From the Beginning to Plato

Figure 8.12
GREEK ARITHMETIC, GEOMETRY AND HARMONICS 281
Obviously our perception of Hippocrates’ reasoning is mediated by both
Simplicius and Eudemus. We might suppose that, once the quotations from
Euclid have been subtracted, the remainder of the material which follows is by
Eudemus, but that still leaves us with the question of distinguishing Eudemian
from Hippocratic material, a question which does not seem to be capable of
being settled. 52 Simplicius’ extract from Eudemus begins as follows:
In the second book of his history of geometry Eudemus says the following.
The quadratures of lunes, which are not considered as superficial
constructions (diagrammata) because of their connection with the circle,
were first described by Hippocrates in a way which was considered to be in
order. Let us therefore touch on this subject in more detail and go through
it. He made himself a starting-point and set out as the first of the things useful
for the quadratures, the proposition that:
(i) similar segments of circles have to one another the same ratio as their
bases have in square. He showed this on the basis of having shown that:
(ii) diameters [of circles] have the same ratios in square as the circles
do, a proposition which Euclid puts second in Book XII of the Elements,
where the proposition says ‘Circles are to one another as the squares on their
diameters’. For as the circles are to one another, so are their similar
segments, since:
(iii) similar segments are those which are the same part of a circle, for
example, a semicircle is similar to a semicircle, a third of a circle to a third
of a circle. Therefore,
(iv) similar segments admit equal angles, at least the angles of all
semicircles are right, and the angles of segments greater than semicircles
are less than right angles and as much less as the segments are greater than
semicircles, and the angles of segments less than semicircles are greater
than right angles and as much greater as the segments are less than
semicircles.
([8.84], 60.30–61.18, my numbers inserted)