From the Beginning to Plato

282 FROM THE BEGINNING TO PLATO
Figure 8.13
Proposition (ii) is, as Simplicius says, equivalent to Elements XII.2, which is
proved by the method of exhaustion (see section 1 of part one). Simplicius says
that Hippocrates ‘showed’ (deiknumi) (ii), but scholars are reluctant to admit that
he could have proved it in a rigorous fashion. Euclid takes (iv) as the definition of
similar segments (III, def. 11), but he himself makes very little use of them.
Hippocrates apparently defined two segments to be similar if they are the same
‘part’ of the circles of which they are segments. In Euclid a part of something is
one nth of it, which would mean that Hippocrates picked out very few of the
similar segments, indeed, none greater than a semicircle, and certainly none of the
incommensurable ones. The way in which Hippocrates gets from (iii) to (iv)
suggests that he wasn’t using much of a notion of proportionality at all, and just
arguing in some sort of loose way. Obviously this looseness bears on the
question whether or not Hippocrates knew about incommensurability. Again we
are faced with standard kinds of choice: blame Simplicius or Eudemus for
misunderstanding; assume that Hippocrates’ argumentation was not entirely
rigorous; assume that incommensurability was discovered after Hippocrates
squared his lunes or at least not long before. Finally, the move from (ii) to (i)
seems to require some proposition about the relationship between the base of a
segment of a circle and the diameter of the circle, perhaps that the segment is to
the semicircle as the square on its base is to the square on the diameter. But we
are not told how Hippocrates might have proved this.
Hippocrates squared in succession three particular lunes, one with a semicircle
as outer circumference, one with an outer circumference greater than a
semicircle, and one with an outer circumference less than a semicircle, and then
a circle plus a particular lune. Simplicius’ report does not make it seem as though
Hippocrates claimed to have shown how to square the circle because he had
shown how to square a circle plus a lune and how to square lunes with an outer
circumference of ‘any size’. Perhaps he did, but it seems equally likely that the
investigations described by Simplicius were an attempt at quadrature which
somehow was interpreted to involve a claim to success.
It is not possible for me to describe here Hippocrates’ four quadratures. 53 I shall,
however, mention one construction described by Simplicius. In the left
configuration in Figure 8.13, EG is parallel to KB, EK≈KB≈BG, EF≈FG, and the
circle segments on EF, FG, EK, KB and BG are all similar.
It should be clear that the lune EKBGF will be squarable if: