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\begin{document}
\title{Abscissas and Ordinates}
\author{David Pierce}
\date{\today, \printtime}
\publishers{Mathematics Department\\
Mimar Sinan Fine Arts University\\
Istanbul\\
\url{dpierce@msgsu.edu.tr}\\
\url{http://mat.msgsu.edu.tr/~dpierce/}}
\maketitle

\begin{abstract}
In the manner of Apollonius of Perga,
but hardly any modern book,
we investigate conic sections \emph{as such.}
We thus discover 
why Apollonius calls a conic section 
a parabola, an hyperbola, or an ellipse;
and we discover the meanings of the terms abscissa and ordinate.
In an education that is liberating and not simply indoctrinating,
the student of mathematics will learn these things.
\end{abstract}

\tableofcontents

\section{The liberation of mathematics}

In the first of the eight books
of the \emph{Conics} \cite{MR1660991}, 
Apollonius of Perga derives properties of the conic sections 
that can be used to write their equations 
in rectangular or oblique coordinates.%%%%%
\footnote{The first four books of the \emph{Conics} survive in Greek; 
the next three, in Arabic translation only.  The last book is lost.
Lucio Russo \cite[p.~8]{MR2038833} uses this and other examples
to show that we cannot expect the best ancient work to have survived.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
This article reviews these properties,
because 
\begin{inparaenum}[(1)]
\item
they have intrinsic mathematical interest,
\item
they are the reason why Apollonius gave to the three conic sections
the names that they now have,
and 
\item
the vocabulary of Apollonius
is a source for many of our technical terms.
\end{inparaenum}

In a modern textbook of analytic geometry,
the two coordinates of a point in the so-called Cartesian plane
may be called the \enquote{abscissa} and \enquote{ordinate.}
Probably the book will not explain why.
But the reader deserves an explanation.
The student should not have to learn meaningless words,
for the same reason that s/he should not be expected 
to memorize the quadratic formula without a derivation.
True education is not indoctrination, but liberation.
Mathematics is liberating when it teaches us our own power 
to decide what is true.
This power comes with a responsibility to justify our decisions 
to anybody who asks;
but this is a responsibility 
that must be shared by all of us who do mathematics.

Mathematical terms \emph{can} be assigned arbitrarily.
This is permissible, but it is not desirable.
The terms \enquote{abscissa} and \enquote{ordinate}
arise quite naturally in Apollonius's development of the conic sections.
This development should be better known,
especially by anybody who teaches analytic geometry.
This is why I write.

\section{Lexica and registers}

Apollonius did not create his terms:
they are just ordinary words, used to refer to mathematical objects.
When we do not \emph{translate} Apollonius,
but simply transliterate his words,
or use their Latin translations,
then we put some distance between ourselves and the mathematics.
When I first learned that a conic section had a \emph{latus rectum,}
I had a sense that there was a whole theory of conic sections
that was not being revealed,
although its existence was hinted at
by this peculiar Latin term.
If we called the \emph{latus rectum}
by its English name of \enquote{upright side,}
then the student could ask, 
\enquote{What is upright about it?}
In turn, textbook writers might feel obliged to answer this question.
In any case, I am going to answer it here.
Briefly, it is called upright because, for good reason, 
it is to be conceived as having one endpoint on the vertex of the conic section,
but as sticking out from the plane of the section.

English does borrow foreign words freely:
this is a characteristic of the language.
A large lexicon is not a bad thing.
A choice from among two or more synonyms
can help establish the register of a piece of speech.
In the 1980s, as I recall,
there was a book called \emph{Color Me Beautiful}
that was on the American bestseller lists week after week.
The \emph{New York Times} blandly said 
the book provided \enquote{beauty tips for women};
the \emph{Washington Post} described it
as offering \enquote{the color-wheel approach to female pulchritude.}
By using an obscure synonym for beauty,
the \emph{Post} mocked the book.

If distinctions between near-synonyms are maintained,
then subtleties of expression are possible.
\enquote{Circle} and \enquote{cycle} are Latin and Greek words
for the same thing, 
but the Greek word is used more abstractly in English,
and it would be bizarre to refer to a finite group of prime order
as being circular rather than cyclic.

To propose or maintain distinctions between near-synonyms
is a \emph{raison d'\^etre} 
of works like Fowler's \emph{Dictionary of Modern English Usage} \cite{MEU}.
Fowler laments, for example, the use of the Italian word \emph{replica} 
to refer to any copy of an art-work,
when the word properly refers to a copy
\emph{made by the same artist.}
In his article on synonyms, 
Fowler sees in language the kind of liberation,
coupled with responsibility, 
that I ascribed to mathematics:
\begin{quote}
Synonym books in which differences are analysed,
engrossing as they may have been to the active party, the analyst,
offer to the passive party, the reader, nothing but boredom.
Every reader must, for the most part, be his own analyst;
\&\ no-one who does not expend, whether expressly \&\ systematically
or as a half-conscious accompaniment of his reading \&\ writing,
a good deal of care upon points of synonymy is likely to write well.  
\end{quote}
The boredom of the reader of a book of synonyms
may be comparable to that of the reader of a mathematics textbook
that begins with a bunch of strange words 
like \enquote{abscissa} and \enquote{ordinate.}

Mathematics can be done in any language.
Greek does mathematics without a specialized vocabulary.
It is worthwhile to consider what this is like.

I shall take Apollonius's terminology 
from Heiberg's edition \cite{Apollonius-Heiberg}
(actually a printout of a \url{pdf} image
downloaded from the Wilbour Hall website, \url{wilbourhall.org}).
Meanings are checked with the big Liddell--Scott--Jones lexicon \cite{LSJ}
(available from the Perseus Digital Library, \url{perseus.tufts.edu},
though I splurged on the print version myself).

I am going to write out Apollonius's terms in Greek letters.
I shall use the customary minuscule forms developed in the Middle Ages.
Apollonius himself would have used only the letters that we now call capital;
but modern mathematics uses minuscule Greek letters freely,
and the reader ought to be able to make sense of them.

\section{The gendered Greek article}

Apollonius's word for \textbf{cone} is \gk{<o k~wnos}, 
meaning originally \enquote{pine-cone.}
Evidently our word comes ultimately from Apollonius's
(and this is confirmed by such resources as \cite{CODoEE}).
I write out the \gk{<o} to indicate the gender of \gk{k~wnos}:
\gk{<o} is the masculine definite article.
The feminine article is \gk{<h}.
In each case, the diacritical mark over the vowel 
indicates the prefixed sound that is spelled in English with the letter H.
Other diacritical marks can be ignored;
I reproduce them because they are in the modern texts.

In the terminology of Apollonius,
all of the nouns that we shall look at will be feminine or masculine.
Greek does however have a neuter gender as well,%%%%%
\footnote{English retains the threefold gender distinction
in \enquote{he/she/it.}}
and the neuter article is \gk{t'o}.
I want to note by the way the economy of expression
that is made possible by gendered articles.
In mathematics, \textbf{point} is \gk{t`o shme~ion}, neuter; 
\textbf{line} is \gk{<h gramm'h}, feminine.
The feminine \gk{<h stigm'h} can also be used for a mathematical point;
it is \emph{not used,}
argues Reviel Netz \cite[p.~113]{MR1683176},
so that an expression like \gk{<h A} 
can unambiguously designate a particular \emph{line} in a diagram, 
while \gk{t`o A} would designate a point.
In Proposition I.43 of the \emph{Elements,}
Euclid can refer to a parallelogram \gk{AEKJ} simply as \gk{t`o EJ}
\cite[p.~100]{Euclid-Heiberg}:
the neuter article is used, because \gk{parallhl'ogrammon} is neuter.
The reader cannot think that \gk{t`o EJ} is a line;
the line would be \gk{<h EJ}.
The English reader \emph{can} make this mistake.
In Heath's translation \cite{MR1932864,MR17:814b},
Euclid says,
\begin{quote}
  Let $ABCD$ be a parallelogram, and $AC$ its diameter; 
and about $AC$ let $EH$, $FG$ be parallelograms, 
and $BK$, $KD$ the so-called complements.
\end{quote}
It is confusing to see both lines and parallelograms 
given two-letter designations.
Perhaps the confusion is easily overcome;
but the Greek reader would not have had it in the first place.
This is one of the few cases where gender in a language is actually useful.

\section{The cone of Apollonius}

For Apollonius, 
a \textbf{cone} (\gk{<o k~wnos} \enquote{pine-cone,} as above) 
is a solid figure determined by
\begin{inparaenum}[(1)]
\item
a \textbf{base} (\gk{<h b'asis}), 
which is a circle, and
\item 
a \textbf{vertex} (\gk{<h koruf'h} \enquote{summit}), 
which is a point that is not in the plane of the base.  
\end{inparaenum}
The surface of the cone contains all of the straight lines 
drawn from the vertex to the circumference of the base.  
A \textbf{conic surface} (\gk{<h kwnik`h >epif'aneia}%%%%%
\footnote{The word \gk{>epif'aneia} means originally \enquote{appearance}	
and is the source of the English \enquote{epiphany.}}%%%%%
) consists of such straight lines, 
not bounded by the base or the vertex, 
but extended indefinitely in both directions.

The straight line drawn from the vertex of a cone to the center of the base 
is the \textbf{axis} (\gk{<o >'axwn} \enquote{axle}) of the cone.  
If the axis is perpendicular to the base, 
then the cone is \textbf{right} (\gk{>orj'os});
otherwise it is \textbf{scalene} (\gk{skalhn'os} \enquote{uneven}).
Apollonius considers both kinds of cones indifferently.

A plane containing the axis intersects the cone in a triangle.
Suppose a cone with vertex $A$ has axial triangle $ABC$.  
Then the base $BC$ of this triangle is a diameter of the base of the cone.  
Let an arbitrary chord%%%%%
\footnote{Although it is the source 
of the English \enquote{cord} and \enquote{chord} \cite{CODoEE}, 
Apollonius does not use the word \gk{<h qord'h},
although he proves in Proposition I.10 
that the straight line joining any two points of a conic section
\emph{is} a chord,
in the sense that it falls within the section.
The Greek \gk{qord'h} means gut, hence \emph{anything} made with gut,
be it a lyre-string or a sausage \cite{LSJ}.}
$DE$ of the base of the cone
cut the base $BC$ of the axial triangle at right angles at a point $F$, 
as in Figure~\ref{fig:ax-base}.
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\caption{Axial triangle and base of a cone}\label{fig:ax-base}
\end{figure}
In the axial triangle, 
let a straight line $FG$ be drawn from the base to the side $AC$.  
This straight line $FG$ may, but need not, be parallel to the side $BA$.
It is not at right angles to $DE$,
unless the plane of the axial triangle 
is at right angles to the plane of the base of the cone. 
In any case, the two straight lines $FG$ and $DE$, meeting at $F$, 
are not in a straight line with one another, 
and so they determine a plane.  
This plane cuts the surface of the cone 
in such a curve $DGE$ as is shown in Figure~\ref{fig:curve}.
\begin{figure}[ht]
\centering
\begin{pspicture}(-1.732,-0.5)(1.732,3.5)
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\psplot{-1.732}{1.732}{3 x x mul sub}
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\uput[d](0,0){$F$}
\uput[u](-0.75,3){$G$}
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\caption{A conic section}\label{fig:curve}
\end{figure}
Apollonius refers to such a curve first (in Proposition I.7)
as a section (\gk{<h tom'h}) in the \emph{surface} of the cone,
and later (I.10) as a section of a cone.
All of the chords of this section that are parallel to $DE$
are bisected by the straight line $GF$.
Therefore Apollonius calls this straight line
a \textbf{diameter} (\gk{<h di'ametros [gramm'h]}) of the section.%%%%%
\footnote{The associated verb is \gk{diametr'e-w} \enquote{measure through};
this is the verb used in Homer's \emph{Iliad} \cite[III.315]{Iliad-Loeb})
for what Hector and Odysseus do in measuring out lists
for the single combat of Paris and Menelaus.  (The reference is in \cite{LSJ}.)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The parallel chords bisected by the diameter
are said to be drawn to the diameter \textbf{in an orderly way.}
The Greek adverb here is \gk{tetagm'enws}, 
from the verb \gk{t'assw}, 
which has meanings like \enquote{to draw up in order of battle} \cite{LSJ}.
A Greek noun derived from this verb is \gk{t'axis},
which is found in English technical terms 
like \enquote{taxonomy} and \enquote{syntax} \cite{CID}.
The Latin adverb corresponding to the Greek \gk{tetagm'enws} 
is \emph{ordinate} from the verb \emph{ordino.}
From the Greek expression for 
\enquote{the straight line drawn in an orderly way,}
Apollonius will elide the middle part, 
leaving \enquote{the in-an-orderly-way.}%%%
\footnote{Heath \cite[p.~clxi]{Heath-Apollonius} 
translates \gk{tetagm'enws} as \enquote{ordinate-wise}; 
Taliaferro \cite[p.~3]{MR1660991}, as \enquote{ordinatewise.}
But this usage strikes me as anachronistic.
The term \enquote{ordinatewise} 
seems to mean \enquote{in the manner of an ordinate};
but ordinates are just what we are trying to define
when we translate \gk{tetagm'enws}.}
This term will refer to \emph{half} of a chord bisected by a diameter.
Similar elision in the Latin leaves us 
with the word \textbf{ordinate} for this half-chord \cite{OED}.
In the \emph{Geometry,} Descartes refers to ordinates
as \emph{[lignes] qui s'appliquent par ordre [au] diametre}
\cite[p.~328]{Descartes-Geometry}.

I do not know whether the classical \emph{orders} of architecture---%
the Doric, Ionic, and Corinthian orders---are so called
because of the mathematical use of the word \enquote{ordinate.}
But we may compare the ordinates of a conic section as in Figure~\ref{fig:ord}
\begin{figure}[ht]
\begin{center}
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  %\psgrid
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    \psline(5,0)(5,2.236)
    \psline(6,0)(6,2.449)
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  \end{center}
  \caption{Ordinates of a conic section}\label{fig:ord} 
\end{figure}
with the row of columns of a Greek temple,
as in Figure~\ref{fig:priene}.
\begin{figure}[ht]
\centering
%\includegraphics[width=6cm]{priene-smaller.eps}
\includegraphics[width=6cm]{priene.eps}
\caption[Columns in the Ionic order at Priene]{Columns in the Ionic order, at Priene, S\"oke, Ayd\i n, Turkey}\label{fig:priene}
\end{figure}

Back in Figure \ref{fig:curve},
the point $G$ at which the diameter $GF$ cuts the conic section $DGE$
is called a \textbf{vertex} (\gk{koruf'hs} as before).
The segment of the diameter between the vertex and an ordinate
has come to be called in English an \textbf{abscissa;} 
but this just the Latin translation of Apollonius's Greek for being cut off (\gk{>apolambanom'enh} \enquote{taken}%%%%%
\footnote{I note the usage of the Greek participle 
in \cite[I.11, p.~38]{Apollonius-Heiberg}.
Its general usage for what we translate as \emph{abscissa}
is confirmed in \cite{LSJ},
although the general sense of the verb is not of cutting, but of taking.}).

Apollonius will show that every point of a conic section
is the vertex for some unique diameter.
If the ordinates corresponding to a particular diameter
are at right angles to it,
then the diameter will be an \textbf{axis} of the section.
Meanwhile, 
in describing the relation between the ordinates and the abscissas of conic section, 
there are three cases to consider.

\section{The parabola}

Suppose the diameter of a conic section is parallel 
to a side of the corresponding axial triangle.
For example, suppose in Figure~\ref{fig:parab} 
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\caption{%Axial triangle and base of a cone from which a parabola is cut
}\label{fig:parab}
\end{figure}
that $FG$ is parallel to $BA$.
The square on the ordinate $DF$ is equal to the rectangle whose sides are $BF$ and $FC$
(by Euclid's Proposition III.35).
More briefly, $DF^2=BF\cdot FC$.
But $BF$ is independent of the choice of the point $D$ on the conic section.
That is, for any such choice
(aside from the vertex of the section),
a plane containing the chosen point
and parallel to the base of the cone
cuts the cone in another circle,
and the axial triangle cuts this circle along a diameter,
and the plane of the section
cuts this diameter at right angles into two pieces,
one of which is equal to $BF$.
The square on $DF$ thus varies as $FC$, which varies as $FG$.
That is, the square on an ordinate varies as the abscissa (Apollonius I.20).
Hence there is a straight line $GH$ such that
\begin{equation*}
DF^2=FG\cdot GH,
\end{equation*}
and $GH$ is independent of the choice of $D$.

This straight line $GH$ can be conceived as being drawn at right angles
to the plane of the conic section $DGE$.
Therefore Apollonius calls $GH$ the \textbf{upright side} 
(\gk{>orj'ia [pleur'a]}),
and Descartes accordingly calls it
\emph{le cost\'e droit}  \cite[p.~329]{Descartes-Geometry}.
Apollonius calls the conic section itself 
\gk{<h parabol'h}; we transliterate this as \textbf{parabola.}
The Greek word is also the origin of the English \enquote{parable,}
but can have various related meanings, 
like \enquote{juxtaposition, comparison, conjunction, application.}
The word is self-descriptive: it can be understood as 
a juxtaposition of the preposition \gk{par'a} \enquote{along, beside}
and the noun \gk{<h bol'h} \enquote{throw.}
Alternatively, \gk{parabol'h} is a noun derived from the verb
\gk{parab'allw}, which is \gk{par'a} plus \gk{b'allw} \enquote{throw.}
In the parabola of Apollonius,
the rectangle bounded by the abscissa and the upright side
is the result of \emph{applying}
the square on the ordinate to the upright side. 
Such an application is made for example 
in Proposition I.44 of Euclid's \emph{Elements,}
where a parallelogram equal to a given triangle 
is \emph{applied} to a given straight line:
that is, the parallelogram is constructed 
on the given straight line as base.%%%%%
\footnote{This proposition is a lemma for Proposition 45,
that if a figure with any number of straight sides be given,
then a rectangle---or indeed a parallelogram in any given angle---%
can be constructed that is equal to this figure.
This is the climax of Book I of the \emph{Elements,}
and it recalls Herodotus's tracing of the origins of geometry
to the measuring of land lost 
in the annual flooding of the Nile in Egypt \cite[II.109]{Herodotus-Loeb}.
Propositions 47 and 48, the Pythagorean Theorem and its converse,
are merely the \emph{d\'enouement} of Book I of Euclid.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The \emph{latus rectum}}

The Latin term for the upright side 
is \emph{latus rectum.}
This term is also used in English.  
In the \emph{Oxford English Dictionary} \cite{OED}, 
the earliest quotation illustrating the use of the term 
is from a mathematical dictionary published in 1702.  
Evidently the quotation refers to Apollonius and gives his meaning:
\begin{quote}
App.\ Conic Sections 11 \ 
In a Parabola the Rectangle of the Diameter, and Latus Rectum, 
is equal to the rectangle of the Segments of the double Ordinate.
\end{quote}
I assume the \enquote{segments of the double ordinate} 
are the two halves of a chord, 
so that each of them is what we are calling an ordinate, 
and the rectangle contained by them is equal to the square on one of them.

The possibility of defining the conic sections
in terms of a \emph{directrix} and \emph{focus}
is shown by Pappus \cite[VII.312--8, pp.~1004--15]{Pappus}
and was presumably known to Apollonius.
Pappus does not use such technical terms though;
there is just a straight line and a point,
as in the following,
a slight modification%%%%%
\footnote{I have put \enquote{the ratio of \gk{GD} to \gk{DE}}
where Thomas has \enquote{the ratio $\gkm{GD}:\gkm{DE}$}
because Pappus uses no special notation for a ratio as such,
but refers merely to \gk{l'ogos\dots t~hs GD pr`os DE}.
The recognition of ratios as individual mathematical objects
(namely numbers) distinguishes modern from ancient mathematics,
although the beginnings of this recognition can be seen in Pappus;
but that is a subject for another article.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 of Thomas's translation \cite[pp.~492--503]{MR13:419a}:
\begin{quote}
  If \gk{AB} be a straight line given in position,
and the point \gk G be given in the same plane,
and \gk{DG} be drawn,
and \gk{DE} be drawn perpendicular [to \gk{AB}],
and if the ratio of \gk{GD} to \gk{DE} be given,
then the point \gk D will lie on a conic section.%%%%%
\footnote{As Heath \cite[pp.~xxxvi--xl]{Heath-Apollonius} explains, 
Pappus proves this theorem
because Euclid did not supply a proof in his treatise on \emph{surface loci.}
(This treatise itself is lost to us.)
Euclid must have omitted the proof because it was already well known;
and Euclid predates Apollonius.
Morris Kline \cite[p.~96]{MR0472307} summarizes all of this 
by saying that the focus-directrix property 
\enquote{was known to Euclid and is stated and proved by Pappus.}
Later (on his page 128), Kline gives a precise reference to Pappus:
it is Proposition 238, in Hultsch's numbering, of Book VII.
Actually this proposition is a recapitulation,
which is incomplete in the extant manuscripts;
one must read a few pages earlier in Pappus for more details,
as in the selection in Thomas's anthology.
In any case, Kline says,  
\enquote{As noted in the preceding chapter, 
Euclid probably knew} the proposition.
According to Boyer however,
\enquote{It appears that Apollonius knew 
of the focal properties for central conics, 
but it is possible that the focus-directrix property for the parabola 
was not known before Pappus} \cite[\S XI.12, p.~211]{Boyer}.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{quote}
A modern textbook may define the parabola
in terms of a directrix and focus,
explicitly so called.
An example is Nelson, Folley, and Borgman,
\emph{Analytic Geometry} \cite{NFB},
a book that I happen to have on hand
because my mother used it in college,
and because I perused it at the age of 12
when I wanted to understand the curves that could be encoded in equations.
Dissatisfaction with such textbooks
leads me back to the Ancients.
According to Nelson \etal,
\begin{quote}
The chord of the parabola 
which contains the focus and is perpendicular to the axis 
is called the \emph{latus rectum.}  
Its length is of value 
in estimating the amount of \enquote{spread} of the parabola.
\end{quote}
The first sentence here defines the \emph{latus rectum}
as a certain line segment
that is indeed equal to Apollonius's upright side.
The second sentence correctly describes 
the significance of the \emph{latus rectum.}  
However, the juxtaposition of the two sentences may mislead somebody 
who knows just a little Latin.  
The Latin adjective \emph{latus, -a, -um} does mean
\enquote{broad, wide; spacious, extensive} \cite{POLD}:
it is the root of the English noun \enquote{latitude.}  
An extensive \emph{latus rectum} does mean a broad parabola.
However, the Latin adjective \emph{latus} 
is unrelated to the noun \emph{latus, \mbox{-eris}} 
\enquote{side; flank,}
which is found in English in the adjective \enquote{lateral}; 
and the noun \emph{latus} 
is what is used in the phrase \emph{latus rectum.}%%%%%
\footnote{In \emph{latus rectum,} the adjective \emph{rectus, -a, -um} 
\enquote{straight, upright} is given the neuter form, 
because the noun \emph{latus} is neuter.  
The plural of \emph{latus rectum} is \emph{latera recta.}  
The neuter plural of the adjective \emph{latus} would be \emph{lata.}  
The dictionary writes the adjective as \emph{l\=atus,} with a long \enquote{a};
but the \enquote{a} in the noun is unmarked and therefore short.  
As far as I can tell, 
the adjective is to be distinguished 
from another Latin adjective with the same spelling 
(and the same long \enquote{a}), 
but with the meaning of \enquote{carried, borne}, 
used for the past participle of the verb 
\emph{fero, ferre, tul\={\i}, l\=atum.}  
This past participle appears in English in words like \enquote{translate,} 
while \emph{fer-} appears in \enquote{transfer.}  
The \emph{American Heritage Dictionary} \cite{CID} 
traces \emph{l\=atus} \enquote{broad} to an Indo-European root \emph{stel-} 
and gives \enquote{latitude} and \enquote{dilate} as English derivatives; 
\emph{l\=atus} \enquote{carried} comes from an Indo-European root \emph{tel-} 
and is found in English words like \enquote{translate} and \enquote{relate,} 
but also \enquote{dilatory.}  
Thus \enquote{dilatory} 
is not to be considered as a derivative of \enquote{dilate.}  
A French etymological dictionary \cite{LDE} implicitly confirms this 
under the adjacent entries \emph{dilater} and \emph{dilatoire.}  
The older Skeat \cite{Skeat} does give \enquote{dilatory} 
as a derivative of \enquote{dilate.}  
However, under \enquote{latitude,} Skeat traces \emph{l\=atus} \enquote{broad} 
to the Old Latin \emph{stl\=atus,} 
while under \enquote{tolerate} he traces \emph{l\=atum} \enquote{borne} 
to \emph{tl\=atum}.  
In his introduction, Skeat says he has collated his dictionary 
\enquote{with the \emph{New English Dictionary} 
[as the \emph{Oxford English Dictionary} was originally called] 
from A to H (excepting a small portion of G).}  
In fact the \emph{OED} distinguishes \emph{two} English verbs \enquote{dilate,} 
one for each of the Latin adjectives \emph{l\=atus.}  
But the dictionary notes, 
``The sense `prolong' comes so near `enlarge', `expand', 
or `set forth at length'\dots 
that the two verbs were probably not thought of as distinct words.''}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Denoting abscissa by $x$, and ordinate by $y$, and \emph{latus rectum} by $\ell$, 
we have for the parabola the modern equation
\begin{equation}\label{eqn:parab}
y^2=\ell x.
\end{equation}
The letters here can be considered as numbers in the modern sense,
or just as line segments, or congruence-classes of segments.

\section{The hyperbola}

The second possibility for a conic section is that the diameter 
meets the other side of the axial triangle
when this side is extended beyond the vertex of the cone.
In Figure~\ref{fig:hyper},
\begin{figure}[ht]
\mbox{}\hfill
\begin{pspicture}(-0.5,-0.5)(4,4)
\psline(1,0)(2,4)(0,0)(4,0)(1,2)
\uput[l](1,2){$A$}
\uput[l](0,0){$B$}
\uput[r](4,0){$C$}
\uput[d](1,0){$F$}
\uput[ur](1.438,1.714){$G$}
\uput[dr](2,4){$K$}
\psset{linestyle=dashed}
\psline(1,0)(2.647,-0.412)(2,4)
\psline(1.438,1.714)(3.076,1.302)(2.647,-0.412)
\psline(2.370,1.479)(1.941,-0.235)
\uput[ur](2.370,1.479){$H$}
\uput[d](2.647,-0.412){$L$}
\uput[ur](3.076,1.302){$M$}
\end{pspicture}
\hfill
\begin{pspicture}(0,-0.5)(4.5,4)
\pscircle(2,2){2}
\psline(0,2)(4,2)
\psline(1,0.268)(1,3.732)
\psline[linestyle=dashed](1,0.268)(2.732,0.268)(2.732,2)
\uput[l](0,2){$B$}
\uput[r](4,2){$C$}
\uput[dl](1,0.268){$D$}
\uput[ul](1,3.732){$E$}
\uput[ur](1,2){$F$}
\end{pspicture}
\hfill\mbox{}
\caption{%Axial triangle and base of a cone from which an hyperbola is cut
}\label{fig:hyper}
\end{figure}
the diameter $FG$,
crossing one side of the axial triangle $ABC$ at $G$,
crosses the other side, extended, at $K$.
Again $DF^2=BF\cdot FC$; but the latter product now varies as $KF\cdot FG$.  
The upright side $GH$ can now be defined so that
\begin{equation*}
BF\cdot FC:KF\cdot FG\as GH:GK.
\end{equation*}
We draw $KH$ and extend to $L$ so that $FL$ is parallel to $GH$, 
and we extend $GH$ to $M$ so that $LM$ is parallel to $FG$.
Then
\begin{align*}
FL\cdot FG:KF\cdot FG
&\as FL:KF\\
&\as GH:GK\\
&\as BF\cdot FC:KF\cdot FG,
\end{align*}
and so $FL\cdot FG=BF\cdot FC$.  Thus
\begin{equation*}
DF^2=FG\cdot FL.
\end{equation*}
Apollonius calls the conic section here an \textbf{hyperbola} 
(\gk{<h <uperbol'h}),
that is, 
an \emph{excess,} an \emph{overshooting,} 
a \emph{throw} (\gk{bol'h}) \emph{beyond} (\gk{<up'er}),
because the square on the ordinate
is equal to a rectangle whose one side is the abscissa,
and whose other side is applied to the upright side:
but this rectangle \emph{exceeds} (\gk{<uperb'allw}),
by another rectangle,
the rectangle contained by the abscissa and the upright side.
The excess rectangle is similar to the rectangle
contained by the upright side $GH$ and $GK$.
Apollonius calls $GK$ 
the \textbf{transverse side} (\gk{<h plag'ia pleur'a}) of the hyperbola.
Denoting it by $a$, and the other segments as before,
we have the modern equation
\begin{equation}\label{eqn:hyperb}
y^2=\ell x+\frac{\ell}ax^2.
\end{equation}

\section{The ellipse}

The last possibility is that the diameter meets the other side of the axial triangle
when this side is extended below the base.
All of the computations will be as for the hyperbola,
except that now, if it is considered as a \emph{directed} segment,
the transverse side is negative,
and so the modern equation is
\begin{equation}\label{eqn:ell}
y^2=\ell x-\frac{\ell}ax^2.
\end{equation}
In this case Apollonius calls the conic section an \textbf{ellipse} (\gk{<h >'elleiyis}),
that is, a \emph{falling short,}
because again the square on the ordinate
is equal to a rectangle whose one side is the abscissa,
and whose other side is applied to the upright side:
but this rectangle now \emph{falls short} (\gk{>elle'ipw})
of the rectangle contained by the abscissa and the upright side
by another rectangle.
Again this last rectangle is similar to the rectangle
contained by the upright and transverse sides.

\section{Descartes}

We have seen that the terms \enquote{abscissa} and \enquote{ordinate}
are ultimately translations of Greek words
that describe certain line segments
determined by points on conic sections.
For Apollonius,
an ordinate and its corresponding abscissa 
are not required to be at right angles to one another.

Descartes generalizes
the use of the terms slightly.
In one example \cite[p.~339]{Descartes-Geometry},
he considers a curve derived from a given conic section
in such a way that,
if a point of the conic section is given by an equation of the form
\begin{equation*}
y^2=\dots x\dots,
\end{equation*}
then a point on the new curve is given by
\begin{equation*}
y^2=\dots x'\dots,
\end{equation*}
where $xx'$ is constant.
But Descartes just describes the new curve in words:
\begin{quote}
toutes les lignes droites appliqu\'ees par ordre a son diametre
estant esgales a celles d'une section conique,
les segmens de ce diametre,
qui sont entre le sommet \&\ ces lignes,
ont mesme proportion a une certaine ligne donn\'ee, 
que cete ligne donn\'ee a aux segmens du diametre de la section conique,
auquels les pareilles lignes sont appliqu\'ees par ordre.%%%%%
\footnote{%
\enquote{All of the straight lines drawn in an orderly way to its diameter 
being equal to those of a conic section,
the segments of this diameter
that are between the vertex and these lines
have the same ratio to a given line
that this given line has to the segments of the diameter of the conic section
to which the parallel lines are drawn in an orderly way.}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{quote}
The new curve has ordinates, namely
\emph{les lignes droites appliqu\'es par ordre a son diametre.}
These ordinates have corresponding abscissas,
\emph{les segmens de ce diametre,
qui sont entre le sommet \&\ ces lignes.}
There is still no notion that an arbitrary point
might have two coordinates,
called abscissa and ordinate respectively.
A point determines an ordinate and abscissa
only insofar as the point belongs to a given curve
with a designated diameter.

%\bibliographystyle{amsplain}
%\bibliography{../references}

\def\rasp{\leavevmode\raise.45ex\hbox{$\rhook$}} \def\cprime{$'$}
  \def\cprime{$'$} \def\cprime{$'$} \def\cprime{$'$}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
% \MRhref is called by the amsart/book/proc definition of \MR.
\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
\begin{thebibliography}{10}

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\bibitem{Apollonius-Heiberg}
\bysame, \emph{Apollonii {P}ergaei qvae {G}raece exstant cvm commentariis
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\bysame, \emph{Conics. {B}ooks {I}--{III}}, revised ed., Green Lion Press,
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\bibitem{Boyer}
Carl~B. Boyer, \emph{A history of mathematics}, John Wiley \& Sons, New York,
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\bibitem{Descartes-Geometry}
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\bibitem{LDE}
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\bibitem{Euclid-Heiberg}
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\bibitem{MEU}
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\bibitem{LSJ}
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\bibitem{OED}
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\bibitem{Pappus}
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\bibitem{Skeat}
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\bibitem{MR13:419a}
Ivor Thomas (ed.), \emph{Selections illustrating the history of {G}reek
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\end{thebibliography}


\end{document}