[239]. In the fragment referred to above (p. 113, [n. 236]), Speusippos speaks of four as the first pyramidal number; but this is taken from Philolaos, so we cannot safely ascribe it to Pythagoras.
[240]. We have ὅροι of a series (ἔκθεσις), then of a proportion, and in later times of a syllogism. The signs :, ::, and ∴ are a survival of the original use. The term χώρα is often used by the later Pythagoreans, though Attic usage required χωρίον for a rectangle. The spaces between the γραμμαί of the abacus and the chess-board were also called χῶραι.
[241]. In his commentary on Euclid i. 44, Proclus tells us on the authority of Eudemos that the παραβολή, ἔλλειψις, and ὑπερβολή of χωρία were Pythagorean inventions. For an account of these and the subsequent application of the terms in Conic Sections, see Milhaud, Philosophes géomètres, pp. 81 sqq.
[242]. The verb ὑποτείνειν is, of course, used intransitively. The explanation suggested in the text seems to me much simpler than that of Max C. P. Schmidt (Kulturhistorische Beiträge, Heft i. pp. 64 sqq.). He explains the hypotenuse as the longest string in a triangular harp; but my view seems more in accordance with analogy. So ἡ κάθετος is, literally, a plumb-line.
[243]. The statement comes from Eudemos; for it is found in Proclus’s commentary on Euclid i. 47. Whether historical or not, it is no Neopythagorean fancy.
[244]. Arist. An. Pr. Α, 23. 41 a 26, ὅτι ἀσύμμετρος ἡ διάμετρος διὰ τὸ γίγνεσθαι τὰ περιττὰ ἴσα τοῖς ἀρτίοις συμμέτρου τεθείσης. The proofs given at the end of Euclid’s Tenth Book (vol. iii. pp. 408 sqq., Heiberg) turn on this very point. They are not Euclidean, and may be substantially Pythagorean. Cf. Milhaud, Philosophes géomètres, p. 94.
[245]. Plato, Theaet. 147 d 3 sqq.
[246]. How novel these consequences were, is shown by the fact that in Laws, 819 d 5, the Athenian Stranger says that he had only realised them late in life.
[247]. This version of the tradition is mentioned in Iamblichos, V. Pyth. 247, and looks older than the other, which we shall come to later ([§ 148]). Hippasos is the enfant terrible of Pythagoreanism, and the traditions about him are full of instruction.
[248]. Plato (Tim. 36 a 3) defines the harmonic mean as τὴν ... ταὐτῷ μέρει τῶν ἄκρων αὐτῶν ὑπερέχουσαν καὶ ὑπερεχομένην. The harmonic mean of 12 and 6 is therefore 8; for 8 = 12 - 12/3 = 6 + 6/3.