190 THE ELEMENTS DOWN TO PLATO’S TIME
hypothesis is impossible; indeed the canons of literary criti
cism seem to exclude it altogether. If this is so, the whole
of Rudio’s elaborate structure falls to the ground.
We can now consider the whole question ah initio. First,
are the sentences in question the words of Eudemus or of
Simplicius ? On the one hand, I t|nnk the whole paragraph
would be much more like the ‘ summary ’ manner of Eudemus
if it stopped at ‘have the same ratio as the circles’, i.e. if the
sentences were not there at all. Taken together, they are
long and yet obscurely argued, while the last sentence is
really otiose, and, I should have said, quite unworthy of
Eudemus. On the other hand, I do not see that Simplicius
had any sufficient motive for interpolating such an explana
tion : he might have added the words ‘ for, as the circles are
to one another, so also are similar segments of them ’, but
there was no need for him to define similar segments; he
must have been familiar enough with the term and its
meaning to take it for granted that his readers would knoAv
them too. I think, therefore, that the sentences, down to ‘ the
same part of the circles respectively ’ at any rate, may be
from Eudemus. In these sentences, then, can ‘ segments ’ mean
segments in the proper sense (and not sectors) after all %
The argument that it cannot rests on the assumption that the
Greeks of Hippocrates’s day would not be likely to speak of
a segment which was one third of the whole circle if they
did not see their way to visualize it by actual construction.
But, though the idea would be of no use to us, it does not
follow that their point of view would be the same as ours.
On the contrary, I agree with Zeuthen that Hippocrates may
well have said, of segments of circles which are in the same
ratio as the circles, that they are ‘ the same part ’ of the circles
respectively, for this is (in an incomplete form, it is true) the
language of the definition of proportion in the only theory of
proportion (the numerical) then known (cf. Eucl. All. Def. 20,
‘ Numbers are proportional when the first is the same multiple,
or the same part, or the same parts, of the second that the
third is of the fourth’, i.e. the two equal ratios are of one
of the following forms m, * or ~ where m, n are integers);
the illustrations, namely the semicircles and the segments