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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{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{'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{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{'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}
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\begin{thebibliography}{10}
\bibitem{Heath-Apollonius}
{Apollonius of Perga}, \emph{Apollonius of {P}erga: Treatise on conic
sections}, University Press, Cambridge, UK, 1896, Edited by {T}. {L}. {H}eath
in modern notation, with introductions including an essay on the earlier
history of the subject.
\bibitem{Apollonius-Heiberg}
\bysame, \emph{Apollonii {P}ergaei qvae {G}raece exstant cvm commentariis
antiqvis}, vol.~I, Teubner, 1974, Edidit et Latine interpretatvs est I. L.
Heiberg.
\bibitem{MR1660991}
\bysame, \emph{Conics. {B}ooks {I}--{III}}, revised ed., Green Lion Press,
Santa Fe, NM, 1998, 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. \MR{MR1660991 (2000d:01005)}
\bibitem{Boyer}
Carl~B. Boyer, \emph{A history of mathematics}, John Wiley \& Sons, New York,
1968.
\bibitem{Descartes-Geometry}
Ren{\'e} Descartes, \emph{The geometry of {R}en{\'e} {D}escartes}, Dover
Publications, Inc., New York, 1954, Translated from the French and Latin by
David Eugene Smith and Marcia L. Latham, with a facsimile of the first
edition of 1637.
\bibitem{LDE}
Jean Dubois, Henri Mitterand, and Albert Dauzat, \emph{Dictionnaire
d'{\'e}tymologie}, Larousse, Paris, 2001, First edition 1964.
\bibitem{Euclid-Heiberg}
Euclid, \emph{Euclidis {E}lementa}, Euclidis Opera Omnia, vol.~I, Teubner,
1883, Edidit et Latine interpretatvs est I. L. Heiberg.
\bibitem{MR17:814b}
\bysame, \emph{The thirteen books of {E}uclid's {E}lements translated from the
text of {H}eiberg. {V}ol. {I}: {I}ntroduction and {B}ooks {I}, {I}{I}. {V}ol.
{I}{I}: {B}ooks {I}{I}{I}--{I}{X}. {V}ol. {I}{I}{I}: {B}ooks
{X}--{X}{I}{I}{I} and {A}ppendix}, Dover Publications Inc., New York, 1956,
Translated with introduction and commentary by Thomas L. Heath, 2nd ed.
\MR{17,814b}
\bibitem{MR1932864}
\bysame, \emph{Euclid's {E}lements}, Green Lion Press, Santa Fe, NM, 2002, All
thirteen books complete in one volume. The Thomas L. Heath translation,
edited by Dana Densmore. \MR{MR1932864 (2003j:01044)}
\bibitem{MEU}
H.~W. Fowler, \emph{A dictionary of modern {E}nglish usage}, Wordsworth
Editions, Ware, Hertfordshire, UK, 1994, reprint of the original 1926
edition.
\bibitem{Herodotus-Loeb}
Herodotus, \emph{The {P}ersian wars, books {I}--{II}}, Loeb Classical Library,
vol. 117, Harvard University Press, Cambridge, Massachusetts and London,
England, 2004, Translation by A. D. Godley; first published 1920; revised,
1926.
\bibitem{CODoEE}
T.~F. Hoad (ed.), \emph{The concise {O}xford dictionary of {E}nglish
etymology}, Oxford University Press, Oxford and New York, 1986, Reissued in
new covers, 1996.
\bibitem{Iliad-Loeb}
Homer, \emph{The {I}liad}, Loeb Classical Library, Harvard University Press and
William Heinemann Ltd, Cambridge, Massachusetts, and London, 1965, with an
English translation by A. T. Murray.
\bibitem{MR0472307}
Morris Kline, \emph{Mathematical thought from ancient to modern times}, Oxford
University Press, New York, 1972. \MR{MR0472307 (57 \#12010)}
\bibitem{LSJ}
Henry~George Liddell and Robert Scott, \emph{A {G}reek-{E}nglish lexicon},
Clarendon Press, Oxford, 1996, revised and augmented throughout by Sir Henry
Stuart Jones, with the assistance of Roderick McKenzie and with the
cooperation of many scholars. With a revised supplement.
\bibitem{CID}
William Morris (ed.), \emph{The {G}rolier international dictionary}, Grolier
Inc., Danbury, Connecticut, 1981, two volumes; appears to be the
\emph{American Heritage Dictionary} in a different cover.
\bibitem{POLD}
James Morwood (ed.), \emph{The pocket {O}xford {L}atin dictionary}, Oxford
University Press, 1995, First edition published 1913 by Routledge \&\ Kegan
Paul.
\bibitem{OED}
Murray et~al. (eds.), \emph{The compact edition of the {O}xford {E}nglish
{D}ictionary}, Oxford University Press, 1973.
\bibitem{NFB}
Alfred~L. Nelson, Karl~W. Folley, and William~M. Borgman, \emph{Analytic
geometry}, The Ronald Press Company, New York, 1949.
\bibitem{MR1683176}
Reviel Netz, \emph{The shaping of deduction in {G}reek mathematics}, Ideas in
Context, vol.~51, Cambridge University Press, Cambridge, 1999, A study in
cognitive history. \MR{MR1683176 (2000f:01003)}
\bibitem{Pappus}
Pappus, \emph{Pappus {A}lexandrini collectionis quae supersunt}, vol.~II,
Weidmann, Berlin, 1877, E libris manu scriptis edidit, Latina interpretatione
et commentariis instruxit Fridericus Hultsch.
\bibitem{MR2038833}
Lucio Russo, \emph{The forgotten revolution}, Springer-Verlag, Berlin, 2004,
How science was born in 300 BC and why it had to be reborn, translated from
the 1996 Italian original by Silvio Levy. \MR{MR2038833 (2004k:01006)}
\bibitem{Skeat}
Walter~W. Skeat, \emph{A concise etymological dictionary of the {E}nglish
language}, Perigee Books, New York, 1980, First edition 1882; original date
of this edition not given.
\bibitem{MR13:419a}
Ivor Thomas (ed.), \emph{Selections illustrating the history of {G}reek
mathematics. {V}ol. {I}. {F}rom {T}hales to {E}uclid}, Loeb Classical
Library, vol. 335, Harvard University Press, Cambridge, Mass., 1951, With an
English translation by the editor. \MR{13,419a}
\end{thebibliography}
\end{document}