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\begin{document}
\title{Ratio Then and Now}
\titlehead{Notes for a talk at
  ``Model Theory and Mathematical Logic:
  Conference in honor of Chris Laskowski's 60th birthday,''
University of Maryland, College Park}
\author{David Pierce}
\date{June 21, 2019\\
Edited July 2, 2019}
\publishers{Matematik B\"ol\"um\"u\\
Mimar Sinan G\"uzel Sanatlar \"Universitesi\\
\url{mat.msgsu.edu.tr/~dpierce/}\\
\url{polytropy.com}}
\maketitle

\chapter{Preface}

So that they might serve as a reference
for my talk a week later in College Park.
I prepared and printed these notes before leaving Istanbul on June 14, 2019.
I assembled them,
in part from earlier notes and a published article:
\begin{compactitem}\sloppy
\item
  \enquote{Euclid Mathematically and Historically,}
  from a 50-minute colloquium talk
  in the mathematics department of Bilkent University,
  Ankara, March 7, 2018;
\item
  \enquote{Conic Sections With and Without Algebra,}
  from a 20-minute contributed talk at Antalya Algebra Days,
  Nesin Mathematics Village, May 15, 2019;
\item
  \enquote{Affine Geometry,} filename \url{affine.tex};
\item
  \enquote{Thales and the Nine-point Conic} \cite{Pierce-Thales-9}.
\end{compactitem}
While in the US,
by hand I wrote out five A5-size pages of notes
that I might actually write on the boards.
In the process, I detected some mistakes,
now corrected, in the present notes.
After the talk,
I prepared a separate typeset document
containing notes of what I said and might have said.

\tableofcontents
\listoffigures

\chapter{Abstract}

Having submitted this as plain text,
I typeset it here as such.

\begin{ttfamily}\raggedright
  \setlength{\parskip}{\baselineskip}
  ``Heraclitus holds that the findings of sense-experience are untrustworthy, and he sets up reason [logos, ratio] as the criterion'' (Sextus Empiricus)

  ``It is necessary to know that war is common and right is strife [eris] and all things happen by strife and necessity''  (Heraclitus, according to Origen)

  1.  Strife has arisen between the historian of mathematics and the mathematician who thinks about the past.  One must be both, to understand Euclid's obscure definition of proportion of numbers.  Proportion is sameness of ratio.  When this occurs between two pairs of numbers, something should be the same about each pair.  In Book VII of the Elements, this can only mean that the Euclidean Algorithm has the same steps when applied to either pair of numbers.  From this, despite modern suggestions to the contrary, Euclid has rigorous proofs, not only of what we call Euclid's Lemma, but also of the commutativity of multiplication.

  2.  Apollonius of Perga gives three ways to characterize a conic section: (i) an equation, involving a latus rectum, that we can express in Cartesian form; (ii) the proportion whereby the square on the ordinate varies as the abscissa or product of abscissas; (iii) an equation of a triangle with a parallelogram or trapezoid.  The latter equation holds in an affine plane.  With the advent of Cartesian methods in 1637, the equation seems to have been forgotten, because it is not readily translated into the lengths (symbolized by single minuscule letters) that Descartes has taught us to work with.  With the affine equation, Apollonius can give a proof-without-words of what today we consider a coordinate change, performed with more or less laborious computations.

  3.  By interpreting the field where algebra is done in the plane where geometry is done, Descartes does inspire new results.  An example still builds on work of an ancient mathematician, Pappus of Alexandria.  The model companion of the theory of Pappian affine spaces of unspecified dimension, considered as sets of points with ternary relation of collinearity and quaternary relation of parallelism, is the theory of Pappian affine planes over algebraically closed fields.

\end{ttfamily}

\chapter{Notes for talk}

\section{Affinity}

\begin{pres}
  If ${\Ver}$ and ${\Vasn}$ are the points $(1,0)$ and $(a,b)$
  on the ellipse or hyperbola with center ${\Koc}$ given by
  \begin{equation}\label{eqn:curve}
    x^2\pm y^2=1,
  \end{equation}
  as in \figref{fig:curves},
  \begin{figure}
    \psset{PointSymbol=none}
    \subfloat[Ellipse]{
      \begin{pspicture}(-2.5,-2.5)(2.5,2.5)
        \pstGeonode[PosAngle={135,-45}]
        (0,0){\Koc}(1.25,0){\Ver}
        \pstCircleOA{\Koc}{\Ver}
        \pstRotation[RotAngle=60,PosAngle=60]{\Koc}{\Ver}[\Vasn]
        \psdot(\Vasn)
        \psline{->}(0,-2.5)(0,2.5)
        \psline{->}(-2.5,0)(2.5,0)
      \end{pspicture}
    }
    \hfill
    \subfloat[Hyperbola]{
      \begin{pspicture}(-2.5,-2.5)(2.5,2.5)
        \pstGeonode[PosAngle={135,-135,120}]
        (0,0){\Koc}(1.25,0){\Ver}
        (! 0.5 dup COSH 1.25 mul exch SINH 1.25 mul){\Vasn}
        \psdot(\Vasn)
        \psline{->}(0,-2.5)(0,2.5)
        \psline{->}(-2.5,0)(2.5,0)
        \parametricplot{-1}{1}{t dup COSH 1.25 mul exch SINH 1.25 mul}
        \parametricplot{-1}{1}{t dup COSH -1.25 mul exch SINH -1.25 mul}
      \end{pspicture}
    }
    \caption{Ellipse and hyperbola}
    \label{fig:curves}
  \end{figure}
  then the curve is preserved under the linear transformation
  that interchanges ${\Ver}$ and ${\Vasn}$.
\end{pres}

\begin{proof}
  The transformation is multiplication by
  \begin{equation*}
    \begin{pmatrix}
      a&c\\b&d
    \end{pmatrix}
  \end{equation*}
  for some $(c,d)$, and then
  \begin{equation*}
    \begin{pmatrix}
      1\\0
    \end{pmatrix}
    =
    \begin{pmatrix}
      a&c\\b&d
    \end{pmatrix}
    \begin{pmatrix}
      a\\b
    \end{pmatrix}
    =
    \begin{pmatrix}
      a^2+cb\\ba+db
    \end{pmatrix},
  \end{equation*}
  so that
  \begin{align*}
    c&=\pm b,
    &d&=-a,
  \end{align*}
  and
  \begin{equation*}
    \begin{pmatrix}
      a&\pm b\\b&-a
    \end{pmatrix}
    \begin{pmatrix}
      x\\y
    \end{pmatrix}
    =
    \begin{pmatrix}
      ax\pm by\\bx-ay
    \end{pmatrix}.
  \end{equation*}
  Since
  \begin{equation*}
    a^2\pm b^2=1,
  \end{equation*}
  we compute
  \begin{equation*}
    (ax\pm by)^2\pm(bx-ay)^2=x^2\pm y^2.
  \end{equation*}
  Thus the image $(ax\pm by,bx-ay)$
  lies on our curve if $(x,y)$ does.
\end{proof}

\begin{corollary}
  If $\vec{\Koc}{\Ver}$ and $\vec{\Koc}{\Lk}$
  are independent vectors,
  as in \figref{fig:1ell}
  \begin{figure}
    \centering
    \begin{pspicture}(-4.6,-3.6)(6.2,3.1)
      %    \psgrid
      \psset{PointSymbol=none}
      \parametricplot[linewidth=2.4pt]0{360}{t cos 4 mul t sin 3 mul}
      \pstGeonode[PosAngle={105,-60,90,75,0,0,-90},
        PointName={default,default,default,default,none,none,default}]
      (0,0){\Koc}
      (! -12 cos 4 mul -12 sin 3 mul){\Ver}
      (! 90 -12 add cos 4 mul 90 -12 add sin 3 mul){\Lk}
      (!  35 cos 4 mul  35 sin 3 mul){\Vasn}
      (! 0 47 cos)x
      (0,1)t
      (! -85 cos 4 mul -85 sin 3 mul){\Pnt}
      \pstTranslation[PointName=none] t{\Ver}x
      \pstInterLL[PosAngle=-105]{x'}x{\Koc}{\Ver}{\Mp}
      \pstTranslation[PointName=none] x{\Ver}t
      \pstInterLL[PosAngle=-60]{t'}t{\Koc}{\Ver}{\Edp}
      \pstTranslation[PointName=none]{\Mp}{\Vasn}{\Ver,\Pnt}[u,v]
      \pstInterLL[PosAngle=75]{\Ver}u{\Koc}{\Vasn}{\Endp}
      \pstInterLL[PosAngle=-60]{\Pnt}v{\Koc}{\Ver}{\Xp}
      \pstInterLL[PosAngle=90]{\Pnt}{\Xp}{\Koc}{\Vasn}{\Ypn}
      \pstTranslation[PointName=none]{\Edp}{\Vasn}{\Pnt}[w]
      \pstInterLL[PosAngle=75]{\Pnt}w{\Koc}{\Ver}{\Yp}
      \pstInterLL[PosAngle=-105]{\Pnt}{\Yp}{\Koc}{\Vasn}{\Xpn}
      \pstTranslation[PosAngle=150]{\Ver}{\Koc}{\Koc}[\Wv]
      \pstTranslation[PosAngle=-150]{\Vasn}{\Koc}{\Koc}[\Wnv]
      \pstInterLL[PosAngle=10]{\Vasn}{\Edp}{\Ver}{\Endp}{\Fp}
      \ncline{\Mp}{\Vasn}
      \ncline{\Ver}{\Endp}
      \ncline{\Vasn}{\Edp}
      \ncline{\Koc}{\Endp}
      \ncline{\Koc}{\Edp}
      \ncline{\Koc}{\Lk}
      \psset{linestyle=dashed}
      \ncline{\Wnv}{\Koc}
      \ncline{\Wv}{\Koc}
      \psset{linestyle=dotted}
      \ncline{\Pnt}{\Ypn}
      \ncline{\Pnt}{\Yp}
    \end{pspicture}
    
    \caption{Ellipse}
    \label{fig:1ell}  
  \end{figure}
  or \figref{fig:1hyp},
  \begin{figure}
    \centering
    \begin{pspicture}(-4,-4.5)(6,9.5)
      \psset{PointSymbol=none,subgriddiv=1}
      %        \psgrid
      \newcommand{\myv}{3.9}
      \newcommand{\myh}{4}
      \parametricplot[linewidth=2.4pt]{-0.8}{1.1}{t SINH \myh\space mul t COSH \myv\space mul}
      \parametricplot[linewidth=2.4pt]{-0.6}{0.6}{t SINH -\myh\space mul t COSH -\myv\space mul}
      \pstGeonode[PosAngle={180,-120,0,-30,0,0,-120},
        PointName={default,default,default,default,none,none,default}]
      (0,0){\Koc}
      (! -0.2 SINH \myh\space mul -0.2 COSH \myv\space mul){\Ver}
      (! -0.2 COSH \myh\space mul -0.2 SINH \myv\space mul){\Lk}
      (! 0.6 SINH \myh\space mul 0.6 COSH \myv\space mul){\Vasn}
      (! 0.8 COSH 0)x
      (1,0)t
      (! -0.75 SINH \myh\space mul -0.75 COSH \myv\space mul){\Pnt}
      \pstTranslation[PointName=none] t{\Ver}x
      \pstInterLL[PosAngle=30]{x'}x{\Koc}{\Ver}{\Mp}
      \pstTranslation[PointName=none] x{\Ver}t
      \pstInterLL[PosAngle=-150]{t'}t{\Koc}{\Ver}{\Edp}
      \pstTranslation[PointName=none]{\Mp}{\Vasn}{\Ver,\Pnt}[u,v]
      \pstInterLL[PosAngle=-30]{\Ver}u{\Koc}{\Vasn}{\Endp}
      \pstInterLL[PosAngle=30]{\Pnt}v{\Koc}{\Ver}{\Xp}
      \pstInterLL[PosAngle=-30]{\Pnt}{\Xp}{\Koc}{\Vasn}{\Ypn}
      \pstTranslation[PointName=none]{\Edp}{\Vasn}{\Pnt}[w]
      \pstInterLL[PosAngle=120]{\Pnt}w{\Koc}{\Ver}{\Yp}
      \pstInterLL[PosAngle=0]{\Pnt}{\Yp}{\Koc}{\Vasn}{\Xpn}
      \pstTranslation[PosAngle=30]{\Ver}{\Koc}{\Koc}[\Wv]
      \pstTranslation[PosAngle=120]{\Vasn}{\Koc}{\Koc}[\Wnv]
      \pstInterLL[PosAngle=-90]{\Vasn}{\Edp}{\Ver}{\Endp}{\Fp}
      \ncline{\Mp}{\Vasn}
      \ncline{\Ver}{\Endp}
      \ncline{\Vasn}{\Edp}
      \ncline{\Koc}{\Xpn}
      \ncline{\Koc}{\Yp}
      \ncline{\Koc}{\Lk}
      \psset{linestyle=dashed}
      \ncline{\Wnv}{\Koc}
      \ncline{\Wv}{\Koc}
      \psset{linestyle=dotted}
      \ncline{\Pnt}{\Ypn}
      \ncline{\Pnt}{\Xpn}
    \end{pspicture}
    
    \caption{Hyperbola}
    \label{fig:1hyp}  
  \end{figure}
  and ${\Vasn}$ lies on the locus of points
  ${\Pnt}$ such that
  \begin{equation*}
    \vec{\Koc}{\Pnt}=x\cdot\vec{\Koc}{\Ver}+y\cdot\vec{\Koc}{\Lk},
  \end{equation*}
  where again \eqref{eqn:curve} holds,
  then the \emph{affinity}
  or affine transformation of the plane that fixes ${\Koc}$
  and interchanges ${\Ver}$ and ${\Vasn}$ fixes the curve.
\end{corollary}

After publication of Descartes's \emph{Geometry}
\cite{Descartes-Geometry,Descartes-Geometrie}
in 1637,
versions of the preservation theorem,
or rather its corollary,
were expressed and proved in
\begin{compactdesc}
\item[1655] by
  Wallis \cite{Wallis},
  \item[1659] by
    de Witt \cite{de-Witt},
  \item[1748] by
    Euler \cite{Euler}, and
  \item[1758] by
    Hugh Hamilton \cite{Hamilton,Hamilton-Eng}.
\end{compactdesc}
In each case \emph{lengths} are used,
or perhaps \emph{vectors.}
Descartes reduced the ancient algebra of areas and volumes
to an algebra of lengths alone.
Nobody seems to follow Apollonius
in proving the preservation theorem using areas,
as in Book \textsc i of the \emph{Conics}
\cite{Apollonius-Heiberg,MR1660991}.
We may thus have lost something.

Apollonius reasons as follows.
Dropping
\begin{compactitem}
\item
  ${\Ver}{\Endp}$ to ${\Koc}{\Vasn}$ and
  \item
  ${\Vasn}{\Mp}$ and ${\Pnt}{\Xp}$ to ${\Koc}{\Ver}$,
\end{compactitem}
all parallel to ${\Koc}{\Lk}$,
writing \eqref{eqn:curve} as
\begin{equation*}
  \pm y^2=(1-x)(1+x),
\end{equation*}
we have
\begin{align}\notag
  \vec{\Xp}{\Pnt}^2
  &\propto\vec{\Xp}{\Ver}\cdot\vec{\Wv}{\Xp}\\\notag
  &\propto\vec{\Xp}{\Ver}\cdot(\vec{\Koc}{\Ver}+\vec{\Koc}{\Xp})\\\notag
  &\propto\vec{\Xp}{\Ver}\cdot(\vec{\Ver}{\Endp}+\vec{\Xp}{\Yp})\\\label{eqn:prop1}
  &\propto{\Ver}{\Xp}{\Ypn}{\Endp}.
\end{align}
Letting ${\Edp}$ on ${\Koc}{\Ver}$ satisfy
\begin{equation*}
  {\Vasn}{\Ver}\parallel{\Endp}{\Edp},
\end{equation*}
equivalently
\begin{equation*}
      \rat{\vec{\Koc}{\Mp}}{\vec{\Koc}{\Ver}}
    \prop\rat{\vec{\Koc}{\Ver}}{\vec{\Koc}{\Edp}},
\end{equation*}
we have
\begin{equation}\label{eqn:1EMV*}
  {\Mp}{\Vasn}{\Edp}={\Ver}{\Mp}{\Vasn}{\Endp}.
\end{equation}
Dropping ${\Pnt}{\Yp}$ to ${\Koc}{\Ver}$ parallel to ${\Vasn}{\Edp}$,
we have
\begin{equation*}
  \vec{\Xp}{\Pnt}^2
  \propto{\Xp}{\Pnt}{\Yp}
\end{equation*}
and therefore, from \eqref{eqn:prop1},
\begin{equation*}
  {\Xp}{\Pnt}{\Yp}
\propto{\Ver}{\Xp}{\Ypn}{\Endp}.
\end{equation*}
Since this becomes an equation,
namely \eqref{eqn:1EMV*}, when ${\Pnt}$ is ${\Vasn}$,
we conclude
\begin{equation}\label{eqn:1XPY}
  {\Xp}{\Pnt}{\Yp}
={\Ver}{\Xp}{\Ypn}{\Endp}.
\end{equation}
This is an alternative defining equation for our curve.
The polygons in \eqref{eqn:1XPY} are oriented,
and the non-parallel sides of the trapezoid may intersect internally.
If either of
\begin{align*}
  \vec{\Pnt}{\Xp}&=\vec{\Xp}{\Pnto},
  &\vec{\Pnt}{\Koc}&=\vec{\Koc}{\Pnto}
\end{align*}
holds,
then ${\Pnto}$ lies on the curve.  Thus, because
\begin{compactitem}
  \item
${\Ver}{\Koc}$ bisects the chords
that are parallel to ${\Ver}{\Endp}$,
it is a \textbf{diameter}
of the curve;
\item
  the curve is symmetric about ${\Koc}$,
  this is its \textbf{center.}
\end{compactitem}
Consequently,
\begin{center}
  every line through ${\Koc}$ is a diameter of the curve.
\end{center}
Moreover,
${\Pnt}{\Yp}$ cutting ${\Vasn}{\Koc}$ at ${\Xpn}$,
if to either side of the defining equation \eqref{eqn:1XPY} we add
the quadrilateral
\begin{equation*}
  {\Yp}{\Xpn}{\Ypn}{\Xp},
\end{equation*}
we obtain
\begin{align*}
  {\Ypn}{\Pnt}{\Xpn}
  &={\Ver}{\Yp}{\Xpn}{\Endp}&&\\
  &={\Edp}{\Yp}{\Xpn}{\Vasn}.&&\text{[by \eqref{eqn:1EMV*}]}
\end{align*}
Thus,
\begin{center}
  with respect to any diameter,\\
the curve has the same equation.
\end{center}
This is the Preservation Theorem.

For Rosenfeld in his commentary
\cite[p.\ 57]{Rosenfeld}
on the theorem,
\begin{quote}
  Apollonius never mentions parabolic, elliptic, and hyperbolic turns,
  but no doubt that he used these transformations
  to generalize the results obtained by his precursors
  in rectangular coordinates for the cases of oblique coordinates.
\end{quote}
A \enquote{turn} is for Coxeter \cite[pp.\ 206--7]{Coxeter-Geometry} a
rotation; it is a kind of affinity.

There are all kinds of doubt
that Apollonius had any notion of an affinity in our technical sense.
This is important because:
\begin{compactenum}[1)]
\item
  Apollonius's proof is more direct than the best modern proof;
\item
  misunderstanding of the ancient use of ratios
  has led some modern mathematicians to accuse Euclid
  of logical error.
\end{compactenum}

\section{Affine plane}

In Book \textsc i of the \emph{Elements,}
Propositions 37 and 39 are that,
in \figref{fig:37-39},
  \begin{figure}
    \subfloat[Proposition 37]{\label{fig:37}
    \begin{pspicture}(5,4)
%      \psgrid
      \psset{PointSymbol=none}
      \pstGeonode[PosAngle={-90,-90,90,90}](1,0.6)C(3,0.6)D
      (0.2,3.4)A
      \pstTranslation[PosAngle=90]CDA[B]
      \pstHomO[HomCoef=1.3,PosAngle=90]AB[E]
      \pstTranslation[PosAngle=90]CDE[F]
      \pstInterLL[PosAngle=180] CEDBG
        \pspolygon(A)(D)(C)
        \psline(C)(F)(D)
        \ncline[linestyle=dashed]AF
        \psset{linestyle=dotted}
        \ncline BD\ncline CE
    \end{pspicture}
    }
    \hfill
    \subfloat[Proposition 39]{\label{fig:39}
    \begin{pspicture}(5,4)
%      \psgrid
      \psset{PointSymbol=none}
      \pstGeonode[PosAngle={-90,-90,90,90}](1,0.6){C}(3,0.6){D}
      (0.2,2.5){A}(4.8,3.4){F}
      \pstTranslation[PointName=none]CDA
      \pstInterLL[PosAngle=-45]{A'}AFDH
        \pspolygon(A)(D)(C)
        \psline(C)(F)(D)
        \ncline[linestyle=dashed]AF
        \psset{linestyle=dotted}
        \ncline AH\ncline HC
    \end{pspicture}
    }    
    \caption{Parallelism}
    \label{fig:37-39}
  \end{figure}
  \begin{equation*}
    AF\parallel CD
    \liff ACD=FCD.
  \end{equation*}
  We can understand Euclid's proof
  as having the following steps,
  the justifications of which
  will be sufficient to axiomatize an
 \textbf{affine plane}
  in the full sense of being acted on transitively and faithfully
  by a $2$-dimensional vector space over a commutative field,
  albeit of characteristic other than $2$.
  \begin{compactenum}
  \item
    First we assume $AF\parallel CD$.
    Using \textbf{Playfair's Axiom,}
    that through a point not on a line passes exactly one parallel,
    we let
    \begin{align*}
      AC&\parallel BD,
      &CE&\parallel DF.
    \end{align*}
  \item\label{ax:prism}
    Because $ABDC$ and $CDFE$ are parallelograms,
    as in \figref{fig:prism-ax},
        \begin{figure}
      \psset{PointSymbol=none}
      \subfloat[Statement]{\label{fig:prism-ax}
        \begin{pspicture}(5,4)
          %\psgrid
          \pstGeonode[PosAngle={135,90,-90,-90}]
          (0.6,1.8)C(1,3.4)A(1.5,0.6)E(4.4,0.6)F
          \pstTranslation[PosAngle=135] EFC[D]
          \pstTranslation[PosAngle=90] EFA[B]
          \psset{linestyle=dashed}
          \ncline FE\ncline BA\ncline DC
          \psset{linestyle=solid,linewidth=2.4pt}
          \pspolygon(A)(E)(C)
          \pspolygon(D)(B)(F)
        \end{pspicture}
      }
      \hfill
      \subfloat[Proof]{\label{fig:prism-thm}
        \begin{pspicture}(5,4)
          %\psgrid
          \pstGeonode[PosAngle={135,90,-90,-90}]
          (0.6,1.8)C(1,3.4)A(1.5,0.6)E(4.4,0.6)F
          \pstTranslation[PosAngle=135] EFC[D]
          \pstTranslation[PosAngle=90] EFA[B]
          \pstHomO[HomCoef=0.8,PosAngle=-90]EF[G]
          \ncline BG\ncline GD
          \psset{linestyle=dashed}
          \ncline FE\ncline BA\ncline DC
          \psset{linestyle=solid,linewidth=2.4pt}
          \pspolygon(A)(E)(C)
          \pspolygon(D)(B)(F)
        \end{pspicture}
      }
      \caption{Prism Theorem}
 
    \end{figure}
    \begin{equation*}
      ACE=BDF.
    \end{equation*}
    \item\label{ax:group}
    Because equality is a congruence
    on the abelian group of polygons,
by Euclid's Common Notions 2 and 3,
\begin{quote}\centering
If equals be added to (or subtracted from) equals,\\
    the wholes (or remainders) are equal,
\end{quote}
we conclude
\begin{align*}
  ACDB
  &=ACGB+CDG\\
  &=ACE-BGE+CDG\\
  &=BDF-BGE+CDG
  =ECDF
\end{align*}
(which is Proposition 35).
\item
  Because a diagonal bisects a parallelogram
  (by Proposition 34),
  \begin{equation*}
    2ACD=2FCD.
  \end{equation*}
\item\label{ax:2}
No element of the group of polygons having order $2$,
  \begin{equation*}
    ACD=FCD.
  \end{equation*}
  \item\label{ax:col}
  Now suppose $AF\nparallel CD$.
  Letting $AH\parallel CD$, we conclude
  \begin{equation*}
    ACD=HCD\neq HCD+FCH=FCD,
  \end{equation*}
by Euclid's Common Notion 5,
\begin{quote}\centering
The whole is greater than the part.
\end{quote}
  \end{compactenum}
      Our axioms formally govern a structure
with two sorts,
  \begin{compactitem}
  \item
    $\points$, of points (\gr{t`a shme~ia}), and
    \item
  $\polygons$, of polygons (\gr{t`a pol'ugwna}).
  \end{compactitem}
  When $n\geqslant3$, there is an $n$-ary map
  \begin{equation*}
    (P_1,\dots,P_n)\mapsto P_1\cdots P_n
  \end{equation*}
  from $\points$ to $\polygons$.
  In Axiom \ref{ax:group},
  the group of polygons satisfies the following rules,
  where $\Gamma$ and $\Delta$
  are strings of letters for points:
      \begin{gather*}\notag
    A\Gamma= A\Gamma A,\\\label{eqn:cycle}
    A\Gamma=\Gamma A,\\\label{eqn:sum}
    A\Gamma B+B\Delta A= A\Gamma B\Delta,\\\label{eqn:neg}
    -A_1\cdots A_n= A_n\cdots A_1.
      \end{gather*}
      We can understand Axiom \ref{ax:col}
      as a definition of collinearity:
      \begin{equation}\label{eqn:col}
        \col ABC\liff ABC=0,
      \end{equation}
  or rather as meaning that
  if each of two distinct points
  is collinear, in the sense of \eqref{eqn:col},
  with two other distinct points,
  the converse holds as well:
  \begin{equation*}
    A\not\equiv B\land ACD=0\land BCD=0\land C\not\equiv D
    \lto ABC=0.
  \end{equation*}
  It is now a theorem that two points determine a line:
  if the two points are $A$ and $B$,
  the line they determine is defined by
  \begin{equation*}
    ABX=0,
  \end{equation*}
  and if $C$ and $D$ are two points on this line,
  then they determine the same line.

  To obtain a companionable theory
  whose model-companion is still a theory of affine planes,
  we let there be a quaternary relation of
  \textbf{parallelism} on $\points$,
  given by the axiom
  \begin{multline*}
    AB\parallel CD\liff
        \Forall X
    \Bigl(A\not\equiv B\land C\not\equiv D\land\\
    (ABX=0\land CDX=0\lto ABC=0\land ABD=0)\Bigr).
  \end{multline*}
  The axiom is $\forall\exists$,
  by the general rule
  \begin{equation*}
    \bigl(\sigma\liff\Forall x\phi(x)\bigr)\liff
    \Forall x\Exists y\Bigl(\bigl(\sigma\lto\phi(x)\bigr)
    \land\bigl(\phi(y)\lto\sigma\bigr)\Bigr).
  \end{equation*}
  Also for companionability,
  Axiom \ref{ax:2} should be that
  the abelian group of polygons
  is either torsion-free
  or an elementary $p$-group
  (thus a vector space)
  for some odd prime $p$.

  The case of \textbf{Desargues's Theorem}
  that I call the \textbf{Prism Theorem,}
  and that is
  shown in \figref{fig:prism-ax}, is now a theorem.
  If $AB$, $CD$, and $EF$ are parallel to one another
  (parallelism being transitive by Playfair's Axiom), then
  \begin{equation*}
    AC\parallel BD\land CE\parallel DF\lto AE\parallel BF.
  \end{equation*}
  We may assume $ACE\neq0$.
  By Axiom \ref{ax:prism},
  \begin{equation*}
    ACE=BDF.
  \end{equation*}
  If $AE\nparallel BF$, let
  \begin{equation*}
    AE\parallel BG.
  \end{equation*}
  By Axiom \ref{ax:prism} again,
  \begin{equation*}
    ACE=BDG.
  \end{equation*}
  By Euclid's Proposition 39 as above,
  \begin{equation*}
    BD\parallel EF,
  \end{equation*}
  contradicting the (implicit) assumption
  that $AB$ and $CD$ are distinct lines.

  Just as Pappus does, using Euclid's Propositions 37 and 39,
  we prove \textbf{Pappus's Theorem}
  \cite{Pappus-Pierce-2,Jones-whole,Pappus}
  that if the vertices of $ABCDEF$ lie alternately
  on two lines,
  as in \figref{fig:VIII},
  \begin{figure}
    \centering
    \begin{pspicture}(5,5)
      \psset{PointSymbol=none}
%      \psgrid
      \pstGeonode[PosAngle={-90,-90,90,90}]
      (0.2,0.6)E(4.8,0.6)D(2,4.6)A(4,4.6)B
    \pstHomO[HomCoef=0.45,PosAngle=150]AE[C]
    \pstTranslation[PointSymbol=none,PointName=none] CDA
    \pstInterLL A{A'}BDF
    \ncline EA\ncline BD
    \psset{linestyle=dotted}
    \ncline BE\ncline CF\ncline AD
    \psset{linewidth=2pt,linestyle=solid}
    \ncline BC\ncline CD\ncline EF\ncline FA
    \psset{linestyle=dashed}
    \ncline DE
    \ncline BA
    %    \psset{linestyle=dotted}
%    \ncline AD\ncline CF\ncline EB
  \end{pspicture}

        \caption{Pappus's Theorem}
        \label{fig:VIII}
    
  \end{figure}
  then
  \begin{equation*}
    BC\parallel EF\land CD\parallel FA
    \lto AB\parallel DE.
  \end{equation*}
  For, under the hypothesis,
  \begin{align*}
    ABE
    &=ABC+CBE\\
    &=ABC+CBF\\
    &=ABF+FAC\\
    &=ABF+FAD\\
    &=ABD,
  \end{align*}
  or more briefly,
  \begin{equation*}
    ABE=ABFC=ABD.
  \end{equation*}
  Selecting now a proper triangle $IOI'$,
  we define a multiplication on $OI$
  as in \figref{fig:mul-def}.
  \begin{figure}
    \psset{PointSymbol=none}
    \subfloat[Definition]{\label{fig:mul-def}
      \begin{pspicture}(5,5)
        \pstGeonode[PosAngle={-90,-90,90}]
          (0.2,0.6)O(4.8,0.6){ab}(2,4.4){b'}
          \pstHomO[HomCoef=0.6,PosAngle=-90]O{ab}[b]
          \pstHomO[HomCoef=0.4,PosAngle=-90]O{ab}[a]
          \pstHomO[HomCoef=0.6,PosAngle=-90]Oa[I]
          \pstTranslation[PointName=none]{ab}{b'}a[x]
          \pstInterLL[PosAngle=135] axO{b'}{I'}
          \ncline O{ab}\ncline O{b'}
          \ncline{ab}{b'}\ncline{b'}b
          \ncline a{I'}\ncline{I'}I
      \end{pspicture}
      }
    \hfill
        \subfloat[Commutativity]{\label{fig:mul-comm}
      \begin{pspicture}(5,5)
        \pstGeonode[PosAngle={-90,-90,90}]
          (0.2,0.6)O(4.8,0.6){ab}(2,4.4){b'}
          \pstHomO[HomCoef=0.6,PosAngle=-90]O{ab}[b]
          \pstHomO[HomCoef=0.4,PosAngle=-90]O{ab}[a]
          \pstHomO[HomCoef=0.6,PosAngle=-90]Oa[I]
          \pstTranslation[PointName=none]{ab}{b'}a[x]
          \pstInterLL[PosAngle=135] axO{b'}{I'}
          \pstTranslation[PointName=none]{b}{b'}a[x]
          \pstInterLL[PosAngle=135] axO{b'}{a'}
          \ncline O{ab}\ncline O{b'}
          \ncline{ab}{b'}\ncline{b'}b
          \ncline a{I'}\ncline{I'}I
          \psset{linestyle=dashed}
          \ncline a{a'}\ncline{a'}{ab}\ncline{I'}b
      \end{pspicture}
      }
\caption{Multiplication}
    
  \end{figure}
  The operation is commutative by Pappus's Theorem,
  as in \figref{fig:mul-comm}.
  For associativity, use \figref{fig:mul-assoc},
  \begin{figure}
    \psset{PointSymbol=none}
    \centering
          \begin{pspicture}(10,6)
        \pstGeonode[PosAngle={-90,-90,90}]
          (0.2,0.6)O(9.8,0.6){ab}(2,5.4){b'}
          \pstHomO[HomCoef=0.6,PosAngle=-90]O{ab}[b]
          \pstHomO[HomCoef=0.4,PosAngle=-90]O{ab}[a]
          \pstHomO[HomCoef=0.6,PosAngle=-90]Oa[I]
          \pstTranslation[PointName=none]{ab}{b'}a[x]
          \pstInterLL[PosAngle=135] axO{b'}{I'}
          \pstHomO[HomCoef=0.7,PosAngle=-90]OI[c]
          \pstTranslation[PointName=none]I{I'}c[x]
          \pstInterLL[PosAngle=135] cxO{b'}{c'}          
          \pstTranslation[PointName=none]{I'}{ab}{c'}[x]
          \pstInterLL[PosAngle=-90]{c'}xO{ab}{abc}          
          \pstTranslation[PointName=none]{I'}{a}{c'}[x]
          \pstInterLL[PosAngle=-90]{c'}xO{ab}{ac}          
          \pstTranslation[PointName=none]{I'}{ac}{b'}[x]
          \pstInterLL[PosAngle=-90,PointName=none]
                     {b'}xO{ab}{acb}
          \ncline O{ab}\ncline O{b'}
          \ncline{ab}{b'}
          \ncline{I'}{ab}\ncline{c'}{abc}
          \ncline{c'}{ac}
          \ncline{I'}{ac}\ncline{b'}{acb}
          \psset{linestyle=dotted}
          \ncline{b'}b\ncline c{c'}
          \ncline{I'}I
          \ncline a{I'}
      \end{pspicture}

          \caption{Associativity}
          \label{fig:mul-assoc}
  \end{figure}
  where is shown
  \begin{equation*}
    (ab)c=(ac)b,
  \end{equation*}
  from which, by commutativity,
  associativity follows.
  
  The Prism Theorem lets us define define
  parallel directed segments
  $\vec AB$ and $\vec CD$ as \textbf{equal}
  if and only if
  \begin{compactitem}
  \item
    the segment are not collinear, and
    $AC\parallel BD$, or
  \item
    they are collinear and equal to a third
    in the sense just defined.
  \end{compactitem}
  A \textbf{vector} is the class of directed segments
  equal to a given one.
  We can add non-parallel vectors by completing the parallelogram;
  parallel, by placing them end to end,
  as in \figref{fig:weak},
\begin{figure}
  \centering
  \psset{PointSymbol=none}
  \begin{pspicture}(10,4)
%      \psgrid
    \pstGeonode[PosAngle={-90,-90,90}](0.2,0.6)A(4,0.6)B(2.4,3.4)D
    \pstHomO[HomCoef=1.5,PosAngle=-90]AB[C]
    \pstTranslation[PosAngle={-90,90}] AC{B,D}[F,E]
    \pstTranslation[PosAngle=90] ADB[G]
%    \pstInterLL[PosAngle=60] CGBEH
    \ncline AD
    \ncline AF\ncline FE\ncline ED
    \ncline DB\ncline CE
    \psset{linestyle=dashed}
    \ncline BG
    \ncline GF\ncline DC
    \psset{linestyle=dotted}
%    \ncline DF\ncline BE\ncline CG
  \end{pspicture}
  \caption{Commutativity of addition}\label{fig:weak}
  
\end{figure}
where
\begin{equation*}
  \vec AB+\vec AC=\vec AF=\vec AC+\vec AB,
\end{equation*}
by definition and by Pappus's Theorem.


\subsection{Associativity}

To prove associativity of addition of vectors,
we have three cases to consider.
\begin{asparaenum}
  \item
When $A$ is not collinear with any two of $B$, $C$, and $D$,
as in \figref{fig:ass},
\begin{figure}
        \psset{PointSymbol=none}
  \centering
  \begin{pspicture}(-1,-0.6)(7,4.5)
%    \psgrid
    \pstGeonode[PosAngle={-90,-90,45,180}] A(5,0)B(2,1.5)C(-1,2.5)D
    \pstTranslation ABC[E]
    \pstTranslation[PosAngle=90] ADE[F]
    \pstTranslation[PosAngle=90] ACD[G]
    \ncline AB\ncline AC\ncline AD
    \psset{linestyle=dashed}
    \ncline BE\ncline EC
    \ncline AE\ncline EF\ncline FD
    \ncline CG\ncline GD\ncline AG
    \ncline[linestyle=dotted] FG
    \ncline[linestyle=dotted] BF
  \end{pspicture}

  \caption{Associativity of addition: easy case}\label{fig:ass}
\end{figure}
then
\begin{align*}
  \vec AB+\vec AC&=\vec AE,&
  \vec AE+\vec AD&=\vec AF,&
  \vec AC+\vec AD&=\vec AG,
\end{align*}
and then
\begin{equation}\label{eqn:AB+AG}
  \vec AB+\vec AG=\vec AF
  \iff AB\parallel GF\And AG\parallel BF.
\end{equation}
They \emph{are} parallel, by the Prism Theorem, as follows.
We can apply the Theorem first to triangles $ACE$ and $DGF$,
so that,
since
\begin{align*}
  CG&\parallel AD\parallel EF,&AC&\parallel DG,&AE&\parallel DF,
\end{align*}
we can conclude $CE\parallel GF$.
Since also $AB\parallel CE$, we obtain
$AB\parallel GF$.
Now we can apply the Prism Theorem to triangles $ACG$ and $BEF$,
obtaining $AG\parallel BF$.
So we have the right-hand side of \eqref{eqn:AB+AG},
and therefore the left-hand side, which means
\begin{equation}\label{eqn:+ass}
  \vec AB+(\vec AC+\vec AD)=
  (\vec AB+\vec AC)+\vec AD.
\end{equation}
\item
When $AB$ contains $C$, but not $D$,
then \eqref{eqn:+ass} still holds,
since in \figref{fig:+as},
\begin{figure}
      \psset{PointSymbol=none}
  \centering
  \begin{pspicture}(-0.2,-0.6)(9.2,3.6)
%      \psgrid
    \pstGeonode[PosAngle={-90,-90,90}] A(3,0)B(1,3)D
    \pstHomO[HomCoef=1.5,PosAngle=-90]AB[C]
    \pstTranslation[PosAngle={-90,90}] AC{B,D}[F,E]
    \pstTranslation[PosAngle=90] ADB[G]
    \pstTranslation[PosAngle=90] ADF[K]
    \ncline AF\ncline FK\ncline KD\ncline DA
    \ncline DB\ncline BG\ncline CE\ncline EF
    \psset{linestyle=dotted}
    \ncline AE\ncline BK
%    \psset{linestyle=dashed}
%    \ncline DF\ncline BE\ncline CG
  \end{pspicture}
  \caption{Associativity of addition: less easy case}\label{fig:+as}
  
\end{figure}
\begin{align*}
  \vec AB+\vec AC&=\vec AF,&
  \vec AF+\vec AD&=\vec AK,&
  \vec AC+\vec AD&=\vec AE,
\end{align*}
so that
\begin{equation*}
  \vec AK=\vec AB+\vec AE\iff AE\parallel BK.
\end{equation*}
The parallelism follows from the Parallel Theorem,
applied to the hexagon $ADBKFE$.
\item
Finally, when $AB$ contains both $C$ and $D$,
then, making use of commutativity,
in \figref{fig:+ass}
\begin{figure}
      \psset{PointSymbol=none}
  \centering
  \begin{pspicture}(0,-0.6)(9.6,4.5)
%    \psgrid
    \pstGeonode[PosAngle={-90,-90,90}] A(3,0)B(1.5,4)E
    \psset{PosAngle=-90}
    \pstHomO[HomCoef=0.67]AB[C]
    \pstHomO[HomCoef=1.5]AB[D]
    \pstTranslation[PosAngle=90] AEB[F]
    \ncline BF\ncline EB
    \pstTranslation ECF[G]
    \ncline FG\ncline EG
    \pstTranslation[PosAngle=90] AED[H]
    \pstTranslation EGH[K]
    \ncline HK
    \pstTranslation ECH[L]
    \ncline HL
    \pstTranslation[PosAngle=90] AEL[M]
    \ncline LM
    \ncline EM\ncline BK
    \psset{linestyle=dashed}
    \ncline MK
    \ncline BH\ncline GM
    \psset{linestyle=dotted}
    \ncline AB\ncline EC\ncline AE\ncline DH
%    \psset{linestyle=dashed}\ncline BM\ncline EK\ncline GH
  \end{pspicture}
  \caption{Associativity of addition: hardest case}\label{fig:+ass}
  
\end{figure}
we have
\begin{gather*}
  \vec AB+\vec AC=\vec AC+\vec AB=\vec AG,\\
\vec AG+\vec AD=\vec AK,\qquad
\vec AC+\vec AD=\vec AL,\\
  \vec AB+\vec AL=\vec AK\iff BE\parallel MK.
\end{gather*}
The Parallel Theorem,
applied to $BFGMLH$,
yields $BH\parallel GM$;
then, applied to $KHBEGM$,
$BE\parallel MK$.
\end{asparaenum}

\subsection{Distributivity}

Addition of vectors makes any line into an abelian group.
Multiplication in the line $OI$
makes that line into a field
and every line through $O$ into a one-dimensional vector space.
Indeed, in \figref{fig:dist},
\begin{figure}
  \centering
  \psset{PointSymbol=none}
  \begin{pspicture}(10,7)
    \pstGeonode[PosAngle={-90,90,-90,-90}]
    (0.2,3)E(6.5,6.4){B'}(9.8,3)B(4.5,0.6){A'}
    \pstInterLL[PosAngle=-45]{A'}{B'}EBO
    \pstTranslation[PointName=none]E{B'}{A'}
    \pstInterLL[PosAngle=90]{A'}{A''}EBD
    \pstTranslation[PointName=none]B{B'}{A'}
    \pstInterLL[PosAngle=90]{A'}{A''}EBA
    \pstTranslation[PosAngle=-90] OAB[C]
    \pstTranslation[PointName=none]B{B'}C[x]
    \pstInterLL[PosAngle=30]CxO{B'}{C'}
    \pstTranslation[PointName=none]{B'}E{C'}
    \pstInterLL[PosAngle=-90]{C'}{C''}OAF
    \pstTranslation[PosAngle=30] CB{C',B'}[H,K]
    \pstTranslation[PosAngle=135] FE{C',B'}[G,L]
    \ncline{A'}{B'}
    \ncline EB\ncline E{B'}\ncline{B'}B
    \ncline{A'}D\ncline{A'}A
    \ncline C{C'}
    \ncline{C'}F
    \psset{linestyle=dotted}
    \ncline GH\ncline HK\ncline KL\ncline LG
    \ncline A{B'}\ncline OK\ncline{A'}H
  \end{pspicture}
  \caption{Distributivity}
  \label{fig:dist}
\end{figure}
where
\begin{align*}
  \vec OA&=\bm a,
  &\vec OB&=\bm b,
  &\vec OC&=\bm a+\bm b,
\end{align*}
and then
\begin{align*}
  \vec OD&=c\bm a,
  &\vec OE&=c\bm b,
  &\vec OF&=c(\bm a+\bm b),
\end{align*}
We have
\begin{align*}
  \vec OA
  &=\vec BC&&\text{[definition of +]}\\
  &=\vec{C'}B&&\text{[Prism Theorem]}\\
  &=\vec{B'}K,&&\text{[Prism Theorem]}
\end{align*}
then
\begin{equation*}
  A{B'}\parallel OK,
\end{equation*}
so by the converse of the Prism Theorem,
\begin{equation*}
  \vec{A'}O=\vec HK=\vec{C'}{B'}.
\end{equation*}
Now the converse of the Prism Theorem yields
\begin{equation*}
\vec OD=\vec L{B'}=\vec G{C'}=\vec EF,  
\end{equation*}
and therefore
\begin{equation*}
  c\bm a+c\bm b=c(\bm a+\bm b).
\end{equation*}


\subsection{Vector space}

To ensure finally that, with $O$ selected,
the plane is a vector space over the field of points along $OI$,
we show that the field is independent of choice of $I'$.
It is independent,
because Desargues's Theorem holds
in the special case shown in
``Thales and the Nine-point Conic'' \cite{Pierce-Thales-9}.
First we need to establish that, in \figref{fig:I.43},
\begin{figure}
  \psset{PointSymbol=none}
  \subfloat[Euclid I.43]{\label{fig:I.43}
  \begin{pspicture}(4,5)
    \pstGeonode[PosAngle={90,-90,-90}](0.2,4.4)B(1,0.6)O(3.8,0.6)D
    \pstTranslation[PosAngle=90] ODB[L]
    \pstHomO[HomCoef=0.4,PosAngle=-90]OD[C]
    \pstHomO[HomCoef=0.4,PosAngle=180]OB[A]
    \pstTranslation[PosAngle=0] ODA[M]
    \pstTranslation[PosAngle=90] OBC[N]
    \pstInterLL[PosAngle=-30] AMCNG
    \pspolygon(O)(B)(L)(D)
    \ncline AM\ncline CN\ncline OG\ncline[linestyle=dotted] GL
    \psset{linestyle=dashed}
    \ncline AC\ncline BD
  \end{pspicture}
  }
  \hfill
  \subfloat[]{\label{fig:trans}
  \begin{pspicture}(6,5)
    \pstGeonode[PosAngle={90,-90,-90,90}]
    (0.2,4.4)B(0.5,0.6)O(3.8,0.6)D(5.8,4.4)P
    \pstTranslation[PosAngle=90] ODB[L]
    \pstHomO[HomCoef=0.46,PosAngle=-90]OD[C]
    \pstHomO[HomCoef=0.46,PosAngle=180]OB[A]
    \pstTranslation[PosAngle=45] ODA[M]
    \pstInterLL[PosAngle=-30] AMDPR
    \pstTranslation[PosAngle=90] OBC[N]
    \pstInterLL[PosAngle=-45] AMCNG
    \pstTranslation[PosAngle=90] DP{O,C}[F,Q]
    \pstInterLL[PosAngle=120] OFARE
    \pstInterLL[PosAngle=120] CQARS
    \pspolygon(O)(B)(P)(D)
    \ncline DP\ncline OF\ncline CQ\ncline DL
    \ncline AR\ncline CN%\ncline OG\ncline[linestyle=dotted] GL
%    \psset{linestyle=dashed}
%    \ncline AC\ncline BD
  \end{pspicture}
}
  \caption{Conditions for parallelism}

\end{figure}
\begin{equation*}
  AC\parallel BD\iff AGNB=GCDM
  \iff OGL=0.
\end{equation*}
Then, in \figref{fig:trans},
when we assume
\begin{align*}
  AC&\parallel BD,
  &AE&\parallel BF\parallel OD.
\end{align*}
Then
\begin{equation*}
CDRS=CDMG=AGNB=ESQF,  
\end{equation*}
so
\begin{equation*}
  EC\parallel FD.
\end{equation*}


\section{Proportion}

At the head of Book \textsc{vii} of the \emph{Elements,}
we are told,
among what are labelled as definitions:
\begin{compactenum}
  \item
\enquote{A number is a multitude of units.}%%%%%
\thinspace\footnote{As he notes his \enquote{Mathematicall Preface} \cite{Dee} 
  to Billingsley's 1570 English translation of the \emph{Elements,}
  John Dee \emph{created} the word \eng{unit}
precisely to translate Euclid's \gr{mon'ac}.
The existing alternative was \eng{unity.}
See my article \enquote{On commensurability and symmetry}
\cite{MR3680853}.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\item
Numbers $A$, $B$, $C$, and $D$ are \textbf{proportional} 
when $A$ is
\begin{compactitem}%[[\thinspace 1\thinspace {]}]
\item  
 the same multiple,
or 
\item
the same part, or 
\item
the same parts,
\end{compactitem}
of $B$ that $C$ is of $D$.
\item
The following are equivalent for numbers.
\begin{compactenum}%[(1)]
  \item
$B$ is a multiple of $A$,
\item
$A$ is a part of $B$,
\item
$A$ \textbf{measures} $B$.
\end{compactenum}
\item
If neither multiple nor part of $B$,
$A$ is \textbf{parts} of $B$.%%%%%
\footnote{The text says only that the \emph{less} is parts of the greater
  when not measuring the greater;
but the definition of proportion implies that 
the greater is parts of the less when not a multiple of the less.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{compactenum}
Measuring will be dividing in \emph{extension,} but not in \emph{intension}:
\begin{compactenum}[(1)]
\item
We can \textbf{measure} 12 apples evenly by 4 apples.
  \item
In the process, we \textbf{divide} the apples into 3 groups.%%%%%
\footnote{Euclid uses dividing, as far as I know,
only to say that an even number can be divided in two.
Alexandre Borovik discusses measuring and dividing apples \cite{Borovik-Meta},
though not with the terminology of measuring.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{compactenum}

It is clear when $A$ is the \textbf{same multiple}
or \textbf{same part}
of $B$ that $C$ is of $D$;
but not \textbf{same parts.} 

If
\begin{equation*}
  A:B::C:D,
\end{equation*}
this should mean at least that
for some numbers $E$ and $F$,
for some multipliers $\coA$ and $\coB$
(either of which can be unity),
\begin{align}\label{eqn:AEk}
&  \begin{gathered}
    A=E\cdot\coA,\\
    B=E\cdot\coB,
   \end{gathered}&
&  \begin{gathered}
    C=F\cdot\coA,\\
    D=F\cdot\coB.
  \end{gathered}
\end{align}
\begin{compactitem}
  \item
Some Moderns call this the 
\textbf{Pythagorean} definition of proportion of numbers.%%%%%
\footnote{Heath thinks
  \begin{inparaenum}[(1)]
    \item
 the theory of Book \textsc{vii} 
is due to the Pythagoreans \cite[Vol.\ 2, p.\ 294]{MR17:814b}, and
\item
its definition of proportion
is the one that we are calling Pythagorean \cite[p.\ 190]{MR654679}.
\end{inparaenum}
In Thomas's first Loeb volume of \emph{Greek Mathematical Works} 
\cite{MR13:419a}, the chapter \enquote{Pythagorean Arithmetic}
gives first the definitions that head Book \textsc{vii} of the \emph{Elements,}
but nothing ensues that requires a careful interpretation 
of the definition of proportion.
That this definition ought \emph{immediately} 
to imply transitivity of sameness of ratio:
this might seem belied by Proposition 11 in Book \textsc v,
which proves the transitivity 
for arbitrary magnitudes under the Eudoxan definition;
however, the proof is trivial.
Nonetheless,
Pengelley and Richman \cite[pp.\ 196, 199]{MR2204484}
accept Heath's judgment,
and Mazur \cite[n.\ 6]{Mazur-Th-pub} accepts \emph{their} judgment;
for convenience, I imitate them in using the term \emph{Pythagorean.}
I have seen no suggestion that the Pythagoreans proved general theorems
like the commutativity of multiplication or Euclid's Lemma;
thus perhaps they had no theoretical need 
for the transitivity of sameness of ratio.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item
For the proper \textbf{Euclidean} definition, I say, we need also
\begin{align}\label{eqn:gcm}
  E&=\gcm(A,B),&F&=\gcm(C,D),
\end{align}
where $\gcm$ means \textbf{greatest common measure;}
equivalently, $\coA$ and $\coB$ in \eqref{eqn:AEk} are coprime.
\end{compactitem}
Without \eqref{eqn:gcm},
\begin{compactenum}[(1)]
  \item
sameness of ratio is not \emph{immediately} transitive;
\item
%implicitly using transitivity, 
thus proofs in Book \textsc{vii} are inadequate;
\item
  Proposition 4 makes little sense,
  the enunciation,
\enquote{Any number is either a part or parts of any number,
the less of the greater,}
only restating a definition,
though the proof is nontrivial.
\end{compactenum}

\section{Anthyphaeresis}

In Book \textsc{vii},
\textbf{Propositions 1--3} show,
for two or more numbers, 
\begin{compactenum}[(1)]
  \item
\emph{how} to find a $\gcm$, and
\item
\emph{that} it is measured by all common measures.
\end{compactenum}
The proofs use the \textbf{Euclidean Algorithm,}
namely,
\begin{compactenum}[(1)]
\item
  replace the greater of two magnitudes with its remainder,
  if there is one,
after measurement by the less;
\item
repeat.
\end{compactenum}
When the greater is measured exactly by the less, this is the $\gcm$.
Thus from
\begin{gather*}
  80=62\cdot1+18,\\
62=18\cdot3+8,\\
18=8\cdot2+2,\\
8=2\cdot4,
\end{gather*}
we have
$\gcm(80,62)=2$,
Also,
\begin{align}\label{eqn:80}
80&=2\cdot40,&62&=2\cdot31,
\end{align}
and the multipliers $40$ and $31$ are automatically coprime.

Euclid proves \textbf{Proposition 4}
(again \enquote{Any number is either a part or parts of any number,
the less of the greater})
by finding $\gcm$'s,
showing implicitly (in my view) 
that he intends what I am calling the Euclidean definition of proportion.
From
\begin{gather*}
  120=93\cdot1+27,\\
93=27\cdot3+12,\\
27=12\cdot2+3,\\
12=3\cdot4,
\end{gather*}
we have $\gcm(120,93)=3$, and also
\begin{align}\label{eqn:120}
  120&=3\cdot40,&93&=3\cdot31.
\end{align}
By the repetition in \eqref{eqn:120} of multipliers from \eqref{eqn:80},
\begin{equation*}
  80:62::120:93.
\end{equation*}
The same follows,
just from the repetition of the multipliers $(1,3,2,4)$
in the steps of the Algorithm.
Indeed, we can write either of the fractions $\nicefrac{80}{62}$
and $\nicefrac{120}{93}$ as the continued fraction
\begin{equation*}
  1+\cfrac1{3+\cfrac1{2+\cfrac14}}.
\end{equation*}
In Greek, the Algorithm is
\textbf{anthyphaeresis} or
\enquote{alternating subtraction.}%%%%%
\footnote{The term derives ultimately from \gr{>anjufair'e-w}
(\emph{anthyphaire-\^o})
\enquote{alternately subtract,}
the verb that Euclid uses to describe his Algorithm.
The analysis is
  \gr{>ant'i}\thinspace+\thinspace\gr{<up'o}\thinspace+\thinspace\gr{a<ir'e-w}
(\emph{anti\thinspace+\thinspace hypo\thinspace+\thinspace haire-\^o}),
the core verb meaning \emph{take.}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{compactitem}
  \item
In Book \textsc v,
for arbitrary magnitudes,
Euclid gives the \textbf{Eudoxan} definition
of proportion,
whereby a ratio is effectively a Dedekind cut.
\item\sloppy
Before this was known,
there was an \textbf{anthyphaeretic} definition,
whereby the proportion
\begin{equation*}
  A:B::C:D
\end{equation*}
means the Euclidean Algorithm has the same steps,
whether applied to $A$ and $B$ or $C$ and $D$.%%%%%
\footnote{See Thomas \cite[pp.~504--9]{MR13:419a} or
Fowler \cite{MR2000c:01004}.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item
  The Euclidean definition is a simplification of this for numbers.
\end{compactitem}
The anthyphaeretic definition 
applies even to incommensurable magnitudes,%%%%%
\footnote{In particular,
there is no reason to think that the Eudoxan theory was
\enquote{developed to handle incommensurable magnitudes.}
Pengelley and Richman \cite[p.\ 199]{MR2204484} suggest that it was, 
even though they cite the book \cite{MR2000c:01004} of Fowler, 
who says, 
\enquote{I now disagree with everything in this line of interpretation}---% 
the line whereby the Pythagoreans based mathematics on commensurable magnitudes,
until the discovery of incommensurability,
whose problems were not resolved until the Eudoxan theory was formulated.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
such as the diagonal and side of a square,
as in Figure \ref{fig:DS},
\begin{figure}
  \centering
\psset{unit=6mm,labelsep=1pt}
  \begin{pspicture}(-1,-1)(11,10)
%\psgrid
\psset{PointSymbol=none}
\psset{PointName=none}
\pstGeonode(0,0)A(0,10)B(10,10)C(10,0)D
%    \pspolygon(0,0)(10,0)(10,10)(0,10)
\ncline AB
{\psset{linestyle=dotted}\ncline CD\ncline DA}
\pstInterLC ACCBEF
\pstInterLC BAEAGH
\pstArcOAB CBA
\pstArcnOAB GEA
\pstInterLC ABGEKL
\ncline AC
{\psset{offset=-1.4}
\ncline%[offset=-1cm]% some kind of conflict here
{|-|}AC
\nbput{$D$}}
\ncline EG\naput{$A$}
\ncline BC\nbput{$S$}
\ncline EC\naput{$S$}
\ncline AE\nbput{$A$}
\ncline AK\naput{$B$}
\ncline KG\naput{$A$}
\ncline GB\naput{$A$}
  \end{pspicture}
  \caption{Anthyphaeresis of diagonal and side of square}
  \label{fig:DS}
\end{figure}
where
\begin{align*}
  D&=S+A,&
S&=A\cdot2+B,
\end{align*}
and ever after, the less goes twice into the greater,
so that
\begin{equation*}
  S:A::A:B,
\end{equation*}
and also the ratio $D:S$ is independent of $D$.

Understanding proportion is important
because Euclid uses it to prove
\begin{compactenum}[(1)]
  \item
commutativity of multiplication, and
\item
\textbf{Euclid's Lemma,}
that a prime measuring a product measures one of the factors.
\end{compactenum}
Under the Euclidean definition,
the proofs are rigorous.

%\chapter{Theorems}

\section{Commutativity}\label{sect:comm}

In Book \textsc{vii} of the \emph{Elements,}
from either the anthyphaeretic 
or the Euclidean definition of proportion of numbers,
we obtain \textbf{Propositions 5--8:}
\begin{equation*}
  A:B::C:D\implies A:B::A\pm C:B\pm D.
\end{equation*}
Repeated application gives \textbf{Proposition 9:}
\begin{equation*}
  E:F::E\cdot\coB:F\cdot\coB.
\end{equation*}
This gives, \emph{by transitivity,} \textbf{Proposition 10:}
\begin{equation}\label{eqn:7.10}
  E\cdot\coA:F\cdot\coA::E\cdot\coB:F\cdot\coB.
\end{equation}
Automatically, if $\coA$ and $\coB$ are coprime,
\begin{equation}\label{eqn:auto}
  E\cdot\coA:E\cdot\coB::F\cdot\coA:F\cdot\coB.
\end{equation}
Since every proportion can be written in this form,
the implication $\eqref{eqn:auto}\Rightarrow\eqref{eqn:7.10}$
is \textbf{Proposition 13,} \textbf{Alternation:}
\begin{equation*}
  A:B::C:D\implies A:C::B:D.
\end{equation*}
Since
\begin{equation}\label{eqn:BA}
  1:A::B:B\times A,
\end{equation}
by Alternation, $1:B::A:B\times A$,
so by symmetry
\begin{equation}\label{eqn:AB}
  1:A::B:A\times B.
\end{equation}
Comparing \eqref{eqn:BA} and \eqref{eqn:AB} 
yields \textbf{Proposition 16,}
\textbf{Commutativity:}
\begin{equation*}
  A\times B=B\times A.
\end{equation*}

\section{Euclid's Lemma}\label{sect:lemma}

\textbf{Proposition 17} is like 9:
\begin{equation}\label{eqn:7.17}
  C:D::C\times A:D\times A.
\end{equation}
Hence \textbf{Proposition 18,} 
$C:D::A\times C:A\times D$,
or with different letters,
\begin{equation}\label{eqn:7.18}
  A:B::C\times A:C\times B.
\end{equation}
From \eqref{eqn:7.17}, \eqref{eqn:7.18},
and \emph{transitivity,}
we get \textbf{Proposition 19,}
\begin{equation*}
  A:B::C:D\iff D\times A=C\times B,
\end{equation*}
the ``Eudoxan'' definition of proportion of numbers.
\textbf{Proposition 20} is that,
if $A$ and $B$ are the least $X$ and $Y$ such that
\begin{equation*}
  X:Y::C:D,
\end{equation*}
then $A$ measures $C$,%%%%%
\footnote{And $B$ measures $D$ the same number of times,
as Euclid says.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
for by \emph{Alternation}
\begin{equation*}
  A:C::B:D,
\end{equation*}
and so $A$ is the same part or parts of $C$
that $B$ is of $D$;
but it cannot be parts, by minimality.%%%%%
\footnote{Mazur says,
\enquote{Now I don't quite follow Euclid's proof of this pivotal proposition, 
and I worry that there may be a tinge of circularity 
in the brief argument given in his text} \cite[p.\ 243]{Mazur-Th-pub};
then he cites Pengelley and Richman \cite{MR2204484}.
Mazur's own proof uses what he calls Propositions 5 and 6,
though 7 and 8 are also needed,
to conclude $A:B::C-A\cdot\coA:D-B\cdot\coA$.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Here $A$ and $B$ are also coprime.%%%%%
\footnote{This is Proposition 22, but we shall not need it.}
%%%%%%%%%%%%%%%%%%%%%%%%
The converse is \textbf{Proposition 21.}

Immediately from the definitions,
\textbf{Proposition 29:}
every prime is coprime with its every non-multiple.

For \textbf{Proposition 30,} \textbf{Euclid's Lemma,}
suppose a prime $P$ measures $A\times B$, 
so that for some $C$,
\begin{equation*}
  P\times C=A\times B.
\end{equation*}
By 19 (the Eudoxan definition),
\begin{equation*}
P:A::B:C.  
\end{equation*}
If $P$ does not measure $A$,
then
\begin{compactitem}
  \item
 $P$ and $A$ are coprime by 29,
\item
they are the least numbers having their ratio by 21,
\item
$P$ measures $B$ by 20.%%%%%
\footnote{For Mazur \cite[p.\ 242]{Mazur-Th-pub},
\enquote{that if a prime divides a product of two numbers, 
it divides\lips one of them, 
is essentially Euclid's Proposition 24 of Book VII.}
Strictly,
this is that the product of numbers prime to a number
is also prime to it.
Like that of 30, 
the proof relies on 20, which again for Mazur is problematic.}
%%%%%%%%%%%%%%%%%%%%
\end{compactitem}

Euclid uses words alone to describe proportions.
This could be because the Ancients were more used to 
\emph{hearing} mathematics than seeing it.
Modern commentators use fractions and the equals sign.
I have tried to preserve the distinction between proportions and equations,
while making Euclid's rigor \emph{visible}
in the way that we Moderns are used to.

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

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

\bibitem{Apollonius-Heiberg}
{Apollonius of Perga}.
\newblock {\em Apollonii {P}ergaei Qvae {G}raece Exstant Cvm Commentariis
  Antiqvis}, volume~I.
\newblock Teubner, 1974.
\newblock Edited with Latin interpretation by I. L. Heiberg. First published
  1891.

\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{Borovik-Meta}
Alexandre Borovik.
\newblock Metamathematics of elementary mathematics.
\newblock \url{www.matematikdunyasi.org/yazokulu/borovik_1b.pdf}, July 2008.
\newblock Lecture at the Nesin Mathematics Village, {\c S}irince, Sel{\c c}uk,
  Izmir, Turkey.

\bibitem{Coxeter-Geometry}
H.~S.~M. Coxeter.
\newblock {\em Introduction to Geometry}.
\newblock John Wiley, New York, second edition, 1969.
\newblock First edition, 1961.

\bibitem{Dee}
John Dee.
\newblock Mathematicall preface.
\newblock In {\em The Elements of Geometrie of the most auncient Philosopher
  {E}uclid of Megara}. Iohn Daye, London, 1570.
\newblock Facsimile in \url{pdf} format.

\bibitem{Descartes-Geometry}
Ren{\'e} Descartes.
\newblock {\em The Geometry of {R}en{\'e} {D}escartes}.
\newblock Dover Publications, 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{Descartes-Geometrie}
Ren{\'e} Descartes.
\newblock {\em La G{\'e}om{\'e}trie}.
\newblock Jacques Gabay, Sceaux, France, 1991.
\newblock Reprint of Hermann edition of 1886.

\bibitem{MR17:814b}
Euclid.
\newblock {\em The Thirteen Books of {E}uclid's \emph{{E}lements}}.
\newblock Dover Publications, New York, 1956.
\newblock Translated from the text of {H}eiberg with introduction and
  commentary by Thomas L. Heath. In three volumes. Republication of the second
  edition of 1925. First edition 1908.

\bibitem{Euler}
Leonard Euler.
\newblock Introductio in analysin infinitorum.
\newblock \url{www.17centurymaths.com/contents/introductiontoanalysisvol1.htm},
  accessed March 31, 2019.
\newblock Translated and annotated by Ian Bruce.

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

\bibitem{de-Witt}
Albert~W. Grootendorst, editor.
\newblock {\em Jan de Witt's Elementa Curvarum Linearum, Liber Primus}.
\newblock Springer, 2000.
\newblock Text, translation, introduction, and commentary. First edition 1659.

\bibitem{Hamilton}
Hugh Hamilton.
\newblock {\em De Sectionibus Conicis. Tractatus Geometricus. In quo, ex natura
  ipsius coni, sectionum affectioens [sic] facillime deducuntur. Methodo nova}.
\newblock London, 1758.
\newblock \url{archive.org/details/desectionibusco00hamigoog/page/n7}, accessed
  April 1, 2019.

\bibitem{Hamilton-Eng}
Hugh Hamilton.
\newblock {\em A geometrical treatise of the conic sections, in which the
  properties of the sections are derived from the nature of the cone, in an
  easy manner, and by a new method}.
\newblock J. Nourse, London, 1773.
\newblock Translated from the Latin original into English.
  \url{catalog.hathitrust.org/Record/000208065}, accessed March 31, 2019.

\bibitem{MR654679}
Thomas Heath.
\newblock {\em A History of {G}reek Mathematics. {V}ol. {I}. {F}rom {T}hales to
  {E}uclid}.
\newblock Dover Publications, New York, 1981.
\newblock Corrected reprint of the 1921 original.

\bibitem{Mazur-Th-pub}
Barry Mazur.
\newblock How did {T}heaetetus prove his theorem?
\newblock In P.~Kalkavage and E.~Salem, editors, {\em The Envisioned Life:
  {E}ssays in honor of {E}va {B}rann}. Paul Dry Books, 2007.
\newblock \url{www.math.harvard.edu/~mazur/preprints/Eva.pdf}, accessed
  September 20, 2012.

\bibitem{Pappus}
{Pappus of Alexandria}.
\newblock {\em Pappus {A}lexandrini Collectionis Quae Supersunt}, volume~II.
\newblock Weidmann, Berlin, 1877.
\newblock E libris manu scriptis edidit, Latina interpretatione et commentariis
  instruxit Fridericus Hultsch.

\bibitem{Jones-whole}
{Pappus of Alexandria}.
\newblock {\em Book 7 of the {\itshape Collection}. {P}art 1. {I}ntroduction,
  Text, and Translation. {P}art 2. {C}ommentary, Index, and Figures}.
\newblock Springer Science+Business Media, New York, 1986.
\newblock Edited With Translation and Commentary by Alexander Jones.

\bibitem{Pappus-Pierce-2}
{Pappus of Alexandria}.
\newblock {{\"O}}klid'in {P}orizmalar'{\i} {\.i}{\c c}in {D}erleme'nin yedinci
  kitab{\i}'n{\i}n 38 lemmas{\i}ndan {\.i}lk 19 lemmas{\i}.
\newblock \url{mat.msgsu.edu.tr/~dpierce/Dersler/Geometriler/}, March 2019.
\newblock The First 19 of the 38 Lemmas for Euclid's Porisms in the Seventh
  Book of the Collection. Turkish translation by David Pierce.

\bibitem{MR2204484}
David Pengelley and Fred Richman.
\newblock Did {E}uclid need the {E}uclidean algorithm to prove unique
  factorization?
\newblock {\em Amer. Math. Monthly}, 113(3):196--205, 2006.

\bibitem{Pierce-Thales-9}
David Pierce.
\newblock Thales and the nine-point conic.
\newblock {\em The De Morgan Gazette}, 8(4):27--78, 2016.
\newblock \url{education.lms.ac.uk/2016/12/thales-and-the-nine-point-conic/},
  accessed June 1, 2017.

\bibitem{MR3680853}
David Pierce.
\newblock On commensurability and symmetry.
\newblock {\em J. Humanist. Math.}, 7(2):90--148, 2017.

\bibitem{Rosenfeld}
Boris Rosenfeld.
\newblock The up-to-date complete scientific {E}nglish translation of the great
  treatise of {A}pollonius of {P}erga with the most comprehensive commentaries.
\newblock \url{http://www.personal.psu.edu/sxk37/Apollonius.html} (hosted by
  Svetlana Katok), accessed April 8, 2019.

\bibitem{MR13:419a}
Ivor Thomas, editor.
\newblock {\em Selections Illustrating the History of {G}reek Mathematics.
  {V}ol. {I}. {F}rom {T}hales to {E}uclid}.
\newblock Number 335 in Loeb Classical Library. Harvard University Press,
  Cambridge, Mass., 1951.
\newblock With an English translation by the editor.

\bibitem{Wallis}
John Wallis.
\newblock {\em De Sectionibus Conicis, Nova Methodo Expositis, Tractatus}.
\newblock Oxford, 1655.
\newblock \url{archive.org/details/bub_gb_03M_AAAAcAAJ/page/n4} accessed April
  1, 2019.

\end{thebibliography}


\end{document}