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\begin{document}
\title{Geometry\\
as made rigorous\\
by Euclid and Descartes}
\author{David Pierce}
\date{October 31, 2013}
%\publishers{Mimar Sinan G\"uzel Sanatlar \"Universitesi\\\url{http://mat.msgsu.edu.tr/~dpierce/}}

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\begin{multicols}{2}
{\relscale{0.9}\tableofcontents}  
\end{multicols}


\newpage
\setcounter{section}{-1}
\section{Introduction}

%\pagebreak

According to one \textbf{textbook} of the subject,
\begin{quote}
\textbf{analytic geometry} is based on the idea that a one-to-one correspondence can be established between the set of points of a straight line and the set of all real numbers.
\end{quote}
%The \textbf{idea} is that:
\begin{itemize}
\item
A straight line is an \textbf{ordered abelian group} in a geometrically natural way.

\mbox{}
\hfill
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\hfill
$AO=CB$
\hfill\mbox{}
\item
This ordered group is \textbf{isomorphic} to $(\mathbb R,+,<)$.
\end{itemize}

\pagebreak

The isomorphism from $(\mathbb R,+,<)$ to a straight line induces a \textbf{multiplication} on that straight line.  

This multiplication has a \textbf{geometric meaning.}

\emph{This,} if anything, is the ``Fundamental Principle of Analytic Geometry.''

Descartes establishes it.

Details can be worked out from Book I of Euclid's \emph{Elements.}

\begin{comment}
%Specifically, the product of two \textbf{lengths} corresponds to the \textbf{area} of a rectangle whose sides have those lengths.

The book above tacitly assumes this.

It proves the Distance Formula by an unexplained appeal to the \textbf{Pythagorean Theorem.}

The Pythagorean Theorem and its converse are the last propositions of \textbf{Book I of Euclid's \emph{Elements.}}

Analytic geometry can be made \textbf{rigorous} by means of this book.
\end{comment}

\pagebreak

\section{Origins of geometry}

%\pagebreak

\textbf{\emph{Geometry}} comes from \gk{gewmetr'ia}, 
formed of \gk{g~h} (\emph{land}) and \gk{m'etron} (\emph{measure}).

According to Herodotus (b. \emph{c.}\ 484 \bce),\nocite{Herodotus-Loeb} 
in Egypt, land was taxed in proportion to size.  
If the Nile's annual flooding robbed you of land, the king
%King Sesostris 
sent \textbf{surveyors} to measure the loss.
\begin{quote}
From this, to my thinking, the Greeks learned the \textbf{art of measuring land} (\gk{gewmetr'ih}); the sunclock and the sundial, and the twelve divisions of the day, came to Hellas not from Egypt but from Babylonia. \hfill[2.109]
\end{quote}

\pagebreak

\begin{optional}

Plato (b.\ 427 \bce) in the \emph{Phaedrus}\nocite{Plato-Loeb-I} has Socrates say of the \textbf{Egyptian god Theuth,}
\begin{quote}
He it was who invented
\begin{itemize}
\item
 \textbf{numbers} (\gk{>arijm'os}) and 
 \item
 \textbf{arithmetic} (\gk{logism'os}) and 
 \item
 \textbf{geometry} (\gk{gewmetr'ia}) and 
 \item
 \textbf{astronomy} (\gk{>astronom'ia}), also 
 \item
 draughts and dice, and, most important of all, 
 \item
 \textbf{letters} (\gk{gr'ammata}). \hfill[274c]
 \end{itemize}
\end{quote}

\pagebreak

\end{optional}

According to Aristotle (b.\ 384 \bce),\nocite{Aristotle-XVII}
\begin{comment}

\begin{quote}
In general the sign of \emph{knowing} (\gk{e>id'ws}) 
is the ability to teach (\gk{did'askw}), 
and for this reason we hold that \emph{skill} (\gk{t'eqnh}) 
rather than \emph{experience} (\gk{>empeir'ia}) 
is \emph{science} (\gk{>epist'hmh}).  \hfill[I.i.12]
For the artists can teach, but the others cannot.
Further, we do not consider any of the senses to be \emph{Wisdom} (\gk{SOFIA}).  
They are indeed our chief sources of knowledge about particulars, 
but they do not tell us the reason for anything, 
as for example why fire is hot, but only that it \emph{is} hot.

It is therefore probable that at first the inventor of any art (\gk{t'eqnh})
which went further than the ordinary sensations 
was admired by his fellow-men, 
not merely because some of his inventions were useful, 
but as being a wise and superior person.  
And 

\end{comment}
%after the other \textbf{sciences} (\gk{>epist'hmai}) were invented,
\begin{quote}
as more and more \emph{skills} (\gk{t'eqnai}) were discovered, 
some relating to the \emph{necessities} (\gk{>anagka~ia})
and some to the pastimes of life, 
the inventors of the latter were always considered wiser than those of the former, 
because their \emph{sciences} (\gk{>epist'hmai}) did not aim at utility.  
Hence when all the discoveries of this kind were fully developed, 
the sciences concerning neither \emph{pleasure} (\gk{<hdon'h}) 
nor necessities were invented, 
and first in those places where men \textbf{had leisure} (\gk{sqol'azw}).  

Thus \textbf{mathematics} (\gk{majhmatika'i}) 
originated in Egypt (\gk{A>'iguptos}), because there 
the \emph{priestly class} (\gk{<ier'ewn >'ejnos}) was allowed leisure.
\mbox{} \hfill\mbox{[\emph{Metaphysics} I.i.16]}
\end{quote}
\pagebreak

\section{Euclid's geometry}

%\pagebreak

The \emph{Elements} (\gk{Stoiqe~ia}) of Euclid (fl.~300 \bce)\nocite{MR1932864,bones,MR17:814b,Euclid-Heiberg} begins with
five \textbf{Postulates} (\gk{A>it'hmata} ``Demands'').

By the first four, we have three tools of a builder:
\begin{itemize}
\item
a \textbf{ruler} or \textbf{chalk line,}
(1) to draw a straight line from one point to another, or
(2) to extend a given straight line;
\item
a \textbf{compass,}
(3) to draw a circle with a given center, passing through a given point;
\item
a \textbf{set square,} whose mere existence ensures
(4) that all right angles are equal to one another.
\end{itemize}

\pagebreak

\begin{optional}

Actually these postulates allude to previous \textbf{Definitions} (\gk{<'Oroi} ``Boundaries''):
\begin{quote}
\begin{minipage}{0.6\textwidth}
When a straight line set up on a straight line makes the adjacent angles \emph{equal} (\gk{>'isos}) to one another, each of the equal angles is \textbf{right} (\gk{>orj'os}).
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\begin{quote}
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A \textbf{circle} (\gk{k'uklos}) is\dots contained by one line such that all the straight lines falling upon it from one point [called the \textbf{center} (\gk{k'entron})] are equal to one another.
\end{minipage}
\end{quote}

\pagebreak

\end{optional}

The \textbf{Fifth Postulate} is that, if
\begin{equation*}
\angle\gkm{BHJ}+\angle\gkm{HJD}<2\text{ right angles,}
\end{equation*}
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then \gk{AB} and \gk{GD}, extended, \textbf{meet.}
\begin{itemize}
\item
This is \textbf{unambiguous} by the 4th postulate.
\item
It tells us what the 2nd postulate can \textbf{achieve.}
\end{itemize}

\pagebreak

After the Postulates come the Axioms or \textbf{Common Notions} (\gk{Koina`i >'ennoiai}):
\begin{enumerate}
\item
\textbf{Equals} to the same are equal to one another.
\item
If equals be \textbf{added} to equals, the wholes are equal.
\item
If equals be \textbf{subtracted} from equals, the remainders are equal.
\item
Things \textbf{congruent} with one another are equal to one another.
\item
The whole is \textbf{greater} than the part.
\end{enumerate}
After the Common Notions come the \textbf{48 propositions} of Book I of the \emph{Elements,} and then the remaining 12 books.

\pagebreak

\section{Equality and proportion}

%\pagebreak

\textbf{Equality} in Euclid is:
\begin{itemize}
\item
not identity, by
\begin{compactitem}
\item
 the definitions of the circle and the right angle,
 \item
 the 4th Postulate;
 \end{compactitem}
\item
symmetric (implicitly);
\item
transitive (Common Notion~1);
\item
implied by congruence (C.N.~4);
\item
implied by congruence of respective parts (C.N.~2);
\item
not universal (C.N.~5).
\end{itemize}

\pagebreak

Equality is congruence of parts only in \textbf{Proposition I.35:}  \emph{Parallelograms on the same base and in the same parallels are equal.}
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\pagebreak

Equality is \emph{not} congruence of parts in \textbf{Proposition XII.7:}
\emph{A triangular prism is divided into three equal triangular pyramids.}
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\vfill
This uses \textbf{Proposition XII.5:}
\emph{Triangular pyramids of the same height 
have to one another the \textbf{same ratio} as their bases.}

\pagebreak

By \textbf{Book V,} 
a magnitude $A$ has to $B$ the \textbf{same ratio} (\gk{a>ut'os l'ogos}) 
that $C$ has to $D$
if, for all positive integers $k$ and $n$,
\begin{equation*}
kA>nB\iff kC>nD.
\end{equation*}
Then the four magnitudes are \textbf{proportional} (\gk{>an'alogos}),
and today we write $A:B:\;:C:D$.
The pair
\begin{equation*}
\left(
\left\{\displaystyle\frac nk\colon kA>nB\right\},
\left\{\displaystyle\frac nk\colon kA\leq nB\right\}
\right)
\end{equation*}
is a \textbf{Dedekind cut.}
Thus, for Dedekind (b.~1831), a ratio is a positive real number.

The theory of proportion is said to be due to \textbf{Eudoxus of Knidos} (b.~408 \bce), a student of Plato.

\pagebreak

By \textbf{Propositions V.9} and \textbf{16,}
\begin{center}
if $A:B\as C:C$, then $A=B$.
\end{center}

\begin{proof}
We use the so-called \textbf{Axiom of Archimedes} (b.~287~\bce), found in Euclid's definition of \textbf{having a ratio} (\gk{l'ogon >'eqw}).  Suppose
\begin{equation*}
A>B.
\end{equation*}
Then for some $n$, we have $n(A-B)>B$.  Consequently
\begin{align*}
nA&>(n+1)B,&
nC&<(n+1)C,
\end{align*}
and therefore
\begin{equation*}
A:B>C:C.\qedhere
\end{equation*}
\end{proof}

\pagebreak

\begin{optional}

If the \textbf{Euclidean algorithm,}
\begin{itemize}
\item
applied to two \textbf{numbers,}
\begin{itemize}
\item
yields a \emph{unit,} the numbers are \emph{prime to one another} \textbf{(Proposition VII.1);}
\item
yields a \emph{number,} this is the \emph{greatest common measure} of the original numbers \textbf{(VII.2);}
\end{itemize}
\item
applied to two \textbf{magnitudes,}
\begin{itemize}
\item
never ends, the two magnitudes are \textbf{incommensurable} (\gk{as'ummetros}) \textbf{(X.2);}
\item
yields a magnitude, this is the \emph{greatest common measure} of the original magnitudes \textbf{(X.3).}
\end{itemize}
\end{itemize}

\pagebreak

The Euclidean algorithm is \textbf{to subtract alternately} (\gk{>anjufair'ew}).

\begin{minipage}{0.45\textwidth}
\raggedright
This yields in the diagram
\begin{center}
\begin{tabular}{cc}
 \gk{AB}&\gk{AG}\\
 \gk{AD}&\gk{AZ}\\
 \gk{AJ}& \dots
 \end{tabular}
 \end{center}
and so the diagonal \gk{AB} and side \gk{AG} of the square are \textbf{incommensurable.}
\end{minipage}
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The Euclidean algorithm is a remnant of an \textbf{earlier theory of proportion.}

\pagebreak

According to Aristotle in the \emph{Topics,}
\begin{quote}
It would seem that in mathematics also some things are not easily proved by lack of a \textbf{definition,} such as %[the proposition] 
that the straight line parallel to the side [of the parallelogram] divides \emph{similarly} (\gk{<omo'iows}) both the line and the area.
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But when the definition is stated, what was stated becomes immediately clear.  For the areas and the lines have the same \textbf{antanaeresis} (\gk{>antana'iresic}); and this is the \emph{definition of the same ratio} (\gk{<orism`oc to~u a>uto~u l'ogou}).
\end{quote}

\pagebreak

Alexander of Aphrodisias (fl.~200 \ce)\ comments:
\begin{quote}
For the \emph{definition of proportions} (\gk{<orism`oc t~wn >anal'ogwn}) that the Ancients used is this:  
\begin{center}
Magnitudes that have the same \textbf{anthyphaeresis} (\gk{anjufa'iresis}) are proportional.  
\end{center}
But [Aristotle] has called anthyphaeresis antanaeresis.
\end{quote}
The connection between the Aristotle passage and the Euclidean algorithm was made by Oskar Becker in 1933.\nocite{MR13:419a}

Heath's second edition of the \emph{Elements} is from 1925; his \emph{History of Greek Mathematics,} 1921.\nocite{MR654679,MR654680}

\pagebreak

Anthyphaeresis yields continued fractions:
\begin{align*}
\phantom{\frac{\surd3}1}
	\surd3&=1+(\surd3-1),\\
	\frac1{\surd3-1}=\frac{\surd3+1}2&=1+\frac{\surd3-1}2,\\
	\frac2{\surd3-1}=\surd3+1&=2+(\surd3-1),
	\end{align*}
	and thus 
	$\surd3=1+\cfrac1{1+\cfrac1{2+\cfrac1{1+\displaystyle\frac1{\dots}}}}
	=[1;\overline{1,2}]$.

\pagebreak

Likewise
\begin{align*}
\surd 5&=[2;\overline 4],        &\surd13&=[3;\overline{1,1,1,1,6}],\\
\surd 7&=[2;\overline{1,1,1,4}], &\surd17&=[4;\overline 8],\\
\surd11&=[3;\overline{3,6}],     &\surd19&=[4;\overline{2,1,3,1,2,8}].
\end{align*}
Plato has Theaetetus say,
\begin{quote}
Theodorus was proving to us a certain thing about square roots, I mean the square roots of $3$ square feet and $5$ square feet, namely, that these roots are not commensurable in length with the foot-length, and he proceeded in this way, taking each case in turn up to the root of $17$ square feet; at this point for some reason he stopped.
\end{quote}

\pagebreak

\end{optional}

\section{Some propositions}

\textbf{Proposition I.1} of the \emph{Elements} is
the \emph{problem} of constructing, 
on a given bounded straight line, an \textbf{equilateral triangle.}
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Does this need an \textbf{axiom of continuity?}

\pagebreak

\begin{optional}

Proposition I.1 is a \textbf{problem} as opposed to a \textbf{theorem.}

Writes Pappus  of Alexandria (fl.~320 \ce):
\begin{quote}
Those who wish to make more skilful distinctions in geometry find it worthwhile to call
\begin{itemize}
\item
a \textbf{problem} (\gk{pr'oblhma}) that in which it is \emph{proposed} (\gk{prob'alletai}) to do or construct something,
\item
a \textbf{theorem} (\gk{je'wrhma}) that in which the consequences and necessary implications of certain hypotheses \emph{are investigated} (\gk{jewre~itai}).
\end{itemize}
But among the ancients some described them all as problems, some as theorems.
\end{quote}\nocite{MR13:419b}

\pagebreak

\textbf{Propositions I.2} and \textbf{3} are the problem of \textbf{cutting off} from a given straight line $AB$ a segment equal to a shorter straight line $CD$.
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\uput[-120](-2,-0.866){$F$}
\uput[-60](2,-0.866){$G$}
\uput{10pt}[r](2.207,0.707){$H$}
\end{pspicture}
\end{center}
Thus our compass will \textbf{hold the gap} between its points.

\pagebreak

\section{Hilbert's geometry}

%\pagebreak

In Euclid, transfer of lengths is \textbf{proved.}

So is transfer of angles \textbf{(Proposition I.23).}

These are \textbf{axioms} for David Hilbert (b.~1862) in \emph{The Foundations of Geometry.}

But \textbf{drawing circles} is not.

In Hilbert's system, constructing an equilateral triangle takes a lot of work.

\pagebreak

Hilbert's axioms for the plane:\nocite{MR0116216}

\begin{enumerate}[I.]
\item
Axiom(s) of connection:
%\begin{itemize}
%\item
Two distinct points lie on a unique straight line.
%\end{itemize}
\item
Axioms of order:
\begin{itemize}
\item
The points of a straight line are densely linearly ordered without extrema.
\item[\textbf{Pasch's Axiom:}]
A straight line intersecting one side of a triangle intersects one of the other~two.
\end{itemize}
\item
Axiom of parallels (Euclid's Axiom): 
Through a given point, exactly one parallel to a given straight line can be drawn.
\linebreak
\item
Axioms of congruence.
\begin{itemize}
\item
Every segment can be uniquely \emph{laid off} upon a given side of a given point of a given straight line.
\item
Congruence of segments is transitive and additive.
\item
Every angle can be uniquely \emph{laid off} upon a given side of a given half-ray.
\item
Congruence of angles is transitive.
\item
Side-Angle-Side.
\end{itemize}
\item
Axiom of continuity.  (Archimedean axiom.)
\item[Axiom of Completeness.] 
\end{enumerate}

\pagebreak

\end{optional}

%\section{First theorems}

Side-Angle-Side is an axiom for David Hilbert (b.~1862).

For Euclid it is \textbf{Proposition I.4,}
the first proper \emph{theorem.}  Suppose

\begin{minipage}{0.5\textwidth}
\mbox{}\hfill$\begin{gathered}
	\gkm{AB}=\gkm{DE},\\
	\gkm{AG}=\gkm{DZ},\\
	\angle\gkm{BAG}=\angle\gkm{EDZ}.
\end{gathered}$\hfill\mbox{}
\end{minipage}
\hfill
\begin{minipage}{4cm}
\begin{pspicture}(0,-0.5)(8,2.5)
\pspolygon(0,0)(3,0)(2.5,2.5)
\pspolygon(5,0)(8,0)(7.5,2.5)
\uput[u](2.5,2.5){\gk A}
\uput[d](0,0){\gk B}
\uput[d](3,0){\gk G}
\uput[u](7.5,2.5){\gk D}
\uput[d](5,0){\gk E}
\uput[d](8,0){\gk Z}
%\pscurve(5,0)(6.5,-0.15)(7,0)
\end{pspicture}
\end{minipage}

Then, by the meaning of equality:
\begin{enumerate}
\item
\gk{AB} can be applied exactly to \gk{DE}.
\item
At the same time, $\angle\gkm{BAG}$ can be applied to $\angle\gkm{EDZ}$.
\item
Then \gk{AG} will be applied exactly to \gk{DZ}.
\item
Consequently \gk{BG} will be applied exactly to \gk{EZ}.
\end{enumerate}

\pagebreak

\begin{optional}

\textbf{Proposition I.5} is that the base angles of an isosceles triangle are equal\dots% (as are the angles below the base).

\begin{minipage}{0.48\textwidth}
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\end{pspicture}
\end{minipage}
\hfill
$\begin{aligned}
\because  \qquad\gkm{AB}&=\gkm{AG}\\
                \gkm{AZ}&=\gkm{AH}\\
\therefore\;[\triangle\gkm{AZG}&\cong\triangle\gkm{AHB}]\\
\qquad\gkm{ZG}&=\gkm{HB}\\
         \angle\gkm{AGZ}&=\angle\gkm{ABH}\\
         \angle\gkm{AZG}&=\angle\gkm{AHB}\\
	              \gkm{BZ}&=\gkm{GH}\\
	        [\triangle\gkm{BZG}&\cong\triangle\gkm{GHB}]\\      
%	       \angle\gkm{ZBG}&=\angle\gkm{HGB}\\
	       \angle\gkm{BGZ}&=\angle\gkm{GBH}\\
	       \angle\gkm{ABG}&=\angle\gkm{AGB}
\end{aligned}$
\hfill\mbox{}

\pagebreak

Is Proposition I.5 the world's first theorem?

From \emph{A Commentary on the First Book of Euclid's \emph{Elements,}} by Proclus (of Byzantium, b.\ early 5th c.\ \ce):\nocite{MR1200456}
\begin{quote}
We are endebted to old \textbf{Thales} [of Miletus, b.\ \emph c.\ 624 \bce]\ for the discovery of [Proposition I.5] and many other theorems.  For he, it is said, was the first to notice and assert that in every isosceles triangle the angles at the base are equal, though in somewhat archaic fashion he called the equal angles similar (\gk{<omo'ios}). \qquad\mbox{}\hfill[p.~250]
\end{quote}

\pagebreak

\end{optional}

\textbf{Proposition I.5} is that the base angles of an isosceles triangle are equal. 

Immanuel Kant (b.~1724) alludes to it
in the \emph{Critique of Pure Reason}:\nocite{Kant}
\begin{quote}
\textbf{Mathematics} has from the earliest times\dots travelled the secure path of a science.  Yet it must not be thought that it was as easy for it as for logic\dots to find that royal path\dots its transformation is to be ascribed to a \textbf{revolution,} brought about by the happy inspiration of a single man\dots a new light broke upon the first person who demonstrated [Proposition I.5]
%[that the base angles of] the isosceles triangle [are equal] 
(whether he was called ``Thales'' or had some other name). \qquad \mbox{}\hfill[\textsc b x--xi]
\end{quote}

\pagebreak

\begin{optional}

\section{Analysis and synthesis}

Again Pappus of Alexandria:
\begin{quote}
The so-called \emph{Treasury of Analysis}\dots is\dots 
\begin{comment}

for the use of those who, after going through the usual elements (\gk{stoiqe'ia}), wish to obtain power to solve problems set to them involving curves, and for this purpose only is it useful.  It is 

\end{comment}
the work of three men, Euclid the writer of the \emph{Elements,} Apollonius of Perga, and Aristaeus the elder, and proceeds by the method of \emph{analysis and synthesis.}

Now \textbf{analysis} (\gk{>an'alusis}) is a way of taking that which is sought as though it were admitted and passing from it through its consequences in order to something which is admitted as a result of synthesis\dots 

But in \textbf{synthesis} (\gk{sunj'esis}) we proceed in the opposite way\dots
\end{quote}

\pagebreak

\end{optional}

Euclid's \textbf{Proposition II.6} is a \emph{synthesis}:
\begin{quote}
If a straight line be bisected and a straight line be added to it in a straight line,
the rectangle contained by the whole with the added straight line
together with the square on the half
is equal to the square on the straight line
made up of the half and the added straight line.
\end{quote}
$\begin{gathered}
\because\quad \gkm{AG}=\gkm{GB}\text{ and }
\gkm{GDZE}\text{ is a square,}\\
\therefore\quad	\gkm{AD}\cdot\gkm{DB}+\gkm{GB}^2=\gkm{GD}^2,\\
	(2a+x)\cdot x+a^2=(a+x)^2.
\end{gathered}$
\hfill
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\uput[dr](9,0){\gk Z}
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\uput[dl](4,4){\gk L}
\uput[r](9,4){\gk M}
\end{pspicture}

\pagebreak

\begin{optional}

Proposition II.6 results from an analysis of a special case of \textbf{Proposition VI.29:}
\begin{quote}
To a given straight line [$2a$] to \emph{apply} (\gk{parab'allw}) a [rectangle] equal to a given [square $b^2$] and \textbf{exceeding} (\gk{<uperb'allw}) by a [square].
\end{quote}

We want $x$ such that\hfill
\psset{unit=7.5mm}
\begin{pspicture}(4,-1.5)(9,1.5)
\psline[linewidth=2pt](0,0)(8,0)
\psset{linewidth=1pt}
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\psline(4,1)(4,0)(8,-3)(8,1)
\psarc[linestyle=dashed](4,0){5}{-36.87}{0}
\uput[d](2,0){$a$}
\uput[d](6,0){$a$}
\uput[l](8,-1.5){$b$}
\uput[d](8.5,0){$x$}
\uput[-126](6,-1.5){$c$}
\end{pspicture}
\begin{gather*}
(2a+x)\cdot x=b^2,\\
(2a+x)\cdot x+a^2=a^2+b^2=c^2.	
\end{gather*}
By II.6 it suffices if $a+x=c$.

\pagebreak

\end{optional}

\section{Descartes's geometry}

Euclid's products are \textbf{areas.}
As Descartes (b.~1596) observes, they can be \textbf{lengths,} 
if a unit length is chosen.

\begin{minipage}{0.3\textwidth}
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\begin{pspicture}(0.5,-0.75)(4,3)
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\uput[d](2,0){$A$}
\uput[d](4,0){$B$}
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\uput[d](1,0){$D$}
\uput[ur](1,2.25){$E$}
\end{pspicture}
\end{minipage}
\hfill
\begin{minipage}{0.65\textwidth}
If $AB$ is the unit, and $DE\parallel AC$, then
\begin{equation*}
BE=BD\cdot BC.
\end{equation*}
Thus any number of lengths can be \textbf{multiplied.}
\end{minipage}

Descartes quotes Pappus (fl.~320 \ce)\ as noting that any number of \textbf{ratios} can be multiplied:
\begin{equation*}
A:B\mathbin{\&}B:C\mathbin{\&}\dots\mathbin{\&}Y:Z\as A:Z.
%A_0:A_1\mathbin{\&}A_1:A_2\mathbin{\&}\dots\mathbin{\&}A_{n-1}:A_n\as A_0:A_n.
%A:B\mathbin{\&} B:C\as A:C.
\end{equation*}

\pagebreak

%Descartes thus suggests a \textbf{geometrical justification} of algebra.

%A full justification would require proving multiplication \textbf{associative and commutative.}

As Hilbert shows, multiplication is commutative by a version of \textbf{Pappus's Hexagon Theorem.}

Let $AD\parallel CF$ and $AE\parallel BF$.  Then
\begin{gather*}
ab=ba\iff BD\parallel CE,\\
BD\parallel CE.
\end{gather*}

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\end{pspicture}
\hfill
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\uput[l](0,2){$E$}
\uput[l](0,3){$F$}
\end{pspicture}


\pagebreak

\begin{optional}

\begin{minipage}{0.33\textwidth}
\psset{unit=1cm}
\begin{pspicture}(-0.5,-0.5)(2.5,3.5)
\psline(3,0)(0,0)(0,3)
\psline(1,0)(0,3)
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\psline(0,2)(0.667,0)
\psline(0.667,0)(0,0.667)
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\psline(0,0.667)(1,0)
\psline(3,0)(0,2)
\uput[dl](0.667,0){$A$}
\uput[dr](1,0){$B$}
\uput[d](3,0){$C$}
\uput[l](0,0.667){$D$}
\uput[l](0,2){$E$}
\uput[l](0,3){$F$}
\uput[dl](0,0){$G$}
\end{pspicture}
\end{minipage}
\hfill
\begin{minipage}{0.55\textwidth}
\raggedright
We have assumed $AD\parallel CF$ and $AE\parallel BF$.  By Euclid's \textbf{Proposition VI.2,}
\begin{gather*}
	GA:GC\as GD:GF,\\
	GB:GA\as GF:GE.
\end{gather*}
Then by \textbf{V.23,}
\end{minipage}
\begin{equation*}
GB:GC\as GD:GE,
\end{equation*}
so $CE\parallel BD$.  But Euclid's proof of V.23 relies on \textbf{V.8,}
\begin{equation*}
A>B\implies A:C>B:C,
\end{equation*}
and the proof of this uses the \textbf{Archimedean Axiom.}

\pagebreak

Hilbert avoids the Archimedean Axiom with a lemma:
\vfill
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\end{pspicture}
\end{minipage}
\hfill
\begin{minipage}{0.55\textwidth}
\begin{align*}
        AE&\perp CD,   &&\\
\angle ADB&=90^{\circ},&&\text{\textbf{[III.31]}}\\
\angle ACB&=90^{\circ},&&\\
\angle DAB&=\angle DCB &&\text{\textbf{[III.27]}}\\
          &=\angle EAC,&&\text{\textbf{[I.32]}}\\
\angle DAE&=\angle BAC.&&
\end{align*}
\end{minipage}
\vfill
Thus, if $AB=c$, $\angle BAC=\beta$, and $\angle CAE=\alpha$, then
\begin{equation*}
(c\cos\beta)\cos\alpha=(c\cos\alpha)\cos\beta.
\end{equation*}

\newpage

\end{optional}

Alternatively, we can establish Hilbert's \textbf{algebra of segments} on the basis of Book I of the \emph{Elements} alone.

\begin{minipage}{0.7\textwidth}
\raggedright
Multiplication as in the diagram is commutive, given that:
\begin{itemize}
\item
the rectangles about the diagonal are equal \textbf{(I.43),}
\item
all rectangles of equal dimensions are congruent \textbf{(I.8, 33).}
\end{itemize}
\end{minipage}
\hfill
\begin{minipage}{0.25\textwidth}
\hfill
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\begin{pspicture}(0,-0.6)(3,6.3)
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\uput[l](0,6){$ab$}
\psline[linestyle=dotted](0,0)(3,6)
\end{pspicture}
\end{minipage}

For associativity, we use I.43 and its converse:

\newpage

\begin{minipage}{0.55\textwidth}
By definition of $ab$, $cb$, and $a(cb)$,
\begin{gather*}
	A+B=E+F+H+K,\\
	C=G,\\	
	A=D+E+G+H.
\end{gather*}
Also $a(cb)=c(ab)$ if and only if
\begin{equation*}
C+D+E=K.	
\end{equation*}
We compute
\begin{gather*}
	D+C+B=F+K.
\end{gather*}
We finish by noting
\begin{equation*}
B=E+F.
\end{equation*}
\end{minipage}
\hfill
\begin{minipage}{0.3\textwidth}
\hfill
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\begin{pspicture}(3,6)
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\uput[l](0,1){$cb$}
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\rput(0.75,4.5){$F$}
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\rput(0.25,2.5){$H$}
\rput(0.25,4.5){$K$}
\end{pspicture}
\end{minipage}

\pagebreak

\section{Conclusion}

We thus \textbf{interpret} an ordered field in the Euclidean plane.

The \textbf{positive elements} of this ordered field 
are congruence-classes of line-segments.

We impose a \textbf{rectangular coordinate system} as usual.

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\end{pspicture}
\end{minipage}
\hfill
\begin{minipage}{0.5\textwidth}
Straight lines are now given by \textbf{linear equations:}
\begin{gather*}
	a\cdot(y-b)=-b\cdot x,\\
	bx+ay=ab.
\end{gather*}
\end{minipage}

\pagebreak

Conversely, let an \textbf{ordered field} $K$ be given.

In $K\times K$, obtain the \textbf{Cauchy--Schwartz Inequality,} 
and then the \textbf{Triangle Inequality.}  Define
\begin{itemize}
\item
\textbf{line segments:} $\bm a\bm b$ is the set
\begin{equation*}
\{\bm x\colon\abs{\bm b-\bm a}=\abs{\bm b-\bm x}+\abs{\bm x-\bm a}\};
\end{equation*}
\item
their \textbf{congruence:} $\bm a\bm b\cong\bm c\bm d$ means
\begin{equation*}
\abs{\bm b-\bm a}=\abs{\bm d-\bm c};
\end{equation*}
\item
\textbf{angle congruence:} $\angle\bm b\bm a\bm c\cong\angle\bm e\bm d\bm f$ means
\begin{equation*}
\frac{(\bm c-\bm a)\cdot(\bm b-\bm a)}{\abs{\bm c-\bm a}\cdot\abs{\bm b-\bm a}}=
\frac{(\bm f-\bm d)\cdot(\bm e-\bm d)}{\abs{\bm f-\bm d}\cdot\abs{\bm e-\bm d}}.
\end{equation*}
\end{itemize}

\pagebreak

Alternatively,
define figures to be \textbf{congruent} 
when they can be transformed into one another by a composition of a \textbf{translation}
\begin{equation*}
\bm x\mapsto\bm x+\bm a
\end{equation*}
 and a \textbf{rotation}
 \begin{equation*}
 \bm x\mapsto\begin{pmatrix}a&-b\\b&a\end{pmatrix}\cdot\bm x,
\end{equation*}
where $a^2+b^2=1$.

$K$ should be \textbf{Euclidean} or at least \textbf{Pythagorean.}

\begin{center}
\larger
One just ought to be clear what one is doing.
\end{center}

%\begin{optional}

\pagebreak

\smaller\smaller

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

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

\bibitem{Aristotle-XVII}
Aristotle.
\newblock {\em The Metaphysics, Books I--IX}, volume XVII of {\em Loeb
  Classical Library}.
\newblock Harvard University Press and William Heinemann Ltd., Cambridge,
  Massachusetts, and London, 1980.
\newblock with an English translation by Hugh Tredennick. First printed 1933.

\bibitem{Euclid-Heiberg}
Euclid.
\newblock {\em Euclidis {E}lementa}, volume~I of {\em Euclidis Opera Omnia}.
\newblock Teubner, 1883.
\newblock Edidit et Latine interpretatvs est I. L. Heiberg.

\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{bones}
Euclid.
\newblock {\em The Bones}.
\newblock Green Lion Press, Santa Fe, NM, 2002.
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\newblock Green Lion Press, Santa Fe, NM, 2002.
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\bibitem{MR654679}
Thomas Heath.
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\newblock From Thales to Euclid, Corrected reprint of the 1921 original.

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\newblock From Aristarchus to Diophantus, Corrected reprint of the 1921
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\newblock Harvard University Press, Cambridge, Massachusetts and London,
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\newblock Translation by A. D. Godley; first published 1920; revised, 1926.

\bibitem{MR0116216}
David Hilbert.
\newblock {\em The foundations of geometry}.
\newblock Authorized translation by E. J. Townsend. Reprint edition. The Open
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\bibitem{Kant}
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\newblock {\em Critique of Pure Reason}.
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\newblock Loeb Classical Library. Harvard University Press and William
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\newblock with an English Translation by Harold North Fowler, first printed
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\bibitem{MR1200456}
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\newblock Translated from the Greek and with an introduction and notes by Glenn
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\bibitem{MR13:419a}
Ivor Thomas, editor.
\newblock {\em Selections illustrating the history of {G}reek mathematics.
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\newblock Harvard University Press, Cambridge, Mass., 1951.
\newblock With an English translation by the editor.

\bibitem{MR13:419b}
Ivor Thomas, editor.
\newblock {\em Selections illustrating the history of {G}reek mathematics.
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\newblock Harvard University Press, Cambridge, Mass, 1951.
\newblock With an English translation by the editor.

\end{thebibliography}

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