Chapter 30

790. Cf. Speusippos in the extract preserved in theTheologumena 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.

792. Arist.Met.Μ, 6. 1080 b 18 sqq., 1083 b 8 sqq.;de Caelo, Γ, 1. 300 a 16 (R. P. 76 a).

793. Zeller, p. 381.

794. We learn from Plato,Theaet.148 b 1, that Theaitetos called surds, what Euclid calls δυνάμει σύμμετρα, by the name of δυνάμεις, while rational square roots were called μήκη. Now inTim.31 c 4 we find a division of numbers into ὄγκοι and δυνάμεις, which seem to mean rational and irrational quantities. Cf. also the use of ὄγκοι inParm.164 d. Zeno in his fourth argument about motion, which, we shall see (§ 163), was directed against the Pythagoreans, used ὄγκοι for points. Aetios, i. 3, 19 (R. P. 76 b), says that Ekphantos of Syracuse was the first of the Pythagoreans to say that their units were corporeal. Probably, however, “Ekphantos” was a personage in a dialogue of Herakleides (Tannery,Arch.xi. pp. 263 sqq.), and Herakleides called the monads ἄναρμοι ὄγκοι (Galen,Hist. Phil.18;Dox.p. 610).

795. Zeller, p. 382.

796. Arist.Met.Α, 8. 990 a 22 (R. P. 81 e). I read and interpret thus: “For, seeing that, according to them, Opinion and Opportunity are in a given part of the world, and a little above or below them Injustice and Separation and Mixture,—in proof of which they allege that each of these is a number,—and seeing that it is also the case (reading συμβαίνῃ with Bonitz) that there is already in that part of the world a number of composite magnitudes (i.e.composed of the Limit and the Unlimited), because those affections (of number) are attached to their respective regions;—(seeing that they hold these two things), the question arises whether the number which we are to understand each of these things (Opinion, etc.) to be is the same as the number in the world (i.e.the cosmological number) or a different one.” I cannot doubt that these are the extended numbers which are composed (συνίσταται) of the elements of number, the limited and the unlimited, or, as Aristotle here says, the “affections of number,” the odd and the even. Zeller’s view that “celestial bodies” are meant comes near this, but the application is too narrow. Nor is it the number (πλῆθος) of those bodies that is in question, but their magnitude (μέγεθος). For other views of the passage, see Zeller, p. 391, n. 1.

797. Zeller, p. 404.

798.Ibid.pp. 467 sqq.

799. All this has been put in its true light by the publication of the extract from Menon’s Ἰατρικά, on which see p. 322,n.742.

800. In Aet. ii. 6, 5 (R. P. 80) the theory is ascribed to Pythagoras, which is an anachronism, as the mention of “elements” shows it must be later than Empedokles. In his extract from the same source, Achilles says οἱ Πυθαγόρειοι, which doubtless represents Theophrastos better. There is a fragment of “Philolaos” bearing on the subject (R. P. 79), where the regular solids must be meant by τὰ ἐν τᾷ σφαίρᾳ σώματα.

801. See above, p. 329,n.767.

802. Plato,Tim.31 b 5.

803. Plato,Tim.54 c 4. It is to be observed that inTim.48 b 5 Plato says of the construction of the elements οὐδείς πω γένεσιν αὐτῶν μεμήνυκεν, which implies that there is some novelty in the theory as he makes Timaios state it. If we read the passage in the light of what has been said in § 141, we shall be inclined to believe that Plato is working out the Pythagorean doctrine on the lines of the discovery of Theaitetos. There is another indication of the same thing in Arist.Gen. Corr.Β, 3. 330 b 16, where we are told that, in the Διαιρέσεις, Plato assumed three elements, but made the middle one a mixture. This is stated in close connexion with the ascription of Fire and Earth to Parmenides.

804. See above, Chap. IV. p. 213,n.462.

805. Aet. ii. 6, 5 (R. P. 80); “Philolaos,” fr. 12 (= 20 M.; R. P. 79). On the ὁλκάς, see Gundermann inRhein. Mus.1904, pp. 145 sqq. I agree with him in holding that the reading is sound, and that the word means “ship,” but I think that it is the structure, not the motion, of a ship which is the point of comparison.

806. Aet. ii. 4, 15, ὅπερ τρόπεως δίκην προϋπεβάλετο τῇ τοῦ παντὸς <σφαίρᾳ> ὁ δημιουργὸς θεός.

807. Cf. the ὑποζώματα of Plato,Rep.616 c 3. As ὕλη generally means “timber” for shipbuilding (when it does not mean firewood), I suggest that we should look in this direction for an explanation of the technical use of the word in later philosophy. Cf. Plato,Phileb.54 c 1, γενέσεως ... ἕνεκα ... πᾶσαν ὕλην παρατίθεσθαι πᾶσιν, which is part of the answer to the question πότερα πλοίων ναυπηγίαν ἕνεκα φῂς γίγνεσθαι μᾶλλον ἢ πλοῖα ἕνεκα ναυπηγίας; (ib.b 2);Tim.69 a 6, οἷα τέκτοσιν ἡμῖν ὕλη παράκειται.

808. Plato,Phd.110 b 6, ὥσπερ οἱ δωδεκάσκυτοι σφαῖραι with Wyttenbach’s note.

809. Plato,Tim.55 c 4. Neither this passage nor the last can refer to the Zodiac, which would be described by a dodecagon, not a dodecahedron. What is implied is the division of the heavens into twelve pentagonal fields.

810. Gow,Short History of Greek Mathematics, pp. 164 sqq.

811. This is pointed out by Kinkel,Gesch. der Phil.vol. i. p. 121.

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).

822. Plato,Tim.40 c 1, (γῆν) φύλακα καὶ δημιουργὸν νυκτός τε καὶ ἡμέρας ἐμηχανήσατο. On the other hand, νὺξ μὲν οὖν ἡμέρα τε γέγονεν οὕτως καὶ διὰ ταῦτα, ἡ τῆς μιᾶς καὶ φρονιμωτάτης κυκλήσεως περίοδος (39 c 1).

823. Arist.de Caelo, Β, 13. 293 b 15 sqq.

824. Boeckh admitted a very slow motion of the heaven of the fixed stars, which he at first supposed to account for the precession of the equinoxes, though he afterwards abandoned that hypothesis (Untersuchungen, p. 93). But, as Dreyer admits (Planetary Systems, p. 49), it is “not ... necessary with Boeckh to suppose the motion of the starry sphere to have been an exceedingly slow one, as it might in any case escape direct observation.”

825. Aet. ii. 20, 13 (Chap. IV. p. 275,n.609); cf.ib.12 (of Philolaos), ὥστε τρόπον τινὰ διττοὺς ἡλίους γίγνεσθαι, τό τε ἐν τῷ οὐρανῷ πυρῶδες καὶ τὸ ἀπ’ αὐτοῦ πυροειδὲς κατὰ τὸ ἐσοπτροειδές· εἰ μή τις καὶ τρίτον λέξει τὴν ἀπὸ τοῦ ἐνόπτρου κατ’ ἀνάκλασιν διασπειρομένην πρὸς ἡμᾶς αὐγήν. Here τὸ ἐν τῷ οὐρανῷ πυρῶδες is the central fire, in accordance with the use of the word οὐρανός explained in another passage of Aetios, Stob.Ecl.i. p. 196, 18 (R. P. 81). It seems to me that these strange notices must be fragments of an attempt to show how the heliocentric hypothesis arose from the theory of Empedokles as to the sun’s light. The meaning is that the central fire really was the sun, but that Philolaos unnecessarily duplicated it by supposing the visible sun to be its reflexion.

826. Chap. VI. §113.

827. Aet. i. 7, 7 (R. P. 81). Procl.in Tim.p. 106, 22, Diehl (R. P. 83 e).

828. On these points, see Staigmüller,Beiträge zur Gesch. der Naturwissenschaften im klassichen Altertume(Progr., Stuttgart, 1899); and“Herakleides Pontikos und das heliokentrische System”(Arch.xv. pp. 141 sqq.). Though, for reasons which will partly appear from the following pages, I should not put the matter exactly as Staigmüller does, I have no doubt that he is substantially right. Diels had already expressed his adhesion to the view that Herakleides was the real author of the heliocentric hypothesis (Berl. Sitzb., 1893, P. 18).

829. In his letter to Pope Paul III., Copernicus quotes Plut.Plac.iii. 13, 2-3 (R. P. 83 a), and adds“Inde igitur occasionem nactus, coepi et ego de terrae mobilitate cogitare.”The whole passage is paraphrased by Dreyer,Planetary Systems, p. 311. Cf. also the passage from the original MS., which was first printed in the edition of 1873, translated by Dreyer,ib.pp. 314 sqq.

830. Arist.Met.Α, 5. 986 a 3 (R. P. 83 b).

831. Aet. ii. 29, 4, τῶν Πυθαγορείων τινὲς κατὰ τὴν Ἀριστοτέλειον ἱστορίαν καὶ τὴν Φιλίππου τοῦ Ὀπουντίου ἀπόφασιν ἀνταυγείᾳ καὶ ἀντιφράξει τοτὲ μὲν τῆς γῆς, τοτὲ δὲ τῆς ἀντίχθονος (ἐκλείπειν τὴν σελήνην).

832. Arist.de Caelo, Β, 13. 293 b 21, ἐνίοις δὲ δοκεῖ καὶ πλείω σώματα τοιαῦτα ἐνδέχεσθαι φέρεσθαι περὶ τὸ μέσον ἡμῖν ἄδηλα διὰ τὴν ἐπιπρόσθησιν τῆς γῆς. διὸ καὶ τὰς τῆς σελήνης ἐκλείψεις πλείους ἢ τὰς τοῦ ἡλίου γίγνεσθαί φασιν· τῶν γὰρ φερομένων ἕκαστον ἀντιφράττειν αὐτήν, ἀλλ’ οὐ μόνον τὴν γῆν.

833. It is not expressly stated that they were Pythagoreans, but it is natural to suppose so. Such, at least, was Alexander’s opinion (Simpl.de Caelo, P. 515, 25).

834. The term οἱ μαθηματικοί is that used by Poseidonios for the Chaldæan astrologers (Berossos). Diels,Elementum, p. 11, n. 3. As we have seen, the Babylonians knew the planets better than the Greeks.

835. Arist.de Caelo, Β, 9. 290 b 12 sqq. (R. P. 82).

836. Alexander,in Met.p. 39, 24 (from Aristotle’s work on the Pythagoreans), τῶν γὰρ σωμάτων τῶν περὶ τὸ μέσον φερομένων ἐν ἀναλογίᾳ τὰς ἀποστάσεις ἐχόντων ... ποιούντων δὲ καὶ ψόφον ἐν τῷ κινεῖσθαι τῶν μὲν βραδυτέρων βαρύν, τῶν δὲ ταχυτέρων ὀξύν. We must not attribute the identification of the seven planets with the seven strings of the heptachord to the Pythagoreans of this date. Mercury and Venus have in the long run the same velocity as the sun, and we must take in the earth and the fixed stars. We can even find room for theantichthonas προσλαμβανόμενος.

837. For the various systems, see Boeckh,Kleine Schriften, vol. iii. pp. 169 sqq., and Carl v. Jan,“Die Harmonie der Sphären”(Philol.1893, pp. 13 sqq.). They vary with the astronomy of their authors, but they bear witness to the fact stated in the text. Many give the highest note to Saturn and the lowest to the Moon, while others reverse this. The system which corresponds best, however, with the Pythagorean planetary system must include the heaven of the fixed stars and the earth. It is that upon which the verses of Alexander of Ephesos quoted by Theon of Smyrna, p. 140, 4, are based:γαῖα μὲν οὖν ὑπάτη τε βαρεῖά τε μέσσοθι ναίει·ἀπλανέων δὲ σφαῖρα συνημμένη ἔπλετο νήτη, κ.τ.λ.The “base of Heaven’s deep Organ” in Milton’s “ninefold harmony” (Hymn on the Nativity, xiii.) implies the reverse of this.

γαῖα μὲν οὖν ὑπάτη τε βαρεῖά τε μέσσοθι ναίει·ἀπλανέων δὲ σφαῖρα συνημμένη ἔπλετο νήτη, κ.τ.λ.

γαῖα μὲν οὖν ὑπάτη τε βαρεῖά τε μέσσοθι ναίει·

ἀπλανέων δὲ σφαῖρα συνημμένη ἔπλετο νήτη, κ.τ.λ.

The “base of Heaven’s deep Organ” in Milton’s “ninefold harmony” (Hymn on the Nativity, xiii.) implies the reverse of this.

838. The difficulty appears clearly in Adam’s note onRepublic, 617 b (vol. ii. p. 452). There the ἀπλανής appears rightly as the νήτη, while Saturn, which comes next to it, is the ὑπάτη. It is inconceivable that this should have been the original scale. Aristotle touches upon the point (de Caelo, Β, 10. 291 a 29 sqq.); and Simplicius sensibly observes (de Caelo, p. 476, 11), οἱ δὲ πάσας τὰς σφαίρας τὴν αὐτὴν λέγοντες κίνησιν τὴν ἀπ’ ἀνατολῶν κινεῖσθαι καθ’ ὑπόληψιν (ought not the reading to be ὑπόλειψιν?), ὥστε τὴν μὲν Κρονίαν σφαῖραν συναποκαθίστασθαι καθ’ ἡμέραν τῇ ἀπλανεῖ παρ’ ὀλίγον, τὴν δὲ τοῦ Διὸς παρὰ πλέον καὶ ἐφεξῆς οὕτως, οὗτοι πολλὰς μὲν ἄλλας ἀπορίας ἐκφεύγουσι, but their ὑπόθεσις is ἀδύνατος. This is what led to the return to the geocentric hypothesis and the exclusion of earth and ἀπλανὴς from the ἁρμονία. The only solution would have been to make the earth rotate on its axis or revolve round the central fire in twenty-four hours, leaving only precession for the ἀπλανής. As we have seen, Boeckh attributed this to Philolaos, but without evidence. If he had thought of it, these difficulties would not have arisen.

839.Tim.39 a 5-b 2, especially the words τὰ τάχιστα περιιόντα ὑπὸ τῶν βραδυτέρων ἐφαίνετο καταλαμβάνοντα καταλαμβάνεσθαι (“they appear to be overtaken, though they overtake”).

840. Plato,Laws, 822 a 4 sqq. The Athenian says of the theory that he had not heard of it in his youth nor long before (821 e 3). If so, it can hardly have been taught by Philolaos, though it may have been by Archytas.

841. Cf. especiallyMet.Α, 6. 787 b 10 (R. P. 65 d). It is not quite the same thing when he says, as in Α, 5. 985 b 23 sqq. (R. P.ib.), that they perceived many likenesses in things to numbers. That refers to the numerical analogies of Justice, Opportunity, etc.

842. Aristoxenosap.Stob. i. pr. 6 (p. 20), Πυθαγόρας ... πάντα τὰ πράγματα ἀπεικάζων τοῖς ἀριθμοῖς.

843. Stob.Ecl.i. p. 125, 19 (R. P. 65 d).

844. Iambl.in Nicom.p. 10, 20 (R. P. 56 c).

845. Plato,Phd.73 a sqq.

846.Ibid.74 a sqq.

847. Cf. especially the words ὃ θρυλοῦμεν ἀεί (76 d 8). The phrases αὐτὸ ὃ ἔστιν, αὐτὸ καθ’ αὑτό, and the like are assumed to be familiar. “We” define reality by means of question and answer, in the course of which “we” give an account of its being (ἧς λόγον δίδομεν τοῦ εἶναι, 78 d 1, where λόγον ... τοῦ εἶναι is equivalent to λόγον τῆς οὐσίας). When we have done this, “we” set the seal or stamp of αὐτὸ ὃ ἔστιν upon it (75 d 2). Technical terminology implies a school. As Diels puts it (Elementum, p. 20), it is in a school that “the simile concentrates into a metaphor, and the metaphor condenses into a term.”

848. Xen.Mem.i. 2, 48.

849. Plato,Soph.248 a 4.

850. See Diels,Elementum, pp. 16 sqq. Parmenides had already called the original Pythagorean “elements” μορφαί (§ 91), and Philistion called the “elements” of Empedokles ἰδέαι. If the ascription of this terminology to the Pythagoreans is correct, we may say that the Pythagorean “forms” developed into the atoms of Leukippos and Demokritos on the one hand (§ 174), and into the “ideas” of Plato on the other.

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