HIPPOCRATES’S QUADRATURE OF LUNES 191
which are one third of the circles respectively, are from this
point of view quite harmless.
Only the transition to the view of similar segments as
segments ‘ containing equal angles ’ remains to be explained.
And here we are in the dark, because we do not know how, for
instance, Hippocrates would have drawn a segment in one
given circle which should be ‘ the same part ’ of that circle
that a given segment of another given circle is of that circle.
(If e.g. he had used the proportionality of the parts into which
the bases of the two similar segments divide the diameters
of the circles which bisect them perpendicularly, he could,
by means of the sectors to which the segments belong, have
proved that the segments, like the sectors, are in the ratio
of the circles, just as Rudio supposes him to have done; and
the equality of the angles in the segments would have followed
as in Rudio’s proof.)
As it is, I cannot feel certain that the sentence Sib kou ktX.
‘ this is the reason why similar segments contain equal angles ’
is not an addition by Simplicius. Although Hippocrates was
fully aware of the fact, he need not have stated it in this
place, and Simplicius may have inserted the sentence in order
to bring Hippocrates’s view of similar segments into relation
with Euclid’s definition. The sentence which follows about
‘angles of’ semicircles and ‘angles of’ segments, greater or
less than semicircles, is out of place, to say the least, and can
hardly come from Eudemus.
W e resume Eudemus’s account.
‘ After proving this, he proceeded to show in what way it
was possible to square a lune the outer circumference of which
is that of a semicircle. This he effected by circumscribing
a semicircle about an isosceles right-angled triangle and
(circumscribing) about the base [= describing on the base]
a segment of a circle similar to those cut off by the sides.’
[This is the problem of Eucl. III. 33,
and involves the knowledge that similar
segments contain equal angles.]
‘Then, since the segment about the
base is equal to the sum of those about
the sides, it follows that, when the part
of the triangle above the segment about the base is added
to both alike, the lune will be equal to the triangle.