ELEMENTS AS KNOWN TO HIPPOCRATES 201 
the idea of the reduction of the problem of duplication may 
have occurred to hiçti through analogy. The problem of 
doubling a square is included in that of finding one mean 
proportional between two lines; he might therefore have 
thought of what would be the effect of finding two mean 
proportionals. Alternatively he may have got the idea from 
the theory of numbers. Plato in the Timaeus has the pro 
positions that between two square numbers there is one mean 
proportional number, but that two cube numbers are connected, 
not by one, but by two mean numbers in continued proportion. 1 
These are the theorems of Eucl. VIII. 11, 12, the latter of 
which is thus enunciated : ‘ Between two cube numbers there 
are two mean proportional numbers, and the cube has to the 
cube the ratio triplicate of that which the side has to the side.’ 
If this proposition was really Pythagorean, as seems prob 
able enough, Hippocrates had only to give the geometrical 
adaptation of it. 
(y) The Elements as known to Hippocrates. 
We can now take stock of the advances made in the 
Elements up to the time when Hippocrates compiled a work 
under that title. We have seen that the Pythagorean geometry 
already contained the substance of Euclid’s Books I and II, 
part of Book IV, and theorems corresponding to a great part 
of Book VI ; but there is no evidence that the Pythagoreans 
paid much attention to the geometry of the circle as we find 
it, e.g., in Eucl., Book III. But, by the time of Hippocrates, 
the main propositions of Book III were also known and used, 
as we see from Eudemus’s account of the quadratures of 
lunes. Thus it is assumed that ‘ similar ’ segments contain 
equal angles, and, as Hippocrates assumes that two segments 
of circles are similar when the obvious thing about the figure 
is that the angles at the circumferences which are the supple 
ments of the angles in the segments are one and the same, 
we may clearly infer, as above stated, that Hippocrates knew 
the theorems of Eucl. III. 20-2. Further, he assumes the 
construction on a given straight line of a segment similar to 
another given segment (cf. Eucl. III. 33). The theorems of 
Eucl. III. 26-9 would obviously be known to Hippocrates, 
1 Plato, Timaeus, 32 A, e.