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\title{Ancient Greek mathematics}
\author{David Pierce}
\date{\today}
\address{Mathematics Dept, Middle East Technical University,
  Ankara 06531, Turkey}
\urladdr{www.metu.edu.tr/~dpierce/}
\email{dpierce@metu.edu.tr}

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\begin{comment}

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

\email{dpierce@metu.edu.tr}
\urladdr{http://www.math.metu.edu.tr/~dpierce/}
\end{comment}

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\tableofcontents

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%\section{Introduction}


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

Some time in the 3rd century \textsc{b.c.e.,} Apollonius of Perga wrote eight
books on \textbf{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}
\cite{MR17:814b,MR1932864,euclid-fitzpatrick}, a work of mathematics
of which
some of 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.

In 2008, getting ready to teach a course on the conic sections\footnote{At the
Nesin Mathematics Village, \c Sirince, Sel\c cuk, \.Izmir, Turkey.}, I
wrote some notes on ancient mathematics.  Using those notes, I have
prepared the present notes, for use in a course called `History of
Mathematical Concepts I' at METU---a course in which participants will
read Euclid and Apollonius.

In the latter sections of these notes, 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, to see
the sorts of improvements that he thought he was making to
mathematical practice of mathematicians like Euclid and Apollonius.

Because I shall occasionally refer to some Greek words, I review the Greek
alphabet in Table~\ref{table:Greek}.
  (I have heard a rumor that students can
improve their mathematics simply by learning this alphabet, assuming
they didn't grow up knowing it.)
\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}

\section{Why read the Ancients?}

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, I think, not because they were considered to
  be the \emph{best} textbooks, but 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 curves called  \textbf{conic
sections} are a standard part of an elementary course of mathematics.
The origin of such curves is in the name: they are obtained by slicing
a cone.  Apollonius treated the curves in this way.
But in math courses today, the conic sections are usually given as the curves
defined by certain equations, such as  
\begin{equation*}
ay=x^2
\end{equation*}
or 
\begin{equation*}
\frac{x^2}{a^2}\pm\frac{y^2}{b^2}=1.  
\end{equation*}
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 METU.}
  writes:
\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 author's figure.)
The conic sections result from intersecting a cone with a plane: this
can be understood as a \emph{definition} of the conic sections.  Let
us call it Definition I.
%I omit the book's Figure 1; but it can serve as one definition of the
%conic sections.  
More precisely, this definition distinguishes three kinds of conic
sections, depending on the angle of the plane with respect to the cone.
One kind of conic section is called the
\emph{parabola,} and 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}
Here then is another definition of the parabola; call it Definition
II.  Definitions I and II are equivalent in that they define the same
objects; but the author does not clearly say so, much less prove it.
I don't think he \emph{needs} to prove the 
equivalence; 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 Definitions I and II.  But this is not his style.  He
is usually eager to give his readers every assistance.  Note for
example that he apparently does not
trust readers to infer for themselves that parabolas are worth 
studying.  Before concluding anything from his definition of
parabolas, 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 \emph{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 the surfaces defined by the equations
\begin{gather*}
ax+by+cz+d=0,\\
(x-ez)^2+y^2=fz^2.
\end{gather*}
\end{enumerate}
What the author means, I think, is that it is convenient to define
certain curves `ana\-lytically'---that is, in a coordinate system such
as Descartes introduced; properties of the curves can then be obtained
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,} as Apollonius considered
them. 


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

It may be said that, in reading Euclid and Apollonius, we are
going to do \textbf{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 \textbf{analytic
  geometry;} by contrast, the geometry of ancient mathematicians like
Euclid and Apollonius is sometimes called 
\textbf{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 \textbf{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 \textbf{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 \textbf{theoretical} (\Gk{jewrhtik'oc}), and the other,
whose object is to find something set for finding, being called
\textbf{problematical} (\Gk{problhmatik'oc}). 
\end{quote}
This passage is not very useful without examples: I shall propose one
presently.  Meanwhile, I 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 \textbf{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 \textbf{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 $CD$ 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 that turns out to be) a 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{Theorems and problems}\label{sect:Proclus}

The text of Apollonius as we have it consists almost entirely of
theorems and problems (in the sense of the last section).  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 (that is, Constantinople, now \.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}
Alternative translations are: for
\Gk{>'ekjesic}, \emph{setting out,} and 
for \Gk{diorism'oc}, \emph{definition of
  goal} \cite[p.~10]{MR1683176}.

For an illustration, we may analyze Proposition 1 of Book I of
Euclid's \emph{Elements} (in Fitzpatrick's translation
\cite{euclid-fitzpatrick}).  The 
proposition is a \emph{problem:} 

\begin{enunciation}
To construct an equilateral triangle on a given finite
straight-line.
\end{enunciation}

\begin{exposition}
    Let AB be the given finite straight-line.
\end{exposition}

\begin{specification}
    So it is required to construct an equilateral triangle on
the straight-line AB.
\end{specification}

\begin{construction}
    Let the circle BCD with center A and radius AB have
been drawn, and again let the circle ACE with
center B and radius BA have been drawn. And
let the straight-lines CA and CB have been joined from
the point C, where the circles cut one another, to the
points A and B (respectively).
\end{construction}

\begin{proof}
    And since the point A is the center of the circle CDB,
AC is equal to AB. Again, since the point
B is the center of the circle CAE, BC is equal to BA. But CA was also
shown (to be) equal to AB. 
Thus, CA and CB are each equal to AB. But things equal
to the same thing are also equal to one another.
Thus, CA is also equal to CB. Thus, the three (straight-
lines) CA, AB, and BC are equal to one another.
\end{proof}

\begin{conclusion}
    Thus, the triangle ABC is equilateral, and has been
constructed on the given finite straight-line AB. (Which
is) the very thing it was required to do.
\end{conclusion}

\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
\textbf{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 \textbf{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}

In the old understanding, a \textbf{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 \textbf{circle} (\Gk{k'ukloc}) is a plane figure contained by one
line (\Gk{gramm'h}) 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 (\Gk{e>uje~ia gramm'h}) does have endpoints;
but the straight line may
be \emph{produced} (extended) beyond these endpoints, as far as desired.

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

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

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\end{thebibliography}


\end{document}
