From the Beginning to Plato

258 FROM THE BEGINNING TO PLATO
Figure 8.2
lines commensurable in length with the one-foot dunamis—or, equivalently, with
a line set out—as ‘rational’ from those which are not.
The comparison between numbers and figures to which Plato’s Theaetetus
refers is quite clear in much Greek arithmetic vocabulary, of which ‘square’ and
‘cube’ are perhaps the most common modern survivors. But we do not find in
Euclid anything genuinely like the representation of a unit as a straight line u
with a corresponding unit square sq(u), and other numbers represented both as
multiples of u and as rectangles contained by such multiples (cf. Figure 8.2). But
this seems to be what lies behind the discussion in the Theaetetus. That is to say,
it looks as though Theodoras was using a unit length u and proving what we would
call the irrationality of n for certain n by showing that the side s n of a square
corresponding to n was incommensurable in length with u. Theaetetus made a
generalization of what he was shown by Theodoras by assuming or proving that:
(i) n is a perfect square if and only if s n is commensurable with u.
It should be clear that there is a big difference between assuming and proving (i).
And although the proof of implication from left to right is quite straightforward,
the proof of right-left implication is far from it. In fact, it is just the proposition
we would assert by saying that the square root of any non-square positive integer
is irrational. There is no extant ancient proof of such an assertion.
We may gain more insight by formulating the question raised by Theaetetus in
the Theaetetus and his answer to it as:
Question: If y is the geometric mean between m·u and n·u (i.e. if the square
on y is equal to the rectangle on m·u and n·u), under what conditions is y
commensurable with u?
Answer: If y is the geometric mean between m·u and n·u, then y is
commensurable with u (if and) only if , for some k.