diff options
| author | Roger Frank <rfrank@pglaf.org> | 2025-10-15 02:16:49 -0700 |
|---|---|---|
| committer | Roger Frank <rfrank@pglaf.org> | 2025-10-15 02:16:49 -0700 |
| commit | 5180e601a8820b1e6701bf3184fa7f4ef2c9e626 (patch) | |
| tree | 5ba22dc735094b10023dc4316050ef057304ef41 | |
35 files changed, 11430 insertions, 0 deletions
diff --git a/.gitattributes b/.gitattributes new file mode 100644 index 0000000..6833f05 --- /dev/null +++ b/.gitattributes @@ -0,0 +1,3 @@ +* text=auto +*.txt text +*.md text diff --git a/25387-pdf.pdf b/25387-pdf.pdf Binary files differnew file mode 100644 index 0000000..d2446be --- /dev/null +++ b/25387-pdf.pdf diff --git a/25387-pdf.zip b/25387-pdf.zip Binary files differnew file mode 100644 index 0000000..c5232ec --- /dev/null +++ b/25387-pdf.zip diff --git a/25387-t.zip b/25387-t.zip Binary files differnew file mode 100644 index 0000000..942dd03 --- /dev/null +++ b/25387-t.zip diff --git a/25387-t/25387-t.tex b/25387-t/25387-t.tex new file mode 100644 index 0000000..a93e4f6 --- /dev/null +++ b/25387-t/25387-t.tex @@ -0,0 +1,8876 @@ +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % +% % +% Project Gutenberg's Mathematical Essays and Recreations, by Hermann Schubert +% % +% This eBook is for the use of anyone anywhere at no cost and with % +% almost no restrictions whatsoever. You may copy it, give it away or % +% re-use it under the terms of the Project Gutenberg License included % +% with this eBook or online at www.gutenberg.org % +% % +% % +% Title: Mathematical Essays and Recreations % +% % +% Author: Hermann Schubert % +% % +% Translator: Thomas J. McCormack % +% % +% Release Date: May 9, 2008 [EBook #25387] % +% % +% Language: English % +% % +% Character set encoding: ISO-8859-1 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK MATHEMATICAL ESSAYS *** % +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{25387} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% memoir: Advanced book class. Required. %% +%% memhfixc: Part of memoir; needed to work with hyperref. 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. %% +%% graphicx: standard graphics inclusion package. Required. %% +%% Driver option needs to be set explicitly. %% +%% wrapfig: Allows placement of graphics inside text cutouts. Required. %% +%% flafter: Stops graphics floating backwards. Required. %% +%% inputenc: Allows accented characters in source. Required. %% +%% perpage: Resets footnote markers every page. Optional. %% +%% lettrine: For drop caps at beginning of chapters. Recommended. %% +%% type1cm: Allows use of CM fonts at arbitrary sizes. Required with %% +%% lettrine package. %% +%% psibycus: For authentic polytonic Greek. Strongly recommended. %% +%% If absent, Greek will be faked using math fonts. %% +%% soul: For sheep-stealing on the title page and advertisements. %% +%% Recommended. %% +%% multicol: To get balanced columns on the final index page. Required. %% +%% varioref: To allow suppression of hyperlinks within a single spread. %% +%% Recommended. %% +%% %% +%% %% +%% Producer's Comments: Mostly text, very little mathematics. %% +%% %% +%% 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 %% +%% graphicx driver can handle PDF graphics (of which there are 15): OK %% +%% Fonts: best results if psibycus Type 1 Greek fonts are installed and %% +%% Latin Modern companion fonts (but code will compensate if %% +%% these are not available) %% +%% Spellcheck: OK %% +%% LaCheck: OK %% +%% Lprep/gutcheck: OK %% +%% PDF page size: 499 x 709pt (b5) %% +%% PDF document info: filled in %% +%% PDF bookmarks: generated but closed %% +%% PDF Pages: 159 %% +%% Alterations (referring to position of floats) still appropriate: %% +%% compile with draft option to see where these are %% +%% Text cutouts around wrapped graphic not crossing page break %% +%% (Fig 36 on p83) %% +%% Two underfull vbox warnings (pp 43 and 59): don't worry, these are %% +%% due to TeX finding it impossible to fit the floats in amongst the %% +%% text according to its rather strict parameters. Output looks OK. %% +%% Two overfull vboxes (both by 14.5pt) for the advertisement pages at %% +%% the end. Not a problem: there really is too much on the originals %% +%% as well. %% +%% %% +%% %% +%% Compile History: %% +%% May 08: dcwilson. %% +%% Compiled with pdfLaTeX TWO times, then makeindex -r, then %% +%% pdfLaTeX another TWO times. %% +%% MiKTeX 2.7, Windows XP Pro %% +%% %% +%% Command block: %% +%% pdflatex x2 %% +%% makeindex -r %% +%% pdflatex x2 %% +%% %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\listfiles + +\makeatletter + +\documentclass[b5paper,12pt,twoside,openany,onecolumn]{memoir}[2005/09/25] +\setlrmarginsandblock{2.3cm}{2.6cm}{*} +\setulmarginsandblock{3.1cm}{2.2cm}{*} +\setlength{\headsep}{1cm} +\setlength{\footskip}{0.6cm} +\fixthelayout +\typeoutlayout + +% +% font issues +% Courier, for the PG licence stuff +\DeclareRobustCommand\ttfamily + {\not@math@alphabet\ttfamily\mathtt + \fontfamily{pcr}\fontencoding{T1}\selectfont} +% to set text numerals as oldstyle, plus a few shorthands +% for single digits, and an oldstyle slashed fraction +\let\Num\oldstylenums +\def\0{\Num0} +\def\1{\Num1} +\def\2{\Num2} +\def\3{\Num3} +\def\4{\Num4} +\def\5{\Num5} +\def\6{\Num6} +\def\7{\Num7} +\def\8{\Num8} +\def\9{\Num9} +\def\Numfrac#1#2{\leavevmode\kern.1em\raise.5ex\hbox + {\scriptsize\Num{#1}}\kern-.1em/\kern-.15em\lower.25ex\hbox + {\scriptsize\Num{#2}}} + +\IfFileExists{lmodern.sty} +{% use the mathcomp symbol for degrees (we don't want to load the whole package though) + \GenericInfo{*** }{*** Using Latin Modern for degree symbol...\@gobble} + \@namedef{TS1:lmr}{0} + \input{ts1enc.def} + \DeclareSymbolFont{TC}{TS1}{lmr}{m}{n} + \DeclareMathSymbol{\tcdegree}{\mathord}{TC}{176} + \let\degrees\tcdegree +}{% else just use the less authentic ^\circ + \def\degrees{^\circ} +} + +% There are a couple of bits of classical Greek: +% [Greek: tetragonizein] +% [Greek: tetragonizousa] +% 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\tetragonizein{\textsl{\textgreek{tetragwni'zein}}} + \def\tetragonizousa{\textsl{\textgreek{tetragwni'zousa}}} +}{% else fake with math greek + \def\tetragonizein{$\tau\epsilon\tau\rho\alpha\gamma\omega\nu + \!\acute\iota\zeta\epsilon\iota\nu$} + \def\tetragonizousa{$\tau\epsilon\tau\rho\alpha\gamma\omega\nu + \!\acute\iota\zeta o\upsilon\sigma\alpha$} + \GenericInfo{*** }{*** Faking breathed Greek using math\MessageBreak\expandafter\@gobble\@gobble} +} +\usepackage[latin1]{inputenc}[2006/05/05] % NB must be loaded *after* ibycus +% for drop caps at chapter starts +\usepackage{lettrine}[2006/03/17] +\usepackage{type1cm}[2002/09/05] + + +% mathematics +\usepackage[reqno]{amsmath}[2000/07/18] +\usepackage[psamsfonts]{amssymb}[2002/01/22] +\AtBeginDocument{\def\th{\textsuperscript{\textit{th}}}} +\let\dotm\centerdot + +% footnotes +\renewcommand{\thefootnote}{\BringhurstX{footnote}} +\footmarkstyle{#1\hfill} +\renewcommand{\foottextfont}{\footnotesize\normalfont} +\setlength{\footmarksep}{\z@} +\setlength{\footmarkwidth}{1.3em} +\usepackage{perpage}[2002/12/20] +\MakePerPage{footnote} +\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} + +% illustrations +% (external files are all .pdf) +\usepackage{flafter}[2000/07/23] +% 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@} +\renewcommand\floatpagefraction{0.5} +% \Legend{caption text for illustration} +\newcommand*\Legend[1]{\DPlabel{fig:#1}\legend{Fig.\ \Num{#1}.}} +% \figref{number} creates hyperlink to that-numbered figure, with anchor text Fig.~number +% use \figref*{number} to suppress leading space (eg inside paranetheses) +\newcommand*\figref{\@ifstar{\figr@fstar}{\figr@f}} +\newcommand*\figr@fstar[1]{\vhyperlink*{fig:#1}{Fig.~\Num{#1}}} +\newcommand*\figr@f[1]{\vhyperlink{fig:#1}{Fig.~\Num{#1}}} + +% allow some automation in references to illustrations +% depending on where LaTeX has actually floated them to +% although it doesn't help with deciding above/below on +% a particular page +\usepackage{varioref}[2006/05/13] +\providecommand{\vp@geref}[1]{\reftextafter} +% suppress a link if it goes to the same page +% requires destination to have been created by \DPlabel +% \vhyperlink{destination}{anchor text} +% \vhyperlink* suppresses leading space; here used internally by \figref* (see above) +\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} + +% for manual modifications to the text (eg if floats have moved) +% \Alteration{new stuff}{the original} +% only the new stuff is shown in the body text +% in draft compilation, original will be shown in the margin +\def\Alteration#1#2{#1\ifdraftdoc + \marginpar{\noindent\raggedright\Small #2}\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 PDF\MessageBreak + graphics, so make sure you use an appropriate workflow!\MessageBreak + \expandafter\@gobble\@gobble} +\RequirePackage{wrapfig}[2003/01/31] +\captionstyle{\centering} +\captiontitlefont{\normalfont\SMALL} +\setlength{\belowcaptionskip}{-10pt}% + +% PDF stuff: links, document info, etc +% 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 pdftex instead and issue a warning +\def\@tempa{hypertex} +\ifx\@tempa\Hy@driver + \GenericWarning{*** }{***\MessageBreak + Inappropriate driver for hyperref specified: assuming pdftex.\MessageBreak + You should amend the source code if using another driver.\MessageBreak + \expandafter\@gobble\@gobble} + \Hy@SetCatcodes\input{hpdftex.def}\Hy@RestoreCatcodes +\fi +\usepackage{memhfixc}[2006/01/21] +\providecommand{\ebook}{2xydw} +\hypersetup{pdftitle=The Project Gutenberg eBook \#\ebook: Mathematical Essays and Recreations, + pdfsubject=Translated from the German by Thomas J. McCormack, + pdfauthor=Hermann Schubert, + pdfkeywords={David Wilson}, + pdfstartview=Fit, + pdfstartpage=1, + pdfpagemode=UseNone, + pdfdisplaydoctitle, + bookmarksopen, + bookmarksopenlevel=1, + linktocpage=false, + pdfpagescrop=0 0 499 709, b5paper, % b5 176x250mm + pdfpagelayout=TwoPageRight, % this is Acrobat 6's "Facing" + plainpages=false, linkcolor=\ifdraftdoc blue\else black\fi, + menucolor=\ifdraftdoc blue\else black\fi, + citecolor=\ifdraftdoc blue\else black\fi, + urlcolor=\ifdraftdoc magenta\else black\fi} + +% for adding explicit destinations, here used only internally in \Legend (see above) +\newcommand*\DPanchor[1]{\rlap{\hyper@anchorstart{#1}\hyper@anchorend}} +\newcommand*\DPlabel[1]{\DPanchor{#1}\label{#1}} + +% A quasi-verbatim environment for boilerplate, slightly less drastic than alltt +% Spaces, linebreaks, $ , %, &, _, ^, and # will appear as typed +% but unlike full verbatim, commands will still be interpreted and long lines will wrap +% (comments documenting the boilerplate text need to use | as the comment character) +% uses slightly non-standard obeylines and active space. +% The optional argument can be used to specify an explicit font size for the boilerplate. +% If no optional argument is provided, a fontsize will be computed to allow--as nearly as +% possible--73 fixed-width characters per \textwidth (the longest line in the PG license +% had 73 characters at some time) +% +% \begin{PGboilerplate}[optional font sizing command] +% "verbatim" text +% \end{PGboilerplate} +% +{\catcode`\^^M=\active % these lines must end with % + \global\def\PGobeylines{\catcode`\^^M\active \def^^M{\null\par}}}% +{\obeyspaces% +\global\def\PGb@ilerplate[#1]{\def\PGb@ilerplateHook{#1}\catcode`\%11\relax\catcode`\_11\relax% +\catcode`\$11\relax\catcode`\#11\relax\catcode`\&11\relax\catcode`\^11\relax\catcode`\|=14\relax% +\pretolerance=\@m\hyphenpenalty=5000% +\rightskip=\z@\@plus20em\relax% +\frenchspacing\ttfamily\PGb@ilerplateHook% +\def {\noindent\null\space}% +\parindent=\z@\PGobeylines\obeyspaces}} +\def\PGboilerplate{% + \@ifnextchar[{\PGb@ilerplate}{\PGb@ilerplate[\PGAutoFit{73}]}} +\let\endPGboilerplate\empty +% \PGAutoFit adjusts the fontsize so a specified number of +% fixed-width characters will fit in the current \textwidth +\def\PGrem@pt#1.#2Q@!!@Q{#1} +\def\PGAutoFit#1{\setbox\z@=\hbox{m}\dimen@=\wd\z@\relax + \multiply\dimen@#1\relax \dimen@i=\dimen@\relax + \dimen@=\textwidth\relax + \dimen@ii=\f@size pt \advance\dimen@ii0.5pt + \expandafter\multiply\expandafter\dimen@\expandafter\PGrem@pt\the\dimen@ii Q@!!@Q + \expandafter\divide\expandafter\dimen@\expandafter\PGrem@pt\the\dimen@i Q@!!@Q + \dimen@i=\dimen@\multiply\dimen@i12\divide\dimen@i10 + \fontsize{\strip@pt\dimen@}{\strip@pt\dimen@i}\ttfamily\selectfont} +{\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}} + +% half-title, title and copyright pages +\aliaspagestyle{title}{empty} +\setlength{\droptitle}{-6pc} +\pretitle{\begin{center}\huge} +\posttitle{\par\end{center}} +\preauthor{\vfill\begin{center}{\tiny BY}\\[1em]\large} +\postauthor{\par\end{center}} +\def\affiliation#1{\renewcommand{\maketitlehookc}{\begin{center}\miniscule + \textsc{#1}\par\end{center}}} +\def\subtitle{\def\SubTitle} +\predate{\vfill\begin{center}\small\SubTitle\par + \vspace{\z@\@plus2.5fill} + \rule{2cm}{.2pt}\par\vspace{\z@\@plus2.5fill} + \Small\itshape} +\postdate{\par\end{center}\vspace*{-2em}} +\let\transcribersnotes\@empty +\let\transcribersNotes\@empty +\newcommand{\transcribersnote}[1]{% + \@ifnotempty{#1}{\g@addto@macro\transcribersnotes{#1\par}% + \@xp\@ifempty\@xp{\transcribersNotes}% + {\renewcommand{\transcribersNotes}{Transcriber's note}} + {\renewcommand{\transcribersNotes}{Transcriber's notes}}}} +\newtoks\PGheader +{\catcode`\L\active\PGboilerplate +\global\PGheader{| +Project Gutenberg's Mathematical Essays and Recreations, by Hermann Schubert + +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: Mathematical Essays and Recreations + +Author: Hermann Schubert + +Translator: Thomas J. McCormack + +Release Date: May 9, 2008 [EBook #25387] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK MATHEMATICAL ESSAYS *** +}} + +\def\makehalftitlepage{% the boilerplate header + \begingroup + \begin{PGboilerplate}[\tiny] % 8pt for B5 + \PGlicencelink + \the\PGheader + \end{PGboilerplate} + \null\vfil + \clearpage + \endgroup} + +\def\makecopyrightpage{% production credits and transcriber's notes + \begingroup + \null\vfil + \begin{center} +Produced by David Wilson + \end{center} + \vfil\vfil + \vbox{\Small\hsize=.75\textwidth\parindent=\z@\parskip=.75em + \textit{\transcribersNotes}\par\medskip\raggedright + \transcribersnotes\par} + \cleartorecto + \endgroup} + +% headers and footers +\def\thePAGE{\expandafter\oldstylenums\expandafter{\number\c@page}} +\copypagestyle{mainstuff}{headings} +\makepsmarks{mainstuff}{% + \let\@mkboth\markboth + \def\chaptermark##1{% + \markboth{\MakeUppercase{##1.}}{\MakeUppercase{##1.}}}% + \def\indexmark{\markboth{\MakeUppercase{\indexname}.}% + {\MakeUppercase{\indexname}.}}% + } +\makeevenhead{mainstuff}{\normalfont\SMALL\thePAGE}{\normalfont + \SMALL\MakeUppercase{\leftmark}}{} +\makeoddhead{mainstuff}{}{\normalfont + \SMALL\MakeUppercase{\rightmark}}{\normalfont\SMALL\thePAGE} +% make it hard to end the next (odd) page with a hyphen +\makeevenfoot{mainstuff}{}{}{\global\brokenpenalty10000} +% make it less hard to end the next (even) page with a hyphen +% (because the other end of the hyphen will still be on the same spread) +\makeoddfoot{mainstuff}{}{}{\global\brokenpenalty150} + +\copypagestyle{chapter}{plain} +\makeevenfoot{chapter}{}{\if@mainmatter\normalfont\SMALL\thePAGE\fi}{} +\makeoddfoot{chapter}{}{\if@mainmatter\normalfont\SMALL\thePAGE\fi}{} + +\copypagestyle{licence}{headings} +\makeevenhead + {licence}{\normalfont\SMALL\thePAGE}{\normalfont\SMALL LICENSING.}{} +\makeoddhead + {licence}{}{\normalfont\SMALL LICENSING.}{\normalfont\SMALL\thePAGE} + +% chapters etc +\makechapterstyle{schubert}{% + \setlength{\beforechapskip}{4pc} + \renewcommand{\printchapternum}{} + \renewcommand{\printchaptername}{\begin{center}} + \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{schubert} + +\renewcommand{\thesection}{\@Roman\c@section} +\setsecnumformat{\csname the#1\endcsname.} +\setsecnumdepth{section} % we want bookmarks down to this level +\maxsecnumdepth{subsection} +\maxtocdepth{chapter} +\setlength{\parindent}{1.8em} +\setlength{\leftmargini}{1.8em} +\setsecheadstyle{\normalfont\small\centering} +\let\sectionmark\@gobble +\def\Sectionformat#1#2{\\*[9pt]#1} + +% table of contents +\setpnumwidth{2em} +\renewcommand{\cftchapterfont}{\normalfont} +\setlength{\cftbeforechapterskip}{0.4em \@plus\p@} +\def\cftchapterpagefont{\spaceskip\fontdimen\tw@\font + \usefont{OML}{\rmdefault}{\f@series}{it}% + \mathgroup\symletters} +\let\chapternumberline\@gobble +\renewcommand{\cftchapterleader}{% + \cftdotfill{10}} +\AtBeginDocument{\addtocontents{toc}{\string + \rightline{\string\textsc{\string\tiny\space page}}}} +\noindexintoc + +% index things--- +% to get oldstyle in the index, and the \ignorespaces allows run-in subitems +\def\hyperpage#1{\oldstylenums{\@hyperpage#1----\\}\ignorespaces} +\def\@hyperpage#1--#2--#3\\{% + \ifx\\#2\\% + \@commahyperpage{#1}% + \else + \HyInd@pagelink{#1}{\fontencoding{OT1}\fontshape{n}\selectfont--}\HyInd@pagelink{#2}% + \fi +} +\setlength\indexcolsep{15pt} +\renewcommand{\indexspace}{\par\penalty-3000 \vskip 12pt plus6pt minus4pt\relax} +\renewcommand{\subitem} {; } % run-in subitems +\renewcommand{\@idxitem} {\par\hangindent 20\p@} % smaller indent than default +\setlength\indexrule{0.4\p@} +% to get balanced columns without doing it manually +\usepackage{multicol}[2006/05/18] +\onecolindextrue +\def\preindexhook{\Small\DPpdfbookmark[0]{Index}{Index*1} + \raggedright\hyphenpenalty=10000\emergencystretch.5\hsize + \setlength{\columnseprule}{\indexrule}% + \setlength{\columnsep}{\indexcolsep}% + \multicols{2}\def\endtheindex + {\def\@currenvir{multicols}\endmulticols}} +% for "et seq" entries in index +\def\etseq#1{\hyperpage{#1}~\textrm{et~seq.}\ignorespaces} +% dodgy index kludges +\def\Arabs#1{\textrm{magic squares of,} \hyperpage{#1}} +\def\Bible#1{\textrm{speaks of four dimensions,} \hyperpage{#1}} +\def\Curvature#1{\textrm{negative,} \etseq{#1}} +\def\Mathematical#1{\textrm{on the nature of,} \hyperpage{#1}} +\def\Number#1{\textrm{notation and definition of,} \hyperpage{#1}} +\def\Numbers#1{\textrm{as symbols,} \hyperpage{#1}} +\def\Problems#1{\textrm{fundamental,} \hyperpage{#1}} +\def\Quadrature#1{\textrm{in Egypt,} \hyperpage{#1}} +\def\Quantities#1{\textrm{four variable,} \hyperpage{#1}} +\def\Series#1{\textrm{infinite,} \etseq{#1}} +\def\Space#1{\textrm{curvature of,} \hyperpage{#1}} +\def\Spirits#1{\textrm{existence of four-dimensional,} \etseq{#1}} +\def\Spiritualistic#1{\textrm{mediums,} \etseq{#1}} +\let\Gobble\@gobble + +% for itemized lists without hanging indent +% +% \begin{Itemize} +% \item +% \item +% \end{Itemize} +% +\def\Itemize{\leftmargini\z@\list{}{\topsep\z@\itemsep\z@\parsep\z@\labelwidth\parindent \itemindent2\parindent + \def\makelabel##1{\hss\llap{##1}}}}% +\let\endItemize\endlist + +% slight modification of hyperref's command, +% for adding explicit bookmarks (and their destinations) +% +% \DPpdfbookmark[bookmark level]{bookmark text}{destination} +% +\newcommand\DPpdfbookmark[3][0]{\rlap{\hyper@anchorstart{#3}\hyper@anchorend + \Hy@writebookmark{}{#2}{#3}{#1}{toc}}} +% hyperref seems to use the most recent section anchor instead of page +% anchors in a \pageref, so we have to fix that; using AtBeginDocument +% so hyperref doesn't clobber it +% plus oldstylenums... +\AtBeginDocument{\def\pageref#1{\Num + {\expandafter\@pagesetref\csname r@#1\endcsname\@empty{#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\CSqr#1{\vbox to\SqHt{\vss\hbox to\SqWd{\small\hss\vphantom + {\SqHtDefault}#1\hss}\vss}} +\def\LSqr#1{\vbox to\SqHt{\vss\hbox to\SqWd{\small\vphantom + {\SqHtDefault}#1\hss}\vss}} +\def\RSqr#1{\vbox to\SqHt{\vss\hbox to\SqWd{\small\hss\vphantom + {\SqHtDefault}#1}\vss}} +\def\TSqr#1{\vbox to\SqHt{\hbox to\SqWd{\small\hss\vphantom + {\SqHtDefault}#1\hss}\vss}} +\def\SqHtDefault{7} +\def\Cell(#1,#2;#3){\put(#1,#2){\CSqr{#3}}} +\def\cell(#1,#2;#3){\put(#1,#2){\CSqr{\Num{#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 + \Grid(#1,#2) + \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}} + \M@gicSqu@reExtr@ + \end{picture}\end} +\def\Grid(#1,#2){\begingroup + \linethickness{\fboxrule} + \@tempcnta=#2 + \@tempcntb\@ne + \loop + \ifnum\@tempcnta>\@ne + \advance\@tempcnta\MSqVertAdvance + \put(0,\@tempcnta){\line(1,0){#1}} + \repeat + \loop + \ifnum\@tempcntb<#1 + \put(\@tempcntb,0){\line(0,1){#2}} + \advance\@tempcntb\MSqHorizAdvance + \repeat + \endgroup} +\def\endMagicSquare{\aftergroup\ignorespaces}\def\MagicSquareExtra{\def\M@gicSqu@reExtr@} +\let\M@gicSqu@reExtr@\empty +\let\MSqHorizAdvance\@ne +\let\MSqVertAdvance\m@ne +\long\def\MSqN@xtCell#1{\advance\@tempcntb\MSqHorizAdvance + \Cell(\@tempcntb,\@tempcnta;\Num{#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 +\@tempdima=2\fboxrule % magic square outline is twice as thick as gridlines +\expandafter\linethickness\expandafter{\the\@tempdima} +\fboxsep\z@ +\def\SqHt{1.5em} +\def\SqWd{1.5em} + +% for the advertisements at the end, and the title page +\usepackage{soul}[2003/11/17] +\providecommand{\sodef}[5]{} +\providecommand{\so}{} +\providecommand{\textso}{} +\providecommand{\textul}{} +% very slightly letterspace the main title +\sodef\textso{}{.1em}{.5em}{\z@} + +% fiddle with page dimensions for adverts +\newdimen\advertsave +\newenvironment{adverts}% + {\topmargin-1.25cm + \headheight\z@ + \headsep\z@ + \footskip\z@ + \advertsave=\textheight + \advance\textheight2.5cm + \vsize\textheight + \advance\textwidth2cm + \leftmargini-1cm + \pagestyle{empty} + \list{}{\listparindent 1.5em% + \itemindent \listparindent + \rightmargin \leftmargin + \parsep\z@\@plus\p@}% + \item[]}% + {\endlist\textheight\advertsave\cleartorecto} + +% to deal with the scanned page breaks +% add an explicit "draft" option to the documentclass invocation +% to see the scan numbers (and \Alteration old text) +\ifdraftdoc +\def\PG seq=#1 Page #2--#3 +{\marginpar{\noindent\null\hfill\Small #2}} +\def\PGx seq=#1 Page #2--#3 +{} +\else +\def\PG seq=#1 Page #2--#3 +{} +\let\PGx\PG +\fi + + +% bits and pieces +\emergencystretch=12pt +\setlength\parskip{0\p@ \@plus 6\p@} +% for rare occasions where a paragraph covers more than one page so +% because there are no stretch points on the page it ends up underfull +% also helps with avoiding widows and orphans +\AtBeginDocument{\advance\baselineskip0pt plus0.5pt\relax} + +\let\Small\footnotesize +\let\SMALL\scriptsize +\newcommand*\IE{i.\;e.} +\newcommand*\ThoughtBreakStars{\fancybreak*{{*}\\[-6pt]{*\kern8em*}}} + +% lprep "comment": anything from \LP to end of line will be stripped by lprep +\let\LP\empty + +\makeatother + +\makeindex + +\begin{document} + +\frontmatter +\pagestyle{empty} +\makehalftitlepage + +\PGx seq=1 Page --[unnumbered] +\PGx seq=2 Page --[unnumbered] +\PGx seq=3 Page --[unnumbered] +\PGx seq=4 Page --[unnumbered] +\PGx seq=5 Page --[unnumbered] +\vspace*{0pt plus1fil} +{\Small\LP\leftmargini1cm +\begin{center} +{\large IN THE SAME SERIES.} + +\rule{2cm}{.2pt} + +\end{center} +\list{}{\LP\labelwidth0pt \itemindent-1.5em\rightmargin\leftmargin + \LP\parsep 10pt \let\makelabel\empty} +\item[ON THE STUDY AND DIFFICULTIES OF MATHEMATICS\@.]\LP\hskip0ptplus1em +By \textsc{Augustus +De Morgan}. Entirely new edition, with portrait of the author, +index, and annotations, bibliographies of modern works on algebra, the +philosophy of mathematics, pan-geometry, etc. Pp., \Num{288}. Cloth, \$\1.\Num{25} (\5s.). + +\item[LECTURES ON ELEMENTARY MATHEMATICS\@.]\LP\hskip0ptplus1em +By \textsc{Joseph Louis Lagrange}. +Translated from the French by \textit{Thomas J. McCormack}. With +photogravure portrait of Lagrange, notes, biography, marginal analyses, +etc. Only separate edition in French or English. Pages, \Num{172}. Cloth, +\$\1.\Num{00} (\5s.). + +\item[HISTORY OF ELEMENTARY MATHEMATICS\@.]\LP\hskip0ptplus1em +By \textsc{Dr.\ Karl Fink}, late +Professor in Tübingen. Translated from the German by Prof.\ \textit{Wooster +Woodruff Beman} and Prof.\ \textit{David Eugene Smith}. (In preparation.) + +\endlist}\begin{center} + +\rule{2cm}{.2pt} + +\medskip +THE OPEN COURT PUBLISHING CO. +\medskip + +\tiny \Num{324} \textsc{dearborn st., chicago.} + +\end{center} + +\PGx seq=6 Page --[unnumbered] + +\clearpage + +\PGx seq=7 Page --[unnumbered] + +\title{\so{MATHEMATICAL ESSAYS}\\[.7cm] +{\tiny AND}\\[.7cm] +\LARGE\so{RECREATIONS}} + +\author{HERMANN SCHUBERT} +\affiliation{PROFESSOR OF MATHEMATICS IN THE JOHANNEUM, HAMBURG, GERMANY} + +\subtitle{{\tiny FROM THE GERMAN BY}\\[1em] +THOMAS J. McCORMACK} + +\date{Chicago, \Num{1898}} + +\maketitle + +\newpage + +\transcribersnote{This e-text was created from scans of the book published at Chicago +in \Num{1898} by the Open Court Publishing Company, and at London by +Kegan Paul, Trench, Truebner \& Co.} + +\transcribersnote{The translator has occasionally chosen unusual forms of words: these have been retained.} + +\transcribersnote{\SMALL Some cross-references have been slightly reworded to take account of changes in the relative position of text and floated figures. +Details are documented in the \LaTeX\ source, along with minor typographical corrections.} + +\makecopyrightpage + +{\Small +\PGx seq=8 Page --[unnumbered] +\chapter*{Translator's Note} + +\lettrine{T}{he} mathematical essays and recreations in this volume are by + one of the most\DPpdfbookmark[0]{Translator's Note}{Preface*1} +successful teachers and text-book writers of Germany. The monistic construction +of arithmetic, the systematic and organic development of all its consequences +from a few thoroughly established principles, is quite foreign to the general run of +American and English elementary text-books, and the first three essays of Professor +Schubert will, therefore, from a logical and esthetic side, be full of suggestions for +elementary mathematical teachers and students, as well as for non-mathematical +readers. For the actual detailed development of the system of arithmetic here +sketched, we may refer the reader to Professor Schubert's volume \textit{Arithmetik +und Algebra}, recently published in the Göschen-Sammlung (Göschen, Leipsic),---an +extraordinarily cheap series containing many other unique and valuable text-books % hyphen retained on frequency grounds +in mathematics and the sciences. + +The remaining essays on ``\hyperlink{chapter.4}{Magic Squares},'' ``The \hyperlink + {chapter.5}{Fourth Dimension},'' and +``The History of the \hyperlink{chapter.6}{Squaring of the Circle},'' will be found to be the most complete +generally accessible accounts in English, and to have, one and all, a distinct +educational and ethical lesson. + +In all these essays, which are of a simple and popular character, and designed +for the general public, Professor Schubert has incorporated much of his original +research. + +\bigskip\bigskip +\begin{flushright} +\textsc{Thomas J. McCormack.}\qquad\null +\end{flushright} + +\textsc{La Salle}, Ill., December, 1898. + +}\cleartorecto +{\Small +\DPpdfbookmark[0]{Table of Contents}{ToC*1} +\tableofcontents* + +\clearpage +} +\PGx seq=9 Page --[unnumbered] +\PGx seq=10 Page --[unnumbered] +\PGx seq=11 Page --[unnumbered] + +\mainmatter +\pagestyle{mainstuff} + +\PGx seq=12 Page 1 ------------------------------------------------------- +\chapter{Notion and Definition of Number} + +\lettrine{M}{any} essays have been written on the definition of number\index + {Number|(Number}. +But most of them contain too many technical expressions, +both philosophical and mathematical, to suit the non-mathematician. +The clearest idea of what counting\index + {Counting|etseq} and numbers mean may +be gained from the observation of children and of nations in the +childhood of civilisation. When children count or add, they use +either their fingers\index + {Fingers|etseq}, or small sticks of wood, or pebbles, or similar +things, which they adjoin singly to the things to be counted or +otherwise ordinally associate with them. As we know from history, +the Romans and Greeks employed their fingers when they counted +or added. And even to-day we frequently meet with people to whom +the use of the fingers is absolutely indispensable for computation. + +Still better proof that the accurate association of such ``other'' +things with the things to be counted is the essential element of + numeration\index{Numeration|etseq} +are the tales of travellers in Africa, telling us how African +tribes sometimes inform friendly nations of the number of the enemies +who have invaded their domain. The conveyance of the information +is effected not by messengers, but simply by placing at spots +selected for the purpose a number of stones exactly equal to the +number of the invaders. No one will deny that the number of the +tribe's foes is thus communicated, even though no name exists for +this number in the languages of the tribes. The reason why the +fingers are so universally employed as a means of numeration is, +that every one possesses a definite number of fingers, sufficiently +large for purposes of computation and that they are always at hand. + +Besides this first and chief element of numeration which, as we +\PG seq=13 Page 2 ------------------------------------------------------ +have seen, is the exact, individual conjunction or association of other +things with the things to be counted, is to be mentioned a second +important element, which in some respects perhaps is not so absolutely +essential; namely, that the things to be counted shall be regarded +as of the same kind. Thus, any one who subjects apples and +nuts collectively to a process of numeration will regard them for the +time being as objects of the same kind, perhaps by subsuming them +under the common notion of fruit. We may therefore lay down provisionally +the following as a definition of counting: to count a group +of things is to regard the things as the same in kind and to associate +ordinally, accurately, and singly with them other things. In writing, +we associate with the things to be counted simple signs, like points, +strokes, or circles. The form of the symbols\index + {Symbols} we use is indifferent. +Neither need they be uniform. It is also indifferent what the spatial +relations or dispositions of these symbols are. Although, of +course, it is much more convenient and simpler to fashion symbols +growing out of operations of counting on principles of uniformity +and to place them spatially near each other. In this manner are +produced what I have called\footnote + {\textit{System der Arithmetik}. (Potsdam: Aug.\ Stein. \Num{1885}.)} +natural number-pictures\label{numberpictures}\index + {Numberp@Number-pictures and signs, natural|etseq}; for example, +\[\makeatletter\let\@arstrut\empty +\begin{tabular}{ccccccccc} +&&&\textbullet~\textbullet&\textbullet~\textbullet + &\textbullet~\textbullet~\textbullet + &\textbullet~\textbullet~\textbullet + &\textbullet~\textbullet~\textbullet~\textbullet\\ +&&&&\textbullet&&\textbullet~\phantom{\textbullet}&\phantom{etc.}\\ +\textbullet&\textbullet~\textbullet + &\textbullet~\textbullet~\textbullet + &\textbullet~\textbullet&\textbullet~\textbullet + &\textbullet~\textbullet~\textbullet + &\textbullet~\textbullet~\textbullet + &\textbullet~\textbullet~\textbullet~\textbullet& ~etc. +\end{tabular} +\] +Now-a-days such natural number-pictures are rarely employed, and +are to be seen only on dominoes, dice, and sometimes, also, on playing-cards. + +It can be shown by arch\ae ological evidence that originally numeral +writing\index + {Numeral writing|etseq} was made up wholly of natural number-pictures. For +example, the Romans in early times represented all numbers, which +were written at all, by assemblages of strokes. We have remnants +of this writing in the first three numerals of the modern Roman system. +If we needed additional evidence that the Romans originally +employed natural number-signs, we might cite the passage in Livy\index{Livy}, +VII.~\3, where we are told, that, in accordance with a very ancient +law, a nail was annually driven into a certain spot in the sanctuary of +\PG seq=14 Page 3 ------------------------------------------------------ +Minerva\index + {Minerva}, the ``inventrix'' of counting, for the purpose of showing the +number of years which had elapsed since the building of the edifice. +We learn from the same source that also in the temple at Volsinii +nails were shown which the Etruscans\index + {Etruscans} had placed there as marks for +the number of years. + +Also recent researches in the civilisation of ancient Mexico\index + {Mexico, ancient} show +that natural number-pictures were the first stage of numeral notation. +Whosoever has carefully studied in any large ethnographical +collection the monuments of ancient Mexico, will surely have remarked +that the nations which inhabited Mexico before its conquest +by the Spaniards, possessed natural number-signs for all numbers +from one to nineteen, which they formed by combinations of circles. +If in our studies of the past of modern civilised peoples, we meet +with natural number-pictures only among the Greeks\index + {Greeks} or Romans, +and some Oriental nations, the reason is that the other nations, as the +Germans, before they came into contact with the Romans and adopted +the more highly developed notation of the latter, were not yet sufficiently +advanced in civilisation to feel any need of expressing numbers +symbolically. But since the most perfect of all systems of numeration, +the Hindu system of ``local value\index + {Local value, Hindu system of},'' was introduced and +adopted in Europe in the twelfth century, the Roman\index + {Romans} numeral system +gradually disappeared, at least from practical computation, and +at present we are only reminded by the Roman characters of inscriptions +of the first and primitive stage of all numeral notation. To-day +we see natural number-pictures, except in the above-mentioned +games, only very rarely, as where the tally-men of wharves or warehouses +make single strokes with a pencil or a piece of chalk, one for +each bale or sack which is counted. + +As in writing it is of consequence to associate with each of the +things to be counted some simple sign, so in speaking it is of consequence +to utter for each single thing counted some short sound. +It is quite indifferent here what this sound is called, also whether +the sounds which are associated with the things to be counted are +the same in kind or not, and finally, whether they are uttered at +equal or unequal intervals of time. Yet it is more convenient and +simpler to employ the same sound and to observe equal intervals in +\PG seq=15 Page 4 ------------------------------------------------------ +their utterance. We arrive thus at natural number-words\index + {Numberw@Number-words, natural}. For example, +utterances like, +\[ +\text{oh, oh-oh, oh-oh-oh, oh-oh-oh-oh, oh-oh-oh-oh-oh,} +\] +are natural number-words for the numbers from one to five. Number-words +of this description are not now to be found in any known +language. And yet we hear such natural number-words constantly, +every day and night of our lives; the only difference being that the +speakers are not human beings but machines---namely, the striking-apparatus +of our clocks\index{Clocks}. + +Word-forms\index{Numeral words|etseq}\index + {Words, numeral|etseq}\index + {Number!denominate} of the kind described are too inconvenient, however, +for use in language, not only for the speaker, on account of +their ultimate length, but also for the hearer, who must be constantly +on the \emph{qui vive} lest he misunderstand a numeral word so formed. It +has thus come about that the languages of men from time immemorial +have possessed numeral words which exhibit no trace of the +original idea of single association. But if we should always select +for every new numeral word some new and special verbal root, we +should find ourselves in possession of an inordinately large number +of roots, and too severely tax our powers of memory. Accordingly, +the languages of both civilised and uncivilised peoples always construct +their words for larger numbers from words for smaller +numbers. What number we shall begin with in the formation of compound\index + {Numeral words!compound} +numeral words is quite indifferent, so far as the idea of number +itself is concerned. Yet we find, nevertheless, in nearly all +languages one and the same number taken as the first station in the +formation of compound numeral words, and this number is ten\index{Ten}. +Chinese and Latins, Fins and Malays, that is, peoples who have no +linguistic relationship, all display in the formation of numeral words +the similarity of beginning with the number ten the formation of +compound numerals. No other reason can be found for this striking +agreement than the fact that all the forefathers of these nations possessed +ten fingers. + +Granting it were impossible to prove in any other way that +people originally used their fingers in reckoning\index + {Fingers!in reckoning|etseq}, the conclusion +could be inferred with sufficient certainty solely from this agreement +with regard to the first resting-point in the formation of compound +\PG seq=16 Page 5 ------------------------------------------------------ +numerals among the most various races. In the Indo-Germanic +tongues the numeral words\index + {Numeral words!in the Indo-Germanic tongues} from ten to ninety-nine are + formed\index{Numeral@Numerals, the formation of|etseq} by +composition from smaller numeral words. Two methods remain +for continuing the formation of the numerals: either to take a new +root as our basis of composition (hundred) or to go on counting +from ninety-nine, saying tenty, eleventy, etc. If we were logically +to follow out this second method we should get tenty-ty for a thousand, +tenty-ty-ty for ten thousand, etc. But in the utterance of such +words, the syllable \emph{ty} would be so frequently repeated that the same +inconvenience would be produced as above in our individual number-pictures. +For this reason the genius which controls the formation +of speech took the first course. + +But this course is only logically carried out in the old Indian +numeral words. In Sanskrit we not only have for ten, hundred, +and thousand a new root, but new bases of composition also exist +for ten thousand, one hundred thousand, ten millions, etc., which +are in no wise related with the words for smaller numbers. Such +roots exist among the Hindus for all numerals up to the number expressed +by a one and fifty-four appended naughts. In no other language +do we find this principle carried so far. In most languages the +numeral words for the numbers consisting of a one with four and +five appended naughts are compounded, and in further formations +use is made of the words million, billion, trillion, etc., which really +exhibit only one root, before which numeral words of the Latin +tongue are placed. + +Besides numeral word-systems based on the number \emph{ten}, only logical +systems are found based on the number five and on the number +twenty. Systems of numeral words which have the basis five\index + {Five, as a numeral basis} occur +in equatorial Africa. (See the language-tables of Stanley's books +on Africa.) The Aztecs\index{Aztecs} and Mayas of ancient Mexico had the base +twenty. In Europe it was mainly the Celts who reckoned with +twenty as base\index + {Twenty, as base}. The French language still shows some few traces +of the Celtic vicenary system\index + {Vicenary system}, as in its word for eighty, \emph{quatre-vingt}. +The choice of five and of twenty as bases is explained simply enough +by the fact that each hand has five fingers, and that hands and feet +together have twenty fingers and toes. +\PG seq=17 Page 6 ------------------------------------------------------ + +As we see, the languages of humanity now no longer possess +natural number-signs and number-words, but employ names and +systems of notation\index + {Notation, systems of} adopted subsequently to this first stage. Accordingly, +we must add to the definition of counting above given a +third factor or element which, though not absolutely necessary, is +yet important, namely, that we must be able to express the results +of the above-defined associating of certain other things with the +things to be counted, by some conventional sign or numeral word. + +Having thus established what counting or \emph{numbering}\index + {Numbering} means, +we are in a position to define also the notion of \emph{number}, which we +do by simply saying that by number we understand \emph{the results} of +counting or numeration. These are naturally composed of two elements. +First, of the ordinary number-word or number-sign; and +secondly, of the word standing for the specific things counted. For +example, eight men, seven trees, five cities. When, now, we have +counted one group of things, and subsequently also counted another +group of things of the same kind, and thereupon we conceive the +two groups of things combined into a single group, we can save +ourselves the labor of counting the things a third time by blending +the number-pictures belonging to the two groups into a single number-picture +belonging to the whole. In this way we arrive on the +one hand at the idea of addition, and on the other, at the notion of +``unnamed'' number\index + {Unnamed number}. Since we have no means of telling from the +two original number-pictures and the third one which is produced +from these, the kind or character of the things counted, we are ultimately +led in our conception of number to abstract wholly from the +nature of the things counted, and to form the definition of unnamed +number. + +We thus see that to ascend from the notion of named number\index + {Named number} +to the notion of unnamed number, the notion of addition, joined to +a high power of abstraction, is necessary. Here again our theory +is best verified by observations of children learning to count and +add. A child, in beginning arithmetic, can well understand what +five pens or five chairs are, but he cannot be made to understand +from this alone what five abstractly is. But if we put beside the +first five pens three other pens, or beside the five chairs three other +\PG seq=18 Page 7 ------------------------------------------------------ +chairs, we can usually bring the child to see that five things plus +three things are always eight things, no matter of what nature the +things are, and that accordingly we need not always specify in +counting what kind of things we mean. At first we always make +the answer to our question of what five plus three is, easy for the +child, by relieving him of the process of abstraction, which is necessary +to ascend from the named to the unnamed number, an end +which we accomplish by not asking first what five plus three is, but +by associating with the numbers words designating things within +the sphere of the child's experience, for example, by asking how +many five pens plus three pens are. + +The preceding reflexions have led us to the notion of unnamed +or abstract numbers\index + {Number!dz@abstract}. The arithmetician calls these numbers positive +whole numbers, or positive integers, as he knows of other kinds +of numbers, for example, negative numbers, irrational numbers, etc. +Still, observation of the world of actual facts, as revealed to us by our +senses, can naturally lead us only to positive whole numbers, such +only, and no others, being results of actual counting. All other kinds +of numbers are nothing but artificial inventions of mathematicians +created for the purpose of giving to the chief tool of the mathematician, +namely, arithmetical notation, a more convenient and more +practical form, so that the solution of the problems which arise in +mathematics may be simplified. All numbers, excepting the results +of counting above defined, are and remain mere symbols\index + {Numbera@Numbers|Numbers}, which, +although they are of incalculable value in mathematics, and, therefore, +can scarcely be dispensed with, yet could, if it were a question +of principle, be avoided. Kronecker\index + {Kronecker} has shown that any problem +in which positive whole numbers are given, and only such are +sought, always admits of solution without the help of other kinds of +numbers, although the employment of the latter wonderfully simplifies +the solution\index{Number|)Number}. + +How these derived species of numbers, by the logical application +of a single principle, flow naturally from the notion of number +and of addition above deduced, I shall show in the \hyperlink + {chapter.2}{next article} entitled +``Monism in Arithmetic.'' +\PG seq=19 Page 8 ------------------------------------------------------ + + + +\chapter{Monism in Arithmetic} + +\lettrine{I}{n his} \textit{Primer of Philosophy}, Dr.\ Paul Carus defines + monism\index{Arithmetic!monism in|(}\index{Monism in arithmetic|(} +as a ``unitary conception of the world.'' Similarly, we shall +understand by monism in a science the unitary conception of that +science. The more a science advances the more does monism dominate +it. An example of this is furnished by physics\index{Physics}. Whereas +formerly physics was made up of wholly isolated branches, like +Mechanics, Heat, Optics, Electricity, and so forth, each of which +received independent explanations, physics has now donned an almost +absolute monistic form, by the reduction of all phenomena to +the \emph{motions} of molecules. For example, optical and electrical phenomena, +we now know, are caused by the undulatory movements +of the ether, and the length of the ether-waves constitutes the sole +difference between light and electricity. + +Still more distinctly than in physics is the monistic element +displayed in pure arithmetic\index{Arithmetic!apure@pure|etseq}\index + {Pure arithmetic}, by which we understand the theory of +the combination of two numbers into a third by addition and the +direct and indirect operations springing out of addition. Pure arithmetic +is a science which has completely attained its goal, and which +can prove that it has, exclusively by internal evidence. For it may +be shown on the one hand that besides the seven familiar operations +of addition, subtraction, multiplication, division, involution, evolution, +and the finding of logarithms, no other operations are definable +which present anything essentially new; and on the other hand +that fresh extensions of the domain of numbers beyond irrational, +imaginary, and complex numbers are arithmetically impossible. +Arithmetic may be compared to a tree that has completed its growth, +\PG seq=20 Page 9 ------------------------------------------------------ +the boughs and branches of which may still increase in size or even +give forth fresh sprouts, but whose main trunk has attained its fullest +development. + +Since arithmetic has arrived at its maturity, the more profound +investigation of the nature of numbers and their combinations shows +that a unitary conception of arithmetic is not only possible but also +necessary. If we logically abide by this unitary conception, we arrive, +starting from the notion of counting and the allied notion of +addition, at all conceivable operations and at all possible extensions of +the notion of number. Although previously expressed by Grassmann\index + {Grassmann}, +Hankel\index{Hankel}, E.~Schröder\index + {Schrod@Schröder, E.}, and Kronecker\index + {Kronecker}, the author of the present article, +in his ``System of Arithmetic,'' Potsdam, \Num{1885}, was the first +to work out the idea referred to, fully and logically and in a form +comprehensible for beginners. This book, which Kronecker in his +``Notion of Number,'' an essay published in Zeller's jubilee work, +makes special mention of, is intended for persons proposing to learn +arithmetic. As that cannot be the object of the readers of these essays, +whose purpose will rather be the study of the logical construction +of the science from some single fundamental principle, the following +pages will simply give of the notions and laws of arithmetic +what is absolutely necessary for an understanding of its development. + +The starting-point of arithmetic is the idea of counting\index + {Counting} and of +number as the result of counting\index + {Number!dzz@as the result of counting}. On this subject, the reader is requested +to read the \hyperlink + {chapter.1}{first essay} of this collection. It is there shown +that the idea of addition springs immediately from the idea of counting. +As in counting it is indifferent in what order we count, so in +addition it is indifferent, for the sum, or the result of the addition, +whether we add the first number to the second or the second to the +first. This law, which in the symbolic language of arithmetic, is +expressed by the formula +\[ +a + b = b + a, +\] +is called the \emph{commutative law} of addition\index + {Addition!commutative law of}\index + {Commutative law of addition}. Notwithstanding this law, +however, it is evidently desirable to distinguish the two quantities +which are to be summed, and out of which the sum is produced, by +special names. As a fact, the two summands\index + {Summands} usually are distinguished +\PG seq=21 Page 10 ------------------------------------------------------ +in some way, for example, by saying $a$ is to be increased by +$b$, or $b$ is to be added to $a$, and so forth. Here, it is plain, $a$ is always +something that is to be increased, $b$ the increase. Accordingly +it has been proposed to call the number which is regarded in addition +as the passive number or the one to be changed, the \emph{augend}\index{Augend}, +and the other which plays the active part, which accomplishes the +change, so to speak, the \emph{increment}\index + {Increment}. Both words are derived from +the Latin and are appropriately chosen. Augend is derived from +\emph{augere}, to increase, and signifies that which is to be increased; +increment comes from \emph{increscere}, to grow, and signifies as in its ordinary +meaning what is added. + +Besides the commutative law one other follows from the idea of +counting---the \emph{associative law} of addition\index + {Addition!associative law of}\index + {Associative law of addition}. This law, which has reference +not to two but to three numbers, states that having a certain +sum, $a + b$, it is indifferent for the result whether we increase the +increment $b$ of that sum by a number, or whether we increase the +sum itself by the same number. Expressed in the symbolic language +of arithmetic this law reads, +\[ +a + (b + c) = (a + b) + c. +\] +To obtain now all the rules of addition we have only to apply the +two laws of commutation and association above stated, though frequently, +in the deduction of the same rule, each must be applied +many times. I may pass over here both the rules and their establishment. + +In addition, two numbers, the augend $a$ and the increment $b$ +are combined into a third number $c$, the sum. From this operation +spring necessarily two inverse operations, the common feature of +which is, that the sum sought in addition is regarded in both as +known, and the difference that in the one the augend also is regarded +as known, and in the other the increment. If we ask what number +added to $a$ gives $c$, we seek the increment. If we ask what number +increased by $b$ gives $c$, we seek the augend. As a matter of reckoning, +the solution of the two questions is the same, since by the commutative +law of addition $a + b = b + a$. Consequently, only one +common name is in use for the two inverses of addition, namely, +\emph{subtraction}\index + {Subtraction|etseq}. But with respect to the notions involved, the two operations +\PG seq=22 Page 11 ------------------------------------------------------ +do differ, and it is accordingly desirable in a logical investigation +of the structure of arithmetic, to distinguish the two by different +names. As in all probability no terms have yet been suggested +for these two kinds of subtraction, I propose here for the +first time the following words for the two operations, namely, \emph + {detraction}\index{Detraction} +to denote the finding of the increment, and \emph{subtertraction}\index + {Subtertraction} to +denote the finding of the augend. We obtain these terms simply +enough by thinking of the augmentation of some object already existing. +For example, the cathedral at Cologne had in its tower an +augend that waited centuries for its increment, which was only +supplied a few decades ago. As the cathedral had originally a +height of one hundred and thirty metres, but after completion was +increased in height twenty-six metres, of the total height of one +hundred and fifty-six metres one hundred and thirty metres is clearly +the augend and twenty-six metres the increment. If, now, we wished +to recover the augend we should have to pull down (Latin, \emph{detrahere}) +the upper part along the whole height. Accordingly, the finding of +the augend is called \emph{detraction}. If we sought the increment, we +should have to pull out the original part from beneath (Latin, \emph{subtertrahere}). +For this reason, the finding of the increment is called \emph{subtertraction}. +Owing to the commutative law, the two inverse operations, +as matters of computation, become one, which bears the name +of \emph{subtraction}. The sign of this operation is the minus sign\index + {Minus sign}, a horizontal +stroke. The number which originally was sum, is called in +subtraction minuend; the number which in addition was increment +is now called detractor; the number which in addition was augend +is now called subtertractor. Comprising the two conceptually different +operations in one single operation, subtraction, we employ +for the number which before was increment or augend, the term subtrahend, +a word which on account of its passive ending is not very +good, and for which, accordingly, E.~Schröder\index + {Schrod@Schröder, E.} proposes to substitute +the word \emph{subtrahent}\index + {Subtrahent}, having an active ending. The result of +subtraction, or what is the same thing, the number sought, is called +the \emph{difference}\index{Difference}. The definition-formula\index + {Definitional formul\ae} of subtraction reads +\[ +a - b + b = a, +\] +that is, $a$ minus $b$ is the number which increased by $b$ gives $a$, or +\PG seq=23 Page 12 ------------------------------------------------------ +the number which added to $b$ gives $a$, according as the one or the +other of the two operations inverse to addition is meant. From the +formula for subtraction, and from the rules which hold for addition, +follow now at once the rules which refer to both addition and subtraction. +These rules we here omit. + +From the foregoing it is plain that the minuend is necessarily +larger than the subtrahent. For in the process of addition the minuend +was the sum, and the sum grew out of the union of two natural +number-pictures.\footnote + {See page \pageref{numberpictures}, \textit{supra}.} +Thus \5 minus \9, or \Num{11} minus \Num{12}, or \8 minus \8, +are combinations of numbers \emph{wholly destitute of meaning}; for no +number, that is, no result of counting, exists that added to \9 gives +the sum \5, or added to \Num{12} gives the sum \Num{11}, or added to \8 gives \8. +What, then, is to be done? Shall we banish entirely from arithmetic +such meaningless combinations of numbers; or, since they +have no meaning, shall we rather invest them with one? If we do +the first, arithmetic will still be confined in the strait-jacket into +which it was forced by the original definition of number as the result +of counting. If we adopt the latter alternative we are forced +to extend our notion of number. But in doing this, we sow the +first seeds of the science of pure arithmetic, an organic body of +knowledge which fructifies all other provinces of science. + +What significance, then, shall we impart to the symbol\index + {Negative numbers|etseq}\index{Quantities!negative|etseq} +\[ +\5-\9? +\] +Since \5 minus \9 possesses no significance whatever, we may, of +course, impart to it any significance we wish. But as a matter +of practical convenience it should be invested with no meaning +that is likely to render it subject to exceptions. As the form of the +symbol $\5-\9$ is the form of a difference, it will be obviously convenient +to give it a meaning which will allow us to reckon with it as +we reckon with every other real difference, that is, with a difference +in which the minuend is larger than the subtrahent. This being +agreed upon, it follows at once that all such symbols in which the +number before the minus sign is less than the number behind it by +the same amount may be put equal to one another. It is practical, +\PG seq=24 Page 13 ------------------------------------------------------ +therefore, to comprise all these symbols under some one single symbol, +and to construct this latter symbol so that it will appear unequivocally +from it by how much the number before the minus sign +is less than the number behind it. This difference, accordingly, is +written down and the minus sign placed before it. + +If the two numbers of such a differential \emph{form}\index + {Differential forms|etseq} are equal, a totally +new sign must be invented for the expression of the fact, having +no relation to the signs which state results of counting. This invention +was not made by the ancient Greeks\index + {Greeks}, as one might naturally +suppose from the high mathematical attainments of that people, but +by Hindu\index{Hindus} Brahman priests at the end of the fourth century after +Christ. The symbol which they invented they called \emph{tsiphra}, empty, +whence is derived the English \emph{cipher}\index{Cipher}. The form of this sign has been +different in different times and with different peoples. But for the +last two or three centuries, since the symbolic language of arithmetic +has become thoroughly established as an international character, +the form of the sign has been $0$\index{Zero} (French \emph{zéro}, German \emph{null}). + +In calling this symbol and the symbols formed of a minus sign +followed by a result of counting, \emph{numbers}, we widen the province of +numbers, which before was wholly limited to results of counting. +In no other way can zero and the negative numbers be introduced +into arithmetic. No man can \emph{prove} that \7 minus \Num{11} is equal to \1 +minus \5. Originally, both are meaningless symbols. And not until +we agree to impart to them a significance which allows us to reckon +with them as we reckon with real differences are we led to a statement +of identity between \7 minus \Num{11} and \1 minus \5. It was a long +time before the negative numbers mentioned acquired the full rights +of citizenship in arithmetic. Cardan\index + {Cardan} called them, in his \textit{Ars Magna}, +\Num{1545}, \emph{numeri ficti}\index + {Numeri ficti} (imaginary numbers), as distinguished from \emph{numeri +veri}\index + {Numeri veri} (real numbers). Not until Descartes, in the first half of the +seventeenth century, was any one bold enough to substitute \emph{numeri +ficti} and \emph{numeri veri} indiscriminately for the same letter of algebraic +expressions. + +We have invested, thus, combinations of signs originally meaningless, +in which a smaller number stood before than after a minus +sign, with a meaning which enables us to reckon with such \emph{apparent} +\PG seq=25 Page 14 ------------------------------------------------------ +differences exactly as we do with ordinary differences. Now it is +just this practical shift of imparting meanings to combinations, which +logically applied deduces naturally the whole system of arithmetic +from the idea of counting and of addition, and which we may characterise, +therefore, as the \emph{foundation-principle}\index + {Foundation-principle} of its whole construction. +This principle, which Hankel\index + {Hankel} once called the \emph{principle of permanence}, +but which I prefer to call the \textsc{principle of no exception}\index + {Principle of no exception}, +may be stated in general terms as follows: + +\emph{In the construction of arithmetic every combination of two previously +defined numbers by a sign for a previously defined operation (plus, minus, +times, etc.) shall be invested with meaning, even where the original definition +of the operation used excludes such a combination; and the meaning +imparted is to be such that the combination considered shall obey the +same formula of definition as a combination having from the outset a signification, +so that the old laws of reckoning shall still hold good and may +still be applied to it.} + +A person who is competent to apply this principle rigorously +and logically will arrive at combinations of numbers whose results +are termed irrational or imaginary with the same necessity and facility +as at the combinations above discussed, whose results are +termed negative numbers and zero. To think of such combinations +as \emph{results} and to call the products reached also ``numbers'' is a misuse +of language. It were better if we used the phrase \emph{forms of numbers}\index + {Number!forms of} +for all numbers that are not the results of counting. But \emph{usus +tyrannus!} + +It will now be my task to show how all numbers at which arithmetic +ever has arrived or ever can arrive naturally flow from the +simple application of the principle of no exception. + +Owing to the commutative and associative laws for addition it +is wholly indifferent for the result of a series of additive processes +in what order the numbers to be summed are added. For example, +\[ +a + (b + c + d) + (e +f) = (a + b + c) + (d + e) + f. +\] +The necessary consequence of this is that we may neglect the consideration +of the order of the numbers and give heed only to what +the quantities are that are to be summed, and, when they are equal, +take note of only two things, namely, of what the quantity which is +\PG seq=26 Page 15 ------------------------------------------------------ +to be repeatedly summed is called and how often it occurs. We +thus reach the notion of multiplication\index + {Multiplication}. To multiply $a$ by $b$ means +to form the sum of $b$ numbers each of which is called $a$. The number +conceived summed is called the multiplicand, the number which +indicates or counts how often the first is conceived summed is called +the multiplier. + +It appears hence, that the multiplier must be a result of counting, +or a number in the original sense of the word, but that the multiplicand +may be any number hitherto defined, that is, may also be +zero or negative. It also follows from this definition that though +the multiplicand may be a concrete number the multiplier cannot. +Therefore, the commutative law of multiplication does not hold +when the multiplicand is concrete. For, to take an example, though +there is sense in requiring four trees to be summed three times, +there is no sense in conceiving the number three summed ``four +trees times.'' When, however, multiplicand and multiplier are unnamed +results of counting, (abstract numbers,) two fundamental +laws hold in multiplication, exactly analogous to the fundamental +laws of addition, namely, the law of commutation\index + {Commutation|(} and the law of +association\index{Law of association}. Thus, +\begin{align*} +a \text{\emph{ times }} b &= b \text{\emph{ times }} a,\\ +\text{and, }a \text{\emph{ times }} (b \text{\emph{ times }} c) + &= (a \text{\emph{ times }} b) \text{\emph{ times }} c. +\end{align*} +The truth and correctness of these laws will be evident, if keeping +to the definition of multiplication as an abbreviated addition of equal +summands, we go back to the laws of addition. Owing to the commutative +law it is unnecessary, for purposes of practical reckoning, +to distinguish multiplicand and multiplier. Both have, therefore, a +common name: \emph{factor}. The result of the multiplication is called the +product; the symbol of multiplication is a dot ($\dotm$) or a cross ($\times$), +which is read ``times.'' Joined with the fundamental formula above +written are a group of subsidiary formul\ae\ which give directions how +a sum or difference is multiplied and how multiplication is performed +with a sum or difference. I need not enter, however, into any discussion +of these rules here. + +As the combination of two numbers by a sign of multiplication +has no significance according to our definition of multiplication, +\PG seq=27 Page 16 ------------------------------------------------------ +when the multiplier is zero or a negative number, it will be seen +that we are again in a position where it is necessary to apply the +above explained principle of no exception. We revert, therefore, to +what we above established, that zero and negative numbers\index + {Negative numbers} are symbols +which have the form of differences, and lay down the rule that +multiplications with zero\index{Zero} and negative numbers shall be performed +exactly as with real differences. Why, then, is minus one times +minus one, for example, equal to plus one? For no other reason +than that minus one can be multiplied with an ordinary difference, +as, for example, \8 minus \5, by first multiplying by \8, then multiplying +by \5, and subtracting the differences obtained, and because +agreeably to the principle of no exception we must say that the multiplication +must be performed according to exactly the same rule +with a symbol which has the \emph{form} of a difference whose minuend is +less by one than its subtrahent. + +As from addition two inverse operations, detraction and subtertraction, +spring, so also from multiplication two inverse operations +must proceed which differ from each other simply in the respect that +in the one the multiplicand is sought and in the other the multiplier. +As matters of computation, these two inverse operations coalesce +in a single operation, namely, division, owing to the validity of the +commutative law in multiplication. But in so far as they are different +ideas, they must be distinguished. As most civilised languages +distinguish the two inverse processes of multiplication in the case +in which the multiplicand is a line, we will adopt for arithmetic a +name which is used in this exception. Let us take this example, +\[ +\4 \text{ \emph{yards}} \times \3 = \Num{12} \text{ \emph{yards}}. +\] +If twelve yards and four yards are given, and the multiplier \3 is +sought, I ask, how many summands, each equal to four yards, give +twelve yards, or, what is the same thing, how often I can lay a +length of four yards on a length of twelve yards? But this is \emph{measuring}. +Secondly, if twelve yards and the number \3 are given, and the +multiplicand four yards is sought, I ask what summand it is which +taken three times gives twelve yards, or, what is the same thing, +what part I shall obtain if I cut up twelve yards into three equal +parts? But this is partition, or \emph{parting}\index + {Parting}. If, therefore, the multiplier +\PG seq=28 Page 17 ------------------------------------------------------ +is sought we call the division \emph{measuring}\index + {Measuring}, and if the multiplicand +is sought, we call it \emph{parting}\index{Parting}. In both cases the number which +was originally the product is called the dividend, and the result the +quotient. The number which originally was multiplicand is called +the measure; the number which originally was multiplier is called +the parter. The common name for measure and parter is divisor. +The common symbol for both kinds of division is a colon, a horizontal +stroke, or a combination of both. Its definitional formula +reads, +\[ +(a\div b)\dotm b = a, \text{ or, }\frac ab\dotm b = a. +\] +Accordingly, dividing $a$ by $b$ means, to find the number which multiplied +by $b$ gives $a$, or to find the number \emph{with} which $b$ must be +multiplied to produce $a$. From this formula, together with the +formul\ae\ relative to multiplication, the well-known rules of division +are derived, which I here pass over. + +In the dividend of a quotient only such numbers can have a +place which are the product of the divisor with some previously defined +number. For example, if the divisor is \5 the dividend can +only be \5, \Num{10}, \Num{15}, and so forth, and \0, $-\5$, $-\Num{10}$ and + so forth. Accordingly, +a stroke of division having underneath it \5 and above it +a number different from the numbers just named is a combination +of symbols having no meaning. For example, $\frac35$ or $\frac{12}5$ are meaningless +symbols. Now, conformably to the principle of no exception +we must invest such symbols which have the form of a quotient +without their dividend being the product of the divisor with any +number yet defined, with a meaning such that we shall be able to +reckon with such apparent quotients as with ordinary quotients. +This is done by our agreeing always to put the product of such a +quotient form with its divisor equal to its dividend. In this way we +reach the definition of broken numbers or \emph{fractions}\index + {Fractions|etseq}, which by the +application of the principle of no exception spring from division exactly +as zero and negative numbers sprang from subtraction. The +latter had their origin in the impossibility of the subtraction; the +former have their origin in the impossibility of the division. Putting +\PG seq=29 Page 18 ------------------------------------------------------ +together now both these extensions of the domain of numbers, we +arrive at \emph{negative fractional numbers}\index + {Fractional numbers, negative}\index{Negative numbers}. + +We pass over the easily deduced rules of computation for fractions +and shall only direct the reader's attention to the connexion +which exists between fractional and non-fractional or, as we usually +say, whole numbers. Since the number \Num{12} lies between the numbers +\Num{10} and \Num{15}, or, what is the same thing, $\Num{10} <\Num{12} < \Num{15}$, and since +$\Num{10}:\5 =\2$, $\Num{15}:\5=\3$, we say also that $\Num{12}:\5$ lies between \2 and \3, or +that +\[ +\2 < \tfrac{12}5 < \3. +\] +In itself, the notion of ``less than'' has significance only for results +of counting. Consequently, it must first be stated what is meant +when it is said that \2 is less than $\frac{12}5$. Plainly, nothing is meant by +this except that \2 times \5 is less than \Num{12}. We thus see that every +broken number can be so interpolated between two whole numbers +differing from each other only by \1 that the one shall be smaller +and the other greater, where smaller and greater have the meaning +above given. + +From the above definitions and the laws of commutation\index{Commutation|)} and +association\index + {Association, laws of} all possible rules of computation follow, which in virtue +of the principle of no exception now hold indiscriminately for all +numbers hitherto defined. It is a consequence of these rules, again, +that the combination of two such numbers by means of any of the +operations defined must in every case lead to a number which has +been already defined, that is, to a positive or negative whole or fractional +number, or to zero. The sole exception is the case where +such a number is to be divided by zero. If the dividend also is +zero, that is, if we have the combination $\frac00$, the expression is one +which stands for any number whatsoever, because any number whatsoever, +no matter what it is, if multiplied by zero gives zero. But +if the dividend is not zero but some other number $a$, be it what it +will, we get a quotient form to which \emph{no} number hitherto defined +can be equated. But we discover that if we apply the ordinary arithmetical +rules to $a\div0$ all such forms may be equated to one another +both when $a$ is positive and also when $a$ is negative. We may therefore +invent two new signs for such quotient forms, namely $+\infty$ and +\PG seq=30 Page 19 ------------------------------------------------------ +$-\infty$. We find, further, that in transferring the notions greater and +less to these symbols, $+\infty$ is greater than any positive number, +however great, and $-\infty$ is smaller than any negative number, however +small. We read these new signs, accordingly, ``plus infinitely +great'' and ``minus infinitely great\index{Infinitely great}.'' + +But even here arithmetic has not reached its completion, although +the combination of as many previously defined numbers as +we please by as many previously defined operations as we please +will still lead necessarily to some previously defined number. Every +science must make every possible advance, and still one step in advance +is possible in arithmetic. For in virtue of the laws of commutation +and association, which also fortunately obtain in multiplication, +just as we advance from addition to multiplication, so here +again we may ascend from multiplication to \emph{an operation of the third +degree}\index{Third degree, operations of the}. +For, just as for $a+a+a+a$ we read $4\dotm a$, so with the same +reason we may introduce some more abbreviated designation for +$a\dotm a\dotm a\dotm a$. The introduction of this new operation is in itself simply +a matter of convenience and not an extension of the ideas of arithmetic. +But if after having introduced this operation we repeatedly +apply the monistic principle of arithmetic, the principle of no exception, +we reach new means of computation which have led to undreamt +of advances not only in the hands of mathematicians but +also in the hands of natural scientists. The abbreviated designation +mentioned, which, fructified by the principle of no exception, can +render science such incalculable services, is simply that of writing +for a product of $b$ factors of which each is called $a$, $a^b$, which we +read $a$ to the $b$\th\ power. Here a new direct operation, that of \emph + {involution}\index{Involution}, +is defined, and from now on we are justified in distinguishing +operations which are not inverses of others, as addition, multiplication, +and involution, by \emph{numbers of degree}\index + {Degreez@Degree, numbers of}. Addition is the direct +operation of the first degree, multiplication that of the second degree, +and involution that of the third degree. In the expression +$a^b$ the passive number $a$ is called the \emph{base}, the active number $b$ the +\emph{exponent}, the result, the \emph{power}. + +But whilst in the direct operations of the first and second degree, +the laws of commutation and association hold, here in involution, +\PG seq=31 Page 20 ------------------------------------------------------ +the operation of third degree, the two laws are inapplicable, +and the result of their inapplicability is that operations of a still +higher degree than the third form no possible advancement of pure +arithmetic. The product of $b$ factors $a$ is not equal to the product +of $a$ factors $b$; that is, the law of commutation does not hold. The +only two different integers for which $a$ to the $b$\th\ power is equal to $b$ +to the $a$\th\ power are \2 and \4, for \2 to the \4\th\ power is \Num{16}, and \4 to +the second power also is \Num{16}. So, too, the law of association as a +general rule does not hold. For it is hardly the same thing whether +we take the ($b^c$)\th\ power of $a$ or the $c$\th\ power of $a^b$. + +From the definition of involution follow the usual rules for reckoning +with powers\index{Powers}, of which we shall only mention one, namely, +that the $(b-c)$\th\ power of $a$ is equal to the result of the division of +$a$ to the $b$\th\ power by $a$ to the $c$\th\ power. If we put here $c$ equal to +$b$, we are obliged, by the principle of no exception, to put $a$ to the +$0$\th\ power equal to \1; a new result not contained in the original notion +of involution, for that implied necessarily that the exponent +should be a result of counting. Again, if we make $b$ smaller than $c$ +we obtain a \emph{negative exponent}\index + {Negative exponents}, which we should not know how to +dispose of if we did not follow our monistic law of arithmetic. According +to the latter, $a$ to the $(b-c)$\th\ power must still remain equal +to $a^b$ divided by $a^c$ even when $b$ is smaller than $c$. Whence follows +that $a$ to the minus $d$\th\ power is equal to \1 divided by $a$ to the +$d$\th\ power, or to take specific numbers, that \3 to the minus \2\textsuperscript{\textit{nd}} power +is equal to $\frac19$. + +At this point, perhaps, the reader will inquire what $a$ raised to a +fractional power is. But this can be explained only when we have +discussed the inverse processes of involution, to which we now pass. + +If $a^b=c$, we may ask two questions: first, what the base is +which raised to the $b$\th\ power gives $c$; the second, what the exponent +of the power is to which $a$ must be raised to produce $c$. In the first +case we seek the base, and term the operation which yields this result +\emph{evolution}\index + {Evolution}; in the second case we seek the exponent and call the +operation which yields this exponent, the \emph{finding of the logarithm}\index + {Logarithm, finding of the}. +In the first case, we write $\sqrt[b]{c}=a$ (which we read, the $b$\th\ root of $c$ is +equal to $a$), and call $c$ the \emph{radicand}\index +{Radicand}, $b$ the \emph{exponent of the root}, and $a$ +\PG seq=32 Page 21 ------------------------------------------------------ +the \emph{root}. In the second case, we write $\log_a c=b$ (which we read, the +logarithm of $c$ to the base $a$ is equal to $b$), and call $c$ the \emph{logarithmand}\index{Logarithmand} +or \emph{number}, $a$ the \emph{base of the logarithm}, and $b$ the \emph{logarithm}. + +While, owing to the validity of the law of commutation in addition +and multiplication, the two inverse processes of those operations +are identical so far as computation is concerned, here in the +case of involution the two inverse operations are in this regard essentially +different, for in this case the law of commutation does not +hold. + +From the definitional formul\ae\index + {Definitional formul\ae} for evolution\index{Evolution} and the finding of +logarithms, namely, +\[ +(\sqrt[b]{c})^b = c,\text{ and }(a)^{\log_a c}=c, +\] +follow, by the application of the laws of involution\index + {Involution}, the rules for +computation with roots and logarithms. These rules we pass over +here, only remarking, first, that for the present $\sqrt[b]{c}$ has meaning +only when $c$ is the $b$\th\ power of some number already defined; and, +secondly, that for the present also $\log_a c$ has meaning only when $c$ +can be produced by raising the number $a$ to some power which is a +number already defined. In the phrase ``has only meaning for the +present'' is contained a possibility of new extensions of the domain +of number. But before we pass to those extensions we shall first +make use of the idea of evolution just defined to extend the notion +of power also to cases in which the exponent is a fractional number. + +According to the original definition of involution\index + {Involution}, $a^b$ was meaningless +except where $b$ was a result of counting. But afterwards, +even powers which had for their exponents zero\index + {Zero exponents} or a negative integer\index{Negative exponents} +could be invested with meaning. Now we have to consider the +arithmetical combination ``$a$ raised to the fractional power $\frac pq$." The +principle of no exception compels us to give to the arithmetical combination +``$a$ to the $\frac pq$\th\ power'' a significance such that all the rules +of computation will hold with respect to it. Now, one rule that +holds is, that the $m$\th\ power of the $n$\th\ power of $a$ is equal to the +$(m\times n)$\th\ power of $a$. Consequently, the $q$\th\ power of $a$ raised to the +$\frac pq$\th\ power must be equal to $a$ raised to a power whose exponent is +equal to $\frac pq$ times $q$. But the last-mentioned product gives, according +to the definition of division, the number $p$. Consequently the symbol +\PG seq=33 Page 22 ------------------------------------------------------ +$a$ to the $\frac pq$\th\ power is so constituted that its $q$\th\ power is equal to +$a$ to the $p$\th\ power; \IE, it is equal to the $q$\th\ root of $a^p$. Similarly, +we find that the symbol ``$a$ to the minus $\frac pq$\th\ power'' must be put +equal to \1 divided by the $q$\th\ root of $a$ to the $p$\th\ power, if we are to +reckon with this symbol as we do with real powers. Again, just as +$a$ to the $b$\th\ power is invested with meaning when $b$ is a fractional +number, so some meaning harmonious with the principle of no exception +must be imparted to the $b$\th\ root of $c$ where $b$ is a positive or +negative fractional number. For example, the three-fourths\th\ root +of \8 is equal to \8 to the $\frac43$ power, that is, to the cube root of \8 to the +\4\th\ power, or \Num{16}. + +The principle underlying arithmetic now also compels us to +give to the symbol the ``$b$\th\ root of $c$'' a meaning when $c$ is not the +$b$\th\ power of any number yet defined. First, let $c$ be any \emph{positive} +integer or fraction. Then always to be able to reckon with the +$b$\th\ root of $c$ in the same way that we do with extractible roots, we +must agree always to put the $b$\th\ power of the $b$\th\ root of $c$ equal to +$c$---for example, $(\sqrt[2]{\3})^2$ always exactly equal to \3. A careful inspection +of the new symbols, which we will also call numbers, shows, that +though no one of them is exactly equal to a number hitherto defined, +yet by a certain extension of the notions greater and less, two numbers +of the character of numbers already defined may be found for +each such new number, such that the new number is greater than the +one and less than the other of the two, and that further, these two +numbers may be made to differ from each other by as small a quantity +as we please. For example, +\[ +(\tfrac75)^3 = \tfrac{343}{125} = 2\tfrac{93}{125} < 3<3\tfrac38 = \tfrac{27}8 = (\tfrac32)^3. +\] +The number \3, as we see, is here included between two limits which +are the third powers of two numbers $\frac75$ and $\tfrac32$ whose difference is $\tfrac1{10}$. +We could also have arranged it so that the difference should be +equal to $\frac1{100}$, or to any specified number, however small. Now, instead +of putting the symbol ``less than'' between $(\frac75)^3$ and \3, and +between \3 and $(\frac32)^3$, let us put it between their third roots; for example, +let us say: +\[ +\tfrac75 < \sqrt[3]{3} < \tfrac32, \text{ meaning by this that }(\tfrac75)^3 < 3 < (\tfrac32)^3. +\] +In this sense we may say that the new numbers always lie \emph{between} +\PG seq=34 Page 23 ------------------------------------------------------ +two old numbers whose difference may be made as small as we +please. Numbers possessing this property are called \emph{irrational} numbers\index +{Irrational numbers}\index{Numbera@Numbers!irrational}, +in contradistinction to the numbers hitherto defined, which are +termed \emph{rational}. The considerations which before led us to negative +rational numbers, now also lead us to negative irrational numbers. +The repeated application of addition and multiplication as of their +inverse processes to irrational numbers, (numbers which though not +exactly equal to previously defined rational numbers may yet be +brought as near to them as we please,) again simply leads to numbers +of the same class. + +A totally new domain of numbers is reached, however, when we +attempt to impart meaning to \emph{the square roots of negative numbers}. +The square root of minus \9 is neither equal to plus \3 nor to minus +\3, since each multiplied by itself gives plus \9, nor is it equal to any +other number hitherto defined. Accordingly, the square root of minus +\9 is a new number-form, to which, harmoniously with the principle +of no exception, we may give the definition that $(\sqrt[2]{-\9})^2$ shall always +be put equal to minus \9.\footnote + {Henceforward we shall use the simpler sign $\sqrt{\phantom{0}}$ for $\sqrt[2]{\phantom{0}}$.} +Keeping to this definition we see +at once that $\sqrt{-a}$, where $a$ is any positive rational or irrational +number, is a symbol which can be put equal to the product of $\sqrt{+a}$ +by $\sqrt{-\1}$. In extending to these new numbers the rights of arithmetical +citizenship, in calling them also ``numbers,'' and so shaping +their definition that we can reckon with them by the same rules as +with already defined numbers, we obtain a fourth extension of the +domain of numbers which has become of the greatest importance +for the progress of all branches of mathematics. The newly defined +numbers are called \emph{imaginary}\index + {Imaginary numbers}\index + {Numbera@Numbers!imaginary}, in contradistinction to all heretofore +defined, which are called \emph{real}\index + {Real numbers}\index{Numbera@Numbers!real}. Since all imaginary numbers can be +represented as products of real numbers with the square root of +minus one, it is convenient to introduce for this one imaginary number +some concise symbol. This symbol is the first letter of the word +imaginary, namely, $i$; so that we can always put for such an expression +as $\sqrt{-\9}$, $\3\dotm i$. + +If we combine real and imaginary numbers by operations of the +\PG seq=35 Page 24 ------------------------------------------------------ +first and second degree, always supposing that we follow in our +reckoning with imaginary numbers the same rules that we do in +reckoning with real numbers, we always arrive again at real or +imaginary numbers, excepting when we join together a real and an +imaginary number by addition or its inverse operations. In this +case \emph{we reach the symbol} $a + i\dotm b$, where $a$ and $b$ stand for real numbers. +Agreeably to the principle of no exception we are permitted +to reckon with $a + ib$ according to the same rules of computation as +with symbols previously defined, if for the second power of $i$ we +always substitute minus \1. + +In the numerical combination $a + ib$, which we also call number, +we have found the most general numerical form to which the +laws of arithmetic can lead, even though we wished to extend the +limits of arithmetic still further. Of course, we must represent to +ourselves here by $a$ and $b$ either zero or positive or negative rational +or irrational numbers. If $b$ is zero, $a + ib$ represents all real numbers; +if $a$ is zero, it stands for all purely imaginary numbers. This +general number $a + ib$ is called a \emph{complex number}\index + {Complex numbers}\index{Number!ez@complex}, so that the complex +number includes in itself as special cases all numbers heretofore +defined. By the introduction of irrational, purely imaginary, +and the still more general complex numbers, all combinations become +invested with meaning which the operations of the third degree +can produce. For example, the fifth root of \5 is an irrational +number, the logarithm of \2 to the base \Num{10} is an irrational number. +The logarithm of minus \1 to the base \2 is a purely imaginary number; +the fourth root of minus \1 is a complex number. Indeed, we +may recognise, proceeding still further, \emph{that every combination of two +complex numbers, by means of any of the operations of the first, second, +or third degree will lead in turn to a complex number}, that is to say, +never furnishes occasion, by application of the principle of no exception, +for inventing new forms of numbers. + +A certain limit is thus reached in the construction of arithmetic. +But such a limit was also twice previously reached. After the investigation +of addition and its inverse operations, we reached no +other numbers except zero and positive and negative whole numbers, +and every combination of such numbers by operations of the +\PG seq=36 Page 25 ------------------------------------------------------ +first degree led to no new numbers. After the investigation of multiplication +and its inverse operations, the positive or negative fractional +numbers and ``infinitely great'' were added, and again we +could say that the combination of two already defined numbers +by operations of the first and second degree in turn also always +led to numbers already defined. Now we have reached a point at +which we can say that the combination of two complex numbers by +all operations of the first, second, and third degree must again +always lead to complex numbers; only that now such a combination +does not necessarily always lead to a single number, but may +lead to many regularly arranged numbers. For example, the combination +``logarithm of minus one to a positive base'' furnishes a +countless number of results which form an arithmetical series of +purely imaginary numbers. \emph{Still, in no case now do we arrive at new +classes of numbers.} But just as before the ascent from multiplication +to involution brought in its train the definition of new numbers, so +it is also possible that \emph{some new operation springing out of involution +as involution sprang from multiplication might furnish the germ of other +new numbers which are not reducible to} $a + ib$. As a matter of fact, +mathematicians have asked themselves this question and investigated +the direct operation of the fourth degree, together with its +inverse processes. The result of their investigations was, that an +operation which springs from involution as involution sprang from +multiplication is incapable of performing any real mathematical service; +the reason of which is, that in involution the laws of commutation +and association do not hold. It also further appeared that +the operations of the fourth degree could not give rise to new numbers. +No more so can operations of still higher degrees. With +respect to quaternions\index{Quaternions}, which many might be disposed to regard as +new numbers, it will be evident that though quaternions are valuable +means of investigation in geometry and mechanics they are not +numbers of arithmetic, because the rules of arithmetic are not unconditionally +applicable to them. + +The building up of arithmetic is thus completed. The extensions +of the domain of number are ended. It only remains to be +asked why the science of arithmetic appears in its structure so logical, +\PG seq=37 Page 26 ------------------------------------------------------ +natural, and unarbitrary; why zero, negative, and fractional +numbers appear as much derived and as little original as irrational, +imaginary, and complex numbers? We answer, wholly and alone in +virtue of the logical application of the monistic principle of arithmetic, +the principle of no exception\index + {Arithmetic!monism in|)}\index{Monism in arithmetic|)}. +\PG seq=38 Page 27 ------------------------------------------------------ + + + +\chapter{On the Nature of Mathematical +Knowledge} + +\lettrine{``M}{athematically} certain and unequivocal'' is a phrase\index + {Mathematical knowledge|(Mathematical} +which is often heard in the sciences and in common life, +to express the idea that the seal of truth is more deeply imprinted +upon a proposition than is the case with ordinary acts of knowledge. +We propose to investigate in this article the extent to which mathematical +knowledge really is more certain and unequivocal than +other knowledge. + +The intrinsic character\index + {Mathematical knowledge!intrinsic character of} of mathematical research and knowledge +is based essentially on three properties: first, on its conservative\index + {Mathematics!most conservative of all sciences} +attitude towards the old truths and discoveries of mathematics; +secondly, on its progressive mode of development, due to the incessant +acquisition of new knowledge on the basis of the old; and thirdly, +on its self-sufficiency and its consequent absolute independence. + +That mathematics is the most conservative of all the sciences +is apparent from the incontestability of its propositions. This last +character bestows on mathematics the enviable superiority that no +new development can undo the work of previous developments or +substitute new in the place of old results. The discoveries that +Pythagoras\index{Pythagoras}, Archimedes\index + {Archimedes}, and Apollonius\index{Apollonius} made are as valid to-day +as they were two thousand years ago. This is a trait which no +other science possesses. The notions of previous centuries regarding +the nature of heat have been disproved. Goethe's theory of +colors is now antiquated. The theory of the binary combination of +salts was supplanted by the theory of substitution, and this, in its +turn, has also given way to newer conceptions. Think of the profound +\PG seq=39 Page 28 ------------------------------------------------------ +changes which the conceptions of theoretical medicine, zoölogy, +botany, mineralogy, and geology have undergone. It is the +same, too, in the other sciences. In philology, comparative linguistics, +and history our ideas are quite different from what they formerly +were. + +In no other science is it so indispensable a condition that whatever +is asserted must be true, as it is in mathematics. Whenever, +therefore, a controversy arises in mathematics, the issue is not +whether a thing is true or not, but whether the proof might not be +conducted more simply in some other way, or whether the proposition +demonstrated is sufficiently important for the advancement of the +science as to deserve especial enunciation and emphasis, or finally, +whether the proposition is not a special case of some other and +more general truth which is just as easily discovered. + +Let me recall the controversy which has been waged in this +century regarding the eleventh axiom of Euclid\index + {Euclid}, that only one line +can be drawn through a point parallel to another straight line\index + {Parallels, theory of}. This +discussion impugned in no wise the truth of the proposition; for +that things are true in mathematics is so much a matter of course +that on this point it is impossible for a controversy to arise. The +discussion merely touched the question whether the axiom was +capable of demonstration solely by means of the other propositions, +or whether it was not a special property, apprehensible only by +sense-experience, of that space of three dimensions in which the +organic world has been produced and which therefore is of all others +alone within the reach of our powers of representation. The truth +of the last supposition affects in no respect the correctness of the +axiom but simply assigns to it, in an epistemological regard, a different +status from what it would have if it were demonstrable, as +was at one time thought, without the aid of the senses, and solely by +the other propositions of mathematics. + +I may recall also a second controversy which arose a few decades +ago as to whether all continuous functions\index + {Continuous functions} were differentiable. +In the outcome, continuous functions were defined that possessed +no differential coefficient, and it was thus learned that certain truths +which were enunciated unconditionally by Newton\index{Newton}, Leibnitz\index{Leibnitz}, and +\PG seq=40 Page 29 ------------------------------------------------------ +their mathematical successors, required qualification. But this did +not invalidate at all the correctness of the method of differentiation, +nor its application in all practical cases; the theoretical speculations +pursued on this subject simply clarified ideas and sifted out +the conditions upon which differentiability depended. Happily the +gifted minds who invent the new methods and open up the new +paths of research in mathematics, are not deterred by the fear that +a subsequent generation gifted with unusual acumen will spy out +isolated cases in which their methods fail. Happily the creators +of the differential calculus\index + {Calculus, Differential} pushed onward without a thought that a +critical posterity would discover exceptions to their results. In +every great advance that mathematics makes, the clarification and +scrutinisation of the results reached are reserved necessarily for a +subsequent period, but with it the demonstration of those results is +more rigorously established. Despite all this, however, in no science +does cognition bear so unmistakably the imprint of truth as in +pure mathematics. And this fact bestows on mathematics its conservative +character. + +This conservative character again is displayed in the \emph{objects} of +mathematical research. The physician, the historian, the geographer, +and the philologist have to-day quite different fields of investigation +from what they had centuries ago. In mathematics, too, +every new age gives birth to new problems, arising partly from the +advance of the science itself, and partly also from the advance of +civilisation, where improvements in the other sciences bring in their +train new problems that are constantly taxing afresh the resources +of mathematics. But despite all this, in mathematics more than in +any other science problems exist that have played a rôle for hundreds, +nay, for thousands of years. + +In the oldest mathematical manuscript which we possess, the +Rhind Papyrus\index + {Rhind Papyrus} of the British Museum, which dates back to the +eighteenth century before Christ, and whose decipherment we owe +to the industry of Eisenlohr\index + {Eisenlohr}, we find an attempt to solve the problem +of converting a circle into a square of equal area, a problem +whose history covers a period of three and a half thousand years. +For it was not until \Num{1882} that a rigorous proof was given of the impossibility +\PG seq=41 Page 30 ------------------------------------------------------ +of solving this problem exactly by the use of straight +edge and compasses alone. (See pp.~\pageref{p:116}, \pageref{p:141}--\pageref{p:143}.) + +It is, of course, the insoluble problems\index + {Insoluble problems} that have the longest +history; partly because it is harder to show that a thing is impossible +than that it is possible, and, on the other hand, because problems +that have long defied solution are ever evoking anew the spirit +of inquiry and the ambition of mathematicians, and because the uncertainty +of insolubility lends to such problems a peculiar charm. +Of the geometrical problems that have occupied competent and incompetent +minds from the time of the ancient Greeks to the present +may be mentioned in addition to the squaring of the circle\index + {Squaring of the circle} two +others that are also perhaps well-known to educated readers, at least +by name: the trisection\index + {Trisection of the angle} of the angle and the Delic % the proper translation would be "Delian" +problem of the duplication of the cube\index + {Duplication of the cube}. All three problems involve the condition, +which is often overlooked by lay readers, that only straight edge +and compasses shall be employed in the constructions. In the trisection +of the angle any angle is assigned, and it is required to find +the two straight lines which divide the angle into three equal parts. +In the Delic problem the edge of a cube is given and the edge of a +second cube is sought, containing twice the volume of the first cube. +In Greece, in the golden age of the sciences, when all scholars had +to understand mathematics, it was a fashionable requisite almost to +have employed oneself on these famous problems. + +Fortunately for us, these problems were insoluble. For in their +ambition to conquer them it came to pass that men busied themselves +more and more with geometry, and in this way kept constantly discovering +new truths and developing new theories, all of which perhaps +might never have been done if the problems had been soluble +and had early received their solutions. Thus is the struggle after +truth often more fruitful than the actual discovery of truth. So, too, +although in a slightly different sense, the apophthegm of Lessing\index{Lessing} is +confirmed here, that the search for truth is to be preferred to its +possession. + +Whilst the three above-named problems are now acknowledged +to be insoluble, and have ceased, therefore, to stimulate mathematical +inquiry, there are of course other problems in mathematics +\PG seq=42 Page 31 ------------------------------------------------------ +whose solution has been sought for a long time, but not yet reached, +and in the case of which there is no reason for supposing that they +are insoluble. Of such problems the two following perhaps have +found their way out of the isolated circles of mathematicians and +have become more or less known to other scholars. I refer to the +astronomical Problem of Three Bodies\index + {Three Bodies, problem of} and to the problem of the +frequency of prime numbers\index + {Prime numbers, problem of the frequency of}. The first of these two problems assumes +three or more heavenly bodies whose movements are mutually +influenced by one another according to Newton's law of gravitation, +and requires the exact determination of the path which each body +describes. The second problem requires the construction of a formula +which shall tell how many prime numbers there are below a +certain given number. So far these two problems have been solved +only approximately, and not with absolute mathematical exactness. + +If the eternal and inviolable correctness of its truths lends to +mathematical research, and therefore also to mathematical knowledge, +a \emph{conservative} character, on the other hand, by the continuous +outgrowth of new truths and methods from the old, \emph{progressiveness} +is also one of its characteristics. In marvellous profusion old knowledge +is augmented by new, which has the old as its necessary condition, +and, therefore, could not have arisen had not the old preceded +it. The indestructibility of the edifice of mathematics renders +it possible that the work can be carried to ever loftier and loftier +heights without fear that the highest stories shall be less solid and +safe than the foundations, which are the axioms, or the lower stories, +which are the elementary propositions. But it is necessary for +this that all the stones should be \emph{properly fitted together}; and it +would be idle labor to attempt to lay a stone that belonged above in +a place below. A good example of a stone of this character belonging +in what is now the uppermost layer of the edifice, is Lindemann's\index{Lindemann} +demonstration of the insolubility of the quadrature of the +circle, a demonstration of which interesting simplifications have +been given by several mathematicians, including Weierstrass\index{Weierstrass} and +Felix Klein\index{Klein, Felix}. Lindemann's demonstration could not have been produced +in the preceding century, because it rests necessarily on theories +whose development falls in the present century. It is true, +\PG seq=43 Page 32 ------------------------------------------------------ +Lambert\index{Lambert} succeeded in \Num{1761} in demonstrating the irrationality of the +ratio of the circumference of a circle to its diameter, or, which is +the same thing, the irrationality of the ratio of the area of a circle +to the area of the square on its radius. Afterwards, Lambert also +supplied a proof that it was impossible for this ratio to be the square +root of a rational number. But this was the first step only in a long +journey. The attempt to prove that the old problem is insoluble +was still destined to fail. An astounding mass of mathematical investigations +were necessary before the demonstration could be successfully +accomplished. + +As we see, the majority of the mathematical truths now possessed +by us presuppose the intellectual toil of many centuries. A +mathematician, therefore, who wishes to-day to acquire a thorough +understanding of modern research in this department, must think +over again in quickened tempo the mathematical labors of several +centuries. This constant dependence of new results on old ones +stamps mathematics as a science of uncommon exclusiveness and +renders it generally impossible to lay open to uninitiated readers a +speedy path to the apprehension of the higher mathematical truths. +For this reason, too, the theories and results of mathematics are +rarely adapted for popular presentation. There is no royal road to +the knowledge of mathematics, as Euclid once said to the first +Egyptian Ptolemy\index + {Ptolemy}. This same inaccessibility of mathematics, although +it secures for it a lofty and aristocratic place among the sciences, +also renders it odious to those who have never learned it, and +who dread the great labor involved in acquiring an understanding +of the questions of modern mathematics. Neither in the languages +nor in the natural sciences are the investigations and results so +closely interdependent as to make it impossible to acquaint the uninitiated +student with single branches or with particular results of +these sciences, without causing him to go through a long course of +preliminary study. + +The third trait which distinguishes mathematical research is its +self-sufficiency\index + {Mathematics!self-sufficiency of}. In philology the field of inquiry is the organic one +of languages, and philology, therefore, is dependent in its investigations +on the mode of development of languages, which is more or +\PG seq=44 Page 33 ------------------------------------------------------ +less accidental. Its task is connected with something which is given +to it from without and which it cannot alter. It is much the same +with the science of history\index + {History}, which must contemplate the history of +mankind as it has actually occurred. Also zoölogy, botany, mineralogy, +geology, and chemistry work with given data. In order not +to become involved in futile speculations the last-mentioned sciences +are constantly and inevitably obliged to revert to observations +by the senses. It is then their task to link together these individual +observations by bonds of causality and in this way to +erect from the single stones an edifice, the view of which will render +it easier for limited human intelligence to comprehend nature. +Physics\index{Physics} of all sciences stands nearest to mathematics in this respect, +because unlike the other sciences she is generally in need of only a +few observations in order to proceed deductively. But physics, +too, must resort to observations of nature, and could not do without +them for any length of time. + +Mathematics alone, after certain premises have been permanently +established, is able to pursue its course of development independently +and unmindful of things outside of it. It can leave +entirely unnoticed, questions and influences emanating from the +outer world, and continue nevertheless its development. +As regards geometry, the first beginnings of this science did +indeed take their origin in the requirements of practical life. But +it was not long before it freed itself from the restrictions of the practical +art to which it owed its birth. Herodotus\index{Herodotus} recounts that geometry +had its origin\index + {Geometry!origin of} in Egypt where the inundations of the Nile obliterated +the boundaries of the riparian estates, and by giving rise +to frequent disputes constantly compelled the inhabitants to compare +the areas of fields of different shapes. But with the early Greek +mathematicians, who were the heirs of the Egyptian art of measurement, +geometry appeared as a science which men pursued for its +own sake without a thought of how their intellectual discoveries +could be turned to practical account. + +Nevertheless, although the workers in the domain of pure mathematics +are not stimulated by the thought that their researches are +likely to be of practical value, yet that result is still frequently realised, +\PG seq=45 Page 34 ------------------------------------------------------ +often after the lapse of centuries. The history of mathematics +shows numerous instances of mathematical results which +were originally the outcome of a mere desire to extend the science, +suddenly receiving in astronomy, mechanics, or in physics practical +applications which their originators could scarce have dreamt of. +Thus Apollonius\index + {Apollonius} erected in ancient times the stately edifice of the +properties of conic sections\index + {Conic sections}, without having any idea that the planets +moved about the sun in conic sections, and that a Kepler\index{Kepler} and a +Newton\index{Newton} were one day to come who should apply these properties to +explaining and calculating the motions of the planets about the +sun. The question of the practical availability of its results in other +fields has at no period exercised more than a subordinate influence +on mathematical inquiry. Particularly is this true of \emph{modern} mathematical +research, whether the same consist in the extended development +of isolated theories or in uniting under a higher point of +view theories heretofore regarded as different.\footnote + {Cf.\ Felix Klein, ``Remarks Given at the Opening of the Mathematical and + Astronomical Congress at Chicago.'' \textit + {The Monist} (Vol.~IV, No.~\1, October, \Num{1893}).} + +This independence of its character has rendered the results of +pure mathematics independent also of the accidental direction which +the development of civilisation has taken on our planet; so that +the remark is not altogether without justification, that if beings endowed +with intelligence existed on other planets, the truths of mathematics +would afford the only basis of an understanding with them. +Uninterruptedly and wholly from its own resources mathematics has +built itself up. It is scarcely credible to a person not versed in the +science, that mathematicians can derive satisfaction from the comfortless +and wearisome operation of heaping up demonstration on +demonstration, of rivetting truth on truth, and of tormenting themselves +with self-imposed problems, whose solution stands no one in +stead, and affords satisfaction to no one but the solver himself. Yet +this self-sufficiency of mathematicians becomes a little more intelligible +when we reflect that the progress which has been made, particularly +in the last few decades, and which is uninfluenced from +without, does not consist solely in the accumulation of new truths +\PG seq=46 Page 35 ------------------------------------------------------ +and in the enunciation of new problems, nor solely in deductions +and solutions, but culminates rather in the discovery of new methods\index + {Methods, discovery of} +and points of view in which the old disconnected and isolated +results appear suddenly in a new connexion or as different interpretations +of a common fundamental truth, or finally, as a single organic +whole. + +Thus, for example, the idea of representing imaginary\index + {Imaginary numbers}\index{Numbera@Numbers!imaginary} and complex +numbers\index{Complex numbers}\index + {Number!ez@complex} in a plane, two apparently different branches, the theory +of dividing the circumference of a circle into any given number of +equal parts, and the theory of the solutions of the equation $x^n=\1$, +have been made to exhibit an extremely simple connexion with one +another which enables us to express many a truth of algebra in a +corresponding truth of geometry and \emph{vice versa}. Another example is +afforded by the discovery which we chiefly owe to Alfred Clebsch\index{Clebsch, Alfred}, +of the relation which subsists between the higher theory of functions\index + {Functions, theory of} +and the theory of algebraic curves\index + {Algebraic curves, theory of}, a relation which led to the +discovery of the condition under which two curves can be co-ordinated +to each other, point for point, and hence also adequately represented +on each other. Of course such combinations and extensions +of view possess a much greater charm for the mathematician +than the mere accumulation of truths and solutions, whose fascination +consists entirely in their truth or correctness. + +From these three cardinal characteristics\index + {Mathematical knowledge!characteristics of}, now, which distinguish +mathematical \emph{research} from research in other fields, we may +gather at once the three positive characteristics that distinguish +mathematical \emph{knowledge} from other knowledge. They may be briefly +expressed as follows; first, mathematical knowledge bears more +distinctly the imprint of truth on all its results than any other +kind of knowledge; secondly, it is always a sure preliminary step to +the attainment of other correct knowledge; thirdly, it has no need +of other knowledge. Naturally, however, there are associated with +these characteristics which place mathematical knowledge high +above all other knowledge, other characteristics which somewhat +counterbalance the great superiority which mathematics thus appears +to have over the other sciences. In order to show more distinctly +the nature of these characteristics, which we prefer to call +\PG seq=47 Page 36 ------------------------------------------------------ +negative, we shall select and confine our remarks to a branch which +is commonly taken to be synonymous with mathematics, namely, to +arithmetic\index{Arithmetic} in the broadest sense of the word. + +The subject of inquiry in arithmetic is numbers and their combinations. +On this account arithmetic is, of all sciences, most free +from what lies outside its boundaries. Perception by the senses is +necessary only in an extremely insignificant measure for the understanding +of its definitions and premises. It is possible to acquaint +a person who lacks both sight and hearing with the fundamental +principles of arithmetic solely by the medium of ``time\index{Time}.'' Such a +person needs only the sense of feeling. By slight excitations of his +skin, induced at equal or unequal intervals of time, he can be led to +the notion of differences of time and hence also to the notion of differences +of number. Uninfluenced by matter and force, independently, +too, of the properties of geometrical magnitudes, arithmetic +could be conducted solely by its own intrinsic potencies to its highest +goals, drawing deductively truth from truth, without a break. + +But what sort of a science should we arrive at by this method +of procedure? Nothing but a gigantic web of self-evident truths. +For, once we admit the first notions and premises to which a man +thus bereft of his senses can be led, we are compelled of necessity +also to admit the derivative results of arithmetic. If the beginnings +of arithmetic appear self-evident, the rest of it, too, bears this +character. Owing to this deductive character of arithmetic, and to +its exemption from influence from without, this science appears to +one person extremely attractive, while to another it appears extremely +repulsive, according as each is constituted. Be that as it +may, however, a finished and complete science of this character +subserves no purpose in the comprehension of the world, or in the +advancement of civilisation. Hence, an arithmetic which heaps up +theorem on theorem with never a thought of how its results are to +be turned to practical account in the acquisition of knowledge in +other fields, resembles an inquisitive physician, who, taking up his +abode in a desert, should arrive there at momentous results in +bacteriology\index + {Bacteriology}, but should bear them with him to his grave, without +their ever redounding to the benefit of humanity. The value of +\PG seq=48 Page 37 ------------------------------------------------------ +arithmetical knowledge lies entirely in its applications. But this +constitutes no reason why many mathematicians, pursuing their +purely deductive bent of mind, should not devote themselves exclusively +to pure arithmetical developments and leave it to others +at the proper time to turn to the material profit of the world the +capital which they have garnered. + +Geometry\index{Geometry}, on the other hand, must have recourse in a much +higher degree than arithmetic to the outside world for its first notions +and premises. The axioms of geometry are nothing but facts of experience +perceived by our senses. The geometry which Bolyai\index + {Bolyai}, Lobachévski\index{Lobachévski}, +Gauss\index{Gauss}, Riemann\index{Riemann}, and Helmholtz\index + {Helmholtz} created and which is +both independent of the eleventh axiom of Euclid\index + {Euclid} and perfectly +free from self-contradictions, has supplied an epistemological demonstration +that geometry is a science that rests on the observation +of nature, and therefore in the correct sense of the word, is a natural +science. + +Yet what a difference there is, for instance, between geometry +and chemistry\index + {Chemistry}! Both derive their constructive materials from sense-perception. +But whilst geometry is compelled to draw only its first +results from observation and is then in a position to move forward +deductively to other results without being under the necessity of +making fresh observations, chemistry on the other hand is still +compelled to make observations and to have recourse to nature. + +It follows, therefore, that a given act of geometrical knowledge +and a given act of chemical knowledge are with respect to the certainty +of the truth they contain not qualitatively but only quantitatively +different. In chemistry the probability of error is greater +than in geometry, because more numerous and more difficult observations +have to be made there than in geometry, where only the +very first premises, which no man with sound senses could ever impugn, +rest on observation. + +The preceding reflexions deprive mathematical knowledge of +that degree of certainty and incontestability which is commonly +attributed to it when we say a thing is ``mathematically certain.'' +This certainty is lessened still more as we pass to the semi-mathematical +sciences, where mechanics has the first claim to our attention. +\PG seq=49 Page 38 ------------------------------------------------------ +All the notions of mechanics\index{Mechanics}, and consequently of all +the other departments of physics\index{Physics}, are composed, by multiplication +or division, of three fundamental notions---length, time, and mass. +That is to say, to the notions of geometry resting on length and its +powers, two other fundamental notions, time and mass, are added, +which, joined to that of length, lead to the notions of force, work, +horse-power, atmospheric pressure, etc. The knowledge of mechanics, +thus, highly certain though it be, is rendered less certain +than that of geometry and \emph{a fortiori} than that of arithmetic. The +uncertainty of knowledge continues to increase in branches which +are still more remote from mathematics, owing to the increasing complexity +of the observational material which must here be put to the +test. + +Still, although mathematical knowledge does not lead to absolutely +certain results, it yet invests known results with incomparably +greater trustworthiness than does the knowledge of the other sciences. +But after all, it remains a useless accumulation of capital +so long as it is not turned to practical account in other sciences, +such as metaphysics, physics, chemistry, biology, political economy, +etc. Hence also arises an obligation on the part of the other sciences, +so to shape their problems and investigations that they can +be made susceptible of mathematical treatment. Then will mathematics +gladly perform her duty. The moment a science has advanced +far enough to permit of the mathematical formulation of its +problems, mathematics will not be slow to treat and to solve these +problems. Mathematical knowledge, aristocratic as it may appear +by the greater certainty of its results, will, so far as the advancement +of human kind is concerned, never be more than a useless +mass of self-evident truths, unless it constantly places itself in the +service of the other sciences\index + {Mathematical knowledge|)Mathematical}. +\PG seq=50 Page 39 ------------------------------------------------------ + + + +\chapter{The Magic Square} +% I. +\section{INTRODUCTORY.} + +\lettrine{A}{mong} the philosophies of modern times there is none which\index{Magic Squares|(} +has emphasised so much the importance of form and formal +thought as the monism of \textit{The Monist}\index + {Monist@\textit{Monist, The}}. An expression of this philosophy +is found in the following passages: + +\begin{quote} +``The order that prevails among the facts of reality is due to the laws of form. +Upon the order of the world depends its cognisability. + +``\dots The laws of form are no less eternal than are matter and energy and +`Verily I say unto you, till heaven and earth pass, one jot or one tittle shall in no +wise pass from the law!' + +``The laws of form and their origin have been a puzzle to all philosophers. +`Ay, there's the rub!' The difficulties of Hume's\index + {Hume} problem of causation, of Kant's\index{Kant} +\emph{a priori}, of Plato's\index{Plato} ideas, of Mill's\index + {Mill, J.\;S.} method of deduction, etc., etc., all arise from a +one-sided view of form and the laws of form and formal thought.'' +\end{quote} + +Considering the great results which engineering and other applied +sciences accomplish through the assistance of mathematics\index{Mathematics}, +we must confess that the forms of thought are wonderful indeed, +and it is not at all astonishing that the primitive thinkers of mankind +when the importance of the laws of formal thought in some +way or another first dawned on their minds, attributed magic powers +to numbers and geometrical figures. + +We shall devote the following pages to a brief review of +magic squares, the consideration of which has made many a man +believe in mysticism\index{Mysticism}. And yet there is no mysticism about them +unless we either consider everything mystical, even that twice two +is four, or join the sceptic in his exclamation that we can truly not +\PG seq=52 Page 41 ------------------------------------------------------ +know whether twice two might not be five in other spheres of the +universe. + +\PGx seq=51 Page 40 ------------------------------------------------------ +\begin{figure*}[p] +\begin{center} +\framebox{\begin{minipage}{.93\textwidth} +\begin{center} +\bigskip +\includegraphics[width=.9\textwidth]{images/melancholia.pdf}\\ +\bigskip +ALBERT DÜRER'S ENGRAVING\\ +\bigskip +{\LARGE MELANCHOLY\index{Melancholy}\\ +\bigskip} +{\tiny OR THE}\\ +\bigskip +\textsc{\small Genius of the Industrial Science of Mechanics.} +\bigskip +\end{center} +\end{minipage} +}\end{center} +\end{figure*} + +The author of the short article on ``Magic Squares'' in the English +Cyclop\ae dia (Vol.~III, p.~415), presumably Prof.\ De Morgan\index + {Demo@De Morgan}, +says: +\begin{quote} +``Though the question of magic squares be in itself of no use, yet it belongs to +a class of problems which call into action a beneficial species of investigation. Without +laying down any rules for their construction, we shall content ourselves with +destroying their magic quality, and showing that the non-existence of such squares +would be much more surprising than their existence.'' +\end{quote} + +This is the point. There obtains a symphonic harmony in +mathematics which is the more startling the more obvious and self-evident +it appears to him who understands the laws that produce this +symphonic harmony. + +\ThoughtBreakStars + +% there is a collision of float and very long footnote otherwise +\LP\interfootnotelinepenalty=-5000 + +On the wood-cut named ``Melancholia''\footnote + {The term melancholy meant in Dürer's\index + {Durer@Dürer, Albert} time, as it did also in Shakespeare's\index{Shakespeare} + and Milton's, ``thought or thoughtfulness.'' Says Milton\index + {Milton} in \textit{Il Penseroso}: + \begin{verse} + \noindent\llap{``}Hail, thou Goddess, sage and holy,\\> + Hail divinest melancholy\\>[2em] + Whose saintly visage is too bright\\>[2em] + To hit the sense of human sight,\\> + And therefore to our weaker view\\> + O'erlaid with black, staid Wisdom's hue.---I, \Num{12}. + \end{verse} + + Thought that does not lead to action produces a gloomy state of mind. Thoughtfulness + which cannot find a way out of itself is that melancholy which engenders + weakness,---a truth which is illustrated in Hamlet\index + {Hamlet}. Shakespeare still uses the words + thought and melancholy as synonyms, saying: + \begin{verse} + \noindent\hphantom{Is sicklied o'er}``The native hue of resolution\\> + Is sicklied o'er with the pale cast of thought.'' + \end{verse} + + Dürer's\index + {Durer@Dürer, Albert} melancholy\index + {Melancholy} does not represent the gloominess of thought, but the power + of invention. Soberness and even a certain sadness are considered only as an element + of this melancholy, but on the whole the genius of thought appears bright, + self-possessed, and strong. + + Dürer\index + {Durer@Dürer, Albert} represents the Science of Mechanical Invention as a winged female figure + musing over some problem. Scattered on the floor around her lie some of the simple + tools used in the sixteenth century. A ladder leans against the house, that assists in + climbing otherwise inaccessible heights. A scale, an hour-glass, a bell, and + the magic square are hanging on the wall behind her. + + At a distance a bat-like creature, being the gloom of melancholy, hovers in the + air like a dark cloud, but the sun rises above the horizon, and at the happy middle + between these two extremes stands the rainbow of serene hope and cheerful confidence.} +of the famous Nuremberg painter, Albrecht Dürer\index + {Durer@Dürer, Albert}, is found among a number of other +\PG seq=53 Page 42 ------------------------------------------------------ +emblems, which the reader will notice in our reproduction of the cut, +the subjoined square. This arrangement of the sixteen natural numbers +\begin{figure*}[htb] +\centering +\begin{MagicSquare}{4} + 1 & {14} & {15} & 4\\ +{12} & 7 & 6 & 9\\ + 8 & {11} & {10} & 5\\ +{13} & 2 & 3 & {16} +\end{MagicSquare} +% [Illustration: Fig. 1.] +\Legend{1} +\end{figure*} +from \1 to \Num{16} possesses the remarkable property that the same +sum \Num{34} will always be obtained whether we add together the four +figures of any of the horizontal rows or the four of any vertical row +or the four which lie in either of the two diagonals. Such an arrangement +of numbers is termed a magic square, and the square +which we have reproduced above is the \emph{first magic square which is +met with in the Christian Occident}. + +% restore normal behaviour +\LP\interfootnotelinepenalty=100 + +Like chess and many of the problems founded on the figure of +the chess-board, the problem of constructing\index + {Magic Squares!aproblem@problem and origin of|etseq} a magic square also +probably traces its origin to Indian soil. From there the problem +found its way among the Arabs\index{Arabs|Arabs}, and by them it was brought to the +Roman Orient. Finally, since Albrecht Dürer's time, the scholars of +Western Europe also have occupied themselves with methods for +the construction of squares of this character. + +The oldest and the simplest magic square consists of the quadratic +arrangement of the nine numbers from \1 to \9 in such a manner +that the sum of each horizontal, vertical, or diagonal row, always +remains the same, namely \Num{15}. This square is the adjoined. +\[ +\begin{minipage}{.5\textwidth} +\centering +\begin{MagicSquare}{3} +2 & 7 & 6\\ +9 & 5 & 1\\ +4 & 3 & 8 +\end{MagicSquare} +% [Illustation: Fig. 2.] +\Legend{2} +\end{minipage} +\] +Here, we will find, \Num{15} always comes out whether we add \2 and \7 and +\6, or \9 and \5 and \1, or \4 and \3 and \8, or \2 and \9 and \4, or \7 and \5 +and \3, or \6 and \1 and \8, or \2 and \5 and \8, or \6 and \5 and \4. + +The question naturally presents itself, whether this condition +of the constant equality of the added sum also remains fulfilled when +the numbers are assigned different places. It may be easily shown +\PG seq=54 Page 43 ------------------------------------------------------ +however that \5 necessarily must occupy the middle place, and that +the even numbers must stand in the corners. This being so, there +are but \7 additional arrangements possible, which differ from the +arrangement above given and from one another only in the respect +that the rows at the top, at the left, at the bottom, and at the right, +exchange places with one another and that in addition a mirror be +imagined present with each arrangement. So too from Dürer's\index + {Durer@Dürer, Albert} +square of \4 times \4 places, by transpositions, a whole set of new +correct squares may be formed. A magic square of the \4 times \4 +numbers from \1 to \Num{16} is formed in the simplest manner as follows. +We inscribe the numbers from \1 to \Num{16} in their natural order in the +squares, thus: +\[ +\begin{minipage}{.5\textwidth} +\centering +\begin{MagicSquare}{4} + 1 & 2 & 3 & 4\\ + 5 & 6 & 7 & 8\\ + 9 &{10}&{11}&{12}\\ +{13}&{14}&{15}&{16} +\end{MagicSquare} +% [Illustration: Fig. 3.] +\Legend{3} +\end{minipage} +\] +We then leave the numbers in the four corner-squares, viz. \1, \4, \Num{13}, +\Num{16}, as well also as the numbers in the four middle-squares, viz. \6, +\7, \Num{10}, \Num{11}, in their original places; and in the place of the remaining +eight numbers, we write the complements of the same with respect +to \Num{17}: thus \Num{15} instead of \2, \Num{14} instead of \3, \Num{12} instead of \5, \9 instead +of \8, \8 instead of \9, \5 instead of \Num{12}, \3 instead of \Num{14}, and \2 instead +of \Num{15}. We obtain thus the magic square +\[ +\begin{minipage}{.5\textwidth} +\hrule height.5em width0pt depth0pt +\centering +\MagicSquareExtra{ +\put(4,3){\LSqr{\tiny\;=\Num{34}}} +\put(4,2){\LSqr{\tiny\;=\Num{34}}} +\put(4,1){\LSqr{\tiny\;=\Num{34}}} +\put(4,0){\LSqr{\tiny\;=\Num{34}}} +\put(0,-1){\TSqr{\tiny\Num{34}}} +\put(1,-1){\TSqr{\tiny\Num{34}}} +\put(2,-1){\TSqr{\tiny\Num{34}}} +\put(3,-1){\TSqr{\tiny\Num{34}}} +\put(4,4){\LSqr{\rotatebox[origin=c]{45}{\tiny=\Num{34}}}} +\put(-1,4){\RSqr{\rotatebox[origin=c]{-45}{\tiny\Num{34}=}}} +} +\begin{MagicSquare}{4} + 1 &{15}&{14}& 4\\ +{12}& 6 & 7 & 9\\ + 8 &{10}&{11}& 5\\ +{13}& 3 &{12}&{16} +\end{MagicSquare} +\hrule height.5em width0pt depth0pt +% [Illustration: Fig. 4.] +\Legend{4} +\end{minipage} +\] +from which the same sum \Num{34} always results. It is an interesting +property of this square that any four numbers which form a rectangle +or square about the centre also always give the same sum \Num{34}; for +example, \1, \4, \Num{13}, \Num{16}, or \6, \7, \Num{10}, \Num{11}, or \Num{15}, \Num{14}, \3, \2, or \Num{12}, \9, \5, \8, +\PG seq=55 Page 44 ------------------------------------------------------ +or \Num{15}, \8, \2, \9, or \Num{14}, \Num{12}, \3, \5. We may easily convince ourselves +that this square is obtainable from the square of Dürer by interchanging +with one another the two middle vertical rows. + +%II. +\section{EARLY METHODS FOR THE CONSTRUCTION OF ODD-NUMBERED +SQUARES.} + +Since early times rules have also been known\index + {Magic Squares!odd-numbered|etseq} for the construction +of magic squares of more than \3 times \3, or \4 times \4 spaces. +In the first place, it is easy to calculate the sum which in the case +of any given number of cells must result from the addition of each +row. We take the determinate number of cells in each side of the +square which we have to fill, multiply that number by itself, add \1, +again multiply the number thus obtained by the number of the cells +in each side, and, finally, divide the product by \2. Thus, with \4 +times \4 cells or squares, we get: \4 times \4 are \Num{16}, \Num{16} and \1 are \Num{17}, +and one half of \Num{17} times \4 is \Num{34}. Similarly, with \5 times \5 squares, +we get: \5 times \5 are \Num{25}, and \1 makes \Num{26}, and the half of \Num{26} times +\5 is \Num{65}. Analogously, for \6 times \6 squares the summation \Num{111} is +obtained, for \7 times \7 squares \Num{175}, for \8 times \8 squares \Num{260}, for \9 +times \9 squares \Num{369}, for \Num{10} times \Num{10} squares \Num{505}, and so on. The +Hindu rule for the construction of magic squares whose roots are +odd, may be enunciated as follows: To start with, write \1 in the +centre of the topmost row, then write \2 in the lowest space of the +\begin{figure*}[hbt] +\hrule height.5em width0pt depth0pt +\centering +\MagicSquareExtra{ +\put(7,0){\LSqr{\tiny\;=\Num{175}}} +\put(7,1){\LSqr{\tiny\;=\Num{175}}} +\put(7,2){\LSqr{\tiny\;=\Num{175}}} +\put(7,3){\LSqr{\tiny\;=\Num{175}}} +\put(7,4){\LSqr{\tiny\;=\Num{175}}} +\put(7,5){\LSqr{\tiny\;=\Num{175}}} +\put(7,6){\LSqr{\tiny\;=\Num{175}}} +\put(0,-1){\TSqr{\tiny\Num{175}}} +\put(1,-1){\TSqr{\tiny\Num{175}}} +\put(2,-1){\TSqr{\tiny\Num{175}}} +\put(3,-1){\TSqr{\tiny\Num{175}}} +\put(4,-1){\TSqr{\tiny\Num{175}}} +\put(5,-1){\TSqr{\tiny\Num{175}}} +\put(6,-1){\TSqr{\tiny\Num{175}}} +\put(7,7){\LSqr{\rotatebox[origin=c]{45}{\tiny=\Num{175}}}} +\put(-1,7){\RSqr{\rotatebox[origin=c]{-45}{\tiny\Num{175}=}}} +} +\begin{MagicSquare}{7} +{30}&{39}&{48}& 1 &{10}&{19}&{28}\\ +{38}&{47}& 7 & 9 &{18}&{27}&{29}\\ +{46}& 6 & 8 &{17}&{26}&{35}&{37}\\ + 5 &{14}&{16}&{25}&{34}&{36}&{45}\\ +{13}&{15}&{24}&{33}&{42}&{44}& 4 \\ +{21}&{23}&{32}&{41}&{43}& 3 &{12}\\ +{22}&{31}&{40}&{49}& 2 &{11}&{20} +\end{MagicSquare} +\hrule height.5em width0pt depth0pt +% [Illustration: Fig. 5.] +\Legend{5} +\end{figure*} +vertical column next adjacent to the right, and then so inscribe the +remaining numbers in their natural order in the squares diagonally +upwards towards the right, that on reaching the right-hand margin +\PG seq=56 Page 45 ------------------------------------------------------ +the inscription shall be continued from the left-hand margin in the +row just above, and on reaching the upper margin shall be continued +from the lower margin in the column next adjacent to the right, +noting that whenever we are arrested in our progress by a square +already occupied we are to fill out the square next beneath the one +we have last filled. In this manner, for example, the last preceding % could be false +square of \7 times \7 cells is formed, in which the reader is requested +to follow the numbers in their natural sequence (\figref*{5}). + +For the next further advancements of the theory of magic +squares and of the methods for their construction we are indebted +to the Byzantian Greek, Moschopulus\index{Moschopulus}, who lived in the fourteenth +century; also, after Albrecht Dürer\index + {Durer@Dürer, Albert} who lived about the year \Num{1500}, +to the celebrated arithmetician Adam Riese\index{Riese, Adam}, and to the mathematician +Michael Stifel\index + {Stifel, Michael}, which two last lived about \Num{1550}. In the seventeenth +century Bachet de Méziriac\index + {Deme@De Méziriac, Bachet}, and Athanasius Kircher\index + {Kircher, Athanasius} employed +themselves on magic squares. About \Num{1700}, finally, the +French mathematicians De la Hire\index{Dela@De la Hire} and Sauveur\index + {Sauveur} made considerable +contributions to the theory. In recent times mathematicians have +concerned themselves much less about magic squares, as they have +indeed about mathematical recreations generally. But quite recently +the Brunswick mathematician Scheffler\index{Scheffler} has put forth his own and +other's studies on this subject in an elegant form. +\begin{figure*}[hbt] +\hrule height4.5em width0pt depth0pt +\centering +\MagicSquareExtra{\linethickness{\fboxrule} + \Cell(1,7;\Num{5}) + \Cell(3,7;\Num{13}) + \Cell(5,7;\Num{21}) + \Cell(2,8;\Num{6}) + \Cell(4,8;\Num{14}) + \Cell(3,9;\Num{7}) + \Cell(1,-1;\Num{29}) + \Cell(3,-1;\Num{37}) + \Cell(5,-1;\Num{45}) + \Cell(2,-2;\Num{36}) + \Cell(4,-2;\Num{44}) + \Cell(3,-3;\Num{43}) + \Cell(-3,3;\Num{1}) + \Cell(-2,2;\Num{8}) + \Cell(-2,4;\Num{2}) + \Cell(-1,1;\Num{15}) + \Cell(-1,3;\Num{9}) + \Cell(-1,5;\Num{3}) + \Cell(7,1;\Num{47}) + \Cell(7,3;\Num{41}) + \Cell(7,5;\Num{35}) + \Cell(8,2;\Num{48}) + \Cell(8,4;\Num{42}) + \Cell(9,3;\Num{49}) + \put(1,7){\line(0,1){1}} + \put(2,7){\line(0,1){2}} + \put(3,7){\line(0,1){3}} + \put(4,7){\line(0,1){3}} + \put(5,7){\line(0,1){2}} + \put(6,7){\line(0,1){1}} + \put(1,0){\line(0,-1){1}} + \put(2,0){\line(0,-1){2}} + \put(3,0){\line(0,-1){3}} + \put(4,0){\line(0,-1){3}} + \put(5,0){\line(0,-1){2}} + \put(6,0){\line(0,-1){1}} + \put(0,6){\line(-1,0){1}} + \put(0,5){\line(-1,0){2}} + \put(0,4){\line(-1,0){3}} + \put(0,3){\line(-1,0){3}} + \put(0,2){\line(-1,0){2}} + \put(0,1){\line(-1,0){1}} + \put(7,6){\line(1,0){1}} + \put(7,5){\line(1,0){2}} + \put(7,4){\line(1,0){3}} + \put(7,3){\line(1,0){3}} + \put(7,2){\line(1,0){2}} + \put(7,1){\line(1,0){1}} + } +\begin{MagicSquare}{7} + 4 &{ }&{12}&{ }&{20}&{ }&{28}\\ +{ }&{11}&{ }&{19}&{ }&{27}&{ }\\ +{10}&{ }&{18}&{ }&{26}&{ }&{34}\\ +{ }&{17}&{ }&{25}&{ }&{33}&{ }\\ +{16}&{ }&{24}&{ }&{32}&{ }&{40}\\ +{ }&{23}&{ }&{31}&{ }&{39}&{ }\\ +{22}&{ }&{30}&{ }&{38}&{ }&{46} +\end{MagicSquare} +\hrule height4.5em width0pt depth0pt +% [Illustration: Fig. 6.] +\Legend{6} +\end{figure*} +\PGx seq=57 Page 46 ------------------------------------------------------ + +The best known of the various methods of constructing magic +squares of an odd number of cells is the following. First write the +numbers in diagonal succession as in the \Alteration{following}{preceding} diagram (\figref*{6}). +After \Num{25} cells of the square of \Num{49} cells which we have to fill out, +have thus been occupied, transfer the six figures found outside each +side of the square, without changing their configuration, into the +empty cells of the side directly opposite. By this method, which +we owe to Bachet de Méziriac\index + {Deme@De Méziriac, Bachet}, we obtain the following magic square +of the numbers from \1 to \Num{49}: +\[ +\begin{minipage}{.5\textwidth} +\centering +\begin{MagicSquare}{7} + 4 &{29}&{12}&{37}&{20}&{45}&{28}\\ +{35}&{11}&{36}&{19}&{44}&{27}& 3 \\ +{10}&{42}&{18}&{43}&{26}& 2 &{34}\\ +{41}&{17}&{49}&{25}& 1 &{33}& 9 \\ +{16}&{48}&{24}& 7 &{32}& 8 &{40}\\ +{47}&{23}& 6 &{31}&{14}&{39}&{15}\\ +{22}& 5 &{30}&{13}&{38}&{21}&{46} +\end{MagicSquare} +% [Illustration: Fig. 7.] +\Legend{7} +\end{minipage} +\] + +% III. +\section{MODERN MODES OF CONSTRUCTION OF ODD-NUMBERED +SQUARES.} + +The reader will justly ask whether there do not exist other correct +magic squares which are constructed after a different method +from that just given, and whether there do not exist modes of construction +which will lead to all the imaginable and possible magic +squares of a definite number of cells. A general mode of construction +of this character was first given for odd-numbered squares by +De la Hire\index + {Dela@De la Hire}, and recently perfected by Professor Scheffler\index + {Scheffler}. + +To acquaint ourselves with this general method, let us select +as our example a square of \5. First we form two auxiliary squares. +In the first we write the numbers from \1 to \5 five times; and in the +second, five times, the following multiples of five, viz.: \0, \5, \Num{10}, \Num{15}, +\Num{20}. It is clear now that by adding each of the numbers of the series +from \1 to \5 to each of the numbers \0, \5, \Num{10}, \Num{15}, \Num{20}, we shall get +all the \Num{25} numerals from \1 to \Num{25}. All that additionally remains to +be done therefore, is, so to inscribe the numbers that by the addition +\PG seq=58 Page 47 ------------------------------------------------------ +of the two numbers in any two corresponding cells each combination +shall come out once and only once; and further that in each +horizontal, vertical, and diagonal row in each auxiliary square each +number shall once appear. Then the required sum of \Num{65} must +necessarily result in every case, because the numbers from \1 to \5 +added together make \Num{15}, and the numbers \0, \5, \Num{10}, \Num{15}, \Num{20} make \Num{50}. + +We effect the required method of inscription by imagining the +numbers \1, \2, \3, \4, \5 (or \0, \5, \Num{10}, \Num{15}, \Num{20}) arranged in cyclical succession, +that is \1 immediately following upon \5, and, starting from +any number whatsoever, by skipping each time either none or one +or two or three etc.\ figures. Cycles are thus obtained of the first, the +second, the third etc.\ orders; for example \3~\4~\5~\1~\2 is a cycle of the +first order, \2~\4~\1~\3~\5 is a cycle of the second order, \1~\5~\4~\3~\2 is a +cycle of the fourth order, etc. The only thing then to be looked out +for in the two auxiliary squares is, that the same ``cycle'' order be +horizontally preserved in all the rows, that the same also happens for +the vertical rows, but that the cycle order in the horizontal and vertical +rows is different. Finally we have only additionally to take +care that to the same numbers of the one auxiliary square not like +numbers but \emph{different} numbers correspond in the other auxiliary +square, that is lie in similarly situated cells. The following auxiliary +squares are, for example, thus possible: +\begin{center} +\begin{minipage}{.33\textwidth} +\centering +\begin{MagicSquare}{5} +3 &4 &5 &1 &2 \\ +5 &1 &2 &3 &4 \\ +2 &3 &4 &5 &1 \\ +4 &5 &1 &2 &3 \\ +1 &2 &3 &4 &5 +\end{MagicSquare} +% [Illustration: Fig. 8.] +\Legend{8} +\end{minipage} +\hbox{ ~and~ } +\begin{minipage}{.33\textwidth} +\centering +\begin{MagicSquare}{5} + 0 &{10}&{20}& 5 &{15}\\ + 5 &{15}& 0 &{10}&{20}\\ +{10}&{20}& 5 &{15}& 0 \\ +{15}& 0 &{10}&{20}& 5 \\ +{20}& 5 &{15}& 0 &{10} +\end{MagicSquare} +% [Illustration: Fig. 9.] +\Legend{9} +\end{minipage} +\end{center} + +Adding in pairs the numbers which occupy similarly situated +cells, we obtain the following correct magic square: +\[ +\begin{minipage}{.5\textwidth} +\centering +\begin{MagicSquare}{5} + 3 &{14}&{25}& 6 &{17}\\ +{10}&{16}& 2 &{13}&{24}\\ +{12}&{23}& 9 &{20}& 1 \\ +{19}& 5 &{11}&{22}& 8 \\ +{21}& 7 &{18}& 4 &{15} +\end{MagicSquare} +% [Illustration: Fig. 10.] +\Legend{10} +\end{minipage} +\PGx seq=59 Page 48 ------------------------------------------------------ +\] + +It will be seen that we are able thus to construct a very large +number of magic squares of \5 times \5 spaces by varying in every +possible manner the numbers in the two auxiliary squares. Furthermore, +the squares thus formed possess the additional peculiarity, +that every \5 numbers which fill out two rows that are parallel to a +diagonal and lie on different sides of the diagonal also give the constant +sum of \Num{65}. For example: \3 and \7, \Num{11}, \Num{20}, \Num{24}; or \Num{10}, \Num{14} and \Num{18}, +\Num{22}, \1. Altogether then the sum \Num{65} is produced out of \Num{20} rows or +pairs of rows. On this peculiarity is dependent the fact that if we +imagine an unlimited number of such squares placed by the side of, +above, or beneath an initial one, we shall be able to obtain as many +quadratic cells as we choose, so arranged that the square composed +of any \Num{25} of these cells will form a correct magic square, as the following +figure will show: +\[ +\begin{minipage}{.8\textwidth} +\centering +\linethickness{0pt} +\MagicSquareExtra{ + \linethickness{0.14em} + \put(2,2){\line(0,1){5}} + \put(2,2){\line(1,0){5}} + \put(2,7){\line(1,0){5}} + \put(7,2){\line(0,1){5}} + \put(5,6){\line(0,1){5}} + \put(5,6){\line(1,0){5}} + \put(5,11){\line(1,0){5}} + \put(10,6){\line(0,1){5}} + } +\begin{MagicSquare}{11}[13] + 2 &{13}&{24}&{10}&{16}& 2 &{13}&{24}&{10}&{16}& 2 \\ + 9 &{20}& 1 &{12}&{23}& 9 &{20}& 1 &{12}&{23}& 9 \\ +{11}&{22}& 8 &{19}& 5 &{11}&{22}& 8 &{19}& 5 &{11}\\ +{18}& 4 &{15}&{21}& 7 &{18}& 4 &{15}&{21}& 7 &{18}\\ +{25}& 6 &{17}& 3 &{14}&{25}& 6 &{17}& 3 &{14}&{25}\\ + 2 &{13}&{24}&{10}&{16}& 2 &{13}&{24}&{10}&{16}& 2 \\ + 9 &{20}& 1 &{12}&{23}& 9 &{20}& 1 &{12}&{23}& 9 \\ +{11}&{22}& 8 &{19}& 5 &{11}&{22}& 8 &{19}& 5 &{11}\\ +{18}& 4 &{15}&{21}& 7 &{18}& 4 &{15}&{21}& 7 &{18}\\ +{25}& 6 &{17}& 3 &{14}&{25}& 6 &{17}& 3 &{14}&{25}\\ + 2 &{13}&{24}&{10}&{16}& 2 &{13}&{24}&{10}&{16}& 2 \\ + 9 &{20}& 1 &{12}&{23}& 9 &{20}& 1 &{12}&{23}& 9 \\ +{11}&{22}& 8 &{19}& 5 &{11}&{22}& 8 &{19}& 5 &{11} +\end{MagicSquare} +% [Illustration: Fig. II.] +\Legend{11} +\end{minipage} +\] + +Every square of every \Num{25} of these numbers, as for example the +two dark-bordered ones, possesses the property that the addition of +the horizontal, vertical, and diagonal rows gives each the same +sum, \Num{65}. + +As an example of a higher number of cells we will append here +a magic square of \Num{11} times \Num{11} spaces formed by the general method +of De la Hire from the two auxiliary squares of Figs.\ \vhyperlink{fig:12}{\Num{12}} and \vhyperlink{fig:13}{\Num{13}}. +From these two auxiliary squares we obtain by the addition of the +\PG seq=60 Page 49 ------------------------------------------------------ +two numbers of every two similarly situated cells, the magic +square, exhibited in \vhyperlink{fig:14}{Diagram \Num{14}}, in which each row gives the same +sum \Num{671}. +\begin{figure*}[htb] +\centering +\hbox to\textwidth{\hss +\begin{minipage}{.55\textwidth} +\centering +\begin{MagicSquare}{11} + 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 &{10}&{11}\\ + 3 & 4 & 5 & 6 & 7 & 8 & 9 &{10}&{11}& 1 & 2 \\ + 5 & 6 & 7 & 8 & 9 &{10}&{11}& 1 & 2 & 3 & 4 \\ + 7 & 8 & 9 &{10}&{11}& 1 & 2 & 3 & 4 & 5 & 6 \\ + 9 &{10}&{11}& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ +{11}& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 &{10}\\ + 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 &{10}&{11}& 1 \\ + 4 & 5 & 6 & 7 & 8 & 9 &{10}&{11}& 1 & 2 & 3 \\ + 6 & 7 & 8 & 9 &{10}&{11}& 1 & 2 & 3 & 4 & 5 \\ + 8 & 9 &{10}&{11}& 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ +{10}&{11}& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 +\end{MagicSquare} +\Legend{12} +\end{minipage}\qquad +\begin{minipage}{.55\textwidth} +\centering +\begin{MagicSquare}{11} + 0 & {11}& {22}& {33}& {44}& {55}& {66}& {77}& {88}& {99}&{110}\\ + {33}& {44}& {55}& {66}& {77}& {88}& {99}&{110}& 0 & {11}& {22}\\ + {66}& {77}& {88}& {99}&{110}& 0 & {11}& {22}& {33}& {44}& {55}\\ + {99}&{110}& 0 & {11}& {22}& {33}& {44}& {55}& {66}& {77}& {88}\\ + {11}& {22}& {33}& {44}& {55}& {66}& {77}& {88}& {99}&{110}& 0 \\ + {44}& {55}& {66}& {77}& {88}& {99}&{110}& 0 & {11}& {22}& {33}\\ + {77}& {88}& {99}&{110}& 0 & {11}& {22}& {33}& {44}& {55}& {66}\\ +{110}& 0 & {11}& {22}& {33}& {44}& {55}& {66}& {77}& {88}& {99}\\ + {22}& {33}& {44}& {55}& {66}& {77}& {88}& {99}&{110}& 0 & {11}\\ + {55}& {66}& {77}& {88}& {99}&{110}& 0 & {11}& {22}& {33}& {44}\\ + {88}& {99}&{110}& 0 & {11}& {22}& {33}& {44}& {55}& {66}& {77} +\end{MagicSquare} +\Legend{13} +\end{minipage}\hss} +\bigskip +\begin{MagicSquare}{11} + 1 & {13}& {25}& {37}& {49}& {61}& {73}& {85}& {97}&{109}&{121}\\ + {36}& {48}& {60}& {72}& {84}& {96}&{108}&{120}& {11}& {12}& {24}\\ + {71}& {83}& {95}&{107}&{119}& {10}& {22}& {23}& {35}& {47}& {59}\\ +{106}&{118}& 9 & {21}& {33}& {34}& {46}& {58}& {70}& {82}& {94}\\ + {20}& {32}& {44}& {45}& {57}& {69}& {81}& {93}&{105}&{117}& 8 \\ + {55}& {56}& {68}& {80}& {92}&{104}&{116}& 7 & {19}& {31}& {43}\\ + {79}& {91}&{103}&{115}& 6 & {18}& {30}& {42}& {54}& {66}& {67}\\ +{114}& 5 & {17}& {29}& {41}& {53}& {65}& {77}& {78}& {90}&{102}\\ + {28}& {40}& {52}& {64}& {76}& {88}& {89}&{101}&{113}& 4 & {16}\\ + {63}& {75}& {87}& {99}&{100}&{112}& 3 & {15}& {27}& {39}& {51}\\ + {98}&{110}&{111}& 2 & {14}& {26}& {38}& {50}& {62}& {74}& {86} +\end{MagicSquare} +\Legend{14} +\end{figure*} + +%IV. +\section{EVEN-NUMBERED SQUARES.} + +Of magic squares having an even number\index + {Magic Squares!even-numbered|etseq} of places we have +hitherto had to deal only with the square of \4. To construct squares +of this description having a higher even number of places, different +and more complicated methods must be employed than for +squares of odd numbers of places. However, in this case also, as +in dealing with the square of \4, we start with the natural sequence +\PG seq=61 Page 50 ------------------------------------------------------ +of the numbers and must then find the complements of the numbers +with respect to some other certain number (as \Num{17} in the square of +\4) and also effect certain exchanges of the numbers with one another. +To form, for example, a magic square of \6 times \6 places, +we inscribe in the \Num{12} diagonal cells the numbers that in the natural +sequence of inscription fall into these places, then in the remaining +cells the complements of the numbers that belong therein with respect +to \Num{37}, and finally effect the following six exchanges, viz.\ of +the numbers \Num{33} and \3, \Num{25} and \7, \Num{20} and \Num{14}, \Num{18} and \Num{13}, \Num{10} and \9, +and \5 and \2. In this way the following magic square is obtained. +\[ +\begin{minipage}{.5\textwidth} +\centering +\begin{MagicSquare}{6} + 1 &{35}&{34}& 3 &{32}& 6 \\ +{30}& 8 &{28}&{27}&{11}& 7 \\ +{24}&{23}&{15}&{16}&{14}&{19}\\ +{13}&{17}&{21}&{22}&{20}&{18}\\ +{12}&{26}& 9 &{10}&{29}&{25}\\ +{31}& 2 & 4 &{33}& 5 &{36} +\end{MagicSquare} +% [Illustration: Fig. 15.] +\Legend{15} +\end{minipage} +\] + +This square may also be constructed by the method of De la +Hire, from two auxiliary squares with the numbers \1, \2, \3, \4, \5, \6 +and \0, \6, \Num{12}, \Num{18}, \Num{24}, \Num{30} respectively. In this case, however, the +vertical rows of the one square and the horizontal rows of the other +must each so contain two numbers three times repeated that the +summation shall always remain \Num{21} and \Num{90} respectively. In this +manner we get the magic square last given above from the two following +auxiliary squares: +\begin{center} +\begin{minipage}{.4\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} +%Fig. 16. +\Legend{16} +\end{minipage} +\hbox{ ~and~ } +\begin{minipage}{.4\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} +%Fig. 17. +\Legend{17} +\end{minipage} +\end{center} + +It is to be noted in connection with this example that here also +as in the case of odd-numbered squares, it is possible so to inscribe +\PG seq=62 Page 51 ------------------------------------------------------ +six times the numbers from \1 to \6 that each number shall appear once +and only once in each horizontal, vertical, and diagonal row; for +example, in the following manner: +\[ +\begin{minipage}{.5\textwidth} +\centering +\begin{MagicSquare}{6} +1&2&3&4&5&6\\ +2&4&6&1&3&5\\ +3&6&5&2&1&4\\ +5&3&1&6&4&2\\ +6&5&4&3&2&1\\ +4&1&2&5&6&3 +\end{MagicSquare} +%Fig. 18. +\Legend{18} +\end{minipage} +\] +But if we attempt so to insert, in a like manner, the other set of +numbers \0, \6, \Num{12}, \Num{18}, \Num{24}, \Num{30} in a second auxiliary square, that each +number of the first auxiliary square shall stand once and once only +in a corresponding cell with each number of the second square, all +the attempts we may make to fulfil coincidently the last named condition +will result in failure. It is therefore necessary to select auxiliary +squares like the two given above. It is noteworthy, that the +fulfilment of the second condition is impossible only in the case of +the square of \6, but that in the case of the square of \4 or of the +square of \8, for example, two auxiliary squares, such as the method +of De la Hire\index + {Dela@De la Hire} requires, are possible. Thus, taking the square of \4 +we get +\begin{center} +\begin{minipage}{.3\textwidth} +\centering +\begin{MagicSquare}{4} +1&2&3&4\\ +4&3&2&1\\ +2&1&4&3\\ +3&4&1&2 +\end{MagicSquare} +%Fig. 19. +\Legend{19} +\end{minipage} +\hbox{ ~and~ } +\begin{minipage}{.3\textwidth} +\centering +\begin{MagicSquare}{4} + 0 & 4 & 8 &{12}\\ + 8 &{12}& 0 & 4 \\ +{12}& 8 & 4 & 0 \\ + 4 & 0 &{12}& 8 +\end{MagicSquare} +%Fig. 20. +\Legend{20} +\end{minipage} +\end{center} + +{% avoid one-line paragraph +\LP\looseness=1 +The reader may form for himself the magic square which these +give. + +}The existence of these two auxiliary squares furnishes a key to +the solution of a pretty problem at cards\index + {Cards, problem at|etseq}. If we replace, namely, +the numbers \1, \2, \3, \4 by the Ace, the King, the Queen, and the +Knave, and the numbers \0, \4, \8, \Num{12} by the four suits, clubs, spades, +hearts, and diamonds, we shall at once perceive that it is possible, +and must be so necessarily, quadratically to arrange in such a manner +the four Aces, the four Kings, the Four Queens, and the four +\PG seq=63 Page 52 ------------------------------------------------------ +Knaves, that in each horizontal, vertical, and diagonal row, each +one of the four suits and each one of the four denominations shall +appear once and once only. The auxiliary squares above given furnish +the appended solution of this problem: +\[ +\begin{minipage}{.8\textwidth} +\centering +\def\SqHt{3.8em} +\def\SqWd{3.8em} +\unitlength=3.8em +\def\Box#1{\begin{minipage}{\SqWd} + \scshape\centering + \tiny\advance\baselineskip3pt + #1\end{minipage}} +\let\Num\empty +\begin{MagicSquare}{4} +{\Box{clubs\\ace}}&{\Box{spades\\king}} + &{\Box{hearts\\queen}}&{\Box{diamonds\\knave}}\\ +{\Box{hearts\\knave}}&{\Box{diamonds\\queen}} + &{\Box{clubs\\king}}&{\Box{spades\\ace}}\\ +{\Box{diamonds\\king}}&{\Box{hearts\\ace}} + &{\Box{spades\\knave}}&{\Box{clubs\\queen}}\\ +{\Box{spades\\queen}}&{\Box{clubs\\knave}} + &{\Box{diamonds\\ace}}&{\Box{hearts\\king}} +\end{MagicSquare} +%Fig. 21. +\Legend{21} +\end{minipage} +\] + +To fix the solution of the problem in the memory, observe that, +starting from the several corners, each suit and each denomination +must be placed in the spots of the move of a Knight\index + {Knight, move of a}. If we fix the +positions of the four cards of any one row, there will be only two +possibilities left of so placing the other cards that the required condition +of having each suit and each denomination once and only +once in each row shall be fulfilled. + +Of magic squares of an even number of places we have up to +this point examined only the squares of \4 and of \6. For the sake of +completeness we append % could be false +here \vhyperlink{fig:22}{one of \8} and \vhyperlink{fig:23}{one of \Num{10}} places. The +mode of construction of these squares is similar to the method above +discussed for the lower even numbers. +\begin{figure*} +\centering +\begin{MagicSquare}{8} + 1 &{63}&{62}& 4 & 5 &{59}&{58}& 8 \\ +{56}&{10}&{11}&{53}&{52}&{14}&{15}&{49}\\ +{48}&{18}&{19}&{45}&{44}&{22}&{23}&{41}\\ +{25}&{39}&{38}&{28}&{29}&{35}&{34}&{32}\\ +{33}&{31}&{30}&{36}&{37}&{27}&{26}&{40}\\ +{24}&{42}&{43}&{21}&{20}&{46}&{47}&{17}\\ +{16}&{50}&{51}&{13}&{12}&{54}&{55}& 9 \\ +{57}& 7 & 6 &{60}&{61}& 3 & 2 &{64} +\end{MagicSquare} +%Fig. 22. +\Legend{22} +\end{figure*} +\PGx seq=64 Page 53 ------------------------------------------------------ +\begin{figure*} +\centering +\begin{MagicSquare}{10} + 1 &{99}& 3 &{97}&{96}& 5 &{94}& 8 &{92}&{10}\\ +{90}&{12}&{88}&{14}&{86}&{85}&{17}&{83}&{19}&{11}\\ +{80}&{79}&{23}&{77}&{25}&{26}&{74}&{28}&{22}&{71}\\ +{31}&{69}&{68}&{34}&{66}&{65}&{37}&{33}&{62}&{40}\\ +{60}&{42}&{58}&{57}&{45}&{46}&{44}&{53}&{49}&{51}\\ +{50}&{52}&{43}&{47}&{55}&{56}&{54}&{48}&{59}&{41}\\ +{61}&{32}&{38}&{64}&{36}&{35}&{67}&{63}&{39}&{70}\\ +{21}&{29}&{73}&{27}&{75}&{76}&{24}&{78}&{72}&{30}\\ +{20}&{82}&{18}&{84}&{15}&{16}&{87}&{13}&{89}&{81}\\ +{91}& 9 &{93}& 4 & 6 &{95}& 7 &{98}& 2&{100}\\ +\end{MagicSquare} +%Fig. 23. +\Legend{23} +\end{figure*} + +The magic squares of even numbers thus constructed are not +the only possible ones. On the contrary, there are very many others +possible, which obey different laws of formation. It has been calculated, +for example, that with the square of \4 it is possible to construct +\Num{880}, and with the square of \6, \emph{several million}, different magic +squares. The number of odd-numbered magic squares constructible +by the method of De la Hire\index + {Dela@De la Hire} is also very great. With the square of +\7, the possible constructions amount to \Num{363},\Num{916},\Num{800}. With the +squares of higher numbers the multitude of the possibilities increases +in the same enormous ratio. + +% v. +\section{MAGIC SQUARES WHOSE SUMMATION GIVES THE NUMBER +OF A YEAR.} + +The magic squares which we have so far considered\index + {Magic Squares!ww@whose summation gives the number of a year|etseq} contain +only the natural numbers from \1 upwards. It is possible, however, +easily to deduce from a correct magic square other squares in which +a different law controls the sequence of the numbers to be inscribed. +Of the squares obtained in this manner, we shall devote our attention +here only to such in which, although formed by the inscription +of successive numbers, the sum obtained from the addition of the +rows is a determinate number which we have fixed upon beforehand, +as \emph{the number of a year}. In such a case we have simply to add to +the numbers of the original square a determinate number so to be +calculated, that the required sum shall each time appear. If this +\PG seq=65 Page 54 ------------------------------------------------------ +sum is divisible by \3, magic squares will always be obtainable with +\3 times \3 spaces which shall give this sum. In such a case we divide +the sum required by \3 and subtract \5 from the result in order +to obtain the number which we have to add to each number of the +original square. If the sum desired is even but not divisible by \4, +we must then subtract from it \Num{34} and take one fourth of the result, +to obtain the number which in this case is to be added in each +place. If, for example, we wish to obtain the number of the year +\Num{1890} as the resulting sum of each row, we shall have to add to each +of the numbers of an ordinary magic square of \4 times \4 spaces the +number \Num{464}; in other words, instead of the numbers from \1 to \Num{16} we +have to insert in the squares the numbers from \Num{465} to \Num{480}. As the +number of the year \Num{1892} is divisible by eleven, it must be possible +to deduce from the magic square constructed by us at the conclusion +of \hyperlink{fig:14}{Section III} a second magic square in which each row of +\Num{11} cells will give the number of the year \Num{1892}. To do this, we subtract +from \Num{1892} the sum of the original square, namely \Num{671}, and divide +the remainder by \Num{11}, whereby we get \Num{111} and thus perceive +that the numbers from \Num{112} to \Num{232} are to be inscribed in the cells of +\begin{figure*}[hbt] +\centering +\MagicSquareExtra{ +\put(11,0){\LSqr{\tiny\;=\Num{1892}}} +\put(11,1){\LSqr{\tiny\;=\Num{1892}}} +\put(11,2){\LSqr{\tiny\;=\Num{1892}}} +\put(11,3){\LSqr{\tiny\;=\Num{1892}}} +\put(11,4){\LSqr{\tiny\;=\Num{1892}}} +\put(11,5){\LSqr{\tiny\;=\Num{1892}}} +\put(11,6){\LSqr{\tiny\;=\Num{1892}}} +\put(11,7){\LSqr{\tiny\;=\Num{1892}}} +\put(11,8){\LSqr{\tiny\;=\Num{1892}}} +\put(11,9){\LSqr{\tiny\;=\Num{1892}}} +\put(11,10){\LSqr{\tiny\;=\Num{1892}}} +\put(0,-1){\TSqr{\tiny\Num{1892}}} +\put(1,-1){\TSqr{\tiny\Num{1892}}} +\put(2,-1){\TSqr{\tiny\Num{1892}}} +\put(3,-1){\TSqr{\tiny\Num{1892}}} +\put(4,-1){\TSqr{\tiny\Num{1892}}} +\put(5,-1){\TSqr{\tiny\Num{1892}}} +\put(6,-1){\TSqr{\tiny\Num{1892}}} +\put(7,-1){\TSqr{\tiny\Num{1892}}} +\put(8,-1){\TSqr{\tiny\Num{1892}}} +\put(9,-1){\TSqr{\tiny\Num{1892}}} +\put(10,-1){\TSqr{\tiny\Num{1892}}} +}\def\SqHt{1.7em}\def\SqWd{1.7em}\unitlength=1.7em +\begin{MagicSquare}{11} +{112}&{124}&{136}&{148}&{160}&{172}&{184}&{196}&{208}&{220}&{232}\\ +{147}&{159}&{171}&{183}&{195}&{207}&{219}&{231}&{122}&{123}&{135}\\ +{182}&{194}&{206}&{218}&{230}&{121}&{133}&{134}&{146}&{158}&{170}\\ +{217}&{229}&{120}&{132}&{144}&{145}&{157}&{169}&{181}&{193}&{205}\\ +{131}&{143}&{155}&{156}&{168}&{180}&{192}&{204}&{216}&{228}&{119}\\ +{166}&{167}&{179}&{191}&{203}&{215}&{227}&{118}&{130}&{142}&{154}\\ +{190}&{202}&{214}&{226}&{117}&{129}&{141}&{153}&{165}&{177}&{178}\\ +{225}&{116}&{128}&{140}&{152}&{164}&{176}&{188}&{189}&{201}&{213}\\ +{139}&{151}&{163}&{175}&{187}&{199}&{200}&{212}&{224}&{115}&{127}\\ +{174}&{186}&{198}&{210}&{211}&{223}&{114}&{126}&{138}&{150}&{162}\\ +{209}&{221}&{222}&{113}&{125}&{137}&{149}&{161}&{173}&{185}&{197} +\end{MagicSquare} +\hrule height.5em width0pt depth0pt +% [Illustration: Fig. 24.] +\Legend{24} +\end{figure*} +the square required. We get in this way the \Alteration{\vhyperlink{fig:24}{following}}{preceding} square, from +which \emph{one and the same sum, namely \Num{1892}, can be obtained \Num{44} times}, +first from each of the \Num{11} horizontal rows, secondly from each of the +\Num{11} vertical rows, thirdly from each of the two diagonal rows, and +\PG seq=66 Page 55 ------------------------------------------------------ +fourthly twenty additional times from each and every pair of any two +rows that lie parallel to a diagonal, have together \Num{11} cells, and lie +on different sides of the diagonal, as for example, \Num{196}, \Num{122}, \Num{158}, \Num{205}, +\Num{131}, \Num{167}, \Num{214}, \Num{140}, \Num{187}, \Num{223}, \Num{149}. + +% VI. +\section{CONCENTRIC MAGIC SQUARES.} + +The acuteness of mathematicians has also discovered magic\index + {Concentric magic squares|etseq}\index + {Magic Squares!concentric|etseq} +squares which possess the peculiar property that if one row after another +be taken away from each side, the smaller inner squares remaining +will still be magical squares, that is to say, all their rows +when added will give the same sum. It will be sufficient to give +two examples here of such squares, (the laws for their construction +being somewhat more complicated,) of which the first has \7 times \7 +and the second \8 times \8 places. The numbers within each of the +dark-bordered frames form with respect to the centre smaller squares +which in their own turn are magical. +\begin{figure*} +\linethickness{0.14em} +\centering +\begin{minipage}[b]{11em} +\MagicSquareExtra{ + \put(1,1){\line(0,1){5}} + \put(1,1){\line(1,0){5}} + \put(6,1){\line(0,1){5}} + \put(1,6){\line(1,0){5}} + \put(2,2){\line(0,1){3}} + \put(2,2){\line(1,0){3}} + \put(5,2){\line(0,1){3}} + \put(2,5){\line(1,0){3}} + \put(3,3){\line(0,1){1}} + \put(3,3){\line(1,0){1}} + \put(4,3){\line(0,1){1}} + \put(3,4){\line(1,0){1}} + } +\begin{MagicSquare}{7} + 4 & 5 & 6 &{43}&{39}&{38}&{40}\\ +{49}&{15}&{16}&{33}&{30}&{31}& 1 \\ +{48}&{37}&{22}&{27}&{26}&{13}& 2 \\ +{47}&{36}&{29}&{25}&{21}&{14}& 3 \\ + 8 &{18}&{24}&{23}&{28}&{32}&{42}\\ + 9 &{19}&{34}&{17}&{20}&{35}&{41}\\ +{10}&{45}&{44}& 7 &{11}&{12}&{46} +\end{MagicSquare} +%[Illustration: Fig. 25.] +\Legend{25} +\end{minipage} +\qquad +\begin{minipage}[b]{13em} +\centering +\MagicSquareExtra{ + \put(1,1){\line(0,1){6}} + \put(1,1){\line(1,0){6}} + \put(7,1){\line(0,1){6}} + \put(1,7){\line(1,0){6}} + \put(2,2){\line(0,1){4}} + \put(2,2){\line(1,0){4}} + \put(6,2){\line(0,1){4}} + \put(2,6){\line(1,0){4}} + } +\begin{MagicSquare}{8} + 1 &{56}&{55}&{11}&{53}&{13}&{14}&{57}\\ +{63}&{15}&{47}&{22}&{42}&{24}&{45}& 2 \\ +{62}&{49}&{25}&{40}&{34}&{31}&{16}& 3 \\ + 4 &{48}&{28}&{37}&{35}&{30}&{17}&{61}\\ + 5 &{44}&{39}&{26}&{32}&{33}&{21}&{60}\\ +{59}&{19}&{38}&{27}&{29}&{36}&{46}& 6 \\ +{58}&{20}&{18}&{43}&{23}&{41}&{50}& 7 \\ + 8 & 9 &{10}&{54}&{12}&{52}&{51}&{64} +\end{MagicSquare} +%[Illustration: Fig. 26.] +\Legend{26} +\end{minipage} +\end{figure*} +In the \vhyperlink{fig:25}{first} of these two squares the internal square of \3 times \3 +places contains the numbers from \Num{21} to \Num{29} in such a manner that +each row gives when added the sum of \Num{75}. This square lies within +a larger one of \5 times \5 spaces, which contains the numbers from +\Num{13} to \Num{37} in such a manner that each row gives the sum of \Num{125}. +Finally, this last square forms part of a square of \7 times \7 places +which contains the numbers from \1 to \Num{49} so that each row gives the +sum of \Num{175}. + +In the \vhyperlink{fig:26}{second} square the inner central square of \4 times \4 places +contains the numbers from \Num{25} to \Num{40} in such a manner that each row +\PG seq=67 Page 56 ------------------------------------------------------ +gives the sum of \Num{130}. This square is the middle of a square of \6 +times \6 places which so contains the numbers from \Num{15} to \Num{50} that +each row gives the sum \Num{195}. Finally, this last square is again the +middle of an ordinary magic square composed of the numbers from +\1 to \Num{64}. + +% VII. +\section{MAGICAL SQUARES WITH MAGICAL PARTS.} + +If we divide a square of \8 times \8 places by means\index + {Magic Squares!with magical parts|etseq} of the two +middle lines parallel to its sides into \4 parts containing each \4 times +\4 spaces, we may propound the problem of so inserting the numbers +from \1 to \Num{64} in these spaces that not only the whole shall form a +magic square, but also that each of the \4 parts individually shall be +magical, that is to say, give the same sum for each row. This problem +also has been successfully solved, as the following diagram will +show. +\[ +\begin{minipage}{.5\textwidth} +\centering +\MagicSquareExtra{ + \put(4,0){\line(0,1){8}} + \put(0,4){\line(1,0){8}} + } +\begin{MagicSquare}{8} + 1 & 4 &{63}&{62}& 5 & 8 &{59}&{58}\\ +{64}&{61}& 2 & 3 &{60}&{57}& 6 & 7 \\ +{42}&{43}&{24}&{21}&{34}&{35}&{32}&{29}\\ +{23}&{22}&{41}&{44}&{31}&{30}&{33}&{36}\\ +{13}&{16}&{55}&{50}& 9 &{12}&{55}&{54}\\ +{52}&{49}&{14}&{15}&{56}&{53}&{10}&{11}\\ +{38}&{39}&{28}&{25}&{46}&{47}&{20}&{17}\\ +{27}&{26}&{37}&{40}&{19}&{18}&{45}&{48} +\end{MagicSquare} +% [Illustration: Fig. 27.] +\Legend{27} +\end{minipage} +\] +The \4 numbers in each row of any one of the sub-squares here, gives +\Num{130}; so that the sum of each one of the rows of the large square +will be \Num{260}. + +Finally, in further illustration of this idea, we will submit to +the consideration of our readers a very remarkable square of the +numbers from \1 to \Num{81}. This square, which will be found \Alteration{below}{on the +following page} +(\figref*{28}), is divided by parallel lines into \9 parts, of +which each contains \9 consecutive numbers that severally make up +a magic square by themselves. + +\begin{figure*}[hbt] +\centering +\MagicSquareExtra{ + \put(3,0){\line(0,1){9}} + \put(6,0){\line(0,1){9}} + \put(0,3){\line(1,0){9}} + \put(0,6){\line(1,0){9}} + } +\begin{MagicSquare}{9} +{31}&{36}&{29}&{76}&{81}&{74}&{13}&{18}&{11}\\ +{30}&{32}&{34}&{75}&{77}&{79}&{12}&{14}&{16}\\ +{35}&{28}&{33}&{80}&{73}&{78}&{17}&{10}&{15}\\ +{22}&{27}&{20}&{40}&{45}&{38}&{58}&{63}&{56}\\ +{21}&{23}&{25}&{39}&{41}&{43}&{57}&{59}&{61}\\ +{26}&{19}&{24}&{44}&{37}&{42}&{62}&{55}&{60}\\ +{67}&{72}&{65}& 4 & 9 & 2 &{49}&{54}&{47}\\ +{66}&{68}&{70}& 3 & 5 & 7 &{48}&{50}&{52}\\ +{71}&{64}&{69}& 8 & 1 & 6 &{53}&{46}&{51} +\end{MagicSquare} +% [Illustration: Fig. 28.] +\Legend{28} +\end{figure*} + +Wonderful as the properties of this square may appear, the law +by which the author constructed it is equally simple. We have +\PG seq=68 Page 57 ------------------------------------------------------ +simply to regard the \9 parts as the \9 cells of a magic square of the +numbers from I to IX, and then to inscribe by the magic prescript +in the square designated as I the numbers from \1 to \9, in the square +designated as II the numbers from \Num{10} to \Num{18}, and so on. In this +way the square above given is obtained from the following base-square: +\[ +\begin{minipage}{.5\textwidth} +\centering +\let\Num\textsc +\begin{MagicSquare}{3} +{iv}&{ix}&{ii}\\ +{iii}&{v}&{vii}\\ +{viii}&{i}&{vi} +\end{MagicSquare} +% [Illustration: Fig. 29.] +\Legend{29} +\end{minipage} +\] + +% VIII. +\section{MAGIC SQUARES THAT INVOLVE THE MOVE OF THE +CHESS-KNIGHT.} + +What one of our readers does not know the problems contained\index + {Chess-knight, magic squares that involve the move of the|etseq}\index + {Magic Squares!zz@that involve the move of the chess-knight|etseq} +in the recreation columns of our magazines, the requirements of +which are to compose into a verse \8 times \8 quadratically arranged +syllables, of which every two successive syllables stand on spots so +situated with respect to each other that a chess-knight can move +from the one to the other? If we replace in such an arrangement +the \Num{64} successive syllables by the \Num{64} numbers from \1 to \Num{64}, we shall +obtain a knight-problem made up of numbers. Methods also exist +indeed for the construction of such dispositions of numbers, which +then form the foundation of the construction of the problems in the +newspapers. But the majority of knight-problems of this class +are the outcome of experiment rather than the product of methodical +\PG seq=69 Page 58 ------------------------------------------------------ +creation. If however it is a severe test of patience to form a +knight-problem\index + {Knight-problem} by experiment, it stands to reason that it is a still +severer trial to effect at the same time the additional result that the +\Num{64}\ numbers which form the knight-problem shall also form a magic +square. + +This trial of endurance was undertaken several decades ago, by +a pensioned Moravian officer named Wenzelides\index + {Wenzelides}, who was spending +the last days of his life in the country. After a series of trials which +lasted years he finally succeeded in so inscribing in the \Num{64} squares +of the chess-board the numbers from \1 to \Num{64} that successive numbers, +as well also as the numbers \Num{64} and \1, were always removed +from one another in distance and direction by the move of a knight, +and that in addition thereto the summation of the horizontal and the +vertical rows always gave the same sum \Num{260}. Ultimately he discovered +several squares of this description, which were published in +the \textit{Berlin Chess Journal}. One of these is here appended: +\[ +\begin{minipage}{.5\textwidth} +\centering +\begin{MagicSquare}{8} +{47}&{10}&{23}&{64}&{49}& 2 &{59}& 6 \\ +{22}&{63}&{48}& 9 &{60}& 5 &{50}& 3 \\ +{11}&{46}&{61}&{24}& 1 &{52}& 7 &{58}\\ +{62}&{21}&{12}&{45}& 8 &{57}& 4 &{51}\\ +{19}&{36}&{25}&{40}&{13}&{44}&{53}&{30}\\ +{26}&{39}&{20}&{33}&{56}&{29}&{14}&{43}\\ +{35}&{18}&{37}&{28}&{41}&{16}&{31}&{54}\\ +{38}&{27}&{34}&{17}&{32}&{55}&{42}&{15} +\end{MagicSquare} +% [Illustration: Fig. 30.] +\Legend{30} +\end{minipage} +\] + +The move of the knight and the equality of the summation of +the horizontal and vertical rows, therefore, are the facts to be noted +here. The diagonal rows do \emph{not} give the sum \Num{260}. Perhaps some +one among our readers who possesses the time and patience will be +tempted to outdo Wenzelides, and to devise a numeral knight-problem +of this kind which will give \Num{260} not only in the horizontal and +vertical but also in the two diagonal rows. +\PG seq=70 Page 59 ------------------------------------------------------ + +% IX. +\section{MAGICAL POLYGONS.} + +So far we have considered only such extensions\index + {Polygons, magical|etseq} of the idea underlying +the construction of the magic square in which the figure of +the square was retained. We may however contrive extensions of +the idea in which instead of a square, a rectangle, a triangle, or a +pentagon, and the like, appear. Without entering into the consideration +of the methods for the construction of such figures, we +will give here of magical polygons simply a few examples, all supplied +by Professor Scheffler\index{Scheffler}: +\begin{Itemize} +\item[\1)] The numbers from \1 to \Num{32} admit of being written in a rectangle +of $\4 \times \8$ in such a manner that the long horizontal rows give +the sum of \Num{132} and the short vertical rows the sum of \Num{66}; thus: +\[ +\begin{minipage}{.7\textwidth} +\centering +\begin{MagicSquare}{8}[4] + 1 &{10}&{11}&{29}&{28}&{19}&{18}&{16}\\ + 9 & 2 &{30}&{12}&{20}&{27}& 7 &{25}\\ +{24}&{31}& 3 &{21}&{13}& 6 &{26}& 8 \\ +{32}&{23}&{22}& 4 & 5 &{14}&{15}&{17} +\end{MagicSquare} +% [Illustration: Fig. 31.] +\Legend{31} +\end{minipage} +\] +\item[\2)] The numbers from \1 to \Num{27} admit of being so arranged in three +regular triangles about a point which forms a common centre, that +each side of the outermost triangle will present \6 numbers of the +total summation \Num{96} and each side of the middle triangle \4 numbers +whose sum is \Num{61}; as the following figure shows: +\[ +\begin{minipage}{.8\textwidth} +\centering +\unitlength=1.65em +\def\SqHt{1.65em} +\def\SqWd{1.65em} +\begin{picture}(11,10) +\cell(0,8.66;26)\cell(2,8.66;3)\cell(4,8.66;6) + \cell(6,8.66;10)\cell(8,8.66;24)\cell(10,8.66;27) +\cell(2.5,7.21;20)\cell(4.17,7.21;9) + \cell(5.83,7.21;11)\cell(7.5,7.21;21) +\cell(1,6.93;18)\cell(9,6.93;2) +\cell(4.12,6.25;16)\cell(5.83,6.25;17) +\cell(3.33,5.77;15)\cell(6.67,5.77;8) +\cell(2,5.2;22)\cell(8,5.2;5) +\cell(5,4.81;12) +\cell(4.17,4.32;7)\cell(5.83,4.32;13) +\cell(3,3.46;4)\cell(7,3.46;23) +\cell(5,2.88;19) +\cell(4,1.73;1)\cell(6,1.73;14) +\cell(5,0;25) +\end{picture} +% [Illustration: Fig. 32.] +\Legend{32} +\end{minipage} +\] +\PGx seq=71 Page 60 ------------------------------------------------------ +\item[\3)] The numbers from \1 to \Num{80} admit of being formed about a +point as common centre into \4 pentagons, such that each side of +the first pentagon from within contains two numbers, each side of +the second pentagon four numbers, each of the third six numbers, +and each side of the fourth, outermost pentagon eight numbers. +The sum of the numbers of each side of the second pentagon is \Num{122}, +the sum of those of each side of the third pentagon is \Num{248}, and that +of those of each side of the fourth pentagon \Num{254}. Furthermore, the +sum of any four corner numbers lying in the same straight line with +the centre, is also the same; namely, \Num{92}. +\begin{figure*}[htb] +\centering +\unitlength=1.25em +\def\SqHt{1.25em} +\def\SqWd{1.25em} +\begin{picture}(24,22) +\cell(12,21.54;1) +\cell(10.38,20.37;26)\cell(13.62,20.37;54) +\cell(8.76,19.19;31)\cell(15.24,19.19;49) +\cell(7.15,18.02;10)\cell(16.85,18.02;80) +\cell(5.53,16.84;76)\cell(18.47,16.84;9) +\cell(3.91,15.67;50)\cell(20.09,15.67;32) +\cell(2.29,14.49;55)\cell(21.71,14.49;27) +\cell(0.67,13.31;5)\cell(23.33,13.31;2) +\cell(1.29,11.41;30)\cell(22.71,11.41;53) +\cell(1.91,9.51;35)\cell(22.09,9.51;48) +\cell(2.53,7.61;6)\cell(21.47,7.61;79) +\cell(3.15,5.71;77)\cell(20.85,5.71;8) +\cell(3.76,3.8;46)\cell(20.24,3.8;33) +\cell(4.38,1.9;51)\cell(19.62,1.9;28) +\cell(5,0;4)\cell(7,0;29)\cell(9,0;34)\cell(11,0;7) + \cell(13,0;78)\cell(15,0;47)\cell(17,0;52)\cell(19,0;3) +\cell(12,18.57;15) +\cell(10.3,17.33;36)\cell(13.7,17.33;44) +\cell(8.6,16.1;70)\cell(15.4,16.1;72) +\cell(6.9,14.86;71)\cell(17.1,14.86;66) +\cell(5.2,13.63;45)\cell(18.8,13.63;37) +\cell(3.51,12.39;11)\cell(20.49,12.39;14) +\cell(4.15,10.4;40)\cell(19.85,10.4;43) +\cell(4.8,8.4;69)\cell(19.2,8.4;73) +\cell(5.45,6.4;75)\cell(18.55,6.4;67) +\cell(6.1,4.41;41)\cell(17.9,4.41;38) +\cell(6.75,2.41;12)\cell(8.85,2.41;39)\cell(10.95,2.41;68) + \cell(13.05,2.41;74)\cell(15.15,2.41;42)\cell(17.25,2.41;13) +\cell(12,15.59;16) +\cell(10.11,14.22;25)\cell(13.89,14.22;65) +\cell(8.22,12.85;61)\cell(15.78,12.85;24) +\cell(6.34,11.47;20)\cell(17.66,11.47;17) +\cell(7.06,9.26;21)\cell(16.94,9.26;64) +\cell(7.78,7.04;62)\cell(16.22,7.04;23) +\cell(8.5,4.82;19)\cell(10.83,4.82;22) + \cell(13.17,4.82;63)\cell(15.5,4.82;18) +\cell(12,12.61;60) +\cell(9.17,10.55;56)\cell(14.83,10.55;59) +\cell(10.25,7.22;57)\cell(13.75,7.22;58) +\end{picture} +% [Illustration: Fig. 33.] +\Legend{33} +\end{figure*} +\item[\4)] The numbers from \1 to \Num{73} admit of being arranged about a +centre, in which the number \Num{37} is written, into three hexagons which +contain respectively \3, \5, and \7 numbers in each side and possess +the following pretty properties. Each hexagon always gives the +same sum, not only when the summation is made along its six sides, +but also when it is made along the six diameters that join its corners +\PG seq=72 Page 61 ------------------------------------------------------ +and along the six that are constructed at right angles to its sides; +this sum, for the first hexagon from within, is \Num{111}, for the second +\Num{185}, and for the third \Num{259}. +\begin{figure*}[hbt] +\unitlength=1.65em +\def\SqHt{1.65em} +\def\SqWd{1.65em} +\centering +\begin{picture}(13,11) +\cell(3,10.38;1)\cell(4,10.38;5)\cell(5,10.38;6)\cell(6,10.38;70) + \cell(7,10.38;60)\cell(8,10.38;59)\cell(9,10.38;58) +\cell(2.5,9.51;63)\cell(9.5,9.51;8) +\cell(2,8.65;62)\cell(4,8.65;19)\cell(5,8.65;53)\cell(6,8.65;46) + \cell(7,8.65;22)\cell(8,8.65;45)\cell(10,8.65;9) +\cell(1.5,7.78;61)\cell(3.5,7.78;20)\cell(8.5,7.78;24)\cell(10.5,7.78;64) +\cell(1,6.92;2)\cell(3,6.92;48)\cell(5,6.92;31)\cell(6,6.92;42) + \cell(7,6.92;38)\cell(9,6.92;49)\cell(11,6.92;57) +\cell(0.5,6.05;3)\cell(2.5,6.05;47)\cell(4.5,6.05;39) + \cell(7.5,6.05;40)\cell(9.5,6.05;44)\cell(11.5,6.05;56) +\cell(0,5.19;67)\cell(2,5.19;51)\cell(4,5.19;41)\cell(6,5.19;37) + \cell(8,5.19;33)\cell(10,5.19;23)\cell(12,5.19;7) +\cell(0.5,4.33;66)\cell(2.5,4.33;50)\cell(4.5,4.33;34) + \cell(7.5,4.33;35)\cell(9.5,4.33;54)\cell(11.5,4.33;11) +\cell(1,3.46;65)\cell(3,3.46;25)\cell(5,3.46;36)\cell(6,3.46;32) + \cell(7,3.46;43)\cell(9,3.46;26)\cell(11,3.46;12) +\cell(1.5,2.6;10)\cell(3.5,2.6;30)\cell(8.5,2.6;27)\cell(10.5,2.6;13) +\cell(2,1.73;17)\cell(4,1.73;29)\cell(5,1.73;21)\cell(6,1.73;28) + \cell(7,1.73;52)\cell(8,1.73;55)\cell(10,1.73;72) +\cell(2.5,0.87;18)\cell(9.5,0.87;71) +\cell(3,0;16)\cell(4,0;69)\cell(5,0;68)\cell(6,0;4) + \cell(7,0;14)\cell(8,0;15)\cell(9,0;73) +\end{picture} +% [Illustration: Fig. 34.] +\Legend{34} +\end{figure*} +\end{Itemize} + +% X. +\section{MAGIC CUBES.} + +Several inquirers, particularly\index{Magic cubes|etseq} Kochansky\index + {Kochansky} (\Num{1686}), Sauveur\index{Sauveur} +(\Num{1710}), Hugel\index{Hugel} (\Num{1859}), and Scheffler\index + {Scheffler} (\Num{1882}), have extended the principle +of the magic squares of the plane to three-dimensioned space. +Imagine a cube divided by planes parallel to its sides and equidistant +from one another, into cubical compartments. The problem is then, +so to insert in these compartments the successive natural numbers +that every row from the right to the left, every row from the front +to the back, every row from the top to the bottom, every diagonal +of a square, and every principal diagonal passing through the centre +of the cube shall contain numbers whose sum is always the same. +For \3 times \3 times \3 compartments, a magic cube of this description +is not constructible. For \4 times % corrected "times," to "times" + \4 times \4 compartments a +cube is constructible such that any row parallel to an edge of the +cube and every principal diagonal give the sum of \Num{130}. To obtain +a magic cube of \Num{64} compartments, imagine the numbers which belong +in the compartments written on the upper surface of the same +\PG seq=73 Page 62 ------------------------------------------------------ +and the numbers then taken off in layers of \Num{16} from the top downwards. +We obtain thus \4 squares of \Num{16} cells each, which together +make up the magic cube; as the following diagrams will show: +\[ +\begin{minipage}{.2\textwidth} +\centering{\tiny +First Layer\\ +from the Top. + +}\begin{MagicSquare}{4} + 1 &{48}&{32}&{49}\\ +{60}&{21}&{37}&{12}\\ +{56}&{25}&{41}& 8 \\ +{13}&{36}&{20}&{61} +\end{MagicSquare} +\end{minipage}\quad +\begin{minipage}{.2\textwidth} +\centering{\tiny +Second Layer\\ +from the Top. + +}\begin{MagicSquare}{4} +{63}&{18}&{34}&{15}\\ + 6 &{43}&{27}&{54}\\ +{10}&{39}&{23}&{58}\\ +{51}&{30}&{46}& 3 +\end{MagicSquare} +\end{minipage}\quad +\begin{minipage}{.2\textwidth} +\centering{\tiny +Third Layer\\ +from the Top. + +}\begin{MagicSquare}{4} +{62}&{19}&{35}&{14}\\ + 7 &{42}&{26}&{55}\\ + 1 &{38}&{22}&{59}\\ +{50}&{31}&{47}& 2 +\end{MagicSquare} +\end{minipage}\quad +\begin{minipage}{.2\textwidth} +\centering{\tiny +Fourth Layer\\ +from the Top. + +}\begin{MagicSquare}{4} + 4 &{45}&{29}&{52}\\ +{57}&{24}&{40}& 9 \\ +{53}&{28}&{44}& 5 \\ +{16}&{33}&{17}&{64} +\end{MagicSquare} +\end{minipage} +\] + +The same sum \Num{130} here comes out not less than \Num{52} times; viz.\ +in the first place from the \Num{16} rows from left to right, secondly from +the \Num{16} rows from the front to the back, thirdly from the \Num{16} rows +counting from the top to the bottom, and lastly from the \4 rows +which join each two opposite corners of the cube, namely from the +rows: \1, \Num{43}, \Num{22}, \Num{64}; \Num{49}, \Num{27}, \Num{38}, \Num{16}; \Num{13}, \Num{39}, \Num{26}, \Num{52}; \Num{61}, \Num{23}, \Num{42}, \4. + +For a cube with \5 compartments in each edge the arrangement +of the figures can so be made that all the \Num{75} rows parallel to any and +every edge, all the \Num{30} rows lying in any diagonal of a square, and +all the \4 rows forming any principal diagonal shall have one and the +same summation, \Num{315}. + +Just as the magic squares of an odd number of cells could be +formed with the aid of \emph{two} auxiliary squares, so also odd-numbered +magic cubes can be constructed with the help of \emph{three} auxiliary cubes. +\begin{figure*}[hbt] +\centering +\begin{minipage}{.3\textwidth} +\centering{\tiny +First Layer from Top.} +\begin{MagicSquare}{5} +{121}&{ 27}&{ 83}&{ 14}&{ 70}\\ + {11}& 6 &{117}&{ 48}&{ 79}\\ + {44}&{100}& 1 & {57}&{113}\\ + {53}&{109}&{ 40}&{ 91}&{ 22}\\ + {87}&{ 18}&{ 74}&{105}&{ 31} +\end{MagicSquare} +\end{minipage}\quad +\begin{minipage}{.3\textwidth} +\centering{\tiny +Second Layer from Top.} +\begin{MagicSquare}{5} + 2 &{ 58}&{114}&{ 45}&{ 96}\\ +{ 36}&{ 92}&{ 23}&{ 54}&{110}\\ +{ 75}&{101}&{ 32}&{ 88}&{ 19}\\ +{ 84}&{ 15}&{ 66}&{122}&{ 28}\\ +{118}&{ 49}&{ 80}& 6 &{ 62} +\end{MagicSquare} +\end{minipage}\quad +\begin{minipage}{.3\textwidth} +\centering{\tiny + Third Layer from Top.} +\begin{MagicSquare}{5} +{ 33}&{ 89}&{ 20}&{ 71}&{102}\\ +{ 67}&{123}&{ 29}&{ 85}&{ 11}\\ +{ 76}& 7 &{ 63}&{119}&{ 50}\\ +{115}&{ 41}&{ 97}& 3 &{ 59}\\ +{ 24}&{ 55}&{106}&{ 37}&{ 93} +\end{MagicSquare} +\end{minipage}\\[1em] +\begin{minipage}{.3\textwidth} +\centering{\tiny +Fourth Layer from Top.} +\begin{MagicSquare}{5} +{ 64}&{120}&{ 46}&{ 77}& 8 \\ +{ 98}& 4 &{ 60}&{111}&{ 42}\\ +{107}&{ 38}&{ 94}&{ 25}&{ 51}\\ +{ 16}&{ 72}&{103}&{ 34}&{ 90}\\ +{ 30}&{ 81}&{ 12}&{ 68}&{124} +\end{MagicSquare} +\end{minipage}\quad +\begin{minipage}{.3\textwidth} +\centering{\tiny +Lowest Layer.} +\begin{MagicSquare}{5} +{ 95}&{ 21}&{ 52}&{108}&{ 39}\\ +{104}&{ 35}&{ 86}&{ 17}&{ 73}\\ +{ 13}&{ 69}&{125}&{ 26}&{ 82}\\ +{ 47}&{ 78}& 9 &{ 65}&{116}\\ +{ 56}&{112}&{ 43}&{ 99}& 5 +\end{MagicSquare} +\end{minipage} +\PGx seq=74 Page 63 ------------------------------------------------------ +\end{figure*} + +In this manner the preceding magic cube of \5 times \5 times \5 % positioning OK +compartments is formed, in which, it may be additionally noticed, +the middle number between \1 and \Num{125}, namely \Num{63}, is placed in the +central compartment; by which arrangement the attainment of the +sum of \Num{315} is assured in the four principal diagonals and the \Num{30} sub-diagonals. +The condition attained in the magic squares, that the +diagonal-pairs parallel to the sub-diagonals also shall give the sum +\Num{315} is not attainable in this case but is so in the case of higher numbers +of compartments. + +\section*{CONCLUSION.} + +Musing on such problems as are the magic squares is fascinating +to thinkers of a mathematical turn of mind. We take delight +in discovering a harmony that abides as an intrinsic quality in +the forms of our thought. The problems of the magic squares are +playful puzzles, invented as it seems for mere pastime and sport. +But there is a deeper problem underlying all these little riddles, and +this deeper problem is of a sweeping significance. It is the philosophical +problem of the world-order. + +The formal sciences are creations of the mind. We build the +sciences of mathematics, geometry, and algebra with our conception +of pure forms which are abstract ideas. And the same order that +prevails in these mental constructions permeates the universe, so that +an old philosopher, overwhelmed with the grandeur of law, imagined +he heard its rhythm in a cosmic harmony of the spheres\index + {Harmony of the spheres}\index{Magic Squares|)}. +\PG seq=75 Page 64 ------------------------------------------------------ + + + +\chapter{The Fourth Dimension} + +\section*{MATHEMATICAL AND SPIRITUALISTIC.} + +% I. +\section{INTRODUCTORY.} + +\lettrine{T}{he} tendency to generalise long ago\index + {Fourth dimension, the|(} led mathematicians to +extend the notion of three-dimensional space, which is the +space of sensible representation, and to define aggregates of points, +or spaces, of more than three dimensions, with the view of employing +these definitions as useful means of investigation. They had +no idea of requiring people to imagine four-dimensional things and +worlds, and they were even still less remote from requiring them +to believe in the real existence of a four-dimensioned space\index + {Space!four-dimensional}. In +the hands of mathematicians this extension of the notion of space +was a mere means devised for the discovery and expression, by +shorter and more convenient ways, of truths applicable to common +geometry and to algebra operating with more than three unknown +quantities. At this stage, however, the spiritualists\index + {Spiritualists} came in, +and coolly took possession of this private property of the mathematicians. +They were in great perplexity as to where they should put +the spirits of the dead. To give them a place in the world accessible +to our senses was not exactly practicable. They were compelled, +therefore, to look around after some \emph{terra incognita}, which +should oppose to the spirit of research inborn in humanity an insuperable +barrier. The abiding-place of the spirits had perforce to be +inaccessible to the senses and full of mystery to the mind. This +property the four-dimensioned space of the mathematicians possessed. +With an intellectual perversity which science has no idea of, +these spiritualists boldly asserted, first, that the whole world was +\PG seq=76 Page 65 ------------------------------------------------------ +situated in a four-dimensioned space as a plane might be situated in +the space familiar to us, secondly, that the spirits of the dead\index + {Spirits!of the dead} lived +in such a four-dimensioned space, thirdly, that these spirits could +accordingly act upon the world and, consequently, upon the human +beings resident in it, exactly as we three-dimensioned creatures can +produce effects upon things that are two-dimensioned; for example, +such effects as that produced when we shatter a lamina of ice, and +so influence some possibly existing two-dimensioned \emph{ice}-world. +Since spiritualism, under the leadership of a Leipsic Professor, +Zöllner\index + {Zoll@Zöllner}, thus proclaimed the existence of a four-dimensioned space, +this notion, which the mathematicians are thoroughly master of,---for +in all their operations with it, though they have forsaken the +path of actual representability, they have never left that of the truth,---this +notion has also passed into the heads of lay persons who have +used it as a catchword, ordinarily without having any clear idea of +what they or any one else mean by it. To clear up such ideas and +to correct the wrong impressions of cultured people who have not a +technical mathematical training, is the purpose of the following +pages. A similar elucidation was aimed at in the tracts which +Schlegel\index{Schlegel} (Riemann, Berlin, \Num{1888}) and Cranz\index + {Cranz} (Virchow-Holtzendorff's +Sammlung, Nos.\ \Num{112} and \Num{113}) have published on the so-called fourth +dimension. Both treatises possess indubitable merits, but their +methods of presentation are in many respects too concise to give to +lay minds a profound comprehension of the subject. The author, +accordingly, has been able to add to the reflections which these excellent +treatises offer, a great deal that appears to him necessary +for a thorough explanation in the minds of non-mathematicians of +the notion of the fourth dimension. + +% II. +\section{THE CONCEPT OF DIMENSION.} + +Many text-books of stereometry begin with\index + {Dimension, concept of|etseq} the words: ``Every +body has three dimensions, length, breadth, and thickness.'' If we +should ask the author of a book of this description to tell us the +length, breadth, and thickness of an apple, of a sponge, or of a cloud +of tobacco smoke, he would be somewhat perplexed and would probably +\PG seq=77 Page 66 ------------------------------------------------------ +say, that the definition in question referred to something different. +A cubical box, or some similar structure, whose angles are +all right angles and whose bounding surfaces are consequently all +rectangles is the only body of which it can at all be unmistakably +asserted that there are three principal directions distinguishable in +it, of which any one can be called the length, any other the breadth, +and any third the thickness. We thus see that the notions of length, +breadth, and thickness are not sufficiently clear and universal to +enable us to derive from them any idea of what is meant when it is +said that every body possesses three dimensions, or that the space +of the world is three-dimensional. + +This distinction may be made sharper and more evident by the +following considerations: We have, let us suppose, a straight line +on which a point is situated, and the problem is proposed to determine +the position of the point on the line in an unequivocal manner. +The simplest way to solve this is, to state how far the point is removed +in the one or the other direction from some given fixed point; +just as in a thermometer the position of the surface of the mercury +is given by a statement of its distance in the direction of cold or heat +from a predetermined fixed point---the point of freezing water. To +state, therefore, the position of a point on a straight line, the sole +datum necessary is a single number, if beforehand we have fixed +upon some standard line, like the centimetre, and some definite +point to which we give the value zero, and have also previously decided +in what direction from the zero-point, points must be situated +whose position is expressed by positive numbers, and also in what +direction those must lie whose position is expressed by negative +numbers. This last-mentioned fact, that a \emph{single} number is sufficient +to determine the place of a point in a straight line, is the real +reason why we attribute to the straight line or to any part of it a +single dimension. + +More generally, we call every totality or system, of infinitely +numerous things, \emph{one}-dimensional, in which \emph{one} number is all that +is requisite to determine and mark out any particular one of these +things from among the entire totality. Thus, time is one-dimensional. +We, as inhabitants of the earth, have naturally chosen as +\PG seq=78 Page 67 ------------------------------------------------------ +our unit of time, the period of the rotation of the earth about its +axis, namely, the day, or a definite portion of a day. The zero-point +of time is regarded in Christian countries as the year of the birth of +Christ, and the positive direction of time is the time \emph{subsequent} to +the birth of Christ. These data fixed, all that is necessary to establish +and distinguish any definite point of time amid the infinite totality +of all the points of time, \emph{is a single number}. Of course this +number need not be a whole number, but may be made up of the +sum of a whole number and a fraction in whose numerator and denominator +we may have numbers as great as we please. We may, +therefore, also say that the totality of all conceivable numerical +magnitudes, or of only such as are greater than one definite number +and smaller than some other definite number, is one-dimensional. + +We shall add here a few additional examples of one-dimensional +magnitudes presented by geometry. First, the circumference +of a circle is a one-dimensional magnitude, as is every curved line, +whether it returns into itself or not. Further, the totality of all +equilateral triangles which stand on the same base is one-dimensional, +or the totality of all circles that can be described through +two fixed points. Also, the totality of all conceivable cubes will be +seen to be one-dimensional, provided they are distinguished, not +with respect to position, but with respect to magnitude. + +In conformity with the fundamental ideas by which we define +the notion of a one-dimensional manifoldness, it will be seen that +the attribute \emph{two}-dimensional\index + {Multi-dimensional magnitudes and spaces|etseq} must be applied to all totalities of +things in which \emph{two} numbers are necessary (and sufficient) to distinguish +any determinate individual thing amid the totality. The +simplest two-dimensioned complex which we know of is the plane. +To determine accurately the position of a point in a plane, the simplest +way is to take two axes at right angles to each other, that is, +fixed straight lines, and then to specify the distances by which the +point in question is removed from each of these axes. + +This method of determining the position of a point in a plane +suggested to the celebrated philosopher and mathematician Descartes\index + {Descartes} +the fundamental idea of analytical geometry\index + {Analytical geometry}, a branch of +mathematics in which by the simple artifice of ascribing to every +\PG seq=79 Page 68 ------------------------------------------------------ +point in a plane two numerical values, determined by its distances +from the two axes above referred to, planimetrical considerations +are transformed into algebraical. So, too, all kinds of curves that +graphically represent the dependence of things on time, make use +of the fact that the totality of the points in a plane is two-dimensional. +For example, to represent in a graphical form the increase +in the population of a city, we take a horizontal axis to represent +the time, and a perpendicular one to represent the numbers which +are the measures of the population. Any two lines, then, whose +lengths practical considerations determine, are taken as the unit of +time, which we may say is a year, and as the unit of population, +which we will say is one thousand. Some definite year, say \Num{1850}, +is fixed upon as the zero point. Then, from all the equally distant +points on the horizontal axis, which points stand for the years, we +proceed in directions parallel to the other axis, that is, in the perpendicular +direction, just so much upwards as the numbers which +stand for the population of that year require. The terminal points +so reached, or the curve which runs through these terminal points, +will then present a graphic picture of the rates of increase of the +population of the town in the different years. The rectangular axes +of Descartes are employed in a similar way for the construction of +barometer curves, which specify for the different localities of a +country the amount of variation of the atmospheric pressure during +any period of time. Immediately next to the plane the surface of +the earth will be recognised as a two-dimensional aggregate of +points. In this case geographical latitude and longitude supply the +two numbers that are requisite accurately to determine the position +of a point. Also, the totality of all the possible straight lines that +can be drawn through any point in space is two-dimensional, as we +shall best understand if we picture to ourselves a plane which is +cut in a point by each of these straight lines and then remember +that by such a construction every point on the plane will belong to +some one line and, \emph{vice versa}, a line to every point, whence it follows +that the totality of all the straight lines, or, as we may call +them, rays\index{Rays}, which pass through the point assigned are of the same +dimensions as the totality of the points of the imagined plane. +\PG seq=80 Page 69 ------------------------------------------------------ + +The question might be asked, In what way and to what extent +in this case is the specification of \emph{two} numbers requisite and sufficient +to determine amid all the rays which pass through the specified +point a definite individual ray? To get a clear idea of the +problem here involved, let us imagine the ray produced far into the +heavens where some quite definite point will correspond to it. Now, +the position of a point in the heavens depends, as does the position +of a point on all spherical surfaces, on two numbers. In the heavens +these two numbers are ordinarily supplied by the two angles called +altitude, or the distance above the plane of the horizon, and azimuth, +or the angular distance between the circle on which the altitude +is measured and the meridian of the observer. It will be seen +thus that the totality of all the luminous rays that an eye, conceived +as a point, can receive from the outer world is two-dimensional, and +also that a luminous point emits a two-dimensional group of luminous +rays. It will also be observed, in connexion with this example, that +the two-dimensional totality of all the rays that can be drawn through +a point in space is something different from the totality of the rays +that pass through a point but are required to lie in a given plane. Such +a group of objects as the last-named one is a one-dimensional totality. + +Now that we have sufficiently discussed the attributes that are +characteristic of one and two-dimensional aggregates, we may, +without any further investigation of the subject, propose the following +definition, that, generally, \emph{an $n$-dimensional totality\index + {Nb@N-dimensional totalities|etseq} of infinitely +numerous things is such that the specification of $n$ numbers is necessary +and sufficient to indicate definitely any individual amid all the infinitely +numerous individuals of that totality}. + +Accordingly, the point-aggregate made up of the world-space +which we inhabit, is a three-dimensional totality. To get our bearings +in this space and to define any determinate point in it, we have +simply to lay through any point which we take as our zero-point +three axes at right angles to each other, one running from right to +left, one backwards and forwards, and one upwards and downwards. +We then join each two of these axes by a plane and are enabled +thus to specify the position of every point in space by the three perpendicular +distances by which the point in question is removed in a +\PG seq=81 Page 70 ------------------------------------------------------ +positive or negative sense from these three planes. It is customary +to denote the numbers which are the measures of these three distances +by $x$, $y$, and $z$, the positive $x$, positive $y$, and positive $z$ ordinarily +being reckoned in the right hand, the hitherward, and the +upward directions from the origin. If now, with direct reference to +this fundamental axial system, any particular specification of $x$, $y$, +and $z$ be made, there will, by such an operation, be cut out and isolated +from the three-dimensional manifoldness of all the points of +space a totality of less dimensions. If, for example, $z$ is equal to +seven units or measures, this is equivalent to a statement that only +the two-dimensional totality of the points is meant, which constitute +the plane that can be laid at right angles to the upward-passing +$z$-axis at a distance of seven measures from the zero-point. Consequently, +every imaginable equation\index + {Equations between $x$, $y$ and $z$} between $x$, $y$, and $z$ isolates and +defines a two-dimensional aggregate of points. If two different equations +obtain between $x$, $y$, and $z$, two such two-dimensional totalities +will be isolated from among all the points of space. But as these +last must have some one-dimensional totality in common, we may +say that the co-existence of two equations between $x$, $y$, and $z$ defines +a one-dimensional totality of points, that is to say a straight line, a +line curved in a plane, or even, perhaps, one curved in space. It +is evident from this that the introduction of the three axes of reference +forms a bridge between the theory of space and the theory of +equations involving three variable quantities, $x$, $y$, $z$. The reason +that the theory of space cannot thus be brought into connection +with algebra in general, that is, with the theory of indefinitely numerous +equations, but only with the algebra of three quantities, $x$, +$y$, $z$, is simply to be sought in the fact that space, as we picture it, +can have only three dimensions. + +We have now only to supply a few additional examples of $n$-dimensional +totalities. All particles of air are four-dimensional in +magnitude when in addition to their position in space we also consider +the variable densities which they assume, as they are expressed +by the different heights of the barometer in the different parts of the +atmosphere. Similarly, all conceivable spheres in space\index + {Spheres in space|etseq} are four-dimensional +magnitudes, for their centres form a three-dimensional +\PG seq=82 Page 71 ------------------------------------------------------ +point-aggregate, and around each centre there may be additionally +conceived a one-dimensional totality of spheres, the radii of which +can be expressed by every numerical magnitude from zero to infinity. +Further, if we imagine a measuring-stick of invariable length to assume +every conceivable position in space, the positions so obtained +will constitute a five-dimensional aggregate. For, in the first place, +one of the extremities of the measuring stick may be conceived to +assume a position at every point of space, and this determines for +one extremity alone of the stick a three-dimensional totality of positions; +and secondly, as we have seen above, there proceeds from +every such position of this extremity a two-dimensional totality of +directions, and by conceiving the measuring-stick to be placed +lengthwise in every one of these directions we shall obtain all the +conceivable positions which the second extremity can assume, and +consequently, the dimensions must be \3 plus \2 or \5. Finally, to +find out how many dimensions the totality of all the possible positions of a square\index + {Squarz@Squares in space|etseq}, invariable in magnitude, possesses, we first give +one of its corners all conceivable positions in space, and we thus +obtain three dimensions. One definite point in space now being +fixed for the position of one corner of the square, we imagine drawn +through this point all possible lines, and on each we lay off the +length of the side of the square and thus obtain two additional dimensions. +Through the point obtained for the position of the second +corner of the square we must now conceive all the possible directions +drawn that are perpendicular to the line thus fixed, and +we must lay off once more on each of these directions the side of +the square. By this last determination the dimensions are only increased +by one, for only one one-dimensional totality of perpendicular +directions is possible to one straight line in one of its points. +Three corners of the square are now fixed and therewith the position +of the fourth also is uniquely determined. Accordingly, the +totality of all equal squares which differ from one another only by +their position in space, constitutes a manifoldness of six dimensions. +\PG seq=83 Page 72 ------------------------------------------------------ + +%III. +\section{THE INTRODUCTION OF THE NOTION OF FOUR-DIMENSIONAL +POINT-AGGREGATES, PERMISSIBLE.} + +In the preceding section it was shown\index + {Four-dimensional point-aggregates|etseq} that we can conceive not +only of manifoldnesses of one, two, and three dimensions, but also +of manifoldnesses of \emph{any} number of dimensions. But it was at the +same time indicated that our world-space, that is, the totality of all +conceivable \emph{points} that differ only in respect of position, cannot in +agreement with our notions of things possess more than three dimensions. +But the question now arises, whether, if the progress of +science tends in such a direction, it is permissible to extend the notion +of space by the introduction of point-aggregates of more than +three dimensions, and to engage in the study of the properties of +such creations, although we know that notwithstanding the fact +that we may conceptually establish and explore such aggregates of +points, yet we cannot picture to ourselves these creations as we do +the spatial magnitudes which surround us, that is, the regular three-dimensional +aggregates of points. + +To show the reader clearly that this question must be answered +in the affirmative, that the extension of our notion of space is permissible, +although it leads to things which we cannot perceive by +our senses, I may call the reader's attention to the fact that in arithmetic +we are accustomed from our youth upwards to extensions of +ideas, which, accurately viewed, as little admit of graphic conception +as a four-dimensional space, that is, a point-aggregate of four +dimensions. By his senses man first reaches only the idea of whole +numbers---the results of counting\index + {Counting|etseq}. The observation of primitive +peoples\footnote + {See the essay \hyperlink{chapter.1}{\textit{Notion and Definition of Number}} in this collection.} +and of children clearly proves that the essential decisive +factors of counting are these three: First, we abstract, in the counting +of things, completely from the individual and characteristic attributes +of these things, that is, we consider them as homogeneous. +Second, we associate individually with the things which we count +\PG seq=84 Page 73 ------------------------------------------------------ +other homogeneous things. These other things are even now, among +uncivilised peoples, the ten fingers of the two hands. They may, +however, be simple strokes, or, as in the case of dice and dominoes, +black points on a white background. Third, we substitute for the +result of this association some concise symbol or word; for example, +the Romans\index{Romans} substituted for three things counted, three strokes placed +side by side, namely: III; but for greater numbers of things they +employed abbreviated signs. The Aztecs\index{Aztecs}, the original inhabitants +of Mexico, had time enough, it seems, to express all the numbers +up to nineteen by equal circles placed side by side. They had abbreviated +signs only for the numbers \Num{20}, \Num{400}, \Num{8000}, and so forth. +In speaking, some one same sound might be associated with the +things counted; but this method of counting is nowadays employed +only by clocks: the languages of men since prehistoric times have +fashioned concise words for the results of the association in question. +From the notion of number, thus fixed as the result of counting, man +reached the notion of the addition of two numbers, and thence the +notion that is the inverse of the last process, the notion of subtraction\index + {Subtraction|etseq}. But at this point it clearly appears that not every problem +which may be propounded is soluble; for there is no number which +can express the result of the subtraction of a number from one +which is equally large or from one which is smaller than itself. The +primary school pupil who says that \8 from \5 ``won't go'' is perfectly +right from his point of view. For there really does not exist +any result of counting which added to eight will give five. + +If humanity had abided by this point of view and had rested +content with the opinion that the problem ``\5 minus \8'' is not solvable, +the science of arithmetic would never have received its full +development, and humanity would not have advanced as far in civilisation +as it has. Fortunately, men said to themselves at this +crisis: ``If \5 minus \8 won't go, we'll \Num{make it go}; if \5 minus \8 does +not possess an intelligible meaning, we will simply give it one.'' As +a fact, things which have not a meaning always afford men a pleasing +opportunity of investing them with one. The question is, then, +what significance is the problem ``\5 minus \8'' to be invested with? +\PG seq=85 Page 74 ------------------------------------------------------ + +The most natural and therefore the most advantageous solution +undoubtedly is to abide by the original notion of subtraction as +the inverse of addition, and to make the significance of \5 minus \8 +such, that for \5 minus \8 plus \8 we shall get our original minuend~\5. +By such a method all the rules of computation which apply to real +differences will also hold good for unreal differences, such as \5 minus~\8. +But it then clearly appears that all forms expressive of differences +in which the numbers that stand before the minus symbol +are less by an equal amount than those which follow it may be regarded +as equal; so that the simplest course seems to be to introduce +as the common characteristic of all equal differential forms\index + {Differential forms} of +this description a common sign, which will indicate at the same +time the difference of the two numbers thus associated. Thus it +came about that for \5 minus \8, as well as for every differential form +which can be regarded as equal thereto, the sign ``$-\3$'' was introduced. +But in calling differential forms of this description numbers, +the notion of number was extended and a new domain was +opened up, namely the domain of negative numbers\index{Negative numbers}. + +In the further development of the science of arithmetic, through +the operation of division viewed as the inverse of multiplication, a +second extension of the idea of number was reached, namely, the +notion of fractional numbers as the outcome of divisions that had +led to numbers hitherto undefined. We find, thus, that the science +of arithmetic throughout its whole development has strictly adhered +to the principle of conformity and consistency and has invested +every association of two numbers, which before had no significance, +by the introduction of new numbers, with a real significance, such +that similar operations in conformity with exactly the same rules +could be performed with the new numbers, viewed as the results of +this association, as with the numbers which were before known and +perfectly defined. Thus the science proceeded further on its way and +reached the notions of irrational, imaginary, and complex numbers. + +The point in all this, which the reader must carefully note, is, +that all the numbers of arithmetic, with the exception of the positive +whole numbers, are artificial products of human thought, invented +to make the language of arithmetic more flexible, and to +\PG seq=86 Page 75 ------------------------------------------------------ +accelerate the progress of science. All these numbers lack the attributes +of representability. + +No man in the world can picture to himself ``minus three +trees.'' It is possible, of course, to know that when three trees +of a garden have been cut down and carried away, three are +missing, and by substituting for ``missing'' the inverse notion of +``added,'' we may say, perhaps, that ``minus three trees'' are added. +But this is quite different from the feat of imagining a negative +number of trees. We can only picture to ourselves a number of +trees that results from actual counting, that is, a positive whole +number. Yet, notwithstanding all this, people had not the slightest +hesitation in extending the notion of number. Exactly so must +it be permitted us in geometry to extend the notion of space, even +though such an extension can only be mentally defined and can never +be brought within the range of human powers of representation. + +In mathematics, in fact, the extension of any notion\index + {Extension of notions in mathematics|etseq} is admissible, +provided such extension does not lead to contradictions with +itself or with results which are well established. Whether such +extensions are necessary, justifiable, or important for the advancement +of science is a different question. It must be admitted, therefore, +that the mathematician is justified in the extension of the notion +of space as a point-aggregate of three dimensions, and in the +introduction of space or point-aggregates of more than three dimensions, +and in the employment of them as means of research. Other +sciences also operate with things which they do not know exist, and +which, though they are sufficiently defined, cannot be perceived by +our senses. For example, the physicist employs the ether\index + {Ether|etseq} as a +means of investigation, though he can have no sensory knowledge +of it. The ether is nothing more than a means which enables us to +comprehend mechanically the effects known as action at a distance +and to bring them within the range of a common point of view. +Without the assumption of a material which penetrates everything, +and by means of whose undulations impulses are transmitted to the +remotest parts of space, the phenomena of light, of heat, of gravitation, +and of electricity would be a jumble of isolated and unconnected +mysteries. The assumption of an ether, however, comprises +\PG seq=87 Page 76 ------------------------------------------------------ +in a systematic scheme all these isolated events, facilitates our mental +control of the phenomena of nature, and enables us to produce +these phenomena at will. But it must not be forgotten in such reflexions +that the ether itself is even a greater problem for man, and +that the ether-hypothesis does not solve the difficulties of phenomena, +but only puts them in a unitary conceptual shape. Notwithstanding +all this, physicists have never had the least hesitation in +employing the ether as a means of investigation. And as little do +reasons exist why the mathematicians should hesitate to investigate +the properties of a four-dimensioned point-aggregate, with the view +of acquiring thus a convenient means of research. + +%IV. +\section{THE INTRODUCTION OF THE IDEA OF FOUR-DIMENSIONED POINT-AGGREGATES +OF SERVICE TO RESEARCH.} + +From the concession that the mathematician has the right to +define and investigate the properties of point-aggregates of more +than three dimensions, it does not necessarily follow that the introduction +of an idea of this description is of value to science. Thus, +for example, in arithmetic, the introduction of operations which +spring from involution\index + {Involution}, as involution and its two inverse operations +proceed from multiplication, is undoubtedly permitted. Just as for +``$a$ times $a$ times $a$'' we write the abbreviated symbol ``$a^3$,'' (which +we read, $a$ to the third power,) and investigate in detail the operation +of involution thus defined, so we might also introduce some +shorthand symbol for ``$a$ to the $a$\th\ power to the $a$\th\ power'' and thus +reach an operation of the fourth degree\index + {Fourth degree, operation of}, which would regard $a$ as +a passive number and the number \3, or any higher number, as the +active number, that is, as the number which indicates how often $a$ +is taken as the base of a power whose exponent may be $a$, or ``$a$ to +the $a$\th, or ``$a$ to the $a$\th\ to the $a$\th\ power.'' + +But the introduction of such an operation of the fourth degree +has proved itself to be of no especial value to mathematics. And +the reason is that in the operation of involution the law of commutation +does not hold good. In addition\index{Addition}, the numbers to be added may +be interchanged and the introduction of multiplication is therefore +\PG seq=88 Page 77 ------------------------------------------------------ +of great value. So, also, in multiplication\index + {Multiplication} the numbers which are +combined, that is, the factors, may be changed about in any way, +and thus the introduction of involution is of value. But in involution +the base and the exponent cannot be interchanged, and consequently +the introduction of any higher operation is almost valueless. + +But with the introduction of the idea of point-aggregates of +multiple dimensions the case is wholly different. The innovation +in question has proved itself to be not only of great importance to +research, but the progress of science has irresistibly forced investigators +to the introduction of this idea, as we shall now set forth in +detail. + +In the first place, algebra, especially the algebraical theory of +systems of equations, % original has "eqations" + derives much advantage from the notion of +multi-dimensioned spaces\index + {Multi-dimensional magnitudes and spaces}. If we have only three unknown quantities, +$x$, $y$, $z$, the algebraical questions which arise from the possible +problems of this class admit, as we have above seen, of geometrical +representation to the eye. Owing to this possibility of geometrical +representation, some certain simple geometrical ideas like ``moving,'' +``lying in,'' ``intersecting,'' and so forth, may be translated +into algebraical events. Now, no reason exists why algebra should +stop at three variable quantities; it must in fact take into consideration +any number of variable quantities. + +For purposes of brevity and greater evidentness, therefore, it is +quite natural to employ geometrical forms of speech in the consideration +of more than three variables. But when we do this, we assume, +perhaps without really intending to do so, the idea of a space +of more than three dimensions. If we have four variable quantities\index + {Fourz@Four variable quantities}\index{Quantities|Quantities}, +$x$, $y$, $z$, $u$, we arrive, by conceiving attributed to each of these four +quantities every possible numerical magnitude, at a four-dimensioned +manifoldness of numerical quantities, which we may just as +well regard as a four-dimensioned aggregate of points. Two equations +which exist on this supposition between $x$, $y$, $z$, and $u$, define +two three-dimensioned aggregates of points, which intersect, as we +may briefly say, in a two-dimensioned aggregate of points, that is, +in a surface; and so on. In a somewhat different manner the determination +of the contents of a square or a cube by the involution +\PG seq=89 Page 78 ------------------------------------------------------ +of a number which stands for the length of its sides, leads to the +notion of four-dimensioned structures, and, consequently, to the +notion of a four-dimensioned point-space. When we note that $a^2$ +stands for the contents of a square, and $a^3$ for the content of a cube, +we naturally inquire after the contents of a structure which is produced +from the cube as the cube is produced from the square and +which also will have the contents $a^4$. We cannot, it is true, clearly +picture to ourselves a structure of this description, but we can, +nevertheless, establish its properties with mathematical exactness.\footnote + {Victor Schlegel\index + {Schlegel}, indeed, has made models of the three-dimensional nets\index + {Models of three-dimensional nets} of all + the six structures which correspond in four-dimensioned space to the five regular + bodies of our space, in an analogous manner to that by which we draw in a plane + the nets of five regular bodies. Schlegel's models are made by Brill of Darmstadt.} +It is bounded by \8 cubes just as the cube is bounded by \6 squares; +it has \Num{16} corners, \Num{24} squares, and \Num{32} edges, so that from every corner +\4 edges, \6 squares, and \4 cubes proceed, and from every edge +\3 squares and \3 cubes. + +Yet despite the great service to algebra of this idea of multi-dimensioned +space, it must be conceded that the conception although +convenient is yet not indispensable. It is true, algebra is +in need of the idea of multiple dimensions, but it is not so absolutely +in need of the idea of \emph{point}-aggregates of multiple dimensions. + +This notion is, however, necessary and serviceable for a profound +comprehension of geometry. The system of geometrical +knowledge which Euclid\index + {Euclid} of Alexandria created about three hundred +years before Christ, supplied during a period of more than two +thousand years a brilliant example of a body of conclusions and +truths which were mutually consistent and logical. Up to the present +century the idea of elementary geometry was indissolubly bound +up with the name of Euclid, so that in England where people adhered +longest to the rigid deductive system of the Grecian mathematician, +the task of ``learning geometry'' and ``reading Euclid'' +were until a few years ago identical. Every proposition of this +Euclidean system rests on other propositions, as one building-stone +in a house rests upon another. Only the very lowest stones, the +foundations, were without supports. These are the axioms or fundamental +\PG seq=90 Page 79 ------------------------------------------------------ +propositions, truths on which all other truths are, directly +or indirectly, founded, but which themselves are assumed without +demonstration as self-evident. + +But the spirit of mathematical research grew in time more and +more critical, and finally asked, whether these axioms might not possibly +admit of demonstration. Especially was a rigid proof sought +for the eleventh\footnote + {Also called the twelfth axiom, also the fifth postulate.---\textit{Tr.}} +axiom of Euclid\index + {Axioms, Euclid's|etseq}\index + {Euclid's axioms|(}, which treats of parallels\index + {Parallels, theory of|etseq}. + +After centuries of fruitless attempts to prove Euclid's eleventh +axiom, Gauss\index{Gauss}, and with him Bolyai\index + {Bolyai} and Lobachévski\index{Lobachévski}, Riemann\index{Riemann}, +and Helmholtz\index + {Helmholtz}, finally stated the decisive reasons why any attempt +to prove the axiom of the parallels must necessarily be futile. These +reasons consist of the fact that though this axiom holds good enough +in the world-space such as we do and can conceive it, yet three-dimensioned +spaces are ideally conceivable though not capable of +mental representation, where the axiom does not hold good. The +axiom was thus shown to be a mere fact of \emph{observation}, and from that +time on there could no longer be any thought of a deductive demonstration +of it. In view of the intimate connection, which both in an +historical and epistemological point of view exists between the extension +of the concept of space and the critical examination of the +axioms of Euclid, we must enter at somewhat greater length into +the discussion of the last mentioned propositions. + +Of the axioms which Euclid lays at the foundation of his +geometry, only the following three are really geometrical axioms: + +\emph{Eighth axiom:} Magnitudes which coincide with one another +are equal to one another. + +\emph{Eleventh axiom:} If a straight line meet two straight lines so as +to make the two interior angles on the same side of it taken together +less than two right angles, these straight lines, being continually +produced, shall at length meet on that side on which are +the angles which are less than two right angles. + +\emph{Twelfth axiom:} Two straight lines cannot inclose a [finite] % yes "inclose" is correct +space. + +The numerous proofs which in the course of time were adduced +\PG seq=91 Page 80 ------------------------------------------------------ +in demonstration of these axioms, especially of the eleventh, all +turn out on close examination to be pseudo-proofs. Legendre\index{Legendre} drew +attention to the fact that either of the following axioms might be +substituted for the eleventh: + +\begin{Itemize} +\item[\textit{a})] Given a straight line, there can be drawn through a point +in the same plane with that line, one and one line only which shall +not intersect the first (parallels) however far the two lines may be +produced; + +\item[\textit{b})] If two parallel lines are cut by a third straight line, the interior +alternate angles will be equal. + +\item[\textit{c})] The sum of the angles of a triangle is equal to two right +angles, that is, to the angle of a straight line or to $\Num{180}\degrees$. +\end{Itemize} + +By the aid of any one of these three assertions, the eleventh +axiom of Euclid may be proved, and, \emph{vice versa}, by the aid of the +latter each of the three assertions may be proved, of course with +the help of the other two axioms, eight and twelve. The perception +that the eleventh axiom does not admit of demonstration without +the employment of one of the foregoing substitutes may best +be gained from the consideration of congruent figures\index + {Congruent figures}. Every +reader will remember from his first instruction in geometry that the +congruence of two triangles is demonstrated by the superposition +of one triangle on the other and by then ascertaining whether the +two completely coincide, no assumptions being made in the determination +except those above mentioned. +\begin{center} +\includegraphics[width=.95\textwidth]{images/illus-080.pdf} +%[Illustration: Fig. 35.] +\Legend{35} +\end{center} + +In the case of triangles which are congruent, as are I and II in +the preceding cut, this coincidence may be effected by the simple +\emph{displacement} of one of the triangles; so that even a two-dimensional +being, supposed to be endowed with powers of reasoning, but only +capable of picturing to itself motions within a plane, also might +convince itself that the two triangles I and II could be made to +coincide. But a being of this description could not convince itself +\PG seq=92 Page 81 ------------------------------------------------------ +in like manner of the congruence of triangles I and III\@. It would +discover the equality of the three sides and the three angles, but it +could never succeed in so superposing the two triangles on each +other as to make them coincide. A three-dimensional being, however, +can do this very easily. It has simply to turn triangle I about +one of its sides and to shove the triangle, thus brought into the position +of its reflexion in a mirror, into the position of triangle III\@. +Similarly, triangles II and III may be made to coincide by moving +either out of the plane of the paper around one of its sides as axis +and turning it until it again falls in the plane of the paper. The +triangle thus turned over can then be brought into the position of +the other. + +Later on we shall revert to these two kinds of congruence: +``congruence by displacement\index + {Displacement, congruence by}'' and ``congruence by circumversion\index + {Circumversion, congruence by}.'' +For the present we will start from the fact that it is always possible +within the limits of a plane to take a triangle out of one position +and bring it into another without altering its sides and angles. The +question is, whether this is only possible in the plane, or whether +it can also be done on other surfaces. + +We find that there are certain surfaces in which this is possible, +and certain others in which it is not. For instance, it is impossible +to move the triangle drawn on the surface of an egg into +some other position on the egg's surface without a distension or +contraction of some of the triangle's parts. On the other hand, it +is quite possible to move the triangle drawn on the surface of a +sphere into any other position on the sphere's surface without a +distension or contraction of its parts. The mathematical reason of +this fact is, that the surface of a sphere, like the plane, has everywhere +the same curvature, but that the surface of an egg at different +places has different curvatures. Of a plane we say that it has +everywhere the curvature zero; of the surface of a sphere we say it +has everywhere a positive curvature\index + {Curvature!positive|etseq}, which is greater in proportion +as the radius is smaller. There are surfaces also which have a +constant negative curvature\index + {Curvature|Curvature}; these surfaces exhibit at every point in +directions proceeding from the same side a partly concave and a +partly convex structure, somewhat like the centre of a saddle. +\PG seq=93 Page 82 ------------------------------------------------------ +There is no necessity of our entering in any detail into the character +and structure of the last-mentioned surfaces. + +Intimately related with the plane, however, are all those surfaces, +which, like the plane, have the curvature zero\index + {Curvature!zero}; in this category +belong especially cylindrical surfaces and conical surfaces. A +sheet of paper of the form of the sector of a circle may, for example, +be readily bent into the shape of a conical surface. If two congruent +triangles, now, be drawn on the sheet of paper, which may +by displacement be translated the one into the other, these triangles +will, it is plain, also remain congruent on the conical surface; that +is, on the conical surface also we may displace the one into the +other; for though a bending of the figures will take place, there +will be no distension or contraction. Similarly, there are surfaces +which, like the sphere, have everywhere a constant positive curvature. +On such surfaces also every figure can be transferred into +some other position without distension or contraction of its parts. +Accordingly, on all surfaces thus related to the plane or sphere, +the assumption which underlies the eighth axiom of Euclid, that it +is possible to transfer into any new position any figure drawn on +such surfaces without distortion, holds good. + +The eleventh axiom in its turn also holds good on all surfaces +of constant curvature, whether the curvature be zero or positive; +only in such instances instead of ``straight'' line we must say +``shortest'' line\index + {Shortest line}. On the surface of a sphere, namely, two shortest +lines, that is, arcs of two great circles, always intersect, no matter +whether they are produced in the direction of the side at which the +third arc of a great circle makes with them angles less than two +right angles, or, in the direction of the other side, where this arc +makes with them angles of more than two right angles. On the +plane, however, two straight lines intersect only on the side where +a third straight line that meets them makes with them interior angles +less than two right angles. + +The twelfth axiom of Euclid, finally, only holds good on the +plane and on the surfaces related to it, but not on the sphere or other +surfaces which, like the sphere, have a constant positive curvature. +This also accounts for the fact that one of the three postulates +\PG seq=94 Page 83 ------------------------------------------------------ +which we regarded as substitutes for the eleventh axiom, though +valid for the plane, is not true for the surface of a sphere; namely, +the postulate that defines the sum of the angles of a triangle\index + {Triangle, sum of the angles of a}. This +sum in a plane triangle is two right angles; in a spherical triangle +it is more than two right angles, the spherical triangle being +greater, the greater the excess the sum of its angles is above +two right angles. It will be seen, from these considerations, that +in geometries in which curved surfaces and not fixed planes are +studied, the axioms of Euclid\index + {Euclid's axioms|)} are either all or partially false. + +The axioms of geometry thus having been revealed as facts of +experience, the question suggested itself whether in the same way +in which it was shown that different two-dimensional geometries +were possible, also different three-dimensional systems of geometry +might not be developed; and consequently what the relations were +in which these might stand to the geometry of the space given by +our senses and representable to our mind. As a fact, a three-dimensional +geometry can be developed, which like the geometry of the +surface of an egg will exclude the axiom that a figure or body can +be transferred from any one part of space to any other and yet remain +congruent to itself. Of a three-dimensional space in which +such a geometry can be developed we say, that it has no constant +measure of curvature. + +The space which is representable to us, and which we shall +henceforth call the \emph{space of experience}\index + {Space!of experience}, possesses, as our experiences +without exception confirm, the especial property that every bodily +thing can be transferred from any one part of it to any other without +suffering in the transference any distension or any contraction. +The space of experience, therefore, has a constant measure of curvature. +The question, however, whether this measure of curvature +is zero or positive, that is, whether the space of experience possesses +the properties which in two-dimensional structures a plane possesses, +or whether it is the three dimensional analogue of the surface +of a sphere is one which future experience alone can answer. If the +space of experience has a constant positive measure of curvature +which is different from zero, be the difference ever so slight, a point +which should move forever onward in a straight line, or, more accurately +\PG seq=95 Page 84 ------------------------------------------------------ +expressed, in a shortest line, would sometime, though perhaps +after having traversed a distance which to us is inconceivable, +ultimately have to arrive from the opposite direction at the place +from which it set out, just as a point which moves forever onward +in the same direction on the surface of a sphere must ultimately arrive +at its starting point, the distance it traverses being longer the +greater the radius of the sphere or the smaller its curvature. + +It will seem, at first blush, almost incredible, that the space of +experience possibly could have this property. But an example, +which is the historical analogue of this modern transformation of our +conceptions, will render the idea less marvellous. Let us transport +ourselves to the age of Homer\index{Homer}. At that time people believed that +the earth was a great disc surrounded on all sides by oceans which +were conceived to be in all directions infinitely great. Indeed, for +the primitive man, who has never journeyed far from the place of +his birth, this is the most natural conception. But imagine now that +some scholar had come, and had informed the Homeric hero Ulysses\index{Ulysses} +that if he would travel forever on the earth in the same direction he +would ultimately come back to the point from which he started; +surely Ulysses would have gazed with as much astonishment upon +this scholar as we now look upon the mathematician who tells us +that it is possible that a point which moves forever onward in space +in the same direction may ultimately arrive at the place from which +it started. But despite the fact that Ulysses would have regarded +the assertion of the scholar as false because contradictory to his +familiar conceptions, that scholar, nevertheless, would have been +right; for the earth is not a plane but a spherical surface. So also +the mathematician may be right who bases this more recent strange +view on the possible fact that the space of experience may have a +measure of curvature which is not exactly zero but slightly greater +than zero. If this were really the case, the \emph{volume} of the space of +experience, though very large, would, nevertheless, be finite; just +as the real spherical surface of the earth as contrasted with the +Homeric plane surface is finite, having so and so many square miles. +When the objection is here made that a finiteness of space is totally +at variance with our modes of thought and conceptions, two ideas, +\PG seq=96 Page 85 ------------------------------------------------------ +``infinitely great\index{Infinitely great}'' and ``unlimited\index + {Unlimited},'' are confounded. All that is at +variance with our practical conceptions is that space can anywhere +have a boundary; not that it may possibly be of tremendous but +finite magnitude. + +It will now be asked if we cannot determine by actual observation +whether the measure of curvature of experiential space\index + {Curvature!zzz@of space}\index{Space|Space} is exactly +zero or slightly different therefrom. The theorem of the sum +of the angles of a triangle and the conclusions which follow from +this theorem do indeed supply us with a means of ascertaining this +fact. And the results of observation have been, that \emph{the measure of +curvature of space is in all probability exactly equal to zero or if it is +slightly different from zero it is so little so that the technical means of +observation at our command and especially our telescopes are not competent +to determine the amount of the deviation}. More, we cannot with +any certainty say. + +All these reflections, to which the criticism of the hypotheses +that underlie geometry long ago led investigators, compel us to institute +a comparison between the space of experience and other +three-dimensional aggregates of points (spaces), which we cannot +mentally represent but can in thought and word accurately define +and investigate. As soon, however, as we are fully implicated in the +task of accurately investigating the properties of three-dimensional +aggregates of points, we find ourselves similarly forced to regard +such aggregates as the component elements of a manifoldness of +more than three dimensions. In this way the exact criticism of +even ordinary geometry leads us to the abstract assumption of a +space of more than three dimensions. And as the extension of every +idea gives a clearer and more translucent form to the idea as it originally +stood, here too the idea of multi-dimensioned aggregates +of points and the investigation of their properties has thrown a new +light on the truths of ordinary geometry and placed its properties +in clearer relief. Among the numerous examples which show how +the notion of a space of multiple dimensions has been of great service\index + {Space!of multiple dimensions, great service to science} +to science in the investigation of three-dimensional space, we +shall give one a place here which is within the comprehension of +non-mathematicians. +\PG seq=97 Page 86 ------------------------------------------------------ + +Imagine in a plane two triangles whose angles are denoted +by pairs of numbers---namely, by \1-\2, \1-\3, \1-\4, and \2-\5, \3-\5, \4-\5. +(See \figref{36}.) Let the two triangles so lie that the three lines which +join the angles \1-\2 and \2-\5, \1-\3 and \3-\5, and \1-\4 and \4-\5 intersect at +a point, which we will call \1-\5. If now we cause the sides of the +triangles which are opposite to these angles to intersect, it will be +found that the points of intersection so obtained possess the peculiar +property of lying all in one and the same straight line. The point +of intersection of the connection \1-\3 and \1-\4 with the connection \4-\5 +and \3-\5 may appropriately be called \3-\4. Similarly, the point of intersection{\LP\spaceskip=.33333em plus.33333em +\begin{wrapfigure}{o}{.6\textwidth} +\includegraphics[height=.6\textwidth,viewport=0 4 63 67]{images/illus-086.pdf} +% [Illustration: Fig. 36] + \Legend{36} +\end{wrapfigure} +\2-\4 is produced +by the meeting of \4-\5, \2-\5 +and \1-\2, \1-\4; and the point +of intersection \2-\3, by the +meeting of \1-\3, \1-\2 and +\3-\5, \2-\5. The statement, +that the three points of +intersection \3-\4, \2-\4, \2-\3, +thus obtained, lie in one +straight line, can be +proved by the principles +of plane geometry only +with difficulty and great +circumstantiality. But by +resorting to the three-dimensional +space of experience, +in which the plane of the drawing lies, the proposition +can be rendered almost self-evident. + +}To begin with, imagine any five points in space which may be +denoted by the numbers \1, \2, \3, \4, \5; then imagine all the possible +ten straight lines of junction drawn between each two of these points, +namely, \1-\2, \1-\3~\dots \4-\5; and finally, also, all the ten planes of +junction of every three points described, namely, the plane \1-\2-\3, +\1-\2-\4,~\dots \3-\4-\5. A spatial figure will thus be obtained, whose ten +straight lines will meet some interposed plane in ten points whose +relative positions are exactly those of the ten points above described. +\PG seq=98 Page 87 ------------------------------------------------------ +Thus, for example, on this plane the points \1-\2, \1-\3, and \2-\3 will lie in +a straight line, for through the three spatial points \1, \2, \3, a plane can +be drawn which will cut the plane of a drawing in a straight line. +The reason, therefore, that the three points \3-\4, \2-\4, \2-\3, also must +ultimately lie in a straight line, consists in the simple fact that the +plane of the three points \2, \3, \4, must cut the plane of the drawing +in a straight line. The figure here considered consists of ten points +of which sets of three so lie ten times in a straight line that conversely +from every point also three straight lines proceed. + +Now, just as this figure is a section of a complete three-dimensional +pentagon, so another remarkable figure, of similar properties, +\begin{figure*}[hbt] +\centering +\includegraphics[height=.6\textwidth]{images/illus-087.pdf} +%[Illustration: Fig. 37.] +\Legend{37} +\end{figure*} +may be obtained from the section of a figure of four-dimensioned +space. Imagine six points, \1, \2, \3, \4, \5, \6, situated in +this four-dimensioned space, and every three of them connected by +a plane, and every four of them by a three-dimensioned space. We +shall obtain thus twenty planes and fifteen three-dimensioned spaces +which will cut the plane in which the figure is to be produced in +twenty points and fifteen rays which so lie that each point sends out +three rays and every ray contains four points. (See \figref{37}.) Figures +of this description, which are so composed of points and rays +that an equal number of rays proceed from every point and an equal +\PG seq=99 Page 88 ------------------------------------------------------ +number of points lie in every ray, are called \emph{configurations}\index + {Configurations}. Other +configurations may, of course, be produced, by taking a different +number of points and by assuming that the points taken lie in a +space of different or even higher dimensions. The author of this +article was the first to draw attention to configurations derived from +spaces of higher dimensions. As we see, then, the notion of a space +of more than three dimensions has performed an important service +in the investigations of common plane geometry. + +In conclusion, I should like to add a remark which Cranz makes +regarding the application of the idea of multi-dimensioned space +to theoretical chemistry\index + {Chemistry!application of the idea of multi-dimensional space to theoretical}. + (See the treatise before cited.) In chemistry, +the molecules of a compound body are said to consist of the +atoms of the elements which are contained in the body, and these are +supposed to be situated at certain distances from one another, and +to be held in their relative positions by certain forces. At first, the +centres of the atoms were conceived to lie in one and the same +plane. But Wislicenus\index{Wislicenus} was led by researches in paralactic acid +to explain the differences of isomeric molecules of the same structural +formul\ae\ by the different positions of the atoms in \emph{space}. (Compare +\textit{La chimie dans l'espace} by van't Hoff\index + {Van't Hoff}, \Num{1875}, preface by J.~Wislicenus). +In fact four points can always be so arranged in space +that every two of them may have any distance from each other; +and the change of one of the six distances does not necessarily involve +the alteration of any other. + +But suppose our molecule consists of five atoms? Four of these +may be so placed that the distance between any two of them can be +made what we please. But it is no longer possible to give the fifth +atom a position such that each of the four distances by which it is +separated from the other atoms may be what we please. On the +contrary, the fourth distance is dependent on the three remaining +distances; for the space of experience has only three dimensions. +If, therefore, I have a molecule which consists of five atoms I cannot +alter the distance between two of them without at least altering +some second distance. But if we imagine the centres of the atoms +placed in a four-dimensioned space, this can be done; all the ten +distances which may be conceived to exist between the five points +\PG seq=100 Page 89 ------------------------------------------------------ +will then be independent of one another. To reach the same result +in the case of six atoms we must assume a five-dimensional space; +and so on. + +Now, if the independence of all the possible distances between +the atoms of a molecule is absolutely required by theoretical chemical +research, the science is really compelled, if it deals with molecules +of more than four atoms, to make use of the idea of a space of +more than three dimensions. This idea is, in this case, simply an +instrument of research, just as are, also, the ideas of molecules and +atoms---means designed to embrace in a perspicuous and systematic +form the phenomena of chemistry and to discover the conditions +under which new phenomena can be evoked. Whether a four-dimensioned +space really exists is a question whose insolubility +cannot prevent research from making use of the idea, exactly as +chemistry has not been prevented from making use of the notion +of atom, although no one really knows whether the things we call +atoms exist or not. + +%V. +\section{REFUTATION OF THE ARGUMENTS ADDUCED TO PROVE THE +EXISTENCE OF A FOUR-DIMENSIONED SPACE INCLUSIVE +OF THE VISIBLE WORLD.} + +The considerations of the preceding section\index + {Four-dimensioned space, refutation of the existence of|etseq} will have convinced +the cultured non-mathematician of the service which the theory of +multi-dimensioned spaces has done, and bids fair to do, for geometrical +research. In addition thereto is the consideration that +every extension of one branch of mathematical science is a constant +source of beneficial and helpful influence to the other branches. The +knowledge, however, that mathematicians can employ the notion of +four-dimensioned space with good results in their researches, would +never have been sufficient to procure it its present popularity; for +every man of intelligence has now heard of it, and, in jest or in +earnest, often speaks of it. The knowledge of a four-dimensioned +space did not reach the ears of cultured non-mathematicians until +the consequences which the spiritualists fancied it was permissible +to draw from this mathematical notion were publicly known. But +it is a tremendous step from the four-dimensioned space of the +\PG seq=101 Page 90 ------------------------------------------------------ +mathematicians to the space from which the spirit-friends of the +spiritualistic mediums\index + {Spiritualistic|Spiritualistic} entertain us with rappings, knockings, and +bad English. Before taking this step we will first discuss the question +of the real existence of a four-dimensional space, not deciding +the question whether this space, if it really does exist, is inhabited +by reasonable beings who consciously act upon the world in which +we exist. + +Among the reasons which are put forward to prove the existence +of a four-dimensional space containing the world, the least reprehensible +are those which are based on the existence of symmetry. +We spoke above of two triangles in the same plane which have all +\begin{figure*}[!htb] +\centering +\includegraphics[width=.8\textwidth]{images/illus-090.pdf} +% [Ilustration: Fig. 38.} +\Legend{38} +\end{figure*} +their sides and angles congruent, but which cannot be made to coincide +by simple displacement within the plane; but we saw that +this coincidence could be effected by holding fast one side of one +triangle and moving it out of its plane until it had been so far turned +round that it fell back into its plane. Now something similar to +this exists in space. Cut two figures, exactly like that of \figref{38}, out +of a piece of paper, and turn the triangle $ABF$ about the side $AB$, +$ACE$ about the side $AC$, $BCD$ about the side $BC$, and in one figure +above and in the other below; then in both cases the points $D$, $E$, $F$ +will meet at a point, because $AE$ is equal to $AF$, $BF$ is equal to +$BD$, $CD$ is equal to $CE$. In this manner we obtain two pyramids +which in all lengths and all angles are congruent, yet which cannot, +\PG seq=102 Page 91 ------------------------------------------------------ +no matter how we try, be made to coincide, that is, be so fitted the +one into the other that they shall both stand as one pyramid. But +the \emph{reflected} image of the one could be brought into coincidence +with the other. Two spatial structures whose sides and angles are +thus equal to each other, and of which each may be viewed as the +reflected image of the other, are called \emph{symmetrical}. For instance, +the right and the left hand are symmetrical; or, a right and a left +glove. Now just as in two dimensions it is impossible by simple +displacement to bring into congruence triangles which like those +above mentioned can only be made to coincide by circumversion, +so also in three dimensions it is impossible to bring into congruence +two symmetrical pyramids. Careful mathematical reflection, however, +declares that this could be effected, if it were possible, while +holding one of the surfaces, to move the pyramid out of the space +of experience, and to turn it round through a four-dimensioned +space until it reached a point at which it would return again into +our experiential space. This process would simply be the four-dimensional +analogue of the three-dimensional circumversion in +the above-mentioned case of the two triangles. Further, the interior +surfaces in this process would be converted into exterior surfaces, +and \emph{vice versa}, exactly as in the circumversion of a triangle the anterior +and posterior sides are interchanged. If the structure which +is to be converted into its symmetrical counterpart is made of a flexible +material, the interchange mentioned of the interior and exterior +surfaces may be effected by simply turning the structure inside out; +for example, a right glove may thus be converted into a left glove. + +Now from this truth, that every structure can be converted, by +means of a four-dimensional space inclusive of the world, into a +structure symmetrical with it, it has been sought to establish the +probability of the real existence of a four-dimensioned space. Yet +it will be evident, from the discussion of the preceding section, that +the only inference which we can here make is, that the idea of a +four-dimensioned space is competent, from a mathematical point of +view, to throw some light upon the phenomena of symmetry\index{Symmetry}. To +conclude from these facts that a space of this kind really exists, +would be as daring as to conclude from the fact that the uniform +\PG seq=103 Page 92 ------------------------------------------------------ +angular velocity of the apparent motions of the fixed stars is explicable +from the assumption of an axial motion of the firmament, that +the fixed stars are really rigidly placed in a celestial sphere rotating +about its axis. It must not be forgotten that our comprehension of +the phenomena of the real world consists of two elements: first, of +that which the things really are; and, second, of that by which we +rationally apprehend the things. This latter element is partly dependent +on the sum of the experiences which we have before acquired, +and partly on the necessity, due to the imperfection of reason, +of our classifying the multitudinous isolated phenomena of the +world into categories which we ourselves have formed, and which, +therefore, are not wholly derived from the phenomena themselves, +but to a great extent are dependent on us. + +Besides geometrical reasons, Zöllner\index + {Zoll@Zöllner|etseq} has also adduced cosmological % original has a separate entry for this page, but it collides with the etseq entry on page 93 in the output so we make this the start of the etseq instead +reasons to prove the existence of a four-dimensional space. +To these reasons belong especially the questions, whether the number +of the fixed stars is infinitely great, whether the world is finite +or infinite in extension, whether the world had a beginning or will +have an end, whether the world is not hastening towards a condition +of equilibrium or dead level by the universal distribution of its matter +and energy; the problems, also, of gravitation and action at a distance; +and finally, the questions concerning the relations between +the phenomena in the world of sense-perception to the unknown +things-in-themselves. All these questions which can be decided in +no definite sense, led Zöllner and his followers to the assumption +that a four-dimensioned space inclusive of the space of experience +must really exist. But more careful reflection will show that this +assumption does not dispose of the difficulties but simply displaces +them into another realm. Furthermore, even if four-dimensioned +space did unravel and make clear all the cosmological problems +which have bothered the human mind, still, its existence would not +be proved thereby; it would yet remain a mere hypothesis, designed +to render more intelligible to a being who can only make experiences +in a three-dimensional space, the phenomena therein which are full +of mystery to it. A four-dimensioned space would in such case possess +for the metaphysician a value similar to that which the ether +\PG seq=104 Page 93 ------------------------------------------------------ +possesses for the physicist. Still more convincing than these cosmological +reasons to the majority of men is the physio-psychological +reason drawn from the phenomena of vision\index + {Vision|etseq} which Zöllner adduces. % really an \index{Zoll@Zöllner|etseq} here +Into this main argument we will enter in more detail. + +When we ``see'' an object, as we all know, the light which proceeds +or is reflected therefrom produces an image on the retina of +our eye; this image is conducted to our consciousness by means of +the optic nerve, and our reason draws therefrom an inference respecting +the object. When, now, we look at a square whose sides +are a decimetre in length, and whose centre is situated at the distance +of a metre from the pupil of our eye, an image is produced on the +retina. But exactly the same image will be produced there if we +look at a square whose sides are parallel to the sides of the first +square but two decimetres in length, and whose centre is situated +at a distance of two metres from the pupil of the eye. Proceeding +thus further, we readily discover that an eye can perceive in any +length or line only the ratio of its magnitude to the distance at which +it is situated from it, and that generally a three-dimensional world +must appear to the eye two-dimensional, because all points which +lie behind each other in the direction outwards from the eye produce +on the retina only one image. This is due to the fact that the +retinal images are themselves two-dimensional; for which reason, +Zöllner says, the world must appear to a child as two-dimensional, +if it be supposed to live in a primitive condition of unconscious mental +activity. To such a child two objects which are moving the one +behind the other, must appear as suffering displacement on a surface, +which we conceive behind the objects, and on which the latter +are projected. In all these apparent displacements, coincidences +and changes of form also are effected. All these things must appear +puzzling to a human being in the first stages of its development, +and the mind thus finds itself, as Zöllner further argues, in the first +years of childhood forced to adopt a hypothesis concerning the constitution +of space and to assume that the world is three-dimensional, +although the eye can really perceive it as only two-dimensional. +Zöllner then further says, that in the explanation of the effects of +the external world, man constantly finds this hypothesis of his childish +\PG seq=105 Page 94 ------------------------------------------------------ +years confirmed, and that in this way it has become in his mind +so profound a conviction that it is no longer possible for him to +think it away. Consonant with this argumentation, also, is Zöllner's +remark, that the same phenomenon has presented itself in +astronomical methods of knowledge. To explain the movements of +the planets, which appear to describe regular paths on the surface +of a celestial sphere, we were compelled in the solution of the riddles +which these motions presented, to assume in the structure of the +heavens a dimension of ``depth\index{Depth},'' and the complicated motions in +the two-dimensioned firmament were converted into very simple +motions in three-dimensioned space. Zöllner also contends that our +conception of the entire visible world as possessed of three dimensions +is a product of our reason, which the mind was driven to form +by the contradictions which would be presented to it on the assumption +of only two dimensions by the perspective distortions, coincidences, +and changes of magnitude of objects. When a child moves +its hand before its eyes, turns it, brings it nearer, or pushes it farther +away, this child successively receives the most various impressions +on the surface of its retina of one and the same object of whose +identity and constancy its feelings offer it a perfect assurance. If +the child regarded the changeable projection of the hand on the surface +of the retina as the real object, and not the hand which lies beyond +it, the child would constantly be met with contradictions in its +experience, and to avoid this it makes the hypothesis that the space +of experience is three-dimensional. Zöllner's contention is, therefore, +that man originally had only a two-dimensional intuition\index + {Space!intuition of} of +space, but was forced by experience to represent to himself the objects +which on the retinal surface appeared two-dimensional, as +three-dimensional, and thus to transform his two-dimensional space-intuition +into a three-dimensional one. Now, in exactly the same +way, according to Zöllner's notion, will man, by the advancement +and increasing exactness of his knowledge of the phenomena of the +outer world, also be compelled to conceive of the material world as +a ``shadow cast by a more real four-dimensional world,'' so that +these conceptions will be just as trivial for the people of the twentieth +century as since Copernicus's\index + {Copernicus} time the explanation of the motions +\PG seq=106 Page 95 ------------------------------------------------------ +of the heavenly bodies by means of a three-dimensional motion +has been. + +Zöllner's arguments from the phenomena of vision may be refuted +as follows: In the first place it is incorrect to say that we see +the things of the external world by means of two-dimensional retinal +images. The light which penetrates the eye causes an irritation +of the optic nerves, and any such effect which, though it be not +powerful, is, nevertheless, a mechanical one, can only take place on +things which are material. But material things are always three-dimensional. +The effect of light on the sensitive plates of photography +can with just as little justice be regarded as two-dimensional. +Our senses can have perception\index + {Perception} of nothing but three-dimensional +things, and this perception is effected by forces which in their turn +act on three-dimensional things, namely our sensory nerves. It is +wrong to call an image two-dimensional, for it is only by abstraction\index + {Abstraction} +that we can conceive of a thickness so growing constantly smaller +and smaller as to admit of our regarding a three-dimensional picture +as two-dimensional, by giving it in mind a vanishingly small thickness. +It is also wrong to say, as Zöllner says, that when we see the +shadow of a hand which is cast upon a wall we see something two-dimensional. +What we really perceive is that no light falls upon +our eye from the region included by the shadow, while from the +entire surrounding region light does fall on our eye. But this light +is reflected from the material particles which form the surface of the +wall, that is, from three-dimensional particles of matter. We must +always remember that our eye communicates to us only three-dimensional +knowledge, and that for the comprehension of anything which +has two, one, or no dimensions, \emph{a purely intellectual act of abstraction +must be added to the act of perception}. When we imagine we have +made a lead-pencil mark on paper, we have, exactly viewed, simply +heaped alongside of each other little particles of graphite in such a +manner that there are by far fewer graphite particles in the lateral +and upward directions than there are in the longitudinal direction, +and thus our reason arrives by abstraction at the notion of a straight +line. When we look at an object, say a cube of wood, we recognise +the object as three-dimensional, and it is only by abstraction that +\PG seq=107 Page 96 ------------------------------------------------------ +we can conceive of its two-dimensional surfaces, of its twelve one-dimensional +edges, and of its eight no-dimensional corners. For we +reach the perception of its surface, for example, solely by reason of +the fact that the material particles which form the cube prevent the +transmission of light, and reflect it, whereby a part of the light reflected +from every material particle strikes our eye. Now, by thinking +exclusively of those material particles which are reflected, in +contrariety to the empty space without and the hidden and therefore +non-reflected particles within, we form the notion of a surface. + +It is evident from this, that all that we perceive is three-dimen\-sion\-al, +that we cannot reach anything two-dimensional without +an intellectual abstraction, and that, therefore, we cannot conceive +of anything two-dimensional exerting effects upon material things. +But this fact is a refutation of the retinal argument of Zöllner. If +vision consisted wholly and exclusively in the creation of a two-dimensional +image, the things which take place in the world could +never come into our consciousness. The child, therefore, does not +originally apprehend the world, as Zöllner says, as two-dimensional; +on the contrary, it apprehends it either not at all, or it apprehends +it as three-dimensional. Of course the child must first ``learn how'' +to see. It is found from the observation of children during the first +months of their lives, and of the congenitally blind who have suddenly +acquired the power of vision by some successful operation, +that seeing does not consist alone in the irritations which arise in +the optic nerves, but also in the correct interpretation of these irritations +by reason. This correct interpretation, however, can be +accomplished only by the accumulation of a considerable stock of +experience. Especially must the recognition of the distance\index + {Distance of objects} of the +object seen be gradually learned. In this, two things are especially +helpful; first, the fact that we have two eyes and, consequently, +that we must feel two irritations of the optic nerves which are not +wholly alike; and, secondly, the fact that we are enabled by our +power of motion and our sense of touch to convince ourselves of +the distance and form of the bodies seen. The question now arises, +what sort of an intuition of space would a creature have that had +only one eye, that could neither move itself nor its eye, and also +\PG seq=108 Page 97 ------------------------------------------------------ +possessed no peripheral nerves. According to Zöllner's view, this +creature could, owing to its two-dimensional retinal images, have +only a two-dimensional intuition of space. The author's opinion, +however, is, that such a creature could not see at all, as it has no +possibility of collecting experiences which are adapted in any way +to interpreting the effects of things on its retina. The light which +proceeded from the objects roundabout and fell on the retina could +produce no other effect on the being than that of a wholly unintelligible +irritation, or perhaps even pain. + +The reflections presented sufficiently show that neither the +phenomena of symmetry nor the retinal images of the objects of +vision necessarily force upon us the assumption of a four-dimensioned +space. If the material world should ever present problems +which could not in the progress of knowledge be solved in a natural +way, the assumption that a four-dimensional space containing the +world exists would also be incompetent to resolve the difficulties +presented; it would simply convert these difficulties into others, +and not dispose of the problems but simply displace them to another +world. Yet the question might be asked, is the existence of +a four-dimensional space really \emph{impossible?} To answer this question +we must first clearly know what we mean by ``exist.'' If existence +means that the intellectual \emph{idea} of a thing can be formed and that +this idea shall not lead to contradictions with other well-established +ideas and with experience, we have only to say that four-dimensioned +space does exist, as the arguments adduced in sections \hyperlink{section.5.3}{III} +and \hyperlink{section.5.4}{IV} have rendered plain. If, namely, the space of four dimensions +did not exist as a clear idea in the minds of mathematicians, +mathematicians could certainly not have been led by this idea to +results which are recognised by the senses as true, and which really +take place in our own representable space. But if existence means +``material actuality,'' we must say that we neither now nor in the +future can know anything about it. For we know material actuality +only as three-dimensional, our senses can only make three-dimensional +experiences, and the inferences of our reason, although +they can well abstract from material things, can never ascend to +the point of explaining a four-dimensional materiality. Just as little, +\PG seq=109 Page 98 ------------------------------------------------------ +therefore, as we can locally fix the idea of a two-dimensional material +world, as little can we ever verify the notion of a four-dimensional +material existence. + +%VI. +\section{EXAMINATION OF THE HYPOTHESIS CONCERNING THE EXISTENCE +OF FOUR-DIMENSIONAL SPIRITS.} + +In connection with the belief\index + {Spirits|Spirits} that the visible world is contained +in a four-dimensioned space, Zöllner and his adherents further hold +that this higher space is inhabited by intelligent beings who can +act consciously and at will on the human beings who live in experiential +space. To invest this opinion with greater strength, Zöllner +appealed to the fact that the greatest thinkers of antiquity and of +modern times were either wholly of this opinion or at least held +views from which his contentions might be immediately derived. +Plato's\index{Plato} dialogue between Socrates\index + {Socrates} and Glaukon\index{Glaukon} in the seventh book +of the Republic, is evidence, says Zöllner, that this greatest philosopher +of antiquity possessed some presentiment of this extension of +the notion of space. Yet any one who has connectedly studied and +understood Plato's system of philosophy must concede that the so-called +``ideas'' of the Platonic system denote something wholly different +from what Zöllner sees in them or pretends to see. Zöllner +says that these Platonic ideas are spatial objects of more than three +dimensions and represent ``real existence'' in the same sense that +the material world, as contrasted with the images on the retina, represents +it. Zöllner similarly deals with the Kantian ``thing-in-itself\index + {Thing-in-itself},'' +which is also regarded as an object of higher dimensions. + +To show Kant\index + {Kant|etseq} in the light of a predecessor, Zöllner quotes the +following passage from the former's ``Träume eines Geistersehers, +erläutert durch Träume der Metaphysik'' (\Num{1766}, \textit{Collected Works}, +Vol.~VII. page \Num{32} et seq.): ``I confess that I am very much inclined +to assert the existence of immaterial beings in the world, +and to rank my own soul as one of such a class. It appears, there +is a spiritual essence existent which is intimately bound up with +matter but which does not act on those forces of the elements by +which the latter are connected, but upon some internal principle +\PG seq=110 Page 99 ------------------------------------------------------ +of its own condition. It will, in the time to come---I know not +when or where---be proved, that the human soul, even in this life, +exists in a state of uninterrupted connection with all the immaterial +natures of the spiritual world; that it alternately acts on +them and receives impressions from them, of which, as a human +soul, it is not, in the normal state of things, conscious. It would +be a great thing, if some such systematic constitution of the spiritual +world, as we conceive it, could be deduced, not exclusively +from our general notion of spiritual nature, which is altogether too +hypothetical, but from some real and universally admitted observations,---or, +for that matter, if it could even be shown to be +probable.'' + +What Kant really asserts here is, first, the partly independent +and partly dependent existence of the soul, and of spiritual beings +generally, on matter, and, second, that spiritual beings have some +common connection with and mutually influence one another. This +contention, which is that of very many thinkers, does not, however, +entail the consequence that the ``transcendental subject of +Kant'' must be four-dimensional, as Zöllner asserts it does. Kant +never even hinted at the theory that the psychical features of the +world owe their connection with the material features to the fact +that they are four-dimensional and, therefore, include the three-dimensional. +Is it a necessary conclusion that if a thing exists and +is not three-dimensional, as is the case with the soul, it is therefore +four-dimensional? Can it not in fact be so constituted that it +is wholly meaningless to speak of dimensions at all in connection +with it? + +Yet still more strangely than the words of Plato and Kant do +certain utterances of the mathematicians Gauss\index{Gauss} and Riemann speak +in favor of Zöllner's hypothesis. S.~v.\ Waltershausen\index % yes "v." is lowercase in original + {Waltershausen} relates of +Gauss in his \textit{Gruss zum Gedächtniss}, (Leipsic, \Num{1856}), that Gauss +had often remarked that the three dimensions of space were only +a specific peculiarity of the human mind. We can think ourselves, +he said, into beings who are conscious of only two dimensions; +similarly, perhaps, beings who are above and outside our world may +look down upon us; and there were, he continued, in a jesting tone, +\PG seq=111 Page 100 ------------------------------------------------------ +a number of problems which he had here indefinitely laid aside, but +hoped to treat in a superior state by superior geometrical methods. +Leaving aside this jest, which quite naturally suggested itself, the +remarks of Gauss are quite correct. We possess the power to abstract +and can think, therefore, what kind of geometry a being that +is only acquainted with a two-dimensional world would have; for +instance, we can imagine that such a being could not conceive of +the possibility of making two triangles coincide which were congruent +in the sense above explained, and so on. So, also, we can +understand that a being who has control of four dimensions can only +conceive of a geometry of four-dimensional space, yet may have the +capacity of thinking itself into spaces of other dimensions. But it +does not follow from this that a four-dimensional space exists, let +alone that it is inhabited by reasonable beings. + +Riemann\index + {Riemann}, on the other hand, speaks directly of a world of spirits. +In his \textit{Neue mathematische Principien der Naturphilosophie} he +puts forth the hypothesis that the space of the world is filled with +a material that is constantly pouring into the ponderable atoms, +there to disappear from the phenomenal world. In every ponderable +atom, he says, at every moment of time, there enters and appears a +determinate amount of matter, proportional to the force of gravitation. +The ponderable bodies, according to this theory, are the +place at which the spiritual world enters and acts on the material +world. Riemann's world of spirits, the sole office of which is to explain +the phenomenon of gravitation as a force governing matter, +is, however, essentially different from the spiritual world of Zöllner, +the function of which is to explain supposed supersensuous phenomena +which stand in the most glaring contradiction with the established +known laws of the material world. + +Besides this appeal to the testimony of eminent men like Plato, +Kant, Gauss, and Riemann, the scientific prophet of modern spiritualism +also bases his theory on the belief, which has obtained at all +times and appeared in various forms among all peoples, that there +exist in the world forces which at times are competent to evoke +phenomena that are exempt from the ordinary laws of nature. We +have but to think of the phenomena of table-turning which once excited +\PG seq=112 Page 101 ------------------------------------------------------ +the Chinese as much as it has aroused, during the last few +decades, the European and American worlds; or of the divining-rod, +by whose help our forefathers sought for water, in fact, as we +do now in parts of Europe and America. + +Cranz, in his essay on the subject, divides spiritualistic phenomena +into physical and intellectual. Of the first class he enumerates +the following: the moving of chairs and tables; the animation +of walking-sticks, slippers, and broomsticks; the miraculous throwing +of objects; spirit-rappings\index{Spirittr@Spirit-rappings} (Luther\index + {Luther} heard a sound in the Wartburg, +``as if three score casks were hurled down the stairs''); the +ecstatic suspension of persons above the floor; the diminution of +the forces of gravity; the ordeals of witches; the fetching of wished-for +objects; the declination of the magnetic needle by persons at a +distance; the untying of knots in a closed string; insensibility to +injury and exemption therefrom when tortured, as in handling red-hot +coals, carrying hot irons, etc.; the music of invisible spirits; +the materialisations of spirits or of individual parts of spirits (the +footprints in the experiments of Slade\index{Slade}, photographed by Zöllner); +the double appearance of the same person; the penetration of matter +(of closed doors, windows, and so forth). As numerous also is +the selection presented by Cranz\index{Cranz} of intellectual phenomena, namely: +spirit-writing\index{Spirittw@Spirit-writing} (Have's\index + {Have, Mr.} instrument for the facilitation of intercourse +with spirits), the clairvoyance and divination of somnambulists, of +visionary, ecstatic, and hypnotised persons, prompted or controlled +by narcotic medicines, by sleeping in temples, by music and dancing, +by ascetic modes of life and residence in barren localities, by the exudations +of the soil and of water, by the contemplation of jewels, +mirrors, and crystal-pure water, and by anointing the finger-nails +with consecrated oil. Also the following additional intellectual phenomena +are cited: increased eloquence or suddenly acquired power +of speaking in foreign languages; spirit-effects at a distance; inability +to move, transferences of the will, and so forth. + +All these phenomena, presented with the aspect of truth, and +associated more or less with trickery, self-deception, and humbug, +are adduced by the spiritualists to substantiate the belief in a world +of spirits which intentionally and consciously take part in the events +\PG seq=113 Page 102 ------------------------------------------------------ +of the material world, and to prove that these phenomena may be +sufficiently and consistently explained by the effects of the activity +of such a world. It is impossible for us to discuss and put to the +test here the explanations of all these supersensuous phenomena. +Anything and everything can be explained by spirits who act at will +upon the world. There are only a few of these phenomena, namely, +clairvoyance and Slade's experiments, whose explanations are so +intimately connected with our main theme, the so-called fourth +dimension, that they cannot be passed over. + +First, with respect to clairvoyance, the American visionary Davis\index + {Davis, the American visionary} +describes the experiences which he claims to have made in this +condition, induced by ``magnetic sleep,'' as follows:\footnote + {Quoted by Cranz.} +``The sphere +of my vision now began to expand. At first, I could only clearly +discern the walls of the house. At the start they seemed to me dark +and gloomy; but they soon became brighter and finally transparent. +I could now see the objects, the utensils, and the persons in the adjoining +house as easily as those in the room in which I sat. But my +perceptions extended further still; before my wandering glance, +which seemed to control a great semi-circle, the broad surface of % yes this "semi-circle" is hyphenated +the earth, for hundreds of miles about me, grew as transparent as +water, and I saw the brains, the entrails, and the entire anatomy of +the beasts that wandered about in the forests of the Eastern Hemisphere, +hundreds and thousands of miles from the room in which I +sat.'' The belief in the possibility of such states of clairvoyance is +by no means new. Alexander Dumas\index + {Dumas, Alexander} made use of it, for example, +in his \textit{Mémoires d'un médicin}, in which Count Balsamo, afterwards +called Cagliostro\index + {Cagliostro}, is said to possess the power to throw suitable +persons into this wonderful condition and thus to find out what +other persons at distant places are doing. Zöllner explains clairvoyance +by means of the fourth dimension thus: + +A man who is accustomed to viewing things on a plain is supposed +to ascend to a considerable height in a balloon. He will +there enjoy a much more extended prospect than if he had remained +on the plain below, and will also be able to signal to greater distances. +\PG seq=114 Page 103 ------------------------------------------------------ +The plain, that is, the two-dimensioned space, is accordingly +viewed by him from points outside of the plain as ``open'' in +all directions. Exactly so, in Zöllner's theory, must three-dimensioned +spaces appear, when viewed from points in four-dimensioned +space, namely, as ``open''; and the more so in proportion as the +point in question is situated at a greater distance from the place of +our body, or in proportion as the soul ascends to a greater height in +this fourth dimension. Zöllner thus explains clairvoyance\index + {Clairvoyance} as a condition +in which the soul has displaced itself out of its three-dimensioned +space and reached a point which with respect to this space +is four-dimensionally situated and whence it is able to contemplate +the three-dimensional world without the interference of intervening +obstacles. + +The following remark is to be made to this explanation. The +reason why we have a better and more extended view from a balloon +than from places on the earth is simply this, that between the +suspended balloon and the objects seen at a distance nothing intervenes +but the air, and air allows the transmission of light, whereas, +at the places below on the earth there are all kinds of material +things about the observer which prevent the transmission of light +and either render difficult or absolutely impossible the sight of +things which lie far away. In the same way, also, from a point in +four-dimensioned space, a three-dimensional object will be visible +only provided there are no obstacles intervening. If, therefore, this +awareness of a distant object is a real, actual vision by means of a +luminous ray which strikes the eye, there is contained in the explanation +of Zöllner the tacit assumption that the medium with +which the four-dimensional world is filled is also pervious to light +exactly as the atmosphere is. + +The theory that there are four-dimensional spirits who produce +the phenomena cited by the spiritualists received special support +from the experiments which the prestidigitateur Slade, who claimed +he was a spiritualistic medium, performed before Zöllner. Of these +experiments we will speak of the two most important, the experiment +with the glass sphere and the experiment with the knots\index + {Knots, experiment with the}. To +explain the connection of the glass sphere experiment with the fourth +\PG seq=115 Page 104 ------------------------------------------------------ +dimension, we must first conceive of two-dimensional reasoning beings, +or, let us say, two-dimensional worms\index + {Worms, two-dimensional}, living and moving in a +plane. For a creature of this kind it will be self-evident that there +are no other paths between two points of its plane than such as lie +within the plane. It must, accordingly, be beyond the range of +conception of this worm, how any two-dimensional object which lies +within a circle in its space can be brought to any other position in +its space outside the circle without the object passing through the +barriers formed by the circle's circumference. But if this worm +could procure the services of a three-dimensional being, the transportation +of the object from a position within the circle to any position +outside it could be effected by the three-dimensional being simply +taking the object \emph{out} of the plane and placing it at the desired +point. This object, therefore, would, in an inexplicable manner, +suddenly disappear before the eyes of the worms who were assembled +as spectators, and after the lapse of an interval of time would +again appear outside the circle without having passed at any point +through the circle's circumference. If now we add another dimension, +we shall derive from this trick, which is wholly removed from +the sense-perception of the flattened worms, the following experiment, +which is wholly beyond the perception of us human beings. +Inside a glass sphere, which is closed all around, a grain of corn is +placed; the problem is to transport the corn to some place outside +the sphere without passing through the glass surface. Now we +should be able to perform this trick if some four-dimensional being +would render us the same aid that we before rendered the two-dimensional +worm. For the four-dimensional being could take the +grain of corn into his four-dimensional space and then replace it in +our space in the desired spot outside of the glass sphere. Slade +performed this trick before Zöllner. Its mere performance sufficed +to convince this latter investigator that Slade had here made use of +a four-dimensional agent, who, in respect of power of motion, controlled +his four-dimensional space as we do our three-dimensional +space. It never occurred to Zöllner that this experiment was the +cleverly executed trick of a prestidigitateur, or, as it would at once +occur to us, that the whole thing was a sensory illusion. The fact +\PG seq=116 Page 105 ------------------------------------------------------ +that we cannot explain a trick easily and naturally does not irrevocably +prove that it is accomplished by other means than those +which the world of matter presents. + +Still better known than this last performance is Slade's experiments +with knots\index + {Knots, experiment with the}. To explain this in connection with the fourth +dimension, we must resort again to the plane and the flat worm inhabiting +it. To two parallel lines in a plane let the two ends of a +third line, which has a double point, that is, intersects itself once, +be attached. Our flat worm would not be able to untie the loop +formed by the doubled third line, which we will call a string, because +it cannot execute motions in three dimensions. If, therefore, +a two-dimensional prestidigitateur should appear and accomplish +the trick of untying this loop without removing the two ends of the +string from the parallel lines, he will have accomplished for our flat +worm a supersensuous experiment. A human being engaged in the +service of the prestidigitateur could execute for him the experiment +by simply lifting the string a little out of the plane, pulling it taut, +and placing it back again. This suggests the following analogous +experiment for three-dimensional beings. The two ends of a string +in which a common knot has been made are sealed to the opposite +walls of a room. The problem is to untie this knot without breaking +the seals at the two ends of the string. Everybody knows that +this problem is not soluble, but it may be calculated mathematically +that the knot in the string can be untied as easily by motions +in a fourth dimension of space as in the experiment above described +the knot in the two-dimensional string was untied by a three-dimensional +motion. Now as Slade untied the knot before Zöllner's eyes +without apparently making any use of the ends fastened in the +walls, Zöllner was still more firmly confirmed in the view that Slade +had power over spirits who performed the experiments for him. + +Still more far-reaching is the theory of Carl du Prel\index + {Duprel@Du Prel, Carl} concerning +the relations of the material and the four-dimensional world. (Compare +his numerous essays in the spiritualistic magazine \textit{Sphinx}.) +Just as the shadows of three-dimensional objects cast on a wall are +controlled in their movements by the things whose projections they +are, in the same way it is claimed does there exist back of everything +\PG seq=117 Page 106 ------------------------------------------------------ +of this sense-perceptible world a real transcendental and four-dimensional +``thing-in-itself'' whose projection in the space of experience +is what we falsely regard as the independent thing. Thus +every man besides existing in his terrestrial self also exists in a +spiritual or astral self which constantly accompanies him in his +walks through life and whose existence is especially proclaimed in +states of profound sleep, of somnambulism, and in the conditions +of mediums. In this way Du Prel explains the wonderful feats of +somnambulists, and the aerial journeys of sorcerers and witches. +Whereas, ordinarily the separation of the material body from the +astral body is only effected at the moment of death; in the case of +somnambulists this separation may take place at any time, or, as Du +Prel says, ``the threshold of feeling may be permanently displaced.'' + +In view of the natural relations of such theories to the dogmas +of Christianity it is explainable that theologians also have raised +their voices for or against spiritualism. While the \textit{Protestant Church +Times} beheld in the ``repulsive thaumaturgic performances which +these coryph\ae i of modern science offer, a lack of comprehension of +real philosophy,'' the magazine \textit{The Proof of Faith} expresses its delight +at the discovery of spiritualism in the following manner: +``Every Christian will surely rejoice at the deep and perhaps mortal +wound which these new discoveries have in all probability administered +to modern materialism.'' + +We shall pass by the childish opinion that the Bible also speaks +of four dimensions\index + {Bible|Bible}, as both in Job (xi.\ \8--\9) and in the Epistle to +the Ephesians (iii.~\Num{18}) only breadth, length, depth, and height, +that is, four directions of extension, are mentioned. Yet we will +still add, as Cranz has done, the reflections which Zöllner, as the +most prominent representative of modern spiritualism, has put forward +respecting its relations to the doctrines of Christianity (\textit{Wissensch.\ +Abhandl.}, Vol.\ III). By the foundation of transcendental +physics on the basis of spiritualistic phenomena, the ``new light'' +has arisen which is spoken of in the New Testament. The rending of +the veil of the Temple on the crucifixion of Christ, the resurrection, +the ascension, the transfiguration, the speaking with many tongues +on the giving out of the Holy Ghost, all these are in Zöllner's view +\PG seq=118 Page 107 ------------------------------------------------------ +spiritualistic phenomena. Similarly, he sees a reference to the +four-dimensional world of spirits in all those sayings of Christ in +which the latter speaks to his Apostles of the impossibility of their +having any image or notion of the place to which when he disappeared +he would go and whence he would return. (Gospel of +St.\ John, xii.\ \Num{33}, \Num{36}; xiv.~\2, \3, \Num{28}; xvi.~5, \Num{13}.) + +Ulrici\index{Ulrici}, however, goes farthest in the mingling of spiritualistic +and Christian beliefs; for he sees in the doctrine of spiritualism a +means of strengthening belief in a supreme moral world-order and +in the immortality of the soul. In answer to Ulrici's tract ``Spiritualism +So-called, a Question of Science'' (\Num{1889}) Wundt\index{Wundt} wrote an +annihilating reply bearing the title ``Spiritualism, a Question of +Science So-called.'' Wundt criticises the future condition of our +soul according to spiritualistic hypotheses in the following sarcastic +yet pertinent words, which Cranz also quotes: ``(\1) Physically, the +souls of the dead come into the thraldom of certain living beings +who are called mediums\index{Mediums}. These mediums are, for the present at +least, a not widely diffused class, and they appear to be almost +exclusively Americans. At the command of these mediums, departed +souls perform mechanical feats which possess throughout +the character of absolute aimlessness. They rap, they lift tables +and chairs, they move beds, they play on the harmonica, and do +other similar things. (\2) Intellectually, the souls of the dead +enter a condition which, if we are to judge from the productions + which they deposit on the slates of the mediums, must be termed +a very lamentable one. These slate-writings belong throughout +in the category of imbecility; they are totally bereft of any contents. +(\3) The most favored, apparently, is the moral condition of +the soul. According to the testimony which we have, its character +cannot be said to be anything else than that of harmlessness. +From brutal performances, such, for instance, as the destruction +of bed-canopies, the spirits most politely refrain.'' Wundt then +laments the demoralising effect which spiritualism exercises on people +who have hitherto devoted their powers to some serious pursuit +or even to the service of science. In fact it is a presumptuous and +flagrant procedure to forsake the path of exact research, which +\PG seq=119 Page 108 ------------------------------------------------------ +slow as it is, yet always leads to a sure extension of knowledge, in +the hope that some aimless, foolhardy venture into the realm of +uncertainty will carry us farther than the path of slow toil, and that +we can ever thus easily lift the veil which hides from man the problems +of the world that are yet unsolved. + +\ThoughtBreakStars + +\indent Now that we have presented the opinions of others respecting +the existence of a four-dimensional world of spirits, the author would +like to develop one or two ideas of his own on the subject. In the +preceding section it was stated that everything that we perceive by +our senses is three-dimensional and that everything that possesses +four or more dimensions can only be regarded as abstractions or fictions +which our reason forms in its constant efforts after an extension +and generalisation of knowledge. To speak of two-dimensional +matter is as self-contradictory as the notion of four-dimensional +matter. But a two or a four-dimensional world might exist in some +other manner than a material manner, and for all we know in one +which to us does not admit of representation. But in such a case, +if it were without the power of affecting the material world, we should +never be able to acquire any knowledge concerning its existence, +and it would be totally indifferent to the people of the three-dimensional +world, whether such a world existed or not. Just as an artist +during his lifetime produces a number of different works of art, so +also the Creator might have created a number of different-dimensioned +worlds which in no wise interfere with one another. In such +a case, any one world would not be able to know anything of any +other, and we must consequently regard the question whether a +four-dimensional world exists which is incapable of affecting ours, +as insoluble. We have only to examine, therefore, the question +whether the phenomenal world perhaps is a single individual in a +great layer of worlds of which every successive one has one more +dimension than the preceding and which are so connected with one +another that each successive world contains and includes the preceding +world, and, therefore, can produce effects in it. For our +reason, which draws its inferences from the phenomena of this world, +tells us, that if outside the three-dimensional world there exists a +\PG seq=120 Page 109 ------------------------------------------------------ +second four-dimensional world, containing ours, there is no reason +why worlds of more or less dimensions should not, with the same +right, also exist. But if now, as Zöllner and his adherents maintain, +four-dimensional spirits exist which can act by the mere efforts +of their own wills on our world, there is surely no reason why we +three-dimensional beings should not also be able to produce effects +on some two-dimensional animated world. Whether we have, generally, +any such influence we do not know, but we certainly do know +that we do not act purposely and consciously on a two-dimensional +world. If, therefore, Zöllner were right, the plan of creation would +possess the wonderful feature that four-dimensional beings are capable +of arbitrarily affecting the three-dimensional world, but that +three-dimensional beings have no right in their turn consciously to +affect a two-dimensional world. + +The supporters of Zöllner's hypothesis will perhaps reply to +the objection just made, that the plan of creation might, after all, +possibly possess this wonderful peculiarity, that we human beings +perhaps, in some higher condition of culture, will be able to act consciously +on two-dimensional worlds, and that at any rate it is simply +an inference by analogy to conclude from the non-existence of a relation +between three and two dimensions that the same relation is also +wanting in the case of four and three dimensions. As a matter of fact, +the objection above made is not intended to refute Zöllner's hypothesis, +but only to stamp it as very improbable. But despite this improbability +Zöllner would still be right if the phenomena of the material +world actually made his hypothesis necessary. That, however, +is by no means the case. Although most of the phenomena to which +the spiritualists appeal are probably founded on sense-illusions, +humbug, and self-deception, it cannot be denied that there possibly +do exist phenomena which cannot be brought into harmony with +the natural laws now known. There always have been mysterious +phenomena, and there always will be. Yet, as we have often seen +that the progress of science has again and again revealed as natural +what former generations held to be supernatural, it is certainly +wholly wrong to bring in for the explanation of phenomena which +now seem mysterious an hypothesis like that of Zöllner, by which +\PG seq=121 Page 110 ------------------------------------------------------ +everything in the world can be explained. If we adopt a point of +view which regards it as natural for spirits arbitrarily to interfere in +the workings of the world, all scientific investigation will cease, for +we could never more trust or rely upon any chemical or physical experiment, +or any botanical or zoölogical culture. If the spirits are +the authors of the phenomena that are mysterious to us, why should +they also not have control of the phenomena which are not mysterious? +The existence of mysterious phenomena justifies in no manner +or form the assumption that spirits exist which produce them. +Would it not be much simpler, if we \emph{must} have supernatural + influences\index{Supernatural influences|etseq}, +to adopt the naïve religious point of view, according to +which everything that happens is traceable to the direct, actual, and +personal interference of a single being which we call God? Things +which formerly stood beyond the sphere of our knowledge and were +regarded as marvellous events, as a storm, for example, now stand +in the most intimate connection with known natural laws. Things +that formerly were mysterious are so no longer. If one hundred and +fifty years ago some scientists were in the possession of our present +knowledge of inductional electricity and had connected Paris and +Berlin with a wire by whose aid one could clearly interpret in +Berlin what another person had at that very moment said in Paris, +people would have regarded this phenomenon as supernatural and +assumed that the originator of this long-distance speaking was in +league with spirits. + +We recognise, thus, that the things which are termed supernatural +depend to a great extent on the stage of culture which humanity +has reached. Things which now appear to us mysterious, +may, in a very few decades, be recognised as quite natural. This +knowledge, however, is not to be obtained by the lazy assumption +of bands of spirits as the authors of mysterious phenomena, but by +performing what in physics and chemistry is called experiment. +But the first and essential condition of all scientific experimenting +is that the experimenter shall be absolutely master of the conditions +under which the experiment is or is not to succeed. Now, this criterion +of scientific experimenting is totally lacking in all spiritualistic +experiments. We can never assign in their case the conditions under +\PG seq=122 Page 111 ------------------------------------------------------ +which they will or will not succeed. When all the preparations +in a spiritualistic \emph{séance}\index + {Spiritualistic!séance} have been properly made, but nothing takes +place, the beautiful excuse is always forthcoming that the ``spirits +were not willing,'' that there were ``too many incredulous persons +present,'' and so forth. Fortunately, in physical experiments these +pretexts are not necessary. By the path of experiment, and not by +that of transcendental speculation, physics has thus made incredible +progress and has piled new knowledge strata on strata upon the +old. Accordingly, the prospect is left that the mysteries which the +conditions and properties of the human soul still present can be +solved more and more by the methods of scientific experiment\index + {Science}. To +this end, however, it is especially necessary that the physio-psychological +experiments in question should only be performed by men +who possess the critical eye of inquiry, who are free from the dangers +of self-illusion, and who are competent to keep apart from their +experiments all superstition and deception. The history of natural +science clearly teaches that it is only by this road that man can arrive +at certain and well-established knowledge. If, therefore, there +really is behind such phenomena as mind-reading, telepathy, and +similar psychical phenomena, something besides humbug and self-illusion, +what we have to do is to study privately and carefully by +serious experiments the success or non-success of such phenomena, +and not allow ourselves to be influenced by the public and dramatic +performances of psychical artists, like Cumberland\index + {Cumberland} and his ilk. + +The high eminence on which the knowledge and civilisation of +humanity now stands was not reached by the thoughtless employment +of fanciful ideas, nor by recourse to four-dimensional worlds, +but by hard, serious labor, and slow, unceasing research. Let all men +of science, therefore, band themselves together and oppose a solid +front to methods that explain everything that is now mysterious to +us by the interference of independent spirits. For these methods, +owing to the fact that they can explain everything, explain nothing, +and thus oppose dangerous obstacles to the progress of real research, +to which we owe the beautiful temple of modern knowledge\index + {Fourth dimension, the|)}. +\PG seq=123 Page 112 ------------------------------------------------------ + + + +\chapter{The Squaring of the Circle} + +\section*{AN HISTORICAL SKETCH OF THE PROBLEM FROM THE REMOTEST TIMES.} + +%I. +\section{UNIVERSAL INTEREST IN THE PROBLEM.} + +\lettrine{F}{or} two and a half thousand years\index + {Squaring of the circle|(}, both competent and incompetent +minds have striven in vain to solve the problem known +as the squaring of the circle. Now that geometers have at last succeeded +in giving a rigorous demonstration of the impossibility of +solving the problem with straight edge and compasses, it seems +fitting and opportune to cast a glance into the nature and history +of this very ancient problem. And this will be found the more justifiable +in view of the fact that the squaring of the circle, at least +in name, is very widely known outside of the narrow circle of professional +mathematicians. + +The \textit{Proceedings of the French Academy}\index + {Academy, French} for the year \Num{1775} contain +at page \Num{61} a resolution of the Academy not to examine from +that time on, any so-called solutions of the quadrature of the circle. +The Academy was driven to this determination by the overwhelming +multitude of professed solutions of the famous problem which +were sent to it every month in the year,---solutions which of course +were an invariable attestation of the ignorance and self-conceit of +their authors, but which suffered collectively from the very important +drawback in mathematics of being \emph{wrong}. Since that time all +professed solutions of the problem received by the Academy find a +sure and permanent resting-place in the waste-basket, and remain +unanswered for all time. The circle-squarer, however, sees in this +\PG seq=124 Page 113 ------------------------------------------------------ +high-handed manner of rejection only the envy of the great and +powerful at his grand intellectual discovery. He is determined to +secure recognition, and appeals therefore to the public. The newspapers +must obtain for him the appreciation that scientific societies +have denied. And every year the old mathematical sea-serpent +more than once disports itself in the columns of our newspapers in +the shape of an announcement that Mr.~N.\;N., of P.\;P., has at last +solved the problem of the quadrature of the circle. + +But what manner of people are these circle-squarers\index + {Circle-squarers|etseq}, when examined +by the light? Almost always they will be found to be imperfectly +educated persons, whose mathematical knowledge does +not exceed that of a modern high-school student. It is seldom that +they know accurately what the requirements of the problem are and +what its nature; they are totally ignorant of the two and a half +thousand years' history of the problem; and they have no idea +whatever of the important investigations which have been made +with regard to it by great and real mathematicians in every century +down to our own time.\footnote + {For the full psychogeny and psychiatry of the circle-squarer see A.~De Morgan\index + {Demo@De Morgan}, + \textit{A Budget of Paradoxes} (London, \Num{1872}).---\textit{Tr.}} + +Yet great as is the quantum of ignorance that circle-squarers +intermix with their intellectual products, the lavish supply of conceit +and egotism with which they season their performances is still +greater. I have not far to go to furnish a verification of this. A +book printed in Hamburg in the year \Num{1840} lies before me, in which +the author thanks Almighty God at every second page that He has +selected him and no one else to solve the ``problem phenomenal'' +of mathematics, ``so long sought for, so fervently desired, and attempted +by millions.'' After this modest author has proclaimed +himself the unmasker of Archimedes's\index{Archimedes} deceit, he says: ``And thus +it hath pleased our mother Nature to withhold this precious mathematical +jewel from the eye of human investigation, until she +thought it fitting to reveal truth to simplicity.'' + +This will suffice to show the great fatuity of the author. But +it does not suffice to prove his ignorance. He has no conception +\PG seq=125 Page 114 ------------------------------------------------------ +of mathematical demonstration; he takes it for granted that things +are so because they seem so to him. Errors of logic, also, abound +in his book. But, minor fallacies apart, wherein does the real error +of this ``unmasker'' of Archimedes consist? It requires considerable +labor to extricate the kernel of the demonstration from the +turgid language and bombastic style in which the author has buried +his conclusions. But it is this. The author inscribes a square in a +circle, circumscribes another about it, then points out that the inside +square is made up of four congruent triangles, whereas the circumscribed +square is made up of eight such triangles; from which fact, +seeing that the circle is larger than the one square and smaller than +the other, he draws the bold conclusion that the circle is equal in +area to six such triangles. It is hardly conceivable that a rational +being could infer that something which is greater than \4 and less +than \8 must necessarily be \6. But with a man that attempts the +squaring of the circle this kind of ratiocination \emph{is} possible. + +It is the same with all the other attempted solutions of the +problem; in all of them either logical fallacies or violations of elementary +arithmetical or geometrical truths can be pointed out. +Only they are not always of such a trivial nature as in the book +just mentioned. + +Let us now inquire into the origin of this propensity which +leads people to occupy themselves with the quadrature of the circle. + +Attention must first be called to the antiquity of the problem. +A quadrature was attempted in Egypt\index + {Quadrature of the circle|Quadrature} \Num{500} years before the exodus +of the Israelites. Among the Greeks the problem never ceased to +play a part that greatly influenced the progress of mathematics. +And in the middle ages also the squaring of the circle sporadically +appears as the philosopher's stone of mathematics. The problem +has thus never ceased to be dealt with and considered. But it +is not by the antiquity of the problem that circle-squarers are enticed, +but by the allurement which everything exerts that is calculated +to raise the individual above the mass of ordinary humanity, +and to bind about his temples the laurel crown of celebrity. +Ambition spurred men on in ancient Greece and still spurs them +on in modern times to crack this primeval mathematical nut. +\PG seq=126 Page 115 ------------------------------------------------------ +Whether they are competent thereto is a secondary consideration. +They look upon the squaring of the circle as the grand prize of a +lottery that can just as well fall to their lot as that of any other +man. They do not remember that--- +\begin{quote} +``Toil before honor is placed by sagacious decrees of Immortals.'' % yes original has full stop +\end{quote} +and that it requires years of consecutive study to gain possession of +the mathematical weapons that are indispensably necessary to attack +the problem, but which even in the hands of the most distinguished +mathematical strategists did not suffice to take the stronghold. + +But why is it, we must further ask, that it happens to be the +squaring of the circle and not some other unsolved mathematical +problem upon which the efforts of people are bestowed who have +no knowledge of mathematics yet busy themselves with mathematical +questions? The question is answered by the fact that the squaring +of the circle is about the only mathematical problem that is +known to the unprofessional world,---at least by name. Even among +the Greeks the problem was very widely known outside of mathematical +circles. In the eyes of the Grecian\index + {Greeks!squaring of the circle among} layman, as at present +among many of his modern brethren, occupation with this problem +was regarded as the most important and essential business of +mathematicians. In fact, they had a special word to designate this +species of activity, namely, \tetragonizein\index + {Tet@\tetragonizein}, which means to busy one's +self with the quadrature. In modern times, also, every educated +person, though he be not a mathematician, knows the problem by +name, and knows that it is insoluble, or at least, that despite the +efforts of the most famous mathematicians it has not yet been +solved. For this reason the phrase ``to square the circle,'' is now +generally used in the sense of attempting the impossible. + +But in addition to the antiquity of the problem, and the fact +also that it is known to the lay world, there is an important third +factor that induces people to occupy themselves with it. This is the +report that has been current for more than a century now, that the +Academies, the Queen of England, or some other influential person +has offered a large prize to be given to the one that first solves the +\PG seq=127 Page 116 ------------------------------------------------------ +problem. As a matter of fact, the hope of obtaining this large +prize of money is with many circle-squarers the principal incitement +to action. And the author of the book above referred to begs his +readers to lend him their assistance in obtaining the prizes\index + {Prizes offered circle-squarers} offered. + +Although the opinion is widely current in the unprofessional +world, that professional mathematicians are still busied with the +solution of the problem, this is by no means the case. On the contrary, +for some two hundred years, the endeavors of many great +mathematicians have been directed solely towards demonstrating +with exactness that the problem is insoluble. It is, as a rule,---and +naturally,---more difficult to prove that a thing is impossible +than to prove that it is possible. And thus it has happened, that +up to within a few years ago, despite the employment of the most +varied and the most comprehensive methods of modern mathematics, +no one succeeded in supplying the wished-for demonstration of +the problem's impossibility\index + {Impossibility of demonstration}\label{p:116}. At last, Professor Lindemann\index + {Lindemann}, of +Königsberg, in June, \Num{1882}, succeeded in furnishing a demonstration,---and +the first demonstration,---that it is impossible by employing +only straight edge and compasses to construct a square % original has "squaer" +that is mathematically exactly equal in area to a given circle. The +demonstration, naturally, was not effected with the help of the old +elementary methods; for if it were, it would have been accomplished +centuries ago; but methods were requisite that were first +furnished by the theory of definite integrals and departments of +higher algebra developed in the last few decades; in other words, +it required the direct and indirect preparatory labor of many centuries +to make finally possible a demonstration of the insolubility +of this historic problem. + +Of course, this demonstration will have no more effect than +the resolution of the Paris Academy of \Num{1775}, in causing the fecund +race of circle-squarers to vanish from the face of the earth. In the +future as in the past, there will be people who know nothing of +this demonstration and will not care to know anything, and who +believe that they cannot help succeeding in a matter in which others +have failed, and that just they have been appointed by Providence +to solve the famous puzzle. But unfortunately the ineradicable +\PG seq=128 Page 117 ------------------------------------------------------ +mania for solving the quadrature of the circle has also its +serious side. Circle-squarers are not always so self-satisfied as +the author of the book above mentioned. They often see, or at +least divine, the insuperable difficulties that tower up before them, +and the conflict between their aspirations and their performances, +the consciousness that the problem they long to solve they are unable +to solve, darkens their soul and, lost to the world, they become +interesting subjects for the science of psychiatry. + +%II. +\section{NATURE OF THE PROBLEM.} + +It is easy to determine the length of the radius of a circle, or +the length of its diameter, which must be double that of the radius; +and the question next arises, what is the number that tells how many +times larger the circumference of the circle, that is the length of the +circular line, is than its radius or its diameter. From the fact that +all circles have the same shape it follows that this proportion will +be the same for all circles both large and small. Now, since the +time of Archimedes\index + {Archimedes}, all civilised nations that have cultivated mathematics +have denoted the number that tells how many times larger +the circumference of a circle is than the diameter by the symbol $\pi$\index + {Pi@$\pi$|etseq},---the +Greek initial letter of the word periphery.\footnote + {The Greek symbol $\pi$ was first employed by W.~Jones\index + {Jones, W.} in \Num{1706} and did not + come into general use until about the middle of the eighteenth century through + the works of Euler\index{Euler}.---\textit{Trans.}} +To compute $\pi$, +therefore, means to calculate how many times larger the circumference +of a circle is than its diameter. This calculation is called +``the numerical rectification of the circle\index + {Rectification of the circle, numerical|etseq}\index + {Quadrature of the circle!numerical|etseq}.'' + +Next to the calculation of the circumference, the calculation of +the superficial contents of a circle by means of its radius or diameter +is perhaps most important; that is, the computation of how +great an area that part of a plane which lies within a circle measures. +This calculation is called the ``numerical quadrature.'' It +depends, however, upon the problem of numerical rectification; +that is, upon the calculation of the magnitude of $\pi$. For it is demonstrated +in elementary geometry, that the area of a circle is +\PG seq=129 Page 118 ------------------------------------------------------ +equal to the area of a triangle produced by drawing in the circle a +radius, erecting at the extremity of the same a tangent,---that is, in +this case, a perpendicular,---cutting off upon the latter the length +of the circumference, measuring from the extremity, and joining +the point thus obtained with the centre of the circle. It follows +from this that the area of a circle is as many times larger than the +square upon its radius as the number $\pi$ amounts to. + +The numerical rectification and numerical quadrature of the +circle based upon the computation of the number $\pi$ are to be clearly +distinguished from problems\index + {Construction, problems of} that require a straight line equal in +length to the circumference of a circle, or a square equal in area to +a circle, to be \emph{constructively} produced from its radius or its diameter; +problems which might properly be called ``constructive rectification'' +or ``constructive quadrature.'' Approximately, of course, by +employing an approximate value for $\pi$, these problems are easily +solvable. But to solve a problem of construction in geometry, +means to solve it with mathematical exactitude. If the value $\pi$ +were exactly equal to the ratio of two whole numbers to each +other, the constructive rectification would present no difficulties. +For example, suppose the circumference of a circle were exactly +$3\frac17$ times greater than its diameter; then the diameter could be divided +into seven equal parts, which could easily be done by the +principles of planimetry with straight edge and compasses; then +by prolonging to the amount of such a part a straight line exactly +three times as long as the diameter, we should obtain a straight +line exactly equal to the circumference of the circle. But as a matter +of fact,---and this has actually been demonstrated,---there do +not exist two whole numbers, be they ever so great, that exactly +represent by their proportion to each other the number $\pi$. Consequently, +a rectification of the kind just described does not attain +the object desired. + +It might be asked here, whether from the demonstrated fact +that the number $\pi$ is not equal to the ratio of two whole numbers +however great, it does not immediately follow that it is impossible +to construct a straight line exactly equal in length to the circumference +of a circle; thus demonstrating at once the impossibility of +\PG seq=130 Page 119 ------------------------------------------------------ +solving the problem. This question is to be answered in the negative. +For in geometry there can easily exist pairs of lines of which +the one can be readily constructed from the other, notwithstanding +the fact that no two whole numbers can be found to represent the +ratio of the two lines. The side and the diagonal of a square, for +instance, are so constituted. It is true the ratio of the latter two +magnitudes is nearly that of \5 to \7. But this proportion is not +exact, and there are in fact no two numbers that represent the ratio +exactly. Nevertheless, either of these two lines can be readily constructed +from the other by employing only straight edge and compasses\index + {Straight edge and compasses, to construct with|etseq}. +This might be the case, too, with the rectification of the +circle; and consequently from the impossibility of representing $\pi$ +by the ratio between two whole numbers the impossibility of the +problem of rectification is not inferable. + +The quadrature of the circle stands and falls with the problem +of rectification\index + {Quadrature of the circle!a@stands and falls with the problem + of rectification}. This rests upon the truth above mentioned, that +a circle is equal in area to a right-angled triangle, in which one +side is equal to the radius of the circle and the other to the circumference. +Supposing, accordingly, that the circumference of the circle +had been rectified, then we could construct this triangle. But every +triangle, as we know from plane geometry, can, with the help of +straight edge and compasses be converted into a square exactly +equal to it in area. So that, supposing the rectification of the circumference +of a circle to have been successfully effected, a square +could be constructed that would be exactly equal in area to the +circle. + +The dependence upon one another of the three problems of the +computation of the number $\pi$, the quadrature of the circle, and its +rectification, thus obliges us, in dealing with the history of the +quadrature, to regard investigations with respect to the value of $\pi$ +and attempts to rectify the circle as of equal importance, and to +consider them accordingly. + +We have used repeatedly in the course of this discussion the +expression ``to construct with straight edge and compasses.'' It +will be necessary to explain what is meant by the specification of +these two instruments. When to a requirement in geometry to +\PG seq=131 Page 120 ------------------------------------------------------ +construct a figure there are so large a number of conditions annexed +that the construction of only \emph{one} figure or a limited number +of figures is possible in accordance with those conditions; such a +full and stated requirement is called a problem of construction, or +briefly a problem. When a problem of this kind is presented for +solution it is necessary to reduce it to simpler problems, already +recognised as solvable; and since these latter depend in their turn +upon other, still simpler problems, we are finally brought back to +certain fundamental problems\index + {Problems|Problems} upon which the rest are based but +which are not themselves reducible to problems less simple. These +fundamental problems are, so to speak, the lowermost stones of the +edifice of geometrical construction. The question next arises as to +what problems may be properly regarded as fundamental; and it +has been found, that the solution of a great part of the problems +that arise in elementary plane geometry rests upon the solution of +only five original problems. They are: +\begin{Itemize} +\item[\1.] The construction of a straight line that shall pass through +two given points. + +\item[\2.] The construction of a circle the centre of which is a given +point and the radius of which has a given length. + +\item[\3.] The determination of the point lying coincidently on two +given straight lines prolonged as far as necessary,---in case such a +point (point of intersection) exists. + +\item[\4.] The determination of the two points that lie coincidently +on a given straight line and a given circle,---in case such common +points (points of intersection) exist. + +\item[\5.] The determination of the two points that lie coincidently on +two given circles,---in case such common points (points of intersection) exist. +\end{Itemize} + +For the solution of the three last of these five problems the +eye alone is needed, while for the solution of the first two problems, +besides pencil, ink, chalk, or the like, additional special instruments +are required: for the solution of the first problem a +straight edge or ruler is most generally used, and for the solution +of the second a pair of compasses. But it must be remembered +that it is no concern of geometry what mechanical instruments\index + {Mechanical instruments of geometry|etseq} are +\PG seq=132 Page 121 ------------------------------------------------------ +employed in the solution of the five problems mentioned. Geometry +simply limits itself to the presupposition that these problems +are solvable, and regards a complicated problem as solved if, upon +a specification of the constructions of which the solution consists, +no other requirements are demanded than the five above mentioned. +Since, accordingly, geometry does not itself furnish the solution of +these five problems, but rather exacts them, they are termed \emph + {postulates}\index{Postulates}.\footnote + {Usually geometers mention only two postulates (Nos.\ \1 and \2). But since to + geometry proper it is indifferent whether only the eye, or additional special mechanical + instruments are necessary, the author has regarded it more correct in point of + method to assume five postulates.} +All problems of plane geometry are not reducible to these +five problems alone. There are problems that can be solved only +by assuming other problems as solvable which are not included in +the five given; for example, the construction of an ellipse, having +given its centre and its major and minor axes. Many problems, +however, possess the property of being solvable with the assistance +of the above-formulated five postulates alone, and where this is the +case they are said to be ``constructible with straight edge and compasses,'' +or ``elementarily'' constructible. + +After these general remarks upon the solvability of problems\index + {Solvability of problems} +of geometrical construction, which an understanding of the history +of the squaring of the circle makes indispensable, the significance +of the question whether the quadrature of the circle is or is not +solvable, that is elementarily solvable, will become intelligible. +But the conception of elementary solvability only gradually took +clear form, and we therefore find among the Greeks as well as +among the Arabs endeavors, successful in some respects, that aimed +at solving the quadrature of the circle with other expedients than +the five postulates. We have also to take these endeavors into +consideration, and especially so as they, no less than the unsuccessful +efforts at elementary solution, have upon the whole advanced +the science of geometry, and contributed much to the clarification +of geometrical ideas. +\PG seq=133 Page 122 ------------------------------------------------------ + +%III. +\section{THE EGYPTIANS, BABYLONIANS, AND GREEKS.} + +In the oldest mathematical work that we possess we find a rule +telling us how to construct a square which is equal in area to a +given circle. This celebrated book, the Rhind Papyrus\index{Rhind Papyrus}\index + {Egyptians, the squaring of the circle among|etseq}\index + {Greeks!squaring of the circle among|etseq} of the British +Museum, translated and explained by Eisenlohr\index{Eisenlohr} (Leipsic, \Num{1877}), +was written, as stated in the work itself, in the thirty-third year of +the reign of King Ra-a-us, by a scribe of that monarch, named +Ahmes\index{Ahmes}. The composition of the work falls accordingly in the period +of the two Hyksos dynasties, that is, in the period between \Num{2000} +and \Num{1700}~B.\;C. But there is another important circumstance to be +noted. Ahmes mentions in his introduction that he composed his +work after the model of old treatises, written in the time of King +Raenmat; whence it appears that the originals of the mathematical +expositions of Ahmes are half a thousand years older still than the +Rhind Papyrus. + +The rule given in this papyrus for obtaining a square equal to +a circle specifies that the diameter of the circle shall be shortened +one-ninth of its length and upon the shortened line thus obtained +a square erected. Of course, the area of a square of this construction +is only approximately equal to the area of the circle. An idea +may be obtained of the degree of exactness of this original, primitive +quadrature by remarking, that if the diameter of the circle in +question is one metre in length, the square that is supposed to be +equal to the circle is a little less than half a square decimetre too +large; an approximation not so accurate as that made by Archimedes, +yet much more correct than many a one later employed. It +is not known how Ahmes or his predecessors arrived at this approximate +quadrature; but it is certain that it was handed down in +Egypt from century to century, and in late Egyptian times it appears +repeatedly. + +In addition to the effort of the Egyptians, we also find in pre-Grecian +antiquity an attempt at circle-computation among the Babylonians\index + {Babylonians, squaring of the circle among|etseq}. +This is not a quadrature, but is intended as a rectification +\PG seq=134 Page 123 ------------------------------------------------------ +of the circumference. The Babylonian mathematicians had +discovered, that if the radius of a circle be successively inscribed +as a chord within its circumference, after the sixth inscription we +arrive at the point from which we set out, and they concluded from +this that the circumference of a circle must be a little larger than a +line which is six times as long as the radius, that is three times as +long as the diameter. A trace of this Babylonian method of computation +may even be found in the Bible\index + {Bible!squaring of the circle in the}; for in \1~Kings vii.\ \Num{23}, +and \2~Chron.\ iv.~\2, the great laver is described, which under the +name of the ``molten sea'' constituted an ornament of the temple +of Solomon; and it is said of this vessel that it measured ten cubits +from brim to brim, and thirty cubits round about. The number \3 +as the ratio of the circumference to the diameter is still more plainly +given in the Talmud\index{Talmud}, where we read that ``that which measures +three lengths in circumference is one length across.'' + +With regard to the earlier Greek mathematicians---as Thales\index{Thales} +and Pythagoras\index{Pythagoras}---we know that they acquired their elementary +mathematical knowledge in Egypt. But nothing has been handed +down to us which shows that they knew of the old Egyptian quadrature, +or that they dealt with the problem at all. But tradition +says, that, subsequently, the teacher of Euripides and Pericles, the +great philosopher and mathematician Anaxagoras\index{Anaxagoras}, whom Plato so +highly praised, ``drew the quadrature of the circle'' in prison, in the +year \Num{434}~B.\;C. This is the account of Plutarch in the seventeenth +chapter of his work \textit{De Exilio}. The method is not told us in which +Anaxagoras is supposed to have solved the problem, and it is not +said whether, knowingly or unknowingly, he gave an approximate +solution after the manner of Ahmes. But at any rate, to Anaxagoras +belongs the merit of having called attention to a problem that +was to bear rich fruit by inciting Grecian scholars to busy themselves +with geometry, and thus more and more to advance that +science. + +Again, it is reported that the mathematician Hippias\index{Hippias} of Elis +invented a curved line that could be made to serve a double purpose: +first, to trisect an angle, and second to square the circle. +This curved line is the \tetragonizousa\ so often mentioned by the % should be slanted greek +\PG seq=135 Page 124 ------------------------------------------------------ +later Greek mathematicians, and by the Romans called ``quadratrix\index{Quadratrix}.'' +Regarding the nature of this curve we have exact knowledge +from Pappus\index{Pappus}. But it will be sufficient here to state that the quadratrix +is not a circle nor a portion of a circle, so that its construction +is not possible by means of the postulates enumerated in the +preceding section. And therefore the solution of the quadrature +of the circle founded on the construction of the quadratrix is not +an elementary solution in the sense discussed in the last section. +We can, it is true, conceive a mechanism that will draw this curve +as well as compasses draw a circle; and with the assistance of a +mechanism of this description the squaring of the circle is solvable +with exactitude. But if it be allowed to employ in a solution an +apparatus especially adapted thereto, every problem may be said to +be solvable. Strictly taken, the invention of the curve of Hippias\index{Hippias} +substitutes for one insuperable difficulty another equally insuperable. +Some time afterwards, about the year \Num{350}~B.\;C., the mathematician +Dinostratus\index{Dinostratus} showed that the quadratrix could also be used +to solve the problem of rectification, and from that time on this +problem plays almost the same rôle in Grecian mathematics as the +related problem of quadrature. + +As these problems gradually became known to the non-mathe\-mat\-icians +of Greece, attempts at solution at once sprang up +that are worthy of a place by the side of the solutions of modern +amateur circle-squarers. The Sophists\index + {Sophists} especially believed themselves +competent by seductive dialectic to take the stronghold that +had defied the intellectual onslaughts of the greatest mathematicians. +With verbal nicety, amounting to puerility, it was said that +the squaring of the circle depended upon the finding of a number +which represented in itself both a square and a circle; a square by +being a square number, a circle in that it ended with the same +number as the root number from which, by multiplication with itself, +it was produced. The number \Num{36}, accordingly, was, as they +thought, the one that embodied the solution of the famous problem. + +Contrasted with this twisting of words the speculations of Bryson +and Antiphon, both contemporaries of Socrates, though inexact, +\PG seq=136 Page 125 ------------------------------------------------------ +appear in a high degree intelligent. Antiphon\index{Antiphon} divided the +circle into four equal arcs, and by joining the points of division obtained +a square; he then divided each arc again into two equal +parts and thus obtained an inscribed octagon; thence he constructed +an inscribed \Num{16}-gon, and perceived that the figure so inscribed +more and more approached the shape of a circle. In this way, he +said, one should proceed, until there was inscribed in the circle a +polygon whose sides by reason of their smallness should coincide +with the circle. Now this polygon could, by methods already +taught by the Pythagoreans, be converted into a square of equal +area; and upon the basis of this fact Antiphon regarded the squaring +of the circle as solved. + +{Nothing can be said against this method except that, however +far the bisection of the arcs is carried, the result still remains an +approximate one.\LP\looseness=1 + +}The attempt of Bryson\index{Bryson} of Heraclea was better still; for this +scholar did not rest content with finding a square that was very +little smaller than the circle, but obtained by means of circumscribed +polygons another square that was very little larger than the +circle. Only Bryson committed the error of believing that the area +of the circle was the arithmetical mean between an inscribed and a +circumscribed polygon of an equal number of sides. Notwithstanding +this error, however, to Bryson belongs the merit---first, of +having introduced into mathematics by his emphasis of the necessity +of a square which was too large and one which was too small, +the conception of upper and lower ``limits\index{Limits}'' in approximations; +and secondly, by his comparison of the regular inscribed and circumscribed +polygons with a circle, of having indicated to Archimedes +the way by which an approximate value of $\pi$ was to be reached. + +Not long after Antiphon and Bryson, Hippocrates\index{Hippocrates} of Chios +treated the problem, which had now become more and more famous, +from a new point of view. Hippocrates was not satisfied +with approximate equalities, and searched for curvilinearly bounded +plane figures which should be mathematically equal to a rectilinearly +bounded figure, and which therefore could be converted by +straight edge and compasses into a square equal in area. First, +\PG seq=137 Page 126 ------------------------------------------------------ +Hippocrates found that the crescent-shaped plane figure produced +by drawing two perpendicular radii in a circle and describing upon +the line joining their extremities a semicircle, is exactly equal in % yes this "semicircle" is unhyphenated +area to the triangle that is formed by this line of junction and the +two radii; and upon the basis of this fact the endeavors of this untiring +scholar were directed towards converting a circle into a crescent. +Naturally he was unable to attain this object, but by his efforts +he discovered many new geometrical truths; among others +being the generalised form of the theorem mentioned, which bears +to the present day the name of \emph{lunulae Hippocratis}, the lunes of +Hippocrates\index + {Lunes of Hippocrates}. Thus, in the case of Hippocrates, it appears in the +plainest light, how precisely the insolvable problems of science are +qualified to advance science; in that they incite investigators to +devote themselves with persistence to its study and thus to fathom +its utmost depths. + +Following Hippocrates in the historical line of the great Grecian +geometricians comes the systematist Euclid\index + {Euclid}, whose rigid formulation +of geometrical principles has remained the standard presentation +down to the present century. The Elements of Euclid, +however, contain nothing relating to the quadrature of the circle +or to circle-computation. Comparisons of surfaces which relate to +the circle are indeed found in the work, but nowhere a computation +of the circumference of a circle or of the area of a circle. This +palpable gap in Euclid's system was filled by Archimedes, the +greatest mathematician of antiquity. + +Archimedes\index{Archimedes|etseq} was born in Syracuse in the year \Num{287}~B.\;C., and +devoted his life, which was spent in that city, to the mathematical +and the physical sciences, which he enriched with invaluable contributions. +He lived in Syracuse till the taking of the town by +Marcellus, in the year \Num{212}~B.\;C., when he fell by the hand of a Roman +soldier whom he had forbidden to destroy the figures he had +drawn in the sand. To the greatest performances of Archimedes +the successful computation of the number $\pi$ unquestionably belongs. +Like Bryson he started with regular inscribed and circumscribed +polygons. He showed how it was possible, beginning with +the perimeter of an inscribed hexagon, which is equal to six radii, +\PG seq=138 Page 127 ------------------------------------------------------ +to obtain by calculation the perimeter of a regular dodecagon, and +then the perimeter of a figure having double the number of sides of +that, and so on. Treating, then, the circumscribed polygons in a +similar manner, and proceeding with both series of polygons up to +a regular \Num{96}-sided polygon, he discovered on the one hand that the +ratio of the perimeter of the inscribed \Num{96}-sided polygon to the +diameter was greater than $\Num{6336}:\Num{2017}\Numfrac{\1}{\4}$, and on the other hand, that +the corresponding ratio with respect to the circumscribed \Num{96}-sided +polygon was smaller than $\Num{14688}:\Num{4673}\Numfrac{\1}{\2}$. He inferred from this, +that the number $\pi$\index{Pi@$\pi$}, the ratio of the circumference to the diameter, +was greater than the fraction $\frac{6336}{2017\frac14}$ and smaller than $\frac{14688}{4673\frac12}$. Reducing +the two limits thus found for the value of $\pi$, Archimedes then +showed that the first fraction was greater than $\3\frac{10}{71}$, and that the +second fraction was smaller than $\3\frac17$, whence it followed with certainty +that the value sought for $\pi$ lay between $\3\frac17$ and $\3\frac{10}{71}$. The +larger of these two approximate values is the only one usually +learned and employed. That which fills us with most astonishment +in the case of Archimedes's computation of $\pi$, is, first, the +great acumen and accuracy displayed by him in all the details of +the computation, and secondly the unwearied perseverance which +he exercised in calculating the limits of $\pi$ without the help of the +Arabian system of numerals and the decimal notation. For it must +be considered that at many stages of the computation what we call +the extraction of roots was necessary, and that Archimedes could +only by extremely tedious calculations obtain ratios that expressed +approximately the roots of given numbers and fractions.\footnote + {For Archimedes's actual researches, see Rudio\index + {Rudio}, \textit{Archimedes, Huygens, Lambert, + Legendre, vier Abhand.\ über die Kreismessung} (Leipsic, \Num{1892}), where + translations of the works of these four authors on cyclometry will be found.---\textit{Tr.}} + +With regard to the mathematicians of Greece that follow Archi\-medes, +all refer to and employ the approximate value of $\3\frac17$ for $\pi$, +without, however, contributing anything essentially new to the +problems of quadrature and of cyclometry\index{Cyclometry|etseq}. Thus Hero of Alexandria, +the father of surveying, who flourished about the year +\Num{100}~B.\;C., employs for purposes of practical measurement sometimes +\PG seq=139 Page 128 ------------------------------------------------------ +the value $\3\frac17$ for $\pi$ and sometimes even the rougher approximation +$\pi =\3$. The astronomer Ptolemy\index + {Ptolemy}, who lived in Alexandria +about the year \Num{150}~A.\;D., and who was famous as being the +author of the planetary system universally recognised as correct +down to the time of Copernicus, was the only one who furnished a +more exact value; this he designated, in the sexagesimal system of +fractional notation which he employed, by \3, \8, \Num{30},---that is \3 and +$\frac8{60}$ and $\frac{30}{3600}$, or as we now say \3 degrees\index + {Degrees}, \8 minutes\index{Minutes} (\textit{partes minutae +primae}), and \Num{30} seconds\index + {Seconds} (\textit{partes minutae secundae}). As a matter of +fact, the expression $\3+\frac8{60}+\frac{30}{3600}=\3\frac{17}{120}$ represents the number $\pi$ +more exactly than $\3\frac17$; but on the other hand, is, by reason of the +magnitude of the numbers \Num{17} and \Num{120} as compared with the numbers +\1 and \7, more cumbersome. + +%IV. +\section{THE ROMANS, HINDUS, CHINESE, ARABS, AND THE CHRISTIAN +NATIONS TO THE TIME OF NEWTON.} + +In the mathematical sciences\index + {Arabs!squaring of the circle among|etseq}\index + {Chinese!squaring of the circle among|etseq}\index + {Christian nations, squaring of the circle among|etseq}\index + {Hindus!squaring of the circle among|etseq}\index + {Romans!squaring of the circle among|etseq}, more than in any other, the Romans +stood upon the shoulders of the Greeks. Indeed, with respect +to cyclometry, they not only did not add anything new to the +Grecian discoveries, but frequently even evinced that they either +did not know of the beautiful result obtained by Archimedes, or at +least could not appreciate it. For instance, Vitruvius\index{Vitruvius}, who lived +during the time of Augustus, computed that a wheel \4 feet in diameter +must measure $\Num{12}\frac12$ feet in circumference; in other words, he +made $\pi$ equal to $\3\frac18$. And, similarly, a treatise on surveying, preserved +to us in the Gudian manuscript\index + {Gudian manuscript} of the library of Wolfenbüttel\index{Wolfenbüttel Library}, +contains the following instructions for squaring the circle: +Divide the circumference of a circle into four parts and make one +part the side of a square; this square will be equal in area to the +circle. Apart from the fact that the rectification of the arc of a +circle is requisite to the construction of a square of this kind, the +Roman quadrature, viewed as a calculation, is more inexact even +than any other known computation; for its result is that $\pi=\4$. + +The mathematical performances of the Hindus\index + {Hindus!mathematical performances of the|etseq} were not only +greater than those of the Romans, but in certain directions surpassed +\PG seq=140 Page 129 ------------------------------------------------------ +even those of the Greeks. In the most ancient source of +the mathematics of India that we know of, the Culvasûtras\index{Culvasûtras}, which +date back to a little before our chronological era, we do not find, it +is true, the squaring of the circle treated of, but the opposite problem +is dealt with, which might fittingly be termed the circling of +the square. The half of the side of a given square is prolonged in +length one third of the excess of half the diagonal over half the +side, and the line thus obtained is taken as the radius of the circle +equal in area to the square. The simplest way to obtain an idea +of the exactness of this construction is to compute how great $\pi$ +would have to be if the construction were exactly correct. We +find out in this way that the value of $\pi$ upon which the Indian circling +of the square is based, is about from five to six hundredths +smaller than the true value, whereas the approximate $\pi$ of Archimedes, +$\3\frac17$, is only from one to two thousandths too large, and that +the old Egyptian value exceeds the true value by from one to two +hundredths. + +Cyclometry very probably made great advances among the +Hindus in the first four or five centuries of our era; for Aryabhatta\index{Aryabhatta}, +who lived about the year \Num{500} after Christ, states, that the ratio of +the circumference to the diameter is $\Num{62832}:\Num{20000}$, an approximation +that in exactness surpasses even that of Ptolemy\index + {Ptolemy}. The Hindu +result gives \3.\Num{1416} for $\pi$, while $\pi$ really lies between \3.\Num{141592} and +\3.\Num{141593}. How the Hindus obtained this excellent value is told by +Gane\c ca, the commentator of Bhâskara\index{Bhâskara}, an author of the twelfth +century. Gane\c ca\index{Gane\c ca} says that the method of Archimedes was carried +still farther by the Hindu mathematicians; that by continually +doubling the number of sides they proceeded from the hexagon to +a polygon of \Num{384} sides, and that by the comparison of the circumferences +of the inscribed and circumscribed \Num{384}-sided polygons they +found that $\pi$ was equal to $\Num{3927}:\Num{1250}$. It will be seen that the value +given by Bhâskara is identical with the value of Aryabhatta. It is +further worthy of remark that the earlier of these two Hindu mathematicians +does not mention either the value $\3\frac17$ of Archimedes or +the value $\3\frac{17}{120}$ of Ptolemy, but that the later one knows of both +values and especially recommends that of Archimedes as the most +\PG seq=141 Page 130 ------------------------------------------------------ +useful for practical applications. Strange to say, the good approximate +value of Aryabhatta does not occur in Brahmagupta\index{Brahmagupta}, the +great Hindu mathematician who flourished in the beginning of the +seventh century; but we find the curious information in this author +that the area of a circle is exactly equal to the square root of \Num{10} +when the radius is unity. The value of $\pi$ as derivable from this +formula,---a value from two to three hundredths too large,---has +unquestionably arisen on Hindu soil. For it occurs in no Grecian +mathematician; and Arabian authors, who were in a better position +than we to know Greek and Hindu mathematical literature, declare +that the approximation which makes $\pi$ equal to the square root of +\Num{10}, is of Hindu origin. It is possible that the Hindu people, who +were addicted more than any other to numeral mysticism, sought +to find in this approximation some connection with the fact that +man has ten fingers, and that accordingly ten is the basis of their +numeral system. + +Reviewing the achievements of the Hindus generally with respect +to the problem of quadrature, we are brought to recognise +that this people, whose talents lay more in the line of arithmetical +computation than in the perception of spatial relations, accomplished +as good as nothing on the purely geometrical side of the +problem, but that the merit belongs to them of having carried the +Archimedean method of computing $\pi$ several stages farther, and of +having obtained in this way a much more exact value for it---a circumstance +that is explainable when we consider that the Hindus +are the inventors of our present system of numeral notation, possessing +which they easily outdid Archimedes, who employed the +awkward Greek system. + +With regard to the Chinese\index{Chinese}, this people operated in ancient +times with the Babylonian value for $\pi$, or \3; but they possessed +knowledge of the approximate value of Archimedes at least since +the end of the sixth century. Besides this, there appears in a number +of Chinese mathematical treatises an approximate value peculiarly +their own, in which $\pi=\3\frac7{50}$; a value, however, which notwithstanding +it is written in larger figures, is no better than that of +\PG seq=142 Page 131 ------------------------------------------------------ +Archimedes. Attempts at the \emph{constructive} quadrature of the circle +are not found among the Chinese. + +Greater were the merits of the Arabians in the advancement of +mathematics; and especially in virtue of the fact that they preserved +from oblivion the results of both Greek and Hindu research +and handed them down to the Christian countries of the West. The +Arabians expressly distinguished between the Archimedean approximate % original has "Archimedian"; corrected on frequency grounds +value and the two Hindu values, the square root of \Num{10} and +the ratio $\Num{62832}:\Num{20000}$. This distinction occurs also in Muhammed +Ibn Musa Alchwarizmî\index + {Alchwarizmî, Muhammed Ibn Musa}, the same scholar who in the beginning of +the ninth century brought the principles of our present system of +numerical notation from India and introduced it into the Mohammedan +world. The Arabians, however, did not study the numerical +quadrature of the circle only, but also the constructive; for instance, +an attempt of this kind was made by Ibn Alhaitam\index{Alhaitam, Ibn}, who +lived in Egypt about the year \Num{1000} and whose treatise upon the +squaring of the circle is preserved in a Vatican codex, which unfortunately +has not yet been edited. + +Christian civilisation, to which we are now about to pass, produced +up to the second half of the fifteenth century extremely insignificant +results in mathematics. Even with regard to our present +problem we have but a single important work to mention; the +work, namely, of Frankos von Lüttich\index + {Von Lüttich, Frankos} on the squaring of the circle, +published in six books, but preserved only in fragments. The +author, who lived in the first half of the eleventh century, was +probably a pupil of Pope Sylvester~II.\index + {Sylvester II.}, who was himself a not inconsiderable +mathematician for his time and the author of the most +celebrated geometrical treatise of the period. + +Greater interest came to be bestowed upon mathematics, and +especially on the problem of the quadrature of the circle, in the +second half of the fifteenth century, when the sciences again began +to revive. This interest was principally aroused by Cardinal Nicolas +de Cusa\index + {Decu@De Cusa, Nicolas}, a man highly esteemed for his astronomical and calendarial +studies. He claimed to have discovered the quadrature of +the circle by employing only straight edge and compasses and thus +attracted the attention of scholars to the historic problem. People +\PG seq=143 Page 132 ------------------------------------------------------ +believed the famous Cardinal, and marvelled at his wisdom, until +Regiomontanus\index{Regiomontanus}, in letters written in \Num{1464} and \Num{1465} and published +in \Num{1533}, rigorously demonstrated that the Cardinal's quadrature +was incorrect. The construction of Cusa was as follows. The radius +of a circle is prolonged a distance equal to the side of the inscribed +square; the line so obtained is taken as the diameter of a +second circle, and in the latter an equilateral triangle is described; +then the perimeter of the latter is equal to the circumference of the +original circle. If this construction, which its inventor regarded as +exact, be considered as a construction of approximation, it will be +found to be more inexact even than the construction resulting from +the value $\pi=\3\frac17$. For by Cusa's method $\pi$ would be from five to +six thousandths smaller than it really is. + +In the beginning of the sixteenth century a certain Bovillius\index{Bovillius} +appears, who also gave the construction of Cusa,---this time without +notice. But about the middle of the sixteenth century a book +was published which the scholars of the time at first received with +interest. It bore the proud title \textit{De Rebus Mathematicis Hactenus +Desideratis}. Its author, Orontius Finaeus\index + {Finaeus, Orontius}, represented that he had +overcome all the difficulties that had ever stood in the way of geometrical +investigators; and incidentally he also communicated to +the world the ``true quadrature'' of the circle. His fame was short-lived. +For soon afterwards, in a book entitled \textit{De Erratis Orontii}, +the Portuguese Petrus Nonius\index + {Nonius, Petrus} demonstrated that Orontius's quadrature, +like most of his other professed discoveries, was incorrect. + +In the succeeding period the number of circle-squarers so increased +that we shall have to limit ourselves to those whom mathematicians +recognise. And particularly is Simon Van Eyck\index{Van Eyck, Simon} to be +mentioned, who towards the close of the sixteenth century published +a quadrature which was so approximate that the value of $\pi$ +derived from it was more exact even than that of Archimedes; and +to disprove it the mathematician Peter Metius\index + {Metius, Peter} was obliged to seek +a still more accurate value than $\3\frac17$. The erroneous quadrature of +Van Eyck was thus the occasion of Metius's discovery that the ratio +$\Num{355}:\Num{113}$, or $\3\frac{16}{113}$, varied from the true value of $\pi$ by less than +one one-millionth, eclipsing accordingly all values hitherto obtained. +\PG seq=144 Page 133 ------------------------------------------------------ +Moreover, it is demonstrable by the theory of continued +fractions, that, admitting figures to four places only, no two numbers +more exactly represent the value of $\pi$ than \Num{355} and \Num{113}. + +In the same way the quadrature of the great philologist Joseph +Scaliger\index{Scaliger, Joseph} led to refutations. Like most circle-squarers who believe +in their discovery, Scaliger also was little versed in the elements of +geometry. He solved the famous problem, however,---at least in +his own opinion,---and published in \Num{1592} a book upon it, which +bore the pretentious title \textit{Nova Cyclometria}, and in which the name +of Archimedes was derided. The baselessness of his supposed discovery +was demonstrated to him by the greatest mathematicians of +his time; namely, Vieta\index{Vieta}, Adrianus Romanus\index + {Romanus, Adrianus}, and Clavius\index{Clavius}. + +Of the erring circle-squarers that flourished before the middle +of the seventeenth century three others deserve particular mention,---Longomontanus\index{Longomontanus} +of Copenhagen, who rendered such great services +to astronomy, the Neapolitan John Porta\index{Porta, John}, and Gregory of St.\ +Vincent\index + {Gregory of St.\ Vincent}. Longomontanus made $\pi=\3\frac{14185}{100000}$ and was so convinced +of the correctness of his result as to thank God fervently, in the +preface to his work \textit{Inventio Quadraturae Circuli}, that He had +granted him in his old age the strength to conquer the celebrated +difficulty. John Porta followed the example of Hippocrates and +endeavored to solve the problem by a comparison of lunes. Gregory +of St.\ Vincent published a quadrature, the error of which was very +hard to detect but was finally discovered by Descartes. + +Of the famous mathematicians who dealt with our problem in +the period between the close of the fifteenth century and the time +of Newton, we first meet with Peter Metius\index{Metius, Peter}, before mentioned, who +succeeded in finding in the fraction $\Num{355}:\Num{113}$ the best approximate +value for $\pi$ involving small numbers only. The problem received +a different advancement at the hands of the famous mathematician +Vieta. Vieta\index{Vieta} was the first to whom the idea occurred of representing +$\pi$ with mathematical exactness by an infinite series\index + {Series|Series} of definitely +prescribed operations. By comparing inscribed and circumscribed +polygons, Vieta found that we approach nearer and nearer to $\pi$ if +we cause the operations of extracting the square root of $\frac12$, and +certain related additions and multiplications, to succeed each other +\PG seq=145 Page 134 ------------------------------------------------------ +in a certain manner, and that $\pi$ must come out exactly, if this series +of operations could be continued indefinitely. Vieta thus found +that to a diameter of \Num{10000} million units a circumference belongs of +from \Num{31415} million \Num{926535} units to \Num{31415} million \Num{926536} units of +the same length. + +But Vieta was outdone by the Netherlander Adrianus Romanus\index{Romanus, Adrianus}, +who added five additional decimal places to the ten of Vieta. To +accomplish this he computed with unspeakable labor the circumference +of a regular circumscribed polygon of \Num{1073741824} sides. +This number is the thirtieth power of \2. Yet great as the labor of +Adrianus Romanus was, that of Ludolf Van Ceulen\index + {Van Ceulen, Ludolf} was still greater; +for the latter calculator succeeded in carrying the Archimedean +process of approximation for the value of $\pi$ to \Num{35} decimal places; +that is, the deviation from the true value was smaller than one one-thousand +quintillionth, a degree of exactness that we can have +scarcely any conception of. Ludolf published the figures of the +tremendous computation that led to his result. His calculation +was carefully examined by the mathematician Griemberger and declared +to be correct. Ludolf was justly proud of his work, and following +the example of Archimedes, requested in his will that the +result of his most important mathematical performance, the computation +of $\pi$ to \Num{35} decimal places, be engraved upon his tombstone; +a request which is said to have been carried out. In honor +of Ludolf, $\pi$ is called to-day in Germany the Ludolfian number. + +Although through the labor of Ludolf a degree of exactness for +cyclometrical operations was now obtained that was more than sufficient +for any practical purpose that could ever arise, neither the +problem of constructive rectification nor that of constructive quadrature +had been in any respect theoretically advanced thereby. The +investigations conducted by the famous mathematicians and physicists +Huygens and Snell about the middle of the seventeenth century, +were more important from a mathematical point of view than +the work of Ludolf. In his book \textit{Cyclometricus} Snell\index + {Snell} took the position +that the method of comparison of polygons, which originated +with Archimedes and was employed by Ludolf, was not necessarily +the best method of attaining the end sought; and he succeeded by +\PG seq=146 Page 135 ------------------------------------------------------ +employing propositions which state that certain arcs of a circle are +greater or smaller than certain straight lines connected with the +circle, in obtaining methods that make it possible to reach results +like the Ludolfian with much less labor of calculation. The beautiful +theorems of Snell were proved a second time, and better +proved, by the celebrated Dutch promoter of the science of optics, +Huygens\index{Huygens} (\textit{Opera Varia}, p.~\Num{365} et seq.; \textit{Theoremata De Circuli et +Hyperbolae Quadratura}, \Num{1651}), as well as perfected in many ways +by him. Snell and Huygens were fully aware that they had advanced +the problem of numerical quadrature only, and not that of +the constructive quadrature. This plainly appeared in Huygens's +case from the vehement dispute which he conducted with the English +mathematician James Gregory\index + {Gregory, James}. This controversy is significant +for the history of our problem, from the fact that Gregory made +the first attempt to prove that the squaring of the circle with straight +edge and compasses was impossible. The result of the controversy, +to which we owe many valuable tracts, was, that Huygens +finally demonstrated in an incontrovertible manner the incorrectness +of Gregory's proof of impossibility, adding that he also was of +opinion that the solution of the problem with straight edge and +compasses was impossible, but nevertheless was not himself able +to demonstrate this fact. And Newton later expressed himself to +the same effect. As a matter of fact a period of over \Num{200} years +elapsed before higher mathematics was far enough advanced to +furnish a rigorous demonstration of impossibility. + +%V. +\section{FROM NEWTON TO THE PRESENT.} + +Before we proceed to consider the promotive influence which +the invention of the differential and the integral calculus exercised +upon our problem, we shall enumerate a few at least of that never-ending +succession of erring quadrators who delighted the world +with the products of their ingenuity from the time of Newton to +the present; and out of a pious and sincere regard for the contemporary +world, we shall omit entirely to speak of the circle-squarers +of our own time. +\PG seq=147 Page 136 ------------------------------------------------------ + +First to be mentioned is the celebrated English philosopher +Hobbes\index{Hobbes}. In his book \textit{De Problematis Physicis}, in which he proposes +to explain the phenomena of gravity and of ocean tides, he +also takes up the quadrature of the circle and gives a very trivial +construction, which in his opinion definitively solved the problem. +It made $\pi=\3\frac15$. In view of Hobbes's importance as a philosopher, +two mathematicians, Huygens\index{Huygens} and Wallis\index + {Wallis}, thought it proper to +refute him at length. But Hobbes defended his position in a special +treatise, where to sustain at least the appearance of being right, +he disputed the fundamental principles of geometry and the theorem +of Pythagoras\index{Pythagoras}. + +In the last century France especially was rich in circle-squarers. +We will mention: Oliver de Serres\index + {Dese@De Serres, Oliver}, who by means of a pair of +scales determined that a circle weighed as much as the square upon +the side of the equilateral triangle inscribed in it, that therefore +they must have the same area, an experiment in which $\pi=\3$; +Mathulon\index + {Mathulon}, who offered in legal form a reward of a thousand dollars +to the person who would point out an error in his solution of +the problem, and who was actually compelled by the courts to pay +the money; Basselin\index{Basselin}, who believed that his quadrature must be +right because it agreed with the approximate value of Archimedes, +and who anathematised his ungrateful contemporaries, in the confidence +that he would be recognised by posterity; Liger\index{Liger}, who +proved that a part is greater than the whole and to whom therefore +the quadrature of the circle was child's play; Clerget\index{Clerget}, who based +his solution upon the principle that a circle is a polygon of a definite +number of sides, and who calculated, also, among other things, +how large the point is at which two circles touch. + +Germany and Poland also furnish their contingent to the army +of circle-squarers. Lieutenant-Colonel Corsonich\index + {Corsonich, Lieutenant-Colonel} produced a quadrature +in which $\pi$ equalled $\3\frac18$, and promised fifty ducats to the person +who could prove that it was incorrect. Hesse\index{Hesse} of Berlin wrote +an arithmetic in \Num{1776}, in which a true quadrature was also ``made +known,'' $\pi$ being exactly equal to $\3\frac{14}{99}$. About the same time Professor +Bischoff\index + {Bischoff, Professor} of Stettin defended a quadrature previously published +by Captain Leistner\index{Leistner}, Preacher Merkel\index + {Merkel}, and Schoolmaster +\PG seq=148 Page 137 ------------------------------------------------------ +Böhm\index + {Bohm@Böhm}, which virtually made $\pi$ equal to the square of $\frac{62}{35}$, not even +attaining the approximation of Archimedes. + +From attempts of this character are to be clearly distinguished +constructions of approximation\index + {Approximation, construction of} in which the inventor is aware that +he has not found a mathematically exact construction, but only an +approximate one. The value of such a construction will depend +upon two things---first, upon the degree of exactness with which it +is numerically expressed, and secondly on whether the construction +can be easily made with straight edge and compasses. Constructions +of this kind, simple in form and yet sufficiently exact for +practical purposes, have been produced for centuries in great numbers. +The great mathematician Euler\index{Euler}, who died in \Num{1783}, did not +think it out of place to attempt an approximate construction of this +kind. A very simple construction for the rectification of the circle +and one which has passed into many geometrical text-books is that +published by Kochansky\index + {Kochansky} in \Num{1685} in the \textit{Leipziger Berichte}. It is as +follows: ``Erect upon the diameter of a circle at its extremities +perpendiculars; with the centre as vertex and the diameter as side +construct an angle of $\Num{30}\degrees$; find the point of intersection of the +line last drawn with the perpendicular, and join this point of intersection +with that point on the other perpendicular which is distant +three radii from the base of the perpendicular. The line of +junction so obtained is very approximately equal to one-half of the +circumference of the given circle.'' Calculation shows that the difference +between the true length of the circumference and the line +thus constructed is less than $\frac3{100000}$ of the diameter. + +Although such constructions of approximation are very interesting +in themselves, they nevertheless play but a subordinate rôle +in the history of the squaring of the circle; for on the one hand +they can never furnish greater exactness for circle-computation +than the thirty-five decimal places which Ludolf\index{Ludolf} found, and on the +other hand they are not adapted to advance in any way the question +whether the exact quadrature of the circle with straight edge % original has "straight-edge" but every other occurrence lacks the hyphen +and compasses is possible. + +The numerical side of the problem, however, was considerably +advanced by the new mathematical methods perfected by Newton +\PG seq=149 Page 138 ------------------------------------------------------ +and Leibnitz, and known as the differential and the integral calculus. + +About the middle of the seventeenth century, before Newton +and Leibnitz represented $\pi$ by series of powers, the English mathematicians +Wallis and Lord Brouncker, Newton's predecessors in +certain lines, succeeded in representing $\pi$\index + {Pi@$\pi$|etseq} by an infinite series of +figures combined according to the first four rules of arithmetic. A +new method of computation was thus opened. Wallis\index{Wallis} found that +the fourth part of $\pi$ is represented by the regularly formed product +\[ +\tfrac23\times\tfrac43\times\tfrac45\times\tfrac65\times\tfrac67\tfrac87\tfrac89\times\text{etc.} +\] +more and more exactly the farther the multiplication is continued, +and that the result always comes out too small if we stop at a proper +fraction but too large if we stop at an improper fraction. Lord +Brouncker\index + {Brouncker, Lord}, on the other hand, represents the value in question by +a continued fraction in which the denominators are all \2 and the +numerators are the squares of the odd numbers. Wallis, to whom +Brouncker had communicated his elegant result without proof, demonstrated +the same in his \textit{Arithmetic of Infinites}. + +The computation of $\pi$ could scarcely have been pushed to a +greater degree of exactness by these results than that to which Ludolf +and others had carried it by the older and more laborious +methods. But the series of powers derived from the differential +calculus of Newton and Leibnitz furnished a means of computing +$\pi$ to hundreds of decimal places. + +Gregory\index{Gregory, James}, Newton\index{Newton}, and Leibnitz\index + {Leibnitz} found that the fourth part of +$\pi$ was equal exactly to +\[ +1-\tfrac13+\tfrac15-\tfrac17+\tfrac19-\tfrac1{11}+\tfrac1{13}-\dots +\] +if we conceive this series, which is called the \emph{Leibnitz series}\index + {Series!Leibnitz's}, continued +indefinitely. This series is wonderfully simple but is not +adapted to the computation of $\pi$, for the reason that entirely too +many members have to be taken into account to obtain $\pi$ accurately +to a few decimal places only. The original formula, however, from +which this series is derived, gives other formul\ae\ which are excellently +adapted to the actual computation. The original formula is +the general series: +\[ +\alpha = a - \tfrac13a^3 + \tfrac15a^5 - \tfrac17a^7+\dots, +\] +\PG seq=150 Page 139 ------------------------------------------------------ +where $\alpha$ is the length of the arc belonging to any central angle in a +circle of radius \1, and $a$ the tangent to this angle. From this we +derive the following: +\begin{multline*} +\frac\pi4 = (a+b+c+\dots) - \tfrac13(a^3+b^3+c^3+\dots)\\ + +\tfrac15(a^5+b^5+c^5+\dots)-\dots, +\end{multline*} +where $a, b, c\dots$ are the tangents of angles whose sum is $\Num{45}\degrees$. Determining, +therefore, the values of $a, b, c\dots$, which are equal to +small and convenient fractions and fulfil the conditions just mentioned, +we obtain series of powers which are adapted to the computation of $\pi$. + +The first to add by the aid of series of this description additional +decimal places to the old \Num{35} in the number $\pi$ was the English +arithmetician Abraham Sharp\index + {Sharp, Abraham}, who, following Halley's\index{Halley} instructions, +in \Num{1700} worked out $\pi$ to \Num{72} decimal places. A little later +Machin\index{Machin}, professor of astronomy in London, computed $\pi$ to \Num{100} +decimal places, by putting, in the series given above, $a=b=c=d +=\frac15$ and $e=-\frac1{239}$; that is, by employing the following series: +\begin{multline*} +\frac\pi4 = 4\dotm\left[\frac15-\frac1{3\dotm5^2}+\frac1{5\dotm5^5}-\frac1{7\dotm5^7}+\dots\right]\\ +-\left[\frac1{239}-\frac1{3\dotm239^3}+\frac1{5\dotm239^5}-\dots\right] +\end{multline*} + +In the year \Num{1819}, Lagny\index{Lagny} of Paris outdid the computation of +Machin, determining in two different ways the first \Num{127} decimal +places of $\pi$. Vega then obtained as many as \Num{140} places, and the +Hamburg arithmetician Zacharias Dase\index + {Dase, Zacharias} went as far as \Num{200} places. +The latter did not use Machin's series in his calculation, but the +series produced by putting in the general series above given $a=\frac12$, +$b=\frac15$, $c=\frac18$. Finally, at a recent date, $\pi$ has been computed to +\Num{500} places.\footnote + {In \Num{1873} the approximation was carried by Shanks\index + {Shanks} to \Num{707} places of decimals.---\textit{Trans.}} + +The computation to so many decimal places may serve as an +illustration of the excellence of the modern methods as contrasted +with those anciently employed, but it has otherwise neither a theoretical +\PG seq=151 Page 140 ------------------------------------------------------ +nor a practical value. That the computation of $\pi$ to say \Num{15} +decimal places more than sufficiently satisfies the subtlest requirements +of practice may be gathered from a concrete example of the +degree of exactness thus obtainable. Imagine a circle to be described +with Berlin as centre, and the circumference to pass through +Hamburg; then let the circumference of the circle be computed by +multiplying its diameter by the value of $\pi$ to 15 decimal places, +and then conceive it to be actually measured. The deviation from +the true length in so large a circle as this even could not be as great +as the \Num{18} millionth part of a millimetre. + +An idea can hardly be obtained of the degree of exactness produced +by \Num{100} decimal places. But the following example may possibly +give us some conception of it. Conceive a sphere constructed +with the earth as centre, and imagine its surface to pass through +Sirius, which is $\Num{134}\frac12$ millions of millions of kilometres distant from +the earth. Then imagine this enormous sphere to be so packed +with microbes that in every cubic millimetre millions of millions of +these diminutive animalcula are present. Now conceive these microbes +to be all unpacked and so distributed singly along a straight +line, that every two microbes are as far distant from each other as +Sirius from us, that is $\Num{134}\frac12$ million million kilometres. Conceive +the long line thus fixed by all the microbes, as the diameter of a +circle, and imagine the circumference of it to be calculated by multiplying +its diameter by $\pi$ to \Num{100} decimal places. Then, in the +case of a circle of this enormous magnitude even, the circumference +so calculated would not vary from the real circumference by a millionth +part of a millimetre. + +This example will suffice to show that the calculation of $\pi$ to +\Num{100} or \Num{500} decimal places is wholly useless. + +Before we close this chapter upon the evaluation of $\pi$, we +must mention the method, less fruitful than curious, which Professor +Wolff\index{Wolff, Professor} of Zurich employed some decades ago to compute the +value of $\pi$ to \3 places.\footnote + {See also A.~De Morgan\index + {Demo@De Morgan}, \textit{A Budget of Paradoxes}, pp.\ \Num{169}--\Num{171}.---\textit{Tr.}} +The floor of a room is divided up into equal +squares, so as to resemble a huge chess-board, and a needle exactly +\PG seq=152 Page 141 ------------------------------------------------------ +equal in length to the side of each of these squares, is cast +haphazard upon the floor. If we calculate, now, the probabilities\index{Probabilities} +of the needle so falling as to lie wholly within one of the squares, +that is so that it does not cross any of the parallel lines forming the +squares, the result of the calculation for this probability will be +found to be exactly equal to $\pi-\3$. Consequently, a sufficient +number of casts of the needle according to the law of large numbers +must give the value of $\pi$ approximately. As a matter of fact, +Professor Wolff, after \Num{10000} trials, obtained the value of $\pi$ correctly +to \3 decimal places. + +Fruitful as the calculus of Newton and Leibnitz was for the +evaluation of $\pi$, the problem of converting a circle into a square +having exactly the same area was in no wise advanced thereby. +Wallis, Newton, Leibnitz, and their immediate followers distinctly +recognised this. The quadrature of the circle could not be solved\label{p:141}; +but it also could not be proved that the problem was insolvable +with straight edge and compasses, although everybody was convinced +of its insolvability. In mathematics, however, a conviction +is only justified when supported by incontrovertible proof; and in +the place of endeavors to solve the quadrature there accordingly +now come endeavors to prove the impossibility of solving the celebrated +problem. + +The first step in this direction, small as it was, was made by +the French mathematician Lambert\index{Lambert}, who proved in the year \Num{1761} +that $\pi$ was neither a rational number nor even the square root of a +rational number; that is, that neither $\pi$ nor the square of $\pi$ could +be exactly represented by a fraction the denominator and numerator +of which are whole numbers, however great the numbers be +taken. Lambert's proof\footnote + {Given in Legendre's\index + {Legendre} \textit{Geometry}, in the Appendix to De Morgan\index + {Demo@De Morgan}, \emph{op.\ cit.}, p.~\Num{495}, + and in Rudio\index{Rudio}, \emph{op.\ cit.}---\textit{Tr.}} +showed, indeed, that the rectification and +the quadrature of the circle could not be accomplished in one particular +simple way, but it still did not exclude the possibility of the +problem being solvable in some other more complicated way, and +without requiring further aids than straight edge and compasses. +\PG seq=153 Page 142 ------------------------------------------------------ + +Proceeding slowly but surely it was next sought to discover +the essential properties which distinguish problems solvable with +straight edge and compasses\index + {Problems!solvable with straight edge and compasses} from problems the construction of +which is elementarily impossible, that is by employing the postulates +only. Slight reflection showed, that a problem, to be elementarily +solvable, must always be such that the unknown lines of its +figure are connected with the known lines by an equation for the +solution of which equations of the first and second degree only are +requisite, and which can be so arranged that the measures of the +known lines will appear as integers only. The conclusion to be +drawn from this was that if the quadrature of the circle and consequently +its rectification were solvable elementarily, the number $\pi$, +which represents the ratio of the unknown circumference to the +known diameter, must be the root of a certain equation, of a very +high degree perhaps, but in which all the numbers are whole numbers; +that is, there would have to exist an equation, made up entirely +of whole numbers, which would be correct if its unknown +quantity were made equal to $\pi$. + +Since the beginning of this century, consequently, the efforts +of a number of mathematicians have been bent upon proving that +$\pi$ generally is not algebraical, that is, that it cannot be the root of +an equation having whole numbers for coefficients. But mathematics +had to make tremendous strides forward before the means +were at hand to accomplish this demonstration. After the French +Academician, Professor Hermite\index + {Hermite, Professor}, had furnished important preparatory +assistance in his treatise \textit{Sur la Fonction Exponentielle}, published +in the seventy-seventh volume of the \textit{Comptes Rendus}, Professor +Lindemann\index{Lindemann}, at that time of Freiburg, now of Munich, finally +succeeded, in June \Num{1882}, in rigorously demonstrating that the number +$\pi$ is not algebraical,\footnote + {For the benefit of my mathematical readers I shall present here the most + important steps of Lindemann's demonstration. M.~Hermite in order to prove the + transcendental character of + \[ + e = 1+\frac11+\frac1{1\dotm2}+ + \frac1{1\dotm2\dotm3}+\frac1{1\dotm2\dotm3\dotm4}+\dots + \] + developed relations between certain definite integrals (\textit{Comptes Rendus} of the + Paris Academy, Vol.~\Num{77}, \Num{1873}). Proceeding from the relations thus established, + Professor Lindemann first demonstrates the following proposition: If the coefficients + of an equation of the $n$th degree are all real or complex whole numbers and + the $n$ roots of this equation $z_1, z_2,\dots, z_n$ are different from zero and from each + other it is impossible for + \[ + e^{z_1} + e^{z_2} + e^{z_3} \dots + e^{z_n} + \] + to be equal to $\frac ab$, where $a$ and $b$ are real or complex whole numbers. It is then + shown that also between the functions + \[ + e^{rz_1} + e^{rz_2} + e^{rz_3} + \dots e^{rz_n}, + \] + where $r$ denotes an integer, no linear equation can exist with rational coefficients + different from zero. Finally the beautiful theorem results: If $z$ is the root of an + irreducible algebraic equation the coefficients of which are real or complex whole + numbers, then $e^z$ cannot be equal to a rational number. Now in reality $e^{\pi\surd-\1}$ is + equal to a rational number, namely, $-\1$. Consequently, $\pi\sqrt{-\1}$, and therefore $\pi$ + itself, cannot be the root of an equation of the $n$th degree having whole numbers + for coefficients, and therefore also not of such an equation having rational coefficients. + The property last mentioned, however, $\pi$ would have if the squaring of the + circle with straight edge and compasses were possible. [The questions involved + in the discussions of the last three pages have been excellently treated by Klein\index{Klein, Felix} in + \textit{Famous Problems of Elementary Geometry} recently translated by Beman\index{Beman, W.\;W.} and + Smith\index{Smith, D.\;E.} (Ginn \& Co., Boston). Lindemann's proof is here presented in a simplified + form, and so brought within the comprehension of students conversant only with + algebra.---\textit{Tr}.]} +and so supplied the first proof that the +\PG seq=154 Page 143 ------------------------------------------------------ +problems of the rectification and squaring of the circle with the +help only of algebraical instruments like straight edge and compasses +are insolvable\index + {Insoluble problems}. Lindemann's proof appeared successively +in the \textit{Reports of the Berlin Academy} (June, \Num{1882}), in the \textit{Comptes +Rendus} of the French Academy (Vol.\ \Num{115}, pp.\ \Num{72} to \Num{74}), and in the +\textit{Mathematische Annalen} (Vol.\ \Num{20}, pp.\ \Num{213} to \Num{225}). + +``It is impossible with straight edge and compasses to construct +a square equal in area to a given circle.'' These are the +words of the final determination of a controversy which is as old as +the history of the human mind. But the race of circle-squarers, +unmindful of the verdict of mathematics, the most infallible of +arbiters, will never die out as long as ignorance and the thirst for +glory remain united\index + {Squaring of the circle|)}\label{p:143}. +\PGx seq=155 Page 144 ------------------------------------------------------ +\PGx seq=156 Page 145 ------------------------------------------------------ +% INDEX. + +\cleartorecto +\printindex + +\PGx seq=157 Page 146 +\PGx seq=158 Page 147 +\PGx seq=159 Page 148 +\PGx seq=160 Page 149 +\PGx seq=161 Page --[unnumbered] +\cleartorecto +\raggedbottom +\begin{adverts} +\DPpdfbookmark[0]{Advertisements}{Adverts*1} +\LP\parskip=0pt +\PG seq=162 Page --[unnumbered] +\begin{center} +\textbf{\fontsize{30}{32}\selectfont %36/39 +\textso{THE MONIST.}} + +{\tiny A QUARTERLY MAGAZINE} + +\textbf{\small Devoted to the Philosophy of Science.} + +{\miniscule PUBLISHED BY} + +\textso{THE OPEN COURT PUBLISHING COMPANY, CHICAGO, ILL.} + +\textsc{\tiny + Monon Building, \Num{324} Dearborn Street. 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These properties, +five in number, give rise in man to intellectual faculties, represented by five senses. There are also five faculties +of emotion. The author teaches a new doctrine of judgments, and carefully analyses them in the five +intellections which he calls sensation, perception, understanding, reflection, and ideation, each of these faculties +being founded on one of the senses. + +Intellectual errors are classified as fallacies of sensation, fallacies of perception, fallacies of understanding, +fallacies of reflection, and fallacies of ideation, and a war is waged against the metaphysic of the +idealists in the interest of the philosophy of science. + +In the chapters on fallacies there is a careful discussion of the theory of ghosts, especially as treated in +the publications of the Society for Psychical Research, and by various other authors on the same subject. + +No student of the sciences can afford to neglect this book. The discussion is clear and entertaining. + +} +\begin{center} +\rule{3cm}{.4pt} +\vspace{10pt plus 1fill} + +\includegraphics{images/demorgan.pdf} +%On the Study and Difficulties of Mathematics + +\large By AUGUSTUS DE MORGAN. + +\end{center} + +\textbf{Just Published}---New corrected and annotated edition, with references +to date, of the work published in \Num{1831} by the Society for the Diffusion of +Useful Knowledge. The original is now scarce. + +With a fine Portrait of the great Mathematical Teacher, Complete +Index, and Bibliographies of Modern Works on Algebra, the Philosophy of +Mathematics, Pangeometry, etc. + +{\Small +\begin{center} + Pp.\ viii + 288. Cloth, \$\1.\Num{25} (\5s.). +\end{center} + +``\textbf{A Valuable Essay.}''---\textsc{Prof.\ Jevons}, in the \textit{Encyclop\ae dia Britannica}. + +``The mathematical writings of De Morgan can be commended unreservedly.''---\textsc{Prof.\ +W.\;W. 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Anyone seeking to utilize +this eBook outside of the United States should confirm copyright +status under the laws that apply to them. diff --git a/README.md b/README.md new file mode 100644 index 0000000..892b3c9 --- /dev/null +++ b/README.md @@ -0,0 +1,2 @@ +Project Gutenberg (https://www.gutenberg.org) public repository for +eBook #25387 (https://www.gutenberg.org/ebooks/25387) |