[812]. Iambl. V. Pyth. 247. Cf. above, Chap. II. p. 117, [n. 247].
[813]. 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 συνθείᾳ λεγόμενον πένταλφα σύμβολον ἦν πρὸς ἀλλήλους Πυθαγορείων ἀναγνωριστικὸν καὶ τούτῳ ἐν ταῖς ἐπιστολαῖς ἐχρῶντο.
[814]. Arist. de An. Α, 3. 407 b 20 (R. P. 86 c).
[815]. Plato, Phd. 85 e sqq.; and for Echekrates, ib. 88 d.
[816]. Plato, Phd. 86 b 7-c 5.
[817]. For the authorities, see R. P. 81-83. The attribution of the theory to Philolaos is perhaps due to Poseidonios. The “three books” were doubtless in existence by his time.
[818]. Plato attributes an axial rotation to the heavenly bodies (Tim. 40 a 7), which must be of this kind. It is quite likely that the Pythagoreans already did so, though Aristotle was unable to see the point. He says (de Caelo, Β, 8. 290 a 24), ἀλλὰ μὴν ὅτι οὐδὲ κυλίεται τὰ ἄστρα, φανερόν· τὸ μὲν γὰρ κυλιόμενον στρέφεσθαι ἀνάγκη, τῆς δὲ σελήνης ἀεὶ δηλόν ἐστι τὸ καλούμενον πρόσωπον. This, of course, is just what proves it does rotate.
[819]. Plato, Phd. 108 e 4 sqq. Simmias assents to this doctrine in the emphatic words Καὶ ὀρθῶς γε.
[820]. The primitive character of the astronomy taught by Demokritos as compared with that of Plato is the best evidence of the value of the Pythagorean researches.
[821]. Arist. de Caelo, Β, 13. 293 a 18 sqq. (R. P. 83).