\documentclass[a4paper,twoside,11pt,reqno,draft]{amsart}
%\documentclass[a4paper,twoside,11pt,draft]{article}
\usepackage{typearea}
\usepackage[polutonikogreek,english]{babel}
\usepackage[oxonia]{psgreek}
\newcommand{\Gk}[1]{\selectlanguage{polutonikogreek}#1\selectlanguage{english}}
\newcommand{\Eng}[1]{\emph{#1}}
\newcommand{\Tur}[1]{\textsf{#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{second draft})}
\author{David Pierce}
\date{\today}

%\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}

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

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


\renewcommand{\theenumi}{\roman{enumi}}
\renewcommand{\labelenumi}{(\theenumi)}

\renewcommand{\land}{\mathrel{\&}}

\theoremstyle{plain}

\newtheorem{theorem}{Theorem}
\newtheorem{axiom}{Axiom}

\newtheorem*{enunciation}{Enunciation}

\theoremstyle{definition}

\newtheorem{proposition}{Proposition}

\theoremstyle{remark}

\newtheorem*{exposition}{Exposition}
\newtheorem*{specification}{Specification}
\newtheorem*{construction}{Construction}
%\newtheorem*{proof}{Proof}
\newtheorem*{conclusion}{Conclusion}

\begin{document}
  \maketitle
\tableofcontents

%\setcounter{section}{-1}
%\setcounter{part}{-1}

\part{Preface}

%\section{Preface}

As an undergraduate, I attended a college\footnote{St.\ John's
  College, with campuses in Annapolis, Maryland, and Santa Fe,
  New Mexico, USA.} where Euclid and Apollonius were used as
  textbooks.  They were so used, not because they were considered to
  be the \emph{best} textbooks, but simply because they \emph{had
  been} textbooks for countless generations of mathematicians:
  therefore (the idea was), one might gain some understanding of
  humanity and oneself by reading these books.  (The same is true for
  Homer, Aeschylus, Plato, and the other great books read at the college.)

Now, having become a professional mathematician, I ask what Euclid and
Apollonius have to offer the mathematician of today.  It is in pursuit
of an answer to this question that I prepare these notes---which
therefore are part of an ongoing project.

I prepare these notes also for the sake of honesty about what students
are asked to learn.  The conic
sections are a standard part of an elementary course of mathematics.
But the sections are usually given as the curves defined by certain
equations, such as $ay=x^2$ or $(x/a)^2\pm(y/b)^2=1$.  Or perhaps the
curves
are given in terms of foci and directrices.  A textbook may \emph{assert}
that the curves so defined can indeed by obtained as sections of cones;
but it is rare that this assertion is justified.

One calculus textbook\footnote{James Stewart, \emph{Calculus}, fifth
  edition, p.~720.  This text is currently in use at Middle East
  Technical University.}
  writes at the beginning of a section:
\begin{quote}
  In this section we give geometric definitions of parabolas,
  ellipses, and hyperbolas and derive their standard equations.  They
  are called \textbf{conic sections}, or \textbf{conics}, because they
  result from intersecting a cone with a plane as shown in Figure 1.
\end{quote}
I omit the book's Figure 1.  The text continues under the heading
\emph{Parabolas}:
\begin{quote}
  A \textbf{parabola} is the set of points in a plane that are
  equidistant from a fixed point $F$ (called the \textbf{focus}) and a
  fixed line (called the \textbf{directrix})\dots In the 16th century
  Galileo showed that the path of a projectile that is shot into the
  air at an angle to the ground is a parabola.  Since then, parabolic
  shapes have been used in designing automobile headlights, reflecting
  telescopes, and suspension bridges\dots We obtain a
  particularly simple equation for a parabola if we place its vertex
  at the origin $O$\dots
\end{quote}
Is the parabola obtained from a cone, or
is it obtained from a focus and a directrix?  It can be obtained in
either way, but the author does not say so.  The author need not
\emph{prove} the equivalence of the two constructions; but at least he
ought to state that he is not going to prove it.  

Perhaps the author expects the reader to \emph{infer} the equivalence
of the two definitions of a parabola.  But this is not like the
author, who is otherwise eager to give the reader every assistance
that he can think of.  He apparently does not think the reader can
be left to infer that parabolas are worth 
studying.  Before concluding anything from the definition, the
author feels the need to tell the reader how \emph{useful} parabolas
are. 

Another textbook\footnote{Robert A. Adams, \emph{Calculus: a complete
    course,} fourth edition, p.~476.  This text was formerly used
 at METU.} follows a similar procedure, first defining the conic
    sections as such, then defining them in terms of foci and
    directrices.  Between the two definitions, the writer observes that the
    intersection of a cone and a plane will be given by a
    second-degree equation.  This suggests that the quadratic
    equations to be derived presently in the book may indeed define conic
    sections.  However, no attempt is made to prove
    that every curve defined by a quadratic equation can be obtained
    as the section of a cone.  The author observes:
\begin{quote}
  After straight lines the conic sections are the simplest of plane
  curves.  They have many properties that make them useful in
  applications of mathematics; that is why we include a discussion of
  them here.  Much of this material is optional from the point of view
  of a calculus course, but familiarity with the properties of conics
  can be very important in some applications.  Most of the properties
  of conics were discovered by the Greek geometer, Apollonius of
  Perga, about 200 BC.  It is remarkable that he was able to obtain
  these properties using only the techniques of classical Euclidean
  geometry; today most of these properties are expressed more
  conveniently using analytic geometry and specific coordinate
  systems. 
\end{quote}
Again, the justification offered for the study of the conic sections
is their usefulness.  But as for `expressing' the properties of conic
sections, which of the following expresses better what a conic section
\emph{is?} 
\begin{enumerate}
  \item
It is the intersection of a cone and a plane.
\item
It is the intersection of two surfaces defined by equations
$ax+by+cz+d=0$ and $(x-ez)^2+y^2=fz^2$ respectively.
\end{enumerate}
What the author means, I think, is that it is convenient to define
certain curves analytically and obtain their properties by further
analysis.  But showing that those curves are conic sections is a whole
other problem, not addressed in the book.

By the way, despite what the last quotation suggests, I am not
sure that obtaining nice results with limited 
mathematical tools is remarkable in itself.  The tools of an artisan
depend on what is available in the physical environment; but the tools
of a mathematician depend only on imagination.  A
mathematician without the imagination to come up with the best tool
for the job would seem to be an unremarkable mathematician.


The first chapter of Hilbert and Cohn-Vossen's \emph{Geometry and the
  Imagination}~\cite{MR0046650} contains a
beautiful account of how various properties of the conic sections
arise from consideration of the cones from which the sections are
obtained.  However, the cones considered by the authors
are all \emph{right} cones.  Apollonius does not make this
restriction.  Hilbert and Cohn-Vossen give an etymology for the names
of the ellipse, the hyperbola, and the parabola: it involves
eccentricity.  The etymology is plausible, but it appears to be
literally incorrect, as a reading of Book I of Apollonius would show.

Mathematics reveals underlying correspondences between seemingly
dissimilar things.  Sometimes we treat these correspondences as
identities.  This can be a mistake.  There is a correspondence between
conic sections and quadratic equations.  But are the sections
\emph{really} the equations?  One cannot answer the question without
considering \emph{conic sections as such.}

\part{Introduction}

%\section{Introduction}\label{sect:intro}

Some time in the 3rd century \textsc{b.c.e.,} Apollonius of Perga wrote eight
books on conic sections.  We have the first four books
\cite{MR1660991,Apollon4} in the original
Greek; the next three books survive in Arabic translation
\cite{MR1053158}; the eighth book is lost.

As Apollonius tells us in an introductory letter, his first four books
are part of an elementary course on the conic sections.
The present notes are for use in such a course.

Before Apollonius, around 300 \textsc{b.c.e.}, Euclid published the
thirteen books of the \emph{Elements,} a work of mathematics of which
some of the parts could well be used as a textbook today.  The
\emph{Elements} provide a good example of mathematical exposition and
of what it means to prove something.

Apollonius assumes some of the concepts and theorems found in Euclid.
Before getting into Apollonius then, I shall in
\S~\ref{sect:prop} have a look at the \emph{Elements,} and
specifically at the
theory of \emph{proportion} developed there.

Before that, in \S\S~\ref{sect:c-imp} and~\ref{sect:lines}, I look at
some general features of ancient mathematics as I understand it.
Meanwhile, in
\S~\ref{sect:syn-anal}, I jump forward in history to Descartes, in
order to have in mind the
sorts of improvements that he thought he was making to
mathematical practice.

Because I shall occasionally refer to some Greek words, 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]{The first letter or two of
  the (Latin) name for a Greek 
letter provides a transliteration
for that letter.  However, upsilon is also transliterated by \Eng{y.}
The diphthong \Gk{ai} often comes into English (\emph{via} Latin)
  as \Eng{ae,} while \Gk{oi} may come as \Eng{oe.}
The second form of the small sigma is used at the ends of words.
  In texts, the rough-breathing mark (\Gk{<}) over 
an initial
vowel (or \Gk r) is transcribed as a preceeding (or following) \letter
h (as in \Gk{<o <r'omboc} \emph{ho rhombos} `the rhombus').  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}
  (I have heard a rumor that students can
improve their mathematics simply by learning this alphabet, assuming
they didn't grow up knowing it.)

\section{Theorems and problems}\label{sect:Proclus}

The text of Apollonius as we have it consists almost entirely of
theorems and problems.  There are some introductory remarks, some
definitions, but nothing else.  The theorems and problems can be
analyzed in a way described by Proclus,\footnote{Proclus was born in
  Byzantium (Constantinople, \.Istanbul), but his parents were from
  Lycia (Likya), and he was educated first in Xanthus.  He moved to
  Alexandria, then Athens, to study philosophy
  \cite[p.~xxxix]{MR1200456}.} in the
fifth century \textsc{c.e.,} in his 
commentaries on Euclid \cite[p.~159]{MR1200456}:
\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'o\-deixic}), 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.
\end{quote}

More is said in \S~\ref{sect:syn-anal} about the distinction
between a theorem and a problem.  In \S~\ref{sect:prop}, I label the
parts of a theorem with the terminology of Proclus.  But what I
present in \S~\ref{sect:conics}
are merely \emph{enunciations} in the sense of Proclus.  They do not
use any special mathematical notation, although they may involve
technical terms.  Sometimes I incorporate the \emph{definitions} of
technical terms in these enunciations.
While each of Apollonius's own enunciations tends to consist of a
single long sentence, I use several sentences.  So these notes should not be
considered as an English translation of Apollonius.
The point here is to do \emph{mathematics,} not history.

\section{Synthesis and analysis}\label{sect:syn-anal}

We may however 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 \defn{pre-Cartesian}{} mathematics: mathematics as done
before (well before) the time of Ren\'e Descartes (1596--1650).

The geometry pioneered by
Ren\'e Descartes is called \defn{analytic
  geometry}; by contrast, the geometry of ancient mathematicians like
Euclid and Apollonius 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} (from \Gk{sun} \Eng{together} and \Gk{t'ijhmi}
  \Eng{put}).  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} (from \Gk{>ana} \Eng{up,} \Gk{l'uw}
  \Eng{loose}).    
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}
This passage is not very useful without examples: I shall propose one
presently.  Meanwhile, I note by
the way that Pappus elsewhere \cite[pp.~564--567]{MR13:419b} says more
about the distinction 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 straight line, Descartes abstracts something called \emph{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.  To be more precise, he `takes that
which is sought as though it were admitted' in the following way.  In
Figure~\ref{fig:Descartes},
\begin{figure}[ht]
    \begin{pspicture}(-2,-4)(6,4)
      \psline(-1.364,-1.364)(3.333,3.333)
      \psline(0,0)(4,0)
      \psline(1.5,-3)(3.714,1.429)
      \psset{linestyle=dotted}
      \psline(4.388,-1.94)(1.5,-3)
      \psline(4.388,-1.94)(3.333,3.333)
      \psline(-1.364,-1.364)(4.388,-1.94)
      \uput[ul](3,0){$A$}
      \uput[ur](4,0){$B$}
      \uput[dr](4.388,-1.94){$C$}
      \uput[dl](1.5,-3){$D$}
      \uput[ul](0,0){$E$}
      \uput[ul](-1.364,-1.364){$F$}
      \uput[l](3.714,1.429){$R$}
      \uput[ul](3.333,3.333){$S$}
    \end{pspicture}
\caption{}\label{fig:Descartes}
\end{figure}
 straight lines $BE$, $DR$, and $FS$ are
given in position (meaning their endpoints themselves are not fixed);
and the sizes of angles $ABC$, $ADC$, and $CFE$ are given.  It is required to
find the point $C$ so that the
rectangle with sides $BC$ and $BD$ has a given ratio to the square on
$CF$.  (This is a simplified version of the problem that Descartes
takes up in the \emph{Geometry.})

In his analytic approach, Descartes assumes that $C$ has already been
found, as in
the figure.  We denote $AB$ by $x$, and $BC$ by $y$.  The ratio
$AB:BR$ is given; call it $z:b$.  Then
\begin{equation*}
  RB=\frac{bx}z,\qquad CR=y+\frac{bx}z=\frac{zy+bx}z.
\end{equation*}
But $CR:CD$ is given; call it $z:c$.  Then
\begin{equation*}
  CD=\frac{czy+bcx}{z^2}.
\end{equation*}
Also $AE$ is given; call it $k$.  And let $BE:BS=z:d$.  Then
\begin{equation*}
  BE=k+x,\quad BS=\frac{dk+dx}z,\quad CS=\frac{zy+dk+dx}z.
\end{equation*}
Finally, if $CS:CF=z:e$, then
\begin{equation*}
  CF=\frac{ezy+dek+dex}{z^2}.
\end{equation*}
So it is given that the ratio
\begin{equation*}
y\cdot\frac{czy+bcx}{z^2}:\left(\frac{ezy+dek+dex}{z^2}\right)^2
\end{equation*}
is constant.  This gives us a quadratic equation in the unknowns $x$ and
$y$.  

Descartes's method does not use explicitly drawn
\emph{axes} with respect to which $x$ and $y$ are measured. Also, the
straight lines called $x$ and $y$ are not 
required to be perpendicular: they are merely not parallel.

Through analysis, we have found an equation
that determines the point $C$.  Since the equation is quadratic, the
point $C$ lies on a curve known as a \defn{conic section}.  When there
are more straight lines in the problem, then the resulting equation
may have a higher degree.

We do not get any sense here for what the curve of $C$ \emph{looks like.}
We might get some sense by analyzing the equation for $C$.
Apollonius will give us a sense for what conic sections look like by
showing \emph{how they are related to the cones that they come from.}

\section{Conversational implicature}\label{sect:c-imp}

One apparent difference between the ancient and modern approaches to
 mathematics may result from a modern habit that is exemplified
in a Russian textbook of the Soviet period \cite[pp.~9 f.]{DPR}:
\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.
\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:} this is 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.

This habit of implicature may be reflected in the ancient
understanding,
according to which \emph{one} (\Gk{<'en}) is not a \emph{number}
(\Gk{>arijm'oc}).
In Book VII of the \emph{Elements,} Euclid somewhat
obscurely defines a
\defn{unit}{} (\Gk{mon'ac}) as that by virtue of which each being is
called `one'. (This English version of the definition is based on the
Greek text supplied in
\cite[Vol.~2, p.279]{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 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.}

Conversational implicature may be seen in Apollonius's treating of the
circle as different from an ellipse.

\section{Lines}\label{sect:lines}

I follow the old understanding whereby a \defn{line}{} need not be
straight.  A line may have endpoints, or it may be, for
example, the circumference of a \emph{circle.}  Indeed, according to
the definition in Euclid's \emph{Elements}
\cite{MR17:814b,bones,MR1932864},
\begin{quote}
A \defn{circle}{} is a plane figure contained by one line such that all
the straight lines falling upon it from one point among those lying
within the figure are equal to one another.
\end{quote}
A \emph{straight} line does have endpoints; but the straight line may
be \emph{produced} (extended) beyond these endpoints, as far as desired.

\section{Magnitudes and proportions}\label{sect:prop}

The things with size are \defn{magnitude}{s.}  These may be straight
 lines; they may be plane
 figures like rectangles; they may perhaps be other things (solids,
 angles,~\dots).  With Descartes, we freely turned straight lines into
 numbers, with which we could do arithmetic.
Henceforth, I want to be more careful.  Here
I shall attempt to suggest a theory of magnitudes based what is found in
 Books V and VI of Euclid.  However, I shall use modern terminology, with
 abbreviative notation that is apparently foreign to Euclid's style.

The class of magnitudes is equipped
with a binary relation, which I shall denote by $<$, so that $A<B$ is
to be read as `$A$ is 
 \defn{less than}{} $B$'.  

 \begin{axiom}
   The relation $<$ is a strict partial ordering: we never have $A<A$;
   we never have both $A<B$ and $B<A$;
but if $A<B$, and $B<C$, then $A<C$.
 \end{axiom}

The reflexive partial ordering corresponding to $<$ is $\leq$.  That
is, $A\leq B$ if and only if $A<B$ or $A=B$. I shall refer to this
relation as \defn{inclusion}.  

Let us say that 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 $\sim$.  

\begin{theorem}
Comparability of magnitudes is an equivalence: always $A\sim A$; if
  $A\sim B$, 
  then $B\sim A$; if $A\sim B$ and $B\sim C$, then $A\sim C$.
\end{theorem}

On every class of comparable magnitudes, there is an operation of
\defn{addition}, which we may denote by $+$.  
If $A\sim B$, then the two magnitudes have a \defn{sum}, $A+B$.

\begin{axiom}
On every class of comparable magnitudes, addition is associative and
commutative: $A+(B+C)=(A+B)+C$, and $A+B=B+A$.
\end{axiom}

\begin{axiom}
If $A\sim B$, then $A<A+B$.
\end{axiom}

\begin{axiom}
If $A<B$ and $A\sim C$, then ($B\sim C$ and) $A+C<B+C$.
\end{axiom}

\begin{theorem}\label{thm:cancel}
If $A+B=A+C$, then $B=C$.
\end{theorem}

\begin{axiom}
If $A<B$, then there is $C$ such that $A+C=B$.  
\end{axiom}

By Theorem~\ref{thm:cancel} then, if $A<B$, then there is exactly one
magnitude $C$ such that $A+C=B$.  We may write $C$ as $B-A$, calling
it the \defn{excess}{} of $B$ over $A$.

If two magnitudes are comparable, then it may happen that the lesser
\defn{measure}s the greater.  If $A$ measures $B$, let us denote this
by $A\measures B$.  In this case, we may say that $B$ is a
\defn{multiple}{} of $A$.

\begin{axiom}
  Measurement is a reflexive partial order.
\end{axiom}

\begin{axiom}
   Measurement refines inclusion: if
  $A\measures B$, then $A\leq B$.
\end{axiom}

It is not clear (to me) whether Euclid allows a magnitude to
be a multiple of itself; but he does allow a magnitude to measure
itself: see his Proposition~X.3.

\begin{axiom}
  $A\measures A$.
\end{axiom}

\begin{axiom}
  If $A\measures B$, then $A\measures A+B$.
\end{axiom}

\begin{axiom}
If $A\sim B$, then there is $C$ such that $B\measures C$ and $A<C$.
\end{axiom}

We need not talk about \emph{how many times} a magnitude measures
another; but we need to be able to say whether $A$ measures $B$
\emph{as many times} as $C$ measures $D$.  If it does, I propose to
write
\begin{equation*}
  \equim ABCD;
\end{equation*}
we may say also, with Euclid, that $B$ and $D$ are
\defn{equimultiples}{} of $A$ and $C$.  Implicit in this notation is
that, if $\equim ABCD$, then $A\measures B$.

\begin{axiom}\mbox{}
  \begin{enumerate}
    \item
If $A\measures B$, then $\equim ABAB$.
\item
$\equim ABCD$ if and only if $\equim CDAB$.
\item
If $\equim ABCD$ and $\equim CDEF$, then $\equim ABEF$.
  \end{enumerate}
\end{axiom}

\begin{axiom}
  Suppose $\equim ABCD$. then $A<C$ if and only if $B<D$.
\end{axiom}

\begin{theorem}
    Suppose $\equim ABCD$.  Then $A=C$ if and only if $B=D$.
\end{theorem}

\begin{axiom}
  If $A\measures B$, and $D$ is arbitrary, then there are magnitudes
  $C$ and $E$ such that $\equim ABCD$ and $\equim ABDE$.
\end{axiom}

In Propositions V.1, 2, and 3, Euclid offers proofs of the following
three axioms. 

\begin{axiom}
  If $\equim ABCD$, then $\equim AB{A+C}{B+D}$.
\end{axiom}

\begin{axiom}
  If $\equim ABDE$ and $\equim ACDF$, then $\equim A{B+C}D{E+F}$.
\end{axiom}

\begin{axiom}\label{axiom:5.3}
  If $\equim ABDE$ and $\equim BCEF$, then $\equim ACDF$.
\end{axiom}

The quaternary relation of equimultiplicity is a special case of
\defn{proportion}, which we can now define.
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 $D$,' or `$A$ has the \defn{same
  ratio}{} to
$B$ that $C$ has to $D$,' provided $A\sim B$ and $C\sim
D$, and moreover, whenever $\equim AECF$ and $\equim BGDH$, we have
\begin{equation*}
  E<G\quad\text{if and only if}\quad F<H.
\end{equation*}

\begin{theorem}
  Having the same ratio is an equivalence, that is:
  \begin{enumerate}
    \item
If $A\sim B$, then $\prop ABAB$.
\item
$\prop ABCD$ if and only if $\prop CDAB$.
\item
If $\prop ABCD$ and $\prop CDEF$, then $\prop ABEF$.
  \end{enumerate}
\end{theorem}

Here are Euclid's next three propositions (V.4, 5, and 6):

\begin{theorem}
  If $\prop ABCD$ and $\equim AECF$ and $\equim BGDH$, then $\prop
  EGFH$. 
\end{theorem}

\begin{proof}
  Suppose $\prop ABCD$ and $\equim AECF$ and $\equim BGDH$.  Let
  $\equim EKFL$ and $\equim GMHN$.  By
  Axiom~\ref{axiom:5.3}, we have $\equim AKCL$ and $\equim BMDN$.  By
  definition of proportionality, we have $K<M$ if and only if $L<N$.
  Therefore, by this definition again, $\prop EGFH$. 
\end{proof}

How can this theory of proportion be used to prove the proportionality
of corresponding sides of similar triangles?  The following theorems are
found in Euclid.  I state them in my own words, and analyze the first
theorem
according to the terminology of Proclus given in \S~\ref{sect:Proclus}.
That part which is labelled `theorem' is just the enunciation.

\begin{theorem}[\emph{Elements} I.35]\label{thm:same-base}
If two parallelograms have a common base, and the sides opposite this
base are part of a common straight line, then the parallelograms are
equal.
\end{theorem}

\begin{exposition}
  Suppose $ABCD$ and $ABEF$ are parallelograms, with the common base
$AB$, such that the sides $CD$ and $EF$ lie in a common straight line.
\end{exposition}

\begin{center}
  \begin{pspicture}(-0.5,-0.5)(4.5,2.5)
    \psline(0,0)(3,0)(2.5,2)(-0.5,2)(0,0)(1,2)(4,2)(3,0)
\uput[d](0,0){$A$}
\uput[d](3,0){$B$}
\uput[u](2.5,2){$C$}
\uput[u](-0.5,2){$D$}
\uput[u](4,2){$E$}
\uput[u](1,2){$F$}
  \end{pspicture}
\end{center}

\begin{specification}
Then $ABCD$ is equal to $ABEF$.
\end{specification}

\begin{construction}
We may assume that $C$ and $F$ are on the same side of $D$.  If $C$ is
between $D$ and $F$, then let $G$ be the intersection of $AF$ and $BC$.
\end{construction}

\begin{center}
    \begin{pspicture}(-0.5,-0.5)(3.5,2.5)
    \psline(0,0)(2,0)(3.8,2)(-1,2)(0,0)
    \psline(0,0)(1.8,2)
    \psline(2,0)(1,2)
\uput[d](0,0){$A$}
\uput[d](2,0){$B$}
\uput[u](1,2){$C$}
\uput[u](-1,2){$D$}
\uput[u](3.8,2){$E$}
\uput[u](1.8,2){$F$}
\uput[r](1.3,1.4){$G$}
  \end{pspicture}
\end{center}
\begin{proof}
The triangle $ADF$ is equal to $BCE$.
If $C$ is \emph{not} between $D$ and $F$, then 
$ABEF$ is obtained from $ABCD$ by
removing triangle $ADF$ and adding the equal triangle $BCE$.  
On the other hand, if $C$ \emph{is} between $D$ and $F$, then the
quadrilaterals $ADCG$ and $BEFG$ are equal, being obtained from the
equal triangles $ADF$ and $BCE$ respectively be subtraction of the
common triangle $CFG$.
Then 
$ABEF$ is obtained from $ABCD$ by
removing the quadrilateral $ADCG$ and adding the equal quadrilateral
$BEFG$.
\end{proof}

\begin{conclusion}
Therefore, if two parallelograms have a common base, and the sides
opposite this
base are part of a common line, then the parallelograms are equal.
\end{conclusion}

The conclusion repeats the enunciation.  The construction is not
always needed, even in principle.  Henceforth I shall follow the usual
custom of treating what is not part of the enunciation as part of the
proof.  I omit some proofs, as being clear from the picture.

\begin{theorem}[I.34]\label{thm:half}
A diagonal divides a parallelogram into two equal triangles.
\end{theorem}

\begin{center}
  \begin{pspicture}(-1,-0.5)(3.5,2.5)
    \psline(0,0)(3,0)(2.5,2)(-0.5,2)(0,0)(2.5,2)
\uput[d](0,0){$A$}
\uput[d](3,0){$B$}
\uput[u](2.5,2){$C$}
\uput[u](-0.5,2){$D$}
  \end{pspicture}
\end{center}

\begin{proof}
  Let $ABCD$ be a parallelogram, with diagonal $AC$.  Then angle $BAC$
  equals angle $ACD$, and angle $BCA$ equals angle $CAD$, and the side
  $AC$ is common: so triangle $ABC$ equals triangle $CDA$.
\end{proof}

\begin{theorem}[I.36]
  Parallelograms between the same parallel straight lines and with
  equal bases are equal.
\end{theorem}

\begin{center}
  \begin{pspicture}(-0.5,-0.5)(7.5,2.5)
    \psline(0,0)(7,0)(6,2)(0.5,2)(0,0)
    \psline(3,0)(3.5,2)(7,0)
    \psline(0.5,2)(4,0)(3,2)
\uput[u](0.5,2){$D$}
\uput[u](3,2){$H$}
\uput[u](3.5,2){$C$}
\uput[u](6,2){$G$}
\uput[d](0,0){$A$}
\uput[d](3,0){$B$}
\uput[d](4,0){$E$}
\uput[d](7,0){$F$}
  \end{pspicture}
\end{center}

\begin{proof}
  Let $ABCD$ and $EFGH$ be parallelograms such that the bases $AB$ and
  $EF$ are equal and are part of the same straight line; and suppose
  $CD$ and $GH$ are in the same straight line.  Draw the straight
  lines  $CF$ and $DE$.  Then $CDEF$ is a parallelogram equal to both
  $ABCD$ and $EFGH$, by Theorem~\ref{thm:same-base}.
\end{proof}

\begin{theorem}[VI.1]
  Parallelograms drawn between the same parallel
  straight lines are to one another as their bases.
\end{theorem}

\begin{center}
  \begin{pspicture}(-0.5,-0.5)(7.5,2.5)
    \psline(0,0)(7,0)(6,2)(0.5,2)(0,0)
    \psline(2,0)(2.5,2)
    \psline(4,0)(3,2)
    \psline(6,0)(5,2)
\uput[u](0.5,2){$D$}
\uput[u](3,2){$H$}
\uput[u](2.5,2){$C$}
\uput[u](6,2){$G$}
\uput[d](0,0){$A$}
\uput[d](2,0){$B$}
\uput[d](4,0){$E$}
\uput[d](7,0){$F$}
\uput[d](6,0){$K$}
\uput[u](5,2){$L$}
  \end{pspicture}
\end{center}

\begin{proof}
Suppose $ABCD$  
 and $EFGH$ are parallelograms such that the bases $AB$ and
  $EF$ are part of the same straight line, and also
  $CD$ and $GH$ are in the same straight line.  We may assume that
 $EF$ is greater than $AB$.  Between $E$ and $F$, let $K$ be chosen so
 that $EF$ is equal to $AB$.  Through $K$, a straight line drawn
 parallel to $EH$ meets $GH$ at a point $L$.  Then parallelograms
 $ABCD$ and $EKLH$ are equal.  And $EFGH$ is greater than $EKLH$;
 therefore it is greater than $ABCD$.  In short, for parallelograms
 between the same parallels, the parallelogram with the greater base
 is the greater.  A multiple of $ABCD$ is equal to another
 parallelogram between the same parallels; likewise for a multiple of
 $EFGH$; and the greater multiple corresponds to the greater base.
 Therefore $ABCD$ is to $EFGH$ as $AB$ is to $EF$.
\end{proof}

\begin{theorem}[VI.1]\label{thm:prop}
  Triangles drawn between the same parallel straight lines are to one
  another as their bases.
\end{theorem}

\begin{center}
  \begin{pspicture}(-0.5,-0.5)(7.5,2.5)
    \psline(0,0)(7,0)(6,2)(0.5,2)(0,0)
    \psline(2,0)(2.5,2)
    \psline(4,0)(3,2)
    \psline(0,0)(2.5,2)
    \psline(4,0)(6,2)
\uput[u](0.5,2){$D$}
\uput[u](3,2){$H$}
\uput[u](2.5,2){$C$}
\uput[u](6,2){$G$}
\uput[d](0,0){$A$}
\uput[d](2,0){$B$}
\uput[d](4,0){$E$}
\uput[d](7,0){$F$}
  \end{pspicture}
\end{center}

\begin{proof}
  Let $ABC$ and $EFG$ be triangles whose bases $AB$ and $EF$ are in a
  straight line, and whose apices $C$ and $G$ are in a straight line $DG$
  parallel to the bases.  Through $A$ draw straight line $AD$ parallel
  to $BC$ and meeting $DG$ at $D$; likewise, through $E$, draw
  straight line $EH$ parallel to $FG$ and meeting $DG$ at $H$.  Then
  parallelograms  $ABCD$ and $EFGH$ are to one another as their bases
  $AB$ and $EF$, which are the bases of $ABC$ and $EFG$; and these
  triangles are half of the corresponding parallelograms, by
  Theorem~\ref{thm:half}.  Therefore the triangles are to one another
  as their bases.
\end{proof}

\begin{theorem}[VI.2]
  A straight line drawn parallel to the base of a triangle cuts the
  two sides in proportion.
\end{theorem}

\begin{center}
  \begin{pspicture}(-0.5,-0.5)(4.5,3.5)
    \psline(0,0)(4,0)(3,3)(0,0)
    \psline(4,0)(1,1)(3.667,1)(0,0)
\uput[d](0,0){$A$}
\uput[d](4,0){$B$}
\uput[u](3,3){$C$}
\uput[ul](1,1){$D$}
\uput[ur](3.667,1){$E$}
  \end{pspicture}
\end{center}

\begin{proof}
  Let $ABC$ be a triangle, and on sides $AC$ and $BC$ respectively let
  $D$ and $E$ be chosen so that $DE$ is parallel to $BC$.  Let
  straight lines $AE$ and $BD$ be drawn.  Then triangles $ADE$ and
  $BDE$ are equal, by Theorem~\ref{thm:prop}.  Add to these the
  common part $CDE$: then the triangles $ACE$ and $BCD$ are equal.
  Therefore $CD$ is to $CA$ as $CDE$ is to $ACE$, that is, as $CDE$ is
  to $BCD$.  But also $CDE$ is
  to $BCD$ as $CE$ is to $CB$.  Therefore $CD$ is to $CA$ as $CE$ is
  to $CB$.  Hence also $CD$ is to $DA$ as $CE$ is to $EB$.
\end{proof}

\begin{theorem}[VI.4]
  If two triangles have corresponding angles equal, then the triangles
  are to one another as their corresponding sides.
\end{theorem}

%Now let us turn to cones.

\part{Conic Sections}

\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}): this
comprises every point of every straight line that passes through both
the apex and the circumference of the base.  One conceives of the cone
as obtained by taking a straight line from the apex to the circumference of
the base, then moving the line around this circumference.
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 arbitrary cones: they may be right or oblique.

A conic surface really comprises two surfaces, on opposite sides of
the apex.  We shall just call these \defn{opposite surface}{s.}  Each
of these has an inside and an outside.

\section{Propositions}\label{sect:conics}

  \begin{proposition}
    Pick a point on a conic surface.  Draw a straight line from the
    apex to this point.  Then this line lies on the conic surface.
  \end{proposition}

  \begin{proposition}
    Pick one of two opposite surfaces.  Pick
    two points on it.  Join them with a straight line.  Either this
    line falls within the surface, and outside when produced; or else,
    when produced, the straight line meets the apex.
  \end{proposition}

  \begin{proposition}
    Let a cone be cut by a plane that contains the apex.  The
    resulting section of the cone is a triangle.
If the cutting plane contains the axis of the cone, then the
section is called an \defn{axial triangle}.
  \end{proposition}

  \begin{proposition}
    Let either of two opposite surfaces be cut by a plane parallel to
    the base of the cone.  The resulting 
    section is the circumference of a circle.  This circle is the base
    of a new cone with the same apex and axis as the original cone.
  \end{proposition}

\begin{proposition}
  In an \emph{oblique} cone, suppose an axial triangle is
  perpendicular to the base.  (How is this possible?)  Cut the cone by
  a plane that:
  \begin{enumerate}
    \item
is not parallel to the base, but
\item
is perpendicular to the axial triangle, and
\item
cuts off from the axial triangle, on the side of the apex, another
triangle. 
  \end{enumerate}
This new triangle may still be similar to the axial triangle, while
lying `subcontrariwise' to it  (about the apex).  In this case,
the section of the cone is a circle.
This section may be called
a \defn{subcontrary}{} section.
\end{proposition}

In a cone, suppose a particular axial triangle is chosen.
In the base of the cone, suppose a straight line is drawn perpendicular
to the base of the axial triangle.  Straight lines drawn parallel to
this straight line will be said to be drawn \defn{ordinate-wise}.

Henceforth, cones will implicitly have distinguished axial triangles.

\begin{proposition}
On the surface of a cone, pick a point that is not on a side of the
  axial triangle.  From this point, draw a straight line
  ordinate-wise towards the axial triangle.  This straight line will
  meet the conic surface again, on 
  the other side of the axial triangle; and the axial triangle will
  bisect the straight line between the two points on the conic surface.
\end{proposition}

  In the plane of the base of a cone, let a straight line be drawn
  perpendicular to the base of the axial triangle, or to this base
  produced.  Let a plane containing this straight line cut the cone.
  This plane cuts the conic surface in a line called a \defn{conic
  section}.   

\begin{proposition}
Choose a point on a conic section that is not on the axial triangle.
A straight line drawn ordinate-wise from this point meets the conic
section again.  The resulting chord of the conic section is bisected
by the axial triangle.  The intersection of the axial triangle and the
plane of the conic section is therefore a \defn{diameter}{} of the
conic section.  A point where the diameter meets the conic section is
a \defn{vertex}{} of the section.  A straight line drawn ordinatewise
from the conic section to the diameter is an called an
\defn{ordinate}.  So an ordinate meets the diameter at some point.
The part of the diameter between this point and a vertex is the
corresponding \defn{abscissa}{} (Latin: \Eng{[line] cut off}).
In a right cone, ordinates are perpendicular to abscissas. In an
oblique cone, they are not, unless the axial triangle is perpendicular
to the base of the cone.
\end{proposition}

\begin{proposition}
  Suppose the diameter of a conic section, when extended beyond a
  vertex, meets the extension of a side of the axial triangle beyond
  the apex.  Then,
  for every straight line, there is an abscissa equal to that straight
  line.  The same is true if the diameter is parallel to a side of the
  axial triangle.
\end{proposition}

\begin{proposition}
  Suppose a conic section has two vertices.  
If the plane of the conic section is not parallel to the base of the
cone, and if the conic section is not subcontrary, then the conic
section is not a circle.
\end{proposition}

\begin{proposition}
  A chord of a conic section falls inside the section (that is, inside
  the conic surface of which the section is a section); produced, the chord
  falls outside.
\end{proposition}

\begin{proposition}
  Suppose the diameter of a conic section is parallel to a side of the
  axial triangle.  From the vertex of the conic section, we shall draw
  a certain straight line perpendicular to the plane of the conic
  section.  This straight line will be called the \defn{upright side}{}
  (\Gk{>orj'ia pleur'a}; in Latin, \emph{latus rectum}).  It is drawn
  so as to have the same ratio to the part of the side of the axial
  triangle between the vertex of the conic section and the apex of the
  triangle that the square on the base of the axial triangle has to
  the rectangle whose sides are the sides of the axial triangle.  The
  square on any ordinate is then equal to the rectangle whose sides
  are the abscissa and the upright side.  The conic section is
  therefore called a \defn{parabola}{} (\Gk{parabol'h}
  \Eng{juxtaposition, comparison, application}, from the verb
  \Gk{parab'allw} \Eng{throw beside,} from \Gk{par'a} \Eng{beside,}
  \Gk{b'allw} \Eng{throw.}) 
\end{proposition}

\begin{proposition}
  Suppose the diameter of a conic section, produced, meets a side of
  the axial triangle 
  produced beyond the apex.  The segment of the diameter produced,
  between the two points where it meets the sides of the axial
  triangle, is the \defn{transverse side}{} of the conic section.  An
  upright side is constructed, as for the parabola; but now the
  upright side is to the \emph{transverse side} as a certain square is
  to a certain rectangle.  Namely, the square is on the straight line
  drawn from the apex of the cone to the base, parallel to the
  diameter of the conic section; this straight line cuts the base into
  two segments, and 
  these are the sides of the rectangle.  Then the square on an
  ordinate \emph{exceeds} the rectangle whose sides are the abscissa
  and the upright side:  the excess is the rectangle whose base is the
  abscissa, and whose height is to its base as the upright side of the
  conic section is to its transverse side.  The conic section is
  therefore called a \defn{hyperbola}{} (\Gk{<uperbol'h} \Eng{excess,
  pre-eminence, throwing beyond,} from \Gk{<uperb'allw} \Eng{throw
  beyond}). 
\end{proposition}

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

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

\bibitem{MR1053158}
{Apollonius of Perga}.
\newblock {\em Conics. {B}ooks {V}--{VII}}, volume~9 of {\em Sources in the
  History of Mathematics and Physical Sciences}.
\newblock Springer-Verlag, New York, 1990.
\newblock The Arabic translation of the lost Greek original in the version of
  the Ban\=u M\=us\=a, Volume I: Introduction, text and translation, Volume II:
  Commentary, figures and indexes, Edited with translation and commentary by G.
  J. Toomer.

\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{Apollon4}
{Apollonius of Perga}.
\newblock {\em Conics. {B}ook {IV}}.
\newblock Green Lion Press, Santa Fe, N.M., 2002.
\newblock First English edition. Translation, introduction, and diagrams by
  Michael N. Fried.

\bibitem{Descartes-Geometry}
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{bones}
Euclid.
\newblock {\em The Bones}.
\newblock Green Lion Press, Santa Fe, NM, 2002.
\newblock A handy where-to-find-it pocket reference companion to {E}uclid's
  {\it {E}lements}.

\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{MR0046650}
D.~Hilbert and S.~Cohn-Vossen.
\newblock {\em Geometry and the imagination}.
\newblock Chelsea Publishing Company, New York, N. Y., 1952.
\newblock Translated by P. Nem\'enyi.

\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}
