188 THE ELEMENTS DOWN TO PLATO’S TIME 
edition of the whole extract. Whereas Diels, Usener, Tannery, 
and Heiberg had all seen in the sentences ‘ For, as the circles 
are to one another . . . less than semicircles ’ an addition by 
Simplicius, like the phrase just preceding (not quoted above), 
‘ a proposition which Euclid placed second in his twelfth book 
with the enunciation “ Circles are to one another as the squares 
on their diameters’”, Rudio maintains that the sentences are 
wholly Eudemian, because ‘ For, as the circles are to one 
another, so are the similar segments’ is obviously connected 
with the proposition that similar segments are as the squares 
on their bases a few lines back. Assuming, then, that the 
sentences are Eudemian, Rudio bases his next argument on 
the sentence defining similar segments, ‘ For similar segments 
are those which are the same part of the circles: thus a semi 
circle is similar to a semicircle, and a third part (of one circle) 
to a third part (of another circle) He argues that a ‘ segment ’ 
in the proper sense which is one third, one fourth, *&c., of the 
circle is not a conception likely to have been introduced into 
Hippocrates’s discussion, because it cannot be visualized by 
actual construction, and so would not have conveyed any clear 
idea. On the other hand, if we divide the four right angles 
about the centre of a circle into 3, 4, or n equal parts by 
means of 3, 4, or n radii, we have an obvious division of the 
circle into equal parts which would occur to any one; that is, 
any one would understand the expression one third or one 
fourth part of a circle if the parts were sectors and not 
segments. (The use of the word r/xriixa in the sense of sector 
is not impossible in itself at a date when mathematical 
terminology was not finally fixed; indeed it means ‘ sector ’ 
in one passage of Aristotle. 1 ) Hence Rudio will have it that 
‘similar segments’ in the second and third places in our passage 
are ‘ similar sectors But the ‘ similar segments ’ in the funda 
mental proposition of Hippocrates enunciated just before are 
certainly segments in the proper sense; so are those in the 
next sentence which says that similar segments contain equal 
angles. There is, therefore, the very great difficulty that, 
under Rudio’s interpretation, the word T/jLrjfxara used in 
successive sentences means, first segments, then sectors, and 
then segments again. However, assuming this to be so, Rudio 
1 Arist. De caelo, ii. 8, 290 a 4.