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\begin{document}
\title{Euclid Mathematically and Historically}
\subtitle{Notes for a colloquium}
\author{David Pierce}
\date{Friday, March 9, 2018\\
Corrected Sunday, June 10, 2018}
\publishers{Mathematics Department\\
Mimar Sinan Fine Arts University\\
Istanbul\\
\url{mat.msgsu.edu.tr/~dpierce/}\\
\url{david.pierce@msgsu.edu.tr}}

  \maketitle

\chapter*{Preface}

These notes were prepared for a general colloquium 
in the mathematics department of Bilkent University,
Ankara, on Wednesday, March 7, 2018, 3:40--4:30 \textsc{pm}.
I had been invited by Sinan Sert\"oz,
who was engaged in translating all of Euclid's \emph{Elements} into Turkish
\cite{Euclid-Sertoz}.

As expected, there was not time during the colloquium for all details.
Omitted entirely were \S\S\ 
\ref{sect:language}, \ref{sect:law}, and \ref{sect:lemma}.

I started the talk by noting the assertion in the abstract
that Euclid was in some ways more rigorous than we.
An example was his proof of commutativity of multiplication of numbers.
Few students today may see this proved,
from the Peano Axioms perhaps.%%%%%
\footnote{If they do see a proof,
it might be as in Landau \cite{MR12:397m}.
I did not go into how Peano himself \cite{Peano}
misunderstood the axioms, 
in the way that Landau discusses in his
\enquote{Preface for the Teacher,} 
and I discuss in \enquote{Induction and Recursion}
\cite{Pierce-IR}. 
Proof by induction alone does not justify definition by recursion.  
Dedekind, writing earlier \cite{MR0159773},
understood this.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Concerning equality in Euclid,
my example was not the parallelograms in the same parallels
of Propositions \textsc i.35 and 36,
as in Figures \ref{fig:I.35} and \ref{fig:equal-bases},
but the \enquote{complementary} parallelograms
in the one large parallelogram of Proposition \textsc i.43, 
as in Figure \ref{fig:I.43}.
I added this example to the present text after the talk.

During the talk,
I used the example for an excuse to write out
Proposition XII.2 of the \emph{Elements} in Greek.
This proposition uses the article in three different forms,
and Proposition \textsc i.43 
illustrates the notational usefulness of the gendered article,
as discussed in my article 
\enquote{Abscissas and Ordinates} \cite{MR3312989}.

I ended the talk with the proof of commutativity in \S\ref{sect:comm}.

I brought some props to the talk:
\begin{compactenum}
  \item
Three chopsticks, marked off into four parts each;
and four chopsticks, cut to three-quarter length and marked off into thirds.
The sticks are supposed to show that it is not absolutely trivial
that three fours are equal to four threes.
\item
A triangle, providing both a straightedge and a right angle
as allowed by Euclid's postulates.
\item
A cord, as allowed by Archimedes's postulate.
\item
Bottle caps, for distinguishing measuring from dividing;
I had used them to illustrate a blog article 
about commutativity of multiplication \cite{Pierce-geo-numbers}.
\end{compactenum}

In the formal question period after the talk,
Laurence Barker asked whether Euclid was rigorous
in the sense of rejecting arguments that he knew, but considered invalid.
On the spot, 
I could only marvel at how Euclid found it worthwhile 
to give what for us would be an epsilon--delta proof 
that circles vary as the squares on their diameters.
The Egyptians defined the area of a quadrilateral
as the product of the averages of the lengths of opposite sides;
Euclid showed how any straight-edged figure
was exactly equal to a rectangle on a given base.

I review how I became interested in all of this.
In high school I wanted to read Euclid,
rather than our textbook by Weeks and Adkins \cite{Weeks-Adkins}.
I read some of the \emph{Elements} \cite{MR17:814b} on my own,
and more of them as a freshman at St John's College
\cite{Pierce-SJC}.
As a sophomore there I read Descartes \cite{Descartes-Geometry},
and I have found his \enquote{digitization} of geometry
to be directly useful for at least one piece of modern research
\cite{MR2505433}.  In Istanbul,
I helped to create, and I often teach, 
a course in which freshmen read and present 
Book \textsc i of the \emph{Elements} \cite{Oklid-2014}.
I began studying Euclid's number-theory
to see if it could be incorporated into a later course for our students.
I concluded that it could not.
Euclid's way of thinking was too different from ours.
I discovered that even modern mathematicians,
as far as I could tell, were misreading Euclid.
My study led to much of what is in the present notes.

When the abstract that I submitted
was included in announcements of the colloquium,
the definite article \eng{The}
had been inexplicably inserted in front of the opening \enquote{Mathematics,}
though the initial capital letter of this had been retained.

A briefer abstract might be that,
when we translate a proportion of numbers into an equation of fractions,
%like $a/b=c/d$, 
we may overlook the subtlety with which Euclid works.
He even proves, rigorously, 
something so \enquote{obvious} 
as the commutativity of multiplication of numbers.
Few of us today may ever do this, 
in our Cartesian drive to get new results.

\tableofcontents
\listoffigures

\addchap{Abstract}

%\minisec{Abstract}
Mathematics can save the world,
not through a theorem or application,
but as an example of an endeavor where
\begin{compactenum}[(1)]
  \item
differences can and must be resolved peacefully;
\item
dissent is encouraged; and 
\item
wealth has to be shared to be recognized, 
and then only gains in value.
\end{compactenum}
Nonetheless, in trying to share in the wealth of mathematical knowledge,
we have to think historically as well as mathematically,
particularly when the wealth comes down to us
from Ancients such as Euclid.

The thirteen books of Euclid's \emph{Elements} 
have given us a paradigm of mathematical exposition,
with axioms and postulates at the beginning,
definitions as needed,
and propositions stated and proved.
We may have improved on the model;
but sometimes we misunderstand it.
In Greek, the root meaning of \gr{gewmetr'ia}, \emph{geometria},
is surveying.
Herodotus of Halicarnassus (today's Bodrum)
said the Greeks had learned \emph{geometria} from the Egyptians.
However, the mathematics that Euclid went on to work out,
presumably in Alexandria,
did not follow naturally from a need to measure land 
lost to the annual flooding of the Nile.
Neither does Euclid's meaning follow naturally 
from a superficial reading of his words today.
\emph{Anistoresy,} a-historicity,
as for example concerning equality and proportion,
can lead to misunderstanding and even misdiagnosis of logical error in Euclid.
In some ways Euclid's mathematics is more rigorous than ours.

\chapter{Circles}

Here are three theorems about circles,
or one theorem expressed three ways.
\begin{compactenum}
  \item
In school today
we learn a formula for the area of a circle:
\begin{equation}\label{eqn:A}
  A=\uppi\,r^2.
\end{equation}
The area $A$ of the circle is said to be the number $\uppi$ (\enquote{pi})
times the \enquote{square} of the radius $r$.%%%%%
\footnote{I was taught the rule in precisely the form \eqref{eqn:A}
from Weeks and Adkins \cite{Weeks-Adkins} in 1981.  
Even in Turkish, where the word for radius is \tur{yar\i\c cap\i}
\enquote{half diameter,}
the area of a circle is given as $\uppi\,r^2$,
at least in the two sources that I consulted
\cite{demirtas,YGS-LYS}.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item
Archimedes \enquote{squares} the circle as in Figure \ref{fig:Archimedes},
\begin{figure}
\psset{unit=13mm}
%  \mbox{}\hfill
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    \pscircle(1,1)1
\psline(1,1)(1,2)
\uput[r](1,1.5){$r$}
\uput[dr](! 2 sqrt 2 div dup 1 add exch 1 sub neg){$C$}
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\uput[r](0,1.5){$r$}
\uput[d](3.14,1){$C$}
\end{pspicture}
%\hfill\mbox{}
  \caption{Triangle equal to circle}
  \label{fig:Archimedes}
\end{figure}
showing what we might write as
\begin{equation*}
  A=\frac12\,r\,C,
\end{equation*}
but he uses words:
\begin{quote}
  Every circle is equal to a right triangle, 
where the radius is equal to one of the legs,
and the circumference to the base.
//
\gr{P~as k'uklos >'isos >est`i trig'wnw| >orjogwn'iw|,
o<~u <h m`en >ek to~u k'entrou >ish mi~a| t~wn per`i t`hn >orj'hn,
<h d`e per'imetros t~h| b'asei.}%%%%%
\footnote{The Greek is from Heiberg's text 
\cite[p.\ 258]{Heiberg-Archimedes-I}; the English is my translation.  
\enquote{The radius} would be more literally \enquote{the from-the-center.}
Also \enquote{where} would be more literally \enquote{of which,}
but then, in English, this would seem to apply to \enquote{the radius}
rather than \enquote{the legs.}
For Archimedes, the legs are \enquote{the [sides] about the right [angle],}
but I do not know why,
having specified one of these,
he says then \enquote{base} rather than \enquote{other leg.}  
Heath's translation is looser,
but perhaps clearer to the modern mind: \enquote{The area of any circle
is equal to a right-angled triangle in which 
one of the sides about the right angle is equal to the radius,
and the other to the circumference, of the circle} 
\cite[p.\ 91]{Heath-Archimedes}.
The introduction of \emph{area} may be misleading.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{quote}
\item
The rule\label{rule} 
of Proposition 2 of Book \textsc{xii} of Euclid's \emph{Elements} is,
%\begin{quote}\centering
%Οἱ κύκλοι πρὸς ἀλλήλους εἰσὶν ὡς τὰ ἀπὸ τῶν διαμέτρων τετράγωνα.
%\gr{O<i k'ukloi pr`os >all'hlous e>is`in\\
%  <ws t`a >ap`o t~wn diam'etrwn tetr'agwna.}
%\end{quote}
\begin{quote}\centering
  \begin{tabular}{*5c}
    \gr{O<i}&\gr{k'ukloi}&\gr{pr`os}&\gr{>all'hlous}&\gr{e>is`in}\\
    {The}&{circles}&{to}&{one another}&{are}
  \end{tabular}

  \begin{tabular}{*6c}
    \gr{<ws}&\gr{t`a}&\gr{>ap`o}&\gr{t~wn}&\gr{diam'etrwn}&\gr{tetr'agwna.}\\
    {as}&{the}&{on}&{the}&{diameters}&{squares.}
  \end{tabular}
\end{quote}
See Figure \ref{fig:12.2}.
\begin{figure}
  \centering
  \mbox{}\hfill
  \begin{pspicture}(-1,-1)(1,1)
    \pscircle(0,0)1
    \pspolygon(-1,-1)(1,-1)(1,1)(-1,1)
  \end{pspicture}
\hfill\psset{unit=1.5cm}
  \begin{pspicture}(-1,-1)(1,1)
    \pscircle(0,0)1
    \pspolygon(-1,-1)(1,-1)(1,1)(-1,1)
  \end{pspicture}
\hfill\mbox{}
  \caption{Circles and the squares on their diameters}
  \label{fig:12.2}
\end{figure}
More smoothly in English, this is,
\begin{quote}\centering
  Circles are to one another\\
as the squares on the diameters.
\end{quote}
Other ways to say this include,
\enquote{Circles have the ratio of the squares on their diameters.}%%%%%
\footnote{The Greek again is from Heiberg \cite{Euclid-Heiberg-IV},
and the smooth, but now still literal, translation is Heath's \cite{bones}.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{compactenum}
To say these three theorems are one theorem
is itself a theorem,
perhaps more sophisticated than is generally understood.
\begin{compactenum}
  \item
The modern theorem is about so-called real numbers.
\item
Archimedes's theorem is about geometrical figures,
except that there is no properly geometrical way to construct the triangle.
\item
Euclid's theorem is purely geometrical.%%%%%
\footnote{By Euclid here,
I mean simply the author or authors of the collection known as
Euclid's \emph{Elements.}
I make no assertion about what is original with Euclid.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{compactenum}
I suggest that Euclid is more rigorous than we are,
in part because he does not make all of the unexamined assumptions
that lie behind our blithe assertion that the three theorems are the same.


  \section{Equations}
  
What is the \emph{meaning} of \eqref{eqn:A}, namely $A=\uppi\,r^2$?
The formula allows us to perform a computation.
Given a radius, in decimal notation, 
we can punch its digits into a pocket calculator,
punch a few more keys, and get a result for the area.

The formula \eqref{eqn:A} assumes 
that geometrical objects can indeed be assigned numbers.
This is like the assumption of some art-museum visitors
that every painting on display has a numerical value in dollars.

The \fbox{$\uppi$} key on the calculator
supplies an exact numerical value, but not that of $\uppi$.
This \emph{has} an exact value,
but it is irrational, even transcendental.
In principle, we can compute it as finely as we want,
as by using the Leibniz formula,
\begin{align*}
  \frac{\uppi}4
&=1-\frac13+\frac15-\frac17+\dotsb\\
&=\frac2{1\cdot3}+\frac2{5\cdot7}+\frac2{9\cdot11}+\dotsb,
\end{align*}
which we can derive from
\begin{equation*}
  \frac{\uppi}4=\arctan(1)=\int_0^1\frac{\mathrm dx}{1+x^2}
=\lim_{t\to1^-}\int_0^t\sum_{k=0}^{\infty}(-1)^kx^{2k}\mathrm dx.
\end{equation*}
As for the formula \eqref{eqn:A} itself,
we may declare that a circle of radius $r$
is defined by the formula
\begin{equation}\label{eqn:x^2}
  x^2+y^2=r^2,
\end{equation}
so that $A$ is by definition
\begin{equation*}
  2\int_{-r}^r\sqrt{r^2-x^2}\;\dee x.
\end{equation*}
We compute this by letting
\begin{align*}
  x&=r\sin u,&\dee x&=r\cos u\;\dee u,
\end{align*}
so that
\begin{equation*}
  A=2r^2\int_{-\uppi/2}^{\uppi/2}\cos^2 u\;\dee u
=r^2\int_{-\uppi/2}^{\uppi/2}(1+\cos2u)\;\dee u
=r^2\,\uppi.
\end{equation*}
Perhaps this assures us more that we have learned calculus properly
than that Euclid and Archimedes were correct.
This is the idea of Russell and Whitehead,
who write in the Preface of the \emph{Principia Mathematica}
\cite[p.\ v]{PM},
\begin{quote}
  the chief reason in favour of any theory on the principles of mathematics
must always be inductive,
\emph{i.e.}\ it must lie in the fact that the theory in question
enables us to deduce ordinary mathematics.
\end{quote}
I propose that Euclid can be understood as ordinary mathematics here.
The \enquote{analytic} style of mathematics
introduced by Descartes \cite{Descartes-Geometrie}
is inductive in the sense that it is justified by \emph{working}---%
working to \enquote{explain} 
such mathematics as is already known from the Ancients.

According to Kline in
\emph{Mathematical Thought from Ancient to Modern Times}
\cite[pp.\ 87--8]{MR0472307},
\begin{quote}
  A critical study of Euclid, 
with, of course, the advantage of present insights,
shows that he uses dozens of assumptions that he never states
and undoubtedly did not recognize.
\end{quote}
Kline seems to assume that we do mathematics better today.
However, who today recognizes all of the assumptions
that go into our saying that the formula \eqref{eqn:x^2},
namely $x^2+y^2=r^2$,
defines a circle of radius $r$?%%%%%
\footnote{I quote Russell and Whitehead with approval,
and I suggest that Euclid is the paradigm for ordinary mathematics.
Kline \cite[p.\ 1005]{MR0472307} however quotes Russell as saying in 1902
\cite{Russell-Euclid},
\enquote{It has been customary when Euclid, considered as a text-book, 
is attacked for his verbosity or his obscurity or his pedantry, 
to defend him on the ground that his logical excellence is transcendent, 
and affords an invaluable training to the youthful powers of reasoning. 
This claim, however, vanishes on a close inspection. 
His definitions do not always define, 
his axioms are not always indemonstrable, 
his demonstrations require many axioms of which he is quite unconscious. 
A valid proof retains its demonstrative force when no figure is drawn, 
but very many of Euclid's earlier proofs fail before this test\lips
the value of his work as a masterpiece of logic 
has been very grossly exaggerated.}
The same criticisms apply to modern mathematics,
at least as it is taught,
though we think we can correct the defects, if we really have to.
The dogmatic assertion about proofs and figures is just that.
\emph{We} think proofs ought be to be \enquote{digital}
(or digitizable, so that a computer can check them);
the Ancients simply do not.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

I propose that all of the hidden assumptions of mathematics \emph{today}
are (or contribute to) why people are often afraid of it.
People equate mathematics with equations.
Perhaps what they fear is equations.
Equations \emph{ought} to be frightening.
They hide a lot of difficulty.
They let you do something, without really knowing what it is.

\section{Decimals}

In equation \eqref{eqn:A}, namely $A=\uppi\,r^2$,
the letters stand for \emph{positive real numbers.}
Such numbers have \emph{decimal expansions.}
We blithely performs computations with them;
and yet, strictly speaking,
there is no proper algorithm for this.

Suppose an \enquote{oracle} feeds us the digits of a number,
one by one, and the first few digits are 0.33333.
Given the job of multiplying by 3,
we shall never know whether the answer starts out as 1.0
or as 0.9,
unless at some point the oracle actually gives us a digit other than 3.

This bothers almost nobody today.%%%%%
\footnote{An exception is David Fowler \cite{MR1186456},
who gives the example (which he attributes to Christopher Zeeman)
  of computing $1.\overline 2\times0.\overline{81}$.
  The factors being $\nicefrac{11}9$ and $\nicefrac{81}{99}$,
  the product is $1$;
  but this digit does not arise from any finite computation
$1.2\dots2\times0.81\dots81$.}
%%%%%%%%%%%%%%%%%%%%%%%%
It bothers Euclid.
For him, there is no decimal system that he declines to use;
but he declines to use anything like what we call $\uppi$.
For him, the ratio of two circles
is the \emph{same} as the ratio of the squares on their diameters,
and therefore on their radii.
We may write this as
\begin{equation*}
  A:A_1::r^2:r_1{}^2.
\end{equation*}
By \textbf{alternation,}
which is Proposition 16 of Book \textsc v of the \emph{Elements,}
\begin{equation*}
  A:r^2::A_1:r_1{}^2.
\end{equation*}
Thus the ratio of a circle to the square on its radius 
is independent of the radius.
We can call this ratio $\uppi$;
but Euclid does not consider it,
presumably because there is no use for it in \emph{proving} anything.

\section{Postulates}

Archimedes does consider the ratio that we call $\uppi$.
He finds%%%%%
\footnote{As Archimedes himself puts it,
\gr{Pant`os k'uklou <h per'imetros t~hs diam'etrou triplas'iwn >est'i,
ka`i >'eti <uper'eqei >el'assoni m`en >`h <ebd'omw| m'erei t~hs diam'etrou,
me'izoni d`e >`h d'eka <ebdomhkostom'onois}
\cite[p.\ 263]{Heiberg-Archimedes-I},
\enquote{Of every circle, the circumference is triple the diameter,
and yet exceeds by less than the seventh part of the diameter,
and by more than ten seventy-first [parts].}
The big Liddell--Scott lexicon cites just this passage under
\gr{<ebdomhkost'omonos}, though other authors are cited
for the words for 72nd, 75th, and 73rd.
The 1925 Preface of Jones acknowledges the contribution of Heath
to the entries for mathematical terminology \cite[p.\ vii]{LSJ}.
For Archimedes's result,
Heath \cite[p.\ 93]{Heath-Archimedes} uses the mixed fractions $3\frac17$
and $3\frac{10}{71}$.
In Smyth's \emph{Greek Grammar}
\cite[\P 347, p.\ 103]{Smyth},
\enquote{seventieth} is \gr{<ebdomhkost'os},
\enquote{twentieth} is \gr{e>ikost'os},
but \enquote{twenty-first} is \gr{pr~wtos ka`i e>ikost'os}.
A reason to look at these words is that \emph{English} is strange
to use the suffix \eng{-first.}
Our word \eng{first} is cognate with \gr{pr~wtos} 
and is used as the ordinal form of \eng{one,}
but is in origin the superlative
of what is now the preposition \eng{for,}
which once meant \emph{before} \cite{CODEE}.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation*}
  \frac17<\uppi-3<\frac{10}{71}.
\end{equation*}
This needs
a \textbf{postulate:}%%%%%
\footnote{Archimedes calls it a 
\gr{lamban'omenon} \cite[p.\ 8--10]{Heiberg-Archimedes-I},
a \emph{thing taken};
Netz \cite[p.\ 36]{MR2093668} translates this as \eng{postulate.}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
not only is the chord shorter than the circular arc,
but the arc is shorter than the circumscribed angle,
so that, in Figure \ref{fig:arc},
\begin{figure}
  \centering
  \psset{unit=45mm,PointSymbol=none}
  \newcommand{\mywidth}{0.8}
  \begin{pspicture}(-\mywidth,0.9)(1.25,1.75)
%    \psgrid
    \pstGeonode[PointName={none,default,default},PosAngle={0,-90,-90}]
    (0,0)O(-\mywidth,1)A(\mywidth,1)B
    \pstArcOAB OBA
    %\ncline AB
\psset{PointName=none}
    \pstRotation[RotAngle=90] AO[N]
    \pstRotation[RotAngle=-90] BO[P]
    \pstInterLL[PosAngle=90,PointName=default] ANBPD
    \pstInterLC[PointNameA=C,PointNameB={},PosAngleA=90] DOOACG
    \ncline AD
    \ncline BD
\pstInterLC[PointNameA={},PointNameB={}] ABADHK
\pstTranslation DKB[L]
\pstInterLC[PointNameA={},PointNameB=E,PosAngleB=-90] ABKLME
\ncline AE
\pstHomO[HomCoef=0.5,PosAngle=-90,PointName=default]BE[F]
  \end{pspicture}
  \caption{Archimedes's postulate}\label{fig:arc}
  
\end{figure}
\begin{equation*}
  AB<ACB<ADB.
\end{equation*}

Four of \emph{Euclid's} five postulates
are more basic.%%%%%
\footnote{Even the word for a postulate is different here;
now it is \gr{a>'ithma} \cite[p.\ 8]{Euclid-Heiberg}, a \emph{demand}
or \emph{request.}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
They give us a toolkit,
comprising ruler, compass, and set square (or triangle), for
\begin{compactenum}[(1)]
\item
  drawing straight lines,
\item
  extending straight lines,
\item 
drawing circles, and
\item
checking whether angles are right.%%%%%
\footnote{Strictly,
the fourth postulate is that all right angles are equal to one another;
this is symbolized by the set square,
which carries a right angle from place to place.
There is no need to postulate that we can draw right angles,
since this can be proved as in Propositions 11 and 12 
of Book \textsc i of the \emph{Elements.}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{compactenum}
Practically speaking,
in place of ruler and compass,
we could use a cord, a string---a \emph{line.}%%%%%
\footnote{The word \eng{line} comes to English by two or three routes,
from Latin words related to \lat{linum} \emph{flax}
\cite{CODEE};
\eng{linen} means \emph{made of flax.}
In Greek, Turkish, and other languages,
a straight line is called for short a \enquote{straight};
in English, perversely, a \enquote{line.}
However, we are now considering this in the proper sense of a 
\emph{flexible} one-dimensional object,
not necessarily pulled taut.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We could use a line for measuring an arc
by wrapping the arc with the line,
then straightening it.
Archi\-medes's postulate tells us what would happen.
In Figure \ref{fig:arc},
by purely Euclidean means,
we can find the point $E$ so that
\begin{equation*}
  AE=AD+DB.
\end{equation*}
With our line, when we find the point $F$
so that
\begin{equation*}
  AF=ACB,
\end{equation*}
Archimedes tells us $F$ will lie between $B$ and $E$.
There is no \emph{construction} for this point,
other than by the method that we have described.
Practically speaking,
the method requires \emph{cutting out} the arc,
to give it an edge for holding the line.

Euclid's fifth postulate tells us what will happen
when we extend two straight lines;
but this is something that we already know how to do.%%%%%
\footnote{This observation tends to justify the assertion of Seidenberg
  \cite{Seidenberg-Euclid} that,
  unlike the method of Archimedes, that of Euclid is not axiomatic.
  But then Seidenberg's closing remark is odd:
  \enquote{\textsc{Bolyai,}
    writing to his father about his work on the theory of parallels, said:
    \senquote{From nothing I have created another wholly new world.}
    \textsc{Euclid}
    might very well have taken this proud declaration as his motto.}
However, earlier in the paper,  
Seidenberg's point seems to be that,
  unlike Archimedes, Euclid is \emph{not} making anything up:
  \enquote{It would never have occurred to him
    that to prove a theorem (\senquote{the arc is greater than the chord}),
  it is all right
  to generalize it, and then assume the generalization.}}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The straight lines will intersect,
so as to form the sides of a triangle,
\emph{provided that} 
the base angles will together be less than two right angles,
as in Figure \ref{fig:5th}.%%%%%
\footnote{The converse,
that any two angles of a triangle are less than two right angles, 
is Proposition 17 
of Book \textsc i of the \emph{Elements.}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
  \centering
\psset{unit=15mm,PointSymbol=none}
  \begin{pspicture}(-0.1,-0.4)(6.8,2.4)
%\psgrid
    \pstGeonode[PosAngle={-90,90}](0,0)A(1,2)B
\ncline AB
\pstRotation[RotAngle=100,PointName=none] BA
\pstRotation[RotAngle=-60,PointName=none] AB
\ncline B{A'}
\ncline A{B'}
\pstInterLL B{A'}A{B'}C
\psset{linestyle=dashed}
\ncline{A'}C
\ncline{B'}C
  \end{pspicture}
  \caption{Euclid's 5th postulate}
  \label{fig:5th}
\end{figure}

\section{Proportions}

Archimedes knows Euclid,
whose rule again (as on page \pageref{rule}) for the area of a circle is,
\begin{quote}\centering
  \begin{tabular}{*5c}
    \gr{O<i}&\gr{k'ukloi}&\gr{pr`os}&\gr{>all'hlous}&\gr{e>is`in}\\
    {The}&{circles}&{to}&{one another}&{are}
  \end{tabular}

  \begin{tabular}{*6c}
    \gr{<ws}&\gr{t`a}&\gr{>ap`o}&\gr{t~wn}&\gr{diam'etrwn}&\gr{tetr'agwna.}\\
    {as}&{the}&{on}&{the}&{diameters}&{squares.}
  \end{tabular}
\end{quote}
In several translations, this is:
\begin{compactdesc}
  \item[English:]\mbox{}
    \begin{compactitem}
      \item
    Circles are to one another as the squares on the diameters
    (Heath \cite{bones}).
      \item
    Circles are to one another as the squares on (their) diameters
    (Fitzpatrick \cite{euclid-fitzpatrick}).
  \end{compactitem}
\item[Turkish:]\mbox{}
    \begin{compactitem}
    \item
      \tur{Daireler, \c caplar\i ndaki kareler gibi birbirinedir} 
(my attempt at literalism).
\item
  \tur{Dairelerin  birbirine  oran\i\
    \c caplar\i\ \"uzerine \c cizilen  karelerin
    oran\i na e\c sittir} (Sert\"oz \cite{Euclid-Sertoz}).
    \end{compactitem}
  \item[Latin:]\mbox{}
    \begin{compactitem} 
\item
\lat{Circuli inter se sunt vt diametrorum quadrata}
(Commandinus \cite{Euclid-Commandinus}).
\item
\lat{Circuli eam inter se rationem habent quam quadrata diametrorum}
(Heiberg \cite{Euclid-Heiberg-IV}).
    \end{compactitem}
\end{compactdesc}
This is highly concentrated, like the formula \eqref{eqn:A},
or $A=\uppi\,r^2$.
However, the language is ordinary.
There are no
\begin{compactenum}[(1)]
  \item
algebraic notation, 
\item
number $\uppi$, or
\item
technical terms,
\end{compactenum}
except the words for circle, diameter, and square;
but these are what the proposition is about.

There is of course a technical meaning 
to the ordinary words.
Euclid can also say
\begin{compactitem}
\item
circles \textbf{are analogously as} the squares on their diameters;%%%%%
\footnote{Alternatively, 
\enquote{circles are \emph{analogous to} 
the squares on their diameters};
but I think this is less literal.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item
the \textbf{ratio} of two circles
is the \textbf{same} as the ratio of the squares on the diameters.
\end{compactitem}
There are two key terms in the Greek:
\begin{compactdesc}
  \item[\gr{L'ogos}]\mbox{}
    \begin{compactitem}
      \item
root meaning of \emph{speech}
\item
in Latin, \lat{ratio,} whence \eng{reason}
\item
gives \gr{>an'alogon} \eng{analogous(ly)} with prefix \gr{>an'a}
\begin{compactitem}
\item
noun form \gr{>analog'ia} \eng{analogy}
\item
in Latin, \lat{pro portione}
\eng{proportional(ly)}
\end{compactitem}
\end{compactitem}
\item[\gr{A>ut'os, -'h, -'o}]\mbox{}
  \begin{compactitem}
    \item
means \emph{same} or \emph{self}
\item
yields \eng{auto-,} as in \eng{automorphism}
\item
first element of
\gr{a>uj'enths}, 
a person who acts on his or her own behalf:
an \emph{authority}
\begin{compactitem}
  \item
\eng{authentic} in English
\item
in Turkish, both \tur{efendi} and \tur{otantik.}%%%%%
\footnote{I use Chantraine \cite{Chantraine} for Greek etymology;
Ni\c sanyan \cite{Nis-Et} for Turkish.
The word \gr{a>uj'enths}
can also mean a murderer.
In this case, the form \gr{a>uto'enths}
is also used, as in
lines 106--7 of \emph{Oedipus Rex},
spoken to Oedipus by his brother-in-law (and unknown uncle) Creon:
\enquote{The god commanded clearly: let some one /
punish with force this dead man's murderers}
\cite{Sophocles-I}---%
the dead man being Laius, killed by Oedipus.
The example is pointed out in the LSJ \cite{LSJ}.}
%%%%%%%%%%%%%%%%%%%%%%%%
\end{compactitem}
  \end{compactitem}
\end{compactdesc}
Not used here is \gr{>'isos -h, -on} \emph{equal,}
appearing in English as in \emph{isomorphism.}
Two ratios are never \emph{equal,}
but they may be the \emph{same.}%%%%%
\footnote{Turkish has the native \tur{oran} for ratio;
but the root meaning seems to be not speaking or thinking, 
but cutting or splitting.
As I understand from Ni\c sanyan \cite{Nis-Et},
\tur{oran} is related to \tur{orak} \enquote{sickle}
and to the verb \tur{yarmak} \enquote{split in two,}
whence \tur{yar\i m} \enquote{half};
also related is \tur{ara} \enquote{interval.}
I suppose the idea is that things can have a relation, a ratio,
only if they have been split apart.}

\section{Language}\label{sect:language}

We can do mathematics in any language.
However,
if we do borrow our technical terminology from other languages,
or even if we just give strange meanings to ordinary words,
then our subject becomes \textbf{esoteric:}
understood only by an inner circle.%%%%%
\footnote{This is a theme of my article
\enquote{Abscissas and Ordinates} \cite{MR3312989}.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Mathematics will always be esoteric, after a certain point.
Students will be able to dig to a certain depth, 
or climb to a certain height,
and then they will be overwhelmed.%%%%%
\footnote{The point was made by the poet Robert Fitzgerald
in his address to the graduating class of St John's College, Annapolis,
1984, at the end of my freshman year.
In 1983, Mary Kay Zuravleff, 
writer-in-residence at St Albans School in Washington,
told us students how 
she had studied mathematics as an undergraduate,
but encountered the Axiom of Choice as a stumbling-block.
In \emph{Mathematics: A Very Short Introduction} \cite[p.\ 132]{MR2147526},
Timothy Gowers observes that, in the learning of our subject,
\enquote{Every so often, 
a new idea is introduced which is very important 
and markedly more sophisticated than those that have come before, 
and each one provides an opportunity to fall behind. 
An obvious example is the use of letters to stand for numbers\lips}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

However, if we do mathematics only with our native tongue,
then the mathematical meaning of our words
is may harmonize better with their ordinary meaning.

Conscious reform may not achieve this.
It would be pointless to replace all uses of the Greek \emph{logos}
with \enquote{native} English words.
Such reform is why English has the needless \eng{foreword}
alongside the perfectly good \eng{preface.}


\chapter{Equality and sameness}

\section{Equality before the law}\label{sect:law}

In ordinary language,
equality is not sameness.
Article 7 of the Universal Declaration of Human Rights reads 
(emphasis mine),%%%%%
\footnote{%\sloppy
From \url{http://www.un.org/en/universal-declaration-human-rights/}, 
accessed September 2, 2016.
Proclaimed by the United Nations General Assembly 
in Paris on December 10, 1948.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{quote}
  All are \textbf{equal} before the law 
and are entitled without any discrimination to 
\textbf{equal} protection of the law. 
All are entitled to \textbf{equal} protection against any discrimination 
in violation of this Declaration 
and against any incitement to such discrimination.
\end{quote}
In Turkish, this is \tur{Madde~7} of the
\tur{\.Insan haklar\i\ evrensel beyannamesi:}%%%%%
\footnote{From 
\url{http://www.ohchr.org/EN/UDHR/Pages/Language.aspx?LangID=trk}, 
accessed September 2, 2016.}
\begin{quote}
\tur{Kanun \"on\"unde herkes \textbf{e\c sittir}
ve farks{\i}z olarak kanunun \textbf{e\c sit} korumas{\i}ndan 
istifade hakk{\i}n{\i} haizdir.
Herkesin i\c sbu Beyannameye ayk{\i}r{\i} 
her t\"url\"u ay{\i}rdedici mualeleye kar\c s{\i} 
ve b\"oyle bir ay{\i}rdedici muamele i\c cin yap{\i}lacak 
her t\"url\"u k{\i}\c sk{\i}rtmaya kar\c s{\i} 
\textbf{e\c sit} korunma hakk{\i} vard{\i}r.}
\end{quote}
The idea goes back to the Funeral Oration of Pericles, in Athens,
as recounted by Thucydides
 \cite[II.37, p.\ 145]{Thucydides}:
\begin{quote}
  Let me say that our system of government does not copy the institutions of our neighbours.
It is more the case of our being a model to others,
than of our imitating anyone else.
Our constitution is called a democracy
because power is in the hands not of a minority but of the whole people.
%καὶ ὄνομα μὲν διὰ τὸ μὴ ἐς ὀλίγους ἀλλ᾽ ἐς πλείονας οἰκεῖν δημοκρατία κέκληται:
%(\gr{ka`i >'onoma m`en di`a t`o m`h >es >ol'igous >all'' >es ple'ionas o>ike~in dhmokrat'ia k'eklhtai}).
\textbf{When it is a question of settling private disputes,
everyone is equal before the law;}
when it is a question of putting one person before another
in positions of public responsibility,
what counts is not membership of a particular class,
but the actual ability which the man possesses.%%%%%
\footnote{Pericles's idea of 
\enquote{what counts in positions of public responsibility}
may be compared with the findings of Jenny White:
\enquote{Hierarchies characterizing Turkish political life are brittle 
because they are not founded in organizational competence, 
in rules and procedures, in merit, 
or even on a relationship of trust between leader and followers. 
These networks instead constitute what I call a spindle autocracy, 
grounded in loyalty and obedience to a single, central person 
instead of the organization itself 
or to the concept of merit as a marker for leadership and promotion}
\cite{White-Jenny}.}
%%%%%%%%%%%%%%%%%%%%%%%%%
\end{quote}
The language of Thucydides is notoriously difficult,
but the English of Rex Warner seems not quite right;
Pericles says power is in the hands,
not of the \emph{whole} people,
but of the majority.%%%%%
\footnote{Thucydides's Greek is,
\gr{%χρώμεθα γὰρ πολιτείᾳ οὐ ζηλούσῃ τοὺς τῶν πέλας νόμους, 
qr'wmeja g`ar polite'ia| o>u zhlo'ush| to`us t~wn p'elas n'omous,
%παράδειγμα δὲ μᾶλλον αὐτοὶ ὄντες τισὶν ἢ μιμούμενοι ἑτέρους.   
par'adeigma d`e m~allon a>uto`i >'ontes tis`in >`h mimo'umenoi >et'erous.
%καὶ ὄνομα μὲν διὰ τὸ μὴ ἐς ὀλίγους   
ka`i >'onoma m`en di`a t`o m`h >es >ol'igous
%ἀλλ᾽ ἐς πλείονας οἰκεῖν δημοκρατία κέκληται:   
>all'' >es ple'ionas o>ike~in dhmokrat'ia k'eklhtai;
%μέτεστι δὲ κατὰ μὲν τοὺς νόμους 
\hlt{m'etesti d`e kat`a m`en to`us n'omous 
%πρὸς τὰ ἴδια διάφορα πᾶσι τὸ ἴσον, 
pr`os t`a >'idia di'afora p~asi t`o >'ison,}
%κατὰ δὲ τὴν ἀξίωσιν, 
kat`a d`e t`hn >ax'iwsin,
%ὡς ἕκαστος ἔν τῳ εὐδοκιμεῖ, 
>ws <'ekastos >'en tw| e>udokime~i,
%οὐκ ἀπὸ μέρους τὸ πλέον ἐς τὰ κοινὰ ἢ ἀπ᾽ ἀρετῆς προτιμᾶται,
o>uk >ap`o m'erous t`o pl'eon >es t`a koin`a >`h >ap'' >aret~hs protim~atai}
\cite{Thucydides-Oxford}.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The Turkish of Furkan Akderin
 \cite[p.\ 82]{Thucydides-Akderin}
seems more faithful in this regard,
though perhaps not in others:
\begin{quote}
\tur{Siyasi yap\i m\i z\i n kom\c sular\i m\i zdan bir fark\i\ yok.
Hatta onlardan \"ust\"un oldu\u gumuzu bile s\"oyleyebiliriz.
\c C\"unk\"u biz onlara g\"ore de\u gil,
onlar bize g\"ore yasalar\i n\i\ yap\i yorlar.
Bizim devletimiz az\i nl\i\u g\i n de\u gil \c co\u gunlu\u gun \c c\i karlar\i n\i\ g\"ozetmektedir.
Bu nedenle de ismi demokrasidir.
\textbf{Herhangi bir anla\c smazl\i k an\i nda herkes yasalar kar\c s\i s\i nda e\c sittir.}
Ancak konu  kamu ya\c sam\i na kat\i lmak oldu\u gunda kim di\u gerlerinden daha \"ust\"unse y\"onetimde o bulunur.}  
\end{quote}
As persons, we are equal before the law;
but we are not all the same person.

\section{Equality in Euclid}

In mathematics today, we confuse equality with sameness.
Equal numbers are the same number.
Euclid does not make this confusion.
Thus for example in Book \textsc i of the \emph{Elements,}
Proposition 35 is that parallelograms on the \textbf{same} base
in the same parallels are \textbf{equal} to one another,
as $A+B$ and $B+D$ in Figure \ref{fig:I.35}.
\begin{figure}
  \centering
\psset{unit=12mm}
  \begin{pspicture}(-1.2,0)(5.5,3)
%\psgrid
\newcommand{\myparam}{1.8}
\newcommand{\myheight}{3}
\psset{PosAngle=90,PointSymbol=none,PointName=none}    
\pstGeonode[PosAngle={-90,-90,90,90}](0,0)S(3,0)T
(! \myparam\space \myheight)V(! \myparam\space 1.5 add \myheight)E
\pstTranslation TVS[R]
\pstTranslation SET[F]
\ncline RS\ncline ST\ncline TV\ncline SE\ncline TF\ncline RF
\pstInterLL[PosAngle=-95] SETVG
\pstMiddleAB RSH
\pstMiddleAB STM
\pstMiddleAB TFP
\psset{PointName=default,PointSymbol=none,PointNameSep=-1pt}    
\pstHomO[HomCoef=0.278]HE[A]
\pstHomO[HomCoef=0.333]MG[B]
\pstHomO[HomCoef=0.778]HE[C]
\pstHomO[HomCoef=0.278]PV[D]
  \end{pspicture}
  \caption{Parallelograms on the same base}
  \label{fig:I.35}
\end{figure}
For $A+C$ and $C+D$ are \textbf{congruent} triangles,
and therefore
\begin{align*}
  A+C&=C+D,&
A&=D,&
A+B&=B+D.
\end{align*}
Congruence \emph{implies} equality, but not conversely.

Another example is Proposition 43,
as in Figure \ref{fig:I.43},
\begin{figure}
  \centering
%\psset{unit=12mm}
  \begin{pspicture}(-1,0)(7,4)
%\psgrid
\psset{PosAngle=90,PointSymbol=none,PointName=none}    
\pstGeonode(0,0)G(7,0)H(-1,4)K
\ncline GH\ncline GK\ncline HK
\pstTranslation GHK[L]
\ncline HL\ncline KL
\pstHomO[HomCoef=0.4]KH[M]
\pstTranslation GHM[N]
\pstInterLL MNGKP
\pstInterLL MNHLQ
\ncline PQ
\pstTranslation GKM[R]
\pstInterLL MRGHS
\pstInterLL MRKLT
\ncline ST
\pstMiddleAB PMU
\pstMiddleAB TMV
\pstMiddleAB SHW
\pstMiddleAB HQX
\psset{PointName=default,PointSymbol=none,PointNameSep=-1pt}    
\pstHomO[HomCoef=0.333]UK[A]
\pstHomO[HomCoef=0.5]PS[B]
\pstHomO[HomCoef=0.333]WM[C]
\pstHomO[HomCoef=0.333]VK[D]
\pstHomO[HomCoef=0.5]TQ[E]
\pstHomO[HomCoef=0.333]XM[F]
  \end{pspicture}
  \caption{Equal \enquote{complementary} parallelograms}
  \label{fig:I.43}
\end{figure}
where by congruence
\begin{align*}
  A+B+C&=D+E+F,&A&=D,&C&=F,
\end{align*}
and therefore $B=E$.

Congruent figures \textbf{coincide.}
This is the word that Heath uses in translating one of the so-called
Common Notions at the beginning of the \emph{Elements}
\cite{MR17:814b}:
\begin{quote}\centering
  Things which coincide with one another\\
are equal to one another.
\end{quote}
However, the Latin translations of Commandinus and Heiberg 
use the verb \lat{congrvo},
whose second element means \enquote{fall} 
and gives us \eng{ruin} in English.

Since parallelograms in the same parallels
on the \emph{same} base are equal,
so are parallelograms on \emph{equal} bases
(Figure \ref{fig:equal-bases}).
\begin{figure}
  \centering
  \begin{pspicture}(-1.2,0)(6.5,3)
%\psgrid
\newcommand{\myparam}{1.8}
\newcommand{\myheight}{3}
\psset{PosAngle=90,PointSymbol=none,PointName=none}    
\pstGeonode[PosAngle={-90,-90,90,90}](0,0)S(3,0)T(3.5,0)W
(! \myparam\space \myheight)V(! \myparam\space 1.5 add \myheight)E
\pstTranslation TVS[R]
\pstTranslation SET[F]
\ncline RS\ncline TV
\ncline RF
\ncline EW
\pstTranslation EWF[X]
\ncline FX
\ncline SX
\psset{linestyle=dashed}
\ncline SE\ncline TF
  \end{pspicture}  
  \caption{Parallelograms on equal bases}
  \label{fig:equal-bases}
\end{figure}
I'm not sure how clearly students today see the distinction,
since they have been trained to assign \emph{numbers}
as lengths to line segments, 
and \enquote{equal} numbers are the same number.

\emph{Ratios} for Euclid are never equal.
One ratio may be greater than another.
Otherwise, the two ratios are not really two at all,
because they are the \emph{same.}

\section{Sameness of ratio}

When are two ratios the same?
We can say when,
as Euclid does in Book \textsc v of the \emph{Elements,}
without actually defining ratios.
Euclid says only,
\begin{quotation}\noindent
  A \textbf{ratio} is a sort of relation in respect of size 
  between two magnitudes of the same kind.
  
Magnitudes are said to \textbf{have a ratio} to one another which are capable,
when multiplied, of exceeding one another.
\end{quotation}
Thus, to have a ratio, 
magnitudes must meet what we call an \textbf{Archimedean} condition,
though Archimedes came later.

Suppose $A$ and $B$ have a ratio,
and $C$ and $D$ have a ratio.
The ratios are the \textbf{same,}
so that the four magnitudes themselves are \textbf{proportional,}
and we may write
\begin{equation*}
  A:B::C:D,
\end{equation*}
provided that,
for all multipliers $\coA$ and $\coB$
(we call them positive integers),
\begin{equation*}
  A\cdot\coA\gtreqqless B\cdot\coB\iff C\cdot\coA\gtreqqless D\cdot\coB.
\end{equation*}
%whenever $*$ is $<$, $=$, or $>$.
This is the \textbf{Eudoxan} definition of proportion,
so called
since a scholium attributes the definition to Eudoxus [of Cnidus],
teacher of Plato.%%%%%
\footnote{Heath \cite[Vol.\ 2, p.\ 112]{MR17:814b}
  quotes the scholium from Heiberg \cite[p.\ 280]{Euclid-Heiberg-V}
and suggests that it may be by Proclus.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

If $A$ and $B$ are \textbf{incommensurable,}
so that $A\cdot\coA=B\cdot\coB$ never,
then the ordered pairs of multipliers are cut into two disjoint sets:
\begin{align*}
  &\{(\coB,\coA)\colon B\cdot\coB<A\cdot\coA\},&
&\{(\coB,\coA)\colon B\cdot\coB>A\cdot\coA\}.
\end{align*}
One may prefer to write $(\coB,\coA)$ as $\nicefrac{\coB}{\coA}$.
The two sets compose what we call a \textbf{Dedekind cut.}

Dedekind was told \cite[pp.~39--40]{MR0159773}
that his idea could already be found in Bertrand's
\emph{\foreignlanguage{french}{Trait\'e d'arithmetique}} \cite{Bertrand}.
Dedekind pointed out that \emph{Bertrand's}
idea could be found in Euclid:
distinct ratios determine distinct cuts.
Dedekind's innovation was to use cuts in place of ratios,
thus obtaining all of what we now call the positive real numbers,
without need for geometry.

By Euclid's definition of \textbf{greater ratio,} 
\begin{equation*}
  3:2>7:5
\end{equation*}
because, for some multipliers $\coA$ and $\coB$,
\begin{align*}
  3\cdot\coA&>2\cdot\coB,&7\cdot\coA=5\cdot\coB.
\end{align*}
Indeed, just let $k=5$ and $m=7$.

Thus we have a test for proportionality of numbers:
\begin{equation}\label{eqn:Eudoxan}
  A:B::C:D\iff A\times D=B\times C.
\end{equation}
Let us call this the \textbf{Eudoxan} definition
of proportion of \emph{numbers.}
It assumes that multiplication of numbers is commutative.
Euclid \emph{proves} this, \emph{using} proportions.
Thus the \textbf{Euclidean} definition of proportion of numbers
will be different.

\chapter{Numbers}

\section{Proportion}

At the head of Book \textsc{vii} of the \emph{Elements,}
we are told:
\begin{compactenum}
  \item
\enquote{A number is a multitude of units.}%%%%%
\thinspace\footnote{John Dee \emph{created} the word \eng{unit}
precisely to translate Euclid's \gr{mon'ac},
as he notes his \enquote{Mathematicall Preface} \cite{Dee} 
to Billingsley's 1570 English translation of the \emph{Elements.}
The existing alternative was \eng{unity.}
See my article \enquote{On commensurability and symmetry} 
\cite{Pierce-on-comm-symm}.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\item
\enquote{Numbers are \textbf{proportional} 
when the first is
\begin{compactitem}%[[\thinspace 1\thinspace {]}]
\item  
 the same multiple,
or 
\item
the same part, or 
\item
the same parts,
\end{compactitem}
of the second that the third is of the fourth.}
\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
When 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 is 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 $\coB$,%%%%%
\footnote{If one of the multipliers $\coA$ and $\coB$ is unity,
then we have \enquote{same part} or \enquote{same multiple};
otherwise, \enquote{same parts.}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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
This much is called the 
\textbf{Pythagorean} definition of proportion of numbers.
\item
For the \textbf{Euclidean} definition, we need
\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 as a whole makes little sense.
\end{compactenum}
Therefore I say that the Euclidean definition 
must be the one that Euclid means.%%%%%
\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.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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 \textbf{Euclidean Algorithm} is used,
namely,
\begin{compactenum}[(1)]
\item
  of two magnitudes, 
replace the greater 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.

The enunciation of \textbf{Proposition 4} is,
\enquote{Any number is either a part or parts of any number,
the less of the greater.}
Euclid \enquote{proves} this by finding $\gcm$'s,
showing implicitly (in my view) 
that Euclid intends 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.}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
There is good evidence that
before the Eudoxan definition,
there was an \textbf{anthyphaeretic} definition
of proportion for arbitrary magnitudes,
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}.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The Euclidean definition is a simplification of this for numbers;
but 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=7mm,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{align*}
  S:A&::A:B,&
D:S&\text{ is constant.}
\end{align*}
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*}
which is \eqref{eqn:Eudoxan},
the Eudoxan definition of proportion of numbers
on page \pageref{eqn:Eudoxan}.
\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.

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\end{document}
