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\begin{document}
\title{Abscissas and Ordinates}
\author{David Pierce}
\date{\today}
\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
\listoffigures
\section{The liberation of mathematics}
In the third century before the Common Era,
Apollonius of Perga (near Antalya in Turkey)
wrote eight books on conic sections.
The first four of these books survive in the original Greek;
the next three survive in Arabic translation only. The last book is lost.
Lucio Russo \cite[page 8]{MR2038833} uses this and other examples
to show that we cannot expect all of the best ancient work
to have come down to us.
In the first book
of the \emph{Conics} \cite{MR1660991},
Apollonius derives properties of the conic sections
that can be used to write their equations
in rectangular or oblique coordinates.
The present 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 a number of our other technical terms as well.
\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 she should not be expected
to memorize the quadratic formula without seeing a derivation of it.
True education is not indoctrination, but liberation.
I elaborate on this point,
if only obliquely, in an article on my college \cite{Pierce-SJC}.
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 read in high school 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,
a book called \emph{Color Me Beautiful}
was on the American bestseller lists.
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.}
(The quotations are from my memory only.)
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}
Fowler's own book \emph{is,} in part, one of the synonym books
that he denigrates here;
so I suppose he is saying,
\enquote{You will not be a good writer, just by reading me:
you must read, and \emph{think,} for yourself.}
What other synonym books does he have in mind?
Perhaps books like \emph{Roget's Thesaurus,}
which I do find occasionally useful,
but which is \emph{not} the
\enquote{indispensible guide to the English language}
that the back cover of \enquote{my} edition \cite{Roget} claims it to be.
Fowler's own book is closer to that description,
for the example it sets of sound thinking.
This is why I prefer to read his original work,
rather than the posthumous second edition,
edited and updated by Gowers \cite{MEU2}.
The boredom, described by Fowler,
of the reader of a \emph{mere} book of synonyms
is 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,
the \enquote{\LSJ} \cite{LSJ}.
The articles of the \LSJ\ are available from the Perseus Digital Library,
\url{perseus.tufts.edu},
though I myself splurged on the print version of the whole book.
In fact the great majority of works in the references,
including all of the language books,
are from the shelves of my personal library;
some might not be the most up-to-date references,
but they are the ones that I am pleased to have at hand.
I am going to write out Apollonius's terms in Greek letters,
using the \textneohellenic{\relscale{0.9}NeoHellenic} font
made available by the Greek Font Society.
I shall use the customary minuscule forms---today's \enquote{lower case}---%
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,
even without studying the Greek language as such.
I have heard a plausible rumor
that it is actually \emph{beneficial} for calculus students
to memorize the Greek alphabet.
\section{The gendered article}
Apollonius's word for \textbf{cone} is \gk{anagka'ia}) for them
because the Nile overflows and obliterates the boundary lines
between their properties.
It is not surprising that the discovery of this and the other sciences
had its origin in necessity (\gk{arq'h}).
\end{quote}
There is no specific mention of necessity,
except that \gk{>arqh} itself can be translated as \enquote{cause.}
While it may be \emph{inspired} by the physical world,
I want to propose that mathematics
has no external \emph{cause.}
Any necessity in its development is internal.
Apparently the Egyptians were \emph{not} required by their physical conditions
to find a formula for quadrilateral areas
that we would recognize today as strictly correct.
Euclid did happen to develop a precise theoretical understanding of areas,
as presented in Book \textsc i of the \emph{Elements};
but nothing \emph{made} him do it but his own internal drive.
The Devil says to the Angel in Blake's \emph{Marriage of Heaven and Hell}
\cite[plate 23]{Blake},
\begin{quote}
\centering
bray a fool in a mortar with wheat\\
yet shall not his folly be beaten out of him.
\end{quote}
Nothing external can separate the fool from his folly.
This might be contradicted
by the experience of George Orwell,
as recounted in the essay called
\enquote{Such, such were the joys \dots} \cite{Orwell-essays}.
The essay happens to take its ironic title
from the middle stanza of \enquote{The Ecchoing Green,}
one of Blake's \emph{Songs of Innocence} \cite{Blake-Innocence}:
\begin{quote}
\settowidth{\versewidth}{In our youth time were seen}
\begin{verse}[\versewidth]
Old John, with white hair,\\
Does laugh away care,\\
Sitting under the oak,\\
Among the old folk.\\
``Such, such were the joys\\
When we all, girls and boys,\\
In our youth time were seen\\
On the Ecchoing Green.''
\end{verse}
\end{quote}
At a boarding school at the age of eight,
Orwell was not pounded by a pestle,
but he was whipped with a riding crop
until he stopped wetting his bed.
Since he did in fact stop,
\enquote{perhaps this barbarous remedy does work,
though at a heavy price, I have no doubt.}
During a lesson,
Orwell might be taken out for a beating,
right in the middle of construing a Latin sentence;
then he would be brought back in, \enquote{red-wealed and smarting,}
to continue.
\begin{quote}
It is a mistake to think that such methods do not work.
They work very well for their special purpose.
Indeed, I doubt whether classical education ever has been
or can be successfully carried on without corporal punishment.
The boys themselves believed in its efficacy.
\end{quote}
Orwell mentions a boy who wished he had been beaten more
before an examination that he failed.
The student did fail though, and I doubt a whipping would have helped him.
Orwell succeeded at examinations;
but he was at school as a \enquote{scholarship boy}
only because he had been thought likely to be successful.
Moreover, this success was of doubtful value,
at least given its price:
\begin{quote}
Over a period of two or three years
the scholarship boys were crammed with learning
as cynically as a goose is crammed for Christmas.
And with what learning!
This business of making a gifted boy's career
depend on a competitive examination,
taken when he is only twelve or thirteen,
is an evil thing at best,
but there do appear to be preparatory schools
which send scholars to Eton, Winchester, etc.,
without teaching them to see everything in terms of marks.
At Crossgates the whole process was frankly a preparation
for a sort of confidence trick.
Your job was to learn exactly those things
that would give the examiner
the impression that you knew more than you did know,
and as far as possible to avoid burdening your brain
with anything else.
\end{quote}
I quote all of this
because it sounds as if American schools today
are becoming like Orwell's \enquote{Crossgates,}
at least with their standardized examinations.
There may be no corporal punishment,
though the threat of school closure or reduced funding
might be comparable to it.
I return to the Devil's assertion that folly cannot be beaten out of you.
Neither can wisdom---or true academic success---be beaten \emph{into} you,
be it by a stick, or the flooding of the Nile,
or frequent examinations for that matter.
If the weapons seem efficacious,
it is only because their victims had it in themselves to perform.
Is there really no better way to encourage this performance?
As one of Blake's \enquote{Proverbs of Hell} runs \cite[plate 7]{Blake},
\begin{quote}
\centering
If the fool would persist in his folly he would become wise.
\end{quote}
In the academic context,
I interpret this as recommending \emph{attention,}
the avoidance of distractions:
distractions such as certain portable electronic devices
are \emph{designed} to be.
Another word for what is wanted is---\emph{application.}
After the finding of the area of an arbitrary polygon
by triangulation and application,
the \emph{d\'enouement} of Book \textsc i of Euclid is
Propositions 47 and 48: the Pythagorean Theorem and its converse.
\section{The cone}
Again, the \textbf{cone} of Apollonius
is \gk{epif'aneia},
the last word meaning originally \enquote{appearance}
and being the source of the English \enquote{epiphany.}
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 the axial triangle $ABC$
shown in Figure \ref{fig:ax-base}.
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Then the base $BC$ of this triangle is a diameter of the base of the cone.
Let an arbitrary chord $DE$ of the base of the cone
cut the base $BC$ of the axial triangle at right angles at a point $F$,
again as in Figure~\ref{fig:ax-base}.
We are going to cut the cone itself
with a plane that contains this chord $DE$,
thus obtaining a \emph{conic section.}
Figure \ref{fig:ax-base} consists of a \enquote{cross section}
and \enquote{floor plan} of a cone,
as if in an architectural drawing.
However,
it is important to note
that the cross section may not be strictly vertical.
I shall not attempt an \enquote{axonometric projection,}
showing the cone itself, with a conic section drawn along its surface.
I am reluctant to spend my own time
figuring out how to draw such a projection,
using \textsf{PSTricks} (as for the present figures) or another program.
I am not \emph{opposed} to making such projections.
I should not mind being able to write a code for drawing them.
I have a dream that a sculptor can be inspired
to make three-dimensional diagrams for the propositions of Apollonius,
using heavy gauge wires perhaps.
I have made them myself, using cardboard,
as in Figure \ref{fig:model}.
\begin{figure}
\includegraphics[width=4.2cm]{cone-oblique.eps}\hfill
\includegraphics[width=5.6cm]{cone-top.eps}
\caption{Cardboard model}\label{fig:model}
\end{figure}
But there may be pedagogic value
in having to construct, in one's own mind,
the vision of a cone and its sections.
One can find wooden models of cones cut by various planes;
but I have seen only \emph{right} cones in this way.
The beauty of Apollonius is that his cones can be scalene.
He is not doing projective geometry though:
there is still a distance between any two points,
and any two distances have a ratio.
Earlier mathematicians knew the ratios
that arose from sections of \emph{right} cones;
Apollonius understood that the restriction was not needed.
We do not know what Apollonius's own diagrams looked like.
This frees the editors of the
\emph{Green Lion} English edition \cite{MR1660991}
to provide the best diagrams according to their own judgment.
However, Netz recognizes the possibility and value
of figuring out what the original diagrams
of the Greek mathematicians looked like.
They do not necessarily look like what we might draw.
Netz himself has initiated the recovery
of the diagrams of Archimedes \cite{MR2093668}.
Given that Heiberg's edition of Apollonius \cite{Apollonius-Heiberg}
does \emph{not} give diagrams in the architectural style
of Figure \ref{fig:ax-base},
we can probably assume that Apollonius did not draw them this way.
Nonetheless, this manner of drawing may clarify some points.
We drew the chord $DE$ of the base of a cone.
Apollonius uses no word for a chord as such,
even though he proves in his Proposition \textsc 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 English words \enquote{cord} and \enquote{chord}
are derived from the Greek \gk{apolambanom'enh}.
This participle is used in Proposition \textsc i.11
\cite[page 38]{Apollonius-Heiberg},
and its general usage for what we translate as \emph{abscissa}
is confirmed in the \LSJ\ lexicon;
however, the root sense of the verb
is actually 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{The parabola in the cone}\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 \textsc{iii}.35).
More briefly,
\begin{equation*}
DF^2=BF\cdot FC.
\end{equation*}
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 \textsc i.20).
By means of Euclid's Proposition \textsc i.43,
used as in \textsc i.44 as discussed earlier,
we obtain 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[page 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 along 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.
Apollonius concludes Book I concludes by showing that
every curve given by an equation of one of the forms
\eqref{eqn:parab}, \eqref{eqn:hyperb}, and \eqref{eqn:ell}
is indeed a conic section.
This may be true for us today, by definition;
but Apollonius finds a cone of which the given curve is indeed a section.
\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[page 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}
\emph{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.}
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}
In particular, the new curve has \textbf{ordinates,} namely
\emph{les lignes droites appliqu\'es par ordre a son diametre.}
These ordinates have corresponding \textbf{abscissas,} which are
\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.
The \emph{Wikipedia} (\url{en.wikipedia.org})
articles \enquote{ordinate} and \enquote{abscissa}
do not explain the origins of these terms (at least as of October 4, 2014).
This is unfortunately true of many \emph{Wikipedia} articles on mathematics.
Of course, anybody who cares may work to change this
(as I did, for example, in adding the section \enquote{Origins}
to the article \enquote{Pappus's hexagon theorem}).
The \emph{Wikipedia} \enquote{ordinate} article
does have a reference to the website
\emph{Earliest Known Uses of Some of the Words of Mathematics}
(maintained by Jeff Miller, \url{jeff560.tripod.com/mathword.html});
but this site currently provides only a \emph{modern} history of
\enquote{ordinate} and \enquote{abscissa.}
It is said that Descartes did not use the latter term,
but did use the former;
however, the quotation above suggests
that he did not even use the term \enquote{ordinate} as such,
but used only the longer phrase, derived from Apollonius,
that was presumably the precursor of this term.
It appears that the use of \enquote{ordinate} and \enquote{abscissa}
as technical terms is due to Leibniz.
The articles on Miller's website
refer to Struik in \emph{A Source Book in Mathematics, 1200--1800,}
where a footnote \cite[page 272, note 1]{MR858706}
to an article of Leibniz reads,
\begin{quote}
Note the Latin term \emph{abscissa.}
This term, which was not new in Leibniz's day,
was made by him into a standard term,
as were so many other technical terms.
In the article \enquote{De linea ex lineis
numero infinitis ordinatim ductis inter se concurrentibus formata \dots,}
\emph{Acta Eruditorum 11} (1692), 168--171
(Leibniz, \emph{Mathematische Schriften,} Abth.\ 2, Band I (1858),
266--269),
in which Leibniz discusses evolutes,
he presents a collection of technical terms.
Here we find
\emph{ordinata, evolutio, differentiare,
parameter, differentiabilis, functio,}
and \emph{ordinata} and \emph{abscissa}
together designated as \emph{coordinatae.}
Here he also points out that ordinates may be given
not only along straight but also along curved lines.
The term \emph{ordinate} is derived from
\emph{rectae ordinatim applicatae,}
\enquote{straight lines designated in order,}
such as parallel lines.
The term \emph{functio} appears in the sentence:
\enquote{the tangent and some other functions depending on it,
such as perpendiculars from the axis conducted to the tangent.}
\end{quote}
Presumably Leibniz (1646--1716)
knew Books I--IV of Apollonius's \emph{Conics}
from Commandino's 1566 Latin translation of them.
Arabic manuscripts of the later books were brought to Europe,
starting later in the 16th century;
but the first proper Latin translation
did not appear until Halley's 1710 edition \cite[xxi--xxv]{Apollonius-V-VII}.
In any case, all we have needed for the present article is Book I.
It was noted in the beginning
that the original Greek of Books V--VII of Apollonius has been lost,
and Book VIII exists in no language at all.
Terminology based on Apollonius survives,
thanks apparently to Leibniz;
but a proper understanding of the terminology has not generally survived.
This is unfortunate, but can change.
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\end{document}