\documentclass[twoside,10pt,openany]{book} \usepackage[a4paper,margin=1.5cm]{geometry} %\special{landscape} \usepackage{cclicenses} \usepackage{verbatim} \usepackage{auto-pst-pdf} \usepackage{dblfnote} % This (from ``yafoot'') caused problems, %perhaps because there are too many footnotes in columns made by the %parcolumns package---or by the multicol package, used in the introduction: more than 20 footnotes seems to be too much. But if the Introduction is broken into two chapters, the problem is solved. %\usepackage[english,all,portrait,draft]{draftcopy} doesn't seem to work with pdflatex % The following also has problems: \begin{comment} \usepackage[printwatermark=true, % for a "Draft" watermark allpages=true, fontfamily=pag, color=gray!12, %grayness=0.9, textmark=Draft, angle=55, fontsize=5cm, width=\paperwidth, fontseries=b, scale=0.8, xcoord=0, ycoord=0]{xwatermark} \end{comment} % So I try an alternative: the needed \begin{comment} \usepackage{color} \usepackage{eso-pic} \definecolor{lightgray}{gray}{.25} \AddToShipoutPicture{% \AtTextCenter{% \makebox(0,0)[c]{\resizebox{\textwidth}{!}{% \rotatebox{45}{\textsf{\textbf{\color{lightgray}DRAFT}}}}} } } \end{comment} \usepackage[polutonikogreek,english]{babel} \usepackage[latin5]{inputenc} %\usepackage[LGR,T1]{fontenc} % adding LGR before T1 does not seem to %make a difference; adding it after changes all letters to Greek \usepackage[T1]{fontenc} %\usepackage{parskip} % did not have the desired effect. \usepackage{pstricks} \usepackage{multicol} \usepackage[neverdecrease]{paralist} \usepackage{hfoldsty} \usepackage{relsize} % Here \smaller scales by 1/1.2; \relscale{X} scales by X \renewenvironment{quote}{\begin{list}{} {\relscale{.90}\setlength{\leftmargin}{0.05\textwidth} \setlength{\rightmargin}{\leftmargin}} \item[]} {\end{list}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\lett}[1]{\textsf{#1}} % For Greek \newcommand\gr[1]{\foreignlanguage{polutonikogreek}{#1}} \newcommand{\strgt}{\textsc{straight}} \newcommand{\Strgt}{\textsc{Straight}} \newcommand{\rgt}{\textsc{right}} % The following is to make ``long lines'' that stretch a line onto a % second when the parallel lines are so. %\newcommand{\lli}{\hspace{0.1\textwidth}\mbox{}}% for making long % lines, to maintain parallelism %\newcommand{\lli}{\quad \mbox{} \quad \mbox{} \quad \mbox{}} %\newcommand{\lli}{\phantom{a a a a a a}} % \newcommand{\lli}{\\\mbox{}} \newcommand{\myqed}{---just what it was necessary to show.} \newcommand{\myqef}{---just what it was necessary to do.} \newcommand{\ozqed}{---g{\"o}sterilmesi gereken tam buydu.} \newcommand{\ozqef}{---yap{\i}lmas{\i} gereken tam buydu.} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{verse} \newenvironment{myverse}{% \begin{verse}% }% {\end{verse}} \usepackage{parcolumns} % documentation under ``sauerj'' \newcommand{\parsen}[3]{% \colchunk{\begin{myverse}#1\end{myverse}}% \colchunk{\begin{myverse}\gr{#2}\end{myverse}}% \colchunk{\begin{myverse}#3\end{myverse}}% \colplacechunks% } \newenvironment{textpart}{ \begin{parcolumns}[rulebetween,distance=1em,sloppy]{3}% \setlength{\leftmargini}{0em}% } {\end{parcolumns}} \newcommand{\prop}[1][]{%\newpage% \section{#1}\setcounter{myfn}0\setcounter{footnote}0} % Proposition %\newenvironment{proposition}[1][]{ %\begin{textpart}[\prop[#1]]} %{\end{textpart}} \newenvironment{proposition}[1][]{\prop[#1] \begin{parcolumns}[rulebetween,distance=1em,sloppy]{3}% \setlength{\leftmargini}{0em}% } {\end{parcolumns}} \newcommand{\parancil}[2]{% for ancillary material \colchunk{#1}% \colchunk{#2}% \colplacechunks% } \newenvironment{ancillary}{% \setcounter{myfn}0\setcounter{footnote}0% \begin{parcolumns}[rulebetween,distance=3em,sloppy]{2}% } {\end{parcolumns}} %\usepackage{footnote} % This didn't work for me %\makesavenoteenv{parsen} %\renewcommand{\thefootnote}{\fnsymbol{footnote}} \newcounter{myfn} \newcommand{\myfntext}[1]{\stepcounter{myfn}% \footnotetext[\value{myfn}]{#1}} \usepackage{url} %\special{!TeXDict begin /landplus90{true}store end} % This makes the % ps file right-side-up in landscape % mode \begin{document} \title{Book I of the Elements\qquad \gr{STOIQEIWN A}\qquad {\"O}{\u g}elerin Birinci Kitab\i} \author{Euclid\qquad\gr{EUKLEIDOS}\qquad {\"O}klid} \maketitle \begin{center} This work is licensed under the\\ Creative Commons\\ Attribution-NonCommercial-ShareAlike 3.0\\ Unported License.\\ To view a copy of this license, visit\\ \url{http://creativecommons.org/licenses/by-nc-sa/3.0/}\\ or send a letter to\\ Creative Commons,\\ 444 Castro Street, Suite 900,\\ Mountain View, California, 94041, USA. \mbox{}\\ Bu \c cal\i\c sma\\ Creative Commons Attribution-Gayriticari-ShareAlike 3.0\\ Unported Lisans\i\ ile lisansl\i.\\ Lisans\i n bir kopyas\i n\i\ g\"orebilmek i\c cin,\\ \url{http://creativecommons.org/licenses/by-nc-sa/3.0/}\\ adresini ziyaret edin ya da mektup at\i n:\\ Creative Commons,\\ 444 Castro Street, Suite 900,\\ Mountain View, California, 94041, USA.\\ \mbox{}\\ \cc \ccby \"Ozer \"Ozt\"urk \&\ David Pierce \ccnc \ccsa\\ \mbox{}\\ Matematik B\"ol\"um\"u\\ Mimar Sinan G\"uzel Sanatlar \"Universitesi\\ Bomonti, \c Si\c sli, \.Istanbul, 34380\\ \mbox{}\\ \url{ozer.ozturk@msgsu.edu.tr}\qquad \url{dpierce@msgsu.edu.tr}\\ \url{http://mat.msgsu.edu.tr/} \end{center} %\begin{multicols}3 %\tableofcontents %\end{multicols} %\listoftables %\chapter*{Preface} \vfill This edition of the first book of Euclid's \emph{Elements} was prepared for a first-year undergraduate course in the mathematics department of Mimar Sinan Fine Arts University. The text has been corrected after its use in the course in the fall of 2011. \vfill \"Oklid'in \"O\u geler'inin bu bask\i s\i\ Mimar Sinan G\"uzel Sanatlar \"Universitesi, Matematik B\"ol\"umnde bir birinci s\i n\i f lisans dersi i\c cin haz\i rlanm\i\c st\i r. 2010--2011 G\"uz d\"oneminde bu notlar ilk defa kullan\i lm\i\c s ve fark edilen hatalar d\"uzeltilmi\c stir. \vfill \input{euclid-introduction} \input{euclid-introduction-tr3} \chapter{Elements}%{Elements I, \gr{STOIQEIWN A}, \"O\u gelerin 1si} \section*{`Definitions'} \begin{textpart} \parsen{ Boundaries\footnotemark } { <'Oroi } { S{\i}n{\i}rlar } \myfntext{The usual translation is `definitions', but what follow are not really definitions in the modern sense.} \parsen{ {}[1] A point is\\ {}[that] whose part is nothing.\footnotemark } { Shme~i'on >estin,\\ o<~u m'eroc o>uj'en. } { Bir nokta,\\ par\c cas\i \ hi{\c c}bir {\c s}ey oland\i r. } \myfntext{Presumably subject and predicate are inverted here, so the sense is that of `A point is that of which nothing is a part.'} \parsen{ {}[2] A line,\\ length without breadth. } { Gramm`h d`e\\ m~hkoc >aplat'ec. } { Bir \c cizgi,\\ ensiz uzunluktur. } \parsen{ {}[3] Of a line,\\ the extremities are points. } { Gramm~hc d`e\\ p'erata shme~ia. } { Bir \c cizginin\\ u\c clar\i ndakiler, noktalard\i r. } \parsen{ {}[4] A straight line is\\ whatever [line] evenly\\ with the points of itself\\ lies. } { E>uje~ia gramm'h >estin,\\ <'htic >ex >'isou\\ to~ic >ef> Epif'aneia d'e >estin,\\ <`o m~hkoc ka`i pl'atoc m'onon >'eqei. } { Bir y\" uzey,\\ sadece eni ve boyu oland\i r. } \parsen{ {}[6] Of a surface,\\ the boundaries are lines. } { >Epifane'iac d`e\\ p'erata gramma'i. } { Bir y\" uzeyin\\ u\c clar\i ndakiler, \c cizgilerdir. } \parsen{ {}[7] A plane surface is\\ what [surface] evenly\\ with the points of itself\\ lies. } { >Ep'ipedoc >epif'anei'a >estin,\\ <'htic >ex >'isou\\ ta~ic >ef> uje'iaic\\ ke~itai. } { Bir d\" uzlem,\\ \" uzerindeki do\u grular{\i}n noktalar{\i}yla hizal\i \ uzanan bir y\" uzeydir. } \parsen{ {}[8] A plane angle is,\\ \dots\footnotemark\\ in a plane,\\ two lines taking hold of one another,\\ and not lying on a \strgt,\\ to one another\\ the inclination of the lines. } { >Ep'ipedoc d`e gwn'ia >est`in\\ en >epip'edw|\\ d'uo gramm~wn all'hlwn\\ ka`i m`h >ep> e>uje'iac keim'enwn\\ pr`oc >all'hlac\\ t~wn gramm~wn kl'isic. } { Bir d\" uzlem a\c c\i s{\i},\\ \mbox{}\\ bir d\"uzlemde\\ kesi\c sen ve ayn\i \ do\u gru \" uzerinde uzanmayan\\ iki \c cizginin birbirine g\" ore e\u gikli\u gidir. } \myfntext{There is no way to put `the' here to parallel the Greek.} \parsen{ {}[9] Whenever the lines containing the angle\\ be straight,\\ rectilineal is called the angle. } { <'Otan d`e auje~iai >~wsin,\\ e>uj'ugrammoc kale~itai uje~ia\\ {}>ep> e>uje~ian staje~isa\\ t`ac >efex~hc gwn'iac\\ {}>'isac >all'hlaic poi~h|,\\ {}>orj`h\\ 'iswn gwni~wn >esti,\\ ka`i\\ efesthku~ia e>uje~ia\\ k'ajetoc kale~itai,\\ {}>ef> <`hn >ef'esthken. } { Bir do\u gru\\ ba{\c s}ka bir do\u grunun \" uzerine yerle\c sip\\ birbirine e\c sit biti{\c s}ik a\c c\i lar olu\c sturdu\u gunda,\\ e\c sit a\c c\i lar\i n her birine dik a\c c\i,\\ ve di\u gerinin \" uzerinde duran do\u gruya da;\\ \" uzerinde durdu\u gu do\u gruya bir dik do\u gru denir. } \myfntext{This definition is quoted in Proposition 12.} \parsen{ {}[11] An obtuse angle is\\ that [which is] greater than a \rgt. } { >Amble~ia gwn'ia >est`in\\ orj~hc. } { Bir geni\c s a\c c\i,\\ b\" uy\" uk oland{\i}r bir dik a\c c\i dan. } \parsen{ {}[12] Acute,\\ that less than a \rgt. } { >Oxe~ia d`e\\ el'asswn >orj~hc. } { Bir dar a\c c\i,\\ k\" u\c c\" uk oland{\i}r bir dik a\c c\i dan. } \parsen{ {}[13] A boundary is\\ whis is a limit of something. } { <'Oroc >est'in, <'o tin'oc >esti p'erac. } { Bir \emph{s\i n\i r,}\\ bir \c seyin ucunda oland\i r. } \parsen{ {}[14] A figure is\\ what is contained by some boundary or boundaries.\footnotemark } { Sq~hm'a >esti\\ t`o 'h tinwn <'orwn perieq'omenon. } { Bir fig\" ur,\\ bir s\i n\i r taraf\i ndan veya s\i n\i rlarca i\c cerilendir. } \myfntext{In Greek what is repeated is not `boundary' but `some'.} \parsen{ {}[15] A circle is\\ a plane figure\\ contained by one line\\ {}[which is called the circumference]\\ to which,\\ from one point\\ of those lying inside of the figure\\ all \strgt s falling\\ {}[to the circumference of the circle]\\ are equal to one another. } { K'ukloc >est`i\\ sq~hma >ep'ipedon\\ af> ent`oc to~u sq'hmatoc keim'enwn\\ p~asai auje~iai\\ {}[pr`oc t`hn to~u k'uklou perif'ereian]\\ {}>'isai >all'hlaic e>is'in. } { Bir daire,\\ d\" uzlemdeki\\ bir \c cizgice i\c cerilen \\ {}[bu {\c c}izgiye {\c c}ember denir]\\ bir fig\" urd\"ur \" oyle ki\\ fig\" ur\" un i\c cerisindeki\\ noktalar\i n birinden \\ \c cizgi \" uzerine gelen\\ t\" um do\u grular,\\ birbirine e\c sittir; } \parsen{ {}[16] A\footnotemark\ center of the circle\\ the point is called. } { K'entron d`e to~u k'uklou\\ t`o shme~ion kale~itai. } { Ve o noktaya, dairenin merkezi denir. } \myfntext{None of the terms defined in this section is preceeded by a definite article. In particular, what is being defined here is not \emph{the} center of a circle, but \emph{a} center. However, it is easy to show that the center of a given circle is unique; also, in Proposition III.1, Euclid finds \emph{the} center of a given circle.} \parsen{ {}[17] A diameter of the circle is\\ some \strgt\\ drawn through the center\\ and bounded\\ to either parts\\ by the circumference of the circle,\\ which also bisects the circle. } { Di'ametroc d`e to~u k'uklou >est`in\\ e>uje~i'a tic\\ di`a to~u k'entrou >hgm'enh\\ ka`i peratoum'enh\\ >ef> esti\\ t`o perieq'omenon sq~hma\\ apolambanom'enhc a>ut~hc perifere'iac.\\ k'entron d`e to~u ut'o,\\ <`o ka`i to~u k'uklou >est'in. } { Bir yar\i daire,\\ bir \c cap\\ ve onun kesti\u gi bir \c cevrece\\ i\c cerilen fig\" urd\" ur, ve yar\i dairenin merkezi, o dairenin merkeziyle ayn\i d\i r. } \parsen{ {}[19] Rectilineal figures are\footnotemark\\ those contained by \strgt s,\\ triangles, by three,\\ quadrilaterals, by four,\\ polygons,\footnotemark\ by more than four\\ \strgt s contained. } { Sq'hmata e>uj'ugramm'a >esti\\ t`a ujei~wn perieq'omena,\\ tr'ipleura m`en t`a `h tess'arwn\\ e>ujei~wn perieq'omena. } { \emph{D\" uzkenar fig\" ur}ler,\\ do\u grularca i\c cerilenlerdir. \emph{\" U\c ckenar} fig\" urler \" u\c c, \emph{d\" ortkenar} fig\" urler d\" ort ve \emph{\c cokkenar} fig\" urler ise d\" ortten daha fazla do\u gruca i\c cerilenlerdir. } \myfntext{As in Turkish, so in Greek, a plural subject can take a singular verb, when the subject is of the neuter gender in Greek, or names inanimate objects in Turkish.} \myfntext{To maintain the parallelism of the Greek, we could (like Heath) use `trilateral', `quadrilateral', and `multilateral' instead of `triangle', `quadrilateral', and `polygon'. Today, triangles and quadrilaterals \emph{are} polygons. For Euclid, they are not: you never call a triangle a polygon, because you can give the more precise information that it is a triangle.} \parsen{ {}[20] There being trilateral figures,\\ an equilateral triangle is\\ that having three sides equal,\\ isosceles, having only two sides equal,\\ scalene, having three unequal sides. } { ~wn d`e triple'urwn sqhm'atwn\\ {}>is'opleuron m`en tr'igwn'on >esti\\ t`o t`ac tre~ic >'isac >'eqon pleur'ac,\\ {}>isoskel`ec d`e t`o t`ac d'uo m'onac >'isac >'eqon pleur'ac,\\ skalhn`on d`e t`o t`ac tre~ic >an'isouc >'eqon pleur'ac. } { \" U\c ckenar fig\" urlerden\\ bir e\c skenar \" u\c cgen,\\ \" u\c c kenar\i \ e\c sit olan,\\ ikizkenar, e\c sit iki kenar\i \ olan\\ \c ce\c sitkenar, \" u\c c kenar\i \ e\c sit olmayand\i r. } \parsen{ {}[21] Yet of trilateral figures,\\ a right-angled triangle is\\ that having a right angle,\\ obtuse-angled, having an obtuse angle,\\ acute-angled, having three acute angles. } { >'Eti d`e t~wn triple'urwn sqhm'atwn\\ {}>orjog'wnion m`en tr'igwn'on >esti\\ t`o >'eqon >orj`hn gwn'ian,\\ {}>amblug'wnion d`e t`o >'eqon >amble~ian gwn'ian,\\ {}>oxug'wnion d`e t`o t`ac tre~ic >oxe'iac >'eqon gwn'iac. } { Ayr\i ca, \" u\c ckenar fig\" urlerden,\\ bir dik \" u\c cgen,\\ bir dik a\c c\i s\i \ olan,\\ geni\c s a\c c\i l\i, bir geni\c s a\c c\i s\i \ olan,\\ dar a\c c\i l\i, \" u\c c a\c c\i s\i \ dar a\c c\i \ oland\i r. } \parsen{ {}[22] Of quadrilateral figures,\\ a square is\\ what is equilateral and right-angled,\\ an oblong,\\ right-angled, but not equilateral,\\ a rhombus,\\ equilateral,\\ but not right-angled,\\ rhomboid,\\ having opposite sides and angles equal,\\ which is neither equilateral nor right-angled;\\ and let quadrilaterals other than these be called trapezia. } { T`wn d`e tetraple'urwn sqhm'atwn\\ tetr'agwnon m'en >estin,\\ <`o >is'opleur'on t'e >esti ka`i >orjog'wnion,\\ orjog'wnion m'en, o>uk >is'opleuron d'e,\\ is'opleuron m'en,\\ o>uk >orjog'wnion d'e,\\ apenant'ion pleur'ac te ka`i gwn'iac >'isac >all'hlaic >'eqon,\\ <`o o>'ute >is'opleur'on >estin o>'ute >orjog'wnion;\\ t`a d`e par`a ta~uta tetr'apleura trap'ezia kale'isjw. } { D\" ortkenar fig\" urlerden\\ bir kare,\\ hem e\c sit kenar hem de dik-a\c c\i l\i \ olan,\\ bir dikd\" ortgen,\\ dik-a{\c c}\i l\i \ olan ama e\c sit kenar olmayan,\\ bir e\c skenar d\" ortgen,\\ e{\c s}it kenar olan\\ ama dik-a\c c\i l\i \ olmayan,\\ bir paralelkenar\\ kar\c s\i l\i kl\i \ kenar ve a\c c\i lar\i \ e\c sit olan\\ ama e{\c s}it kenar ve dik-a\c c\i l\i \ olmayand\i r.\\ Ve bunlar\i n d\i\c s\i nda kalan d\" ortkenarlara yamuk denilsin. } \parsen{ {}[23] Parallels are\\ \strgt s, whichever,\\ being in the same plane,\\ and extended to infinity\\ to either parts,\\ to neither [parts] fall together with one another. } { Par'allhlo'i e>isin\\ e>uje~iai, a<'itinec\\ {}>en t~w| a>ut~w| >epip'edw| o>~usai\\ ka`i >ekball'omenai e>ic >'apeiron\\ {}>ef> ep`i mhd'etera sump'iptousin >all'hlaic. } { Paraleller,\\ ayn\i \ d\" uzlemde bulunan\\ ve her iki y\" onde de\\ s\i n\i rs\i zca uzat\i ld\i kla\-r\i nda\\ hi\c cbir noktada kesi\c smeyen\\ do\u grulard\i r. } \end{textpart} \newpage \section*{Postulates} \begin{textpart} \parsen{ Postulates } { A>it'hmata } {Postulatlar } \parsen{ Let it have been postulated\\ from any point\\ to any point\\ a straight line\\ to draw. } { >Hit'hsjw\\ {}>ap`o pant`oc shme'iou\\ {}>ep`i p~an shme~ion\\ e>uje~ian gramm`hn\\ {}>agage~in. } { Postulat olarak kabul edilsin\\ herhangi bir noktadan\\ herhangi bir noktaya\\ bir do\u gru\\ \c cizilmesi. } \parsen{ Also, a bounded \strgt\\ continuously\\ in a straight\\ to extend. } { Ka`i peperasm'enhn e>uje~ian\\ kat`a t`o suneq`ec\\ {}>ep> e>uje'iac\\ {}>ekbale~in. } {Ve sonlu bir do\u grunun\\ kesiksiz {\c s}ekilde\\ bir do{\u g}ruda\\ uzat{\i}lmas{\i}. } \parsen{ Also, to any center\\ and distance\\ a circle\\ to draw. } { Ka`i pant`i k'entrw|\\ ka`i diast'hmati\\ k'uklon\\ gr'afesjai. } {Ve her merkez\\ ve uzunlu{\u g}a\\ bir daire\\ \c cizilmesi. } \parsen{ Also, all right angles\\ equal to one another\\ to be. } { Ka`i p'asac t`ac >orj`ac gwn'iac\\ {}>'isac >all'hlaic\\ e>~inai. } {Ve b\" ut\" un dik a\c c\i lar\i n\\ bir birine e\c sit\\ oldu\u gu. } \parsen{ Also, if in two straight lines\\ falling\\ the interior angles to the same parts\\ less than two \rgt s make,\\ the two \strgt s, extended\\ to infinity,\\ fall together,\\ to which parts are\\ the less than two \rgt s. } { Ka`i >e`an e>ic d'uo e>uje'iac e>uje~ia\\ {}>emp'iptousa\\ t`ac >ent`oc ka`i >ep`i t`a a>ut`a m'erh gwn'iac\\ d'uo >orj~wn >el'assonac poi~h|,\\ {}>ekballom'enac t`ac d'uo e>uje'iac\\ {}>ep> >'apeiron\\ sump'iptein,\\ {}>ef> <`a m'erh e>is`in\\ aorj~wn >el'assonec. } { Ve \. iki do\u gruyu\\ kesen bir do\u grunun\\ ayn\i \ tarafta olu\c sturdu\u gu\\ i\c c a\c c\i lar iki dik a\c c\i dan k\" u\c c\" ukse,\\ bu iki do\u grunun,\\ s\i n\i rs\i zca uzat\i ld\i klar\i nda\\ a\c c\i lar\i n\\ iki dik a\c c\i dan k\" u\c c\" uk oldu\u gu tarafta\\ kesi\c sece\u gi. } \end{textpart} \newpage \section*{Common Notions} \begin{textpart} \parsen{ Common notions } { Koina`i >'ennoiai } {Genel Kavramlar } \parsen{ Equals to the same\\ also to one another are equal. } { T`a t~w| a>ut~w| >'isa\\ ka`i >all'hloic >est`in >'isa. } {Ayn\i \ \c seye e\c sitler\\ birbirlerine de e\c sittir. } \parsen{ Also, if to equals\\ equals be added,\\ the wholes are equal. } { Ka`i >e`an >'isoic\\ {}>'isa prostej~h|,\\ t`a <'ola >est`in >'isa. } {E\u ger e\c sitlere\\ e\c sitler eklenirse,\\ elde edilenler de e\c sittir. } \parsen{ Also, if from equals\\ equals be taken away,\\ the remainders are equal. } { a`i >e`an >ap`o >'iswn\\ {}>'isa >afairej~h|,\\ t`a kataleip'omen'a >estin >'isa. } {E\u ger e\c sitlerden\\ e\c sitler \c c\i kart\i l\i rsa,\\ kalanlar e\c sittir. } \parsen{ Also things applying to one another\\ are equal to one another. } { Ka`i t`a >efarm'ozonta >ep> >all'hla\\ {}>'isa >all'hloic >est'in. } {Birbiriyle \c cak\i \c san \c seyler\\ birbirine e\c sittir. } \parsen{ Also, the whole\\ than the part is greater. } { Ka`i t`o <'olon\\ to~u m'erouc me~iz'on [>estin]. } {B\" ut\" un,\\ par\c cadan b\" uy\" ukt\" ur. } \end{textpart} \newpage \begin{proposition}%Proposition I.1 \parsen{ On\\ the\footnotemark\ given bounded \strgt\\ {}for\footnotemark\ an equilateral triangle\\ to be constructed. } { {}>Ep`i\\ t~hc doje'ishc e>uje'iac peperasm'enhc\\ tr'igwnon >is'opleuron\\ sust'hsasjai. } { Verilmi{\c s} s{\i}n{\i}rlanm{\i}{\c s} do{\u g}ruya\\ e{\c s}kenar {\"u}{\c c}gen\\ in{\c s}a edilmesi. } \myfntext{\label{note:the}Heath's translation has the indefinite article `a' here, in accordance with modern mathematical practice. However, Euclid does use the Greek \emph{definite} article here, just as in the \emph{exposition} (see \S\ref{sect:analysis}). In particular, he uses the definite article as a \emph{generic} article, which `makes a single object the representative of the entire class' \cite[\P1123, p.~288]{Smyth}. English too has a generic use of the definite article, `to indicate the class or kind of objects, as in the well-known aphorism: \emph{\textbf{The} child is the father of the man}' \cite[p.~76]{Harman}. (However, the enormous \emph{Cambridge Grammar} does not discuss the generic article in the obvious place \cite[5.6.1, pp.~568--71]{CGEL}. By the way, the `well-known aphorism' is by Wordsworth; see \url{http://en.wikisource.org/wiki/Ode:_Intimations_of_Immortality_from_Recollections_of_Early_Childhood} [accessed July 27, 2011].) See note \ref{note:gen} to Proposition 9 below.} \myfntext{The Greek form of the enunciation here is an infinitive clause, and the subject of such a clause is generally in the accusative case \cite[\P1972, p.~438]{Smyth}. In English, an infinitive clause with expressed subject (as here) is always preceded by `for' \cite[14.1.3, p.~1178]{CGEL}. Normally such a clause, in Greek or English, does not stand by itself as a complete sentence; here evidently it is expected to. Note that the Greek infinitive is thought to be originally a noun in the dative case \cite[\P1969, p.~438]{Smyth}; the English infinitive with `to' would seem to be formed similarly.} \parsen{ Let be\footnotemark\\ the given bounded {\strgt}\\ \gr{AB}. } { >'Estw\\ uje~ia peperasm'enh\\ uje~ia gramm`h uje~ia gramm'h}, `the straight line \gr{AB}'. In Proposition XIII.4, Euclid says, \gr{>'Estw e>uje~ia 'Estw e>uje~ia ep`i t~hc AB e>uje'iac\\ tr'igwnon >is'opleuron\\ sust'hsasjai. } { {\c S}imdi gereklidir\\ \gr{AB} do{\u g}rusuna\\ e{\c s}kenar {\"u}{\c c}genin\\ in{\c s}a edilmesi. } \myfntext{Slightly less literally, `It is necessary that on the \strgt{} \gr{AB}, an equilateral triangle be constructed.'} \parsen{ To center \gr A\\ at distance \gr{AB}\\ suppose a circle has been drawn,\\ {}[namely] \gr{BGD},\\ and moreover,\\ to center \gr B\\ at distance \gr{BA}\\ suppose a circle has been drawn,\\ {}[namely] \gr{AGE},\\ and from the point \gr G,\\ where the circles cut one another,\\ to the points \gr A and \gr B,\\ suppose there\footnotemark\ have been joined\\ the {\strgt}s \gr{GA} and \gr{GB}. } { K'entrw| m`en t~w| A\\ diast'hmati d`e t~w| AB\\ k'ukloc gegr'afjw\\ ap`o to~u G shme'iou,\\ kaj> <`o t'emnousin >all'hlouc oep'i t`a A, B shme~ia\\ {}>epeze'uqjwsan\\ e>uje~iai aepe`i t`o A shme~ion\\ k'entron >est`i to~u GDB k'uklou,\\ {}>'ish >est`in epe`i t`o B shme~ion\\ k'entron >est`i to~u GAE k'uklou,\\ {}>'ish >est`in ede'iqjh d`e ka`i 'ish;\\ 'ara t~wn GA, GB t~h| AB\\ {}>estin >'ish. \\ t`a d`e t~w| a>ut~w| >'isa\\ ka`i >all'hloic >est`in >'isa;\\ ka`i 'ara t~h| GB >estin >'ish;\\ a'ara a'isai >all'hlaic e>is'in.} } { Ve \gr{A} noktas{\i}\\ \gr{GDB} {\c c}emberinin merkezi oldu{\u g}u i{\c c}in, \\ \gr{AG}, \gr{AB} do{\u g}rusuna e{\c s}ittir.\\ Dahas{\i}\\ \gr{B} noktas{\i} \gr{GAE} {\c c}emberinin merkezi oldu{\u g}u i{\c c}in, \\ \gr{BG}, \gr{BA} do{\u g}rusuna e{\c s}ittir.\\ Ve \gr{GA} do{\u g}rusunun, \gr{AB} do{\u g}rusuna e{\c s}it oldu{\u g}u g{\"o}sterilmi{\c s}ti.\\ O zaman \gr{GA}, \gr{GB} do{\u g}rular{\i}n{\i}n her biri \gr{AB} do{\u g}rusuna e{\c s}ittir.\\ Ama ayn{\i} {\c s}eye e{\c s}it olanlar\\ birbirine e{\c s}ittir.\\ O zaman \gr{GA}, \gr{GB} do{\u g}rusuna e{\c s}ittir.\\ O zaman o {\"u}{\c c} do{\u g}ru, \gr{GA}, \gr{AB}, \gr{BG},\\ birbirine e{\c s}ittir. } \parsen{ Equilateral therefore\\ is triangle \gr{ABG}.\\ Also, it has been constructed\\ on the given bounded {\strgt}\\ \gr{AB};\\ \myqef } { \gr{>Is'opleuron >'ara\\ {}>est`i t`o ABG tr'igwnon. \\ ka`i sun'estatai\\ {}>ep`i t~hc doje'ishc e>uje'iac peperasm'enhc\\ t~hc AB.\footnotemark \\ <'oper >'edei poi~hsai.} } { E{\c s}kenard{\i}r dolay{\i}s{\i}yla,\\ \gr{ABG} {\"u}{\c c}geni \\ ve in{\c s}a edilmi{\c s}tir\\ verilmi{\c s} s{\i}n{\i}rlanm{\i}{\c s},\\ \gr{AB} do{\u g}rusuna;\\ \ozqef } \myfntext{Normally Heiberg puts a semicolon at this position. Perhaps he has a period here only because he has bracketed the following words (omitted here): `Therefore, on a given bounded \strgt, an equilateral triangle has been constructed.' According to Heiberg, these words are found, not in the manuscripts of Euclid, but in Proclus's commentary \cite[p.~210]{MR1200456} alone.} \begin{center} \psset{unit=2cm} \begin{pspicture}(3,2) \pscircle(1,1)1 \pscircle(2,1)1 \uput[l](1,1){\gr A} \uput[r](2,1){\gr B} \uput[u](1.5,1.866){\gr G} \uput[l](0,1){\gr D} \uput[r](3,1){\gr E} \pspolygon(1,1)(2,1)(1.5,1.866) \end{pspicture} \end{center} \end{proposition} \begin{proposition}%Proposition I.2 \parsen{ At the given point,\\ equal to the given \strgt,\\ for a \strgt\ to be placed. } { \gr{Pr`oc t~w| doj'enti shme'iw|\\ t~h| doje'ish| e>uje'ia| >'ishn\\ e>uje~ian j'esjai.} } { Verilmi{\c s} noktaya\\ verilmi{\c s} do{\u g}ruya e{\c s}it olan\\ bir do{\u g}runun konulmas{\i}. } \parsen{ Let be\\ the given point \gr A, \\ and the given {\strgt}, \gr{BG}. } { >'Estw\\ t`o m`en doj`en shme~ion t`o A, \\ uje~ia uje'ia| t~h| BG >'ishn\\ e>uje~ian j'esjai. } { Gereklidir\\ \gr{A} noktas{\i}na,\\ \gr{BG} do{\u g}rusuna e{\c s}it olan\\ bir do{\u g}runun konulmas{\i}. } \parsen{ For, suppose there has been joined\\ from the point \gr A to the point \gr B\\ a \strgt, \gr{AB},\\ and there has been constructed on it\\ an equilateral triangle, \gr{DAB},\\ and suppose there have been extended\\ on a \strgt\footnotemark\ with \gr{DA} and \gr{DB}\\ the \strgt s \gr{AE} and \gr{BZ},\\ and to the center \gr B\\ at distance \gr{BG}\\ suppose a circle has been drawn,\\ \gr{GHJ},\\ and again to the center \gr D\\ at distance \gr{DH}\\ suppose a circle has been drawn,\\ \gr{HKL}.\\ } { >Epeze'uqjw g`ar\\ {}>ap`o to~u A shme'iou >ep'i t`o B shme~ion\\ e>uje~ia ep> a>ut~hc\\ tr'igwnon >is'opleuron t`o DAB, \\ ka`i >ekbebl'hsjwsan\\ {}>ep> e>uje'iac ta~ic DA, DB\\ e>uje~iai aep> e>uje'iac} will recur a number of times. The adjective, which is feminine here, appears to be a genitive singular, though it could be accusative plural.} \parsen{ Since then the point \gr B is the center\\ of \gr{GHJ},\\ \gr{BG} is equal to \gr{BH}.\\ Moreover,\\ since the point \gr D is the center\\ of the circle \gr{KHL},\\ equal is \gr{DL} to \gr{DH};\\ of these, the [part] \gr{DA} to \gr{DB}\\ is equal.\\ Therefore the remainder \gr{AL}\\ to the remainder \gr{BH}\\ is equal.\\ But \gr{BG} was shown equal to \gr{BH}.\\ Therefore either of \gr{AL} and \gr{BG} to \gr{BH}\\ is equal.\\ But equals to the same\\ also are equal to one another.\\ And therefore \gr{AL} is equal to \gr{BG}. } { >Epe`i o>~un t`o B shme~ion k'entron >est`i\\ to~u GHJ,\\ {}>'ish >est`in epe`i t`o D shme~ion k'entron >est`i\\ to~u HKL k'uklou,\\ {}>'ish >est`in 'ish >est'in.\\ loip`h >'ara estin >'ish.\\ {}>ede'iqjh d`e ka`i 'ish;\\ 'ara t~wn AL, BG t~h| BH\\ {}>estin >'ish. \\ t`a d`e t~w| a>ut~w| >'isa\\ ka`i >all'hloic >est`in >'isa; \\ ka`i 'ara t~h| BG >estin >'ish. } { \gr{B} noktas{\i} \gr{GHJ} {\c c}emberinin merkezi oldu{\u g}u i{\c c}in,\\ \gr{BG}, \gr{BH} do{\u g}rusuna e{\c s}ittir.\\ Yine,\\ \gr{D} noktas{\i} \gr{HKL} {\c c}emberinin merkezi oldu{\u g}u i{\c c}in,\\ \gr{DL}, \gr{DH} do{\u g}rusuna e{\c s}ittir,\\ ve (birincinin) \gr{DA} par{\c c}as{\i},\\ (ikincinin) \gr{DB} par{\c c}as{\i}na e{\c s}ittir.\\ Dolay{\i}s{\i}yla \gr{AL} kalan{\i},\\ \gr{BH} kalan{\i}na\\ e{\c s}ittir.\\ Ve \gr{BG} do{\u g}rusunun, \gr{BH} do{\u g}rusuna e{\c s}it oldu{\u g}u g{\"o}sterilmi{\c s}ti.\\ Dolay{\i}s{\i}yla \gr{AL}, \gr{BG} do{\u g}rular{\i}n{\i}n her biri \gr{BH} do{\u g}rusuna e{\c s}ittir.\\ Ama ayn{\i} {\c s}eye e{\c s}it olanlar birbirine e{\c s}ittir.\\ Ve dolay{\i}s{\i}yla \gr{AL} da, \gr{BG} do{\u g}rusuna e{\c s}ittir. } \parsen{ Therefore at the given point \gr A\\ equal to the given {\strgt} \gr{BG}\\ the {\strgt} \gr{AL} is laid down;\\ \myqef } { Pr`oc >'ara t~w| doj'enti shme'iw|\\ t~w| A t~h| doje'ish| e>uje'ia| t~h| BG >'ish\\ e>uje~ia ke~itai 'edei poi~hsai. } { Dolay{\i}s{\i}yla verilmi{\c s} \gr{A} noktas{\i}na\\ verilmi{\c s} \gr{BG} do{\u g}rusuna e{\c s}it olan\\ \gr{AL} do{\u g}rusu konulmu{\c s}tur;\\ \ozqef } \begin{center} \begin{pspicture}(-3,-3.5)(3,3) \pscircle(0,0)3 \psline(-0.5,-0.866)(0.5,-0.866) \pscircle(0.5,-0.866)2 \psline(0.5,-0.866)(1.914,0.548) \psline(0,0)(2,-3.464) \psline(0,0)(-2,-3.464) \uput[l](-0.5,-0.866){\gr A} \uput[-5](0.5,-0.866){\gr B} \uput[ur](1.914,0.548){\gr G} \uput[u](0,0){\gr D} \uput[d](-2,-3.464){\gr E} \uput[d](2,-3.464){\gr Z} \uput[-15](1.5,-2.598){\gr H} \uput[u](-0.5,0.866){\gr J} \uput[u](-1.5,2.598){\gr K} \uput[195](-1.5,-2.598){\gr L} \end{pspicture} \end{center} \end{proposition} \begin{proposition}%Proposition I.3 \parsen{ Two unequal {\strgt}s being given,\\ from the greater,\\ equal to the less,\\ a \strgt\ to take away. } { D'uo dojeis~wn e>ujei~wn >an'iswn\\ {}>ap`o t~hc me'izonoc\\ t~h| >el'assoni >'ishn\\ e>uje~ian >afele~in. } { {\.I}ki e{\c s}it olmayan do{\u g}ru verilmi{\c s} ise, \\ daha b{\"u}y{\"u}kten\\ daha k{\"u}{\c c}{\"u}{\u g}e e{\c s}it olan\\ bir do{\u g}ru kesmek. } \parsen{ Let be\\ the two given unequal {\strgt}s\\ \gr{AB} and \gr G,\footnotemark\\ of which let the greater be \gr{AB}. } { >'Estwsan\\ auje~iai >'anisoi\\ a'estw ap`o t~hc me'izonoc t~hc AB\\ t~h| >el'assoni t~h| G >'ishn\\ e>uje~ian >afele~in. } { Gereklidir\\ daha b{\"u}y{\"u}k olan \gr{AB} do{\u g}rusundan\\ daha k{\"u}{\c c}{\"u}k olan \gr G do{\u g}rusuna e{\c s}it olan\\ bir do{\u g}ru kesmek. } \parsen{ Let there be laid down\\ at the point \gr A,\\ equal to the line \gr G,\\ \gr{AD};\\ and to center \gr A\\ at distance \gr{AD}\\ suppose circle \gr{DEZ} has been drawn. } { Ke'isjw\\ pr`oc t~w| A shme'iw|\\ t~h| G e>uje'ia| >'ish\\ epe`i t`o A shme~ion\\ k'entron >est`i to~u DEZ k'uklou,\\ {}>'ish >est`in all`a ka`i estin >'ish.\\ 'ara t~wn AE, G\\ t~h| AD >estin >'ish;\\ <'wste ka`i estin >'ish. } { Ve \gr A noktas{\i}\\ \gr{DEZ} {\c c}emberinin merkezi oldu{\u g}u i{\c c}in,\\ \gr{AE}, \gr{AD} do{\u g}rusuna e{\c s}ittir. \\ Ama \gr G, \gr{AD} do{\u g}rusuna e{\c s}ittir.\\ Dolay{\i}s{\i}yla \gr{AE}, \gr G do{\u g}rular{\i}n{\i}n her biri\\ \gr{AD} do{\u g}rusuna e{\c s}ittir.\\ Sonu{\c c} olarak,\\ \gr{AE}, \gr G do{\u g}rusuna e{\c s}ittir. } \parsen{ Therefore, two unequal \strgt s being given, \gr{AB} and \gr G,\\ from the greater, \gr{AB},\\ an equal to the less, \gr G,\\ has been taken away, [namely] \gr{AE};\\ \myqef } { D'uo >'ara dojeis~wn e>ujei~wn >an'iswn t~wn AB, G\\ {}>ap`o t~hc me'izonoc t~hc AB\\ t~h| >el'assoni t~h| G >'ish\\ {}>af'h|rhtai 'edei poi~hsai. } { Dolay{\i}s{\i}yla iki e{\c s}it olmayan \gr{AB}, \gr G do{\u g}rusu verilmi{\c s} ise,\\ daha b{\"u}y{\"u}k olan \gr{AB} do{\u g}rusundan\\ daha k{\"u}{\c c}{\"u}k olan \gr G do{\u g}rusuna e{\c s}it olan\\ \gr{AE} do{\u g}rusu kesilmi{\c s}ti;\\ \ozqef } \begin{center} \begin{pspicture}(-2,-2)(5.5,2) \psline(-1.414,1.414)(0,0)(3,0) \psline(3.5,1)(5.5,1) \pscircle(0,0)2 \uput[d](0,0){\gr A} \uput[d](3,0){\gr B} \uput[u](4.5,1){\gr G} \uput[ul](-1.414,1.414){\gr D} \uput[ur](2,0){\gr E} \end{pspicture} \end{center} \end{proposition} \begin{proposition}%Proposition I.4 \parsen{ If two triangles\\ two sides\\ to two sides\\ have equal,\footnotemark\\ either [side] to either,% \footnotemark\\ and angle to angle have equal,\\ ---that which is by the equal {\strgt}s% \footnotemark\\ contained,\\ also% \footnotemark\ base to base\\ they will have equal,\\ and the triangle to the triangle\\ will be equal,\\ and the remaining angles\\ to the remaining angles\\ will be equal,\\ either to either,\\ ---those that the equal sides subtend. } { >E`an d'uo tr'igwna\\ t`ac d'uo pleur`ac\\ {}[ta~ic] dus`i pleura~ic\\ {}>'isac >'eqh|\\ 'ishn >'eqh| \\ t`hn 'iswn e>ujei~wn\\ perieqom'enhn, \\ ka`i t`hn b'asin t`h| b'asei\\ {}>'ishn <'exei,\\ ka`i t`o tr'igwnon t~w| trig'wnw|\\ {}>'ison >'estai, \\ ka`i a'isai >'esontai\\ <`ac a'isai pleura`i 'Estw\\ d'uo tr'igwna t`a ABG, DEZ\\ t`ac d'uo pleur`ac t`ac AB, AG\\ ta~ic dus`i pleura~ic ta~ic DE, DZ\\ {}>'isac >'eqonta\\ 'ishn. } { Verilmi{\c s} olsun,\\ \gr{ABG} ve \gr{DEZ} (adlar{\i}nda) iki {\"u}{\c c}gen,\\ iki kenar{\i} \gr{AB}, \gr{AG}\\ \gr{DE}, \gr{DZ} iki kenar{\i}na\\ e{\c s}it olan\\ her biri birine,\\ ({\c s}{\"o}yle ki) \gr{AB}, \gr{DE} kenar{\i}na ve \gr{AG}, \gr{DZ} kenar{\i}na,\\ ve \gr{BAG} (taraf{\i}ndan i{\c c}erilen) a{\c c}{\i}s{\i}\\ \gr{EDZ} a{\c c}{\i}s{\i}na\\ e{\c s}it olan. } \parsen{ I say that\\ the base \gr{BG} is equal to the base \gr{EZ}, \\ and triangle \gr{ABG}\\ will be equal to triangle \gr{DEZ},\\ and the remaining angles\\ to the remaining angles\\ will be equal,\\ either to either,\\ those that equal sides subtend, \\ {}[namely] \gr{ABG} to \gr{DEZ},\\ and \gr{AGB} to \gr{DZE}. } {l'egw, <'oti\\ ka`i b'asic 'ish >est'in, \\ ka`i t`o ABG tr'igwnon\\ t~w| DEZ trig'wnw| >'ison >'estai, \\ ka`i a'isai >'esontai\\ <`ac a'isai pleura`i Efarmozom'enou g`ar\\ to~u ABG trig'wnou\\ {}>ep`i t`o DEZ tr'igwnon \\ ka`i tijem'enou\\ to~u m`en A shme'iou >ep`i t`o D shme~ion\\ t~hc d`e AB e>uje'iac >ep`i t`hn DE,\\ {}>efarm'osei ka`i t`o B shme~ion >ep`i t`o E\\ di`a t`o >'ishn e>~inai t`hn AB t~h| DE;\\ {}>efarmos'ashc d`h t~hc AB >ep`i t`hn DE\\ {}>efarm'osei ka`i uje~ia >ep`i t`hn DZ \\ di`a t`o >'ishn e>~inai\\ t`hn ep`i t`o Z shme~ion\\ {}>efarm'osei\\ di`a t`o >'ishn p'alin e>~inai t`hn AG t~h| DZ.\\ {}>all`a m`hn ka`i t`o B >ep`i t`o E >efhrm'okei; \\ <'wste b'asic ep`i b'asin t`hn EZ\\ {}>efarm'osei. \\ e>i g`ar\\ to~u m`en B >ep`i t`o E >efarm'osantoc \\ to~u d`e G >ep`i t`o Z \\ ep`i t`hn EZ o>uk >efarm'osei, \\ d'uo e>uje~iai qwr'ion peri'exousin;\\ <'oper >est`in >ad'unaton.\\ {}>efarm'osei >'ara\\ ep`i t`hn EZ\\ ka`i >'ish a>ut~h| >'estai;\\ <'wste ka`i <'olon t`o ABG tr'igwnon\\ {}>ep`i <'olon t`o DEZ tr'igwnon\\ {}>efarm'osei\\ ka`i >'ison a>ut~w| >'estai,\\ ka`i aep`i t`ac loip`ac gwn'iac\\ {}>efarm'osousi\\ ka`i >'isai a>uta~ic >'esontai,\\ E`an >'ara d'uo tr'igwna\\ t`ac d'uo pleur`ac\\ {}[ta~ic] d'uo pleura~ic\\ {}>'isac >'eqh|\\ 'ishn >'eqh|\\ t`hn 'iswn e>ujei~wn\\ perieqom'enhn,\\ ka`i t`hn b'asin t`h| b'asei\\ {}>'ishn <'exei,\\ ka`i t`o tr'igwnon t~w| trig'wnw|\\ {}>'ison >'estai,\\ ka`i a'isai >'esontai\\ <`ac a'isai pleura`i 'edei de~ixai. } { Dolay{\i}s{\i}yla, e{\u g}er,\\ iki {\"u}{\c c}genin, varsa iki kenar{\i} e{\c s}it olan iki kenara, \\ her bir (kenar) birine,\\ ve varsa a{\c c}{\i}ya e{\c s}it a{\c c}{\i}s{\i},\\% (yani) e{\c s}it do{\u g}rularca i{\c c}erilen,\\ hem tabana e{\c s}it tabanlar{\i} olacak,\\ hem {\"u}{\c c}gen e{\c s}it olacak {\"u}{\c c}gene,\\ hem de geriye kalan a{\c c}{\i}lar e{\c s}it olacak geriye kalan a{\c c}{\i}lar{\i}n,\\ her biri birine,\\% (yani) e{\c s}it kenarlar{\i} g{\"o}renler;\\ \ozqed } \begin{center} \begin{pspicture}(0,-0.5)(7,2.5) \pspolygon(0,0)(3,0)(2.5,2.5) \pspolygon(4,0)(7,0)(6.5,2.5) \uput[u](2.5,2.5){\gr A} \uput[d](0,0){\gr B} \uput[d](3,0){\gr G} \uput[u](6.5,2.5){\gr D} \uput[d](4,0){\gr E} \uput[d](7,0){\gr Z} \pscurve(4,0)(5.5,-0.15)(7,0) \end{pspicture} \end{center} %\epsfysize=1.35in %\centerline{\epsffile{Book01/fig04g.eps}} \end{proposition} \begin{proposition}%Proposition I.5 \parsen{ In\footnotemark\ isosceles triangles,\\ the angles at the base\\ are equal to one another,\\ and,\\ the equal \strgt s being extended,\\ the angles under the base\\ will be equal to one another. } {T~wn >isoskel~wn trig'wnwn\\ a'isai >all'hlaic e>is'in,\\ ka`i\\ prosekblhjeis~wn t~wn >'iswn e>ujei~wn\\ a'isai >all'hlaic >'esontai. } { {\.I}kizkenar {\"u}{\c c}genlerde,\\ tabandaki a{\c c}{\i}lar,\\ birbirine e{\c s}ittir,\\ ve,\\ e{\c s}it do{\u g}rular uzat{\i}ld{\i}{\u g}{\i}nda,\\ taban{\i}n alt{\i}nda kalan a{\c c}{\i}lar,\\ birbirine e{\c s}it olacaklar.\\ } \myfntext{More literally, `of'.} \parsen{ Let there be\\ an isosceles triangle, \gr{ABG}\\ having equal\\ side \gr{AB} to side \gr{AG},\\ and suppose have been extended\\ on a \strgt\ with \gr{AB} and \gr{AG}\\ the {\strgt}s \gr{BD} and \gr{GE}. } { >'Estw\\ tr'igwnon >isoskel`ec t`o ABG\\ {}>'ishn >'eqon\\ t`hn AB pleur`an t~h| AG pleur~a|,\\ ka`i prosekbebl'hsjwsan\\ {}>ep> e>uje'iac ta~ic AB, AG\\ e>uje~iai a'ish >est'in,\\ il'hfjw g`ar\\ {}>ep`i t~hc BD tuq`on shme~ion t`o Z,\\ ka`i >afh|r'hsjw\\ {}>ap`o t~hc me'izonoc t~hc AE\\ t~h| >el'assoni t~h| AZ\\ {}>'ish epeze'uqjwsan\\ auje~iai. } { {\c C}{\"u}nk{\"u}, kabul edelim ki, se{\c c}ilmi{\c s} olsun,\\ rastgele bir \gr Z noktas{\i} \gr{BD} {\"u}zerinnde,\\ ve \gr{AH},\\ b{\"u}y{\"u}k olan \gr{AE} do{\u g}rusundan\\ k{\"u}{\c c}{\"u}k olan \gr{AZ} do{\u g}rusunun kesilmi{\c s}i olsun,\\ ve \gr{ZG} ile \gr{HB} birle{\c s}tirilmi{\c s} olsun. } \parsen{ Since then \gr{AZ} is equal to \gr{AH},\\ and \gr{AB} to \gr{AG},\\ so the two \gr{AZ} and \gr{AG}\\ to the two \gr{HA}, \gr{AB},\\ will be equal,\\ either to either;\\ and they bound a common angle,\\ {}[namely] \gr{ZAH};\\ therefore the base \gr{ZG} to the base \gr{HB}\\ is equal,\\ and triangle \gr{AZG} to triangle \gr{AHB}\\ will be equal,\\ and the remaining angles\\ to the remaining angles\\ will be equal,\\ either to either,\\ those that the equal sides subtend,\\ \gr{AGZ} to \gr{ABH},\\ and \gr{AZG} to \gr{AHB}.\\ And since \gr{AZ} as a whole\\ to \gr{AH} as a whole\\ is equal,\\ of which the [part] \gr{AB} to \gr{AG} is equal,\\ therefore the remainder \gr{BZ}\\ to the remainder \gr{GH}\\ is equal.\\ And \gr{ZG} was shown equal to \gr{HB}.\\ Then the two \gr{BZ} and \gr{ZG}\\ to the two \gr{GH} and \gr{HB}\\ are equal,\\ either to either,\\ and angle \gr{BZG}\\ to angle \gr{GHB}\\ {}[is] equal,\\ and the common base of them is \gr{BG};\\ and therefore triangle \gr{BZG}\\ to triangle \gr{GHB}\\ will be equal,\\ and the remaining angles\\ to the remaining angles\\ will be equal,\\ either to either,\\ which the equal sides subtend.\\ Equal therefore is\\ \gr{ZBG} to \gr{HGB},\\ and \gr{BGZ} to \gr{GBH}.\\ Since then angle \gr{ABH} as a whole\\ to angle \gr{AGZ} as a whole\\ was shown equal,\\ of which the [part] \gr{GBH} to \gr{BGZ}\\ is equal,\\ therefore the remainder \gr{ABG}\\ to the remainder \gr{AGB}\\ is equal;\\ and they are at the base\\ of the triangle \gr{ABG}.\\ And was shown also\\ \gr{ZBG} equal to \gr{HGB};\\ and they are under the base. } { >Epe`i o>~un >'ish >est`in 'isai e>is`in\\ 'ara 'ish >est'in,\\ ka`i t`o AZG tr'igwnon t~w| AHB trig'wnw|\\ {}>'ison >'estai,\\ ka`i a'isai >'esontai\\ <`ac a'isai pleura`i epe`i <'olh estin >'ish,\\ <~wn estin >'ish,\\ loip`h >'ara estin >'ish.\\ {}>ede'iqjh d`e ka`i 'ish;\\ d'uo d`h a'isai e>is`in\\ 'ish,\\ ka`i b'asic a>ut~wn koin`h 'ara tr'igwnon\\ t~w| GHB trig'wnw|\\ {}>'ison >'estai,\\ ka`i a'isai >'esontai\\ <`ac a'isai pleura`i 'ish >'ara >est`in\\ epe`i o>~un <'olh ede'iqjh >'ish,\\ <~wn 'ish,\\ loip`h >'ara estin >'ish;\\ ka'i e>isi pr`oc t~h| b'asei\\ to~u ABG trig'wnou.\\ {}>ede'iqjh d`e ka`i\\ 'ish;\\ ka'i e>isin isoskel~wn trig'wnwn\\ a'isai >all'hlaic e>is'in,\\ ka`i\\ prosekblhjeis~wn t~wn >'iswn e>ujei~wn\\ a'isai >all'hlaic >'esontai;\\ <'oper >'edei de~ixai. } { Dolay{\i}s{\i}yla bir ikizkenar {\"u}{\c c}genin taban{\i}ndaki a{\c c}{\i}lar birbirine e{\c s}ittir,\\ ve, e{\c s}it do{\u g}rular uzat{\i}ld{\i}{\u g}{\i}nda,\\ taban{\i}n alt{\i}nda kalan a{\c c}{\i}lar birbirine e{\c s}it olacaklar;\\ \ozqed } \begin{figure}[h] \centering \begin{pspicture}(0,-3)(8,3.5) \pspolygon(0,0)(2,0)(1,3) \uput[u](1,3){\gr A} \uput[l](0,0){\gr B} \uput[r](2,0){\gr G} \psline(0,0)(-1,-3) \uput[l](-1,-3){\gr D} \psline(0,0)(2.5,-1.5) \uput[r](2.5,-1.5){\gr H} \psline(2,0)(3,-3) \uput[r](3,-3){\gr E} \psline(2,0)(-0.5,-1.5) \uput[l](-0.5,-1.5){\gr Z} \end{pspicture} \end{figure} %\epsfysize=1.35in %\centerline{\epsffile{Book01/fig04g.eps}} \end{proposition} \begin{proposition}%Proposition I.6 \parsen{ If in a triangle\\ two angles be equal to one another,\\ also the sides that subtend the equal angles\\ will be equal to one another. } {>E`an trig'wnou\\ a'isai >all'hlaic >~wsin,\\ ka`i a'isac gwn'iac 'isai >all'hlaic >'esontai.} { E{\u g}er bir {\"u}{\c c}gende\\ birbirine e{\c s}it iki a{\c c}{\i} varsa,\\ e{\c s}it a{\c c}{\i}lar{\i}n g{\"o}rd{\"u}{\u g}{\"u} kenarlar da\\ birbirine e{\c s}it olacaklar.} \parsen{ Let there be\\ a triangle, \gr{ABG},\\ having equal\\ angle \gr{ABG}\\ to angle \gr{AGB}.\\ } { >'Estw\\ tr'igwnon t`o ABG\\ {}>'ishn >'eqon\\ t`hn estin >'ish. } { {\.I}ddia ediyorum ki\\ \gr{AB} kenar{\i} da \gr{AG} kenar{\i}na \\ e{\c s}ittir. } \parsen{ For if unequal is \gr{AB} to \gr{AG},\\ one of them is greater.\\ Suppose \gr{AB} be greater,\\ and there has been taken away\\ from the greater, \gr{AB},\\ to the less, \gr{AG},\\ an equal, \gr{DB},\\ and there has been joined \gr{DG}. } { E>i g`ar >'anis'oc >estin ut~wn me'izwn >est'in.\\ {}>'estw me'izwn afh|r'hsjw\\ {}>ap`o t~hc me'izonoc t~hc AB\\ t~h| >el'attoni t~h| AG\\ {}>'ish epeze'uqjw Epe`i o>~un >'ish >est`in 'isai e>is`in\\ estin >'ish;\\ b'asic >'ara 'ish >est'in,\\ ka`i t`o DBG tr'igwnon t~w| AGB trig'wnw|\\ {}>'ison >'estai,\\ t`o >'elasson t~w| me'izoni;\\ <'oper >'atopon;\\ o>uk >'ara >'anis'oc >estin 'ish >'ara. } { O zaman \gr{DB} e{\c s}ittir \gr{AG} kenar{\i}na,\\ ve \gr{BG} ortakt{\i}r,\\ b{\"o}ylece \gr{DB}, \gr{BG} ikilisi e{\c s}ittirler \gr{AG}, \gr{BG} ikilisinin,\\ her biri birine,\\ ve \gr{DBG} a{\c c}{\i}s{\i} e{\c s}ittir \gr{AGB} a{\c c}{\i}s{\i}na;\\ dolay{\i}s{\i}yla \gr{DG} taban{\i} e{\c s}ittir \gr{AB} taban{\i}na,\\ ve \gr{DBG} {\"u}{\c c}geni e{\c s}it olacak \gr{AGB} {\"u}{\c c}genine,\\ daha k{\"u}{\c c}{\"u}k daha b{\"u}y{\"u}{\u g}e;\\ ki bu sa{\c c}mad{\i}r.\\ dolay{\i}s{\i}yla \gr{AB} de{\u g}ildir e{\c s}it de{\u g}il \gr{AG} kenar{\i}na;\\ dolay{\i}s{\i}yla e{\c s}ittir. } \parsen{ If therefore in a triangle\\ two angles be equal to one another,\\ also the sides that subtend the equal angles\\ will be equal to one another;\\ \myqed } {>E`an trig'wnou\\ a'isai >all'hlaic >~wsin,\\ ka`i a'isac gwn'iac 'isai >all'hlaic >'esontai;\\ <'oper >'edei de~ixai. } { Dolay{\i}s{\i}yla e{\u g}er bir {\"u}{\c c}genin birbirine e{\c s}it iki a{\c c}{\i}s{\i} varsa,\\ e{\c s}it a{\c c}{\i}lar{\i}n g{\"o}rd{\"u}{\u g}{\"u} kenarlar e{\c s}ittir;\\ \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-2.031875)(4.3690624,2.031875) \psline[linewidth=0.05cm](0.2940625,-1.811875)(3.8940625,-1.811875) \psline[linewidth=0.05cm](2.2740624,1.608125)(0.3140625,-1.811875) \psline[linewidth=0.05cm](3.8740625,-1.791875)(2.2540624,1.608125) \psline[linewidth=0.05cm](3.8940625,-1.811875)(1.5940624,0.388125) \usefont{T1}{ptm}{m}{n} \rput(2.440625,1.853125){\gr A} \usefont{T1}{ptm}{m}{n} \rput(0.10125,-1.886875){\gr B} \usefont{T1}{ptm}{m}{n} \rput(4.195156,-1.846875){\gr G} \usefont{T1}{ptm}{m}{n} \rput(1.4340625,0.533125){\gr D} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.7 \parsen{ On the same \strgt,\\ to the same two \strgt s,\\ two other \strgt s,\\ {}[which are] equal,\\ either to either,\\ will not be constructed\\ to one and another point,\footnotemark\\ to the same parts,\footnotemark\\ having the same extremities\\ as\footnotemark\ the original lines.} { >Ep`i t~hc a>ut~hc e>uje'iac\\ d'uo ta~ic a>uta~ic e>uje'iaic\\ {}>'allai d'uo e>uje~iai\\ {}>'isai\\ u sustaj'hsontai\\ pr`oc >'allw| ka`i >'allw| shme'iw|\\ {}>ep`i t`a a>ut`a m'erh\\ t`a a>ut`a p'erata >'eqousai\\ ta~ic >ex >arq~hc e>uje'iaic. } { Ayn{\i} do{\u g}ru {\"u}zerinde,\\ verilmi{\c s} iki do{\u g}ruya,\\ e{\c s}it iki ba{\c s}ka do{\u g}ru,\\ her biri birine,\\ in{\c s}a edilmeyecek\\ bir ve ba{\c s}ka bir noktaya,\\ ayn{\i} tarafta, \\ ayn{\i} u{\c c}lar{\i} olan\\ ba{\c s}lang{\i}{\c c}taki do{\u g}rularla.} \myfntext{Literally `another and another point'; more clearly in English, `to different points'.} \myfntext{In English as apparently in Greek, \emph{parts} can mean `region'---in this case, more precisely, `side'.} \myfntext{\label{note:F}According to Fowler (\cite[\textbf{as 8,} p.~34]{MEU} and \cite[\textbf{as 9,} p.~38]{MEU2}), `\emph{As} is never to be regarded as a preposition'. This is unfortunate, since it means that the two constructions `Equal to \emph X' and `Same as \emph X' are not grammatically parallel. (We have `equal to him', but `same as he'.) The constructions are parallel in Greek: \gr{>'isoc} + \textsc{dative} and \gr{a>ut'oc} + \textsc{dative}.} \parsen{ For if possible,\\ on the same {\strgt} \gr{AB}\\ to two given {\strgt}s \gr{AG}, \gr{GB},\\ two other {\strgt}s \gr{AD}, \gr{DB},\\ equal\\ either to either\\ suppose have been constructed\footnotemark\\ to one and another point\\ \gr G and \gr D,\\ to the same parts,\\ having the same extremities,\\ so that \gr{GA} is\footnotemark\ equal to \gr{DA},\\ having the same extremity as it, \gr A,\\ and \gr{GB} to \gr{DB},\\ having the same extremity as it, \gr B,\\ and suppose there has been joined\\ \gr{GD}. } { E>i g`ar dunat'on,\\ {}>ep`i t~hc a>ut~hc e>uje'iac t~hc AB\\ d'uo ta~ic a>uta~ic e>uje'iaic ta~ic AG, GB\\ {}>'allai d'uo e>uje~iai a'isai\\ 'allw| ka`i >'allw| shme'iw|\\ t~w| te G ka`i D\\ {}>ep`i t`a a>ut`a m'erh\\ t`a a>ut`a p'erata >'eqousai,\\ <'wste >'ishn e>~inai t`hn m`en GA t~h| DA\\ t`o a>ut`o p'erac >'eqousan a>ut~h| t`o A,\\ t`hn d`e GB t~h| DB\\ t`o a>ut`o p'erac >'eqousan a>ut~h| t`o B,\\ ka`i >epeze'uqjw\\ Epe`i o>~un >'ish >est`in 'ish >est`i\\ ka`i gwn'ia 'ara\\ 'ara\\ est'i t~hc epe`i >'ish >est`in 'ish >est`i ka`i\\ gwn'ia ede'iqjh d`e a>ut~hc ka`i\\ poll~w| me'izwn;\\ <'oper >est`in >ad'unaton. } { {\c C}{\"u}nk{\"u} \gr{AG} e{\c s}ittir \gr{AD} do{\u g}rusuna,\\ yine e{\c s}ittir\\ \gr{AGD}, \gr{ADG} a{\c c}{\i}s{\i}na;\\ dolay{\i}s{\i}yla \gr{ADG} b{\"u}y{\"u}kt{\"u}r \gr{DGB} a{\c c}{\i}s{\i}ndan;\\ dolay{\i}s{\i}yla \gr{GDB} {\c c}ok daha b{\"u}y{\"u}kt{\"u}r \gr{DGB} a{\c c}{\i}s{\i}ndan.\\ {\"U}stelik \gr{GB} e{\c s}it oldu{\u g}u i{\c c}in \gr{DB} do{\u g}rusuna,\\ \gr{GDB} a{\c c}{\i}s{\i} e{\c s}ittir \gr{DGB} a{\c c}{\i}s{\i}na.\\ Ama ondan {\c c}ok daha b{\"u}y{\"u}k oldu{\u g}u g{\"o}sterilmi{\c s}ti;\\ ki bu sa{\c c}mad{\i}r. } \myfntext{Fowler (\cite[\textbf{than 6,} p.~629]{MEU} and \cite[\textbf{than 6,} p.~619]{MEU2}) does grant the possibility of construing `than' as a preposition, though he disapproves. Then English cannot exactly mirror the Greek \gr{me'izwn} + \textsc{genitive.} Turkish does mirror it with \emph{-den b\"uy\"uk}. See note~\ref{note:F} above.} \myfntext{Here one must refer to the diagram.} \parsen{ Not, therefore,\\ on the same {\strgt},\\ to the same two {\strgt}s,\\ two other {\strgt}s\\ {}[which are] equal,\\ either to either,\\ will be constructed\\ to one and another point\\ to the same parts\\ having the same extremities\\ as the original lines;\\ \myqed } { O>uk >'ara\\ {}>ep`i t~hc a>ut~hc e>uje'iac\\ d'uo ta~ic a>uta~ic e>uje'iaic\\ {}>'allai d'uo e>uje~iai\\ {}>'isai\\ 'allw| ka`i >'allw| shme'iw|\\ {}>ep`i t`a a>ut`a m'erh\\ t`a a>ut`a p'erata >'eqousai\\ ta~ic >ex >arq~hc e>uje'iaic;\\ <'oper >'edei de~ixai. } { {\c S}{\"o}yle olmaz, dolay{\i}s{\i}yla; ayn{\i} do{\u g}ru {\"u}zerinde,\\ verilmi{\c s} iki do{\u g}ruya,\\ iki ba{\c s}ka do{\u g}ru, e{\c s}it,\\ her biri birine,\\ in{\c s}a edilecek\\ ba{\c s}ka bir noktaya\\ ayn{\i} tarafta \\ ayn{\i} u{\c c}lar{\i} olan\\ ba{\c s}lang{\i}{\c c}taki do{\u g}rularla.\\ \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.311875)(3.9775,1.311875) \psline[linewidth=0.05cm](0.2771875,-1.051875)(3.6771874,-1.051875) \psline[linewidth=0.05cm](0.2971875,-1.051875)(2.0771875,0.948125) \psline[linewidth=0.05cm](3.6371875,-1.051875)(3.6571875,-1.031875) \psline[linewidth=0.05cm](3.6571875,-1.031875)(3.0571876,0.788125) \psline[linewidth=0.05cm](3.0571876,0.768125)(2.0571876,0.948125) \psline[linewidth=0.05cm](0.3171875,-1.031875)(3.0771875,0.748125) \psline[linewidth=0.05cm](3.6571875,-1.031875)(2.0571876,0.948125) \usefont{T1}{ptm}{m}{n} \rput(0.12375,-1.086875){\gr A} \usefont{T1}{ptm}{m}{n} \rput(3.824375,-1.166875){\gr B} \usefont{T1}{ptm}{m}{n} \rput(3.2371874,0.933125){\gr D} \usefont{T1}{ptm}{m}{n} \rput(1.8582813,1.133125){\gr G} \end{pspicture} } \end{center} \end{proposition} \newpage \begin{proposition}%Proposition I.8 \parsen{ If two triangles\\ two sides\\ to two sides\\ have equal,\\ either to either,\\ and have also base equal to base,\\ also angle to angle\\ they will have equal,\\ {}[namely] that by the equal \strgt s\\ subtended. } { >E`an d'uo tr'igwna\\ t`ac d'uo pleur`ac\\ {}[ta~ic] d'uo pleura~ic\\ {}>'isac >'eqh|\\ 'eqh| d`e ka`i t`hn b'asin t~h| b'asei >'ishn,\\ ka`i t`hn gwn'ian t~h| gwn'ia|\\ {}>'ishn <'exei\\ t`hn 'iswn e>ujei~wn\\ perieqom'enhn. } { E{\u g}er iki {\"u}{\c c}genin, varsa iki kenar{\i} e{\c s}it olan iki kenara, \\ her bir (kenar) birine,\\ ve varsa tabana e{\c s}it taban{\i},\\% ayr{\i}ca olacak a{\c c}{\i}ya e{\c s}it a{\c c}{\i}lar{\i},\\ (yani) e{\c s}it kenarlar{\i} g{\"o}renler. } \parsen{ Let there be\\ two triangles, \gr{ABG} and \gr{DEZ},\\ the two sides \gr{AB} and \gr{AG}\\ to the two sides \gr{DE} and \gr{DZ}\\ having equal,\\ either to either,\\ \gr{AB} to \gr{DE},\\ and \gr{AG} to \gr{DZ};\\ and let them have\\ base \gr{BG} equal to base \gr{EZ}. } { >'Estw\\ d'uo tr'igwna t`a ABG, DEZ\\ t`ac d'uo pleur`ac t`ac AB, AG\\ ta~ic d'uo pleura~ic ta~ic DE, DZ\\ {}>'isac >'eqonta\\ eq'etw d`e\\ ka`i b'asin t`hn BG b'asei t~h| EZ >'ishn; } { Verilmi{\c s} olsun\\ iki {\"u}{\c c}gen, \gr{ABG} ve \gr{DEZ},\\ iki kenar{\i} \gr{AB}, \gr{AG} e{\c s}it olan \gr{DE}, \gr{DZ} iki kenar{\i}n{\i}n\\ her biri birine,\\ \gr{AB}, \gr{DE} kenar{\i}na,\\ ve \gr{AG}, \gr{DZ} kenar{\i}na;\\ ve onlar{\i}n\\ \gr{BG} taban{\i} e{\c s}it olsun \gr{EZ} taban{\i}na. } \parsen{ I say that\\ also angle \gr{BAG}\\ to angle \gr{EDZ}\\ is equal. } { l'egw, <'oti\\ ka`i gwn'ia estin >'ish. } { {\.I}ddia ediyorum ki\\ \gr{BAG} a{\c c}{\i}s{\i} da\\ e{\c s}ittir \gr{EDZ} a{\c c}{\i}s{\i}na. } \parsen{ For, there being applied\\ triangle \gr{ABG}\\ to triangle \gr{DEZ},\\ and there being placed\\ the point \gr B on the point \gr E,\\ and the \strgt{} \gr{BG} on \gr{EZ},\\ also the point \gr G will apply to \gr Z,\\ by the equality of \gr{BG} to \gr{EZ}.\\ Then, \gr{BG} applying to \gr{EZ},\\ also will apply\\ \gr{BA} and \gr{GA} to \gr{ED} and \gr{DZ}.\\ For if base \gr{BG} to the base \gr{EZ}\\ apply,\\ and sides \gr{BA}, \gr{AG} to \gr{ED}, \gr{DZ}\\ do not apply,\\ but deviate,\\ as \gr{EH}, \gr{HZ},\\ there will be constructed\\ on the same \strgt,\\ to two given \strgt s,\\ two other \strgt s equal,\\ either to either,\\ to one and another point\\ to the same parts\\ having the same extremities.\\ But they are not constructed;\\ therefore it is not [the case] that,\\ there being applied\\ the base \gr{BG} to the base \gr{EZ},\\ there do not apply\\ sides \gr{BA}, \gr{AG} to \gr{ED}, \gr{DZ}.\\ Therefore they apply.\\ So angle \gr{BAG}\\ to angle \gr{EDZ}\\ will apply\\ and will be equal to it. } { >Efarmozom'enou g`ar\\ to~u ABG trig'wnou\\ {}>ep`i t`o DEZ tr'igwnon\\ ka`i tijem'enou\\ to~u m`en B shme'iou >ep`i t`o E shme~ion\\ t~hc d`e BG e>uje'iac >ep`i t`hn EZ\\ {}>efarm'osei ka`i t`o G shme~ion >ep`i t`o Z\\ di`a t`o >'ishn e>~inai t`hn BG t~h| EZ;\\ {}>efarmos'ashc d`h t~hc BG >ep`i t`hn EZ\\ {}>efarm'osousi ka`i\\ aep`i t`ac ED, DZ.\\ e>i g`ar b'asic m`en ep`i b'asin t`hn EZ\\ {}>efarm'osei,\\ aep`i t`ac ED, DZ\\ o>uk >efarm'osousin\\ {}>all`a parall'axousin\\ ep`i t~hc a>ut~hc e>uje'iac\\ d'uo ta~ic a>uta~ic e>uje'iaic\\ {}>'allai d'uo e>uje~iai >'isai\\ 'allw| ka`i >'allw| shme'iw|\\ {}>ep`i t`a a>ut`a m'erh\\ t`a a>ut`a p'erata >'eqousai.\\ o>u sun'istantai d'e;\\ o>uk >'ara\\ {}>efarmozom'enhc\\ t~hc BG b'asewc >ep`i t`hn EZ b'asin\\ o>uk >efarm'osousi\\ ka`i aep`i t`ac ED, DZ.\\ {}>efarm'osousin >'ara;\\ <'wste ka`i gwn'ia ep`i gwn'ian t`hn efarm'osei\\ ka`i >'ish a>ut~h| >'estai. } { {\c C}{\"u}nk{\"u}, {\"u}st{\"u}ne koyulursa\\ \gr{ABG} {\"u}{\c c}geni \gr{DEZ} {\"u}{\c c}geninin,\\ ve yerle{\c s}tirilirse\\ \gr B noktas{\i} \gr E noktas{\i}na,\\ ve \gr{BG}, \gr{EZ} do{\u g}rusuna,\\ \gr G noktas{\i} da yerle{\c s}ecek \gr Z noktas{\i}na,\\ sayesinde e{\c s}itli{\u g}inin \gr{BG} do{\u g}rusunun \gr{EZ} do{\u g}rusuna.\\ O zaman, \gr{BG} yerle{\c s}tirilince \gr{EZ} do{\u g}rusuna,\\ \gr{BA} ve \gr{GA} do{\u g}rular{\i} da yerle{\c s}ecekler \gr{ED} ve \gr{DZ} do{\u g}rular{\i}na.\\ {\c C}{\"u}nk{\"u} e{\u g}er \gr{BG} yerle{\c s}irse \gr{EZ} taban{\i}na,\\ ve \gr{BA}, \gr{AG} kenarlar{\i} yerle{\c s}mezse \gr{ED}, \gr{DZ} kenarlar{\i}na,\\ ama saparsa,\\ \gr{EH} ve \gr{HZ} olarak \\ in{\c s}a edilmi{\c s} olacak\\ ayn{\i} do{\u g}ru {\"u}zerinde,\\ verilmi{\c s} iki do{\u g}ruya,\\ iki ba{\c s}ka do{\u g}ru e{\c s}it,\\ her biri birine,\\ ba{\c s}ka bir noktaya\\ ayn{\i} tarafta \\ ayn{\i} u{\c c}lar{\i} olan.\\ Ama in{\c s}a edilmediler;\\ dolay{\i}s{\i}yla (durum) {\c s}{\"o}yle de{\u g}il;,\\ \gr{BG} taban{\i} yerle{\c s}tirilince \gr{EZ} taban{\i}na,\\ \gr{BA}, \gr{AG} kenarlar{\i} yerle{\c s}mez \gr{ED}, \gr{DZ} kenarlar{\i}na.\\ Dolay{\i}s{\i}yla yerle{\c s}irler.\\ B{\"o}ylece \gr{BAG} a{\c c}{\i}s{\i} yerle{\c s}ecek \gr{EDZ} a{\c c}{\i}s{\i}na\\ ve ona e{\c s}it olacak. } \parsen{ If, therefore, two triangles\\ two sides\\ to two sides\\ have equal,\\ either to either,\\ and have also base equal to base,\\ also angle to angle\\ they will have equal,\\ {}[namely] that by the equal \strgt s\\ subtended;\\ \myqed } { >E`an d'uo tr'igwna\\ t`ac d'uo pleur`ac\\ {}[ta~ic] d'uo pleura~ic\\ {}>'isac >'eqh|\\ 'eqh| d`e ka`i t`hn b'asin t~h| b'asei >'ishn,\\ ka`i t`hn gwn'ian t~h| gwn'ia|\\ {}>'ishn <'exei\\ t`hn 'iswn e>ujei~wn\\ perieqom'enhn;\\ <'oper >'edei de~ixai. } { E{\u g}er, dolay{\i}s{\i}yla, iki {\"u}{\c c}genin, \\ varsa iki kenar{\i}\\ e{\c s}it olan\\ iki kenara,\\ her bir (kenar) birine,\\ ve varsa tabana e{\c s}it taban{\i},\\ ayr{\i}ca olacak a{\c c}{\i}ya e{\c s}it a{\c c}{\i}lar{\i},\\ (yani) e{\c s}it kenarlar{\i} g{\"o}renler;\\ \ozqed } \begin{center} { \begin{pspicture}(0,-2.061875)(6.339375,2.061875) \psline[linewidth=0.05cm](0.2940625,-1.741875)(2.6540625,-0.421875) \psline[linewidth=0.05cm](2.6540625,-0.441875)(1.0740625,1.698125) \psline[linewidth=0.05cm](1.0740625,1.698125)(0.3140625,-1.721875) \psline[linewidth=0.05cm](3.6940625,-1.721875)(6.0540624,-0.401875) \psline[linewidth=0.05cm](6.0540624,-0.421875)(4.4740624,1.718125) \psline[linewidth=0.05cm](4.4740624,1.718125)(3.7140625,-1.701875) \psline[linewidth=0.05cm](3.7140625,-1.701875)(5.1140623,1.678125) \psline[linewidth=0.05cm](6.0540624,-0.401875)(5.1140623,1.658125) \usefont{T1}{ptm}{m}{n} \rput(1.240625,1.883125){\gr A} \usefont{T1}{ptm}{m}{n} \rput(0.10125,-1.796875){\gr B} \usefont{T1}{ptm}{m}{n} \rput(2.7751563,-0.276875){\gr G} \usefont{T1}{ptm}{m}{n} \rput(4.2740626,1.803125){\gr D} \usefont{T1}{ptm}{m}{n} \rput(3.5760937,-1.916875){\gr E} \usefont{T1}{ptm}{m}{n} \rput(5.2567186,1.863125){\gr H} \usefont{T1}{ptm}{m}{n} \rput(6.20375,-0.556875){\gr Z} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.9 \parsen{ The\footnotemark\ given rectilineal angle\\ to cut in two.\footnotemark } { T`hn doje~isan gwn'ian e>uj'ugrammon\\ d'iqa teme~in. } { Verilen d{\"u}zkenar a{\c c}{\i}y{\i}\\ ikiye kesmek. } \myfntext{\label{note:gen}Here the generic article (see note~\ref{note:the} to Proposition 1 above) is particularly appropriate. Suppose we take a straight line with a point $A$ on it and draw a circle with center $A$ cutting the line at $B$ and $C$. Then the straight line $BC$ has been bisected at $A$. In particular, \emph{a} line has been bisected. But this does not mean we have solved the problem of the present proposition. In modern mathematical English, the proposition could indeed be `To bisect a rectilineal angle'; but then `a' must be understood as `an arbitrary' or `a given'. Of course, Euclid does supply this qualification in any case.} \myfntext{For `cut in two' we could say `bisect'; but in at least one place, in Proposition 12, \gr{d'iqa teme~in} will be separated.} \parsen{ Let be\\ the given rectilineal angle\\ \gr{BAG}. } { >'Estw\\ uj'ugrammoc\\ ut`hn d'iqa teme~in. } { {\c S}imdi gereklidir\\ onun ikiye kesilmesi. } \parsen{ Suppose there has been chosen\\ on \gr{AB} at random a point \gr D,\\ and there has been taken from \gr{AG}\\ \gr{AE}, equal to \gr{AD},\\ and \gr{DE} has been joined,\\ and there has been constructed on \gr{DE}\\ an equilateral triangle, \gr{DEZ},\\ and \gr{AZ} has been joined. } { E>il'hfjw\\ {}>ep`i t~hc AB tuq`on shme~ion t`o D,\\ ka`i >afh|r'hsjw >ap`o t~hc AG\\ t~h| AD >'ish epeze'uqjw ep`i t~hc DE\\ tr'igwnon >is'opleuron t`o DEZ,\\ ka`i >epeze'uqjw uje'iac.\\ {}>Epe`i g`ar >'ish >est`in 'isai e>is`in\\ 'ish >est'in;\\ gwn'ia >'ara 'ish >est'in. } { {\.I}ddia ediyorum ki\\ \gr{BAG} a{\c c}{\i}s{\i} ikiye kesilmi{\c s} oldu\\ \gr{AZ} do{\u g}rusu taraf{\i}ndan.\\ {\c C}{\"u}nk{\"u}, oldu{\u g}undan, \gr{AD} e{\c s}it \gr{AE} kenar{\i}na,\\ ve \gr{AZ} ortak,\\ \gr{DA}, \gr{AZ} ikilisi e{\c s}ittirler \gr{EA}, \gr{AZ} ikilisinin\\ her biri birine ,\\ ve \gr{DZ} taban{\i} \gr{EZ} taban{\i}na e{\c s}ittir;\\ dolay{\i}s{\i}yla \gr{DAZ} a{\c c}{\i}s{\i} \gr{EAZ} a{\c c}{\i}s{\i}na e{\c s}ittir. } \parsen{ Therefore the given rectilineal angle\\ \gr{BAG}\\ has been cut in two\\ by the \strgt{} \gr{AZ};\\ \myqef } { 'ara doje~isa gwn'ia e>uj'ugrammoc\\ uje'iac;\\ <'oper >'edei poi~hsai. } { Dolay{\i}s{\i}yla verilen d{\"u}zkenar a{\c c}{\i} \gr{BAG}\\ kesilmi{\c s} oldu ikiye\\ \gr{AZ} do{\u g}rusunca;\\ \ozqef } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.911875)(2.8290625,1.911875) \psline[linewidth=0.05cm](1.4740624,1.608125)(0.2740625,-1.611875) \psline[linewidth=0.05cm](1.4740624,1.588125)(2.4740624,-1.591875) \psline[linewidth=0.05cm](0.7340625,-0.411875)(2.0940626,-0.411875) \psline[linewidth=0.05cm](0.7340625,-0.431875)(1.3940625,-1.611875) \psline[linewidth=0.05cm](1.3940625,-1.611875)(2.0940626,-0.431875) \psline[linewidth=0.05cm](1.4740624,1.588125)(1.3940625,-1.631875) \usefont{T1}{ptm}{m}{n} \rput(0.10125,-1.666875){\gr B} \usefont{T1}{ptm}{m}{n} \rput(1.640625,1.733125){\gr A} \usefont{T1}{ptm}{m}{n} \rput(2.6551561,-1.706875){\gr G} \usefont{T1}{ptm}{m}{n} \rput(0.5540625,-0.366875){\gr D} \usefont{T1}{ptm}{m}{n} \rput(2.2960937,-0.306875){\gr E} \usefont{T1}{ptm}{m}{n} \rput(1.56375,-1.766875){\gr Z} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.10 \parsen{ The given bounded \strgt\\ to cut in two. } { T`hn doje~isan e>uje~ian peperasm'enhn\\ d'iqa teme~in. } { Verilen s{\i}n{\i}rl{\i} do{\u g}ruyu\\ ikiye kesmek.\\ } \parsen{ Let be\\ the given bounded straight line \gr{AB}.\\ } { >'Estw\\ uje~ia peperasm'enh uje~ian peperasm'enhn d'iqa teme~in. } { Gereklidir,\\ verilmi{\c s} \gr{AB} s{\i}n{\i}rl{\i} do{\u g}rusunu, kesmek ikiye.\\ } \parsen{ Suppose there has been constructed\\ on it\\ an equilateral triangle, \gr{ABG},\\ and suppose has been cut in two\\ the angle \gr{AGB} by the \strgt{} \gr{GD}. } { Sunest'atw >ep> a>ut~hc\\ tr'igwnon >is'opleuron t`o ABG,\\ ka`i tetm'hsjw\\ uje'ia|; } { Kabul edelim ki {\"u}zerinde in{\c s}a edilmi{\c s} olsun\\ bir e{\c s}kenar {\"u}{\c c}gen, \gr{ABG},\\ ve \gr{AGB} a{\c c}{\i}s{\i} kesilmi{\c s} olsun ikiye\\ \gr{GD} do{\u g}rusunca.\\ } \parsen{ I say that\\ the \strgt{} \gr{AB} has been cut in two\\ at the point \gr D.\\ For, because \gr{AG} is equal to \gr{AB},\\ and \gr{GD} is common,\\ the two, \gr{AG} and \gr{GD},\\ to the two, \gr{BG}, \gr{GD},\\ are equal,\\ either to either,\\ and angle \gr{AGD}\\ to angle \gr{BGD}\\ is equal;\\ therefore the base \gr{AD} to the base \gr{BD}\\ is equal. } {l'egw, <'oti\\ uje~ia d'iqa t'etmhtai\\ kat`a t`o D shme~ion.\\ {}>Epe`i g`ar >'ish >est`in 'isai e>is`in\\ 'ish >est'in;\\ b'asic >'ara 'ish >est'in. } { {\.I}ddia ediyorum ki\\ \gr{AB} do{\u g}rusu ikiye kesilmi{\c s} oldu\\ \gr D noktas{\i}nda. {\c C}{\"u}nk{\"u}, \gr{AG} e{\c s}it oldu{\u g}undan \gr{AB} kenar{\i}na,\\ ve \gr{GD} ortak,\\ \gr{AG} ve \gr{GD} ikilisi, e{\c s}ittirler \gr{BG}, \gr{GD} ikilisinin,\\ her biri birine,\\ ve \gr{AGD} a{\c c}{\i}s{\i} e{\c s}ittir \gr{BGD} a{\c c}{\i}s{\i}na;\\ dolay{\i}s{\i}yla \gr{AD} taban{\i}, \gr{BD} taban{\i}na,\\ e{\c s}ittir. } \parsen{ Therefore the given bounded\qquad \strgt,\\ \gr{AB},\\ has been cut in two at \gr D;\\ \myqef } { 'ara doje~isa e>uje~ia peperasm'enh\\ 'edei poi~hsai. } { Dolay{\i}s{\i}yla verilmi{\c s} s{\i}n{\i}rl{\i} \gr{AB} do{\u g}rusu\\ \gr D noktas{\i}nda ikiye kesilmi{\c s} oldu;\\ \ozqef } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.991875)(5.2575,1.991875) \psline[linewidth=0.05cm](0.3171875,-1.591875)(4.9171877,-1.591875) \psline[linewidth=0.05cm](0.3171875,-1.591875)(2.7371874,1.808125) \psline[linewidth=0.05cm](2.7371874,1.788125)(4.8971877,-1.591875) \psline[linewidth=0.05cm](2.7371874,1.788125)(2.7371874,-1.571875) \usefont{T1}{ptm}{m}{n} \rput(0.12375,-1.706875){\gr A} \usefont{T1}{ptm}{m}{n} \rput(5.104375,-1.686875){\gr B} \usefont{T1}{ptm}{m}{n} \rput(2.9782813,1.813125){\gr G} \usefont{T1}{ptm}{m}{n} \rput(2.8171875,-1.846875){\gr D} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.11 \parsen{ To the given \strgt\\ from the given point on it\\ at right angles\\ to draw\footnotemark\ a straight line.\footnotemark } { T~h| doje'ish| e>uje'ia|\\ {}>ap`o to~u pr`oc a>ut~h| doj'entoc shme'iou\\ pr`oc >orj`ac gwn'iac\\ e>uje~ian gramm`hn >agage~in. } { Verilen bir do{\u g}ruya \\ {\"u}zerinde verilen bir noktada\\ dik a{\c c}{\i}larda\\ bir do{\u g}ru {\c c}izmek.} \myfntext{This is the first time among the propositions that Euclid writes out \emph{straight line} (\gr{e>uje~ia gramm`h}) and not just \emph{straight} (\gr{e>uje~ia}).} \myfntext{Literally `lead, conduct'.} \parsen{ Let be\\ the given \strgt\ \gr{AB},\\ and the given point on it, \gr G. } { >'Estw\\ uje~ia ep> a>ut~hc t`o G; } { Verilmi{\c s} olsun\\ bir do{\u g}ru, \gr{AB},\\ ve {\"u}zerinde bir nokta, \gr G.\\ } \parsen{ It is necessary then\\ from the point \gr G\\ to the \strgt{} \gr{AB}\\ at right angles\\ to draw a straight line. } { de~i d`h\\ {}>ap`o to~u G shme'iou\\ t~h| AB e>uje'ia|\\ pr`oc >orj`ac gwn'iac\\ e>uje~ian gramm`hn >agage~in. } { Gereklidir\\ \gr G noktas{\i}nda\\ \gr{AB} do{\u g}rusuna\\ dik a{\c c}{\i}larda\\ bir do{\u g}ru.\\ } \parsen{ Suppose there has been chosen\\ on \gr{AG} at random a point \gr D,\\ and there has been laid down\\ an equal to \gr{GD}, [namely] \gr{GE},\\ and there has been constructed\\ on \gr{DE}\\ an equilateral triangle, \gr{ZDE},\\ and there has been joined \gr{ZG}. } { E>il'hfjw\\ {}>ep`i t~hc AG tuq`on shme~ion t`o D,\\ ka`i ke'isjw\\ t~h| GD >'ish ep`i t~hc DE tr'igwnon >is'opleuron\\ t`o ZDE,\\ ka`i >epeze'uqjw uje'ia| t~h| AB\\ {}>ap`o to~u pr`oc a>ut~h| doj'entoc shme'iou to~u G\\ pr`oc >orj`ac gwn'iac\\ e>uje~ia gramm`h >~hktai Epe`i g`ar >'ish >est`in 'isai e>is`in\\ 'ish >est'in;\\ gwn'ia >'ara 'ish >est'in;\\ ka'i e>isin >efex~hc.\\ <'otan d`e e>uje~ia\\ {}>ep> e>uje~ian staje~isa\\ t`ac >efex~hc gwn'iac\\ {}>'isac >all'hlaic\\ poi~h|,\\ {}>orj`h 'iswn gwni~wn >estin;\\ {}>orj`h >'ara >est`in 'ara doje'ish| e>uje'ia| t~h| AB\\ {}>ap`o to~u pr`oc a>ut~h| doj'entoc shme'iou\\ to~u G\\ pr`oc >orj`ac gwn'iac\\ e>uje~ia gramm`h >~hktai 'edei poi~hsai. } { Dolay{\i}s{\i}yla, verilen \gr{AB} do{\u g}rusuna,\\ {\"u}zerinde verilmi{\c s} \gr G noktas{\i}nda,\\ dik a{\c c}{\i}larda,\\ bir \gr{GZ} do{\u g}rusu {\c c}izilmi{\c s} oldu;\\ \ozqef } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.6092187)(5.1775,1.6092187) \pstriangle[linewidth=0.04,dimen=outer](2.4371874,-1.2092187)(3.2,2.6) \psline[linewidth=0.04cm](0.0371875,-1.2092187)(5.1171875,-1.2092187) \usefont{T1}{ptm}{m}{n} \rput(2.7398438,1.4357812){\gr Z} \usefont{T1}{ptm}{m}{n} \rput(0.12375,-1.4642187){\gr A} \usefont{T1}{ptm}{m}{n} \rput(5.024375,-1.4442188){\gr B} \usefont{T1}{ptm}{m}{n} \rput(0.8971875,-1.4442188){\gr D} \usefont{T1}{ptm}{m}{n} \rput(2.3582811,-1.4042188){\gr G} \usefont{T1}{ptm}{m}{n} \rput(4.019219,-1.4642187){\gr E} \psline[linewidth=0.04cm](2.4371874,1.3507812)(2.3971875,-1.2092187) \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.12 \parsen{ To the given unbounded \strgt,\\ from the given point,\\ which is not on it,\\ to draw a perpendicular straight line. } { >Ep`i t`hn doje~isan e>uje~ian >'apeiron\\ {}>ap`o to~u doj'entoc shme'iou,\\ <`o m'h >estin >ep> a>ut~hc,\\ k'ajeton e>uje~ian gramm`hn >agage~in. } { Verilen s{\i}n{\i}rlanmam{\i}{\c s} do{\u g}ruya,\\ verilen bir noktadan,\\ {\"u}zerinde olmayan,\\ bir dik do{\u g}ru {\c c}izmek.\\ } \parsen{ Let be\\ the given unbounded \strgt\ \gr{AB},\\ and the given point,\\ which is not on it,\\ \gr G. } { >'Estw\\ uje~ia >'apeiroc estin >ep> a>ut~hc,\\ t`o G;\\ } { Verilmi{\c s} olsun\\ bir s{\i}n{\i}rlanmam{\i}{\c s} do{\u g}ru, \gr{AB},\\ ve bir nokta,\\ {\"u}zerinde olmayan, \gr G.\\ } \parsen{ It is necessary then\\ to the given unbounded \strgt,\\ \gr{AB}\\ from the given point \gr G,\\ which is not on it,\\ to draw a perpendicular straight line. } { de~i d`h\\ {}>ep`i t`hn doje~isan e>uje~ian >'apeiron\\ t`hn AB\\ {}>ap`o to~u doj'entoc shme'iou to~u G,\\ <`o m'h >estin >ep> a>ut~hc,\\ k'ajeton e>uje~ian gramm`hn >agage~in. } { Gereklidir\\ verilmi{\c s} \gr{AB} s{\i}n{\i}rlanmam{\i}{\c s} do{\u g}rusuna\\ verilmi{\c s} \gr G noktas{\i}ndan,\\ {\"u}zerinde olmayan,\\ bir dik do{\u g}ru {\c c}izmek.\\ } \parsen{ For suppose there has been chosen\\ on the other parts\\ of the \strgt{} \gr{AB}\\ at random a point \gr D,\\ and to the center \gr G,\\ at the distance \gr{GD},\\ a circle has been drawn, \gr{EZH},\\ and has been cut\\ the \strgt{} \gr{EH}\\ in two at \gr J,\\ and there have been joined\\ the \strgt s \gr{GH}, \gr{GJ}, and \gr{GE}. } { E>il'hfjw g`ar\\ {}>ep`i t`a <'etera m'erh\\ t~hc AB e>uje'iac\\ tuq`on shme~ion t`o D,\\ ka`i k'entrw| m`en t~w| G\\ diast'hmati d`e t~w| GD\\ k'ukloc gegr'afjw uje~ia\\ d'iqa kat`a t`o J,\\ ka`i >epeze'uqjwsan\\ auje~iai; } { {\c C}{\"u}nk{\"u} kabul edelim ki se{\c c}ilmi{\c s} olsun\\ \gr{AB} do{\u g}rusunun di{\u g}er taraf{\i}nda\\ rastgele bir \gr D noktas{\i},\\ ve \gr G merkezinde,\\ \gr{GD} uzakl{\i}{\u g}{\i}nda,\\ bir {\c c}ember {\c c}izilmi{\c s} olsun, \gr{EZH},\\ ve \gr{EH} do{\u g}rusu \gr J noktas{\i}nda ikiye kesilmi{\c s} olsun,\\ ve birle{\c s}tirilmi{\c s} olsun\\ \gr{GH}, \gr{GJ}, ve \gr{GE} do{\u g}rular{\i}.\\ } \parsen{ I say that\\ to the given unbounded \strgt\\ \gr{AB},\\ from the given point \gr G,\\ which is not on it,\\ has been drawn a perpendicular, \gr{GJ}.\\ For, because \gr{HJ} is equal to \gr{JE},\\ and \gr{JG} is common,\\ the two, \gr{HJ} and \gr{JG},\\ to the two, \gr{EJ} and \gr{JG}, are equal,\\ either to either;\\ and the base \gr{GH} to the base \gr{GE}\\ is equal;\\ therefore angle \gr{GJH}\\ to angle \gr{EJG}\\ is equal;\\ and they are adjacent.\\ Whenever a \strgt,\\ standing on a \strgt,\\ the adjacent angles\\ equal to one another make,\\ right\\ either of the equal angles is,\\ and\\ the \strgt{} that has been stood\\ is called perpendicular\\ to that on which it has been stood. } { l'egw, <'oti\\ {}>ep`i t`hn doje~isan e>uje~ian >'apeiron\\ t`hn AB\\ {}>ap`o to~u doj'entoc shme'iou to~u G,\\ <`o m'h >estin >ep> a>ut~hc,\\ k'ajetoc >~hktai Epe`i g`ar >'ish >est`in 'isai eestin >'ish;\\ gwn'ia >'ara estin >'ish.\\ ka'i e>isin >efex~hc.\\ <'otan d`e e>uje~ia\\ {}>ep> e>uje~ian staje~isa\\ t`ac >efex~hc gwn'iac\\ {}>'isac >all'hlaic poi~h|,\\ {}>orj`h\\ 'iswn gwni~wn >estin,\\ ka`i\\ efesthku~ia e>uje~ia\\ k'ajetoc kale~itai\\ {}>ef> <`hn >ef'esthken. } { {\.I}ddia ediyorum ki\\ verilen s{\i}n{\i}rlanmam{\i}{\c s} \gr{AB} do{\u g}rusuna,\\ verilen \gr G noktas{\i}ndan,\\ {\"u}zerinde olmayan,\\ {\c c}izilmi{\c s} oldu dik \gr{GJ} do{\u g}rusu.\\ {\c C}{\"u}nk{\"u}, \gr{HJ} e{\c s}it oldu{\u g}undan \gr{JE} do{\u g}rusuna,\\ ve \gr{JG} ortak,\\ \gr{HJ} ve \gr{JG} ikilisi,\\ e{\c s}ittirler \gr{EJ} ve \gr{JG} ikilisinin,\\ her biri birine;\\ ve \gr{GH} taban{\i} e{\c s}ittir \gr{GE} taban{\i}na;\\ dolay{\i}s{\i}yla \gr{GJH} a{\c c}{\i}s{\i} e{\c s}ittir \gr{EJG} a{\c c}{\i}s{\i}na.\\ Ve onlar biti{\c s}iktirler.\\ Ne zaman bir do{\u g}ru,\\ bir do{\u g}ru {\"u}zerinde dikildi{\u g}inde,\\ biti{\c s}ik a{\c c}{\i}lar{\i} birbirine e{\c s}it yaparsa,\\ a{\c c}{\i}lar{\i}n her biri e{\c s}ittir,\\ ve dikiltilen do{\u g}ru\\ {\"u}zerinde dikildi{\u g}i do{\u g}ruya diktir denir.\\ } \parsen{ Therefore, to the given unbounded \strgt\ \gr{AB},\\ from the given point \gr G,\\ which is not on it,\\ a perpendicular \gr{GJ} has been drawn;\\ \myqef } { >Ep`i t`hn doje~isan >'ara e>uje~ian >'apeiron t`hn AB\\ {}>ap`o to~u doj'entoc shme'iou to~u G,\\ <`o m'h >estin >ep> a>ut~hc,\\ k'ajetoc ~>hktai 'edei poi~hsai. } { Dolay{\i}s{\i}yla, verilen \gr{AB} s{\i}n{\i}rland{\i}r{\i}lmam{\i}{\c s} do{\u g}ruya,\\ verilen \gr G noktas{\i}ndan,\\ {\"u}zerinde olmayan,\\ bir dik, \gr{GJ}, {\c c}izilmi{\c s} oldu;\\ \ozqef } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.9492188)(5.4203124,1.9892187) \pscircle[linewidth=0.04,dimen=outer](2.51,-0.15921874){1.79} \pstriangle[linewidth=0.04,dimen=outer](2.49,-0.94921875)(3.14,0.8) \psline[linewidth=0.04cm](3.94,-0.92921877)(5.34,-0.94921875) \psline[linewidth=0.04cm](1.04,-0.92921877)(0.0,-0.92921877) \psline[linewidth=0.04cm](2.48,-0.18921874)(2.48,-0.9292187) \usefont{T1}{ptm}{m}{n} \rput(0.1465625,-1.1442188){\gr A} \usefont{T1}{ptm}{m}{n} \rput(5.2671876,-1.1842188){\gr B} \usefont{T1}{ptm}{m}{n} \rput(2.6010938,-0.00421875){\gr G} \usefont{T1}{ptm}{m}{n} \rput(3.86,-1.6842188){\gr D} \usefont{T1}{ptm}{m}{n} \rput(4.282031,-1.2242187){\gr E} \usefont{T1}{ptm}{m}{n} \rput(2.9826562,1.8157812){\gr Z} \usefont{T1}{ptm}{m}{n} \rput(0.783125,-1.1642188){\gr H} \usefont{T1}{ptm}{m}{n} \rput(2.4653125,-1.2242187){\gr J} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.13 \parsen{ If a \strgt,\\ stood on a \strgt,\\ make angles,\\ either two \rgt s\\ or equal to two \rgt s\\ it will make [them]. } { >E`an e>uje~ia\\ {}>ep> e>uje~ian staje~isa\\ gwn'iac poi~h|,\\ {}>'htoi d'uo >orj`ac\\ {}>`h dus`in >orja~ic >'isac\\ poi'hsei. } { E{\u g}er bir do{\u g}ru,\\ dikiltilmi{\c s} bir do{\u g}runun {\"u}zerine,\\ yaparsa a{\c c}{\i}lar,\\ ya iki dik\\ ya da iki dike e{\c s}it \\ yapacak (onlar{\i}).\\ } \parsen{ For, some \strgt{}, \gr{AB},\\ stood on the \strgt{} \gr{GD},\\ ---suppose it makes\footnotemark\ angles\\ \gr{GBA} and \gr{ABD}. } { E>uje~ia g'ar tic ep> e>uje~ian t`hn GD staje~isa\\ gwn'iac poie'itw\\ t`ac 'htoi d'uo >orja'i e>isin\\ {}>`h dus`in >orja~ic >'isai. } { {\.I}ddia ediyorum ki\\ \gr{GBA} ve \gr{ABD} a{\c c}{\i}lar{\i}\\ ya iki dik a{\c c}{\i}d{\i}r\\ ya da iki dik a{\c c}{\i}ya e{\c s}ittir(ler).\\ } \parsen{ If equal is\\ \gr{GBA} to \gr{ABD},\\ they are two \rgt s. } { E>i m`en o>~un >'ish >est`in\\ orja'i e>isin. } { E{\u g}er \gr{GBA} e{\c s}itse \gr{ABD} a{\c c}{\i}s{\i}na,\\ iki dik a{\c c}{\i}d{\i}rlar.\\ } \parsen{ If not,\\ suppose there has been drawn,\\ from the point \gr B,\\ to the [\strgt] \gr{GD},\\ at right angles,\\ \gr{BE}. } { e>i d`e o>'u,\\ {}>'hqjw\\ {}>ap`o to~u B shme'iou\\ t~h| GD [e>uje'ia|]\\ pr`oc >orj`ac\\ 'ara orja'i e>isin;\\ ka`i >epe`i 'ish >est'in,\\ koin`h proske'isjw 'ara 'isai e>is'in.\\ p'alin,\\ {}>epe`i 'ish >est'in,\\ koin`h proske'isjw 'ara 'isai e>is'in.\\ {}>ede'iqjhsan d`e ka`i auta~ic >'isai;\\ t`a d`e t~w| a>ut~w| >'isa\\ ka`i >all'hloic >est`in >'isa;\\ ka`i a'ara\\ ta~ic 'isai e>is'in;\\ {}>all`a aorja'i e>isin;\\ ka`i a'ara\\ dus`in >orja~ic >'isai e>is'in. } { Dolay{\i}s{\i}yla \gr{GBE} ve \gr{EBD} iki diktir;\\ ve oldu{\u g}undan \gr{GBE}\\ e{\c s}it \gr{GBA} ve \gr{ABE} ikilisine,\\ \gr{EBD} her birine eklenmi{\c s} olsun.\\ Dolay{\i}s{\i}yla \gr{GBE} ve \gr{EBD}\\ e{\c s}ittirler, \gr{GBA}, \gr{ABE} ve \gr{EBD} {\"u}{\c c}l{\"u}s{\"u}ne.\\ Dahas{\i},\\ oldu{\u g}undan \gr{DBA}\\ e{\c s}it, \gr{DBE} ve \gr{EBA} ikilisine,\\ \gr{ABG} her birine eklenmi{\c s} olsun;\\ dolay{\i}s{\i}yla \gr{DBA} ve \gr{ABG}\\ e{\c s}ittirler, \gr{DBE}, \gr{EBA} ve \gr{ABG} {\"u}{\c c}l{\"u}s{\"u}ne.\\ Ve \gr{GBE} ve \gr{EBD} a{\c c}{\i}lar{\i}n{\i}n g{\"o}s\-te\-ril\-mi{\c s}\-ti\\ e{\c s}itli{\u g}i ayn{\i} {\"u}{\c c}l{\"u}ye.\\ Ve ayn{\i} {\c s}eye e{\c s}it olanlar birbirine e{\c s}ittir;\\ ve, dolay{\i}s{\i}yla, \gr{GBE} ve \gr{EBD}\\ e{\c s}ittirler \gr{DBA} ve \gr{ABG} a{\c c}{\i}lar{\i}na;\\ ama \gr{GBE} ve\gr{EBD} iki diktir;\\ ve dolay{\i}s{\i}yla \gr{DBA} ve \gr{ABG}\\ iki dike e{\c s}ittirler.\\ } \parsen{ If, therefore, a \strgt,\\ stood on a \strgt,\\ make angles,\\ either two \rgt s\\ or equal to two \rgt s\\ it will make;\\ \myqed } { >E`an >'ara e>uje~ia\\ {}>ep> e>uje~ian staje~isa\\ gwn'iac poi~h|,\\ {}>'htoi d'uo >orj`ac\\ {}>`h dus`in >orja~ic >'isac\\ poi'hsei [them];\\ <'oper >'edei de~ixai. } { E{\u g}er, dolay{\i}s{\i}yla, bir do{\u g}ru,\\ dikiltilmi{\c s} bir do{\u g}runun {\"u}zerine,\\ yaparsa a{\c c}{\i}lar,\\ ya iki dik\\ ya da iki dike e{\c s}it\\ olacak (onlar{\i}).\\ \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.291875)(5.02875,1.291875) \psline[linewidth=0.04cm](0.01375,-0.8865625)(4.83375,-0.8865625) \psline[linewidth=0.04cm](2.21375,-0.8865625)(2.21375,0.9134375) \psline[linewidth=0.04cm](2.21375,-0.8665625)(3.13375,0.7734375) \usefont{T1}{ptm}{m}{n} \rput(3.3603125,0.7584375){\gr A} \usefont{T1}{ptm}{m}{n} \rput(2.0809374,-1.1015625){\gr B} \usefont{T1}{ptm}{m}{n} \rput(4.8548436,-1.1415625){\gr G} \usefont{T1}{ptm}{m}{n} \rput(2.1557813,1.1184375){\gr E} \usefont{T1}{ptm}{m}{n} \rput(0.11375,-1.1215625){\gr D} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.14 \parsen{ If to some \strgt,\\ and at the same point,\\ two \strgt s,\\ not lying to the same parts,\\ the adjacent angles\\ to two \rgt s\\ make equal,\\ on a \strgt\\ will be with one another\\ the \strgt s. } { >E`an pr'oc tini e>uje'ia|\\ ka`i t~w| pr`oc a>ut~h| shme'iw|\\ d'uo e>uje~iai\\ m`h >ep`i t`a a>ut`a m'erh ke'imenai\\ t`ac >efex~hc gwn'iac\\ dus`in >orja~ic >'isac\\ poi~wsin,\\ {}>ep> e>uje'iac\\ {}>'esontai >all'hlaic\\ auje~iai. } { E{\u g}er bir do{\u g}ruya,\\ ve ayn{\i} noktas{\i}nda,\\ iki do{\u g}ru,\\ ayn{\i} taraf{\i}nda kalmayan,\\ biti{\c s}ik a{\c c}{\i}lar{\i},\\ yaparsa\\ iki dik a{\c c}{\i}ya e{\c s}it\\ bir do{\u g}ruda\\ olacaklar birbirleriyle,\\ do{\u g}rular. } \parsen{ For, to some \strgt, \gr{AB},\\ and at the same point, \gr B,\\ two \strgt s \gr{BG} and \gr{BD},\\ not lying to the same parts,\\ the adjacent angles\\ \gr{ABG} and \gr{ABD}\\ equal to two \rgt s\\ ---suppose they make. } { Pr`oc g'ar tini e>uje'ia| t~h| AB\\ ka`i t~w| pr`oc a>ut~h| shme'iw| t~w| B\\ d'uo e>uje~iai aep`i t`a a>ut`a m'erh ke'imenai\\ t`ac >efex~hc gwn'iac\\ t`ac orja~ic >'isac\\ poie'itwsan; } {Bir \gr{AB} do{\u g}rusuna,\\ ve bir \gr B noktas{\i}nda,\\ ayn{\i} taraf{\i}nda kalmayan,\\ iki \gr{BG} ve \gr{BD} do{\u g}rular{\i}n{\i}n,\\ \gr{ABG} ve \gr{ABD}\\ biti{\c s}ik a{\c c}{\i}lar{\i}n{\i}n iki dik a{\c c}{\i}\\ ---oldu{\u g}u kabul edilsin. } \parsen{ I say that\\ on a \strgt\\ with \gr{GB} is \gr{BD}. } { l'egw, <'oti\\ {}>ep> e>uje'iac\\ {}>est`i t~h| GB i g`ar m'h >esti\\ t~h| BG >ep> e>uje'iac\\ 'estw\\ t~h| GB >ep> e>uje'iac\\ Epe`i o>~un e>uje~ia ep> e>uje~ian t`hn GBE >ef'esthken,\\ a'ara orja~ic >'isai e>is'in;\\ e>is`i d`e ka`i aorja~ic >'isai;\\ a'ara 'isai e>is'in.\\ koin`h\\ {}>afh|r'hsjw\\ 'ara estin >'ish,\\ el'asswn t~h| me'izoni;\\ <'oper >est`in >ad'unaton.\\ o>uk >'ara\\ {}>ep> e>uje'iac >est`in ud`e >'allh tic pl`hn t~hc BD;\\ {}>ep> e>uje'iac >'ara\\ {}>est`in E`an >'ara pr'oc tini e>uje'ia|\\ ka`i t~w| pr`oc a>ut~h| shme'iw|\\ d'uo e>uje~iai\\ m`h >ep`i a>ut`a m'erh ke'imenai\\ t`ac >efex~hc gwn'iac\\ dus`in >orja~ic >'isac\\ poi~wsin,\\ {}>ep> e>uje'iac\\ {}>'esontai >all'hlaic\\ auje~iai;\\ <'oper >'edei de~ixai. } { E{\u g}er, dolay{\i}s{\i}yla, bir do{\u g}ruya,\\ ve ayn{\i} noktas{\i}nda,\\ iki do{\u g}ru,\\ ayn{\i} taraf{\i}nda kalmayan,\\ biti{\c s}ik a{\c c}{\i}lar{\i},\\ yaparsa\\ iki dik a{\c c}{\i}ya e{\c s}it\\ bir do{\u g}ruda\\ olacaklar birbirleriyle,\\ do{\u g}rular. \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.351875)(4.373125,1.351875) \psline[linewidth=0.04cm](0.046875,-0.9865625)(4.266875,-1.0065625) \psline[linewidth=0.04cm](2.066875,-0.9865625)(0.866875,0.5934375) \psline[linewidth=0.04cm](2.066875,-0.9665625)(3.846875,0.9934375) \usefont{T1}{ptm}{m}{n} \rput(0.6934375,0.7384375){\gr A} \usefont{T1}{ptm}{m}{n} \rput(2.0140624,-1.2015625){\gr B} \usefont{T1}{ptm}{m}{n} \rput(0.0940625,-1.2015625){\gr G} \usefont{T1}{ptm}{m}{n} \rput(4.206875,-1.2015625){\gr D} \usefont{T1}{ptm}{m}{n} \rput(3.9489062,1.1784375){\gr E} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.15 \parsen{ If two \strgt s cut one another,\\ the vertical\footnotemark\ angles\\ they make equal to one another. } { >E`an d'uo e>uje~iai t'emnwsin >all'hlac,\\ t`ac kat`a koruf`hn gwn'iac\\ {}>'isac >all'hlaic poio~usin. } { E{\u g}er iki do{\u g}ru keserse birbirini,\\ ters a{\c c}{\i}lar\\ olu{\c s}turlar e{\c s}it bir birine. } \myfntext{The Greek is \gr{kat`a koruf`hn}, which might be translated as `at a head', just as, in the conclusion of I.10, \gr{AB} has been cut in two `at \gr D', \gr{kat`a t`o D}. But \gr{koruf'h} and the Latin \emph{vertex} can both mean \emph{crown of the head,} and in anatomical use, the English \emph{vertical} refers to this crown. Apollonius uses \gr{koruf'h} for the vertex of a cone~\cite[pp.~286--7]{MR13:419b}.} \parsen{ For, let the \strgt s \gr{AB} and \gr{GD}\\ cut one another\\ at the point \gr E. } { D'uo g`ar e>uje~iai aall'hlac\\ kat`a t`o E shme~ion; } { {\c C}{\"u}nk{\"u}, \gr{AB} ve \gr{GD} do{\u g}rular{\i} \\ kessinler birbirlerini\\ \gr E noktas{\i}nda. } \parsen{ I say that\\ equal are\\ angle \gr{AEG} to \gr{DEB},\\ and \gr{GEB} to \gr{AED}. } { l'egw, <'oti\\ {}>'ish >est`in\\ Epe`i g`ar e>uje~ia ep> e>uje~ian t`hn GD >ef'esthke\\ gwn'iac poio~usa t`ac 'ara orja~ic >'isai e>is'in.\\ p'alin,\\ {}>epe`i e>uje~ia ep> e>uje~ian t`hn AB >ef'esthke\\ gwn'iac poio~usa t`ac 'ara orja~ic >'isai e>is'in.\\ {}>ede'iqjhsan d`e ka`i aorja~ic >'isai;\\ a'ara 'isai e>is'in.\\ koin`h\\ {}>afh|r'hsjw\\ 'ara 'ish >est'in;\\ 'isai e>is'in. } { {\c C}{\"u}nk{\"u}, \gr{AE} do{\u g}rusu\\ yerle{\c s}mi{\c s}ti \gr{GD} do{\u g}rusuna,\\ olu{\c s}turur \gr{GEA} ve \gr{AED} a{\c c}{\i}lar{\i}n{\i},\\ dolay{\i}s{\i}yla \gr{GEA} ve \gr{AED} a{\c c}{\i}lar{\i}\\ e{\c s}ittirler iki dik a{\c c}{\i}ya.\\ Dahas{\i},\\ \gr{DE} do{\u g}rusu\\ dikiltilmi{\c s}ti \gr{AB} do{\u g}rusuna,\\ olu{\c s}turarak \gr{AED} ve \gr{DEB} a{\c c}{\i}lar{\i}n{\i},\\ dolay{\i}s{\i}yla \gr{AED} ve \gr{DEB} a{\c c}{\i}lar{\i}\\ e{\c s}ittirler iki dik a{\c c}{\i}ya.\\ Ve \gr{GEA} ve \gr{AED} a{\c c}{\i}lar{\i}n{\i}n g{\"o}sterilmi{\c s}ti\\ e{\c s}itli{\u g}i iki dik a{\c c}{\i}ya,\\ dolay{\i}s{\i}yla \gr{GEA} ve \gr{AED}\\ e{\c s}ittirler \gr{AED} ve \gr{DEB} a{\c c}{\i}lar{\i}na.\\ Ortak \gr{AED} a{\c c}{\i}s{\i}n{\i}n {\c c}{\i}kart{\i}lm{\i}{\c s} oldu{\u g}u kabul edilsin;\\ dolay{\i}s{\i}yla \gr{GEA} kalan{\i}\\ e{\c s}ittir \gr{BED} kalan{\i}na;\\ benzer {\c s}ekilde g{\"o}sterilecek ki\\ \gr{GEB} a{\c c}{\i}s{\i} da e{\c s}ittir \gr{DEA} a{\c c}{\i}s{\i}na.} \myfntext{This is a rare moment when two things are said to be equal \emph{simply,} and not equal \emph{to one another.}} \parsen{ If, therefore,\\ two \strgt s cut one another,\\ the vertical angles\\ they make equal to one another;\\ \myqed } { >E`an >'ara\\ d'uo e>uje~iai t'emnwsin >all'hlac,\\ t`ac kat`a koruf`hn gwn'iac\\ {}>'isac >all'hlaic poio~usin;\\ <'oper >'edei de~ixai. } { E{\u g}er, dolay{\i}s{\i}yla,\\ iki do{\u g}ru keserse bir birini,\\ ters a{\c c}{\i}lar\\ olu{\c s}turlar e{\c s}it birbirine\\ \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.131875)(4.22875,1.131875) \psline[linewidth=0.04cm](0.09375,-0.071875)(3.87375,-0.071875) \psline[linewidth=0.04cm](0.47375,0.868125)(3.51375,-0.971875) \usefont{T1}{ptm}{m}{n} \rput(0.2403125,0.953125){\gr A} \usefont{T1}{ptm}{m}{n} \rput(3.6809375,-0.986875){\gr B} \usefont{T1}{ptm}{m}{n} \rput(4.054844,-0.106875){\gr G} \usefont{T1}{ptm}{m}{n} \rput(0.11375,-0.266875){\gr D} \usefont{T1}{ptm}{m}{n} \rput(1.9357812,-0.306875){\gr E} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.16 \parsen{ One of the sides of any triangle\\ being extended,\\ the exterior angle\\ than either\\ of the interior and opposite angles\\ is greater. } { Pant`oc trig'wnou mi~ac t~wn pleur~wn\\ prosekblhje'ishc\\ ekt`oc gwn'ia\\ ent`oc ka`i >apenant'ion gwni~wn\\ me'izwn >est'in. } { Herhangi bir {\"u}{\c c}genin kenarlar{\i}ndan biri\\ uzat{\i}l{\i}d{\i}{\u g}{\i}nda,\\ d{\i}{\c s} a{\c c}{\i}\\ her bir\\ i{\c c} ve kar{\c s}{\i}t a{\c c}{\i}dan\\ b{\"u}y{\"u}kt{\"u}r. } \parsen{ Let there be\\ a triangle, \gr{ABG},\\ and let there have been extended\\ its side \gr{BG}, to \gr D. } { >'Estw\\ tr'igwnon t`o ABG,\\ ka`i prosekbebl'hsjw\\ a>uto~u m'ia pleur`a ep`i t`o D; } { Olsun,\\ bir \gr{ABG} {\"u}{\c c}geni\\ ve uzat{\i}lm{\i}{\c s} olsun\\ onun \gr{BG} kenar{\i} \gr D noktas{\i}na. } \parsen{ I say that\\ the exterior angle \gr{AGD}\\ is greater\\ than either\\ of the two interior and opposite angles, \gr{GBA} and \gr{BAG}. } { l`egw, <'oti\\ ekt`oc gwn'ia est`in\\ ent`oc ka`i >apenant'ion t~wn epizeuqje~isa ekbebl'hsjw\\ {}>ep> e>uje'iac >ep`i t`o Z,\\ ka`i ke'isjw\\ t~h| BE >'ish epeze'uqjw\\ ep`i t`o H. } { \gr{AG} kenar{\i}, E noktas{\i}ndan ikiye kesilmi{\c s} olsun,\\ ve birle{\c s}tirilen \gr{BE},\\ ---uzat{\i}lm{\i}{\c s} olsun\\ \gr Z noktas{\i}na bir do{\u g}ruda\\ ve yerle{\c s}tirilmi{\c s} olsun,\\ \gr{BE} do{\u g}rusuna e{\c s}it olan \gr{EZ},\\ ve birle{\c s}tirilmi{\c s} olsun\\ \gr{ZG},\\ ve {\c c}izilmi{\c s} olsun \\ \gr{AG} do{\u g}rusu \gr H noktas{\i}na kadar. } \parsen{ Since equal are\\ \gr{AE} to \gr{EG},\\ and \gr{BE} to \gr{EZ},\\ the two, \gr{AE} and \gr{EB}\\ to the two, \gr{GE} and \gr{EZ},\\ are equal,\\ either to either;\\ and angle \gr{AEB}\\ is equal to angle \gr{ZEG};\\ for they are vertical;\\ therefore the base \gr{AB}\\ is equal to the base \gr{ZG},\\ and triangle \gr{ABE}\\ is equal to triangle \gr{ZEG},\\ and the remaining angles\\ are equal to the remaining angles,\\ either to either,\\ which the equal sides subtend.\\ Therefore equal are\\ \gr{BAE} and \gr{EGZ}.\\ but greater is\\ \gr{EGD} than \gr{EGZ};\\ therefore greater\\ {}[is] \gr{AGD} than \gr{BAE}.\\ Similarly\\ \gr{BG} having been cut in two,\\ it will be shown that \gr{BGH},\\ which is \gr{AGD},\\ {}[is] greater than \gr{ABG}. } { >Epe`i o>~un >'ish >est`in\\ 'isai e>is`in\\ 'ish >est'in;\\ kat`a koruf`hn g'ar;\\ b'asic >'ara 'ish >est'in,\\ ka`i t`o ABE tr'igwnon\\ t~w| ZEG trig'wnw| >est`in >'ison,\\ ka`i a'isai e>is`in\\ <`ac a'isai pleura`i 'ish >'ara >est`in\\ estin\\ 'ara\\ 'ara trig'wnou\\ mi~ac t~wn pleur~wn\\ prosekblhje'ishc\\ ekt`oc gwn'ia \\ ekat'erac\\ t~wn >ent`oc ka`i >apenant'ion gwni~wn\\ me'izwn >est'in;\\ <'oper >'edei de~ixai. } { Dolay{\i}s{\i}yla, herhangi bir {\"u}{\c c}genin,\\ kenarlar{\i}ndan biri\\ uzat{\i}ld{\i}{\u g}{\i}nda,\\ d{\i}{\c s} a{\c c}{\i}\\ her bir\\ i{\c c} ve kar{\c s}{\i}t a{\c c}{\i}dan\\ b{\"u}y{\"u}kt{\"u}r;\\ \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-2.001875)(4.8603125,2.001875) \psline[linewidth=0.04cm](1.8540626,1.618125)(0.2540625,-0.981875) \psline[linewidth=0.04cm](0.2540625,-0.981875)(4.6740627,-0.981875) \psline[linewidth=0.04cm](1.8540626,1.598125)(3.5340624,-1.861875) \psline[linewidth=0.04cm](0.2540625,-0.981875)(4.4940624,1.618125) \psline[linewidth=0.04cm](4.4740624,1.598125)(3.1340625,-0.981875) \usefont{T1}{ptm}{m}{n} \rput(1.980625,1.823125){\gr A} \usefont{T1}{ptm}{m}{n} \rput(0.10125,-1.156875){\gr B} \usefont{T1}{ptm}{m}{n} \rput(2.9551563,-1.196875){\gr G} \usefont{T1}{ptm}{m}{n} \rput(4.6940627,-1.196875){\gr D} \usefont{T1}{ptm}{m}{n} \rput(2.5560937,0.723125){\gr E} \usefont{T1}{ptm}{m}{n} \rput(4.6767187,1.663125){\gr Z} \usefont{T1}{ptm}{m}{n} \rput(3.7571876,-1.856875){\gr H} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.17 \parsen{ Two angles of any triangle\\ are less than two \rgt s\\ ---taken anyhow. } { Pant`ovc trig'wnou aorj~wn >el'asson'ec e>isi\\ p'ant~h| metalamban'omenai. } { Herhangi bir {\"u}{\c c}genin iki a{\c c}{\i}s{\i}\\ k{\"u}{\c c}{\"u}kt{\"u}r iki dik a{\c c}{\i}dan\\ ---nas{\i}l al{\i}n{\i}rsa al{\i}nan. } \parsen{ Let there be\\ a triangle, \gr{ABG}. } { >'Estw\\ tr'igwnon t`o ABG; } { Olsun\\ bir \gr{ABG} {\"u}{\c c}geni. } \parsen{ I say that\\ two angles of triangle \gr{ABG}\\ are less than two \rgt s\\ ---taken anyhow. } { >l'egw, <'oti\\ to~u ABG trig'wnou aorj~wn >el'atton'ec e>isi\\ p'anth| metalamban'omenai. } { {\.I}ddia ediyorum ki\\ \gr{ABG} {\"u}{\c c}geninin iki a{\c c}{\i}s{\i}\\ k{\"u}{\c c}{\"u}kt{\"u}r iki dik a{\c c}{\i}dan\\ ---nas{\i}l al{\i}n{\i}rsa al{\i}ns{\i}n. } \parsen{ For, suppose there has been extended\\ \gr{BG} to \gr D. } { >Ekbebl'hsjw g`ar\\ ep`i t`o D. } { {\c C}{\"u}nk{\"u}, uzat{\i}lm{\i}{\c s} olsun,\\ \gr{BG}, \gr D noktas{\i}na. } \parsen{ And since, of triangle \gr{ABG},\\ \gr{AGD} is an exterior angle,\\ it is greater\\ than the interior and opposite \gr{ABG}.\\ Let \gr{AGB} be added in common;\\ therefore \gr{AGD} and \gr{AGB}\\ are greater than \gr{ABG} and \gr{BGA}.\\ But \gr{AGD} and \gr{AGB}\\ are equal to two \rgt s;\\ therefore \gr{ABG} and \gr{BGA}\\ are less than two \rgt s.\\ Similarly we shall show that\\ also \gr{BAG} and \gr{AGB}\\ are less than two \rgt s,\\ and yet [so are] \gr{GAB} and \gr{ABG}. } { Ka`i >epe`i trig'wnou to~u ABG\\ {}>ekt'oc >esti gwn'ia est`i\\ t~hc >ent`oc ka`i >apenant'ion t~hc 'ara isin.\\ {}>all> aorja~ic >'isai e>is'in;\\ a'ara orj~wn >el'asson'ec e>isin.\\ orj~wn >el'asson'ec e>isi\\ ka`i >'eti a'ara trig'wnou aorj~wn >el'ass\-on'ec e>isi\\ p'ant~h| metalamban'omenai;\\ <'oper >'edei de~ixai. } { Dolay{\i}s{\i}yla herhangi bir {\"u}{\c c}genin iki a{\c c}{\i}s{\i}\\ k{\"u}{\c c}{\"u}kt{\"u}r iki dik a{\c c}{\i}dan\\ ---nas{\i}l al{\i}n{\i}rsa al{\i}ns{\i}n;\\ \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.2645313)(5.0434375,1.2645313) \psline[linewidth=0.04cm](1.0571876,-0.89921874)(4.6771874,-0.89921874) \psline[linewidth=0.04cm](1.0771875,-0.89921874)(0.2771875,0.9007813) \psline[linewidth=0.04cm](0.2771875,0.9007813)(3.0571876,-0.89921874) \usefont{T1}{ptm}{m}{n} \rput(0.12375,1.0857812){\gr A} \usefont{T1}{ptm}{m}{n} \rput(0.924375,-1.1142187){\gr B} \usefont{T1}{ptm}{m}{n} \rput(3.0582812,-1.1142187){\gr G} \usefont{T1}{ptm}{m}{n} \rput(4.8771877,-0.9142187){\gr D} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.18 \parsen{ Of any triangle,\\ the greater side\\ subtends the greater angle.\footnotemark } { Pant`oc trig'wnou\\ 'Estw g`ar\\ tr'igwnon t`o ABG\\ me'izona >'eqon t`hn AG pleur`an t~hc AB; } { {\c C}{\"u}nk{\"u}, verilmi{\c s} olsun\\ bir \gr{ABG} {\"u}{\c c}geni,\\ \gr{AG} kenar{\i} daha b{\"u}y{\"u}k olan, \gr{AB} kenar{\i}ndan. } \parsen{ I say that\\ also angle \gr{ABG}\\ is greater than \gr{BGA}. } { l'egw, <'oti\\ ka`i gwn'ia est`i t~hc Epe`i g`ar me'izwn >est`in 'ish\\ epeze'uqjw epe`i trig'wnou to~u BGD\\ {}>ekt'oc >esti gwn'ia est`i\\ t~hc >ent`oc ka`i >apenant'ion t~hc 'ish d`e epe`i ka`i pleur`a estin >'ish;\\ me'izwn >'ara\\ ka`i 'ara\\ est`i\\ t~hc 'ara trig'wnou\\ 'edei de~ixai. } { Dolay{\i}s{\i}yla, herhangi bir {\"u}{\c c}gende\\ daha b{\"u}y{\"u}k bir kenar,\\ daha b{\"u}y{\"u}k bir a{\c c}{\i}y{\i} kar{\c s}{\i}lar;\\ \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.1345313)(5.5321875,1.1345313) \psline[linewidth=0.04cm](5.2771873,-0.80921876)(1.0571876,-0.80921876) \psline[linewidth=0.04cm](1.0571876,-0.80921876)(0.2771875,0.79078126) \psline[linewidth=0.04cm](0.2971875,0.7707813)(5.2571874,-0.80921876) \psline[linewidth=0.04cm](1.0571876,-0.7892187)(2.0571876,0.19078125) \usefont{T1}{ptm}{m}{n} \rput(0.12375,0.9557812){\gr A} \usefont{T1}{ptm}{m}{n} \rput(0.864375,-0.96421874){\gr B} \usefont{T1}{ptm}{m}{n} \rput(5.358281,-0.9842188){\gr G} \usefont{T1}{ptm}{m}{n} \rput(2.2171874,0.37578124){\gr D} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.19 \parsen{ Of any triangle,\\ under the greater angle\\ the greater side subtends.\footnotemark } { Pant`oc trig'wnou\\ 'Estw\\ tr'igwnon t`o ABG\\ me'izona >'eqon t`hn est'in. } { {\.I}ddia ediyorum ki\\ \gr{AG} kenar{\i} da\\ daha b{\"u}y{\"u}kt{\"u}r \gr{AB} kenar{\i}ndan. } \parsen{ For if not,\\ either \gr{AG} is equal to \gr{AB}\\ or less;\\ {}[but] \gr{AG} is not equal to \gr{AB};\\ for [if it were],\\ also \gr{ABG} would be\footnotemark\ equal to \gr{AGB};\\ but it is not;\\ therefore \gr{AG} is not equal to \gr{AB}.\\ Nor is \gr{AG} less than \gr{AB};\\ for [if it were],\\ also angle \gr{ABG} would be [less]\\ than \gr{AGB};\\ but it is not;\\ therefore \gr{AG} is not less than \gr{AB}.\\ And it was shown that\\ it is not equal.\\ Therefore \gr{AG} is greater than \gr{AB}. } { E>i g`ar m'h,\\ {}>'htoi >'ish >est`in `h >el'asswn;\\ {}>'ish m`en o>~un o>uk >'estin 'ish g`ar >`an\\ {}>~hn ka`i gwn'ia uk >'esti d'e;\\ o>uk >'ara >'ish >est`in ud`e m`hn >el'asswn >est`in el'asswn g`ar\\ {}>`an >~hn ka`i gwn'ia uk >'esti d'e;\\ o>uk >'ara >el'asswn >est`in ede'iqjh d'e, <'oti\\ o>ud`e >'ish >est'in.\\ me'izwn >'ara >est`in 'ara trig'wnou\\ 'edei de~ixai. } { Dolay{\i}s{\i}yla, herhangi bir {\"u}{\c c}gende,\\ daha b{\"u}y{\"u}k bir a{\c c}{\i},\\ daha b{\"u}y{\"u}k bir kenarca kar{\c s}{\i}lan{\i}r;\\ \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.8145312)(2.17,1.8145312) \psline[linewidth=0.04cm](1.8140625,1.4907813)(1.8140625,-1.5092187) \psline[linewidth=0.04cm](1.8140625,-1.5092187)(0.2740625,0.45078126) \psline[linewidth=0.04cm](0.2740625,0.45078126)(1.8140625,1.4907813) \usefont{T1}{ptm}{m}{n} \rput(2.000625,1.6357813){\gr A} \usefont{T1}{ptm}{m}{n} \rput(0.10125,0.41578126){\gr B} \usefont{T1}{ptm}{m}{n} \rput(1.9751563,-1.6642188){\gr G} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.20 \parsen{ Two sides of any triangle\\ are greater than the remaining one\\ ---taken anyhow. } { Pant`oc trig'wnou aisi\\ p'anth| metalamban'omenai. } { Herhangi bir {\"u}{\c c}genin iki kenar{\i}\\ daha b{\"u}y{\"u}kt{\"u}r geriye kalandan\\ ---nas{\i}l se{\c c}ilirse se{\c c}ilsin. } \parsen{ For, let there be\\ a triangle, \gr{ABG}. } { >'Estw g`ar\\ tr'igwnon t`o ABG; } { {\c C}{\"u}nk{\"u} verilmi{\c s} olsun\\ bir \gr{ABG} {\"u}{\c c}geni. } \parsen{ I say that\\ two sides of triangle \gr{ABG}\\ are greater than the remaining one,\\ ---taken anyhow,\\ \gr{BA} and \gr{AG}, than \gr{BG},\\ \gr{AB} and \gr{BG}, than \gr{AG},\\ \gr{BG} and \gr{GA}, than \gr{AB}. } { l'egw, <'oti\\ to~u ABG trig'wnou aisi\\ p'anth| metalamban'omenai,\\ aep`i t`o D shme~ion,\\ ka`i ke'isjw\\ t~h| GA >'ish epeze'uqjw\\ Epe`i o>~un >'ish >est`in 'ish >est`i ka`i\\ gwn'ia 'ara epe`i tr'igwn'on >esti t`o DGB\\ me'izona >'eqon t`hn 'ara t~hc BG >esti me'izwn.\\ {}>'ish d`e 'ara aisin,\\ a'ara trig'wnou aisi\\ p'anth| metalamban'omenai;\\ <'oper >'edei de~ixai. } { Dolay{\i}s{\i}yla, herhangi bir {\"u}{\c c}genin iki kenar{\i}\\ daha b{\"u}y{\"u}kt{\"u}r geriye kalandan\\ ---nas{\i}l se{\c c}ilirse se{\c c}ilsin;\\ \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.5592188)(3.2903125,1.5592188) \usefont{T1}{ptm}{m}{n} \rput(0.09796875,1.3857813){\gr Z} \usefont{T1}{ptm}{m}{n} \rput(1.581875,-0.31421876){\gr A} \usefont{T1}{ptm}{m}{n} \rput(1.1225,-1.4142188){\gr B} \usefont{T1}{ptm}{m}{n} \rput(2.6553125,0.68578124){\gr D} \usefont{T1}{ptm}{m}{n} \rput(3.1164062,-1.3942188){\gr G} \psline[linewidth=0.04cm](1.7953125,-0.41921875)(2.9753125,-1.1992188) \psline[linewidth=0.04](2.4953125,0.66078126)(1.2953125,-1.2192187)(2.9953125,-1.2192187)(2.4753125,0.6407812) \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.21 \parsen{ If, of a triangle,\\ on one of the sides,\\ from its extremities,\\ two \strgt s\\ be constructed within,\footnotemark\\ the constructed [\strgt s],\\ than the remaining two sides of the triangle\\ will be less,\\ but will contain the a greater angle. } { >E`an trig'wnou\\ {}>ep`i mi~ac t~wn pleur~wn\\ {}>ap`o t~wn per'atwn\\ d'uo e>uje~iai\\ {}>ent`oc sustaj~wsin,\\ ael'attonec m`en >'esontai,\\ me'izona d`e gwn'ian peri'exousin. } { E{\u g}er bir {\"u}{\c c}gende,\\ kenarlardan birinin\\ u{\c c}lar{\i}ndan,\\ iki do{\u g}ru\\ i{\c c}eride in{\c s}a edilirse,\\ in{\c s}a edilen do{\u g}rular,\\ {\"u}{\c c}genin geriye kalan iki kenar{\i}ndan\\ daha k{\"u}{\c c}{\"u}k olacak,\\ ama daha b{\"u}y{\"u}k bir a{\c c}{\i}y{\i} i{\c c}erecekler. } \myfntext{Here the Greek verb, \gr{sun'isthmi}, is the same one used in I.1 for the contruction of a \emph{triangle} on a given straight line. Is it supposed to be obvious to the reader, even \emph{without} a diagram, that now the two constructed straight lines are supposed to meet at a point? See also I.2 and note.} \parsen{ For, of a triangle, \gr{ABG},\\ on one of the sides, \gr{BG},\\ from its extremities, \gr B and \gr G,\\ suppose two \strgt s have been constructed within,\\ \gr{BD} and \gr{DG}. } { Trig'wnou g`ar to~u ABG\\ {}>ep`i mi~ac t~wn pleur~wn t~hc BG\\ {}>ap`o t~wn per'atwn t~wn B, G\\ d'uo e>uje~iai >ent`oc sunest'atwsan\\ ael'assonec m'en e>isin,\\ me'izona d`e gwn'ian peri'eqousi\\ t`hn ep`i t`o E. } { {\c C}{\"u}nk{\"u}, \gr{BD} {\c c}izilmi{\c s} olsun \gr E noktas{\i}na do{\u g}ru. } \parsen{ And since, of any triangle,\\ two sides than the remaining one\\ are greater,\\ of the triangle \gr{ABE},\\ the two sides \gr{AB} and \gr{AE}\\ are greater than \gr{BE};\\ suppose has been added in common\\ \gr{EG};\\ therefore \gr{BA} and \gr{AG} than \gr{BE} and \gr{EG}\\ are greater.\\ Moreover,\\ since, of the triangle \gr{GED},\\ the two sides \gr{GE} and \gr{ED}\\ are greater than \gr{GD},\\ suppose has been added in common\\ \gr{DB};\\ therefore \gr{GE} and \gr{EB} than \gr{GD} and \gr{DB}\\ are greater.\\ But than \gr{BE} and \gr{EG}\\ \gr{BA} and \gr{AG} were shown greater;\\ therefore by much\\ \gr{BA} and \gr{AG} than \gr{BD} and \gr{DG}\\ are greater. } { ka`i >epe`i pant`oc trig'wnou\\ aisin,\\ to~u ABE >'ara trig'wnou\\ aisin;\\ koin`h proske'isjw\\ 'ara BA, AG t~wn BE, EG\\ me'izon'ec e>isin.\\ p'alin,\\ {}>epe`i to~u GED trig'wnou\\ aisin,\\ koin`h proske'isjw\\ 'ara t~wn GD, DB\\ me'izon'ec e>isin.\\ {}>all`a t~wn BE, EG\\ me'izonec >ede'iqjhsan a'ara\\ aisin. } { Ve herhangi bir {\"u}{\c c}genin\\ iki kenar{\i}, geriye kalandan\\ b{\"u}y{\"u}k oldu{\u g}undan,\\ \gr{ABE} {\"u}{\c c}geninin,\\ iki kenar{\i}, \gr{AB} ve \gr{AE}\\ b{\"u}y{\"u}kt{\"u}r \gr{BE} kenar{\i}ndan;\\ ortak olarak eklenmi{\c s} olsun\\ \gr{EG};\\ dolay{\i}s{\i}yla \gr{BA} ve \gr{AG}, \gr{BE} ve \gr{EG} kenarlar{\i}ndan\\ b{\"u}y{\"u}kt{\"u}rler.\\ Dahas{\i},\\ \gr{GED} {\"u}{\c c}geninin,\\ iki kenarlar{\i}, \gr{GE} ve \gr{ED}\\ b{\"u}y{\"u}kt{\"u}r \gr{GD} kenar{\i}ndan,\\ ortak olarak eklenmi{\c s} olsun\\ \gr{DB};\\ dolay{\i}s{\i}yla \gr{GE} ve \gr{EB}, \gr{GD} ve \gr{DB} kenarlar{\i}ndan\\ b{\"u}y{\"u}kt{\"u}rler.\\ Ama \gr{BE} ve \gr{EG} kenarlar{\i}ndan\\ \gr{BA} ve \gr{AG} kenarlar{\i}n{\i}n g{\"o}sterilmi{\c s}ti b{\"u}y{\"u}kl{\"u}{\u g}{\"u};\\ dolay{\i}s{\i}yla {\c c}ok daha b{\"u}y{\"u}kt{\"u}r\\ \gr{BA} ve \gr{AG}, \gr{BD} ve \gr{DG} kenarlar{\i}ndan.\\ } \parsen{ Again,\\ since of any triangle\\ the external angle\\ than the interior and opposite angle\\ is greater,\\ therefore, of the triangle \gr{GDE}\\ the exterior angle \gr{BDG}\\ is greater than \gr{GED}.\\ For the same [reason] again,\\ of the triangle \gr{ABE},\\ the exterior angle \gr{GEB}\\ is greater than \gr{BAG}.\\ But than \gr{GEB}\\ \gr{BDG} was shown greater;\\ therefore by much\\ \gr{BDG} is greater than \gr{BAG}.} {P'alin,\\ {}>epe`i pant`oc trig'wnou\\ ekt`oc gwn'ia\\ t~hc >ent`oc ka`i >apenant'ion\\ me'izwn >est'in,\\ to~u GDE >'ara trig'wnou\\ ekt`oc gwn'ia est`i t~hc ut`a to'inun\\ ka`i to~u ABE trig'wnou\\ ekt`oc gwn'ia est`i t~hc all`a t~hc ede'iqjh 'ara\\ est`i t~hc E`an >'ara trig'wnou\\ {}>ep`i mi~ac t~wn pleur~wn\\ {}>ap`o t~wn per'atwn\\ d'uo e>uje~iai\\ {}>ent`oc sustaj~wsin,\\ ael'attonec m'en e>isin,\\ me'izona d`e gwn'ian peri'eqousin;\\ <'oper >'edei de~ixai. } { E{\u g}er, dolay{\i}s{\i}yla, bir {\"u}{\c c}genin,\\ kenarlardan birinin\\ u{\c c}lar{\i}ndan,\\ iki do{\u g}ru\\ i{\c c}eride in{\c s}a edilirse,\\ in{\c s}a edilen do{\u g}rular,\\ {\"u}{\c c}genin geriye kalan iki kenar{\i}ndan\\ daha k{\"u}{\c c}{\"u}k olacak,\\ ama daha b{\"u}y{\"u}k bir a{\c c}{\i}y{\i} i{\c c}erecekler;\\ \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.651875)(4.7290626,1.651875) \psline[linewidth=0.04cm](0.1940625,-1.291875)(4.3940625,-1.271875) \psline[linewidth=0.04cm](4.3940625,-1.271875)(1.0140625,1.308125) \psline[linewidth=0.04cm](1.0140625,1.308125)(0.2140625,-1.271875) \psline[linewidth=0.04cm](0.1940625,-1.291875)(1.8140625,0.708125) \psline[linewidth=0.04cm](4.3740625,-1.251875)(1.3540626,0.148125) \usefont{T1}{ptm}{m}{n} \rput(1.040625,1.473125){\gr A} \usefont{T1}{ptm}{m}{n} \rput(0.10125,-1.506875){\gr B} \usefont{T1}{ptm}{m}{n} \rput(4.555156,-1.386875){\gr G} \usefont{T1}{ptm}{m}{n} \rput(1.1740625,0.253125){\gr D} \usefont{T1}{ptm}{m}{n} \rput(1.9760938,0.853125){\gr E} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.22 \parsen{ From three \strgt s,\\ which are equal\\ to three given [\strgt s],\\ a triangle to be constructed;\\ and it is necessary\\ for two than the remaining one\\ to be greater\\ {}[because of any triangle,\\ two sides\\ are\footnotemark\ greater than the remaining one\\ taken anyhow].} { >Ek tri~wn e>ujei~wn,\\ a<'i e>isin >'isai\\ tris`i ta~ic doje'isaic [e>uje'iaic],\\ tr'igwnon sust'hsasjai;\\ de~i d`e\footnotemark\\ t`ac d'uo t~hc loip~hc\\ me'izonac e>~inai\\ p'anth| metalambanom'enac\\ {}[di`a t`o ka`i pant`oc trig'wnou\\ t`ac d'uo pleur`ac\\ t~hc loip~hc me'izonac e>~inai\\ p'anth| metalambanom'enac]. } { {\"U}{\c c} do{\u g}rudan,\\ e{\c s}it olan\\ verilmi{\c s} {\"u}{\c c} do{\u g}ruya,\\ bir {\"u}{\c c}gen olu{\c s}turulmas{\i};\\ ve gereklidir\\ ikisinin, kalandan\\ daha b{\"u}y{\"u}k olmas{\i}\\ ({\c c}{\"u}nk{\"u} herhangi bir {\"u}{\c c}genin,\\ iki kenar{\i}\\ b{\"u}y{\"u}kt{\"u}r geriye kalandan\\ nas{\i}l se{\c c}ilirse se{\c c}ilsin).} \myfntext{In the Greek this is the infinitive \gr{e>~inai} `to be', as in the previous clause.} \myfntext{According to Heiberg, the manuscripts have \gr{de~i d'h} here, as at the beginnings of specifications (see \S\ref{sect:analysis}); but Proclus and Eutocius have \gr{de~i d'e} in their commentaries.} \parsen{ Let be\\ the given three \strgt s\\ \gr A, \gr B, and \gr G,\\ of which two than the remaining one\\ are greater,\\ taken anyhow,\\ \gr A and \gr B than \gr G,\\ \gr A and \gr G than \gr B,\\ and \gr B and \gr G than \gr A. } { >'Estwsan\\ auje~iai\\ a'estwsan\\ p'anth| metalamban'omenai,\\ a'eti aek t~wn >'iswn ta~ic A, B, G\\ tr'igwnon sust'hsasjai. } { Gereklidir\\ \gr A, \gr B ve \gr G do{\u g}rular{\i}na e{\c s}it olanlardan\\ bir {\"u}{\c c}genin in{\c s}a edilmesi. } \parsen{ Suppose there is laid out\\ some straight line, \gr{DE},\\ bounded at \gr D,\\ but unbounded at \gr E,\\ and there is laid down\\ \gr{DZ} equal to \gr A,\\ \gr{ZH} equal to \gr B,\\ and \gr{HJ} equal to \gr G;\\ and to center \gr Z\\ at distance \gr{ZD}\\ a circle has been drawn, \gr{DKL};\\ moreover,\\ to center \gr H,\\ at distance \gr{HJ},\\ circle \gr{KLJ} has been drawn,\\ and \gr{KZ} and \gr{KH} have been joined. } { >Ekke'isjw\\ tic e>uje~ia 'apeiroc d`e kat`a t`o E,\\ ka`i ke'isjw\\ t~h| m`en A >'ish 'ish 'ish epeze'uqjwsan aek tri~wn e>ujei~wn\\ t~wn >'iswn ta~ic A, B, G\\ tr'igwnon sun'estatai t`o KZH. } { {\.I}ddia ediyorum ki\\ {\"u}{\c c} do{\u g}rudan\\ \gr A, \gr B ve \gr G do{\u g}rular{\i}na e{\c s}it olan\\ bir \gr{KZH} {\"u}{\c c}geni in{\c s}a edilmi{\c s}tir. } \parsen{ For, since the point \gr Z\\ is the center of circle \gr{DKL},\\ \gr{ZD} is equal to \gr{ZK};\\ but \gr{ZD} is equal to \gr A.\\ And \gr{KZ} is therefore equal to \gr A.\\ Moreover,\\ since the point \gr H\\ is the center of circle \gr{LKJ},\\ \gr{HJ} is equal to \gr{HK};\\ but \gr{HJ} is equal to \gr G;\\ and \gr{KH} is therefore equal to \gr G.\\ and \gr{ZH} is equal to \gr B;\\ therefore the three \strgt s,\\ \gr{KZ}, \gr{ZH}, and \gr{HK}\\ are equal to the three, \gr A, \gr B, and \gr G. } { >Epe`i g`ar t`o Z shme~ion\\ k'entron >est`i to~u DKL k'uklou,\\ {}>'ish >est`in all`a estin >'ish.\\ ka`i 'ara t~h| A >estin >'ish.\\ p'alin,\\ {}>epe`i t`o H shme~ion\\ k'entron >est`i to~u LKJ k'uklou,\\ {}>'ish >est`in all`a estin >'ish;\\ ka`i 'ara t~h| G >estin >'ish.\\ {}>est`i d`e ka`i 'ish;\\ a'ara e>uje~iai\\ a'isai e>is'in. } { {\c C}{\"u}nk{\"u} merkezi oldu{\u g}undan \gr Z noktas{\i},\\ \gr{DKL} {\c c}emberinin,\\ \gr{ZD} e{\c s}ittir \gr{ZK} do{\u g}rusuna;\\ ama \gr{ZD} e{\c s}ittir \gr A do{\u g}rusuna.\\ Ve \gr{KZ} dolay{\i}s{\i}yla \gr A do{\u g}rusuna e{\c s}ittir.\\ Dahas{\i},\\ merkezi oldu{\u g}undan \gr H noktas{\i}\\ \gr{LKJ} {\c c}emberinin,\\ \gr{HJ} e{\c s}ittir \gr{HK} do{\u g}rusuna;\\ ama \gr{HJ} e{\c s}ittir \gr G do{\u g}rusuna;\\ ve \gr{KH} dolay{\i}s{\i}yla \gr G do{\u g}rusuna e{\c s}ittir.\\ ve \gr{ZH} e{\c s}ittir \gr B do{\u g}rusuna;\\ dolay{\i}s{\i}yla {\"u}{\c c} do{\u g}ru,\\ \gr{KZ}, \gr{ZH} ve \gr{HK}\\ e{\c s}ittirler \gr A, \gr B ve \gr G {\"u}{\c c}l{\"u}s{\"u}ne. } \parsen{ Therefore, from the three \strgt s\\ \gr{KZ}, \gr{ZH}, and \gr{HK},\\ which are equal\\ to the three given \strgt s\\ \gr A, \gr B, and \gr G,\\ a triangle has been constructed, \gr{KZH};\\ \myqed } { >Ek tri~wn >'ara e>ujei~wn\\ t~wn KZ, ZH, HK,\\ a<'i e>isin >'isai\\ tris`i ta~ic doje'isaic e>uje'iaic\\ ta~ic A, B, G,\\ tr'igwnon sun'estatai t`o KZH;\\ <'oper >'edei poi~hsai. } { Dolay{\i}s{\i}yla, {\"u}{\c c} do{\u g}rudan;\\ \gr{KZ}, \gr{ZH} ve \gr{HK},\\ e{\c s}it olan\\ verilmi{\c s} {\"u}{\c c} do{\u g}ruya\\ \gr A, \gr B ve \gr G,\\ bir \gr{KZH} {\"u}{\c c}geni in{\c s}a edilmi{\c s}tir;\\ \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.91)(8.07375,1.91) \pscircle[linewidth=0.04,dimen=outer](2.16375,0.0){1.91} \psline[linewidth=0.04cm](0.29375,-0.01)(5.67375,-0.01) \psline[linewidth=0.04cm](2.15375,-0.01)(3.67375,1.15) \psline[linewidth=0.04cm](3.65375,1.13)(3.65375,-0.01) \pscircle[linewidth=0.04,dimen=outer](3.64375,0.08){1.11} \psline[linewidth=0.04cm](6.45375,-0.01)(7.67375,-0.01) \psline[linewidth=0.04cm](6.45375,0.39)(8.05375,0.39) \psline[linewidth=0.04cm](6.45375,-0.41)(7.37375,-0.41) \usefont{T1}{ptm}{m}{n} \rput(6.3203125,0.375){\gr A} \usefont{T1}{ptm}{m}{n} \rput(6.3009377,-0.045){\gr B} \usefont{T1}{ptm}{m}{n} \rput(6.2948437,-0.425){\gr G} \usefont{T1}{ptm}{m}{n} \rput(0.11375,-0.125){\gr D} \usefont{T1}{ptm}{m}{n} \rput(5.6557813,-0.265){\gr E} \usefont{T1}{ptm}{m}{n} \rput(2.1364062,-0.225){\gr Z} \usefont{T1}{ptm}{m}{n} \rput(3.656875,-0.245){\gr H} \usefont{T1}{ptm}{m}{n} \rput(3.7590625,1.375){\gr K} \usefont{T1}{ptm}{m}{n} \rput(5.0023437,0.215){\gr J} \usefont{T1}{ptm}{m}{n} \rput(3.89625,-1.225){\gr L} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.23 \parsen{ At the given \strgt,\\ and at the given point on it,\\ equal to the given rectilineal angle,\\ a rectilineal angle to be constructed. } { Pr`oc t~h| doje'ish| e>uje'ia|\\ ka`i t~w| pr`oc a>ut~h| shme'iw|\\ t~h| doje'ish| gwn'ia| e>ujugr'ammw| >'ishn\\ gwn'ian e>uj'ugrammon sust'hsasjai. } { Verilmi{\c s} bir do{\u g}ruda,\\ ve {\"u}zerinde verilmi{\c s} noktada,\\ verilmi{\c s} d{\"u}zkenar a{\c c}{\i}ya e{\c s}it olan,\\ bir d{\"u}zkenar a{\c c}{\i} in{\c s}a edilmesi. } \parsen{ Let be\\ the given \strgt\ \gr{AB},\\ the point on it, \gr A,\\ the given rectilineal angle,\\ \gr{DGE}. } { >'Estw\\ uje~ia ut~h| shme~ion t`o A,\\ uj'ugrammoc\\ uje'ia| t~h| AB\\ ka`i t~w| pr`oc a>ut~h| shme'iw| t~w| A\\ t~h| doje'ish| gwn'ia| e>ujugr'ammw|\\ t~h| 'ishn\\ gwn'ian e>uj'ugrammon\\ sust'hsasjai. } { Gereklidir {\c s}imdi,\\ verilmi{\c s} \gr{AB} do{\u g}rusunda,\\ ve {\"u}zerindeki \gr A noktas{\i}nda,\\ verilmi{\c s} d{\"u}zkenar\\ \gr{DGE} a{\c c}{\i}s{\i}na\\ e{\c s}it,\\ bir d{\"u}zkenar a{\c c}{\i}n{\i}n\\ in{\c s}a edilmesi. } \parsen{ Suppose there have been chosen\\ on either of \gr{GD} and \gr{GE}\\ random points \gr D and \gr E,\\ and \gr{DE} has been joined,\\ and from three \strgt s,\\ which are equal to the three,\\ \gr{GD}, \gr{DE}, and \gr{GE},\\ triangle \gr{AZH} has been constructed,\\ so that equal are\\ \gr{GD} to \gr{AZ},\\ \gr{GE} to \gr{AH},\\ and \gr{DE} to \gr{ZH}. } { E>il'hfjw\\ {}>ef> epeze'uqjw ek tri~wn e>ujei~wn,\\ a<'i e>isin >'isai tris`i\\ ta~ic GD, DE, GE,\\ tr'igwnon sunest'atw t`o AZH,\\ <'wste >'ishn e>~inai\\ t`hn m`en GD t~h| AZ,\\ t`hn d`e GE t~h| AH,\\ ka`i >'eti t`hn DE t~h| ZH. } { Se{\c c}ilmi{\c s} olsun\\ \gr{GD} ve \gr{GE} do{\u g}rular{\i}n{\i}n her birinden\\ rastgele \gr D ve \gr E noktalar{\i},\\ ve \gr{DE} birle{\c s}tirilmi{\c s} olsun,\\ ve {\"u}{\c c} do{\u g}rudan,\\ e{\c s}it olan verilmi{\c s} {\"u}{\c c}\\ \gr{GD}, \gr{DE} ve \gr{GE} do{\u g}rular{\i}na,\\ bir \gr{AZH} {\"u}{\c c}gen in{\c s}a edilmi{\c s} olsun\\ {\"o}yle ki, e{\c s}it olsun\\ \gr{GD}, \gr{AZ} do{\u g}rusuna,\\ \gr{GE}, \gr{AH} do{\u g}rusuna,\ ve \gr{DE}, \gr{ZH} do{\u g}rusuna. } \parsen{ Since then the two, \gr{DG} and \gr{GE},\\ are equal to the two, \gr{ZA} and \gr{AH},\\ either to either,\\ and the base \gr{DE} to the base \gr{ZH}\\ is equal,\\ therefore the angle \gr{DGE}\\ is equal to \gr{ZAH}. } { >Epe`i o>~un d'uo a'isai e>is`in\\ 'ish,\\ gwn'ia >'ara estin >'ish. } { O zaman \gr{DG} ve \gr{GE} ikilisi,\\ e{\c s}it oldu{\u g}undan \gr{ZA} ve \gr{AH} ikilisinin,\\ her biri birine,\\ ve \gr{DE} taban{\i}, \gr{ZH} taban{\i}na\\ e{\c s}it,\\ dolay{\i}s{\i}yla \gr{DGE} a{\c c}{\i}s{\i}\\ e{\c s}ittir \gr{ZAH} a{\c c}{\i}s{\i}na. } \parsen{ Therefore, on the given \strgt,\\ \gr{AB},\\ and at the point \gr A on it,\\ equal to the given rectilineal angle, \gr{DGE},\\ the rectilineal angle \gr{ZAH} has been constructed;\\ \myqef } { Pr`oc >'ara t~h| doje'ish| e>uje'ia|\\ t~h| AB\\ ka`i t~w| pr`oc a>ut~h| shme'iw| t~w| A t~h|\\ doje'ish| gwn'ia| e>ujugr'ammw| t~h| 'ish\\ gwn'ia e>uj'ugrammoc sun'estatai 'edei poi~hsai.\\ } { Dolay{\i}s{\i}yla, \\ \gr{AB} do{\u g}rusunda,\\ ve {\"u}zerindeki \gr A noktas{\i}nda,\\ verilen d{\"u}zkenar \gr{DGE} a{\c c}{\i}s{\i}na e{\c s}it,\\ \gr{ZAH} d{\"u}zkenar a{\c c}{\i}s{\i} in{\c s}a edilmi{\c s}tir;\\ \ozqef } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.4192188)(7.5771875,1.4192188) \psline[linewidth=0.04cm](3.8771875,-1.0792187)(0.2771875,-1.0792187) \psline[linewidth=0.04cm](0.2771875,-1.0792187)(2.0971875,1.1007812) \psline[linewidth=0.04cm](2.0971875,1.1007812)(2.9371874,-1.0792187) \psline[linewidth=0.04cm](7.4571877,1.2207812)(5.0971875,0.04078125) \psline[linewidth=0.04cm](5.0971875,0.04078125)(7.5571876,-1.1392188) \psline[linewidth=0.04cm](6.9971876,1.0007813)(6.5371876,-0.6592187) \usefont{T1}{ptm}{m}{n} \rput(0.12375,-1.1742188){\gr A} \usefont{T1}{ptm}{m}{n} \rput(4.124375,-1.0342188){\gr B} \usefont{T1}{ptm}{m}{n} \rput(4.898281,0.06578125){\gr G} \usefont{T1}{ptm}{m}{n} \rput(6.8971877,1.1857812){\gr D} \usefont{T1}{ptm}{m}{n} \rput(6.459219,-0.87421876){\gr E} \usefont{T1}{ptm}{m}{n} \rput(2.2398438,1.2457813){\gr Z} \usefont{T1}{ptm}{m}{n} \rput(2.8603125,-1.2742188){\gr H} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.24 \parsen{ If two triangles\\ two sides\\ to two sides\\ have equal\\ either to either,\\ but angle\\ than angle\\ have greater,\\ {}[namely] that by the equal sides\\ contained,\\ also base\\ than base\\ they will have greater. } { >E`an d'uo tr'igwna\\ t`ac d'uo pleur`ac\\ {}[ta~ic] d'uo pleura~ic\\ {}>'isac >'eqh|\\ 'eqh|\\ t`hn 'iswn e>ujei~wn\\ perieqom'enhn,\\ ka`i t`hn b'asin\\ t~hc b'asewc\\ me'izona <'exei. } { E{\u g}er iki {\"u}{\c c}genin\\ (birinin) iki kenar{\i}\\ (di{\u g}erinin) iki kenar{\i}na\\ e{\c s}itse, \\ her biri birine,\\ ama a{\c c}{\i}s{\i}\\ a{\c c}{\i}s{\i}ndan\\ b{\"u}y{\"u}kse,\\ {}[yani] e{\c s}it kenarlarca\\ i{\c c}erilen(ler),\\ taban{\i} da\\ taban{\i}ndan\\ b{\"u}y{\"u}k olacak. } \parsen{ Let there be\\ two triangles, \gr{ABG} and \gr{DEZ},\\ ---two sides, \gr{AB} and \gr{AG},\\ to two sides, \gr{DE} and \gr{DZ},\\ having equal,\\ either to either,\\ \gr{AB} to \gr{DE},\\ and \gr{AG} to \gr{DZ},\\ ---and the angle at \gr A,\\ than the angle at \gr D,\\ let it be greater. } { >'Estw\\ d'uo tr'igwna t`a ABG, DEZ\\ t`ac d'uo pleur`ac t`ac AB, AG\\ ta~ic d'uo pleura~ic ta~ic DE, DZ\\ {}>'isac >'eqonta\\ 'estw; } { Verilmi{\c s} olsun\\ iki \gr{ABG} ve \gr{DEZ} {\"u}{\c c}geni, \\ --- iki \gr{AB} ve \gr{AG} kenar{\i},\\ iki \gr{DE} ve \gr{DZ} kenar{\i}na,\\ e{\c s}it olan,\\ her biri birine,\\ \gr{AB}, \gr{DE} kenar{\i}na,\\ ve \gr{AG}, \gr{DZ} kenar{\i}na,\\ ---ve \gr A noktas{\i}ndaki a{\c c}{\i}s{\i},\\ \gr D doktas{\i}ndakinden,\\ b{\"u}y{\"u}k olsun. } \parsen{ I say that\\ also the base \gr{BG}\\ than the base \gr{EZ}\\ is greater. } { l'egw, <'oti\\ ka`i b'asic est'in. } { {\.I}ddia ediyorum ki\\ \gr{BG} taban{\i} da\\ \gr{EZ} taban{\i}ndan\\ b{\"u}y{\"u}kt{\"u}r. } \parsen{ For since [it] is greater,\\ {}[namely] angle \gr{BAG}\\ than angle \gr{EDZ},\\ suppose has been constructed\\ on the \strgt, \gr{DE},\\ and at the point \gr D on it,\\ equal to angle \gr{BAG},\\ \gr{EDH},\\ and suppose is laid down,\\ to either of \gr{AG} and \gr{DZ} equal,\\ \gr{DH},\\ and suppose have been joined\\ \gr{EH} and \gr{ZH}. } { >Epe`i g`ar me'izwn\\ uje'ia|\\ ka`i t~w| pr`oc a>ut~h| shme'iw| t~w| D\\ t~h| 'ish\\ 'ish\\ epeze'uqjwsan\\ aEpe`i o>~un >'ish >est`in\\ 'isai e>is`in\\ 'ish;\\ b'asic >'ara estin >'ish.\\ p'alin,\\ {}>epe`i >'ish >est`in\\ 'ish >est`i ka`i\\ 'ara\\ 'ara me'izwn >est`in\\ epe`i tr'igwn'on >esti t`o EZH\\ me'izona >'eqon\\ t`hn 'ara ka`i\\ pleur`a 'ish d`e 'ara ka`i E`an >'ara d'uo tr'igwna\\ t`ac d'uo pleur`ac\\ dus`i pleura~ic\\ {}>'isac >'eqh|\\ 'eqh|\\ t`hn 'iswn e>ujei~wn\\ perieqom'enhn,\\ ka`i t`hn b'asin\\ t~hc b'asewc\\ me'izona <'exei;\\ <'oper >'edei de~ixai. } { E{\u g}er, dolay{\i}s{\i}yla, iki {\"u}{\c c}genin\\ (birinin) iki kenar{\i}\\ (di{\u g}erinin) iki kenar{\i}na\\ e{\c s}itse \\ her biri birine,\\ ama a{\c c}{\i}s{\i}\\ a{\c c}{\i}s{\i}ndan\\ b{\"u}y{\"u}kse,\\ {}[yani] e{\c s}it kenarlarca\\ i{\c c}erilen(ler),\\ taban{\i} da\\ taban{\i}ndan\\ b{\"u}y{\"u}k olacak;\\ \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.561875)(5.85875,1.561875) \psline[linewidth=0.04cm](1.4471875,1.198125)(0.2471875,-1.181875) \psline[linewidth=0.04cm](0.2671875,-1.181875)(2.4471874,-0.801875) \psline[linewidth=0.04cm](2.4271874,-0.801875)(1.4471875,1.178125) \psline[linewidth=0.04cm](4.6471877,1.198125)(3.4671874,-0.961875) \psline[linewidth=0.04cm](3.4671874,-0.961875)(5.5671873,-0.721875) \psline[linewidth=0.04cm](5.5671873,-0.721875)(4.6471877,1.198125) \psline[linewidth=0.04cm](3.4871874,-0.961875)(4.1271877,-1.221875) \psline[linewidth=0.04cm](4.1271877,-1.221875)(5.5871873,-0.721875) \psline[linewidth=0.04cm](4.6671877,1.178125)(4.1271877,-1.241875) \usefont{T1}{ptm}{m}{n} \rput(1.59375,1.383125){\gr A} \usefont{T1}{ptm}{m}{n} \rput(2.594375,-0.896875){\gr B} \usefont{T1}{ptm}{m}{n} \rput(0.10828125,-1.336875){\gr G} \usefont{T1}{ptm}{m}{n} \rput(4.7471876,1.383125){\gr D} \usefont{T1}{ptm}{m}{n} \rput(5.709219,-0.756875){\gr E} \usefont{T1}{ptm}{m}{n} \rput(3.3298438,-1.036875){\gr H} \usefont{T1}{ptm}{m}{n} \rput(4.1303124,-1.416875){\gr Z} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.25 \parsen{ If two triangles\\ two sides\\ to two sides\\ have equal,\\ either to either,\\ but base\\ than base\\ have greater,\\ also angle\\ than angle\\ they will have greater\\ ---that by the equal \strgt s\\ contained. } { >E`an d'uo tr'igwna\\ t`ac d'uo pleur`ac\\ dus`i pleura~ic\\ {}>'isac >'eqh|\\ 'eqh|,\\ ka`i t`hn gwn'ian\\ t~hc gwn'iac\\ me'izona <'exei\\ t`hn 'iswn e>ujei~wn\\ perieqom'enhn. } { E{\u g}er iki {\"u}{\c c}genin\\ (birinin) iki kenar{\i}\\ (di{\u g}erinin) iki kenar{\i}na\\ e{\c s}itse \\ her biri birine,\\ ama taban{\i}\\ taban{\i}ndan\\ b{\"u}y{\"u}kse,\\ a{\c c}{\i}s{\i} da\\ a{\c c}{\i}s{\i}ndan\\ b{\"u}y{\"u}k olacak\\ ---(yani) e{\c s}it do{\u g}rularca\\ i{\c c}erilenler. } \parsen{ Let there be\\ two triangles, \gr{ABG} and \gr{DEZ},\\ two sides, \gr{AB} and \gr{AG},\\ to two sides, \gr{DE} and \gr{DZ},\\ having equal,\\ either to either,\\ \gr{AB} to \gr{DE}\\ and \gr{AG} to \gr{DZ};\\ and the base \gr{BG}\\ than the base \gr{EZ}\\ ---let it be greater.\\ I say that\\ also the angle \gr{BAG}\\ than the angle \gr{EDZ}\\ is greater. } { >'Estw\\ d'uo tr'igwna t`a ABG, DEZ\\ t`ac d'uo pleur`ac t`ac AB, AG\\ ta~ic d'uo pleura~ic ta~ic DE, DZ\\ {}>'isac >'eqonta\\ 'estw;\\ l'egw, <'oti\\ ka`i gwn'ia est'in. } { Verilmi{\c s} olsun\\ \gr{ABG} ve \gr{DEZ} {\"u}{\c c}genleri,\\ iki \gr{AB} ve \gr{AG} kenar{\i},\\ iki \gr{DE} ve \gr{DZ} kenar{\i}na,\\ e{\c s}it olan,\\ her biri birine,\\ \gr{AB}, \gr{DE} kenar{\i}na\\ ve \gr{AG}, \gr{DZ} kenar{\i}na;\\ ve \gr{BG} taban{\i}\\ \gr{EZ} taban{\i}ndan\\ ---b{\"u}y{\"u}k olsun.\\ {\.I}ddia ediyorum ki\\ \gr{BAG} a{\c c}{\i}s{\i} da\\ \gr{EDZ} a{\c c}{\i}s{\i}ndan\\ b{\"u}y{\"u}kt{\"u}r. } \parsen{ For if not,\\ {}[it] is either equal to it, or less;\\ but it is not equal\\ ---\gr{BAG} to \gr{EDZ};\\ for if it is equal,\\ also the base \gr{BG} to \gr{EZ};\\ but it is not.\\ Therefore it is not equal,\\ angle \gr{BAG} to \gr{EDZ};\\ neither is it less,\\ \gr{BAG} than \gr{EDZ};\\ for if it is less,\\ also base \gr{BG} than \gr{EZ};\\ but it is not;\\ therefore it is not less,\\ \gr{BAG} than angle \gr{EDZ}.\\ And it was shown that\\ it is not equal;\\ therefore it is greater,\\ \gr{BAG} than \gr{EDZ}. } { E>i g`ar m'h,\\ {}>'htoi >'ish >est`in a>ut~h| >`h >el'asswn;\\ {}>'ish m`en o>~un o>uk >'estin\\ 'ish g`ar >`an >~hn\\ ka`i b'asic uk >'esti d'e.\\ o>uk >'ara >'ish >est`i\\ gwn'ia ud`e m`hn >el'asswn >est`in\\ el'asswn g`ar >`an >~hn\\ ka`i b'asic uk >'esti d'e;\\ o>uk >'ara >el'asswn >est`in\\ ede'iqjh d'e, <'oti\\ o>ud`e >'ish;\\ me'izwn >'ara >est`in\\ E`an >'ara d'uo tr'igwna\\ t`ac d'uo pleur`ac\\ dus`i pleura~ic\\ {}>'isac >'eqh|\\ 'eqh|,\\ ka`i t`hn gwn'ian\\ t~hc gwn'iac\\ me'izona <'exei\\ t`hn 'iswn e>ujei~wn\\ perieqom'enhn;\\ <'oper >'edei de~ixai. } { E{\u g}er, dolay{\i}s{\i}yla, iki {\"u}{\c c}genin\\ (birinin) iki kenar{\i}\\ (di{\u g}erinin) iki kenar{\i}na\\ e{\c s}itse \\ her biri birine,\\ ama taban{\i}\\ taban{\i}ndan\\ b{\"u}y{\"u}kse,\\ a{\c c}{\i}s{\i} da\\ a{\c c}{\i}s{\i}ndan\\ b{\"u}y{\"u}k olacak\\ ---(yani) e{\c s}it do{\u g}rularca\\ i{\c c}erilenler;\\ \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.421875)(7.7440624,1.421875) \psline[linewidth=0.04cm](1.2540625,1.078125)(0.2940625,-1.101875) \psline[linewidth=0.04cm](0.2940625,-1.101875)(3.0540626,-1.101875) \psline[linewidth=0.04cm](3.0540626,-1.101875)(1.2540625,1.078125) \psline[linewidth=0.04cm](5.6540623,1.078125)(4.4940624,-1.121875) \psline[linewidth=0.04cm](4.4940624,-1.121875)(7.3940625,-1.121875) \psline[linewidth=0.04cm](7.3940625,-1.121875)(5.6340623,1.078125) \usefont{T1}{ptm}{m}{n} \rput(1.360625,1.243125){\gr A} \usefont{T1}{ptm}{m}{n} \rput(0.10125,-1.256875){\gr B} \usefont{T1}{ptm}{m}{n} \rput(3.2151563,-1.216875){\gr G} \usefont{T1}{ptm}{m}{n} \rput(5.8140626,1.223125){\gr D} \usefont{T1}{ptm}{m}{n} \rput(4.316094,-1.276875){\gr E} \usefont{T1}{ptm}{m}{n} \rput(7.596719,-1.236875){\gr Z} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.26 \parsen{ If two triangles\\ two angles\\ to two angles\\ have equal,\\ either to either,\\ and one side\\ to one side\\ equal,\\ either that near the equal sides\\ or that subtending\\ one of the equal sides,\\ also the remaining sides\\ to the remaining sides\\ they will have equal,\\ %either to either,\\ also the remaining angle\\ to the remaining angle. } { {}>E`an d'uo tr'igwna\\ t`ac d'uo gwn'iac\\ dus`i gwn'iaic\\ {}>'isac >'eqh|\\ 'ishn\\ {}>'htoi t`hn pr`oc ta~ic >'isaic gwn'iaic\\ {}>`h t`hn 'iswn gwni~wn,\\ ka`i t`ac loip`ac pleur`ac\\ ta~ic loipa~ic pleura~ic\\ {}>'isac <'exei\\ %{}['Estw\\ d'uo tr'igwna t`a ABG, DEZ\\ t`ac d'uo gwn'iac t`ac 'isac >'eqonta\\ eq'etw d`e\\ ka`i m'ian pleur`an\\ mi~a| pleur~a|\\ {}>'ishn,\\ pr'oteron t`hn pr`oc ta~ic >'isaic gwn'iaic\\ t`hn BG t~h| EZ;} { Verilmi{\c s} olsun\\ iki \gr{ABG} ve \gr{DEZ} {\"u}{\c c}geni\\ iki \gr{ABG} ve \gr{BGA} a{\c c}{\i}lar{\i}\\ iki \gr{DEZ} ve \gr{EZD} a{\c c}{\i}lar{\i}na\\ e{\c s}it olan,\\ her biri birine,\\ \gr{ABG}, \gr{DEZ} a{\c c}{\i}s{\i}na\\ ve \gr{BGA}, \gr{EZD} a{\c c}{\i}s{\i}na;\\ ayr{\i}ca olsun\\ bir kenar{\i}\\ bir kenar{\i}na\\ e{\c s}it,\\ {\"o}nce esit a{\c c}{\i}lar{\i}n yan{\i}nda olan,\\ \gr{BG}, \gr{EZ} kenar{\i}na; } \parsen{ I say that\\ the remaining sides\\ to the remaining sides\\ they will have equal,\\ either to either,\\ \gr{AB} to \gr{DE}\\ and \gr{AG} to \gr{DZ},\\ also the remaining angle\\ to the remaining angle,\\ \gr{BAG} to \gr{EDZ}. } { l'egw, <'oti\\ ka`i t`ac loip`ac pleur`ac\\ ta~ic loipa~ic pleura~ic\\ {}>'isac <'exei\\ i g`ar >'anis'oc >estin\\ ut~wn me'izwn >est'in.\\ {}>'estw me'izwn\\ 'ish\\ epeze'uqjw\\ Epe`i o>~un >'ish >est`in\\ 'isai e>is`in\\ 'ish >est'in;\\ b'asic >'ara 'ish >est'in,\\ ka`i t`o HBG tr'igwnon\\ t~w| DEZ trig'wnw|\\ {}>'ison >est'in,\\ ka`i a'isai >'esontai,\\ <`ac a'isai pleura`i 'ish >'ara all`a 'ish;\\ ka`i 'ara\\ t~h| 'ish >est'in,\\ el'asswn t~h| me'izoni;\\ <'oper >ad'unaton.\\ o>uk >'ara >'anis'oc >estin\\ 'ish >'ara.\\ {}>'esti d`e ka`i\\ 'ish;\\ d'uo d`h a'isai e>is`in\\ estin >'ish;\\ b'asic >'ara 'ish >est'in,\\ ka`i loip`h gwn'ia 'ish >est'in. } { {\c C}{\"u}nk{\"u} o zaman e{\c s}ittir,\\ \gr{BH}, \gr{DE} kenar{\i}na\\ ve \gr{BG}, \gr{EZ} kenar{\i}na,\\ \gr{BH} ve \gr{BG} ikilisi\\ \gr{DE} ve \gr{EZ} ikilisine\\ e{\c s}ittirler,\\ her biri birine,\\ ve \gr{HBG} a{\c c}{\i}s{\i}\\ \gr{DEZ} a{\c c}{\i}s{\i}na\\ e{\c s}ittir;\\ dolay{\i}s{\i}yla \gr{HG} taban{\i}\\ \gr{DZ} taban{\i}na\\ e{\c s}ittir,\\ ve \gr{HBG} {\"u}{\c c}geni\\ \gr{DEZ} {\"u}{\c c}genine\\ e{\c s}ittir,\\ ve kalan a{\c c}{\i}lar\\ kalan a{\c c}{\i}lara\\ e{\c s}it olacaklar,\\ e{\c s}it kenarlar{\i}n kar{\c s}{\i}lad{\i}klar{\i}.\\ E{\c s}ittir dolay{\i}s{\i}yla \gr{BGH} a{\c c}{\i}s{\i}\\ \gr{DZE} a{\c c}{\i}s{\i}na.\\ Ama \gr{DZE},\\ \gr{BGA} a{\c c}{\i}s{\i}na\\ e{\c s}it kabul edilmi{\c s}ti\\ dolay{\i}s{\i}yla \gr{BGH} de\\ \gr{BGA} a{\c c}{\i}s{\i}na\\ e{\c s}ittir,\\ daha k{\"u}{\c c}{\"u}k olan daha b{\"u}y{\"u}k olana,\\ ki bu imkans{\i}zd{\i}r.\\ Dolay{\i}s{\i}yla de{\u g}ildir e{\c s}it de{\u g}il,\\ \gr{AB}, \gr{DE} kenar{\i}na.\\ Dolay{\i}s{\i}yla e{\c s}ittir.\\ Ayr{\i}ca durum {\c s}{\"o}yledir;\\ \gr{BG}, \gr{EZ} kenar{\i}na e{\c s}ittir;\\ o zaman \gr{AB} ve \gr{BG} ikilisi\\ \gr{DE} ve \gr{EZ} ikilisine\\ e{\c s}ittirler,\\ her biri birine;\\ \gr{ABG} a{\c c}{\i}s{\i} da\\ \gr{DEZ} a{\c c}{\i}s{\i}na\\ e{\c s}ittir;\\ dolay{\i}s{\i}yla \gr{AG} taban{\i}\\ \gr{DZ} taban{\i}na\\ e{\c s}ittir,\\ ve kalan \gr{BAG} a{\c c}{\i}s{\i}\\ kalan \gr{EDZ} a{\c c}{\i}s{\i}na\\ e{\c s}ittir. } \myfntext{Fitzpatrick considers this way of denoting the line to be a `mistake'; apparently he thinks Euclid should (and perhaps did originally) write \gr{HB}, for parallelism with \gr{DE}. But \gr{HB} and \gr{BH} are the same line, and for all we know, Euclid preferred to write \gr{BH} because it was in alphabetical order. Netz \cite[Ch.~2]{MR1683176} studies the general Greek mathematical practice of using the letters in different order for the same mathematical object. He concludes that changes in order are made on purpose, though he does not address examples like the present one.} \parsen{ But then again let them be\\ ---[those angles] equal sides\\ subtending---\\ equal,\\ as \gr{AB} to \gr{DE};\\ I say again that\\ also the remaining sides\\ to the remaining sides\\ will be equal,\\ \gr{AG} to \gr{DZ},\\ and \gr{BG} to \gr{EZ},\\ and also the remaining angle \gr{BAG}\\ to the remaining angle \gr{EDZ}\\ is equal. } { >All`a d`h p'alin >'estwsan\\ a'isac gwn'iac pleura`i\\ 'isai,\\ 'isai >'esontai,\\ 'eti 'ish >est'in. } { Ama o zaman, yine olsunlar\\ --- kenarlar e{\c s}it [a{\c c}{\i}lar{\i}]\\ kar{\c s}{\i}layan---\\ e{\c s}it,\\ \gr{AB}, \gr{DE} kenar{\i}na gibi;\\ Yine iddia ediyorum ki\\ kalan kenarlar da\\ kalan kenarlara\\ e{\c s}it olacaklar,\\ \gr{AG}, \gr{DZ} kenar{\i}na\\ ve \gr{BG}, \gr{EZ} kenar{\i}na\\ ve kalan \gr{BAG} a{\c c}{\i}s{\i} da\\ kalan \gr{EDZ} a{\c c}{\i}s{\i}na\\ e{\c s}ittir. } \parsen{ For, if it is unequal,\\ \gr{BG} to \gr{EZ},\\ one of them is greater.\\ Let be greater,\\ if possible,\\ \gr{BG},\\ and let there be cut\\ to \gr{EZ} equal\\ \gr{BJ},\\ and suppose there has been joined\\ \gr{AJ}.\\ Because also it is equal\\ ---\gr{BJ} to \gr{EZ}\\ and \gr{AB} to \gr{DE},\\ then the two \gr{AB} and \gr{BJ}\\ to the two \gr{DE} and \gr{EZ}\\ are equal,\\ either to either;\\ and they contain equal angles;\\ therefore the base \gr{AJ}\\ to the base \gr{DZ}\\ is equal,\\ and the triangle \gr{ABJ}\\ to the triangle \gr{DEZ}\\ is equal,\\ and the remaining angles\\ to the remaining angles\\ are equal,\\ which the equal sides\\ subtend.\\ Therefore equal is\\ angle \gr{BJA}\\ to \gr{EZD}.\\ But \gr{EZD}\\ to \gr{BGA}\\ is equal;\\ then of triangle \gr{AJG}\\ the exterior angle \gr{BJA}\\ is equal\\ to the interior and opposite\\ \gr{BGA};\\ which is impossible.\\ Therefore it is not unequal,\\ \gr{BG} to \gr{EZ};\\ therefore it is equal.\\ And it is also,\\ \gr{AB},\\ to \gr{DE},\\ equal.\\ Then the two \gr{AB} and \gr{BG}\\ to the two \gr{DE} and \gr{EZ}\\ are equal,\\ either to either;\\ and equal angles\\ they contain;\\ therefore the base \gr{AG}\\ to the base \gr{DZ}\\ is equal,\\ and triangle \gr{ABG}\\ to triangle \gr{DEZ}\\ is equal,\\ and the remaining angle \gr{BAG}\\ to the remaining angle \gr{EDZ}\\ is equal. } { E>i g`ar >'anis'oc >estin\\ ut~wn me'izwn >est'in.\\ {}>'estw me'izwn,\\ e>i dunat'on,\\ 'ish\\ epeze'uqjw\\ ep`ei >'ish >est`in\\ 'isai e>is`in\\ 'isac peri'eqousin;\\ b'asic >'ara 'ish >est'in,\\ ka`i t`o ABJ tr'igwnon\\ t~w| DEZ trig'wnw|\\ {}>'ison >est'in,\\ ka`i a'isai >'esontai,\\ <`ac a'isac pleura`i\\ 'ish >'ara >est`in\\ all`a estin >'ish;\\ trig'wnou d`h to~u AJG\\ ekt`oc gwn'ia 'ish >est`i\\ t~h| >ent`oc ka`i >apenant'ion\\ t~h| ad'unaton.\\ o>uk >'ara >'anis'oc >estin\\ 'ish >'ara.\\ {}>est`i d`e ka`i\\ 'ish.\\ d'uo d`h a'isai e>is`in\\ 'isac\\ peri'eqousi;\\ b'asic >'ara 'ish >est'in,\\ ka`i t`o ABG tr'igwnon\\ t~w| DEZ trig'wnw|\\ {}>'ison\\ ka`i loip`h gwn'ia 'ish. } { {\c C}{\"u}nk{\"u}, e{\u g}er e{\c s}it de{\u g}il ise,\\ \gr{BG}, \gr{EZ} kenar{\i}na,\\ biri daha b{\"u}y{\"u}kt{\"u}r.\\ Daha b{\"u}y{\"u}k olsun,\\ e{\u g}er m{\"u}mk{\"u}nse,\\ \gr{BG},\\ ve kesilmi{\c s} olsun\\ \gr{EZ} kenar{\i}na e{\c s}it\\ \gr{BJ},\\ ve kabul edilsin birle{\c s}tirilmi{\c s} oldu{\u g}u\\ \gr{AJ} kenar{\i}n{\i}n.\\ Ayr{\i}ca e{\c s}it oldu{\u g}undan\\ ---\gr{BJ}, \gr{EZ} kenar{\i}na\\ ve \gr{AB}, \gr{DE} kenar{\i}na\\ \gr{AB} ve \gr{BJ} ikilisi\\ \gr{DE} ve \gr{EZ} ikilisine\\ e{\c s}ittirler,\\ her biri birine;\\ ama i{\c c}erirler e{\c s}it a{\c c}{\i}lar{\i};\\ dolay{\i}s{\i}yla \gr{AJ} taban{\i}\\ \gr{DZ} taban{\i}na\\ e{\c s}ittir,\\ ve \gr{ABJ} {\"u}{\c c}geni\\ \gr{DEZ} {\"u}{\c c}genine\\ e{\c s}ittir,\\ ve kalan a{\c c}{\i}lar\\ kalan a{\c c}{\i}lara\\ e{\c s}ittirler,\\ e{\c s}it kenarlar{\i}n\\ kar{\c s}{\i}lad{\i}klar{\i}.\\ Dolay{\i}s{\i}yla e{\c s}ittir\\ \gr{BJA},\\ \gr{EZD} a{\c c}{\i}s{\i}na.\\ Ama \gr{EZD},\\ \gr{BGA} a{\c c}{\i}s{\i}na\\ e{\c s}ittir;\\ o zaman \gr{AJG} {\"u}{\c c}geninin\\ \gr{BJA} d{\i}{\c s} a{\c c}{\i}s{\i}\\ e{\c s}ittir\\ i{\c c} ve kar{\c s}{\i}t\\ \gr{BGA} a{\c c}{\i}s{\i}na;\\ ki bu imkans{\i}zd{\i}r.\\ Dolay{\i}s{\i}yla e{\c s}it de{\u g}il de{\u g}ildir,\\ \gr{BG}, \gr{EZ} kenar{\i}na;\\ dolay{\i}s{\i}yla e{\c s}ittir.\\ Ve yine\\ \gr{AB},\\ \gr{DE} kenar{\i}na,\\ e{\c s}ittir.\\ O zaman \gr{AB} ve \gr{BG} ikilisi\\ \gr{DE} ve \gr{EZ} ikilisine\\ e{\c s}ittirler,\\ her biri birine;\\ e{\c s}it a{\c c}{\i}lar\\ i{\c c}erirler;\\ dolay{\i}s{\i}yla \gr{AG} taban{\i}\\ \gr{DZ} taban{\i}na\\ e{\c s}ittir,\\ ve \gr{ABG} {\"u}{\c c}geni\\ \gr{DEZ} {\"u}{\c c}genine\\ e{\c s}ittir,\\ ve kalan \gr{BAG} a{\c c}{\i}s{\i}\\ kalan \gr{EDZ} a{\c c}{\i}s{\i}na\\ e{\c s}ittir. } \parsen{ If therefore two triangles\\ two angles\\ to two angles\\ have equal,\\ either to either,\\ and one side\\ to one side\\ equal,\\ either that near the equal sides\\ or that subtending\\ one of the equal sides,\\ also the remaining sides\\ to the remaining sides\\ they will have equal,\\ %either to either,\\ also the remaining angle\\ to the remaining angle;\\ \myqed } { >E`an >'ara d'uo tr'igwna\\ t`ac d'uo gwn'iac\\ dus`i gwn'iaic\\ {}>'isac >'eqh|\\ 'ishn\\ {}>'htoi t`hn pr`oc ta~ic >'isaic gwn'iaic,\\ {}>`h t`hn 'iswn gwni~wn,\\ ka`i t`ac loip`ac pleur`ac\\ ta~ic loipa~ic pleura~ic\\ {}>'isac <'exei\\ ka`i t`hn loip`hn gwn'ian\\ t~h| loip~h| gwn'ia|;\\ <'oper >'edei de~ixai. } { E{\u g}er, dolay{\i}s{\i}yla, iki {\"u}{\c c}genin\\ iki a{\c c}{\i}s{\i}\\ iki a{\c c}{\i}s{\i}na\\ e{\c s}itse,\\ her biri birine,\\ ve bir kenar\\ bir kenara\\ e{\c s}itse,\\ ya e{\c s}it a{\c c}{\i}lar{\i}n aras{\i}nda olan\\ ya da kar{\c s}{\i}layan\\ e{\c s}it a{\c c}{\i}lardan birini;\\ kalan kenarlar{\i} da\\ kalan kenarlar{\i}na\\ e{\c s}it olacak,\\ %either to either,\\ kalan a{\c c}{\i}lar{\i} da\\ kalan a{\c c}{\i}lar{\i}na;\\ \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.4426563)(7.7495313,1.4426563) \psline[linewidth=0.04cm](1.266875,1.0989063)(0.306875,-1.0810938) \psline[linewidth=0.04cm](0.306875,-1.0810938)(3.066875,-1.0810938) \psline[linewidth=0.04cm](3.066875,-1.0810938)(1.266875,1.0989063) \psline[linewidth=0.04cm](5.666875,1.0989063)(4.506875,-1.1010938) \psline[linewidth=0.04cm](4.506875,-1.1010938)(7.406875,-1.1010938) \psline[linewidth=0.04cm](7.406875,-1.1010938)(5.646875,1.0989063) \usefont{T1}{ptm}{m}{n} \rput(1.32,1.2639062){\gr A} \usefont{T1}{ptm}{m}{n} \rput(0.10125,-1.2360938){\gr B} \usefont{T1}{ptm}{m}{n} \rput(3.2190626,-1.1960938){\gr G} \usefont{T1}{ptm}{m}{n} \rput(5.816875,1.2439063){\gr D} \usefont{T1}{ptm}{m}{n} \rput(4.3209376,-1.2560937){\gr E} \usefont{T1}{ptm}{m}{n} \rput(7.6021876,-1.2160938){\gr Z} \psline[linewidth=0.04cm](1.26,1.0826563)(2.38,-1.0773437) \psline[linewidth=0.04cm](3.04,-1.0773437)(1.0,0.46265626) \usefont{T1}{ptm}{m}{n} \rput(0.783125,0.5676563){\gr H} \usefont{T1}{ptm}{m}{n} \rput(2.3453126,-1.2923437){\gr J} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.27 \parsen{ If on two \strgt s\\ a \strgt\ falling\\ the alternate angles\\ equal to one another\\ make,\\ parallel will be to one another\\ the \strgt s. } { >E`an e>ic d'uo e>uje'iac\\ e>uje~ia >emp'iptousa\\ t`ac >enall`ax gwn'iac\\ {}>'isac >all'hlaic\\ poi~h|,\\ par'allhloi >'esontai >all'hlaic\\ auje~iai. } { E{\u g}er iki do{\u g}ru {\"u}zerine\\ d{\"u}{\c s}en bir do{\u g}ru\\ ters a{\c c}{\i}lar{\i}\\ birbirine e{\c s}it\\ yaparsa\\ birbirine paralel olacak\\ do{\u g}rular. } \parsen{ For, on the two \strgt s\\ \gr{AB} and \gr{GD}\\ {}[suppose] the \strgt\ falling,\\ {}[namely] \gr{EZ},\\ the alternate angles\\ \gr{AEZ} and \gr{EZD}\\ equal to one another\\ make. } { E>ic g`ar d'uo e>uje'iac\\ t`ac AB, GD\\ e>uje~ia >emp'iptousa\\ enall`ax gwn'iac\\ t`ac 'isac >all'hlaic\\ poie'itw; } { {\c C}{\"u}nk{\"u}, iki do{\u g}ru {\"u}zerine,\\ \gr{AB} ve \gr{GD},\\ {}[kabul edilsin] d{\"u}{\c s}en,\\ \gr{EZ} do{\u g}rusunun,\\ ters \\ \gr{AEZ} ve \gr{EZD} a{\c c}{\i}lar{\i}n{\i}\\ birbirine e{\c s}it\\ olu{\c s}turdu{\u g}unu. } \parsen{ I say that\\ parallel is \gr{AB} to \gr{GD}. } { l'egw, <'oti\\ par'allhl'oc >estin i g`ar m'h,\\ {}>ekball'omenai\\ a'htoi >ep`i t`a B, D m'erh\\ {}>`h >ep`i t`a A, G.\\ {}>ekbebl'hsjwsan\\ ka`i sumpipt'etwsan\\ {}>ep`i t`a B, D m'erh\\ kat`a t`o H.\\ trig'wnou d`h to~u HEZ\\ ekt`oc gwn'ia 'ish >est`i\\ t~h| >ent`oc ka`i >apenant'ion\\ t~h| est`in >ad'unaton;\\ o>uk >'ara\\ aekball'omenai\\ sumpeso~untai >ep`i t`a B, D m'erh.\\ ud`e >ep`i t`a A, G;\\ aep`i mhd'etera t`a m'erh\\ sump'iptousai\\ par'allhlo'i e>isin;\\ par'allhloc >'ara >est`in E`an >'ara e>ic d'uo e>uje'iac\\ e>uje~ia >emp'iptousa\\ t`ac >enall`ax gwn'iac\\ {}>'isac >all'hlaic\\ poi~h|,\\ par'allhloi >'esontai\\ auje~iai;\\ <'oper >'edei de~ixai. } { E{\u g}er, dolay{\i}s{\i}yla, iki do{\u g}ru {\"u}zerine\\ d{\"u}{\c s}en bir do{\u g}ru\\ ters a{\c c}{\i}lar{\i}\\ birbirine e{\c s}it\\ yaparsa\\ birbirine paralel olacak\\ do{\u g}rular;\\ \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.23)(5.9296875,1.23) \psline[linewidth=0.04cm](0.3771875,0.77)(3.9771874,0.77) \psline[linewidth=0.04cm](3.9771874,0.77)(5.5771875,0.17) \psline[linewidth=0.04cm](5.5771875,0.17)(3.9771874,-0.43) \psline[linewidth=0.04cm](3.9771874,-0.43)(0.3371875,-0.43) \psline[linewidth=0.04cm](2.9571874,1.21)(0.9971875,-1.21) \usefont{T1}{ptm}{m}{n} \rput(0.12375,0.815){\gr A} \usefont{T1}{ptm}{m}{n} \rput(4.184375,0.935){\gr B} \usefont{T1}{ptm}{m}{n} \rput(0.104375,-0.445){\gr G} \usefont{T1}{ptm}{m}{n} \rput(3.9971876,-0.645){\gr D} \usefont{T1}{ptm}{m}{n} \rput(5.7603126,0.175){\gr H} \usefont{T1}{ptm}{m}{n} \rput(2.4192188,0.995){\gr E} \usefont{T1}{ptm}{m}{n} \rput(1.7798438,-0.685){\gr Z} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.28 \parsen{ If on two \strgt s\\ a \strgt\ falling\footnotemark\\ the exterior angle\\ to the interior and opposite\\ and in the same parts\\ make equal,\\ or the interior and in the same parts\\ to two \rgt s\\ equal,\\ parallel will be to one another\\ the \strgt s. } { >E`an e>ic d'uo e>uje'iac\\ e>uje~ia >emp'iptousa\\ t`hn >ekt`oc gwn'ian\\ t~h| >ent`oc ka`i >apenant'ion\\ ka`i >ep`i t`a a>ut`a m'erh\\ {}>'ishn poi~h|\\ {}>`h t`ac >ent`oc ka`i >ep`i t`a a>ut`a m'erh\\ dus`in >orja~ic\\ {}>'isac,\\ par'allhloi >'esontai >all'hlaic\\ auje~iai. } { E{\u g}er iki do{\u g}ru {\"u}zerine\\ d{\"u}{\c s}en bir do{\u g}ru,\\ d{\i}{\c s} a{\c c}{\i}y{\i},\\ i{\c c} ve kar{\c s}{\i}t\\ ve ayn{\i} tarafta kalan a{\c c}{\i}ya,\\ e{\c s}it yaparsa,\\ veya i{\c c} ve ayn{\i} tarafta kalanlar{\i},\\ iki dik a{\c c}{\i}ya\\ e{\c s}it,\\ birbirine paralel olacak\\ do{\u g}rular. } \myfntext{It is perhaps impossible to maintain the Greek word order comprehensibly in English. The normal English order would be, `If a straight line, falling on two straight lines'. But the proposition is ultimately about the \emph{two} straight lines; perhaps that is why Euclid mentions them before the one straight line that falls on them.} \parsen{ For, on the two \strgt s \gr{AB} and \gr{GD},\\ the \strgt\ falling---\gr{EZ}---\\ the exterior angle \gr{EHB}\\ to the interior and opposite angle\\ \gr{HJD}\\ equal\\ ---suppose it makes,\\ or the interior and in the same parts,\\ \gr{BHJ} and \gr{HJD},\\ to two \rgt s\\ equal. } { E>ic g`ar d'uo e>uje'iac t`ac AB, GD\\ e>uje~ia >emp'iptousa ekt`oc gwn'ian t`hn ent`oc ka`i >apenant'ion gwn'ia|\\ t~h| 'ishn\\ poie'itw\\ {}>`h t`ac >ent`oc ka`i >ep`i t`a a>ut`a m'erh\\ t`ac orja~ic\\ {}>'isac; } { {\c C}{\"u}nk{\"u}, \gr{AB} ve \gr{GD} do{\u g}rular{\i} {\"u}zerine\\ d{\"u}{\c s}en \gr{EZ} do{\u g}rusu\\ \gr{EHB} d{\i}{\c s} a{\c c}{\i}s{\i}n{\i}\\ i{\c c} ve kar{\c s}{\i}t\\ \gr{HJD} a{\c c}{\i}s{\i}na\\ e{\c s}it\\ ---yapt{\i}{\u g}{\i} varsay{\i}ls{\i}n,\\ veya i{\c c} ve ayn{\i} tarafta kalan,\\ \gr{BHJ} ve \gr{HJD} a{\c c}{\i}lar{\i}n{\i}n,\\ iki dik a{\c c}{\i}ya\\ e{\c s}it oldu{\u g}u. } \parsen{ I say that\\ parallel is\\ \gr{AB} to \gr{GD}. } { l'egw, <'oti\\ par'allhl'oc >estin\\ Epe`i g`ar >'ish >est`in\\ all`a estin >'ish,\\ ka`i 'ara t~h| estin >'ish;\\ ka'i e>isin >enall'ax;\\ par'allhloc >'ara >est`in epe`i aorja~ic\\ {}>'isai e>is'in,\\ e>is`i d`e ka`i aorja~ic\\ {}>'isai,\\ a'ara 'isai e>is'in;\\ koin`h >afh|r'hsjw\lli\\ 'ara estin >'ish;\\ ka'i e>isin >enall'ax;\\ par'allhloc >'ara >est`in E`an >'ara e>ic d'uo e>uje'iac\\ e>uje~ia >emp'iptousa\\ t`hn >ekt`oc gwn'ian\\ t~h| >ent`oc ka`i >apenant'ion\\ ka`i >ep`i t`a a>ut`a m'erh\\ {}>'ishn poi~h|\\ {}>`h t`ac >ent`oc ka`i >ep`i t`a a>ut`a m'erh\\ dus`in >orja~ic\\ {}>'isac,\\ par'allhloi >'esontai\\ auje~iai;\\ <'oper >'edei de~ixai. } { E{\u g}er dolay{\i}s{\i}yla iki do{\u g}ru {\"u}zerine\\ d{\"u}{\c s}en bir do{\u g}ru,\\ d{\i}{\c s} a{\c c}{\i}y{\i},\\ i{\c c} ve kar{\c s}{\i}t\\ ve ayn{\i} tarafta kalan a{\c c}{\i}ya,\\ e{\c s}it yaparsa,\\ veya i{\c c} ve ayn{\i} tarafta kalanlar{\i},\\ iki dik a{\c c}{\i}ya\\ e{\c s}it,\\ birbirine paralel olacak\\ do{\u g}rular; \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.2592187)(5.1175,1.2592187) \psline[linewidth=0.04cm](0.3371875,0.6407812)(4.7371874,0.66078126) \psline[linewidth=0.04cm](0.3171875,-0.35921875)(4.7371874,-0.35921875) \psline[linewidth=0.04cm](0.8371875,1.1207813)(4.1571875,-1.0792187) \usefont{T1}{ptm}{m}{n} \rput(0.12375,0.6457813){\gr A} \usefont{T1}{ptm}{m}{n} \rput(4.964375,0.66578126){\gr B} \usefont{T1}{ptm}{m}{n} \rput(0.11828125,-0.35421875){\gr G} \usefont{T1}{ptm}{m}{n} \rput(4.9371877,-0.39421874){\gr D} \usefont{T1}{ptm}{m}{n} \rput(0.61921877,1.0857812){\gr E} \usefont{T1}{ptm}{m}{n} \rput(4.3398438,-1.1142187){\gr Z} \usefont{T1}{ptm}{m}{n} \rput(1.7403125,0.8857812){\gr H} \usefont{T1}{ptm}{m}{n} \rput(3.1825,-0.13421875){\gr J} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.29 \parsen{ The \strgt\ falling on parallel \strgt s\\ the alternate angles\\ makes equal to one another,\\ and the exterior\\ to the interior and opposite\\ equal,\\ and the interior and in the same parts\\ to two \rgt s equal. } { ic t`ac parall'hlouc e>uje'iac e>uje~ia >emp'iptousa\\ t'ac te >enall`ax gwn'iac\\ {}>'isac >all'hlaic poie~i\\ ka`i t`hn >ekt`oc\\ t~h| >ent`oc ka`i >apenant'ion\\ {}>'ishn\\ ka`i t`ac >ent`oc ka`i >ep`i t`a a>ut`a m'erh\\ dus`in >orja~ic >'isac. } { Paralel do{\u g}rular {\"u}zerine d{\"u}{\c s}en bir do{\u g}ru\\ ters a{\c c}{\i}lar{\i}\\ birbirine e{\c s}it yapar,\\ ve d{\i}{\c s} a{\c c}{\i}y{\i}\\ i{\c c} ve kar{\c s}{\i}t a{\c c}{\i}ya\\ e{\c s}it,\\ ve i{\c c} ve ayn{\i} tarafta kalanlar{\i}\\ iki dik a{\c c}{\i}ya e{\c s}it. } \parsen{ For, on the parallel \strgt s\\ \gr{AB} and \gr{GD}\\ let the \strgt\ \gr{EZ} fall. } { E>ic g`ar parall'hlouc e>uje'iac\\ t`ac AB, GD\\ e>uje~ia >empipt'etw enall`ax gwn'iac\\ t`ac 'isac\\ poie~i\\ ka`i t`hn >ekt`oc gwn'ian t`hn ent`oc ka`i >apenant'ion t~h| 'ishn\\ ka`i t`ac >ent`oc ka`i >ep`i t`a a>ut`a m'erh\\ t`ac orja~ic >'isac. } { {\.I}ddia ediyorum ki\\ ters\\ \gr{AHJ} ve \gr{HJD} a{\c c}{\i}lar{\i}n{\i}\\ e{\c s}it\\ yapar,\\ ve \gr{EHB} d{\i}{\c s} a{\c c}{\i}s{\i}n{\i}\\ i{\c c} ve kar{\c s}{\i}t \gr{HJD} a{\c c}{\i}s{\i}na\\ e{\c s}it,\\ ve i{\c c} ve ayn{\i} taraftaki\\ \gr{BHJ} ile \gr{HJD} a{\c c}{\i}lar{\i}n{\i}\\ iki dik a{\c c}{\i}ya e{\c s}it. } \parsen{ For, if it is unequal,\\ \gr{AHJ} to \gr{HJD},\\ one of them is greater.\\ Let the greater be \gr{AHJ};\\ let be added in common\\ \gr{BHJ};\\ therefore \gr{AHJ} and \gr{BHJ}\\ than \gr{BHJ} and \gr{HJD}\\ are greater.\\ However, \gr{AHJ} and \gr{BHJ}\\ to two \rgt s\\ equal are.\\ Therefore [also] \gr{BHJ} and \gr{HJD}\\ than two \rgt s\\ less are.\\ And [\strgt s] from [angles] that are less\\ than two \rgt s,\\ extended to the infinite,\\ fall together.\\ Therefore \gr{AB} and \gr{GD},\\ extended to the infinite,\\ will fall together.\\ But they do not fall together,\\ by their being assumed parallel.\\ Therefore is not unequal\\ \gr{AHJ} to \gr{HJD}.\\ Therefore it is equal.\\ However, \gr{AHJ} to \gr{EHB}\\ is equal;\\ therefore also \gr{EHB} to \gr{HJD}\\ is equal;\\ let \gr{BHJ} be added in common;\\ therefore \gr{EHB} and \gr{BHJ}\\ to \gr{BHJ} and \gr{HJD}\\ is equal.\\ But \gr{EHB} and \gr{BHJ}\\ to two \rgt s\\ are equal.\\ Therefore also \gr{BHJ} and \gr{HJD}\\ to two \rgt s\\ are equal. } { E>i g`ar >'anis'oc >estin\\ ut~wn me'izwn >est'in.\\ {}>'estw me'izwn 'ara isin.\\ {}>all`a aorja~ic\\ {}>'isai e>is'in.\\ {}[ka`i] a'ara orj~wn\\ {}>el'asson'ec e>isin.\\ aap> >elass'onwn\\ {}>`h d'uo >orj~wn\\ {}>ekball'omenai\\ e>ic >'apeiron\\ sump'iptousin;\\ a'ara AB, GD\\ {}>ekball'omenai e>ic >'apeiron\\ sumpeso~untai;\\ o>u sump'iptousi d`e\\ di`a t`o parall'hlouc a>ut`ac uk >'ara >'anis'oc >estin\\ 'ish >'ara.\\ {}>all`a estin >'ish;\\ ka`i 'ara t~h| estin >'ish;\\ koin`h proske'isjw 'ara 'isai e>is'in.\\ {}>all`a aorja~ic\\ {}>'isai e>is'in;\\ ka`i a'ara\\ d'uo >orja~ic\\ {}>'isai e>is'in. } { {\c C}{\"u}nk{\"u}, e{\u g}er e{\c s}it de{\u g}ilse\\ \gr{AHJ}, \gr{HJD} a{\c c}{\i}s{\i}na,\\ biri b{\"u}y{\"u}kt{\"u}r.\\ B{\"u}y{\"u}k olan \gr{AHJ} olsun;\\ eklenmi{\c s} olsun her ikisine de\\ \gr{BHJ};\\ dolay{\i}s{\i}yla \gr{AHJ} ve \gr{BHJ},\\ \gr{BHJ} ve \gr{HJD} a{\c c}{\i}lar{\i}ndan\\ b{\"u}y{\"u}kt{\"u}rler.\\ Fakat, \gr{AHJ} ve \gr{BHJ}\\ iki dik a{\c c}{\i}ya\\ e{\c s}ittirler.\\ Dolay{\i}s{\i}yla \gr{BHJ} ve \gr{HJD} [da]\\ iki dik a{\c c}{\i}dan\\ k{\"u}{\c c}{\"u}kt{\"u}rler.\\ Ve k{\"u}{\c c}{\"u}k olanlardan,\\ iki dik a{\c c}{\i}dan,\\ sonsuza uzat{\i}lanlar [do{\u g}rular],\\ birbirinin {\"u}zerine d{\"u}{\c s}erler.\\ Dolay{\i}s{\i}yla \gr{AB} ve \gr{GD},\\ uzat{\i}l{\i}nca sonsuza,\\ birbirinin {\"u}zerine d{\"u}{\c s}ecekler.\\ Ama onlar birbirinin {\"u}zerine d{\"u}{\c s}mezler,\\ paralel olduklar{\i} kabul edildi{\u g}inden.\\ Dolay{\i}s{\i}yla e{\c s}it de{\u g}il de{\u g}ildir\\ \gr{AHJ}, \gr{HJD} a{\c c}{\i}s{\i}na.\\ Dolay{\i}s{\i}yla e{\c s}ittir.\\ Ancak, \gr{AHJ}, \gr{EHB} a{\c c}{\i}s{\i}na\\ e{\c s}ittir;\\ dolay{\i}s{\i}yla \gr{EHB} da \gr{HJD} a{\c c}{\i}s{\i}na\\ e{\c s}ittir;\\ eklenmi{\c s} olsun her ikisine de \gr{BHJ};\\ dolay{\i}s{\i}yla \gr{EHB} ve \gr{BHJ},\\ \gr{BHJ} ve \gr{HJD} a{\c c}{\i}lar{\i}na\\ e{\c s}ittir.\\ Ama \gr{EHB} ve \gr{BHJ}\\ iki dik a{\c c}{\i}ya\\ e{\c s}ittirler.\\ Dolay{\i}s{\i}yla \gr{BHJ} ve \gr{HJD} da\\ iki dik a{\c c}{\i}ya\\ e{\c s}ittirler. } \parsen{ Therefore the on-parallel-\strgt s \strgt\\ falling\\ the alternate angles\\ makes equal to one another,\\ and the exterior\\ to the interior and opposite\\ equal,\\ and the interior and in the same parts\\ to two \rgt s equal;\\ \myqed } { 'ara e>ic t`ac parall'hlouc e>uje'iac e>uje~ia\\ {}>emp'iptousa\\ t'ac te >enall`ax gwn'iac\\ {}>'isac >all'hlaic poie~i\\ ka`i t`hn >ekt`oc\\ t~h| >ent`oc ka`i >apenant'ion\\ >'ishn\\ ka`i t`ac >ent`oc ka`i >ep`i t`a a>ut`a m'erh\\ dus`in >orja~ic >'isac;\\ <'oper >'edei de~ixai. } { Dolay{\i}s{\i}yla paralel do{\u g}rular {\"u}zerine, do{\u g}ru\\ d{\"u}{\c s}erken\\ ters a{\c c}{\i}lar{\i}\\ e{\c s}it yapar birbirine,\\ ve d{\i}{\c s} a{\c c}{\i}y{\i}\\ i{\c c} ve kar{\c s}{\i}ta\\ e{\c s}it,\\ ve i{\c c} ve ayn{\i} taraftakileri s\\ iki dik a{\c c}{\i}ya e{\c s}it;\\ \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.2592187)(5.1175,1.2592187) \psline[linewidth=0.04cm](0.3371875,0.6407812)(4.7371874,0.66078126) \psline[linewidth=0.04cm](0.3171875,-0.35921875)(4.7371874,-0.35921875) \psline[linewidth=0.04cm](0.8371875,1.1207813)(4.1571875,-1.0792187) \usefont{T1}{ptm}{m}{n} \rput(0.12375,0.6457813){\gr A} \usefont{T1}{ptm}{m}{n} \rput(4.964375,0.66578126){\gr B} \usefont{T1}{ptm}{m}{n} \rput(0.11828125,-0.35421875){\gr G} \usefont{T1}{ptm}{m}{n} \rput(4.9371877,-0.39421874){\gr D} \usefont{T1}{ptm}{m}{n} \rput(0.61921877,1.0857812){\gr E} \usefont{T1}{ptm}{m}{n} \rput(4.3398438,-1.1142187){\gr Z} \usefont{T1}{ptm}{m}{n} \rput(1.7403125,0.8857812){\gr H} \usefont{T1}{ptm}{m}{n} \rput(3.1825,-0.13421875){\gr J} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.30 \parsen{ {}[\Strgt s] to the same \strgt\\ parallel\\ also to one another are parallel. } { Aut~h| e>uje'ia| par'allhloi\\ ka`i >all'hlaic e>is`i par'allh\-loi. } { Ayn{\i} do{\u g}ruya\\ paralel do{\u g}rular\\ birbirlerine de paraleldir. } \parsen{ Let be\\ either of \gr{AB} and \gr{GD}\\ to \gr{GD} parallel. } { >'Estw\\ esti par'allhloc. } { {\.I}ddia ediyorum ki\\ \gr{AB} da \gr{GD} do{\u g}rusuna paraleldir. } \parsen{ For let fall on them a \strgt, \gr{HK}. } { >Empipt'etw g`ar e>ic a>ut`ac e>uje~ia epe`i e>ic parall'hlouc e>uje'iac\\ t`ac AB, EZ\\ e>uje~ia >emp'eptwken 'ish >'ara epe`i e>ic parall'hlouc e>uje'iac\\ t`ac EZ, GD\\ e>uje~ia >emp'eptwken 'ish >est`in ede'iqjh d`e ka`i\\ 'ish.\\ ka`i 'ara t~h| estin >'ish;\\ ka'i e>isin >enall'ax.\\ par'allhloc >'ara >est`in 'ara\\ t~h| a>ut~h| e>uje'ia|\\ par'allhloi\\ ka`i >all'hlaic e>is`i par'allhloi;]\\ <'oper >'edei de~ixai. } { Dolay{\i}s{\i}yla \\ ayn{\i} do{\u g}ruya\ paraleller\\ birbirlerine de paraleldir;\\ \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.5)(5.1971874,1.5) \psline[linewidth=0.04cm](0.4071875,0.88)(4.8071876,0.9) \psline[linewidth=0.04cm](0.3671875,-0.92)(4.7871876,-0.92) \usefont{T1}{ptm}{m}{n} \rput(0.19375,0.885){\gr A} \usefont{T1}{ptm}{m}{n} \rput(5.034375,0.905){\gr B} \usefont{T1}{ptm}{m}{n} \rput(0.10828125,-0.975){\gr G} \usefont{T1}{ptm}{m}{n} \rput(5.0271873,-0.915){\gr D} \usefont{T1}{ptm}{m}{n} \rput(0.18921874,-0.155){\gr E} \usefont{T1}{ptm}{m}{n} \rput(5.049844,-0.135){\gr Z} \usefont{T1}{ptm}{m}{n} \rput(2.9103124,1.085){\gr H} \usefont{T1}{ptm}{m}{n} \rput(2.4325,0.105){\gr J} \psline[linewidth=0.04cm](0.3871875,-0.12)(4.8071876,-0.12) \psline[linewidth=0.04cm](3.4071875,1.48)(1.8671875,-1.48) \usefont{T1}{ptm}{m}{n} \rput(2.0157812,-0.735){\gr K} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.31 \parsen{ Through the given point\\ to the given \strgt\ parallel\\ a straight line to draw. } { Di`a to~u doj'entoc shme'iou\\ t~h| doje'ish| e>uje'ia| par'allhlon\\ e>uje~ian gramm`hn >agage~in. } { Verilen bir noktadan\\ verilen bir do{\u g}ruya paralel\\ bir do{\u g}ru {\c c}izmek. } \parsen{ Let be\\ the given point \gr A,\\ and the given \strgt\ \gr{BG}. } { >'Estw\\ t`o m`en doj`en shme~ion t`o A,\\ uje~ia uje'ia| par'allhlon\\ e>uje~ian gramm`hn >agage~in. } { {\c S}imdi gereklidir\\ \gr A noktas{\i}ndan\\ \gr{BG} do{\u g}rusuna paralel\\ bir do{\u g}ru {\c c}izmek. } \parsen{ Suppose there has been chosen\\ on \gr{BG}\\ a random point \gr D,\\ and there has been joined \gr{AD}.\\ and there has been constructed,\\ on the \strgt\ \gr{DA},\\ and at the point \gr A of it,\\ to the angle \gr{ADG} equal,\\ \gr{DAE};\\ and suppose there has been extended,\\ in \strgt s with \gr{EA},\\ the \strgt\ \gr{AZ}. } { E>il'hfjw\\ {}>ep`i t~hc BG\\ tuq`on shme~ion t`o D,\\ ka`i >epeze'uqjw uje'ia|\\ ka`i t~w| pr`oc a>ut~h| shme'iw| t~w| A\\ t~h| 'ish\\ ekbebl'hsjw\\ {}>ep> e>uje'iac t~h| EA\\ e>uje~ia epe`i\\ e>ic d'uo e>uje'iac t`ac BG, EZ\\ e>uje~ia >emp'iptousa enall`ax gwn'iac\\ t`ac 'isac >all'hlaic pepo'ihken,\\ par'allhloc >'ara >est`in 'ara shme'iou to~u A\\ t~h| doje'ish| e>uje'ia| t~h| BG par'allhloc\\ e>uje~ia gramm`h >~hktai 'edei poi~hsai. } { Dolay{\i}s{\i}yla, verilen \gr A noktas{\i}ndan,\\ verilen \gr{BG} do{\u g}rusuna paralel,\\ bir do{\u g}ru \gr{EAZ}, {\c c}izilmi{\c s} oldu;\\ \ozqef } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-2.401875)(8.692187,2.401875) \psline[linewidth=0.04cm](0.3340625,1.998125)(4.7340627,2.018125) \psline[linewidth=0.04cm](0.2940625,0.498125)(4.7140627,0.498125) \usefont{T1}{ptm}{m}{n} \rput(3.080625,2.223125){\gr A} \usefont{T1}{ptm}{m}{n} \rput(0.10125,0.463125){\gr B} \usefont{T1}{ptm}{m}{n} \rput(4.8951564,0.483125){\gr G} \usefont{T1}{ptm}{m}{n} \rput(2.1540625,0.283125){\gr D} \usefont{T1}{ptm}{m}{n} \rput(0.13609375,2.003125){\gr E} \usefont{T1}{ptm}{m}{n} \rput(4.996719,2.003125){\gr Z} \psline[linewidth=0.04cm](2.9740624,1.998125)(2.2340624,0.498125) \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.32 \parsen{ Of any triangle\\ one of the sides being extended,\\ the exterior angle\\ to the two opposite interior angles\\ is equal,\\ and the triangle's three interior angles\\ to two \rgt s equal are. } { Pant`oc trig'wnou\\ mi~ac t~wn pleur~wn prosekblhje'ishc\\ ekt`oc gwn'ia\\ dus`i ta~ic >ent`oc ka`i >apenant'ion\\ {}>'ish >est'in,\\ ka`i aent`oc to~u trig'wnou tre~ic gwn'iai\\ dus`in >orja~ic >'isai e>is'in. } { Herhangi bir {\"u}{\c c}genin\\ kenarlar{\i}ndan biri uzat{\i}ld{\i}{\u g}{\i}nda,\\ d{\i}{\c s} a{\c c}{\i}\\ iki kar{\c s}{\i}t i{\c c} a{\c c}{\i}ya\\ e{\c s}ittir,\\ ve {\"u}{\c c}genin {\"u}{\c c} i{\c c} a{\c c}{\i}s{\i}\\ iki dik a{\c c}{\i}ya e{\c s}ittir. } \parsen{ Let there be\\ the triangle \gr{ABG},\\ and suppose there has been extended\\ its one side, \gr{BG}, to \gr D;\\ } { >'Estw\\ tr'igwnon t`o ABG,\\ ka`i prosekbebl'hsjw\\ a>uto~u m'ia pleur`a ep`i t`o D; } { Verilmi{\c s} olsun\\ \gr{ABG} {\"u}{\c c}geni,\\ ve varsay{\i}ls{\i}n uzat{\i}lm{\i}{\c s} oldu{\u g}u\\ bir \gr{BG} kenar{\i}n{\i}n \gr D noktas{\i}na.\\ } \parsen{ I say that\\ the exterior angle \gr{AG} is equal\\ to the two interior and opposite angles\\ \gr{GAB} and \gr{ABG},\\ and the triangle's three interior angles\\ \gr{ABG}, \gr{BGA}, and \gr{GAB}\\ to two \rgt s equal are. } { l'egw, <'oti\\ ekt`oc gwn'ia 'ish >est`i\\ dus`i ta~ic >ent`oc ka`i >apenant'ion\\ ta~ic ent`oc to~u trig'wnou tre~ic gwn'iai\\ aorja~ic >'isai e>is'in. } { {\.I}ddia ediyorum ki\\ \gr{AGD} d{\i}{\c s} a{\c c}{\i}s{\i} e{\c s}ittir\\ iki i{\c c} ve kar{\c s}{\i}t\\ \gr{GAB} ve \gr{ABG} a{\c c}{\i}s{\i}na,\\ ve {\"u}{\c c}genin {\"u}{\c c} i{\c c} a{\c c}{\i}s{\i}\\ \gr{ABG}, \gr{BGA} ve \gr{GAB},\\ iki dik a{\c c}{\i}ya e{\c s}ittir. } \parsen{ For, suppose there has been drawn\\ through the point \gr G\\ to the \strgt\ \gr{AB} parallel\\ \gr{GE}. } { >'Hqjw g`ar\\ di`a to~u G shme'iou\\ t~h| AB e>uje'ia| par'allhloc\\ epe`i par'allhl'oc >estin ic a>ut`ac >emp'eptwken enall`ax gwn'iai a'isai >all'hlaic e>is'in.\\ p'alin, >epe`i par'allhl'oc >estin\\ ic a>ut`ac >emp'eptwken\\ e>uje~ia ekt`oc gwn'ia 'ish >est`i\\ t~h| >ent`oc ka`i >apenant'ion t~h| ede'iqjh d`e ka`i 'ish;\\ <'olh >'ara 'ish >est`i\\ dus`i ta~ic >ent`oc ka`i >apenant'ion\\ ta~ic 'ara 'isai e>is'in.\\ {}>all> aorja~ic {}>'isai e>is'in;\\ ka`i a'ara\\ dus`in >orja~ic >'isai e>is'in. } { Eklenmi{\c s} olsun \gr{AGB} ortak olarak;\\ Dolay{\i}s{\i}yla \gr{AGD} ve \gr{AGB} a{\c c}{\i}lar{\i}\\ \gr{ABG}, \gr{BGA} ve \gr{GAB} {\"u}{\c c}l{\"u}s{\"u}ne\\ e{\c s}ittir.\\ Fakat, \gr{AGD} ve \gr{AGB} a{\c c}{\i}lar{\i}\\ iki dik a{\c c}{\i}ya e{\c s}ittir;\\ \gr{AGB}, \gr{GBA} ve \gr{GAB} da dolay{\i}s{\i}yla\\ iki dik a{\c c}{\i}ya e{\c s}ittir. } \parsen{ Therefore, of any triangle\\ one of the sides being extended,\\ the exterior angle\\ to the two opposite interior angles\\ is equal,\\ and the triangle's three interior angles\\ to two \rgt s equal are;\\ \myqed } { Pant`oc >'ara trig'wnou\\ mi~ac t~wn pleur~wn prosekblhje'ishc\\ ekt`oc gwn'ia\\ dus`i ta~ic >ent`oc ka`i >apenant'ion\\ {}>'ish >est'in,\\ ka`i aent`oc to~u trig'wnou tre~ic gwn'iai\\ dus`in >orja~ic >'isai e>is'in;\\ <'oper >'edei de~ixai. } { Dolay{\i}s{\i}yla, herhangi bir {\"u}{\c c}genin\\ kenarlar{\i}ndan biri uzat{\i}ld{\i}{\u g}{\i}nda,\\ d{\i}{\c s} a{\c c}{\i}\\ iki kar{\c s}{\i}t i{\c c} a{\c c}{\i}ya\\ e{\c s}ittir,\\ ve {\"u}{\c c}genin {\"u}{\c c} i{\c c} a{\c c}{\i}s{\i}\\ iki dik a{\c c}{\i}ya e{\c s}ittir;\\ \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.4145312)(6.3403125,1.4145312) \psline[linewidth=0.04cm](1.8940625,1.0107813)(3.4340625,-1.0692188) \psline[linewidth=0.04cm](0.2940625,-1.0692188)(5.8940625,-1.0692188) \usefont{T1}{ptm}{m}{n} \rput(1.960625,1.2357812){\gr A} \usefont{T1}{ptm}{m}{n} \rput(0.10125,-1.1042187){\gr B} \usefont{T1}{ptm}{m}{n} \rput(3.3951561,-1.2642188){\gr G} \usefont{T1}{ptm}{m}{n} \rput(6.1740627,-1.1042187){\gr D} \usefont{T1}{ptm}{m}{n} \rput(5.1560936,1.1357813){\gr E} \psline[linewidth=0.04cm](1.8940625,1.0307813)(0.3140625,-1.0692188) \psline[linewidth=0.04cm](3.4340625,-1.0492188)(4.9940624,0.97078127) \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.33 \parsen{ \Strgt s joining equals and parallels to the same parts\\ also themselves equal and parallel are. } { A'isac te ka`i parall'hlouc >ep`i t`a a>ut`a m'erh >epizeugn'uousai e>uje~iai\\ ka`i a>uta`i >'isai te ka`i par'allhlo'i e>isin. } { E{\c s}it ve paralellerin ayn{\i} taraflar{\i}n{\i} birle{\c s}tiren do{\u g}rular{\i}n\\ kendileri de e{\c s}it ve paraleldirler. } \parsen{ Let be\\ equals and parallels\\ \gr{AB} and \gr{GD},\\ and let join these\\ in the same parts\\ \strgt s \gr{AG} and \gr{BD}. } { >'Estwsan\\ {}>'isai te ka`i par'allhloi\\ aepizeugn'utwsan a>ut`ac\\ {}>ep`i t`a a>ut`a m'erh\\ e>uje~iai a'isai te ka`i par'allhlo'i e>isin. } { {\.I}ddia ediyorum ki\\ \gr{AG} ve \gr{BD} da\\ e{\c s}it ve paraleldirler. } \parsen{ Suppose there has been joined \gr{BG}.\\ And since parallel is \gr{AB} to \gr{GD},\\ and on these has fallen \gr{BG},\\ the alternate angles \gr{ABG} and \gr{BGD}\\ equal to one another are.\\ And since equal is \gr{AB} to \gr{GD},\\ and common [is] \gr{BG},\\ then the two \gr{AB} and \gr{BG}\\ to the two \gr{BG} and \gr{GD}\\ equal are;\\ also angle \gr{ABG}\\ to angle \gr{BGD}\\ {}[is] equal;\\ therefore the base \gr{AG}\\ to the base \gr{BD}\\ is equal,\\ and the triangle \gr{ABG}\\ to the triangle \gr{BGD}\\ is equal,\\ and the remaining angles\\ to the remaining angles\\ equal will be,\\ either to either,\\ which the equal sides subtend;\\ equal therefore\\ the \gr{AGB} angle to \gr{GBD}.\\ And since on the two \strgt s\\ \gr{AG} and \gr{BD}\\ the \strgt\ falling---\gr{BG}---\\ alternate angles equal to one another\\ has made,\\ parallel therefore is \gr{AG} to \gr{BD}.\\ And it was shown to it also equal. } { >Epeze'uqjw epe`i par'allhl'oc >estin ic a>ut`ac >emp'eptwken enall`ax gwn'iai a'isai >all'hlaic e>is'in.\\ ka`i >epe`i >'ish >est`in 'isai e>is'in;\\ ka`i gwn'ia 'ish;\\ b'asic >'ara estin >'ish,\\ ka`i t`o ABG tr'igwnon\\ t~w| BGD trig'wnw|\\ {}>'ison >est'in,\\ ka`i a'isai >'esontai\\ <`ac a'isai pleura`i 'ish >'ara\\ epe`i e>ic d'uo e>uje'iac\\ t`ac AG, BD\\ e>uje~ia >emp'iptousa enall`ax gwn'iac >'isac >all'hlaic\\ pepo'ihken,\\ par'allhloc >'ara >est`in ede'iqjh d`e a>ut~h| ka`i >'ish. } { Varsay{\i}ls{\i}n birle{\c s}tirilmi{\c s} oldu{\u g}u \gr{BG} do{\u g}rusunun.\\ Ve paralel oldu{\u g}undan \gr{AB}, \gr{GD} do{\u g}rusuna,\\ ve bunlar{\i}n {\"u}zerine d{\"u}{\c s}t{\"u}{\u g}{\"u}nden \gr{BG},\\ ters \gr{ABG} ve \gr{BGD} a{\c c}{\i}lar{\i}\\ birbirlerine e{\c s}ittirler.\\ Ve e{\c s}it oldu{\u g}undan \gr{AB}, \gr{GD} do{\u g}rusuna,\\ ve \gr{BG} ortak,\\ \gr{AB} ve \gr{BG} ikilisi\\ \gr{BG} ve \gr{GD} ikilisine\\ e{\c s}ittir;\\ \gr{ABG} a{\c c}{\i}s{\i} da\\ \gr{BGD} a{\c c}{\i}s{\i}na\\ e{\c s}ittir;\\ dolay{\i}s{\i}yla \gr{AG} taban{\i}\\ \gr{BD} taban{\i}na\\ e{\c s}ittir,\\ ve \gr{ABG} {\"u}{\c c}geni\\ \gr{BGD} {\"u}{\c c}genine\\ e{\c s}ittir,\\ ve kalan a{\c c}{\i}lar\\ kalan a{\c c}{\i}lara\\ e{\c s}it olacaklar,\\ her biri birine,\\ e{\c s}it kenarlar{\i} g{\"o}renler;\\ e{\c s}ittir dolay{\i}s{\i}yla\\ \gr{AGB}, \gr{GBD} a{\c c}{\i}s{\i}na.\\ Ve {\"u}zerine iki\\ \gr{AG} ve \gr{BD} do{\u g}rular{\i}n{\i}n,\\ d{\"u}{\c s}en do{\u g}ru---\gr{BG}---\\ birbirine e{\c s}it ters a{\c c}{\i}lar\\ yapm{\i}{\c s}t{\i}r,\\ paraleldir dolay{\i}s{\i}yla \gr{AG}, \gr{BD} do{\u g}rusuna.\\ Ve e{\c s}it oldu{\u g}u da g{\"o}sterilmi{\c s}ti. } \parsen{ Therefore \strgt s joining equals and parallels to the same parts\\ also themselves equal and parallel are. \myqed } { A'ara t`ac >'isac te ka`i parall'hlouc >ep`i t`a a>ut`a m'erh >epizeugn'uousai e>uje~iai\\ ka`i a>uta`i >'isai te ka`i par'allhlo'i e>isin;\\ <'oper >'edei de~ixai. } { Dolay{\i}s{\i}yla e{\c s}it ve paralellerin ayn{\i} taraflar{\i}n{\i} birle{\c s}tiren do{\u g}rular{\i}n\\ kendileri de e{\c s}it ve paraleldirler; \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.391875)(4.5696874,1.391875) \psline[linewidth=0.04cm](4.09375,1.2334375)(0.69375,1.2334375) \psline[linewidth=0.04cm](0.69375,1.2334375)(0.29375,-1.1465625) \psline[linewidth=0.04cm](0.29375,-1.1465625)(3.67375,-1.1465625) \psline[linewidth=0.04cm](3.67375,-1.1465625)(4.09375,1.2334375) \psline[linewidth=0.04cm](3.67375,-1.1465625)(0.69375,1.2334375) \usefont{T1}{ptm}{m}{n} \rput(0.4409375,1.2184376){\gr B} \usefont{T1}{ptm}{m}{n} \rput(4.4003124,1.1584375){\gr A} \usefont{T1}{ptm}{m}{n} \rput(0.11375,-1.1815625){\gr D} \usefont{T1}{ptm}{m}{n} \rput(3.8748438,-1.2415625){\gr G} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.34 \parsen{ Of parallelogram areas,\\ opposite sides and angles\\ are equal to one another,\\ and the diameter cuts them in two. } { T~wn parallhlogr'ammwn qwr'iwn\\ aapenant'ion pleura'i te ka`i gwn'iai\\ {}>'isai >all'hlaic e>is'in,\\ ka`i ut`a d'iqa t'emnei. } { Paralelkenar alanlar{\i}n,\\ kar{\c s}{\i}t kenar ve a{\c c}{\i}lar{\i}\\ e{\c s}ittir birbirine,\\ ve k{\"o}{\c s}egen onlar{\i} ikiye b{\"o}ler. } \parsen{ Let there be\\ a parallelogram area\\ \gr{AGDB};\\ a diameter of it, \gr{BG}. } { >'Estw\\ parallhl'ogrammon qwr'ion\\ t`o AGDB,\\ di'amet\-roc d`e a>uto~u apenant'ion pleura'i te ka`i gwn'iai\\ {}>'isai >all'hlaic e>is'in,\\ ka`i ut`o d'iqa t'emnei. } { iddia ediyorum ki\\ \gr{AGDB} paralelkenar{\i}n{\i}n\\ kar{\c s}{\i}t kenar ve a{\c c}{\i}lar{\i}\\ e{\c s}ittir birbirine,\\ ve \gr{BG} k{\"o}{\c s}egeni onu ikiye b{\"o}ler. } \parsen{ For, since parallel is\\ \gr{AB} to \gr{GD},\\ and on these has fallen\\ a \strgt, \gr{BG},\\ the alternate angles \gr{ABG} and \gr{BGD}\\ equal to one another are.\\ Moreover, since parallel is\\ \gr{AG} to \gr{BD},\\ and on these has fallen\\ \gr{BG},\\ the alternate angles \gr{AGB} and \gr{GBD}\\ equal to one another are.\\ Then two triangles there are,\\ \gr{ABG} and \gr{BGD},\\ the two angles \gr{ABG} and \gr{BGA}\\ to the two \gr{BGD} and \gr{GBD}\\ equal having,\\ either to either,\\ and one side to one side equal,\\ that near the equal angles,\\ their common \gr{BG};\\ also then the remaining sides\\ to the remaining sides\\ equal they will have,\\ either to either,\\ and the remaining angle\\ to the remaining angle;\\ equal, therefore,\\ the \gr{AB} side to \gr{GD},\\ and \gr{AG} to \gr{BD},\\ and yet equal is the \gr{BAG} angle\\ to \gr{GDB}.\\ And since equal is the \gr{ABG} angle\\ to \gr{BGD},\\ and \gr{GBD} to \gr{AGB},\\ therefore the whole \gr{ABD}\\ to the whole \gr{AGD}\\ is equal.\\ And was shown also\\ \gr{BAG} to \gr{GDB} equal. } { >Epe`i g`ar par'allhl'oc >estin\\ ic a>ut`ac >emp'eptwken\\ e>uje~ia enall`ax gwn'iai a'isai >all'hlaic e>is'in.\\ p'alin >epe`i par'allhl'oc >estin\\ ic a>ut`ac >emp'eptwken\\ enall`ax gwn'iai a'isai >all'hlaic e>is'in.\\ d'uo d`h tr'igwn'a >esti\\ t`a ABG, BGD\\ t`ac d'uo gwn'iac t`ac 'isac >'eqonta\\ 'ishn\\ t`hn pr`oc ta~ic >'isaic gwn'iaic\\ koin`hn a>ut~wn t`hn BG;\\ ka`i t`ac loip`ac >'ara pleur`ac\\ ta~ic loipa~ic\\ {}>'isac <'exei\\ 'ish >'ara\\ 'eti >'ish >est`in epe`i >'ish >est`in 'ara estin >'ish.\\ {}>ede'iqjh d`e ka`i\\ 'ish. } { {\c C}{\"u}nk{\"u}, paralel oldu{\u g}undan\\ \gr{AB}, \gr{GD} do{\u g}rusuna,\\ ve bunlar{\i}n {\"u}zerine d{\"u}{\c s}t{\"u}{\u g}{\"u}nden\\ bir \gr{BG} do{\u g}rusu,\\ ters \gr{ABG} ve \gr{BGD} a{\c c}{\i}lar{\i}\\ e{\c s}ittir birbirlerine.\\ Dahas{\i}, paralel oldu{\u g}undan\\ \gr{AG}, \gr{BD} do{\u g}rusuna,\\ ve bunlar{\i}n {\"u}zerine d{\"u}{\c s}t{\"u}{\u g}{\"u}nden\\ \gr{BG},\\ ters a{\c c}{\i}lar \gr{AGB} ve \gr{GBD}\\ e{\c s}ittir birbirlerine.\\ {\c S}imdi iki {\"u}{\c c}gen vard{\i}r;\\ \gr{ABG} ve \gr{BGD},\\ iki \gr{ABG} ve \gr{BGA} a{\c c}{\i}lar{\i}\\ iki \gr{BGD} ve \gr{GBD} a{\c c}{\i}lar{\i}na\\ e{\c s}it olan,\\ her biri birine,\\ ve bir kenar{\i}, bir kenar{\i}na e{\c s}it olan,\\ e{\c s}it a{\c c}{\i}lar{\i}n yan{\i}nda olan,\\ onlar{\i}n ortak \gr{BG} kenar{\i};\\ o zaman kalan kenarlar{\i} da\\ kalan kenarlar{\i}na\\ e{\c s}it olacaklar,\\ her biri birine,\\ ve kalan a{\c c}{\i}\\ kalan a{\c c}{\i}ya;\\ e{\c s}it, dolay{\i}s{\i}yla,\\ \gr{AB} kenar{\i} \gr{GD} kenar{\i}na,\\ ve \gr{AG}, \gr{BD} kenar{\i}na,\\ ve e{\c s}ittir \gr{BAG} a{\c c}{\i}s{\i}\\ \gr{GDB} a{\c c}{\i}s{\i}na.\\ Ve e{\c s}it oldu{\u g}undan \gr{ABG},\\ \gr{BGD} a{\c c}{\i}s{\i}na,\\ ve \gr{GBD}, \gr{AGB} a{\c c}{\i}s{\i}na,\\ dolay{\i}s{\i}yla a{\c c}{\i}n{\i}n tamam{\i} \gr{ABD},\\ a{\c c}{\i}n{\i}n tamam{\i}na, \gr{AGD}\\ e{\c s}ittir.\\ Ve g{\"o}sterilmi{\c s}ti ayr{\i}ca\\ \gr{BAG} ile \gr{GDB} a{\c c}{\i}s{\i}n{\i}n e{\c s}itli{\u g}i. } \parsen{ Therefore, of parallelogram areas,\\ opposite sides and angles\\ equal to one another are. } { T~wn >'ara parallhlogr'ammwn qwr'iwn\\ aapenant'ion pleura'i te ka`i gwn'iai\\ {}>'isai >all'hlaic e>is'in. } { Dolay{\i}s{\i}yla,paralelkenar alanlar{\i}n,\\ kar{\c s}{\i}t kenar ve a{\c c}{\i}lar{\i}\\ e{\c s}ittir birbirlerine. } \parsen{ I say then that\\ also the diameter them cuts in two. } { L'egw d'h, <'oti\\ ka`i ut`a d'iqa t'emnei. } { {\c S}imdi iddia ediyorum ki\\ k{\"o}{\c s}egen de onlar{\i} ikiye keser. } \parsen{ For, since equal is \gr{AB} to \gr{GD},\\ and common [is] \gr{BG},\\ the two \gr{AB} and \gr{BG}\\ to the two \gr{GD} and \gr{BG}\\ equal are,\\ either to either;\\ and angle \gr{ABG}\\ to angle \gr{BGD}\\ equal.\\ Therefore also the base \gr{AG}\\ to the base \gr{DB}\\ equal.\\ Therefore also the \gr{ABG} triangle\\ to the \gr{BGD} triangle\\ is equal. } { >epe`i g`ar >'ish >est`in 'isai e>is`in\\ 'ish.\\ ka`i b'asic >'ara 'ish.\\ ka`i t`o ABG [>'ara] tr'igwnon\\ t~w| BGD trig'wnw|\\ {}>'ison >est'in. } { {\c C}{\"u}nk{\"u}, e{\c s}it oldu{\u g}undan \gr{AB}, \gr{GD} kenar{\i}na,\\ ve \gr{BG} ortak,\\ \gr{AB} ve \gr{BG} ikilisi\\ - \gr{GD} ve \gr{BG} ikilisine\\ e{\c s}ittirler,\\ her biri birine;\\ ve \gr{ABG} a{\c c}{\i}s{\i}\\ \gr{BGD} a{\c c}{\i}s{\i}na\\ e{\c s}ittir.\\ Dolay{\i}s{\i}yla \gr{AG} taban{\i} da\\ \gr{DB} taban{\i}na\\ e{\c s}ittir.\\ Dolay{\i}s{\i}yla \gr{ABG} {\"u}{\c c}geni de\\ \gr{BGD} {\"u}{\c c}genine\\ e{\c s}ittir. } \parsen{ Therefore the \gr{BG} diameter cuts in two\\ the \gr{ABGD} parallelogram;\\ \myqed } { 'ara BG di'ametroc d'iqa t'emnei\\ t`o ABGD parallhl'ogrammon;\\ <'oper >'edei de~ixai. } { Dolay{\i}s{\i}yla \gr{BG} k{\"o}{\c s}egeni ikiye b{\"o}ler\\ \gr{ABGD} paralelkenar{\i}n{\i};\\ \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.441875)(4.563125,1.441875) \psline[linewidth=0.04cm](4.1671877,1.198125)(0.7671875,1.198125) \psline[linewidth=0.04cm](0.7671875,1.198125)(0.3671875,-1.181875) \psline[linewidth=0.04cm](0.3671875,-1.181875)(3.7471876,-1.181875) \psline[linewidth=0.04cm](3.7471876,-1.181875)(4.1671877,1.198125) \psline[linewidth=0.04cm](4.1671877,1.198125)(0.3871875,-1.161875) \usefont{T1}{ptm}{m}{n} \rput(0.514375,1.183125){\gr A} \usefont{T1}{ptm}{m}{n} \rput(4.39375,1.263125){\gr B} \usefont{T1}{ptm}{m}{n} \rput(3.9671874,-1.296875){\gr D} \usefont{T1}{ptm}{m}{n} \rput(0.10828125,-1.236875){\gr G} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.35 \parsen{ Parallelograms\\ on the same base being\\ and in the same parallels\\ equal to one another are. } { T`a parallhl'ogramma\\ t`a >ep`i t~hc a>ut~hc b'asewc >'onta\\ ka`i >en ta~ic a>uta~ic parall'hloic\\ {}>'isa >all'hloic >est'in. } { Paralelkenarlar;\\ ayn{\i} tabanda olan\\ ve ayn{\i} paralellerde olanlar,\\ birbirlerine e{\c s}ittir. } \parsen{ Let there be\\ parallelograms\\ \gr{ABGD} and \gr{EBGD}\\ on the same base, \gr{GB},\\ and in the same parallels,\\ \gr{AZ} and \gr{BG}. } { >'Estw\\ parallhl'ogramma\\ t`a ABGD, EBGZ\\ {}>ep`i t~hc a>ut~hc b'asewc t~hc BG\\ ka`i >en ta~ic a>uta~ic parall'hloic\\ ta~ic AZ, BG; } { Verilmi{\c s} olsun\\ paralelkenarlar,\\ \gr{ABGD} ve \gr{EBGD},\\ ayn{\i} \gr{GB} taban{\i}nda,\\ ve ayn{\i}\\ \gr{AZ} ve \gr{BG} paralellerinde. } \parsen{ I say that\\ equal is\\ \gr{ABGD}\\ to the parallelogram \gr{EBGZ}. } { l'egw, <'oti\\ {}>'ison >est`i\\ t`o ABGD\\ t~w| EBGZ parallhlogr'ammw|. } { {\.I}ddia ediyorum ki\\ e{\c s}ittir\\ \gr{ABGD}\\ \gr{EBGZ} paralelkenar{\i}na. } \parsen{ For, since\\ a parallelogram is \gr{ABGD},\\ equal is \gr{AD} to \gr{BG}.\\ Similarly then also,\\ \gr{EZ} to \gr{BG} is equal;\\ so that also \gr{AD} to \gr{EZ} is equal;\\ and common [is] \gr{DE};\\ therefore \gr{AE}, as a whole,\\ to \gr{DZ}, as a whole,\\ is equal.\\ Is also \gr{AB} to \gr{DG} equal.\\ Then the two \gr{EA} and \gr{AB}\\ to the two \gr{ZD} and \gr{DG}\\ equal are\\ either to either;\\ also angle \gr{ZDG}\\ to \gr{EAB}\\ is equal,\\ the exterior to the interior;\\ therefore the base \gr{EB}\\ to the base \gr{ZG}\\ is equal,\\ and triangle \gr{EAB}\\ to triangle \gr{DZG}\\ equal will be;\\ suppose has been removed, commonly,\\ \gr{DHE};\\ therefore the trapezium \gr{ABHD} that remains\\ to the trapezium \gr{EHGZ} that remains\\ is equal;\\ let be added in common\\ the triangle \gr{HBG};\\ therefore the parallelogram \gr{ABGD} as a whole\\ to the parallelogram \gr{EBGZ} as a whole\\ is equal. } { >Epe`i g`ar\\ parallhl'ogramm'on >esti t`o ABGD,\\ {}>'ish >est`in ut`a d`h ka`i\\ estin >'ish;\\ <'wste ka`i estin >'ish;\\ ka`i koin`h 'ara estin >'ish.\\ {}>'esti d`e ka`i 'ish;\\ d'uo d`h a'isai e>is`in\\ estin >'ish\\ ekt`oc t~h| >ent'oc;\\ b'asic >'ara 'ish >est'in,\\ ka`i t`o EAB tr'igwnon\\ t~w| DZG trig'wnw|\\ {}>'ison >'estai;\\ koin`on >afh|r'hsjw t`o DHE;\\ loip`on >'ara t`o ABHD trap'ezion\\ loip~w| t~w| EHGZ trapez'iw|\\ {}>est`in >'ison;\\ koin`on proske'isjw t`o HBG tr'igwnon;\\ <'olon >'ara t`o ABGD parallhl'ogrammon\\ <'olw| t~w| EBGZ parallhlogr'ammw|\\ {}>'ison >est'in. } { {\c C}{\"u}nk{\"u}\\ bir paralelkenar oldu{\u g}undan \gr{ABGD},\\ e{\c s}ittir \gr{AD}, \gr{BG} kenar{\i}na.\\ Benzer {\c s}ekilde o zaman,\\ \gr{EZ}, \gr{BG} kenar{\i}na e{\c s}ittir;\\ b{\"o}ylece \gr{AD} da \gr{EZ} kenar{\i}na e{\c s}ittir;\\ ve ortakt{\i}r \gr{DE};\\ dolay{\i}s{\i}yla \gr{AE}, bir b{\"u}t{\"u}n olarak,\\ \gr{DZ} kenar{\i}na\\ e{\c s}ittir.\\ \gr{AB} da \gr{DG} kenar{\i}na e{\c s}ittir.\\ O zaman \gr{EA} ve \gr{AB} ikilisi\\ \gr{ZD} ve \gr{DG} ikilisine\\ e{\c s}ittirler\\ her biri birine;\\ ve \gr{ZDG} a{\c c}{\i}s{\i} da\\ \gr{EAB} a{\c c}{\i}s{\i}na\\ e{\c s}ittir,\\ d{\i}{\c s} a{\c c}{\i}, i{\c c} a{\c c}{\i}ya;\\ dolay{\i}s{\i}yla \gr{EB} taban{\i}\\ \gr{ZG} taban{\i}na\\ e{\c s}ittir,\\ ve \gr{EAB} {\"u}{\c c}geni\\ \gr{DZG} {\"u}{\c c}genine\\ e{\c s}it olacak;\\ kald{\i}r{\i}lm{\i}{\c s} olsun, ortak olarak,\\ \gr{DHE};\\ dolay{\i}s{\i}yla kalan \gr{ABHD} yamu{\u g}u\\ kalan \gr{EHGZ} yamu{\u g}una\\ e{\c s}ittir;\\ eklenmi{\c s} olsun her ikisine birden\\ \gr{HBG} {\"u}{\c c}geni;\\ dolay{\i}s{\i}yla \gr{ABGD} paralelkenar{\i}n{\i}n tamam{\i}\\ \gr{EBGZ} paralelkenar{\i}n{\i}n tamam{\i}na\\ e{\c s}ittir. } \parsen{ Therefore parallelograms\\ on the same base being\\ and in the same parallels\\ equal to one another are;\\ \myqed } { T`a >'ara parallhl'ogramma\\ t`a >ep`i t~hc a>ut~hc b'asewc >'onta\\ ka`i >en ta~ic a>uta~ic parall'hloic\\ {}>'isa >all'hloic >est'in;\\ <'oper >'edei de~ixai. } { Dolay{\i}s{\i}syla paralelkenarlar;\\ ayn{\i} tabanda olan\\ ve ayn{\i} paralellerde olanlar,\\ birbirlerine e{\c s}ittir;\\ \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.051875)(4.8871875,1.051875) \usefont{T1}{ptm}{m}{n} \rput(0.504375,-0.9015625){\gr B} \usefont{T1}{ptm}{m}{n} \rput(0.12375,0.6784375){\gr A} \usefont{T1}{ptm}{m}{n} \rput(2.0971875,0.8784375){\gr D} \usefont{T1}{ptm}{m}{n} \rput(2.6782813,-0.9015625){\gr G} \psline[linewidth=0.04cm](0.7171875,-0.7265625)(2.5371876,-0.7265625) \psline[linewidth=0.04cm](0.3171875,0.6934375)(2.1371875,0.6934375) \psline[linewidth=0.04cm](0.7171875,-0.7265625)(0.3171875,0.7134375) \psline[linewidth=0.04cm](2.5171876,-0.7265625)(2.1171875,0.7134375) \psline[linewidth=0.04cm](0.7171875,-0.7065625)(2.8371875,0.6934375) \psline[linewidth=0.04cm](2.5171876,-0.7265625)(4.5371876,0.6734375) \psline[linewidth=0.04cm](4.5171876,0.6734375)(2.0971875,0.6934375) \usefont{T1}{ptm}{m}{n} \rput(2.8792188,0.8684375){\gr E} \usefont{T1}{ptm}{m}{n} \rput(4.739844,0.6584375){\gr Z} \usefont{T1}{ptm}{m}{n} \rput(2.5203125,0.1584375){\gr H} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.36 \parsen{ Parallelograms\\ that are on equal bases\\ and in the same parallels\\ are equal to one another. } { T`a parallhl'ogramma\\ t`a >ep`i >'iswn b'asewn >'onta\\ ka`i >en ta~ic a>uta~ic parall'hloic\\ {}>'isa >all'hloic >est'in. } { Paralelkenarlar;\\ e{\c s}it tabanlarda olanlar\\ ve ayn{\i} paralellerde olanlar\\ e{\c s}ittirler birbirlerine. } \parsen{ Let there be\\ parallelograms\\ \gr{ABGD} and \gr{EZHJ}\\ on equal bases,\\ \gr{BG} and \gr{ZH},\\ and in the same parallels,\\ \gr{AJ} and \gr{BH}. } { >'Estw\\ parallhl'ogramma\\ t`a ABGD, EZHJ\\ {}>ep`i >'iswn b'asewn >'onta\\ t~wn BG, ZH\\ ka`i >en ta~ic a>uta~ic parall'hloic\\ ta~ic AJ, BH; } { Verilmi{\c s} olsun\\ paralelkenarlar\\ \gr{ABGD} ve \gr{EZHJ}\\ e{\c s}it\\ \gr{BG} ve \gr{ZH} tabanlar{\i}nda,\\ ve ayn{\i}\\ \gr{AJ} ve \gr{BH} paralellerinde. } \parsen{ I say that\\ equal is\\ parallelogram \gr{ABGD}\\ to \gr{EZHJ}. } { l'egw, <'oti\\ {}>'ison >est`i\\ t`o ABGD parallhl'ogrammon\\ t~w| EZHJ. } { {\.I}ddia ediyorum ki\\ e{\c s}ittir\\ \gr{ABGD},\\ \gr{EZHJ} paralelkenar{\i}na. } \parsen{ For, suppose have been joined\\ \gr{BE} and \gr{GJ}. } { >Epeze'uqjwsan g`ar\\ aepe`i >'ish >est`in all`a estin >'ish,\\ ka`i 'ara t~h| EJ >estin >'ish.\\ e>is`i d`e ka`i par'allhloi.\\ ka`i >epizeugn'uousin a>ut`ac a'isac te ka`i parall'hlouc >ep`i t`a a>ut`a m'erh >epizeugn'uousai\\ {}>'isai te ka`i par'allhlo'i e>isi\\ {}[ka`i a'ara\\ {}>'isai t'e e>isi ka`i par'allhloi].\\ parallhl'ogrammon >'ara >est`i t`o EBGJ.\\ ka'i >estin >'ison t~w| ABGD;\\ b'asin te g`ar a>ut~w| t`hn a>ut`hn >'eqei\\ t`hn BG,\\ ka`i >en ta~ic a>uta~ic parall'hloic\\ {}>est`in a>ut~w| ta~ic BG, AJ.\\ d`ia t`a a>ut`a d`h\\ ka`i t`o EZHJ t~w| a>ut~w| t~w| EBGJ\\ {}>estin >'ison;\\ <'wste ka`i t`o ABGD parallhl'ogrammon\\ t~w| EZHJ >estin >'ison. } { Ve e{\c s}it oldu{\u g}undan \gr{BG} ile \gr{ZH},\\ ama \gr{ZH}, \gr{EJ} kenar{\i}na e{\c s}ittir,\\ dolay{\i}s{\i}yla \gr{BG} da \gr{EJ} kenar{\i}na e{\c s}ittir.\\ Ve paraleldirler de.\\ Ayr{\i}ca \gr{EB} ve \gr{JG} onlar{\i} birle{\c s}tirir.\\ Ve e{\c s}it ve paralelleri ayn{\i} tarafta birle{\c s}tiren do{\u g}rular\\ e{\c s}it ve paraleldirler.\\ {}[Yine dolay{\i}s{\i}yla \gr{EB} ve \gr{HJ} \\e{\c s}it ve paraleldirler.]\\ Dolay{\i}s{\i}yla \gr{EBGJ} bir paralelkenard{\i}r.\\ Ve e{\c s}ittir \gr{ABGD} paralelkenar{\i}na.\\ {\c C}{\"u}nk{\"u} onunla ayn{\i},\\ \gr{BG} taban{\i} vard{\i}r,\\ ve onunla ayn{\i},\\ \gr{BG} ve \gr{AJ} paralellerindedir.\\ Ayn{\i} sebeple o {\c s}imdi,\\ \gr{EZHJ} da ona, [yani] \gr{EBGJ} paralelkenar{\i}na,\\ e{\c s}ittir;\\ b{\"o}ylece \gr{ABGD},\\ \gr{EZHJ} paralelkenar{\i}na e{\c s}ittir. } \parsen{ Therefore parallelograms\\ that are on equal bases\\ and in the same parallels\\ are equal to one another;\\ \myqed } { T`a >'ara parallhl'ogramma\\ t`a >ep`i >'iswn b'asewn >'onta\\ ka`i >en ta~ic a>uta~ic parall'hloic\\ {}>'isa >all'hloic >est'in;\\ <'oper >'edei de~ixai. } { Dolay{\i}s{\i}yla paralelkenarlar;\\ e{\c s}it tabanlarda olanlar\\ ve ayn{\i} paralellerde olanlar\\ e{\c s}ittirler birbirlerine;\\ \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.981875)(8.106563,1.981875) \usefont{T1}{ptm}{m}{n} \rput(0.10125,-1.7715625){\gr B} \usefont{T1}{ptm}{m}{n} \rput(0.560625,1.7684375){\gr A} \usefont{T1}{ptm}{m}{n} \rput(2.7140625,1.7684375){\gr D} \usefont{T1}{ptm}{m}{n} \rput(2.2751563,-1.8315625){\gr G} \usefont{T1}{ptm}{m}{n} \rput(4.296094,1.8084375){\gr E} \usefont{T1}{ptm}{m}{n} \rput(5.516719,-1.8115625){\gr Z} \usefont{T1}{ptm}{m}{n} \rput(7.9371877,-1.8115625){\gr H} \psline[linewidth=0.04cm](0.7340625,1.5834374)(6.5540624,1.5834374) \psline[linewidth=0.04cm](0.3340625,-1.6165625)(7.7340627,-1.6165625) \psline[linewidth=0.04cm](0.3340625,-1.6165625)(0.7540625,1.5834374) \psline[linewidth=0.04cm](7.7340627,-1.6165625)(6.5340624,1.6034375) \psline[linewidth=0.04cm](2.7140625,1.5634375)(2.3340626,-1.5965625) \psline[linewidth=0.04cm](2.3340626,-1.5965625)(6.5540624,1.5834374) \psline[linewidth=0.04cm](5.5340624,-1.6165625)(4.3540626,1.5834374) \psline[linewidth=0.04cm](0.3340625,-1.5965625)(4.3540626,1.5634375) \usefont{T1}{ptm}{m}{n} \rput(6.719375,1.7684375){\gr J} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.37 \parsen{ Triangles\\ that are on the same base\\ and in the same parallels\\ are equal to one another. } { T`a tr'igwna\\ t`a >ep`i t~hc a>ut~hc b'asewc >'onta\\ ka`i >en ta~ic a>uta~ic parall'hloic\\ {}>'isa >all'hloic >est'in. } { {\"U}{\c c}genler;\\ ayn{\i} tabanda\\ ve ayn{\i} paralellerde olanlar,\\ e{\c s}ittir birbirlerine. } \parsen{ Let there be\\ triangles \gr{ABG} and \gr{DBG},\\ on the same base \gr{BG}\\ and in the same parallels\\ \gr{AD} and \gr{BG}. } { >'Estw\\ tr'igwna t`a ABG, DBG\\ {}>ep`i t~hc a>ut~hc b'asewc t~hc BG\\ ka`i >en ta~ic a>uta~ic parall'hloic\\ ta~ic AD, BG; } { Verilmi{\c s} olsun\\ \gr{ABG} ve \gr{DBG} {\"u}{\c c}genleri,\\ ayn{\i} \gr{BG} taban{\i}nda\\ ve ayn{\i}\\ \gr{AD} ve \gr{BG} paralellerinde. } \parsen{ I say that\\ equal is\\ triangle \gr{ABG}\\ to triangle \gr{DBG}. } { l'egw, <'oti\\ {}>'ison >est`i\\ t`o ABG tr'igwnon\\ t~w| DBG trig'wnw|. } { {\.I}ddia ediyorum ki\\ e{\c s}ittir\\ \gr{ABG} {\"u}{\c c}geni\\ \gr{DBG} {\"u}{\c c}genine. } \parsen{ Suppose has been extended\\ \gr{AD} on both sides to \gr E and \gr Z,\\ and through \gr B,\\ parallel to \gr{GA}\\ has been drawn \gr{BE},\\ and through \gr G\\ parallel to \gr{BD}\\ has been drawn \gr{GZ}. } { >Ekbebl'hsjw\\ ef> ep`i t`a E, Z,\\ ka`i di`a m`en to~u B\\ t~h| GA par'allhloc\\ {}>'hqjw 'hqjw 'ara\\ {}>est`in isin >'isa;\\ {}>ep'i te g`ar t~hc a>ut~hc b'ase'wc e>isi\\ t~hc BG\\ ka`i >en ta~ic a>uta~ic parall'hloic\\ ta~ic BG, EZ;\\ ka'i >esti\\ to~u m`en EBGA parallhlogr'ammou\\ <'hmisu\\ t`o ABG tr'igwnon;\\ ut`o d'iqa t'emnei;\\ to~u d`e DBGZ parallhlogr'ammou\\ <'hmisu\\ t`o DBG tr'igwnon;\\ ut`o d'iqa t'emnei.\\ {}[t`a d`e t~wn >'iswn 'isa >all'hloic >est'in].\\ {}>'ison >'ara >est`i\\ t`o ABG tr'igwnon t~w| DBG trig'wnw|. } { Dolay{\i}s{\i}yla birer paralelkenard{\i}r\\ \gr{EBGA} ile \gr{DBGZ};\\ ve bunlar e{\c s}ittir;\\ ayn{\i} \\ \gr{BG} taban{\i}nda,\\ ve ayn{\i},\\ \gr{BG} ve \gr{EZ} paralellerinde olduklar{\i} i{\c c}in;\\ ve\\ \gr{EBGA} paralelkenar{\i}n{\i}n\\ yar{\i}s{\i}\\ --- \gr{ABG} {\"u}{\c c}genidir;\\ \gr{AB} k{\"o}{\c s}egeni onu ikiye kesti{\u g}i i{\c c}in;\\ ve \gr{DBGZ} paralelkenar{\i}n{\i}n\\ yar{\i}s{\i}\\ --- \gr{DBG} {\"u}{\c c}genidir;\\ \gr{DG} k{\"o}{\c s}egeni onu ikiye kesti{\u g}i i{\c c}in.\\ {}[Ve e{\c s}itlerin yar{\i}lar{\i}\\ e{\c s}ittirler birbirlerine.]\\ Dolay{\i}s{\i}yla e{\c s}ittir\\ \gr{ABG} {\"u}{\c c}geni \gr{DBG} {\"u}{\c c}genine. } \parsen{ Therefore triangles\\ that are on the same base\\ and in the same parallels\\ are equal to one another;\\ \myqed } { T`a >'ara tr'igwna\\ t`a >ep`i t~hc a>ut~hc b'asewc >'onta\\ ka`i >en ta~ic a>uta~ic parall'hloic\\ {}>'isa >all'hloic >est'in;\\ <'oper >'edei de~ixai. } { Dolay{\i}s{\i}yla {\"u}{\c c}genler;\\ ayn{\i} tabanda\\ ve ayn{\i} paralellerde olanlar,\\ e{\c s}ittir birbirlerine;\\ \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.801875)(6.5225,1.801875) \usefont{T1}{ptm}{m}{n} \rput(1.6196876,-1.5915625){\gr B} \usefont{T1}{ptm}{m}{n} \rput(2.3790624,1.5884376){\gr A} \usefont{T1}{ptm}{m}{n} \rput(3.7125,1.6284375){D} \usefont{T1}{ptm}{m}{n} \rput(4.5735936,-1.6515625){\gr G} \usefont{T1}{ptm}{m}{n} \rput(0.09453125,1.5084375){\gr E} \usefont{T1}{ptm}{m}{n} \rput(6.3751564,1.5084375){\gr Z} \psline[linewidth=0.04cm](0.3325,1.4034375)(6.1525,1.4034375) \psline[linewidth=0.04cm](1.7325,-1.3965625)(4.5525,-1.3965625) \psline[linewidth=0.04cm](4.5525,-1.3965625)(6.1525,1.4034375) \psline[linewidth=0.04cm](1.7525,-1.3965625)(0.3525,1.4034375) \psline[linewidth=0.04cm](1.7525,-1.3765625)(2.3325,1.4034375) \psline[linewidth=0.04cm](2.3325,1.4034375)(4.5525,-1.3965625) \psline[linewidth=0.04cm](4.5525,-1.3965625)(3.6725,1.4034375) \psline[linewidth=0.04cm](3.6925,1.4034375)(1.7525,-1.4165626) \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.38 \parsen{ Triangles\\ that are on equal bases\\ and in the same parallels\\ are equal to one another. } { T`a tr'igwna\\ t`a >ep`i >'iswn b'asewn >'onta\\ ka`i >en ta~ic a>uta~ic parall'hloic\\ >'isa >all'hloic >est'in. } { {\"U}{\c c}genler;\\ e{\c s}it tabanlarda\\ ve ayn{\i} paralelerde olanlar,\\ e{\c s}ittir birbirlerine. } \parsen{ Let there be\\ triangles \gr{ABG} and \gr{DEZ}\\ on equal bases \gr{BG} and \gr{EZ}\\ and in the same parallels\\ \gr{BZ} and \gr{AD}. } { >'Estw\\ tr'igwna t`a ABG, DEZ\\ >ep`i >'iswn b'asewn t~wn BG, EZ\\ ka`i >en ta~ic a>uta~ic parall'hloic\\ ta~ic BZ, AD; } { Verilmi{\c s} olsun\\ \gr{ABG} ve \gr{DEZ} {\"u}{\c c}genleri\\ e{\c s}it \gr{BG} ve \gr{EZ} tabanlar{\i}nda\\ ve ayn{\i} \\ \gr{BZ} ve \gr{AD} paralellerinde. } \parsen{ I say that\\ equal is\\ triangle \gr{ABG}\\ to triangle \gr{DEZ}. } { l'egw, <'oti\\ {}>'ison >est`i\\ t`o ABG tr'igwnon\\ t~w| DEZ trig'wnw|. } { {\.I}ddia ediyorum ki\\ e{\c s}ittir\\ \gr{ABG} {\"u}{\c c}geni\\ \gr{DEZ} {\"u}{\c c}genine. } \parsen{ For, suppose has been extended\\ \gr{AD} on both sides to \gr H and \gr J,\\ and through \gr B,\\ parallel to \gr{GA},\\ has been drawn \gr{BH},\\ and through \gr Z,\\ parallel to \gr{DE},\\ has been drawn \gr{ZJ}. } { >Ekbebl'hsjw g`ar\\ ef> ep`i t`a H, J,\\ ka`i di`a m`en to~u B\\ t~h| GA par'allhloc\\ >'hqjw 'hqjw 'ara\\ {}>est`in 'ison t`o HBGA t~w| DEZJ;\\ {}>ep'i te g`ar >'iswn b'ase'wn e>isi\\ t~wn BG, EZ\\ ka`i >en ta~ic a>uta~ic parall'hloic\\ ta~ic BZ, HJ;\\ ka'i >esti\\ to~u m`en HBGA parallhlogr'ammou\\ <'hmisu\\ t`o ABG tr'igwnon.\\ ut`o d'iqa t'emnei;\\ to~u d`e DEZJ parallhlogr'ammou\\ <'hmisu\\ t`o ZED tr'igwnon;\\ ut`o d'iqa t'emnei\\ {}[t`a d`e t~wn >'iswn 'isa >all'hloic >est'in].\\ {}>'ison >'ara >est`i\\ t`o ABG tr'igwnon t~w| DEZ trig'wnw|. } { Dolay{\i}s{\i}yla birer paralelkenard{\i}r\\ \gr{HBGA} ile \gr{DEZJ};\\ ve \gr{HBGA} e{\c s}ittir \gr{DEZJ} paralelkenar{\i}na;\\ e{\c s}it,\\ \gr{BG} ve \gr{EZ} tabanlar{\i}nda,\\ ve ayn{\i},\\ \gr{BZ} ve \gr{HJ} paralellerinde olduklar{\i} i{\c c}in;\\ ve\\ \gr{HBGA} paralelkenar{\i}n{\i}n\\ yar{\i}s{\i}\\ ---\gr{ABG} {\"u}{\c c}genidir.\\ \gr{AB} k{\"o}{\c s}egeni onu ikiye kesti{\u g}i i{\c c}in;\\ ve \gr{DEZJ} paralelkenar{\i}n{\i}n\\ yar{\i}s{\i}\\ --- \gr{ZED} {\"u}{\c c}genidir;\\ \gr{DZ} k{\"o}{\c s}egeni onu ikiye kesti{\u g}i i{\c c}in.\\ {}[Ve e{\c s}itlerin yar{\i}lar{\i}\\ e{\c s}ittirler birbirlerine.]\\ Dolay{\i}s{\i}yla e{\c s}ittir\\ \gr{ABG} {\"u}{\c c}geni \gr{DEZ} {\"u}{\c c}genine. } \parsen{ Therefore triangles\\ that are on equal bases\\ and in the same parallels\\ are equal to one another;\\ \myqed } { T`a >'ara tr'igwna\\ t`a >ep`i >'iswn b'asewn >'onta\\ ka`i >en ta~ic a>uta~ic parall'hloic\\ {}>'isa >all'hloic >est'in;\\ <'oper >'edei de~ixai. } { Dolay{\i}s{\i}yla {\"u}{\c c}genler;\\ e{\c s}it tabanlarda\\ ve ayn{\i} paralelerde olanlar,\\ e{\c s}ittir birbirlerine;\\ \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.7645313)(5.7840624,1.7645313) \usefont{T1}{ptm}{m}{n} \rput(0.10125,-1.5742188){\gr B} \usefont{T1}{ptm}{m}{n} \rput(2.360625,1.5857812){\gr A} \usefont{T1}{ptm}{m}{n} \rput(2.9140625,1.5857812){\gr D} \usefont{T1}{ptm}{m}{n} \rput(1.8551563,-1.6142187){\gr G} \usefont{T1}{ptm}{m}{n} \rput(3.5160937,-1.5942187){\gr E} \usefont{T1}{ptm}{m}{n} \rput(5.6367188,-1.5742188){\gr Z} \psline[linewidth=0.04cm](0.6740625,1.4007813)(4.8940625,1.4207813) \psline[linewidth=0.04cm](0.3140625,-1.3992188)(5.4940624,-1.3792187) \psline[linewidth=0.04cm](0.6940625,1.4007813)(0.3340625,-1.3992188) \psline[linewidth=0.04cm](0.3340625,-1.3992188)(2.2740624,1.3807813) \psline[linewidth=0.04cm](2.2740624,1.3807813)(1.8940625,-1.3992188) \psline[linewidth=0.04cm](4.8940625,1.4207813)(5.4740624,-1.3592187) \psline[linewidth=0.04cm](5.4740624,-1.3592187)(2.8740625,1.4207813) \psline[linewidth=0.04cm](3.5140624,-1.3792187)(2.8940625,1.4207813) \usefont{T1}{ptm}{m}{n} \rput(5.079375,1.5057813){\gr J} \usefont{T1}{ptm}{m}{n} \rput(0.4971875,1.5257813){\gr H} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.39 \parsen{ Equal triangles\\ that are on the same base\\ and in the same parts\\ are also in the same parallels. } { T`a >'isa tr'igwna\\ t`a >ep`i t~hc a>ut~hc b'asewc >'onta\\ ka`i >ep`i t`a a>ut`a m'erh\\ ka`i >en ta~ic a>uta~ic parall'hloic >est'in. } { E{\c s}it {\"u}{\c c}genler;\\ ayn{\i} tabanda \\ ve onun ayn{\i} taraf{\i}nda olan,\\ ayn{\i} paralellerdedirler de. } \parsen{ Let there be\\ equal triangles \gr{ABG} and \gr{DBG},\\ being on the same base\\ and on the same side of \gr{BG}. } { >'Estw\\ {}>'isa tr'igwna t`a ABG, DBG\\ {}>ep`i t~hc a>ut~hc b'asewc >'onta\\ ka`i >ep`i t`a a>ut`a m'erh t~hc BG; } { Verilmi{\c s} olsun\\ \gr{ABG} ve \gr{DBG} e{\c s}it {\"u}{\c c}genleri,\\ ayn{\i} \gr{BG} taban{\i}nda\\ ve onun ayn{\i} taraf{\i}nda olan . } \parsen{ I say that\\ they are also in the same parallels. } { l'egw, <'oti\\ ka`i >en ta~ic a>uta~ic parall'hloic >est'in. } { {\.I}ddia ediyorum ki\\ ayn{\i} paralellerdedirler de. } \parsen{ For suppose has been joined \gr{AD}. } { >Epeze'uqjw g`ar estin i g`ar m'h,\\ {}>'hqjw\\ di`a to~u A shme'iou\\ t~h| BG e>uje'ia| par'allhloc\\ epeze'uqjw 'ison >'ara >est`i\\ t`o ABG tr'igwnon\\ t~w| EBG trig'wnw|;\\ {}>ep'i te g`ar t~hc a>ut~hc b'ase'wc\\ {}>estin a>ut~w| t~hc BG\\ ka`i >en ta~ic a>uta~ic parall'hloic.\\ {}>all`a t`o ABG t~w| DBG >estin >'ison;\\ ka`i t`o DBG >'ara t~w| EBG >'ison >est`i\\ t`o me~izon t~w| >el'assoni;\\ <'oper >est`in >ad'unaton;\\ o>uk >'ara par'allhl'oc >estin ud> >'allh tic pl`hn t~hc AD;\\ 'ara t~h| BG >esti par'allhloc. } { {\c C}{\"u}nk{\"u} e{\u g}er de{\u g}il ise,\\ {\c c}izilmi{\c s} oldu{\u g}u varsay{\i}ls{\i}n\\ \gr A noktas{\i}ndan\\ \gr{BG} do{\u g}rusuna paralel\\ \gr{AE} do{\u g}rusunun,\\ ve birle{\c s}tirildi{\u g}i \gr{EG} do{\u g}rusunun.\\ E{\c s}ittir dolay{\i}s{\i}yla\\ \gr{ABG} {\"u}{\c c}geni\\ \gr{EBG} {\"u}{\c c}genine;\\ onunla ayn{\i}\\ \gr{BG} taban{\i}nda,\\ ve ayn{\i} paralellerde oldu{\u g}u i{\c c}in.\\ Ama \gr{ABG} e{\c s}ittir \gr{DBG} {\"u}{\c c}genine.\\ Ve dolay{\i}s{\i}yla \gr{DBG}, \gr{EBG} {\"u}{\c c}genine e{\c s}ittir,\\ b{\"u}y{\"u}k k{\"u}{\c c}{\"u}{\u g}e;\\ ki bu imkans{\i}zd{\i}r.\\ Dolay{\i}s{\i}yla paralel de{\u g}ildir \gr{AE}, \gr{BG} do{\u g}rusuna.\\ Benzer {\c s}ekilde o zaman g{\"o}sterece{\u g}iz ki\\ \gr{AD} d{\i}{\c s}{\i}ndakiler de paralel de{\u g}ildid ;\\ dolay{\i}s{\i}yla \gr{AD}, \gr{BG} do{\u g}rusuna paaraleldir. } \parsen{ Therefore equal triangles\\ that are on the same base\\ and in the same parts\\ are also in the same parallels;\\ \myqed } { T`a >'ara >'isa tr'igwna\\ t`a >ep`i t~hc a>ut~hc b'asewc >'onta\\ ka`i >ep`i t`a a>ut`a m'erh\\ ka`i >en ta~ic a>uta~ic parall'hloic >est'in;\\ <'oper >'edei de~ixai. } { Dolay{\i}s{\i}yla e{\c s}it {\"u}{\c c}genler;\\ ayn{\i} tabanda \\ ve onun ayn{\i} taraf{\i}nda olan,\\ ayn{\i} paralellerdedirler de;\\ \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.721875)(5.7890625,1.721875) \usefont{T1}{ptm}{m}{n} \rput(0.10125,-1.576875){\gr B} \usefont{T1}{ptm}{m}{n} \rput(0.440625,1.543125){\gr A} \usefont{T1}{ptm}{m}{n} \rput(5.0940623,1.543125){\gr D} \usefont{T1}{ptm}{m}{n} \rput(5.615156,-1.536875){\gr G} \usefont{T1}{ptm}{m}{n} \rput(3.8560936,0.483125){\gr E} \psline[linewidth=0.04cm](0.6740625,1.398125)(4.8940625,1.418125) \psline[linewidth=0.04cm](0.3140625,-1.401875)(5.4940624,-1.381875) \psline[linewidth=0.04cm](0.6940625,1.398125)(0.3340625,-1.401875) \psline[linewidth=0.04cm](4.8940625,1.418125)(5.4740624,-1.361875) \psline[linewidth=0.04cm](0.3340625,-1.381875)(4.9140625,1.418125) \psline[linewidth=0.04cm](5.4740624,-1.361875)(0.6940625,1.398125) \psline[linewidth=0.04cm](0.7140625,1.398125)(3.9140625,0.798125) \psline[linewidth=0.04cm](3.9140625,0.798125)(5.4740624,-1.381875) \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.40 \parsen{ Equal triangles\\ that are on equal bases\\ and in the same parts\\ are also in the same parallels. } { T`a >'isa tr'igwna\\ t`a >ep`i >'iswn b'asewn >'onta\\ ka`i >ep`i t`a a>ut`a m'erh\\ ka`i >en ta~ic a>uta~ic parall'hloic >est'in. } { E{\c s}it {\"u}{\c c}genler,\\ e{\c s}it tabanlarda\\ ve ayn{\i} tarafta olan,\\ ayn{\i} paralelerdedirler de. } \parsen{ Let there be\\ equal triangles \gr{ABG} and \gr{GDE},\\ on equal bases \gr{BG} and \gr{GE},\\ and in the same parts. } { >'Estw\\ {}>'isa tr'igwna t`a ABG, GDE\\ {}>ep`i >'iswn b'asewn t~wn BG, GE\\ ka`i >ep`i t`a a>ut`a m'erh. } { Verilmi{\c s} olsun\\ e{\c s}it \gr{ABG} ve \gr{GDE} {\"u}{\c c}genleri,\\ e{\c s}it \gr{BG} ve \gr{GE} tabanlar{\i}nda,\\ ve ayn{\i} tarafta olan. } \parsen{ I say that\\ they are also in the same parallels. } { l'egw, <'oti\\ ka`i >en ta~ic a>uta~ic parall'hloic >est'in. } { {\.I}ddia ediyorum ki\\ ayn{\i} paralellerdedirler de. } \parsen{ For suppose \gr{AD} has been joined. } { >Epeze'uqjw g`ar estin i g`ar m'h,\\ {}>'hqjw\\ di`a to~u A\\ t~h| BE par'allhloc\\ epeze'uqjw 'ison >'ara >est`i\\ t`o ABG tr'igwnon\\ t~w| ZGE trig'wnw|;\\ {}>ep'i te g`ar >'iswn b'ase'wn e>isi\\ t~wn BG, GE\\ ka`i >en ta~ic a>uta~ic parall'hloic\\ ta~ic BE, AZ.\\ {}>all`a t`o ABG tr'igwnon\\ {}>'ison >est`i t~w| DGE [tr'igwnw|];\\ ka`i t`o DGE >'ara [tr'igwnon]\\ {}>'ison >est`i t~w| ZGE trig'wnw|\\ t`o me~izon t~w| >el'assoni;\\ <'oper >est`in >ad'unaton;\\ o>uk >'ara par'allhloc ud> >'allh tic pl`hn t~hc AD;\\ 'ara t~h| BE >esti par'allhloc. } { {\c C}{\"u}nk{\"u} e{\u g}er de{\u g}il ise,\\ varsay{\i}ls{\i}n birle{\c s}tirildi{\u g}i\\ \gr A noktas{\i}ndan,\\ \gr{BE} do{\u g}rusuna paralel,\\ \gr{AZ} do{\u g}rusunun,\\ ve birle{\c s}tirildi{\u g}i \gr{ZE} do{\u g}rusunun.\\ Dolay{\i}s{\i}yla e{\c s}ittir\\ \gr{ABG} {\"u}{\c c}geni\\ \gr{ZGE} {\"u}{\c c}genine;\\ e{\c s}it,\\ \gr{BG} ve \gr{GE} tabanlar{\i}nda,\\ ve ayn{\i},\\ \gr{BE} ve \gr{AZ} paralellerinde olduklar{\i} i{\c c}in.\\ Fakat \gr{ABG} {\"u}{\c c}geni\\ e{\c s}ittir \gr{DGE} {\"u}{\c c}genine;\\ ve dolay{\i}s{\i}yla \gr{DGE} {\"u}{\c c}genini\\ e{\c s}ittir \gr{ZGE} {\"u}{\c c}genine,\\ b{\"u}y{\"u}k k{\"u}{\c c}{\"u}{\u g}e;\\ ki bu imkans{\i}zd{\i}r.\\ Dolay{\i}s{\i}yla paralel de{\u g}ildir \gr{AZ}, \gr{BE} do{\u g}rusuna.\\ Benzer {\c s}ekilde o zaman g{\"o}sterece{\u g}iz ki\\ \gr{AD} d{\i}{\c s}{\i}ndakiler de paralel de{\u g}ildir;\\ dolay{\i}s{\i}yla \gr{AD}, \gr{BE} do{\u g}rusuna paraleldir. } \parsen{ Therefore equal triangles\\ that are on equal bases\\ and in the same parts\\ are also in the same parallels;\\ \myqed } { T`a >'ara >'isa tr'igwna\\ t`a >ep`i >'iswn b'asewn >'onta\\ ka`i >ep`i t`a a>ut`a m'erh\\ ka`i >en ta~ic a>uta~ic parall'hloic >est'in;\\ <'oper >'edei de~ixai. } { Dolay{\i}s{\i}yla e{\c s}it {\"u}{\c c}genler,\\ e{\c s}it tabanlarda\\ ve ayn{\i} tarafta olan,\\ ayn{\i} paralelerdedirler de;\\ \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.7245313)(5.845625,1.7245313) \usefont{T1}{ptm}{m}{n} \rput(0.10125,-1.5742188){\gr B} \usefont{T1}{ptm}{m}{n} \rput(0.440625,1.5457813){\gr A} \usefont{T1}{ptm}{m}{n} \rput(5.0940623,1.5457813){\gr D} \usefont{T1}{ptm}{m}{n} \rput(2.8751562,-1.5742188){\gr G} \usefont{T1}{ptm}{m}{n} \rput(5.6960936,-1.5142188){\gr E} \psline[linewidth=0.04cm](0.6740625,1.4007813)(4.8940625,1.4207813) \psline[linewidth=0.04cm](0.3140625,-1.3992188)(5.4940624,-1.3792187) \psline[linewidth=0.04cm](0.6940625,1.4007813)(0.3340625,-1.3992188) \psline[linewidth=0.04cm](4.8940625,1.4207813)(5.4740624,-1.3592187) \psline[linewidth=0.04cm](2.9140625,-1.3992188)(4.9140625,1.4207813) \psline[linewidth=0.04cm](2.8940625,-1.3792187)(0.6940625,1.4007813) \psline[linewidth=0.04cm](0.7140625,1.4007813)(4.4740624,0.82078123) \psline[linewidth=0.04cm](4.4740624,0.82078123)(5.4740624,-1.3592187) \usefont{T1}{ptm}{m}{n} \rput(4.036719,0.6457813){\gr Z} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.41 \parsen{ If a parallelogram\\ have the same base as a triangle,\\ and be in the same parallels,\\ double is\\ the parallelogram of the triangle. } { >E`an parallhl'ogrammon\\ trig'wnw| b'asin te >'eqh| t`hn a>ut`hn\\ ka`i >en ta~ic a>uta~ic parall'hloic >~h|,\\ dipl'asi'on >est'i\\ t`o parallhl'ogrammon to~u trig'wnou. } { E{\u g}er bir paralelkenar\\ bir {\"u}{\c c}genle ayn{\i} tabana sahipse,\\ ve ayn{\i} paralelerdeyse,\\ iki kat{\i}d{\i}r\\ paralelkenar, {\"u}{\c c}genin. } \parsen{ For, the parallelogram \gr{ABGD}\\ as the triangle \gr{EBG},\\ ---suppose it has the same base, \gr{BG},\\ and is in the same parallels,\\ \gr{BG} and \gr{AE}. } { Parallhl'ogrammon g`ar t`o ABGD\\ trig'wnw| t~w| EBG\\ b'asin te >eq'etw t`hn a>ut`hn t`hn BG\\ ka`i >en ta~ic a>uta~ic parall'hloic >'estw\\ ta~ic BG, AE; } { {\c C}{\"u}nk{\"u} \gr{ABGD} paralelkenar{\i}n{\i}n\\ \gr{EBG} {\"u}{\c c}geniyle,\\ ---ayn{\i} \gr{BG} taban{\i} oldu{\u g}u varsay{\i}ls{\i}n,\\ ve ayn{\i}\\ \gr{BG} ve \gr{AE} paralelerinde olduklar{\i}. } \parsen{ I say that\\ double is\\ the parallelogram \gr{ABGD}\\ of the triangle \gr{BEG}. } { l'egw, <'oti\\ dipl'asi'on >esti\\ t`o ABGD parallhl'ogrammon\\ to~u BEG trig'wnou. } { {\.I}ddia ediyorum ki\\ iki kat{\i}d{\i}r\\ \gr{ABGD} paralelkenar{\i}\\ \gr{BEG} {\"u}{\c c}geninin. } \parsen{ For, suppose \gr{AG} has been joined. } { >Epeze'uqjw g`ar 'ison d'h >esti t`o ABG tr'igwnon\\ t~w| >EBG trig'wnw|;\\ {}>ep'i te g`ar t~hc a>ut~hc b'ase'wc >estin a>ut~w|\\ t~hc BG\\ ka`i >en ta~ic a>uta~ic parall'hloic\\ ta~ic BG, AE.\\ {}>all`a t`o >ABGD parallhl'ogrammon\\ dipl'asi'on >esti to~u ABG trig'wnou;\\ AG di'ametroc a>ut`o d'iqa t'emnei;\\ <'wste t`o ABGD parallhl'ogrammon\\ {}>ka`i to~u EBG trig'wnou >est`i dipl'asion. } { E{\c s}ittir \gr{ABG} {\"u}{\c c}geni\\ \gr{EBG} {\"u}{\c c}genine;\\ onunla ayn{\i},\\ \gr{BG} taban{\i}na sahip,\\ ve ayn{\i}\\ \gr{BG} ve \gr{AE} paralelerinde oldu{\u g}u i{\c c}in.\\ Fakat \gr{ABGD} paralelkenar{\i}\\ iki kat{\i}d{\i}r \gr{ABG} {\"u}{\c c}geninin;\\ \gr{AG} k{\"o}{\c s}egeni onu ikiye kesti{\u g}inden;\\ b{\"o}ylece \gr{ABGD} paralelkenar{\i} da\\ \gr{EBG} {\"u}{\c c}geninin iki kat{\i}d{\i}r. } \parsen{ Therefore, if a parallelogram\\ have the same base as a triangle,\\ and be in the same parallels,\\ double is\\ the parallelogram of the triangle;\\ \myqed } { >E`an >'ara parallhl'ogrammon\\ trig'wnw| b'asin te >'eqh| t`hn a>ut`hn\\ ka`i >en ta~ic a>uta~ic parall'hloic >~h|,\\ dipl'asi'on >est'i\\ t`o parallhl'ogrammon to~u trig'wnou;\\ <'oper >'edei de~ixai. } { Dolay{\i}s{\i}yla, e{\u g}er bir paralelkenar\\ bir {\"u}{\c c}genle ayn{\i} tabana sahipse,\\ ve ayn{\i} paralelerdeyse,\\ iki kat{\i}d{\i}r\\ paralelkenar, {\"u}{\c c}genin;\\ \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.521875)(5.44875,1.521875) \usefont{T1}{ptm}{m}{n} \rput(0.924375,-1.3515625){\gr B} \usefont{T1}{ptm}{m}{n} \rput(0.12375,1.3084375){\gr A} \usefont{T1}{ptm}{m}{n} \rput(3.1171875,1.3484375){\gr D} \usefont{T1}{ptm}{m}{n} \rput(4.278281,-1.3715625){\gr G} \usefont{T1}{ptm}{m}{n} \rput(5.2992187,1.2884375){\gr E} \psline[linewidth=0.04cm](1.0971875,-1.2165625)(4.0971875,-1.1965625) \psline[linewidth=0.04cm](4.0971875,-1.1965625)(3.0971875,1.1834375) \psline[linewidth=0.04cm](3.0971875,1.1834375)(0.2971875,1.1834375) \psline[linewidth=0.04cm](0.2971875,1.1834375)(1.0971875,-1.2165625) \psline[linewidth=0.04cm](1.0971875,-1.2165625)(5.0971875,1.1834375) \psline[linewidth=0.04cm](5.0971875,1.1834375)(3.0571876,1.1834375) \psline[linewidth=0.04cm](4.0771875,-1.1965625)(5.0971875,1.1834375) \psline[linewidth=0.04cm](4.0971875,-1.1765625)(0.2971875,1.1834375) \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.42 \parsen{ To the given triangle equal,\\ a parallelogram to construct\\ in the given rectilineal angle. } { T~w| doj'enti trig'wnw| >'ison\\ parallhl'ogrammon sust'hsasjai\\ {}>en t~h| doje'ish| gwn'ia| e>ujugr'ammw|. } { Verilen bir {\"u}{\c c}gene e{\c s}it,\\ bir paralelkenar{\i}\\ verilen bir d{\"u}zkenar a{\c c}{\i}da in{\c s}a etmek. } \parsen{ Let be\\ the given triangle \gr{ABG},\\ and the given rectilineal angle, \gr D. } { >'Estw\\ t`o m`en doj`en tr'igwnon t`o ABG,\\ uj'ugrammoc 'ison\\ parallhl'ogrammon sust'hsasjai\\ {}>en t~h| D gwn'ia| e>ujugr'ammw|. } { {\c S}imdi gerklidir\\ \gr{ABG} {\"u}{\c c}genine e{\c s}it\\ bir paralelkenar{\i}n\\ \gr D d{\"u}zkenar a{\c c}{\i}s{\i}na in{\c s}a edilmesi. } \parsen{ Suppose \gr{BG} has been cut in two at \gr E,\\ and there has been joined \gr{AE},\\ and there has been constructed\\ on the \strgt\ \gr{EG},\\ and at the point \gr E on it,\\ to angle \gr D equal,\\ \gr{GEZ},\\ also, through \gr A, parallel to \gr{EG},\\ suppose \gr{AH} has been drawn,\\ and through \gr G, parallel to \gr{EZ},\\ suppose \gr{GH} has been drawn;\\ therefore a parallelogram is \gr{ZEGH}. } { Tetm'hsjw epeze'uqjw uje'ia|\\ ka`i t~w| pr`oc a>ut~h| shme'iw| t~w| E\\ t~h| D gwn'ia| >'ish\\ 'hqjw 'hqjw 'ara >est`i t`o ZEGH. } { Varsay{\i}ls{\i}n \gr{BG} kenar{\i}n{\i}n \gr E noktas{\i}nda ikiye kesildi{\u g}i\\ ve \gr{AE} do{\u g}rusunun birle{\c s}tirildi{\u g}i,\\ ve in{\c s}a edildi{\u g}i\\ \gr{EG} do{\u g}rusunda,\\ ve {\"u}zerindeki\gr E noktas{\i}nda,\\ \gr D a{\c c}{\i}s{\i}na e{\c s}it,\\ \gr{GEZ} a{\c c}{\i}s{\i}n{\i}n,\\ ayr{\i}ca, \gr A noktas{\i}ndan, \gr{EG} do{\u g}rusuna paralel,\\ \gr{AH} do{\u g}rusunun {\c c}izilmi{\c s} oldu{\u g}u varsay{\i}ls{\i}n,\\ ve \gr G noktas{\i}ndan, \gr{EZ} do{\u g}rusuna paralel,\\ \gr{GH} do{\u g}rusunun {\c c}izilmi{\c s} oldu{\u g}u varsay{\i}ls{\i}n;\\ dolay{\i}s{\i}yla \gr{ZEGH} bir paralelkenard{\i}r. } \parsen{ And since equal is \gr{BE} to \gr{EG},\\ equal is also\\ triangle \gr{ABE} to triangle \gr{AEG};\\ for they are on equal bases,\\ \gr{BE} and \gr{EG},\\ and in the same parallels,\\ \gr{BG} and \gr{AH};\\ double therefore is\\ triangle \gr{ABG} of triangle \gr{AEG}.\\ also is\\ parallelogram \gr{ZEGH}\\ double of triangle \gr{AEG};\\ for it has the same base as it,\\ and\\ is in the same parallels as it;\\ therefore is equal\\ the parallelogram \gr{ZEGH}\\ to the triangle \gr{ABG}.\\ And it has angle \gr{GEZ}\\ equal to the given \gr D. } { ka`i >epe`i >'ish >est`in 'ison >est`i ka`i\\ t`o ABE tr'igwnon t~w| AEG trig'wnw|;\\ {}>ep'i te g`ar >'iswn b'ase'wn e>isi\\ t~wn BE, EG\\ ka`i >en ta~ic a>uta~ic parall'hloic\\ ta~ic BG, AH;\\ dipl'asion >'ara >est`i\\ t`o ABG tr'igwnon to~u AEG trig'wnou.\\ {}>'esti d`e ka`i\\ t`o ZEGH parallhl'ogrammon\\ dipl'asion to~u AEG trig'wnou;\\ b'asin te g`ar a>ut~w| t`hn a>ut`hn >'eqei\\ ka`i\\ {}>en ta~ic a>uta~ic >estin a>ut~w| parall'hloic;\\ {}>'ison >'ara >est`i\\ t`o ZEGH parallhl'ogrammon\\ t~w| ABG trig'wnw|.\\ ka`i >'eqei t`hn 'ishn t~h| doje'ish| t~h| D. } { Ve e{\c s}it oldu{\u g}undan \gr{BE}, \gr{EG} do{\u g}rusuna,\\ e{\c s}ittir\\ \gr{ABE} {\"u}{\c c}geni de \gr{AEG} {\"u}{\c c}genine;\\ tabanlar{\i}\\ \gr{BE} ve \gr{EG} e{\c s}it,\\ ve ayn{\i} \\ \gr{BG} ve \gr{AH} paralelerinde olduklar{\i} i{\c c}in;\\ iki kat{\i}d{\i}r dolay{\i}s{\i}yla\\ \gr{ABG} {\"u}{\c c}geni \gr{AEG} {\"u}{\c c}geninin,\\ ayr{\i}ca\\ \gr{ZEGH} paralelkenar{\i}\\ iki kat{\i}d{\i}r \gr{AEG} {\"u}{\c c}geninin;\\ onunla ayn{\i} taban{\i} oldu{\u g}u,\\ ve\\ onunla ayn{\i} paralellerde oldu{\u g}u i{\c c}in;\\ dolay{\i}s{\i}yla e{\c s}ittir\\ \gr{ZEGH} paralelkenar{\i}\\ \gr{ABG} {\"u}{\c c}genine.\\ Ve onun \gr{GEZ} a{\c c}{\i}s{\i}\\ e{\c s}ittir verilen \gr D a{\c c}{\i}s{\i}na. } \parsen{ Therefore, to the given triangle \gr{ABG}\\ equal,\\ a parallelogram has been constructed,\\ \gr{ZEGH},\\ in the angle \gr{GEZ},\\ which is equal to \gr D;\\ \myqef } { T~w| >'ara doj'enti trig'wnw| t~w| ABG\\ {}>'ison\\ parallhl'o\-gram\-mon sun'estatai\\ t`o ZEGH\\ {}>en gwn'ia| t~h| est`in >'ish t~h| D;\\ <'oper >'edei poi~hsai. } { Dolay{\i}s{\i}yla, verilen \gr{ABG} {\"u}{\c c}genine\\ e{\c s}it,\\ bir paralelkenar,\\ \gr{ZEGH}, in{\c s}a edilmi{\c s} oldu\\ \gr{GEZ} a{\c s}{\i}s{\i}nda,\\ \gr D a{\c s}{\i}s{\i}na e{\c s}it olan;\\ \ozqef } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.3792187)(4.2665625,1.3792187) \usefont{T1}{ptm}{m}{n} \rput(2.7967188,1.2057812){\gr Z} \usefont{T1}{ptm}{m}{n} \rput(1.100625,1.1657813){\gr A} \usefont{T1}{ptm}{m}{n} \rput(0.10125,-1.1942188){\gr B} \usefont{T1}{ptm}{m}{n} \rput(2.2340624,-0.5342187){\gr D} \usefont{T1}{ptm}{m}{n} \rput(3.1351562,-1.1942188){\gr G} \pspolygon[linewidth=0.04](1.2540625,1.0407813)(0.2740625,-1.0192188)(3.0340624,-1.0192188)(3.9340625,1.0007813)(2.1740625,1.0207813) \psline[linewidth=0.04](3.0340624,-1.0192188)(1.2740625,1.0207813)(1.8340625,-1.0192188)(2.6340625,1.0207813) \usefont{T1}{ptm}{m}{n} \rput(1.8760937,-1.2342187){\gr E} \usefont{T1}{ptm}{m}{n} \rput(4.0971875,1.1857812){\gr H} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.43 \parsen{ Of any parallelogram,\\ of the parallelograms about the diameter,\\ the complements\\ are equal to one another. } { Pant`oc parallhlogr'ammou\\ t~wn per`i t`hn di'ametron parallhlogr'ammwn\\ t`a paraplhr'wmata\\ {}>'isa >all'hloic >est'in. } { Herhangi bir paralelkenar{\i}n,\\ k{\"o}{\c s}egeni etraf{\i}ndaki paralelkenarlar{\i}n,\\ t{\"u}mleyenleri\\ e{\c s}ittir birbirlerine. } \parsen{ Let there be\\ a parallelogram \gr{ABGD},\\ and its diameter, \gr{AG},\\ and about \gr{AG}\\ let be parallelograms,\\ \gr{EJ} and \gr{ZH},\footnotemark\\ and the so-called\footnotemark\ complements,\\ \gr{BK} and \gr{KD}. } { >'Estw\\ parallhl'ogrammon t`o ABGD,\\ di'ametroc d`e a>uto~u 'estw\\ t`a EJ, ZH,\\ t`a d`e leg'omena paraplhr'wmata\\ t`a BK, KD; } { Verilmi{\c s} olsun\\ bir \gr{ABGD} paralelkenar{\i},\\ ve onun \gr{AG} k{\"o}{\c s}egeni,\\ ve \gr{AG} etraf{\i}nda\\ paralelkenarlar,\\ \gr{EJ} ve \gr{ZH},\\ ve bunlar{\i}n t{\"u}mleyenleri,\\ \gr{BK} ile \gr{KD}. } \myfntext{Here Euclid can use two letters without qualification for a parallelogram, because they are not unqualified in the Greek: they take the neuter article, while a line takes the feminine article.} \myfntext{This is Heath's translation. The Greek does not require anything corresponding to `so-'. The LSJ lexicon \cite{LSJ} gives the present proposition as the original geometrical use of \gr{parapl'hrwma}---other meanings are `expletive' and a certain flowering herb.} \parsen{ I say that\\ equal is the complement \gr{BK}\\ to the complement \gr{KD}. } { l'egw, <'oti\\ {}>'ison >est`i t`o BK parapl'hrwma\\ t~w| KD paraplhr'wmati. } { {\.I}ddia ediyorum\\ e{\c s}ittir \gr{BK} t{\"u}mleyeni\\ \gr{KD} t{\"u}mleyenine. } \parsen{ For, since a parallelogram is\\ \gr{ABGD},\\ and its diameter, \gr{AG},\\ equal is\\ triangle \gr{ABG} to triangle \gr{AGD}.\\ Moreover, since a parallelogram is\\ \gr{EJ},\\ and its diameter, \gr{AK},\\ equal is\\ triangle \gr{AEK} to triangle \gr{AJK}.\\ Then for the same [reasons] also\\ triangle \gr{KZG} to \gr{KHG} is equal.\\ Since then triangle \gr{AEK}\\ is equal to triangle \gr{AJK},\\ and \gr{KZG} to \gr{KHG},\\ triangle \gr{AEK} with \gr{KHG}\\ is equal\\ to triangle \gr{AJK} with \gr{KZG};\\ also is triangle \gr{ABG}, as a whole,\\ equal to \gr{ADG}, as a whole;\\ therefore the complement \gr{BK} remaining\\ to the complement \gr{KD} remaining\\ is equal. } { >Epe`i g`ar parallhl'ogramm'on >esti\\ t`o ABGD,\\ di'ametroc d`e a>uto~u 'ison >est`i\\ t`o ABG tr'igwnon t~w| AGD trig'wnw|.\\ p'alin, >epe`i parallhl'ogramm'on >esti\\ t`o EJ,\\ di'ametroc d`e a>uto~u >estin 'ison >est`i\\ t`o AEK tr'igwnon t~w| AJK trig'wnw|.\\ di`a t`a a>ut`a d`h ka`i\\ t`o KZG tr'igwnon t~w| KHG >estin >'ison.\\ {}>epe`i o>~un t`o m`en AEK tr'igwnon\\ t~w| AJK trig'wnw| >est`in >'ison,\\ t`o d`e KZG t~w| KHG,\\ t`o AEK tr'igwnon met`a to~u KHG\\ {}>'ison >est`i\\ t~w| AJK trig'wnw| met`a to~u KZG;\\ {}>'esti d`e ka`i <'olon t`o ABG tr'igwnon\\ <'olw| t~w| ADG >'ison;\\ loip`on >'ara t`o BK parapl'hrwma\\ loip~w| t~w| KD paraplhr'wmat'i\\ {}>estin >'ison. } { {\c C}{\"u}nk{\"u}, bir paralelkenar oldu{\u g}undan\\ \gr{ABGD},\\ ve \gr{AG}, onun k{\"o}{\c s}egeni,\\ e{\c s}ittir\\ \gr{ABG} {\"u}{\c c}geni \gr{AGD} {\"u}{\c c}genine.\\ Dahas{\i}, bir paralelkenar oldu{\u g}undan\\ \gr{EJ},\\ \gr{AK},onun k{\"o}{\c s}egeni,\\ e{\c s}ittir\\ \gr{AEK} {\"u}{\c c}geni \gr{AJK}{\"u}{\c c}genine.\\ {\c S}imdi ayn{\i} nedenle\\ \gr{KZG} e{\c s}ittir \gr{KHG} {\"u}{\c c}genine.\\ O zaman \gr{AEK}\\ e{\c s}it oldu{\u g}undan \gr{AJK} {\"u}{\c c}genine,\\ ve \gr{KZG}, \gr{KHG} {\"u}{\c c}genine,\\ \gr{AEK} ile \gr{KHG} {\"u}{\c c}genleri\\ e{\c s}ittirl\\ \gr{AJK} ile \gr{KZG} {\"u}{\c c}genlerine;\\ ayr{\i}ca \gr{ABG} {\"u}{\c c}geninin t{\"u}m{\"u}\\ e{\c s}ittir \gr{ADG} {\"u}{\c c}geninin t{\"u}m{\"u}ne;\\ dolay{\i}s{\i}yla geriye kalan \gr{BK} t{\"u}mleyeni,\\ geriye kalan \gr{KD} t{\"u}mleyenine\\ e{\c s}ittir. } \parsen{ Therefore, of any parallelogram area,\\ of the about-the-diameter\\ parallelograms,\\ the complements\\ are equal to one another;\\ \myqed } { Pant`oc >'ara parallhlogr'ammou qwr'iou\\ t~wn per`i t`hn di'ametron\\ parallhlogr'ammwn\\ t`a paraplhr'wmata\\ {}>'isa >all'h\-loic >est'in;\\ <'oper >'edei de~ixai. } { Dolay{\i}s{\i}yla, herhangi bir paralelkenar{\i}n,\\ k{\"o}{\c s}egeni etraf{\i}ndaki paralelkenarlar{\i}n,\\ t{\"u}mleyenleri\\ e{\c s}ittir birbirlerine;\\ \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.591875)(5.3603125,1.591875) \usefont{T1}{ptm}{m}{n} \rput(0.10125,-1.3215625){\gr B} \usefont{T1}{ptm}{m}{n} \rput(0.840625,1.3784375){\gr A} \usefont{T1}{ptm}{m}{n} \rput(5.1940627,1.3584375){\gr D} \usefont{T1}{ptm}{m}{n} \rput(4.3151565,-1.3615625){\gr G} \usefont{T1}{ptm}{m}{n} \rput(0.5960938,0.7184375){\gr E} \usefont{T1}{ptm}{m}{n} \rput(5.036719,0.5984375){Z} \psline[linewidth=0.04cm](1.0140625,1.2334375)(5.0140624,1.2334375) \psline[linewidth=0.04cm](5.0140624,1.2334375)(4.2140627,-1.1665626) \psline[linewidth=0.04cm](4.1940627,-1.1665626)(0.2140625,-1.1465625) \psline[linewidth=0.04cm](0.2140625,-1.1465625)(1.0340625,1.2534375) \psline[linewidth=0.04cm](1.0340625,1.2134376)(4.2140627,-1.1665626) \psline[linewidth=0.04cm](0.8340625,0.6334375)(4.8140626,0.6534375) \psline[linewidth=0.04cm](2.0140624,1.2134376)(1.1740625,-1.1465625) \usefont{T1}{ptm}{m}{n} \rput(2.0771875,1.4184375){\gr J} \usefont{T1}{ptm}{m}{n} \rput(1.059375,-1.4415625){\gr H} \usefont{T1}{ptm}{m}{n} \rput(2.0826561,0.7984375){\gr K} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.44 \parsen{ Along the given \strgt,\\ equal to the given triangle,\\ to apply a parallelogram\\ in the given rectilineal angle. } { Par`a t`hn doje~isan e>uje~ian\\ t~w| doj'enti trig'wnw| >'ison\\ parallhl'ogrammon parabale~in\\ {}>en t~h| doje'ish| gwn'ia| e>ujugr'am\-mw|. } { Verilen bir do{\u g}ru boyunca\\ verilen bir {\"u}{\c c}gene e{\c s}it,\\ bir paralel kenar{\i} yerle{\c s}tirmek\\ verilen bir d{\"u}z kenar a{\c c}{\i}da. } \parsen{ Let be\\ the given \strgt\ \gr{AB},\\ and the given triangle, \gr G,\\ and the given rectilineal angle, \gr D. } { >'Estw\\ uje~ia uj'ugrammoc uje~ian t`hn AB\\ t~w| doj'enti trig'wnw| t~w| G >'ison\\ parallhl'ogrammon parabale~in\\ {}>en >'ish| t~h| D gwn'ia|. } { {\c S}imdi gereklidir\\ verilen \gr{AB} do{\u g}rusu boyunca\\ \gr G {\"u}{\c c}genine e{\c s}it\\ bir paralelkenar{\i}\\ \gr D a{\c c}{\i}s{\i}nda yerle{\c s}tirmek. } \parsen{ Suppose has been constructed\\ equal to triangle \gr G,\\ a parallelogram \gr{BEZH}\\ in angle \gr{EBH},\\ which is equal to \gr D;\\ and let it be laid down\\ so that on a \strgt\ is \gr{BE}\\ with \gr{AB},\\ and suppose has been drawn through\\ \gr{ZH} to \gr J,\\ and through \gr A,\\ parallel to either of \gr{BH} and \gr{EZ},\\ suppose there has been drawn\\ \gr{AJ},\\ and suppose there has been joined\\ \gr{JB}. } { Sunest'atw\\ t~w| G trig'wnw| >'ison\\ parallhl'ogrammon t`o BEZH\\ {}>en gwn'ia| t~h| estin >'ish t~h| D;\\ ka`i ke'isjw\\ <'wste >ep> e>uje'iac e>~inai t`hn BE\\ t~h| AB,\\ ka`i di'hqjw\\ ep`i t`o J,\\ ka`i di`a to~u A\\ 'hqjw epeze'uqjw epe`i e>ic parall'hlouc t`ac AJ, EZ\\ e>uje~ia >en'epesen 'ara orja~ic e>isin >'isai.\\ a'ara orj~wn >el'asson'ec e>isin;\\ aap`o >elass'onwn >`h d'uo >orj~wn e>ic >'apeiron >ekball'omenai\\ sump'iptousin;\\ a'ara >ekball'omenai\\ sumpeso~untai. } { Ve \gr{AJ} ile \gr{EZ} paralellerinin {\"u}zerine\\ d{\"u}{\c s}t{\"u}{\u g}{\"u}nden \gr{JZ} do{\u g}rusu,\\ \gr{AJZ} ve \gr{JZE} a{\c c}{\i}lar{\i}\\ e{\c s}ittir iki dik a{\c c}{\i}ya.\\ Dolay{\i}s{\i}yla \gr{BJH} ve\gr{HZE}\\ k{\"u}{\c c}{\"u}kt{\"u}r iki dik a{\c c}{\i}dan.\\ Ve k{\"u}{\c c}{\"u}k olanlardan\\ iki dik a{\c c}{\i}dan,\\ uzat{\i}ld{\i}klar{\i}nda sonsuza,\\ birbirlerine d{\"u}{\c s}erler do{\u g}rular.\\ Dolay{\i}s{\i}yla \gr{JB} ve \gr{ZE}, uzat{\i}l{\i}rsa,\\ birbirlerine d{\"u}{\c s}erler. } \parsen{ Suppose they have been extended,\\ and they have fallen together at \gr K,\\ and through the point \gr K,\\ parallel to either of \gr{EA} and \gr{ZJ},\\ suppose has been drawn \gr{KL},\\ and suppose have been extended \gr{JA} and \gr{HB}\\ to the points \gr L and \gr M. } { >ekbebl'hsjwsan\\ ka`i sumpipt'etwsan kat`a t`o K,\\ ka`i di`a to~u K shme'iou\\ 'hqjw ekbebl'hsjwsan aep`i t`a L, M shme~ia. } { Varsay{\i}ls{\i}n uzat{\i}ld{\i}klar{\i},\\ ve \gr K noktas{\i}nda kesi{\c s}tikleri,\\ ve \gr K noktas{\i}ndan,\\ paralel olan \gr{EA} veya \gr{ZJ} do{\u g}rusuna,\\ {\c c}izilmi{\c s} olsun \gr{KL},\\ ve uzat{\i}lm{\i}{\c s} olsunlar \gr{JA} ve \gr{HB} do{\u g}rular{\i}\\ \gr L ve \gr M noktalar{\i}ndan. } \parsen{ A parallelogram therefore is \gr{JLKZ},\\ a diameter of it is \gr{JK},\\ and about \gr{JK} [are]\\ the parallelograms \gr{AH} and \gr{ME},\\ and the so-called complements,\\ \gr{LB} and \gr{BZ};\\ equal therefore is \gr{LB} to \gr{BZ}.\\ But \gr{BZ} to triangle \gr G is equal.\\ Also therefore \gr{LB} to \gr G is equal.\\ And since equal is\\ angle \gr{HBE} to \gr{ABM},\\ but \gr{HBE} to \gr D is equal,\\ also therefore \gr{ABM} to \gr D\\ is equal. } { parallhl'ogrammon >'ara >est`i t`o JLKZ,\\ di'ametroc d`e a>uto~u 'ison >'ara >est`i t`o LB t~w| BZ.\\ {}>all`a t`o BZ t~w| G trig'wnw| >est`in >'ison;\\ ka`i t`o LB >'ara t~w| G >estin >'ison.\\ ka`i >epe`i >'ish >est`in\\ all`a estin >'ish,\\ ka`i 'ara t~h| D gwn'ia|\\ {}>est`in >'ish. } { Bir paralelkenard{\i}r dolay{\i}s{\i}yla \gr{JLKZ},\\ ve onun k{\"o}{\c s}egeni \gr{JK},\\ ve \gr{JK} etraf{\i}ndad{\i}r \\ \gr{AH} ve \gr{ME} paralelkenarlar{\i},\\ ve bunlar{\i}n t{\"u}mleyenleris,\\ \gr{LB} ile\gr{BZ};\\ e{\c s}ittirler dolay{\i}s{\i}yla \gr{LB} ile \gr{BZ} t{\"u}mleyenlerine.\\ Ama \gr{BZ}, \gr G {\"u}{\c c}genine e{\c s}ittir.\\ Dolays{\i}s{\i}yla \gr{LB} da \gr G {\"u}{\c c}genine e{\c s}ittir.\\ Ve e{\c s}it oldu{\u g}undan\\ \gr{HBE}, \gr{ABM} a{\c c}{\i}s{\i}na,\\ fakat \gr{HBE}, \gr D a{\c c}{\i}s{\i}na e{\c s}it,\\ dolay{\i}s{\i}yla \gr{ABM} de \gr D a{\c c}{\i}s{\i}na\\ e{\c s}ittir. } \parsen{ Therefore, along the given \strgt,\\ \gr{AB},\\ equal to the given triangle, \gr G,\\ a parallelogram has been applied,\\ \gr{LB},\\ in the angle \gr{ABM},\\ which is equal to \gr D;\\ \myqef } { Par`a t`hn doje~isan >'ara e>uje~ian\\ t`hn AB\\ t~w| doj'enti trig'wnw| t~w| G >'ison\\ parallhl'ogrammon parab'eblhtai\\ t`o LB\\ {}>en gwn'ia| t~h| estin >'ish t~h| D;\\ <'oper >'edei poi~hsai. } { Dolays{\i}syla, verilen bir,\\ \gr{AB} do{\u g}rusu boyunca,\\ verilen bir \gr G {\"u}{\c c}genine e{\c s}it,\\ bir,\\ \gr{LB} paralelkenar{\i} yerle{\c s}tirilmi{\c s} oldu,\\ \gr{ABM} a{\c s}{\i}s{\i}nda,\\ e{\c s}it olan \gr D a{\c c}{\i}s{\i}na;\\ \ozqef } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.5592188)(8.543125,1.5592188) \usefont{T1}{ptm}{m}{n} \rput(4.3471875,-0.41421875){\gr B} \usefont{T1}{ptm}{m}{n} \rput(4.3465624,-1.4142188){\gr A} \usefont{T1}{ptm}{m}{n} \rput(5.222031,1.3857813){\gr E} \usefont{T1}{ptm}{m}{n} \rput(4.0226564,1.3457812){\gr Z} \psline[linewidth=0.04cm](4.16,1.2007812)(8.16,1.2007812) \psline[linewidth=0.04cm](8.16,1.2007812)(7.36,-1.1992188) 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\end{proposition} \begin{proposition}%Proposition I.45 \parsen{ To the given rectilineal [figure] equal\\ a parallelogram to construct\\ in the given rectilineal angle. } { T~w| doj'enti e>ujugr'ammw| >'ison\\ parallhl'ogrammon sust'hsasjai\\ {}>en t~h| doje'ish| gwn'ia| e>ujugr'ammw|. } { Verilen bir d{\"u}zkenar [fig{\"u}re] e{\c s}it\\ bir paralelkenar in{\c s}a etmek,\\ verilen d{\"u}zkenar a{\c c}{\i}da. } \parsen{ Let be\\ the given rectilineal [figure] \gr{ABGD},\\ and the given rectilineal angle, \gr E. } { >'Estw\\ t`o m`en doj`en e>uj'ugrammon t`o ABGD,\\ uj'ugrammoc uju\-gr'ammw| >'ison\\ parallhl'ogrammon sust'hsasjai\\ {}>en t~h| doje'ish| gwn'ia| t~h| E. } { Gereklidir {\c s}imdi\\ \gr{ABGD} d{\"u}zkenar{\i}na e{\c s}it\\ bir paralelkenar in{\c s}a etmek,\\ verilen \gr E a{\c c}{\i}s{\i}nda. } \parsen{ Suppose has been joined \gr{DB},\\ and suppose has been constructed,\\ equal to the triangle \gr{ABD},\\ a parallelogram, \gr{ZJ},\\ in the angle \gr{JKZ},\\ which is equal to \gr E;\\ and suppose there has been applied\\ along the \strgt\ \gr{HJ},\\ equal to triangle \gr{DBG},\\ a parallelogram, \gr{HM},\\ in the angle \gr{HJM},\\ which is equal to \gr E. } { >Epeze'uqjw 'ison\\ parallhl'ogrammon t`o ZJ\\ {}>en t~h| estin >'ish t~h| E;\\ ka`i parabebl'hsjw\\ par`a t`hn HJ e>uje~ian\\ t~w| DBG trig'wnw| >'ison\\ parallhl'ogrammon t`o HM\\ {}>en t~h| estin >'ish t~h| E. } { Birle{\c s}tirilmi{\c s} oldu{\u g}u \gr{DB} do{\u g}rusunun,\\ ve in{\c s}a edilmi{\c s} olsun,\\ \gr{ABD} {\"u}{\c c}genine e{\c s}it,\\ bir \gr{ZJ} paralelkenar{\i},\\ \gr{JKZ} a{\c c}{\i}s{\i}nda,\\ e{\c s}it olan \gr E a{\c c}{\i}s{\i}na;\\ ve yerle{\c s}tirilmi{\c s} olsun\\ \gr{HJ} do{\u g}rusu boyunca ,\\ \gr{DBG} {\"u}{\c c}genine e{\c s}it,\\ bir \gr{HM} paralelkenar{\i},\\ \gr{HJM} a{\c c}{\i}s{\i}nda,\\ e{\c s}it olan \gr E a{\c c}{\i}s{\i}na. } \parsen{ And since angle \gr E\\ to either of \gr{JKZ} and \gr{HJM}\\ is equal,\\ therefore also \gr{JKZ} to \gr{HJM}\\ is equal.\\ Let \gr{KJH} be added in common;\\ therefore \gr{ZKJ} and \gr{KJH}\\ to \gr{KJH} and \gr{HJM}\\ are equal.\\ But \gr{ZKJ} and \gr{KJH}\\ are equal to two \rgt s;\\ therefore also \gr{KJH} and \gr{HJM}\\ are equal to two \rgt s.\\ Then to some \strgt, \gr{HJ},\\ and at the same point, \gr J,\\ two \strgt s, \gr{KJ} and \gr{JM},\\ not lying in the same parts,\\ the adjacent angles\\ make equal to two \rgt s.\\ In a \strgt\ then are \gr{KJ} and \gr{JM};\\ and since on the parallels \gr{KM} and \gr{ZH}\\ fell the \strgt\ \gr{JH},\\ the alternate angles \gr{MJH} and \gr{JHZ}\\ are equal to one another.\\ Let \gr{JHL} be added in common;\\ therefore \gr{MJH} and \gr{JHL}\\ to \gr{JHZ} and \gr{JHL}\\ are equal.\\ But \gr{MJH} and \gr{JHL}\\ are equal to two \rgt s;\\ therefore also \gr{JHZ} and \gr{JHL}\\ are equal to two \rgt s;\\ therefore on a \strgt\ are \gr{ZH} and \gr{HL}.\\ And since \gr{ZK} to \gr{JH}\\ is equal and parallel,\\ but also \gr{JH} to \gr{ML},\\ therefore also \gr{KZ} to \gr{ML}\\ is equal and parallel;\\ and join them\\ \gr{KM} and \gr{ZL}, which are \strgt s;\\ therefore also \gr{KM} and \gr{ZL}\\ are equal and parallel;\\ a parallelogram therefore is \gr{KZLM}.\\ And since equal is\\ triangle \gr{ABD}\\ to the parallelogram \gr{ZJ},\\ and \gr{DBG} to \gr{HM},\\ therefore, as a whole,\\ the rectilineal \gr{ABGD}\\ to parallelogram \gr{KZLM} as a whole\\ is equal. } { ka`i >epe`i estin >'ish,\\ ka`i 'ara t~h| estin >'ish.\\ koin`h proske'isjw 'ara 'isai e>is'in.\\ {}>all> aorja~ic >'isai e>is'in;\\ ka`i a'ara\\ d'uo >orja~ic >'isai e>is'in.\\ pr`oc d'h tini e>uje~ia| t~h| HJ\\ ka`i t~w| pr`oc a>ut~h| shme'iw| t~w| J\\ d'uo e>uje~iai aep`i t`a a>ut`a m'erh ke'imenai\\ t`ac >efex~hc gwn'iac\\ d'uo >orja~ic >'isac poio~usin;\\ {}>ep> e>uje'iac >'ara >est`in epe`i e>ic parall'hlouc t`ac KM, ZH\\ e>uje~ia >en'epesen enall`ax gwn'iai a'isai\\ {}>all'hlaic e>is'in.\\ koin`h proske'isjw 'ara 'isai e>isin.\\ >all> aorja~ic >'isai e>is'in;\\ ka`i a'ara\\ d'uo >orja~ic >'isai e>is'in;\\ {}>ep> e>uje'iac >'ara >est`in epe`i 'ish te ka`i par'allhl'oc >estin,\\ {}>all`a ka`i 'ara t~h| ML\\ {}>'ish te ka`i par'allhl'oc >estin;\\ ka`i >epizeugn'uousin a>ut`ac e>uje~iai a'ara\\ {}>'isai te ka`i par'allhlo'i e>isin;\\ parallhl'ogrammon >'ara >est`i t`o KZLM.\\ ka`i >epe`i >'ison >est`i\\ t`o m`en ABD tr'igwnon t~w| ZJ parallhlogr'ammw|,\\ t`o d`e DBG t~w| HM,\\ <'olon >'ara t`o ABGD e>uj'ugrammon\\ <'olw| t~w| KZLM parallhlogr'ammw|\\ {}>est`in >'ison. } { Ve \gr E a{\c c}{\i}s{\i}\\ \gr{JKZ} ve \gr{HJM} a{\c c}{\i}lar{\i}n{\i}n her birine\\ e{\c s}it oldu{\u g}undan,\\ \gr{JKZ} da \gr{HJM} a{\c c}{\i}s{\i}na\\ e{\c s}ittir.\\ Eklenmi{\c s} olsun \gr{KJH} ortak olarak;\\ dolay{\i}s{\i}yla \gr{ZKJ} ve \gr{KJH},\\ \gr{KJH} ve \gr{HJM} a{\c c}{\i}lar{\i}na\\ e{\c s}ittirler.\\ Fakat \gr{ZKJ} ve \gr{KJH}\\ e{\c s}ittirler iki dik a{\c c}{\i}ya;\\ dolay{\i}s{\i}yla \gr{KJH} ve \gr{HJM} a{\c c}{\i}lar{\i}da\\ e{\c s}ittirler iki dik a{\c c}{\i}ya.\\ {\c S}imdi bir \gr{HJ} do{\u g}rusuna,\\ ve ayn{\i} \gr J noktas{\i}nda,\\ iki \gr{KJ} ve \gr{JM} do{\u g}rular{\i},\\ ayn{\i} tarafta kalmayan,\\ kom{\c s}u a{\c c}{\i}lar{\i}\\ iki dik a{\c c}{\i}ya e{\c s}it yapar.\\ O zaman bir do{\u g}rudad{\i}r \gr{KJ} ve \gr{JM};\\ ve \gr{KM} ve \gr{ZH} paralelleri {\"u}zerine\\ d{\"u}{\c s}t{\"u}{\u g}{\"u}nden \gr{JH} do{\u g}rusu,\\ ters \gr{MJH} ve \gr{JHZ} a{\c c}{\i}lar{\i}\\ e{\c s}ittir birbirine.\\ eklenmi{\c s} olsun \gr{JHL} ortak olarak;\\ dolay{\i}s{\i}yla \gr{MJH} ve \gr{JHL},\\ \gr{JHZ} ve \gr{JHL} a{\c c}{\i}lar{\i}na\\ e{\c s}ittirler.\\ Fakat \gr{MJH} ve \gr{JHL}\\ e{\c s}ittirler iki dik a{\c c}{\i}ya;\\ dolay{\i}s{\i}yla \gr{JHZ} ve \gr{JHL} da\\ e{\c s}ittirler iki dik a{\c c}{\i}ya;\\ dolays{\i}syla bir do{\u g}ru {\"u}zerindedir \gr{ZH} ve \gr{HL}.\\ Ve oldu{\u g}undan \gr{ZK}, \gr{JH} do{\u g}rusuna\\ e{\c s}it ve paralel,\\ ve de \gr{JH}, \gr{ML} do{\u g}rusuna,\\ dolay{\i}s{\i}yla \gr{KZ} da \gr{ML} do{\u g}rusuna\\ e{\c s}it ve paraleldir;\\ ve birle{\c s}tirir onlar{\i}\ \gr{KM} ile \gr{ZL}, ki bunlarda do{\u g}rulard{\i}r;\\ dolay{\i}s{\i}yla \gr{KM} ve \gr{ZL} da\\ e{\c s}it ve paraleldirler;\\ dolay{\i}s{\i}yla \gr{KZLM} bir paralelkenard{\i}r.\\ Ve e{\c s}it oldu{\u g}undan\\ \gr{ABD} {\"u}{\c c}geni\\ \gr{ZJ} paralelkenar{\i}na,\\ ve \gr{DBG}, \gr{HM} paralelkenar{\i}na,\\ dolay{\i}s{\i}syla, bir b{\"u}t{\"u}n olarak,\\ \gr{ABGD} d{\"u}zkenar{\i}\\ bir b{\"u}t{\"u}n olarak \gr{KZLM} paralelkenar{\i}na\\ e{\c s}ittir. } \parsen{ Therefore, to the given rectilineal [figure], \gr{ABGD}, equal,\\ a parallelogram has been constructed,\\ \gr{KZLM},\\ in the angle \gr{ZKM},\\ which is equal to the given \gr E;\\ \myqef } { T~w| >'ara doj'enti e>ujugr'ammw| t~w| ABGD >'ison\\ parallhl'ogrammon sun'estatai\\ t`o KZLM\\ {}>en gwn'ia| t~h| estin >'ish t~h| doje'ish| t~h| E;\\ <'oper >'edei poi~hsai. } { Dolay{\i}s{\i}yla, verilen d{\"u}zkenar \gr{ABGD} fig{\"u}r{\"u}ne e{\c s}it,\\ bir \gr{KZLM} paralelkenar{\i} in{\c s}a edilmi{\c s} oldu,\\ \gr{ZKM} a{\c c}{\i}s{\i}nda,\\ e{\c s}it olan verilmi{\c s} \gr E a{\c c}{\i}s{\i}na;\\ \ozqef } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.5392188)(9.369375,1.5392188) \usefont{T1}{ptm}{m}{n} \rput(1.224375,-0.7342188){\gr B} \usefont{T1}{ptm}{m}{n} \rput(0.12375,1.0257813){\gr A} \usefont{T1}{ptm}{m}{n} \rput(3.1192188,0.00578125){\gr E} \psline[linewidth=0.04cm](3.2371874,0.62078124)(2.6371875,-0.15921874) \psline[linewidth=0.04cm](2.6371875,-0.15921874)(3.4971876,-0.29921874) \usefont{T1}{ptm}{m}{n} \rput(4.8998437,1.3057812){\gr Z} \psline[linewidth=0.04cm](5.0571876,1.2007812)(9.057187,1.2007812) \psline[linewidth=0.04cm](9.057187,1.2007812)(8.257188,-1.1992188) \psline[linewidth=0.04cm](8.237187,-1.1992188)(4.2571874,-1.1792188) \psline[linewidth=0.04cm](4.2571874,-1.1792188)(5.0771875,1.2207812) \psline[linewidth=0.04cm](7.8571873,1.1807812)(6.9971876,-1.1792188) \usefont{T1}{ptm}{m}{n} \rput(6.9403124,-1.3942188){\gr J} \usefont{T1}{ptm}{m}{n} \rput(7.8825,1.3657813){\gr H} \usefont{T1}{ptm}{m}{n} \rput(4.105781,-1.3142188){\gr K} \usefont{T1}{ptm}{m}{n} \rput(9.219687,1.3057812){\gr L} \usefont{T1}{ptm}{m}{n} \rput(8.490469,-1.3542187){\gr M} \usefont{T1}{ptm}{m}{n} \rput(1.6771874,1.3257812){\gr D} \usefont{T1}{ptm}{m}{n} \rput(2.2982812,0.54578125){\gr G} \psline[linewidth=0.04cm](1.4771875,1.2007812)(2.1771874,0.38078126) \psline[linewidth=0.04cm](2.1771874,0.38078126)(1.3571875,-0.55921876) \psline[linewidth=0.04cm](1.3571875,-0.55921876)(0.0771875,0.8407813) \psline[linewidth=0.04cm](0.0971875,0.82078123)(1.4971875,1.1807812) \psline[linewidth=0.04cm](1.4971875,1.1807812)(1.3571875,-0.5392187) \end{pspicture} } \end{center} \end{proposition} \newpage \begin{proposition}%Proposition I.46 \parsen{ On the given \strgt\\ to set up a square. } { >Ap`o t~hc doje'ishc e>uje'iac\\ tetr'agwnon >anagr'ayai. } { Verilen bir do{\u g}ruda\\ bir kare kurmak. } \parsen{ Let be\\ the given \strgt\ \gr{AB}. } { >'Estw\\ uje~ia ap`o t~hc AB e>uje'iac\\ tetr'agwnon >anagr'ayai. } { {\c S}imdi gereklidir\\ \gr{AB} do{\u g}rusunda\\ bir kare kurmak. } \parsen{ Suppose there has been drawn\\ to the \strgt\ \gr{AB},\\ at the point \gr A of it,\\ at a \rgt,\\ \gr{AG},\\ and suppose there has been laid down,\\ equal to \gr{AB},\\ \gr{AD};\\ and through the point \gr D,\\ parallel to \gr{AB},\\ suppose there has been drawn \gr{DE};\\ and through the point \gr B,\\ parallel to \gr{AD},\\ suppose there has been drawn \gr{BE}.\\ } { >'Hqjw\\ t~h| AB e>uje'ia|\\ {}>ap`o to~u pr`oc a>ut~h| shme'iou to~u A\\ pr`oc >orj`ac\\ 'ish\\ 'hqjw 'hqjw 'ara >est`i t`o ADEB;\\ {}>'ish >'ara >est`in all`a estin >'ish;\\ a'ara\\ a'isai >all'hlaic e>is'in;\\ {}>is'opleuron >'ara\\ {}>est`i t`o ADEB parallhl'ogrammon. } { Bir paralelkenard{\i}r dolay{\i}s{\i}yla \gr{ADEB};\\ e{\c s}ittir dolay{\i}s{\i}yla \gr{AB}, \gr{DE} do{\u g}rusuna,\\ ve \gr{AD}, \gr{BE} do{\u g}rusuna.\\ Ama \gr{AB}, \gr{AD} do{\u g}rusuna e{\c s}ittir.\\ Dolays{\i}syla {\c s}u d{\"o}rd{\"u}\\ \gr{BA}, \gr{AD}, \gr{DE} ve \gr{EB}\\ birbirlerine e{\c s}ittirler;\\ e{\c s}kenard{\i}r dolay{\i}s{\i}yla\\ \gr{ADEB} paralelkenar{\i}. } \parsen{ I say then that\\ it is also right-angled.\\ } { l'egw d'h, <'oti\\ ka`i >orjog'wnion. } { {\c S}imdi iddia ediyorum ki\\ ayn{\i} zamanda dik a{\c c}{\i}l{\i}d{\i}r.\\ } \parsen{ For, since on the parallels \gr{AB} and \gr{DE}\\ fell the \strgt\ \gr{AD},\\ therefore the angles \gr{BAD} and \gr{ADE}\\ are equal to two \rgt s.\\ And \gr{BAD} is right;\\ right therefore is \gr{ADE}.\\ And of parallelogram areas\\ the opposite sides and angles\\ are equal to one another.\\ Right therefore is either\\ of the opposite angles \gr{ABE} and \gr{BED};\\ right-angled therefore is \gr{ADEB}.\\ And it was shown also equilateral. } { >epe`i g`ar e>ic parall'hlouc t`ac AB, DE\\ e>uje~ia >en'epesen 'ara orja~ic >'isai e>is'in.\\ {}>orj`h d`e orj`h >'ara ka`i apenant'ion pleura'i te ka`i gwn'iai\\ {}>'isai >all'hlaic e>is'in;\\ {}>orj`h >'ara ka`i apenant'ion t~wn orjog'wnion >'ara >est`i t`o ADEB.\\ {}>ede'iqjh d`e ka`i >is'opleuron. } { {\c C}{\"u}nk{\"u}, \gr{AB} ve \gr{DE} paralellerinin {\"u}zerine\\ d{\"u}{\c s}t{\"u}{\u g}{\"u}nden \gr{AD} do{\u g}rusu,\\ e{\c s}ittir dolays{\i}yla \gr{BAD} ve \gr{ADE}\\ iki dik a{\c c}{\i}ya.\\ Ve \gr{BAD} diktir;\\ diktir dolay{\i}s{\i}yla \gr{ADE}.\\ Ve paralelkenar alanlar{\i}n\\ kar{\c s}{\i}t kenar ve a{\c c}{\i}lar{\i}\\ e{\c s}ittir birbirlerine.\\ Diktir dolay{\i}s{\i}yla her bir\\ kar{\c s}{\i}t a{\c c}{\i} \gr{ABE} ve \gr{BED};\\ dik a{\c c}{\i}l{\i}d{\i}r dolay{\i}s{\i}yla \gr{ADEB}.\\ Ve g{\"o}sterilmi{\c s}ti ki e{\c s}kenard{\i}r da. } \parsen{ A square therefore it is;\\ and it is on the \strgt\ \gr{AB}\\ set up;\\ \myqef } { Tetr'agwnon >'ara >est'in;\\ ka'i >estin >ap`o t~hc AB e>uje'iac\\ {}>anagegramm'enon;\\ <'oper >'edei poi~hsai. } { Bir karedir dolay{\i}s{\i}yla o;\\ ve o \gr{AB} do{\u g}rusu {\"u}zerine\\ kurulmu{\c s}tur;\\ \ozqef } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.871875)(2.88875,1.871875) \usefont{T1}{ptm}{m}{n} \rput(2.724375,-1.726875){\gr B} \usefont{T1}{ptm}{m}{n} \rput(0.12375,-1.706875){\gr A} \usefont{T1}{ptm}{m}{n} \rput(2.7392187,0.493125){\gr E} \usefont{T1}{ptm}{m}{n} \rput(0.1571875,0.413125){\gr D} \usefont{T1}{ptm}{m}{n} \rput(0.17828125,1.693125){\gr G} \psline[linewidth=0.04cm](0.3771875,1.628125)(0.3771875,-1.571875) \psline[linewidth=0.04cm](0.3771875,-1.571875)(2.5771875,-1.571875) \psline[linewidth=0.04cm](2.5771875,-1.571875)(2.5771875,0.428125) \psline[linewidth=0.04cm](2.5771875,0.428125)(0.3771875,0.428125) \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.47 \parsen{ In right-angled triangles,\\ the square on the side that subtends the right angle\\ is equal\\ to the squares on the sides that contain the right angle. } { >En to~ic >orjogwn'ioic trig'wnoic\\ t`o >ap`o t~hc t`hn >orj`hn gwn'ian 'ison >est`i\\ to~ic >ap`o t~wn t`hn >orj`hn gwn'ian perieqous~wn pleur~wn tetrag'wnoic. } { Dik a{\c c}{\i}l{\i} {\"u}{\c c}genlerde,\\ dik a{\c c}{\i}n{\i}n g{\"o}rd{\"u}{\u g}{\"u} kenar {\"u}zerindeki kare\\ e{\c s}ittir\\ dik a{\c c}{\i}y{\i} i{\c c}eren kenarlar{\i}n {\"u}zerindekilere. } \parsen{ Let be\\ a right-angled triangle, \gr{ABG},\\ having the angle \gr{BAG} right. } { >'Estw\\ tr'igwnon >orjog'wnion t`o ABG\\ {}>orj`hn >'eqon t`hn ap`o t~hc BG tetr'agwnon\\ {}>'ison >est`i\\ to~ic >ap`o t~wn BA, AG tetrag'wnoic. } { {\.I}ddia ediyorum ki\\ \gr{GB} {\"u}zerindeki kare\\ e{\c s}ittir\\ \gr{BA} ve \gr{AG} {\"u}zerlerindeki karelere. } \parsen{ For, suppose there has been set up\\ on \gr{BG}\\ a square, \gr{BDEG},\\ and on \gr{BA} and \gr{AG},\\ \gr{HB} and \gr{JG},\\ and through \gr A,\\ parallel to either of \gr{BD} and \gr{GE},\\ suppose \gr{AL} has been drawn;\\ and suppose have been joined\\ \gr{AD} and \gr{ZG}. } { >Anagegr'afjw g`ar\\ {}>ap`o m`en t~hc BG\\ tetr'agwnon t`o BDEG,\\ {}>ap`o d`e t~wn BA, AG\\ t`a HB, JG,\\ ka`i di`a to~u A\\ 'hqjw epeze'uqjwsan\\ aepe`i >orj'h >estin\\ uje'ia| t~h| BA\\ ka`i t~w| pr`oc a>ut~h| shme'iw| t~w| A\\ d'uo e>uje~iai aep`i t`a a>ut`a m'erh ke'imenai\\ t`ac >efex~hc gwn'iac\\ dus`in >orja~ic >'isac poio~usin;\\ {}>ep> e>uje'iac >'ara >est`in ut`a d`h\\ ka`i estin >ep> e>uje'iac.\\ ka`i >epe`i >'ish >est`in\\ orj`h g`ar 'ara estin >'ish.\\ ka`i >epe`i >'ish >est`in\\ 'isai e>is`in\\ 'ish;\\ b'asic >'ara estin] >'ish,\\ ka`i t`o ABD tr'igwnon\\ t~w| ZBG trig'wnw|\\ {}>est`in >'ison;\\ ka'i [>esti] to~u m`en ABD trig'wnou\\ dipl'asion t`o BL parallhl'ogrammon;\\ b'asin te g`ar t`hn a>ut`hn >'eqousi t`hn BD\\ ka`i >en ta~ic a>uta~ic e>isi parall'hloic\\ ta~ic BD, AL;\\ to~u d`e ZBG trig'wnou\\ dipl'asion t`o HB tetr'agwnon;\\ b'asin te g`ar p'alin t`hn a>ut`hn >'eqousi\\ t`hn ZB\\ ka`i >en ta~ic a>uta~ic e>isi parall'hloic\\ ta~ic ZB, HG.\\ {}[t`a d`e t~wn >'iswn\\ dipl'asia >'isa >all'hloic >est'in;]\\ {}>'ison >'ara >est`i\\ ka`i t`o BL parallhl'ogrammon\\ t~w| HB tetrag'wnw|.\\ epizeugnum'enwn t~wn AE, BK\\ deiqj'hsetai\\ ka`i t`o GL parallhl'ogrammon\\ {}>'ison t~w| JG tetrag'wnw|;\\ <'olon >'ara t`o BDEG tetr'agwnon\\ dus`i to~ic HB, JG tetrag'wnoic\\ {}>'ison >est'in.\\ ka'i >esti\\ t`o m`en BDEG tetr'agwnon >ap`o t~hc BG >anagraf'en,\\ t`a d`e HB, JG >ap`o t~wn BA, AG.\\ t`o >'ara >ap`o t~hc BG pleur~ac tetr'agwnon\\ {}>'ison >est`i\\ to~ic >ap`o t~wn BA, AG pleur~wn tetrag'wnoic. } { Ve dik oldu{\u g}undan\\ \gr{BAG} ve \gr{BAH} a{\c c}{\i}lar{\i}n{\i}n her biri,\\ bir \gr{BA} do{\u g}rusunda,\\ {\"u}zerindeki \gr A noktas{\i}na,\\ \gr{AG} ve \gr{AH} do{\u g}rular{\i},\\ ayn{\i} tarafta kalmayan,\\ biti{\c s}ik a{\c c}{\i}lar\\ olu{\c s}tururlar e{\c s}it iki dik a{\c c}{\i}ya;\\ bir do{\u g}rudad{\i}r dolay{\i}s{\i}syla \gr{GA} ile \gr{AH}.\\ Sonra ayn{\i} nedenle\\ \gr{BA} ile \gr{AJ} da bir do{\u g}rudad{\i}r.\\ Ve e{\c s}it oldu{\u g}undan\\ \gr{DBG}, \gr{ZBA} a{\c c}{\i}s{\i}na;\\ her ikiside diktir;\\ eklenmi{\c s} olsun \gr{ABG} her ikisine de;\\ dolay{\i}s{\i}yla \gr{DBA} a{\c c}{\i}s{\i}n{\i}n tamam{\i}\\ \gr{ZBG} a{\c c}{\i}s{\i}n{\i}n tamam{\i}na \\ e{\c s}ittir.\\ Ve e{\c s}it oldu{\u g}undan\\ \gr{DB}, \gr{BG} do{\u g}rusuna,\\ ve \gr{ZB}, \gr{BA} do{\u g}rusuna\\ \gr{DB} ve \gr{BA} ikilisi\\ \gr{ZB} ve \gr{BG} ikilisine\footnotemark\\ e{\c s}ittirler,\\ her biri birine;\\ ve \gr{DBA} a{\c c}{\i}s{\i}\\ \gr{ZBG} a{\c c}{\i}s{\i}na\\ e{\c s}ittir;\\ dolay{\i}s{\i}yla \gr{AL} taban{\i}\\ \gr{ZG} taban{\i}na\\ e{\c s}ittir,\\ ve \gr{ABD} {\"u}{\c c}geni\\ \gr{ZBG} {\"u}{\c c}genine\\ e{\c s}ittir;\\ ve \gr{ABD} {\"u}{\c c}geninin\\ \gr{BL} paralelkenar{\i} iki kat{\i}d{\i}r;\\ ayn{\i} \gr{BL} tabanlar{\i} oldu{\u g}u,\\ ve ayn{\i}\\ \gr{BD} ve \gr{AL} parallerinde olduklar{\i} i{\c c}in;\\ ve \gr{ZBG} {\"u}{\c c}geninin\\ \gr{HB} karesi iki kat{\i}d{\i}r;\\ yine ayn{\i}\\ \gr{ZB} tabanlar{\i} oldu{\u g}u\\ ve ayn{\i}\\ \gr{ZB} ve \gr{HG} parallerinde olduklar{\i} i{\c c}in.\\ {}[Ve e{\c s}itlerin,\\ iki katlar{\i} birbirlerine e{\c s}ittirler.]\\ E{\c s}ittir dolays{\i}yla\\ \gr{BL} paralelkenar{\i} da\\ \gr{HB} karesine.\\ {\c S}imdi benzer {\c s}ekilde,\\ birle{\c s}tirildi{\u g}inde \gr{AE} ve \gr{BK},\\ g{\"o}sterilecek ki\\ \gr{GL} paralelkenar{\i} da\\ e{\c s}ittir \gr{JG} karesine.\\ Dolay{\i}s{\i}yla \gr{DBEG} bir b{\"u}t{\"u}n olarak\\ \gr{HB} ve \gr{JG} iki karesine\\ e{\c s}ittir.\\ Ayr{\i}ca\\ \gr{BDEG} karesi \gr{BG} {\"u}zerine kurulmu{\c s}tur,\\ ve \gr{HB} ve \gr{JG}, \gr{BA} ve \gr{AG} {\"u}zerine.\\ Dolay{\i}s{\i}yla \gr{BG} kenar{\i}ndaki kare\\ e{\c s}ittir\\ \gr{BA} ve \gr{AG} kenarlar{\i}ndaki karelere. } \myfntext{Fitzpatrick considers this ordering of the two straight lines to be `obviously a mistake'. But if it is a mistake, how could it have been made?} \parsen{ Therefore in right-angled triangles\\ the square on the side subtending the right angle\\ is equal\\ to the squares on the sides subtending the right [angle];\\ \myqed } { >En >'ara to~ic >orjogwn'ioic trig'wnoic\\ t`o >ap`o t~hc t`hn >orj`hn gwn'ian 'ison >est`i\\ to~ic >ap`o t~wn t`hn >orj`hn [gwn'ian] perieqous~wn pleur~wn tetrag'wnoic;\\ <'oper >'edei de~ixai. } { Dolay{\i}s{\i}yla dik a{\c c}{\i}l{\i} {\"u}{\c c}genlerde,\\ dik a{\c c}{\i}n{\i}n g{\"o}rd{\"u}{\u g}{\"u} kenar {\"u}zerindeki kare\\ e{\c s}ittir\\ dik a{\c c}{\i}y{\i} i{\c c}eren kenarlar{\i}n {\"u}zerindekilere;\\ \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-2.6892188)(5.2784376,2.6892188) \usefont{T1}{ptm}{m}{n} \rput(1.0425,-0.36421874){\gr B} \usefont{T1}{ptm}{m}{n} \rput(2.041875,1.0757812){\gr A} \usefont{T1}{ptm}{m}{n} \rput(3.8373437,-2.5442188){\gr E} \usefont{T1}{ptm}{m}{n} \rput(1.1153125,-2.5042188){\gr D} \usefont{T1}{ptm}{m}{n} \rput(2.0153125,-2.6042188){\gr L} \usefont{T1}{ptm}{m}{n} \rput(3.8764062,-0.34421876){\gr G} \psline[linewidth=0.04cm](2.0753126,0.79078126)(1.2753125,-0.20921876) \psline[linewidth=0.04cm](1.2753125,-0.20921876)(3.6753125,-0.20921876) \psline[linewidth=0.04cm](3.6753125,-0.20921876)(1.0353125,1.4707812) \psline[linewidth=0.04cm](1.0353125,1.4707812)(0.2753125,0.43078125) \psline[linewidth=0.04cm](0.2753125,0.43078125)(1.2753125,-0.22921875) \psline[linewidth=0.04cm](1.2753125,-0.22921875)(1.2753125,-2.3892188) \psline[linewidth=0.04cm](1.2753125,-2.3692188)(3.6753125,-2.3892188) \psline[linewidth=0.04cm](3.6553125,-2.3892188)(3.6753125,-0.20921876) \psline[linewidth=0.04cm](3.6753125,-0.20921876)(4.8953123,1.2707813) \psline[linewidth=0.04cm](4.8953123,1.2707813)(3.2353125,2.3707812) \psline[linewidth=0.04cm](3.2353125,2.3707812)(2.0553124,0.7707813) \psline[linewidth=0.04cm](3.6953125,-0.20921876)(0.2753125,0.43078125) \psline[linewidth=0.04cm](1.2753125,-0.22921875)(4.9153123,1.2707813) \psline[linewidth=0.04cm](2.0753126,0.79078126)(1.2953125,-2.3492188) \psline[linewidth=0.04cm](2.0953126,0.7707813)(3.6353126,-2.3892188) \psline[linewidth=0.04cm](2.0753126,0.81078124)(2.0553124,-2.3692188) \usefont{T1}{ptm}{m}{n} \rput(0.09796875,0.39578125){\gr Z} \usefont{T1}{ptm}{m}{n} \rput(1.0584375,1.6157813){\gr H} \usefont{T1}{ptm}{m}{n} \rput(3.260625,2.5157812){\gr J} \usefont{T1}{ptm}{m}{n} \rput(5.103906,1.3157812){\gr K} \end{pspicture} } \end{center} \end{proposition} \begin{proposition}%Proposition I.48 \parsen{ If of a triangle\\ the square on one of the sides\\ be equal\\ to the squares on the remaining sides of the triangle,\\ the angle contained\\ by the two remaining sides of the triangle\\ is right. } { >E`an trig'wnou\\ t`o >ap`o mi~ac t~wn pleur~wn tetr'agwnon\\ {}>'ison >~h|\\ to~ic >ap`o t~wn loip~wn to~u trig'wnou d'uo pleur~wn tetrag'wnoic,\\ orj'h >estin. } { E{\u g}er bir {\"u}{\c c}gende\\ bir kenar{\i}n {\"u}zerindeki kare\\ e{\c s}itse\\ {\"u}{\c c}genin geriye kalan kenarlar{\i}ndaki karelere,\\ {\"u}{\c c}genin geriye kalan kenarlar{\i}nca i{\c c}erilen\\ a{\c c}{\i}\\ diktir. } \parsen{ For, of the triangle \gr{ABG}\\ the square on the one side \gr{BG}\\ ---suppose it is equal\\ to the squares on the sides \gr{BA} and \gr{AG}. } { Trig'wnou g`ar to~u ABG\\ t`o >ap`o mi~ac t~hc BG pleur~ac tetr'agwnon\\ {}>'ison >'estw\\ to~ic >ap`o t~wn BA, AG pleur~wn tetrag'wnoic; } { {\c C}{\"u}nk{\"u}, \gr{ABG} {\"u}{\c c}geninin\\ bir \gr{BG} kenar{\i}ndaki karesi\\ ---varsay{\i}ls{\i}n e{\c s}it\\ \gr{BA} ve \gr{AG} kenarlar{\i}ndaki karelere. } \parsen{ I say that\\ right is the angle \gr{BAG}. } { l'egw, <'oti\\ {}>orj'h >estin 'Hqjw g`ar\\ {}>ap`o to~u A shme'iou\\ t~h| AG e>uje'ia|\\ pr`oc >orj`ac\\ 'ish\\ epeze'uqjw epe`i >'ish >est`in 'ison >est`i\\ ka`i t`o >ap`o t~hc DA tetr'agwnon\\ t~w| >ap`o t~hc AB tetrag'wnw|.\\ koin`on proske'isjw\\ t`o >ap`o t~hc AG tetr'agwnon;\\ t`a >'ara >ap`o t~wn DA, AG tetr'agwna\\ {}>'isa >est`i\\ to~ic >ap`o t~wn BA, AG tetrag'wnoic.\\ {}>all`a to~ic m`en >ap`o t~wn DA, AG\\ {}>'ison >est`i\\ t`o >ap`o t~hc DG;\\ {}>orj`h g'ar >estin ap`o t~wn BA, AG\\ {}>'ison >est`i\\ t`o >ap`o t~hc BG;\\ 'ara >ap`o t~hc DG tetr'agwnon\\ {}>'ison >est`i\\ t~w| >ap`o t~hc BG tetrag'wnw|;\\ <'wste ka`i pleur`a\\ estin >'ish;\\ ka`i >epe`i >'ish >est`in 'isai e>is'in;\\ ka`i b'asic 'ish;\\ gwn'ia >'ara estin] >'ish.\\ {}>orj`h d`e orj`h >'ara ka`i E`an >ar`a trig'wnou\\ t`o >ap`o mi~ac t~wn pleur~wn tetr'agwnon\\ {}>'ison >~h|\\ to~ic >ap`o t~wn loip~wn to~u trig'wnou d'uo pleur~wn tetrag'wnoic,\\ orj'h >estin;\\ <'oper >'edei de~ixai. } { E{\u g}er dolay{\i}s{\i}yla bir {\"u}{\c c}gende\\ bir kenar{\i}n {\"u}zerindeki kare\\ e{\c s}itse\\ {\"u}{\c c}genin geriye kalan kenarlar{\i}ndaki karelere,\\ {\"u}{\c c}genin geriye kalan kenarlar{\i}nca i{\c c}erilen\\ a{\c c}{\i}\\ diktir;\\ \ozqed } \begin{center} \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.521875)(2.6940625,1.521875) \psline[linewidth=0.04cm](1.35375,1.218125)(0.35375,-1.161875) \psline[linewidth=0.04cm](0.37375,-1.161875)(2.35375,-1.181875) \psline[linewidth=0.04cm](2.35375,-1.181875)(1.35375,1.218125) \psline[linewidth=0.04cm](1.35375,1.218125)(1.33375,-1.161875) \usefont{T1}{ptm}{m}{n} \rput(1.3003125,-1.376875){\gr A} \usefont{T1}{ptm}{m}{n} \rput(2.5409374,-1.276875){\gr B} \usefont{T1}{ptm}{m}{n} \rput(1.4748437,1.343125){\gr G} \usefont{T1}{ptm}{m}{n} \rput(0.11375,-1.236875){\gr D} \end{pspicture} } \end{center} \end{proposition} %\loadgeometry{ancillary} %\begin{multicols}2 %\bibliographystyle{amsplain} %\bibliography{../Public/references} \def\cprime{$'$} \def\cprime{$'$} \def\cprime{$'$} \providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} \providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR } % \MRhref is called by the amsart/book/proc definition of \MR. \providecommand{\MRhref}[2]{% \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2} } \providecommand{\href}[2]{#2} \begin{thebibliography}{10} \bibitem{Euclid-Heiberg} Euclid, \emph{Euclidis {E}lementa}, Euclidis Opera Omnia, Teubner, 1883, Edited, and with Latin translation, by I. L. Heiberg. \bibitem{MR17:814b} \bysame, \emph{The thirteen books of {E}uclid's {E}lements translated from the text of {H}eiberg. {V}ol. {I}: {I}ntroduction and {B}ooks {I}, {I}{I}. {V}ol. {I}{I}: {B}ooks {I}{I}{I}--{I}{X}. {V}ol. {I}{I}{I}: {B}ooks {X}--{X}{I}{I}{I} and {A}ppendix}, Dover Publications Inc., New York, 1956, Translated with introduction and commentary by Thomas L. Heath, 2nd ed. \MR{17,814b} \bibitem{MR1932864} \bysame, \emph{Euclid's {E}lements}, Green Lion Press, Santa Fe, NM, 2002, All thirteen books complete in one volume, the Thomas L. Heath translation, edited by Dana Densmore. \MR{MR1932864 (2003j:01044)} \bibitem{MEU2} H.~W. Fowler, \emph{A dictionary of modern {E}nglish usage}, second ed., Oxford University Press, 1982, revised and edited by Ernest Gowers. \bibitem{MEU} \bysame, \emph{A dictionary of modern {E}nglish usage}, Wordsworth Editions, Ware, Hertfordshire, UK, 1994, reprint of the original 1926 edition. \bibitem{Harman} Homer~C House and Susan~Emolyn Harman, \emph{Descriptive english grammar}, second ed., Prentice-Hall, Englewood Cliffs, N.J., USA, 1950, Revised by Susan Emolyn Harman. Twelfth printing, 1962. \bibitem{CGEL} Rodney Huddleston and Geoffrey~K. Pullum, \emph{The {C}ambridge grammar of the {E}nglish language}, Cambridge University Press, Cambridge, UK, 2002, reprinted 2003. \bibitem{MR1384401} Wilbur~R. Knorr, \emph{The wrong text of {E}uclid: on {H}eiberg's text and its alternatives}, Centaurus \textbf{38} (1996), no.~2-3, 208--276. \MR{1384401 (97c:01007)} \bibitem{MR1892196} \bysame, \emph{On {H}eiberg's {E}uclid}, Sci. Context \textbf{14} (2001), no.~1-2, 133--143, Intercultural transmission of scientific knowledge in the middle ages: Graeco-Arabic-Latin (Berlin, 1996). \MR{1892196 (2003b:01009)} \bibitem{LSJ} Henry~George Liddell and Robert Scott, \emph{A {G}reek-{E}nglish lexicon}, Clarendon Press, Oxford, 1996, revised and augmented throughout by Sir Henry Stuart Jones, with the assistance of Roderick McKenzie and with the cooperation of many scholars. With a revised supplement. \bibitem{OCG} James Morwood and John Taylor (eds.), \emph{The pocket {O}xford classical {G}reek dictionary}, Oxford University Press, Oxford, 2002. \bibitem{MR1683176} Reviel Netz, \emph{The shaping of deduction in {G}reek mathematics}, Ideas in Context, vol.~51, Cambridge University Press, Cambridge, 1999, A study in cognitive history. \MR{MR1683176 (2000f:01003)} \bibitem{Chapman-Iliad} Allardyce Nicoll (ed.), \emph{{C}hapman's {H}omer: The {I}liad}, paperback ed., Princeton University Press, Princeton, New Jersey, 1998, With a new preface by Garry Wills. Original publication, 1956. \bibitem{MR1200456} Proclus, \emph{A commentary on the first book of {E}uclid's \emph{{E}lements}}, Princeton Paperbacks, Princeton University Press, Princeton, NJ, 1992, Translated from the Greek and with an introduction and notes by Glenn R. Morrow, Reprint of the 1970 edition, With a foreword by Ian Mueller. \MR{MR1200456 (93k:01008)} \bibitem{Ozkirimli} Atilla~\"Ozk\i r\i ml\i, \emph{T\"urk dili, dil ve anlat\i m [the turkish language, language, and expression]}, \.Istanbul Bilgi \"Universitesi Yay\i nlar\i, 2001, Ya\c sayan {T}\"urk\c ce \"Uzerine Bir Deneme [An Essay on Living Turkish]. \bibitem{Smyth} Herbert~Weir Smyth, \emph{Greek grammar}, Harvard University Press, Cambridge, Massachussets, 1980, Revised by Gordon M. Messing, 1956. Eleventh Printing. Original edition, 1920. \bibitem{MR13:419b} Ivor Thomas (ed.), \emph{Selections illustrating the history of {G}reek mathematics. {V}ol. {II}. {F}rom {A}ristarchus to {P}appus}, Harvard University Press, Cambridge, Mass, 1951, With an English translation by the editor. \MR{13,419b} \bibitem{Thoreau} Henry~David Thoreau, \emph{Walden [1854] and other writings}, Bantam Books, New York, 1962, Edited and with an introduction by Joseph Wood Krutch. \end{thebibliography} %\end{multicols} \end{document}