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\begin{document}
\title{Analytic geometry}
\author{David Pierce}
\date{June 2, 2014\\
corrected \usdate\today, \currenttime}
\publishers{Mathematics Department\\
Mimar Sinan Fine Arts University\\
Istanbul\\
\url{dpierce@msgsu.edu.tr}\\
\url{http://mat.msgsu.edu.tr/~dpierce/}}
\maketitle
\tableofcontents
\listoffigures
\addchap{Introduction}
The writing of this report%%%%%
\footnote{I call this document a report
simply because I have used for it the \LaTeX\ document class called \enquote{report}
(strictly the \textsc{koma}-script class corresponding to this).}
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.
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.\label{stuto}
Students today%%%%%
\footnote{And if it is true for the students,
must it not also be so for their teachers?}
suppose 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 supposition is quite astounding,
since the positive real numbers compose an uncountable set.
Euclidean geometry can in fact be done in a countable space,
as David Hilbert pointed out.
I made notes on some of these matters.
The notes grew into this report
as I found more and more things that were worth saying.
There are many avenues to explore.
Some 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.
Analytic geometry can be built up
either from \enquote{synthetic} geometry
or from an ordered field.
When the chosen foundations are unclear, proof becomes meaningless.
This is illustrated by the example of \enquote{proving analytically}
that the base angles of an isosceles triangle are equal.
\item
Rigor is not an absolute notion,
but must be defined in terms of the audience being addressed.
As modern examples of failures in rigor,
I consider the failure to distinguish between
\begin{compactitem}
\item
the two kinds of completeness
possessed by the ordered field of real numbers;
\item
induction and well ordering
as properties of the natural numbers.
\end{compactitem}
\item
Ancient mathematicians like Euclid and Archimedes
still set the standard for rigor
\begin{compactitem}
\item
in the theory of proportion,
which ultimately made possible
Dedekind\index{Dedekind}'s rigorous definition of the real numbers;
\item
in the justification of infinitesimal methods,
as for example in the proof that circles are to one another
as the squares on their diameters.
\end{compactitem}
\item
\emph{Why} are the Ancients rigorous? I don't know.
But we ourselves 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.
If we don't expect this, we ought to.
\item
There is an old analytic geometry textbook
that I learned something from as a child,
but that I now find mathematically sloppy or extravagant,
for not laying out clearly
\begin{compactitem}
\item
its geometrical assumptions and
\item
and its \enquote{analytic} assumption
of a one-to-one correspondence between the positive real numbers
and the abstractions called \emph{lengths.}
\end{compactitem}
It would be better not to encourage the fantasy
of a universal ruler that can measure every line segment.
This should be, not an assumption, but a theorem,
which can be established by means of the concept of \emph{congruence}
and the \emph{comparability} of any two line segments.
\item
Because the text considered in the previous chapter
uses the technical terms \enquote{abscissa} and \enquote{ordinate}
without explaining their origin,
I provide an explanation of their origins
in the conic sections
as studied by Apollonius.
\item
\begin{compactenum}
\item
How Apollonius himself works out his theorems remains mysterious.
For example,
Descartes's methods do not seem to illuminate
the theorem of Apollonius
that every straight line
that is parallel to the axis of a parabola
is a \emph{diameter} of the parabola
(in the sense of bisecting each chord
that is parallel to the tangent of the parabola
at the vertex of the diameter).
\item
The conic sections may have been discovered by Menaechmus
for the sake of his solution to the problem of duplicating the cube.
The solution can be found, if curves exist with certain properties.
Such curves turn out to exist, in a geometric sense:
they are sections of cones.
\item
Both the ancient geometer Pappus and the modern geometer Descartes
are leery of curves like the quadratrix,
for not having a geometric definition.
\item
Descartes is able to give a geometric description
of a curve given analytically by a cubic equation.
Pappus was mathematically equipped to understand cubic equations
and indeed equations of any degree.
So Descartes did make progress
with a kind of problem that made sense to the Ancients.
\end{compactenum}
\item
I look at an analytic geometry textbook that I once taught from.
It is more sophisticated than the textbook from my childhood
considered in Chapter \ref{ch:NFB}.
This makes its failures of rigor more dangerous for the student.
The book is nominally founded on
the \enquote{Fundamental Principle of Analytic Geometry,}
elsewhere called the \enquote{Cantor--Dedekind\index{Dedekind} Axiom}:
an infinite straight line is,
after choice of a neutral point and a direction,
an ordered group
isomorphic to the ordered group of real numbers.
This principle or axiom is neither sufficient nor necessary
for doing analytic geometry:
\begin{compactitem}
\item
it is true in an arbitrary Riemannian manifold
with no closed geodesics,
\item
analytic geometry can be done over a countable ordered field.
\end{compactitem}
\item
I give Hilbert's axioms for geometry
and note the essential point for analytic geometry:
when an infinite straight line is conceived as an ordered additive group,
then this group can be made into an ordered field
by a geometrically meaningful definition of multiplication.
Descartes, Hilbert, and Hartshorne work this out,
though Descartes omits details
and assumes that the ordered field will be Archimedean.
I work out a definition of multiplication
solely on the basis of Book I of Euclid's \emph{Elements.}
Thus does algebra receive a geometrical justification.
\item
In the other direction,
I review how the algebra of certain ordered fields
can be used to obtain a Euclidean plane.
\item[\ref{app:fields}.]
The example of completeness from Chapter \ref{ch:failures}
is worked out at a more elementary level in the appendix.
\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,
both 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.
For the last three years,
I have been 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.
I am reminded 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.
Then they write down more assertions,
and they end up with something that is supposed to be a proof,
although it has the appearance of a sequence or array or jumble of statements
whose logical interconnections are unclear.%%%%%
\footnote{Unfortunately some established mathematicians
use the same style in their own lectures.}
Euclid does not write this way,
\emph{except} in one small respect.
He begins each of his propositions with a bare assertion.
He does not preface this enunciation or \emph{protasis} (\gk{pr'otasis})
with the word \enquote{theorem} or \enquote{problem,} as we might today
(and as I shall do in this report).
Euclid does not have the typographical means that Heiberg uses,
in his own edition \cite{Euclid-Heiberg} of Euclid,%%%%%
\footnote{Bracketed numbers refer to the bibliography.
Some books there, like Heiberg's, I possess only as electronic files,
obtained from somewhere on the web.
Heiberg uses increases the letter-spacing for Euclid's protases.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
to distinguish the protasis
from the rest of the proposition.
No, the protasis just sits there,
not even preceded by the \enquote{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}\label{ch: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
and 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 of the Common Era,
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 university mathematics department in Istanbul,
students of analytic geometry
have had a semester of calculus,
and a semester of synthetic geometry from its own original source,
namely Book I of Euclid's \emph{Elements} \cite{MR1932864,bones}.
Such students 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,%
\label{geoapp}
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, the following,
which is actually Proposition I.5 of the \emph{Elements.}%%%%%
\footnote{I had a memory that this problem
was assigned in an analytic geometry course
that I once taught with two senior colleagues.
However, I cannot find the problem in my files.
I do find similar problems, such as
\begin{inparaenum}[(1)]
\item
to prove that the line segment bisecting two sides of triangle
is parallel to the third side and is half its length,
or
\item
to prove that, in an isosceles triangle,
the median drawn to the third side is just its perpendicular bisector.
\end{inparaenum}
In each case, the student is explicitly required
to use analytic methods.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{thm:I.5}
The base angles of an isosceles triangle are equal.
\end{theorem}
To prove this,
perhaps students would be expected to come up with 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}
Then $\beta$ and $\gamma$ are given by the equations
\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 we compute
\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,
then 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 at $B$ 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 $\size b/\sqrt{a^2+b^2}$;
and so the angles themselves are equal.
\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 explicitly
as an independent proposition.
For, the equality of angles that it establishes
can be proved and reproved as needed
by the method attributed to Pappus
by Proclus \cite[pp.~249--50]{1873procli}:
\begin{proof}[Proof 5.]
In triangle $ABC$,
if $AB=AC$, then
the triangle is congruent to its mirror image $ACB$
by means of Euclid's Proposition I.4,
the Side-Angle-Side theorem;
in particular, $\angle ABC=\angle ACB$.
\end{proof}
Thus one can see clearly that Theorem \ref{thm:I.5} is true,
without needing to resort to any of the analytic methods
of the first four proofs.
\chapter{Failures of rigor}\label{ch:failures}
The root meaning of the word \enquote{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 kind of questioning might be done.
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,
an \enquote{honors} course that had been quite rigorous,
being 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 \emph{different} mathematics of a later course,
then the two instances of mathematics ought to be equally rigorous.
Otherwise the older student might assume, wrongly,
that the mathematics of the earlier course
could in fact stand up to the same scrutiny
that the mathematics of the later course 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.
I have extracted the foregoing rule
from the examples that I am going to work out in this chapter.
By the standard of rigor that I propose,
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,
I suppose 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 \emph{alternative} definition of limits.
How many calculus teachers,
ignorant of Abraham Robinson's so-called \enquote{nonstandard} analysis
\cite{MR1373196},
will try to give their students some notion of epsilons and deltas,
out of a misguided conception 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 \emph{valued} field has a completion,
and every \emph{ordering} of a field
gives rise to a valuation,
it is possible to suppose wrongly
that every \emph{ordered} field as such has a completion.
This confusion might be due to a lack of rigor in education,
somewhere along the way.%%%%%
\footnote{Here arises the usefulness
of Spivak's final chapter,
\enquote{Uniqueness of the real numbers} \cite[ch.~29]{0458.26001}.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
I spell out the relevant distinctions,
both in the next section and,
in more detail, in Appendix \ref{app:fields} (page \pageref{app:fields}).
\section{Analysis}\label{sect:analysis}
The real numbers can be defined from the rational numbers
either as \emph{Dedekind\index{Dedekind} cuts}
or as equivalence-classes of \emph{Cauchy sequences.}
The former definition
yields $\R$ as the \emph{completion} of $\Q$ as an ordered set.
It so happens that the field structure of $\Q$ extends to $\R$.
This is because $\Q$ is \emph{Archimedean} as an ordered field.
Applied to a \nonarchimedean\ ordered field,
the Dedekind\index{Dedekind} construction still yields a complete ordered \emph{set,}
but not an ordered \emph{field.}
Applied to an arbitrary subfield of $\R$,
the construction yields (a field isomorphic to) $\R$.
Thus $\R$ is unique (up to isomorphism) as a complete ordered field.
(Again, formal definitions of terms can be found in,
or at least inferred from, Appendix \ref{app:fields}.)
By the alternative construction,
$\R$ is $S/I$, where $S$ is the ring of Cauchy sequences of $\Q$,
and $I$ is the maximal ideal comprising the sequences that converge to $0$.
If we replace $\Q$ with a possibly-\nonarchimedean\ ordered field $K$,
we can still define the absolute-value function $\abs{{}\cdot{}}$ on $K$ by
\begin{equation*}
\abs x=\max(x,-x).
\end{equation*}
The Cauchy-sequence construction of $\R$ from $\Q$
can be applied to $K$,
yielding an ordered field $\completion K$
in which every Cauchy sequence converges.
But if $K$ is \nonarchimedean,
then $\completion K$ is not isomorphic to $\R$.
Alternatively, if $K$ is \nonarchimedean,
we may observe that the ring of \emph{finite} elements of $K$
is a \emph{valuation ring} $\vr$ of $K$,
and the maximal ideal $\maxi$ of $\vr$
consists of the \emph{infinitesimal} elements of $K$.
Then the quotient $\units K/\units{\vr}$ is ordered by the rule
\begin{equation*}
a\units{\vr}**mB\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,\label{equality}
but the \emph{identity,} of two ratios.
Equality is a possible property of two nonidentical magnitudes.
Magnitudes are geometrical 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:
\enquote{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} on the proposition
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 \textbf{cut}\label{Ded-cut}
of positive rational numbers
in the sense of Dedekind\index{Dedekind} \cite[p.~13]{MR0159773}.
Dedekind\index{Dedekind} traces his definition of irrational numbers to the idea 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
That an irrational number is to be considered
as fully defined by the specification just described,
this conviction certainly long before the time of Bertrand
was the common property of all mathematicians
who concerned themselves with the irrational\dots
[I]f\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.~39--40]{MR0159773}}
\end{quote}
In saying this, Dedekind\index{Dedekind} intends
to \emph{distinguish} his account of the completeness or continuity
of the real number line from other accounts.
Dedekind\index{Dedekind} does \emph{not} define an irrational number\label{Dedekind-not-def}
as a ratio of two \enquote{measurable quantities}:
the definition of cuts as above
does not require the use of magnitudes such as $A$ and $B$.
Dedekind\index{Dedekind} observes moreover that Euclid's geometrical constructions
do not require continuity of lines.
\enquote{If any one should say,} writes Dedekind\index{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 \enquote{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\index{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,
\enquote{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.
\section{Ratios of circles}
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 in-\linebreak scribed in circles are to one another
as the squares on the diameters.
\end{theorem}
And yet Euclid himself 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_1ut'oc}) base
and in the same parallels
are equal to one another;
Parallelograms that are on \emph{equal} (\gk{>'isoc}) bases
and in the same parallels
are equal to one another.
\end{quote}
Equality here is what we also call \emph{congruence};
and indeed the fourth of the Common Notions
in Heiberg's edition of Euclid can be translated as,
\begin{quote}
Things that \emph{are congruent} (\gk{>efarm'ozw}) to one another
are equal to one another.%%%%%
\footnote{Heath has \enquote{coincide with} in place of \enquote{are congruent to.}}
%%%%%%%%%
\end{quote}
The distinction between identity and congruence
may help to clarify analytic geometry.
\section{Geometry first}
I consider now analytic geometry as presented in an old textbook,
which I possess, only because my mother used it in college:
Nelson, Folley, and Borgman's 1949 volume
\emph{Analytic Geometry} \cite{NFB}.
The Preface (pp.~iii-iv) opens with this paragraph:
\begin{quote}
This text has been prepared for use in an undergraduate course
in analytic geometry which is planned as preparation for the\linebreak
\mbox{calculus} rather than as a study of geometry. In order that it may
be of maximum value to the future student of the calculus, the
basic sciences, and engineering, considerable attention is given to
two important problems of analytic geometry. They are (a) given
the equation of a locus, to draw the curve, or describe it geomet\-%
rically; (b) given the geometric description of a locus, to find its
equation, that is, to translate a verbal description of a locus into
a mathematical equation.
\end{quote}
These \enquote{two important problems}
are why I was interested in this book at the age of 12:
I wanted to understand the curves that could be encoded in equations.
The third paragraph of the Preface is as follows:
\begin{quote}
Inasmuch as the student's ability to use analytic geometry as
a~tool depends largely on his understanding of the coordinate sys\-%
tem, particular attention has been given to producing as thorough
a grasp as possible. He must appreciate, for example, that the
point $(a,b)$ is not necessarily located in the first quadrant, and
that the equation of a curve may be made to take a simple form~if
the coordinate axes are placed with forethought\dots
\end{quote}
By referring to judicious placement of axes,
the authors reveal their working hypothesis
that there is already a geometric plane, before any coordinatization.
It is not clear what students are expected to know about this plane.
\section{The ordered group of directed segments}
The book proper begins on page 3 as follows:
\begin{quotation}
\textbf{1.\ Directed Line Segments.} If $A$, $B$, and $C$ (Fig.\ 1) are three
points which are taken in that order on an infinite straight line,
then in conformity with the principles of plane geometry we may
write
\begin{equation}\tag{1}\label{eqn:ABC}
AB+BC=AC.
\end{equation}
For the purposes of analytic geometry it is convenient to have
equation (1) valid regardless of the order of the points $A$, $B$, and~$C$
on the infinite line.
\begin{figure}[h!]
\relscale{.90}
\centering
\setlength{\unitlength}{1cm}
\begin{picture}(7.2,0.5)
\put(0,0.425){\vector(1,0){7}}
\put(0.5,0.35){\line(0,1){0.15}}
\put(4,0.35){\line(0,1){0.15}}
\put(6,0.35){\line(0,1){0.15}}
\put(0,0){\makebox(1,0.3)[t]{$A$}}
\put(3.5,0){\makebox(1,0.3)[t]{$B$}}
\put(5.5,0){\makebox(1,0.3)[t]{$C$}}
\end{picture}
\\
\textsc{Fig.}~1
\end{figure}
The conventional way of accomplishing this %figure comes here
is to select a positive direction on the line and then define the
symbol $AB$ to mean the number of linear units between $A$ and~$B$,
or the negative of that number, according as we associate with the
segment $AB$ the positive or the negative direction. With this
understanding the segment $AB$ is called a \emph{directed line segment.}
In any given problem such a segment possesses an intrinsic sign
decided in advance through the arbitrary selection of a positive
direction for the infinite line of which the segment is a part%%%%%
\dots\footnote{This quotation has almost exactly the same visual appearance
as in the original text.
The line breaks are the same.
The figure should be placed after
\enquote{The conventional way of accomplishing this.}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{quotation}
Thus it is assumed
that the student knows what a \enquote{number of linear units} means.
I suppose the student
has been trained to believe that\label{train}
\begin{inparaenum}[(1)]
\item
every line segment has a length and
\item
this length is a number of some unit.
\end{inparaenum}
But probably the student has no idea
of how \emph{numbers} in the original sense%
---natural numbers---%
can be used to create all of the numbers that might be needed
to designate geometrical lengths.
The student can express lengths as rational numbers of a unit
by means of a ruler;
but Nelson \etal\ will have the students consider lengths
that are irrational and even transcendental.
Instead of \emph{length,}
we can take \emph{congruence} as the fundamental notion.
Without defining length itself,
we can say that congruent line segments have the \emph{same length.}
Somebody who knows about equivalence relations
can then define a length itself as a congruence class of segments;%%%%%
\footnote{This would seem to be
the idea behind \emph{motivic integration}
as described in the expository article \cite{MR2133307}.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
but this need not be made explicit.
Alternatively,
%and perhaps more in line with the approach of Nelson \etal,
we can fix a unit line segment
in the manner of Descartes in the \emph{Geometry} \cite{Descartes-Geometry}.
Then, by using the definition of proportion
found in Book V of Euclid's \emph{Elements}
and discussed in Chapter~\ref{ch:standard} above,
we can define the length of an arbitrary line segment
as the ratio of this segment to the unit segment.
This gives us lengths rigorously as positive real numbers,
if we use Dedekind\index{Dedekind}'s definition of the latter.
As Dedekind\index{Dedekind} observed though,
and as we repeated (page \pageref{Dedekind-not-def}),
there is no need to assume that \emph{every} positive real number
is the length of some segment.
These details need not be rehearsed with the student.
But neither is it necessary to introduce lengths at all
in order to justify equation \eqref{eqn:ABC}, namely $AB+BC=AC$.
It need only be said
that an expression like $AB$ no longer represents merely a line segment,
but a \emph{directed} line segment.
Then $BA$ is the \emph{negative} of $AB$, and we can write
\begin{align}\tag{2}\label{eqn:BA=-AB}
BA&=-AB,&AB&=-BA,&AB+BA&=0,
\end{align}
as indeed Nelson \etal\ do later in their \S1, on page 4.
\section{Notation}
In its entirety, their \S2 is as follows:
\begin{quotation}
\textbf{2.\ Length, Distance.}
The \emph{length} of a directed line segment~is
the number of linear units which it contains. The symbol $\lvert AB\rvert$
will be used to designate the length of the segment $AB$, or the
\emph{distance between} the points $A$ and $B$.
Occasionally the symbol $AB$ will be used to represent the line
segment as a geometric entity, but if a numerical measure is implied
then it stands for the directed segment $AB$ or the \emph{directed distance
from $A$ to $B$.}
Two directed segments of the same line, or of parallel lines, are
\emph{equal} if they have equal lengths and the same intrinsic signs.
\end{quotation}
Again we see the unexamined assumption
that the reader knows what a \enquote{number of linear units} means.
It would be more rigorous to say
that $\abs{AB}$ is the congruence-class of the segment $AB$;
but I think there is an even better alternative.
In the second paragraph of the quotation,
Nelson \etal\ suggest that the expression $AB$ will usually stand,
not for a segment, but for a \emph{directed} segment.
Then it can \emph{always} so stand,
and the expression $\abs{AB}$ can stand for the undirected segment,
so that $\abs{BA}$ stands for the same thing.
The last paragraph of the quotation can be understood
as corresponding to Common Notion 4 of Euclid quoted above:
equality of directed line segments is just
congruence of undirected segments
that is established by translation only,
without rotation or reflection.
Then an equation like
\begin{equation}\label{eqn:BC=DE}
BC=DE
\end{equation}
means, as in Figure \ref{fig:BCDE}, either
\begin{compactitem}
\item
$BCED$ is a parallelogram,
or
\item
there is a directed segment $FG$
such that $BCFG$ and $DEFG$ are both parallelograms.
\end{compactitem}
\begin{figure}[ht]
\mbox{}\hfill
\begin{pspicture}(0,-0.5)(1.5,1.5)
\pspolygon(0,0)(1,0)(1.5,1)(0.5,1)
\uput[d](0,0){$B$}
\uput[d](1,0){$C$}
\uput[u](0.5,1){$D$}
\uput[u](1.5,1){$E$}
\end{pspicture}
\hfill
\begin{pspicture}(0,-0.5)(2.5,1.5)
\pspolygon(0,0)(1,0)(1.5,1)(0.5,1)
\pspolygon(1.5,0)(2.5,0)(1.5,1)(0.5,1)
\psline(1,0)(1.5,0)
\uput[d](0,0){$B$}
\uput[d](1,0){$C$}
\uput[u](0.5,1){$G$}
\uput[u](1.5,1){$F$}
\uput[d](1.5,0){$D$}
\uput[d](2.5,0){$E$}
\uput[r](1.167,0.333){$H$}
\end{pspicture}
\hfill\mbox{}
\caption{Congruence of directed segments}\label{fig:BCDE}
\end{figure}
Given $BC=DE$,
we can use Euclid's Common Notion 2
(\enquote{If equals be added to equals,
the wholes are equal})%%%%%
\footnote{Actually we use a special case:
If the \emph{same} be added to equals, the wholes are equal.
Equality is implicitly a \emph{reflexive} relation
in the sense that a thing is equal to itself.
The proof of Euclid's Proposition I.35 quoted above
uses this special case:
in the right-hand part of Figure~\ref{fig:BCDE},
we have $BD=BC+CD=CD+DE=CE$, so $BDG=CEF$ (as triangles),
and hence
$BCFG=BDG-CDH+GFH=CEF-CDH+GFH=DEFG$.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
to conclude
\begin{equation*}
AB+BC=AB+DE;
\end{equation*}
then, by applying Common Notion 1
(\enquote{Equals to the same are also equal to one another})
to this and \eqref{eqn:ABC}, we obtain
\begin{equation}\label{eqn:+}
AC=AB+DE.
\end{equation}
Thus sums of directed segments can be defined;
we need not even require them to be segments of the same straight line,
though we may.
More precisely, \emph{congruence} of sums of directed segments can be defined
so that every sum of two directed segments
is congruent to a single directed segment.
Governed by the relations
given by \eqref{eqn:ABC} and \eqref{eqn:BA=-AB},
the congruence classes of directed segments of a given infinite straight line
compose an abelian group.
Although nothing is said in the text of Nelson \etal\
about the commutativity or associativity of addition of segments,
these properties might be understood to follow
from the \enquote{principles of plane geometry}
mentioned as justifying equation \eqref{eqn:ABC}
in the earlier quotation.
Equation \eqref{eqn:ABC} could be understood to hold
for arbitrary directed segments of a plane,
so that congruence classes of these would compose an abelian group.
Evidently Nelson \etal\ do not wish to consider this group,
and that is fine.
There is also no need to talk to students about congruence classes and groups.
All that need be established
is that there is an \enquote{algebra} of directed segments
that resembles the algebra of numbers studied in school.
There is also nothing wrong with confusing directed segments
with their congruence classes.
According to the derivation of \eqref{eqn:+},
the sum of arbitrary directed segments of a straight line
can be \emph{equal to} a directed segment;
we may just say the sum \emph{is} a directed segment.
This is like saying that the integers compose a group of order $n$,
provided equality is understood to be congruence \emph{modulo} $n$.
This example is from Mazur, who observes \cite[p.~223]{MR2452082}:
\begin{quote}
Few mathematical concepts enter our repertoire
in a manner other than ambiguously a \emph{single object}
and at the same time an \emph{equivalence class of objects.}
\end{quote}
If a positive direction is fixed for the straight line containing $A$ and $B$,
then $AB$ itself is understood as positive,
if $B$ is further than $A$ in the positive direction;
otherwise $AB$ is negative.
Thus the abelian group of directed segments of a given straight line
becomes an ordered group.
Where Nelson \etal\ say $\lvert AB\rvert$ means the length of the segment $AB$,
or the distance between $A$ and $B$,
we can understand it to be simply the greater of $AB$ and $-AB$,
in the usual sense of \enquote{absolute value.}
What do we mean by \enquote{directed segment} in the first place?
We could say formally that, as an undirected segment,
$AB$ is just the set $\ell$ of points between $A$ and $B$ inclusive.
Then, as a directed segment,
$AB$ could be understood formally as the ordered pair $(\ell,A)$.
There is an alternative.
Our notation distinguishes the fraction $1/2$
from the ordered pair $(1,2)$ of numbers,
so that we can have $1/2=2/4$, although $(1,2)\neq(2,4)$.
Likewise, since the expression $BC$ is distinct from $(B,C)$,
we can understand $BC$
to denote the equivalence class
consisting of the pairs $(D,E)$
such that the equation \eqref{eqn:BC=DE} holds as defined above.
Then $\abs{BC}$
will be the union of the equivalence classes denoted by $BC$ and $CB$
(assuming all segments are segments of the same infinite straight line).
According to the last-quoted passage of Nelson \etal,
an expression like $AB$ can have any of three meanings.
It can mean
\begin{compactenum}[(i)]
\item\label{item:seg}
the segment bounded by $A$ and $B$,
\item\label{item:dseg}
the directed segment from $A$ to $B$, or
\item\label{item:ddist}
the directed distance from $A$ to $B$.
\end{compactenum}
We can understand the \enquote{directed distance} in \eqref{item:ddist}
to be the appropriate equivalence class of $(A,B)$ just mentioned.
I propose to take this equivalence class as the official meaning of $AB$.
Writing the equivalence class as $AB$ allows us to infer
that one representative of this class
is indeed the directed segment from $A$ to $B$ indicated in \eqref{item:dseg}.
Similarly,
indicating an arbitary rational number $x$ as $a/b$
allows us to infer that the multiple $bx$ is the integer $a$.
The rational number $x$ has no unique numerator and denominator;
but nonetheless we speak of the numerator $a$ and denominator $b$ of $a/b$.
Likewise we can speak of the initial point $A$ and terminal point $B$ of $AB$,
even if, strictly, $AB$ is only an equivalence class.
Finally, $\abs{AB}$ can be understood formally
as the union of the two equivalence classes $AB$ and $BA$.
One representative of this class is the segment indicated in \eqref{item:seg}.
In fact Nelson \etal\ will give yet another possible meaning
to the expression $AB$,
a meaning that will be used without comment in a quotation given below:
$AB$ can mean
\begin{compactenum}[(i)]\setcounter{enumi}{3}
\item
the infinite straight line containing $A$ and $B$.
\end{compactenum}
Meanwhile,
let us note how the text actually uses expressions like $AB$ and $\abs{AB}$.
Here are some examples from the first set of exercises, on page 7:
\begin{quotation}
\textbf{1.}
For Fig.~2 [omitted] verify that $AC+CB+BA=0$.
Show also that $\abs{AC}-\abs{CB}+\abs{BA}=0$.
\textbf{2.}
In Fig.~3 [omitted] let $M$ be the mid-point of the segment $AB$.
Verify that $\frac12(OA+OB)=OM$ and that $\frac12\abs{OA-OB}=\abs{AM}$.
\end{quotation}
The equation $AC+CB+BA=0$ holds by two applications of \eqref{eqn:ABC},
regardless of the relative positions of the points.
If the second equation is going to be true,
then $A$ must lie between $B$ and $C$, so that
\begin{equation*}
\abs{AC}+\abs{BA}=\abs{AC+BA}=\abs{BA+AC}=\abs{BC}=\abs{CB}.
\end{equation*}
Here it does not matter whether $AB$ is a particular directed segment
or an equivalence class.
In later exercises, it does matter:
\begin{quotation}
In Problems 5--8, the consecutive points $A$, $B$, $C$, $D$, $E$, $F$, $G$ are spaced
one inch apart on an infinite line which is positively directed from $A$
to $B$.
\textbf{5.}
Verify that $BD+GA=BA+GD$.
\textbf{6.}
Verify that $DB+GA=DA+GB$.
\textbf{7.}
Verify that $BG+FC=BC+FG=2DE$.
\textbf{8.}
Verify that $\frac12(EA+EG)=ED$.
\end{quotation}
Of the five equations here,
the first three can be understood as equations of directed segments,
as in \eqref{item:dseg};
the remaining two must be understood as equations of directed distances,
as in \eqref{item:ddist}, or of directed equivalence classes of segments.
For example,
\begin{multline*}
BD+GA
=BD+(GD+DB+BA)\\
=BD+GD-BD+BA
=GD+BA
=BA+GD,
\end{multline*}
regardless of the relative positions of the points;
but $BC+FG=2DE$
only because it is given that $BC=DE=FG$.
The preamble to the problems here
refers not just to abstract units,
but to \emph{inches,}
which have no mathematical definition.
The reference might as well have been to some particular segment,
even $AB$ itself.
In another problem,
the authors display their assumption
that numbers of units can be irrational and even transcendental:\label{transc}
\begin{quotation}
\textbf{11.}
On a coordinate axis where the unit of measure is one inch, plot
the points whose coordinates are $2$, $-\frac53$, $\pi$, $3-\sqrt5$, and $\sqrt[3]{-16}$, re\-%
spectively.
\end{quotation}
There is no properly \emph{geometric} reason
to introduce transcendental or even nonquadratic lengths.
However, as noted in the quotation from the preface,
the book is for students of the calculus.
The book has a chapter called
\enquote{Graphs of Single-Valued Transcendental Functions}.
\section{Coordinatization}
Meanwhile,
a coordinatization of a straight line is considered in \S3 of Chapter 1.
\begin{quotation}
\textbf{3.\ Coordinates on a Straight Line.}
The locations of points on a given infinite straight line
may be described with the help of directed segments.
However, preliminary agreements must be made with regard to
(a) a point of reference or \emph{origin,} (b) a unit of length,
(c) a positive direction on the infinite straight line.
Then the location of any point on the line may be given by a single number,
or \emph{coordinate,}
which is defined as the directed distance from the origin to the point\dots
On the line of Fig.\ 4
suppose that the points $P_1$, $P_2$
have the coordinates $x_1$, $x_2$ respectively. Then
\begin{equation*}
P_1P_2=P_1O+OP_2=OP_2-OP_1=x_2-x_1.
\end{equation*}
That is, on any straight line,
\emph{the directed distance from one point to another
is equal to the coordinate of the terminal point
minus the coordinate of the initial point.}
\begin{figure}[ht]
\relscale{.90}
\centering
\setlength{\unitlength}{1cm}
\begin{picture}(7.5,1)
\put(0,0.5){\vector(1,0){7.2}}
\put(2.8,0.425){\line(0,1){0.15}}
\put(5,0.425){\line(0,1){0.15}}
\put(6.5,0.425){\line(0,1){0.15}}
\put(2.3,0.65){\makebox(1,0.3)[b]{$P_1$}}
\put(4.5,0.65){\makebox(1,0.3)[b]{$O$}}
\put(6,0.65){\makebox(1,0.3)[b]{$P_2$}}
\put(2.3,0.05){\makebox(1,0.3)[t]{$x_1$}}
\put(4.5,0.05){\makebox(1,0.3)[t]{$0$}}
\put(6,0.05){\makebox(1,0.3)[t]{$x_2$}}
\put(0.4,0.65){Points}
\put(1.4,0.75){\vector(1,0){0.7}}
\put(0.0,0.05){\makebox(1.7,0.3)[t]{Coordinates}}
\put(1.7,0.2){\vector(1,0){0.4}}
\end{picture}
\\
\textsc{Fig.\ 4}
\end{figure}
\end{quotation}
This all makes sense without reference to units or numbers.
When an origin $O$ is chosen on the straight line,
and points $P_1$ and $P_2$ of the line are labelled as $x_1$ and $x_2$,
this means $x_1$ and $x_2$ are abbreviations for $OP_1$ and $OP_2$,
so that the given equations hold.
Next,
the plane is considered:
\begin{quotation}
\textbf{4.\ Rectangular Coordinates.} The coordinate system of the
preceding article may be generalized
\begin{figure}[ht]
\relscale{.90}
\centering
\begin{pspicture}(-2.5,-2.5)(3,3)
\psline{->}(-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 again 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 that I was expected to learn technical terms
without their etymologies.
In a literature class,
learning \emph{zeugma}
would have been easier,
if only we had recognized
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}.
Such an example requires the use of \emph{multiplication.}
\section{Multiplication}
In the text of Nelson \etal,
multiplication first appears in \S5 of Chapter 1,
in the derivation of the distance formula.
Two points $P_1(x_1,y_1)$ and $P_2(x_2,y_2)$ are given,
orthogonal projections onto the coordinate axes are taken,
and ultimately the point $Q$ is found whose coordinates are $(x_2,y_1)$,
although this is not said.
What we are told is that,
\begin{quote}
Since the angle $P_1QP_2$ is a right angle, it follows that
\begin{equation*}
(P_1P_2)^2=(P_1Q)^2+(QP_2)^2=(x_2-x_1)^2+(y_2-y_1)^2.
\end{equation*}
If $d$ be the unsigned distance between $P_1$ and $P_2$,
we have, by extracting square roots,
\begin{equation*}
\bm{d=\sqrt{\smash{(x_2-x_1)^2+(y_2-y_1)^2}\vphantom{x^2}}.}
\end{equation*}
In words, this formula states that
\emph{the distance between two points
equals the positive square root of
the sum of the squares of the differences in the coordinates of the points.}
\end{quote}
There is no reference to the Pythagorean Theorem
or any other theorem.
Expressions like $(P_1P_2)^2$ are not defined.
Presumably,
since the text has explained an expression like $P_1P_2$
as a number of linear units,
$(P_1P_2)^2$ is supposed to be a number of \enquote{square} units,
the number being the product of the original number with itself.
But $(P_1P_2)^2$ can be understood alternatively
as the equivalence class of squares
whose sides make up the equivalence class $P_1P_2$.
In this case,
the quotation implicitly invokes Euclid's Proposition I.47.
For Euclid,
\begin{quote}
A \emph{number} (\gk{>arijm'oc})
is a \emph{multitude} (\gk{pl~hjoc})
composed of \emph{units} (\gk{mon'ac, mon'ad-}).
\end{quote}
This is the definition in Book VII of the \emph{Elements.}
In this sense, numbers by themselves are elliptical:
\enquote{four} can only mean four things, or four of something.
Today, a number of units has two separable parts:
the number, and the unit.
The number can be multiplied by other numbers,
without regard to any associated units.
This is true for Euclid as well,
but only because his numbers are our positive integers
(usually with the exception of one: one thing is not a multitude).
But for us today, and in particular for Nelson \etal,
a number is a so-called real number,
although the reality of these numbers
would appear not to have been well established
until the work of Dedekind\index{Dedekind}
(mentioned in \S\ref{sect:analysis}, page \pageref{sect:analysis};
described more fully in \S\ref{sect:prop}, page \pageref{sect:prop}).
\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 \enquote{abscissa} and \enquote{ordinate.}
\end{inparaenum}
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,}
it might be easier for the student to ask,
\enquote{What is an upright side?}
In turn, textbook writers might feel more obliged to explain what it is.
In any case, I am going to give an explanation here.
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 the book called \emph{Color Me Beautiful}
as offering \enquote{the color-wheel approach to female pulchritude.}
The \emph{New York Times} just said
the book provided \enquote{beauty tips for women.}
(I draw the quotations from memory;
they were in the newspapers' lists of bestsellers for the week, for many weeks.)
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.
\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.
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 \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: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 (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 $DGE$.
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 \enquote{abscissa} and \enquote{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{\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}
Thus it appears that,
for Descartes, there is still no notion that an arbitrary point
might have two coordinates,
called abscissa and ordinate respectively;
at any rate, he is not interested in inculcating such a notion in his readers.
\chapter{The geometry of the conic sections}\label{ch:conics}
\section{Diameters}
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{compactenum}[(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{compactenum}
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 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, \enquote{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 \enquote{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 \enquote{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.
\section{Duplication of the cube}
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,
\enquote{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 Isidore of Miletus \cite[p.~25]{MR654680},
who along with Anthemius of Tralles
was a master-builder of Justinian's Hagia Sophia \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,
we shall have~\eqref{eqn:Men} as desired if
\begin{compactenum}[(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;
\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 point with coordinates \gk A and \gk E.
\end{compactenum}
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 \cite[pp.~104--5]{Boyer},
\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.%%%%%
\footnote{In fact what Boyer refers to as \enquote{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,
\enquote{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 \enquote{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.
\section{Quadratrix}
Did Descartes have a \enquote{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 $AD$.
If there is a point $H$ where the quadratrix meets the lower edge of the square,
then
\begin{equation*}
BD:AB\as AB:AH,
\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.
\begin{comment}
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}.}
%%%%%%%%%%%%%%%%
\end{comment}
Is \enquote{the general concept of an equation in unknown quantities}
something that is \enquote{alien to Greek thought}?
Perhaps it is alien to our own thought.
According to Boyer as quoted above,
\enquote{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\index{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.
\section{Locus problems}
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, the cone; 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,
namely, given four equally spaced parallel straight lines
and a fifth straight line perpendicular to them,
to find the locus of points,
the product of whose distances to three of the parallel lines
is equal to the product of three other distances:
\begin{inparaenum}[(1)]
\item
to the remaining parallel,
\item
to the fifth line, and
\item
between adjacent parallels.
\end{inparaenum}
Descartes expresses the problem with the equation
\begin{equation*}
(2a-y)(a-y)(y+a)=axy,
\end{equation*}
and he finds the solution as the curve,
each of whose points is the intersection
of a certain parabola and straight line.
The parabola slides,
and the straight line
passes through a fixed point and a point that moves 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 senior 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.
Chapter 1 of the text, \enquote{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 below in Chapter \ref{ch:Hilbert}.
As spelled out on pages 15 and 16 of 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 $x0$
are \textbf{positive.}
Another way to say that an additive group is ordered
is that the set of positive elements is closed under addition.
An \textbf{ordered field}
is a field $K$ whose additive group is so ordered
that its positive elements constitute an ordered group $\pos K$
with respect to multiplication.
Every ordered field may be assumed to include $\Q$.
This is not true for fields in general,
since for example each field $\F_p$, being finite,
cannot include $\Q$.
Moreover, some fields, like $\C$, that do include $\Q$
cannot be ordered.
The field $\Q(X)$ can be ordered
by letting the positive elements
be those $f$ such that $\lim_{x\to\infty}f(x)>0$,
equivalently, the values of $f$ are all positive on some interval $(r,\infty)$.
Every ordered field
has the \textbf{absolute-value} operation $\abs{{}\cdot{}}$,
given by
\begin{equation*}
\abs x=\max\{x,-x\}.
\end{equation*}
An element of an ordered field
that is greater in absolute value than every element of $\Q$
is called \textbf{infinite.}
Non-infinite elements are \textbf{finite.}
The element $0$
and the reciprocals of infinite elements
are \textbf{infinitesimal.}
An ordered field with no infinite elements is \textbf{Archimedean;}
with infinite elements, \textbf{\nonarchimedean.}%%%%%
\footnote{See page \pageref{Arch-ax}
for Archimedes's axiom.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Thus $\R$ is Archimedean,
but the field $\Q(X)$,
ordered as above, is \nonarchimedean,
with $X$ being infinite, and $\inv X$ being a positive infinitesimal.
If $\vr$ is a sub-ring of an arbitrary field $K$,
and the reciprocal of every element of $K\setminus\vr$ belongs to $\vr$,
then $\vr$ is called a \textbf{valuation ring} of $K$,
and $K$ or more precisely $(K,\vr)$ is a \textbf{valued field.}
For example, the finite elements of an ordered field
constitute a valuation ring of the field.
Thus every ordered field \enquote{is} a valued field.
Using the ordering above,
let us denote the ring of finite elements of $\Q(X)$
by $\vr_{\infty}$.
Then
\begin{equation*}
\vr_{\infty}
=\left\{f\in\Q(X)\colon\left|\lim_{x\to\infty}f(x)\right|<\infty\right\}.
\end{equation*}
This equation still gives us a valuation ring
if we replace $\infty$ with an element $a$ of $\Q$, obtaining
\begin{equation*}
\vr_a=\left\{f\in\Q(X)\colon\left|\lim_{x\to a}f(x)\right|<\infty\right\}.
\end{equation*}
If $a\in\Q$, then $\vr_a$ arises from the ordering of $\Q(X)$
according to which the positive elements
are those that, as functions,
are positive on some interval $(a,a+\delta)$.
The valuation rings $\vr_{\infty}$ and $\vr_a$ of $\Q(X)$ that we have defined
can be defined without use of the ordering of $\Q$.
Indeed, we can replace $\Q$ with an arbitrary field $K$.
We extend the field operations on $K$ partially to $K\cup\{\infty\}$
by defining
\begin{gather*}
a\neq\infty\implies a\pm\infty=\infty\And\frac a{\infty}=0,\\
a\neq0\implies a\cdot\infty=\infty\And\frac a0=\infty.
\end{gather*}
We leave $\infty\pm\infty$, $\infty/\infty$, $0\cdot\infty$,
and $0/0$ undefined.
However,
for every $f$ in $K(X)$ and every $a$ in $K\cup\{\infty\}$,
there is a well-defined element $f(a)$ of $K\cup\{\infty\}$,
and so we can define
\begin{equation*}
\vr_a=\{f\in K(X)\colon f(a)\neq\infty\}.
\end{equation*}
This is a valuation ring of $K(X)$,
regardless of whether $K$ has an ordering.
A \textbf{unit} of a ring $R$
is a nonzero element whose reciprocal also belongs to $R$.
Then the units of $R$ compose a multiplicative group,
denoted by $\units R$.
For example, $\units{\Z}=\{1,-1\}$,
but if $K$ is a field, then $\units K=K\setminus\{0\}$.
An additive subgroup $I$ of $R$ is an \textbf{ideal} of $R$
if every product of an element of $I$ by an element of $R$ is in $I$,
but $I$ is not all of $R$.
In this case $R/I$ is also a well-defined ring.
Let $(K,\vr)$ be a valued field.
One shows that $\vr\setminus\units{\vr}$ is an ideal of $\vr$.
Then it must be a maximal ideal of $\vr$
and indeed the only maximal ideal of $\vr$;%%%%%
\footnote{Therefore $\vr$ is called a \emph{local ring;}
but not every local ring is a valuation ring.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
let us denote it by $\maxi$.
The multiplicative quotient $\units K/\units{\vr}$
becomes an \emph{ordered} group
by the rule whereby the \enquote{negative} elements---the elements
less than $1$---are just those cosets $a\units{\vr}$ such that $a\in\maxi$.
We define
\begin{equation*}
\val x=
\begin{cases}
x\units{\vr},&\text{ if }x\in\units K,\\
0,&\text{ if }x=0,
\end{cases}
\end{equation*}
with $0$ being understood
as the least element of $\{0\}\cup(\units K/\units{\vr})$.
Then the function $\val{{}\cdot{}}$ is a \textbf{valuation} of $K$,
and $\vr$ can be recovered from this
as $\{x\in K\colon\val x\leq1\}$.
The absolute-value function on an Archimedean ordered field
may be called an \textbf{Archimedean valuation,}
and then a valuation in the earlier sense
is called a \textbf{\nonarchimedean\ valuation.}
Any field equipped with a valuation,
be it Archimedean or not,
can be called a \textbf{valued field.}
However, we shall see an important ambiguity in this terminology.
From $\Q$, we can obtain the field $\R$ of real numbers
in (at least) two ways:
\begin{asparaenum}
\item
Let $\upomega$ be the set $\{0,1,2,\dots\}$ of natural numbers.
(See also page \pageref{omega}.)
We define a \textbf{Cauchy sequence} in $\Q$ as usual in calculus:
it is a sequence $(a_n\colon n\in\upomega)$ of rational numbers
such that for every positive rational number $\epsilon$,
there is a natural number $k$ such that,
for all natural numbers $m$ and $n$,
\begin{equation*}
k\leq m\leq n\implies\abs{a_m-a_n}<\epsilon.
\end{equation*}
The Cauchy sequences in $\Q$ compose a ring $S$,
and the sequences that converge to $0$
compose a maximal ideal $\maxi[]$ of $S$.
We define $\R$ as the quotient $S/\maxi[]$.
This is an ordered field,
and its every Cauchy sequence of converges,
and so it is said to be \textbf{complete}
as a valued field.
Also $\Q$ embeds densely in $\R$
as the set of cosets of constant sequences;
so $\R$ is a \textbf{completion} of $\Q$.
\item
Alternatively,
instead of Cauchy sequences,
we start with the notion of a \textbf{Dedekind\index{Dedekind} cut} of $\Q$,
namely a pair $(A,B)$,
where
\begin{compactenum}
\item
$A$ and $B$ are nonempty disjoint subsets of $\Q$,
\item
every rational number belongs to one of $A$ and $B$,
\item
every number in $A$ is less than every number in $B$,
\item
$A$ contains no greatest number.
\end{compactenum}
(See also page \pageref{Ded-cut}.)
It will be simpler to think of the set $A$ by itself as the cut,
since $B$ can be recovered from $A$ as $\Q\setminus A$.
We define $\R$ as the set of cuts of $\Q$.
Each rational number $a$ can be identified with the cut (in the second sense)
comprising the rational numbers that are less than $a$.
Then $\R$ is ordered by inclusion,
%the embedding of $\Q$ is dense,
and every subset of $\R$ with an upper bound
has a supremum, namely the union of the elements of the subset;
so $\R$ is said to be \textbf{complete}
as a linearly ordered set.
Moreover, addition and multiplication on $\R$ can be defined in a natural way.
Then these operations are continuous
by the usual definition from calculus,
and so the usual properties follow,
making $\R$ a \textbf{complete ordered field.}
\end{asparaenum}
The two methods of completing $\Q$ yield isomorphic results,
but should still be distinguished,
for the following reasons.
Dedekind\index{Dedekind}'s construction can be applied to an arbitrary
linearly ordered set.
In particular, it can be applied to an arbitrary ordered field.
when the ordering is Archimedean,
then the resulting completion is also a field, isomorphic to $\R$;
but if the ordering is \nonarchimedean,
then the completion is not a field
or even an additive group.
The Cauchy-sequence construction
can also be applied to an arbitrary ordered field,
but now it always yields an ordered field whose Cauchy sequences converge.
If the original ordering is Archimedean,
again the result is a field isomorphic to $\R$.
But suppose we start with $\Q(X)$,
with the \nonarchimedean\ ordering described above,
where $f>0$ means $\lim_{x\to\infty}f(x)>0$.
Cauchy sequences of $\Q$ are not Cauchy sequences of $\Q(X)$
unless they are eventually constant,
since no positive rational number
is less than the positive infinitesimal $\inv X$.
However,
for \emph{every} sequence $(a_n\colon n\in\upomega)$ of rational numbers,
the sequence
\begin{equation*}
(a_0+a_1X^{-1}+\dots+a_nX^{-n}\colon n\in\upomega)
\end{equation*}
of polynomials is a Cauchy sequence of $\Q(X)$.
The quotient of the ring of these sequences
by the ideal of sequences that converge to $0$
is the field $\Q((X^{-1}))$,
consisting of power series
\begin{equation*}
a_0X^m+a_1X^{m-1}+a_2X^{m-2}+\cdots,
\end{equation*}
where the coefficients $a_k$ are in $\Q$, and $m$ ranges over $\Z$.
In the construction of $\Q((\inv X))$,
the role of the absolute-value function $\abs{{}\cdot{}}$
could have been played by the valuation $\val[\vr_{\infty}]{{}\cdot{}}$,
with no effect on the result.
The same is true for an arbitrary \nonarchimedean\ ordered field;
whether we complete it by using $\abs{{}\cdot{}}$
or $\val[\vr_{\infty}]{{}\cdot{}}$ makes no difference.
Using $\val[\vr_{\infty}]{{}\cdot{}}$ relies not on its origin in an ordering,
but only on its being a valuation.
Thus we can obtain a completion of an arbitrary valued field $(K,\vr)$.
For the commonly seen valued fields $(K,\vr)$,
the \textbf{value group} $\units K/\units{\vr}$ embeds in $\pos{\R}$
and is therefore \textbf{Archimedean,}
even though the valuation $\val{{}\cdot{}}$ itself is called \nonarchimedean.
But \nonarchimedean\ value groups are possible.
What is \enquote{worse,}
possibly there is no sequence $(\epsilon_k\colon k\in\upomega\}$
of values in $\units K/\units{\vr}$ that converges to $0$.
In this case, every Cauchy sequence of $K$ is eventually constant,
and in particular it converges.
We may then wish to consider sequences $(a_{\alpha}\colon\alpha<\kappa)$,
where $\kappa$ is the least of the infinite cardinals $\lambda$
such that some sequences of length $\lambda$
in the value group do converge to $0$.
The point for now is that the notion of completeness for ordered fields
is ambiguous.
Dedekind\index{Dedekind} himself describes his construction as achieving
the \emph{continuity} of $\R$ as an ordered field \cite{MR0159773};
this is echoed in the term \enquote{continuum.}
Then $\R$ is unique as a continuous ordered field or continuum,
but not as a complete valued field.
%\bibliographystyle{plain}
%\bibliography{../references}
\def\rasp{\leavevmode\raise.45ex\hbox{$\rhook$}} \def\cprime{$'$}
\def\cprime{$'$} \def\cprime{$'$} \def\cprime{$'$}
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\printindex
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