163
PYTHAGOREAN GEOMETRY
shows the equal sides of the five isosceles triangles of the type
referred to and also the points at which they are divided in
extreme and mean ratio. (I should perhaps add that the
pentagram is said to be found on the vase of Aristonophus
found at Caere and supposed to belong to the seventh
century B.C., while the finds at Mycenae include ornaments of
pentagonal form.)
It would be easy to conclude that the dodecahedron is in-
seribable in a sphere, and to find the centre of it, without
constructing both in the elaborate mannei^of Eucl. XIII. 17
and working out the relation between an edge of the dodeca
hedron and the radius of the sphere, as is there done: an
investigation probably due to Theaetetus. It is right to
mention here the remark in scholium No. 1 to Eucl. XIII
that the book is about
‘the five so-called Platonic figures, which, however, do not
belong to Plato, three of the five being due to the Pytha
goreans, namely the cube, the pyramid, and the dodeca
hedron, while the octahedron and icosahedron are due to
Theaetetus
This statement (taken probably from Geminus) may per
haps rest on the fact that Theaetetus was the first to write
at any length about the two last-mentioned solids, as he was
probably the first to construct all five theoretically and in
vestigate fully their relations to one another and the circum
scribing spheres.
(£) Pythagorean astronomy.
Pythagoras and the Pythagoreans occupy an important place
in the history of astronomy. (1) Pythagoras was one of the first
to maintain that the universe and the earth are spherical
in form. It is uncertain what led Pythagoras to conclude
that the earth is a sphere. One suggestion is that he inferred
it from the roundness of the shadow cast by the earth in
eclipses of the moon. But it is certain that Anaxagoras was
the first to suggest this, the true, explanation of eclipses.
The most likely supposition is that Pythagoras’s ground was
purely mathematical, or mathematico-aesthetical; that is, he
Heiberg’s Euclid, vol. v, p. 654.
