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\begin{document}
\title{Analytic geometry}
\author{David Pierce}
\date{\today, \printtime}
\publishers{Mathematics Department\\
Mimar Sinan Fine Arts University\\
\url{dpierce@msgsu.edu.tr}\\
\url{http://mat.msgsu.edu.tr/~dpierce/}}
\maketitle
\tableofcontents
\listoffigures
\addchap{Introduction}
The writing of this report was originally provoked
both by frustration with the lack of rigor in analytic geometry texts
and by a belief that this problem can be remedied
by attention to mathematicians like Euclid and Descartes,
who are the original sources of our collective understanding of geometry.
Analytic geometry arose
with the importing of algebraic notions and notations into geometry.
Descartes, at least, justified the algebra geometrically.
Now it is possible to go the other way,
using algebra to justify geometry.
Textbook writers of recent times do not make it clear which way they are going.
This makes it impossible for a student of analytic geometry
to get a correct sense of what a \emph{proof} is.
I find this unacceptable.
If it be said that analytic geometry is not concerned with proof,
I would respond that in this case the subject
pushes the student back to a time before Euclid,
but armed with many more unexamined presuppositions.
Students today have the idea that
every line segment has a length,
which is a positive real number of units, and conversely
every positive real number is the length of some line segment.
The latter presupposition is quite astounding,
since the real numbers compose an uncountable set.
Euclidean geometry can in fact be done in a countable space,
as David Hilbert points out.
I made notes on some of these matters.
The notes grew into this report
as I found more and more things that seemed worth saying.
There are still many more avenues to explore.
Some of the notes here are just indications
of what can be investigated further,
either in mathematics itself
or in the existing literature about it.
Meanwhile, the contents of the numbered chapters of this report
might be summarized as follows.
\begin{compactenum}
\item
The logical foundations of analytic geometry
as it is often taught are unclear.
The subject is not presented rigorously.
\item
What is rigor anyway?
I consider some modern examples where rigor is lacking.
Rigor is not an absolute notion,
but must be defined in terms of the audience being addressed.
\item
Ancient mathematicians like Euclid and Archimedes
still set the standard for rigor.
\item
I have suggested \emph{how} they are rigorous;
but \emph{why} are they rigorous? I don't know.
But we still expect rigor from our students,
if only because we expect them to be able to justify their answers
to the problems that we assign to them%
---or if we don't expect this, we ought to.
\item
I look at an old analytic geometry textbook
that I learned something from as a child,
but that I now find mathematically sloppy.
\item
Because that book uses the odd terms \emph{abscissa} and \emph{ordinate}
without explaining their origin, I provide an explanation,
\linebreak which involves the conic sections
as presented by Apollonius.
\item
The material on conic sections spills over to another chapter.
Here we start to look in more detail at the geometry of Descartes.
How Apollonius himself works out his theorems remains mysterious:
Descartes's methods do not seem to illuminate those theorems.
\item
I look at an analytic geometry textbook that I once taught from.
It is more sophisticated than the textbook from my childhood.
This makes its failures of rigor more frustrating to me
and possibly more dangerous for the student.
\item
I spell out more details of the justification
of the use of algebra in geometry.
Descartes acknowledges the need to provide this justification.
\item
Finally, I review how the algebra of certain ordered fields
can be used to obtain a Euclidean plane.
\end{compactenum}
My scope here is the whole history of mathematics.
Obviously I cannot give this a thorough treatment.
I am not prepared to \emph{try} to do this.
To come to some understanding of a mathematician,
one must \emph{read} him or her;
but I think one must read
with a sense of what it means to do mathematics,
\emph{and}
with an awareness that this sense may well differ
from that of the mathematician whom one is reading.
This awareness requires experience,
in addition to the mere will to have it.
I have been fortunate to read old mathematics,
both as a student and as a teacher,
in classrooms where \emph{everybody}
is working through this mathematics and presenting it to the class.
I am currently in the third year of seeing
how new undergraduate mathematics students
respond to Book I of Euclid's \emph{Elements.}
I continue to be surprised by what the students have to say.
Mostly what I learn from the students themselves
is how strange the notion of proof can be to some of them.
This impresses on me how amazing it is
that the \emph{Elements} was produced in the first place.
It may remind me that what Euclid even \emph{means} by a proof
may be quite different from what we mean today.
But the students alone may not be able to impress on me some things.
Some students are given to writing down assertions
whose correctness has not been established.
These students' proofs may end up as a sequence of statements
whose logical interconnections are unclear.%%%%%
\footnote{Unfortunately some established mathematicians
use the same style in their own lectures.}
This is not the case with Euclid.
And yet Euclid does begin each of his propositions with a bare assertion.
He does not preface this enunciation or \emph{protasis} (\gk{pr'otasis})
with the word ``theorem'' or ``problem'' as we might today.
He does not have the typographical means that Heiberg uses
in his own edition of Euclid
to distinguish the protasis
from the rest of the proposition.
No, the protasis just sits there,
not even preceded by the ``I say that'' (\gk{l'egw <'oti})
that may be seen further down in the proof.
For me to notice this,
na\"{\i}ve students were apparently not enough,
but I had also to read Fowler's
\emph{Mathematics of Plato's Academy} \cite[10.4(e), pp.~385--6]{MR2000c:01004}.
\chapter{The problem}
Textbooks of analytic geometry
do not make their logical foundations clear.
Of course I can speak only of the books that I have been able to consult:
these are from the last century or so.
Descartes's original presentation \cite{Descartes-Geometry}
in the 17th century
\emph{is} clear enough.
In an abstract sense,
Descartes may be no more rigorous than his successors.
He does get credit for actually inventing his subject,
or at least for introducing the notation we use today:
minuscule letters for lengths,
with letters from the beginning of the alphabet used for known lengths,
and letters from the end for unknown lengths.
As for his mathematics itself,
Descartes explicitly bases it
on an ancient tradition
that culminates in the 4th century with Pappus of Alexandria.
More recent analytic geometry books
start in the middle of things,
but they do not make it clear what those things are.
I think this is a problem.
The chief aim of these notes
is to identify this problem and its solution.
How can analytic geometry be presented rigorously?
Rigor is not a fixed standard, but depends on the audience.
Still, it puts some requirements on any work of mathematics,
as I shall discuss in Chapter~\ref{ch:failures}.
In my own mathematics department,
students of analytic geometry
have had a semester of calculus,
and a semester of synthetic geometry from its own original source,
Book I of Euclid's \emph{Elements} \cite{MR1932864,bones}.
They are the audience that I especially have in mind
in my considerations of rigor.
But I would suggest that any students of analytic geometry
ought to come to the subject similarly prepared,
at least on the geometric side.
Plane analytic geometry can be seen as the study of the Euclidean plane
with the aid of a sort of rectangular grid
that can be laid over the plane as desired.
Alternatively, the subject can be seen
as a discovery of geometric properties
in the set of ordered pairs of real numbers.
I propose to call these two approaches
the \emph{geometric} and the \emph{algebraic,} respectively.
Either approach can be made rigorous.
But a course ought to be clear \emph{which} approach is being taken.
Probably most courses of analytic geometry take the geometric approach,
relying on students to know something of synthetic geometry already.
Then the so-called Distance Formula can be justified
by appeal to the Pythagorean Theorem.
However, even in such a course, students might be asked
to use algebraic methods to prove, for example,
that \emph{the base angles of an isosceles triangle are equal.}%%%%%
\footnote{I had a memory that this problem
was assigned in an analytic geometry course that I was once involved with.
However, I cannot find the problem in my files.
I do find similar problems, such as
to prove that the line segment bisecting two sides of triangle
is parallel to the third side and is half its length,
or to prove that, in an isosceles triangle,
the median drawn to the third side is just its perpendicular bisector.
In each case, the student is explicitly required
to use analytic methods.}
Perhaps what would be expected is something like the following.
\begin{proof}[Proof 1.]
Suppose the vertices of a triangle are $\bm a$, $\bm b$, and $\bm c$,
and the angles at $\bm b$ and $\bm c$
are $\beta$ and $\gamma$ respectively as in Figure~\ref{fig:abc}.
\begin{figure}[ht]
\centering
\begin{pspicture}(2,2.5)
\pspolygon(0,0)(2,0)(1,2)
\uput[dl](0,0){$\bm b$}
\uput[32](0,0){$\beta$}
\uput[dr](2,0){$\bm c$}
\uput[148](2,0){$\gamma$}
\uput[u](1,2){$\bm a$}
\end{pspicture}
\caption{An isosceles triangle in a vector space}\label{fig:abc}
\end{figure}
These angles are given by
\begin{gather*}
(\bm a-\bm b)\cdot(\bm c-\bm b)
=\size{\bm a-\bm b}\cdot\size{\bm c-\bm b}\cdot\cos\beta,\\
(\bm a-\bm c)\cdot(\bm b-\bm c)
=\size{\bm a-\bm c}\cdot\size{\bm b-\bm c}\cdot\cos\gamma,
\end{gather*}
We assume the triangle is isosceles, and in particular
\begin{equation*}
\size{\bm a-\bm b}=\size{\bm a-\bm c}.
\end{equation*}
Then
\begin{align*}
(\bm a-\bm c)\cdot(\bm b-\bm c)
&=(\bm a-\bm c)\cdot(\bm b-\bm a+\bm a-\bm c)\\
&=(\bm a-\bm c)\cdot(\bm b-\bm a)+(\bm a-\bm c)\cdot(\bm a-\bm c)\\
&=(\bm a-\bm c)\cdot(\bm b-\bm a)+(\bm a-\bm b)\cdot(\bm a-\bm b)\\
&=(\bm c-\bm a)\cdot(\bm a-\bm b)+(\bm a-\bm b)\cdot(\bm a-\bm b)\\
&=(\bm c-\bm b)\cdot(\bm a-\bm b),
\end{align*}
and so $\cos\beta=\cos\gamma$.
\end{proof}
If one has the Law of Cosines, the argument is simpler:
\begin{proof}[Proof 2.]
Suppose the vertices of a triangle are $\bm a$, $\bm b$, and $\bm c$,
and the angles at $\bm b$ and $\bm c$
are $\beta$ and $\gamma$ respectively, again as in Figure~\ref{fig:abc}.
By the Law of Cosines,
\begin{gather*}
\size{\bm a-\bm c}^2
=\size{\bm a-\bm b}^2+\size{\bm c-\bm b}^2
-2\cdot\size{\bm a-\bm b}\cdot\size{\bm c-\bm b}\cdot\cos\beta,\\
\cos\beta
=\frac{\size{\bm c-\bm b}}{2\cdot\size{\bm a-\bm b}},
\end{gather*}
and similarly
\begin{equation*}
\cos\gamma
=\frac{\size{\bm b-\bm c}}{2\cdot\size{\bm a-\bm c}}.
\end{equation*}
If $\size{\bm a-\bm b}=\size{\bm a-\bm c}$,
then $\cos\beta=\cos\gamma$, so $\beta=\gamma$.
\end{proof}
In this last argument though, the vector notation is a needless complication.
We can streamline things as follows.
\begin{proof}[Proof 3.]
In a triangle $ABC$,
let the sides opposite $A$, $B$, and $C$
have lengths $a$, $b$, and $c$ respectively,
and let the angles as $B$ and $C$ be $\beta$ and $\gamma$ respectively,
as in Figure~\ref{fig:ABC}.
\begin{figure}[ht]
\centering
\begin{pspicture}(2,2.5)
\pspolygon(0,0)(2,0)(1,2)
\uput[dl](0,0){$B$}
\uput[32](0,0){$\beta$}
\uput[dr](2,0){$C$}
\uput[148](2,0){$\gamma$}
\uput[u](1,2){$A$}
\uput[ul](0.5,1){$c$}
\uput[ur](1.5,1){$b$}
\uput[d](1,0){$a$}
\end{pspicture}
\caption{An isosceles triangle}\label{fig:ABC}
\end{figure}
If $b=c$, then
\begin{gather*}
b^2=c^2+a^2-2ca\cos\beta,\\
\cos\beta=\frac a{2c}=\frac a{2b}=\cos\gamma.\qedhere
\end{gather*}
\end{proof}
Possibly this is not considered \emph{analytic} geometry though,
since coordinates are not used, even implicitly.
We can use coordinates explicitly, laying down our grid conveniently:
\begin{proof}[Proof 4.]
Suppose a triangle has vertices $(0,a)$, $(b,0)$, and $(c,0)$,
as in Figure~\ref{fig:abc0}.
\begin{figure}[ht]
\centering
\begin{pspicture}(0,-0.5)(2,3)
\psline(0,0)(1,2)(2,0)
\psline{->}(-1,0)(3,0)
\psline{->}(1,-0.5)(1,3)
\uput[d](0,0){$b$}
\uput[d](2,0){$c$}
\uput[ur](1,2){$a$}
\end{pspicture}
\caption{An isosceles triangle in a coordinate plane}\label{fig:abc0}
\end{figure}
We assume $a^2+b^2=a^2+c^2$, and so $b=-c$.
In this case the cosines
of the angles at $(b,0)$ and $(c,0)$ must be the same,
namely $\size b/\sqrt{a^2+b^2}$.
\end{proof}
In any case, as a proof of what is actually Euclid's Proposition I.5,
this whole exercise is logically worthless,
assuming we have taken the geometric approach to analytic geometry.
By this approach, we shall have had to show how to erect perpendiculars
to given straight lines,
as in Euclid's Proposition I.11,
whose proof relies ultimately on I.5.
One could perhaps develop analytic geometry
on Euclidean principles without proving Euclid's I.5
as an independent proposition.
For, the equality of angles that it establishes
can be established immediately by means of I.4,
the Side-Angle-Side theorem,
by the method attributed to Pappus
by Proclus \cite[pp.~249--50]{1873procli}.
In this case, one can still see clearly that I.4 is true,
without needing to resort to any of the analytic methods suggested above.
\chapter{Failures of rigor}\label{ch:failures}
The root meaning of the word \emph{rigor} is stiffness.
Rigor in a piece of mathematics
is what makes it able to stand up to questioning.
Rigor in mathematics \emph{education} requires helping students
to see what questions might be asked about a piece of mathematics.
An education in mathematics
will take the student through several passes over the same subjects.
With each pass, the student's understanding should deepen.%%%%%
\footnote{It might be counted as a defect in my own education
that I did not have undergraduate courses in algebra and topology
before taking graduate versions of these courses.
Graduate analysis was for me a continuation of my high school course,
which had been based on Spivak's \emph{Calculus} \cite{0458.26001}
and (in small part) Apostol's \emph{Mathematical Analysis} \cite{MR49:9123}.}
At an early stage,
the student need not and cannot be told
all of the questions that might be raised at a later stage.
But if the mathematics of an early course
resembles that of a different later course,
it ought to be equally rigorous.
Otherwise the older student might assume, wrongly,
that the earlier mathematics could in fact stand up to the same scrutiny
that the later mathematics stands up to.
Concepts in an earlier course
must not be presented in such a way
that they will be misunderstood in a later course.
By this standard,
students of calculus
need not master the epsilon-delta definition of limit.
If the students later take an analysis course,
then they will fill in the logical gaps
from the calculus course.
The students are not going to think
that everything was already proved in calculus class,
so that epsilons and deltas are a needless complication.
They may think there is no \emph{reason} to prove everything,
but that is another matter.
If students of calculus never study analysis,
but become engineers perhaps,
or teachers of school mathematics,
they are not likely to have false beliefs
about what theorems can be proved in mathematics;
they just will not have a highly developed notion of proof.
By introducing and using the epsilon-delta definition of limit
at the very beginning of calculus,
the teacher might actually violate the requirements of rigor,
if he or she instills the false notion
that there is no rigorous alternative definition of limits.
How many calculus teachers,
ignorant of Robinson's ``nonstandard'' analysis \cite{MR1373196},
will try to give their students some notion of epsilons and deltas,
out of a misguided notion of rigor,
when the intuitive approach by means of infinitesimals
can be given full logical justification?
On the other hand, in mathematical circles,
I have encountered disbelief
that the real numbers constitute the unique complete ordered field.
Since every valued field has a completion,
it is possible to suppose wrongly
that every ordered field has a completion.
This confusion might be due to a lack of rigor in education,
somewhere along the way.\footnote{I may have been saved from this confusion by
Spivak's final chapter, ``Uniqueness of the real numbers'' \cite[ch.~29]{0458.26001}.}
There are (at least) two ways to obtain $\R$ from $\Q$.
One way is to take the quotient
of the ring of Cauchy sequences of rational numbers
by the ideal of such sequences that converge to $0$.
Another way is to complete the ordering of $\Q$,
as by taking Dedekind cuts,
and then to show that this completion can be made into a field.
The first construction can be generalized
to apply to non-Archimedean ordered fields;
the second cannot.
More precisely, the second construction
can be applied to a non-Archimedean field,
but the result is not a field.
It might be better to say that
while the first construction achieves
the completeness of $\R$ as a metric space,
the second achieves the \emph{continuity} of $\R$ as an ordered field.
At any rate, continuity is the word Dedekind uses
for what his construction is supposed to accomplish \cite{MR0159773}.
In an elementary course, the student may learn a theorem
according to which
certain conditions on certain structures
are logically equivalent.
But the theorem may use assumptions that are not spelled out.
This is a failure of rigor.
In later courses, the student learns logical equivalences
whose assumptions \emph{are} spelled out.
The student may then assume that the earlier theorem is like the later ones.
It may not be, and failure to appreciate this
may cause the student to overlook some lovely pieces of mathematics.
The word \emph{student} here may encompass all of us.
The supposed theorem that I have in mind is that, in number theory,
the principles of induction and well ordering
are equivalent.\footnote{``Either principle may be considered as a basic assumption about the natural numbers'' \cite[ch.~2, p.~23]{MR1373196}. I use this book as an example because it is otherwise so admirable.}
Proofs of two implications may be offered to back up this claim,
though one of the proofs may be left as an exercise.
The proofs will be of the standard form.
They will look like other proofs.
And yet, strictly speaking,
they will make no logical sense, because:
\begin{compactitem}
\item
Induction is a property of algebraic structures
in a signature with a constant, such as $1$,
and a singulary function symbol such as ${}'$ (``prime'')
for the operation of adding $1$.
\item
Well ordering is a property of ordered structures.
\end{compactitem}
When well ordering is used to prove induction,
a set $A$ is taken that contains $1$ and is closed under adding $1$,
and it is shown that the complement of $A$ cannot have a least element.
For, the least element cannot be $1$,
and if the least element is $n+1$,
then $n\in A$, so $n+1\in A$, contradicting $n+1\notin A$.
It is assumed here that $nmB\liff kC>mD.
\end{equation*}
In this case we may write
\begin{equation*}
A:B\as C:D,
\end{equation*}
though Euclid uses no such notation.
What is expressed by this notation is not the equality,
but the \emph{identity,} of two ratios.
Equality is a possible property of two nonidentical magnitudes.
Magnitudes are geometric things,
ratios are not.
Euclid never draws a ratio or assigns a letter to it.%%%%%
\footnote{I am aware of one possible counterexample to this claim.
The last proposition---number 39---in Book VII is
\emph{to find the number that is the least of those that will have given parts.}
The meaning of this is revealed in the proof, which begins:
``Let the given parts be \gk A, \gk B, and \gk G.
Then it is required to find the number
that is the least of those that will have the parts \gk A, \gk B, and \gk G.
So let \gk D, \gk E, and \gk Z be numbers
homonymous with the parts \gk A, \gk B, and \gk G,
and let the least number \gk H measured by \gk D, \gk E, and \gk Z be taken.''
Thus \gk H is the least common multiple of \gk D, \gk E, and \gk Z,
which can be found by Proposition VII.36.
Also, if for example \gk D is the number $n$,
then \gk A is an $n$th, considered abstractly:
it is not given as an $n$th part of anything in particular.
Then \gk A might be considered as the ratio of $1$ to $n$.
Possibly VII.39 was added later to Euclid's original text,
although Heath's note \cite[p.~344]{MR17:814b} suggests no such possibility.
If indeed VII.39 is a later addition, then so, probably,
are the two previous propositions, on which it relies:
they are
that if $n\divides r$, then $r$ has an $n$th part,
and conversely.
But Fowler mentions Propositions 37 and 38, seemingly being as typical
or as especially illustrative examples of propositions from Book VII \cite[p.~359]{MR2000c:01004}.}
In any case, in the definition, it is assumed
that $A$ and $B$ \textbf{have a ratio} in the first place,
in the sense that some multiple of either of them exceeds the other;
and likewise for $C$ and $D$.
In this case, the pair
\begin{equation*}
\left(
\left\{\displaystyle\frac mk\colon kA>mB\right\},
\left\{\displaystyle\frac mk\colon kA\leq mB\right\}
\right)
\end{equation*}
is a \emph{cut} (of positive rational numbers)
in the sense of Dedekind \cite[p.~13]{MR0159773}.
Dedekind traces his definition of irrational numbers to the conviction that%%%%%
\begin{quote}
an irrational number is defined by the specification of all rational numbers that are less and all those that are greater than the number to be defined\dots
if\dots one regards the irrational number as the ratio of two measurable quantities
then is this manner of determining it already set forth in the clearest possible way
in the celebrated definition which Euclid gives of the equality of two ratios. \hfill \mbox{\cite[pp.~40--1]{MR0159773}}
\end{quote}
In saying this, Dedekind intends
to \emph{distinguish} his account of the completeness or continuity
of the real number line from some other accounts.
Dedekind does not literally define an irrational number
as a ratio of two ``measurable quantities'':
the definition of cuts such as the one above
does not require the use of magnitudes such as $A$ and $B$.
Dedekind observes moreover that Euclid's geometrical constructions
do not require continuity of lines.
``If any one should say'' writes Dedekind
\begin{quote}
that we cannot conceive of space as anything else than continuous,
I should venture to doubt it and
to call attention to the fact
that a far advanced, refined scientific training
is demanded in order to perceive clearly the essence of continuity
and to comprehend that besides rational quantitative relations,
also irrational, and besides algebraic, also transcendental quantitative relations
are conceivable. \hfill \mbox{\cite[pp.~38]{MR0159773}}
\end{quote}
Modern geometry textbooks (as in Chapter~\ref{ch:NFB} below)
assume continuity in this sense,
but without providing the ``refined scientific training''
required to understand what it means.
Euclid does provide something of this training,
starting in Book V of the \emph{Elements};
before this, he makes no use of continuity in Dedekind's sense.
In the Muslim and Christian worlds,
Euclid has educated mathematicians for centuries.
He shows the world what it means to prove things.
One need not read \emph{all} of the \emph{Elements} today.
But Book~I lays out the basics of geometry in a beautiful way.
If you want students to learn what a proof is,
I think you can do no better then tell them,
``A proof is something like what you see in Book I of the \emph{Elements.}''
I have heard of textbook writers who,
informed of errors,
decide to leave them in their books anyway,
to keep the readers attentive.
The perceived flaws in Euclid can be considered this way.
The \emph{Elements} must not be treated as a holy book.
If it causes the student to think how things might be done better,
this is good.
The \emph{Elements} is not a holy book;
it is one of the supreme achievements of the human intellect.
It is worth reading for this reason,
just as, say, Homer's \emph{Iliad} is worth reading.
The rigor of Euclid's \emph{Elements} is astonishing.
Students in school today learn formulas,
like $A=\uppi r^2$ for the area of a circle.
This formula encodes the following.
\begin{theorem}[Proposition XII.2 of Euclid]\label{thm:XII.2}
Circles are to one another as the squares on the diameters.
\end{theorem}
One might take this to be an obvious corollary of:
\begin{theorem}[Proposition XII.1 of Euclid]\label{thm:XII.1}
Similar polygons inscribed in circles are to one another
as the squares on the diameters.
\end{theorem}
And yet Euclid gives an elaborate proof of XII.2
by what is today called the Method of Exhaustion:
\begin{proof}[Euclid's proof of Theorem~\ref{thm:XII.2}, in modern notation.]
Suppose a circle $C_1$ with diameter $d_1$ is to a circle $C_2$ with diameter $d_2$ in a \emph{lesser} ratio than $d_1{}^2$ is to $d_2{}^2$.
Then $d_1{}^2$ is to $d_2{}^2$ as $C_1$ is to some fourth proportional $R$ that is \emph{smaller} than $C_2$. More symbolically,
\begin{gather*}
C_1:C_2C_2$. Then
\begin{align*}
rmC_1&<(rk-1)C_2,& rmd_1{}^2&\geq rkd_2{}^2.
\end{align*}
Assuming $2^{n-1}\geq rk$, let $P_1$ be the $2^n$-gon inscribed in $C_1$, and $P_2$ in $C_2$. Then
\begin{gather*}
C_2-P_2<\frac1{2^{n-1}}C_2\leq\frac1{rk}C_2,\\
rmP_1}(-2.5,0)(2.5,0)
\psline{->}(0,-2.5)(0,2.5)
\uput[r](2.5,0){$X$}
\uput[u](0,2.5){$Y$}
\uput[dr](0,0){$O$}
\psline(-1.9,0)(-1.9,1.3)(0,1.3)
\psdots(-1.9,1.3)
\uput[u](-1.9,1.3){$P$}
\psset{linewidth=1pt}
\psline{->}(0,0)(0,1.3)
\psline{->}(0,0)(-1.9,0)
\uput[d](-1,0){$x$}
\uput[r](0,0.7){$y$}
\rput(1.5,2.2){\parbox{2cm}{\centering First\\Quadrant}}
\rput(-1.3,2.2){\parbox{2cm}{\centering Second\\Quadrant}}
\rput(-1.3,-1.7){\parbox{2cm}{\centering Third\\Quadrant}}
\rput(1.5,-1.7){\parbox{2cm}{\centering Fourth\\Quadrant}}
\end{pspicture}
\\
\textsc{Fig.}~5
\end{figure}
so as to enable us to describe %figure goes here
the location of a point in a the plane. Through any point $O$ (Fig.~5)~se\-%
lect two mutually perpendicular directed infinite straight lines $OX$
and $OY$, thus dividing the plane into four parts called \emph{quadrants,}
which are numbered as shown in the figure. The point $O$ is the
\emph{origin} and the directed lines are called the \emph{$x$-axis} and the \emph{$y$-axis,}
respectively. A unit of measure is selected for each axis. Unless the
contrary is stated, the units selected will be the same for both axes.
The directed distance from $OY$ to any point $P$ in the plane is
the \emph{$x$-coordinate,} or \emph{abscissa,} of $P$; the directed distance from $OX$
to the point is its \emph{$y$-coordinate,} or \emph{ordinate.} Together, the abscissa
and ordinate of a point are called its \emph{rectangular coordinates.}
When a letter is necessary to represent the abscissa, $x$ is most
frequently used; $y$ is used to represent the ordinate\dots
\dots Consequently, we may represent a point by its coordinates placed in parentheses (the abscissa always first), and refer to this symbol as the point itself. For example, we may refer to the point $P_1$ of [the omitted figure] as the point $(3,5)$. Sometimes it is convenient to use both designations; we then write $P_1(3,5)$\dots When a coordinate of a point is an irrational number, a decimal approximation is used in plotting the point\dots
\end{quotation}
Here $OX$ and $OY$ are not segments, but infinite straight lines.
Nelson \etal\ evidently do not want to give a name such as $\R$
to the set of all numbers under consideration.
Hence they cannot say that they identify the geometrical plane with $\R\times\R$;
they can say only that they identify individual points with pairs of numbers.
This is fine, except that it leaves unexamined
the assumption that lengths are numbers of units.
How many generations of students
have had to learn the words \emph{abscissa} and \emph{ordinate}
without being given their etymological meanings?%%%%%%%%%%
\footnote{One complaint I have about my own education
is having had to learn technical terms without their etymology.
In a literature class in high school,
how much easier it would have been to learn what \emph{zeugma} was,
if only it had been pointed out to us
that the Greek word was cognate
with the Latin-derived \emph{join} and the Anglo-Saxon \emph{yoke.}}
Nelson \etal\ do not discuss them,
even in their chapter on conic sections,
although the terms are the Latin translations
of Greek words used by Apollonius of Perga
in the \emph{Conics} \cite{MR1660991}.
I consider their original meaning in Chapter \ref{ch:abscissa} below.
Meanwhile, I just have to wonder
whether an analytic geometry textbook
cannot be more enticing than that of Nelson \etal{}
If the purpose of the subject is to solve problems,
why not present some of the actual problems
that the subject was invented to solve?
A possible example is the duplication of the cube,
discussed at the end of Chapter \ref{ch:conics}.
\chapter{Abscissas and ordinates}\label{ch:abscissa}
In the first of the eight books%%%%%
\footnote{The first four books survive in Greek,
the next three in Arabic translation; the last book is lost.}
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.
I review these properties here,
because
\begin{inparaenum}[(1)]
\item
they have intrinsic 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,
including \emph{abscissa} and \emph{ordinate.}
\end{inparaenum}
Apollonius did not invent any of his terms:
these were just ordinary words, used in a certain way.
When we carry those words%
---or their Latin versions---%
over into our own language,
we create some distance between ourselves and mathematics.
When I first learned that a conic section had a \emph{latus rectum,}
I understood 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.
Had the \emph{latus rectum} been called an upright side
as in Apollonius,
it would have been easier to ask
``What is an upright side?''
In turn, textbook writers might have felt more obliged to explain what it was.
In any case, I am going to give an explanation below.
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.%%%
\footnote{In the 1980s, the \emph{Washington Post}
described a best-selling book called \emph{Color Me Beautiful}
as offering ``the color-wheel approach to female pulchritude.''
The \emph{New York Times} just said
the book provided ``beauty tips for women.''
The register of the \emph{Post} was mocking; the \emph{Times,} neutral.}
If distinctions between near-synonyms are carefully maintained,
then subtlety of expression is possible.
\emph{Circle} and \emph{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.''
However, mathematics can be done in any language.
Greek does mathematics without a specialized vocabulary.
It is worthwhile to consider what this is like.
For Apollonius,
a \textbf{cone} (\gk{epif'aneia}%%%%%
\footnote{The word \gk{>epif'aneia} means originally ``appearance''
and is the source of the English ``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} ``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} ``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 \emph{cord} and \emph{chord} \cite{CODoEE},
Apollonius does not use the word \gk{apolambanom'enh} ``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:ax-base} 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 (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 $DGE$.
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 a \textbf{parabola} (\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.
Thus the terms \emph{abscissa} and \emph{ordinate}
are ultimately translations of Greek words
that merely describe certain line segments
that can be used to describe points on conic sections.
For Apollonius, they 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{``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}
There is still no notion that an arbitrary point
might have two coordinates,
called abscissa and ordinate respectively.
\chapter{The geometry of the conic sections}\label{ch:conics}
For an hyperbola or ellipse, the \textbf{center} (\gk{k'entron})
is the midpoint of the transverse side.
In Book I of the \emph{Conics,}
Apollonius shows that the diameters of
\begin{inparaenum}[(1)]
\item
an ellipse are the straight lines through its center,
\item
an hyperbola are the straight lines through its center
that actually cut the hyperbola,%%%%%
\footnote{If the hyperbola is considered together with its conjugate hyperbola,
then all straight lines through the center are diameters, except the asymptotes.}
\item
a parabola are the straight lines that are parallel to the axis.
\end{inparaenum}
Moreover, with respect to a new diameter,
the relation between ordinates and abscissas is as before,
except that the upright and transverse sides may be different.
I do not know of an efficient way to prove these theorems
by Cartesian, analytic methods.
Descartes opens his \emph{Geometry} by saying,
\begin{quote}
All problems in geometry can easily be reduced to such terms
that one need only know the lengths of certain straight lines
in order to solve them.
\end{quote}
However, Apollonius proves his theorems about diameters
by means of \emph{areas.}
Areas can be reduced to products of straight lines,
but the reduction in the present context seems not to be particularly easy.
For example, to shift the diameter of a parabola,
Apollonius will use the following.
\begin{lemma}[Proposition I.42 of Apollonius]
In Figure~\ref{fig:A.42},
\begin{figure}[ht]
\centering
\psset{unit=8cm}
\begin{pspicture}(-0.625,-0.25)(0,0.25)
\psplot{-0.5}{0}{0 x x mul sub}
\pspolygon(-0.5,-0.25)(0,-0.25)(0,0.25)
\psline(-0.5,-0.25)(-0.5,0)(0,0)
\psline(0,-0.0625)(-0.5,-0.0625)
\psline(-0.25,-0.0625)(0,0.1875)
\uput[r](0,0.25){$A$}
\uput[r](0,0){$B$}
\uput[l](-0.5,-0.25){$G$}
\uput[d](-0.25,-0.0625){$D$}
\uput[r](0,0.1875){$E$}
\uput[r](0,-0.0625){$Z$}
\uput[ul](-0.5,0){$H$}
\uput[r](0,-0.25){$J$}
\end{pspicture}
\caption{Proposition I.42 of Apollonius}\label{fig:A.42}
\end{figure}
it is assumed that
\begin{inparaenum}[(1)]
\item
the parabola $GDB$ has diameter $AB$,
\item
$AG$ is tangent to the parabola at $G$,
\item
$GJ$ is an ordinate, and
\item
$GJBH$ is a parallelogram.
Moreover
\item
the point $D$ is chosen at random on the parabola, and
\item
triangle $EDZ$ is drawn similar to $AGJ$.
\end{inparaenum}
It follows that
\begin{center}
the triangle $EDZ$ is equal to the parallelogram $HZ$.
\end{center}
\end{lemma}
\begin{proof}
The proof relies on knowing (from I.35)
that $AB=BJ$.
Therefore $AGJ=HJ$.
Thus the claim follows when $D$ is just the point $G$.
In general we have
\begin{align*}
EDZ:HJ
&\as EDZ:AGJ&&\text{[Euclid V.7]}\\
&\as DZ^2:GJ^2&&\text{[Euclid VI.19]}\\
&\as BZ:BJ&&\text{[Apollonius I.20]}\\
&\as HZ:HJ,&&\text{[Euclid VI.1]}
\end{align*}
and so $EDZ=HZ$ by Euclid V.8. The relative positions of $D$ and $G$ on the parabola are irrelevant to the argument.
\end{proof}
Then the diameter of a parabola can be shifted by the following.
\begin{theorem}[Proposition I.49 of Apollonius]
In Figure~\ref{fig:A.49},
\begin{figure}[ht]
\psset{unit=5cm}
\begin{pspicture}(-0.85,-0.6)(0,0.25)
\psplot{-0.75}{0}{0 x x mul sub}
\psline(-0.5,-0.25)(0,0.25)(0,-0.5625)
\psline(-0.75,-0.5625)(-0.5,-0.3125)%(0,0.1875)
\psline(-0.5,0)(-0.5,-0.5625)
\psline(0,0)(-0.5,-0.5)
\uput[r](0,-0.5625){$M$}
\uput[r](0,0){$B$}
\uput[r](0,0.25){$G$}
\uput[ul](-0.5,-0.25){$D$}
\uput[l](-0.5,0){$Z$}
\uput[d](-0.5,-0.5625){$N$}
\uput[l](-0.75,-0.5625){$K$}
\uput[dr](-0.5,-0.3125){$L$}
%\uput[r](0,0.1875){$P$}
\uput[l](-0.5,-0.5){$R$}
\end{pspicture}
\hfill
\begin{pspicture}(-0.75,-0.6)(0.1,0.25)
\psplot{-0.75}{0}{0 x x mul sub}
\psline(-0.5,-0.25)(0,0.25)(0,-0.5625)(-0.75,-0.5625)(0,0.1875)
\psline(0,0)(-0.5,0)(-0.5,-0.5625)
\psline(0,-0.25)(-0.5,-0.25)
\psline(0,0)(-0.5,-0.5)
\uput[r](0,-0.5625){$M$}
\uput[r](0,0){$B$}
\uput[r](0,0.25){$G$}
\uput[ul](-0.5,-0.25){$D$}
\uput[l](-0.5,0){$Z$}
\uput[d](-0.5,-0.5625){$N$}
\uput[l](-0.75,-0.5625){$K$}
\uput[dr](-0.5,-0.3125){$L$}
\uput[r](0,0.1875){$P$}
\uput[l](-0.5,-0.5){$R$}
\uput[r](0,-0.25){$X$}
\uput[ul](-0.25,0){$E$}
\end{pspicture}
\caption{Proposition I.49 of Apollonius}\label{fig:A.49}
\end{figure}
it is assumed that
\begin{inparaenum}[(1)]
\item
$KDB$ is a parabola,
\item
its diameter is $MBG$,
\item
$GD$ is tangent to the parabola, and
\item
through $D$, parallel to $BG$, straight line $ZDN$ is drawn.
Moreover
\item
the point $K$ is chosen at random on the parabola,
\item
through $K$, parallel to $GD$, the straight line $KL$ is drawn, and
\item
$BR$ is drawn parallel to $GD$.
\end{inparaenum}
It follows that%%%%%
\footnote{Apollonius also finds the upright side corresponding to the new diameter $DN$: it is $H$ such that
$ED:DZ\as H:2GD$.}
\begin{equation*}
KL^2:BR^2\as DL:DR.
\end{equation*}
\end{theorem}
\begin{proof}
Let ordinate $DX$ be drawn, and let $BZ$ be drawn parallel to it. Then
\begin{align*}
GB
&=BX&&\text{[Apollonius I.35]}\\
&=ZD,&&\text{[Euclid I.34]}
\end{align*}
and so (by Euclid I.26 \&\ 29)
\begin{equation*}
\triangle EGB=\triangle EZD.
\end{equation*}
Let ordinate $KNM$ be drawn. Adding to either side of the last equation the pentagon $DEBMN$, we have the trapezoid $DGMN$ equal to the parallelogram $ZM$ (that is, $ZBMN$).
Let $KL$ be extended to $P$. By the lemma above, the parallelogram $ZM$ is equal to the triangle $KPM$. Thus
\begin{equation*}
DGMN=KPM.
\end{equation*}
Subtracting the trapezoid $LPMN$ gives
\begin{equation*}
KLN=LG.
\end{equation*}
We have also
\begin{equation*}
BRZ=RG
\end{equation*}
(as by adding the trapezoid $DEBR$ to the equal triangles $EZD$ and $EGB$).
Therefore
\begin{align*}
KL^2:BR^2
&\as KLN:BRZ\\
&\as LG:RG\\
&\as LD:RD.\qedhere
\end{align*}
\end{proof}
The proof given above works when $K$ is to the left of $D$. The argument can be adapted to the other case. Then, as a corollary, we have that $DN$ bisects all chords parallel to $DG$. In fact Apollonius proves this independently, in Proposition I.46.
Again, I do not see how the foregoing arguments can be improved
by expressing all of the areas involved
in terms of lengths.
%Perhaps the reader can find a way.
Rule Four in Descartes's \emph{Rules for the Direction of the Mind} \cite{Descartes-Eng}
is, ``We need a method if we are to investigate the truth of things.''
Descartes elaborates:
\begin{quote}
\dots So useful is this method that without it the pursuit of learning would,
I think, be more harmful than profitable.
Hence I can readily believe that the great minds of the past
were to some extent aware of it,
guided to it even by nature alone\dots
This is our experience in the simplest of sciences,
arithmetic and geometry:
we are well aware that the geometers of antiquity
employed a sort of analysis which they went on to apply
to the solution of every problem,
though they begrudged revealing it to posterity.
At the present time a sort of arithmetic called ``algebra'' is flourishing,
and this is achieving for numbers what the ancients did for figures\dots
But if one attends closely to my meaning,
one will readily see that
ordinary mathematics is far from my mind here,
that it is quite another discipline I am expounding,
and that these illustrations are more its outer garments
than its inner parts\dots
Indeed, one can even see some traces of this true mathematics, I think,
in Pappus and Diophantus who,
though not of that earliest antiquity,
lived many centuries before our time.
But I have come to think that these writers themselves,
with a kind of pernicious cunning,
later suppressed this mathematics
as, notoriously, many inventors are known to have done
where their own discoveries are concerned\dots
In the present age some very gifted men
have tried to revive this method,
for the method seems to me to be none other
than the art which goes by the outlandish name of ``algebra''%
---or at least it would be
if algebra were divested
of the multiplicity of numbers and imprehensible figures which overwhelm it
and instead possessed that abundance of clarity and simplicity
which I believe true mathematics ought to have.
\end{quote}
Descartes does not mention Apollonius among the ancient mathematicians,
and I do not believe that in his \emph{Geometry}
he has managed to recover the method
whereby Apollonius proves all of his theorems.
On the other hand, Descartes may have recovered \emph{one} method
used by ancient mathematicians,
because perhaps some of these mathematicians \emph{did} solve problems
by considering equations of polynomial functions of lengths only.
An example is Menaechmus,
``a pupil of Eudoxus and a contemporary of Plato'' \cite[p.~xix]{Heath-Apollonius}.
Apollonius did not discover the conic sections;
Menaechmus is thought to have done this,
if only because his is the oldest name associated with the conic sections.
According to the commentary by Eutocius%%%%%
\footnote{Eutocius flourished around 500 \ce,
and his commentary was revised by Isadore of Miletus \cite[p.~25]{MR654680},
who along with Anthemius of Tralles
was a master-builder of Justinian's Ayasofya \cite{Procopius-Buildings}.}
on Archimedes,
Menaechmus had two methods
for finding two mean proportionals to two given straight lines;
each of these methods uses conic sections.
One of the methods is illustrated by Figure~\ref{fig:Men};
\begin{figure}[ht]
\centering
\begin{pspicture}(0,-0.5)(4,3)
\psline(0,3)(0,0)(4,0)
\psline(0,2)(1,2)(1,0)
\psplot{0.67}{4}{2 x div}
\psplot{0}{2.25}{4 x mul sqrt}
\uput[80](1,2){\gk J}
\uput[l](0,2){\gk K}
\uput[d](0,0){\gk D}
\uput[d](1,0){\gk Z}
\uput[d](4,0){\gk H}
\end{pspicture}
\caption{Menaechmus's finding of two mean proportionals}\label{fig:Men}
\end{figure}
apparently Menaechmus's own diagram was just like this \cite[p.~288]{MR2093668}.
Given the lengths \gk A and \gk E, we want to find \gk B and \gk G so that
\begin{equation*}
\gkm A:\gkm B\as \gkm B:\gkm G\as \gkm G:\gkm E,
\end{equation*}
or equivalently
\begin{align}\label{eqn:Men}
\gkm B^2&=\gkm A\cdot\gkm G,&
\gkm B\cdot\gkm G&=\gkm A\cdot\gkm E.
\end{align}
In the special case where \gk A is twice \gk E,
we shall have that the cube with side \gk G is double the cube with side \gk E.
In any case, it is sufficient if
\begin{inparaenum}[(1)]
\item
\gk B is an ordinate, and \gk G the corresponding abscissa,
of the parabola with upright side \gk A whose axis is \gk{DH} in the diagram, and
\item
\gk B and \gk G are the coordinates of a point on the hyperbola
whose asymptotes are \gk{DK} and \gk{DH} in the diagram
and which also passes through the points with coordinates \gk A and \gk E.
\end{inparaenum}
For then we shall have~\eqref{eqn:Men} as desired.
Thus, if \gk J is the intersection of the parabola and hyperbola,
we can let \gk B be \gk{ZJ} and let \gk G be \gk{DZ}.
We have used the property proved by Apollonius in his Proposition II.12,
that the rectangle bounded by the straight lines
drawn from a point on an hyperbola to the asymptotes
has constant area.
Heath has an idea of how Menaechmus proved this \cite[xxv--xxviii]{Heath-Apollonius}.
In any case, by the report of Eutocius,
Menaechmus's other method of finding two mean proportionals
was to use two parabolas with orthogonal axes.
I referred to \gk B and \gk G as coordinates, but this is an anachronism.
According to one historian,
\begin{quote}
Since this material has a strong resemblance to the use of coordinates,
as illustrated above,
it has sometimes been maintained that
Menaechmus had analytic geometry.
Such a judgment is warranted only in part,
for certainly Menaechmus was unaware that any equation in two unknown quantities
determines a curve.
In fact, the general concept of an equation in unknown quantities
was alien to Greek thought.
It was shortcomings in algebraic notations that,
more than anything else,
operated against the Greek achievement of a full-fledged coordinate geometry.
\hfill \mbox{\cite[pp.~104--5]{Boyer}}%%%%%
\footnote{In fact what Boyer refers to as ``this material''
is the properties of the conic sections given by equations \eqref{eqn:parab}, \eqref{eqn:hyperb} and \eqref{eqn:ell} in the previous chapter.
Boyer will presently give the method of cube-duplication using two parabolas,
and then say,
``It is probable that Menaechmus knew that the duplication could be achieved
also by the use of a rectangular hyperbola and a parabola.''
It is not clear why he says ``It is probable that,''
unless he questions the authority of Eutocius.}
\end{quote}
Boyer evidently considers analytic geometry
as the study of the graphs of arbitrary equations;
but this would seem to be within the purview of calculus
rather than geometry.
The book of Nelson \etal\ discussed in Chapter~\ref{ch:NFB}
does have chapters on
graphs of single-valued algebraic functions,
single-valued transcendental functions, and
multiple-valued functions, as well as on
parametric equations; but
this fits the explicit purpose of the text as a preparation for calculus.
Did Descartes have a ``full-fledged analytic geometry'' in the sense of Boyer?
In the \emph{Geometry} \cite[pp.~315--7]{Descartes-Geometry},
Descartes rejects the study of curves like the quadratrix,
which today can be defined by the equation
\begin{equation*}
\tan\left(\frac{\uppi}2\cdot y\right)=\frac yx,
\end{equation*}
or more elaborately by the pair of equations
\begin{align*}
\frac{\theta}y&=\frac{\uppi}2,&
\tan\theta=\frac yx,
\end{align*}
the variables being as in Figure~\ref{fig:quad}.
\begin{figure}[ht]
\centering
\psset{unit=2.5cm}
\begin{pspicture}(-0.2,-0.2)(1.2,1.2)
\pspolygon(0,0)(1,0)(1,1)(0,1)
\parametricplot{0.001}{1}{t 90 t mul tan div t}
\psarc[linestyle=dotted](0,0){1}{0}{90}
\psset{linestyle=dashed}
\psline(0,0)(0.577,1)
\psline(0,0.667)(1,0.667)
\uput[d](0.193,0.667){$x$}
\uput[l](0,0.333){$y$}
\uput[30](0,0){$\theta$}
\uput[dl](0,0){$A$}
\uput[ul](0,1){$B$}
\uput[ur](1,1){$G$}
\uput[dr](1,0){$D$}
\uput[d](0.637,0){$H$}
\end{pspicture}
\caption{The quadratrix}\label{fig:quad}
\end{figure}
Descartes does not write down an equation for the quadratrix;
but an equation is not needed for proving theorems about this curve.
Pappus \cite[pp.~336--47]{MR13:419a} defines the quadratrix
as being traced in a square
by the intersection of two straight lines,
one horizontal and moving from the top edge $BG$ to the bottom edge $AD$,
the other swinging about the lower left corner $A$
from the left edge $AB$ to the bottom edge $AD$.
Assuming there is a point $H$ where the quadratrix meets the lower edge of the square,
we have
\begin{equation*}
BD:AB\as AB:BH,
\end{equation*}
where $BD$ is the circular arc centered at $A$.
Then a straight line equal to this arc can be found,
and so the circle can be squared.
This is why the curve is called the quadratrix (\gk{tetragwn'izousa}).
Pappus demonstrates this property, while pointing out
that we have no way to construct the quadratrix
without knowing where the point $H$ is in the first place.%%%%%
\footnote{Pappus attributes this criticism to one Sporus,
about whom we apparently have no source
but Pappus himself \cite[p.~285, n.~78]{MR2093668}.}
Today we have a notation for its position:
if $D$ is one unit away from $A$,
then the length of $AH$ is what we call $2/\uppi$.
However, this notation does not give us the location of $H$
any better than Pappus's description of the quadratrix does.
Today we can prove that $\uppi$ (and hence $2/\uppi$) is transcendental;
but this is not a topic of analytic geometry.%%%%%
\footnote{It is however a topic included in Spivak's \emph{Calculus} \cite[ch.~20]{0458.26001}.}
Is ``the general concept of an equation in unknown quantities''
something that is ``alien to Greek thought''?
Perhaps it is alien to our own thought.
According to Boyer, ``any equation in two unknown quantities determines a curve.''
But this would seem to be an exaggeration,
unless an arbitrary subset $S$ of the plane $\R\times\R$ is to be considered a curve.
For, if $\upchi_S$ is the characteristic function of $S$,
then $S$ is the solution-set of the equation
\begin{equation*}
\upchi_S(x,y)=1.
\end{equation*}
Probably Boyer does not have in mind
equations with parameters like $S$,
but equations whose only parameters are real numbers,
and in particular equations
that are expressed by means of
polynomial, trigonometric, logarithmic, and exponential functions.
If Menaechmus neglects to study all such functions,
it is not for lack of adequate algebraic notation,
but lack of interest.
He solves the problem
of finding two mean proportionals
to two given line segments.
If a numerical approximation is wanted,
this can be found, as close as desired;
therefore, by the continuity of the real line
established by Dedekind,
an exact solution exists.
But Menaechmus wants a \emph{geometric} solution,
and he finds one,
evidently by using the kind of mathematics
that we refer to today as analytic geometry.
Indeed, Heath suspects
that Menaechmus first came up with the equations \eqref{eqn:Men}
and \emph{then} discovered
that curves defined by these equations
could be obtained as conic sections \cite[p.~xxi]{Heath-Apollonius}.
Figure~\ref{fig:Men} could appear at the beginning of any analytic geometry text,
as an illustration of what the subject is about.
Pappus \cite[pp.~346--53]{MR13:419a} reports three kinds of geometry problem:
\textbf{plane,} as being solved by means straight lines and circles only,
which lie in a plane;
\textbf{solid,} as requiring also the use of conic sections, which in particular are sections of a solid figure; and
\textbf{linear,} as involving more complicated \emph{lines,} that is, curves,
such as the quadratrix.
Perhaps justly, Descartes criticizes this analysis as simplistic.
He shows that curves given by polynomial equations
have a heirarchy determined by the degrees of the polynomials.
This hierarchy could have been meaningful for Pappus,
since lower-degree curves can be used to construct higher-degree curves
by methods more precise than the construction of the quadratrix.
One solid problem described by Pappus \cite[pp.~486--9]{MR13:419a}
is the four-line locus problem:
find the locus of points such that
the rectangle whose dimensions are the distances to two given straight lines
bears a given ratio to
the rectangle whose dimensions are the distances to two more given straight lines.
According to Pappus, theorems of Apollonius were needed to solve this problem;
but it is not clear whether Pappus thinks
Apollonius actually did work out a full solution.
By the last three propositions, namely 54--6,
of Book III of the \emph{Conics} of Apollonius,
it is implied that the conic sections are three-line loci, that is,
solutions to the four-line locus problem when two of the lines are identical.
Taliaferro \cite[pp.~267--75]{MR1660991} works out the details
and derives the theorem that the conic sections are four-line loci.
Descartes works out a full solution to the four-line locus problem.
He also solves a particular \emph{five}-line locus problem:
the solution is a curve obtained as the intersection of
a sliding parabola
and a straight line through two points,
one fixed,
the other sliding along with the parabola.
Thus Descartes would seem to have made progress along an ancient line of research,
rather than just heading off in a different direction.
As Descartes observes,
Pappus \cite[pp.~600-3]{MR13:419b} could \emph{formulate}
the $2n$-line locus problem for arbitrary $n$.
If $n>3$, the ratio of the product of $n$ segments with the product of $n$ segments
can be understood as the ratio compounded of the respective ratios of segment to segment.
That is, given $2n$ segments $A_1$, \dots, $A_n$, $B_1$, \dots, $B_n$,
we can understand the ratio of the product of the $A_k$ to the product of the $B_k$ as the ratio of $A_1$ to $C_n$, where
\begin{gather*}
A_1:C_1\as A_1:B_1,\\
C_1:C_2\as A_2:B_2,\\
C_2:C_3\as A_3:B_3,\\
\makebox[3cm]{\dotfill},\\
C_{n-1}:C_n\as A_n:B_n.
\end{gather*}
Descartes expresses the solution of the $2n$-line locus problem
as an $n$th-degree polynomial equation in $x$ and $y$, where
$y$ is the distance from the point to one of the given straight lines,
and $x$ is the distance from a given point on that line
to the foot of the perpendicular from the point of the locus.
In fact Descartes does not use the perpendicular as such,
but a straight line drawn at an arbitrarily given angle to the given line.
For, the original $2n$-line problem literally involves
not distances to the given lines,
but lengths of straight lines drawn at given angles to the given lines.
For the methods of Descartes, the distinction is trivial.
For Apollonius, the distinction would seem not to be trivial.
The question remains: If Descartes can express the solution of a locus problem
in terms that would make sense to Apollonius or Pappus,
would the ancient mathematician accept Descartes's \emph{proof,}
a proof that involves algebraic manipulations of symbols?
\chapter{A book from the 1990s}\label{ch:Kar}
In 2006 in Ankara, with two colleagues,
I taught a first-year, first-semester
undergraduate analytic geometry course
from a locally published text that was undated,
but had apparently been produced in 1994 \cite{Karakas}.
The preface of that text begins:
\begin{quotation}
This book is meant as a basic text book for a course in Analytic Geometry.
Throughout the book, the connections and interrelations between algebra and geometry are emphasized. the notions of Linear Algebra are introduced and applied simultaneously with more traditional topics of Analytic Geometry. Some of the notions of Linear Algebra are used without mentioning them explicitly.
\end{quotation}
The preface continues with brief descriptions of the eight chapters and two appendices,
and it concludes with acknowledgements.
The text's Chapter 1, ``Fundamental Principle of Analytic Geometry'', has five sections:
\begin{compactenum}
\item
Set Theory
\item
Relations
\item
Functions
\item
Families of Sets
\item
Fundamental Principle of Analytic Geometry
\end{compactenum}
Thus the book appears more sophisticated than the 1949 book discussed in Chapter~\ref{ch:NFB}.
Possibly this shows the influence of the intervening New Math in the US,
if the text draws on American sources;
but here I am only speculating.
The author's acknowledgements
include no written sources,
and the book has no bibliography.
The introduction to Chapter 1 reads:
\begin{quotation}
Analytic Geometry is a branch of mathematics which studies geometry through the use of algebra. It was \emph{Rene Descartes} (1596--1650) who introduced the subject for the first time. Analytic geometry is based on the observation that there is a one-to-one correspondence between the points of a straight line and the real numbers (see \S5). This fact is used to introduce coordinate systems in the plane or in three space, so that a geometric object can be viewed as a set of pairs of real numbers or as a set of triples of real numbers.
In this chapter, we list notations, review set theoretic notions and give the fundamental principle of analytic geometry.
\end{quotation}
The reference to Descartes is too vague to be meaningful.
Descartes does not \emph{observe,}
but he tacitly \emph{assumes,}
that there is a one-to-one correspondence
between lengths and \emph{positive} numbers.
He assumes too that numbers can be multiplied by one another;
but in case there is any question about this assumption,
he \emph{proves} that this multiplication
is induced by a geometrically meaningful notion.
His proof is discussed further below in Chapter \ref{ch:Hilbert}.
As spelled out on pages 15 and 16 in the book under review,
the \textbf{Fundamental Principle of Analytic Geometry}
is that for every straight line $\ell$
there is a function $P$ from $\R$ to $\ell$ such that:
\begin{compactenum}[a)]
\item
$P(0)\neq P(1)$;
\item
for every positive integer $n$, the points $P(\pm n)$ are $n$ times as far away from $P(0)$ as $P(1)$ is, and are on the same and opposite sides of $P(0)$ respectively;
\item
similarly for the points $P(\pm k)$ and $P(k/n)$, when $k$ is also a positive integer;
\item
if $x