Chapter 29

The central fire.

150. The planetary system which Aristotle attributes to “the Pythagoreans” and Aetios to Philolaos is sufficiently remarkable.[817]The earth is no longer in the middle of the world; its place is taken by a central fire, which is not to be identified with the sun. Round this fire revolve ten bodies. First comes theAntichthonor Counter-earth, and next the earth, which thus becomes one of the planets. After the earth comes the moon, then the sun, the five planets, and the heaven of the fixed stars. We do not see the central fire and theantichthonbecause the side of the earth on which we live is always turned away fromthem. This is to be explained by the analogy of the moon. That body always presents the same face to us; and men living on the other side of it would never see the earth. This implies, of course, that all these bodies rotate on their axes in the same time as they revolve round the central fire.[818]

It is not very easy to accept the view that this system was taught by Philolaos. Aristotle nowhere mentions him in connexion with it, and in thePhaedoPlato gives a description of the earth and its position in the world which is entirely opposed to it, but is accepted without demur by Simmias the disciple of Philolaos.[819]It is undoubtedly a Pythagorean theory, however, and marks a noticeable advance on the Ionian views then current at Athens. It is clear too that Plato states it as something of a novelty that the earth does not require the support of air or anything of the sort to keep it in its place. Even Anaxagoras had not been able to shake himself free of that idea, and Demokritos still held it.[820]The natural inference from thePhaedowould certainly be that the theory of a spherical earth, kept in the middle of the world by its equilibrium, was that of Philolaos himself. If so, the doctrine of the central fire would belong to a somewhat later generation of the school, and Plato mayhave learnt it from Archytas and his friends after he had written thePhaedo. However that may be, it is of such importance that it cannot be omitted here.

It is commonly supposed that the revolution of the earth round the central fire was intended to account for the alternation of day and night, and it is clear that an orbital motion of the kind just described would have the same effect as the rotation of the earth on its axis. As the same side of the earth is always turned to the central fire, the side upon which we live will be turned towards the sun when the earth is on the same side of the central fire, and turned away from it when the earth and sun are on opposite sides. This view appears to derive some support from the statement of Aristotle that the earth “being in motion round the centre, produces day and night.”[821]That remark, however, would prove too much; for in theTimaeusPlato calls the earth “the guardian and artificer of night and day,” while at the same time he declares that the alternation of day and night is caused by the diurnal revolution of the heavens.[822]That is explained, no doubt quite rightly, by saying that, even if the earth were regarded as at rest, it could still be said to produce day and night; for night is due to the intervention of the earth between the sun and the hemisphere opposite to it. If we remember how recent was the discovery that night was the shadow of the earth, we shall see how it may have been worth while to say this explicitly.

In any case, it is wholly incredible that the heavenof the fixed stars should have been regarded as stationary. That would have been the most startling paradox that any scientific man had yet propounded, and we should have expected the comic poets and popular literature generally to raise the cry of atheism at once. Above all, we should have expected Aristotle to say something about it. He made the circular motion of the heavens the very keystone of his system, and would have regarded the theory of a stationary heaven as blasphemous. Now he argues against those who, like the Pythagoreans and Plato, regarded the earth as in motion;[823]but he does not attribute the view that the heavens are stationary to any one. There is no necessary connexion between the two ideas. All the heavenly bodies may be moving as rapidly as we please, provided that their relative motions are such as to account for the phenomena.[824]

It seems probable that the theory of the earth’s revolution round the central fire really originated in the account given by Empedokles of the sun’s light. The two things are brought into close connexion by Aetios, who says that Empedokles believed in two suns, while Philolaos believed in two or even in three.[825]The theory of Empedokles is unsatisfactory in so far as it gives two inconsistent explanations of night. It is, we have seen, the shadow of the earth; but at the same time Empedokles recognised a fiery diurnal hemisphere and a nocturnal hemisphere with only a little fire in it.[826]All this could be simplified by the hypothesis of a central fire which is the true source of light. Such a theory would, in fact, be the natural issue of the recent discoveries as to the moon’s light and the cause of eclipses, if that theory were extended so as to include the sun.

The central fire received a number of mythological names. It was called the Hestia or “hearth of the universe,” the “house” or “watch-tower” of Zeus, and the “mother of the gods.”[827]That was in the manner of the school; but these names must not blind us to the fact that we are dealing with a real scientific hypothesis. It was a great thing to see that the phenomena could best be “saved” by a central luminary, and that the earth must therefore be a revolving sphere like the planets. Indeed, we are almost tempted to say that the identification of the central fire with the sun, which was suggested for the first time in the Academy, is a mere detail in comparison. The great thing was that the earth should definitely take its place among the planets; for once it has done so, we can proceed to search for the true “hearth” of the planetary system at our leisure. It is probable, at any rate, that it was this theory which made it possiblefor Herakleides of Pontos and Aristarchos of Samos to reach the heliocentric hypothesis,[828]and it was certainly Aristotle’s reversion to the geocentric theory which made it necessary for Copernicus to discover the truth afresh. We have his own word for it that the Pythagorean theory put him on the right track.[829]

Theantichthon.

151. The existence of theantichthonwas also a hypothesis intended to account for the phenomena of eclipses. In one place, indeed, Aristotle says that the Pythagoreans invented it in order to bring the number of revolving bodies up to ten;[830]but that is a mere sally, and Aristotle really knew better. In his work on the Pythagoreans, we are told, he said that eclipses of the moon were caused sometimes by the intervention of the earth and sometimes by that of theantichthon; and the same statement was made by Philip of Opous, a very competent authority on the matter.[831]Indeed, Aristotle shows in another passage exactly how the theory originated. He tells us that some thought there might be a considerable number of bodies revolving round the centre, though invisibleto us because of the intervention of the earth, and that they accounted in this way for there being more eclipses of the moon than of the sun.[832]This is mentioned in close connexion with theantichthon, so there is no doubt that Aristotle regarded the two hypotheses as of the same nature. The history of the theory seems to be this. Anaximenes had assumed the existence of dark planets to account for the frequency of lunar eclipses (§ 29), and Anaxagoras had revived that view (§ 135). Certain Pythagoreans[833]had placed these dark planets between the earth and the central fire in order to account for their invisibility, and the next stage was to reduce them to a single body. Here again we see how the Pythagoreans tried to simplify the hypotheses of their predecessors.

Planetary motions.

152. We must not assume that even the later Pythagoreans made the sun, moon, and planets, including the earth, revolve in the opposite direction to the heaven of the fixed stars. It is true that Alkmaion is said to have agreed with “some of the mathematicians”[834]in holding this view, but it is never ascribed to Pythagoras or even to Philolaos. The old theory was, as we have seen (§ 54), that all the heavenly bodies revolved in the same direction, from east to west, but that the planets revolved more slowly the further they were removedfrom the heavens, so that those which are nearest the earth are “overtaken” by those that are further away. This view was still maintained by Demokritos, and that it was also Pythagorean, seems to follow from what we are told about the “harmony of the spheres.” We have seen (§ 54) that we cannot attribute this theory in its later form to the Pythagoreans of the fifth century, but we have the express testimony of Aristotle to the fact that those Pythagoreans whose doctrine he knew believed that the heavenly bodies produced musical notes in their courses. Further, the velocities of these bodies depended on the distances between them, and these corresponded to the intervals of the octave. He distinctly implies that the heaven of the fixed stars takes part in the concert; for he mentions “the sun, the moon, and the stars, so great in magnitude and in number as they are,” a phrase which cannot refer solely or chiefly to the remaining five planets.[835]Further, we are told that the slower bodies give out a deep note and the swifter a high note.[836]Now the prevailing tradition gives the high note of the octave to the heaven of the fixed stars,[837]from which it followsthat all the heavenly bodies revolve in the same direction, and that their velocity increases in proportion to their distance from the centre.

The theory that the proper motion of the sun, moon, and planets is from west to east, and that they also share in the motion from east to west of the heaven of the fixed stars, makes its first appearance in the Myth of Er in Plato’sRepublic, and is fully worked out in theTimaeus. In theRepublicit is still associated with the “harmony of the spheres,” though we are not told how it is reconciled with that theory in detail.[838]In theTimaeuswe read that the slowest of the heavenly bodies appear the fastest andvice versa; and, as this statement is put into the mouth of a Pythagorean, we might suppose the theory of a composite movement to have been anticipated by some members at least of that school.[839]That is, of course, possible; for thePythagoreans were singularly open to new ideas. At the same time, we must note that the theory is even more emphatically expressed by the Athenian Stranger in theLaws, who is in a special sense Plato himself. If we were to praise the runners who come in last in the race, we should not do what is pleasing to the competitors; and in the same way it cannot be pleasing to the gods when we suppose the slowest of the heavenly bodies to be the fastest. The passage undoubtedly conveys the impression that Plato is expounding a novel theory.[840]

Things likenesses of numbers.

153. We have still to consider a view, which Aristotle sometimes attributes to the Pythagoreans, that things were “like numbers.” He does not appear to regard this as inconsistent with the doctrine that thingsarenumbers, though it is hard to see how he could reconcile the two.[841]There is no doubt, however, that Aristoxenos represented the Pythagoreans as teaching that things werelikenumbers,[842]and there are other traces of an attempt to make out that this was the original doctrine. A letter was produced, purporting to be by Theano, the wife of Pythagoras, in which she says that she hears many of the Hellenes think Pythagoras said things were madeofnumber, whereashe really said they were madeaccording tonumber.[843]It is amusing to notice that this fourth-century theory had to be explained away in its turn later on, and Iamblichos actually tells us that it was Hippasos who said number was the exemplar of things.[844]

When this view is uppermost in his mind, Aristotle seems to find only a verbal difference between Plato and the Pythagoreans. The metaphor of “participation” was merely substituted for that of “imitation.” This is not the place to discuss the meaning of Plato’s so-called “theory of ideas”; but it must be pointed out that Aristotle’s ascription of the doctrine of “imitation” to the Pythagoreans is abundantly justified by thePhaedo. The arguments for immortality given in the early part of that dialogue come from various sources. Those derived from the doctrine of Reminiscence, which has sometimes been supposed to be Pythagorean, are only known to the Pythagoreans by hearsay, and Simmias requires to have the whole psychology of the subject explained to him.[845]When, however, we come to the question what it is that our sensations remind us of, his attitude changes. The view that the equal itself is alone real, and that what we call equal things are imperfect imitations of it, is quite familiar to him.[846]He requires no proof of it, and is finally convinced of the immortality of the soul just because Sokrates makes him see that the theory of forms implies it.

It is also to be observed that Sokrates does not introduce the theory as a novelty. The reality of the“ideas” is the sort of reality “we are always talking about,” and they are explained in a peculiar vocabulary which is represented as that of a school. The technical terms are introduced by such formulas as “we say.”[847]Whose theory is it? It is usually supposed to be Plato’s own, though nowadays it is the fashion to call it his “early theory of ideas,” and to say that he modified it profoundly in later life. But there are serious difficulties in this view. Plato is very careful to tell us that he was not present at the conversation recorded in thePhaedo. Did any philosopher ever propound a new theory of his own by representing it as already familiar to a number of distinguished living contemporaries? It is not easy to believe that. It would be rash, on the other hand, to ascribe the theory to Sokrates, and there seems nothing for it but to suppose that the doctrine of “forms” (εἴδη, ἰδέαι) originally took shape in Pythagorean circles, perhaps under Sokratic influence. There is nothing startling in this. It is a historical fact that Simmias and Kebes were not only Pythagoreans but disciples of Sokrates; for, by a happy chance, the good Xenophon has included them in his list of true Sokratics.[848]We have also sufficient ground for believing that the Megarians had adopted a like theory under similar influences, and Plato states expressly that Eukleides and Terpsion ofMegara were present at the conversation recorded in thePhaedo. There were, no doubt, more “friends of the ideas”[849]than we generally recognise. It is certain, in any case, that the use of the words εἴδη and ἰδέαι to express ultimate realities is pre-Platonic, and it seems most natural to regard it as of Pythagorean origin.[850]

We have really exceeded the limits of this work by tracing the history of Pythagoreanism down to a point where it becomes practically indistinguishable from the earliest form of Platonism; but it was necessary to do so in order to put the statements of our authorities in their true light. Aristoxenos is not likely to have been mistaken with regard to the opinions of the men he had known personally, and Aristotle’s statements must have had some foundation. We must assume, then, a later form of Pythagoreanism which was closely akin to early Platonism. That, however, is not the form of it which concerns us here, and we shall see in the next chapter that the fifth-century doctrine was of the more primitive type already described.

732. For Philolaos, see Plato,Phd.61 d 7; e 7; and for Lysis, Aristoxenos in Iambl.V. Pyth.250 (R. P. 59 b).

733. Diog. viii. 79-83 (R. P. 61). Aristoxenos himself came from Taras. For the political activity of the Tarentine Pythagoreans, see Meyer,Gesch. des Alterth.v. § 824. The story of Damon and Phintias (told by Aristoxenos) belongs to this time.

734. Diog. viii. 46 (R. P. 62).

735. Compare the way in which theTheaetetusis dedicated to the school of Megara.

736. See Aristoxenosap.Val. Max. viii. 13, ext. 3; and Souidass.v.

737. See below,§ 150–152.

738. Plato,Phd.61 d 6.

739. This appears to follow at once from the remark of Simmias inPhd.64 b. The whole passage would be pointless if the words φιλόσοφος, φιλοσοφεῖν, φιλοσοφία had not in some way become familiar to the ordinary Theban of the fifth century. Now Herakleides Pontikos made Pythagoras invent the word, and expound it in a conversation with Leon, tyrant of Sikyonor Phleious. Cf. Diog. i. 12 (R. P. 3), viii. 8; Cic.Tusc.v. 3. 8; Döring inArch.v. pp. 505 sqq. It seems to me that the way in which the term is introduced in thePhaedois fatal to the view that this is a Sokratic idea transferred by Herakleides to the Pythagoreans. Cf. also the remark of Alkidamas quoted by Arist.Rhet.Β, 23. 1398 b 18, Θήβησιν ἅμα οἱ προστάται φιλόσοφοι ἐγένοντο καὶ εὐδαιμόνησεν ἡ πόλις.

740. For reasons which will appear, I do not attach importance in this connexion to Philolaos, fr. 14 Diels = 23 Mullach (R. P. 89), but it does seem likely that the μυθολογῶν κομψὸς ἀνήρ ofGorg.493 a 5 (R. P. 89 b) is responsible for the whole theory there given. He is certainly, in any case, the author of the τετρημένος πίθος, which implies the same general view. Now he is called ἴσως Σικελός τις ἢ Ἰταλικός, which means he was an Italian; for the Σικελός τις is merely an allusion to the Σικελὸς κομψὸς ἀνὴρ ποτὶ τὰν ματέρ’ ἔφα of Timokreon. We do not know of any Italian from whom Plato could have learnt these views except Philolaos or one of his disciples. They may, however, be originally Orphic for all that (cf. R. P. 89 a).

741. See above, Chap. II. p. 113,n.236.

742. It is a good illustration of the defective character of our tradition (Introd.§ XIII.) that this was quite unknown till the publication of the extracts from Menon’sIatrikacontained in the Anonymus Londinensis. The extract referring to Philolaos is given and discussed by Diels inHermes, xxviii. pp. 417 sqq.

743.Hermes,loc. cit.

744. Plato,Phileb.16 c sqq.

745. Diog. iii. 37. For similar charges, cf. Zeller,Plato, p. 429, n. 7.

746. Iambl.V. Pyth.199. Diels is clearly right in ascribing the story to Aristoxenos (Arch.iii. p. 461, n. 26).

747. Timonap.Gell. iii. 17 (R. P. 60 a).

748. For Hermippos and Satyros, see Diog. iii. 9; viii. 84, 85.

749. So Iambl.in Nicom.p. 105, 11; Proclus,in Tim.p. 1, Diehl.

750. Diels,Vors.p. 269.

751. They are τὰ θρυλούμενα τρία βιβλία (Iambl.V. Pyth.199), τὰ διαβόητα τρία βιβλία (Diog. viii. 15).

752. As Mr. Bywater says (J. Phil.i. p. 29), the history of this work “reads like the history, not so much of a book, as of a literaryignis fatuusfloating before the minds of imaginative writers.”

753. Diels,“Ein gefälschtes Pythagorasbuch”(Arch.iii. pp. 451 sqq.).

754. Diog. viii. 85 (R. P. 63 b). Diels reads πρῶτον ἐκδοῦναι τῶν Πυθαγορικῶν <βιβλία καὶ ἐπιγράψαι Περὶ> Φύσεως.

755. Diog. viii. 7.

756. Proclus,in Eucl.p. 22, 15 (Friedlein). Cf. Boeckh,Philolaos, pp. 36 sqq. Boeckh refers to a sculptured group ofthreeBakchai, whom he supposes to be Ino, Agaue, and Autonoe.

757. The passage is given in R. P. 68. For a full discussion of this and the other fragments, see Bywater, “On the Fragments attributed to Philolaus the Pythagorean” (J. Phil.i. pp. 21 sqq.).

758. Boeckh,Philolaos, p. 38. Diels (Vors.p. 246) distinguishes theBakchaifrom the three books Περὶ φύσιος (ib.p. 239). As, however, he identifies the latter with the “three books” bought from Philolaos, and regards it as genuine, this does not seriously affect the argument.

759. See Diels inArch.iii. pp. 460 sqq.

760. On the Achaian dialect, see O. Hoffmann in Collitz and Bechtel,Dialekt-Inschriften, vol. ii. p. 151. How slowly Doric penetrated into the Chalkidian states may be seen from the mixed dialect of the inscription of Mikythos of Rhegion (Dial.-Inschr.iii. 2, p. 498), which is later than 468-67B.C.There is no reason to suppose that the Achaian dialect of Kroton was less tenacious of life.

761. The scanty fragments contain one Doric form, ἔχοντι (fr. 1), but Alkmaion calls himself Κροτωνιήτης, which is very significant; for Κροτωνιάτας is the Achaian as well as the Doric form. He did not, therefore, write a mixed dialect like that referred to in the last note. It seems safest to assume with Wachtler,De Alcmaeone Crotoniata, pp. 21 sqq., that he used Ionic.

762.Arch.iii. p. 460.

763. He is distinctly called a Krotoniate in the extracts from Menon’s Ἰατρικά (cf. Diog. viii. 84). It is true that Aristoxenos called him and Eurytos Tarentines (Diog. viii. 46), but this only means that he settled at Taras after leaving Thebes. These variations are common in the case of migratory philosophers. Eurytos is also called a Krotoniate and a Metapontine (Iambl.V. Pyth.148, 266). Cf. also p. 380,n.921on Leukippos, and p. 406,n.988on Hippon.

764. For Androkydes, see Diels,Vors.p. 281. As Diels points out (Arch.iii. p. 461), even Lucian has sufficient sense of style to make Pythagoras speak Ionic.

765. Cf. fr. 12 = 20 M. (R. P. 79), τὰ ἐν τᾷ σφαίρᾳ σώματα πέντε ἐντί.

766. Plato,Rep.528 b.

767. Heiberg’s Euclid, vol. v. p. 654, 1, Ἐν τούτῳ τῷ βιβλίῳ, τουτέστι τῷ ιγ’, γράφεται τὰ λεγόμενα Πλάτωνος ε̄ σχήματα, ἃ αὐτοῦ μὲν οὐκ ἔστιν, τρία δὲ τῶν προειρημένων ε̄ σχημάτων τῶν Πυθαγορείων ἐστίν, ὅ τε κύβος καὶ ἡ πυραμὶς καὶ τὸ δωδεκάεδρον, Θεαιτήτου δὲ τό τε ὀκτάεδρον καὶ τὸ εἰκοσάεδρον. It is no objection to this that, as Newbold points out (Arch.xix. p. 204), the inscription of the dodecahedron is more difficult than that of the octahedron and icosahedron. The Pythagoreans were not confined to strict Euclidean methods. It may further be noted that Tannery comes to a similar conclusion with regard to the musical scale described in the fragment of Philolaos. He says:“Il n’y a jamais eu, pour la division du tétracorde, une tradition pythagoricienne; on ne peut pas avec sûreté remonter plus haut que Platon ou qu’Archytas”(Rev. de Philologie, 1904, p. 244).

768. Aristotle says distinctly (Met.Α, 6. 987 b 25) that “to set up a dyad instead of the unlimited regarded as one, and to make the unlimited consist of the great and small, is distinctive of Plato.” Zeller seems to make an unnecessary concession with regard to this passage (p. 368, n. 2; Eng. trans. p. 396, n. 1).

769. Zeller, p. 369 sqq. (Eng. trans. p. 397 sqq.).

770. For the doctrine of “Philolaos,” cf. fr. 1 = 2 Ch. (R. P. 64); and for the unknowable ἐστὼ τῶν πραγμάτων, see fr. 3 = 4 Ch. (R. P. 67). It has a suspicious resemblance to the later ὕλη, which Aristotle would hardly have failed to note if he had ever seen the passage. He is always on the lookout for anticipations of ὕλη.

771. Arist.Met.Α, 8. 989 b 29 (R. P. 92 a).

772. Arist.Met.Α, 8. 990 a 3, ὁμολογοῦντες τοῖς ἄλλοις φυσιολόγοις ὅτι τό γ’ ὂν τοῦτ’ ἐστὶν ὅσον αἰσθητόν ἐστὶ καὶ περιείληφεν ὁ καλούμενος οὐρανός.

773.Met. ib.990 a 5, τὰς δ’ αἰτίας καὶ τὰς ἀρχάς, ὥσπερ εἴπομεν, ἱκανὰς λέγουσιν ἐπαναβῆναι καὶ ἐπὶ τὰ ἀνωτέρω τῶν ὄντων, καὶ μᾶλλον ἢ τοῖς περὶ φύσεως λόγοις ἁρμοττούσας.

774.Met.Α, 5. 986 a 1, τὰ τῶν ἀριθμῶν στοιχεῖα τῶν ὄντων στοιχεῖα πάντων ὑπέλαβον εἶναι; Ν, 3. 1090 a 22, εἶναι μὲν ἀριθμοὺς ἐποίησαν τὰ ὄντα, οὐ χωριστοὺς δέ, ἀλλ’ ἐξ ἀριθμῶν τὰ ὄντα.

775.Met.Μ, 6. 1080 b 2, ὡς ἐκ τῶν ἀριθμῶν ἐνυπαρχόντων ὄντα τὰ αἰσθητά;ib.1080 b 17, ἐκ τούτου (τοῦ μαθηματικοῦ ἀριθμοῦ) τὰς αἰσθητὰς οὐσίας συνεστάναι φασίν.

776.Met.Μ, 8. 1083 b 11, τὰ σώματα ἐξ ἀριθμῶν εἶναι συγκείμενα;ib.b 17, ἐκεῖνοι δὲ τὸν ἀριθμὸν τὰ ὄντα λέγουσιν· τὰ γοῦν θεωρήματαπροσάπτουσιπροσάπτουσιτοῖς σώμασιν ὡς ἐξ ἐκείνων ὄντων τῶν ἀριθμῶν; Ν, 3. 1090 a 32, κατὰ μέντοι τὸ ποιεῖν ἐξ ἀριθμῶν τὰ φυσικὰ σώματα, ἐκ μὴ ἐχόντων βάρος μηδὲ κουφότητα ἔχοντα κουφότητα καὶ βάρος.

777.Met.Α, 5. 986 a 2, τὸν ὅλον οὐρανὸν ἁρμονίαν εἶναι καὶ ἀριθμόν; Α, 8. 990 a 21, τὸν ἀριθμὸν τοῦτον ἐξ οὗ συνέστηκεν ὁ κόσμος; Μ, 6. 1080 b 18, τὸν γὰρ ὅλον οὐρανὸν κατασκευάζουσιν ἐξ ἀριθμῶν;de Caelo, Γ, 1. 300 a 15, τοῖς ἐξ ἀριθμῶν συνιστᾶσι τὸν οὐρανόν· ἔνιοι γὰρ τὴν φύσιν ἐξ ἀριθμῶν συνιστᾶσιν, ὥσπερ τῶν Πυθαγορείων τινές.

778.Met.Ν, 3. 1091 a 18, κοσμοποιοῦσι καὶ φυσικῶς βούλονται λέγειν.

779.Met.Μ, 6. 1080 b 16; Ν, 3. 1090 a 20.

780. Arist.Met.Α, 5. 987 a 15.

781.Met. ib.986 a 15 (R. P. 66).

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.

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