\documentclass[a4paper,twoside,12pt,reqno,draft]{amsart}
%\documentclass[a4paper,twoside,12pt,draft]{article}
\usepackage[headings]{fullpage}
\usepackage[polutonikogreek,english]{babel}
\usepackage[oxonia]{psgreek}
%\usepackage{psgreek}
\newcommand{\Gk}[1]{\selectlanguage{polutonikogreek}#1\selectlanguage{english}}
\newcommand{\Eng}[1]{\emph{#1}}
\newcommand{\Tur}[1]{\texttt{#1}}
\newcommand{\letter}[1]{\textsf{#1}}  %for letters as such
%\usepackage{showlabels}  % This doesn't show all labels; there must
                          % be a conflict with another package 

%\usepackage{showkeys}
%\usepackage[integers]{layout}

\title{Conic sections as such \emph{(first draft)}}
\author{David Pierce}
\date{March 20, 2008}

%\usepackage{verbatim}

%\begin{comment}
\address{Mathematics Department\\
Middle East Technical University\\
Ankara 06531, Turkey}

\email{dpierce@metu.edu.tr}
\urladdr{http://metu.edu.tr/~dpierce/}

%\end{comment}


\usepackage{amssymb,amsmath,amsthm}
\usepackage{amscd}     % commutative diagram
\usepackage[mathscr]{euscript}
\usepackage{url}

%\usepackage{wrapfig}

\usepackage{pstricks}
%\usepackage{pst-3d}
%\usepackage{pst-plot}


%\input{../../../TeX/abbreviations}
%\input{../../../TeX/sectionless-theorems}

\usepackage{amssymb,amsmath,amsthm}
\usepackage{amscd}     % commutative diagram
\usepackage[mathscr]{euscript}
\usepackage{url}
\usepackage{verbatim}  % allows a comment environment:

%\usepackage{parskip}   % paragraphs not indented, but separated by spaces

\usepackage{hfoldsty}  % gives old-style numerals

\newcommand{\defn}[2]{\textbf{#1#2}}
\newcommand{\tech}[2]{\textsl{#1#2}}
%\newcommand{\size}[1]{\lvert#1\rvert}  % cardinality
%\newcommand{\mpi}{\mathrm{\pi}}
\renewcommand{\leq}{\leqslant}
\renewcommand{\geq}{\geqslant}
\renewcommand{\phi}{\varphi}

\newcommand{\ratio}[2]{#1:#2}
\newcommand{\psign}{\mathrel{:\,:}}
\newcommand{\eqr}[2]{#1\psign#2}
\newcommand{\prop}[4]{\eqr{\ratio{#1}{#2}}{\ratio{#3}{#4}}}

\begin{document}
  \maketitle

\section{Introduction}

The following notes investigate the conic
sections in a manner inspired by Apollonius of Perga and other ancient
mathematicians.  The point here is to do \emph{mathematics,} not
history; but we may be aided in our own mathematics by trying to
understand how others have done it.  It may be said that we are
going to do \emph{pre-Cartesian} mathematics: mathematics as done
before (well before) the time of Ren\'e Descartes (1596--1650).

As I shall be referring to some works in Greek; as Greek is the origin
of much of our technical vocabulary; as mathematicians use Greek
letters all the time: for these reasons, I review the Greek alphabet in
Table~\ref{table:Greek}.
\begin{table}[ht]
  \begin{center}
      \begin{tabular}{| c  l | c  l | c l | c l |} \hline
\Gk{A a} & \textbf alpha  & \Gk{H h} & \textbf{\=e}ta   & \Gk{N n} & \textbf nu
& \Gk{T t} & \textbf tau \\ 
\Gk{B b} & \textbf beta   & \Gk{J j} & \textbf{th}eta & \Gk{X x} & \textbf xi
& \Gk{U u} & \textbf upsilon \\ 
\Gk{G g} & \textbf gamma  & \Gk{I i} & \textbf iota  & \Gk{O o} & \textbf
omicron& \Gk{F f} & \textbf{ph}i\\ 
\Gk{D d} & \textbf delta  & \Gk{K k} & \textbf kappa & \Gk{P p} & \textbf pi
& \Gk{Q q} & \textbf{ch}i\\ 
\Gk{E e} & \textbf epsilon& \Gk{L l} & \textbf lambda& \Gk{R r} & \textbf{r}ho
& \Gk{Y y} & \textbf {ps}i\\ 
\Gk{Z z} & \textbf zeta   & \Gk{M m} & \textbf mu    & \Gk{S s,c} & \textbf
sigma  & \Gk{W w} & \textbf{\=o}mega\\ \hline 
  \end{tabular}
  \end{center}
\caption[The Greek alphabet]{In this table, the first letter or two of
  the (Latin) name for a Greek 
letter provides a transliteration
for that letter.  In texts, the rough-breathing mark (\Gk{<}) over
an initial
vowel (or \Gk r) is transcribed as a preceeding (or following) \letter
h; the smooth-breathing mark~(\Gk{>}) and the three tonal accents
%(\Gk{'{\ }}, \Gk{~}, \Gk{`{\ }}) 
(\Gk{'a}, \Gk{~a}, \Gk{`a}) 
can be ignored.  Especially in the dative case (the Turkish \Tur{-e
  hali}), some long vowels may be given the iota 
subscript (\Gk{a|}, \Gk{h|}, \Gk{w|}), representing what was once a
following iota (\Gk{ai}, \Gk{hi}, \Gk{wi}).
}
\label{table:Greek}
\end{table}

\section{Synthesis and analysis}

The geometry pioneered by
Ren\'e Descartes is called \defn{analytic
  geometry}; by contrast, the geometry of ancient mathematicians like
Euclid and Apollonius of Perge is sometimes called 
\defn{synthetic geometry}.  But what does this \emph{mean?}
  The word \Eng{synthetic} comes from the Greek \Gk{sunjetik'oc} meaning
  \Eng{skilled in putting together} or \Eng{constructive.}  This Greek
  adjective derives from the verb \Gk{sunt'ijhmi} \Eng{put together,
  construct.}  The word \Eng{analytic} is the English form of
  \Gk{>analutik'oc}, which derives from the verb \Gk{>anal'uw}
  \Eng{undo, set free, dissolve.}  
Although we refer to ancient geometry as synthetic, the Ancients
evidently recognize both analytic and synthetic
methods.  Around 320 \textsc{c.e.,} Pappus of Alexandria
writes \cite[p.~597]{MR13:419b}:
\begin{quote}
  Now \defn{analysis}{} (\Gk{>an'alusic}) is a method of taking that
  which is sought as
  though it were admitted and passing from it through its consequences
  in order to something which is admitted as a result of
  synthesis; for in analysis we suppose that which is sought to be
  already done, and we inquire what it is from which this comes about,
  and again what is the antecedent cause of the latter, and so on
  until, by retracing our steps,  we light upon something already
  known or ranking as a first principle; and such a method we call
  analysis, as being a \emph{reverse solution}
  (\Gk{>an'apalin l'usic}).

But in \defn{synthesis}{} (\Gk{sunj'esic}), proceeding in the opposite
  way, we suppose to be already done that which was last reached in
  the analysis, and arranging in their natural order as consequents
  what were formerly antecedents and linking them one with another, we
  finally arrive at the construction of what was sought; and this we
  call synthesis. 

Now analysis is of two kinds, one, whose object is to seek the truth,
being called \defn{theoretical}{} (\Gk{jewrhtik'oc}), and the other,
whose object is to find something set for finding, being called
\defn{problematical}{} (\Gk{problhmatik'oc}). 
\end{quote}
By the way, Pappus elsewhere \cite[pp.~564--567]{MR13:419b} says more
about the distinction brought up here between theorems and problems:
  \begin{quote}
    Those who favor a more technical terminology in geometrical
    research use \defn{problem}{} (\Gk{pr'oblhma}) to
    mean a [proposition\footnote{Ivor Thomas \cite[p.~567]{MR13:419b}
    uses \Eng{inquiry} here in his translation;
    but there is \emph{no} word in the Greek original corresponding to
    this or to \Eng{proposition}.}] in which it is proposed to do or construct
    [something]; and \defn{theorem}{} (\Gk{je'wrhma}), a [proposition]
    in which the
    consequences and necessary implications of certain hypotheses are
    investigated; but among the ancients some described them all as
    problems, some as theorems.
  \end{quote}

What really distinguishes Cartesian geometry from what came before is
perhaps suggested by the first sentence of Descartes's \emph{Geometry}
\cite[p.~2]{Descartes-Geometry}:
\begin{quote}
  Any problem in geometry can easily be reduced to such terms that a
  knowledge of the lengths of certain straight lines is sufficient for
  its construction.
\end{quote}
From a line, Descartes abstracts something called length.  A length is
something that we might today call a positive real number.  Descartes
takes the edifice of geometry that has been built up or
`synthesized' over the centuries, and reduces or `analyzes' its study
into the manipulation of numbers.  This is just what we are \emph{not}
going to do in these notes.

Again, my main interest will be in the conic sections.
If one has
time and interest, one should just read 
Apollonius on the subject, if not in the original Greek,\footnote{Of
  the eight books of Apollonius's treatise on the conic sections, the
  last has been lost, and the fifth, sixth, and seventh come down to
  us only in Arabic translation.} then in a
proper translation \cite{MR0053831, 
MR1660991} as opposed to a paraphrase \cite{Heath-Apollonius}. 
However, it may be that some features of the text of Apollonius are
dictated by customs of mathematical exposition that have no great
connexion with the mathematics itself.  
The style of Apollonius's text as we have it is reflected in the
observations of Proclus, in the fifth century \textsc{c.e.,} in his
commentaries on Euclid:
\begin{quote}
  Every problem and every theorem that is furnished with all its parts
  should contain the following elements: an \emph{enunciation}
  (\Gk{pr'otasic}), an \emph{exposition} (\Gk{>'ekjesic}), a
  \emph{specification}
  (\Gk{diorism'oc}), a \emph{construction} (\Gk{kataskeu'h}), a \emph{proof}
  (\Gk{>ap'odeixic}), and a \emph{conclusion} (\Gk{sump'erasma}).  Of these,
  the enunciation states what is given and what is being sought from
  it, for a perfect enunciation consists of both these parts.  The
  exposition takes separately what is given and prepares it in advance
  for use in the investigation.  The specification takes separately
  the thing that is sought and makes clear precisely what it is.  The
  construction adds what is lacking in the given for finding what is
  sought.  The proof draws the proposed inference by reasoning
  scientifically from the propositions that have been admitted.  The
  conclusion reverts to the enunciation, confirming what has been
  proved. 

So many are the parts of a problem or a theorem.  The most essential
ones, and those which are always present, are enunciation, proof, and
conclusion. \cite[p.~159]{MR1200456}.
\end{quote}
The propositions of Apollonius can be analyzed in these terms; but
such analysis is not my concern here.

\section{Conversational implicature}

Another difference between my approach and the ancient approach to
mathematics may result from a modern habit exemplified, for example,
in a Russian textbook of the Soviet period:
\begin{quote}
The student of mathematics must at all times have a clear-cut
  understanding of all fundamental mathematical concepts\dots
  The student will also recall the signs of weak inequalities: $\leq$
  (less than or equal to) and~$\geq$ (greater than or equal to).  The
  student usually finds no difficulty when using them in formal
  transformations, but examinations have shown that many students do
  not fully comprehend their meaning.

To illustrate, a frequent answer to: ``\emph{Is the inequality $2\leq
  3$ true?}'' is ``No, since the number $2$ is less than $3$.''  Or,
  say, ``\emph{Is the inequality $3\leq3$ true?}'' the answer is often
  ``No, since $3$ is equal to $3$.''  Nevertheless, students who
  answer in this fashion are often found to write the result of a
  problem as $x\leq3$.  Yet their understanding of the sign $\leq$
  between concrete numbers signifies that not a single specific number
  can be substituted in place of $x$ in the inequality $x\leq3$, which
  is to say that the sign $\leq$ cannot be used to relate any numbers
  whatsoever. \cite[pp.~9 f.]{DPR}
\end{quote}
The students referred to, who will not allow that $2\leq3$, are following
a habit of ordinary language, whereby the \emph{whole} truth must be
told.  According to this habit, one does not say $2\leq3$, because one
can make a stronger, more informative statement, namely $2<3$.  This
habit would appear to be an instance of \emph{conversational
  implicature:} the ability of people to convey or \emph{implicate}
statements that are not logically \emph{implied} by their words
\cite[ch.~1, \S5, pp.~36--40]{CGEL}.  In saying \Eng{$A$ or $B$ [is true],} one
usually implicates that one does not know \emph{which} is true.

I believe this habit of implicature may be reflected in the Greek
understanding,
according to which \emph{one} (\Gk{<'en}) is not a \emph{number}
(\Gk{>arijm'oc}).
In the \emph{Elements} \cite{MR1932864}, Book VII, Euclid somewhat
obscurely defines a
\defn{unit}{} (\Gk{mon'ac}) as that by virtue of which each being is
called `one'~\cite{MR17:814b}.  Then a \defn{number}{} is defined as a
\emph{multitude}
(\Gk{pl~hjoc}) composed of units.  In particular, a unit is not a
number, because it is not a multitude: it is one.  Euclid does not
bother to state explicitly this distinction between units and numbers,
but it can be inferred, for example,
from his presentation of what we now call the Euclidean algorithm.
Proposition VII.1 of the \emph{Elements} involves 
a pair of numbers such that the Euclidean algorithm, when applied to
them, yields a unit (\Gk{mon'ac}).  Then
this unit is \emph{not} considered as a greatest common divisor of the
numbers; the numbers do not \emph{have} a greatest common divisor; the
numbers are simply relatively prime.  If the numbers are
\emph{not} relatively prime, then the \emph{same} algorithm yields
their greatest common divisor.  This observation appears to be the
contrapositive of the first, but Euclid distinguishes it as
Proposition VII.2 of the \emph{Elements.}

In these notes, I shall treat one as a number.  Finally,
I shall use special symbolism more freely
than the Ancients did.  However, I shall \emph{not} abstract numbers
from the things numbered, or sizes distinct from 
the things with size.

\section{Magnitude and proportion}

The things with size are \defn{magnitude}{s.}  These may be straight
 lines (what we call \emph{line segments;} they may be plane
 figures like rectangles; they may perhaps be other things (solids,
 angles,~\dots).
The class of magnitudes is equipped
with a binary relation, denoted by $<$, where $A<B$ is read as `$A$ is
 less than $B$'.  There is also the
partial binary operation of \defn{addition}, denoted by $+$, where
 $A+B$ is the \defn{sum}{} of $A$ and $B$.  Axioms
governing magnitudes might be laid out as follows, starting with:
\newcounter{axioms}
\begin{enumerate}
  \item
The relation $<$ is a partial ordering.
\setcounter{axioms}{\value{enumi}}
\end{enumerate}
Two magnitudes are \defn{comparable}{} if they are equal, or if one is
less than the other.  Let us denote the
relation of {comparability} by $\sim$.  Then we can state some more axioms:
\begin{enumerate}
\setcounter{enumi}{\value{axioms}}
\item
$\sim$ is an equivalence-relation.
\item
Two magnitudes have a sum if and only if they are comparable.
\item
On every class of comparable magnitudes, addition is associative and
commutative.  
\setcounter{axioms}{\value{enumi}}
\end{enumerate}
We can now speak simply of \defn{multiple}s of a magnitude; we may
abbreviate these as follows:
\begin{equation*}
  A+A=2A,\quad A+A+A=3A,\quad\dots,
  \quad\underbrace{A+\dotsb+A}_n=nA,\quad\dots
\end{equation*}
\begin{enumerate}
\setcounter{enumi}{\value{axioms}}
\item
If $A\sim B$, then $A<A+B$.
\item
If $A<B$ and $A\sim C$, then $A+C<B+C$.
\setcounter{axioms}{\value{enumi}}
\end{enumerate}
Consequently, $A+C=B+C\implies A=B$.
\begin{enumerate}
\setcounter{enumi}{\value{axioms}}
\item
If $A<B$, then there is $C$ such that $A+C=B$.  
\setcounter{axioms}{\value{enumi}}
\end{enumerate}
This $C$ is unique by the
previous observation and can be denoted by $B-A$: it is the
\defn{difference}{} 
  of $B$ from $A$.
\begin{enumerate}
\setcounter{enumi}{\value{axioms}}
\item
(Archimedean Axiom)  
If $A\sim B$, then $A<nB$ for some multiple $nB$ of $B$.
\end{enumerate}
We define the quaternary relation of \defn{proportionality}{} as
follows.  This relation holds amongst the magnitudes $A$, $B$, $C$,
and $D$, and we write
\begin{equation*}
  \prop ABCD,
\end{equation*}
saying `$A$ is to $B$ as $C$ is to $D$,' provided $A\sim B$ and $C\sim
D$, and moreover
\begin{equation*}
  mA<nB\iff mC<nD
\end{equation*}
for all equimultiples $mA$ and $mC$ of $A$ and $C$, and 
%all equimultiples 
$nB$ and $nD$ of $B$ and~$D$.

\section{Definitions}

A \defn{cone}{} (\Gk{k~wnoc}) has, and is determined by:
\begin{enumerate}
\item
  a \defn{base}{}  (\Gk{b'asic}), which is a circle;
\item
an \defn{apex}{} (\Gk{koruf'hc}), which is a point not in the plane of
the base.
\end{enumerate}
Indeed, the base and the apex determine a \defn{conic surface}{}
(\Gk{kwnik`h >epif'aneia}), which
comprises every point of every straight line that passes through both
the apex and the circumference of the base.
The cone itself is the solid figure bounded by the base and the
conic surface.

The straight line through the apex and the center of the base is the
\defn{axis}{} (\Gk{>'axwn}) of the cone.  If this is perpendicular to
the base, then
the cone is \defn{right}{} (\Gk{orj'oc}); otherwise, the cone is
\defn{oblique}{} (\Gk{skalhn'oc}).  We 
shall work with an arbitrary cone.

Let the apex of the cone be $A$.
The conic surface has two parts, connected at $A$.  The cone
itself is bounded by one of these.  Suppose a plane, not containing
$A$, and not parallel to the base, cuts the part of the conic
surface that bounds the cone.  We shall study the \defn{conic
  section}{} (\Gk{tom'h}) so made.  We
may assume that the plane cuts the base in some 
chord $BC$, with midpoint $D$.  There is a diameter $EF$ of the
base that is perpendicular to $BC$; these two chords meet at $D$.  We
have a triangle $AEF$, called an
\defn{axial triangle}{} (\Gk{di`a tou >'axonoc trig'wnoc}), since it
contains the axis of the cone.  See Figure~\ref{fig:tb}. 

\begin{figure}[ht]
  \begin{pspicture}(-1,-3)(14,5)
    \psline(0,0)(4,0)(3,4)(0,0)
    \psline(2,2.6667)(2.6667,0)
    \psline(1.5,2)(3.5,2)
    \uput[u](3,4){$A$}
    \uput[d](0,0){$E$}
    \uput[d](4,0){$F$}
    \uput[d](2.6667,0){$D$}
    \uput[ul](2,2.6667){$G$}
    \uput[ur](2.1667,2){$L$}
    \uput[ul](1.5,2){$M$}
    \uput[ur](3.5,2){$N$}
    \pscircle(8,0){2}
    \psline(6,0)(10,0)
    \psline(8.6667,1.8856)(8.6667,-1.8856)
    \uput[l](6,0){$E$}
    \uput[r](10,0){$F$}
    \uput[dr](8.6667,0){$D$}
    \uput[d](8.6667,-1.8856){$B$}
    \uput[u](8.6667,1.8856){$C$}
    \pscircle(12,2){1}
    \psline(11,2)(13,2)
    \psline(11.6667,2.9428)(11.6667,1.0572)
    \uput[d](11.6667,1.0572){$H$}
    \uput[u](11.6667,2.9428){$K$}
    \uput[dr](11.6667,2){$L$}
    \uput[l](11,2){$M$}
    \uput[r](13,2){$N$}
  \end{pspicture}
\caption{Axial triangle, base, and section parallel to the base}
\label{fig:tb}
\end{figure}

We may
assume that the conic section contains a point $G$ of $AE$ (extended
as necessary).  Then $G$
is the \defn{vertex}{} (\Gk{koruf'hc}) of the section, and $GD$ is the
\defn{diameter}{} (\Gk{di'ametroc}) of the section.
Indeed, let a point $H$ be chosen at random on the section.  If $H$ is
not $G$, then the section has a chord $HK$ that is parallel to $BC$,
and $HK$ is bisected by $GD$, say at $L$.

Let the straight line through $L$ parallel to $EF$ meet $AE$ at $M$
and $AF$ at $N$.  We shall refer to $HL$ as an \defn{ordinate}{}
(\Gk{katag'omenh tetagm'enwc}), and to
$GL$ as the corresponding \defn{abscissa}.  Our first aim is to relate
ordinates and abscissas.  We may note first of all
\begin{equation*}
  HL^2=ML\cdot LN;\qquad BD^2=ED\cdot DF.
\end{equation*}

\section{Kinds of conic sections}

There are three cases to consider.  Suppose first of all that
$GD\parallel AF$.  Then $LN=DF$, so that
\begin{equation*}
  \prop{HL^2}{BD^2}{ML}{ED}.
\end{equation*}
Thus abscissas are to one another as the squares on the ordinates.
Draw $GP$ perpendicular to $GD$ so that
\begin{equation*}
  PG\cdot GD=BD^2=ED\cdot DF
\end{equation*}
and therefore also $PG\cdot GL=KL^2$.  The segment $PG$ is called, in
Latin, the \defn{\emph{latus rectum}}, and in English, the \defn{upright
  side}{} (\Gk{>orj'ia}).  The conic section itself is a
\defn{parabola}{} (\Gk{parabol'h} \Eng{juxtaposition, comparison,
  application}), from the verb \Gk{parab'allw} \Eng{throw beside,}
because the square on the ordinate is equal to a rectangle on the
abscissa that can be applied to the upright side.
For an alternative characterization of the upright side, we have
\begin{equation*}
  \prop{PG}{DF}{ED}{GD}\psign\ratio{EF}{AF}.
\end{equation*}
But also $\prop{DF}{GA}{EF}{EA}$.  Compounding ratios, we have
\begin{equation*}
  \text{\fbox{$\prop{PG}{GA}{EF^2}{EA\cdot AF}$}}.
\end{equation*}

In the second case, $DG$ extended meets $FA$ extended at a point $Q$.

%\bibliographystyle{plain}
%\bibliography{../../../TeX/references}

\def\cprime{$'$}
\begin{thebibliography}{10}

\bibitem{Heath-Apollonius}
{Apollonius of Perga}.
\newblock {\em Apollonius of Perga: Treatise on Conic Sections}.
\newblock University Press, Cambridge, UK, 1896.
\newblock Edited by {T}. {L}. {H}eath in modern notation, with introductions
  including an essay on the earlier history of the subject.

\bibitem{MR0053831}
{Apollonius of Perga}.
\newblock {\em On conic sections}.
\newblock Great Books of the Western World, no. 11. Encyclopaedia Britannica,
  Inc., Chicago, London, Toronto, 1952.

\bibitem{MR1660991}
{Apollonius of Perga}.
\newblock {\em Conics. {B}ooks {I}--{III}}.
\newblock Green Lion Press, Santa Fe, NM, revised edition, 1998.
\newblock Translated and with a note and an appendix by R. Catesby Taliaferro,
  With a preface by Dana Densmore and William H. Donahue, an introduction by
  Harvey Flaumenhaft, and diagrams by Donahue, Edited by Densmore.

\bibitem{Descartes-Geometry}
Ren{\'e} Descartes.
\newblock {\em The Geometry of {R}en{\'e} {D}escartes}.
\newblock Dover Publications, Inc., New York, 1954.
\newblock Translated from the French and Latin by David Eugene Smith and Marcia
  L. Latham, with a facsimile of the first edition of 1637.

\bibitem{DPR}
G.~Dorofeev, M.~Potapov, and N.~Rozov.
\newblock {\em Elementary Mathematics: Selected Topics and Problem Solving}.
\newblock Mir Publishers, Moscow, 1973.
\newblock Translated from the Russian by George Yankovsky.

\bibitem{MR17:814b}
Euclid.
\newblock {\em 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}.
\newblock Dover Publications Inc., New York, 1956.
\newblock Translated with introduction and commentary by Thomas L. Heath, 2nd
  ed.

\bibitem{MR1932864}
Euclid.
\newblock {\em Euclid's {\it {E}lements}}.
\newblock Green Lion Press, Santa Fe, NM, 2002.
\newblock All thirteen books complete in one volume, the Thomas L. Heath
  translation, edited by Dana Densmore.

\bibitem{CGEL}
Rodney Huddleston and Geoffrey~K. Pullum.
\newblock {\em The Cambridge Grammar of the English Language}.
\newblock Cambridge University Press, Cambridge, UK, 2002.
\newblock reprinted 2003.

\bibitem{MR1200456}
Proclus.
\newblock {\em A commentary on the first book of {E}uclid's {\it {E}lements}}.
\newblock Princeton Paperbacks. Princeton University Press, Princeton, NJ,
  1992.
\newblock 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.

\bibitem{MR13:419b}
Ivor Thomas, editor.
\newblock {\em Selections illustrating the history of {G}reek mathematics.
  {V}ol. {II}. {F}rom {A}ristarchus to {P}appus}.
\newblock Harvard University Press, Cambridge, Mass, 1951.
\newblock With an English translation by the editor.

\end{thebibliography}


\end{document}
