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