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\title{\gk{Summetr'ia} in Aristotle}
\author{David Pierce}
\date{January 28, 2006; revised, May 31, 2007}

\begin{document}
  \maketitle
The Greek abstract noun \gk{summetr'ia} is evidently the source of our
\eng{symmetry.}  To what extent do the two words share meaning,
especially in Aristotle?

The Greek adjective \gk{s'ummetroc, -a, -on}
comes to us \emph{via} Latin as \eng{commensurable,} and the latter
can be used at least sometimes to translate the former.  For example,
Definition 1 of Book X of Euclid's \emph{Elements}
\cite[vol.~3, p.~11]{MR17:814b} is: 
\begin{quote}
  \gk{\look{S'ummetra} meg'ejh l'egetai t`a t~w| a>ut~w| m'etrw| metro'umena,
  \look{>as'ummetra} d'e, ~<wn mhd`en >endeqetai koin`on m'etron gen'esjai.}

Magnitudes measured by the same measure are called
\textbf{commensurable;} those that admit no common measure,
\textbf{incommensurable.} 
\end{quote}
(All translations here are my own.)
In the \emph{Metaphysics,} XIII.\textsc{iii}.10
(1078${}^{\text{a}}$35), Aristotle makes a general statement about
\gk{summetr'ia}:
\begin{quote}
  \gk{to~u d`e kalo~u m'egista e>'idh t'axic ka`i \look{summetr'ia} ka`i t`o
    <wrism'enon, 
<`a m'alista deikn'uousin a<i majhmatika`i >epist~hmai.}

The greatest shapes of the beautiful are arrangement, symmetry, and
the delimited, which the mathematical sciences show especially.
\end{quote}
(Aristotle's texts can be found at \url{http://www.perseus.tufts.edu}.
  I am using \eng{shape} here for 
  \gk{e>~idoc}, rather than \eng{form} or \eng{species.})  It is not
  clear here whether mathematics \emph{is} symmetric, or only concerns
  symmetrical (and arranged, well-defined) things.
Aristotle's comment is preceeded by:
\begin{quote}
 Since the good and the beautiful are different (for, the former is
 always in deeds, but the beautiful is also in motionless things),
 those who say that the mathematical sciences are not about the
 beautiful and the good are wrong\dots 
\end{quote}
The passage does not suggest what symmetry is.  Elsewhere in the
\emph{Metaphysics,} IV.\textsc{ii}.18, (1004${}^{\text{b}}$11,) we
find:
\begin{quote}
  there are particular properties (\gk{>'idia p'ajh}) of number
  \emph{qu\^a} number, such as oddness/evenness,
  commensurability/equality, excess/defect\dots
\end{quote}
Here Aristotle seems to be naming properties in correlative pairs.
Some of the properties belong to single numbers; others, to pairs.
What is the meaning of the middle pair of properties,
\gk{summetr'ia}/\gk{>is'othc}? 
Possibly a single number can be more or less `symmetric', depending on
how many factors it has.  But a single number cannot be `equal';
equality is a property of pairs of numbers.  Possibly, for Aristotle,
every pair of unequal numbers is symmetric in the sense of
commensurable: for, the
numbers have a common measure, what we call their greatest common
divisor.  If the numbers are equal, then we would not refer to them
as (merely) commensurable; we would say they were equal.

However, the possibility of `asymmetric' or incommensurable pairs of
mathematical objects 
is suggested at XI.\textsc{iii}.7 (1061${}^{\text{a}}$28):
\begin{quote}
  Just as the mathematician brings about a theory concerning [things
  obtained] by taking away [\emph{i.e.}\ abstraction]---for, he theorizes,
  having stripped away all that can be sensed, such as weight and
  lightness\dots; he leaves only the how-much [\emph{i.e.}\ quantity] and
  the holding-together [\emph{i.e.}\ continuity]\dots,   and the
  properties of things insofar as they are so much and
  continuous\dots; for some things, he investigates their placement
  regarding one another, and what belongs to them; for others, their
  commensurabilities and incommensurabilities, or their ratios
  (\gk{l'ogoi})\dots 
\end{quote}
Symmetry/commensurability in a more practical context arises in the
\emph{Nichomachean Ethics} (V.\textsc{v}, 1133${}^{\text{b}}$16):
\begin{quote}
  Money, as a measure, making [things] commensurable, equalizes
  (\gk{>is'azei}) [them].  For,  without commerce, there 
  would be no community;---no commerce, without equality; no equality,
  without commensurability. 
\end{quote}
The passage does not make much sense to me unless the
verb \gk{>isazw} here means something like `make comparable' rather
than `make equal'; the
dictionary suggests also `balance'.  Symmetry in the sense of balance is
mentioned in the \emph{Physics} (VII.\textsc{iii}, 246${}^{\text
  b}$4):
\begin{quote}
  We say that all excellences (\gk{>aret'ai}) are in \emph{holding}
  somehow with respect to something.  For, the [excellences] of the
  body, such as health or vigor, we place in the mixture or balance
  (\gk{symmetr'ia}) of hot things and cold things---these with respect
  to themselves, or the environment.
\end{quote}
(Most passages quoted here are cited in \cite{LSJ} under \gk{summetr'ia}.)

\begin{thebibliography}{1}

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

\bibitem{LSJ}
Henry~George Liddell and Robert Scott.
\newblock {\em A Greek-English Lexicon}.
\newblock Clarendon Press, Oxford, 1940.
\newblock Revised and augmented throughout by Sir Henry Stuart Jones.

\end{thebibliography}


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