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\begin{document}
\title{Geometry as made rigorous\\
by Euclid and Descartes}
\subtitle{Seminar at Mimar Sinan}
\author{David Pierce}
\date{October 31, 2013}
\maketitle
\begin{abstract}
\noindent
\emph{Abstract:}
For Immanuel Kant (born 1724), the discovery of mathematical proof by Thales of Miletus (born around 624 \bce) is a revolution in human thought.
Modern textbooks of analytic geometry 
often seem to represent a return to prerevolutionary times.  
The counterrevolution is attributed to Ren{\'{e}} Descartes (born 1596).
But Descartes understands ancient Greek geometry and adds to it.  
He makes algebra rigorous by interpreting its operations geometrically.

The definition of the real numbers by Richard Dedekind (born 1831) 
makes a rigorous converse possible.  
David Hilbert (born 1862) spells it out: 
geometry can be interpreted in the ordered field of real numbers, and even in certain countable ordered fields.

In modern textbooks, these ideas are often entangled, 
making the notion of proof practically meaningless.
I propose to disentangle the ideas 
by means of Book I of Euclid's \emph{Elements} and Descartes's \emph{Geometry.}
\end{abstract}

\listoffigures

\section*{Mathematicians and commentators}

\begin{center}
\begin{tabular}{lrr}
Thales of Miletus&b.&\emph c.~624\\
Herodotus&b.&\emph c.~484\\
Eudoxus &b.&408\\
Aristotle&b.&384\\
Menaechmus&b.&380\\
Euclid&fl.&300\\
Archimedes&b.&287\\
Apollonius&b.&262\\\hline
Pappus&fl.&320\\
Proclus&b.&412\\
Eutocius&fl.&500\\
Isidore of Miletus&fl.&532--7\\
Ren\'e Descartes&b.&1596\\
Immanuel Kant&b.&1724\\
Richard Dedekind&b.&1831\\
David Hilbert&b.&1862
\end{tabular}
\end{center}

\section*{Introduction}

\textbf{Rigor} in mathematics is \emph{ability to stand up under questioning.}

Rigor in \textbf{education} has an extra component: 
\emph{teaching what questions should be asked.}

This talk is inspired or rather provoked by \textbf{two books} of analytic geometry
that fail to be rigorous.

One is an old book \cite{NFB} used by my mother in college.
When \textbf{young} I used this book in order to sketch the graphs
of conic sections and of trigonometric and logarithmic functions.

But this book is not a book that one can sit and read for \textbf{pleasure.}
I think Spivak's \emph{Calculus} \cite{0458.26001} is such a book.
But the analytic geometry book begins with uninspiring exercises about coordinates,
with no motivation.

Possible motivation can be found in the problem of \textbf{duplicating the cube,}
as solved by Menaechmus.

The problem is to find two \textbf{mean proportionals} to a unit length and its double.
In modern symbolic terms, this is to solve the system
\begin{equation*}
\frac1x=\frac xy=\frac y2.
\end{equation*}
From this system we obtain
\begin{equation*}
\frac1{x^3}=\frac12.
\end{equation*}
Geometrically then, $x$ is the side of a cube that has twice the volume of the unit cube.

Many solutions of this problem are reported by Eutocius (flourished around 500 \ce)\
in his commentary \cite{MR2093668} on Archimedes.
This commentary has been revised by Isidore of Miletus,
who, with Anthemius of Tralles,
is one of the master-builders of the \textbf{Ayasofya.}
Menaechmus was a student of Eudoxus of Knidos and a contemporary of Plato \cite{Heath-Apollonius}.
Eudoxus invented the theory of proportion
found in Euclid's \emph{Elements} \cite{MR1932864,MR17:814b};
we shall talk about this later.

Menaechmus's solution to the problem above%
---in fact one of his two solutions---%
can be understood as follows.
We obtain two equations
\begin{align*}
xy&=2,& 2x&=y^2.
\end{align*}
It is known that these are the equations of certain \textbf{conic sections,}
which Apollonius \cite{MR1660991} would later call the hyperbola and the parabola.
The point is that Menaechmus knows that the curves really can
be obtained by slicing a cone.
The hyperbola can be given asymptotes as in 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}
Then the axis of the parabola will be the horizontal asymptote of the hyperbola.
The \textbf{coordinates} of the intersection of the two conic sections
solve the original problem.

Was Menaechmus doing analytic geometry as we understand it?
Perhaps not.
Today we would just calculate the solution to the original system as
\begin{equation*}
(x,y)=(\sqrt[3]2,\sqrt[3]4).
\end{equation*}
But a point with these coordinates cannot be found 
with the usual tools of straightedge and compass.
Menaechmus gives us reason to believe that this point \textbf{exists anyway.}
The reason he gives is geometric.

Two thousand years later,
Ren\'e Descartes \cite{Descartes-Geometry} seems to share the view that
solutions to equations should be \textbf{understood geometrically.}
For example, in Figure~\ref{fig:Des},
\begin{figure}[ht]
\centering
\psset{unit=5mm}
\begin{pspicture}*(-5,-2)(3,7)
\psline(-4,-2)(-4,7)
\psline(-2,-2)(-2,7)
\psline(0,-2)(0,7)
\psline(2,-2)(2,7)
\psline(-5,0)(3,0)
\uput[dr](0,0){$A$}
\uput[ur](0,2.536){$B$}
\psdots(-1.464,2.536)
\uput[ur](-1.464,2.536){$C$}
\uput[ul](-2,2.536){$D$}
\uput[dl](-2,0){$E$}
\uput[ul](-4,2.536){$F$}
\uput[dl](-4,0){$G$}
\uput[ur](2,2.536){$H$}
\uput[dr](2,0){$I$}
\uput[dr](-1.464,0){$M$}
\uput[r](-1.464,1.268){$x$}
\uput[d](-0.732,2.536){$y$}
\psset{linestyle=dashed}
\psline(-4,2.536)(2,2.536)
\psline(-1.464,0)(-1.464,2.536)
\end{pspicture}
\caption{Descartes's locus problem}\label{fig:Des}
\end{figure}
assuming $GE=EA=AI=a$,
suppose we want the locus of points $C$ such that
\begin{equation*}
CF\cdot CD\cdot CH=CB\cdot CM\cdot AI,
\end{equation*}
that is,
\begin{equation*}
(2a-y)(a-y)(a+y)=yxa.
\end{equation*}
Given any value of $y$, we can compute $x$ and thus sketch the curve as in Figure~\ref{fig:locus}.
\begin{figure}[ht]
\centering
\psset{unit=5mm}
\begin{pspicture}*(-5,-2)(3,7)
\psline(-4,-2)(-4,7)
\psline(-2,-2)(-2,7)
\psline(0,-2)(0,7)
\psline(2,-2)(2,7)
\psline(-5,0)(3,0)
\psset{linestyle=dashed}
\psplot{-5}{-0.1}{x 4 add x x mul 4 sub mul 2 div x div}
\psplot{0.1}{3}{x 4 add x x mul 4 sub mul 2 div x div}
\end{pspicture}
\caption{The locus itself}\label{fig:locus}
\end{figure}
But Descartes finds it worthwhile to do more.
He shows that the point $C$ lies on the intersection, shown in Figure~\ref{fig:sol},
\begin{figure}[ht]
\centering
\psset{unit=5mm}
\begin{pspicture}*(-5,-2)(3,7)
\psline(-4,-2)(-4,7)
\psline(-2,-2)(-2,7)
\psline(0,-2)(0,7)
\psline(2,-2)(2,7)
\psline(-5,0)(3,0)
\psset{linestyle=dashed}
\psplot{-5}{3}{x x mul 4 div 2 add}
\psplot{-4}{0}{x 4 add}
\uput[dr](0,2){$K$}
\uput[ur](0,4){$L$}
\uput[dl](-4,0){$G$}
\end{pspicture}
\caption{Descartes's geometrical solution}\label{fig:sol}
\end{figure}
 of:
\begin{compactitem}
\item
a parabola with axis $AB$ and \emph{latus rectum} $a$ 
whose vertex $K$ slides along $AB$,
\item
the straight line through $GL$, where $KL=a$.
\end{compactitem}
Thus the curve given by the cubic equation above becomes geometrically meaningful.

Again, we think the problem of duplicating the cube 
is solved simply by taking the cube root of $2$.
But how is this taken?
There is an algorithm for finding \textbf{decimal approximations.}
But why do we think these approximations have a limit?
We can just declare that $\sqrt[3]2$ is some infinite decimal expansion.
But why do we think that infinite decimal expansions like this compose a field?

Richard Dedekind \cite{MR0159773} claims that,
before he gave a rigorous definition of the rational numbers,
the theorem
\begin{equation*}
\sqrt2\cdot\sqrt3=\sqrt6
\end{equation*}
had not been proved.
David Fowler (author of \emph{The Mathematics of Plato's Academy} \cite{MR2000c:01004})
seems to be correct that Dedekind is correct.

There is \textbf{no algorithm} for computing with infinite decimals.
For example, what is the following sum?
\begin{equation*}
3.1415926535\dots{}+0.8584073464\dots
\end{equation*}
It is either $3.9\dots$ or $4.0\dots$, 
but we cannot specify a number of digits that are sufficient to tell us which.
Fowler gives the example
\begin{equation*}
1.222\dots{}\times0.818181\dots
\end{equation*}
which is
\begin{gather*}
	\left(1+\frac29\right)\times\frac{81}{99}=\frac{11}9\times\frac{81}{99}=1;
\end{gather*}
but no amount of multiplying finite decimal approximations 
tells us that the product is not required to begin as $0.9$.

Dedekind's definition of the real numbers 
explicitly avoids making use of geometric notions.
Therefore we can use the set of ordered pairs of real numbers 
as a \emph{model} for geometric axioms,
thus showing that these axioms are consistent.
David Hilbert \cite{MR0116216} does this.

Conversely, the Euclidean plane can be used to turn a straight line
into a model of axioms for an ordered field.
Descartes suggests this.
David Hilbert\cite{MR0116216} fills in the missing details.
More recently, Robin Hartshorne \cite{MR1761093} does the same,
using theorems about circles from Book III of Euclid's \emph{Elements.}
In fact Book I of the \emph{Elements} is enough.

Thus there are two complementary approaches to analytic geometry.
Either geometry or algebra can be taken as fundamental.
But textbooks assume both of these foundations.
I think this is a defect of rigor.

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

\def\rasp{\leavevmode\raise.45ex\hbox{$\rhook$}} \def\cprime{$'$}
  \def\cprime{$'$} \def\cprime{$'$} \def\cprime{$'$}
\begin{thebibliography}{10}

\bibitem{Heath-Apollonius}
{Apollonius of Perga}.
\newblock {\em Apollonius of {P}erga: Treatise on Conic Sections}.
\newblock University Press, Cambridge, UK, 1896.
\newblock Edited by {T}. {L}. {H}eath in modern notation, with introductions
  including an essay on the earlier history of the subject.

\bibitem{MR1660991}
{Apollonius of Perga}.
\newblock {\em Conics. {B}ooks {I}--{III}}.
\newblock Green Lion Press, Santa Fe, NM, revised edition, 1998.
\newblock Translated and with a note and an appendix by R. Catesby Taliaferro,
  With a preface by Dana Densmore and William H. Donahue, an introduction by
  Harvey Flaumenhaft, and diagrams by Donahue, Edited by Densmore.

\bibitem{MR2093668}
Archimedes.
\newblock {\em The two books \emph{{O}n the sphere and the cylinder}}, volume~I
  of {\em The works of {A}rchimedes}.
\newblock Cambridge University Press, Cambridge, 2004.
\newblock Translated into English, together with Eutocius' commentaries, with
  commentary, and critical edition of the diagrams, by Reviel Netz.

\bibitem{MR0159773}
Richard Dedekind.
\newblock {\em Essays on the theory of numbers. {I}: {C}ontinuity and
  irrational numbers. {II}: {T}he nature and meaning of numbers}.
\newblock authorized translation by Wooster Woodruff Beman. Dover Publications
  Inc., New York, 1963.

\bibitem{Descartes-Geometry}
Ren{\'e} Descartes.
\newblock {\em The Geometry of {R}en{\'e} {D}escartes}.
\newblock Dover Publications, Inc., New York, 1954.
\newblock Translated from the French and Latin by David Eugene Smith and Marcia
  L. Latham, with a facsimile of the first edition of 1637.

\bibitem{MR17:814b}
Euclid.
\newblock {\em The thirteen books of {E}uclid's {E}lements translated from the
  text of {H}eiberg. {V}ol. {I}: {I}ntroduction and {B}ooks {I}, {I}{I}. {V}ol.
  {I}{I}: {B}ooks {I}{I}{I}--{I}{X}. {V}ol. {I}{I}{I}: {B}ooks
  {X}--{X}{I}{I}{I} and {A}ppendix}.
\newblock Dover Publications Inc., New York, 1956.
\newblock Translated with introduction and commentary by Thomas L. Heath, 2nd
  ed.

\bibitem{MR1932864}
Euclid.
\newblock {\em Euclid's {E}lements}.
\newblock Green Lion Press, Santa Fe, NM, 2002.
\newblock All thirteen books complete in one volume. The Thomas L. Heath
  translation, edited by Dana Densmore.

\bibitem{MR2000c:01004}
David Fowler.
\newblock {\em The mathematics of {P}lato's academy}.
\newblock Clarendon Press, Oxford, second edition, 1999.
\newblock A new reconstruction.

\bibitem{MR1761093}
Robin Hartshorne.
\newblock {\em Geometry: {E}uclid and beyond}.
\newblock Undergraduate Texts in Mathematics. Springer-Verlag, New York, 2000.

\bibitem{MR0116216}
David Hilbert.
\newblock {\em The foundations of geometry}.
\newblock Authorized translation by E. J. Townsend. Reprint edition. The Open
  Court Publishing Co., La Salle, Ill., 1959.
\newblock Project Gutenberg edition released December 23, 2005
  (\url{www.gutenberg.net}).

\bibitem{NFB}
Alfred~L. Nelson, Karl~W. Folley, and William~M. Borgman.
\newblock {\em Analytic Geometry}.
\newblock The Ronald Press Company, New York, 1949.

\bibitem{0458.26001}
Michael Spivak.
\newblock {\em {Calculus. 2nd ed.}}
\newblock {Berkeley, California: Publish or Perish, Inc. XIII, 647 pp.}, 1980.

\end{thebibliography}




\end{document}
