We must not take it for granted, however, that the Pythagorean construction of the elements was exactly the same as that which we find in Plato’s _Timaeus_. It has been mentioned already that there is good reason for believing they only knew three of the regular solids, the cube, the pyramid (tetrahedron), and the dodecahedron.[801] Now it is very significant that Plato starts from fire and earth,[802] and in the construction of the elements proceeds in such a way that the octahedron and the icosahedron can easily be transformed into pyramids, while the cube and the dodecahedron cannot. From this it follows that, while air and water pass readily into fire, earth cannot do so,[803] and the dodecahedron is reserved for another purpose, which we shall consider presently. This would exactly suit the Pythagorean system; for it would leave room for a dualism of the kind outlined in the Second Part of the poem of Parmenides. We know that Hippasos made Fire the first principle, and we see from the _Timaeus_ how it would be possible to represent air and water as forms of fire. The other element is, however, earth, not air, as we have seen reason to believe that it was in early Pythagoreanism. That would be a natural result of the discovery of atmospheric air by Empedokles and of his general theory of the elements. It would also explain the puzzling fact, which we had to leave unexplained above, that Aristotle identifies the two “forms” spoken of by Parmenides with Fire and Earth.[804] All this is, of course, problematical; but it will not be found easy to account otherwise for the facts. Footnote 801: See above, p. 329, _n._ 767. Footnote 802: Plato, _Tim._ 31 b 5. Footnote 803: Plato, _Tim._ 54 c 4. It is to be observed that in _Tim._ 48 b 5 Plato says of the construction of the elements οὐδείς πω γένεσιν αὐτῶν μεμήνυκεν, which implies that there is some novelty in the theory as he makes Timaios state it. If we read the passage in the light of what has been said in § 141, we shall be inclined to believe that Plato is working out the Pythagorean doctrine on the lines of the discovery of Theaitetos. There is another indication of the same thing in Arist. _Gen. Corr._ Β, 3. 330 b 16, where we are told that, in the Διαιρέσεις, Plato assumed three elements, but made the middle one a mixture. This is stated in close connexion with the ascription of Fire and Earth to Parmenides. Footnote 804: See above, Chap. IV. p. 213, _n._ 462. [Sidenote: The dodecahedron.]