is clear that all these passages refer to the same thing, and that can hardly be anything else than those arrangements of “terms” in patterns with which we are already familiar (§ 47). If we think of these, we shall see in what sense it is true that bipartition goes on _ad infinitum_. However high the number may be, the number of ways in which it can be equally divided will also increase. Footnote 783: __Met.__ Α, 5. 986 a 17 (R. P. 66); _Phys._ Γ, 4. 203 a 10 (R. P. 66 a). Footnote 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.” Footnote 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. Footnote 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. Footnote 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.