^{page 1 note 1} Iambl, . V.P. 82Google Scholar τ στι τ ν Δελφοῖς μαντεῖον; τετρακτς, ὅπορ στν ρμονα ν ᾖ αἱ Σειρνες. Diels, Vors. ³ 45c 4.

^{page 1 note 2} Plut. Is. et Os. 75.

^{page 1 note 3} Theon Smyrn. π. τʵτρακτος, 154 (ed. Dupuis).

^{page 1 note 4} Ar. Met. A 5, 986a 8. Aet. 1. 3. 8. Hippol. Ref. VI. 23. Ten is the perfect number to Pythagoras, τ γρ ἓνδʵκα κα δώδεκα προσθκην κα παναποδισμν τς δεκδες οὐκ ἄλλου τινς ριθμο γννησω τὂ προστιθμενον.

^{page 1 note 5} Met. A 5, 986a 15. Τοῖ δ ριθμο στοιχεῖα τ τ' ἄρτιον κα τδ περιττν (τοτων δ τ μν ἄπειρον, τ δ πεπερασμνον), τ δ' ἓν ξ μφοτρων εἶναι τοτων (κα γρ ἄρτιον εἶναι κα περιττν), τν δ ριθμν κ το νς, ριθμοὐς δ, καθπερ εἴρηται, τδν λον οὐρανν.

^{page 2 note 1} Met. A 5, 990a 8, has πρας (not πεπερασμνον) and ἄπειρον as the equivalents of περιττν and ἄρτιον, Πρας (περαῖνον, Philolaus) is correct.

^{page 2 note 2} Diels, Vors. ³ 45 B 2, who compares Ar. Met. M 8, 1083b 28, ἔτι αἱ ν τῇ τριδι αὐτῇ (μον-δες) πς; μα γρ περιττ. λλ δι τοτο ἴσως αὐτ τ ἓν ποιοσιν ν τῷ μσον. For Explanations and other definitions see Heath, , The Thirteen Books of Euclid's Elements (1908), Vol. II., p. 281Google Scholar. The curious and unique use of ἰσοσκελς = ἄρτιος and σκαληνς = περιττς in Plato, Euthyphro 12Google Scholar D may be explained by the diagrams ·]·,:]:, ⋮ ] ⋮, etc., and · ]:,:] ⋮, ⋮ [ ⋮, etc., which show even numbers when divided as ‘equal-legged,’ odd numbers as having one leg longer than the other.

^{page 2 note 3} de E. ap. Delphos, 388 A. On this subject see Heidel, W. A., πρας and ἅπειρον in the Pythagorean philosophy, Arch. Gesch. Phil. N. F. VII. 384Google Scholar.

^{page 2 note 4} Plutarch (Diels, , Dox. 96Google Scholar) ap. Stob. Ecl. Phys. I. I. 10, p. 22, Wachsmuth.

^{page 2 note 5} Ar. de caelo α 1. 268a 10 καθπερ φασ κα οἰ Πυθαγρειοι, τ πν κα τ πντα τοῖς τρισν ὣρισται τελευτ γρ κα ρχ τν ριθμν ἔχει τν το παντς, τατα δ τν τς τριδος. Aristoxenus (Stob. I. I pr. 6) οὕτως ν περισσαῖς μραις αἱ κρσεις τν νοσημτων γγνεσθαι δοκοσιν κα αἱ μεταβολα, ὅτι περιττς κα ρχν καἰ τελευτν κα μσον ἔχει, ρχς καἰ κμς κα παρακμς χμεναι. This sounds primitive.

^{page 2 note 6} Aristoxenus ap. Stob. I. I pr. 6. Diels, Vors. ³ 45 B2.

^{page 3 note 1} Hence in the above passage from Aristotle (Met. A 5, 986a 19) I translate τ δ ἓν ξ μφο-τρων εἷναι τοτων ‘the One consists of both of these’ (odd and even), not (with Ross, e.g.) ‘the I proceeds from both of these.’ [So Alexander (on 985b 26, p. 30, 16 Bz.): τν δ ριθμν τν μονδα ρχν εἷναι, σ υ γ κ ε ι μ ν η ν ἔκτε το ρτου κα περιττο εἶναι γρ τν μονδα ἅμα ρτιοπριττον, ὃ δεκνυε δι τ γεννητικν αὐτν εἷναι κα το περιττο κα το ρτου ριθμο]. It is true that ‘proceeds’ is appropriate to the following words, τν δ' ριθμν κ το νς, but in any case the relation here expressed by κ cannot be the same as in ξ μφοτρων εἶναι. It may, however, be doubted whether Aristotle himself clearly understood.

^{page 3 note 2} Ed. Dupuis (1892), p. 164.

^{page 3 note 3} Cf. Apoll. Rhod. I. 494, Ὀρφες … ἤειδεν δ' (I) ὡς γαῖα κα οὐρανς ἠδ θλασσα | τ πρν π' λλλοισι μιῇ συναρηρτα μορφῇ | (2) νεκεος ξ λοοῖο δικριθεν μφς ἕκαστα | ἠδ' ὡς ἔμπεδον αἰν ν αἰθρι τκμαρ ἔχουσιν | ἄστρα σεληναη τε κα ἠελοιο κλενθοι | οὔρε θ' ὡς ντειλε, κα ὡς ποταμο κελδοντες | αὐτῇσιν νμφῃσι (3) κα ρπετ πντ' γνοντο. For the separation of Father Heaven and Mother Earth out of a primal unity and their subsequent marriage see Tylor, , Primitive Culture ⁴ (1903) I. 325Google Scholar (parallels from New Zealand, China, etc.), and Grimble, A., Myths from the Gilbert Islands, Folklore xxxiii. (1922), 91 ffGoogle Scholar.

^{page 3 note 4} So Aristotle, Phys. α 4, 187a 20, οἱ δ κ το νς ν ο σ α ς τς ναντιτητας κκρνεσθαι, ὥσπερ Ἀναξμανδρς φησι.

^{page 4 note 1} [Ar.] M. Mor. α 1. 1182a 11, δικαιοσνη ριθμς ἰσκις ἴσος. This interpretation of 4 in the Decad occurs in a Paris MS. published by Delatte, Études sur la lit. Pyth. p. 167, τετρς δικαιοσνη δι τ ἰσκις ἰσον.

^{page 4 note 2} Aesch. Danaids 44, N², ρᾷ μν γνς οὐρανς τρώσαι χθνα, | ἔρως δ γαῖαν λαμβνει γμου τυχεῖν | μβρος δ' π' εὐρανο πεσὼν | ἔδευσε γαῖαν ἣ δ τκτεται βροτοῖς | μλων τε βοσκᾰς κα βον Δημτριον.

^{page 4 note 3} Cf. Empedocles 17. 26 of his elements: τατα γἄρ ἶσ τε πντα κα ἥλικα γνναν ἔασι, | τιμς δ' ἄλλης ἄλλο μδει, πρα δ' ἦθος κστῳ, | ν δ μρει κρατουσι περιπλομνοιο χρνοιο. Alex. Poly-histor ap. Diog. L. VIII. 26 (Pythagorean doctrine): ἰσμιρ τ' εἶναι ν τῷ κσμῳ φς κα σκτος, κα θερμν κα ψυχρν κα ξηρν κα ὑγρν ὧν κτ' πικρτειαν θερμο μν θρος γνεσθαι, ψυχρο δ χειμνα, ξηρο δ' ἔαρ, κα ὑγρο φθινπωρον. ν δ ἰσομοιρῇ, τ κλλιστα εἷναι το ἔτους …

^{page 4 note 4} Ar. E.N. E 5, 1132b 21.

^{page 4 note 5} Cf. Plato's description of δικαιοσνη above quoted (Vol. XVI., p. 147).

^{page 4 note 6} Alex. on Ar. Met. 987a 9 (p. 36, 18 Bz). The saying φιλτης ἰστης is attributed to Pythagoras by Iambl. V.P. 162 and Porphyry V.P. 20 (probably following Timaeus, Delatte, , Études sur la lit. Pyth. 253Google Scholar).

^{page 5 note 1} I agree with Gilbert, O. (Arch. Gesch. Phil. XXII. 155)Google Scholar against Zeller that to the mystical Pythagoreans the Monad was God (Aetius 1. 3. 8).

^{page 6 note 1} Aristotle's obscure remark as to the Pythagorean κοσμποια (Met. N. 3 1091a 12) refers, I believe, to the later system of Number-atomism discussed below, see p. 9. At 990a 8 Aristotle remarks that, though the Pythagoreans γεννσι τν οὐρανν, they have no explanation how there is to be motion when only Limit and Unlimited or Odd Even are posited.

^{page 6 note 2} Iambl, . V.P. 89Google Scholar καλεῖτο δ γεωμετρα πρς Πυθαγρου στορα.

^{page 6 note 3} Later mysticism regards the emergence of the Dyad as an act of rebellious audacity: Theol. Arith. II. 10 πρώτη γρ δυς διεχώρισεν αὑτν κ τς μονδος, ὃθεν κα τλμα καλεῖται. So Plotinus, Enn. V. 1. 1, ascribes the fall of the soul to τλμα. Proclus on Plato, Alcib, I. 104 E attributes this use of τλμα to the Pythagoreans.

^{page 6 note 4} Simplicius, Phys. 181 (quoting Eudorus): ‘According to their highest teaching we must say that the Pythagoreans hold the One to be the principle of all things; according to a secondary teaching (δετερος λγος) they hold that there are two principles of created things, the One and the nature opposed to it.’

^{page 8 note 1} Nicomachus I, 7, I combines several definitions of number: (I) πλθος ὡρισμένον (cf. Ar. Met. 1020a 13 πλθος τ πεπερασμένον); (2) μονδων σύστημα (cf. Ar. 1053a 30 πλθος μονάδων; 1039a 12 σύνθεσις μονάδων, ὥσπερ λέγεται ὑπό τινων; 207b 7 ἔνα πλεω); (3) ποσότητος ϰύμα κ μονδων σνγκε μενον (cf. Moderatus ap. Stob. Ecl. I. I, pr. 8 σύστημα μονάδων, ἢ προποδισμς πλθους π μονάδος ρϰόμενος κα ναποδισμς ες μονδα καταλγων. So also Theon Smyrn. p. 28, Dupuis). Euclid, Book VII. def. 2, has only τ κ μονάδων σνγκεμενον πλθος. (So Aristoxenus, Diels, Vors.³ 45 B 2). The first of the above definitions, πλθος ὡρισμένον, is given as the view of Eudoxus the Pythagorean by Iamblichus, (Comm. on Nicom. Pistelli, p. 10)Google Scholar, who contrasts it with the second, μονάδων σύστημα, which he attributes to Thales, following the Egyptian view. It may be suggested that πλθος ὡρισμένον or πεπερασμένον goes back to the characteristically Pythagorean conception of number as the product of the Union of πέρας and ἄπειρον; whereas σύστημα μονάδων is goes the crude, and so to say materialistic, view which may well have been shared by the Egyptians and the Pythagorean mathematicians or number-atomists now under consideration.

^{page 10 note 1} Plutarch, (Stob. Ecl. I, pr. 2)Google Scholar, ἠ μονς γον ὑπ Τιμαου το Αοκρο προσαγορεεται, ὡ;ς ἄρχουσα τς τν ριθμ γενσεως. Hermes, (Stob. Ecl. I. 10. 15)Google Scholar, γρ μονς, οὗσα πντων ρχ κα ῥζα, ν πσν ὡς ν ῥζα κα ρϰή. Cf. also Ar. Met. N 5, 1092a 23, τνα τρόπον ριθμός στιν κ τν ρϰν … (32) λλ' ὡς π σπέρματος: λλ' οὐχ οἶόν τε το διαιρέτον τι πελθεῖν. Theon (p. 158, Dupuis), ἓκτη δ (τετρακτὺς) τν Φεομνων. τ μν σπρμα νλογον μονδι κα σημεω.

^{page 10 note 2} Speusippus, (Theol. Arith., p. 61 sqq.Google Scholar; Diels, , Vors.³, p. 304, 19)Google Scholar, ἔν τε πιπέδοις κα στερεοῖς πρτά στι τατα · στιγμή, γραμμή, τργωνον, πνραμς.

^{page 10 note 3} There is no evidence for attributing more than πρ ρϰή to Hippasus, though the story (countenanced by Heath, , Greek Mathematics, I. 160)Google Scholar connecting his name with the constrution of a regular solid may be recalled. Simplicius, ad loc., does not know to whom to attribute the doctrine mentioned by Aristotle.

^{page 11 note 1} The Pythagorean Ecphantus of Syracuse is said to have been the first who regarded the Pythagorean monads as bodily (σωματικάς) or as άδιαρετα σὠματα of which sensible things consist (Aet. I. 3. 19; Hippol, . Ref. I. 15Google Scholar). His date is unknown; but the testimony supports the view that this number-atomism was no part of the original doctrine, and that the view that things are related to numbers by μμησις is older than the identification of bodies with numbers.

^{page 11 note 2} Themistius observes that he does not know which Pythagoreans are meant (π. ψνϰς, I. 2, p. 17, Spengel). The doctrine was evidently obsolete.

^{page 11 note 3} Ar. de caelo III, 5, 304a 7, ὅσοι δ πρ ύποτθɛνται τ στοιϰῖον … ο μν … σϰμα περιάπ τονσι τῷ πνρ, καθπερ ο τν πνραμδα ποιοντες, κα τούτων ο μν άπλονστέρως λέγοντες ὅτι τν μν σϰημάτων τμητικώτατον πνραμς, τν δ σωμάτων τ πρ Cf. de anim. 404a I sqq. Democritus and Leucippus made soul consist of spherical atoms δι τ μάλιστα δι παντς δύνασθαι διαδύειν τούς τοιούτονς ῤνσμούς

^{page 12 note 1} Plato, , Parm. 128 cGoogle Scholar. The imaginary date of the dialogue is about 450 B.C. Zeno is ‘about 40 years old’ (127 B), and he speaks of his treatise as having been written ‘when he was young’ ὑπ νον ντος μοῖ γρφη 128 D). This suggests a date about 470, which would be too early for an attack on Atomism proper.