should be or +not be a leap year according as the multiple was or was not +divisible by four% +\index{Calendar, the Julian@\nobreak--- the Julian|)}% +\index{Julian Calendar|)}% +\index{Leap-year|)}. + +The fundamental idea of the reform was due to Lilius\index +{Lilius on the Calendar}, who +died before it was carried into effect. The work of framing +the new calendar\index{Gregorian Calendar} was entrusted to Clavius\index +{Clavius on Calendar}, who explained the +principles and necessary rules in a prolix but accurate work\footnote +{\textit{Romani Calendarii a Greg.~XIII, restituti Explicatio}, Rome, 1603.} +of over 700 folio pages. The plan adopted was due to a suggestion +of Pitatus\index{Pitatus on the Calendar} made in 1552 or perhaps 1537: +the alternative and +more accurate proposal of Stöffler\index +{Stoffler@Stöffler on the Calendar}, made in 1518, to omit one +day in every 134 years being rejected by Lilius and Clavius for +reasons which are not known. + +Clavius believed the year to contain 365.2425432 days, +but he framed his calendar so that a year, on the average, +\PG----File: 360.png---------------------------------------------------- +contained 365.2425 days, which he thought to be wrong by +one day in 3323 years: in reality it is a trifle more accurate +than this, the error amounting to one day in about 3600 years. + +The change was unpopular, but Riccioli\index +{Riccioli on the Calendar}\footnote +{\textit{Chronologia Reformata}, Bonn, 1669, vol.~\textsc{ii}, p.~206.} +tells us that, as +those miracles which take place on fixed dates---\Eg\ the +liquefaction of the blood of S.~Januarius---occurred according +to the new calendar, the papal decree was presumed to have a +divine sanction---Deo ipso huic correctioni Gregorianae subscribente---and +was accepted as a necessary evil. + +In England a bill to carry out the same reform was introduced +in 1584, but was withdrawn after being read a second +time; and the change was not finally effected till 1752, when +eleven days were omitted from that year. In Roman Catholic +countries the new style was adopted in 1582. In Scotland +the change was made in 1600. In the German Lutheran +States it was made in 1700. In England, as I have said +above, it was introduced in 1752; and in Ireland it was made +in 1782. It is well known that the Greek Church still adheres +to the Julian calendar\index +{Calendar, the Civil@\textsc{Calendar}, the Civil|)}\index{Year, Civil|)}. + +The Mohammedan year\index{Year, Mohammedan} contains 12 lunar months, +or $354\frac{1}{3}$ +days, and thus has no connection with the seasons. + +\phantomsection +\addcontentsline{toc}{section}{The Ecclesiastical Calendar (date of Easter)} +The Gregorian\index{Calendar, the Ecclesiastical@\nobreak--- the Ecclesiastical|(}% +\index{Calendar, the Gregorian@\nobreak--- the Gregorian|(}% +\index{Gregorian Calendar} change in the calendar was introduced in +order to keep Easter at the right time of year. The date of +Easter\index{Easter, Date of|(} depends on that of the vernal equinox, and +as the Julian calendar made the year of an average length of 365.25 +days instead of 365.242216 days, the vernal equinox came +earlier and earlier in the year, and in 1582 had regreded to +within about ten days of February. + +The rule for determining Easter is as follows\footnote +{De~Morgan\index{DeMorgan@De Morgan, A.}, \textit{Companion to the Almanac}, +London, 1845, pp. 1--36; \Ibid, 1846, pp. 1--10.}. In 325 the +Nicene Council\index{Nicene Council on Easter} decreed that the Roman +practice should be +followed; and after 463 (or perhaps, 530) the Roman practice +required that Easter-day should be the first Sunday after +\PG----File: 361.png---------------------------------------------------- +the full moon which occurs on or next following the vernal +equinox---full moon being assumed to occur on the fourteenth +day from the day of the preceding new moon (though as a +matter of fact it occurs on an average after an interval of +rather more than $14\frac{3}{4}$ days), and the vernal equinox being +assumed to fall on March~21 (though as a matter of fact it +sometimes falls on March~22). + +This rule and these assumptions were retained by Gregory\index +{Gregory@Gregory XIII|)} +on the ground that it was inexpedient to alter a rule with which +so many traditions were associated; but, in order to save disputes +as to the exact instant of the occurrence of the new +moon, a mean sun and a mean moon defined by Clavius\index +{Clavius on Calendar} were +used in applying the rule. One consequence of using this +mean sun and mean moon and giving an artificial definition of +full moon is that it may happen, as it did in 1818 and 1845, +that the actual full moon occurs on Easter Sunday. In the +British Act, 24 Geo.~II. cap.~23, the explanatory clause which +defines full moon is omitted, but practically full moon has +been interpreted to mean the Roman ecclesiastical full moon; +hence the Anglican and Roman rules are the same. Until +1774 the German Lutheran States employed the actual sun +and moon. Had full moon been taken to mean the fifteenth +day of the moon, as is the case in the civil calendar, then the +rule might be given in the form that Easter-day is the Sunday +on or next after the calendar full moon which occurs next +after March~21. + +Assuming that the Gregorian calendar and tradition are +used, there still remains one point in this definition of Easter +which might lead to different nations keeping the feast at +different times. This arises from the fact that local time is +introduced. For instance the difference of local time between +Rome and London is about 50 minutes. Thus the instant +of the first full moon next after the vernal equinox might +occur in Rome on a Sunday morning, say at 12.30~a.m., while +in England it would still be Saturday evening, 11.40~p.m., +in which case our Easter would be one week earlier than at +\PG----File: 362.png------------------------------------------------------ +Rome. Clavius foresaw the difficulty, and the Roman Communion +all over the world keep Easter on that day of the +month which is determined by the use of the rule at Rome. +But presumably the British Parliament intended time to be +determined by the Greenwich meridian, and if so the Anglican +and Roman dates for Easter might differ by a week; whether +such a case has ever arisen or been discussed I do not know, +and I leave to ecclesiastics to say how it should be settled. + +The usual method of calculating the date on which Easter-day +falls in any particular year is involved, and possibly the +following simple rule\footnote +{It is due to Gauss\index{Gauss}, and his proof is given in Zach's \textit +{Monatliche Correspondenz}, August, 1800, vol.~\textsc{ii}, pp. 221--230.} +may be unknown to some of my readers. + +Let $m$ and $n$ be numbers as defined below\label{easter:rule}. (i)~Divide +the number of the year by $4$, $7$, $19$; and let the remainders be +$a$, $b$, $c$, respectively. (ii)~Divide $19c + m$ by $30$, and let $d$ be +the remainder. (iii)~Divide $2a + 4b + 6d + n$ by $7$, and let $e$ be +the remainder. (iv)~Then the Easter full moon occurs $d$ days +after March~21; and Easter-day is the $(22 + d + e)$th of March +or the $(d + e - 9)$th day of April, except that if the calculation +gives $d = 29$ and $e = 6$ (as happens in 1981) then Easter-day is +on April~19 and not on April~26, and if the calculation gives +$d = 28$, $e = 6$, and also $c > 10$ (as happens in 1954) then Easter-day +is on April~18 and not on April~25, that is, in these two +cases Easter falls one week earlier than the date given by the +rule. These two exceptional cases cannot occur in the +Julian calendar, and in the Gregorian calendar they occur only very +rarely. It remains to state the values of $m$ and $n$ for the particular +period. In the Julian calendar we have $m=15$, $n = 6$. +In the Gregorian calendar we have, from 1582 to 1699 inclusive, +$m = 22$, $n = 2$; from 1700 to 1799, $m = 23$, $n = 3$; +from 1800 to 1899, $m =23$, $n = 4$; from 1900 to 2099, $m = 24$, +$n = 5$; from 2100 to 2199, $m = 24$, $n = 6$; from 2200 to 2299, +$m = 25$, $n = 0$; from 2300 to 2399, $m = 26$, $n=1$; and from +2400 to 2499, $m = 25$, $n=1$. Thus for the year 1908 we +\PG----File: 363.png------------------------------------------------------ +have $m = 24$, $n = 5$; hence $a = 0$, $b = 4$, $c=8$; $d =26$; and +$e = 2$: therefore Easter Sunday will be on the 19th of April. +After the year 4200 the form of the rule will have to be +slightly modified. + +The dominical letter\index{Dominical Letter} and the golden number\index +{Golden Number} of the ecclesiastical +calendar can be at once determined from the values of +$b$ and $c$. The epact, that is, the moon's age at the beginning of +the year, can be also easily calculated from the above data +in any particular case; the general formula was given by +Delambre\index{Delambre on Calendar}, but its value is required so rarely +by any but +professional astronomers and almanack-makers that it is unnecessary +to quote it here. + +We can evade the necessity of having to recollect the +values of $m$ and $n$ by noticing that, if $N$ is the given year, +and if $\{N/x\}$ denotes the integral part of the quotient when +$N$ is divided by $x$, then $m$ is the remainder when $15 + \xi$ is +divided by 30, and $n$ is the remainder when $6 + \eta$ is divided +by $7$: where, in the Julian calendar, $\xi = 0$, and $\eta = 0$; and, in +the Gregorian calendar, $\xi = \{N/100\}\allowbreak - \{N/400\} - \{N/300\}$, +and $\eta = \{N/100\}\allowbreak - \{N/400\} - 2$\index +{Calendar, the Gregorian@\nobreak--- the Gregorian|)}. + +If we use these values of $m$ and $n$, and if we put for +$a$, $b$, $c$, their values, namely, $a = N - 4 \{N/4\}$, +$b = N - 7 \{N/7\}$, +$c = N - 19 \{N/19\}$, the rule given \vpageref[above]{easter:rule} takes +the following form. +``Divide $19N - \{N/19\}\allowbreak + 15 + \xi$ by $30$, and let +the remainder be $d$. Next divide $6 (N + d + 1)\allowbreak - \{N/4\}+\eta$ +by $7$, and let the remainder be $e$. Then Easter full +moon is on the $d$th day after March~21, and Easter-day +is on the ($22 + d + e$)th of March or the ($d + e - 9$)th +of April as the case may be; except that if the calculation +gives $d = 29$, and $e = 6$, or if it gives $d = 28$, +$e = 6$, and $c > 10$, then Easter-day is on the ($d + e - 16$)th +of April.'' + +Thus, if $N = 1899$, we divide $19 (1899) - 99\allowbreak + 15 + (18-4-6)$ +by $30$, which gives $d = 5$, and then we proceed to divide +$6(1899 + 5 + 1)\allowbreak - 474\allowbreak + (18-4-2)$ by $7$, which gives +$e = 6$: therefore Easter-day is on April~2\index +{Calendar, the Ecclesiastical@\nobreak--- the Ecclesiastical|)}% +\index{Easter, Date of|)}. + +\PG----File: 364.png------------------------------------------------------ +The above rules cover all the cases worked out with so +much labour by Clavius\index{Clavius on Calendar} and others\footnote +{Most of the above-mentioned facts about the calendar are taken +from Delambre's\index{Delambre on Calendar} \textit{Astronomie}, Paris, 1814, +vol.~\textsc{iii}, chap.~xxxviii; and his +\textit{Histoire de l'astronomie moderne}, Paris, 1821, vol.~\textsc{i}, +chap.~i: see also +A.~De~Morgan, \textit{The Book of Almanacs}, London, 1851; +S.~Butcher\index{Butcher on the Calendar}, \textit{The +Ecclesiastical Calendar}, Dublin, 1877; and C.~Zeller, \textit +{Acta Mathematica}, Stockholm, 1887, vol.~\textsc{ix}, pp.~131--136: +on the chronological details see J.L.~Ideler\index{Ideler on the Calendar}, +\textit{Lehrbuch der Chronologie}, Berlin, 1831.}. + +\phantomsection +\addcontentsline{toc}{section}{Day of the week corresponding to a given date} +I may add here a rule, quoted by Zeller\index{Zeller}, for determining the +day of the week corresponding to any given date\index +{Days of Week from date}\index{Week, Days of, from date}. Suppose that +the $p$th day of the $q$th month of the year $N$~\emph{anno domini} is the +$r$th day of the week, reckoned from the preceding Saturday. Then +$r$ is the remainder when $p + 2q\allowbreak + \{3 (q + 1)/5\}\allowbreak + + N + \{N/4\}-\eta$ +is divided by $7$; provided January and February are reckoned +respectively as the $13$th and $14$th months of the preceding year. + +For instance, Columbus\index{Columbus} first landed in the New World on +Oct.~12, 1492. Here $p = 12$, $q = 10$, $N=1492$, $\eta = 0$. If we +divide $12 + 20 + 6\allowbreak + 1492 + 373$ by $7$ we get $r = 6$; hence it +was on a Friday. Again, Charles~I\index{Charles I} was executed on Jan.~30, +1649. Here $p=30$, $q = 13$, $N=1648$, $\eta = 0$, and we find +$r = 3$; hence it was on a Tuesday. As another example, +the battle of Waterloo\index{Waterloo, Battle of} was fought on June~18, +1815. Here $p = 18$, $q = 6$, $N = 1815$, $\eta = 12$, and we find $r = 1$; +hence it took place on a Sunday. + +\phantomsection +\addcontentsline{toc}{section}{Means of measuring Time} +I proceed now to give a short account of some of the +means of measuring time which were formerly in use. + +\phantomsection +\addcontentsline{toc}{subsection}{Styles, Sun-dials, Sun-rings} +Of devices for measuring time, the earliest of which we +have any positive knowledge are the \emph{styles}\index{Styles} +or \emph{gnomons}\index{Gnomons} erected +in Egypt and Asia Minor. These were sticks placed vertically +in a horizontal piece of ground, and surrounded by three +concentric circles, such that every two hours the end of the +shadow of the stick passed from one circle to another. Some +of these have been found at Pompeii and Tusculum. + +The \emph{sun-dial}\index{Dials, Sun-|(}\index{Sun-dials|(} is not very +different in principle. It consists +of a rod or style fixed on a plate or dial; usually, but not +\PG----File: 365.png------------------------------------------------------ +necessarily, the style is placed so as to be parallel to the axis +of the earth. The shadow of the style cast on the plate by the +sun falls on lines engraved there which are marked with the +corresponding hours. + +The earliest sun-dial, of which I have read, is that made +by Berosus\index{Berosus} in 540~\textsc{b.c.} One was erected by +Meton\index{Meton} at Athens +in 433~\textsc{b.c.} The first sun-dial at Rome was constructed by +Papirius Cursor\index{Cursor} in 306~\textsc{b.c.} Portable sun-dials, with +a compass fixed in the face, have been long common in the East as well +as in Europe. Other portable instruments of a similar kind +were in use in medieval Europe, notably the sun-rings, hereafter +described, and the sun-cylinders\index +{Cylinders, Sun-}\index{Sun-cylinders}\footnote +{Thus Chaucer\index{Chaucer on the Sun-cylinder} in the \textit +{Shipman's Tale}, ``by my chilindre it is prime +of day,'' and Lydgate\index{Lydgate on the Sun-cylinder} in the \textit +{Siege of Thebes}, ``by my chilyndre I gan +anon to see\textellipsis that it drew to nine.''}. % NB tight ellipsis matches original + +I believe it is not generally known that a sun-dial can be +so constructed that the shadow will, for a short time near +sunrise and sunset, move backwards on the dial\footnote +{Ozanam\index{Ozanam@Ozanam's \textit{Récréations}}, 1803 edition, +vol.~\textsc{iii}, p.~321; 1840 edition, p.~529.}. This +was discovered by Nonez\index{Nonez on Sun-dials}. The explanation is as +follows. Every day the sun appears to describe a circle round the +pole, and the line joining the point of the style to the sun +describes a right cone whose axis points to the pole. The +section of this cone by the dial is the curve described by the +extremity of the shadow, and is a conic. In our latitude the +sun is above the horizon for only part of the twenty-four +hours, and therefore the extremity of the shadow of the +style describes only a part of this conic. Let $QQ'$ be the arc +described by the extremity of the shadow of the style from +sunrise at $Q$ to sunset at $Q'$, and let $S$ be the point of the style +and $F$ the foot of the style, \IE\ the point where the style meets +the plane of the dial. Suppose that the dial is placed so that +the tangents drawn from $F$ to the conic $QQ'$ are real, and that +$P$ and $P'$, the points of contact of these tangents, lie on the arc +$QQ'$. If these two conditions are fulfilled, then the shadow +will regrede through the angle $QFP$ as its extremity moves +\PG----File: 366.png------------------------------------------------------ +from $Q$ to $P$, it will advance through the angle $PFP'$ as its +extremity moves from $P$ to $P'$, and it will regrede through +the angle $P'FQ'$ as its extremity moves from $P'$ to $Q'$. + +If the sun's apparent diurnal path crosses the horizon---as +always happens in temperate and tropical latitudes---and +if the plane of the dial is horizontal, the arc $QQ'$ will consist +of the whole of one branch of a hyperbola, and the above conditions +will be satisfied if $F$ is within the space bounded by this +branch of the hyperbola and its asymptotes. As a particular +case, in a place of latitude $12^{\circ}$~N.\ on a day when the sun is in +the northern tropic (of Cancer) the shadow on a dial whose +face is horizontal and style vertical will move backwards for +about two hours between sunrise and noon. + +If, in the case of a given sun-dial placed in a certain +position, the conditions are not satisfied, it will be possible to +satisfy them by tilting the sun-dial through an angle properly +chosen. This was the rationalistic explanation, offered by the +French encyclopaedists, of the miracle recorded in connection +with Isaiah\index{Isaiah} and Hezekiah\index{Hezekiah}\footnote +{2 Kings, chap.~\textsc{xx}, \textit{vv}.~9--11.}. +Suppose, for instance, that the +style is perpendicular to the face of the dial. Draw the +celestial sphere. Suppose that the sun rises at $M$ and culminates +at $N$, and let $L$ be a point between $M$ and $N$ on the +sun's diurnal path. Draw a great circle to touch the sun's +diurnal path $MLN$ at $L$, let this great circle cut the celestial +meridian in $A$ and $A'$, and of the arcs $AL$, $A'L$ suppose that +$AL$ is the less and therefore is less than a quadrant. If the +style is pointed to $A$, then, while the sun is approaching $L$, the +shadow will regrede, and after the sun passes $L$ the shadow +will advance. Thus if the dial is placed so that a style which +is normal to it cuts the meridian midway between the equator +and the tropic, then between sunrise and noon on the longest +day the shadow will move backwards through an angle +\[ + \sin^{-1} (\cos \omega \sec \tfrac{1}{2} \omega) +- \cot^{-1} \{ \sin \omega \cos(l - \tfrac{1}{2} \omega) + (\cos^2 l - \sin^2 \omega)^{-\frac{1}{2}} \}\,, +\] +where $l$ is the latitude of the place and $\omega$ is the obliquity of +the ecliptic. + +\PG----File: 367.png------------------------------------------------------ +The above remarks refer to the sun-dials in ordinary use. +In 1892 General Oliver\index{Oliver on Sun-dials} brought out in London +a dial with a +solid style, the section of the style being a certain curve whose +form was determined empirically by the value of the equation +of time as compared with the sun's declination\footnote +{An account of this sun-dial with a diagram was given in \textit{Knowledge}, +July~1, 1892, pp.~133, 134.}. The shadow +of the style on the dial gives the local mean time, though of +course in order to set the dial correctly at any place the +latitude of the place must be known: the dial may be also set +so as to give the mean time at any other locality whose longitude +relative to the place of observation is known\index +{Dials, Sun-|)}\index{Sun-dials|)}. + +\begin{figure*}[!ht] +\ifPaper\vspace*{1cm}\fi +\centerline{\includegraphics +[height=\ifPaper 8cm\else.7\textheight\fi]{./images/illus367}} +\label{illus:367} +\end{figure*} +The \emph{sun-ring}\index{Sun-rings|(} or \emph{ring-dial}\index +{Ring-Dial|(} is another instrument for measuring +solar time\footnote +{See Ozanam\index{Ozanam@Ozanam's \textit{Récréations}}, 1803 edition, +vol.~\textsc{iii}, p.~317; 1840 edition, p.~526.}. +One of the simplest type is figured in the +% original reads "diagram below" but in print format we hit a +% varioref page-break loop :( +\ifPaper accompanying diagram\else +diagram \vpageref[below]{illus:367}\fi. +The sun-ring consists of a thin brass band, +about a quarter of an inch wide, bent into the shape of a +circle, which slides between two fixed circular rims---the radii +of the circles being about one inch. At one point of the band +there is a hole; and when the ring is suspended from a fixed +\PG----File: 368.png------------------------------------------------------ +point attached to the rims so that it hangs in a vertical plane +containing the sun, the light from the sun shines through this +hole and makes a bright speck on the opposite inner or concave +surface of the ring. On this surface the hours are +marked, and, if the ring is properly adjusted, the spot of light +will fall on the hour which indicates the solar time. The +adjustment for the time of year is made as follows. The rims +between which the band can slide are marked on their outer +or convex side with the names of the months, and the band +containing the hole must be moved between the rims until +the hole is opposite to that month for which the ring is being +used. + +For determining times near noon the instrument is reliable, +but for other hours in the day it is accurate only if the time +of year is properly chosen, usually near one of the equinoxes. +This defect may be corrected by marking the hours on a +curved brass band affixed to the concave surface of the rims. +I possess two specimens of rings of this kind. These rings +were distributed widely. Of my two specimens, one was +bought in the Austrian Tyrol and the other in London. +Astrolabes and sea-rings can be used as sun-rings\index +{Ring-Dial|)}\index{Sun-rings|)}. + +\phantomsection +\addcontentsline{toc}{subsection}{Water-clocks, Sand-clocks, +Graduated Candles} +\emph{Clepsydras}\index{Clepsydras} or water-clocks\index +{Water-clocks}\index{Clocks|(}, and \emph{hour-glasses}\index{Hour-glasses} +or sand-clocks\index{Sand-clocks}, +afford other means of measuring time. The time occupied +by a given amount of some liquid or sand in running +through a given orifice under the same conditions is always +the same, and by noting the level of the liquid which has +run through the orifice, or which remains to run through it, +a measure of time can be obtained. + +The burning of graduated candles gives another way of +measuring time, and we have accounts of those used by Alfred\index +{Alfred the Great} +the Great for the purpose. Incense sticks were used by the +Chinese in a similar way. + +\phantomsection +\addcontentsline{toc}{subsection}{Clocks and Watches} +Modern \emph{clocks} and \emph{watches}\index{Watches}\footnote +{See \textit{Clock and Watch Making} by Lord Grimthorpe\index +{Grimthorpe on Clocks}, 7th edition, London, 1883.} +comprise a train of wheels +\PG----File: 369.png------------------------------------------------------ +turned by a weight, spring, or other motive power, and regulated +by a pendulum, balance, fly-wheel, or other moving body +whose motion is periodic and time of vibration constant. +The direction of rotation of the hands of a clock was selected +originally so as to make the hands move in the same direction +as the shadow on a sun-dial whose face is horizontal---the dial +being situated in our hemisphere. + +The invention of clocks with wheels is attributed by tradition +to Pacificus\index{Pacificus on Clocks} of Verona, circ.\ 850, +and also to Gerbert\index{Gerbert}, who +is said to have made one at Magdeburg in 996: but there is +reason to believe that these were sun-clocks. The earliest +wheel-clock of which we have historical evidence was one sent +by the Sultan of Egypt in 1232 to the Emperor Frederick~II\index +{Frederick II of Germany}, +though there seems to be no doubt that they had been made +in Italy at least fifty years earlier. + +The oldest clock in England of which we know anything +was one erected in 1288 in or near Westminster Hall out of a +fine imposed on a corrupt Lord Chief Justice. The bells, and +possibly the clock, were staked by Henry~VIII\index{Henry VIII of England} +on a throw of +dice and lost, but the site was marked by a sun-dial, destroyed +about sixty years ago, and bearing the inscription \emph{Discite justiciam +moniti}. In 1292 a clock was erected in Canterbury +Cathedral at a cost of \pounds30. One erected at Glastonbury Abbey +in 1325 is at present in the Kensington Museum and is in +regular action. Another made in 1326 for St Alban's Abbey +showed the astronomical phenomena, and seems to have been +one of the earliest clocks that did so. One put up at Dover in +1348 is still in good working order. The clocks at Peterborough +and Exeter were of about the same date, and portions of them +remain \emph{in situ}. Most of these early clocks were regulated +by horizontal balances: pendulums being then unknown. Of +the elaborate clocks of a later date, that at Strasburg made +by Dasypodius\index{Dasypodius} in 1571, and that at Lyons constructed by +Lippeus\index{Lippeus} in 1598, are especially famous: the former was +restored in 1842, though in a manner which destroyed most of +the ancient works. + +\PG----File: 370.png------------------------------------------------------ +In 1370, Vick\index{Vick on Clocks} constructed a clock for +Charles~V\index{Charles V of Germany} with +a weight as motive power and a vibrating escapement---a +great improvement on the rough time-keepers of an +earlier date. + +The earliest clock regulated by a pendulum seems to have +been made in 1621 by a clockmaker named Harris\index +{Harris on pendulum clock}, of Covent +Garden, London, but the theory of such clocks is due to +Huygens\index{Huygens}\footnote +{\textit{Horologium Oscillatorium}, Paris, 1673.}. +Galileo\index{Galileo on Pendulum} had discovered previously the isochronism +of a pendulum, but did not apply it to the regulation of the +motion of clocks. Hooke\index{Hooke on Timepieces} made such clocks, +and possibly discovered independently this use of the pendulum: he +invented or re-invented the anchor pallet. + +A watch\index{Watches} may be defined as a clock which will go in any +position. Watches, though of a somewhat clumsy design, were +made at Nuremberg by P.~Hele\index{Hele, P.} early in the sixteenth +century---the motive power being a ribbon of steel, wound round a +spindle, and connected at one end with a train of wheels which +it turned as it unwound---and possibly a few similar time-pieces +had been made in the previous century. By the end of +the sixteenth century they were not uncommon. At this time +they were usually made in the form of fanciful ornaments +such as skulls, or as large pendants, but about 1620 the +flattened oval form was introduced, rendering them more +convenient to carry in a pocket or about the person. In the +seventeenth century their construction was greatly improved, +notably by the introduction of the spring balance by Huygens\index{Huygens} +in 1674, and independently by Hooke\index{Hooke on Timepieces} in 1675---both +mathematicians having discovered that small vibrations of a coiled +spring, of which one end is fixed, are practically isochronous. +The fusee had been used by R.~Zech\index{Zech, R.} of Prague in 1525, but +was re-invented by Hooke\index{Hooke on Timepieces}. + +Clocks and watches are usually moved and regulated in +the manner indicated above. Other motive powers and other +\PG----File: 371.png------------------------------------------------------ +devices for regulating the motion may be met with occasionally. +Of these I may mention a clock in the form of a cylinder, +usually attached to another weight as in Atwood's machine, +which rolls down an inclined plane so slowly that it takes +twelve hours to roll down, and the highest point of the face +always marks the proper hour\footnote +{Ozanam\index{Ozanam@Ozanam's \textit{Récréations}}, 1803 edition, +vol.~\textsc{ii}, p.~39; 1840 edition, p.~212; or \textit{La +Nature}, Jan.~23, 1892, pp.~123, 124.}. + +A water-clock\index{Water-clocks} made on a somewhat similar plan is +described by Ozanam\footnote +{Ozanam\index{Ozanam@Ozanam's \textit{Récréations}}, 1803 edition, +vol.~\textsc{ii}, p.~68; 1840 edition, p.~225.} as one of the sights of +Paris at the beginning of +the last century. It was formed of a hollow cylinder divided +into various compartments each containing some mercury, so +arranged that the cylinder descended with uniform velocity +between two vertical pillars on which the hours were marked +at equidistant intervals. + +Other ingenious ways of concealing the motive power +have been described in the columns of \textit{La Nature}\footnote +{See especially the volumes issued in 1874, 1877, and 1878.}. Of +such mysterious timepieces the following are not uncommon +examples, and probably are known to most readers of this +book. One kind of clock consists of a glass dial suspended +by two thin wires; the hands however are of metal, and the +works are concealed in them or in the pivot. Another kind +is made of two sheets of glass in a frame containing a spring +which gives to the hinder sheet a very slight oscillatory +motion--imperceptible except on the closest scrutiny--and +each oscillation moves the hands through the requisite angles. +Some so-called perpetual motion timepieces were described +above \vpagerefrange{page:PerpClockStart}{page:PerpClockEnd}. +Lastly, I have seen in France a clock +the hands of which were concealed at the back of the dial, and +carried small magnets; pieces of steel in the shape of insects +were placed on the dial, and, following the magnets, served to +indicate the time\index{Clocks|)}. + +\phantomsection +\addcontentsline{toc}{section}{Watches as Compasses} +The position of the sun relative to the points of the compass +\PG----File: 372.png------------------------------------------------------ +determines the solar time. Conversely, if we take the time +given by a watch as being the solar time---and it will differ +from it by only a few minutes at the most---and we observe +the position of the sun, we can find the points of the compass\index +{Compasses, Watches as|(}\index +{Watches as Compasses@\nobreak--- as Compasses|(}\footnote +{The rule is given by W.H.~Richards\index{Richards on use of compass}, +\textit{Military Topography}, London, 1883, p.~31, though it is not stated +quite correctly. I do not know who first enunciated it.}. +To do this it is sufficient to point the hour-hand to the sun, +and then the direction which bisects the angle between the +hour and the figure \textsc{xii} will point due south. For instance, +if it is four o'clock in the afternoon, it is sufficient to point +the hour-hand (which is then at the figure \textsc{iiii}) to the sun, and +the figure \textsc{ii} on the watch will indicate the direction of south. +Again, if it is eight o'clock in the morning, we must point the +hour-hand (which is then at the figure \textsc{viii}) to the sun, and +the figure \textsc{x} on the watch gives the south point of the +compass. + +Between the hours of six in the morning and six in the +evening the angle between the hour and \textsc{xii} which must be +bisected is less than $180^{\circ}$, but at other times the angle to be +bisected is greater than $180^{\circ}$; or perhaps it is simpler to say +that at other times the rule gives the north point and not the +south point. + +The reason is as follows. At noon the sun is due south, +and it makes one complete circuit round the points of the +compass in $24$ hours. The hour-hand of a watch also makes +one complete circuit in $12$ hours. Hence, if the watch is held +in the plane of the ecliptic with its face upwards, and the +figure \textsc{xii} on the dial is pointed to the south, both the hour-hand +and the sun will be in that direction at noon. Both +move round in the same direction, but the angular velocity +of the hour-hand is twice as great as that of the sun. Hence +the rule. The greatest error due to the neglect of the equation +of time is less than $2^{\circ}$. Of course in practice most people, +instead of holding the face of the watch in the ecliptic, would +\PG----File: 373.png------------------------------------------------------ +hold it horizontal, and in our latitude no serious error would +be thus introduced. + +In the southern hemisphere where at noon the sun is due +north the rule requires modification. In such places the hour-hand +of a watch (held face upwards in the plane of the ecliptic) +and the sun move in opposite directions. Hence, if the watch +is held so that the figure \textsc{xii} points to the sun, +then the direction which bisects the angle between the hour of the day +and the figure \textsc{xii} will point due north\index +{Compasses, Watches as|)}\index{Watches as Compasses@\nobreak--- as Compasses|)}. + +\PG----File: 374.png------------------------------------------------------ + + + +% CHAPTER XIV. +\UseChapterXIVHeadings + +\chapter{Matter and Ether Theories.} + +\textsc{Matter}, like space and time, cannot be defined, but either\chapindex +{Matter, Constitution of} +the statement that matter is whatever occupies space or the\chapindex +{Atomic@\textsc{Atomic Theories}} +statement that it is anything which can be seen, touched, or +weighed, suggests its more important characteristics to anyone +already familiar with it. + +The means of measuring matter and some of its properties +are treated in most text-books on mechanics, and I do not +propose to discuss them. I confine the chapter to an account +of some of the hypotheses by physicists as to the +ultimate constitution of matter, but I exclude metaphysical +conjectures, which from their nature are mere assertions incapable +of proof and are not subject to mathematical analysis. +The question is intimately associated with the explanation of +the phenomena of attraction, light, chemistry, electricity, and +other branches of physics. + +I commence with a list of some of the more plausible of +the hypotheses formerly proposed which accounted for the +obvious properties of matter, and shall then discuss how far +they explain or are consistent with other facts\footnote +{I have based my account mainly on \textit{Recent Advances in Physical +Science}, by P.G.~Tait\index{Tait}, Edinburgh, 1876 (chaps,~\textsc{xii}, +\textsc{xiii}); and on the article \textit{Atom} by J.~Clerk Maxwell\index +{Maxwell, J. Clerk} in the Encyclopaedia Britannica or +his \textit{Collected Works}, vol.~\textsc{ii}, pp.~445--484: see also +W.M.~Hicks's\index{Hicks on Matter} +address, Report of the British Association (Ipswich meeting), 1895, +vol.~\textsc{lxv}, pp.~595--606. For the more recent speculations see +J.J.~Thomson\index{Thomson, J.J.}, \textit{Electricity and Matter}, +Westminster, 1904, and J.~Larmor\index{Larmor on Electrons}, \textit{Aether +and Matter}, Cambridge, 1900.}. The interest +\PG----File: 375.png------------------------------------------------------ +of the list is largely historical, for within the last few years +new views as to the constitution of matter have been propounded, +but the details of these more recent hypotheses are +so complicated and technical that only professional mathematicians +can understand them. Accordingly I allude to them +only briefly. + +\section[Hypothesis of Continuous Matter][Matter and Ether Theories]% +{Hypothesis of Continuous Matter} +It may be\index{Continuity of Matter}% +\index{Matter, Hyp@\nobreak--- Hypotheses on|(} +supposed that matter is homogeneous and continuous, in +which case there is no limit to the infinite divisibility of +bodies. This view was held by Descartes\index{Descartes}\footnote +{Descartes, \textit{Principia}, vol.~\textsc{ii}, pp.~18, 23.}. + +This conjecture is consistent with the facts deducible by +untrained observation, but there are many other phenomena +for which it does not account; moreover there seems to be no +way of reconciling such a structure of matter either with the +facts of chemical changes or with the results of spectrum +analysis\index{Spectrum Analysis}. At any rate the theory must be regarded +as extremely improbable. + +\section[Atomic Theories][Matter and Ether Theories]{Atomic Theories} +If matter is not continuous we +must suppose that every body is composed of aggregates of +molecules. If so, it seems probable that each such molecule is +built up by the association of two or more atoms, that the +number of kinds of atoms is finite, and that the atoms of any +particular kind are alike. As to the nature of the atoms the +following hypotheses have been made. + +\subsection[Popular Atomic Hypothesis][Matter and Ether Theories]% +{Popular Atomic Hypothesis} The popular view is that +every atom of any particular kind is a minute indivisible +article possessing definite qualities, everlasting in its form and +properties, and infinitely hard. + +This statement is plausible, but the difficulties to which it +leads appear to be insuperable. In fact we have reason to +think that the atoms which form a molecule are composite +systems in incessant vibration at a rate characteristic of the +molecule, and it is most probable that they are elastic. + +\PG----File: 376.png------------------------------------------------------ +Newton\index{Newton} seems to have hazarded a conjecture of this kind +when he suggested\footnote +{Newton, \textit{Principia}, bk.~\textsc{ii}, prop.~50.} +that the difficulties, connected with the +fact that the velocity of sound\index{SoundVel@\nobreak--- Velocity of} was +one-ninth greater than that +required by theory, might be overcome if the particles of air +were little rigid spheres whose distance from one another under +normal conditions was nine times the diameter of any one of +them. This was ingenious, but obviously the view is untenable, +because, if such a structure of air existed, the air could not +be compressed beyond a certain limit, namely, about $1/1021$st +part of its original volume, which has been often exceeded. +The true explanation of the difficulty noticed by Newton was +given by Laplace\index{Laplace on velocity of sound}. + +\subsection[Boscovich's Hypothesis][Matter and Ether Theories]% +{Boscovich's Hypothesis} +In 1759 Boscovich suggested\index{Boscovich on Matter}\footnote +{\textit{Philosophiae naturalis Theoria redacta ad unicam Legem Virium}, +Vienna, 1759.} +that the facts might be explained by supposing that an atom +was an infinitely small indivisible mass which was a centre of +force---the law of force being attractive for sensible distances, +alternately attractive and repulsive for minute distances, and +repulsive for infinitely small distances. In this theory all +action between bodies is action at a distance. + +He explained the apparent extension of bodies by saying +that two parts are consecutive (or similarly that two bodies are +in contact) when the nearest pair of atoms in them are so close +to one another that the repulsion at any point between them +is sufficiently great to prevent any other atom coming between +them. It is essential to the theory that the atom shall have +a mass but shall not have dimensions. + +This hypothesis is not inconsistent with any known facts, +but it has been described, perhaps not unjustly, as a mere +mathematical fiction, and certainly it is opposed to the apparent +indications of our senses. At any rate it is artificial, though +it may be a prejudice to regard that as an argument against +its adoption. To some extent this view was accepted by +Faraday\index{Faraday on Matter}. + +\PG----File: 377.png------------------------------------------------------ +Lord Kelvin\index{Kelvin}, better known as Sir William Thomson, has +shown\footnote +{\textit{Proceedings of the Royal Society of Edinburgh}, April~21, 1862, +vol.~\textsc{iv}, pp.~604--606.} +that, if we assume the existence of gravitation, then +each of the above hypotheses will account for cohesion. + +\subsection[Hypothesis of an Elastic Solid Ether. Labile Ether]% +[Matter and Ether Theories]{Hypothesis of an Elastic Solid Ether} +Some physicists\index{Ether Theories|(} +have tried to explain the known phenomena by properties of +the medium through which our impressions are derived. By +postulating that all space is filled with a medium possessed of +many of the characteristics of an elastic solid, it has been +shown by Fresnel\index{Fresnel on Ether}, Green\index{Green on Ether}, +Cauchy\index{Cauchy}, Neumann\index{Neumann on Ether}, +MacCullagh\index{MacCullagh on Ether}, and +others that a large number of the properties of light and +electricity may be explained. In spite of the difficulties to +which this hypothesis necessarily leads, and of its inherent +improbability, it has been discussed by Stokes\index{Stokes on Ether}, +Lamé\index{Lame@Lamé}, Boussinesq\index{Boussinesq on Ether}, +Sarrau\index{Sarrau on Ether}, Lorenz\index{Lorentz on Ether}, +Lord Rayleigh\index{Rayleigh}, and Kirchhoff\index{Kirchhoff on Ether}. + +This hypothesis has been modified and rendered somewhat +more plausible by von~Helmholtz\index{Helmholz}\index{Von Helmholz}, +Lommel\index{Lommel on Ether}, Ketteler\index{Ketteler on Ether}\footnote +{\textit{Theoretische Optik}, Braunschweig, 1885.}, and +Voigt\index{Voigt on Ether}, who based their researches on the assumption of +a mutual reaction between the ether and the material molecules +located in it: on this view the problems connected with refraction +and dispersion have been simplified. Finally, Sir +William Thomson\index{Kelvin}, in his Baltimore Lectures, 1885, suggested +a mechanical analogue to represent the relations between +matter and this ether, by which a possible constitution of the +ether can be realized. He also suggested later a form of +\emph{labile ether}\index{Labile Ether}, from whose properties most of the +more familiar physical phenomena can be deduced, provided the arrangement +can be considered stable; a labile ether is an elastic solid, +and its properties in two dimensions may be compared with +those of a soap-bubble film, in three dimensions. + +It is, however, difficult to criticise any of these hypotheses +as a theory of the constitution of matter until the arrangement +of the atoms or their nature is more definitely expressed. + +\PG----File: 378.png------------------------------------------------------ + + +\section[Dynamical Theories][Matter and Ether Theories]{Dynamical Theories} +In recent years the suggestion +has been made that the so-called atoms may be forms of +motion (\Eg~permanent eddies) in one elementary material +known as the ether; on this view all the atoms are constituted +of the same matter, but the physical conditions are +different for the different kinds of atoms. It has been said +that there is an initial difficulty in any such hypothesis, since +the all-pervading elementary fluid must possess inertia, so that +to explain matter we assume the existence of a fluid possessing +one of the chief characteristics of matter. This is true as far +as it goes, but it is not more unreasonable than to attribute +all the fundamental properties of matter to the atoms themselves, +as is done by many writers. The next paragraph +contains a statement of one of the earliest attempts to +formulate a dynamical atomic hypothesis. + +\subsection[The Vortex Ring Hypothesis][Matter and Ether Theories]% +{The Vortex Ring Hypothesis} This hypothesis assumes +that each atom is a vortex ring in an incompressible frictionless +homogeneous fluid. + +Vortex rings\index{Vortex rings}---though, since friction is brought into +play, of an imperfect character---can be produced in air by many +smokers. Better specimens can be formed by taking a +cardboard box in one side of which a circular hole is cut, +filling it with smoke, and hitting the opposite side sharply. +The tendency of the particles forming a ring to maintain their +annular connection may be illustrated by placing such a box +on one side of a room in a direct line with the flame of a +lighted candle on the other side. If properly aimed, the ring +will travel across the room and put out the flame. If the box +is filled only with air, so that the ring is not visible, the +experiment is more effective. + +In 1858 von Helmholtz\index{Helmholtz}\index{Von Helmholz}\footnote +{\textit{Crelle's Journal}, 1858, vol.~\textsc{lv}, pp.~25--55; translated +by Tait\index{Tait} in the \textit{Philosophical Magazine}, June, 1867, +supplement, series~4, vol.~\textsc{xxxiii}, pp.~485--512.} +showed that a closed vortex filament +in a perfect fluid is indestructible and retains certain +\PG----File: 379.png------------------------------------------------------ +characteristics always unaltered. In 1867 Sir William Thomson\index{Kelvin} +propounded\footnote +{\textit{Proceedings of the Royal Society of Edinburgh}, Feb.~18, 1867, +vol.~\textsc{vi}, pp.~94--105.} the idea that matter consists of vortex +rings in a fluid which fills space. If the fluid is perfect we could +neither create new vortex rings nor destroy those already +created, and thus the permanence of the atoms is explained. +Moreover the atoms would be flexible, compressible, and in +incessant vibration at a definite fundamental rate. This rate +is very rapid, and Sir William Thomson gave the number of +vibrations per second of a sodium ring as probably being +greater than $10^{14}$. + +By a development of this hypothesis Prof.\ J.J.~Thomson\index +{Thomson, J.J.}\footnote +{\textit{A Treatise on the Motion of Vortex Rings}\index{Vortex rings}, +Cambridge, 1883.} showed, some years ago, that chemical combination may be +explained. He supposed that a molecule of a compound is +formed by the linking together of vortex filaments representing +atoms of different elements: this arrangement may be +compared with that of helices on an anchor ring. For stability +not more than six filaments may be combined together, and +their strengths must be equal. Another way of explaining +chemical combination on the vortex atom hypothesis has been +suggested by W.M.~Hicks\index{Hicks on Matter}. It is known\footnote +{See a memoir by M.J.M.~Hill\index{Hill, M.J.M.} in the \textit{Philosophical +Transactions of the Royal Society}, London, 1894, part~i, pp.~213--246.} +that a spherical +mass of fluid, whose interior possesses vortex motion, can +move through liquid like a rigid sphere, and he has shown +that one of these spherical vortices\index{Vortex spheres@\nobreak--- Spheres} +can swallow up another, thus forming a compound element. + +\subsection[The Vortex Sponge Hypothesis][Matter and Ether Theories]% +{The Vortex Sponge Hypothesis} +Any vortex\index{Vortex sponges@\nobreak--- Sponges} atom +hypothesis labours under the difficulty of requiring that +the density of the fluid ether shall be comparable with +that of ordinary matter. In order to obviate this and +at the same time to enable it to transmit transversal +radiations Sir William Thomson suggested what has been +\PG----File: 380.png------------------------------------------------------ +termed, not perhaps very happily, the vortex sponge hypothesis\index +{Vortex sponges@\nobreak--- Sponges}\footnote +{\textit{Philosophical Magazine}, London, October, 1887, series 5, +vol.~\textsc{xxiv}, pp.~342--353.}: +this rests on the assumption that laminar motion +can be propagated through a turbulently moving inviscid +liquid. The mathematical difficulties connected with such +motion have prevented an adequate discussion of this hypothesis, +and I therefore confine myself to merely mentioning it. + +These hypotheses, of vortex motion in a fluid, account for +the indestructibility of matter and for many of its properties. +But in order to explain statical electrical attraction it would +seem necessary to suppose that the ether is elastic; in other +words, that an electric field must be a field of strain. If so, +complete fluidity in the ether would be impossible, and hence +the above theories are now regarded as untenable. + +\subsection[The Ether-Squirts Hypothesis][Matter and Ether Theories]% +{The Ether-Squirts Hypothesis} +Prof.\ Karl Pearson\index{Pearson on Ether-Squirts}\index +{Ether-Squirts}\footnote +% switched with footnote on next page as per errata sheet +{\textit{American Journal of Mathematics}, 1891, vol. \textsc{xiii}, +pp.~309--362.} +has suggested another dynamical theory in which an atom is +conceived as a point at which ether is pouring into our space +from space of four dimensions. + +If an observer lived in two dimensional space filled with +ether and confined by two parallel and adjacent surfaces, and +if through a hole in one of these surfaces fresh ether were +squirted into this space, the variations of pressure thereby +produced might give the impression of a hard impenetrable +body. Similarly an ether-squirt from space of four dimensions +into our space might give us the impression of matter. + +It seems necessary on this hypothesis to suppose that there +are also ether-sinks, or atoms of negative mass; but as ether-squirts +and ether-sinks would repel one another we may suppose +that the latter have moved out of the universe known to +our senses. + +By defining the mass of an atom as the mean rate at which +\PG----File: 381.png--------------------------------------------------- +ether is squirting into our space at that point, we can deduce +the Newtonian law of gravitation, and by assuming certain +periodic variations in the rate of squirting we can deduce +some of the phenomena of cohesion, of chemical action, and of +electromagnetism and light. But of course the hypothesis rests +on the assumption of the existence of a world beyond our senses\index +{Ether Theories|)}. + +\subsection[The Electron Hypothesis][Matter and Ether Theories]% +{The Electron Hypothesis} MacCullagh, in 1837 and\index{Electrons|(} +1839, proposed to account for optical phenomena on the +assumption of an elastic ether possessing elasticity of the type +required to enable it to resist rotation. This suggestion has +been recently modified and extended by Dr~J.~Larmor\index +{Larmor on Electrons}\footnote +% switched with 2nd footnote on previous page as per errata sheet +{\textit{Philosophical Transactions of the Royal Society}, London, 1894, +pp.~719--822; 1895, pp.~695--743.}, and, +as now enunciated, it accounts for many of the electrical and +magnetic (as well as the optical) properties of matter. + +The hypothesis is however very artificial. The assumed +ether is a rotationally elastic incompressible fluid. In this +fluid Larmor\index{Larmor on Electrons} introduces monad electric +elements or \emph{electrons}, +which are nuclei of radial rotational strain. He supposes +that these electrons constitute the basis of matter. He +further supposes that an electrical current consists of a procession +of these electrons, and that a magnetic particle is +one in which these entities are revolving in minute orbits. +Dynamical considerations applied to such a system lead to +an explanation of nearly all the more obvious phenomena. +By further postulating that the orbital motion of electrons in +the atom constitute it a fluid vortex it is possible to apply the +hydrodynamical pulsatory theory of Bjerknes\index{Bjerknes} +or Hicks\index{Hicks on Matter} and +obtain an explanation of gravitation. + +Thus on this view mass is explained as an electrical manifestation. +Electricity in its turn is explained by the existence +of electrons, that is, of nuclei of strain in the ether, which +are supposed to be in incessant and rapid motion. Whilst, to +render this possible, properties are attributed to the ether +which are apparently inconsistent with our experience of the +space it fills. Put thus, the hypothesis seems very artificial. +\PG----File: 382.png--------------------------------------------------- +Perhaps the utmost we can say for it is that, from some points +of view, it may, so far as analysis goes, be an approximation +to the true theory; in any case much work will have to be +done before it can be considered established even as a working +hypothesis. + +\phantomsection +\addcontentsline{toc}{subsection}{Speculations due to +investigations on Radio-activity} +Most of the above was written in 1891. Since then +investigations on radio-activity have opened up new avenues +of conjecture which tend to strengthen the electron theory as +a working hypothesis. More than thirty years ago Clerk +Maxwell\index{Maxwell, J. Clerk} had shown that light and electricity +were closely +connected phenomena. It was then believed that both were +due to waves in the hypothetical ether, but it was supposed +that the phenomena of matter on the one side and of light +and electricity on the other were sharply distinguished one +from the other. The differences, however, between matter and +light tend to disappear as investigations proceed. In 1895 +Röntgen established the existence of rays which could produce +light, which had the same velocity as light, which were not +affected by a magnet, and which could traverse wood and certain +other opaque substances like glass. A year later Becquerel\index +{Becquerel Rays} +showed that uranium was constantly emitting rays which, +though not affecting the eye as light, were capable of producing +an image on a photographic plate. Like Röntgen +rays\index{Rontgen@Röntgen Rays} they can go through thin sheets of metal; +like heat rays +they burn the skin; like electricity they generate ozone from +oxygen. Passed into the air they enable it to conduct the +electric current. Their speed has been measured and found +to be rather more than half that of light and electricity. +It was soon found that thorium possessed a similar property, +but in 1903 Prof.\ Curie\index{Curie on radio-activity} showed that +radium possessed radio-activity +to an extent previously unsuspected in any body, and +in fact the rays were so powerful as to make the substance +directly visible. Further experiments showed that numerous +bodies are radio-active, but the effects are so much more +marked in radium that it is convenient to use that substance +for most experimental purposes. + +\PG----File: 383.png--------------------------------------------------- +Radium gives off no less than three kinds of rays besides +a radio-active emanation. In these discharges there appears +to be a gradual change from what had been supposed to be +an elementary form of matter to another. This leads to the +belief that of the known forms of matter some, perhaps even +all, are not absolutely stable. On the other hand, it may be +that only radio-active bodies are unstable, and that in their +disintegration we are watching the final stage in the evolution +of stable and constant forms of matter. It may, however, in +any case turn out that some, or perhaps all, of the so-called +elements may be capable of resolution into different combinations +of electrons or electricity. + +At an earlier date J.J.~Thomson\index{Thomson, J.J.} had concluded that the +glow, seen when an electric current passes through a high +vacuum tube, is due to a rush of minute particles across the +tube. He calculated their weight, their velocity, and the +charge of electricity transported by or represented by them, +and found these to be constant. They were deflected like +Becquerel rays\index{Becquerel Rays}. All space seems to contain them, +and electricity, +if not identical with them, is at least carried by them. +This suggested that these minute particles might be electrons. +If so, they might thus give the ultimate explanation of electricity +as well as matter, and the atom of the chemist would +be not an irreducible unit of matter, but a system comprising +numerous such minute particles. These conclusions are consistent +with those subsequently deduced from experiments +with radium. In 1904 the hypothesis was carried one stage +further. In that year J.J.~Thomson\index{Thomson, J.J.} investigated +the conditions +of stability of certain systems of revolving particles; +and on the hypothesis that an atom of matter consists of a +number of particles carrying negative charges of electricity +revolving in orbits within a sphere of positive electrification\index +{Electrons|)} +he deduced many of the properties of the different chemical +atoms corresponding to different possible stable systems of +this kind. His scheme led to results agreeing closely with +the results of Mendeléeff's\index{Mendel@Mendeléeff} periodic hypothesis. +An interesting +\PG----File: 384.png------------------------------------------------------ +consequence of this view is that Franklin's\index{Franklin, B.} description +of electricity as subtle particles pervading all bodies, may turn +out to be substantially correct. It is also remarkable that +corpuscles somewhat analogous to those whose existence was +suggested in Newton's\index{Newton} corpuscular theory of light should be +now supposed to exist in cathode and Becquerel rays\index{Becquerel Rays}. + +\subsection[The Bubble Hypothesis][Matter and Ether Theories]% +{The Bubble Hypothesis\footnote +{O.~Reynolds\index{Reynolds, O.}, \textit{Submechanics of the Universe}, +Cambridge, 1903.}} +The difficulty of conceiving\index{Bubble Theory of Matter} +the motion of matter through a solid elastic medium has been +met in another way, namely, by suggesting that what we call +matter is a deficiency of the ether, and that this region of +deficiency can move through the ether in a manner somewhat +analogous to that in which a bubble can move in a liquid. +To express this in technical language we may suppose the +ether to consist of an arrangement of minute uniform spherical +grains piled together so closely that they cannot change their +neighbours, although they can move relatively one to another. +Places where the number of grains is less or greater than the +number necessary to render the piling normal, move through +the medium, as a wave moves through water, though the +grains do not move with them. Places where the ether is in +excess of the normal amount would repel one another and +move away out of our ken, but places where it is below the +normal amount would attract each other according to the law +of gravity, and constitute particles of matter which would +be indestructible. It is alleged that the theory accounts for +the known phenomena of gravity, electricity, and light, provided +the size of its grains is properly chosen. Reynolds\index{Reynolds, O.} +has calculated that for this purpose their diameter should be +rather more than $5 \times 10^{-18}$ centimetres, and that the pressure +in the medium would be about $10^4$ tons per square centimetre. +This theory is in itself more plausible than the electron +hypothesis, but its consequences have not yet been fully +worked out\index{Matter, Hyp@\nobreak--- Hypotheses on|)}. +\PG----File: 385.png--------------------------------------------------- + +\ThoughtBreakSpace +\phantomsection +\addcontentsline{toc}{section}{Conjectures as to the cause of Gravity} +\markright{The cause of Gravity} +Returning from these novel hypotheses to the classical +theories of matter, we may now proceed a step further. +Before a hypothesis on the structure of matter can be ranked +as a scientific theory we may reasonably expect it to afford +some explanation of three facts. These are (\emph{a})~the Newtonian +law of attraction; (\emph{b})~the fact that there are only a finite +number of ultimate kinds of matter---such as oxygen, iron, etc.---which +can be arranged in a series such that the properties of +the successive members are connected by a regular law; and +(\emph{c})~the main results of spectrum analysis. + +In regard to the first point (\emph{a}), we can say only that none\index +{Attraction, Law of|(}\index{Gravity@\textsc{Gravity}, Hypotheses on|(} +of the above theories are inconsistent with the known laws of +attraction; and as far as the ether-squirts, the electron, and +the bubble hypotheses are concerned, they have been elaborated +into a form from which the gravitational law of attraction +can be deduced. But we may still say that as to the cause +of gravity---or indeed of force---we know nothing. + +Newton\index{Newton}, in his Letters to Bentley\index{Bentley, Newton to}, +while declaring his +ignorance of the cause of gravity, refused to admit the possibility +of force acting at a finite distance through a vacuum. +``You sometimes speak of gravity,'' said he\footnote +{Letter dated Jan.~17, 1693. I quote from the original, which is in +the Library of Trinity College, Cambridge; it is printed in the \textit +{Letters to Bentley}, London, 1756, p.~20.\label{ibid:17}}, ``as essential +and inherent to matter: pray do not ascribe that notion to +me, for the cause of gravity is what I do not pretend to +know.'' And in another place he wrote\footnote +{Letter dated Feb.~25, 1693; \ibidref{ibid:17}{\textit +Letters to Bentley}, pp.~25, 26. }, ``'Tis inconceivable, +that inanimate brute matter should (without the mediation of +something else which is not material) operate upon and affect +other matter without mutual contact; as it must if gravitation +in the sense of Epicurus\index{Epicurus on Gravitation}, be essential +and inherent in it\textellipsis\ That gravity should be innate, +inherent, and essential to +matter, so that one body may act upon another at a distance +thro' a vacuum, without the mediation of anything else, by +and through which their action and force may be conveyed +\PG----File: 386.png--------------------------------------------------- +from one to another, is to me so great an absurdity, that I +believe no man who has in philosophical matters a competent +faculty of thinking can ever fall into it. Gravity must be +caused by an agent acting constantly according to certain +laws, but whether this agent be material or immaterial, I have +left to the consideration of my readers.'' + +I have already alluded to conjectural explanations of +gravity dependent on the ether-squirts, the electron, and the +bubble hypotheses. Of other conjectures as to the cause of +gravity, three, which do not involve the idea of force acting +at a distance, may be here mentioned: + +(1)\quad The first of these conjectures was propounded by +Newton\index{Newton} in the \emph{Queries} at the end of his \textit +{Opticks}, where he +suggested as a possible explanation the existence of a stress +in the ether surrounding a particle of matter\footnote +{Quoted by S.P.~Rigaud\index{Rigaud, S.P.} in his \textit{Essay} on the +\textit{Principia}, Oxford, +1838, appendix, pp.~68--70. On other guesses by Newton see Rigaud, +text, pp.~61--62, and references there given.}. + +This has been elaborated on a statical basis by Maxwell\index +{Maxwell, J. Clerk}, who showed\footnote +{Article \emph{Attraction}, in \textit{Encyclopaedia Britannica}, +or \textit{Collected Works}, vol.~\textsc{ii}, p.~489. } +that the stress would have to be at least 3000 +times greater than that which the strongest steel would +support. Sir William Thomson (Lord Kelvin\index{Kelvin}) has +suggested\footnote +{\textit{Proceedings of the Royal Society of Edinburgh}, Feb.~7, 1870, +vol.~\textsc{vii}, pp.~60--63.} +a dynamical way of producing the stress by supposing that +space is filled with an incompressible fluid, constantly being +annihilated by each atom of matter at a rate proportional to +its mass, a constant supply being kept up at an infinite +distance. It is true that this avoids Maxwell's difficulty, but +we have no right to introduce such sinks and sources of fluid +unless we have other grounds for believing in their existence. +The conclusion is that Newton's conjecture is very improbable +unless we adopt the ether-squirts theory: on that hypothesis +it is a plausible explanation. + +\PG----File: 387.png------------------------------------------------------ +I should add that Maclaurin\index{Maclaurin on Newton} implies\footnote +{\textit{An Account of Sir Isaac Newton's Philosophical Discoveries}, London, +1748, p.~111.} that though the +above explanation was Newton's\index{Newton} early opinion, yet his final +view was that he could not devise any tenable hypothesis +about the cause of gravitation. + +(2)\quad In 1782 Le~Sage\index{LeSage@Le Sage on Gravity} of Geneva +suggested\footnote +{\textit{Mémoires de l'Académie des Sciences} for 1782, Berlin, 1784, +pp.~404--432: see also the first two books of his \textit +{Traité de Physique}, Geneva, 1818.} that gravity +was caused by the bombardment of streams of ultramundane +corpuscles. These corpuscles are supposed to come in all +directions from space and to be so small that inter-collisions +are rare. + +A body by itself in space would receive on an average as +many blows on one side as on another, and therefore would +have no tendency to move. But, if there are two bodies, each +will screen the other from some of the bombarding corpuscles. +Thus each body will receive more blows on the side remote from +the other body than on the side turned towards it. Hence +the two bodies will be impelled each towards the other. + +In order to make this force between two particles vary +directly as the product of their masses and inversely as the +square of the distance between them, Le~Sage\index +{LeSage@Le Sage on Gravity} showed that +it was sufficient to suppose that the mass of a body was proportional +to the area of a section at right angles to the +direction in which it was attracted. This requires that the +constitution of a body shall be molecular, and that the +distances between consecutive molecules shall be very large +compared with the sizes of the molecules. On the vortex +hypothesis we may suppose that the ultramundane corpuscles +are vortex rings. + +This is ingenious, and it is possible that if the corpuscles +were perfectly elastic the theory might be tenable\footnote +{See a paper by Sir William Thomson (Lord Kelvin\index{Kelvin}) in the +\textit{Proceedings of the Royal Society of Edinburgh}, Dec.~18, 1871, +vol.~\textsc{vii}, pp.~577--589.}. But +\PG----File: 388.png------------------------------------------------ +the results of Maxwell's\index{Maxwell, J. Clerk} numerical calculation show, +first, that the particles must be imperfectly elastic; second, that +merely to produce the effect of the attraction of the earth +on a mass of one pound would require that Le~Sage's\index +{LeSage@Le Sage on Gravity} corpuscles +should expend energy at the rate of at least billions\footnote +{I use billion with the English (and not the French) meaning, that +is, a billion $=10^{12}$.} +of foot-pounds per second; and third, that it is probable that +the effect of such a bombardment would be to raise the +temperature of all bodies beyond a point consistent with our +experience. Finally, it seems probable that the distance +between consecutive molecules would have to be considerably +greater than is compatible with the results given below. + +Tait\index{Tait} summed up the objections to these two hypotheses +by saying\footnote +{\textit{Properties of Matter}, London, 1885, art.~164.}, +``One common defect of these attempts is\textellipsis that % NB tight ellipsis matches original +they all demand some prime mover, working beyond the +limits of the visible universe or inside each atom: creating or +annihilating matter, giving additional speed to spent corpuscles, +or in some other way supplying the exhaustion +suffered in the production of gravitation. Another defect is +that they all make gravitation a mere difference-effect, as it +were; thereby implying the presence of stores of energy absolutely +gigantic in comparison with anything hitherto observed, +or even suspected to exist, in the universe; and therefore +demanding the most delicate adjustments, not merely to maintain +the conservation of energy which we observe, but to +prevent the whole solar and stellar systems from being instantaneously +scattered in fragments through space. In fact, +the cause of gravitation remains undiscovered.'' + +(3)\quad There is another conjecture on the cause of gravity +which I may mention\footnote +{See an article by myself in the \textit{Messenger of Mathematics}, +Cambridge, 1891, vol.~\textsc{xxi}, pp.~20--24.}. +It is possible that the attraction of +one particle on another might be explained if both of them +rested on a homogeneous elastic body capable of transmitting +\PG----File: 389.png------------------------------------------------ +energy. This is the case if our three-dimensional universe rests +in the direction of a fourth dimension on a four-dimensional +homogeneous elastic body (which we may call the ether) whose +thickness in the fourth dimension is small and constant. + +The results of spectrum analysis lead us to suppose that +every molecule of matter in our universe is in constant vibration. +On the above hypothesis these vibrations would cause a +disturbance in the supporting space, \IE\ in the ether. This +disturbance would spread out uniformly in all directions; the +intensity diminishing as the square of the distance from the +centre of vibration, but the rate of vibration remaining unaltered. +The transmission of light and radiant heat may be +explained by such vibrations transversal to the direction of +propagation. It is possible that gravity may be caused by +vibrations in the supporting space which are wholly longitudinal +or are compounded of vibrations which are partly longitudinal +and partly transversal in any of the three directions at +right angles to the direction of propagation. If we define the +mass of a molecule as proportional to the intensity of these +vibrations caused by it, then at any other point in space the +intensity of the vibration there would vary as the mass of the +molecule and inversely as the square of the distance from the +molecule; hence, if we may assume that such vibrations of the +medium spreading out from any centre would draw to that +centre a particle of unit mass at any other point with a force +proportional to the intensity of the vibration there, then the +Newtonian law of attraction would follow. This conjecture +is consistent either with Boscovich's hypothesis\index{Boscovich on Matter} +or with the vortex theory. It would be interesting if the results of a +branch of pure mathematics so abstract as the theory of hyperspace +should be found to be closely connected with one of the +most fundamental problems of material science% +\index{Attraction, Law of|)}% +\index{Gravity@\textsc{Gravity}, Hypotheses on|)}. + +I should sum up the effect of this discussion on gravity +on the relative probabilities of the hypotheses as to the constitution +of matter enumerated above, by saying that it does +not enable us to discriminate between them. +\PG----File: 390.png------------------------------------------------ + +\ThoughtBreakSpace +\phantomsection +\addcontentsline{toc}{section}{Conjectures to explain the finite number +of species of Atoms}\markright{Finite number of kinds of Matter} +The fact that the number of kinds of matter\index +{Matter, Kind@\nobreak--- Kinds of, limited|(} (chemical +elements) is finite and the consequences of spectrum analysis +are closely related and may be treated together. The results of +spectrum analysis show that every molecule of any species of +matter, such as hydrogen, vibrates with (so far as we can tell) +exactly equal sets of periods of vibration. This then is one +of the characteristics of the particular kind of matter, and it +is probable that any explanation of why the molecules of each +kind have a definite set of periods of vibration will account +also for the fact that the number of kinds of matter is finite. + +Various attempts to explain why the molecules of matter +are capable only of certain definite periods of vibration have +been made, and it may be interesting if I give them +briefly. + +(1)\quad To begin with, I may note the conjecture that it +depends on properties of time. This, however, is impossible, for +the continuity of certain spectra proves that in these cases there +is nothing which prevents the period of vibration from taking +any one of millions of different values: thus no explanation +dependent on the nature of time is permissible. + +(2)\quad It has been suggested that there may have been a +sorting agency, and only selected specimens of the infinite +number of species formed originally have got into our universe. +The objection to this is that no explanation is offered as to +what has become of the excluded molecules. + +(3)\quad The finite number\label{page:373} of species might be explained by +supposing a physical connection to exist between all the molecules +in the universe, just as two clocks whose rates are nearly +the same tend to go at the same rate if their cases are connected. + +Maxwell's\index{Maxwell, J. Clerk} objection to this is that we have no +other reason for supposing that such a connection exists, but if we are +living in a space of four dimensions as suggested above in +\hyperlink{chapter.12}{chapter~\textsc{xii}}, this connection does exist, +for all the molecules rest on one and the same body. This body is capable of +transmitting vibrations, hence, no matter how the molecules were +set vibrating originally, they would fall into certain groups, +\PG----File: 391.png------------------------------------------------ +and all the members of each group would vibrate at the same +rate. It was the possibility of obtaining thus a physical +connection between the various particles in our universe that +first suggested to me the idea of a supporting medium in a +fourth dimension. + +(4)\quad If we accept Boscovich's hypothesis\index{Boscovich on Matter} or +that of an elastic solid ether, and if we may lay it down as axiomatic +that the mass of every sub-atom is the same, we may conceive +that the number of ways of combining the sub-atoms into a +permanent system is limited, and that the period of vibration +depends on the form in which the sub-atoms are combined +into an atom. This view is not inconsistent with any known +facts. I may add that it is probable that the chemical atom +is the essential vibrating system, for the sodium spectrum, to +take one instance, is the same as that of all its compounds. + +(5)\quad In the same way we may suppose that the vortex +rings are formed so that they can have only a definite number +of stable forms produced by interlinking or kinking. + +(6)\quad Similarly we may modify the popular hypothesis by +treating the atoms as indivisible aggregates of sub-atoms which +are in all respects equal and similar, and can be combined in +only a limited number of forms which are permanent. But +most of the old difficulties connected with the atoms arise +again in connection with the sub-atoms. + +(7)\quad I am not aware that Maxwell\index{Maxwell, J. Clerk} discussed any +other hypotheses in connection with this point, but it has been +suggested recently that, if the various forms of matter were +evolved originally out of some one primitive material, then +there may have been periodic disturbances in this matter when +the atoms were being formed, such that they were produced +only at some definite phase in the period\footnote +{See Crookes\index{Crookes@Crookes on Mendeléeff's Laws} on +Mendeléeff's\index{Mendel@Mendeléeff} periodic law, \textit{Nature}, +Sept.~2, 1886, vol.~\textsc{xxxiv}, pp.~423--432.}. + +Thus, if the disturbance is represented by the swinging of +a pendulum in a resisting medium, it might be supposed that +\PG----File: 392.png------------------------------------------------ +the atoms were formed at the points of maximum amplitude, +and we should expect that the atoms successively thrown off +would form a series having the properties of its successive +members connected by a regular periodic law. This conjecture, +when worked out in some detail, led to the conclusion +that some elements which ought to have appeared in the series +were missing, but it was possible to predict their properties +and to suggest the substances with which they were most +likely to be found in combination. Guided by these theoretical +conclusions a careful chemical analysis revealed the fact that +such elements did exist. + +That this hypothesis has led to new discoveries is something +in its favour, but I do not wish to be understood to say +that it is a theory which leads to results that have been verified +subsequently. I should say rather that we have obtained an +analogy which is sufficiently like the truth to suggest new +discoveries. Such analogies are often the precursors of laws, +so that it is not unreasonable to hope that ere long our knowledge +of this border-land of chemistry and physics may be +more definite, and thus that molecular physics may be brought +within the domain of mathematics. It is however very remarkable +that J.J.~Thomson's\index{Thomson, J.J.} conclusions on the stability of +the orbital systems he devised should agree so closely with +Mendeléeff's\index{Mendel@Mendeléeff} periodic law. + +On the whole Maxwell thought that the phenomena point +to a common origin of all molecules of the same kind, that +this was an event not belonging to that order of nature under +which we live, but must have originated when or before the +existing order was established, and that so long as the present +order exists it is immutable. + +This is equivalent to saying that we have arrived at a +point beyond which our limited experience does not enable +us to carry the explanation\index{Matter, Kind@\nobreak--- Kinds of, limited|)}. + +\ThoughtBreakSpace +\phantomsection +\markright{The Size of Molecules of Matter.} +\addcontentsline{toc}{section}{Size of the molecules of bodies} +That we should be able to form an approximate idea of\index +{Molecules@\textsc{Molecules, Size of}|(}% +\index{Matter, Size@\nobreak--- Size of Molecules|(} +the size of the molecules of matter is a testimony to the +\PG----File: 393.png------------------------------------------------ +extraordinary advance which mathematical physics has made +recently. + +Sir William Thomson, now Lord Kelvin\index{Kelvin}, whose ingenuity +seems to know no limits---has suggested\footnote +{See \textit{Nature}, March~31, 1870, vol.~\textsc{i}, pp.~531--553; and +Tait's\index{Tait} \textit{Recent +Advances}, pp.~303--318. The fourth method had been proposed by +Loschmidt\index{Loschmidt on Molecules} in 1863.} four distinct methods +of attacking the problem. They lead to results which are not +very different. + +The first of these rests on an assertion of Cauchy\index{Cauchy} that +the\index{Atoms, Size of} +phenomena of prismatic colours show that the distance between +consecutive molecules of matter is comparable with the +wave-lengths of light. Taking the most unfavourable case +this would seem to indicate that in a transparent homogeneous +solid or liquid medium there are not more than $64 \times 10^{24}$ +molecules in a cubic inch, that is, that the distance between +consecutive molecules is greater than $1/(4 \times 10^8)$th of an +inch. + +The second method is founded on the amount of work +required to draw out a film of liquid, such as a soap-bubble, +to a given thickness. This can be calculated from experiments +in a capillary tube, and it is found that, if a soap-bubble could +be drawn out to a thickness of $1/10^8$th of an inch there +would be but a few molecules in its thickness. This method +is not quantitative. + +Thirdly, Sir William Thomson proved that the contact +phenomena of electricity require that in an alloy of brass +the distance between two molecules, one of zinc and one of +copper, shall be greater than $1/(7 \times 10^8)$th of an inch; hence +the number of molecules in a cubic inch of zinc or copper is +not greater than $35 \times 10^{25}$. + +Lastly, the kinetic theory of gases\index{Gases, Theory of|(}% +\index{Kinetic Theory of Gases|(} +leads to the conclusion +that certain phenomena of temperature and viscosity depend, +\emph{inter alia}, on inter-molecular collisions, and so on the sizes +and velocities of the molecules, while the average velocity +\PG----File: 394.png------------------------------------------------------ +with which the molecules move increases with the temperature. +This leads to the conclusion that the distance +between two consecutive molecules of a gas at normal pressure +and temperature is greater than $1/(6 \times 10^6)$th of an inch, +and is less than $1/10^7$th of an inch; while the actual size +of the molecule is a trifle greater than $1/(3 \times 10^{20})$th of a +cubic inch; and the number of molecules in a cubic inch is +about $3 \times 10^{20}$\index{Gases, Theory of|)}% +\index{Kinetic Theory of Gases|)}. + +Thus it would seem that a cubic inch of gas at ordinary +pressure and temperature contains about $3 \times 10^{20}$ molecules, +all similar and equal, and each molecule has a volume of about +$1/(3 \times 10^{25})$th of a cubic inch; while a cubic inch of the simplest +solid or liquid contains rather less than $10^{27}$ molecules, and +perhaps each molecule has a volume of about $1/(3 \times 10^{26})$th of +a cubic inch. For instance, if a pea or a drop of water whose +radius is $1/16$th inch was magnified to the size of the earth, +then there would be about thirty molecules in every cubic foot +of it, and probably the size of a molecule would be about the +same as that of a fives-ball. The average size of the minute +drops of water in a very light cloud can be calculated from +the coloured rings produced when the sun or moon shines +through it. The radius of a drop is about $1/30000$th of an +inch. Such a drop therefore would contain about $2 \times 10^{13}$ +separate molecules. In gases and vapours, the number of +atoms required to make up one of these molecules can be +estimated, but in liquids the number is not as yet known. + +Loschmidt\index{Loschmidt on Molecules} asserted that a cube whose side is +$1/4000$th of a millimetre is the smallest object which can be made visible +at the present time. Such a cube of oxygen or nitrogen +would contain from 60 to 100 millions of molecules of the +gas. Also on an average about 50 elementary molecules of +the so-called elements are required to constitute one molecule +of organic matter. At least half of every living organism +consists of water, and we may for the moment suppose that +the remainder consists of organic matter. Hence the smallest +living being which is visible under the microscope contains +\PG----File: 395.png------------------------------------------------------ +from 30 to 50 millions of elementary molecules which are +combined in the form of water, and from 30 to 50 millions +of elementary molecules which are combined so as to make +not more than one million organic molecules. + +Hence a very simple organism might be built up out of as +few as a million similar organic molecules. Maxwell\index +{Maxwell, J. Clerk} did not +consider that this was sufficient to justify the current conclusions +of physiologists, and said that they must not suppose +that structural details of infinitely small dimensions can furnish +by themselves an explanation of the variety known to exist +in the properties and functions of the most minute organisms; +but physiologists have replied that whether their conjectures +be right or wrong Maxwell's argument is vitiated by his non-consideration +of differences due to the physical (as opposed to +the chemical) structure of the organism and the consequent +motions of the component parts\index +{Molecules@\textsc{Molecules, Size of}|)}% +\index{Matter, Size@\nobreak--- Size of Molecules|)}. + +\PGx---File: 396.png------------------------------------------------ +\PGx---File: 397.png---------------------------------------------------- +\PGx---File: 398.png---------------------------------------------------- +\PGx---File: 399.png------------------------------------------------------ +\PGx---File: 400.png------------------------------------------------------ +\PGx---File: 401.png------------------------------------------------------ +\PGx---File: 402.png------------------------------------------------------ +\PGx---File: 403.png----------------------------------------------------- +\PGx---File: 404.png----------------------------------------------------- +\PGx---File: 405.png----------------------------------------------------- +\PGx---File: 406.png----------------------------------------------------- + +\index{Clerk Maxwell|see {Maxwell}} +\index{Durations|see {Time}} +\index{Meziriac@Méziriac|see {Bachet}} +\index{Morgan, A. De|see {De Morgan}} +\index{P@$\pi$|see {table of contents}} +\index{Smith, RC@\nobreak--- R.C.|see {Raphael}} +\index{Thomson, Sir Wm.|see {Kelvin}} + +\PrintTheIndex + +\Printer{CAMBRIDGE: PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS.} +% note implicit page throw in previous macro + +\pagestyle{adverts} +\begingroup +\setlength\parskip{1ex plus2pt minus1pt} +% because we don't want the attributions of the quotes as widows +\widowpenalty=40000 +\phantomsection +\DPaddcontentsline{toc}{chapter}{\protect\tocsecbox{\protect\textsc +{Notices of some works---chiefly historico-mathematical}}}% +\begin{center} +\large\textbf{A SHORT ACCOUNT OF THE} \\[1em] +\LARGE\textbf{HISTORY OF MATHEMATICS} \\[1em] +\normalsize \textsc{By W.W.~ROUSE BALL}.\\[1.5em] +\large[\textit{Third Edition.\qquad Pp.} xxiv + 527.\qquad + \textit{Price $10$s.\ net.}]\\[1em] +\small\textsc{MACMILLAN AND CO. Ltd., LONDON AND NEW YORK.} +\end{center} +\bigskip +\hrule +\bigskip + +\noindent\textsc{This} book gives an account of the lives and discoveries of +those mathematicians to whom the development of the subject is mainly +due. The use of technicalities has been avoided and the work is +intelligible to any one acquainted with the elements of mathematics. + +The author commences with an account of the origin and progress +of Greek mathematics, from which the Alexandrian, the Indian, +and the Arab schools may be said to have arisen. Next the +mathematics of medieval Europe and the renaissance are described. +The latter part of the book is devoted to the history of modern +mathematics (beginning with the invention of analytical geometry +and the infinitesimal calculus), the account of which is brought +down to the present time. + +\bigskip +\hrule +\bigskip +{\Small +This excellent summary of the history of mathematics supplies a want +which has long been felt in this country. The extremely difficult question, +how far such a work should be technical, has been solved with great +tact\textellipsis. +The work contains many valuable hints, and is thoroughly readable. The +biographies, which include those of most of the men who played important +parts in the development of culture, are full and general enough to interest +the ordinary reader as well as the specialist. Its value to the latter is +much increased by the numerous references to authorities, a good table of +contents, and a full and accurate index.---\textit{The Saturday Review.} + +Mr.~Ball's book should meet with a hearty welcome, for though we possess +other histories of special branches of mathematics, this is the first serious +attempt that has been made in the English language to give a systematic +account of the origin and development of the science as a whole. It is +\PG----File: 407.png----------------------------------------------------- +written too in an attractive style. Technicalities are not too numerous or +obtrusive, and the work is interspersed with biographical sketches and +anecdotes likely to interest the general reader. Thus the tyro and the +advanced mathematician alike may read it with pleasure and +profit.---\textit{The Athenæum.} + +A wealth of authorities, often far from accordant with each other, renders +a work such as this extremely formidable; and students of mathematics have +reason to be grateful for the vast amount of information which has been +condensed into this short account\textellipsis. In a survey of so wide extent +it is of course impossible to give anything but a bare sketch of the various +lines of research, and this circumstance tends to render a narrative scrappy. +It says much for Mr.~Ball's descriptive skill that his history reads more +like a continuous story than a series of merely consecutive +summaries.---\textit{The Academy}. + +We can heartily recommend to our mathematical readers, and to others +also, Mr.~Ball's \textit{History of Mathematics}. The history of what might +be supposed a dry subject is told in the pleasantest and most readable style, +and at the same time there is evidence of the most careful +research.---\textit{The Observatory}. + +All the salient points of mathematical history are given, and many of the +results of recent antiquarian research; but it must not be imagined that the +book is at all dry. On the contrary the biographical sketches frequently +contain amusing anecdotes, and many of the theorems mentioned are very +clearly explained so as to bring them within the grasp of those who are only +acquainted with elementary mathematics.---\textit{Nature}. + +Le style de M.~Ball est clair et élégant, de nombreux aperçus rendent +facile de suivre le fil de son exposition et de fréquentes citations +permettent à celui qui le désire d'approfondir les recherches que +l'auteur n'a pu qu'effleurer\textellipsis. +Cet ouvrage pourra devenir très utile comme manuel d'histoire +des mathématiques pour les étudiants, et il ne sera pas déplacé dans +les bibliotheques des savants.---\textit{Bibliotheca Mathematica}. + +The author modestly describes his work as a compilation, but it is +thoroughly well digested, a due proportion is observed between the various +parts, and when occasion demands he does not hesitate to give an independent +judgment on a disputed point. His verdicts in such instances appear to +us to be generally sound and reasonable\textellipsis. To many readers who +have not the courage or the opportunity to tackle the ponderous volumes of +Montucla or the (mostly) ponderous treatises of German writers on special +periods, it may be somewhat of a surprise to find what a wealth of human +interest attaches to the history of so ``dry'' a subject as mathematics. We +are brought into contact with many remarkable men, some of whom have +played a great part in other fields, as the names of Gerbert, Wren, Leibnitz, +Descartes, Pascal, D'Alembert, Carnot, among others may testify, and with +at least one thorough blackguard (Cardan); and Mr.~Ball's pages abound +with quaint and amusing touches characteristic of the authors under +consideration, or of the times in which they lived.---\textit +{Manchester Guardian}. + +There can be no doubt that the author has done his work in a very excellent +way\textellipsis. There is no one interested in almost any part of +mathematical science who will not welcome such an exposition as the present, +at once popularly written and exact, embracing the entire +subject\textellipsis. Mr.~Ball's work is destined to become a standard one +on the subject.---\textit{The Glasgow Herald}. + +A most interesting book, not only for those who are mathematicians, but +for the much larger circle of those who care to trace the course of general +scientific progress. It is written in such a way that those who have only an +elementary acquaintance with the subject can find on almost every page +something of general interest.---\textit{The Oxford Magazine}. + +} +\PG----File: 408.png----------------------------------------------------- +\clearpage +\begin{center} +\large\textbf{A PRIMER OF THE} \\[1em] +\LARGE\textbf{HISTORY OF MATHEMATICS} \\[1em] +\normalsize \textsc{By W.W.~ROUSE BALL}.\\[1.5em] +\large[\textit{Second Edition.\qquad Pp.} iv + 148.\qquad + \textit{Price $2$s.\ net.}]\\[1em] +\small\textsc{MACMILLAN AND CO. Ltd., LONDON AND NEW YORK.} +\end{center} +\par +\bigskip +\hrule +\bigskip + +\noindent\textsc{This} book contains a sketch in popular language of the +history of mathematics; it includes some notice of the lives and surroundings +of those to whom the development of the subject is mainly +due as well as of their discoveries. + +\bigskip +\hrule +\bigskip +{\Small +This Primer is written in the agreeable style with which the author has +made us acquainted in his previous essays; and we are sure that all readers +of it will be ready to say that Mr.~Ball has succeeded in the hope he has +formed, that ``it may not be uninteresting'' even to those who are +unacquainted with the leading facts. It is just the book to give an +intelligent young student, and should allure him on to the perusal of +Mr.~Ball's ``Short Account.'' The present work is not a mere +\emph{réchauffé} of that, though +naturally most of what is here given will be found in equivalent form in the +larger work\textellipsis. The choice of material appears to us to be such as +should lend interest to the study of mathematics and increase its educational +value, which has been the author's aim. The book goes well into the pocket, +and is excellently printed.---\textit{The Academy}. + +We have here a new instance of Mr.~Rouse Ball's skill in giving in a small +space an intelligible account of a large subject. In 137 pages we have a +sketch of the progress of mathematics from the earliest records up to the +middle of this century, and yet it is interesting to read and by no means a +mere catalogue.---\textit{The Manchester Guardian}. + +It is not often that a reviewer of mathematical works can confess that he +has read one of them through from cover to cover without abatement of +interest or fatigue. But that is true of Mr.~Rouse Ball's wonderfully +entertaining little ``History of Mathematics,'' which we heartily recommend +to even the quite rudimentary mathematician. The capable mathematical +master will not fail to find a dozen interesting facts therein to season his +teaching.---\textit{The Saturday Review}. + +A fascinating little volume, which should be in the hands of all who do +not possess the more elaborate \textit{History of Mathematics} by the same +author.---\textit{The Mathematical Gazette}. + +This excellent sketch should be in the hands of every student, whether he +is studying mathematics or no. In most cases there is an unfortunate lack +\PG----File: 409.png----------------------------------------------------- +of knowledge upon this subject, and we welcome anything that will help to +supply the deficiency. The primer is written in a concise, lucid and easy +manner, and gives the reader a general idea of the progress of mathematics +that is both interesting and instructive.---\textit{The Cambridge Review}. + +Mr.~Ball has not been deterred by the existence and success of his larger +``History of Mathematics'' from publishing a simple compendium in about a +quarter of the space\textellipsis. Of course, what he now gives is a bare +outline of the subject, but it is ample for all except the most advanced +proficients. There is no question that, as the author says, a knowledge of +the history of a science lends interest to its study, and often increases +its educational value. +We can imagine no better cathartic for any mathematical student who has +made some way with the calculus than a careful perusal of this little +book.---\textit{The Educational Times}. + +The author has done good service to mathematicians by engaging in work +in this special field\textellipsis. The Primer gives, in a brief compass, +the history of +the advance of this branch of science when under Greek influence, during the +Middle Ages, and at the Renaissance, and then goes on to deal with the +introduction of modern analysis and its recent developments. It refers to the +life and work of the leaders of mathematical thought, adds a new and +enlarged value to well-known problems by treating of their inception and +history, and lights up with a warm and personal interest a science which +some of its detractors have dared to call dull and cold.---\textit +{The Educational Review}. + +It is not too much to say that this little work should be in the possession +of every mathematical teacher\textellipsis. The Primer gives in a small +compass the leading events in the development of mathematics\textellipsis. +At the same time, it is no dry chronicle of facts and theorems. +The biographical sketches of the +great workers, if short, are pithy, and often amusing. Well-known +propositions will attain a new interest for the pupil as he traces their +history long before the time of Euclid.---\textit{The Journal of Education}. + +This is a work which all who apprehend the value of ``mathematics'' +should read and study\textellipsis, and those who wish to learn how to think +will find advantage in reading it.---\textit{The English Mechanic}. + +The subject, so far as our own language is concerned, is almost Mr.~Ball's +own, and those who have no leisure to read his former work will find in this +Primer a highly-readable and instructive chapter in the history of education. +The condensation has been skilfully done, the reader's interest being +sustained by the introduction of a good deal of far from tedious +detail.---\textit{The Glasgow Herald}. + +Mr.~W.W.~Rouse Ball is well known as the author of a very clever history +of mathematics, besides useful works on kindred subjects. His latest +production is \textit{A Primer of the History of Mathematics}, a book of one +hundred and forty pages, giving in non-technical language a full, concise, +and readable narrative of the development of the science from the days of the +Ionian Greeks until the present time. Anyone with a leaning towards algebraic +or geometrical studies will be intensely interested in this account of +progress from primitive usages, step by step, to our present elaborate +systems. The lives of the men who by their research and discovery helped +along the good work are described briefly, but graphically\textellipsis. +The Primer should become a standard text-book.---\textit{The Literary World}. + +This is a capital little sketch of a subject on which Mr.~Ball is an +acknowledged authority, and of which too little is generally known. Mr.~Ball, +moreover, writes easily and well, and has the art of saying what he has to +say in an interesting style.---\textit{The School Guardian}. + +} +\PG----File: 410.png----------------------------------------------------- +\clearpage +\begin{center} +\large\textbf{MATHEMATICAL} \\[1em] +\LARGE\textbf{RECREATIONS AND ESSAYS} \\[1em] +\normalsize \textsc{By W.W.~ROUSE BALL}.\\[1.5em] +\large[\textit{Fourth Edition.\qquad Pp.} xvi + 402.\qquad + \textit{Price $7$s.\ net.}]\\[1em] +\small\textsc{MACMILLAN AND CO. Ltd., LONDON AND NEW YORK.} +\end{center} +\par +\bigskip +\hrule + +\noindent\textsc{This} work is divided into two parts; the first is on +mathematical recreations and puzzles, the second includes some miscellaneous +essays and an account of some problems of historical interest. In +both parts questions which involve advanced mathematics are +excluded. + +The mathematical recreations include numerous elementary +questions and paradoxes, as well as problems such as the proposition +that to colour a map not more than four colours are necessary, +the explanation of the effect of a cut on a tennis ball, the fifteen +puzzle, the eight queens problem, the fifteen school-girls, the construction +of magic squares, the theory and history of mazes, and +the knight's path on a chess-board. + +The second part commences with sketches of the history of the +Mathematical Tripos at Cambridge, of the three famous classical +problems in geometry (namely, the duplication of the cube, the +trisection of an angle, and the quadrature of the circle) and of +Mersenne's Numbers. These are followed by essays on Astrology +and Ciphers. The last three chapters are devoted to an account of +the hypotheses as to the nature of Space and Mass, and the means +of measuring Time. + +\bigskip +\hrule +\bigskip +{\Small +Mr.~Ball has already attained a position in the front rank of writers on +subjects connected with the history of mathematics, and this brochure will +add another to his successes in this field. In it he has collected a mass of +information bearing upon matters of more general interest, written in a style +which is eminently readable, and at the same time exact. He has done his +work so thoroughly that he has left few ears for other gleaners. The nature +of the work is completely indicated to the mathematical student by its title. +Does he want to revive his acquaintance with the \textit{Problèmes Plaisans +et Délectables} of Bachet, or the \textit{Récréations Mathématiques +et Physiques} of +\PG----File: 411.png----------------------------------------------------- +Ozanam? Let him take Mr.~Ball for his companion, and he will have the +cream of these works put before him with a wealth of illustration quite +delightful. Or, coming to more recent times, he will have full and accurate +discussion of `the fifteen puzzle,' `Chinese rings,' `the fifteen school-girls +problem' \emph{et id genus omne}. Sufficient space is devoted to accounts of +magic squares and unicursal problems (such as mazes, the knight's path, and +geometrical trees). These, and many other problems of equal interest, come +under the head of `Recreations.' The problems and speculations include an +account of the Three Classical Problems; there is also a brief sketch of +Astrology; and interesting outlines of the present state of our knowledge of +hyper-space and of the constitution of matter. This enumeration badly +indicates the matter handled, but it sufficiently states what the reader may +expect to find. Moreover for the use of readers who may wish to pursue the +several heads further, Mr.~Ball gives detailed references to the sources from +whence he has derived his information. These \textit{Mathematical Recreations} +we can commend as suited for mathematicians and equally for others who wish +to while away an occasional hour.---\textit{The Academy}. + +The idea of writing some such account as that before us must have been +present to Mr.~Ball's mind when he was collecting the material which he has +so skilfully worked up into his \textit{History of Mathematics}. We think +this because \textellipsis\ many bits of ore which would not suit the earlier +work find a fitting niche in this. Howsoever the case may be, we are sure +that non-mathematical, as well as mathematical, readers will derive amusement, +and, we venture to think, profit withal, from a perusal of it. The author has +gone very exhaustively over the ground, and has left us little opportunity of +adding to or correcting what he has thus reproduced from his note-books. The +work before us is divided into two parts: mathematical recreations and +mathematical problems and speculations. All these matters are treated +lucidly, and with sufficient detail for the ordinary reader, and for others +there is ample store of references\textellipsis. Our analysis shows how great +an extent of ground is covered, and the account is fully pervaded by the +attractive charm Mr.~Ball knows so well how to infuse into what many persons +would look upon as a dry subject.---\textit{Nature}. + +A fit sequel to its author's valuable and interesting works on the history of +mathematics. There is a fascination about this volume which results from a +happy combination of puzzle and paradox. There is both milk for babes and +strong meat for grown men\textellipsis. A great deal of the information is +hardly accessible in any English books; and Mr.~Ball would deserve the +gratitude of mathematicians for having merely collected the facts. But he has +presented them with such lucidity and vivacity of style that there is not a +dull page in the book; and he has added minute and full bibliographical +references which greatly enhance the value of his work.---\textit +{The Cambridge Review}. + +Mathematicians with a turn for the paradoxes and puzzles connected with +number, space, and time, in which their science abounds, will delight in +\textit{Mathematical Recreations and Problems of Past and Present +Times}.---\textit{The Times}. + +Mathematicians have their recreations; and Mr.~Ball sets forth the +humours of mathematics in a book of deepest interest to the clerical +reader, and of no little attractiveness to the layman. The notes attest an +enormous amount of research.---\textit{The National Observer}. + +Mr.~Ball, to whom we are already indebted for two excellent Histories of +Mathematics, has just produced a book which will be thoroughly appreciated +by those who enjoy the setting of the wits to work\textellipsis. He has +collected a vast amount of information about mathematical quips, tricks, +cranks, and puzzles---old and new; and it will be strange if even the most +learned do not find something fresh in the assortment.---\textit +{The Observatory}. + +\PG----File: 412.png----------------------------------------------------- +Mr.~Rouse Ball has the true gift of story-telling, and he writes so pleasantly +that though we enjoy the fulness of his knowledge we are tempted to forget +the considerable amount of labour involved in the preparation of his book. +He gives us the history and the mathematics of many problems \textellipsis\ and +where the limits of his work prevent him from dealing fully with the points +raised, like a true worker he gives us ample references to original +memoirs\textellipsis. The book is warmly to be recommended, and should find a +place on the shelves of every one interested in mathematics and on those of +every public library.---\textit{The Manchester Guardian}. + +A work which will interest all who delight in mathematics and mental +exercises generally. The student will often take it up, as it contains many +problems which puzzle even clever people.---\textit{The English Mechanic and +World of Science}. + +This is a book which the general reader should find as interesting as the +mathematician. At all events, an intelligent enjoyment of its contents +presupposes no more knowledge of mathematics than is now-a-days possessed by +almost everybody.---\textit{The Athenæum}. + +An exceedingly interesting work which, while appealing more directly to +those who are somewhat mathematically inclined, it is at the same time +calculated to interest the general reader\textellipsis. Mr.~Ball writes in a +highly interesting manner on a fascinating subject, the result being a work +which is in every respect excellent.---\textit{The Mechanical World}. + +É um livro muito interessante, consagrado a recreios mathematicos, alguns +dos quaes s\^ao muito bellos, e a problemas interessantes da mesma sciencia, +que n\^ao exige para ser lido grandes conhecimentos mathematicos e que tem +em gráo elevado a qualidade de instruir, deleitando ao mesmo +tempo.---\textit{Journal de sciencias mathematicas, Coimbra}. + +The work is a very judicious and suggestive compilation, not meant mainly +for mathematicians, yet made doubly valuable to them by copious references. +The style in the main is so compact and clear that what is central in a long +argument or process is admirably presented in a few words. One great merit +of this, or any other really good book on such a subject, is its +suggestiveness; and in running through its pages, one is pretty sure to think +of additional problems on the same general lines.---\textit +{Bulletin of the New York Mathematical Society}. + +A book which deserves to be widely known by those who are fond of solving +puzzles \textellipsis\ and will be found to contain an admirable classified +collection of ingenious questions capable of mathematical analysis. As the +author is himself a skilful mathematician, and is careful to add an analysis +of most of the propositions, it may easily be believed that there is food for +study as well as amusement in his pages\textellipsis. Is in every way worthy +of praise.---\textit{The School Guardian}. + +Once more the author of a \textit{Short History of Mathematics} and a \textit +{History of the Study of Mathematics at Cambridge} gives evidence of the +width of his reading and of his skill in compilation. From the elementary +arithmetical puzzles which were known in the sixteenth and seventeenth +centuries to those modern ones the mathematical discussion of which has taxed +the energies of the ablest investigator, very few questions have been left +unrepresented. The sources of the author's information are indicated with +great fulness\textellipsis. The book is a welcome addition to English +mathematical literature.---\textit{The Oxford Magazine}. + +} +\PG----File: 413.png----------------------------------------------------- +\clearpage +\begin{center} +\large\textbf{A HISTORY OF THE STUDY OF} \\[1em] +\LARGE\textbf{MATHEMATICS AT CAMBRIDGE} \\[1em] +\normalsize \textsc{By W.W.~ROUSE BALL}.\\[1.5em] +\large[\textit{Pp.} xvi + 264.\qquad \textit{Price $6$s.}]\\[1em] +\small\textsc{THE UNIVERSITY PRESS, CAMBRIDGE.} +\end{center} +\par +\bigskip +\hrule +\bigskip + +\noindent\textsc{This} work contains an account of the development of the +study of mathematics in the university of Cambridge from the twelfth century +to the middle of the nineteenth century, and a description of +the means by which proficiency in that study was tested at various +times. + +The first part of the book is devoted to a brief account of the +more eminent of the Cambridge mathematicians, the subject matter +of their works, and their methods of exposition. The second part +treats of the manner in which mathematics was taught, and of the +exercises and examinations required of students in past times. A +sketch is given of the origin and history of the Mathematical +Tripos; this includes the substance of the earlier parts of the +author's work on that subject, Cambridge, 1880. To explain the +relation of mathematics to other departments of study an outline +of the general history of the university and the organization of +education therein is added. + +\bigskip +\hrule +\bigskip +{\Small +The present volume is very pleasant reading, and though much of it necessarily +appeals only to mathematicians, there are parts---\eg\ the chapters on +Newton, on the growth of the tripos, and on the history of the +university---which are full of interest for a general reader\textellipsis. +The book is well written, the style is crisp and clear, and there is a +humorous appreciation of some of the curious old regulations which have been +superseded by time and change of custom. Though it seems light, it must +represent an extensive study and investigation on the part of the author, the +essential results of which are skilfully given. We can most thoroughly +commend Mr.~Ball's volume to all readers who are interested in mathematics or +in the growth and the position of the Cambridge school of +mathematicians.---\textit{The Manchester Guardian.} + +\PG----File: 414.png----------------------------------------------------- +Voici un livre dont la lecture inspire tout d'abord le regret que des travaux +analogues n'aient pas été faits pour toutes les Écoles célèbres, et +avec autant de soin et de clarté\textellipsis. Toutes les parties du livre +nous out vivement intéressé.---\textit +{Bulletin des sciences mathématiques.} + +A book of pleasant and useful reading for both historians and mathematicians. +Mr.~Ball's previous researches into this kind of history have already +established his reputation, and the book is worthy of the reputation of its +author. It is more than a detailed account of the rise and progress of +mathematics, for it involves a very exact history of the University of +Cambridge from its foundation.---\textit{The Educational Times.} + +Mr.~Ball is far from confining his narrative to the particular science of +which he is himself an acknowledged master, and his account of the study of +mathematics becomes a series of biographical portraits of eminent professors +and a record not only of the intellectual life of the \emph{élite} but of +the manners, habits, and discussions of the great body of Cambridge men from +the sixteenth century to our own\textellipsis. He has shown how the University +has justified its liberal reputation, and how amply prepared it was for the +larger freedom which it now enjoys.---\textit{The Daily News.} + +Mr.~Ball has not only given us a detailed account of the rise and progress of +the science with which the name of Cambridge is generally associated but has +also written a brief but reliable and interesting history of the university +itself from its foundation down to recent times\textellipsis. The book is +pleasant reading alike for the mathematician and the student of +history.---\textit{St.~James's Gazette.} + +A very handy and valuable book containing, as it does, a vast deal of +interesting information which could not without inconceivable trouble be +found elsewhere\textellipsis. It is very far from forming merely a +mathematical biographical dictionary, the growth of mathematical science +being skilfully traced in connection with the successive names. There are +probably very few people who will be able thoroughly to appreciate the +author's laborious researches in all sorts of memoirs and transactions of +learned societies in order to unearth the material which he has so agreeably +condensed\textellipsis. Along with this there is much new matter which, while +of great interest to mathematicians, and more especially to men brought up at +Cambridge, will be found to throw a good deal of new and important light on +the history of education in general.---\textit{The Glasgow Herald.} + +Exceedingly interesting to all who care for mathematics\textellipsis. After +giving an account of the chief Cambridge Mathematicians and their works in +chronological order, Mr.~Rouse Ball goes on to deal with the history of +tuition and examinations in the University \textellipsis\ and recounts the steps +by which the word ``tripos'' changed its meaning ``from a thing of wood to a +man, from a man to a speech, from a speech to two sets of verses, from verses +to a sheet of coarse foolscap paper, from a paper to a list of names, and from +a list of names to a system of examination.''---Never did word undergo so many +alterations.---\textit{The Literary World.} + +In giving an account of the development of the study of mathematics in the +University of Cambridge, and the means by which mathematical proficiency +was tested in successive generations, Mr.~Ball has taken the novel plan of +devoting the first half of his book to \textellipsis\ the more eminent Cambridge +mathematicians, and of reserving to the second part an account of how at +various times the subject was taught, and how the result of its study was +tested\textellipsis. +Very interesting information is given about the work of the students during +the different periods, with specimens of problem-papers as far back as 1802. +The book is very enjoyable, and gives a capital and accurate digest of many +excellent authorities which are not within the reach of the ordinary +reader.---\textit{The Scots Observer.} + +} +\PG----File: 415.png------------------------------------------------------ +\clearpage +\begin{center} +\large\textbf{AN ESSAY ON} \\[1em] +\textbf{THE GENESIS, CONTENTS, AND HISTORY OF} \\[1em] +\LARGE\textbf{NEWTON'S ``PRINCIPIA''} \\[1em] +\normalsize \textsc{By W.W.~ROUSE BALL}.\\[1.5em] +\large[\textit{Pp.} x + 175.\qquad \textit{Price $6$s.\ net.}]\\[1em] +\small\textsc{MACMILLAN AND CO. Ltd., LONDON AND NEW YORK.} +\end{center} +\par +\bigskip +\hrule + +\noindent\textsc{This} work contains an account of the successive discoveries +of Newton on gravitation, the methods he used, and the history of his +researches. + +It commences with a review of the extant authorities dealing +with the subject. In the next two chapters the investigations +made in 1666 and 1679 are discussed, some of the documents dealing +therewith being here printed for the first time. The fourth +chapter is devoted to the investigations made in 1684: these are +illustrated by Newton's professorial lectures (of which the original +manuscript is extant) of that autumn, and are summed up in the +almost unknown memoir of February, 1685, which is here reproduced +from Newton's holograph copy. In the two following chapters +the details of the preparation from 1685 to 1687 of the +\textit{Principia} are described, and an analysis of the work is given. The +seventh chapter comprises an account of the researches of Newton +on gravitation subsequent to the publication of the first edition of +the \textit{Principia}, and a sketch of the history of that work. + +In the last chapter, the extant letters of 1678--1679 between +Hooke and Newton, and of those of 1686--1687 between Halley +and Newton, are reprinted, and there are also notes on the extant +correspondence concerning the production of the second and third +editions of the \textit{Principia}. + +\bigskip +\hrule +\bigskip +\PG----File: 416.png----------------------------------------------------- +{\Small +For the essay which we have before us, Mr~Ball should receive the thanks +of all those to whom the name of Newton recalls the memory of a great man. +The \textit{Principia}, besides being a lasting monument of Newton's life, is +also to-day the classic of our mathematical writings, and will be so for some +time to come\textellipsis. The value of the present work is also enhanced by +the fact that, besides containing a few as yet unpublished letters, there are +collected in its pages quotations from all documents, thus forming a complete +summary of everything that is known on the subject\textellipsis. The author +is so well known a writer on anything connected with the history of +mathematics, that we need make no mention of the thoroughness of the essay, +while it would be superfluous for us to add that from beginning to end it is +pleasantly written and delightful to read. Those well acquainted with the +\textit{Principia} will find much that will interest them, while those not so +fully enlightened will learn much by reading through the account of the +origin and history of Newton's greatest work.---\textit{Nature}. + +\textit{An Essay on Newton's Principia} will suggest to many something solely +mathematical, and therefore wholly uninteresting. No inference could be +more erroneous. The book certainly deals largely in scientific technicalities +which will interest experts only; but it also contains much historical +information which might attract many who, from laziness or inability, would be +very willing to take all its mathematics for granted. Mr.~Ball carefully +examines the evidence bearing on the development of Newton's great discovery, +and supplies the reader with abundant quotations from contemporary +authorities. Not the least interesting portion of the book is the appendix, or +rather appendices, containing copies of the original documents (mostly +letters) to which Mr.~Ball refers in his historical criticisms. Several of +these bear upon the irritating and unfounded claims of +Hooke.---\textit{The Athenæum.} + +La savante monographie de M.~Ball est rédigée avec beaucoup de soin, et +à plusieurs égards elle peut servir de modèle pour des écrits de la +m\^eme nature.---\textit{Bibliotheca Mathematica}. + +Newton's \textit{Principia} has world-wide fame as a classic of mathematical +science. But those who know thoroughly the contents and the history of the +book are a select company. It was at one time the purpose of Mr.~Ball to +prepare a new critical edition of the work, accompanied by a prefatory +history and notes, and by an analytical commentary. Mathematicians will +regret to hear that there is no prospect in the immediate future of seeing +this important book carried to completion by so competent a hand. They will +at the same time welcome Mr.~Ball's \textit{Essay on the Principia} for the +elucidations which it gives of the process by which Newton's great work +originated and took form, and also as an earnest of the completed +plan.---\textit{The Scotsman}. + +In this essay Mr.~Ball presents us with an account highly interesting to +mathematicians and natural philosophers of the origin and history of that +remarkable product of a great genius \textit{Philosophiae Naturalis Principia +Mathematica}, `The Mathematical Principles of Natural Philosophy,' better +known by the short term \textit{Principia}\textellipsis. Mr.~Ball's essay is +one of extreme interest to students of physical science, and it is sure to be +widely read and greatly appreciated.---\textit{The Glasgow Herald}. + +To his well-known and scholarly treatises on the \textit{History of +Mathematics} Mr.~W.W.~Rouse Ball has added \textit{An Essay on Newton's +Principia}. Newton's \textit{Principia}, as Mr.~Ball justly observes, is the +classic of English mathematical writings; and this sound, luminous, and +laborious essay ought +\PG----File: 417.png----------------------------------------------------- +to be the classical account of the \textit{Principia}. The essay is the +outcome of a critical edition of Newton's great work, which Mr.~Ball tells us +that he once contemplated. It is much to be hoped that he will carry out his +intention, for no English mathematician is likely to do the work better or in +a more reverent spirit\textellipsis. It is unnecessary to say that Mr.~Ball +has a complete knowledge of his subject. He writes with an ease and clearness +that are rare.---\textit{The Scottish Leader.} + +Le volume de M.~Rouse Ball renferme tout ce que l'on peut désirer savoir +sur l'histoire des \textit{Principes}; c'est d'ailleurs l'\oe{}uvre d'un +esprit clair, judicieux, et méthodique.---\textit +{Bulletin des Sciences Mathématiques.} + +Mr.~Ball has put into small space a very great deal of interesting matter, +and his book ought to meet with a wide circulation among lovers of Newton +and the \textit{Principia}.---\textit{The Academy.} + +Admirers of Mr.~W.W.~Rouse Ball's \textit{Short Account of the History of +Mathematics} will be glad to receive a detailed study of the history of the +\textit{Principia} from the same hand. This book, like its predecessors, +gives a very lucid account of its subject. We find in it an account of +Newton's investigations in his earlier years, which are to some extent +collected in the tract \textit{de Motu} (the germ of the \textit{Principia}) +the text of which Mr.~Rouse Ball gives us in full. In a later chapter there +is a full analysis of the \textit{Principia} itself, and +after that an account of the preparation of the second and third editions. +Probably the part of the book which will be found most interesting by the +general reader is the account of the correspondence of Newton with Hooke, +and with Halley, about the contents or the publication of the +\textit{Principia}. This correspondence is given in full, so far as it is +recoverable. Hooke does not appear to advantage in it. He accuses Newton of +stealing his ideas. +His vain and envious disposition made his own merits appear great in his +eyes, and be-dwarfed the work of others, so that he seems to have believed +that Newton's great performance was a mere expanding and editing of the +ideas of Mr.~Hooke---ideas which were meritorious, but after all mere guesses +at truth. This, at all events, is the most charitable view we can take of his +conduct. Halley, on the contrary, appears as a man to whom we ought to +feel most grateful. It almost seems as though Newton's physical insight and +extraordinary mathematical powers might have been largely wasted, as was +Pascal's rare genius, if it had not been for Halley's single-hearted and +self-forgetful efforts to get from his friend's genius all he could for the +enlightenment of men. It was probably at his suggestion that the writing of +the \textit{Principia} was undertaken. When the work was presented to the +Royal Society, they undertook its publication, but, being without the +necessary funds, the expense fell upon Halley. When Newton, stung by Hooke's +accusations, wished to withdraw a part of the work, Halley's tact was +required to avert the catastrophe. All the drudgery, worry, and expense fell +to his share, and was accepted with the most generous good nature. It will +be seen that both the technical student and the general reader may find +much to interest him in Mr.~Rouse Ball's book.---\textit +{The Manchester Guardian.} + +Une histoire très bien faite de la genèse du livre immortel de % silently correcting typo trés +Newton\textellipsis. Le livre de M.~Ball est une monographie précieuse sur +un point important de l'histoire des mathématiques. Il contribuera à +accroître, si c'est possible, la gloire de Newton, en révélant à +beaucoup de lecteurs, avec quelle merveilleuse rapidité l'illustre +géomètre anglais a élevé à la science ce monument immortel, +les \textit{Principia}.---\textit{Mathesis.} + +} +\PG----File: 418.png----------------------------------------------------- +\clearpage +\begin{center} +\large\textbf{NOTES ON THE HISTORY OF} \\[1em] +\LARGE\textbf{TRINITY COLLEGE, CAMBRIDGE} \\[1em] +\normalsize \textsc{By W.W.~ROUSE BALL}.\\[1.5em] +\large[\textit{Pp.} xiv + 183.\qquad \textit{Price $2$s.~$6$d.\ net.}]\\[1em] +\small\textsc{MACMILLAN AND CO. Ltd., LONDON AND NEW YORK.} +\end{center} +\par +\bigskip +\hrule + +This booklet gives a popular account of the History of Trinity +College, Cambridge, and so far as the author knows, it is as yet +(1905) the only work published on the subject. It was written +mainly for the use of his pupils, and contains such information and +gossip about the College and life there in past times as he believed +would be interesting to most undergraduates and members of the +House. + +\bigskip +\hrule +\bigskip + +{\Small +This modest and unpretending little volume seems to us to do more for its +subject than many of the more formal volumes \textellipsis\ treating of the +separate colleges of the English universities\textellipsis. In nine short, +extremely readable, and truly informing chapters it gives the reader a very +vivid account at once of the origin and development of the University of +Cambridge, of the rise and gradual supremacy of the colleges, of King's Hall +as founded by Edward~II, of the suppression of King's Hall by Henry~VIII on +December~17, 1546, the foundation of Trinity College by royal charter on +December~19, and the subsequent fortunes of the premier college of Cambridge. +The subject is in a way treated under the successive heads of the college, +but this is quite subordinate to the handling and characterisation of the +subject under four great periods---namely, that during the Middle Ages, that +during the Renaissance, that under the Elizabethan statutes, and that during +the last half-century. +The colleges arose from the determination of the University to prevent +students who were very young from seeking lodging, whether under the wing +of one or other of the religious orders---a circumstance which shows this +University to have been an essentially lay corporation. Early in the sixteenth +century the college had absorbed all the members of the University, and +henceforth the University was little more than % original reads "that" +the degree-granting body to +students who lived and moved and had their educational being under the +colleges\textellipsis. The University finally took the form of an aggregate +of separate and independent corporations, with a federal constitution +analogous in a rough sort of way to that of the United States of America, and +different from similar corporations at Paris by the fact that these latter +were always subject to University supervision\textellipsis. There is a good +account of the effort now going on to re-assert the University at the expense +of the colleges. No one who begins Mr.~Ball's book will lay it down till he +has read it from beginning to end.---\textit{The Glasgow Herald.} + +\PG----File: 419.png----------------------------------------------------- +It is a sign of the times, and a very satisfactory one, when \textellipsis\ a +tutor \textellipsis\ takes the trouble to make the history of his college known +to his pupils. Considering the lack of good books about the Universities, we +may thank Mr.~Ball that he has been good enough to print for a larger circle. +Though he modestly calls his book only ``Notes,'' yet it is eminently +readable, and there is plenty of information, as well as abundance of good +stories, in its pages.---\textit{The Oxford Magazine.} + +Mr.~Ball has put not only the pupils for whom he compiled these notes, but +the large world of Trinity men, under a great obligation by this compendious +but lucid and interesting history of the society to whose service he is +devoted. The value of his contribution to our knowledge is increased by the +extreme simplicity with which he tells his story, and the very suggestive +details which, without much comment, he has selected, with admirable +discernment, out of the wealth of materials at his disposal. His initial +account of the development of the University is brief but extremely clear, +presenting us with facts rather than theories, but establishing, with much +distinctness, the essential difference between the hostels, out of which the +more modern colleges grew, and that monastic life which poorer students were +often tempted to join.---\textit{The Guardian.} + +An interesting and valuable book\textellipsis. It is described by its author +as ``little more than an orderly transcript'' of what, as a Fellow and Tutor +of the College, he has been accustomed to tell his pupils. But while it does +not pretend either to the form or to the exhaustiveness of a set history, it +is scholarly enough to rank as an authority, and far more interesting and +readable than most academic histories are. It gives an instructive sketch of +the development of the University and of the particular history of Trinity, +noting its rise and policy in the earlier centuries of its existence, until, +under the misrule of Bentley, it came into a state of disorder which nearly +resulted in its dissolution. The subsequent rise of the College and its +position in what Mr.~Ball calls the Victorian renaissance, are drawn in lines +no less suggestive; and the book, as a whole, cannot fail to be welcome to +every one who is closely interested in the progress of the College.---\textit +{The Scotsman.} + +Mr.~Ball has succeeded very well in giving in this little volume just what an +intelligent undergraduate ought and probably often does desire to know +about the buildings and the history of his College\textellipsis. The debt of +the ``royal and religious foundation'' to Henry VIII is explained with +fulness, and there is much interesting matter as to the manner of life and +the expenses of students in the sixteenth century.---\textit +{The Manchester Guardian.} + +}\endgroup % restore widowpenalty, parskip +\PG----File: 420.png----------------------------------------------------- +%%**[Blank Page] +% we *do* want the licence to start recto, to emphasise it is an addition +\ifPaper\cleartorecto\else\clearpage\fi +\pagestyle{licence} +\setlength\parskip{0pt}\raggedbottom +\phantomsection +\addtocontents{toc}{\protect\bigskip} +\DPaddcontentsline{toc}{chapter}{\protect\textsc{Project Gutenberg Licensing Information}} +\hypertarget{PGlicence}{ }\par +\begin{verbatim} +*** END OF THE PROJECT GUTENBERG EBOOK MATHEMATICAL RECREATIONS *** + +***** This file should be named 26839-pdf.pdf or 26839-pdf.zip ***** +This and all associated files of various formats will be found in: + https://www.gutenberg.org/2/6/8/3/26839/ + +Updated editions will replace the previous one--the old editions will +be renamed. + +Creating the works from print editions not protected by U.S. copyright +law means that no one owns a United States copyright in these works, +so the Foundation (and you!) can copy and distribute it in the +United States without permission and without paying copyright +royalties. 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