[782]. Met. Α, 6. 987 b 27, ὁ μὲν (Πλάτων) τοὺς ἀριθμοὺς παρὰ τὰ αἰσθητά, οἱ δ’ (οἱ Πυθαγόρειοι) ἀριθμοὺς εἶναί φασιν αὐτὰ τὰ αἰσθητά.

[783]. Met. Α, 5. 986 a 17 (R. P. 66); Phys. Γ, 4. 203 a 10 (R. P. 66 a).

[784]. Simpl. Phys. p. 455, 20 (R. P. 66 a). I owe the passages which I have used in illustration of this subject to W. A. Heidel, “Πέρας and ἄπειρον in the Pythagorean Philosophy” (Arch. xiv. pp. 384 sqq.). The general principle of my interpretation is also the same as his, though I think that, by bringing the passage into connexion with the numerical figures, I have avoided the necessity of regarding the words ἡ γὰρ εἰς ἴσα καὶ ἡμίση διαίρεσις ἐπ’ ἄπειρον as “an attempted elucidation added by Simplicius.”

[785]. Aristoxenos, fr. 81, ap. Stob. i. p. 20, 1, ἐκ τῶν Ἀριστοξένου Περὶ ἀριθμητικῆς ... τῶν δὲ ἀριθμῶν ἄρτιοι μέν εἰσιν οἱ εἰς ἴσα διαιρούμενοι, περισσοὶ δὲ οἱ εἰς ἄνισα καὶ μέσον ἔχοντες.

[786]. [Plut.] ap. Stob. i. p. 22, 19, καὶ μὴν εἰς δύο διαιρουμένων ἴσα τοῦ μὲν περισσοῦ μονὰς ἐν μέσῳ περιέστι, τοῦ δὲ ἀρτίου κενὴ λείπεται χώρα καὶ ἀδέσποτος καὶ ἀνάριθμος, ὡς ἂν ἐνδεοῦς καὶ ἀτελοῦς ὄντος.

[787]. Plut. de E apud Delphos, 388 a, ταῖς γὰρ εἰς ἴσα τομαῖς τῶν ἀριθμῶν, ὁ μὲν ἄρτιος πάντῃ διϊστάμενος ὑπολείπει τινὰ δεκτικὴν ἀρχὴν οἷον ἐν ἑαυτῷ καὶ χώραν, ἐν δὲ τῷ περιττῷ ταὐτὸ παθόντι μέσον ἀεὶ περίεστι τῆς νεμήσεως γόνιμον. The words which I have omitted in translating refer to the further identification of Odd and Even with Male and Female. The passages quoted by Heidel might be added to. Cf., for instance, what Nikomachos says (p. 13, 10, Hoche), ἔστι δὲ ἄρτιον μὲν ὃ οἷόν τε εἰς δύο ἴσα διαιρεθῆναι μονάδος μέσον μὴ παρεμπιπτούσης, περιττὸν δὲ τὸ μὴ δυνάμενον εἰς δύο ἴσα μερισθῆναι διὰ τὴν προειρημένην τῆς μονάδος μεσιτείαν. He significantly adds that this definition is ἐκ τῆς δημώδους ὑπολήψεως.

[788]. Arist. Phys. Γ, 4. 204 a 20 sqq., especially a 26, ἀλλὰ μὴν ὥσπερ ἀέρος ἀὴρ μέρος, οὕτω καὶ ἄπειρον ἀπείρου, εἴ γε οὐσία ἐστὶ καὶ ἀρχή.

[789]. See Chap. II. [§ 53].

[790]. Cf. Speusippos in the extract preserved in the Theologumena arithmetica, p. 61 (Diels, Vors. p. 235), τὸ μὴν γὰρ ᾱ στιγμή, τὸ δὲ β̄ γραμμή, τὸ δὲ τρία τρίγωνον, τὸ δὲ δ̄ πυραμίς. We know that Speusippos is following Philolaos here. Arist. Met. Ζ, 11. 1036 b 12, καὶ ἀνάγουσι πάντα εἰς τοὺς ἀριθμούς, καὶ γραμμῆς τὸν λόγον τὸν τῶν δύο εἶναί φασιν. The matter is clearly put in the Scholia on Euclid (p. 78, 19, Heiberg), οἱ δὲ Πυθαγόρειοι τὸ μὲν σημεῖον ἀνάλογον ἐλάμβανον μονάδι, δυάδι δὲ τὴν γραμμήν, καὶ τριάδι τὸ ἐπίπεδον, τετράδι δὲ τὸ σῶμα. καίτοι Ἀριστοτέλης τριαδικῶς προσεληλυθέναι φησὶ τὸ σῶμα, ὡς διάστημα πρῶτον λαμβάνων τὴν γραμμήν.

[791]. The identification of the point with the unit is referred to by Aristotle, Phys. Ε, 3. 227 a 27.