"My reaction: I also wonder what Aristotle meant, and no, there is no
evidence I can see before Alexander of Aphrodisias that he meant, or that
he did not mean, a square. I don't see any reason to believe that it should
always refer to a square, once the question has been raised."
Roger Cooke then said:
"It could certainly have meant the diagonal and side of a pentagon, whose
incommensurability is even easier to prove from the Euclidean algorithm
than that of the diagonal and side of a square."
I have very serious doubts about such ideas because of the (literally)
prosaic fact that the texts in question do not contain a word
that means "diagonal". As far as I can tell, there was no such word
at the time; "diago:nios" seems to come into use first in the Roman
period.
The word that does appear is "diameter". Here is a quick
summary of its mathematical uses, as far as I can find from
just a quarter-hour with books I have at home.
1) Its most common use is (of course) for circles. This is
probably also its earliest mathematical sense; for it is
said to bisect the circle at the start of Euclid, and that
statement is plausibly reported as much older than Euclid.
2) It is used for the line joining the opposite vertices of a
square in the famous dialogue between Socrates and the slave
in Plato's _Meno_. It is not introduced until the end, and the
point seems to be that "The square on the diameter is twice the
original square" would be the familiar mathematical form of
the statement.
3) It is used for the lines joining opposite vertices of a
parallelogram in Euclid I.34, where (here too) it is proved
that these lines bisect the parallelogram.
4) It continues to be used in that sense by Archimedes
(Plane Equilibria), though now (in finding the center of
gravity of a parallelogram) the significant fact is that
each diameter bisects all the segments parallel to the
other diameter.
5) That sense is then further generalized by Apollonius, who
is thinking of conics but more broadly defines a diameter
of an arbitrary curve to be a line bisecting all chords
in a fixed direction.
No book I can find gives even one example of "diameter"
used to mean a line joining two arbitrary (nonadjacent)
vertices of a polygon. The diagonal of a regular pentagon
is constructed as part of Euclid's construction of that
polygon (IV.11), but it is not given a name. The general
definition in Apollonius will give a diameter starting at
one vertex of a regular pentagon, but it will (obviously)
join that vertex not to another vertex but to the middle
of the opposite side.
In summary, we can be almost certain that the "side and
diameter" to which Aristotle so casually refers are
(as we would have presumed in the first place) the same as
the ones in Plato.
William C. Waterhouse
Penn State