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You may copy it, give it away or % +% re-use it under the terms of the Project Gutenberg License included % +% with this eBook or online at www.gutenberg.org % +% % +% % +% Title: Euclid's Book on Divisions of Figures % +% % +% Author: Raymond Clare Archibald % +% % +% Release Date: January 21, 2012 [EBook #38640] % +% % +% Language: English % +% % +% Character set encoding: ISO-8859-1 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK EUCLID'S BOOK ON DIVISIONS ***% +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{38640} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Standard DP encoding. Required. %% +%% %% +%% ifthen: Logical conditionals. Required. %% +%% %% +%% amsmath: AMS mathematics enhancements. Required. %% +%% amssymb: Additional mathematical symbols. Required. %% +%% %% +%% alltt: Fixed-width font environment. Required. %% +%% indentfirst: Indent first word of each sectional unit. Optional. %% +%% %% +%% fancyhdr: Enhanced running headers and footers. Required. %% +%% (fancyhdr includes extramarks) %% +%% %% +%% graphicx: Standard interface for graphics inclusion. Required. %% +%% %% +%% geometry: Enhanced page layout package. Required. %% +%% hyperref: Hypertext embellishments for pdf output. Required. %% +%% %% +%% %% +%% Producer's Comments: %% +%% %% +%% Changes are noted in this file in two ways. %% +%% 1. \DPnote{} for in-line `placeholder' notes. %% +%% 2. \DPtypo{}{} for typographical corrections, showing original %% +%% and replacement text side-by-side. %% +%% %% +%% %% +%% Compilation Flags: %% +%% %% +%% The following behavior may be controlled by boolean flags. %% +%% %% +%% 1) ForPrinting: %% +%% Compile a print-optimized PDF file. Set to true for print- %% +%% optimized file (pages uncropped, two-sided, black hyperlinks). %% +%% Set to false for screen-optimized file (pages cropped, %% +%% one-sided, blue hyperlinks). %% +%% %% +%% 2) BLUEBOXES (false by default) %% +%% Draw a thin blue border around the bounding box of each %% +%% graphic figure. %% +%% %% +%% Things to Check: %% +%% %% +%% %% +%% Spellcheck: .................................. OK %% +%% Smoothreading pool: ......................... yes %% +%% %% +%% lacheck: ..................................... OK %% +%% Numerous false positives from commented code %% +%% %% +%% PDF pages: 111 (94+10) (if ForPrinting set to true) %% +%% PDF page size: US Letter %% +%% PDF bookmarks: created, point to ToC entries %% +%% PDF document info: filled in %% +%% %% +%% Images: 49 pdf diagrams %% +%% (48 geometrical figures plus the Cambridge logo) %% +%% %% +%% Summary of log file: %% +%% * In addition, several overfull hboxes. %% +%% %% +%% %% +%% Compile History: %% +%% %% +%% May 2010: <ralphpdas, Ralph Carmichael> %% +%% texlive2009, GNU/Linux (Ubuntu) %% +%% %% +%% August 2010: new version without wrapfig %% +%% January 2012: final cleanup, texLive2010 Mac OS X %% +%% %% +%% Command block for creating euclid.pdf: %% +%% %% +%% 1) Convert original image source files in folder images/source %% +%% from Encapsulated PostScript format to Portable Document %% +%% Format (EPS -> PDF), using makepdf.bat or a Unix equivalent. %% +%% %% +%% 2) pdflatex euclid x3 (run pdflatex three times) %% +%% pdflatex x3 %% +%% %% +%% %% +%% January 2012: pglatex. %% +%% Compile this project with: %% +%% pdflatex 38640-t.tex ..... THREE times %% +%% %% +%% pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) %% +%% %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\listfiles +\documentclass[12pt,letterpaper]{book}[2007/09/16] +%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PACKAGES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\usepackage[latin1]{inputenc}[2008/03/30] +\usepackage{ifthen}[2001/05/26] %% Logical conditionals +\usepackage{amsmath}[2000/07/18] %% Displayed equations +\usepackage{amssymb}[2009/06/22] %% and additional symbols +\usepackage{alltt}[1997/06/16] %% boilerplate, credits, license +\usepackage{indentfirst}[1995/11/23] %% indent first paragraph +\usepackage{fancyhdr} %% no date; for running heads +\usepackage{extramarks} %% no date +\usepackage[T1]{fontenc} %% no date +\usepackage{lmodern}[2009/10/30] %% v1.3 Latin Modern Fonts +\usepackage[greek,english]{babel}[2008/07/06] %% the babel package +\usepackage{graphicx}[1999/02/16] %% For diagrams +%%% \usepackage{wrapfig}[2003/01/31] %% and wrapping text around them +\usepackage{color}[2005/11/14] %% v1.0j Standard LaTeX Color (DPC) +\usepackage{yfonts}[2003/01/08] %% for gothfrak font v1.3 (WaS) + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%% Interlude: Set up PRINTING (default) or SCREEN VIEWING %%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% ForPrinting=true (default) false +% Asymmetric margins Symmetric margins +% Black hyperlinks Blue hyperlinks +% Start Preface, ToC, etc. recto No blank verso pages +% +\newboolean{ForPrinting} +\setboolean{ForPrinting}{true} % make it true by default +%% UNCOMMENT the next line for a SCREEN-OPTIMIZED VERSION of the text %% +%\setboolean{ForPrinting}{false} + +%% Initialize values to ForPrinting=false (screen-optimized) +\newcommand{\Margins}{hmarginratio=1:1} % Symmetric margins +\newcommand{\HLinkColor}{blue} % Hyperlink color +\newcommand{\PDFPageLayout}{SinglePage} +\newcommand{\TransNote}{Transcriber's Note} +\newcommand{\TransNoteCommon}{% + This book was produced from images provided by the Cornell + University Library: Historical Mathematics Monographs collection. + + A number of typographical errors in the original book have been + corrected without comment. The changes may be examined in the \LaTeX\ + source file by searching for DPtypo. + + Three footnotes that were labelled 60a,107a, and 118a in the original + text have been renamed 601,1071, and 1181 respectively. + \bigskip +} + +\newcommand{\TransNoteText}{% + \TransNoteCommon + + This PDF file is optimized for screen viewing, but may easily be + recompiled for printing. Please see the preamble of the \LaTeX\ + source file for instructions. +} +%% Re-set if ForPrinting=true +\ifthenelse{\boolean{ForPrinting}}{% + \renewcommand{\Margins}{hmarginratio=2:3} % Asymmetric margins + \renewcommand{\HLinkColor}{black} % Hyperlink color + \renewcommand{\PDFPageLayout}{TwoPageRight} + \renewcommand{\TransNote}{Transcriber's Note} + \renewcommand{\TransNoteText}{% + \TransNoteCommon + + This PDF file is optimized for printing, but may easily be + recompiled for screen viewing. Please see the preamble of the + \LaTeX\ source file for instructions. + } +}{% If ForPrinting=false, don't skip to recto + \renewcommand{\cleardoublepage}{\clearpage} +} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%% End of PRINTING/SCREEN VIEWING code; back to packages %%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\usepackage[body={5.25in,8.50in},\Margins]{geometry}[2010/07/13] +\providecommand{\ebook}{00000} % Overridden during white-washing +\usepackage[pdftex, + hyperref, + hyperfootnotes=false, + pdftitle={The Project Gutenberg eBook \#\ebook: Euclid's Book on Division of Figures}, + pdfauthor={Archibald, Raymond Clare}, + pdfkeywords={Ralph Carmichael, Joshua Hutchinson, + Project Gutenberg Online Distributed Proofreading Team}, + pdfstartview=Fit, % default value + pdfstartpage=1, % default value + pdfpagemode=UseNone, % default value + bookmarks=true, % default value + linktocpage=false, % default value + pdfpagelayout=\PDFPageLayout, + pdfdisplaydoctitle, + pdfpagelabels=true, + bookmarksopen=true, + bookmarksopenlevel=1, + colorlinks=true, + linkcolor=\HLinkColor]{hyperref}[2010/06/18] + +%% END of packages. The hyperref package should be the LAST loaded. %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% Re-crop screen-formatted version, accommodating wide displays +\ifthenelse{\boolean{ForPrinting}} + {} + {\hypersetup{pdfpagescrop= 81 72 533 762}} % all margins 0.5in + + +%%%% Fixed-width environment to format PG boilerplate %%%% +%%%%%%# Adjust pt sizes so your project's title fits the text block%%%%%% +\newenvironment{PGtext}{% +\begin{alltt} +\fontsize{9.2}{10.5}\ttfamily\selectfont}% +{\end{alltt}} + +% Cross-referencing: anchors %%%%%% IS THIS USED??? +\newcommand{\Pagelabel}[1] + {\phantomsection\label{page:#1}} + +% and links +\newcommand{\Pageref}[2]{\hyperref[page:#2]{#1~\pageref{page:#2}}} + +%% Changes are noted in this file %% +\newcommand{\DPtypo}[2]{#2} +\newcommand{\DPnote}[1]{} + +%%% Graphic images. BLUEBOXES allows one to draw a thin blue border +%%% about the bounding box of each graphic. This is very helpful when +%%% diagnosing problems associated with wrapped figures. +\newboolean{BLUEBOXES} %% default is false +%% Uncomment the following command to draw the blue outlines +%\setboolean{BLUEBOXES}{true} % to draw bounding boxes in blue +\ifthenelse{\boolean{BLUEBOXES}}{ +\newcommand\bluebox[1]{\textcolor{blue}{% + \setlength\fboxsep{0pt}\fbox{\textcolor{black}{#1}}}}} {} + +% \Graphic{50mm}{063} +\newcommand{\Graphic}[2]{% + \ifthenelse{\boolean{BLUEBOXES}} + {\bluebox{\includegraphics[width=#1]{./images/#2.pdf}}}% + {\includegraphics[width=#1]{./images/#2.pdf}}% +} + +%% Some illustrations are centered; others are right-adjusted and wrapped +%% This is no longer true. All are now centered. +%% The \WrappedGraphic command has been modified to be identical +%% to \CenteredGraphic. +% \CenteredGraphic{50mm}{063} +\newcommand{\CenteredGraphic}[2]{% + \begin{center}% + \Graphic{#1}{#2}% + \end{center}% +} + +% \WrappedGraphic{50mm}{063}{11}{r} +\newcommand{\WrappedGraphic}[4]{% +%%% \begin{wrapfigure}[#3]{#4}{#1}% + \begin{center}% + \Graphic{#1}{#2}% +%%% \end{wrapfigure}% + \end{center}% +} + +%% This book is divided into four chapters, identified by upper-case %% +%% roman numerals. Chapters I and II are also subdivided into %% +%% sections, which are not numbered, but appear in the table of %% +%% contents. The book is further divided into 57 logical paragraphs %% +%% that are printed as bold arabic numerals. Chapter III contains the %% +%% 36 Propositions that make up the restoration of Euclid's work and %% +%% each Proposition is contained in its own paragraph. Chapter IV %% +%% has no sections or paragraphs and contains a bibliography and an %% +%% Index of Names. The index is unusual in the entries do not refer %% +%% to page numbers, but to paragraphs or bibliography items. %% +%% %% +%% This book has page headers that reflect the content of the current %% +%% page being printed. Throught the entire book, the title of the %% +%% book followed by the chapter number is centered on each %% +%% even-numbered page. In chapters I and II, the odd pages have +%% centered text that reflects the current section name. The page %% +%% numbers are printed in the page headers left on even pages and %% +%% right on odd-numbered pages. The current paragraph numbers are %% +%% printed in the page header right on even pages and left on odd %% +%% pages. Commands follow that perform this operation. In section IV, %% +%% there are no paragraph numbers, so these fields are blank. %% +%% In chapter III, the current Proposition %% +%% numbers are printed as centered text on the odd page headers. %% +%% In chapter IV, the centered text on the odd page headers reflects %% +%% the range of entries in the current pages. Making these entries %% +%% automatically match the original book has proven difficult. %% +%% The first page of each chapter is on an odd-numbered page and has %% +%% no header. %% +%% %% +%% Various commands for getting the page headers in the style of %% +%% the original book. Uses package fancyhdr. %% +%% Define \Heading to set font, size, case of headers. %% +\pagestyle{fancy} +\setlength{\headheight}{16pt} +\renewcommand{\headrulewidth}{0pt} +\newcommand{\Heading}[1]{{\normalfont\footnotesize\MakeUppercase{#1}}} + + +%% DEFINITIONS of \Chapter and \Section %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\newcommand{\Chapter}[1]{% + \clearpage\fancyhf{} % Clear stale headings + \ifthenelse{\boolean{ForPrinting}}{\cleardoublepage}{\clearpage} + \section*{\LARGE\bfseries\centering #1.} + \label{chapter:#1.}% + \thispagestyle{empty}% + \fancyhead[CE]{\Heading{EUCLID'S BOOK ON DIVISION OF FIGURES #1}} + \fancyhead[LE,RO]{\thepage}% +} + +\newcommand{\Section}[1]{% + \subsection*{\normalfont\normalsize\centering{#1}} +} +%% END CHAPTER AND SECTION DEFINITIONS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%% PROPOSITION DEFINITIONS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\newcommand{\CurrentProp}{} +\newcommand{\CurrentPar}{} + +\newcommand{\MyPropMarks}{% + \ifthenelse{\equal{\firstleftxmark}{\lastleftxmark}}{% + \lastleftxmark% + }{% + \firstleftxmark--\lastleftxmark}% +} + +% This command will not be executed until Chapter III +\newcommand{\Proposition}[1]{% + \Section{\textsc{Proposition} \textbf{#1.} \nopagebreak} + \fancyhead[CO]{\Heading{Proposition \protect\MyPropMarks}}% + \renewcommand{\CurrentProp}{#1}% + \extramarks{#1}{\CurrentPar}% prop # left; par # right + \nopagebreak% +} +%% END PROPOSITION DEFINITIONS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%% PARAGRAPH DEFINITIONS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\newcommand{\MyParMarks}{% + \ifthenelse{\equal{\firstrightxmark}{\lastrightxmark}}{% + \lastrightxmark% + }{% + \firstrightxmark--\lastrightxmark}% +} + +% \Paragraph has two arguments. The second is the paragraph number +% (without the dot) and the first is the short name of the paragraph that +% will be printed as the odd-page centered header if and when this +% paragraph is current when the header is printed. Note that some +% of these short headers are never printed, but they might be if the +% page size were changed.} +% \Paragraph will be redefined at the beginning of Chapter III +% to put the proposition numbers centered on odd-numbered pages.} +% Example: \Paragraph[MSS. of Muhammed Bagdedinus and Dee]{2} (note omission of dot) +\newcommand{\Paragraph}[2]{% + \paragraph{\hspace*{\parindent}#2.} % puts the dot in + \fancyhead[CO]{\Heading{#1}}% puts paragraph title on odd page center header + \fancyhead[RE]{[\MyParMarks}% + \fancyhead[LO]{\MyParMarks]}% + \renewcommand{\CurrentPar}{#2}% + \extramarks{\CurrentProp}{#2}% + \label{par#2}% defines symbols par1, par2,... for hyperrefing + %%%%%%\pdfbookmark[1]{#1}{} +} +\newcommand{\ParRef}[1]{\hyperref[par#1]{#1}} +%% END PARAGRAPH DEFINITIONS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + +%% FOOTNOTE DEFINITIONS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Adds a space (1em) between footnote mark and footnote text. +\newcommand{\Footnotetext}[2]{% arg 1 is the footnote number; arg 2 is the text + \phantomsection\label{FNN:#1}% + \footnotetext[#1]{\;#2}% +} +\newcommand{\Footnote}[2]{% arg 1 is the footnote number; arg 2 is the text + \phantomsection\label{FNN:#1}% + \footnote[#1]{\;#2}% +} + +\newcommand{\FnRef}[1]{\hyperref[FNN:#1]{\textit{n}.~#1}} +%% END FOOTNOTE DEFINITIONS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + + +%% SPECIAL CONSTRUCTS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% \SmallCenteredNote puts a small size comment centered on the page and +% \SmallLeftNote puts a small sized comment left adjusted ; similarly for Large +\newcommand{\SmallCenteredNote}[1]{\centerline{\small #1\normalsize}} +\newcommand{\SmallLeftNote}[1]{\small #1\normalsize} +\newcommand{\LargeCenteredNote}[1]{\centerline{\large #1\normalsize}} +\newcommand{\LargeLeftNote}[1]{\large #1\normalsize} + +\newcommand{\ItalicGreekSnippet}[1]{\foreignlanguage{greek}{\textit{#1}} } + +%% Define \Rect to draw a rectangle symbol to complement the +%% triangle symbol +\setlength{\fboxsep}{0pt} +\newcommand{\Rect}{\fbox{$\phantom{\rule{18pt}{9pt}}$}} + +\setlength{\fboxsep}{-0.4pt} % to suppress the strange overfull hbox warnings + +%% END SPECIAL CONSTRUCTS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + + +%% DEFINITION OF APPENDIX COMMANDS \MyAppMarks, \CurrentAppItem, \AppItem, \AppRef +\newcommand{\MyAppMarks} {\firstrightxmark--\lastrightxmark} + +\newcommand{\CurrentAppItem}[1]{} + +\newenvironment{Description} + {\begin{list}{}{\let\makelabel\Descriptionlabel + \setlength\listparindent{1em}}}% + {\end{list}} +\newcommand*\Descriptionlabel[1]{\hspace{\labelsep}\normalfont #1} + +\newcommand{\AppItem}[2]{% first arg is LaTeX label; second arg is list label + \item[#1---\hspace{-0.5em}]% + \phantomsection\label{App:#2}% + \renewcommand{\CurrentAppItem}{#1}% + \extramarks{#1}{#1}% + \fancyhead[CO]{\Heading{APPENDIX \protect\MyAppMarks}} +} + +\newcommand{\AppRef}[2]{\hyperref[App:#1]{\, App. #2}} +%%\newcommand{\App0Ref}[0]{\hyperref[Appendix]{App.}} +%% END APPENDIX DEFINITIONS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + +%% INDEX DEFINITIONS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\makeatletter +\renewenvironment{theindex} + {\if@twocolumn + \@restonecolfalse + \else + \@restonecoltrue + \fi\columnsep 35\p@ + \twocolumn[\@makeschapterhead{\indexname}\indexhook]% + \@mkboth{\MakeUppercase\indexname}% + {\MakeUppercase\indexname}% + \thispagestyle{plain}\parindent\z@ + \parskip\z@ \@plus .3\p@\relax + \columnseprule \z@ + \let\item\@idxitem} + {\if@restonecol\onecolumn\else\clearpage\fi} +\makeatother +\let\indexhook\empty +\renewcommand\indexname{\centering{INDEX OF NAMES}} +%% END INDEX DEFINITIONS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + + +%% DEFINITION OF DPalign, DPgather for formatting condensed intertext. %% +%% (copied from the PGDP Wiki ) %% +\makeatletter +\providecommand\shortintertext\intertext +\newcount\DP@lign@no +\newtoks\DP@lignb@dy +\newif\ifDP@cr +\newif\ifbr@ce +\def\f@@zl@bar{\null} +\def\addto@DPbody#1{\global\DP@lignb@dy\@xp{\the\DP@lignb@dy#1}} +\def\parseb@dy#1{\ifx\f@@zl@bar#1\f@@zl@bar + \addto@DPbody{{}}\let\@next\parseb@dy + \else\ifx\end#1 + \let\@next\process@DPb@dy + \ifDP@cr\else\addto@DPbody{\DPh@@kr&\DP@rint}\@xp\addto@DPbody\@xp{\@xp{\the\DP@lign@no}&}\fi + \addto@DPbody{\end} + \else\ifx\intertext#1 + \def\@next{\eat@command0}% + \else\ifx\shortintertext#1 + \def\@next{\eat@command1}% + \else\ifDP@cr\addto@DPbody{&\DP@lint}\@xp\addto@DPbody\@xp{\@xp{\the\DP@lign@no}&\DPh@@kl} + \DP@crfalse\fi + \ifx\begin#1\def\begin@stack{b} + \let\@next\eat@environment + \else\ifx\lintertext#1 + \let\@next\linter@text + \else\ifx\rintertext#1 + \let\@next\rinter@text + \else\ifx\\#1 + \addto@DPbody{\DPh@@kr&\DP@rint}\@xp\addto@DPbody\@xp{\@xp{\the\DP@lign@no}&\\}\DP@crtrue + \global\advance\DP@lign@no\@ne + \let\@next\parse@cr + \else\check@braces#1!Q!Q!Q!\ifbr@ce\addto@DPbody{{#1}}\else + \addto@DPbody{#1}\fi + \let\@next\parseb@dy + \fi\fi\fi\fi\fi\fi\fi\fi\@next} +\def\process@DPb@dy{\let\lintertext\@gobble\let\rintertext\@gobble + \@xp\start@align\@xp\tw@\@xp\st@rredtrue\@xp\m@ne\the\DP@lignb@dy} +\def\linter@text#1{\@xp\DPlint\@xp{\the\DP@lign@no}{#1}\parseb@dy} +\def\rinter@text#1{\@xp\DPrint\@xp{\the\DP@lign@no}{#1}\parseb@dy} +\def\DPlint#1#2{\@xp\def\csname DP@lint:#1\endcsname{\text{#2}}} +\def\DPrint#1#2{\@xp\def\csname DP@rint:#1\endcsname{\text{#2}}} +\def\DP@lint#1{\ifbalancedlrint\@xp\ifx\csname +DP@lint:#1\endcsname\relax\phantom + {\csname DP@rint:#1\endcsname}\else\csname DP@lint:#1\endcsname\fi + \else\csname DP@lint:#1\endcsname\fi} +\def\DP@rint#1{\ifbalancedlrint\@xp\ifx\csname +DP@rint:#1\endcsname\relax\phantom + {\csname DP@lint:#1\endcsname}\else\csname DP@rint:#1\endcsname\fi + \else\csname DP@rint:#1\endcsname\fi} +\def\eat@command#1#2{\ifcase#1\addto@DPbody{\intertext{#2}}\or + \addto@DPbody{\shortintertext{#2}}\fi\DP@crtrue + \global\advance\DP@lign@no\@ne\parseb@dy} +\def\parse@cr{\new@ifnextchar*{\parse@crst}{\parse@crst{}}} +\def\parse@crst#1{\addto@DPbody{#1}\new@ifnextchar[{\parse@crb}{\parseb@dy}} +\def\parse@crb[#1]{\addto@DPbody{[#1]}\parseb@dy} +\def\check@braces#1#2!Q!Q!Q!{\def\dp@lignt@stm@cro{#2}\ifx + \empty\dp@lignt@stm@cro\br@cefalse\else\br@cetrue\fi} +\def\eat@environment#1{\addto@DPbody{\begin{#1}}\begingroup + \def\@currenvir{#1}\let\@next\digest@env\@next} +\def\digest@env#1\end#2{% + \edef\begin@stack{\push@begins#1\begin\end \@xp\@gobble\begin@stack}% + \ifx\@empty\begin@stack + \@checkend{#2} + \endgroup\let\@next\parseb@dy\fi + \addto@DPbody{#1\end{#2}} + \@next} +\def\lintertext{lint}\def\rintertext{rint} +\newif\ifbalancedlrint +\let\DPh@@kl\empty\let\DPh@@kr\empty +\def\DPg@therl{&\omit\hfil$\displaystyle} +\def\DPg@therr{$\hfil} + +\newenvironment{DPalign*}[1][a]{% + \if m#1\balancedlrintfalse\else\balancedlrinttrue\fi + \global\DP@lign@no\z@\DP@crfalse + \DP@lignb@dy{&\DP@lint0&}\parseb@dy +}{% + \endalign +} +\newenvironment{DPgather*}[1][a]{% + \if m#1\balancedlrintfalse\else\balancedlrinttrue\fi + \global\DP@lign@no\z@\DP@crfalse + \let\DPh@@kl\DPg@therl + \let\DPh@@kr\DPg@therr + \DP@lignb@dy{&\DP@lint0&\DPh@@kl}\parseb@dy +}{% + \endalign +} +\makeatother +%% END DEFINITION OF DPalign, DPgather. %% + +% Some words that did not hyphenate properly. This may be fixed +% by now and this would be unneccessary. } +\hyphenation{wis-sen-schaf-ten dies-ter-weg tri-go-nor-um Mu-ham-med} + +%%%%%% END OF PREAMBLE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{document} + +\pagenumbering{alph} +\pagestyle{empty} % change to fancy before introductory + +\phantomsection +\pdfbookmark[-1]{Front Matter}{Front Matter} + +%%%% PG BOILERPLATE %%%% +\Pagelabel{PGBoilerplate} +\phantomsection +\pdfbookmark[0]{PG Boilerplate}{Project Gutenberg Boilerplate} + +\begin{center} +\begin{minipage}{\textwidth} +\small +\begin{PGtext} +The Project Gutenberg EBook of Euclid's Book on Divisions of Figures, by +Raymond Clare Archibald + +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: Euclid's Book on Divisions of Figures + +Author: Raymond Clare Archibald + +Release Date: January 21, 2012 [EBook #38640] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK EUCLID'S BOOK ON DIVISIONS *** +\end{PGtext} +\end{minipage} +\end{center} +\clearpage + +%%%% Credits and transcriber's note %%%% +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by Joshua Hutchinson, Ralph Carmichael and the +Online Distributed Proofreading Team at http://www.pgdp.net +(This file was produced from images from the Cornell +University Library: Historical Mathematics Monographs +collection.) +\end{PGtext} +\end{minipage} +\end{center} +\vfill + +\begin{minipage}{0.85\textwidth} +\small +\phantomsection +\pdfbookmark[0]{Transcriber's Note}{Transcriber's Note} +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% + +\raggedright +\TransNoteText +\end{minipage} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% + +% -----File: 001.png------------------------------------------------------ +% [Production Note] +% -----File: 002.png------------------------------------------------------ +%[Blank Page] +% -----File: 003.png------------------------------------------------------ +%[Sage Endowment Fund] +% -----File: 004.png------------------------------------------------------ +%[Blank Page] +% -----File: 005.png------------------------------------------------------ +\frontmatter % this should begin numbering pages with lower case roman} + +\Large +\begin{center} +\vspace*{75mm} % the * says to honor vspace even though this is a pagebreak} +\textbf{EUCLID'S BOOK} +\vspace{1em} + +\textbf{ON DIVISIONS OF FIGURES} +\end{center} +\normalsize + +\newpage + +% -----File: 006.png------------------------------------------------------ +\begin{center} +\vspace*{50mm} + +\large \textsc{cambridge university press} + +\textsc{c.~f.~clay}, \normalsize\textsc{Manager} + +$\textgoth{London:}$ FETTER LANE, E.C. + +$\textgoth{Edinburgh:}$ 100 PRINCES STREET + +\smallskip +\CenteredGraphic{25mm}{cup1915} % Cambridge University Press Logo +\smallskip +\small +\textgoth{New York:} G.~P.~PUTNAM'S SONS + +\textgoth{Bombay, Calcutta\; and\; Madras:} MACMILLAN AND CO., \textsc{Ltd.} + +\textgoth{Toronto:} J.~M.~DENT AND SONS, \textsc{Ltd.} + +\textgoth{Tokyo:} THE MARUZEN-KABUSHIKI-KAISHA + +\vfill +\emph{All rights reserved} +\end{center} +\normalsize +\newpage + +% -----File: 007.png------------------------------------------------------ + +% This the title page, with special formatting to match the book. +\begin{center} +\Huge +\textbf{EUCLID'S BOOK} + +\textbf{ON DIVISIONS OF FIGURES} + +\medskip + +\large +(\ItalicGreekSnippet{per`i diair'esewn bibl'ion}) + +\bigskip + +\Large +WITH A RESTORATION BASED ON + +WOEPCKE'S TEXT + +\bigskip +\normalsize +AND \ ON \ THE +\bigskip + +\Large +\textit{PRACTICA \ GEOMETRIAE} +\medskip + +OF LEONARDO PISANO\\[27mm] +\normalsize +BY + +\Large +RAYMOND CLARE ARCHIBALD, \textsc{Ph.D.} +\medskip + +\small +ASSISTANT PROFESSOR OF MATHEMATICS IN BROWN + +UNIVERSITY, PROVIDENCE, RHODE ISLAND + +\vfill + +\Large +Cambridge: + +at the University Press + +1915. + +\normalsize +\end{center} +\newpage + +% -----File: 008.png------------------------------------------------------ + +\vspace*{60mm} +\begin{center} +$\textgoth{Cambridge:}$ +\medskip + +\textsc{printed by john clay, m.~a.} +\medskip + +\textsc{at the university press} +\end{center} +\newpage + +% -----File: 009.png------------------------------------------------------ +\vspace*{40mm} +\pdfbookmark[0]{Dedication}{Dedication} +\begin{center} +\Large +TO +\bigskip + +MY OLD TEACHER AND FRIEND +\bigskip +\huge + +ALFRED DEANE SMITH + +\bigskip + +\large +PROFESSOR OF GREEK AND LATIN +\bigskip + +AT MOUNT ALLISON UNIVERSITY +\bigskip + +FOR FORTY-FOUR YEARS +\bigskip + +SCHOLAR OF GREAT ATTAINMENTS +\bigskip + +THE WONDER OF ALL WHO KNOW HIM +\bigskip + +THESE PAGES ARE AFFECTIONATELY DEDICATED + +\normalsize +\end{center} +\newpage + +% -----File: 010.png------------------------------------------------------ +% This is the introductory page. Convert to fancy page style, +% but make page one of the Introductory of empty page style.} +\pagestyle{fancy} +\fancyhead[LE,RO]{\thepage}% +\fancyhead[C]{\Heading{INTRODUCTORY}} +\fancyfoot{} +\LargeCenteredNote{INTRODUCTORY} +\thispagestyle{empty} +\pdfbookmark[0]{Introductory}{Introductory} + +Euclid, famed founder of the Alexandrian School of +Mathematics, was the author of not less than nine +works. Approximately complete texts, all carefully edited, +of four of these, +(1)~the \textit{Elements}, +(2)~the \textit{Data}, +(3)~the \textit{Optics}, +(4)~the \textit{Phenomena}, are now our possession. +In the case of +(5)~the \textit{Pseudaria}, +(6)~the \textit{Surface-Loci}, +(7)~the \textit{Conics}, our +fragmentary knowledge, derived wholly from Greek sources, +makes conjecture as to their content of the vaguest nature. +On~(8) the \textit{Porisms}, Pappus gives extended comment. +As to~(9), the book \textit{On Divisions} (\textit{of figures}), Proclus alone among +Greeks makes explanatory reference. But in an Arabian +MS., translated by Woepcke into French over sixty years ago, +we have not only the enunciations of all of the propositions +but also the proofs of four of them. + +Whilst elaborate restorations of the \textit{Porisms} by Simson +and Chasles have been published, no previous attempt has +been made (the pamphlet of Ofterdinger is not forgotten) to +restore the proofs of the book \textit{On Divisions} (\textit{of figures}). And, +except for a short sketch in Heath's monumental edition of +Euclid's \textit{Elements}, nothing but passing mention of Euclid's +book \textit{On Divisions} has appeared in English. + +In this little volume I have attempted: + +\renewcommand{\labelenumi}{(\arabic{enumi})} +\begin{enumerate} +\item to give, with necessary commentary, a restoration of Euclid's +work based on the Woepcke text and on a thirteenth century geometry +of Leonardo Pisano. + +\item to take due account of the various questions which arise in +connection with (\textit{a}) certain MSS. of ``Muhammed Bagdedinus,'' +(\textit{b}) the Dee-Commandinus book on divisions of figures. + +\item to indicate the writers prior to 1500 who have dealt with +propositions of Euclid's work. + +% -----File: 011.png---Folio vii------------------------------------------------------ +\item to make a selection from the very extensive bibliography of the +subject during the past 400 years. +\end{enumerate} + +In the historical survey the MSS. of ``Muhammed Bagdedinus'' +play an important rôle, and many recent historians, +for example Heiberg, Cantor, Hankel, Loria, Suter, and +Steinschneider, have contributed to the discussion. As it is +necessary for me to correct errors, major and minor, of all of +these writers, considerable detail has to be given in the first +part of the volume; the brief second part treats of writers on +divisions before 1500; the third part contains the restoration +proper, with its thirty-six propositions. The Appendix deals +with literature since 1500. + +A score of the propositions are more or less familiar as +isolated problems of modern English texts, and are also to be +found in many recent English, German and French books +and periodicals. But any approximately accurate restoration +of the work as a whole, in Euclidean manner, can hardly fail +of appeal to anyone interested in elementary geometry or in +Greek mathematics of twenty-two centuries ago. + +In the spelling of Arabian names, I have followed Suter. + +It is a pleasure to have to acknowledge indebtedness to +the two foremost living authorities on Greek Mathematics. +I refer to Professor J.~L. Heiberg of the University of +Copenhagen and to Sir Thomas~L.\ Heath of London. +Professor Heiberg most kindly sent me the proof pages of +the forthcoming concluding volume of Euclid's \textit{Opera Omnia}, +which contained the references to Euclid's book \textit{On Divisions +of Figures}. To Sir Thomas my debt is great. On nearly +every page that follows there is evidence of the influence of +his publications; moreover, he has read this little book in +proof and set me right at several points, more especially in +connection with discussions in Note~113 and Paragraph~50. + +\hfill R.~C.~A. \qquad\qquad + +\quad\textsc{Brown University}, + +\quad\phantom{\textsc{Brown}}\textit{June}, 1915. +\newpage +% -----File: 012.png------------------------------------------------------ + +\thispagestyle{empty} +\LargeCenteredNote{CONTENTS} +\pdfbookmark[0]{Contents}{Table of Contents} +\small +\noindent +\begin{tabular}{clr} +&&\textsc{page}\\ + & \textsc{Introductory} & vi \\ +\multicolumn{3}{c}{I} \\ +\multicolumn{2}{l}{\textsc{paragraph}} & \\ +\multicolumn{2}{l}{\textsc{ numbers}} & \\ + +1 & \textsc{Proclus, and Euclid's Book} \textit{On Divisions of Figures} & \pageref{par1} \\ +2--6 & \textit{De Divisionibus} + \textsc{by ``Muhammed Bagdedinus'' and} & \\ + & \textsc{the Dee MS.} & \pageref{par2} \\ +7--9 & \textsc{The Woepcke-Euclid} MS. & \pageref{par7} \\ +10--13 & \textit{Practica Geometriae} \textsc{of Leonardo Pisano (Fibonaci)} & \pageref{par10} \\ +14--17 & \textsc{Summary:} & \\ + 14 & \qquad \textsc{Synopsis of Muhammed's Treatise} & \pageref{par14} \\ + 15 & \qquad \textsc{Commandinus's Treatise} & \pageref{par15} \\ + 16 & \qquad \textsc{Synopsis of Euclid's Treatise} & \pageref{par16} \\ + 17 & \qquad \textsc{Analysis of Leonardo's Work} & \pageref{par17} \\ + &&\\ % blank line +\multicolumn{3}{c}{II} \\ +18 & \textsc{Abraham Savasorda, Jordanus Nemorarius}, & \\ + & \textsc{Luca Paciuolo} & \pageref{par18} \\ +19 & ``\textsc{Muhammed Bagdedinus}'' \textsc{and other} & \\ + & \textsc{Arabian writers on Divisions of Figures} & \pageref{par19} \\ +20 & \textsc{Practical Applications of the problems on Divisions} & \\ + & \textsc{of Figures; the} + \ItalicGreekSnippet{metrik'a} + \textsc{of Heron of Alexandria} & \pageref{par20} \\ +21 & \textsc{Connection between Euclid's Book} \textit{On Divisions}, & \\ + & \textsc{Apollonius's treatise} \textit{On Cutting off a Space} \textsc{and a} & \\ + & \textsc{Pappus-lemma to Euclid's book of} \textit{Porisms} & \pageref{par21} \\ + &&\\ % blank line +\multicolumn{3}{c}{III} \\ +22--57 & \textsc{Restoration of Euclid's} + \ItalicGreekSnippet{per`i diair'esewn bibl'ion} & \pageref{par22}\\ + &&\\ % blank line +\multicolumn{3}{c}{IV} \\ + & \textsc{Appendix} & \pageref{appendix} \\ + & \textsc{Index of Names} & \pageref{indexpage} +\end{tabular} + + + +% -----File: 013.png----Folio 1----------------------------------------------------- +\mainmatter +\phantomsection\pdfbookmark[-1]{Main Matter}{Main Matter} +\pagestyle{fancy} +\Chapter{I} % Chapter I +\Section{\textit{Proclus, and Euclid's book On Divisions.}} +\Paragraph{PROCLUS, AND EUCLID'S BOOK ON DIVISIONS.}{1} % Paragraph 1 +\phantomsection\pdfbookmark[0]{Proclus, and Euclid's book On Divisions.}{hyperPar1} +%%%%%% {Proclus, and Euclid's book On Divisions.} +Last in a list of Euclid's works ``full of admirable +diligence and skilful consideration,'' Proclus mentions, without +comment, +%[DPStyle Greek: peri diaireseôn biblion] +\ItalicGreekSnippet{per`i diair'esewn bibl'ion}\Footnote{1}{ % Footnote 1 +\textit{Procli Diadochi in primum Euclidis elementorum + librum commentarii} ex rec. + G.~Friedlein, Leipzig, 1873, p.~69. Reference to this work will be + made by ``Proclus.''}. % end footnote 1 +But a little later\Footnote{2}{ % Footnote 2 +Proclus\footnotemark[1], p.~144.} % end footnote 2 +in speaking of the conception or definition of \textit{figure} and of the +divisibility of a figure into others differing from it in kind, +Proclus adds: ``For the circle is divisible into parts unlike +in definition or notion, and so is each of the rectilineal figures; +this is in fact the business of the writer of the Elements in +his Divisions, where he divides given figures, in one case into +like figures, and in another into unlike\Footnote{3}{ % Footnote 3 + In this translation I have followed \textsc{T.~L.\ Heath}, + \textit{The Thirteen Books of Euclid's Elements}, 1, + Cambridge, 1908, p.~8. To Heath's account (pp.~8--10) of + Euclid's book \textit{On Divisions} I shall refer by ``Heath.'' + + ``Like'' and ``unlike'' in the above quotation mean, + not ``similar'' and + ``dissimilar'' in the technical sense, + but ``like'' or ``unlike \emph{in definition} or \emph{notion}'': + thus to divide a triangle into triangles would be to divide it + into ``like'' figures, to + divide a triangle into a triangle and a quadrilateral would be + to divide it into ``unlike'' figures. (Heath.)}.'' % end footnote 3 + + + +\Section{\textit{De Divisionibus by Muhammed Bagdedinus and the Dee MS.}} + +\Paragraph{MSS. OF MUHAMMED BAGDEDINUS AND DEE}{2} % Paragraph 2 +\phantomsection\pdfbookmark[0]{MSS of Muhammed Bagdedinus and Dee}{hyperPar2} +This is all we have from Greek sources, but the +discovery of an Arabian translation of the treatise supplies +the deficiency. In histories of Euclid's works (for example +% -----File: 014.png---- Folio 2 ----------------------------------------------------- +those by Hankel\Footnote{4}{ % Footnote 4 + \textsc{H.~Hankel}, \textit{Zur Geschichte der Mathematik}, + Leipzig, 1874, p.~234.}, % end footnote 4 +Heiberg\Footnote{5}{ % Footnote 5 + \textsc{J.~L.~Heiberg}, + \textit{Litterargeschichtliche Studien über Euklid}, Leipzig, 1882, + pp.~13--16, 36--38. + Reference to this work will be made by ``Heiberg.''}, % end footnote 5 +Favaro\Footnote{6}{ % Footnote 6 + \textsc{E.~A.~Favaro}. + ``Preliminari ad una Restituzione del libro di Euclide sulla + divisione delle figure piane,'' + \textit{Atti del reale Istituto Veneto di Scienze, Lettere ed Arti}, + \textsc{i}$_{6}$, 1883, pp.~393--6. + ``Notizie storico-critiche sulla Divisione delee Aree'' + (Presentata li 28~gennaio, 1883), + \textit{Memorie del reale Istituto Veneto di Scienze, Lettere ed Arti}, + \textsc{xxii}, 129--154. + This is by far the most elaborate consideration of the subject up to + the present. Reference to it will be made by ``Favaro.'' + }, % end footnote 6 +Loria\Footnote{7}{ % Footnote 7 + \textsc{G.~Loria}, + ``Le Scienze esatte nell' antica Grecia, Libro \textsc{ii}, + Il periodo aureo della geometria Greca.'' + \textit{Memorie della regia Accademia di Scienze, + \DPtypo{Lettre}{Lettere} ed Arti in Modena}, \DPnote{Typo Corrected} + \textsc{xi}$_{2}$, 1895, pp.~68--70, 220--221. + \textit{Le Scienze esatte nell' antica Grecia}, + Seconda edizione. Milano, 1914, pp.~250--252, 426--427.}, % end footnote 7 +Cantor\Footnote{8}{ % Footnote 8 + \textsc{M.~Cantor}, + \textit{Vorlesungen über Geschichte der Mathematik}, + \textsc{i}$_{3}$, 1907, pp.~287--8; + \textsc{ii}$_{2}$, 1900, p.~555.}, % end footnote 8 +Hultsch\Footnote{9}{ % Footnote 9 + \textsc{F.~Hultsch}, Article ``Eukleides'' in Pauly-Wissowa's + \textit{Real-Encyclopädie der Class. Al\-tert\-ums\-wis\-sen\-schaf\-ten}, + \textsc{vi}, Stuttgart, 1909, especially Cols.~1040--41.}, % end footnote 9 +Heath\footnotemark[3]) +prominence is given to a treatise \textit{De Divisionibus}, by +one ``Muhammed Bagdedinus.'' Of this in 1563\Footnote{10}{ % Footnote 10 + When Dee was in Italy visiting Commandinus at Urbino.} % end footnote 10 +a copy (in Latin) was given by John Dee to Commandinus who published +it in Dee's name and his own in 1570\Footnote{11}{ % Footnote 11 + \textit{De superficierum divisionibus liber Machometo Bagdedino + ascriptus nunc primum Joannis Dee Londinensis \& Federici Commandini + Urbinatis opera in lucem editus}. Federici Commandini de eadem re + libellus. Pisauri, \textsc{mdlxx}. In the same year appeared an + Italian translation: \textit{Libro del modo di dividere le superficie + attribuito a Machometo Bagdedino. Mandato in luce la prima volta da + M.~G.\ Dee\ldots{}e da M.~F.\ldots Commandino\ldots{}Tradotti dal Latino + in volgare da F.~Viani de' Malatesti,\ldots} + In Pesaro, del \textsc{mdlxx}\ldots 4 unnumbered leaves + and 44 numbered on one side. + + An English translation from the Latin, with the following title-page, + was published in the next century: + \textit{A Book of the Divisions of Superficies: ascribed + to Machomet Bagdedine. + Now put forth, by the pains of John Dee of London, and + Frederic Commandine of Urbin. + As also a little Book of Frederic Commandine, + concerning the same matter. + London Printed by R.~\& W.\ Leybourn}, 1660. + Although this work has a separate title page and the above date, + it occupies the last fifty pages (601--650) of a work dated a year + later: \textit{Euclid's Elements of Geometry in XV Books\ldots{}to + which is added a Treatise of Regular Solids by Campane and Flussas + likewise Euclid's Data and Marinus Preface thereunto annexed. + Also a Treatise of the Divisions of Superficies ascribed to Machomet + Bagdedine, but published by Commandine, at the request of John Dee + of London; whose Preface to the said Treatise declares it to be the + Worke of Euclide, the Author of the Elements. + Published by the care and Industry of John Leeke and George Serle, + Students in the Mathematics}. London\ldots\textsc{mdclxi}. + + A reprint of simply that portion of the Latin edition which is the + text of Muhammed's work appeared in: + \ItalicGreekSnippet{EUKLEIDOU TA SWZOMENA} + \textit{Euclidis quae supersunt omnia. + Ex rescensione Davidis Gregorii}\ldots{}Oxoniae\ldots\textsc{mdcciii}. + Pp.~665--684: + \ItalicGreekSnippet{EUKLEIDOU WS OIONTAI TINES, PERI DIAIRESEWN BIBLOS} + Euclidis, ut quidam arbitrantur, + de divisionibus liber---vel ut alii volunt, Machometi Bagdedini + liber de divisionibus superficierum.''}. % end footnote 11 +Recent writers whose +publications appeared before 1905 have generally supposed +that Dee had somewhere discovered an Arabian original of +Muhammed's work and had given a Latin translation to +Commandinus. Nothing contrary to this is indeed explicitly +% -----File: 015.png---- Folio 3 ----------------------------------------------------- +stated by Steinschneider when he writes in 1905\Footnote{12}{ % Footnote 12 + \textsc{M.~Steinschneider}, + ``Die Europäischen Übersetzungen aus dem Arabischen + bis Mitte des 17.~Jahrhunderts.'' + \textit{Sitzungsberichte der Akademie der Wis\-sen\-schaf\-ten in Wien} + (Philog.-histor.\ Klasse) \textsc{cli}, Jan.~1905, Wien, 1906. + Concerning ``171.\ Muhammed'' \textit{cf}.~pp.~41--2. + Reference to this paper will be made by ``Steinschneider.''}, % end footnote 12 +``Machomet Bagdadinus (=aus Bagdad) heisst in einem alten MS.\ +Cotton (jetzt im Brit.\ Mus.) der Verfasser von: de Superficierum +divisione (22~Lehrsätze); Jo.~Dee aus London entdeckte es +und übergab es T.~Commandino\ldots.'' For this suggestion as +to the place where Dee found the MS.\ Steinschneider gives +no authority. He does, however, give a reference to +Wenrich\Footnote{13}{ % Footnote 13 + \textsc{J.~G.~Wenrich}, \textit{De auctorum Graecorum versionibus}. + Lipsiae, \textsc{mdcccxlii}, p.~184.}, % end footnote 13 +who in turn refers to a list of the printed books +(``Impressi'') of John Dee, in a life of Dee by Thomas +Smith\Footnote{14}{ % Footnote 14 + \textsc{T.~Smith}, + \textit{Vitae quorundam eruditissimorum et illustrium virorum\ldots} + {}Londini\ldots{}\textsc{mdccvii}, + p.~56. It was only the first 55 pages of this + ``Vita Joannis Dee, Mathematici Angli,'' + which were translated into English by W.~A.\ Ayton, London, 1908.} % end footnote 14 +(1638--1710). We here find as the third in the list, +``Epistola ad eximium Ducis Urbini Mathematicum, +Fredericum Commandinum, praefixa libello Machometi Bagdedini +de superficierum divisionibus\ldots\textit{Pisauri}, 1570. Exstat +MS.\ in Bibliotheca Cottoniana sub Tiberio B \textsc{ix}.'' + +Then come the following somewhat mysterious sentences +which I give in translation\Footnote{15}{ % Footnote 15 + ``Post praefationem haec habet D.\ \textit{Usserius} + Archiepiscopus Armachanus. + \textit{Notandum est autem, Auctorem hunc} Euclide + \textit{usum in Arabicam linguam converso, quem postea} Campanus + \textit{Latinum fecit. Auctor igitur propositionum videtur fuisse} + Euclides: \textit{demonstrationum, in quibus} Euclides + \textit{in Arabico codice citatur}, + Machometus Bagded \textit{sive} Babylonius.'' + + It has been stated that Campanus (13.~cent.) did not translate + Euclid's Elements into Latin, but that the work published as his + (Venice, 1482---the first printed edition of the \textit{Elements}) + was the translation made about 1120 by the English monk + Athelhard of Bath. \textit{Cf.} \textsc{Heath}, + \textit{Thirteen Books of Euclid's Elements}, + \textsc{i}, 78, 93--96.}: % end footnote 15 +``After the preface Lord Ussher +[1581--1656], Archbishop of Armagh, has these lines: It is to +be noted that the author uses Euclid's Elements translated +into the Arabic tongue, which Campanus afterwards turned +into Latin. Euclid therefore seems to have been the author +of the Propositions [of \textit{De Divisionibus}] though not of the +demonstrations, which contain references to an Arabic edition +of the Elements, and which are due to Machometus of Bagded +or Babylon.'' This quotation from Smith is reproduced, with +various changes in punctuation and typography, by Kästner\Footnote{16}{ % Footnote 16 + \textsc{A.~G.~Kästner}, + \textit{Geschichte der Mathematik}\ldots{} + Erster Band\ldots{}Göttingen, 1796, + pp.~272--3. See also ``Zweyter'' Band, 1797, pp.~46--47.}. % end footnote 16 +Consideration of the latter part of it I shall postpone to a later +article (5). + +% -----File: 016.png-----Folio 4---------------------------------------------------- + +\Paragraph{MSS. OF MUHAMMED BAGDEDINUS AND DEE}{3} % Paragraph 3 +Following up the suggestion of Steinschneider, Suter +pointed out\Footnote{17}{ % Footnote 17 + \textsc{H.~Suter}, + ``Zu dem Buche `De Superficierum divisionibus' des Muhammed Bagdedinus.'' + \textit{Bibliotheca Mathematica}, + \textsc{vi}$_3$, 321--2, 1905.}, % end footnote 17 +without reference to Smith\footnotemark[14] or Kästner\footnotemark[16], +that in Smith's catalogue of the Cottonian Library there was an +entry\Footnote{18}{ % Footnote 18 + \textsc{T.~Smith}, + \textit{Catalogus Librorum Manuscriptorum Bibliothecae Cottonianae\ldots} + Oxonii,\ldots \textsc{mdcxcvi}, p.~24.} % end footnote 18 +under ``Tiberius\Footnote{19}{ % Footnote 19 + The original Cottonian library was contained in 14 presses, + above each of which was a bust; 12 of these busts were of Roman + Emperors. Hence the + classification of the MSS.\ in the catalogue.} % end footnote 19 +B \textsc{ix}, 6'': ``Liber Divisionum +Mahumeti Bag-dadini.'' As this MS.\ was undoubtedly in +Latin and as Cottonian MSS.\ are now in the British Museum, +Suter inferred that Dee simply made a copy of the above +mentioned MS.\ and that this MS.\ was now in the British +Museum. With his wonted carefulness of statement, Heath +does not commit himself to these views although he admits +their probable accuracy. + +\Paragraph{MSS. OF MUHAMMED BAGDEDINUS AND DEE}{4} % Paragraph 4 +As a final settlement of the question, I propose to +show that Steinschneider and Suter, and hence also many +earlier writers, have not considered all facts available. Some +of their conclusions are therefore untenable. In particular: + +(1) In or before 1563 Dee did \emph{not} make a copy of any +Cottonian MS.; + +(2) The above mentioned MS.\ (Tiberius, B.\ \textsc{ix}, 6) was +never, in its entirety, in the British Museum; + +(3) The inference by Suter that this MS.\ was probably +the Latin translation of the tract from the Arabic, made by +Gherard of Cremona (1114--1187)---among the lists of whose +numerous translations a ``liber divisionum'' occurs---\emph{should +be accepted with great reserve}; + +(4) The MS.\ which Dee used can be stated with absolute +certainty and this MS.\ did not, in all probability, afterwards +become a Cottonian MS. + +(1) Sir Robert Bruce Cotton, the founder of the Cottonian +Library, was born in 1571. The Cottonian Library was not, +therefore, in existence in 1563 and Dee could not then have +copied a Cottonian MS. + +(2) The Cottonian Library passed into the care of the +nation shortly after 1700. In 1731 about 200 of the MSS.\ +% -----File: 017.png---Folio 5------------------------------------------------------ +were damaged or destroyed by fire. As a result of the +parliamentary inquiry Casley reported\Footnote{20}{ % Footnote 20 + \textsc{D.~Casley}, p.~15\,ff.\ of + \textit{A Report from the Committee appointed to view the + Cottonian Library\ldots{}Published by order of the House of Commons}. + London, + \textsc{mdccxxxii} (British Museum MSS.\ 24932). + \textit{Cf.}\ also the page opposite that numbered 120 in + \textit{A Catalogue of the Manuscripts in the Cottonian Library\ldots{}with + an Appendix containing an account of the damage sustained by the Fire in 1731; + by S.~Hooper\ldots} London:\ldots\textsc{mdcclxxvii}.} % end footnote 20 +on the MSS.\ destroyed +or injured. Concerning Tiberius \textsc{ix}, he wrote, ``This volume +burnt to a crust.'' He gives the title of each tract and the +folios occupied by each in the volume. ``Liber Divisionum +Mahumeti Bag-dadini'' occupied folios 254--258. When the +British Museum was opened in \emph{1753}, what was left of the +Cottonian Library was immediately placed there. Although +portions of all of the leaves of our tract are now to be seen +in the British Museum, practically none of the writing is +decipherable. + +(3) Planta's catalogue\Footnote{21}{ % Footnote 21 + \textsc{J.~Planta}, + \textit{A Catalogue of the Manuscripts in the Cottonian Library + deposited in the British Museum. Printed by command of his Majesty + King George III\ldots} 1802. + + In the British Museum there are three MS.\ catalogues of the + Cottonian Library: + + (1) \textit{Harleian MS.} 6018, a catalogue made in 1621. + At the end are memoranda of loaned books. + On a sheet of paper bearing date Novem.~23, 1638, Tiberius B + \textsc{ix} is listed (folio~187) with its art.~4: + ``liber divisione Machumeti Bagdedini.'' + The paper is torn so that the name of the person to whom the work + was loaned is missing. + The volume is not mentioned in the main catalogue. + + (2) \textit{MS.~No.}~36789, made after Sir Robert Cotton's death + in 1631 and before 1638 + (\textit{cf.\ Catalogue of Additions to the MSS.\ in British Museum, + 1900--1905\ldots} London, 1907, pp.~226--227), contains, + apparently, no reference to ``Muhammed.'' + + (3) \textit{MS.~No.}~36682~A, of uncertain date but earlier than + 1654 (\textit{Catalogue of Additions\ldots{}l.c.}\ pp.~188--189). + On folio 78 \textit{verso} we find Tiberius B \textsc{ix}, Art.\ 4: + ``Liber divisione Machumeti Bagdedini.'' + + A ``Muhammed'' MS.\ was therefore in the Cottonian Library in 1638. + + The anonymously printed (1840?) ``Index to articles printed from + the Cotton MSS., \& where they may be found'' which may be seen + in the British Museum, only gives references to the MSS.\ + in ``Julius.''} % end footnote 21 +has the following note concerning +Tiberius \textsc{ix}: ``A volume on parchment, which once consisted +of 272 leaves, written about the XIV.\ century [not the +XII.\ century, when Gherard of Cremona flourished], containing +eight tracts, the principal of which was a `Register +of William Cratfield, abbot of St Edmund'\,'' [d.~1415]. +Tracts 3, 4, 5 were on music. + +(4) On ``$A^\circ$ 1583, 6 Sept.''\ Dee made a catalogue of the +MSS.\ which he owned. This catalogue, which is in the +Library of Trinity College, Cambridge\Footnote{22}{ % Footnote 22 + A transcription of the Trinity College copy, by Ashmole, + is in MS.\ Ashm.\ 1142. Another autograph copy is in the + British Museum: Harleian MS.\ 1879.}, % end footnote 22 +has been published\Footnote{23}{ % Footnote 23 + \textit{Camden Society Publications}, + \textsc{xix}, London, \textsc{m.dccc.xlii}.} % end footnote 23 +% -----File: 018.png---Folio 6------------------------------------------------------ +under the editorship of J.~O.~Halliwell. The 95th item +described is a folio parchment volume containing 24 tracts +on mathematics and astronomy. The 17th tract is entitled +``Machumeti Bagdedini liber divisionum.'' As the contents +of this volume are entirely different from those of Tiberius \textsc{ix} +described above, in~(3), it seems probable that there were two +copies of ``Muhammed's'' tract, while the MS.\ which Dee +used for the 1570 publication was undoubtedly his own, as we +shall presently see. If the two copies be granted, there is no +evidence against the Dee copy having been that made by +Gherard of Cremona. + +\Paragraph{MSS. OF MUHAMMED BAGDEDINUS AND DEE}{5} % Paragraph 5 +There is the not remote possibility that the Dee MS.\ +was destroyed soon after it was catalogued. For in the same +month that the above catalogue was prepared, Dee left his +home at Mortlake, Surrey, for a lengthy trip in Europe. +Immediately after his departure ``the mob, who execrated +him as a magician, broke into his house and destroyed a great +part of his furniture and books\Footnote{24}{ % Footnote 24 + \textit{Dictionary of National Biography}, Article, + ``Dee, John.''}\ldots''\ % end footnote 24 +many of which ``were the +written bookes\Footnote{25}{ % Footnote 25 + ``The compendious rehearsall of John Dee his dutifull declaration A.~1592'' + printed in \textit{Chetham Miscellanies}, + vol.~\textsc{i}, Manchester, \textsc{1851}, p.~27.}.'' % end footnote 25 +Now the Dee catalogue of his MSS.\ +(MS.\ O.\ iv.~20), in Trinity College Library, has numerous +annotations\Footnote{26}{ % Footnote 26 + Although Halliwell professed to publish the Trinity MS., he makes not + the slightest reference to these annotations.} % end footnote 26 +in Dee's handwriting. They indicate just what +works were (1) destroyed or stolen (``Fr.'')\Footnote{27}{ % Footnote 27 + ``Fr.''\ is no doubt an abbreviation for \textit{Furatum}.}\ % end footnote 27 +and (2) left(``T.'')\Footnote{28}{ % Footnote 28 + ``T.'', according to Ainsworth (\textit{Latin Dictionary}), + was put after the name of a + soldier to indicate that he had survived (\textit{superstes}). + Whence this abbreviation?} % end footnote 28 +after the raid. Opposite the titles of the tracts in the volume +including the tract ``liber divisionum,'' ``Fr.''\ is written, and +opposite the title ``Machumeti Bagdedini liber divisionum'' +is the following note: ``Curavi imprimi Urbini in Italia per +Federicum Commandinum exemplari descripto ex vetusto isto +monumento(?) per me ipsum.'' Hence, as stated above, it is +now definitely known (1) that the MS.\ which Dee used was +his own, and (2) that some 20 years after he made a copy, the +MS.\ was stolen and probably destroyed\Footnote{29}{ % Footnote 29 + The view concerning the theft or destruction of the MS.\ is borne + out by the fact that in a catalogue of Dee's Library + (British Museum MS.~35213) made early in the seventeenth century + (\textit{Catalogue of Additions and Manuscripts}\ldots{}1901, + p.~211), Machumeti Bagdedini is not mentioned.}. % end footnote 29 + +On the other hand we have the apparently contradictory +% -----File: 019.png---Folio 7------------------------------------------------------ +evidence in the passage quoted above (Art.~2) from the life +of Dee by Smith\footnotemark[14] +who was also the compiler of the Catalogue +of the Cottonian Library. Smith was librarian when he +wrote both of these works, so that any definite statement +which he makes concerning the library long in his charge is +not likely to be successfully challenged. Smith does not +however say that Dee's ``Muhammed'' MS.\ was in the +Cottonian Library, and if he knew that such was the case +we should certainly expect some note to that effect in the +catalogue\footnotemark[18]; +for in three other places in his catalogue +(Vespasian B \textsc{x}, A \textsc{ii}$_{13}$, +Galba E \textsc{viii}), Dee's original +ownership of MSS.\ which finally came to the Cottonian +Library is carefully remarked. Smith does declare, however, +that the Cottonian MS.\ bore, ``after the preface,'' certain +notes (which I have quoted above) by Archbishop Ussher +(1581--1656). Now it is not a little curious that these notes +by Ussher, who was not born till after the Dee book was +printed, should be practically identical with notes in the +printed work, just after Dee's letter to Commandinus (Art.~3). +For the sake of comparison I quote the notes in question\Footnote{30}{ % Footnote 30 + This quotation from the Leeke-Serle Euclid\footnotemark[11] + is an exact translation of the original.}; % end footnote 30 +\DPnote{The following quote contains instances of the long-s.} +\label{longs} +``To the Reader.---I am here to advertise thee (kinde Reader) +that this author which we present to thee, made use of Euclid +translated into the Arabick Tongue, whom afterwards Campanus +made to speake Latine. This I thought fit to tell thee, that +so in searching or examining the Propositions which are cited +by him, thou mightest not sometime or other trouble thy selfe +in vain, Farewell.'' + +The Dee MS.\ as published did not have any preface. +We can therefore only assume that Ussher wrote in a MS.\ +which \emph{did} have a preface the few lines which he may have +seen in Dee's printed book. + +\Paragraph{MSS. OF MUHAMMED BAGDEDINUS AND DEE}{6} % Paragraph 6 +Other suggestions which have been made concerning +``Muhammed's'' tract should be considered. Steinschneider +asks, ``Ob identisch de Curvis superficiebus, von einem +Muhammed, MS.\ Brit.\ Mus.\ Harl.\ 623\footnotemark[6] +(\textsc{i}, 191)\Footnote{31}{ % Footnote 31 + This should be 625\footnotemark[6] + (\textsc{i}, 391).}?'' % end footnote 31 +I have examined this MS.\ and found that it has nothing to do with +the subject matter of the Dee tract. + +But again, Favaro states\Footnote{32}{ % Footnote 32 + Favaro, p.~140. \textit{Cf.}\ Heiberg, p.~14. + This suggestion doubtless originated + with Ofterdinger\footnotemark[38], +p.~[1].}: % end footnote 32 +``Probabilmente il manoscritto +% -----File: 020.png---Folio 8------------------------------------------------------ +del quale si \DPtypo{servi}{servì}\DPnote{[**sic but should be servì]. Typo Corrected} +il Dee è lo stesso indicato dall'Heilbronner\Footnote{33}{ % Footnote 33 + \textsc{J.~C.~Heilbronner}, + \textit{Historia matheseos Universae\ldots}Lipsiae, \textsc{mdccxlii}, + p.~620: (``Manuscripta mathematica in Bibliotheca Bodlejana'') + ``34 Mohammedis + Bagdadeni liber de superficierum divisionibus, cum Notis H.~S.''} % end footnote 33 +\DPtypo{comme}{come}\DPnote{[**sic but should be come] Typo Corrected} +esistente nella Biblioteca Bodleiana di Oxford.'' Under +date ``6.~3.~1912'' Dr A.~Cowley, assistant librarian in the +Bodleian, wrote me as follows: ``We do not possess a copy +of Heilbronner's Hist.\ Math.\ Univ. In the old catalogue of +MSS.\ which he would have used, the work you mention is included---but +is really a printed book and is only included in the +catalogue of MSS.\ because it contains some manuscript notes--- + +``Its shelf-mark is Savile T 20. + +``It has 76 pages in excellent condition. The title page +has: De Superficierum $|$ divisionibus liber $|$ Machometo Bagdedino $|$ +ascriptus $|$ nunc primum Joannis Dee $|$ \ldots $|$ opera in +lucem editus $|$ \ldots{}Pisauri \textsc{mdlxx}. + +``The MS.\ notes are by Savile, from whom we got the +collection to which this volume belongs.'' + +The notes were incorporated into the Gregory edition\footnotemark[11] +of the Dee tract. Here and elsewhere\Footnote{34}{ % Footnote 34 + \textsc{H.~Savile}, \textit{Praelectiones tresdecim in principium + elementorum Evclidis, Oxonii habitae M.DC.XX}. Oxonii\ldots, + 1621, pp.~17--18.} % end footnote 34 +Savile objected to +attributing the tract to Euclid as author\Footnote{35}{ % Footnote 35 + Dee's statement of the case in his letter to Commandinus + (Leeke-Serle Euclid\footnotemark[11], \textit{cf.}\ note~30) + is as follows: ``As for the authors name, I would have you + understand, that to the very old Copy from whence I writ it, + the name of \textsc{Machomet Bagdedine} was put in ziphers or + Characters, (as they call them) who whether he were that + \textit{Albategnus} whom \textit{Copernicus} often cites as a very + considerable Author in Astronomie; or that Machomet who is said to + have been \textit{Alkindus's} scholar, and is reported to have + written somewhat of the art of Demonstration, I am not yet certain of: + or rather that this may be deemed a Book of our \textit{Euclide}, + all whose Books were long since turned out of the Greeke into the + Syriack and Arabick Tongues. Whereupon, It being found some time or + other to want its Title with the \textit{Arabians} or + \textit{Syrians}, was easily attributed by the transcribers + to that most famous Mathematician among them, Machomet: + which I am able to prove by many testimonies, to be often done in + many Moniments of the Ancients; + \ldots{}yea further, we could not yet perceive so great acuteness of + any \textit{Machomet} in the Mathematicks, from their moniments + which we enjoy, as everywhere appears in these Problems. + Moreover, that \textit{Euclide} also himself wrote one Book + \ItalicGreekSnippet{peri diair'esewn} + that is to say, \textit{of Divisions}, as may be evidenced from + Proclus's Commentaries upon his first of \textit{Elements}: + and we know none other extant under this title, nor can we find any, + which for excellencie of its treatment, may more rightfully or + worthily be ascribed to \textit{Euclid}. Finally, I remember that in + a certain very ancient piece of Geometry, I have read a place cited + out of this little Book in expresse words, even as from amost + (\textit{sic}) certain work of \textit{Euclid}. Therefore we have + thus briefly declared our opinions for the present, which we desire + may carry with them so much weight, as they have truth in + them\ldots. But whatsoever that Book of \textit{Euclid} was + concerning Divisions, certainly this is such an one as may be both + very profitable for the studies of many, and also bring much honour + and renown to every most noble ancient Mathematician; for the most + excellent acutenesse of the invention, and the most accurate + discussing of all the Cases in each Probleme\ldots.''}. % end footnote 35 +His arguments +% -----File: 021.png---Folio 9------------------------------------------------------ +are summed up, for the most part, in the conclusions of +Heiberg followed by Heath: ``the Arabic original could not +have been a direct translation from Euclid, and probably was +not even a direct adaptation of it; it contains mistakes and +unmathematical expressions, and moreover does not contain +the propositions about the division of a circle alluded to by +Proclus. Hence it can scarcely have contained more than +a fragment of Euclid's work.'' + +\Section{\textit{The Woepcke-Euclid MS.}} +\Paragraph{THE WOEPCKE-EUCLID MS.}{7} % Paragraph 7 +\phantomsection\pdfbookmark[0]{The Woepcke-Euclid MS.}{hyperPar7} +On the other hand Woepcke found in a MS.\ (No.~952.~2 +Arab.\ Suppl.)\ of the Bibliothèque nationale, Paris, a treatise +in Arabic on the division of plane figures, which he translated, +and published in 1851\Footnote{36}{ % Footnote 36 + \textsc{F.~Woepcke}, + ``Notice sur des traductions Arabes de deux ouverages perdus d'Euclide'' + \textit{Journal Asiatique}, Septembre--Octobre, 1851, + \textsc{xviii}$_4$, 217--247. + Euclid's work \textit{On the division (of plane figures)}: pp.~233--244. + Reference to this paper will be made by ``Woepcke.'' + In \textit{Euclidis opera omnia}, vol.~8, now in the press, there are + ``Fragmenta collegit et disposuit J.~L.\ Heiberg,'' through whose + great courtesy I have been enabled to see the proof-sheets. + First among the fragments, on pages + 227--235, are (1) the Proclus references to + \ItalicGreekSnippet{peri diair'esewn} +and (2) the Woepcke + translation mentioned above. In the article on Euclid in the last edition of the + \textit{Encyclopaedia Britannica} no reference is made to this work or to the writings + of Heiberg, Hultsch, Steinschneider and Suter.}. % end footnote 36 +``It is expressly attributed to Euclid +in the MS.\ and corresponds to the description of it by Proclus. +Generally speaking, the divisions are divisions into figures of +the same kind as the original figures, e.~g.\ of triangles into +triangles; but there are also divisions into `unlike' figures, +e.~g.\ that of a triangle by a straight line parallel to the base. +The missing propositions about the division of a circle are +also here: `to divide into two equal parts a given figure +bounded by an arc of a circle and two straight lines including +a given angle' and `to draw in a given circle two parallel +straight lines cutting off a certain part of a circle.' Unfortunately +the proofs are given of only four propositions (including the +two last mentioned) out of 36, because the Arabian translator +found them too easy and omitted them.'' That the omission +is due to the translator and did not occur in the original is +indicated in two ways, as Heiberg points out. Five auxiliary +propositions (Woepcke 21, 22, 23, 24, 25) of which no use is +made are introduced. Also Woepcke 5 is: ``\ldots{}and we divide +the triangle by a construction analogous to the preceding +construction''; but no such construction is given. + +The four proofs that are given are elegant and depend +% -----File: 022.png---Foilio 10------------------------------------------------------ +only on the propositions (or easy deductions from them) of +the \textit{Elements}, while Woepcke~18 has the true Greek ring: +``to apply to a straight line a rectangle equal to the rectangle +contained by $\mathit{AB}$, $\mathit{AC}$ and \emph{deficient by a square}.'' + +\Paragraph{THE WOEPCKE-EUCLID MS.}{8} % Paragraph 8 + +To no proposition in the Dee MS.\ is there word for +word correspondence with the propositions of Woepcke but +in content there are several cases of likeness. Thus, Heiberg +continues, +\begin{verse} + Dee 3 = Woepcke 30 (a special case is Woepcke 1);\\ + Dee 7 = Woepcke 34 (a special case is Woepcke 14);\\ + Dee 9 = Woepcke 36 (a special case is Woepcke 16);\\ + Dee 12 = Woepcke 32 (a special case is Woepcke 4). +\end{verse} + +Woepcke~3 is only a special case of Dee~2; Woepcke~6, +7, 8, 9 are easily solved by Dee~8. And it can hardly be +chance that the proofs of exactly these propositions in Dee +should be without fault. That the treatise published by +Woepcke is no fragment but the complete work which was +before the translator is expressly stated\Footnote{37}{ % Footnote 37 + Woepcke, p.~244.}, % end footnote 37 +``fin du traité.'' It +is moreover a well ordered and compact whole. Hence we +may safely conclude that Woepcke's is not only Euclid's own +work but the whole of it, except for proofs of some propositions. + +\Paragraph{THE WOEPCKE-EUCLID MS.}{9} % Paragraph 9 +For the reason just stated the so-called +\textit{Wiederherstellung} +of Euclid's work by Ofterdinger\Footnote{38}{ % Footnote 38 + \textsc{L.~F.~Ofterdinger}, + \textit{Beiträge zur Wiederherstellung der Schrift des Euklides + über der Theilung der Figuren}, Ulm, 1853.}, % end footnote 38 +based mainly on +Dee, is decidedly misnamed. A more accurate description of +this pamphlet would be, ``A translation of the Dee tract with +indications in notes of a certain correspondence with 15 of +Woepcke's propositions, the whole concluding with a translation +of the enunciations of 16 of the remaining 21 propositions +of Woepcke not previously mentioned.'' Woepcke~30, 31, 34, +35, 36 are not even noticed by Ofterdinger. Hence the claim +I made above (``Introductory'') that the first real restoration +of Euclid's work is now presented. Having introduced +Woepcke's text as one part of the basis of this restoration, +the other part demands the consideration of the + +\Section{\textit{Practica Geometriae of Leonardo Pisano (Fibonaci).}} + +\Paragraph{PRACTICA GEOMETRIAE OF LEONARDO PISANO}{10} % Paragraph 10 +\phantomsection\pdfbookmark[0]{Practica Geometriae of Leonardo Pisano}{hyperPar10} +It was in the year 1220 that Leonardo Pisano, who +occupies such an important place in the history of mathematics +% -----File: 023.png---Folio 11------------------------------------------------------ +of the thirteenth century\Footnote{39}{ % Footnote 39 + \textsc{M.~Cantor}, + \textit{Vorlesungen über Geschichte der Mathematik}, + \textsc{ii}$_{2}$, 1900, pp.~3--53; + ``Practica Geometriae,'' pp.~35--40.}, % end footnote 39 +wrote his \textit{Practica Geometriae}, and +the MS.\ is now in the Vatican Library. Although it was +known and used by other writers, nearly six and one half +centuries elapsed before it was finally published by Prince +Boncompagni\Footnote{40}{ % Footnote 40 + \textit{Scritti di Leonardo Pisano matematico del secolo decimoterzo + \DPtypo{publicati}{pubblicati} + da \DPtypo{Baldasarre}{Baldassarre} +%% [**sic but should be Baldassarre -- see books.google.com] + Boncompagni}. Volume \textsc{ii} + (Leonardi Pisani Practica Geometriae ed opuscoli). Roma\ldots{}1862. + Practica Geometriae, pp.~1--224.}. % end footnote 40 +Favaro was the first\footnotemark[6] +to call attention to +the importance of Section IIII\Footnote{41}{ % Footnote 41 + \textit{Scritti di Leonardo Pisano\ldots}\textsc{ii}, pp.~110--148.} % end footnote 41 +of the \textit{Practica Geometriae} in +connection with the history of Euclid's work. This section +is wholly devoted to the enunciation and proof and numerical +exemplification of propositions concerning the divisions of +figures. Favaro reproduces the enunciations of the propositions +and numbers them 1 to 57\Footnote{42}{ % Footnote 42 + These numbers I shall use in what follows. Favaro omits some auxiliary + propositions and makes slips in connection with 28 and 40. + Either 28 should have been more general in statement or another number + should have been introduced. + Similarly for 40. Compare Articles 33--34, 35.}. % end footnote 42 +He points out that in both +enunciation and proof Leonardo 3, 10, 51, 57 are identical +with Woepcke 19, 20, 29, 28 respectively. But considerably +more remains to be remarked. + +\Paragraph{PRACTICA GEOMETRIAE OF LEONARDO PISANO}{11} % Paragraph 11 +No less than twenty-two of Woepcke's propositions +are practically identical in statement with propositions in +Leonardo; the solutions of eight more of Woepcke are either +given or clearly indicated by Leonardo's methods, and all six +of the remaining Woepcke propositions (which are auxiliary) +are assumed as known in the proofs which Leonardo gives of +propositions in Woepcke. Indeed, these two works have +a remarkable similarity. Not only are practically all of the +Woepcke propositions in Leonardo, but the proofs called +for by the order of the propositions and by the auxiliary +propositions in Woepcke are, with a possible single exception\footnotemark[91], +invariably the kind of proofs which Euclid might have given---no +other propositions but those which had gone before or +which were to be found in the \textit{Elements} being required in +the successive constructions. + +Leonardo had a wide range of knowledge concerning +Arabian mathematics and the mathematics of antiquity. His +\textit{Practica Geometriae} contains many references to Euclid's +\textit{Elements} +and many uncredited extracts from this work\Footnote{43}{ % Footnote 43 + For example, on pages 15--16, 38, 95, 100--1, 154.}. % end footnote 43 +% -----File: 024.png---Folio 12------------------------------------------------------ +Similar treatment is accorded works of other writers. But +in the great elegance, finish and rigour of the whole, +originality of treatment is not infrequently evident. If +Gherard of Cremona made a translation of Euclid's book +\textit{On Divisions}, it is not at all impossible that this may have +been used by Leonardo. At any rate the conclusion seems +inevitable that he must have had access to some such MS.\ of +Greek or Arabian origin. + +Further evidence that Leonardo's work was of Greek-Arabic +extraction can be found in the fact that, in connection +with the 113 figures, of the section On Divisions, of Leonardo's +work, the lettering in only 58 contains the letters \textit{c} or \textit{f}; that +is, the Greek-Arabic succession \textit{a, b, g, d, e, z \ldots} is used almost +as frequently as the Latin \textit{a, b, c, d, e, f, g,\ldots}; elimination of +Latin letters added to a Greek succession in a figure, for the +purpose of numerical examples (in which the work abounds), +makes the balance equal. + +\Paragraph{PRACTICA GEOMETRIAE OF LEONARDO PISANO}{12} % Paragraph 12 +My method of restoration of Euclid's work has been +as follows. Everything in Woepcke's text (together with +his notes) has been translated literally, reproduced without +change and enclosed by quotation marks. To all of Euclid's +enunciations (unaccompanied by constructions) which corresponded +to enunciations by Leonardo, I have reproduced +Leonardo's constructions and proofs, with the same lettering +of the figures\Footnote{44}{ % Footnote 44 + This is done in order to give indication of the possible origin of the construction + in question (Art.~11).}, % end footnote 44 +but occasional abbreviation in the form of +statement; that is, the extended form of Euclid in Woepcke's +text, which is also employed by Leonardo, has been sometimes +abridged by modern notation or briefer statement. Occasionally +some very obvious steps taken by Leonardo have been left +out but all such places are clearly indicated by explanation +in square brackets, [\quad]. Unless stated to the contrary, and +indicated by different type, no step is given in a construction +or proof which is not contained in Leonardo. When there +is no correspondence between Woepcke and Leonardo I have +exercised care to reproduce Leonardo's methods in other propositions, +as closely as possible. If, in a given proposition, +the method is extremely obvious on account of what has gone +before, I have sometimes given little more than an indication +of the propositions containing the essence of the required +% -----File: 025.png---Folio 13------------------------------------------------------ +construction and proof. In the case of the six auxiliary +propositions, the proofs supplied seemed to be readily suggested +by propositions in Euclid's \textit{Elements}. + +\Paragraph{PRACTICA GEOMETRIAE OF LEONARDO PISANO}{13} % Paragraph 13 +Immediately after the enunciations of Euclid's +problems follow the statements of the correspondence with +Leonardo; if exact, a bracket encloses the number of the +Leonardo proposition, according to Favaro's numbering, and +the page and lines of Boncompagni's edition where Leonardo +enunciates the same proposition. + +The following is a comparative table of the Euclid and, in +brackets, of the corresponding Leonardo problems: 1~(5); +2~(14); 3~(2,~1); 4~(23); 5~(33); 6~(16); 7~(20)\Footnote{45}{ % Footnote 45 + Leonardo considers the case of ``one third'' instead of Euclid's + ``a certain fraction,'' but in the case of 20 he concludes that in + the same way the figure may be divided + ``into four or many equal parts.'' + \textit{Cf.}\ Article~28.}; % end footnote 45 +8 (27)\Footnote{46}{ % Footnote 46 + Woepcke~8 may be considered as a part of Leonardo~27 or better as an + unnumbered proposition following Leonardo~25.}; % end footnote 46 +9 (30, 31)\Footnote{47}{ % Footnote 47 + Leonardo's propositions 30--32 consider somewhat more general problems + than Euclid's~9 and 13. \textit{Cf.}\ Articles~30 and 34.}; % end footnote 47 +10~(18); 11~(0); 12~(28)\footnotemark[42]; +13~(32)\footnotemark[47]; 14~(36); +15~(40); 16~(37); 17~(39); 18~(0); 19~(3); 20~(10); 21~(0); +22~(0); 23~(0); 24~(0); 25~(0); 26~(4); 27~(11); 28~(57); +29 (51)\footnotemark[45]; +30~(0); 31~(0); 32~(29); 33~(35); 34~(40)\footnotemark[42]; +35~(0); +36~(0). + +\Section{\textit{Summary}} + +It will be instructive, as a means of comparison, to set +forth in synoptic fashion: (1) the Muhammed-Commandinus +treatise; (2) the Euclid treatise; (3) Leonardo's work. In +(1) and (2) I follow Woepcke closely\Footnote{48}{ % Footnote 48 + Woepcke, pp.~245--246.}. % end footnote 48 + +\medskip + +\Paragraph{SYNOPSIS OF MUHAMMED'S TREATISE}{14} % Paragraph 14 +\phantomsection\pdfbookmark[0]{Summary} + {Summary} +\phantomsection\pdfbookmark[1]{Synopsis of Muhammed's Treatise}{hyperPar14} +\textit{Synopsis of Muhammed's Treatise---} +\renewcommand{\labelenumi}{\Roman{enumi}.} +\renewcommand{\labelenumii}{\Alph{enumii}.} +\renewcommand{\labelenumiii}{\arabic{enumiii}.} +\begin{enumerate} +\item In all the problems it is required to divide the proposed +figure into two parts having a given ratio. +\item The figures divided are: the triangle (props.~1--6); +the parallelogram (11); the trapezium\footnotemark[89] +(8, 12, 13); the quadrilateral +(7, 9, 14--16); the pentagon (17, 18, 22); a pentagon +with two parallel sides (19), a pentagon of which a side is +parallel to a diagonal (20). +% -----File: 026.png---Folio 14------------------------------------------------------ +\item The transversal required to be drawn: +\begin{enumerate} +\item passes through a given point and is situated: +\begin{enumerate} +\item at a vertex of the proposed figure (1, 7, 17); +\item on any side (2, 9, 18); +\item on one of the two parallel sides (8). +\end{enumerate} + +\item is parallel: +\begin{enumerate} +\item to a side (not parallel) (3, 13, 14, 22); +\item to the parallel sides (11, 12, 19); +\item to a diagonal (15, 20); +\item to a perpendicular drawn from a vertex of the +figure to the opposite side (4); +\item to a transversal which passes through a vertex of the figure (5); +\item to any transversal (6, 16). +\end{enumerate} +\end{enumerate} +\item Prop.~10: Being given the segment $\mathit{AB}$ and two +lines which pass through the extremities of this segment and +form with the line $\mathit{AB}$ any angles, draw a line parallel to $AB$ +from one or the other side of $\mathit{AB}$ and such as to produce +a trapezium of given size. + +Prop.~21. Auxiliary theorem regarding the pentagon. +\end{enumerate} + +\bigskip +\Paragraph{COMMANDINUS'S TREATISE}{15} % Paragraph 15 +\phantomsection\pdfbookmark[1]{Commandinus's Treatise}{hyperPar15} +\textit{Commandinus's Treatise}---Appended to the first +published edition of Muhammed's work was a short treatise\Footnote{49}{ % Footnote 49 + Commandinus\footnotemark[11], pp.~54--76.} % end footnote 49 +by Commandinus who said\Footnote{50}{ % Footnote 50 + Commandinus\footnotemark[11], p.~[ii]; Leeke-Serle Euclid, p.~603.} % end footnote 50 +of Muhammed: ``for what things +the author of the book hath at large comprehended in many +problems, I have compendiously comprised and dispatched in +two only.'' This statement repeated by Ofterdinger\Footnote{51}{ % Footnote 51 + Ofterdinger\footnotemark[38], p.~11, note.} % end footnote 51 +and Favaro\Footnote{52}{ % Footnote 52 + Favaro\footnotemark[6], p.~139.} % end footnote 52 +is somewhat misleading. + +The ``two problems'' of Commandinus are as follows: + +``Problem I\@. To divide a right lined figure according to +a proportion given, from a point given in any part of the +ambitus or circuit thereof, whether the said point be taken in +any angle or side of the figure.'' + +``Problem II\@. To divide a right lined figure $\mathit{GABC}$, +% -----File: 027.png---Folio 15------------------------------------------------------ +according to a proportion given, $E$ to $F$, by a right line +parallel to another given line $D$.'' + +But the first problem is divided into 18 cases: 4 for the +triangle, 6 for the quadrilateral, 4 for the pentagon, 2 for the +hexagon and 2 for the heptagon; and the second problem, as +Commandinus treats it, has 20 cases: 3 for the triangle, 7 for +the quadrilateral, 4 for the pentagon, 4 for the hexagon, 2 for +the heptagon. + +\Paragraph{SYNOPSIS OF EUCLID'S TREATISE}{16} % Paragraph 16 +\phantomsection\pdfbookmark[1]{Synopsis of Euclid's Treatise}{hyperPar16} +%%%%%% {Synopsis of Euclid's Treatise} +\textit{Synopsis of Euclid's Treatise---} +\renewcommand{\labelenumi}{\Roman{enumi}.} +\renewcommand{\labelenumii}{\arabic{enumii}.} +\renewcommand{\labelenumiii}{\arabic{enumiii}.} + +\begin{enumerate} +\item The proposed figure is divided: +\begin{enumerate} +\item into two equal parts (1, 3, 4, 6, 8, 10, 12, 14, 16, +19, 26, 28); +\item into several equal parts (2, 5, 7, 9, 11, 13, 15, 17, 29); +\item into two parts, in a given ratio (20, 27, 30, 32, 34, 36); +\item into several parts, in a given ratio (31, 33, 35, 36). +\end{enumerate} + +The construction 1 or 3 is always followed by the construction +of 2 or 4, except in the propositions 3, 28, 29. + +\item The figures divided are: +\renewcommand{\labelenumii}{ } +\begin{enumerate} +\item the triangle (1, 2, 3, 19, 20, 26, 27, 30, 31); +\item the parallelogram (6, 7, 10, 11); +\item the trapezium (4, 5, 8, 9, 12, 13, 32, 33); +\item the quadrilateral (14, 15, 16, 17, 34, 35, 36); +\item a figure bounded by an arc of a circle and two lines (28); +\item the circle (29). +\end{enumerate} + +\renewcommand{\labelenumi}{\Roman{enumi}.} +\renewcommand{\labelenumii}{\Alph{enumii}.} +\renewcommand{\labelenumiii}{\arabic{enumiii}.} +%\renewcommand{\labelenumiiii}{\arabic{enumiiii}.} + +\item It is required to draw a transversal: +\begin{enumerate} +\item passing through a point situated: +\begin{enumerate} +\item at a vertex of the figure (14, 15, 34, 35); +\item on any side (3, 6, 7, 16, 17, 36); +\item on one of two parallel sides (8, 9); +\item at the middle of the arc of the circle (28); +\item in the interior of the figure (19, 20); +\item outside the figure (10, 11, 26, 27); +\item in a certain part of the plane of the figure (12, 13) +\end{enumerate} + +% -----File: 028.png---Folio 16------------------------------------------------------ +\item parallel to the base of the proposed figure (1, 2, +4, 5, 30--33). + +\item parallel to one another, the problem is indeterminate (29). +\end{enumerate} +\item Auxiliary propositions: + +18.~To apply to a given line a rectangle of given size +and deficient by a square. + +21, 22, when $a\centerdot d \gtrless b\centerdot c$, it follows that $a: b\gtrless c: d$; + +%$x\gtrless y z \lessgtr q$ + +23, 24, when $a:b > c:d$, it follows that +\[ (a \mp b) :b > (c \mp d): d;\] + +25, when $a: b < c: d$, it follows that $(a-b): b<(c-d): d$. +\end{enumerate} + +In the synopsis of the last five propositions I have +changed the original notation slightly. + +\Paragraph{ANALYSIS OF LEONARDO'S WORK}{17} % Paragraph 17 +\phantomsection\pdfbookmark[1]{Analysis of Leonardo's Work}{hyperPar17} +%%%%%% {Analysis of Leonardo's Work} +\textit{Analysis of Leonardo's Work}. I have not thought +it necessary to introduce into this analysis the unnumbered +propositions referred to above\footnotemark[42]. + +\begin{enumerate} +\item The proposed figure is divided: +\begin{enumerate} +\item into two equal parts (1--5, 15--18, 23--28, 36--38, +42--46, 53--55, 57); + +\item into several equal parts (6, 7, 9, 13, 14, 19, 21, +33, 47--50, 56); + +\item into two parts in a given ratio (8, 10--12, 20, 29--32, + 34, 39, 40, 51, 52); + +\item into several parts in a given ratio (22, 35, 41). +\end{enumerate} + +The construction 1 or 3 is always followed by the +construction of 2 or 4 except in the propositions 42--46, +51, 54, 57. + +\item The figures divided are: + +the triangle (1--14); + +the parallelogram (15--22); + +the trapezium (23--35); + +the quadrilateral (36--41); + +the pentagon (42--43); + +the hexagon (44); + +the circle and semicircle (45--56); + +a figure bounded by an arc of a circle and two lines (57). + +% -----File: 029.png---Folio 17------------------------------------------------------ +\renewcommand{\labelenumii}{(\roman{enumii})} +\renewcommand{\labelenumiii}{\Alph{enumiii}.} +\renewcommand{\labelenumiv}{\arabic{enumiv}.} + +\item \; +\begin{enumerate} +\item It is required to draw a transversal: +\begin{enumerate} +\item passing through a point situated: +\begin{enumerate} +\item at a vertex of the figure (1, 6, 26, 31, 34, 36, 41--44); +\item on a side not produced (2, 7, 8, 16, 20, 37, 39); +\item at a vertex or a point in a side (40); +\item on one of two parallel sides (24, 25, 27, 30); +\item on the middle of the arc of the circle (53, 55, 57); +\item on the circumference or outside of the circle (45); +\item inside of the figure (3, 10, 15, 17, 46); +\item outside of the figure (4, 11, 12, 18); +\item either inside or outside of the figure (38); +\item either inside or outside or on a side of the figure (32); +\item in a certain part of the plane of the figure (28). +\end{enumerate} +\item parallel to the base of the proposed figure (5, 14, 19, 21--23, 29, 33, 35, 54); +\item parallel to a diameter of the circle (49, 50). +\end{enumerate} +\item It is required to draw more than one transversal +(\textit{a}) through one point (9, 47, 48, 56); (\textit{b}) through +two points (13); (\textit{c}) parallel to one another, the +problem is indeterminate (51). +\item It is required to draw a circle (52). +\end{enumerate} +% -----File: 030.png---Folio 18------------------------------------------------------ +\item Auxiliary Propositions: + +Although not explicitly stated or proved, Leonardo makes +use of four out of six of Euclid's auxiliary propositions\footnotemark[113]. +On the other hand he proves two other propositions which Favaro +does not number: (1) Triangles with one angle of the one +equal to one angle of the other, are to one another as the +rectangle formed by the sides about the one angle is to that +formed by the sides about the equal angle in the other; +(2) the medians of a triangle meet in a point and trisect one +another. +\end{enumerate} + + + +% -----File: 031.png---Folio 19--------------------------------------------------- + +\Chapter{II} % Chapter II +\thispagestyle{empty} % this may be redundant + +\Paragraph{JORDANUS NEMORARIUS}{18} % Paragraph 18 +\phantomsection\pdfbookmark[0]{Abraham Savasorda, Jordanus Nemorarius, + Luca Paciuolo}{hyperPar18} +\textit{Abraham Savasorda, Jordanus Nemorarius, Luca +Paciuolo.}---In earlier articles (10, 11) incidental reference was +made to Leonardo's general indebtedness to previous writers +in preparing his \textit{Practica Geometriae}, and also to the debt +which later writers owe to Leonardo. Among the former, +perhaps mention should be made of Abraham bar Chijja ha +Nasi\Footnote{53}{ % Footnote 53 + That is, Abraham son of Chijja the prince. + \textit{Cf.}\ \textsc{Steinschneider}, + \textit{Bibliotheca Mathematica}, 1896, (2), + \textsc{x}, 34--38, and \textsc{Cantor}, + \textit{Vorlesungen über Geschichte d.~Math.} + \textsc{i}$_3$, 797--800, 907.} % end footnote 53 +of Savasorda and his \textit{Liber embadorum} known through +the Latin translation of Plato of Tivoli. Abraham was a +learned Jew of Barcelona who probably employed Plato of +Tivoli to make the translation of his work from the Hebrew. +This translation, completed in 1116, was published by Curtze, +from fifteenth century MSS., in 1902\Footnote{54}{ % Footnote 54 + \textsc{M.~Curtze}, + ``Urkunden zur Geschichte der Mathematik im Mittelalter und + der Renaissance\ldots'' Erster Teil + (\textit{Abhandlung zur Geschichte der Mathematischen + Wis\-sen\-schaf\-ten\ldots} XII.\ Heft), Leipzig, 1902, pp.~3--183.}. % end footnote 54 +Pages 130--159 of this +edition contain ``capitulum tertium in arearum divisionum +explanatione'' with Latin and German text, and among the +many other propositions given by Savasorda is that of +Proclus-Euclid (= Woepcke 28 = Leonardo 57). Compared +with Leonardo's treatment of divisions Savasorda's seems +rather trivial. But however great Leonardo's obligations to +other writers, his originality and power sufficed to make +a comprehensive and unified treatise. + +Almost contemporary with Leonardo was Jordanus Nemorarius +(d.~1237) who was the author of several works, all +probably written before 1222. +% -----File: 032.png---- Folio 20 ----------------------------------------------------- +Among these is \textit{Geometria vel De Triangulis}\Footnote{55}{ % Footnote 55 + Edited with Introduction by \textsc{Max Curtze}, + \textit{Mitteilungen des Copernicus--Vereins + für Wis\-sen\-schaf\-ten und Kunst zu Thorn}. + \textsc{vi}.\ Heft, 1887. In his discussion + of the second book, \textsc{Cantor} + (\textit{Vorlesungen ü.\ Gesch.\ d.\ Math.} \textsc{ii}$_2$, 75) + is misleading and inaccurate. + One phase of his inaccuracy has been referred to by \textsc{Eneström} + (\textit{Bibliotheca Mathematica}, + Januar, 1912, (3), \textsc{xii}, 62).} % end footnote 55 +in four books. The second book is principally +devoted to problems on divisions: Propositions 1--7 to the +division of lines and Propositions 8, 13, 17, 18, 19 to the +division of rectilineal figures. The enunciations of Propositions +8, 13, 17, 19 correspond, respectively, to Euclid 3, 26, 19, 14 +and to Leonardo 2, 4, 3, 36. But Jordanus's proofs are quite +differently stated from those of Euclid or Leonardo. Both +for themselves and for comparison with the Euclidean proofs +which have come down to us, it will be interesting to reproduce +propositions 13 and 17 of Jordanus. + +``13. \textit{Triangulo dato et puncto extra ipsum signato +lineam per punctum transeuntem designare, que triangulum +per equalia parciatur}'' [pp.~15--16]. + +\CenteredGraphic{80mm}{032} % Graphic in Paragraph 18 + +``Let $\mathit{abc}$ be the triangle and $d$ the point outside but +contained within the lines $\mathit{aef, hbl}$, which are lines dividing +the triangle equally and produced. For if $d$ be taken in any +such place, draw $\mathit{dg}$ parallel to $\mathit{ca}$ meeting $\mathit{cb}$ produced in $g$. +Join $\mathit{cd}$ and find $\mathit{mn}$ such that +\[ \triangle\mathit{cdg}:\triangle\mathit{aec}\; + (= \tfrac{1}{2}\, \triangle \mathit{abc})=\mathit{cg:mn}. \] +% -----File: 033.png---Folio 21------------------------------------------------------ +Then divide $\mathit{cg}$ in $k$ such that +\[\mathit{gk:kc}=\mathit{kc:mn}. \] +Produce $\mathit{dk}$ to meet $\mathit{ca}$ in $p$. Then I say that $\mathit{dp}$ divides the +triangle $\mathit{abc}$ into equal parts. + +For, since the triangle $\mathit{ckp}$ is similar to the triangle $\mathit{kdg}$, +by 4 of sixth\Footnote{56}{ % Footnote 56 + That is, Euclid's \textit{Elements}, \textsc{vi}.~4.} % end footnote 56 +and parallel lines and 15 of first and definitions +of similar areas, +\[ +\triangle\mathit{ckp}:\triangle\mathit{kdg}=\mathit{mn:kg} +\] +by corollary to 17 of sixth\Footnote{57}{ % Footnote 57 + I do not know the MS.\ of Euclid here referred to; but manifestly it is the + Porism of \textit{Elements} \textsc{vi}.~19 which is quoted: + ``If three straight lines be proportional, + then as the first is to the third, so is the figure described on the first to that + which is similar and similarly described on the second.''}. % end footnote 57 +But +\[ +\triangle\mathit{kdg}:\triangle\mathit{cdg}=\mathit{kg:cg}. +\] + +Therefore, by equal proportions, +\begin{DPalign*} + \triangle\mathit{ckp}&:\triangle\mathit{cdg}=\mathit{mn:cg}. \\ + \therefore \triangle\mathit{ckp}&:\triangle\mathit{cdg}= + \triangle\mathit{aec}:\triangle\mathit{cdg}. \\ +\lintertext{And} \triangle\mathit{ckp}&=\triangle\mathit{aec}\; (= + \tfrac{1}{2}\triangle\mathit{abc}) +\end{DPalign*} +by 9 of fifth, and this is the proposition. + +\noindent +And by the same process of deduction we may be led to +an absurdity, namely, that all may equal a part if the point $k$ +be otherwise than between $e$ and $b$ or the point $p$ be otherwise +than between $h$ and $a$; the part cut off must always be either +all or part of the triangle $\mathit{aec}$.'' + +``17. \textit{Puncto infra propositum trigonum dato lineam per +ipsum deducere, que triangulum secet per equalia}'' [pp.~17--18]. + +\CenteredGraphic{80mm}{034} % Graphic in Paragraph 18 + +``Let $\mathit{abc}$ be the triangle and $d$ the point inside and +contained within the part between $\mathit{ag}$ and $\mathit{be}$ which divide +two sides and triangle into equal parts. Through $d$ draw $\mathit{fdh}$ +parallel to $\mathit{ac}$ and draw $\mathit{db}$. +Then by 12 of this book\Footnote{58}{ % Footnote 58 + That is, \textit{De Triangulis}, Book~2, Prop.~12: + ``Data recta linea aliam rectam + inuenire, ad quam se habeat prior sicut quilibet datus triangulus ad quemlibet + datum triangulum'' [p.~15].} % end footnote 58 +draw $\mathit{mn}$ such that +\[ +\mathit{bf:mn}=\triangle\mathit{bdf}:\triangle\mathit{bec}\;(= \tfrac{1}{2}\triangle\mathit{abc}). +\] +% -----File: 034.png---Folio 22--------------------------------------------------- + +\begin{DPalign*} + \lintertext{Also find $\mathit{ty}$ such that} & \\ + \mathit{bf:ty} &= \triangle\mathit{bfh}:\triangle\mathit{bec}. \\ + \lintertext{And since} \triangle\mathit{bfh} >& + \triangle\mathit{bdf},\quad \mathit{mn} > \mathit{ty} +\end{DPalign*} +by 8 and 10 of fifth. +\begin{DPalign*} +&\lintertext{Now} \mathit{bf:bc} = \mathit{bc:ty} +\end{DPalign*} +by corollary to 17 of sixth\Footnote{581}{ % Footnote 581 (was 58a) + Rather is it the converse of this corollary, which is quoted in note~57. It + follows at once, however: +\[ +\mathit{bf:ty}=\triangle\mathit{bfh}:\triangle\mathit{bec}= + \mathit{bf}^2:\mathit{bc}^2,\quad + \therefore \mathit{bf}\centerdot\mathit{ty} = + \mathit{bc}^2 \text{ or }\mathit{bf:bc}=\mathit{bc:ty}. + \]},\DPnote{the comma follows the footnote in main text} % end footnote 581 +and $\triangle\mathit{bfh} < \triangle\mathit{bec}$ since $\mathit{fh, ce}$ +are parallel lines. + + + +\begin{DPalign*} +\lintertext{\indent But} \mathit{bc:ty} > \mathit{bc:mn} +\end{DPalign*} +by second part of 8 of fifth. +\begin{gather*} + \therefore \mathit{bf:bc} > \mathit{bc:mn}; \\ + \therefore \mathit{fc} < \tfrac{1}{4} \mathit{mn} +\end{gather*} +by 6 of this book\Footnote{59}{ % Footnote 59 + ``Cum sit linee breuiori adiecte major proporcio ad compositam, quam + composite ad longiorem, breuiorem quarta longioris minorem esse necesse est + [p.~13].}.\\ % end footnote 59 +% -----File: 035.png---Folio 23------------------------------------------------------ +Add then to the line $\mathit{cf}$, from $f$, a line $\mathit{fz}$, +by 5 of this book\Footnote{60}{ % Footnote 60 + ``Duabus lineis propositis, quarum una sit minor quarta alterius uel equalis, + minori talem lineam adiungere, ut, que adiecte ad compositam, eadem sit composite + ad reliquam propositarum proporcio'' [p.~12].}, % end footnote 60 +such that +\[ \mathit{fz:zc} = \mathit{zc:mn}; \] +and $\mathit{fz}$ will be less than $\mathit{fb}$ by the first part of the premise. +[Supposition with regard to $d$?] + +Join $\mathit{zd}$ and produce it to meet $\mathit{ac}$ in $k$; then I say that +the line $\mathit{zdk}$ divides the triangle $\mathit{abc}$ into equal parts. For +\[ + \triangle\mathit{bdf} : \triangle\mathit{zdf}=\mathit{bf:zf} +\] +by 1 of sixth. +\begin{DPalign*} +\lintertext{\indent But} \triangle\mathit{zdf} : \triangle\mathit{zkc}=\mathit{zf:mn} +\end{DPalign*} +by corollary to 17 of sixth\footnotemark[57] and similar triangles.\\ +\indent Therefore by 1 and by equal proportions +\begin{DPalign*} + \triangle\mathit{bdf} &: \triangle\mathit{zkc}=\mathit{bf:mn}.\\ +\lintertext{\indent But} \triangle\mathit{bdf} &: \triangle\mathit{bec}=\mathit{bf:mn}. +\end{DPalign*} +\indent Therefore by the second part of 9 of fifth +\begin{DPalign*} +\triangle\mathit{zkc} &= \triangle\mathit{bec} = \tfrac{1}{2} \triangle\mathit{abc}." \tag*{\textsc{q.e.f.}\quad} +\end{DPalign*} + +Proposition 18 of Jordanus is devoted to finding the centre +of gravity of a triangle\Footnote{601}{ % Footnote 601 (was 60a) + Archimedes proved (\textit{Works of Archimedes}, Heath ed., 1897, p.~201; + \textit{Opera omnia} iterum edidit J.~L.\ Heiberg, + \textsc{ii}, 150--159, 1913) in Propositions 13--14, Book~\textsc{i} + of ``On the Equilibrium of Planes'' that + \textit{the centre of gravity of any triangle is at + the intersection of the lines drawn from any two angles + to the middle points of the + opposite sides respectively}.} % end footnote 601 +and it is stated in the form of a problem +on divisions. In Leonardo this problem is treated\footnotemark[109] +by showing that the medians of a triangle are concurrent; but +in Jordanus (as in Heron\footnotemark[83]) +the question discussed is, ``to find a +point in a triangle such that when it is joined to the angular +points, the triangle will be divided into three equal parts''(p.~18). + +A much later work, \textit{Summa de Arithmetica Geometria +Proportioni et Proportionalita\ldots} by Luca Paciuolo (b.~about +1445) was published at Venice in 1494\Footnote{61}{ % Footnote 61 + A new edition appeared at Toscolano in 1523, + and in the section which we are + discussing there does not appear to be any material change.}. % end footnote 61 +In the geometrical +section (the second, and separately paged) of the work, +pages 35 \textit{verso}--43 \textit{verso}, problems on divisions of figures are +solved, and in this connection the author acknowledges great +debt to Leonardo's work. Although the treatment is not as +% -----File: 036.png---Folio 24------------------------------------------------------ +full as Leonardo's, yet practically the same figures are employed. +The Proclus-Euclid propositions which have to do +with the division of a circle are to be found here. + +\Paragraph{ARABIAN WRITERS ON DIVISIONS OF FIGURES}{19} % Paragraph 19 +\phantomsection\pdfbookmark[0]{Arabian writers on Divisions of Figures}{hyperPar19} +\textit{``Muhammed Bagdedinus'' and other Arabian writers +on Divisions of Figures.}---We have not considered so far +who ``Muhammed Bagdedinus'' was, other than to quote the +statement of Dee\footnotemark[35] +that he may have been ``that \textit{Albategnus} +whom Copernicus often cites as a very considerable author, or +that \textit{Machomet} who is said to have been Alkindus's scholar.'' +Albategnius or Muhammed b.\ Gâbir b.\ Sinân, Abû `Abdallâh, +el Battânî who received his name from Battân, in Syria, +where he was born, lived in the latter part of the ninth and in +the early part of the tenth century\Footnote{62}{ % Footnote 62 + \textsc{M.~Cantor}, + \textit{Vorlesungen ü.\ Gesch.\ d.~Math}. \textsc{i}$_{3}$, 736.}. % end footnote 62 +El-Kindî (d. about 873) the +philosopher of the Arabians was in his prime about 850\Footnote{63}{ % Footnote 63 + \textsc{M.~Cantor}, + \textit{Vorlesungen ü.~Gesch.\ d.~Math}. \textsc{i}$_{3}$, 718.}. % end footnote 63 +``Alkindus's scholar'' would therefore possibly be a contemporary +of Albategnius. It is probably because of these +suggestions of Dee\Footnote{64}{ % Footnote 64 + \textit{Cf}.\ \textsc{Steinschneider}\footnotemark[12].} % end footnote 64 +that Chasles speaks\Footnote{65}{ % Footnote 65 + \textsc{Chasles}, \textit{Aper\c{c}u historique\ldots} + $3^{e}$ éd., Paris, 1889, p.~497.} % end footnote 65 +of ``Mahomet +Bagdadin, géomètre du $\textrm{x}^e$ siècle.'' + +It would be scarcely profitable to do more than give +references to the recorded opinions of other writers such +as Smith\Footnote{66}{ % Footnote 66 + \textsc{T.~Smith}, \textit{Vitae quorumdam\ldots virorum}, +1707, p.~56. \textit{Cf}.\ notes 14, 15.}, % end footnote 66 +Kästner\Footnote{67}{ % Footnote 67 + \textsc{A.~G.~Kästner}, + \textit{Geschichte der Mathematik\ldots}, + Band \textsc{i}, Göttingen, 1796, p.~273. + See also his preface to \textsc{N.~Morville}, + \textit{Lehre von der geometrischen und ökonomischen + Vertheilung der Felder, nach der dänischen Schrift bearbeitet + von J.~W.~Christiani, begleitet mit einer Vorrede\ldots + von A.~G.\ Kästner}, Göttingen, 1793.}, % end footnote 67 +Fabricius\Footnote{68}{ % Footnote 68 + \textsc{J.~A.\ Fabricius}, + \textit{Bibliotheca Graeca\ldots } Editio nova. Volumen quartum, + Hamburgi, \textsc{mdcclxxxv}, p.~81.}, % end footnote 68 +Heilbronner\Footnote{69}{ % Footnote 69 + \textsc{J.~C.\ Heilbronner}, + \textit{Historia Matheseos universae\ldots} + Lipsiae, \textsc{mdccxlii}, p.~438, 163--4.}, % end footnote 69 +Montucla\Footnote{70}{ % Footnote 70 + \textsc{J.~F.\ Montucla}, + \textit{Histoire des mathématiques\ldots}\ + éd.\ nouv. Tome \textsc{i}, An \textsc{vii}, p.~216.}, % end footnote 70 +Hankel\Footnote{71}{ % Footnote 71 + \textsc{H.~Hankel}, + \textit{Zur Geschichte der Math.\ in Alterthum u.\ Mittelalter}, + Leipzig, 1874, p.~234.}, % end footnote 71 +Grunert\Footnote{72}{ % Footnote 72 + \textsc{J.~A.~Grunert}, + \textit{Math. Wörterbuch \ldots von G.S. Klügel, fortgesetzt + von C.~B.\ Mollweide und beendigt von J.A. Grunert\ldots}Erste + Abteilung, die reine Math., + fünfter Theil, erster Band, Leipzig, 1831, p. 76.}--- whose % end footnote 72 +results Favaro summarizes\Footnote{73}{ % Footnote 73 + \textsc{Favaro}, pp.~141--144.}. % end footnote 73 + +The latest and most trustworthy research in this connection +seems to be due to Suter who first surmised\Footnote{74}{ % Footnote 74 + \textsc{H.~Suter}, + ``Die Mathematiker und Astronomen der Araber und ihre Werke'' + (\textit{Abh.\ z.\ Gesch.\ d.\ Math.\ Wiss.} + \textsc{x}. Heft, Leipzig, 1900), p.~202, No.~517.} % end footnote 74 +that the author +% -----File: 037.png---Folio 25------------------------------------------------------ +of the Dee book On Divisions was +Mu\d{h}.\ b.\ Mu\d{h}.\ el-Ba\.{g}dâdî +who wrote at Cairo a table of sines for every minute. A little +later\Footnote{75}{ % Footnote 75 + \textsc{H.~Suter}, \textit{idem}, ``Nachträge und Berichtigung'' + (\textit{Abh.\ z.\ Gesch.\ d.\ Math.\ + Wiss.}\ \textsc{xiv}.\ Heft, 1902), p.~181; + also \textit{Bibliotheca Mathematica}, + \textsc{iv}$_3$, 1903, pp.~22--27.}, % end footnote 75 +however, Suter discovered facts which led him to +believe that the true author was +Abû Mu\d{h}ammed b.\ `Abdelbâqî el-Ba\.{g}dâdî +(d.~1141 at the age of over 70 years) to whom +an excellent commentary on Book \textsc{x} of the \textit{Elements} +has been ascribed. Of a MS.\ by this author Gherard of Cremona +(1114--1187) may well have been a translator. + +Euclid's book \textit{On Divisions} was undoubtedly the ultimate +basis of all Arabian works on the same subject. We have +record of two or three other treatises. + +1. \b{T}âbit b.\ Qorra (826--901) translated parts of the works +of Archimedes and Apollonius, revised Ishâq's translation of +Euclid's \textit{Elements} and \textit{Data} and also revised the work +\textit{On Divisions of Figures} translated by an anonymous +writer\Footnote{76}{ % Footnote 76 + \textsc{H.~Suter}, ``Die Mathematiker\ldots,'' pp.~34--38.}. % end footnote 76 + +2. Abû Mu\d{h}.\ el-Hasan b.\ `Obeidallâh b.\ Soleimân b.\ +Wahb (d.~901) was a distinguished geometer who wrote +``A Commentary on the difficult parts of the work of Euclid'' +and ``The Book on Proportion.'' Suter thinks\Footnote{77}{ % Footnote 77 + \textsc{H.~Suter}, + ``Die Mathematiker\ldots,'' pp.\ 48 and 211, note 23.} % end footnote 77 +that another +reading is possible in connection with the second title, and +that it may refer to Euclid's work \textit{On Divisions}. + +3. Abû'l Wefâ el-Bûz\v{g}ânî (940--997) one of the greatest +of Arabian mathematicians and astronomers spent his later +life in Bagdad, and is the author of a course of Lectures on +geometrical constructions. Chapters \textsc{vii--ix} of the Persian +form of this treatise which has come down to us in roundabout +fashion were entitled: ``On the division of triangles,'' ``On +the division of quadrilaterals,'' ``On the division of circles'' +respectively. Chapter \textsc{vii} and the beginning of Chapter \textsc{viii} +are, however, missing from the Bibliothèque nationale Persian +MS.\ which has been described by Woepcke\Footnote{78}{ % Footnote 78 + \textsc{F.~Woepcke}, + ``Recherches sur l'histoire des Sciences mathématiques chez + les orientaux, d'après des traités inedits Arabes et Persans. + Deuxième article. + Analyse et extrait d'un recueil de constructions géométriques + par Aboûl Wafâ,'' + \textit{Journal asiatique}, + Fevrier--Avril, 1855, (5), V, 218--256, 309--359; reprint, Paris, + 1855, pp.~89.}. % end footnote 78 +This MS., +which gives constructions without demonstrations, was made +from an Arabian text, by one Abû Ishâq b.\ `Abdallâh with +% -----File: 038.png---Folio 26------------------------------------------------------ +the assistance of four pupils and the aid of another translation. +The Arabian text was an abridgment of Abû'l Wefâ's lectures +prepared by a gifted disciple. + +The three propositions of Chapter \textsc{ix}\Footnote{79}{ % Footnote 79 + \textsc{F.~Woepcke}, + \textit{idem}, pp.~340--341; reprint, pp.~70--71.} % end footnote 79 +are practically +identical with Euclid (Woepcke) 28, 29. +In Chapter \textsc{viii}\Footnote{80}{ % Footnote 80 + \textsc{F.~Woepcke}, + \textit{idem}, pp.~338--340; reprint, pp.~68--70.} % end footnote 80 +there are 24 propositions. About a score are given, in substance, +by both Leonardo and Euclid. + +In conclusion, it may be remarked that in Chapter \textsc{xii} +of Abû'l Wefâ's work are 9 propositions, with various solutions, +for dividing the surface of a sphere into equiangular and +equilateral triangles, quadrilaterals, pentagons and hexagons. + +\Paragraph{HERON OF ALEXANDRIA, APOLLONIUS OF PERGA}{20} % Paragraph 20 +\phantomsection\pdfbookmark[0]{Practical applications}{hyperPar20} + +\textit{Practical applications of the problems +On Divisions of Figures; +the \ItalicGreekSnippet{metrik'a} +of Heron of Alexandria}.---The popularity +of the problems of Euclid's book \textit{On Divisions} among Arabians, +as well as later in Europe, was no doubt largely due to the +possible practical application of the problems in the division +of parcels of land of various shapes, the areas of which, +according to the Rhind papyrus, were already discussed in +empirical fashion about 1800 \textsc{b.~c.} In the first century before +Christ\Footnote{81}{ % Footnote 81 + This date is uncertain, + but recent research appears to place it not earlier + than 50~\textsc{b.~c.}\ nor later than 150~\textsc{a.~d.} + \textit{Cf.}~\textsc{Heath}, + \textit{Thirteen Books of Euclid's Elements}, + \textsc{i}, 20--21; or perhaps better still, Article + ``Heron~5'' by K.~Tittel in Pauly-Wissowa's + \textit{Real-Encyclopädie der class. Al\-tert\-ums\-wis\-sen\-schaf\-ten}, + \textsc{viii}, Stuttgart, + 1913, especially columns 996--1000.} % end footnote 81 +we find that Heron of Alexandria dealt with the +division of surfaces and solids in the third book of his +\textit{Surveying} (\ItalicGreekSnippet{metrik'a})\Footnote{82}{ % Footnote 82 + \textit{Heronis Alexandrini opera quae supersunt omnia}, + Vol.~\textsc{iii}, \textit{Rationes Dimetiendi et commentatio + Dioptrica recensuit Hermannus Schoene}, Lipsiae, + \textsc{mcmiii}. Third + book, pp.~140--185. \textit{Cf.}\ \textsc{Cantor}, + \textit{Vorlesungen\ldots}, \textsc{i}$_3$, 380--382.}. % end footnote 82 +Although the enunciations of the +propositions in this book are, as a whole, similar\Footnote{83}{ % Footnote 83 + Only two are exactly the same: + \textsc{ii--iii} (= Euclid~30), \textsc{vii} (= Euclid~32), + the problem considered in \textsc{x} is practically Euclid~27 (Art.~48), + while \textsc{xviii} is closely + related to Euclid~29 (Art.~50). + In \textsc{xix} Heron finds in a triangle a point such that + when it is joined to the angular points, the triangle will be + divided into three equal parts. The divisions of solids of which + Heron treats are of a sphere (\textsc{xxiii}) and the division in a + given ratio, by a plane parallel to the base, of a Pyramid + (\textsc{xx}) and of a Cone (\textsc{xxi}). + For proof of Proposition \textsc{xxiii}: + \textit{To cut a sphere by a plane so that the volumes of the segments + are to one another in a given ratio}, Heron refers to Proposition~4, + Book~\textsc{ii} of ``On the Sphere and Cylinder'' of Archimedes; the + third proposition in the same book of the Archimedean work is + (Heron~\textsc{xvii}): + \textit{To cut a given sphere by a plane so that the surfaces + of the segments may have to one another a given ratio}. + (\textit{Works of Archimedes}, Heath ed., 1897, pp.~61--65; + \textit{Opera omnia} + iterum edidit J.~L.\ Heiberg, \textsc{i}, 184--195, 1910.) + + Propositions \textsc{ii} and \textsc{vii} are also given in Heron's +%[DPStyle Greek: peri dioptras] + \ItalicGreekSnippet{per`i di'optras} + (Schoene's edition, pp.~278--281). + \textit{Cf}.\ ``Extraits des Manuscrits relatifs à la géométrie grecs'' +% typo corrected: single quote should be double. + par A.~J.~C.\ Vincent, + \textit{Notices et extraits des Manuscrits de la bibliothèque impériale}, + Paris, 1858, \textsc{xix}, pp.\ 157, 283, 285.} % end footnote 83 +to those +% -----File: 039.png---Folio 27------------------------------------------------------ +in Euclid's book \textit{On Divisions}, Heron's discussion consists +almost entirely of ``analyses'' and approximations. For +example, \textsc{ii}: ``To divide a triangle in a given ratio by a +line drawn parallel to the base''---while Euclid gives the +general construction, Heron considers that the sides of the +given triangle have certain known numerical lengths and +thence finds the approximate +distance of the angular points of +the triangle to the points in the sides where the +required line +parallel to the base intersects them, because, as he expressly +states, in a field with uneven surface it is difficult to draw a +line parallel to another. +Most of the problems are discussed +with a variety of numbers although theoretical analysis sometimes +enters. +Take as an example Proposition \textsc{x}\Footnote{84}{ % Footnote 84 + \textsc{Heron}, \textit{idem}, p.~160f.}: % end footnote 84 +``\textit{To divide +a triangle in a given ratio by a line drawn from a point in a +side produced.}'' + +\WrappedGraphic{60mm}{039}{}{r} % Graphic in Paragraph 20 + +``Suppose the construction made. Then the ratio of triangle +$\mathit{AEZ}$ to quadrilateral $\mathit{ZEB\Gamma}$ is known; +also the ratio of the triangle $\mathit{AB\Gamma}$ to +the triangle $\mathit{AZE}$. But the triangle +$\mathit{AB\Gamma}$ is known, therefore so is the +triangle $\mathit{AZE}$. Now $\Delta$ is given. +Through a known point $\Delta$ there is +therefore drawn a line which, with two +lines $\mathit{AB}$ and $\mathit{A\Gamma}$ intersecting in $A$, +encloses a known area. + +Therefore the points $E$ and $Z$ are +given. This is shown in the second +book of \textit{On Cutting off a Space}. Hence the required proof. + +If the point $\Delta$ be not on $\mathit{B\Gamma}$ but anywhere this will make +no difference.'' + +\Paragraph{HERON OF ALEXANDRIA, APOLLONIUS OF PERGA}{21} % Paragraph 21 +\phantomsection\pdfbookmark[0]{Apollonius of Perga}{hyperPar21} + +\textit{Connection between Euclid's book On Divisions, +Apollonius's treatise On Cutting off a Space and a Pappus-lemma +to Euclid's book of Porisms}.---Although the name of +the author of the above-mentioned work is not given by +Heron, the reference is clearly to Apollonius's lost work. +According to Pappus it consisted of two books which contained +124 propositions treating of the various cases of the +% -----File: 040.png---Folio 28------------------------------------------------------ +following problem: \textit{Given two coplanar straight lines +$\mathit{A_1P_1,B_2P_2}$, on which $A_1$ and $B_2$ are fixed points; it is required to +draw through a fixed point $\Delta$ of the plane, a transversal $\mathit{\Delta ZE}$ +forming on $\mathit{A_1P_1, B_2P_2}$ the two segments $\mathit{A_1Z, B_2E}$ such that +$\mathit{A_1Z}\centerdot\mathit{B_2E}$ is equal to a given rectangle.} + +Given a construction for the particular case when +$\mathit{A_1P_1,B_2P_2}$ meet in $A$, +and when $A_1$ and $B_2$ coincide with $A$---Heron's +reasoning becomes clear. The solution of this particular +case is practically equivalent to the solution of Euclid's +Proposition 19 or 20 or 26 or 27. References to restorations +of Apollonius's work are given in note 111. + +To complete the list of references to writers before 1500, +who have treated +\WrappedGraphic{70mm}{040}{11}{r} % Graphic in Paragraph 21 +of Euclid's +problems here under discussion, +I should not fail to mention the +last of the 38 lemmas which +Pappus gives as useful in connection +with the 171 theorems +of Euclid's lost book of +\textit{Porisms: +Through a given point $E$ in $\mathit{BD}$ +produced to draw a line cutting +the parallelogram $\mathit{AD}$ such that +the triangle $\mathit{Z\Gamma H}$ is equal to the parallelogram $\mathit{AD}$.} + +After ``Analysis'' Pappus has the following + +``Synthesis. Given the parallelogram $\mathit{AD}$ and the point $E$. Through $E$ +draw the line $\mathit{EZ}$ such that the +rectangle $\mathit{\Gamma Z\centerdot \Gamma H}$ equals twice the +rectangle $\mathit{A\Gamma\centerdot \Gamma D}$. Then +according to the above analysis [which contains a reference +to an earlier lemma discussed a little later\footnotemark[88] +in this book] the +triangle $\mathit{Z\Gamma H}$ equals the parallelogram $\mathit{AD}$. Hence $\mathit{EZ}$ +satisfies the problem and is the only line to do so\Footnote{85}{ % Footnote 85 + Pappus ed.\ by Hultsch, Vol.~2, Berlin, 1877, pp.~917--919. + In Chasles's restoration + of Euclid's \textit{Porisms}, + this lemma is used in connection with ``Porism \textsc{clxxx}: + Given two lines $\mathit{SA,SA'}$, a point $P$ and a space $\nu$: + points $I$ and $J'$ can be found in + a line with $P$ and such that if one take on $\mathit{SA}$, + $\mathit{SA'}$ two points $m$, $m'$, bound by the + equation $\mathit{Im}\centerdot\mathit{J'm'}= \nu$, + the line $\mathit{mm'}$ will pass through a given point.'' + \textit{Les trois livres de Porismes d'Euclide}, + Paris, 1860, p.~284. See also the restoration by + R.~Simson, pp.~527--530 of ``De porismatibus tractatus,'' + \textit{Opera quaedam reliqua\ldots} + Glasguae, \textsc{m.dcc.lxxvi.}}.'' % end footnote 85 + +The tacit assumption here made, that the equivalent of +a proposition of Euclid's book \textit{On Divisions (of Figures)} was +well known, is noteworthy. + + +% -----File: 041.png---Folio 29------------------------------------------------------ + +%% Beginning of Chapter III +%% \Paragraph is re-defined so that it no longer sets the center heading of the +%% odd numbered pages. This will be done by the \Proposition command in Chapter III. +\renewcommand{\Paragraph}[2]{% + \paragraph{\hspace*{\parindent}#2.}% + \fancyhead[RE]{[\MyParMarks}% + \fancyhead[LO]{\MyParMarks]}% + \renewcommand{\CurrentPar}{#2}% + \extramarks{\CurrentProp}{#2}% + \label{par#2}% +} + + +\Chapter{III} % Chapter III +%% Setting this chapter as \raggedbottom alleviates many of the +%% problems associated with unpleasantly large amounts of whitespace. +\raggedbottom +\phantomsection\pdfbookmark[0]{Restoration of Euclid's MS.}{hyperCh3} + +\bigskip + +\LargeCenteredNote{``\textit{The Treatise of Euclid on the Division} (\textit{of plane Figures}).''} + +\thispagestyle{empty} +%%%%%%\newpage + +% -----File: 042.png---Folio 30------------------------------------------------------ + +\bigskip\bigskip\bigskip + +\Proposition{1} % Proposition 1 +\Paragraph{}{22} % Paragraph 22 +\textit{``To divide\Footnote{86}{ % Footnote 86 + Literally, the original runs, according to Woepcke, + ``We propose to ourselves to demonstrate how to divide, etc.'' + I have added all footnotes except those attributed to Woepcke.} % end footnote 86 +a given triangle into equal parts by +a line parallel to its base.''} +[Leonardo 5, p.~119, ll.~7--9.] + +Let $\mathit{abg}$ be the given triangle which it is required to bisect by a line +parallel to $\mathit{bg}$. Produce $\mathit{ba}$ to +$d$ till $\mathit{ba = 2ad}$. Then in $\mathit{ba}$ find a point +$e$ such that +\[ +\mathit{ba:ae} = \mathit{ae:ad}. +\] +Through $e$ draw $\mathit{ez}$ parallel to $\mathit{bg}$; then +the triangle $\mathit{abg}$ is divided by the line +$\mathit{ez}$ into two equal parts, of which one +is the triangle $\mathit{aez}$, and the other the +quadrilateral $\mathit{ebgz}$. + +\WrappedGraphic{45mm}{042}{}{r} % Graphic in Proposition 1 + +\SmallLeftNote{Leonardo then gives three proofs, but as the +first and second are practically equivalent, I shall +only indicate the second and third.} + +I. When three lines are proportional, as the first is to the +third so is a figure on the first to the similar and similarly +situated figure described on the second [\textsc{vi}.~19, ``Porism'']\Footnote{87}{% +Throughout the restoration I have added % Footnote 87 + occasional references of this kind to Heath's edition of Euclid's + \textit{Elements}; \textsc{vi}.~19 refers to Proposition~19 of + Book~\textsc{vi}. \textit{Cf.}\ note~57.}. % end footnote 87 +\[ +\therefore\mathit{ba:ad}=\text{figure on }\mathit{ba}: \text{similar and similarly situated +figure on }\mathit{ae}. +\] +\begin{DPalign*} +\lintertext{\indent Hence} \mathit{ba:ad} &= \triangle\mathit{abg} : \triangle\mathit{aez}\\ + &= 2:1. +\end{DPalign*} +\[ + \therefore \triangle \mathit{abg} = 2 \triangle\mathit{aez}. +\] +\begin{DPalign*} +\lintertext{\indent II.} \mathit{ba:\;} \mathit{ae} &= \mathit{ae:ad}. \\ +\therefore \mathit{ba} & \centerdot\mathit{ad}=\mathit{ae}^2, +\end{DPalign*} +% -----File: 043.png---Folio 31------------------------------------------------------ +and since $\mathit{ad}$ is one-half of $\mathit{ba}$, +\[ +ba^2 = 2\,ae^2. +\] +And since $\mathit{bg}$ is parallel to $\mathit{ez}$, +\begin{flalign*} +&& ba : ae &= ga : az. &&\\ +&& \therefore ba^2 : ae^2 &= ga^2 : az^2. \qquad &&\text{ [\textsc{vi}.~22]} \quad\\ +&\text{\indent But } & ba^2 &= 2\,ae^2. &&\\ +&& \therefore ga^2 &= 2\,az^2. &&\\ +&\text{\indent Then } +& ba\centerdot ag &= 2\,ae\centerdot az, &&\text{ [\textsc{vi}.~22]} \quad\\ +&& \therefore \triangle\mathit{abg} &= 2\,\triangle\mathit{aez}\,\footnotemark[88]. && +\end{flalign*} + +\SmallLeftNote{Then follows a numerical example.} + +\Footnotetext{88}{The theorem here assumed is enunciated by % Footnote 88 (long) + Leonardo (p.~111, ll.~24--27) as follows: +%% The graphic is part of the footnote. There are several of these. +\WrappedGraphic{49mm}{043}{}{r} % Graphic in Proposition 1 + \textit{Et si á trigono + recta protracta fuerit secans duo latera trigonj, que cum ipsis + duobus lateribus faciant trigonum habentem angulum unum comunem cum + ipso trigono, erit proportio unius trigoni ad alium, sicut facta ex + lateribus continentibus ipsum angulum}. This is followed by the + sentence ``Ad cuius rei euidentiam.'' Then come the construction and + proof: + + Let $\mathit{abc}$ be the given triangle and $\mathit{de}$ the line + across it, meeting the sides $\mathit{ca}$ and $\mathit{cb}$ in the points + $d$, $e$, respectively. I say that + \[ + \triangle\mathit{abc}:\triangle\mathit{dec} = + \mathit{ac}\centerdot\mathit{cb:dc}\centerdot\mathit{ce}. + \] + + \textit{Proof:} + To $\mathit{ac}$ apply the triangle $\mathit{afc}= \triangle\mathit{dec}$. [\textsc{i}.~44] + + Since the triangles $\mathit{abc, afc}$ are of the same altitude, + \begin{flalign*} + && \mathit{bc} &: \mathit{fc} = \triangle\mathit{abc} : \triangle\mathit{afc}. + &&\text{ [\textsc{vi}.~1]\quad}\\ + &\text{\indent But} + & \mathit{bc} &: \mathit{fc} = \mathit{ac}\centerdot\mathit{bc}:\mathit{ac}\centerdot\mathit{fc}, + &&\text{ [\textsc{v}.~15]\quad} \\ + && \therefore \triangle\mathit{abc} &: \triangle\mathit{afc} = + \mathit{ac}\centerdot\mathit{bc} : \mathit{ac}\centerdot\mathit{fc},\qquad\qquad&&\\ + &\text{and since } \triangle\mathit{dec} = \triangle\mathit{acf},&&&&\\ + && \triangle\mathit{acb} &: \triangle\mathit{dce} = + \mathit{ac}\centerdot\mathit{bc}:\mathit{ac}\centerdot\mathit{cf}. && + \end{flalign*} + + Again, since the triangles $\mathit{acf, dce}$ are equal and have a common angle, as in + the fifteenth theorem of the sixth book of Euclid, the sides are mutually proportional. + \begin{gather*} + \therefore \mathit{ac:dc}=\mathit{ce:cf},\qquad + \therefore \mathit{ac}\centerdot\mathit{cf}=\mathit{dc}\centerdot\mathit{ce},\\ + \therefore \triangle\mathit{acb}:\triangle\mathit{dce} = + \mathit{ac}\centerdot\mathit{cb}:\mathit{dc}\centerdot\mathit{ce}.\\ + \end{gather*} + \hfill ``quod oportebat ostendere.''\qquad + + It is to be observed that the Latin letters are used with the above figure. + This suggests the possibility of the proof being due to Leonardo. + + The theorem is assumed in Euclid's proof of proposition~19 (Art.~40) + and it occurs, directly or indirectly, in more than one of his + works. A proof, depending on the proposition that the area of a + triangle is equal to one-half the product of its base and altitude, + is given by Pappus (pp.~894--897) in connection with one of his + lemmas for Euclid's book of \textit{Porisms}: \textit{Triangles + which have one angle of the one equal or supplementary to one angle + of the other are in the ratio compounded of the +% -----File: 044.png---Folio 32------------------------------------------------------ + ratios of the sides about the equal or supplementary angles}. + (\textit{Cf.}\ \textsc{R.~Simson}, + ``De Porismatibus Tractatus'' in + \textit{Opera quaedam reliqua\ldots} 1776, + p.~515 ff.---\textsc{P.~Breton} (de Champ), + ``Recherches nouvelles sur les porismes d'Euclide,'' + \textit{Journal de mathématiques pures et appliquées}, + \textsc{xx}, 1855, p.~233 ff. Reprint, p.~25 ff.---\textsc{M.~Chasles}, + \textit{Les trois livres de Porismes d'Euclide\ldots} + Paris, 1860, pp.\ 247, 295, 307.) + + The first part of this lemma is practically equivalent to either + (1) [\textsc{vi}.~23]: + \textit{Equiangular parallelograms have to one another the ratio + compounded of the ratio of their sides}; + or (2) the first part of Prop.~70 of the \textit{Data} + (\textit{Euclidis Data\ldots} edidit H.~Menge, Lipsiae, 1896, p.~130f.): + \textit{If in two equiangular parallelograms the sides + containing the equal angles have a given ratio to one another} + [i.~e.\ one side in one to one side in the other], + \textit{the parallelograms themselves will also have a given ratio to + one another.} \textit{Cf.}\ \textsc{Heath}, \textit{Thirteen Books of + Euclid's Elements}, \textsc{ii}, 250. + + The proposition is stated in another way by Pappus\footnotemark[85] + (p.~928) who proves that + \textit{a parallelogram is to an equiangular parallelogram as the + rectangle contained by the adjacent sides of the first is to the + rectangle contained by the adjacent sides of the second.} + + The above theorem of Leonardo is precisely the first of those + theorems which Commandinus adds to \textsc{vi}.~17 of \emph{his} + edition of Euclid's \textit{Elements} and concerning which he writes + ``à nobis elaborata'' (``fatti da noi''): + \textit{Euclidis Elementorum Libri XV\ldots + A Federico Commandino\ldots}Pisauri, + \textsc{mdlxxii}, p.~81 + \textit{recto (Degli Elementi d' Euclide libri quindici con gli + scholii antichi tradotti prima in lingua latina da } + M.~Federico Commandino + \textit{da Urbino, et con commentarii illustrati, et hora d' ordine + dell' istesso transportati nella nostra vulgare, et da lui riveduti.} + In Urbino, \textsc{m.d.lxxv}, p.~88 + \textit{recto}). } % end footnote 88 (long) + +\Proposition{2} % Proposition 2 +\Paragraph{}{23} \textit{``To divide a given triangle into three equal % Paragraph 23 +parts by two lines parallel to its base.''} +[Leonardo 14, p.~122, l.~8.] + +\WrappedGraphic{45mm}{044}{}{r} % Graphic in Proposition 2 + +Let $\mathit{abg}$ be the given triangle with base $\mathit{bg}$. +Produce $\mathit{ba}$ +to $d$ till $\mathit{ba} = 3\mathit{ad}$, and produce $\mathit{ad}$ to +$e$ till $\mathit{ad}=\mathit{de}$; then $ae = \frac{2}{3} ba$. Find +$\mathit{az}$, a mean proportional between $\mathit{ba}$ +and $\mathit{ad}$, and $\mathit{ia}$ a mean proportional +between $\mathit{ba}$ and $\mathit{ae}$. Then through +$z$ and $i$ draw $\mathit{zt}$, $\mathit{ik}$ parallel to $\mathit{bg}$ +and I say that the triangle $\mathit{abg}$ is +divided into three equal parts of +which one is the triangle $\mathit{azt}$, another +the quadrilateral $\mathit{zikt}$, the third +the quadrilateral $\mathit{ibgk}$. + +\textit{Proof:} Since +\begin{gather*} + \mathit{ba:az} = \mathit{az:ad}, \\ + \mathit{ba:ad} = \triangle\mathit{abg}:\triangle\mathit{azt}, \tag*{[\textsc{vi}.~19, Porism]} +\end{gather*} +for these triangles are similar. +% -----File: 045.png---Folio 33------------------------------------------------------ +\begin{DPalign*} +\lintertext{\indent Now } ba = 3ad; \quad\therefore &\triangle\mathit{abg} = + 3\triangle\mathit{azt}. \\ + \therefore\triangle\mathit{azt} &= \tfrac{1}{3}\triangle\mathit{abg}. && \\ +\lintertext{\indent Again,} ba:ia &= ia:ae; +\end{DPalign*} +$\therefore ba: ae = \triangle$ on $\mathit{ea}$: similar and similarly situated $\triangle$ on $\mathit{ai}$. + +But triangles $\mathit{aik, abg}$ are similar and similarly described +on $\mathit{ai}$ and $\mathit{ab}$; and +\begin{align*} +\mathit{ea:ab} &= 2: 3. \\ + \therefore\triangle\mathit{aik} &= \tfrac{2}{3}\triangle\mathit{abg}. +\end{align*} + +And since $\triangle\mathit{azt} = \tfrac{1}{3}\triangle\mathit{abg}$, there remains the quadrilateral +$\mathit{zikt} = \tfrac{1}{3} \triangle\mathit{abg}$. +We see that the quadrilateral $\mathit{ibgk}$ will be the +other third part; hence the triangle $\mathit{abg}$ has been divided into +three equal parts; ``quod oportebat facere.'' + +\SmallLeftNote{Leonardo continues:} +``Et sic per demonstratos modos omnia +genera trigonorum possunt diuidi in quatuor partes uel +plures.'' +\SmallLeftNote{Cf.~note 45.} + +\Proposition{3} % Proposition 3 +\Paragraph{}{24} \textit{``To divide a given triangle into two equal % Paragraph 24 +parts by a line drawn from a given +point situated on one of the sides +of the triangle.''} +[Leonardo 1, 2, +p.~110, l.~31; p.~111, ll.~41--43.] + +\WrappedGraphic{45mm}{045}{}{r} % Graphic in Proposition 3 + +Given the triangle $\mathit{bgd}$; if $a$ +be the middle point of $\mathit{gd}$ the line +$\mathit{ba}$ will divide the triangle as required; +either because the triangles +are on equal bases and of +the same altitude [\textsc{i}.~38; Leonardo +1], or because +\begin{flalign*} + && \triangle\mathit{bgd}: \triangle\mathit{bad} +&= \mathit{bd}\centerdot\mathit{dg}: \mathit{bd}\centerdot\mathit{da}\footnotemark[88].\qquad &&\\ +\text{\indent Whence } && \triangle\mathit{bgd} &= 2\triangle\mathit{bad}. && +\end{flalign*} + +% -----File: 046.png---Folio 34------------------------------------------------------ + +But if the given point be not the middle point of any side, +let $\mathit{abg}$ be the triangle and +$d$ the given point nearer to $b$ +than to $g$. Bisect $\mathit{bg}$ at $e$ and +draw $\mathit{ad,ae}$. +Through $e$ draw $\mathit{ez}$ parallel to $\mathit{da}$; join $\mathit{dz}$. +Then the triangle $\mathit{abg}$ is bisected by $\mathit{dz}$. + +\WrappedGraphic{60mm}{046}{7}{r} % Graphic in Proposition 3 + +\textit{Proof:} Since +\[ +\mathit{ad}\parallel\mathit{ez},\quad \triangle\mathit{adz} = \triangle\mathit{ade}. +\] + +To each add $\triangle\mathit{abd}$. Then +\begin{DPalign*} + \text{quadl.}\mathit{abdz} &= + \triangle\mathit{abd} + \triangle\mathit{ade}, \\ + &= \triangle\mathit{abe}. \\ + \lintertext{\indent But} \triangle\mathit{abe} &= \tfrac{1}{2} \triangle\mathit{abg}; \\ + \therefore\, \text{quadl.}\mathit{abdz} &= \tfrac{1}{2} \triangle\mathit{abg}; +\end{DPalign*} +and the triangle $\mathit{zdg}$ is the other half of the triangle $\mathit{abg}$. +Therefore the triangle $\mathit{abg}$ is divided into two equal parts by +the line $\mathit{dz}$ drawn from the point $d$; + +\hfill ``ut oportebat facere.''\qquad\qquad + +\SmallCenteredNote{Then follows a numerical example.} + +\Proposition{4} % Proposition 4 +\Paragraph{}{25} \textit{``To divide a given % Paragraph 25 + trapezium\Footnote{89}{% % Footnote 89 + Here, and in what follows, this word is used to refer to a + quadrilateral two of whose sides are parallel.} % end footnote 89 +into two equal parts by a line parallel to its base.''} +[Leonardo~23, p.~125, ll.~37--38.] + +Let $\mathit{abgd}$ be the given trapezium with parallel sides +$\mathit{ad,bg,ad}$ being the lesser. It is required to bisect the trapezium +by a line parallel to the base $\mathit{bg}$. +Let $\mathit{gd,ba}$, produced, meet in a point $e$. +Determine $z$ such that +\[ +ze^2=\tfrac{1}{2}(eb^2+ea^2).\footnotemark[90] % Footnotemark 90 +\] +\Footnotetext{90}{ % Footnotetext 90 + The point $z$ is easily found by constructions which twice make + use of \textsc{i}.~47.} % end of footnote 90 +Through $z$ draw $\mathit{zi}$ parallel to $\mathit{gb}$. +I say that the trapezium $\mathit{abgd}$ is divided into two equal +parts by the line $\mathit{zi}$ parallel to the base $\mathit{bg}$. + +% -----File: 047.png---Folio 35------------------------------------------------------ + +\WrappedGraphic{60mm}{047}{16}{r} % Graphic in Proposition 4 + +\textit{Proof:} For since +\[ +2ze^2 = eb^2 + ea^2, +\] +and all the triangles are similar, +\[ +2\triangle\mathit{ezi} = \triangle\mathit{ebg} + \triangle\mathit{ead}. + \tag*{\text{[\textsc{vi}. 19]}} +\] + +From the triangle $\mathit{ebg}$ take +away the triangle $\mathit{ezi}$. Then +\[ +\triangle\mathit{ezi} = \text{quadl.}\mathit{zbgi} + \triangle\mathit{eda}. +\] + +And taking away from the +equals the triangle $\mathit{eda}$, we get +\[ +\text{quadl.}\mathit{ai} = \text{quadl.}\mathit{zg}. +\] + +Therefore the trapezium +$\mathit{abgd}$ is divided into two equal +parts by the line $\mathit{zi}$ parallel to its base. \qquad\textsc{q.~o.~f.} + +\SmallCenteredNote{A numerical example then follows.} + +\Proposition{5} % Proposition 5 +\Paragraph{}{26} \textit{``And we divide the given trapezium into three % Paragraph 26 +equal parts as we divide the triangle, by a construction analogous to +the preceding construction\footnotemark[91].''} +[Leonardo~33, p.~134, ll.~14--15.] + +Let $\mathit{abgd}$ be the trapezium with parallel sides $\mathit{ad,bg}$ and +other sides $\mathit{ba, gd}$ produced to meet in $e$. Let $\mathit{zti}$ be a line +such that +\[ +\mathit{zi:it} = eb^2:ea^2.\footnotemark[92] +\] + +\Footnotetext{91}{ % Footnote 91 + It is to be noticed that Leonardo's discussion of this proposition + is hardly ``analogous to the preceding construction'' which is + certainly simpler than if it had been similar to that of Prop.~5. + A construction for Prop.~4 along the same lines, + which may well have been Euclid's method, would obviously be as follows: + + Let $\mathit{zti}$ be a line such that + \[ + \mathit{zi:it} = eb^2:ea^2. + \] + + Divide $\mathit{tz}$ into two equal parts, $\mathit{tk,kz}$. Find $m$ such that + \[ + em^2:eb^2 = ki:zi. + \] + + Then $m$ leads to the same solution as before. [For, in brief, + \begin{align*} + em^2 &= eb^2\left(\frac{ki}{zi}\right) + = eb^2\left(\frac{\frac{zt}{2} + ti}{zi}\right) + = \frac{eb^2}{2}\left(\frac{zi + it}{zi}\right) + = \frac{eb^2}{2}\left(\frac{ea^2 + eb^2}{eb^2}\right)\\ + &= \tfrac{1}{2}\left(ea^2 + eb^2\right).\text{]} + \end{align*}} % end footnote 91 + +\Footnotetext{92}{ % Footnote 92 + From \textsc{vi.}~19, Porism, it is clear that the construction + here is to find a line $x$ which is a third proportional to $\mathit{eb}$ + and $\mathit{ea}$. Then $zi:it = eb:x$.} % end footnote 92 + +% -----File: 048.png---Folio 36------------------------------------------------------ + +Divide $\mathit{tz}$ into three equal parts $\mathit{tk, kl, lz}$. + +Find $m$ and $n$ in $\mathit{be}$ such that +\begin{DPalign*} + em^2 : eb^2 &= ik : zi, \\ + \lintertext{and} en^2 : eb^2 &= il : zi. +\end{DPalign*} + +Through $m$ and $n$ draw $\mathit{mo,np}$ parallel to the base $\mathit{bg}$. +Then I say that the quadrilateral $\mathit{ag}$ is divided into three +equal parts: $\mathit{ao,mp,ng}$. +\begin{DPalign*} + \lintertext{\indent \textit{Proof:} For} + eb^2 : ea^2 & = \triangle\mathit{ebg} : \triangle\mathit{ead}. + \tag*{[\text{\textsc{vi.}~19}]\quad} \\ + \therefore zi : it &= \triangle\mathit{ebg} : \triangle\mathit{ead}. + \tag*{\ldots\ldots\ldots\ldots\ldots[1]\quad} +\end{DPalign*} +\CenteredGraphic{60mm}{048} % Graphic in Proposition 5 +\begin{DPalign*} + \lintertext{\indent But } \mathit{zi:ik} &= eb^2 : em^2, \\ + \therefore \mathit{zi:ik} &= \triangle\mathit{ebg} : \triangle\mathit{emo}. + \tag*{\dots[2]\quad}\\ + \lintertext{\indent So also} + \mathit{zi:il} &= \triangle\mathit{ebg} : \triangle\mathit{enp}. + \tag*{\dots[3]\quad}\\ + \lintertext{Whence} + \mathit{it:tk} &= \triangle\mathit{ead} : \text{quadl.}\mathit{ao},\footnotemark[93] \\ + \lintertext{and therefore} + \mathit{tk:kl} &= \text{quadl.}\mathit{ao} : \text{quadl.}\mathit{mp}.\footnotemark[94] +\end{DPalign*} +\Footnotetext{93}{ % Footnote 93 + This may be obtained by combining [1] and [2], + and applying \textsc{v.}\ 11, 16, 17.} % end footnote 93 +\Footnotetext{94}{ % Footnote 94 + Relations [1], [2] and [3] may be employed, as in the preceding, + to give, + \[ + \mathit{it:kl} = \triangle\mathit{ead} : \text{quadl.}\mathit{mp}; + \] + combining this with $\mathit{it:tk} = \triangle\mathit{ead} : \text{quadl.}\mathit{ao}$, + we get the required result, + \[ + \mathit{tk:kl} = \text{quadl.}\mathit{ao} : \text{quadl.}\mathit{mp}. + \]} % end footnote 94 +% -----File: 049.png---Folio 37------------------------------------------------------ +\begin{DPgather*} + \lintertext{\indent But} \mathit{tk} = \mathit{kl}. \quad \therefore \text{quadl.}\mathit{ao} + = \text{quadl.}\mathit{mp}. \\ + \lintertext{So also} \mathit{kl:lz} = \text{quadl.}\mathit{mp}: \text{ quadl.}\mathit{ng}; \\ + \lintertext{and} \mathit{kl} = \mathit{lz}. \quad \therefore \text{quadl.}\mathit{mp} + = \text{quadl.}\mathit{ng}. +\end{DPgather*} + +Therefore the quadrilateral is divided into equal quadrilaterals +$\mathit{ao, mp, ng}$; \hfill ``ut prediximus.'' + +\SmallCenteredNote{Then follows a numerical example.} + +\Proposition{6} % Proposition 6 +\Paragraph{}{27} \textit{``To divide a parallelogram into two equal parts % Paragraph 27 +by a straight line drawn from a given point situated on one of the +sides of the parallelogram.''} +[Leonardo 16, p.~123, ll.~30--31.] + +Let $\mathit{abcd}$ be the parallelogram and $i$ any point in the side $\mathit{ad}$. +Bisect $\mathit{ad}$ +\WrappedGraphic{70mm}{049}{}{r} % Graphic in Proposition 6 +in $f$ and $\mathit{bc}$ in $e$. +Join $\mathit{fe}$. Then the parallelogram +$\mathit{ac}$ is divided into equal +parallelograms $\mathit{ae}$, $\mathit{fc}$ on equal bases. + +Cut off $\mathit{eh=fi}$. Join $\mathit{hi}$. +Then this is the line required. + +\SmallLeftNote{Leonardo gives two proofs:} + +I. Let $\mathit{hi}$ meet $\mathit{fe}$ in $k$. +Then [$\triangle$s $\mathit{fki, hke}$ are equal; +add to each the pentagon $\mathit{kfabh}$, etc.] + +II. Since $\mathit{ae, fc}$ are \Rect \,s, +$\mathit{af=be}$ and $\mathit{fd=ec}$. But +\begin{gather*} %%\PGnote this could just be a gather + \mathit{fd} = \tfrac{1}{2} \mathit{ad}. \\ + \therefore \mathit{fd} = \mathit{af} = \mathit{ec}. +\end{gather*} +\begin{DPgather*} + \lintertext{\indent And since } \mathit{fi} = \mathit{he}, \quad \mathit{ai} = \mathit{ch}. \\ + \lintertext{\indent So also } + di = bh, \text{ and } hi \text{ is common.} \\ + \therefore \text{quadl.}\mathit{iabh} = \text{quadl.}\mathit{ihcd}.\footnotemark[95] +\end{DPgather*} + +\Footnotetext{95}{ % Footnote 95 + The first rather than the second proof is Euclidean. + There is no proposition of the \textit{Elements} with regard to the + equality of quadrilaterals whose sides and angles, taken in the same + order, are equal. Of course the result is readily deduced from + \textsc{i}.~4, if we make certain suppositions with regard to order. + Cf.\ the proof of Prop.~10.} % end of footnote 95 + +% -----File: 050.png---Folio 38------------------------------------------------------ + +Similarly if the given point were between $a$ and $f$, [etc.; +or on any other side]. And thus a parallelogram can be +divided into two equal parts by a straight line drawn from a +given point situated on any one of its sides. + +\Proposition{7} % Proposition 7 +\Paragraph{}{28} \textit{``To cut off a certain fraction from a given % Paragraph 28 +parallelogram by a straight line drawn from a given point situated +on one of the sides of the parallelogram.''} +[Leonardo 20 (the case where the fraction is one-third), p.~124, ll.~24--26.] + +Let $\mathit{abcd}$ be the given parallelogram. Suppose it be +required to cut off a third of this parallelogram, by a straight +line drawn from $i$, in the side $\mathit{ad}$. + +\CenteredGraphic{70mm}{050} % Graphic in Proposition 7 + +\SmallCenteredNote{(The figure here is a combination of two in the original.)} + +Trisect $\mathit{ad}$ in $e$ and $f$ and through $e$, $f$ draw $\mathit{eg, fh}$ parallel +to $\mathit{dc}$; +[then these lines trisect the \Rect\,. +If the point $i$ be in +the line $\mathit{ad}$, at either $e$ or $f$, then the problem is solved. But +if it be between $a$ and $e$, +draw $\mathit{ik}$ to bisect the \Rect \,$\mathit{ah}$ +(Prop.~6), etc. Similarly if $i$ were between $e$ and $f$, or +between $f$ and $d$]. + +\SmallLeftNote{After finishing these cases Leonardo concludes:} + +``eodem modo potest \DPtypo{omnem}{omne} +paralilogramum diuidi in +quatuor uel plures partes equales\footnotemark[45].'' + +\small +The construction in this proposition is limited to the case where ``a certain +fraction'' is the reciprocal of an integer. But more generally, if the fraction +were $\mathit{m:n}$ (the ratio of the lengths of two given lines), +we could proceed in a +very similar way: Divide $\mathit{ad}$ in $e$, internally, +so that $\mathit{ae : ed = m:n-m \quad (n > m)}$. +In $\mathit{ad}$ cut off $\mathit{ef}=\mathit{ae}$ and +through $f$ draw $\mathit{fh}$ parallel to $\mathit{ab}$. +Then, as before, the problem is reduced to Proposition~6. + +If the point $e$ should fall at $i$ or in the interval $\mathit{ai}$ the +part cut off from the +parallelogram by the required line would be in the form of a triangle which +might be determined by \textsc{i}.~44. +\normalsize + +% -----File: 051.png---Folio 39------------------------------------------------------ + +\Proposition{8} % Proposition 8 +\Paragraph{}{29} \textit{``To divide a given trapezium into two equal % Paragraph 29 +parts by a straight line drawn from a given point situated on the +longer of the sides of the trapezium.''} +[Part of Leonardo~27\Footnote{96}{ % Footnote 96 + Leonardo~27: ``Quomodo quadrilatera duorum laterum equidistantium + diuidantur á puncto dato super quodlibet latus ipsius'' + [p.~129, ll.~2--3]. \textit{Cf}.\ note 46.}, % end footnote 96 +p.~127, ll.~2--3.] + +\SmallLeftNote{ +This enunciation means, apparently, ``from a given point situated on the +longer of the [parallel] sides.'' At any rate Leonardo gives constructions for +the cases when the given point is on any side. These I shall take up +successively. The figure is made from more than one of Leonardo's, and +there is a slight change in the lettering.} + +\CenteredGraphic{110mm}{051} % Graphic in Proposition 8 + +Let $\mathit{ad}$ be the shorter of the parallel sides $\mathit{ad, bg}$, which +are bisected in $t$ and $k$ respectively. Join $\mathit{tk}$. Then if $\mathit{bt,gt}$ +be joined, [it is clear, from triangles on equal bases and +between the same parallels, that $\mathit{tk}$ bisects the trapezium]. +[This is Leonardo~24, p.~126, l.~31.] + +\SmallLeftNote{Next consider the given point as any point on the shorter side} +[Leonardo~25, p.~127, ll.~2--3]. + +First let the point be at the angle $a$. Cut off $\mathit{kl}$ in $\mathit{kg}$, +equal to $\mathit{at}$. +Join $\mathit{al}$, meeting $\mathit{tk}$ in $m$; then the quadrilateral +is divided as required by $\mathit{al}$. For [the triangles $\mathit{atm, mkl}$ are +equal in all respects, etc.]. + +Similarly if $d$ were the given point; in $\mathit{kb}$ cut off $\mathit{kn}$ equal +to $\mathit{td}$, and $\mathit{dn}$ divides the quadrilateral into two equal parts +which is proved as in the preceding case. + +% -----File: 052.png---Folio 40------------------------------------------------------ + +[Were the given point anywhere between $a$ and $t$ the +other end of the bisecting line would be between $k$ and $l$. +Similarly if the given point were between $t$ and $d$, the corresponding +point would be between $k$ and $n$.] + +\SmallLeftNote{Although not observed by Favaro, Leonardo now considers:} + +If the given point be in the side $\mathit{bg}$; either $l$, or $n$, or a +point between $l$ and $n$, then the above construction is at once +applicable. + +Suppose, however, that the given point were at $b$ or in +the segment $\mathit{bn}$, at $g$ or in the segment $\mathit{lg}$. +First consider the given point at $b$. +Join $\mathit{bd}$ and through $n$ draw $\mathit{nc}$ parallel +to $\mathit{bd}$ to meet $\mathit{gd}$ in $c$. Join $\mathit{bc}$. +Then $\mathit{bc}$ bisects the trapezium. +For [$\mathit{abnd}$ is half of the trapezium $\mathit{ag}$, and the triangle $\mathit{bnd}$ +equals the triangle $\mathit{bdc}$ etc.]. + +Similarly from a given point between $b$ and $n$, a line could +be drawn meeting $\mathit{gd}$ between $c$ and $d$, and dividing the +quadrilateral into two equal parts. + +So also from $g$ a line $\mathit{gf}$ could be drawn [etc.]; and +similarly for a given point between $g$ and $l$. + +\SmallLeftNote{Leonardo then concludes (p.~127, ll.~37--40):} + +``Jam ostensum est quomodo in duo equa quadrilatera +duorum equidistantium laterum diuidi debeant á linea protracta +ab omni dato puncto super lineas equidistantes ipsius; +nunc uero ostendamus quomodo diuidantur á linea egrediente +á dato puncto super reliqua latera.'' + +\SmallLeftNote{ +This is overlooked by Favaro, though implied in his 27 [Leonardo, p.~129, l.~4]. +I may add Leonardo's discussion of the above proposition although it +does not seem to be called for by Euclid.} + +\WrappedGraphic{60mm}{052}{}{r} % Graphic in Proposition 8 + +Let the point be in the side $\mathit{gd}$. For $g$ or $c$ or $d$ or any +point between $c$ and $d$ the above +constructions clearly suffice. Let +us, then, now consider the given +point $h$ as between $c$ and $g$. +Draw the line $\mathit{iz}$ parallel to $\mathit{gb}$ +to bisect the trapezium (Prop.~4). +Suppose $h$ were between $g$ and $i$. +Join $\mathit{zh}$. Through $i$ draw $\mathit{ik}$ +parallel to $\mathit{hz}$, and meeting $\mathit{ab}$ +in $k$. + +\SmallLeftNote{(The lettering of the original figure is somewhat changed.)} + +% -----File: 053.png---Folio 41------------------------------------------------------ +Join $\mathit{hk}$, then [this is the line required; since +\[ + \triangle\mathit{izh} = \triangle\mathit{kzh}, \text{ etc.]}. +\] + +[Similarly if $h$ were between $i$ and $d$.] + +[So also for points on the line $\mathit{ab}$.] + +\Proposition{9} % Proposition 9 +\Paragraph{}{30} \textit{``To cut off a certain fraction from a given % Paragraph 30 +trapezium by a straight line drawn from a given point situated on the +longer side of the trapezium.''} +[Leonardo 30, 31\Footnote{97}{ % Footnote 97 + As 30, Favaro quotes, + ``Per rectam protractam super duo latera equidistantia + quadrilaterum abscisum in data aliqua proportione dividere''; as 31: + ``Divisionem in eadem proportione ab angulis habere.''}, % end of footnote 97 +p.~133, ll.~17--19, 31.] + +\small +I shall interpret ``longer side'' as in Proposition~8, and lead up to the +consideration of any given point on $\mathit{bg}$ after discussing the cases of points on +the shorter side $\mathit{ad}$. +\normalsize + +\CenteredGraphic{95mm}{053} % Graphic in Proposition 9 + +\SmallCenteredNote{(This figure is made from three of Leonardo's.)} + +Suppose it be required to divide the trapezium in the ratio +$\mathit{ez:zi}\,$\Footnote{98}{ % Footnote 98 + Here, as well as in 15 and 36, Leonardo introduces the representation of + numbers by straight lines, and in considering these lines he invariably writes the + word number in connection with them; + \textit{e.~g.}\ `number $\mathit{ez}$: number $\mathit{zi}$,' not $\mathit{ez:zi}$. + Euclidean MSS.\ of the \textit{Elements}, + Books \textsc{vii} to \textsc{ix}, adopt this same method. + In what follows, I shall use the abbreviated form.}. % end of footnote 98 + +Divide $\mathit{ad}$, $\mathit{bg}$ in the points $t$, $k$, respectively, such that +\[ + \mathit{at:td} = \mathit{ez:zi} = \mathit{bk:kg}. +\] +% -----File: 054.png---Folio 42------------------------------------------------------ +Join $\mathit{tk}$. +Then by joining $\mathit{bt}$ and $\mathit{gt}$ [it is easily seen by \textsc{vi}.~1 +and \textsc{v}.~12, that the trapezium $\mathit{ag}$ is +divided by $\mathit{tk}$ in the ratio $\mathit{ez:zi}$]. + +If the given point be at $a$ or $d$, +make $\mathit{kl}=\mathit{at}$ and $\mathit{gn}=\mathit{bl}$. +Join $\mathit{al,dn}$. [Adding the quadrilateral $\mathit{ak}$ to the congruent +triangles with equal sides $\mathit{at,kl}$, we find $\mathit{al}$ divides the +trapezium in the required ratio. Then from \textsc{vi}.~1, $\mathit{dn}$ does +the same.] + +\SmallLeftNote{ +As in Proposition 8, for any point $t'$ between $a$ and $t$, or $t$ and $d$, we have +a corresponding point $k'$ between $l$ and $k$ or $n$ and $k$, such that the line $t'k'$ +divides the trapezium in the given ratio. + +If the given point be in $\mathit{bg}$ at $l$ or $n$ or between $l$ and $n$, the above +reasoning suffices.} + +Suppose however that the given point were at $b$. Join $\mathit{bd}$. +Through $n$ draw $\mathit{nc}$ parallel to $\mathit{bd}$. +Join $\mathit{bc}$. Then $\mathit{bc}$ divides +the trapezium in the required ratio. Similarly for the point $g$ +and for any point between $b$ and $n$, or between $g$ and $l$. + +\SmallLeftNote{Some of the parts which I have filled in above are covered by the general +final statement:} +``nec non et diuidemus ipsum quadrilaterum ab +omni puncto dato super aliquod laterum ipsius\ldots\ldots'' +\SmallLeftNote{(Page 134, ll.~10--11. Compare Proposition 13.)} + +\Proposition{10} % Proposition 10 +\Paragraph{}{31} \textit{``To divide a parallelogram into two equal parts % Paragraph 31 +by a straight line drawn from a given point outside the parallelogram.''} +[Leonardo 18, p.~124, ll.~5--7.] + +\CenteredGraphic{90mm}{054} % Graphic in Proposition 10 + +Let $\mathit{abcd}$ be the given parallelogram and $e$ the point outside. +Join $\mathit{bd}$ and bisect it in $g$. +Join $\mathit{eg}$ meeting $\mathit{bc}$ in $k$ +% -----File: 055.png---Folio 43------------------------------------------------------ +and produce it to meet $\mathit{ad}$ in $f$. Then the parallelogram has +been divided into two equal parts by the line drawn through $e$, +as may be proved by superposition; and one half is the quadrilateral +$\mathit{fabk}$, +the other, the quadrilateral $\mathit{fkcd}$\,\Footnote{99}{ % Footnote 99 + The proof also follows from the equality of the triangles + $\mathit{fgd,bgk}$, by \textsc{i}.~26 and of the triangles + $\mathit{abd,bdc}$ by + \textsc{i}.~4. This problem is possible for all positions of the + point $e$.}. % end footnote 99 + +\newpage +\Proposition{11} % Proposition 11 +\Paragraph{}{32} \textit{``To cut off a certain fraction from a % Paragraph 32 +parallelogram by a straight line drawn from a given point +outside of the parallelogram.''} + +\small +This proposition is not explicitly formulated by Leonardo; but the general +method he would have employed seems obvious from what has gone before. + +Suppose it were required to cut off one-third of the given parallelogram $\mathit{ac}$ +by a line drawn through a point $e$ outside of the parallelogram. Then by the +method of Proposition 7, form a parallelogram two-thirds of $\mathit{ac}$. There are four +such parallelograms with centres $g_1, g_2, g_3, g_4$. Lines $l_1, l_2, l_3, l_4$ through +each one of these points and $e$ will bisect a parallelogram (Proposition 10). + +There are several cases to consider with regard to the position of $e$ but it +may be readily shown that, in one case at least, there is a line $l_i(i=1, 2, 3, 4)$, +which will cut off a third of the parallelogram $\mathit{ac}$. + +Similarly for one-fourth, one-fifth, or any other fraction such as $\mathit{m:n}$ which +represents the ratio of lengths of given lines. +\normalsize + +\Proposition{12} % Proposition 12 +\Paragraph{}{33} \textit{``To divide a given trapezium into two equal % Paragraph 33 +parts by a straight line drawn from a point which is not situated +on the longer side of the trapezium. It is necessary that the point +be situated beyond the points of concourse of the two sides of the +trapezium.''} +[Leonardo 28, p.~129, ll.~2--4, and another, unnumbered\Footnote{100}{ % Footnote 100 + As Leonardo 28 Favaro gives, +``Qualiter quadrilatera duorum laterum +equidistantium \DPtypo{dividi}{diuidi} +debeant a dato puncto extra figuram'' +and entirely ignores the paragraph headed, +``De diuisione eiusdem generis, qua quadrilaterorum +per rectam transeuntem per punctum datum infra ipsum'' +[p.~131, ll.~13--14].}.] % end footnote 100 + +\Proposition{13} % Proposition 13 +\Paragraph{}{34} \textit{``To cut off a certain fraction from a % Paragraph 34 +(parallel-) trapezium by a straight line which passes through a +given point lying inside or outside the trapezium but so that a +straight line can be drawn through it cutting both the parallel +% -----File: 056.png---Folio 44------------------------------------------------------ +sides of the trapezium}\Footnote{101}{ % Footnote 101 + The final clauses of Propositions 12 and 13, in Woepcke's rendering, + are the same. I have given a literal translation in Proposition~12. + Heath's translation and interpretation (after Woepcke) are given in 13. + Concerning 12 and 13 Woepcke adds the following note: + ``Suppose it were required to cut off the $n$th part of the + trapezium $\mathit{ABDC}$; make $\mathit{A\alpha}$ and + $\mathit{C\gamma}$ respectively equal + to the $n$th parts of $\mathit{AB}$ and of $\mathit{CD}$; + then $\mathit{A\alpha\gamma C}$ will be the $n$th part of the trapezium, + for $\mathit{\gamma\alpha}$ produced + will pass through the intersection of $\mathit{CA}$, $\mathit{DB}$ produced. + Now to draw through a given point $E$ the transversal which cuts off + a certain fraction of the trapezium, join the middle point $\mu$ of + the segment $\mathit{\alpha\gamma}$, and the point $E$, by a line; + this line $\mathit{EFG}$ will be the transversal required to be drawn, + since the triangle $\mathit{\alpha F\mu}$ + equals the triangle $\mathit{\gamma G\mu}$. + +\CenteredGraphic{110mm}{056} % Graphic in Proposition 13 +% again, the graphic is part of the footnote!} + + ``But when the given point is situated as $E'$ or $E''$ such + that the transversal drawn through $\mu$ no longer meets the two + parallel sides but one of the parallel sides and one of the two + other sides, or the other two sides; then the construction indicated + is not valid since $\mathit{CG'\mu\gamma}$ is not equal to $\mathit{BF'\mu\alpha}$. + It appears that this is the idea which the text is intended to express. + The `points of concourse' are the vertices where a parallel side + and one of the two other sides intersect; and the expression + `beyond' refers to the movement of the transversal represented as + turning about the point $\mu$.''}.'' % end footnote 101 +[Part of Leonardo 32\Footnote{102}{ % Footnote 102 + ``Quadrilaterum [trapezium] ab omni puncto dato super aliquod laterum + ipsius, et etiam ab omni puncto dato infra, + uel extra diuidere in aliqua data proportioni.''}% % end footnote 102 +, p.~134, ll.~11--12.] + +\SmallLeftNote{We first take up Leonardo's discussion of Proposition 12.} + +In the figure of Proposition 8, suppose $\mathit{al}$ to be produced +in the directions of the points $e$ and $r$; +$\mathit{tk}$ in the directions of $q$ and $v$, +$\mathit{dn}$ of $z$ and $h$, +$\mathit{cb}$ of $i$ and $o$, +$\mathit{gf}$ of $s$ and $p$. +Then for [any such exterior points +$e$, $q$, $z$, $i$, $s$, $r$, $v$, $h$, $o$, $p$, +lines are drawn bisecting the trapezium]. + +If the given point, $x$, were anywhere in the section of the +plane above $\mathit{ad}$ and between $\mathit{ea}$ and $\mathit{dz}$, +the line joining $x$ to $m$ would [by the same reasoning as in Proposition~8] +bisect the trapezium. Similarly for all points below $\mathit{nl}$ and between +% -----File: 057.png---Folio 45------------------------------------------------------ +$\mathit{hn}$ and $\mathit{lr}$ +[\ldots\ldots so also for all points within the triangles $\mathit{amd, nml}$]. + +\small +This seems to be all that Euclid's Proposition 12 calls for. But just as +Leonardo considers Proposition 8 for the general case with the given point +anywhere on the perimeter of the trapezium, so here, he discusses the constructions +for drawing a line from any point inside or outside of a trapezium +to divide it into two equal parts. + +Leonardo does not give any details of the discussion of Euclid's Proposition +13, but after presentation of the cases given in Proposition 9 +concludes: +\normalsize +``et diuidemus ipsum quadrilaterum ab omni puncto +dato super aliquod laterum ipsius, et etiam ab omni puncto +dato infra uel extra'' +\small +[Leonardo 32, p.~134, ll.~10--12]. + +From Leonardo's discussion in Propositions 8, 9, 12, not only are the +necessary steps for the construction of 13 (indicated in the Woepcke note +above\footnotemark[101]) +evident, but also those for the more general cases, not considered +by Euclid, where restrictions are not imposed on the position of the given +point. +\normalsize + +\Proposition{14} % Proposition 14 +\Paragraph{}{35} \textit{``To divide a given quadrilateral into two % Paragraph 35 +equal parts by a straight line drawn from a given vertex of the +quadrilateral.''} [Leonardo 36, p.~138, ll.~10--11.] + +Let $\mathit{abcd}$ be the quadrilateral and $a$ the given vertex. +Draw the diagonal $\mathit{bd}$, meeting the diagonal $\mathit{ac}$ in $e$. +If $\mathit{be}$, $\mathit{ed}$ +are equal, [$\mathit{ac}$ divides the quadrilateral as required]. + +If $\mathit{be}$ be not equal to $\mathit{ed}$, +make $\mathit{bz} = \mathit{zd}$. + +Draw $\mathit{zi}\parallel\mathit{ac}$ to meet $\mathit{dc}$ in $i$. +Join $\mathit{ai}$. Then the quadrilateral +$\mathit{abcd}$ is divided as required by the line $\mathit{ai}$. + +\CenteredGraphic{75mm}{057} % Graphic in Proposition 14 + +\emph{Proof}: Join $\mathit{az}$ and $\mathit{zc}$. Then the triangles $\mathit{abz, azd}$ are +respectively equal to the triangles $\mathit{cbz, cdz}$. + +% -----File: 058.png---Folio 46------------------------------------------------------ + +Therefore the quadrilateral $\mathit{abcz}$ is one-half the quadrilateral $\mathit{abcd}$. + +And since the triangles $\mathit{azc,aic}$ are on the same base and +between the same parallels $\mathit{ac,zi}$, they are equal. + +To each add the triangle $\mathit{abc}$. + +Then the quadrilateral $\mathit{abcz}$ is equal to the quadrilateral +$\mathit{abci}$. +But the quadrilateral $\mathit{abcz}$ is one-half of the quadrilateral $\mathit{abcd}$. +Therefore $\mathit{abci}$ is one-half of the quadrilateral +$\mathit{abcd}$; \hfill ``ut~oportet.''\qquad\qquad + +\Proposition{15} % Proposition 15 +\Paragraph{}{36} \textit{``To cut off a certain fraction from a given % Paragraph 36 +quadrilateral by a line drawn from a given vertex of the +quadrilateral.''} [Leonardo 40, p.~140, ll.~36--37.] + +Let the given fraction be as $\mathit{ez:zi}$, and let the +quadrilateral be $\mathit{abcd}$ and +the given vertex $d$. +Divide $\mathit{ac}$ in $t$ such that +\[ + \mathit{at:tc} = \mathit{ez:zi}. +\] +If $\mathit{bd}$ pass through $t$ [then $\mathit{bd}$ is +the line required]. + +\WrappedGraphic{60mm}{058}{}{r} % Graphic in Proposition 15 + +But if $\mathit{bd}$ do not pass through +$t$ it will intersect either $\mathit{ct}$ or $\mathit{ta}$; +let it intersect $\mathit{ct}$. Join $\mathit{bt}$, $\mathit{td}$. + +Then +\[ + \text{quadl.}\mathit{tbcd} : \text{quadl.}\mathit{tbad}=\mathit{ct:ta}=\mathit{ez:zi}. +\] + +Draw $\mathit{tl}$ parallel to the diagonal $\mathit{bd}$, and join $\mathit{dl}$. +Then the quadrilaterals $\mathit{lbcd, tbcd}$ are equal and the construction has +been made as required; for +\[ + \mathit{ct:ta}=\mathit{ez:zi} = \text{quadl.}\mathit{lbcd} : \triangle\mathit{dal}. +\] + +And if $\mathit{bd}$ intersect $\mathit{ta}$ [a similar construction may be given +to divide the given quadrilateral, by a line through $d$, into a +quadrilateral and triangle in the required ratio]. + +\SmallLeftNote{Leonardo then gives the construction for dividing a quadrilateral in +a given ratio by a line drawn through a point which divides a side of the +quadrilateral in the given ratio.} + +% -----File: 059.png---Folio 47------------------------------------------------------ + +\Proposition{16} % Proposition 16 +\Paragraph{}{37} \textit{``To divide a given quadrilateral into two % Paragraph 37 +equal parts by a straight line drawn from a given point situated on +one of the sides of the quadrilateral.''} +[Leonardo 37, p.~138, ll.~28--29.] + +Let $\mathit{abcd}$ be the given quadrilateral, $e$ the given point. +Divide $\mathit{ac}$ into two equal parts by the line $\mathit{dt}$ [Prop.~14]. +Join $\mathit{et}$. The line $\mathit{et}$ either is, or is not, +parallel to $\mathit{dc}$. + +\CenteredGraphic{90mm}{059} % Graphic in Proposition 16 + +\SmallCenteredNote{(Two of Leonardo's figures are combined in one, here.)} + +If $\mathit{et}$ be parallel to $\mathit{dc}$, join $\mathit{ec}$. +Then the quadrilateral $\mathit{ac}$ [is bisected by the line $\mathit{ec}$, etc.]. + +If $\mathit{et}$ be not parallel to $\mathit{dc}$, +draw $\mathit{dz} \parallel \mathit{et}$. Join $\mathit{ez}$. +Then $\mathit{ac}$ [is bisected by the line $\mathit{ez}$, etc.]. + +\SmallLeftNote{ +Leonardo does not consider the case of failure of this construction, namely +when $\mathit{dz}$ falls outside the quadrilateral. Suppose in such a case that the +problem were solved by a line joining $e$ to a point $z'$ (not shown in the figure) +on $\mathit{dc}$. Through $t$, draw $\mathit{tt'} \parallel \mathit{cd}$. +Join $\mathit{ct'}$. +Then $\triangle\mathit{ct'd} = \triangle\mathit{ctd}= \triangle\mathit{edz'}$. +Whence $\triangle \mathit{et'c} = \triangle\mathit{ez'c}$, +or $\mathit{t'z'} \parallel \mathit{ce}$. +Therefore from $t'$, $z'$ may be found and the +solution in this case is also possible, indeed in more than one way, but it is +not in Euclid's manner to consider this question.} + +Should the diagonal $\mathit{db}$ bisect the quadrilateral $\mathit{ac}$, +the discussion is similar to the above. + +But if the line drawn from $d$ to bisect the quadrilateral +meet the side $\mathit{ab}$ in $i$, +draw $\mathit{bk}$ bisecting the quadrilateral $\mathit{ac}$. + +If $k$ be not the given point, it will be between $k$ and $d$ or +between $k$ and $a$. +% -----File: 060.png---Folio 48------------------------------------------------------ + +In the first case join $\mathit{be}$ and through $k$ draw +$\mathit{kl}\parallel\mathit{eb}$. +Join $\mathit{el}$ [then $\mathit{el}$ is the required bisector]. + +\CenteredGraphic{120mm}{160} % Graphic in Proposition 16 + +If the point $e$ be between $a$ and $k$ [a similar construction +with the line through $k$ parallel to $\mathit{be}$, and meeting $\mathit{bc}$ in $m$, +leads to the solution by the line $\mathit{em}$]. + +\CenteredGraphic{75mm}{260} % Graphic in Proposition 16 + +Were $e$ at the middle of a side such as $\mathit{ab}$, +draw $\mathit{dz}\parallel\mathit{ab}$ +and bisect $\mathit{dz}$ in $i$. Join $\mathit{ei}$, $\mathit{ci}$ and $\mathit{ec}$. +Through $i$ draw $\mathit{it}\parallel\mathit{ec}$. +Join $\mathit{et}$; then $\mathit{et}$ [bisects the quadrilateral + $\mathit{ac}$, +since $\triangle\mathit{itc}=\triangle\mathit{ite}$, +etc.]. + +\SmallLeftNote{If $\mathit{dz}$ were to fall outside the quadrilateral, +draw from $c$ the parallel to $\mathit{ba}$; +and so on.} + +% The page breaks generated by LaTeX often lead to unusual +% blocks of white space. It was found experimentally that forcing +% a page break at this point made the overall appearance much better. +% I realize that this is a brittle fix. +% Perhaps this will prove unneccessary in the future.} +\newpage +\Proposition{17} % Proposition 17 +\Paragraph{}{38} \textit{``To cut off a certain fraction from a % Paragraph 38 +quadrilateral by a straight line drawn from a given point situated on +one of the sides of the quadrilateral.''} +[Leonardo~39, p.~140, ll.~11--12.] + +% -----File: 061.png---Folio 49------------------------------------------------------ + +Let $\mathit{abcd}$ be the given quadrilateral and suppose it be +required to cut off one-third by a line drawn from the point $e$ +in the side $\mathit{ad}$. + +\CenteredGraphic{90mm}{161} % Graphic in Proposition 17 + +Draw $\mathit{dz}$ cutting off one-third of $\mathit{ac}$ [Prop.~15]. + +Join $\mathit{ez}$, $\mathit{ec}$. + +If $\mathit{ez}\parallel\mathit{dc}$, then $\mathit{ecd}$ [is the required part cut off, etc.]. + +But if $\mathit{ez}$ be not parallel to $\mathit{dc}$, +draw $\mathit{di}\parallel\mathit{ez}$ and join $\mathit{ei}$. +[Then this is the line required, etc.] + +\SmallLeftNote{ +The case when $\mathit{ei}$ cuts $\mathit{dc}$ is not taken up but it may be considered as in +the last proposition. + +So also to divide $\mathit{ac}$ into any ratio: draw $\mathit{dz}$ dividing it in that ratio +(Prop.~15), and then proceed as above. + +A particular case which Leonardo gives may be added.} + +Let $\mathit{ab}$ be divided into three equal parts $\mathit{ae, ef, fb}$; draw +$\mathit{dg} \parallel \mathit{ab}$ and cut off $\mathit{gh} = \tfrac{1}{3}gd$. +Join $\mathit{fc}$ and through $h$ draw +$\mathit{hi} \parallel \mathit{fc}$, meeting $\mathit{dc}$ in $i$. +Join $\mathit{fi}$; and the quadrilateral $\mathit{fbci}$ +will be one-third of the quadrilateral $\mathit{ac}$. [As in latter part of +Prop.~16.] + +\CenteredGraphic{90mm}{261} % originally 95mm % Graphic in Proposition 17 + +% -----File: 062.png---Folio 50------------------------------------------------------ + +Then $\mathit{ek}$ may be drawn to bisect the quadrilateral $\mathit{afid}$ +[Prop.~16], and thus the quadrilateral $\mathit{abcd}$ will be divided +into three equal portions which are the quadrilaterals $\mathit{ak, ei, fc}$. + + +\Proposition{18} % Proposition 18 +\nopagebreak +\Paragraph{}{39} \textit{``To apply to a straight line a rectangle equal % Paragraph 39 +to the rectangle contained by $\mathit{AB}$, $\mathit{AC}$ and deficient by a +square\footnotemark[103].''} + +\Footnotetext{103}{ % Footnote 103 (long) + This proposition is interesting as illustrating the method of + \emph{application of areas} which was + ``one of the most powerful methods on which Greek Geometry relied.'' + The method first appears in the \textit{Elements} in \textsc{i}.~44: + \textit{To a given straight line to apply, in a given rectilineal + angle, a parallelogram equal to a given triangle}---a proposition + which Heath characterises as + ``one of the most impressive in all geometry'' + while the ``marvellous ingenuity of the solution is indeed worthy + of the `godlike men of old' as Proclus calls the discoverers of the + method of `application of areas'; and there would seem to be no + reason to doubt that the particular solution, like the whole theory, + was Pythagorean, and not a new solution due to Euclid himself.'' + + [I continue to quote mainly from Heath who may be consulted for much + greater detail: \textsc{Heath}, + \textit{Thirteen Books of Euclid's Elements}, + \textsc{i}, 9, 36, 343--7, 383--8; + \textsc{ii}, 187, 257--67---\textsc{Heath}, + \textit{Apollonius of Perga Treatise on Conic Sections}, Cambridge, + 1896, pp.~lxxxi--lxxxiv, cii--cxi---\textsc{Heath}, + \textit{The Works of Archimedes}, Cambridge, 1897, + pp.~xl--xlii, 110 and ``Equilibrium of Planes,'' + Bk~\textsc{ii}, Prop.~1, and ``On conoids and spheroids,'' + Props.\ 2, 25, 26, 29. + See also: \textsc{Cantor}, + \textit{Vorlesungen über Geschichte der Math.}\ \textsc{i}$_3$, 289--291, etc. + (under index heading `Flächenanlegung')---\textsc{H.~G.~Zeuthen}, + \textit{Geschichte der Mathematik im Alterthum und Mittelalter}, + Kopenhagen, 1896, pp.~45--52 (French ed.~Paris, + 1902, pp.~36--44)--- \textsc{C.~Taylor}, + \textit{Geometry of Conics}\ldots, Cambridge, 1881, + pp.~\textsc{xliii--xliv}.] + + The simple \emph{application} of a parallelogram of given area to a + given straight line as one of its sides is what we have in the + \textit{Elements} \textsc{i}.~44 and 45; the general form + of the problem with regard to \emph{exceeding} and + \emph{falling-short} may be stated thus: + + ``To apply to a given straight line a rectangle + (or, more generally, a parallelogram) + equal to a given rectilineal figure and (1) \emph{exceeding} + or (2) \emph{falling-short} by a square (or, in the more general + case, a parallelogram similar to a given parallelogram).'' + + What is meant by saying that the applied parallelogram + (1) \emph{exceeds} or (2) \emph{falls short} is that, while its + base coincides and is coterminous \emph{at one end} with the + straight line, the said base (1) overlaps or (2) falls short of + the straight line \emph{at the other end}, and the portion by + which the applied parallelogram exceeds a parallelogram of the + same angle and height on the given straight line (exactly) as + base is a parallelogram similar to a given parallelogram + (or, in particular cases, a square). In the case where the + parallelogram is to \emph{fall short}, some such remark + as Woepcke's (note 104) is necessary to express the condition + of possibility of solution. For the other case see note 116. + + The solution of the problems here stated is equivalent to the + solution of a quadratic equation. By means of + \textsc{ii}.~5 and 6 we can solve the equations + \begin{align*} + ax \pm x^2 &= b^2,\\ + x^2 - ax &= b^2, + \end{align*} + but in \textsc{vi}.~28,~29 Euclid gives the equivalent of + the solution of the general equations \[ax \pm px^2 = A. \] + \textsc{vi}.~28 is: + \textit{To a given straight line to apply a parallelogram equal + to a given rectilineal figure and deficient by a parallelogrammic + figure similar to a given one: + thus the given rectilineal figure must not be greater than the parallelogram + described on the half of the straight line and similar to the defect.} + + The Proposition~18 of Euclid under consideration is a particular case + of this problem and as the fragment of the text and Woepcke's note + (note~104) are contained in it, doubt may well be entertained as to + whether Euclid gave any construction in his book \textit{On Divisions}. + The problem can be solved without the aid of Book \textsc{vi} of the + \textit{Elements} and by means of \textsc{ii}.~5 and + \textsc{ii}.~14 only, as indicated in the text above. + + The appropriation of the terms parabola (\emph{application}), + hyperbola (\emph{exceeding}) and ellipse (\emph{falling-short}) + to conic sections was first introduced by Apollonius as expressing + in each case the fundamental properties of curves as stated by him. + This fundamental property is the geometrical equivalent of the + Cartesian equation referred to any diameter of the conic and the + tangent at its extremity as (in general, oblique) axes. + More particulars in this connection are given by Heath. + + The terms ``parabolic,'' ``hyperbolic'' and ``elliptic,'' + introduced by Klein for the three main divisions of Geometry, + are appropriate to systems in which a straight angle equals, + exceeds and falls short of the angle sum of any triangle. + \textit{Cf.}\ \textsc{W.~B.~Frankland}, + \textit{The First Book of Euclid's Elements with a Commentary + based principally upon that of Proclus Diadochus\dots}Cambridge, 1905, p.~122. + } % end of footnote 103 + +% -----File: 063.png---Folio 51------------------------------------------------------ + +``After having done what was required, if some one ask, +How is it possible to apply to the line $\mathit{AB}$ a rectangle such +that the rectangle $\mathit{AE}\centerdot\mathit{EB}$ +\CenteredGraphic{85mm}{063} % Graphic in Proposition 18 +is equal to the +rectangle $\mathit{AB}\centerdot\mathit{AC}$ +and deficient by a square---we say that it is impossible, because +$\mathit{AB}$ is greater than $\mathit{BE}$ and $\mathit{AC}$ greater +than $\mathit{AE}$, and consequently +the rectangle $\mathit{BA}\centerdot\mathit{AC}$ greater than the rectangle +$\mathit{AE}\centerdot\mathit{EB}$. +Then when one applies to the line $\mathit{AB}$ a parallelogram +equal to the rectangle $\mathit{AB}\centerdot\mathit{AC}$ the rectangle +$\mathit{AZ}\centerdot\mathit{ZB}$ is\dots\dots\footnotemark[104].'' + +\SmallLeftNote{ +In this problem it is required to find in the given line $\mathit{AB}$ a +point $Z$ such that +\[ +\mathit{AB}\centerdot\mathit{ZB} - \mathit{ZB}^2 + [= \mathit{AZ}\centerdot\mathit{ZB} \text{ by \textsc{ii}.~3; cf.~x.~16 lemma}] = + \mathit{AB}\centerdot\mathit{AC}\footnotemark[105]. +\] + +Find, by \textsc{ii}.~14, the side, $b$, of a square equal in area +to the rectangle $\mathit{AB}\centerdot\mathit{AC}$, then the problem +is exactly equivalent to that of which a simple solution was given +by Simson\footnotemark[106]:} + + +\Footnotetext{104}{ % Footnote 104 + Woepcke here remarks: ``Evidently if $a$ denote the length of the + line to which the rectangle is to be applied, Problem 18 is only + possible when + \[\mathit{AB}\centerdot\mathit{AC} < \left( \dfrac{a}{2} \right)^2.\] + Then if $a$ be taken as $\mathit{AB}$ one of the two sides of the + given rectangle, relatively to the other side, $AC < \dfrac{AB}{4}$. + It is probably the demonstration of this which was + given in the missing portion of the text.''} % end footnote 104 + +% This footnote was labelled 106 in the 1915 Cambridge edition. Typo corrected. +\Footnotetext{\DPtypo{106}{105}}{ % Footnote 105 + If $\mathit{AB} = a$, $\mathit{ZB} = x$, + $\mathit{AB}\centerdot\mathit{AC} = b^2$, the problem is to find + a geometric solution of the equation $ax-x^2=b^2$. + Ofterdinger\footnotemark[38] (p.~15) seems to have quite missed the + meaning of this problem. He thought, apparently, that it was + equivalent to \textsc{x}.~16, lemma, of the \textit{Elements}.} % end footnote 105 + +\Footnotetext{106}{ % Footnote 106 + \textsc{R.~Simson}, \textit{Elements of Euclid}, + ninth~ed., Edinburgh, 1793, pp.~335--6.} % end footnote 106 +% -----File: 064.png---Folio 52------------------------------------------------------ + +\small +To apply a rectangle which shall be equal to a given square, to a given +straight line, deficient by a square: but the given square must not be greater +than that upon the half of the given line. + +\CenteredGraphic{95mm}{064} % Graphic in Proposition 18 + +Bisect $\mathit{AB}$ in $D$, and if the square on $\mathit{AD}$ be equal to the square on $b$, +the thing required is done. But if it be not equal to it, $\mathit{AD}$ must be greater +than $b$ according to the determination. Then draw $\mathit{DO}$ perpendicular to $\mathit{AB}$ +and equal to $b$; produce $\mathit{OD}$ to $N$ so that $\mathit{ON}=\mathit{DB}$ +(or $\tfrac{1}{2}a$); and with $O$ as +centre and radius $\mathit{ON}$ describe a circle cutting $\mathit{DB}$ in $Z$. + +Then $\mathit{ZB}$ (or $x$) is found, and therefore the required rectangle $\mathit{AH}$. + +For the rectangle $\mathit{AZ}\centerdot\mathit{ZB}$ together with the square +on $\mathit{DZ}$ is equal to the +square on $\mathit{DB}$, \hfill[\textsc{ii}.~5]\quad + +\hspace{10em}i.~e.\ to the square on $\mathit{OZ}$, + +\hspace{10em}i.~e.\ to the squares on $\mathit{OD, DZ}$. \hfill [\textsc{i}.~47]\quad + +\noindent +Whence the rectangle $\mathit{AZ}\centerdot\mathit{ZB}$ is equal to the square on $\mathit{OD}$. + +Wherefore the rectangle $\mathit{AH}$ equals the given square upon $b$ (i.~e.\ the +rectangle $\mathit{AB}\centerdot\mathit{AC}$) and has been applied to the +given straight line $\mathit{AB}$, +deficient by the square $\mathit{HB}$\footnotemark[107]. + +\Footnotetext{107}{ % Footnote 107 + It is not in the manner of Euclid to take account of the two solutions + found by considering $(F)$, as well as $Z$, determined by the circle + with centre $O$. + + Although Leonardo's construction for Problem 19 is identical with that + of Euclid who makes use of Problem 18, Leonardo does not seem to have + anywhere formulated Problem 18. He may have considered it + sufficiently obvious from \textsc{vi}.~28, or from \textsc{ii}.~5 and + \textsc{ii}.~6, of which he gives the enunciations in the early + pages (15--16) of his \textit{Practica Geometriae}; he also + considers (p.~60) the roots of a resulting quadratic equation, + $ax-x^2 = b$ (\textit{cf.}\ \textsc{Cantor}, \textit{Vorlesungen\dots}, + \textsc{ii}$_2$, 39), but does not give \textsc{ii}.~14. + \textit{Cf.\ Bibliotheca Mathematica}, (3), 1907--8, + \textsc{viii},~190; and also \textsc{ix},~245.} % end footnote 107 + + +\normalsize + +\Proposition{19} % Proposition 19 +\Paragraph{}{40} \textit{``To divide a given triangle into two equal % Paragraph 40 +parts by a line which passes through a point situated in the +interior of the triangle.''} +[Leonardo~3, p.~115, ll.~7--10.] + +% -----File: 065.png---Folio 53------------------------------------------------------ + +``Let the given triangle be $\mathit{ABC}$, and the given point in the interior +of this triangle, $D$. + +\WrappedGraphic{50mm}{065}{15}{r} % Graphic in Proposition 19 + +It is required to draw through +$D$ a straight line which divides +the triangle $\mathit{ABC}$ into two equal +parts. + +Draw from the point $D$ a line +parallel to the line $\mathit{BC}$, as $\mathit{DE}$, and\DPnote{yes, starting a new paragraph} + +Apply to $\mathit{DE}$ a rectangle equal +to half of the rectangle $\mathit{AB}\centerdot\mathit{BC}$, +such as +\[ +\mathit{TB}\centerdot\mathit{DE} +\left[\mathit{TB}=\dfrac{\mathit{AB}\centerdot\mathit{BC}}{2\mathit{DE}}\right]. +\] + +Apply to the line $\mathit{TB}$ a +parallelogram equal to the rectangle $\mathit{BT}\centerdot\mathit{BE}$ +and deficient by a square\footnotemark[1071].\hfill [Prop.~18] \qquad +% We cannot use alphanumeric footnote marker 107a used in original book. + +Let the rectangle applied be +\[ +\mathit{BH}\centerdot\mathit{HT}[(\mathit{TB-HT})\centerdot\mathit{HT}= + \mathit{TB}\centerdot\mathit{BE}]. +\] + +Draw the line $\mathit{HD}$ and produce it to $Z$. + +Then this is the line required and the triangle $\mathit{ABC}$ is +divided into two equal parts $\mathit{HBZ}$ and $\mathit{HZCA}$. + +\textit{Demonstration}. +The rectangle $\mathit{TB}\centerdot\mathit{BE}$ is equal to the +rectangle $\mathit{TH}\centerdot\mathit{HB}$, whence it follows that +\begin{DPalign*} + BT: TH &= HB: BE; \\ + \lintertext{then \textit{dividendo}\footnotemark[108]} TB: BH &= BH: HE. \\ + \lintertext{But} BH: HE &= BZ: ED; \tag*{[\textsc{vi}.~2]} \\ + \lintertext{therefore} TB: BH &= BZ: ED. +\end{DPalign*} +Consequently the rectangle $\mathit{TB}\centerdot\mathit{ED}$ is equal +to the rectangle $\mathit{BH}\centerdot\mathit{BZ}$. +But the rectangle $\mathit{TB}\centerdot\mathit{ED}$ is equal to half the +rectangle $\mathit{AB}\centerdot\mathit{BC}$; and +\[\mathit{BH}\centerdot\mathit{BZ:AB}\centerdot\mathit{BC} += \triangle\mathit{HBZ}: \triangle\mathit{ABC}\footnotemark[88], +\] +% -----File: 066.png---Folio 54------------------------------------------------------ +since the angle $B$ is common. The triangle $\mathit{HBZ}$ is, then, +half the triangle $\mathit{ABC}$. + +\Footnotetext{1071}{ % Footnote 1071 (was 107a) + The corresponding sentence in Leonardo is (p.~115, ll.~15--17): + ``Deinde linee $\mathit{gz}$ applicabis paralilogramum deficiens + figura tetragona, quod sit equale + superficies $\mathit{ge}$ in $\mathit{gz}$.''} % end footnote 1071 +\Footnotetext{108}{ % Footnote 108 + ``Elements, Book~\textsc{v}, definition~16'' (Woepcke). + This is definition~15 in \textsc{Heath}, + \textit{The Thirteen Books of Euclid's Elements}, + \textsc{ii},~135.} % end footnote 108 + +Therefore the triangle $\mathit{ABC}$ is divided into two equal +parts $\mathit{BHZ}$ and $\mathit{AHZC}$. + +If, in applying to $\mathit{TB}$ a parallelogram equal to the rectangle +$\mathit{TB}\centerdot\mathit{BE}$ and of which the complement +is a square, we obtain the rectangle +$\mathit{AB}\centerdot\mathit{AT}$\Footnote{109}{ % Footnote 109 + ``In other words when $H$ coincides with $A$. + This can only be the case + when $D$ is situated on the line which joins $A$ to the middle of + the base $\mathit{BC}$'' + (Woepcke). If $D$ were at the centre of gravity of the triangle, + three lines could be + drawn through $D$ dividing the triangle into two equal parts. + As introductory to his Prop.~3, Leonardo proved that the medians of a + triangle meet in a point, and trisect one another---results known + to Archimedes\footnotemark[601],\DPnote{* was 60a} + but no complete, strictly geometric proof has come down to us from + the Greeks. Leonardo then proves that if a point be taken on any + one of the medians, or on one of the medians produced, the line + through this point and the corresponding angular point of the + triangle will divide the triangle into two equal parts. + He next shows that lines through the vertices of a triangle and + any point within not on one of the medians, will divide the given + triangle into triangles whose areas are each either greater than or + less than the area of half of the original triangle. + This leads Leonardo to the consideration of the problem, to draw + through a point, within a triangle and not on one of the medians, + a line which \emph{will} bisect the area of the triangle. + (Euclid, Prop.~19.) + + The last paragraph of Euclid's proof, as it has come down to us + through Arabian sources, does not ring true, and it was not in + the Euclidean manner to consider special cases. + + After Leonardo's proof of Proposition~19, + a numerical example is given.}% % end footnote 109 +, we may demonstrate in +an analogous manner, by drawing the line $\mathit{AD}$ and prolonging +it to $K$, that the triangle $\mathit{ABK}$ is one-half of the triangle +$\mathit{ABC}$. And this is what was required to be demonstrated.'' + +\Proposition{20} % Proposition 20 +\Paragraph{}{41} \textit{``To cut off a certain fraction from a given % Paragraph 41 +triangle by a line drawn from a given point situated in the interior +of the triangle.''} [Leonardo~10, p.~121, ll.~1--2.] + +``Let $\mathit{ABC}$ be the given triangle and $D$ the given point in the interior +of the triangle. +It is required to pass through +the point $D$ a straight line +which cuts off a certain fraction +of the triangle $\mathit{ABC}$. + +\WrappedGraphic{50mm}{066}{}{r} % Graphic in Proposition 20 + +``Let the certain fraction +be one-third. Draw from the +point $D$ a line parallel to the +line $\mathit{BC}$, as $\mathit{DE}$, and apply +to $\mathit{DE}$ a rectangle equal to +one-third of the rectangle +$\mathit{AB}\centerdot\mathit{BC}$. Let this be +\[ + \mathit{BZ}\centerdot\mathit{ED} + \left[\mathit{BZ}=\frac{\mathit{AB}\centerdot\mathit{BC}}{3\centerdot\mathit{ED}}\right]. +\] +% -----File: 067.png---Folio 55------------------------------------------------------ +Then apply to $\mathit{ZB}$ a rectangle equal to the rectangle +$\mathit{ZB}\centerdot\mathit{BE}$ +and deficient by a square. [Prop.~18.] Let the rectangle +applied be the rectangle +\[ +\mathit{BH}\centerdot\mathit{HZ} \left[\left(ZB-HZ\right) HZ=ZB\centerdot BE\right]. +\] +Draw the line $\mathit{HD}$ and produce it to $T$. + +``On proceeding as above we may demonstrate that the +triangle $\mathit{HTB}$ is one-third of the triangle $\mathit{ABC}$; and by +means of an analogous construction to this we may divide +the triangle in any ratio. But this is what it is required +to do\Footnote{110}{ % Footnote 110 + Leonardo gives the details of the proof for the case of one-third + and does not refer to any other fraction. + If, however, the ``certain fraction'' were the ratio + of the lengths of two given lines, $m:n$, we could readily + construct a rectangle equal to + $\dfrac{m}{n}\centerdot AB\centerdot BC$, and then find the + rectangle $\mathit{BZ}\centerdot\mathit{ED}$ equal to it. + The rest of the construction is the same as given above. + + According to the conditions set forth in Proposition~18, + there will be two, one, or no solutions of Propositions~19 and 20. + Leonardo considers only the Euclidean cases. + \textit{Cf.}\ notes 104 and 107. + + The case where there is no solution may be readily indicated. + Suppose, in the above figure, that $\mathit{BE}=\mathit{EH}$, then of all triangles + formed by lines drawn through $D$ to meet $\mathit{AB}$ and $\mathit{BC}$, + the triangle $\mathit{HBT}$ has the minimum area. (Easily shown synthetically + as in \textsc{D.~Cresswell}, + \textit{An Elementary Treatise on the Geometrical and + Algebraical Investigations of Maxima and Minima}. + Second edition, Cambridge, 1817, pp.~15--17.) + Similar minimum triangles may be found in connection with the pairs + of sides $\mathit{AB},\mathit{AC}$ and $\mathit{AC},\mathit{CB}$. + Suppose that neither of these + triangles is less than the triangle $\mathit{HBT}$. Then if + \[\triangle\mathit{HBT}: \triangle\mathit{ABC} > m:n,\] + the solution of the problem is impossible.} % end footnote 110 +.'' + +\Proposition{21} % Proposition 21 +\Paragraph{}{42} \textit{``Given the four lines $A$, $B$, $C$, $D$ and that the % Paragraph 42 +product of $A$ and $D$ is greater than the product of $B$ and $C$; +I say that the ratio of $A$ to $B$ will be greater than the ratio +of $C$ to $D$''}\Footnote{111}{ % Footnote 111 + This and the next four auxiliary propositions for which I supply % a really long footnote + possible proofs, seem to be neither formally stated nor proved by + Leonardo. + At least some of the results are nevertheless assumed in his + discussion of Euclid's later propositions, as we shall presently see. + Although these auxiliary propositions are not given in the + \textit{Elements}, they are assumed as known by Archimedes, Ptolemy + and Apollonius. + + For example, in Archimedes' ``On Sphere and Cylinder,'' + \textsc{ii}.~9 (Heiberg, ed.~\textsc{i}, 1910, p.~227; + Heath, ed.\ 1897, p.~90), Woepcke~21 is used. + See also Eutocius' Commentary (Archimedis \textit{Opera omnia} + ed.~Heiberg, \textsc{iii}, 1881, p.~257, etc.), and + \textsc{Heiberg}, \textit{Quaestiones Archimedeae}, Hauniae, 1879, + p.~45~f. + For a possible application by Archimedes (in his + \textit{Measurement of a circle}) of what is practically equivalent + to Woepcke~24, see Heath's \textit{Archimedes\dots}, 1897, p.~xc. + + The equivalent of Woepcke~24 is assumed in the proof of a proposition + given by Ptolemy (87--165~A.~D.) in his \textit{Syntaxis}, + vol.~\textsc{i}, Heiberg edition, Leipzig, 1898, pp.~43--44. + This in turn is tacitly assumed by Aristarchus of Samos (\textit{circa} + 310--230~B.~C.) in his work + \textit{On the Sizes and Distances of the Sun and Moon} + (see Heath's edition \textit{Aristarchus of Samos the Ancient Copernicus}, + Oxford, 1913, pp.~367, 369, 377, 381, 389, 391). + + As to the use of the auxiliary propositions in the two works + \textit{Proportional Section} and \textit{On Cutting off a Space}, + of Apollonius, we must refer to Pappus' account + (Pappi Alexandrini \textit{Collectionis\dots} ed.~Hultsch, + vol.~\textsc{ii}, 1877, pp.~684 ff.). + Woepcke~21, 22 occur on pp.~696--697; Woepcke~24 enters on pp.~684--687; + Woepcke~23, 25 are given on pp.~687, 689. Perhaps this last statement + should be modified; for whereas Euclid's propositions affirm that if + \[ a:b \gtrless c:d, \quad a-b:b \gtrless c-d:d, \] + Pappus shows that if \[ a:b \gtrless c:d, \quad a:a-b \lessgtr c:c-d; \] + but these propositions are immediately followed by others which state + that if \[ a:b \gtrless c:d, \text{ then } b:a \lessgtr d:c. \] + + Below is given a list of the various restorations of the above-named + works of Apollonius, based on the account of Pappus. + By reference to these restorations the way in which the auxiliary + propositions are used or avoided may be observed. + We have already (Art.~21) noticed a connection of Apollonius' work + \textit{On Cutting off a Space} with our subject under discussion. + Some of these titles will therefore supplement the list given in the Appendix. + + \textit{Wilebrordi Snellii R.~F.} + \ItalicGreekSnippet{per`i l'ogou `apotomês ka`i per`i chwr'iou `apotomhs} + \textit{(Apollonii) resuscitata geometria.} Lugodini, + \textit{ex officina Platiniana} Raphelengii, \textsc{md.cvii} pp.~23. + +More or less extensive abridgment of Snellius's work is given in: + +\renewcommand{\labelenumi}{(\alph{enumi})} +\begin{enumerate} +\item \textit{Universae geometriae mixtaeque mathematicae synopsis et bini +refractionum demonstratarum tractatus. Studio et opera F.~M.\ Mersenni.} +Parisiis, \textsc{m.dc.xliv}, p.~382. +\item \textit{Cursus mathematicus}, P.~Herigone. Paris, 1634, tome~\textsc{i}, pp.~899--904; +also Paris, 1644. +\end{enumerate} + +\textit{Apollonii Pergaei de sectione rationis libri duo ex Arabico $MS^{to}$ Latine versi +accedunt ejusdem de sectione spatii libri duo restituti\dots opera \& studio Edmundi +Halley}\dots Oxonii,\dots \textsc{mdccvi}, pp.~8 + liii + 168. + +\begin{enumerate} +\item \textit{Die Bücher des Apollonius von Perga De sectione rationis nach dem +Lateinischen des Edm.\ Halley frey bearbeitet, und mit einem Anhange +versehen von W.~A.\ Diesterweg}, Berlin, 1824, pp.~xvi + 218 + 9~pl. +\item \textit{Des Apollonius von Perga zwei Bücher vom Verhältnissschnitt} +(\textit{de sectione rationis}) \textit{aus dem Lateinischen des Halley übersetzt und mit +Anmerkungen begleitet und einem Anhang versehen von August Richter} +\DPnote{[**parentheses ( ) should not be italicized]} +\ldots Elbing, 1836, pp.~xxii + 143 + 4~pl. +\end{enumerate} + +\textit{Die Bücher des Apollonius von Perga de sectione +spatii wiederhergestellt von +Dr~W.~A.~Diesterweg\dots} Elberfeld, 1827\dots pp.~vi + 154 + 5~pl. + +\textit{Des Apollonius von Perga zwei Bücher vom Raumschnitt. Ein Versuch in der +alten Geometrie von A.~Richter.} Halberstadt, 1828, pp.~xvi + 105 + 9~pl. + +\textit{Die Bücher des Apollonius von Perga de sectione spatii, analytisch bearbeitet +und mit einem Anhange von mehreren Aufgaben ähnlicher Art versehen +von M.~G.~Grabow\dots} Frankfurt a.~M., 1834, pp.~80 + 3~pl. + +\textit{Geometrische Analysis enthaltend des Apollonius von Perga sectio rationis, +spatii und determinata, nebst einem Anhange zu der letzten, neu bearbeitet vom +Prof.~Dr~Georg Paucker}, Leipzig, 1837, pp.~xii + 167 + 9~pl. + +M.~Chasles discovered that by means of the theory of involution a single +method of solution could be applied to the main problem of the three books of +Apollonius above mentioned. This solution was first published in \textit{The Mathematician}, +vol.~\textsc{iii}, Nov.~1848, pp.~201--202. This is reproduced by A.~Wiegand in +his \textit{Die schwierigeren geometrischen Aufgaben aus des Herrn Prof.~C.~A.~Jacobi +Anhängen zu Van Swinden's Elementen der Geometrie. Mit Ergänzungen +englischer Mathematiker\dots} Halle, 1849, pp.~148--149, and it appears at greater +length in Chasles' \textit{Traité de Géométrie supérieure}, +Paris, 1852, pp.~216--218; $2^\text{e}$ éd. +1880, pp.~202--204. It was no doubt Chasles who inspired \textit{Die Elemente der +projectivischen Geometrie in synthetischer Behandlung. Vorlesungen von H.~Hankel}, +(Leipzig, 1875), ``Vierter Abschnitt, Aufgaben des Apollonius,'' pp.~128--145; +``sectio rationis,'' pp.~128--138; ``sectio spatii,'' pp.~138--140. + +The \textit{``Three Sections,'' the ``Tangencies'' and a ``Loci + Problem'' of Apollonius\ldots +by M.~Gardiner}, Melbourne, 1860. Reprinted from the \textit{Transactions of the Royal +Society of Victoria}, 1860--1861, \textsc{v},~19--91 + 10~pl. + +\textit{Die sectio rationis, sectio spatii und sectio determinata des Apollonius nebst +einigen verwandten geometrischen Aufgaben von Fr.\ von Lühmann}. Progr.\ +Königsberg in d.~N.~1882, pp.~16 + 1~pl. + + ``Ueber die fünf Aufgaben des Apollonius,'' + \textit{von L.~F.~Ofterdinger. Jahreshefte des Vereines + für Math.\ u.\ Naturwiss.\ in Ulm a.~D.}\ 1888, \textsc{i},~21--38; + ``Verhältnissschnitt,'' pp.~23--25; + ``Flächenschnitt,'' pp.~26--27.}% % end footnote 111 (whew!) +. % this period is the end of the sentence pointing to footnote 111 (140 lines back!) + +% -----File: 068.png---Folio 56------------------------------------------------------ + +\normalsize +\small +Given $A\centerdot D > B\centerdot C$. To prove $A:B > C:D$. + +Let the lines $A$, $D$ be adjacent sides of a rectangle; and let there be +another rectangle with side $B$ lying along $A$ and side $C$ along $D$. Then +either $A$ is greater than $B$, or $D$ greater than $C$, for otherwise the rectangle +$A\centerdot D$ would not be greater than the rectangle $B\centerdot C$. + +% -----File: 069.png---Folio 57------------------------------------------------------ +Let then $A > B$. To $D$ apply the rectangle $B\centerdot C$ and we get a rectangle +\begin{DPalign*} + A'\centerdot D &= B\centerdot C; \tag*{[\textsc{i}.~44--45] \quad} \\ +\lintertext{then} A':B &= C:D. \tag*{\text{[\textsc{vii}.~19] \quad}} \\ +\lintertext{\indent But since } A &> A', \\ + A:B &> A':B; \tag*{\text{[\textsc{v}.~8] \quad}} \\ + \therefore A:B &> C:D. \tag*{\text{[\textsc{v}.~13] \quad}}\\ + \tag*{\textsc{q.~e.~d.}\quad\quad} +\end{DPalign*} + +Pappus remarks: Conversely if $A:B > C:D$, $A\centerdot D > B\centerdot C$. The proof +follows at once. +\begin{DPalign*} + \lintertext{\indent For, find $A'$ such that} A': B &= C:D; \\ + \lintertext{then} A:B &> A':B, +\end{DPalign*} +and $A>A'$. But $A'\centerdot D = B\centerdot C.\quad +\therefore A\centerdot D > B\centerdot C.$ \hfill \textsc{q.~e.~d.} \qquad\qquad +\normalsize + +\Proposition{22} % Proposition 22 +\Paragraph{}{43} \textit{``And when the product of $A$ and $D$ is less than % Paragraph 43 +the product of $B$ and $C$, then the ratio of $A$ to $B$ is less than the +ratio of $C$ to $D$.''} + + +% -----File: 070.png---Folio 58------------------------------------------------------ +\SmallLeftNote{ +From the above proof we evidently have +\[C:D > A:B,\] +that is, +\[A:B < C:D.\] +Conversely, as above, if $A:B < C:D$, $A\centerdot B < C\centerdot D$. + +It is really this converse, and not the proposition, which Euclid uses in +Proposition~26. Proclus remarks (page~407) that the converses of Euclid's +Elements, \textsc{i}.~35,~36, about parallelograms, are unnecessary ``because it is easy +to see that the method would be the same, and therefore the reader may +properly be left to prove them for himself.'' No doubt similar comment is +justifiable here.} + +\Proposition{23} % Proposition 23 +\Paragraph{}{44} \textit{``Given any two straight lines and on these % Paragraph 44 +lines the points $A$,~$B$, and $D$,~$E$; and let the ratio of +$\mathit{AB}$ to $\mathit{BC}$ be greater than the ratio of +$\mathit{DE} : \mathit{EZ}$;} +\textit{I say that \emph{dividendo} the +ratio of $\mathit{AC}$ to $\mathit{CB}$ will be greater than the ratio of +$\mathit{DZ}$ to $\mathit{ZE}$.''} + +\CenteredGraphic{90mm}{070} % Graphic in Proposition 23 + +\small +\begin{DPgather*} + \lintertext{\indent Given} AB:BC > DE:EZ.\\ + \lintertext{\indent To prove } AC:CB > DZ:ZE. +\end{DPgather*} +\indent To $\mathit{AB}, \mathit{BC}, \mathit{DE}$ find a fourth proportional $\mathit{EW}$. \hfill [\textsc{vi.}~12] + +\begin{DPgather*} + \lintertext{\indent Then} AB:BC = DE:EW. \tag*{\ldots\ldots\ldots(1)}\\ + \lintertext{\indent But } AB:BC > DE:EZ; \\ + \therefore DE:EW > DE:EZ.\tag*{[\textsc{v.}~13]} \\ + \therefore EW < EZ. \tag*{[\textsc{v.}~8]} \\ + \lintertext{\indent From (1)} AC:CB = DW:WE; \tag*{\ldots\ldots\ldots(2)~[\textsc{v.}~17]}\\ + \lintertext{\indent since} DW>DZ,\quad DW:WE > DZ:WE. \tag*{[v.~8]} \\ + \therefore AC:CB > DZ:WE. \tag*{[\textsc{v.}~13]} \\ + \lintertext{\indent But} WE < ZE; \quad\therefore DZ:WE > DZ:ZE. \tag*{[\textsc{v.}~8]} \\ + \therefore AC:CB > DZ:ZE. \tag*{From~(2)~and~[\textsc{v.}~17]}\\ + \tag*{\textsc{q.~e.~d.}\quad} +\end{DPgather*} + +\normalsize + +% -----File: 071.png---Folio 59------------------------------------------------------ +\Proposition{24} % Proposition 24 +\Paragraph{}{45} \textit{``And in an exactly analogous manner I say that % Paragraph 45 +when the ratio of $\mathit{AC}$ to $\mathit{CB}$ is greater than the ratio of +$\mathit{DZ}$ to $\mathit{ZE}$, we shall have} componendo\Footnote{112}{ % Footnote 112 + ``Elements, Book \textsc{v}, definition 15'' (Woepcke). + This is definition 14 in \textsc{Heath}, + \textit{The Thirteen Books of Euclid's Elements}, \textsc{ii}, 135.} % end footnote 112 +\textit{the ratio of $\mathit{AB}$ to $\mathit{BC}$ is +greater than the ratio of $\mathit{DE}$ to $\mathit{EZ}$.''} + +\small +\begin{DPgather*} + \lintertext{\indent Given} AC:CB > DZ:ZE. \\ + \lintertext{\indent To prove} AB:BC > DE:EZ. +\end{DPgather*} + \indent Determine $W$, as before, such that +\begin{DPgather*} + AB:BC = DE:EW. \\ + \lintertext{\indent Then} AC:CB = DW:WE. \tag*{[\textsc{v}. 17]} \\ + \therefore \; DW:WE > DZ:ZE. \tag*{\ldots\ldots (1) \qquad [\textsc{v}. 13]} \\ + \lintertext{\indent Now either} EW>EZ \text{ or } EW<EZ. +\end{DPgather*} +\indent If $EW>EZ$,\quad $DW<DZ$, and +\begin{DPalign*} + DW:EW&<DZ:EW. \tag*{[\textsc{v}. 8]} \\ + \lintertext{\indent So much the more is} & \\ + DW:EW&<DZ:EZ \tag*{ [\textsc{v}. 8]} \\ +\lintertext{which contradicts (1).} +\end{DPalign*} +\begin{DPalign*} + \therefore EW &< EZ. \\ + \lintertext{\indent But} AB:BC &= DE:EW, \\ + \lintertext{and} DE:EW &> DE:EZ; \tag*{[\textsc{v}. 8]} \\ + \therefore AB:BC &> DE:EZ. \tag*{ [\textsc{v}. 13]} \\ + & \tag*{\textsc{q.~e.~d.}\quad} +\end{DPalign*} + +\iffalse +\begin{math} +\begin{array}{llr} +\text{\indent Given} & \mathit{AC:CB}>\mathit{DZ:ZE}.& \\ +\text{\indent To prove} & AB:BC>DE:EZ. & \\ +\multicolumn{3}{l}{\text{\indent Determine $W$, as before, such that}} \\ + & AB:BC=DE:EW. & \\ +\text{\indent Then} & AC:CB=DW:WE. & [\textsc{v}. 17] \\ +& \therefore DW:WE>DZ:ZE. \ldots\ldots (1) & [\textsc{v}. 13] \\ +\text{\indent Now either} & EW>EZ\text{ or }EW<EZ. & \\ +\multicolumn{3}{l}{\text{\indent If }EW>EZ,\quad DW<DZ, \text{ and}} \\ +& DW:EW<DZ:EW. & [\textsc{v}. 8]\\ +\multicolumn{3}{l}{\text{\indent So much the more is}} \\ +& DW:EW<DZ:EZ & [\textsc{v}. 8] \\ +\multicolumn{3}{l}{\text{which contradicts (1).}} \\ +& \therefore EW<EZ. \\ +\text{\indent But} & AB:BC=DE:EW, \\ +\text{and} & DE:EW>DE:EZ; & [\textsc{v}. 8] \\ +& \therefore AB:BC>DE:EZ. & [\textsc{v}. 13] \\ +& & \textsc{q.~e.~d.}\quad +\end{array} +\end{math} +\fi +\normalsize + +\Proposition{25} % Proposition 25 +\Paragraph{}{46} \textit{``Suppose again that the ratio of % Paragraph 46 +$\mathit{AB}$ to $\mathit{BC}$ were less than the} +\CenteredGraphic{85mm}{071} % Graphic in Proposition 25 +\textit{ratio of $\mathit{DE}$ to $\mathit{EZ}$; \emph{dividendo} +the ratio of $\mathit{AC}$ to $\mathit{CB}$ will be less than the +ratio of $\mathit{DE}$ to $\mathit{ZE}$}\Footnote{113}{ % Footnote 113 + The auxiliary propositions are introduced, apparently, to assist + in rendering, with faultless logic, the remarkable proof of + Proposition 26. In this proof it will be observed that we are + referred back to Proposition 21, to the converse of Proposition 22 + and to Proposition 25 only, although 23 is really the same as 25. + But no step in the reasoning has led to Proposition 24. If this + is unnecessary, why has it been introduced? + + To answer this question, let us inspect the auxiliary propositions + more closely. In a sense Propositions~21 and 22 go together: + If $ad \gtrless bc$, then $a:b \gtrless c:d$. + So also for Propositions~23 and 25: + If $a:b \gtrless c:d$, then $a-b:b \gtrless c-d:d$. + Proposition~24 is really the converse of 23: + If $a:b > c:d$, then $a+b:b > c+d:d$. + Had Euclid given another proposition: + If $a:b < c:d$ then $a+b:b < c+d:d$, + we should have had two groups of propositions 21, 22, and 23, 25 + with their converses. Now the converses of 21 and 22 are exceedingly + evident in both statement and proof. But this can hardly be said + of the proof of 24, the converse of 23. The converse of 23 having + been given the formulation of the statement and proof of the + converse of 25 is obvious and unnecessary to state, according + to Euclid's ideals (cf.\ Art.~43). + It might therefore seem that Proposition~24 is merely given to + complete what is not altogether obvious, in connection with the + statement of the four propositions 21 and 22, 23 and 25, and their + converses. In Pappus' discussion some support is given to this view, + since Propositions 21 and 22 and converses are treated as a single + proposition; Propositions 23, 25 as another proposition, while the + converses of 23 and 25 are dealt with separately. + + The more probable explanation is, however, that Propositions 23 and + 24 were given by Euclid because they were necessary for the + discussion of other cases of Proposition~26 (assuming that the first + case of Leonardo was that given by Euclid), for it was not his + manner to consider different cases. Indeed if we take $\mathit{be}$ less than + $\mathit{ge}$ in the first part of Leonardo's discussion exactly Propositions + 23 and 24 are necessary.}.'' % end of footnote 113 + +% -----File: 072.png---Folio 60------------------------------------------------------ +\SmallLeftNote{Just as the proof of Proposition~22 was contained in that for Proposition~21, +so here, the proof required is contained in the proof of Proposition~23. +Similarly the converse of Proposition~25 flows out of 24.} + +\Proposition{26} % Proposition 26 +\Paragraph{}{47} \textit{``To divide a given triangle into two equal % Paragraph 47 +parts by a line drawn from a given point situated outside the triangle.''} +[Leonardo 4, p.~116, ll.~35--36.] + +Let the triangle be $\mathit{abg}$ and $d$ the point outside. + +Join $\mathit{ad}$ and let $\mathit{ad}$ meet $\mathit{bg}$ in $e$. +If $\mathit{be}=\mathit{eg}$, what was required is done. +For the triangles $\mathit{abe, aeg}$ being on equal +bases and of the same altitude are equal in area. + +But if $\mathit{be}$ be not equal to $\mathit{eg}$, let it be greater, +and draw through $d$, parallel to $\mathit{bg}$, +a line meeting $\mathit{ab}$ produced in $z$. +\begin{flalign*} + &\text{\indent Since } & \mathit{be} &> \tfrac{1}{2}\mathit{bg}, && \\ + && \text{area } \mathit{ab}\centerdot\mathit{be} &> + \tfrac{1}{2}\:\text{area } \mathit{ab}\centerdot\mathit{bg}; + & \text{[\textit{Cf.}\ \textsc{vii.}~17]} &\\ + &\text{much more then is}&&&&\\ + && \text{area } \mathit{ab}\centerdot\mathit{zd} + &> \tfrac{1}{2}\:\text{area } \mathit{ab}\centerdot\mathit{bg}, + & \text{since } \mathit{zd} > \mathit{be}. & +\end{flalign*} +\begin{flalign*} + &\text{\indent Now take }&&&&\\ + && \text{area } \mathit{ib}\centerdot\mathit{zd} + &= \tfrac{1}{2}\:\text{area } \mathit{ab}\centerdot\mathit{bg}; + & \text{[\textsc{i.}~44]} & \\ + &\text{then } + & \text{area } \mathit{ab}\centerdot\mathit{be} &> + \text{ area } \mathit{ib}\centerdot\mathit{zd}, && \\ + &\text{and } + & \mathit{zd}:\mathit{be} &< \mathit{ba}:\mathit{bi}\footnotemark[114]. + & \text{[Prop.~21 or 22]} & +\end{flalign*} +\begin{flalign*} + &\text{\indent But} & \mathit{zd}:\mathit{be}&=\mathit{za}:\mathit{ab}, & [\textsc{vi}.\;4]\quad& \\ + && \therefore \mathit{zb:ba} &< \mathit{ai:ib}; & \textsc{[v.}\ \text{13 and Prop. 25]}& \\ + &\text{or} & \text{area } \mathit{zb}\centerdot\mathit{bi} &< + \text{area } \mathit{ba}\centerdot\mathit{ai}. & \text{[Converse of Prop. 22]}& +\end{flalign*} + +\Footnotetext{114}{ % Footnote 114 + Therefore $\mathit{bi}<\mathit{ba}$, + and if $\mathit{bi}$ be measured along $\mathit{ba}$, + $i$ will fall between $b$ and $a$.} % end footnote 114 + +% -----File: 073.png---Folio 61------------------------------------------------------ + +Apply a rectangle equal to the rectangle +$\mathit{zb}\centerdot\mathit{bi}$ to the line $\mathit{bi}$, +but exceeding by a square\Footnote{115}{ % Footnote 115 + Here again we have an expression with the true Greek ring: + ``adiungatur quidem recte + $\centerdot\mathit{bi}\centerdot$ + paralilogramum superhabundans figura tetragona equale superficiei + $\centerdot\mathit{zb}\centerdot$ in + $\centerdot\mathit{bi}\centerdot$''}; % end footnote 115 +that is to $\mathit{bi}$ apply a line such +that when multiplied by itself and by $\mathit{bi}$ the sum will be +equal to the product of $\mathit{zb}$ and $\mathit{bi}$; +let $\mathit{ti}$ be the side of the square\Footnote{116}{ % Footnote 116 + We we have seen that $i$ lies between $b$ and $a$. + And since it has been shown that + $\mathit{zb}\centerdot\mathit{bi} < \mathit{ba}\centerdot\mathit{ai}$, + we now have + $\mathit{ba}\centerdot\mathit{ai} > \mathit{bt}\centerdot\mathit{ti}$. + If $\mathit{bt}>\mathit{ba}$, $\mathit{ti}$ is also greater than $\mathit{ai}$, + and $\mathit{bt}\centerdot\mathit{ti} \nless \mathit{ba}\centerdot\mathit{ai}$. + Therefore $\mathit{bt}<\mathit{ba}$ and $t$ falls between $b$ and $a$. + But it also falls between $a$ and $i$ by reason of the construction + (always possible) which is called for. + + In his book on \textit{Divisions} (\textit{of figures}) + Euclid does not + formulate the proposition here quoted, possibly because of its + similarity to Proposition 18 (see note 103). + +\CenteredGraphic{70mm}{073} % Graphic in Proposition 26 + + If we let the rectangle $\mathit{zb}\centerdot\mathit{bi} = c^2$, + $\mathit{ti} = x$, and $\mathit{bi} = a$, + we have to solve geometrically the quadratic equation: + \[ + ax+x^2=c^2. + \] + + Heath points out (\textit{Elements}, vol.\ \textsc{i}, pp.~386--387) + that the solution of a problem theoretically equivalent to the + solution of a quadratic equation of this kind is presupposed in + the fragment of Hippocrates' \textit{Quadrature of lunes} + (5th century~\textsc{b.~c.})\ preserved in a quotation by Simplicius + (fl.~500~\textsc{a.~d.}) from Eudemus' \textit{History of Geometry} + (4th century \textsc{b.~c.}). See Simplicius' + \textit{Comment.\ in Aristot.\ Phys.} ed.\ H.~Diels, Berlin, 1882, + pp.~61--68; see also \textsc{F.~Rudio}, + \textit{Der Bericht des Simplicius über die Quadratur des + Antiphon und Hippokrates}, Leipzig, 1907. + + Moreover as Proposition 18 is suggested by the \textit{Elements}, + \textsc{ii}.\ 5, so here this problem is suggested by \textsc{ii}.\ 6: + \textit{If a straight line be bisected and a straight line be + added to it in a straight line, the rectangle contained by the + whole with the added straight line and the added straight line + together with the square on the half is equal to the square on + the straight line made up of the half and the added straight line.} + + + + If $\mathit{AB}$ is the straight line bisected at $C$ and $\mathit{BD}$ is the + straight line added, then by \textsc{ii}.~6, +\[ + AD\centerdot DB + CB^2 = CD^2. +\] + + In his solution of our problem, Robert Simson proceeds, in effect, + as follows (\textit{Elements of Euclid}, ninth ed., Edinburgh, + 1793, p.~336): Draw $\mathit{BQ}$ at right angles to $\mathit{AB}$ and equal to $c$. + Join $\mathit{CQ}$ and describe a circle with centre $C$ and radius $\mathit{CQ}$ + cutting $\mathit{AB}$ produced in $D$. + Then $\mathit{BD}$ or $x$ is found. For, by \textsc{ii}.~6, + \begin{flalign*} + && AD\centerdot DB + CB^2 &= CD^2,&& \\ + && &= CQ^2,&& \\ + && &= CB^2 + BQ^2, &&\\ + && \therefore AD\centerdot DB &= BQ^2,&&\qquad \\ + &\text{whence} & (a+x)x &= c^2&& \\ + &\text{or} & ax+x^2 &= c^2. && + \end{flalign*} + It was not Euclid's manner to consider more than one solution + in this case. +}. % end footnote 116 + +Draw the straight line $\mathit{tkd}$. Since + +\WrappedGraphic{40mm}{074}{}{r} % Graphic in Proposition 26 +\[ +\text{area }\mathit{zb}\centerdot\mathit{bi}= + \mathit{bi}\centerdot\mathit{ti} + \mathit{ti}^2 = + \text{area }\mathit{bt}\centerdot\mathit{ti}, +\] + +% -----File: 074.png---Folio 62------------------------------------------------------ + +\begin{flalign*} + & & zb:bt &=ti:ib, & \textsc{[vii.~19]}& \\ + &\text{or} & zt:bt &=bt:bi. & \textsc{[v.~18]}& \\ + &\text{But}& zt:bt &=zd:bk, & \textsc{[vi.~4]}& \\ + & & \therefore zd:bk &=bt:bi, && \\ + &\text{and}& \text{area } kb\centerdot bt &= \text{area } zd\centerdot bi. && \\ + &\text{But}& \text{area } zd\centerdot bi&= \tfrac{1}{2}\:\text{area } ab\centerdot bg, && \\ + & & \therefore \triangle\mathit{tbk} &= \tfrac{1}{2}\:\triangle\mathit{abg}\footnotemark[88]. & \\ +\end{flalign*} + + +Therefore the triangle $\mathit{abg}$ is +divided by a line drawn from the +point $d$, that is, by the line $\mathit{tkd}$, into +two equal parts one of which is +the triangle $\mathit{tbk}$, and the other the quadrilateral $\mathit{tkga}$. + +\hfill \textsc{q.~e.~f.}\qquad\qquad + +\SmallLeftNote{Leonardo now gives a numerical example. He then continues:} +% -----File: 075.png---Folio 63------------------------------------------------------ + +[If the point $d$ were on one +side, $\mathit{ab}$, produced at say, $z$], +through $z$ draw $\mathit{ze}$ parallel to +$\mathit{bg}$ and meeting $\mathit{ag}$ produced +in $e$. + +\WrappedGraphic{45mm}{175}{}{r} % Graphic in Proposition 26 + +Make +\[ +\text{area } \mathit{ze}\centerdot\mathit{gi} = + \tfrac{1}{2}\:\text{area } \mathit{ag}\centerdot\mathit{gb}, +\] +and apply a rectangle, equal to +the rectangle $\mathit{eg}\centerdot\mathit{gi}$, to the line +$\mathit{gi}$, but exceeded by a square; +\begin{DPgather*} + \lintertext{then} \mathit{eg}\centerdot\mathit{gi}=\mathit{gt}\centerdot\mathit{ti}. +\end{DPgather*} + +Join $\mathit{tz}$, then [this is the required line. The proof is step +for step as in the first case]. + +\SmallLeftNote{Leonardo then remarks: ``Que etiam demonstrentur in numeris,'' and +proceeds to a numerical example. Thereafter he continues:} + +\WrappedGraphic{50mm}{275}{}{r} % Graphic in Proposition 26 + +But let the sides $\mathit{ab},\mathit{gb}$ of the triangle be produced to $d$ and $e$ +respectively; and let $i$ +be the given point in the angle +$\mathit{ebd}$ from which a line is to be +drawn dividing the triangle +$\mathit{abg}$ into two equal parts. Join +$\mathit{ib}$ and produce it to meet $\mathit{ag}$ in +$z$. If $\mathit{az}=\mathit{zg}$, the triangle $\mathit{abg}$ +is divided into two equal parts +by the line $\mathit{iz}$. But [if $\mathit{az}>\mathit{zg}$,] +let $\mathit{za}$ produced meet, in the +point $t$, the line drawn through +$i$ parallel to $\mathit{ab}$. + +\begin{DPgather*} + \lintertext{\indent Since} \mathit{za}>\tfrac{1}{2}\mathit{ag}, + \qquad\text{area } \mathit{ab}\centerdot\mathit{az} > + \tfrac{1}{2}\:\text{area } \mathit{ba}\centerdot\mathit{ag}. \\ +\lintertext{Make} \text{area } \mathit{it}\centerdot\mathit{ak} = + \tfrac{1}{2}\text{area }\mathit{ba}\centerdot\mathit{ag}; \\ +\lintertext{then make} \text{area } \mathit{al}\centerdot\mathit{kl}= + \text{area } \mathit{ta}\centerdot\mathit{ak}. +\end{DPgather*} + +Join $\mathit{il}$. Then as above the triangle $\mathit{abg}$ is divided into +two equal parts by the line $\mathit{il}$, one part the triangle $\mathit{lac}$, the +other the quadrilateral $\mathit{lcbg}$. + +\SmallLeftNote{To this statement Leonardo adds nothing further. The proof that $k$ lies +between $a$ and $z$, and $l$ between $k$ and $z$, follows as in the first part.} + +\Proposition{27} % Proposition 27 +\Paragraph{}{48} \textit{``To cut off a certain fraction of a triangle % Paragraph 48 +by a straight line drawn from a given point situated outside of the +triangle}\Footnote{117}{ % Footnote 117 + Some generalizations of the triangle problems in Propositions + 19, 20, 26 and 27 may be remarked. Steiner, in 1827, solved the + problem: \textit{through a given point on a sphere to draw an arc + of a great circle cutting two given great circles such that + the intercepted area is equal to a given area.} (\textsc{J.~Steiner}, + ``Verwandlung und Theilung sphärischen Figuren durch Construction,'' + \textit{Crelle Jl}, \textsc{ii} (1827), pp.~56 f. + \textit{Cf.}\ Syllabus of Townsend's course at Dublin Univ., 1846, + in \textit{Nouvelles Annales de Mathématiques}, Sept.~1850, + \textsc{ix}, 364; also Question 427(7) proposed by Vannson + in \textit{Nouvelles Annales\ldots} Jan.~1858, \textsc{xvii}, 45; + answered Aug.\ 1859, \textsc{xviii}, 335--6.) + See also \textsc{Gudermann}, ``Über die niedere Sphärik,'' + \textit{Crelle Jl}, 1832, \textsc{viii}, 368. + + In the next year Bobillier solved, by means of planes and spheres + only, the problem, + \textit{to draw through a given line a plane which shall cut off + from a given cone of revolution a required volume}. + (\textit{Correspondance Math.\ et Physique} + (Quetelet), \textsc{vi}$^e$ livraison, + \textsc{iv}, 2--3, Bruxelles, 1828.)} % end footnote 117 +.'' +[Leonardo 11, p.~121, ll.~22--23.] + +Let $\mathit{abg}$ be the given triangle and $d$ the given point outside. +It is required to cut off from the triangle a certain fraction, say +one-third, by a line drawn +through $d$. Join $\mathit{ad}$, cutting $\mathit{bg}$ in $c$. +If either $\mathit{bc}$ or $\mathit{cg}$ be one-third of $\mathit{bg}$, +then the line $\mathit{ad}$ through the point $d$ +cuts off one-third of the triangle $\mathit{abg}$. +But if this be not the case produce $\mathit{ab}$, +$\mathit{ag}$ to meet in $z$ and $e$ respectively +the line drawn through $d$ parallel +to $\mathit{bg}$. + +\WrappedGraphic{53mm}{076}{}{r} % Graphic in Proposition 27 + +Make +\[ + \text{area } de\centerdot gi + = \tfrac{1}{3} \text{ area } ag\centerdot gb, +\] +and apply to the line $\mathit{gi}$ a rectangle +equal to the rectangle $\mathit{eg}\centerdot\mathit{gi}$, +but exceeded by a square; then +\[ + \mathit{eg}\centerdot\mathit{gi} = \mathit{ik}\centerdot\mathit{kg}. +\] +Draw the line $\mathit{kmd}$. I say that the triangle $\mathit{kmg}$ is one-third +of the triangle $\mathit{abg}$. + +\textit{Proof:} For since +\begin{flalign*} + && \text{area } eg\centerdot gi &= \text{ area } gk\centerdot ki, && \\ + && eg: gk &= ki: ig. && \\ + &\text{\indent Hence}& ek: gk &= gk: gi. & \textsc{[v.~17]}&\\ +% -----File: 077.png---Folio 65------------------------------------------------------ + &\text{\indent But }& ek:kg &= de:gm; & \text{[\textsc{vi}.~2]} &\\ + && \therefore ed:gm &= gk:gi. && \\ + && \therefore \text{ area } gk\centerdot gm + &= \text{ area } de\centerdot gi. && \\ + &\text{\indent But } + & \text{ area } de\centerdot gi + &= \tfrac{1}{3}\text{ area } ag\centerdot gb; &&\\ + && \therefore \text{ area } gk\centerdot gm + &= \tfrac{1}{3} \text{ area } ag\centerdot gb. && +\end{flalign*} +And since +\begin{flalign*} + && \text{ area } gk\centerdot gm : \text{ area } ag\centerdot gb + &= \triangle\mathit{kgm}:\triangle\mathit{agb}\footnotemark[88], &&\\ + && \triangle\mathit{kgm} &= \tfrac{1}{3} \triangle\mathit{agb}. && +\end{flalign*} + +In a similar manner any part of a triangle may be cut off +by a straight line drawn from a given point, on a side of the +triangle produced, or within two produced sides. + +\Proposition{28} % Proposition 28 +\Paragraph{}{49} \textit{``To divide into two equal parts a given figure % Paragraph 49 +bounded by an arc of a circle and by two straight lines which form a +given angle.''} +[Leonardo 57, p.~148, ll.~13--14.] + +``Let $\mathit{ABC}$ be the given figure bounded by the arc $\mathit{BC}$ and by the two +lines $\mathit{AB, AC}$ +which form the angle $\mathit{BAC}$. +It is required to draw a straight +line which will divide the figure +$\mathit{ABC}$ into two equal parts. + +\WrappedGraphic{66mm}{077}{13}{r} % Graphic in Proposition 28 + +``Draw the line $\mathit{BC}$ and +bisect it at $E$. Through the +point $E$ draw a line perpendicular +to $\mathit{BC}$, as $\mathit{EZ}$, and +draw the line $\mathit{AE}$. Then because +$\mathit{BE}$ is equal to $\mathit{EC}$, the +area $\mathit{BZE}$ is equal to the area +$\mathit{EZC}$, and the triangle $\mathit{ABE}$ +is equal to the triangle $\mathit{AEC}$. Then the figure $\mathit{ABZE}$ will +equal the figure $\mathit{ZCAE}$. If the line $\mathit{AE}$ lie in $\mathit{EZ}$ produced, +the figure will be divided into two equal parts $\mathit{ABZE}$ and +$\mathit{CAEZ}$. But if the line $\mathit{AE}$ be not in the line $\mathit{ZE}$ produced, +join $A$ to $Z$ by a straight line and through the point $E$ draw +a line, as $\mathit{ET}$, parallel to the line $\mathit{AZ}$. Finally draw the line +% -----File: 078.png---Folio 66------------------------------------------------------ +$\mathit{TZ}$. I say, that the line $\mathit{TZ}$ is that which it is required to +find, and that the figure $\mathit{ABC}$ is divided into two equal parts +$\mathit{ABZT}$ and $\mathit{ZCT}$. + +``For since the two triangles $\mathit{TZA}$ and $\mathit{EZA}$ are constructed +on the same base $\mathit{AZ}$ and contained between the +same parallels $\mathit{AZ}$, $\mathit{TE}$: the triangle $\mathit{ZTA}$ is equal to the +triangle $\mathit{AEZ}$. Then, adding to each the common part +$\mathit{AZB}$, we have $\mathit{TZBA}$ equal to $\mathit{ABZE}$. But this latter figure +was half of the figure $\mathit{ABC}$; consequently the line $\mathit{ZT}$ is +the line sought and $\mathit{BZCA}$ is divided into two equal parts +$\mathit{ABZT, TZC}$, which was to be demonstrated.'' + +\SmallLeftNote{ +Leonardo's proof is practically word for word as the above. He gives two +figures and in each he uses the \emph{Greek} succession of letters. + +It is doubtless to this Proposition and the next that reference is made in +the account of Proclus [Art.~\textsc{i}].} + +\Proposition{29} % Proposition 29 +\Paragraph{}{50} \textit{``To draw in a given circle two parallel lines % Paragraph 50 +cutting off a certain fraction from the circle.''} +[Leonardo 51 (the case where the fraction is one-third), p.~146, ll.~37--38.] + +``Let the certain fraction be one-third, and the circle be +$\mathit{ABC}$. It is required to do that which is about to be explained. + +\CenteredGraphic{70mm}{078} % Graphic in Proposition 29 + +``Construct the side of the triangle (regular) inscribed in +this circle. Let this be $\mathit{AC}$. Draw the two lines $\mathit{AD}$, $\mathit{DC}$ +and draw through the point $D$ a line parallel to the line $\mathit{AC}$, +such as $\mathit{DB}$. Draw the line $\mathit{CB}$. Divide the arc $\mathit{AC}$ into +% -----File: 079.png---Folio 67------------------------------------------------------ +two equal parts at the point $E$, and draw from the point $E$ +the chord $\mathit{EZ}$ parallel to the line $\mathit{BC}$. Finally draw the line +$\mathit{AB}$. I say that we have two parallel lines $\mathit{EZ, CB}$ cutting +off a third of the circle $\mathit{ABC}$, viz. the figure $\mathit{ZBCE}$. + +``\textit{Demonstration}. The line $\mathit{AC}$ being parallel to the line +$\mathit{DB}$, the triangle $\mathit{DAC}$ will be equal to the triangle $\mathit{BAC}$; +add to each the segment of the circle $\mathit{AEC}$; the whole figure +$\mathit{DAEC}$ will be equal to the whole figure $\mathit{BAEC}$. But the +figure $\mathit{DAEC}$ is one-third of the circle. Consequently the +figure $\mathit{BAEC}$ is also one-third of the circle. Since $\mathit{EZ}$ is +parallel to $\mathit{CB}$, the arc $\mathit{EC}$ will be equal to the arc $\mathit{BZ}$; but +$\mathit{EC}$ is equal to $\mathit{EA}$, hence $\mathit{EA}$ equals $\mathit{ZB}$. Add to these +equal parts the arc $\mathit{ECB}$; the whole arc $\mathit{AB}$ will equal the +whole arc $\mathit{EZ}$. Consequently the line $\mathit{AB}$ will be equal to +the line $\mathit{EZ}$, and the segment of the circle $\mathit{AECB}$ will be +equal to the segment of the circle $\mathit{ECBZ}$. Taking away the +common segment $\mathit{BC}$, there remains the figure $\mathit{EZBC}$ equal +to the figure $\mathit{BAEC}$. But the figure $\mathit{BAEC}$ was one-third +of the circle $\mathit{ABC}$. Then the figure $\mathit{EZBC}$ is one-third of +the circle $\mathit{ABC}$; which was to be demonstrated. + +``When it is required to cut off a quarter of a circle, or a +fifth or any other definite fraction, by means of two parallel +lines, we construct in this circle the side of a square or of the +pentagon (regular) inscribed in the circle and we draw from +the centre to the extremities of this side the two straight lines +as above. (The remainder of) the construction will be +analogous to that which has gone before\Footnote{118}{ % Footnote 118 + This problem is clearly not susceptible of solution with ruler and + compasses, in such a case as when the ``certain fraction,'' + $\frac{1}{n}$, is one-seventh. + In fact the only cases in which the problem is possible, + for a fraction of this kind, is when $n$ is of the form + \[2^p(2^{2^{s_1}}+1)(2^{2^{s_2}}+1)\dots (2^{2^{s_m}}+1),\] + where $p$, and $s$'s (all different), are positive integers or zero, + and $2^{2^{s_m}}+1(m=1, 2, \dots m)$ is a prime number. + (\textit{Cf.}\ C.~F.\ Gauss, + \textit{Disquisitiones Arithmeticae}, Lipsiae, 1801, French ed., + Paris, 1807, p.~489.)} % end footnote 118 + +\small\DPnote{continues for 60 lines} +The statement and form of discussion of this proposition are not wholly +satisfactory. For ``a certain fraction'' in the enunciation we should rather +expect ``one-third,'' as in Leonardo; while at the conclusion of the proof might +possibly occur a remark to the effect that a similar construction would apply +when the certain fraction was one-quarter [by means of \textsc{iv}.~6], +one-fifth [\textsc{iv}.~11], +one-sixth [\textsc{iv}.~15], or one-fifteenth [\textsc{iv}.~16], +but is it conceivable that Euclid +added ``or any other definite fraction''? Moreover the lack of definition of +$D$ and certain matters of form seem to further indicate that modification of +the original has taken place in its passage through Arabian channels. + +% -----File: 080.png---Folio 68------------------------------------------------------ +On the other hand Leonardo presents the proposition as if drawn from +the pure well of Euclid undefiled. Here is his discussion. +(I have substituted $C$ for his $b$, and $B$ for his $g$.) + +``And if, by means of two parallel lines, we wish to cut off from a circle +$\mathit{ACB}$, whose centre is $D$, a given part which is one-third, draw the line $\mathit{AC}$, +the side of an equilateral triangle inscribed in the circle $\mathit{abg}$. Through the +centre $D$ draw $\mathit{DB}$ parallel to this line and join $\mathit{CB}$. +Bisect the arc $\mathit{AC}$ at +$E$ and draw $\mathit{EZ}$ parallel to $\mathit{bg}$. +I say that the figure contained between the lines $\mathit{CB}$ and $\mathit{EZ}$ and +the arcs $\mathit{EC}$ and $\mathit{BZ}$ is one-third part of the circle $\mathit{ACB}$. + +``\textit{Proof:} Draw the lines $\mathit{DA}$ and $\mathit{DB}$ and $\mathit{AB}$. + +``The triangles $\mathit{BAC}$ and $\mathit{DAC}$ are equal. To each add the portion $\mathit{ABE}$. +Then the figure bounded by the lines $\mathit{BA}$ and $\mathit{BC}$ and the arc $\mathit{AEC}$ is +equal to the sector $\mathit{DAEC}$ which is a third part of the circle $\mathit{ABC}$. + +``Therefore the figure bounded by the lines $\mathit{BA}$ and $\mathit{BC}$ and the +arc $\mathit{AEC}$ is a third part of the circle. + +``And since the lines $\mathit{CB}$ and $\mathit{EZ}$ are parallel, +the arcs $\mathit{EC}$ and $\mathit{BZ}$ are equal. +But arc $\mathit{EC}$ is equal to arc $\mathit{AE}$. Therefore arc $\mathit{AE}$ is equal to the +arc $\mathit{BZ}$. +To each add the arc $\mathit{EB}$, and then the arc $\mathit{AECB}$ will be equal to +the arc $\mathit{ECBZ}$. + +``Hence the portion $\mathit{EZBC}$ of the circle is equal to the portion $\mathit{ABCE}$. +Take away the common part between the line $\mathit{CB}$ and the arc $\mathit{BC}$ and there +remains the figure, bounded by the lines $\mathit{BC}$ and $\mathit{EZ}$ and the arcs $\mathit{CE}$ and +$\mathit{BZ}$, which is the third part of the circle since it is equal to the figure bounded +by the lines $\mathit{BA}$ and $\mathit{BC}$ and the arc $\mathit{AEC}$; +quod oportebat ostendere.'' + +In his \ItalicGreekSnippet{metrik'a} +(\textsc{iii}.~18) Heron of Alexandria considers the problem: +\textit{To divide the area of a circle into three equal parts by two straight lines}. He +remarks that ``it is clear that the problem is not rational''; nevertheless ``on +account of its practical use'' he proceeds to give an approximate solution. +By discussion similar to that above he finds the figure $\mathit{BCEA}$, formed by +the triangle $\mathit{BCA}$ and the segment $\mathit{CEA}$, to be one-third of the circle. +Neglecting the smaller segment with chord $\mathit{BC}$, we have, that $\mathit{BA}$ cuts off +``approximately'' one-third of the circle. Similarly a second chord from $B$ +might be drawn to cut off another third of the circle, and the approximate +solution be completed. +\normalsize + +% -----File: 081.png---Folio 69------------------------------------------------------ + +\Proposition{30} % Proposition 30 +\Paragraph{}{51} \textit{''To divide a given triangle into two parts by % Paragraph 51 +a line parallel to its base, such that the ratio of one of the two +parts to the other is equal to a given ratio.''} + +\WrappedGraphic{53mm}{181}{12}{r} % Graphic in Proposition 30 + +\SmallLeftNote{\DPnote{both Prop 30 and 31 are set \small. Not sure why.??} +Although Leonardo does not explicitly formulate this problem or the +next, the method to be employed is clearly +indicated in the discussion of Proposition~5 (Art.~26). + +Let $\mathit{abg}$ be the triangle which is to be +divided in the given ratio $\mathit{ez:zi}$, by a line +parallel to $\mathit{bg}$. Divide $\mathit{ab}$ in $h$ such that +\[ + \mathit{ah}^2 : \mathit{ab}^2 = \mathit{ez} : \mathit{ei}\footnotemark[92]. +\] + +Draw $\mathit{hk}\parallel\mathit{bg}$ and meeting $\mathit{ag}$ in $k$. Then +the triangles $\mathit{ahk}$ and $\mathit{abg}$ are similar and +\[ + \triangle\mathit{ahk} : \triangle\mathit{abg} = \mathit{ah}^2: \mathit{ab}^2. + \tag*{[\textsc{vi}.~19]} +\] +\begin{flalign*} +&\text{\indent But}& ah^2: ab^2 &= \mathit{ez:ei}, && \\ +&& \therefore \triangle\mathit{ahk} : \triangle\mathit{abg} &= \mathit{ez:ei};&& \\ +&\text{whence}& \triangle\mathit{ahk} : \text{quadl.}\mathit{hbgk} &= ez: zi; &[\textsc{v}.~16, 17]& +\end{flalign*} +and the triangle $\mathit{abg}$ has been divided as required.} + + +\Proposition{31} % Proposition 31 +\Paragraph{}{52} \textit{``To divide a given triangle by lines parallel % Paragraph 52 +to its base into parts which have given ratios to one another.''} + +\WrappedGraphic{50mm}{281}{10}{r} % Graphic in Proposition 31 + +\SmallLeftNote{ +Again in the manner of Proposition~5, suppose it be required to divide +the triangle $\mathit{abg}$ into three parts in the ratio\\ +$\mathit{ez:zt:ti}$. Then determine the points $h$, $l$ +in $\mathit{ab}$ such that +\begin{flalign*} + && ah^2: ab^2 &= ez: ei\footnotemark[92]; && \\ + &\text{and} & al^2: ab^2 &= et: ei. \\ + &\text{Then} & ah^2: al^2 &= ez: et. &&[\textsc{v}.~16, 20] +\end{flalign*} + +\begin{flalign*} + && \therefore\; & \triangle\mathit{ahk} : \triangle\mathit{alm} = \mathit{ez:et}, &&\\ + &\text{and} & \therefore\triangle\mathit{ahk} & : \text{quadl.} \mathit{hlmk} = \mathit{ez:zt}.&&\\ + &\text{\indent Similarly} &&&&\\ + && \triangle\mathit{alm} & : \text{quadl.}\mathit{lbgm} = \mathit{et:ti}. &&\\ + &\text{\indent But} & \triangle\mathit{ahk} & : \triangle\mathit{alm} = \mathit{ez:et}. &&\\ + && \therefore \triangle\mathit{ahk} & : \text{quadl.}\mathit{lbgm} = \mathit{ez:ti}. + &[\textsc{v}.~20]&\\ + &\text{\indent Hence,}& & \triangle\mathit{ahk}: \text{quadl.}\mathit{hlmk}: + \text{quadl.}\mathit{lbgm} = \mathit{ez:zt:ti}, && +\end{flalign*} +and the triangle $\mathit{abg}$ has been divided into three parts in a given ratio to one +another. So also for any number of parts which have given ratios to one +another.} +% -----File: 082.png---Folio 70------------------------------------------------------ + +\Proposition{32} % Proposition 32 +\Paragraph{}{53} \textit{``To divide a given trapezium by a line parallel % Paragraph 53 +to its base, into two parts such that the ratio of one of these parts +to the other is equal to a given ratio.''} +[Leonardo 29, p.~131, ll.~41--42.] + +Let $\mathit{abgd}$ be the trapezium which is to be divided in the ratio $\mathit{ez:zi}$ +by a line parallel to +the base. Produce the sides $\mathit{ba}$, +$\mathit{gd}$ through $a$ and $d$ to meet in $t$. + +\WrappedGraphic{53mm}{082}{16}{r} % First Graphic in Proposition 32 + +\begin{DPalign*} + \lintertext{\indent Make} tl^2 : at^2 &= zi : ez\footnotemark[92], \\ + \lintertext{and} ht^2 : (bt^2 + tl^2) &= ez : ei. +\end{DPalign*} + +Through $l$, $h$, draw $\mathit{lm}$, $\mathit{hk}$ +parallel to $\mathit{bg}$ and $\mathit{ad}$. Then I say +that the quadrilateral $\mathit{ag}$ is divided +in the given ratio, $\mathit{ez:zi}$, by the +line $\mathit{hk}$. + +\textit{Proof:} For since the triangles +$\mathit{tlm, tad}$ are similar +\[ + tl^2 : at^2 = \triangle\mathit{tlm} : \triangle\mathit{tad}; +\] + +\begin{flalign*} + &\text{but} & tl^2 : at^2 &= zi : ez; && \\ + && \therefore zi : ez &= \triangle\mathit{tlm} : \triangle\mathit{tad}. && \\ + &\text{Whence} & ei : ez &= (\triangle\mathit{tlm} + \triangle\mathit{tad}) : \triangle\mathit{tad}, & \textsc{[v.~18]}& \\ + &\text{or} & ez : ei &= \triangle\mathit{tad} : (\triangle\mathit{tlm} + \triangle\mathit{tad}). & \textsc{[v.~16]}& +\end{flalign*} + +But by construction +\begin{flalign*} + && \mathit{ez:ei} &= \mathit{ht}^2 : (\mathit{bt}^2 + \mathit{tl}^2), && \\ + &\text{and} & ht^2 : (bt^2 + tl^2) &=\triangle\mathit{thk} : (\triangle\mathit{tbg} + + \triangle\mathit{tlm}). & \textsc{[vi.~19]}& \\ + && \therefore \mathit{ez:ei} &= \triangle\mathit{thk} : (\triangle\mathit{tbg} + + \triangle\mathit{tlm}). &&\\ + &\text{\indent But}& + \triangle\mathit{thk}&=\triangle\mathit{tad}+\text{quadl.}\mathit{ak}. && +\end{flalign*} + +Similarly +\begin{gather*} + \triangle\mathit{tbg} + \triangle\mathit{tlm} = \text{quadl.}\mathit{ag} + \triangle\mathit{tad} + + \triangle\mathit{tlm}.\\ + \therefore ez : ei = (\text{quadl.}\mathit{ak} + \triangle\mathit{tad}) : + (\text{quadl.}\mathit{ag} + \triangle\mathit{tad} + \triangle\mathit{tlm}). +\end{gather*} +% -----File: 083.png---Folio 71------------------------------------------------------ +\begin{flalign*} + & \text{\indent But } + & \mathit{ez:ei} &= \triangle\mathit{tad}: (\triangle\mathit{tad} + \triangle\mathit{tlm}); &&\\ + && \therefore \mathit{ez:ei} &= \text{quadl.}\mathit{ak}:\text{quadl.}\mathit{ag}; + & \text{[\textsc{v}.~11,~19]}&\\ + & \text{whence} & \mathit{ez:zi} &= \text{quadl.}\mathit{ak}:\text{quadl.}\mathit{hg}. && +\end{flalign*} +And the trapezium has been divided in the given ratio. + +\SmallLeftNote{Then follows a numerical example and this alternative construction and proof:} + +Draw $\mathit{mls}$ such that, +\[ + \mathit{ms:ls} = \mathit{tb}^2:\mathit{ta}^2,\footnotemark[92] +\] +and divide $\mathit{ml}$ in $n$, such that $\mathit{ln}$ is to $\mathit{nm}$ in the given proportion. + +\CenteredGraphic{70mm}{083} % Second Graphic in Proposition 32 + +In $\mathit{tb}$ determine $h$ such that +\[ + \mathit{th}^2: \mathit{tb}^2 = \mathit{ns:sm}. +\] + +Draw $\mathit{hk}\parallel\mathit{bg}$. Then, +\[ + \text{quadl.}\mathit{ak}:\text{quadl.}\mathit{hg} = \mathit{ln:nm}. +\] +\begin{flalign*} + & \text{\indent \textit{Proof:} For, } + & tb^2: ta^2 &= \triangle\mathit{tbg}: \triangle\mathit{tad}; + & \text{[\textsc{vi}.~19]} & \\ + & \text{and} & \mathit{ms:ls} &= \mathit{tb}^2: \mathit{ta}^2. &&\\ + && \therefore\mathit{ms:ls} &= \triangle\mathit{tbg}: \triangle\mathit{tad}. & + \ldots\ldots\ldots\text{[1]} &\\ + & \text{Again, since} & \mathit{tb}^2: \mathit{th}^2 &= \mathit{ms:sn}, &&\\ + & \text{and} & \triangle\mathit{tbg}: \triangle\mathit{thk} &= tb^2: th^2, &&\\ + && \therefore\mathit{ms:ns} &= \triangle\mathit{tbg}: \triangle\mathit{thk}; & + \ldots\ldots\ldots\text{[2]} &\\ + && \therefore\mathit{sm:nm} &= \triangle\mathit{tbg}: \text{quadl.}\mathit{hg}. + & \text{[\textsc{v}.~16,~21] \ldots\ldots[3]}&\\ +% -----File: 084.png---Folio 72------------------------------------------------------ + &\text{But } + & \mathit{sm:ls} &= \triangle\mathit{tbg}: \triangle\mathit{tad}\footnotemark[1181], &&\\ + &\text{or} & \mathit{ms}: \triangle\mathit{tbg} &= \mathit{ls}: \triangle\mathit{tad}, &&\\ + &\text{while } & \mathit{ms}: \triangle\mathit{tbg} &= \mathit{ns}: \triangle\mathit{thk}; + &\text{[from [2]]} &\\ + && \therefore\mathit{ls : ns} &= \triangle\mathit{tad}:\triangle\mathit{thk}. & + \ldots\ldots\ldots\text{[4]} &\\ + &\text{\indent From [3]} + & \mathit{ms}:\triangle\mathit{tbg} &=\mathit{nm}: \text{quadl.}\mathit{hg}. &&\\ + &\text{\indent But from [4]} + & \mathit{sl : ln} &= \triangle\mathit{tad}:\text{quadl.}\mathit{ak}, + &\text{[\textsc{v}.~16, 21]} & \\ +%% The next four lines were originally set in a smaller font. + && \therefore\mathit{sl}:\triangle\mathit{tad} &= \mathit{ln}:\text{quadl.}\mathit{ak}. &&\\ + &\text{\indent But from [1]} + & \mathit{ms}:\triangle\mathit{tbg} &= \mathit{sl}:\triangle\mathit{tad}, &&\\ + && \therefore\mathit{ms}:\triangle\mathit{tbg} &= \mathit{ln}: \text{quadl.}\mathit{ak}; && \\ + && \therefore\mathit{mn}: \text{quadl.}\mathit{hg} &= \mathit{ln}:\text{quadl.}\mathit{ak}; &&\\ + && \therefore\mathit{ln : nm} &= \text{quadl.}\mathit{ak}: \text{quadl.}\mathit{hg}. && +\end{flalign*} +\Footnotetext{1181}{ % Footnote 1181 (was 118a) + Such mixed ratios as these (ratios of lines to areas), + and others of like kind which follow in this proof, + are very un-Greek in their formation. This is sufficient to stamp + the second proof as of origin other than Greek. The first proof, on + the other hand, is distinctly Euclidean in character.} % end footnote 1181 + +Hence the quadrilateral $\mathit{ag}$ is divided by the line $\mathit{hk}$, +parallel to the base $\mathit{bg}$, in the given proportion as the +number $\mathit{ln}$ is to the number $\mathit{nm}$. Which was to be done. + +\SmallLeftNote{Then follows a numerical example.} + +\Proposition{33} % Proposition 33 +\Paragraph{}{54} \textit{``To divide a given trapezium, by lines parallel % Paragraph 54 +to its base, into parts which have given ratios to one another.''} +[Leonardo 35, p.~137, ll.~6--7.] + +Let $\mathit{abgd}$ be the given trapezium and [let $\mathit{ez:zi:it}$ denote +\WrappedGraphic{65mm}{084}{}{r} % Graphic in Proposition 33 +the ratios of the three parts +into which the trapezium is to +be divided by lines parallel to +the base $\mathit{bg}$]. Let $\mathit{ba,gd}$ produced +meet in $k$ and find $l$ such +that +\[ + bk^2:ak^2 = \mathit{tl:el}. +\] +Then determine $m$ and $n$ such +that +\begin{flalign*} + & & \mathit{bk}^2: \mathit{km}^2 &= \mathit{tl:lz}, &&\\ + &\text{and} & \mathit{bk}^2: \mathit{kn}^2 &= \mathit{tl:il}. && +\end{flalign*} + +% -----File: 085.png---Folio 73------------------------------------------------------ + +Through $\mathit{m,n}$ draw lines $\mathit{mo,np}$ parallel to $\mathit{bg}$. In the +same manner as above +\begin{flalign*} + & & \text{quadl.}\mathit{ao}:\text{quadl.}\mathit{mp} &=\mathit{ez:zi}; &&\\ + &\text{and}& \text{quadl.}\mathit{mp}:\text{quadl.}\mathit{ng} &=\mathit{zi:it}. && +\end{flalign*} + +\small\DPnote{continues for 40 lines} +Then follows a numerical example in which the line $\mathit{kfhrs}$, perpendicular +to $\mathit{bg}$, is introduced into the figure. + +Here is a proof of the Proposition: + +By construction, \textsc{v}.~16 and \textsc{vi}.~19, +\begin{flalign*} + & &&\triangle\mathit{kbg}:\triangle\mathit{kad}=\mathit{tl:el}. + &\ldots\ldots\ldots\ldots\ldots\ldots\text{[1]}& \\ + &\text{\indent So also} &&\triangle\mathit{kbg}:\triangle\mathit{kmo}=\mathit{tl:lz}, + &\ldots\ldots\ldots\ldots\ldots\ldots\text{[2]}& \\ + &\text{and } &&\triangle\mathit{kbg}:\triangle\mathit{knp}=\mathit{tl:il} + &\ldots\ldots\ldots\ldots\ldots\ldots\text{[3]}&\\ + &\text{\indent From [1] } &&\triangle\mathit{kad}:\triangle\mathit{kbg}=\mathit{el:tl}. &&\\ + &\text{\indent But from [2] }&&\triangle\mathit{kbg}:\triangle\mathit{kmo}=\mathit{tl:lz}; &&\\ + &\text{hence, by [\textsc{v}.~20],}&&\triangle\mathit{kad}:\triangle\mathit{kmo}=\mathit{el:lz}, + &\ldots\ldots\ldots\ldots\ldots\ldots\text{[4]}&\\ + &\text{or alternately}& &\triangle\mathit{kmo}:\triangle\mathit{kad}=\mathit{lz:el}. &&\\ + &\text{\indent Hence, + \textit{separando},}&&\text{quadl.}\mathit{ao}:\triangle\mathit{kad}=\mathit{ez:el}. + &\ldots\ldots\ldots\ldots\ldots\ldots\text{[5]}&\\ + &\rlap{\text{\indent So also from [2] and [3]}}& & &&\\ + && &\triangle\mathit{kmo}:\triangle\mathit{knp}=\mathit{lz:il}; &&\\ + &\text{and}& &\triangle\mathit{kmo}:\text{quadl.}\mathit{mp}=\mathit{lz:iz}. &&\\ + &\text{\indent But from [4]}&&\triangle\mathit{kad}:\triangle\mathit{kmo}=\mathit{el:lz}; &&\\ + &\text{therefore, by [\textsc{v}.~20], } + &&\triangle\mathit{kad}:\text{quadl.}\mathit{mp}=\mathit{el:iz} + &\ldots\ldots\ldots\ldots\ldots\ldots\text{[6]}&\\ +%%%\end{flalign*} +%%%\begin{flalign*} + &\rlap{\text{\indent Hence from [5], by [\textsc{v}.~20],}} &&&&\\ + && &\text{quadl.}\mathit{ao}:\text{quadl.}\mathit{mp}=\mathit{ez:zi}. &&\\ + &\text{\indent Again, from [3],}&&\text{quadl.}\mathit{ng}:\triangle\mathit{kbg}=\mathit{ti:tl}; &&\\ + &\text{and since from [1], }&&\quad\triangle\mathit{kbg}:\triangle\mathit{kad}=\mathit{tl:el}, &&\\ + &\text{we have}&&\text{quadl.}\mathit{ng}:\triangle\mathit{kad}=\mathit{it:el}. &&\\ + &\intertext{\indent Hence from [6], by [\textsc{v}.~20], we get} \\ + & &&\text{quadl.}\mathit{ng}:\text{quadl.}\mathit{mp}=\mathit{it:zi}, &&\\ + &\text{or alternately}&&\text{quadl.}\mathit{mp}:\text{quadl.}\mathit{ng}=\mathit{zi:it}. && +\end{flalign*} + +And since $\text{quadl.}\mathit{ao}:\text{quadl.}\mathit{mp}=\mathit{ez:zi}$, +the trapezium $\mathit{ag}$ has been divided +by lines parallel to the base $\mathit{ag}$, into three parts which are in the required +ratios to one another. \hfill \textsc{q.~e.~f.}\qquad +\normalsize +% -----File: 086.png---Folio 74------------------------------------------------------ + +\Proposition{34} % Proposition 34 +\Paragraph{}{55} \textit{``To divide a given quadrilateral, by a line % Paragraph 55 +drawn from a given vertex of the quadrilateral, into two parts such +that the ratio of one of these parts to the other is equal to a +given ratio.''} +[Leonardo 40, p.~140, ll.~36--37.] + +Let $\mathit{abcd}$ be the given quadrilateral, and $\mathit{ez:zi}$ the given ratio. +It is required to draw from the +angle $d$ a line to divide the +quadrilateral in the ratio $\mathit{ez:zi}$. +\WrappedGraphic{70mm}{086}{}{r} % Graphic in Proposition 34 + +Draw the diagonal $\mathit{ac}$ and +on it find $t$ such that +\[ + \mathit{ct:at}=\mathit{ez:zi}. +\] + +Draw the diagonal $\mathit{bd}$. Then +if $\mathit{bd}$ pass through $t$ the quadrilateral +is divided as required, +in the ratio $\mathit{ez:zi}$. + +For, +\begin{flalign*} + && \triangle\mathit{dct}:\triangle\mathit{dta} &= \mathit{ct:ta}, &&\\ + && &= \triangle\mathit{cbt}:\triangle\mathit{tba}. &&\\ + && \therefore\mathit{ct:ta} &= \triangle\mathit{dcb}:\triangle\mathit{abd}. + &&[\textsc{v}.~18]\\ + &\text{\indent But} &\mathit{ct:ta}&= \mathit{ez:zi}; &&\\ + && \therefore\mathit{ez:zi} &= \triangle\mathit{bdc}:\triangle\mathit{bda}; && +\end{flalign*} +and the quadrilateral $\mathit{ac}$ is divided, by a line drawn from a +given angle, in a given ratio. + +But if $\mathit{bd}$ do not pass through $t$, it will cut $\mathit{ca}$ either +between $c$ and $t$ or between $t$ and $a$. Consider first when +$\mathit{bd}$ cuts $\mathit{ct}$. Join $\mathit{bt}$ and $\mathit{td}$. Then, +\[ + \text{quadl.}\mathit{tbcd}:\text{quadl.}\mathit{tbad} = \mathit{ct:ta}=\mathit{ez:zi}. +\] + +Draw $\mathit{tk}\parallel\mathit{bd}$ and join $\mathit{dk}$. Then +\begin{DPalign*} + \text{quadl.}\mathit{kbcd} &= \text{quadl.}\mathit{tbcd}; \\ + \therefore\text{quadl.}\mathit{kbcd}:\triangle\mathit{dak} &=\mathit{ez:zi}, +\end{DPalign*} +and the line $\mathit{dk}$ has been drawn as required. + +% -----File: 087.png---Folio 75------------------------------------------------------ + +If the diagonal $\mathit{bd}$ cut $\mathit{at}$, through $t$ draw $\mathit{tl}$ +parallel to the +diagonal $\mathit{bd}$. Join $\mathit{dl}$. Then as before, +\[ + \mathit{ct:ta}=\mathit{ez:zi}=\triangle\mathit{dcl}:\text{quadl.}\mathit{abld}. +\] + +\CenteredGraphic{68mm}{187} % Graphic in Proposition 34 + +\SmallLeftNote{ +Hence in every case the quadrilateral has been divided as required by a +line drawn from $d$. Similarly for any other vertex of the quadrilateral.} + +\Proposition{35} % Proposition 35 +\Paragraph{}{56} \textit{``To divide a given quadrilateral by lines % Paragraph 56 +drawn from a given vertex of the quadrilateral into parts which +are in given ratios to one another.''} + +\SmallLeftNote{ +Although Leonardo does not explicitly formulate this problem, the +method he would have followed is clear from his discussion of the last +Proposition. Let $\mathit{abcd}$ be the quadrilateral to be divided, by lines drawn +from $d$, into three parts in the ratios to one another of $\mathit{ez:zi:it}$. + +\CenteredGraphic{72mm}{287} % Graphic in Proposition 35 + +Divide $\mathit{ca}$ at points $\mathit{r,t}$ so that +\[ + \mathit{cr:rt:ta}=\mathit{ez:zi:it}. +\] + +Through $\mathit{r,t}$ draw lines parallel to $\mathit{bd}$, and meeting $\mathit{bc}$ +(or $\mathit{ab}$) in $l$ and +$\mathit{ab}$ (or $\mathit{bc}$) in $m$. + +Then as above $\mathit{dl}$, $\mathit{dm}$ divide the quadrilateral as required. + +We may proceed in a similar manner to divide the quadrilateral $\mathit{abcd}$, by +lines drawn from the angular point $d$, into any number of parts in given ratios +to one another.} + +% -----File: 088.png---Folio 76------------------------------------------------------ + +\Proposition{36} % Proposition 36 +\Paragraph{}{57} \textit{``Having resolved those problems which % Paragraph 57 +have gone before, we are in a position to divide a given quadrilateral +in a given ratio or in given ratios by a line or by lines drawn +from a given point situated on one of the sides of the quadrilateral, +due regard being paid to the conditions mentioned above.''} + +\small\DPnote{continues for 92 lines} +This problem, also, is not formulated by Leonardo; but from his discussion +of Euclid's Propositions 16, 17 and of his own 41, the methods of construction +which Euclid might have employed are clearly somewhat as follows. + +Let $\mathit{abcd}$ be the given quadrilateral and $g$ the given point. + +\CenteredGraphic{85mm}{188} % Graphic in Proposition 36 + +(1) Let it be required to divide $\mathit{abcd}$ into two parts in the ratio $\mathit{ez:zi}$ by +a line drawn through a point $g$ in the side $\mathit{ad}$. + +Draw $\mathit{dl}$ such that + $\triangle\mathit{cld}:\text{quadl.}\mathit{lbad}=\mathit{ez:zi}$. \hfill [Prop.~34] \qquad\newline +Join $\mathit{gl}$. If $\mathit{gl}\parallel\mathit{dc}$, join $\mathit{gc}$, +then this line divides the quadrilateral as required. + +If $\mathit{gl}$ be not parallel to $\mathit{dc}$ draw $\mathit{dh}\parallel\mathit{gl}$, +and meeting $\mathit{bc}$ in $h$. Join $\mathit{gh}$. +Then $\mathit{gh}$ divides the quadrilateral as required. + +If $\mathit{dh}$ fell outside the quadrilateral +draw $\mathit{ll'}\parallel\mathit{cd}$ (not indicated in the figure) +to meet $\mathit{ad}$ in $\mathit{l'}$. +Draw $\mathit{l'z'}\parallel\mathit{gc}$ to meet $\mathit{dc}$ in $\mathit{z'}$. +Then $\mathit{gz'}$ is the line required. + +The above reasoning is on the assumption that $\mathit{dl}$ meets $\mathit{bc}$ in $l$. Suppose +now it meet $\mathit{ab}$ in $l$. Join $\mathit{bd}$ and +draw $\mathit{bk}$ such that +\[ +\text{quadl.}\mathit{bcdk}:\triangle\mathit{kab}=\mathit{ez:zi}. +\] + +\WrappedGraphic{85mm}{288}{}{r} % Graphic in Proposition 36 + +If $k$ do not coincide with $g$ there +are two cases to consider according +as $k$ is between $g$ and $d$ or between +$g$ and $a$. Consider the former case. + +Through $k$ draw $\mathit{km}$ parallel to $\mathit{gb}$ +and meeting $\mathit{bc}$ in $m$. Join $\mathit{gm}$. Then +this is the required line dividing the +quadrilateral $\mathit{ac}$ in such a way that +\[ +\text{quadl.}\mathit{abmg}:\text{quadl.}\mathit{gmcd}=\mathit{ez:zi}. +\] +% -----File: 089.png---Folio 77------------------------------------------------------ + +Similarly if $k$ were between $g$ and $a$. + +(2) Let it be required to divide $\mathit{abcd}$ into, say, three parts in the ratios +$\mathit{ez:zi:it}$, by lines through any point $g$ in the side $\mathit{ad}$ (first figure). + +Draw $\mathit{dl,dm}$ dividing the quadrilateral $\mathit{ac}$ into three parts such that +\begin{equation*} +\triangle\mathit{amd}:\text{quadl.}\mathit{dmbl}:\triangle\mathit{dlc}=\mathit{ez:zi:it}. +\end{equation*} + +There are various cases to consider according as $l$ and $m$ are both +on $\mathit{bc}$, both on $\mathit{ab}$, or one on $\mathit{ab}$ and one on $\mathit{bc}$. +The method will be obvious from working +out one case, say the last. + +Join $\mathit{gc,gl}$. If $\mathit{gl}$ be parallel to $\mathit{cd}$, +$\mathit{gc}$ cuts off the triangle $\mathit{gdc}$ such that +\begin{flalign*} + && \triangle\mathit{gdc}:\text{quadl.}\mathit{abcg}&=\mathit{it:ei \qquad(=ez+zi)}. + &\text{ [\textsc{v}.~24]}\qquad& +\end{flalign*} + +If $\mathit{gl}$ be not parallel to $\mathit{dc}$, +draw $\mathit{dh}$ parallel to $\mathit{gl}$ and meeting $\mathit{bc}$ in $h$; +then $\mathit{gh}$ divides the quadrilateral in such a way that +\[ + \text{quadl.}\mathit{gdch}: \text{quadl.}\mathit{ghba}=\mathit{it:ei}. +\] + +Then apply Proposition 34 to draw from $g$ a line to divide the quadrilateral +$\mathit{abhg}$ in the ratio of $\mathit{ez:zi}$. + +Hence from $g$ are drawn two lines which divide the quadrilateral $\mathit{abcd}$ +into three parts whose areas are in the ratios of $\mathit{ez:zi:it}$. + +The case when $\mathit{dh}$ meets $\mathit{bc}$ produced may be considered as above. + +We could proceed in a similar manner if the quadrilateral $\mathit{abcd}$ were to be +divided by lines drawn from $g$, into a greater number of parts in given ratios. + +The enunciation of this proposition is a manifest corruption of what Euclid +may have given. Such clauses as those at the beginning and end he would +only have included in the discussion of the construction and proof. +\normalsize + +\bigskip +After the enunciation of Proposition 36, Woepcke's translation +of the Arabian MS. concludes as follows: + +``End of the treatise. We have confined ourselves to +giving the enunciations without the demonstrations, because +the demonstrations are easy.'' +\ + +% -----File: 090.png---Folio 78------------------------------------------------------ + + +\Chapter{IV} % Chapter IV +\phantomsection\pdfbookmark[0]{Appendix} + {Appendix} +\flushbottom\DPnote{Chapter III was set raggedbottom; return to flushbottom.} +\thispagestyle{empty} + +\label{appendix} +\label{App:appendix} +\medskip + +\LargeCenteredNote{APPENDIX} +\small +\medskip + +In the earlier pages I have referred to works on Divisions of Figures +written before 1500. Several of these were not published till later; for +example, that of Muhammed Bagdedinus in 1570, of Leonardo Pisano in +1862 and the second edition of Luca Paciuolo's ``Summa'' +in 1523\Footnote{119}{ % Footnote 119 + A synopsis of the portion of the work on divisions of figures is + given on pages 106 and 275--284 of + \textit{Scritti inediti del P.~D.\ Pietro Cossali\ldots}pubblicati + da Baldassarre Boncompagni, + Roma, 1857. \textit{Cf}.\ note 61.}. % end footnote 119 +It has been remarked that Fra Luca's treatment of the subject was based +entirely upon that of Leonardo. But, on account of priority in publication, +to Paciuolo undoubtedly belongs the credit of popularizing the problems on +Divisions of Figures. + +While few publications treat of the subject in the early part, their number +increases in the latter part, of the sixteenth century. In succeeding centuries +the tale of titles is enormous and no useful purpose would be served by the +publication here of an even approximately complete list. It would seem, +however, as if the subject matter were of sufficient interest to warrant, as +completion of the history of the problems, a selection of such references in +this period, (1) to standard or popular works, (2) to the writings of eminent +scientists like Tartaglia, Huygens, Newton, Kepler and Euler; (3) to special +articles, pamphlets or books which treat parts of the subject; (4) to discussions +of division problems requiring other than Euclidean methods for +their solution. + +No account is taken of the extensive literature on the division of the +circumference of a circle, from which corresponding divisions of its area +readily flow. Considerations along this line may be found in: +\textsc{P.~Bachmann}, +\textit{Die Lehre von der Kreisteilung und ihre Beziehungen zur Zahlentheorie}, +Leipzig, 1872, 12 + 300 pp.; and in \textsc{A.~Mitzscherling}, +\textit{Das Problem der Kreisteilung}, +Leipzig und Berlin, 1913, 6 + 214 pp. + +Except for about a dozen titles, all the books or papers mentioned have +been personally examined. In many cases it will be found that only a single +problem (often Euclid's Propositions 19, 20, 26 or 27) is treated in the place +referred to. + +Some titles in note 111 may also be regarded as forming a supplement to +this list. + +\noindent + +\begin{Description} +\AppItem{1539}{1539} +\textsc{W.~Schmid}. +\textit{Das Erst} [Zweit, Dritt und Viert] \textit{Buch der Geometria}. +Nürnberg. + +{\footnotesize ``Dritter Theil, von mancherley Art der Flächen, wie dieselben gemacht und +ausgetheilt werden, auch wie eine Fläche in die andern für sich selbst, oder gegen +einer andern in vorgenommener Proporz, geschätzt, verändert mag werden. Theilungen +und Zeichnungen von Winkeln, Figuren, ordentlichen Vielecken, die letzten, wie man +leicht denken kann, nicht alle geometrisch richtig. Verwandlung von Figuren.''} + +\AppItem{1547}{1547a} +\textsc{L.~Ferrari}. A ``Cartello'' which begins: +``Messer N.~Tartaglia, \DPtypo{gia}{già} +\DPnote{[** cf. http://edit16.iccu.sbn.it/scripts/iccu_ext.dll?fn=10&i=40964]} +otto giorni, cioè alli 16 di Maggio, in risposta della mia replica io riceuetti +la uostra tartagliata, etc.'' [Milan.] + +{\footnotesize Dated June 1, 1547; a challenge to a mathematical disputation from L.~F.\ to +N.~Tartaglia.} + +% -----File: 091.png---Folio 79------------------------------------------------------ + +\AppItem{1547}{1547b} +\textsc{N.~Tartaglia.} +\textit{Terza Risposta data da +N.~Tartalea\ldots al eccellente +M.~H. Cardano\ldots et al eccellente Messer L. Ferraro\ldots con la resolutione, +ouer risposta de 31 quesiti, ouer quistioni da quelli a lui proposti.} +[Venice, 1547.] + +\footnotesize +Dated July 9, 1547. For the discussion between Ferrari (1522--1565) and Tartaglia +(1500--1557) 6 ``cartelli'' by Ferrari and 6 ``Risposte'' by Tartaglia were published at +Milan, Venice and Brescia in 1547--48\Footnote{120}{ % Footnote 120 +\textit{Cf.}\ \textsc{Cantor}, \textit{Vorlesungen über Gesch.\ d.\ Math.} +Bd \textsc{ii}, 2te Aufl., 1900, pp.~490--491, +where the exact dates are given.}. % end footnote 120 +They contained the problems and their +solutions. These publications are of excessive rarity. Only about a dozen copies +(which are in the British Museum and Italian Libraries) are known to exist. But they +have been reprinted in: +\textit{I sei cartelli di matematica disfida primamente intorno alla +generale risoluzione delle equazioni cubiche, di Ludovico Ferrari, +coi sei contro-cartelli in +risposta di Nicolo Tartaglia\ldots comprendenti le soluzioni de' quesiti dall' una e dall' altra +parte proposti\ldots Raccolti, autografati e pubblicati da Enrico Giordani.}\ldots Milano 1876. + +On pages 6--7 of the \textsc{iii}$^\circ$ cartello (Giordani's edition pp.~94--95), Questions 5 and +14, proposed by Ferrari, are:--- + +``5. To bisect, by a straight line, an equilateral, but not equiangular, heptagon.'' + +``14. Through a point without a triangle to draw a line which will cut off a third.'' + +On pages 12 and 20 of the \textsc{iii}$^\circ$ Risposta Tartaglia gives the solutions and assigns +due credit to the treatment of problems on the Division of Figures by Luca Paciuolo. +The general subject was treated much more at length by Tartaglia in a part of his +``General trattato'' published in 1560. +\small + +\AppItem {1560}{1560} +\textsc{N.~Tartaglia.} +\textit{La quinta parte del general trattato de' numeri et misure.} +Venetia. + +{\footnotesize On folio 6 \textit{recto} we have a section entitled ``Del modo di +saper dividere una \DPtypo{figure}{figura} +cioè pigliar, over formar una parte di quella in forma propria.'' The division of figures +is treated on folios 23 +\textit{verso}--44 \textit{recto} +(23--32, triangles; 32--34, parallelograms; 34--44, +quadrilateral, pentagon, hexagon, heptagon, circle without the Euclid-Proclus case). + +\textit{Cf.}\ the synopsis in +\textit{Scritti inediti del P.~D.\ Pietro Cossali chierico regolare teatino} +pubblicati da Baldassarre Boncompagni, Roma, 1857, pp.~299--300.} + +\AppItem{1574}{1574a} +\textsc{J.~Gutman.} +\textit{Feldmessung gewiss, richtig und kurz gestellt.} Heidelberg. + +\AppItem{1574}{1574b} +\textsc{E.~Reinhold.} +\textit{Gründlicher und wahrer Bericht vom Feldmessen, +samt allem, was dem anhängig, darinn alle die Irrthum, so bis daher im +Messen fürgeloffen, entdeckt werden. Dessgleichen vom Markscheiden, +kurzer und gründlicher Unterricht.} Erffurdt. + +{\footnotesize ``Der dritte Theil von Theilung der Aecker. Theilungen aller Figuren, auch des +Kreises mit Exemplen und Tafeln erläutert.''} + +\AppItem{1585}{1585} +\textsc{G.~B.~Benedetti.} +\textit{Diversarum speculationum mathematicarum et +physicarum liber.} Taurini. + +{\footnotesize Pages 304--307.} + +\AppItem{1604}{1604} +\textsc{C.~Clavius.} +\textit{Geometria Practica.} Romae. + +{\footnotesize Pages 276--297.} + +\AppItem{1609}{1609} +\textsc{J.~Kepler.} +\textit{Astronomia nova\ldots commentariis de motibus stellæ +Martis}\ldots [Pragae]. + +\footnotesize +``Kepler's Problem'' occurs on p.~300 of this work +(\textit{Opera Kepleri} ed.~Frisch, +\textsc{iii}, 401). +It is: ``To divide the area of a semicircle in a certain ratio by a straight +line drawn through a given point on the diameter or on the diameter produced.'' +(\textit{Cf.}\ +\textsc{A.~G.~Kästner}, \textit{Geschichte der Math}. + \textsc{iv}, 256, Göttingen, 1800; +\textsc{M.~Cantor}, \textit{Vorlesungen} +etc., \textsc{ii}, 708, Leipzig, 1900). Kepler was led to this problem in his +theory of the path of the planets. It has been attacked by many mathematicians, +notably by Wallis, Hermann, Cassini, D.~Gregory, T.~Simpson, Clairaut, Lagrange, +% -----File: 092.png---Folio 80------------------------------------------------------ +Bossut, and Laplace. (\textit{Cf}.\ \textsc{G.~S.~Klügel}, +\textit{Mathematisches Wörterbuch}\ldots Erste Abtheilung, +Dritter Theil, Leipzig, 1808; Article, ``Kepler's Aufgabe.'' See also +\textsc{C.~Hutton}, \textit{Philosophical and Mathematical Dictionary}. +New edition, London, 1815, 1, 703.) +\small + +\AppItem{1612}{1612} +\textsc{Sybrandt Cardinael.} +\textit{Hondert geometrische questien met hare solutien}. Amsterdam. + +{\footnotesize +This work is also to be found at the end of \textsc{Johan Sems} ende \textsc{Ian Dou} +\textit{Practijck des landmetens}. Amsterdam, 1616. Another edition: Tractatus geometricus. +Darinnen hundert schöne \ldots Questien [übersetzt] durch Sebastianum Curtium. +Amsterdam, 1617; Questions 78, 90--93. + +With these problems Huygens (1629--1695) busied himself when about 17 or 18 +years of age. \textit{Cf.\ Oeuvres complètes de Chr.\ Huygens}, Amsterdam, XI, 24 and 29, 1908. + +I have elsewhere (\textit{Nieuw Archief}, 1914) shown that Sybrandt Cardinael's work +was translated into English, rearranged and published as an original work by +Thomas Rudd (1584?-1656): \textit{A hundred geometrical Questions with their solutions +and demonstrations}. London, M.DC.L.} + +\AppItem{1615}{1615} +\textsc{Ludolph van Ceulen.} +\textit{Fundamenta arithmetica et geometrica cum +eorundem usu \ldots e vernaculo in Latinum translata a W.~S}[\textit{nellio}], +\textit{R.~F.} Lugduni Batavorum. + +{\footnotesize Contains several problems on Change, and Division, of Figures.} + +\AppItem{1616}{1616} +\textsc{J.~Speidell.} +\textit{A geometricall Extraction or a compendiovs collection of +the chiefe and choyse Problemes, collected out of the best, and latest writers. +Wherevnto is added about 30 Problemes of the Authors Invention, being for +the most part, performed by a better and briefer way, than by any former +writer}. London. + +{\footnotesize Another edition, 1617; second edition ``corrected and enlarged,'' London, 1657; +``Now followeth [pp.~84--125] a compleat Instruction of the diuision of all right lined +figures\ldots. Very pleasant and full of delight in practise: Also, most profitable to all +surveighers, or others that are desirous to make any Inclosure.''} + +\AppItem{1619}{1619} +\textsc{A.~Anderson.} +\textit{Exercitationum mathematicarum Decas prima. Continens, +Questionum aliquot, quae nobilissimorum tum hujus tum veteris +aevi, Mathematicorum ingenia exercuere, Enodationem}. Parisiis. + +{\footnotesize Problems in division of a triangle, with reference to Clavius (1604). +\textit{Cf.\ The Ladies' Diary}, London, 1840, pp.~55--56.} + +\AppItem{1645}{1645} +\textsc{C.~Huygens.} +\textit{Oeuvres Complètes}, \textsc{xi}, 1908, pp.~26--27; 219--225. + +{\footnotesize Solution of ``Datum triangulum, ex puncto in latere dato, bifariam secare'' (1645); +two solutions of ``Triang.\ ABC, sectus utcumque lineâ DE, dividendus est aliâ lineâ, +FG, ita ut utraque pars DBE et ADEC bifariam dividatur'' (1650--1668). See also +note under 1687---J.~Bernoulli.} + +\AppItem{1657}{1657} +\textsc{F.~van Schooten.} +\textit{Exercitationvm mathematicarum liber primus +continens propositionum arithmeticarvm et geometricarvm centuriam}. +Lugd.\ Batav. + +{\footnotesize Prop.~L, pp.~107--110. Dutch edition, Amsterdam, 1659. pp.~107--110. Concerning +a Schooten MS. of 1645--6, used by Huygens and of interest in this connection, +\textit{cf.}\ \textsc{C.~Huygens,} \textit{Oeuvres Complètes}, tome \textsc{xi}, 1908, p.~13~ff.} + +\AppItem{1667}{1667} +\textsc{D.~Schwenter.} +\textit{Geometriae Practicae novae et auctae Libri IV \ldots +mit vielen nutzlichen Additionen und neuen Figuren vermehret durch +G.~A.~Böcklern}. Nürnberg. + +{\footnotesize +``Von Austheilung der Figuren in gleiche und ungleiche Theil,'' pp.~269--279; +p.~350; the problem on this last page is taken from \textsc{B.~Bramer}, \textit{Trigonometria +planarum mechanica}, Marpurg, 1617, p.~99. ``Von Austheilungen der Aecker +Wiesen,\ldots '' pp.~567--583.} + +% -----File: 093.png---Folio 81------------------------------------------------------ + +\AppItem{1674}{1674} +\textsc{C.~F.~M.~Deschales.} +\textit{Cursus seu mundus mathematicus}. Lugduni. + +{\footnotesize +``De figurarum planarum divisione,'' \textsc{i},~371--381; second edition, 1690.} + +\AppItem{1676}{1676} +\textsc{I.~Newton.} +\textit{Arithmetica Universalis}. Cantabrigiae, \textsc{MDCCVII}. + +{\footnotesize +Prob.\ \textsc{X}, p.~126 (Prob.\ \textsc{XX}, pp.~254--255 of the 1769 edition). This problem was +discussed in a lecture delivered October, 1676 (see \textit{Correspondence of Sir Isaac Newton +and Professor Cotes}\ldots by J.~Edleston, London, 1850, p.~xciii).} + +\AppItem{1684}{1684} +\textsc{T.~Strode.} +\textit{A Discourse of Combinations, Alternations and Aliquod +Parts} by John Wallis. London, 1685. + +{\footnotesize +On pages 163--164 is printed a letter, dated Nov.~1684, from Strode to Wallis. It +discusses two problems on divisions of a triangle.} + +\AppItem{1687}{1687} +\textsc{J.~Bernoulli.} ``Solutio algebraica problematis de +quadrisectione trianguli scaleni, per duas normales rectas.'' +\textit{Acta Eruditorum}, 1687, pp.~617--623. + +\footnotesize +Also in \textit{Opera}, Genevae, 1744, I,~328--335; see further II,~671. In the solution of this +question Bernoulli is led to the intersection of a conic and a curve of the fourth degree, +that is, to an equation of the eighth degree. And yet, in the seventh edition of +\textsc{Rouché} et \textsc{Comberousse}, +\textit{Traité de Géométrie}, Paris, 1900, we find Problem 453 is: +``Partager un triangle quelconque en quatre parties équivalentes par deux droites +perpendiculaires entre elles!'' The problem was solved by L'Hospital before 1704, +the year of his death, in a posthumous work, \textit{Traité analytique des Sections coniques}, +Paris, 1707, pp.~400--407. As the result of correspondence in \textit{L'Intermédiaire des +Math}., tomes \textsc{i--vii}, 1894--1900, Questions 3 and 587, Loria wrote the history of the +problem: ``Osservazioni sopra la storia di un problema pseudo-elementare.'' \textit{Bibl.\ +Math}., 1903 (3), \textsc{iv}, 48--51. Leibnitz's name appears in this connection. See note on +1645---C.~Huygens. +\small + +\AppItem{1688}{1688} +\textsc{J.~Ozanam.} +\textit{L'usage du compas de proportion expliqué et demontré d'une +manière \DPtypo{court}{courte} +et facile, et augmenté d'un Traité de la division des champs}. +Paris. + +{\footnotesize +``Division des champs,'' pp.~89--138. \DPtypo{Edition}{Ouvrage} %% see books.google.com +revu, corrigé et entièrement refondu +par J.~G.~Garnier. Paris, 1794, pp.~165--257.} + +\AppItem{1694}{1694} +\textsc{S.~Le Clerc.} \textit{Traité de Géométrie sur le terrain} at end of +\textit{Géométrie pratique, ou pratique de la géométrie sur le papier et sur le terrain}. +Amsterdam. + +\AppItem{1699}{1699} +\textsc{J.~Ozanam.} +\textit{Cours de mathématique}, nouv.\ éd. tome 3. Paris. + +{\footnotesize +Pages 23--64. German translation: \textit{Anweisung, die geradlinichten Figuren nach +einen gegebenen Verhältniss ohne Rechnung zu theilen}. Frankfurt u. Leipzig, 1776.} + +\AppItem{1704}{1704} +\textsc{Guisnée.} +\textit{Application de l'algèbre à la géométrie}. Paris. +\nopagebreak + +{\footnotesize +Although the ``approbation'' signed by Fontenelle is dated ``15 Juillet 1704'' the +work was first published in 1710; second edition ``revûe, corrigée et considérablement +augmentée par l'auteur,'' Paris, 1733, pp.~42--47; analytic discussion only.} + +\AppItem{1739}{1739} +\textit{l'abbé}~\textsc{Deidier.} +\textit{La science des géométres} (\textit{sic}) +\textit{ou la théorie +et la pratique de la géométrie}. Paris. + +{\footnotesize +``De la géodésie ou division des champs,'' pp.~279--320; divisions of triangles, rectangles, +trapeziums, polygons.} + +\AppItem{1740}{1740} +\textsc{N.~Saunderson.} +\textit{Elements of Algebra in ten books}, vol.~2. Cambridge. + +{\footnotesize Pages 546--554.} + +\AppItem{1747}{1747} +\textsc{T.~Simpson.} \textit{Elements of Plane Geometry}. London. + +{\footnotesize Pages~151--152; new ed., London, 1821, pp.~207--208; taken from Newton (1676).} + +% -----File: 094.png---Folio 82------------------------------------------------------ +\AppItem{1748}{1748} +\textsc{L.~Euler.} +\textit{Introductio in analysin infinitorum}. Tomus secundus. Lausanne. + +{\footnotesize +Chapter 22: ``Solutio nonnullorum problematum ad circulum pertinentium.'' +Three of the eight problems which Euler here discusses by the method of trial and +error, and tables of circular arcs and logarithmic sines and tangents, are of interest to +us. These are: Problem 2, ``To find the sector of the circle $\mathit{ACB}$ which is divided +by the chord $\mathit{AB}$ into two equal parts, so that the triangle $\mathit{ACB}$ shall be equal to the +segment $\mathit{AEB}$.'' Problem 4, ``Given the semi-circle $\mathit{AEDB}$, to draw from the point +$A$ a chord $\mathit{AD}$ which will divide the semi-circle into two equal parts.'' Problem 5, +``From a point $A$ of the circumference of a circle, to draw two chords $\mathit{AB}$, $\mathit{AC}$ which +shall divide the area of the circle into three equal parts.'' (Heron, \textit{cf.}\ Art.~50.) +Gregory (1840) considers these problems at the close (pp.~186--188) of his Appendix. + +For other editions of Euler's ``Introductio,'' tomus 2, see +\textit{Verzeichnis der Schrif\-ten Leonhard Eulers.} +Bearbeitet von G.~Eneström. Erste Lieferung, Leipzig,~1910.} + + +\AppItem{1752}{1752} +\textsc{T.~Simpson.} +\textit{Select Exercises for young proficients in the mathematics.} London. + +{\footnotesize +Problem \textsc{xlii}, pp.~145--6; new ed.\ by J.~H.~Hearding. London, 1810, pp.~148--9.} + +\AppItem{1754}{1754} +\textsc{J.~le~R.~D'Alembert.} +\textit{Encyclopédie ou Dictionnaire raisonné des +sciences\ldots mis en ordre et publié par M.~Diderot\ldots; et quant à la partie +mathématique par M.~d'Alembert.} Paris. + +{\footnotesize +Article ``Géodésie''; +mostly descriptive of methods of Guisnée (1704) and Clerc (1694).} + +\AppItem{1768}{1768} +\textsc{J.~A.~Euler.} ``Auflösung einiger geometrischen Aufgaben,'' +\textit{Abhandlungen der Churfürstlich-baierischen Akademie der Wis\-sen\-schaf\-ten}, +v, 165--196. + +{\footnotesize +Erste Aufgabe, pp.~167--182: ``Man soll zeigen, wie eine jede geradlinichte Figur +durch Parallellinien in eine gegebene Anzahl gleicher Theile zerschnitten werden +kann.'' Zweite Aufgabe, pp.~182--187: ``Eine Zirkel-fläche durch parallellinien in +eine gegebene Anzahl gleicher Theile zu zerschneiden.'' Dritte Aufgabe, pp.~187--196: +``Die Höhe und Grundlinie einer aufrecht-stehenden geschlossenen Parabelfläche ist +gegeben, man soll dieselbe durch Parallellinien in $n$ gleiche Theile zerschneiden.'' +Discussion mostly analytic.} + +\AppItem{1772(?)}{1772} +\textit{J.~H.~Lamberts deutscher gelehrter Briefwechsel.} +Herausgegeben von Joh.\ Bernoulli. Band 2, Berlin, 1782. + +{\footnotesize +Pages 412--13, undated fragment of a letter from Lambert to J.~E.~Silberschlag. +Analytic solution by quadratic equation of the problem: ``Ein Feld $\mathit{ABCD}$ welches +in $\mathit{ABFE}$ Wiesen, in $\mathit{EFCD}$ Ackerfeld ist, +soll durch eine gerade Linie $\mathit{KM}$ so +getheilt werden, dass so wohl die Wiesen als das Ackerfeld in beliebiger Verhältniss +getheilt werde.'' +[$\mathit{ABCD}$ is a quadrilateral and $\mathit{EF}$ is a straight line segment joining +points on the opposite sides $\mathit{AD}$, $\mathit{BC}$ respectively.] + +In the \textit{Journal of the Indian Mathematical Society}, 1914, +\textsc{vi}, 159, N.~P. Pan\-dya +proposed as Question 563: ``Given two quadrilaterals in the plane of the paper show +how to draw a straight line bisecting them both.'' A solution by means of common +tangents to hyperbolas was offered in 1915, \textsc{vii}, 176.} + +\AppItem{1783}{1783} +\textsc{J.~T.~Mayer.} +\textit{Gründlicher und ausführlicher Unterricht zur praktischen Geometrie,} +3.~Teil. Göttingen. + +{\footnotesize +Pages 215--303: ``Theilung der Felder durch Rechnung, Theilung der Felder durch +blose Zeichnung, Anwendung der Theilungsmethoden auf mancherley, in gemeinen +Leben vorkommende Fälle''; dritte Auflage, 1804, pp.~232--337.} + +\AppItem{1793}{1793} +\textsc{J.~W.~Christiani.} +\textit{Die Lehre von der geometrischen und ökonomischen +Vertheilung der Felder, nach der Dänischen Schrift des N.~Morville +bearbeitet von J.~W.~Christiani.} Preface by A.~G.~Kästner. Göttingen. + +\AppItem{1795}{1795} +\textit{Gentleman's Diary}, London. + +{\footnotesize +No.~54, 1794, p.~47, Question 691 by J.~Rodham: ``Within a given triangle to find +% -----File: 095.png---Folio 83------------------------------------------------------ +a point thus, that if lines be drawn from it to cut each side at right angles, the three +parts into which the triangle thus becomes divided, shall obtain a given ratio.'' Solution +by hyperbolas in No.~55, 1795, pp.~37--38. +See also Davis's edition of the \textit{Gentleman's Diary}, vol.~3, London, 1814, pp.~233--4.} + +\AppItem{1801}{1801} +\textsc{L.~Puissant.} +\textit{Recueil de \DPtypo{divers}{diverses} +propositions de géométrie résolues ou +démontrées par l'analyse algébrique suivant les principes de Monge et de +Lacroix}. Paris. + +{\footnotesize +Pages 33--36; German ed., Berlin, 1806; second French ed., Paris, 1809, +pp.~107--111; third ed., Paris, 1824, pp.~139--142.} + +\AppItem{1805}{1805} +\textsc{M.~Hirsch.} +\textit{Sammlung geometrischer Aufgaben}, Erster Theil. Berlin. + +{\footnotesize +``Theilung der Figuren durch Zeichnung,'' pp.~14--25; ``Theilung der Figuren +durch Rechnung,'' pp.~42--53; Reprint, 1855; English edition translated by J.~A.~Ross +and edited by J.~M.~F.~Wright. London, 1827.} + +\AppItem{1807}{1807a} +\textsc{A.~Bratt.} +\textit{Problema geometricum triangulum datum a dato puncto in +2 partes aequales secandi}. Greifswald. + +{\footnotesize +This title is taken from \textsc{C.~G.~Kayser}, +\textit{Bücher-Lexicon}, Erster Teil, Leipzig, 1834.} + +\AppItem{1807}{1807b} +\textsc{J.~P.~Carlmark.} +\textit{Triangulus datus a dato puncto in 2 partes aequales secandus}. Greifswald. + +{\footnotesize +This title and the next two are taken from \textsc{E.~Wölffing}, +\textit{Math.\ Bücherschatz}, 1903.} + +\AppItem{1809}{1809} +\textsc{J.~Kullberg.} +\textit{Problema geometricum triangulum datum e quovis dato +puncto in 2 partes aequales secandi}. Diss.\ Lund. + +\AppItem{1810}{1810} +\textsc{J.~Kullberg.} +\textit{Problema geometricum triangulum quodcunque datum in +2 aequales divisum iterum in partes aequales ita secandi, ut rectae secantes +angulum constituant rectum}. Diss.\ Upsala. + +\AppItem{1811}{1811} +\textsc{J.~P.~Grüson.} +\textit{Geodäsie oder vollständige Anleitung zur geometrischen +und ökonomischen Feldertheilung}. Halle. + +\AppItem{1819}{1819} +\textsc{L.~Bleibtreu.} +\textit{Theilungslehre oder ausführliche Anleitung}, +\textit{jede Grundfläche auf die zweckmässigste Art} \ldots +\textit{geometrisch zu theilen}. Frankfurt am Main. + +\AppItem{1821}{1821} +\textsc{J.~Leslie.} +\textit{Geometrical Analysis and Geometry of Curve Lines} +Edinburgh. + +{\footnotesize Pages 64--66.} + +\AppItem{1823}{1823} +\textsc{A.~K.~P.~von Forstner.} +\textit{Sammlung systematisch geordneter und +synthetisch \DPtypo{aufgelöseter}{aufgelöster} +geometrischer Aufgaben}. Berlin. + +{\footnotesize ``Theilung der Flächen, mittelst der Proportion und der Aehnlichkeit,'' +pp.~310--371.} + +\AppItem{1827}{1827} +\textit{Correspondance mathématique et physique} +publié par A.~Quetelet, tome \textsc{iii}. + +{\footnotesize +Page 180: ``On donne dans un plan un angle et un point, +et l'on demande de faire +passer par le point une droite qui coupe les cotés de l'angle, +de manière que l'aire +interceptée soit de grandeur donnée.'' Solution by Verhulst, +pp.~269--270. Answer +by Bobillier, tome \textsc{iv}, pp.~2--3. +Generalizing his solution, he gets the result: ``tous +les plans tangens d'un hyperboloïde à deux nappes, +interceptent sur le cône asymptotique +des volumes équivalens.'' Compare note 117.} + +\AppItem{1831}{1831a} +\textsc{P.~L.~M.~Bourdon.} +\textit{Application de l'algèbre à la géométrie comprenant +la géométrie analytique à deux et à trois dimensions}, troisième édition. +Paris. + +{\footnotesize +Pages 46--54; 5$^e$~éd., Paris, 1854, pp.~33--41; 8$^e$~éd.\ rev.\ par +Darboux, Paris, 1875, pp.~30--38. Analytic discussion only.} +% -----File: 096.png---Folio 84------------------------------------------------------ + +\AppItem{1831}{1831b} +\textsc{H.~v.~Holleben}, und \textsc{P.~Gerwien.} +\textit{Geometrische Analysis}. Berlin, 2~Bde, 1831--1832. + +{\footnotesize ``Theilungen,'' \textsc{i},~184--191; \textsc{ii},~144--151.} + +\AppItem{1837}{1837} +\textsc{G.~Ritt.} +\textit{Problèmes d'applications de l'algèbre à la +géométrie avec les solutions développées}, 2$^e$~partie. Paris. + +{\footnotesize Pages 108--109.} + +\AppItem{1840}{1840} +\textsc{O.~Gregory.} +\textit{Hints theoretical, elucidatory and practical, for the use +of Teachers of elementary Mathematics and of self-taught students; with +especial reference to the first volume of Hutton's course and Simson's Euclid, +as Text-Books. Also a selection of miscellaneous tables, and an Appendix +on the geometrical division of plane surfaces}. London. + +{\footnotesize ``Appendix: Problems relative to the division of Fields and other surfaces,'' +pp.~158--188; partly taken from Hirsch~(1805). See also Euler~(1748).} + +\AppItem{1844}{1844} +\textsc{Dreser.} \textit{Die Teilung der Figuren}. Darmstadt. + +{\footnotesize This title is taken from \textsc{E.~Wölffing}, \textit{Math.\ Bücherschatz}, 1903.} + +\AppItem{1847}{1847} +\textsc{R.~Potts.} \textit{An appendix to the larger edition of +Euclid's Elements of Geometry; containing\ldots Hints for the solution +of the Problems}\ldots Cam\-bridge and London.\DPnote{Would not hyphenate Cambridge without the hint. ??} + +{\footnotesize Ex.~91, pp.~72--73.} + +\AppItem{1852}{1852a} +\textsc{H.~Ch.~de~La Frémoire.} +\textit{Théorèmes et Problèmes de Géométrie +élémentaire}, \DPtypo{second}{seconde} +éd.\ revue et corrigée par E.~Catalan. Paris. + +{\footnotesize +Pages 107--108; 6$^e$ éd.\ par Catalan, Paris, 1879, pp.~190--191.} + +\AppItem{1852}{1852b} +\textsc{F.~Rummer.} +\textit{Die Verwandlung und Theilung der Flächen in einer +Reihe von Constructions- u.\ Berechnungs-Aufgaben}. Mit 3 Steintafeln. +Heidelberg. 6~+~90 pp. + +\AppItem{1855}{1855} +\textsc{P.~Kelland.} ``On Superposition.'' +\textit{Transactions of the Royal Society of Edinburgh}, +1885, \textsc{xxi}, 271--273 + 1 pl. + +{\footnotesize +This paper deals, for the most part, with solutions of the following problem proposed +to Professor Kelland by Sir John Robison: ``From a given square one quarter is cut +off, to divide the remaining gnomon into four such parts that they shall be capable of +forming a square.'' In the \textit{Transactions}, 1891, +\textsc{xxxvi}, 91--95, + 2 pls.,\ Robert Brodie +has a paper entitled ``Professor Kelland's Problem on Superposition.''} + +\AppItem{1857}{1857} +\textsc{E.~Catalan.} +\textit{Manuel des Candidats \DPtypo{a}{à} +l'école polytechnique}. Paris, +Tome I. + +\footnotesize +Pages 233--4: ``To divide a circle into two equal parts by means of an arc with its +centre, \textit{A}, on the circumference of the given circles.'' +This is stated by \textsc{A.~Rebière} +(\textit{Mathématique et Mathématiciens}, +2$^e$~éd.,\ Paris, 1893, p.~519) under the form: +``Quelle doit être la longueur de la longe d'un cheval pour qu'en +la fixant au contour d'un pré circulaire l'animal ne puisse +tondre que la moitié du pré?'' + +The solution of this problem leads to a transcendental equation + +\begin{equation*} +\sin x - x \cos x = \frac{\pi}{2}, +\end{equation*} + +where $x$ is the angle under which the points of section of the circumferences are seen +from \textit{A}. Catalan finds $x = 109^\circ 11' 18''$, correct to within a second of arc. + +Cf.\ \textit{L'Intermédiaire des Mathématiciens}, 1914, +Question 4327, \textsc{xxi}, 5, 69, 90, 115, 180. +\small + +\AppItem{1863}{1863} +\textsc{J.~ McDowell.} +\textit{Exercises on Euclid and in Modern Geometry}. +Cambridge. + +{\footnotesize No.~157, pp.~145--6; 3rd ed.\ 1881, p.~118.} + +% -----File: 097.png---Folio 85------------------------------------------------------ +\AppItem{1864}{1864a} +\textit{Educational Times Reprint}, +Vols.\ 1, 40, 44, 66, 68, 69; new series, Vol.\ 1; 1864--1910. + +{\footnotesize +The problems here solved are Euclid's 19, 20, 26, 27: No.~1457 (\textsc{i}, 49, old edition, +1864) proposed by R.~Palmer; solution by Rutherford who states that it was also +published in Thomas Bradley's \textit{Elements of Geometrical Drawing}, 1861---Nos.~7336 and +7369 (\textsc{xl}, 39, 1884) proposed by W.~H.~Blythe and A.~H.~Curtis; solutions by +G.~Heppel and Matz---No.~8272 (\textsc{xliv}, 92, 1886) proposed by E.~Perrin; solution by +D.~Biddle---No.~12973 (\textsc{lxvi}, 29, 1897) proposed by Radhakrishnan; solution by +I.~Arnold---No.~13460 (\textsc{lxviii}, 35, 1898) proposed by I.~Arnold; solution by +W.~S.~Cooney, etc.---No.~13647 (\textsc{lxix}, 42, 1898) proposed by I.~Arnold; solution by +W.~C.~Stanham---No.~16747 (new series \textsc{xviii}, 46, 1910) proposed by I.~Arnold; +solution by proposer, by Euclid's \textit{Elements} Bk~\textsc{i}.} + +\AppItem{1864}{1864b} +\textsc{H.~Hölscher.} +\textit{Anleitung zur Berechnung und Teilung der Polygone +bei rechtwinkligen Koordinaten}. Berlin and Charlottenburg. + +{\footnotesize +This work and the two following are representative of those which treat of +Divisions of Figures by computation, rather than by graphical methods: +(1) \textsc{F.~G.~Gauss}, +\textit{Die Teilung der Grundstücke, insbesondere unter Zugrundelegung Koordinaten}, +2 Auflage, Berlin, 1890; +(2) \textsc{L.~Zimmerman}, \textit{Tafeln für die Teilung der Dreiecke, +Vierecke, und Polygone}, Zweite vermehrte und verbesserte Auflage, Liebenwerda, +1896; 118+64 pp.} + +\AppItem{1870}{1870} +\textsc{F.~Lindman.} ``Problema geometricum.'' +\textit{Archiv der Math.\ u.\ Phys}. +(Grunert), Bd~51, 1870, pp.~247--252. + +\AppItem{1879}{1879} +\textsc{P.~M.~H.~Laurent.} +\textit{Traité d'algèbre à l'usage des candidats\ldots.} +Troi\-si\-ème édition. Paris. + +{\footnotesize +Tome I, p.~191: +``To divide a triangle into two equal parts by the shortest possible line.'' +Solutions in \textit{L'Intermédiaire des Mathématiciens}, 1902, IX, 194--5. See also +F.~G.~M., \textit{Exercices de Géométrie}, Cinquième édition. Tours et Paris, 1912, p. 802.} + +\AppItem{1892}{1892} +\textsc{H.~S.~Hall} and \textsc{F.~H.~Stevens.} +\textit{Key to the Exercises and Examples contained in a Text-Book of Euclid's Elements}. +London. + +{\footnotesize Ex.\ 7, 8, 10, 11, pp.~163--164.} + +\AppItem{1894}{1894} +\textsc{G.~E.~Crawford.} ``Geometrical Problem.'' +\textit{Proc.\ Edinb.\ Math.\ Soc.}, Vol.~13, 1895, p.~36. + +{\footnotesize Paper read Dec.~14, 1894.} + +\AppItem{1899}{1899} +\textsc{W.~J.~Dilworth.} \textit{A New Sequel to Euclid}. London. + +{\footnotesize Ex. \textsc{xxxv}, p.~190.} + +\AppItem{1901}{1901} +\textsc{A.~Larmor.} +\textit{Geometrical Exercises from Nixon's `Euclid Revised.'} +Oxford. + +{\footnotesize Ex.~15, p.~122.} + +\AppItem{1902}{1902} +\textsc{C.~Smith.} +\textit{Solution of the Problems and Theorems in Smith and +Bryant's Elements of Geometry}. London. + +{\footnotesize Ex.~121, pp.~177--178; T.~Simpson's solution and another.} + +\AppItem{1910}{1910a} +\textsc{H.~Flükiger.} +\textit{Die Flächenteilung des Dreiecks mit Hilfe der Hyperbel.} +Diss.\ Bern. 50 pp. + 3 plates. + +\AppItem{1910}{1910b} +\textsc{R.~Zdenek.} ``Halbierung der Dreiecksfläche.'' Wien, +\textit{Zeitschrift für das Realwesen}, Jahrgang \textsc{xxxv}, Heft~10, 8~pp. + +{\footnotesize Discussion by projective geometry leading to hyperbolic arcs.} + +\AppItem{1911}{1911} +\textsc{D.~Biddle.} Problem 17197, +\textit{Educational Times}, London, November, +\textsc{lxiv}, 475. + +{\footnotesize +``Divide a square into five right-angled triangles, the areas of which shall be in +arithmetic progression.'' Solutions in the Educational Times Reprint, new series, +\textsc{xxvi, iii}, 1914.} + +\end{Description} + +% -----File: 098.png---Folio 86------------------------------------------------------ +\backmatter +\label{indexpage} +\phantomsection\pdfbookmark[0]{Index of Names} + {Index of Names} + + + +\fancyhead[C]{\Heading{INDEX OF NAMES}} +\thispagestyle{empty} +\renewcommand{\indexname}{\large{\centerline{INDEX OF NAMES}}} +\renewcommand{\indexhook}{\textit{In the following list, references are + given to paragraph +and footnote} (\textit{=~n.})\ \textit{numbers, except in the case of +the Appendix} (\textit{=~App.})\ \textit{where the numbers +are the years in the chronological list. \; App.,\ without a number, + refers to +the introductory paragraphs on page \pageref{appendix}.}\bigskip} +\begin{theindex} +\footnotesize + +\item Abraham bar Chijja ha Nasi or Ab\-ra\-ham Sa\-va\-sor\-da \ParRef{18}, \FnRef{53} +\item Abû Ishâq b.\ `Abdallâh \ParRef{19} +\item Abû Jûsuf Ja`qûb b.\ Ishâq b.\ el-Sabbah el-Kindî, \textit{see} el-Kindî +\item Abû'l Wefâ \ParRef{19}, \FnRef{78} +\item Abû Muh.\ b.\ Abdalbâqî el Ba\.{g}dâdî el-Faradî, \textit{see} el-Ba\.{g}dâdî +\item Abû Muh.\ el-Hasan b.\ `Obeidallâh b.\ Soleimân, \textit{see} el-Hasan +\item Ainsworth, R. \FnRef{28} +\item Albategnius = Al-Battânî = el-Bat\-tânî, \textit{see there} +\item Al-Kindî = el-Kindî +\item Anderson, A. \AppRef{1619}{\DPtypo{1616}{1619}} +\item Antiphon \FnRef{116} +\item Apollonius of Perga \ParRef{19}, \ParRef{21}, \FnRef{103}, \FnRef{111} +\item Archimedes \ParRef{19}, \FnRef{601}, \FnRef{83}, \FnRef{101}, + \FnRef{103}, \FnRef{109}, \FnRef{111} +\item Aristarchus of Samos \FnRef{111} +\item Aristotle \FnRef{116} +\item Armagh, Archbishop of, \textit{see} Ussher, J. +\item Arnold, I. \AppRef{1864a}{1864} +\item Ashmole \FnRef{22} +\item Athelhard of Bath \FnRef{15} +\item Ayton, W.~A. \FnRef{14} +\indexspace +\item Bachmann, P. \AppRef{appendix}{} +\item Benedetti, G.~B. \AppRef{1585}{1585} +\item Bernoulli, J. \AppRef{1645}{1645},\AppRef{1687}{1687},\AppRef{1772}{1772(?)} +\item Biddle, D. \AppRef{1864a}{1864},\AppRef{1911}{1911} +\item Bleibtreu, L. \AppRef{1819}{1819} +\item Blythe, W.~H. \AppRef{1864a}{1864} +\item Bobillier, E.~E. \FnRef{117}, \AppRef{1827}{1827} +\item Böcklern, G.~A. \AppRef{1667}{1667} +\item Boncompagni, B. \ParRef{10}, \ParRef{13}, + \FnRef{40}, \FnRef{119}, \AppRef{appendix}{}, \AppRef{1560}{1560} +\item Bossut, C. \AppRef{1609}{1609} +\item Bourdon, P.~L.~M. \AppRef{1831a}{1831} +\item Bradley, T. \AppRef{1864a}{1864} +\item Bramer, B. \AppRef{1667}{1667} +\item Bratt, A. \AppRef{1807a}{1807} +\item Breton (de Champ), P. \FnRef{88} +\item Brodie, R. \AppRef{1855}{1855} +\item Bryant, S. \AppRef{1902}{1902} +\indexspace +\item Campanus, J. \ParRef{2}, \ParRef{5}, \FnRef{11}, \FnRef{15} +\item Candale, \textit{see} Flussates +\item Cantor, M. \ParRef{2}, \FnRef{8}, \FnRef{39}, \FnRef{53}, + \FnRef{55}, \FnRef{62}, \FnRef{63}, \FnRef{82}, + \FnRef{103}, \FnRef{107}, \FnRef{120}, \AppRef{1547b}{1547}, \AppRef{1609}{1609} +\item \DPtypo{Cardan, G.}{Cardano, M.~H.} \AppRef{1547b}{1547} +\item Cardinael, S. \AppRef{1612}{1612} +\item Carlmark, J.~P. \AppRef{1807b}{1807} +\item Casley, D. \ParRef{4}, \FnRef{20} +\item Cassini, J. \AppRef{1609}{1609} +\item Catalan, E. \AppRef{1852a}{1852}, \AppRef{1857}{1857} +\item Chasles, M. 19, \FnRef{65}, \FnRef{85}, \FnRef{88}, + \FnRef{111} +\item Christiani, J.~W. \FnRef{67}, \AppRef{1793}{1793} +\item Clairaut, A.~C. \AppRef{1609}{1609} +\item Clavius, C. \AppRef{1604}{1604} +\item Clerc, S. le \AppRef{1694}{1694}, \AppRef{1754}{1754} +\item Comberousse, C. de \AppRef{1687}{1687} +\item Commandinus, F. \ParRef{2}, \ParRef{5}, \ParRef{1}5, \FnRef{10}, \FnRef{11}, + \FnRef{35}, \FnRef{49}, \FnRef{50}, \FnRef{88} +\item Copernicus 19, \FnRef{35}, \FnRef{111} +\item Cossali, P. \FnRef{119}, \AppRef{appendix}{}, \AppRef{1560}{1560} +\item Cotes, R. \AppRef{1676}{1676} +\item Cotton, R.~B. \ParRef{4}, \FnRef{21} +\item Cottonian MSS. \ParRef{2}, \ParRef{3}, \ParRef{4}, \ParRef{5}, + \FnRef{18}, \FnRef{19}, + \FnRef{20}, \FnRef{21} +\item Cowley, A. \ParRef{6} +\item Cratfield, W. \ParRef{4} +\item Crawford, G.~E. \AppRef{1894}{1894} +\item Crelle \FnRef{117} +\item Cresswell, D. \FnRef{110} +\item Curtis, A.~H. \AppRef{1864a}{1864} +\item Curtius, S. \AppRef{1612}{1612} +\item Curtze, M. \ParRef{18}, \FnRef{54}, \FnRef{55} +\indexspace +\item D'Alembert, J. le R. \AppRef{1754}{1754} +\item Darboux, G. \AppRef{1831a}{1831} +\item Davis, A. \AppRef{1795}{1795} +\item Dee, J. \ParRef{2}, \ParRef{3}, \ParRef{4}, \ParRef{5}, \ParRef{6}, \ParRef{8}, + \ParRef{9}, \ParRef{18}, \ParRef{19}, \FnRef{10}, \FnRef{11}, + \textit{n}.~14, \FnRef{24}, \FnRef{25}, + \FnRef{29}, \FnRef{35} +\item Deidier, \textit{l'abbé} \AppRef{1739}{1739} +\item Deschales, C.~F.~M. \AppRef{1674}{1674} +\item Diderot, D. \AppRef{1754}{1754} +\item Diels, H. \FnRef{116} +\item Diesterweg, W.~A. \FnRef{111} +\item Dilworth, W.~J. \AppRef{1899}{1899} +\item Dou, I. \AppRef{1612}{1612} +\item Dreser \AppRef{1844}{1844} +\indexspace +\item \DPtypo{Edelston}{Edleston}, J. \AppRef{1676}{1676} +\item el-Ba\.{g}dâdî\footnote{ +All the ``Muhammeds'' of Bagdad referred to in this volume are here supposed to be +indicated by this single name.} % end footnote + \ParRef{2}, \ParRef{3}, \ParRef{4}, \ParRef{5}, \ParRef{6}, + \ParRef{13}, \ParRef{14}, \ParRef{15}, \ParRef{19}, + \FnRef{11}, +% -----File: 099.png---Folio 87------------------------------------------------------ + \FnRef{12}, \FnRef{15}, \FnRef{17}, \FnRef{21}, + \FnRef{29}, \FnRef{33}, \FnRef{35}, \AppRef{appendix}{} +\item el-Battânî \ParRef{19}, \FnRef{35} +\item el-Bûz\v{g}ânî, \textit{see} Abû'l Wefâ +\item el-Hasan \ParRef{19} +\item el-Kindî \ParRef{19}, \FnRef{35} +\item Eneström,~G. \FnRef{55}, \AppRef{1748}{1748} +\item Euclid \ParRef{1}, \ParRef{2}, \ParRef{5}, \ParRef{6}, \ParRef{7}, \ParRef{8}, + \ParRef{9}, \ParRef{10}, \ParRef{11}, \ParRef{12}, \ParRef{13}, \ParRef{16}, + \ParRef{17}, \ParRef{18}, \ParRef{19}, \ParRef{20}, \ParRef{21}, \ParRef{34}, + \ParRef{50}, \ParRef{57}, + \FnRef{1}, \FnRef{3}, \FnRef{5}, \FnRef{6}, \FnRef{9}, \FnRef{11}, + \FnRef{15}, \FnRef{30}, \FnRef{34}, \FnRef{35}, \FnRef{36}, \FnRef{38}, + \FnRef{45}, \FnRef{47}, \FnRef{50}, \FnRef{56}, \FnRef{57}, \FnRef{81}, + \FnRef{83}, \FnRef{85}, \FnRef{87}, \FnRef{88}, \FnRef{91}, \FnRef{94}, + \FnRef{95}, \FnRef{98}, \FnRef{103}, \FnRef{105}, \FnRef{106}, + \FnRef{108}, \FnRef{109}, \FnRef{111}, \FnRef{113}, \FnRef{116}, + \FnRef{1181}, \AppRef{appendix}{}, \AppRef{1560}{1560}, \AppRef{1840}{1840}, \AppRef{1864a}{1864} +\item Eudemus \FnRef{116} +\item Euler, J.~A. \AppRef{1768}{1768} +\item Euler, L. \AppRef{appendix}{}, \AppRef{1748}{1748}, \AppRef{1840}{1840} +\item Eutocius \FnRef{111} +\indexspace +\item Fabricius, J.A. \ParRef{19}, \FnRef{68} +\item Favaro, E.~A. \ParRef{2}, \ParRef{6}, \ParRef{10}, \ParRef{13}, \ParRef{15}, + \ParRef{19}, \ParRef{29}, + \FnRef{6}, \FnRef{32}, \FnRef{42}, \FnRef{52}, \FnRef{73}, \FnRef{97}, \FnRef{100} +\item Ferrari \AppRef{1547a}{1547} +\item F.~G.~M. \AppRef{1879}{1879} +\item Fibonaci, \textit{see} Pisano Leonardo +\item Flükiger, H. \AppRef{1910a}{1910} +\item Flussates or Foix, \textit{i.~e.} François de Foix-Candalle \FnRef{11} +\item Fontenelle \AppRef{1704}{1704} +\item Forstner, A.~K.~P. von \AppRef{1823}{1823} +\item Frankland, W.~B. \FnRef{103} +\item Friedlein, G. \FnRef{1} +\item Frisch, C. \AppRef{1609}{1609} +\indexspace +\item Gardiner, M. \FnRef{111} +\item Garnier, J.~G. \AppRef{1688}{1688} +\item Gauss, C.~F. \FnRef{118} +\item Gauss, F.~G. \AppRef{1864b}{1864} +\item Gerwien, P. \AppRef{1831b}{1831} +\item Gherard of Cremona \ParRef{4}, \ParRef{11}, \ParRef{19} +\item Giordani, E. \AppRef{1547b}{1547} +\item Grabow, M.~G. \FnRef{111} +\item Gregory, D. 6, \FnRef{11}, \AppRef{1609}{1609} +\item Gregory, O. \AppRef{1748}{1748}, \AppRef{1840}{1840} +\item Grüson, J.~P. \AppRef{1811}{1811} +\item Grunert, J.~A. \ParRef{19}, \FnRef{72}, \AppRef{1870}{1870} +\item Gudermann, C. \FnRef{117} +\item Guisnée \AppRef{1704}{1704}, \AppRef{1754}{1754} +\item Gutman, J. \AppRef{1574a}{1574} +\indexspace +\item Hall, H.~S. \AppRef{1892}{1892} +\item Halley, E. \FnRef{111} +\item Halliwell, J.O. 4, \FnRef{26} +\item Hankel, H. \ParRef{2}, \ParRef{19}, \FnRef{4}, \FnRef{71}, \FnRef{111} +\item Harleian MSS. \ParRef{6}, \FnRef{21}, \FnRef{22}, \FnRef{31} +\item Hearding, J.~H. \AppRef{1752}{1752} +\item Heath, T.L. \ParRef{2}, \ParRef{3}, \ParRef{6}, + \FnRef{3}, \FnRef{15}, \FnRef{601}, + \FnRef{81}, \FnRef{83}, \FnRef{87}, \FnRef{88}, \FnRef{99}, + \FnRef{101}, \FnRef{103}, \FnRef{108}, \FnRef{111}, + \FnRef{112}, \FnRef{116} +\item Heiberg, J.~L. \ParRef{2}, \ParRef{6}, \ParRef{7}, \ParRef{8}, + \FnRef{5}, \FnRef{32}, \FnRef{36}, + \FnRef{601}, \FnRef{83}, \FnRef{111} +\item Heilbronner, J.~C. \ParRef{6}, \ParRef{19}, \FnRef{33}, \FnRef{69} +\item Heppel, G. \AppRef{1864a}{1864} +\item Herigone, P. \FnRef{111} +\item Hermann, J. \AppRef{1609}{1609} +\item Heron of Alexandria \ParRef{18}, \ParRef{20}, \ParRef{21}, + \FnRef{81}, \FnRef{82}, \FnRef{83}, \FnRef{84} +\item Hippocrates \FnRef{116} +\item Hirsch, M. \AppRef{1805}{1805} +\item Hölscher, H. \AppRef{1864b}{1864} +\item Holleben, H. von \AppRef{1831b}{1831} +\item Hooper, S. \FnRef{20} +\item Hultsch, F. \ParRef{2}, \FnRef{9}, \FnRef{36}, \FnRef{85}, \FnRef{109} +\item Hutton, C. \AppRef{appendix}{}, \AppRef{1840}{1840} +\item Huygens, C. \AppRef{appendix}{}, \AppRef{1612}{1612}, \linebreak[2] + \AppRef{1645}{1645}, \AppRef{1657}{1657}, \AppRef{1687}{1687} +\indexspace +\item Ishâq b. Hunein b. Ishâq el-`Ibâdî, Abû Ja`qûb \ParRef{19} +\indexspace +\item Jacobi, C.~J.~A. \FnRef{111} +\item Jordanus Nemorarius \ParRef{18}, \FnRef{58} +\indexspace +\item Kästner, A.~G. \ParRef{2}, \ParRef{3}, \ParRef{19}, \FnRef{16}, \FnRef{19}, + \FnRef{67}, \AppRef{1609}{1609}, \AppRef{1793}{1793} +\item Kayser, C.~G. \AppRef{1807a}{1807} +\item Kelland, P. \AppRef{1855}{1855} +\item Kepler, J. \AppRef{appendix}{}, \AppRef{1609}{1609} +\item Klein, F. \FnRef{103} +\item Klügel, G.~S. \FnRef{72}, \AppRef{1609}{1609} +\item Kullberg, J. \AppRef{1809}{1809}, \AppRef{1810}{1810} +\indexspace +\item La Frémoire, H.~C. de \AppRef{1852a}{1852} +\item Lagrange, J.~L. \AppRef{1609}{1609} +\item Lambert, J.~H. \AppRef{1772}{1772(?)} +\item Laplace, P.~S. \AppRef{1609}{1609} +\item Larmor, A. \AppRef{1901}{1901} +\item Laurent, P.~M.~H. \AppRef{1879}{1879} +\item Leeke, J. \FnRef{11}, \FnRef{30}, \FnRef{35}, \FnRef{50} +\item Leibnitz, G.~W. \AppRef{1687}{1687} +\item Leonardo Pisano, \textit{see} Pisano +\item Leslie, J. \AppRef{1821}{1821} +\item Leybourn, R. and W. \FnRef{11} +\item L'Hospital, G.~F. de \AppRef{1687}{1687} +\item Lindman, C.~F. \AppRef{1870}{1870} +\item Loria, G. \ParRef{2}, \FnRef{7}, \AppRef{1687}{1687} +\item Ludolph van Ceulen \AppRef{1615}{1615} +\item Lühmann, F. von \FnRef{111} +\indexspace +\item McDowell, J. \AppRef{1863}{1863} +\item Marinus \FnRef{11} +\item Matz, F.~P. \AppRef{1864a}{1864} +\item Mayer, J.~T. \AppRef{1783}{1783} +\item Menge, H. \FnRef{88} +\item Mersenne, M. \FnRef{111} +\item Mitzscherling, A. \AppRef{appendix}{} +\item Mollweide, C.B. \FnRef{72} +% -----File: 100.png---Folio 88------------------------------------------------------ +\item Montucla, J.~F. \ParRef{19}, \FnRef{70} +\item Morville, N. \FnRef{67}, \AppRef{1793}{1793} +\item Muhammed Bagdedinus, \textit{see} el-Ba\.{g}\-dâdî +\item Muhammed b. `Abdelbâqî el-Ba\.{g}\-dâdî el-Fardî, \textit{see} el-Ba\.{g}\-dâdî +\item Muhammed b. Gâbir b. Sinân, Abû `Abdallâh el-Battânî, \textit{see} el-Battânî +\item Muhammed b. Muhammed el-Ba\.{g}\-dâdî, \textit{see} el-Ba\.{g}\-dâdî +\item Muhammed b. Muhammed el-Hasib Abû'l Wefâ, \textit{see} Abû'l Wefâ +\indexspace +\item Nemorarius, \textit{see} Jordanus +\item Newton, I. \AppRef{appendix}{}, \AppRef{1676}{1676}, \linebreak[1] \AppRef{1747}{1747} +\item Nixon, R.~C.~J. \AppRef{1901}{1901} +\indexspace +\item Ofterdinger, L.F. \ParRef{9}, \ParRef{15}, \FnRef{32}, \FnRef{38}, + \FnRef{51}, \FnRef{105}, \FnRef{111} +\item Ozanam, J. \AppRef{1688}{1688}, \AppRef{1699}{1699} +\indexspace +\item Pacioli, \textit{see} Paciuolo, L. +\item Paciuolo, L. \ParRef{18}, \AppRef{appendix}{} +\item Palmer, R. \AppRef{1864a}{1864} +\item Pandya, N.~P. \AppRef{1772}{1772(?)} +\item Pappus of Alexandria \ParRef{21}, \FnRef{85}, \FnRef{88}, + \FnRef{111}, \FnRef{113} +\item Paucker, G. \FnRef{111} +\item Pauly-Wissowa \FnRef{9}, \FnRef{81} +\item Perrin, E. \AppRef{1864a}{1864} +\item Pisano, Leonardo \ParRef{10}, \ParRef{11}, \ParRef{12}, \ParRef{13}, + \ParRef{17}, \ParRef{18}, \ParRef{19}, \ParRef{22}, \ParRef{23}, + \ParRef{24}, \ParRef{25}, \ParRef{26}, \ParRef{27}, \ParRef{28}, + \ParRef{29}, \ParRef{30}, \ParRef{31}, \ParRef{32}, \ParRef{33}, + \ParRef{34}, \ParRef{35}, \ParRef{36}, \ParRef{37}, \ParRef{38}, + \ParRef{40}, \ParRef{41}, \ParRef{47}, \ParRef{48}, \ParRef{49}, + \ParRef{50}, \ParRef{51}, \ParRef{52}, \ParRef{53}, \ParRef{54}, + \ParRef{55}, \ParRef{56}, \ParRef{57}, + \FnRef{40}, \FnRef{41}, \FnRef{45}, \FnRef{46}, + \FnRef{47}, \FnRef{88}, \FnRef{96}, \FnRef{98}, + \FnRef{100}, \FnRef{107}, \FnRef{109}, \FnRef{110}, + \FnRef{111}, \FnRef{113}, \AppRef{appendix}{} +\item Planta, J. \ParRef{4}, \FnRef{21} +\item Plato of Tivoli \ParRef{18} +\item Potts, R. \AppRef{1847}{1847} +\item Proclus Diadochus \ParRef{1}, \ParRef{6}, \ParRef{7}, \ParRef{18}, + \ParRef{49}, \FnRef{1}, \FnRef{2}, + \FnRef{35}, \FnRef{36}, \FnRef{103}, \AppRef{1560}{1560} +\item Ptolemy \FnRef{111} +\item Puissant, L. \AppRef{1801}{1801} +\indexspace +\item Quetelet, A. \FnRef{117}, \AppRef{1827}{1827} +\indexspace +\item Radhakrishnan \AppRef{1864a}{1864} +\item Rebière, A. \AppRef{1857}{1857} +\item Reinhold, E. \AppRef{1574b}{1574} +\item Rhind Papyrus \ParRef{20} +\item Richter, A. \FnRef{111} +\item Ritt, G. \AppRef{1837}{1837} +\item Robison, J. \AppRef{1855}{1855} +\item Rodham, J. \AppRef{1795}{1795} +\item Ross, J.~A. \AppRef{1805}{1805} +\item Rouché, E. \AppRef{1687}{1687} +\item Rudd, T. \AppRef{1612}{1612} +\item Rudio, F. \FnRef{116} +\item Rummer, F. \AppRef{1852b}{1852} +\item Rutherford, W. \AppRef{1864a}{1864} +\indexspace +\item Saunderson, N. \AppRef{1740}{1740} +\item Savile, H. 6, \FnRef{34} +\item Schmid, W. \AppRef{1539}{1539} +\item Schoene, H. \FnRef{82}, \FnRef{83} +\item Schooten, F. van \AppRef{1657}{1657} +\item Schwenter, D. \AppRef{1667}{1667} +\item Sems, J. \AppRef{1612}{1612} +\item Serle, G. \FnRef{11}, \FnRef{30}, \FnRef{50} +\item Silberschlag, J. \AppRef{1772}{1772(?)} +\item Simplicius \FnRef{116} +\item Simpson, T. \AppRef{1609}{1609}, \AppRef{1747}{1747}, + \AppRef{1752}{1752}, \AppRef{1902}{1902} +\item Simson, R. \ParRef{39}, \FnRef{85}, \FnRef{88}, \FnRef{106}, + \FnRef{116}, \AppRef{1840}{1840} +\item Smith, C. \AppRef{1902}{1902} +\item Smith, T. \ParRef{2}, \ParRef{3}, \ParRef{5}, \ParRef{19}, + \FnRef{14}, \FnRef{18}, \FnRef{66} +\item Snellius, W. \FnRef{111}, \AppRef{1615}{1615} +\item Speidell, J. \AppRef{1616}{1616} +\item Stanham, W.~C. \AppRef{1864a}{1864} +\item Steiner, J. \FnRef{117} +\item Steinschneider, M. \ParRef{2}, \ParRef{3}, \ParRef{4}, \ParRef{6} + \FnRef{12}, \FnRef{36}, \FnRef{53}, \FnRef{64} +\item Stevens, F.~H. \AppRef{1892}{1892} +\item Strode, T. \AppRef{1684}{1684} +\item Suter, H. \ParRef{3}, \ParRef{4}, \ParRef{19} + \FnRef{17}, \FnRef{36}, \FnRef{74}, + \FnRef{75}, \FnRef{76}, \FnRef{77} +\item Swinden, G.~H. van \FnRef{111} +\indexspace +\item \b{T}âbit b. Qorra \ParRef{19} +\item Tartaglia, N. \AppRef{appendix}{}, \AppRef{1547a}{1547}, \AppRef{1560}{1560} +\item Taylor, C. \FnRef{103} +\item Tittel, K. \FnRef{81} +\item Townsend, R. \FnRef{117} +\indexspace +\item Urbin, Duke of \ParRef{2} +\item Ussher, J. \ParRef{2}, \ParRef{5}, \FnRef{15} +\indexspace +\item Vannson \FnRef{111} +\item Verhulst, P.~F. \AppRef{1827}{1827} +\item Viani de' Malatesti, F. \FnRef{11} +\item Vincent, A.~J.~C. \FnRef{83} +\indexspace +\item Wallis, J. \AppRef{1609}{1609}, \AppRef{1684}{1684} +\item Wenrich, J.~G. \ParRef{2}, \FnRef{13} +\item Wiegand, A. \FnRef{111} +\item Wissowa \FnRef{9}, \FnRef{81} +\item Wölffing, E. \AppRef{1807b}{1807}, \AppRef{1844}{1844} +\item Woepcke, F. \ParRef{7}, \ParRef{8}, \ParRef{9}, \ParRef{10}, + \ParRef{11}, \ParRef{12}, \ParRef{13}, \ParRef{18}, \ParRef{19}, \ParRef{57}, + \FnRef{36}, \FnRef{37}, \FnRef{46}, \FnRef{48}, + \FnRef{78}, \FnRef{79}, \FnRef{80}, \FnRef{86}, + \FnRef{101}, \FnRef{103}, \FnRef{104}, \FnRef{108}, + \FnRef{109}, \FnRef{111}, \FnRef{112} +\item Wright, J.~M.~F. \AppRef{1805}{1805} +\indexspace +\item Zdenek, R. \AppRef{1910b}{1910} +\item Zeuthen, H.G. \FnRef{103} +\item Zimmerman, L. \AppRef{1864b}{1864} +\renewcommand{\footrulewidth}{1pt} +\fancyfoot[C]{\small\textsc{cambridge: printed by john clay, m.a. at the university press}} +\end{theindex} +\pagestyle{empty} + +%%%%%%%%%%%%%%%%%%%%%%%%% GUTENBERG LICENSE %%%%%%%%%%%%%%%%%%%%%%%%%% + +\ifthenelse{\boolean{ForPrinting}}{\cleardoublepage}{\clearpage} + + +\backmatter +\phantomsection +\pdfbookmark[-1]{Back Matter}{Back Matter} +\phantomsection +\pdfbookmark[0]{PG License}{Project Gutenberg License} +\fancyhead[C]{\textsc{LICENSING}} + +\begin{PGtext} +End of the Project Gutenberg EBook of Euclid's Book on Divisions of Figures, by +Raymond Clare Archibald + +*** END OF THIS PROJECT GUTENBERG EBOOK EUCLID'S BOOK ON DIVISIONS *** + +***** This file should be named 38640-pdf.pdf or 38640-pdf.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/3/8/6/4/38640/ + +Produced by Joshua Hutchinson, Ralph Carmichael and the +Online Distributed Proofreading Team at http://www.pgdp.net +(This file was produced from images from the Cornell +University Library: Historical Mathematics Monographs +collection.) + + +Updated editions will replace the previous one--the old editions +will be renamed. + +Creating the works from public domain print editions means that no +one owns a United States copyright in these works, so the Foundation +(and you!) can copy and distribute it in the United States without +permission and without paying copyright royalties. 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show diff --git a/38640-t/images/source/064.eps b/38640-t/images/source/064.eps new file mode 100644 index 0000000..498f170 --- /dev/null +++ b/38640-t/images/source/064.eps @@ -0,0 +1,43 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%BoundingBox: 48 98 475 342 +475 % third BoundingBox parameter + 48 % first BoundingBox parameter +sub 4 div +95 % width in mm specified in \includegraphics +div /myscale exch def + +myscale setlinewidth newpath + 72 293 moveto + 451 293 lineto + 451 184 lineto + 72 184 lineto +closepath stroke + 263 293 moveto + 451 293 lineto + 451 108 lineto + 263 108 lineto +closepath stroke + 341 293 moveto + 341 108 lineto + + 263 108 moveto + 451 293 lineto + + 263 125 moveto + 341 293 lineto + + 263 293 moveto + 263 310 lineto +stroke + 263 125 moveto + 263 125 185 65 115 arc stroke + +/Helvetica findfont 16 myscale mul scalefont setfont + 58 291 moveto (A) show + 453 300 moveto (B) show + 248 277 moveto (D) show + 175 302 moveto (\(F\)) show + 346 171 moveto (H) show + 266 319 moveto (N) show + 248 122 moveto (O) show + 343 300 moveto (Z) show diff --git a/38640-t/images/source/065.eps b/38640-t/images/source/065.eps new file mode 100644 index 0000000..362149f --- /dev/null +++ b/38640-t/images/source/065.eps @@ -0,0 +1,35 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%BoundingBox: 53 84 297 389 +297 % third BoundingBox parameter + 53 % first BoundingBox parameter +sub 4 div +50 % width in mm specified in \includegraphics +div /myscale exch def + +myscale setlinewidth newpath + 72 108 moveto + 230 303 lineto + 275 108 lineto +closepath stroke + 230 303 moveto + 172 108 lineto + + 238 108 moveto + 175 236 lineto + + 140 192 moveto + 197 192 lineto + + 230 303 moveto + 277 362 lineto +stroke +/Helvetica findfont 16 myscale mul scalefont setfont + 217 307 moveto (A) show + 58 99 moveto (B) show + 273 92 moveto (C) show + 201 188 moveto (D) show + 124 188 moveto (E) show + 166 239 moveto (H) show + 170 92 moveto (K) show + 276 365 moveto (T) show + 231 92 moveto (Z) show diff --git a/38640-t/images/source/066.eps b/38640-t/images/source/066.eps new file mode 100644 index 0000000..1c27034 --- /dev/null +++ b/38640-t/images/source/066.eps @@ -0,0 +1,31 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%BoundingBox: 58 87 320 361 +320 % third BoundingBox parameter + 58 % first BoundingBox parameter +sub 4 div +50 % width in mm specified in \includegraphics +div /myscale exch def + +myscale setlinewidth newpath + 151 269 moveto + 72 108 lineto + 298 108 lineto +closepath stroke + 151 269 moveto + 182 333 lineto +stroke + 131 228 moveto + 165 108 lineto +stroke + 96 157 moveto + 151 157 lineto +stroke +/Helvetica findfont 16 myscale mul scalefont setfont + 138 267 moveto (A) show + 63 92 moveto (B) show + 301 92 moveto (C) show + 156 153 moveto (D) show + 81 150 moveto (E) show + 117 225 moveto (H) show + 166 94 moveto (T) show + 182 342 moveto (Z) show diff --git a/38640-t/images/source/070.eps b/38640-t/images/source/070.eps new file mode 100644 index 0000000..4f531ea --- /dev/null +++ b/38640-t/images/source/070.eps @@ -0,0 +1,45 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%BoundingBox: 65 98 466 183 +466 % third BoundingBox parameter + 65 % first BoundingBox parameter +sub 4 div +90 % width in mm specified in \includegraphics +div /myscale exch def + +myscale setlinewidth newpath + 113 161 moveto + 427 161 lineto + + 72 108 moveto + 452 108 lineto + + 113 159 moveto + 113 163 lineto + + 212 159 moveto + 212 163 lineto + + 427 159 moveto + 427 163 lineto + + 72 106 moveto + 72 110 lineto + + 190 106 moveto + 190 110 lineto + + 329 106 moveto + 329 110 lineto + + 452 106 moveto + 452 110 lineto +stroke + +/Helvetica findfont 16 myscale mul scalefont setfont + 423 167 moveto (A) show + 107 167 moveto (B) show + 217 167 moveto (C) show + 449 113 moveto (D) show + 70 113 moveto (E) show + 184 115 moveto ([W]) show + 329 113 moveto (Z) show diff --git a/38640-t/images/source/071.eps b/38640-t/images/source/071.eps new file mode 100644 index 0000000..462573f --- /dev/null +++ b/38640-t/images/source/071.eps @@ -0,0 +1,41 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%BoundingBox: 65 98 466 193 +466 % third BoundingBox parameter + 65 % first BoundingBox parameter +sub 4 div +85 % width in mm specified in \includegraphics +div /myscale exch def + +myscale setlinewidth newpath + 113 152 moveto + 427 152 lineto + + 72 108 moveto + 452 108 lineto + + 113 149 moveto + 113 154 lineto + + 319 149 moveto + 319 154 lineto + + 427 149 moveto + 427 154 lineto + + 72 106 moveto + 72 110 lineto + + 190 106 moveto + 190 110 lineto + + 452 106 moveto + 452 110 lineto +stroke + +/Helvetica findfont 16 myscale mul scalefont setfont + 423 162 moveto (A) show + 107 162 moveto (B) show + 311 162 moveto (C) show + 449 113 moveto (D) show + 70 113 moveto (E) show + 187 113 moveto (Z) show diff --git a/38640-t/images/source/073.eps b/38640-t/images/source/073.eps new file mode 100644 index 0000000..f41b477 --- /dev/null +++ b/38640-t/images/source/073.eps @@ -0,0 +1,37 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%BoundingBox: 62 98 414 488 +414 % third BoundingBox parameter + 62 % first BoundingBox parameter +sub 4 div +70 % width in mm specified in \includegraphics +div /myscale exch def + +myscale setlinewidth newpath + 72 305 moveto + 395 305 lineto + 395 225 lineto + 72 225 lineto +closepath stroke + 198 305 moveto + 395 305 lineto + 395 108 lineto + 198 108 lineto +closepath stroke + 198 305 moveto + 315 463 lineto + + 315 463 moveto + 315 108 lineto + + 198 108 moveto + 395 305 lineto +stroke + 198 305 moveto + 198 305 197 0 60 arc stroke + +/Helvetica findfont 16 myscale mul scalefont setfont + 72 308 moveto (A) show + 301 308 moveto (B) show + 184 308 moveto (C) show + 381 308 moveto (D) show + 315 468 moveto (Q) show diff --git a/38640-t/images/source/074.eps b/38640-t/images/source/074.eps new file mode 100644 index 0000000..c635bbb --- /dev/null +++ b/38640-t/images/source/074.eps @@ -0,0 +1,34 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%BoundingBox: 56 92 292 375 +292 % third BoundingBox parameter + 56 % first BoundingBox parameter +sub 4 div +40 % width in mm specified in \includegraphics +div /myscale exch def + +myscale setlinewidth newpath + 168 358 moveto + 99 178 lineto + 275 178 lineto +closepath stroke + 168 358 moveto + 72 108 lineto + 237 108 lineto +closepath stroke + 237 108 moveto + 147 304 lineto +stroke + 137 285 moveto + 141 283 lineto +stroke + +/Helvetica findfont 16 myscale mul scalefont setfont + 163 361 moveto (a) show + 86 174 moveto (b) show + 241 103 moveto (d) show + 219 181 moveto (e) show + 278 174 moveto (g) show + 131 277 moveto (i) show + 194 164 moveto (k) show + 138 300 moveto (t) show + 61 97 moveto (z) show diff --git a/38640-t/images/source/076.eps b/38640-t/images/source/076.eps new file mode 100644 index 0000000..36cccbe --- /dev/null +++ b/38640-t/images/source/076.eps @@ -0,0 +1,35 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%BoundingBox: 53 89 278 310 +278 % third BoundingBox parameter + 53 % first BoundingBox parameter +sub 4 div +53 % width in mm specified in \includegraphics +div /myscale exch def + +myscale setlinewidth newpath + 185 292 moveto + 72 108 lineto + 254 108 lineto +closepath stroke + 108 167 moveto + 232 167 lineto + + 185 292 moveto + 163 108 lineto + 204 243 lineto + + 215 207 moveto + 219 209 lineto +stroke + +/Helvetica findfont 16.000 scalefont setfont + 187 295 moveto (a) show + 98 164 moveto (b) show + 161 169 moveto (c) show + 163 94 moveto (d) show + 255 96 moveto (e) show + 238 164 moveto (g) show + 222 204 moveto (i) show + 212 237 moveto (k) show + 184 155 moveto (m) show + 63 96 moveto (z) show diff --git a/38640-t/images/source/077.eps b/38640-t/images/source/077.eps new file mode 100644 index 0000000..2bd953e --- /dev/null +++ b/38640-t/images/source/077.eps @@ -0,0 +1,39 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%BoundingBox: 53 89 339 352 +339 % third BoundingBox parameter + 53 % first BoundingBox parameter +sub 4 div +66 % width in mm specified in \includegraphics +div /myscale exch def + +myscale setlinewidth newpath + 274 334 moveto + 323 176 lineto + 72 176 lineto +closepath stroke + 197 108 moveto + 197 176 lineto + + 197 176 moveto + 243 309 lineto + + 243 309 moveto + 197 108 lineto + + 197 108 moveto + 274 334 lineto + + 274 334 moveto + 197 176 lineto +stroke + 197 258 moveto + newpath + 197 258 150 213 327 arc stroke + +/Helvetica findfont 16 myscale mul scalefont setfont + 273 337 moveto (A) show + 325 174 moveto (B) show + 58 174 moveto (C) show + 182 162 moveto (E) show + 231 307 moveto (T) show + 197 94 moveto (Z) show diff --git a/38640-t/images/source/078.eps b/38640-t/images/source/078.eps new file mode 100644 index 0000000..cb064bc --- /dev/null +++ b/38640-t/images/source/078.eps @@ -0,0 +1,41 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%BoundingBox: 53 89 360 399 +360 % third BoundingBox parameter + 53 % first BoundingBox parameter +sub 4 div +70 % width in mm specified in \includegraphics +div /myscale exch def + +myscale setlinewidth newpath + 212 248 moveto + newpath + 212 248 140 0 360 arc stroke + 212 248 moveto + 153 376 lineto + + 212 248 moveto + 200 109 lineto + + 212 248 moveto + 132 133 lineto + + 153 376 moveto + 132 133 lineto + + 200 109 moveto + 132 133 lineto + + 153 376 moveto + 200 109 lineto + + 72 248 moveto + 317 155 lineto +stroke + +/Helvetica findfont 16 myscale mul scalefont setfont + 149 377 moveto (A) show + 191 94 moveto (B) show + 114 127 moveto (C) show + 58 246 moveto (E) show + 217 246 moveto (D) show + 318 138 moveto (Z) show diff --git a/38640-t/images/source/082.eps b/38640-t/images/source/082.eps new file mode 100644 index 0000000..702f50a --- /dev/null +++ b/38640-t/images/source/082.eps @@ -0,0 +1,48 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%BoundingBox: 58 94 288 427 +288 % third BoundingBox parameter + 58 % first BoundingBox parameter +sub 4 div +53 % width in mm specified in \includegraphics +div /myscale exch def + +myscale setlinewidth newpath + 72 392 moveto + 83 108 lineto + 259 108 lineto +closepath stroke + 76 291 moveto + 139 291 lineto + + 78 240 moveto + 172 239 lineto + + 79 218 moveto + 187 218 lineto + + 265 412 moveto + 265 187 lineto + + 263 412 moveto + 267 412 lineto + + 263 356 moveto + 267 356 lineto + + 263 187 moveto + 267 187 lineto +stroke + +/Helvetica findfont 16 myscale mul scalefont setfont + 67 396 moveto (t) show + 65 288 moveto (a) show + 72 103 moveto (b) show + 142 286 moveto (d) show + 269 410 moveto (e) show + 262 103 moveto (g) show + 67 237 moveto (h) show + 269 185 moveto (i) show + 177 234 moveto (k) show + 72 217 moveto (l) show + 191 213 moveto (m) show + 269 354 moveto (z) show diff --git a/38640-t/images/source/083.eps b/38640-t/images/source/083.eps new file mode 100644 index 0000000..96f862c --- /dev/null +++ b/38640-t/images/source/083.eps @@ -0,0 +1,47 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%BoundingBox: 62 94 372 399 +372 % third BoundingBox parameter + 62 % first BoundingBox parameter +sub 4 div +70 % width in mm specified in \includegraphics +div /myscale exch def + +myscale setlinewidth newpath + 231 376 moveto + 141 108 lineto + 348 108 lineto +closepath stroke + 189 250 moveto + 286 250 lineto + + 159 161 moveto + 325 161 lineto + + 72 372 moveto + 72 108 lineto + + 70 372 moveto + 74 372 lineto + + 70 312 moveto + 74 312 lineto + + 70 169 moveto + 74 169 lineto + + 70 108 moveto + 74 108 lineto +stroke + +/Helvetica findfont 16 myscale mul scalefont setfont + 177 246 moveto (a) show + 130 103 moveto (b) show + 290 246 moveto (d) show + 353 103 moveto (g) show + 146 155 moveto (h) show + 332 155 moveto (k) show + 77 167 moveto (l) show + 77 370 moveto (m) show + 77 310 moveto (n) show + 77 106 moveto (s) show + 231 382 moveto (t) show diff --git a/38640-t/images/source/084.eps b/38640-t/images/source/084.eps new file mode 100644 index 0000000..1158055 --- /dev/null +++ b/38640-t/images/source/084.eps @@ -0,0 +1,63 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%BoundingBox: 53 94 339 380 +339 % third BoundingBox parameter + 53 % first BoundingBox parameter +sub 4 div +65 % width in mm specified in \includegraphics +div /myscale exch def + +myscale setlinewidth newpath + 233 348 moveto + 146 127 lineto + 320 127 lineto +closepath stroke + 199 262 moveto + 266 262 lineto + + 176 204 moveto + 289 204 lineto + + 168 182 moveto + 298 182 lineto + + 233 348 moveto + 235 127 lineto + + 72 354 moveto + 72 108 lineto + + 70 354 moveto + 74 354 lineto + + 70 306 moveto + 74 306 lineto + + 70 253 moveto + 74 253 lineto + + 70 218 moveto + 74 218 lineto + + 70 108 moveto + 74 108 lineto +stroke + +/Helvetica findfont 16 myscale mul scalefont setfont + 187 260 moveto (a) show + 131 122 moveto (b) show + 271 260 moveto (d) show + 77 305 moveto (e) show + 236 266 moveto (f) show + 325 122 moveto (g) show + 236 209 moveto (h) show + 77 213 moveto (i) show + 231 354 moveto (k) show + 77 351 moveto (l) show + 161 204 moveto (m) show + 154 178 moveto (n) show + 297 204 moveto (o) show + 304 178 moveto (p) show + 236 188 moveto (r) show + 234 113 moveto (s) show + 77 103 moveto (t) show + 77 248 moveto (z) show diff --git a/38640-t/images/source/086.eps b/38640-t/images/source/086.eps new file mode 100644 index 0000000..63273bd --- /dev/null +++ b/38640-t/images/source/086.eps @@ -0,0 +1,50 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%BoundingBox: 67 98 320 352 +320 % third BoundingBox parameter + 67 % first BoundingBox parameter +sub 4 div +70 % width in mm specified in \includegraphics +div /myscale exch def + +myscale setlinewidth newpath + 75 286 moveto + 301 286 lineto + 221 108 lineto + 84 127 lineto +closepath stroke + 75 286 moveto + 221 108 lineto +stroke + 81 185 moveto + 128 222 lineto + 301 286 lineto +closepath stroke + 84 127 moveto + 128 222 lineto + + 84 127 moveto + 301 286 lineto + + 72 328 moveto + 300 328 lineto + + 72 331 moveto + 72 326 lineto + + 219 331 moveto + 219 326 lineto + + 300 331 moveto + 300 326 lineto +stroke + +/Helvetica findfont 16 myscale mul scalefont setfont + 70 291 moveto (a) show + 73 122 moveto (b) show + 222 101 moveto (c) show + 306 279 moveto (d) show + 67 337 moveto (e) show + 299 337 moveto (i) show + 71 178 moveto (k) show + 128 227 moveto (t) show + 217 337 moveto (z) show diff --git a/38640-t/images/source/087.eps b/38640-t/images/source/087.eps new file mode 100644 index 0000000..5da0538 --- /dev/null +++ b/38640-t/images/source/087.eps @@ -0,0 +1,45 @@ +%!PS-Adobe-3.0 +%%BoundingBox: 43 79 372 352 +372 % third BoundingBox parameter + 43 % first BoundingBox parameter +sub 4 div +67 % width in mm specified in \includegraphics +div /myscale exch def + +1 setlinewidth +newpath +72.0 269.0 moveto +210.1 325.6 lineto +336.0 108.0 lineto +87.9 108.0 lineto +closepath stroke +newpath +72.0 269.0 moveto +336.0 108.0 lineto +stroke +newpath +87.9 108.0 moveto +210.1 325.6 lineto +112.2 108.0 lineto +169.3 209.6 lineto +stroke +/Helvetica findfont 16.000 scalefont setfont +62.6 264.8 moveto +(a) show +79.0 101.0 moveto +(b) show +338.8 103.3 moveto +(c) show +214.7 318.6 moveto +(d) show +109.4 94.0 moveto +(l) show +172.6 213.3 moveto +(t) show + 1.0 0.0 0.0 setrgbcolor +newpath +43.9 79.9 moveto +43.9 351.4 lineto +371.5 351.4 lineto +371.5 79.9 lineto +closepath stroke diff --git a/38640-t/images/source/160.eps b/38640-t/images/source/160.eps new file mode 100644 index 0000000..d64bd0b --- /dev/null +++ b/38640-t/images/source/160.eps @@ -0,0 +1,32 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%BoundingBox: 53 94 578 375 +578 % third BoundingBox parameter + 53 % first BoundingBox parameter +sub 4 div +120 % width in mm specified in \includegraphics +div /myscale exch def + +myscale setlinewidth newpath + 72 359 moveto + 550 257 lineto + 526 108 lineto + 129 108 lineto +closepath stroke + 115 171 moveto + 550 257 lineto + 129 108 lineto + 467 275 lineto + 123 133 lineto + 433 282 lineto + 129 108 lineto +stroke + +/Helvetica findfont 16 myscale mul scalefont setfont + 63 361 moveto (a) show + 119 99 moveto (b) show + 528 103 moveto (c) show + 554 260 moveto (d) show + 465 279 moveto (e) show + 105 167 moveto (i) show + 430 288 moveto (k) show + 109 127 moveto (l) show diff --git a/38640-t/images/source/161.eps b/38640-t/images/source/161.eps new file mode 100644 index 0000000..7a3d728 --- /dev/null +++ b/38640-t/images/source/161.eps @@ -0,0 +1,38 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%BoundingBox: 53 89 452 347 +452 % third BoundingBox parameter + 53 % first BoundingBox parameter +sub 4 div +90 % width in mm specified in \includegraphics +div /myscale exch def + +myscale setlinewidth newpath + 72 323 moveto + 428 333 lineto + 381 108 lineto + 81 122 lineto +closepath stroke + 283 329 moveto + 180 117 lineto + + 180 117 moveto + 428 333 lineto + + 428 333 moveto + 320 111 lineto + + 381 108 moveto + 283 329 lineto + + 320 111 moveto + 283 329 lineto +stroke + +/Helvetica findfont 16 myscale mul scalefont setfont + 63 323 moveto (a) show + 72 110 moveto (b) show + 381 94 moveto (c) show + 432 333 moveto (d) show + 280 331 moveto (e) show + 318 96 moveto (i) show + 177 106 moveto (z) show diff --git a/38640-t/images/source/175.eps b/38640-t/images/source/175.eps new file mode 100644 index 0000000..8796beb --- /dev/null +++ b/38640-t/images/source/175.eps @@ -0,0 +1,32 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%BoundingBox: 56 92 297 328 +297 % third BoundingBox parameter + 56 % first BoundingBox parameter +sub 4 div +45 % width in mm specified in \includegraphics +div /myscale exch def + +myscale setlinewidth newpath + 196 313 moveto + 72 108 lineto + 279 108 lineto +closepath stroke + 105 162 moveto + 257 162 lineto + + 72 108 moveto + 223 247 lineto + + 234 213 moveto + 238 216 lineto +stroke + +/Helvetica findfont 16 myscale mul scalefont setfont + 194 316 moveto (a) show + 93 160 moveto (b) show + 283 103 moveto (e) show + 264 157 moveto (g) show + 241 211 moveto (i) show + 124 164 moveto (l) show + 226 248 moveto (t) show + 61 97 moveto (z) show diff --git a/38640-t/images/source/181.eps b/38640-t/images/source/181.eps new file mode 100644 index 0000000..a87d229 --- /dev/null +++ b/38640-t/images/source/181.eps @@ -0,0 +1,38 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%BoundingBox: 58 94 274 282 +274 % third BoundingBox parameter + 58 % first BoundingBox parameter +sub 4 div +53 % width in mm specified in \includegraphics +div /myscale exch def + +myscale setlinewidth newpath + 149 260 moveto + 72 111 lineto + 205 111 lineto +closepath stroke + 96 157 moveto + 188 157 lineto + + 254 266 moveto + 254 108 lineto + + 251 266 moveto + 256 266 lineto + + 251 214 moveto + 256 214 lineto + + 251 108 moveto + 256 108 lineto +stroke + +/Helvetica findfont 16 myscale mul scalefont setfont + 142 265 moveto (a) show + 63 103 moveto (b) show + 259 265 moveto (e) show + 208 103 moveto (g) show + 86 157 moveto (h) show + 259 106 moveto (i) show + 194 157 moveto (k) show + 259 211 moveto (z) show diff --git a/38640-t/images/source/187.eps b/38640-t/images/source/187.eps new file mode 100644 index 0000000..c9eba0f --- /dev/null +++ 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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..19c4757 --- /dev/null +++ b/README.md @@ -0,0 +1,2 @@ +Project Gutenberg (https://www.gutenberg.org) public repository for +eBook #38640 (https://www.gutenberg.org/ebooks/38640) |