From the Beginning to Plato

GREEK ARITHMETIC, GEOMETRY AND HARMONICS 277
developing a rigorous geometric disguise for carrying on with Babylonian
computation. If we assume that when Eudemus says that the discovery of
application of areas was ancient and Pythagorean, he intends to refer to what we
call early Pythagoreans, the whole history has to be moved back at least to the
early fifth century. The way Burkert ([8.79], 465) avoids this difficulty is to
leave the word ‘ancient’ out of account and argue that the discovery of
incommensurability is ‘not far from Theodoras of Cyrene’, 45 and hence to make
(c) apply to ‘late’ Pythagoreans. My preference is to give up the whole idea that
Greek mathematics is essentially computational and hence the idea that it rests
on Babylonian achievements. With this point of view the question of the
discovery of incommensurability becomes independent of positions on the nature
and origins of Greek mathematics and can be approached on its own. I wish I
could say that approaching the question this way made one answer or another
probable, but, if we abandon Proclus’ statement about Pythagoras, the evidence
for dating is unsatisfactory. The terminus ante quem is provided by references to
irrationality in Plato; and if we believe that the mathematics lesson of the
Theaetetus gives an indication of the state of mathematical knowledge in the
410s, we will be struck by the fact that Theodoras starts his case-by-case
treatment with 3, and not 2; we might, then, take 410 as the terminus.
The discovery of incommensurability: Hippasus of
Metapontum
The version of the Becker proof of the irrationality of which I presented in
section 2 of this part is just a reformulation of the argument to which Aristotle
refers (Prior Analytics I.23.41a26–7) when he illustrates reductio ad absurdum
by referring to the proof that ‘the diagonal of the square is incommensurable
because odd numbers become equal to evens if it is supposed commensurable’. A
version of this proof occurs in our manuscripts as the last proposition of
Elements X, but is printed in an appendix by Heiberg ([8.30] 3:408.1ff.). That
version differs from the proof I gave in avoiding direct reference to the
impossibility of an infinitely descending sequence of numbers by assuming that n
and m are the least numbers such that n2 =2m2 and inferring that, since n is even,
m must be odd; but then the argument shows that m must be even. It is frequently
assumed that some such proof was the first proof of incommensurability, partly
because of the passage in Aristotle and partly because side and diagonal of a
square are the standard Greek example of incommensurability. In its Euclidean
form the proof presupposes a fairly sophisticated understanding of how to deal
with ratios in least terms—an important subject of Elements VII, which is
dependent on anthuphairesis. There are various ways of minimizing this
presupposition, but in general those who believe that the Euclidean proof is a
version of the original proof have used its relative sophistication to argue either
that the proof must be late or that Greek mathematics must have been
sophisticated relatively early.