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PYTHAGOREAN GEOMETRY 
(e) The jive regular solids. 
The same parenthetical sentence in Proclus which attributes 
to Pythagoras the discovery of the theory of irrationals 
(or proportions) also states that he discovered the ‘ putting 
together (arvo-raa-Ls) of the cosmic figures’ (the five regular 
solids). As usual, there has been controversy as to the sense 
in which this phrase is to be taken, and as to the possibility 
of Pythagoras having done what is attributed to him, in any 
sense of the words. I do not attach importance to the 
argument that, whereas Plato, presumably ‘ Pythagorizing ’, 
assigns the first four solids to the four elements, earth, fire, 
air, and water, Empedocles and not Pythagoras was the 
first to declare these four elements to be the material princi 
ples from which the universe was evolved ; nor do I think 
it follows that, because the elements are four, only the first 
four solids had been discovered at the time when the four 
elements came to be recognized, and that the dodecahedron 
must therefore have been discovered later. I see no reason 
why all five should not have been discovered by the early 
Pythagoreans before any question of identifying them with 
the elements arose. The fragment of Philolaus, indeed, says 
that 
‘ there are five bodies in the sphere, the fire, water, earth, 
and air in the sphere, and the vessel of the sphere itself 
making the fifth ’, x 
but as this is only to be understood of the elements in the 
sphere of the universe, not of the solid figures, in accordance 
with Diels’s translation, it would appear that Plato in the 
Timaeus 2 is the earliest authority for the allocation, and 
it may very well be due to Plato himself (were not the solids 
called the ‘ Platonic figures ’ h), although put into the mouth 
of a Pythagorean. At the same time, the fact that the 
Timaeus is fundamentally Pythagorean may have induced 
Aetius’s authority (probably Theophrastus) to conclude too 
1 Stobaeus, Ed. I, proem. 3 (p. 18. 5 Wachsmuth); Diels, Vors. i 3 , 
p. 814. The Greek of the last phrase is koi o ras acpnipas 6Xt<ds, ire/mTov, 
but oXkus is scarcely an appropriate word, and von Wilamowitz (Platon, 
vol. ii, 1919, pp. 91-2) proposes 6 ras- a-cfialpas oXkos, taking 6Xk6s (which 
implies ‘winding’) as volumen. We might then translate by ‘the spherical 
envelope ’. 
2 Timaeus, 53c-55c.