From the Beginning to Plato

284 FROM THE BEGINNING TO PLATO
compass foundation of the Elements. That is to say, it looks as though by the
later fifth century Greek geometry has moved close to what became a permanent
foundation. It seems to me most plausible to imagine this concern with the
equivalent of foundations as the outcome of a rather lengthy history of geometric
demonstration.
Of course, the problems involving Hippocrates’ use of the theory of
proportion, his neusis construction, and his ‘showing’ of the equivalent of
Elements XII.2 remain unsolved. My suggestions on these questions are made
with no great confidence. I see no way to make good sense of the passage on similar
segments which follows Simplicius’ citation of XII.2, and prefer to treat it as
Simplicius’ unsatisfactory attempt to provide a derivation of (i) from (ii) using
(iii), and then to connect (iii) with Euclid’s definition of similar segments. If this
is correct, then Simplicius had no more information on these questions than what
he says before he cites Euclid. In general I accept the standard view that only in
the fourth century did the Greeks develop techniques for dealing with
proportions involving incommensurables. But I am also inclined to put the date of
the discovery of incommensurability back to the time of Hippasus of
Metapontum, whether Hippasus himself discovered it in connection with the
pentagon or it was discovered in something like the way Becker has suggested. I
infer that the Greeks worked for more than half a century using laws of
proportion which they were not able to prove in a rigorous way. Hence I also
infer that the interest in providing a rigorous foundation for the treatment of
proportionality is a fourth-century interest. If this is correct, then we need not
suppose that Hippocrates’ ‘elements’ included any explicit theory of proportion.
Similarly, in the case of XII.2, I think we should assume that Hippocrates
could not have proved this in the Euclidean way, and that, if he did, indeed,
‘show’ it, he did so in some intuitive way. The neusis construction offers us the
alternative of assigning to Hippocrates either a full development of the method
of application of areas or the use of an intuitively based construction which
cannot in general be done with unmarked ruler and compass alone. Simplicius’
silence on Hippocrates’ technique makes it seem to me likely that he did not
know which alternative Hippocrates adopted, and that Eudemus did not say. My
inclination is to assume that Hippocrates used the intuitive construction. Of
course, to say that Hippocrates used a neusis construction in his quadrature is not
to say that he did or did not do the same kind of thing in his ‘elements’. And
even if he did use such constructions there, he may also have been interested in
carrying out as many constructions as possible using some kind of compass and
straight edge. In any case, it seems clear from Hippocrates’ quadratures that he
knew a good deal of the elementary geometry in Euclid’s Elements, and had put
it into some kind of reasonably rigorous order.