HIPPOCRATES’S QUADRATURE OF LUNES 189 
is able to make the argument hang together, in the following 
way. The next sentence says, ‘ For this reason also (Slo kol) 
similar segments contain equal angles ’; therefore this must be 
inferred from the fact that similar sectors are the same part 
of the respective circles. The intermediate steps are not given 
in the text; but, since the similar sectors are the same part 
of the circles, they contain equal angles, and it follows that the 
angles in the segments which form part of the sectors are 
equal, since they are the supplements of the halves of the 
angles of the sectors respectively (this inference presupposes 
that Hippocrates knew the theorems of Euel. III. 20-22, which 
is indeed clear from other passages in the Eudemus extract). 
Assuming this to be the line of argument, Rudio infers that in 
Hippocrates’s time similar segments were not defined as in 
Euclid (namely as segments containing equal angles) but were 
regarded as the segments belonging to ‘ similar sectors ’, which 
would thus be the prior conception. Similar sectors would 
be sectors having their angles equal. The sequence of ideas, 
then, leading up to Hippocrates’s proposition would be this. 
Circles are to one another as the squares on their diameters or 
radii. Similar sectors, having their angles equal, are to one 
another as the whole circles to which they belong. (Euclid has 
not this proposition, but it is included in Theon’s addition to 
VI. 33, and would be known long before Euclid’s time.) 
Hence similar sectors are as the squares on the radii. But 
so are the triangles formed by joining the extremities of the 
bounding radii in each sector. Therefore (cf. Eucl. V. 19) 
the differences between the sectors and the corresponding 
triangles respectively, i.e. the corresponding segments, are in 
the same ratio as (1) the similar sectors, or (2) the similar 
triangles, and therefore are as the squares on the radii. 
We could no doubt accept this version subject to three ifs, 
(1) if the passage is Eudemian, (2) if we could suppose 
Tfj.rjfx.aTa to be used in different senses in consecutive sentences 
without a word of explanation, (3) if the omission of the step 
between the definition of similar ‘ segments ’ and the inference 
that the angles in similar segments are equal could be put 
down to Eudemus’s ‘ summary ’ style. The second of these 
ifs is the crucial one; and, after full reflection, I feel bound 
to aeree with the o-reat scholars who have held that this