148. The most interesting point in the theory is, perhaps, the use made of the dodecahedron. It was identified, we are told, with the “sphere of the universe,” or, as it is put in the Philolaic fragment, with the “hull of the sphere.”[805] Whatever we may think of the authenticity of the fragments, there is no reason to doubt that this is a genuine Pythagorean expression, and it must be taken in close connexion with the word “keel” applied to the central fire.[806] The structure of the world was compared to the building of a ship, an idea of which there are other traces.[807] The key to what we are told of the dodecahedron is given by Plato. In the _Phaedo_ we read that the “true earth,” if looked at from above, is “many-coloured like the balls that are made of twelve pieces of leather.”[808] In the _Timaeus_ the same thing is referred to in these words: “Further, as there is still one construction left, the fifth, God made use of it for the universe when he painted it.”[809] The point is that the dodecahedron approaches more nearly to the sphere than any other of the regular solids. The twelve pieces of leather used to make a ball would all be regular pentagons; and, if the material were not flexible like leather, we should have a dodecahedron instead of a sphere. This points to the Pythagoreans having had at least the rudiments of the “method of exhaustion” formulated later by Eudoxos. They must have studied the properties of circles by means of inscribed polygons and those of spheres by means of inscribed solids.[810] That gives us a high idea of their mathematical attainments; but that it is not too high, is shown by the fact that the famous lunules of Hippokrates date from the middle of the fifth century. The inclusion of _straight_ and _curved_ in the “table of opposites” under the head of Limit and Unlimited points in the same direction.[811] Footnote 805: Aet. ii. 6, 5 (R. P. 80); “Philolaos,” fr. 12 (= 20 M.; R. P. 79). On the ὁλκάς, see Gundermann in _Rhein. Mus._ 1904, pp. 145 sqq. I agree with him in holding that the reading is sound, and that the word means “ship,” but I think that it is the structure, not the motion, of a ship which is the point of comparison. Footnote 806: Aet. ii. 4, 15, ὅπερ τρόπεως δίκην προϋπεβάλετο τῇ τοῦ παντὸς <σφαίρᾳ> ὁ δημιουργὸς θεός. Footnote 807: Cf. the ὑποζώματα of Plato, _Rep._ 616 c 3. As ὕλη generally means “timber” for shipbuilding (when it does not mean firewood), I suggest that we should look in this direction for an explanation of the technical use of the word in later philosophy. Cf. Plato, _Phileb._ 54 c 1, γενέσεως ... ἕνεκα ... πᾶσαν ὕλην παρατίθεσθαι πᾶσιν, which is part of the answer to the question πότερα πλοίων ναυπηγίαν ἕνεκα φῂς γίγνεσθαι μᾶλλον ἢ πλοῖα ἕνεκα ναυπηγίας; (_ib._ b 2); _Tim._ 69 a 6, οἷα τέκτοσιν ἡμῖν ὕλη παράκειται. Footnote 808: Plato, _Phd._ 110 b 6, ὥσπερ οἱ δωδεκάσκυτοι σφαῖραι with Wyttenbach’s note. Footnote 809: