From Chapter VII., The Pythagoreans
148. The Dodecahedron
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
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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