Early Greek Philosophy by John Burnet, with Burnet's notes
147. The Numbers and the Elements 149. The Soul a "Harmony"

From Chapter VII., The Pythagoreans

148. The Dodecahedron
The most interesting point in the theory is, however, 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."74 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.75 The structure of the world was compared to the building of a ship, an idea of which there are other traces.76 The key to what we are told of the dodecahedron is also given by Plato. In the Phaedo, which must have been written before the doctrine of the regular solids was fully established, 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."77 In the Timaeus the same thing is referred to in those words: "Further, as there is still one construction left, the fifth, God made use of it for the universe when he painted it."78 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. That proves that the dodecahedron was well known before Theaitetos, and we may infer that it was regarded as forming the "timbers" on which the spherical hulk of the heavens was built.

The tradition confirms in an interesting way the importance of the dodecahedron in the Pythagorean system. According to one account, Hippasos was drowned at sea for revealing "the sphere formed out of the twelve pentagons."79 The Pythagorean construction of the dodecahedron we may partially infer from the fact that they adopted the pentagram or pentalpha as their symbol. The use of this figure in later magic is well known; and Paracelsus still employed it as a symbol of health, which is exactly what the Pythagoreans called it.80

Burnet's Notes


74. 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. In the Pythagorean myth of Plato's Politicus, the world is regarded as a ship, of which God is the κυβερνήτης (272 a sqq.). The πόντος τῆς ἀνομοιότητος (273 d) is just the ἄπειρον.

75. Aet. ii. 4, 15, ὅπερ τρόπεως δίκην προϋπεβάλετο τῇ τοῦ παντὸς <σφαίρᾳ> ὁ δημιουργὸς θεός.

76. 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.

77. Plato, Phaed. 110 b 6, ὥσπερ οἱ δωδεκάσκυτοι σφαῖραι, the meaning of which phrase is quite correctly explained by Plutarch, Plat. q. 1003 b καὶ γὰρ μάλιστα τῷ πλήθει τῶν στοιχείων ἀμβλύτητι δὲ τῶν γωνιῶν τὴν εὐθύτητα διαφυγὸν εὐκαμπές ἐστι [τὸ δωδεκάεδρον], καὶ τῇ περιτάσει ὥσπερ αἱ δωδεκάσκυτοι σφαῖρα κυκλοτερὲς γίγνεται καὶ περιληπτικόν.

78. Plato, Tim. 55 c 4. Neither this passage nor the last can refer to the Zodiac, which would be described by a dodecagon, not a dodecahedron. What is implied is the division of the heavens into twelve pentagonal fields, in which the constellations were placed. For the history of such methods see Newbold in Arch. xix. pp. 198 sqq.

79. Iambl. V. Pyth. 247. Cf. above, Chap. II. p.106, n. 1.

80. See Gow, Short History of Greek Mathematics, p. 151, and the passages there referred to, adding Schol. Luc. p. 234, 21, Rabe, τὸ πεντάγραμμον] ὅτι τὸ ἐν τῇ συνηθείᾳ λεγόμενον πεντάλφα σύμβολον ἦν πρὸς ἀλλήλους Πυθαγορείων ἀναγνωριστικὸν καὶ τούτῳ ἐν ταῖς ἐπιστολαῖς ἐχρῶντο. The Pythagoreans may quite well have known the method given by Euclid iv. 11 of dividing a line in extreme and mean ratio, the so-called "golden section."

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