I agree with the use in English, just so long as we don't think that
the uses of "diameter" in Greek would have the same range of
meaning as the English word "diagonal". I don't think there's any special
terminological problem in generalization to parallelograms.
The "rational diameter" in Plato (Rep. 546c) and the "side and
diameter" numbers in Theon and in Proclus (on the Republic) of
course deal with squares, but the extension of "diameter" to
the parallelogram is securely in Euclid.
It's very interesting (as he also wrote) that
>...The word diagonios does occur in the Elements, twice in
> XI Proposition 28 ...
I don't have access to Heiberg right now, but I verified this
in Mugler's Dictionnaire. It's descibed as an isolated example,
with no known predecessors or successors until Heron. LSJ did
not notice it and has its first citation from Strabo, and
Heath in his edition of Euclid also overlooked it (the index
reference leads only to a place in the introduction where it
is described as a "later" term, with citation of Heron).
And finally, he wrote
>...What we could have here is a word that originally had two senses
> which then splits into two words...Can anyone give another example,
> in Greek or in English, of such a process?
In current mathematical English, "circle" is splitting into
"circle" (the curve) and "disc" (the plane region). In roughly
the first half of this century (I think), "series" split into "series"
(a summation) and "sequence" (one after another). My (later)
edition of Hardy's _Course in Pure Mathematics_ has an index entry
for "sequence" which merely says "see Functions of an Integral
Variable".
William C. Waterhouse
Penn State