CHAPTER VII. The Pythagoreans

THE PYTHAGOREANS

138. We have seen (§ 40) how the Pythagoreans, after losing their supremacy at Kroton, concentrated themselves at Rhegion; but the school founded there was soon broken up. Archippos stayed behind in Italy; but Philolaos and Lysis, the latter of whom had escaped as a young man from the massacre of Kroton, betook themselves to continental Hellas, settling finally at Thebes. We know from Plato that Philolaos was there some time during the latter part of the fifth century, and Lysis was afterwards the teacher of Epameinondas. Some of the Pythagoreans, however, were able to return to Italy later on. Philolaos certainly did so, and Plato implies that he had left Thebes some time before 399 B.C., the year in which Sokrates was put to death. In the fourth century, the chief seat of the school is at Taras, and we find the Pythagoreans heading the opposition to Dionysios of Syracuse. It is to this period that Archytas belongs. He was the friend of Plato, and almost realised, if he did not suggest, the ideal of the philosopher king. He ruled Taras for years, and Aristoxenos tells us that he was never defeated in the field of battle. He was also the inventor of mathematical mechanics. At the same time, Pythagoreanism had taken root in Hellas. Lysis, we have seen, remained at Thebes, where Simmias and Kebes had heard Philolaos, and there was an important community of Pythagoreans at Phleious. Aristoxenos was personally acquainted with the last generation of the school, and mentioned by name Xenophilos the Chalkidian from Thrace, with Phanton, Echekrates, Diokles, and Polymnestos of Phleious. They were all, he said, disciples of Philolaos and Eurytos. Plato was on friendly terms with these men, and dedicated the Phaedo to them. Xenophilos was the teacher of Aristoxenos, and lived in perfect health at Athens till the age of a hundred and five.

Footnote 732:

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

Footnote 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.

Footnote 734:

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

Footnote 735:

Compare the way in which the Theaetetus is dedicated to the school of Megara.

Footnote 736:

See Aristoxenos ap. Val. Max. viii. 13, ext. 3; and Souidas s.v.

139. This generation of the school really belongs, however, to a later period, and cannot be profitably studied apart from Plato; it is with their master Philolaos we have now to deal. The facts we know about his teaching from external sources are few in number. The doxographers, indeed, ascribe to him an elaborate theory of the planetary system, but Aristotle never mentions his name in connexion with this. He gives it as the theory of “the Pythagoreans” or of “some Pythagoreans.” It seems natural to suppose, however, that the Pythagorean elements of Plato’s Phaedo and Gorgias come mainly from Philolaos. Plato makes Sokrates express surprise that Simmias and Kebes had not learnt from him why it is unlawful for a man to take his life, and it seems to be implied that the Pythagoreans at Thebes used the word “philosopher” in the special sense of a man who is seeking to find a way of release from the burden of this life. It is extremely probable that Philolaos spoke of the body (σῶμα) as the tomb (σῆμα) of the soul. In any case, we seem to be justified in holding that he taught the old Pythagorean religious doctrine in some form, and it is likely that he laid special stress upon knowledge as a means of release. That is the impression we get from Plato, and he is by far the best authority we have on the subject.

Footnote 737:

See below, § 150–152.

Footnote 738:

Plato, Phd. 61 d 6.

Footnote 739:

This appears to follow at once from the remark of Simmias in Phd. 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 Sikyon or Phleious. Cf. Diog. i. 12 (R. P. 3), viii. 8; Cic. Tusc. v. 3. 8; Döring in Arch. v. pp. 505 sqq. It seems to me that the way in which the term is introduced in the Phaedo is 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.

Footnote 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 of Gorg. 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).

We know further that Philolaos wrote on “numbers”; for Speusippos followed him in the account he gave of the Pythagorean theories on that subject. It is probable that he busied himself mainly with arithmetic, and we can hardly doubt that his geometry was of the primitive type described in an earlier chapter. Eurytos was his disciple, and we have seen (§ 47) that his views were still very crude.

Footnote 741:

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

We also know now that Philolaos wrote on medicine, and that, while apparently influenced by the theories of the Sicilian school, he opposed them from the Pythagorean standpoint. In particular, he said that our bodies were composed only of the warm, and did not participate in the cold. It was only after birth that the cold was introduced by respiration. The connexion of this with the old Pythagorean theory is obvious. Just as the Fire in the macrocosm draws in and limits the cold dark breath which surrounds the world (§ 53), so do our bodies inhale cold breath from outside. Philolaos made bile, blood, and phlegm the causes of disease; and, in accordance with the theory just mentioned, he had to deny that the phlegm was cold, as the Sicilian school held it was. Its etymology proved that it was warm. As Diels says, Philolaos strikes us as an “uninteresting eclectic” so far as his medical views are concerned. We shall see, however, that it was just this preoccupation with the medicine of the Sicilian school that gave rise to some of the most striking developments of later Pythagoreanism.

Footnote 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’s Iatrika contained in the Anonymus Londinensis. The extract referring to Philolaos is given and discussed by Diels in Hermes, xxviii. pp. 417 sqq.

Footnote 743:

Hermes, loc. cit.

140. Such, so far as we can see, was the historical Philolaos, and he is a sufficiently remarkable figure. He is usually, however, represented in a different light, and has even been spoken of as a “precursor of Copernicus.” To understand this, we shall have to consider for a little the story of what can only be called a literary conspiracy. Not till this has been exposed will it be possible to estimate the real importance of Philolaos and his immediate disciples.

As we can see from the Phaedo and the Gorgias, Plato was intimate with these men and was deeply impressed by their religious teaching, though it is plain too that he did not adopt it as his own faith. He was still more attracted by the scientific side of Pythagoreanism, and to the last this exercised a great influence on him. His own system in its final form had many points of contact with it, as he is careful to mark in the Philebus. But, just because he stood so near it, he is apt to develop Pythagoreanism on lines of his own, which may or may not have commended themselves to Archytas, but are no guide to the views of Philolaos and Eurytos. He is not careful, however, to claim the authorship of his own improvements in the system. He did not believe that cosmology could be an exact science, and he is therefore quite willing to credit Timaios the Lokrian, or “ancient sages” generally, with theories which certainly had their birth in the Academy.

Footnote 744:

Plato, Phileb. 16 c sqq.

Now Plato had many enemies and detractors, and this literary device enabled them to bring against him the charge of plagiarism. Aristoxenos was one of these enemies, and we know he made the extraordinary statement that most of the Republic was to be found in a work by Protagoras. He seems also to be the original source of the story that Plato bought “three Pythagorean books” from Philolaos and copied the Timaeus out of them. According to this, the “three books” had come into the possession of Philolaos; and, as he had fallen into great poverty, Dion was able to buy them from him, or from his relatives, at Plato’s request, for a hundred minae. It is certain, at any rate, that this story was already current in the third century; for the sillographer Timon of Phleious addresses Plato thus: “And of thee too, Plato, did the desire of discipleship lay hold. For many pieces of silver thou didst get in exchange a small book, and starting from it didst learn to write Timaeus.” Hermippos, the pupil of Kallimachos, said that “some writer” said that Plato himself bought the books from the relatives of Philolaos for forty Alexandrian minae and copied the Timaeus out of it; while Satyros, the Aristarchean, says he got it through Dion for a hundred minae. There is no suggestion in any of these accounts that the book was by Philolaos himself; they imply rather that what Plato bought was either a book by Pythagoras, or at any rate authentic notes of his teaching, which had come into the hands of Philolaos. In later times, it was generally supposed that the work entitled The Soul of the World, by Timaios the Lokrian, was meant; but it has now been proved beyond a doubt that this cannot have existed earlier than the first century A.D. We know nothing of Timaios except what Plato tells us himself, and he may even be a fictitious character like the Eleatic Stranger. His name does not occur among the Lokrians in the Catalogue of Pythagoreans preserved by Iamblichos. Besides this, the work does not fulfil the most important requirement, that of being in three books, which is always an essential feature of the story.

Footnote 745:

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

Footnote 746:

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

Footnote 747:

Timon ap. Gell. iii. 17 (R. P. 60 a).

Footnote 748:

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

Footnote 749:

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

Footnote 750:

Diels, Vors. p. 269.

Footnote 751:

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

Not one of the writers just mentioned professes to have seen the famous “three books”; but at a later date there were at least two works which claimed to represent them. Diels has shown how a treatise in three sections, entitled Παιδευτικόν, πολιτικόν, φυσικόν, was composed in the Ionic dialect and attributed to Pythagoras. It was largely based on the Πυθαγορικαὶ ἀποφάσεις of Aristoxenos, but its date is uncertain. In the first century B.C., Demetrios Magnes was able to quote the opening words of the work published by Philolaos. That, however, was written in Doric. Demetrios does not actually say it was by Philolaos himself, though it is no doubt the same work from which a number of extracts are preserved under his name in Stobaios and later writers. If it professed to be by Philolaos, that was not quite in accordance with the original story; but it is easy to see how his name may have become attached to it. We are told that the other book which passed under the name of Pythagoras was really by Lysis. Boeckh has shown that the work ascribed to Philolaos probably consisted of three books also, and Proclus referred to it as the Bakchai, a fanciful title which recalls the “Muses” of Herodotos. Two of the extracts in Stobaios bear it. It must be confessed that the whole story is very suspicious; but, as some of the best authorities still regard the fragments as partly genuine, it is necessary to look at them more closely.

Footnote 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 literary ignis fatuus floating before the minds of imaginative writers.”

Footnote 753:

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

Footnote 754:

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

Footnote 755:

Diog. viii. 7.

Footnote 756:

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

141. Boeckh argued with great learning and skill that all the fragments preserved under the name of Philolaos were genuine; but no one will now go so far as this. The lengthy extract on the soul is given up even by those who maintain the genuineness of the rest. It cannot be said that this position is plausible on the face of it. Boeckh saw there was no ground for supposing that there ever was more than a single work, and he drew the conclusion that we must accept all the remains as genuine or reject all as spurious. As, however, Zeller and Diels still maintain the genuineness of most of the fragments, we cannot ignore them altogether. Arguments based, on the doctrine contained in them would, it is true, present the appearance of a vicious circle at this stage. It is only in connexion with our other evidence that these can be introduced. But there are two serious objections to the fragments which may be mentioned at once. They are sufficiently strong to justify us in refusing to use them till we have ascertained from other sources what doctrines may fairly be attributed to the Pythagoreans of this date.

Footnote 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.).

Footnote 758:

Boeckh, Philolaos, p. 38. Diels (Vors. p. 246) distinguishes the Bakchai from 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.

In the first place, we must ask a question which has not yet been faced. Is it likely that Philolaos should have written in Doric? Ionic was the dialect of all science and philosophy till the time of the Peloponnesian War, and there is no reason to suppose that the early Pythagoreans used any other. Pythagoras was himself an Ionian, and it is by no means clear that in his time the Achaian states in which he founded his Order had already adopted the Dorian dialect. Alkmaion of Kroton seems to have written in Ionic. Diels says, it is true, that Philolaos and then Archytas were the first Pythagoreans to use the dialect of their homes; but Philolaos can hardly be said to have had a home, and the fragments of Archytas are not written in the dialect of Taras, but in what may be called “common Doric.” Archytas may have found it convenient to use that dialect; but he is at least a generation later than Philolaos, which makes a great difference. There is evidence that, in the time of Philolaos and later, Ionic was still used even by the citizens of Dorian states for scientific purposes. Diogenes of Apollonia in Crete and the Syracusan historian Antiochos wrote in Ionic, while the medical writers of Dorian, Kos and Knidos, continue to use the same dialect. The forged work of Pythagoras referred to above, which some ascribed to Lysis, was in Ionic; and so was the work on the Akousmata attributed to Androkydes, which shows that, even down to Alexandrian times, it was still believed that Ionic was the proper dialect for Pythagorean writings.

Footnote 759:

See Diels in Arch. iii. pp. 460 sqq.

Footnote 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 468-67 B.C. There is no reason to suppose that the Achaian dialect of Kroton was less tenacious of life.

Footnote 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.

Footnote 762:

Arch. iii. p. 460.

Footnote 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. 921 on Leukippos, and p. 406, n. 988 on Hippon.

Footnote 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.

In the second place, there can be no doubt that one of the fragments refers to the five regular solids, four of which are identified with the elements of Empedokles. Now Plato gives us to understand, in a well-known passage of the Republic, that stereometry had not been adequately investigated at the time he wrote, and we have express testimony that the five “Platonic figures,” as they were called, were discovered in the Academy. In the Scholia to Euclid we read that the Pythagoreans only knew the cube, the pyramid (tetrahedron), and the dodecahedron, while the octahedron and the icosahedron were discovered by Theaitetos. This sufficiently justifies us in regarding the “fragments of Philolaos” with something more than suspicion. We shall find more anachronisms as we go on.

Footnote 765:

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

Footnote 766:

Plato, Rep. 528 b.

Footnote 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).

142. We must look, then, for other evidence. From what has been said, it will be clear that we cannot safely take Plato as our guide to the original meaning of the Pythagorean theory, though it is certainly from him alone that we can learn to regard it sympathetically. Aristotle, on the other hand, was quite out of sympathy with Pythagorean ways of thinking, but took a great deal of pains to understand them. This was just because they played so great a part in the philosophy of Plato and his successors, and he had to make the relation of the two doctrines as clear as he could to himself and his disciples. What we have to do, then, is to interpret what Aristotle tells us in the spirit of Plato, and then to consider how the doctrine we arrive at in this way is related to the systems which had preceded it. It is a delicate operation, no doubt, but it has been made much safer by recent discoveries in the early history of mathematics and medicine.

Zeller has cleared the ground by eliminating the purely Platonic elements which have crept into later accounts of the system. These are of two kinds. First of all, we have genuine Academic formulae, such as the identification of the Limit and the Unlimited with the One and the Indeterminate Dyad; and secondly, there is the Neoplatonic doctrine which represents it as an opposition between God and Matter. It is not necessary to repeat Zeller’s arguments here, as no one will any longer attribute these doctrines to the Pythagoreans of the fifth century.

Footnote 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).

Footnote 769:

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

This simplifies the problem very considerably, but it is still extremely difficult. According to Aristotle, the Pythagoreans said Things are numbers, though that does not appear to be the doctrine of the fragments of “Philolaos.” According to them, things have number, which make them knowable, while their real essence is something unknowable. That would be intelligible enough, but the formula that things are numbers seems meaningless. We have seen reason for believing that it is due to Pythagoras himself (§ 52), though we did not feel able to say very clearly what he meant by it. There is no such doubt as to his school. Aristotle says they used the formula in a cosmological sense. The world, according to them, was made of numbers in the same sense as others had said it was made of “four roots” or “innumerable seeds.” It will not do to dismiss this as mysticism. Whatever we may think of Pythagoras, the Pythagoreans of the fifth century were scientific men, and they must have meant something quite definite. We shall, no doubt, have to say that they used the words Things are numbers in a somewhat non-natural sense, but there is no difficulty in such a supposition. We have seen already how the friends of Aristoxenos reinterpreted the old Akousmata (§ 44). The Pythagoreans had certainly a great veneration for the actual words of the Master (αὐτὸς ἔφα); but such veneration is often accompanied by a singular licence of interpretation. We shall start, then, from what Aristotle tells us about the numbers.

Footnote 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.

143. In the first place, Aristotle is quite decided in his opinion that Pythagoreanism was intended to be a cosmological system like the others. “Though the Pythagoreans,” he tells us, “made use of less obvious first principles and elements than the rest, seeing that they did not derive them from sensible objects, yet all their discussions and studies had reference to nature alone. They describe the origin of the heavens, and they observe the phenomena of its parts, all that happens to it and all it does.” They apply their first principles entirely to these things, “agreeing apparently with the other natural philosophers in holding that reality was just what could be perceived by the senses, and is contained within the compass of the heavens,” though “the first principles and causes of which they made use were really adequate to explain realities of a higher order than the sensible.”

Footnote 771:

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

Footnote 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.

Footnote 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.

The doctrine is more precisely stated by Aristotle to be that the elements of numbers are the elements of things, and that therefore things are numbers. He is equally positive that these “things” are sensible things, and indeed that they are bodies, the bodies of which the world is constructed. This construction of the world out of numbers was a real process in time, which the Pythagoreans described in detail.

Footnote 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.

Footnote 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.

Footnote 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 θεωρήματα προσάπτουσι τοῖς 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.

Footnote 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.

Footnote 778:

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

Further, the numbers were intended to be mathematical numbers, though they were not separated from the things of sense. On the other hand, they were not mere predicates of something else, but had an independent reality of their own. “They did not hold that the limited and the unlimited and the one were certain other substances, such as fire, water, or anything else of that sort; but that the unlimited itself and the one itself were the reality of the things of which they are predicated, and that is why they said that number was the reality of everything.” Accordingly the numbers are, in Aristotle’s own language, not only the formal, but also the material, cause of things. According to the Pythagoreans, things are made of numbers in the same sense as they were made of fire, air, or water in the theories of their predecessors.

Footnote 779:

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

Lastly, Aristotle notes that the point in which the Pythagoreans agreed with Plato was in giving numbers an independent reality of their own; while Plato differed from the Pythagoreans in holding that this reality was distinguishable from that of sensible things. Let us consider these statements in detail.

Footnote 780:

Arist. Met. Α, 5. 987 a 15.

Footnote 781:

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

Footnote 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.

144. Aristotle speaks of certain “elements” (στοιχεῖα) of numbers, which were also the elements of things. That, of course, is only his own way of putting the matter; but it is clearly the key to the problem, if we can discover what it means. Primarily, the “elements of number” are the Odd and the Even, but that does not seem to help us much. We find, however, that the Odd and Even were identified in a somewhat violent way with the Limit and the Unlimited, which we have seen reason to regard as the original principles of the Pythagorean cosmology. Aristotle tells us that it is the Even which gives things their unlimited character when it is contained in them and limited by the Odd, and the commentators are at one in understanding this to mean that the Even is in some way the cause of infinite divisibility. They get into great difficulties, however, when they try to show how this can be. Simplicius has preserved an explanation, in all probability Alexander’s, to the effect that they called the even number unlimited “because every even is divided into equal parts, and what is divided into equal parts is unlimited in respect of bipartition; for division into equals and halves goes on ad infinitum. But, when the odd is added, it limits it; for it prevents its division into equal parts.” Now it is plain that we must not impute to the Pythagoreans the view that even numbers can be halved indefinitely. They had carefully studied the properties of the decad, and they must have known that the even numbers 6 and 10 do not admit of this. The explanation is really to be found in a fragment of Aristoxenos, where we read that “even numbers are those which are divided into equal parts, while odd numbers are divided into unequal parts and have a middle term.” This is still further elucidated by a passage which is quoted in Stobaios and ultimately goes back to Poseidonios. It runs: “When the odd is divided into two equal parts, a unit is left over in the middle; but when the even is so divided, an empty field is left, without a master and without a number, showing that it is defective and incomplete.” Again, Plutarch says: “In the division of numbers, the even, when parted in any direction, leaves as it were within itself ... a field; but, when the same thing is done to the odd, there is always a middle left over from the division.” It 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:

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.

145. In this way, then, the Odd and the Even were identified with the Limit and the Unlimited, and it is possible, though by no means certain, that Pythagoras himself had taken this step. In any case, there can be no doubt that by his Unlimited he meant something spatially extended, and we have seen that he identified it with air, night, or the void, so we are prepared to find that his followers also thought of the Unlimited as extended. Aristotle certainly regarded it so. He argues that, if the Unlimited is itself a reality, and not merely the predicate of some other reality, then every part of it must be unlimited too, just as every part of air is air. The same thing is implied in his statement that the Pythagorean Unlimited was outside the heavens. Further than this, it is hardly safe to go. Philolaos and his followers cannot have regarded the Unlimited in the old Pythagorean way as Air; for, as we shall see, they adopted the theory of Empedokles as to that “element,” and accounted for it otherwise. On the other hand, they can hardly have regarded it as an absolute void; for that conception was introduced by the Atomists. It is enough to say that they meant by the Unlimited the res extensa, without analysing that conception any further.

Footnote 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.

Footnote 789:

See Chap. II. § 53.

As the Unlimited is spatial, the Limit must be spatial too, and we should naturally expect to find that the point, the line, and the surface were regarded as all forms of the Limit. That was the later doctrine; but the characteristic feature of Pythagoreanism is just that the point was not regarded as a limit, but as the first product of the Limit and the Unlimited, and was identified with the arithmetical unit. According to this view, then, the point has one dimension, the line two, the surface three, and the solid four. In other words, the Pythagorean points have magnitude, their lines breadth, and their surfaces thickness. The whole theory, in short, turns on the definition of the point as a unit “having position.” It was out of such elements that it seemed possible to construct a world.

Footnote 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.

Footnote 791:

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

146. It is clear that this way of regarding the point, the line, and the surface is closely bound up with the practice of representing numbers by dots arranged in symmetrical patterns, which we have seen reason for attributing to the Pythagoreans (§ 47). The science of geometry had already made considerable advances, but the old view of quantity as a sum of units had not been revised, and so a doctrine such as we have indicated was inevitable. This is the true answer to Zeller’s contention that to regard the Pythagorean numbers as spatial is to ignore the fact that the doctrine was originally arithmetical rather than geometrical. Our interpretation takes full account of that fact, and indeed makes the peculiarities of the whole system depend upon it. Aristotle is very decided as to the Pythagorean points having magnitude. “They construct the whole world out of numbers,” he tells us, “but they suppose the units have magnitude. As to how the first unit with magnitude arose, they appear to be at a loss.” Zeller holds that this is only an inference of Aristotle’s, and he is probably right in this sense, that the Pythagoreans never felt the need of saying in so many words that points had magnitude. It does seem probable, however, that they called them ὄγκοι.

Footnote 792:

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

Footnote 793:

Zeller, p. 381.

Footnote 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 in Tim. 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 in Parm. 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).

Nor is Zeller’s other argument against the view that the Pythagorean numbers were spatial any more inconsistent with the way in which we have now stated it. He himself allows, and indeed insists, that in the Pythagorean cosmology the numbers were spatial, but he raises difficulties about the other parts of the system. There are other things, such as the Soul and Justice and Opportunity, which are said to be numbers, and which cannot be regarded as constructed of points, lines, and surfaces. Now it appears to me that this is just the meaning of a passage in which Aristotle criticises the Pythagoreans. They held, he says, that in one part of the world Opinion prevailed, while a little above it or below it were to be found Injustice or Separation or Mixture, each of which was, according to them, a number. But in the very same regions of the heavens were to be found things having magnitude which were also numbers. How can this be, since Justice has no magnitude? This means surely that the Pythagoreans had failed to give any clear account of the relation between these more or less fanciful analogies and their quasi-geometrical construction of the universe. And this is, after all, really Zeller’s own view. He has shown that in the Pythagorean cosmology the numbers were regarded as spatial, and he has also shown that the cosmology was the whole of the system. We have only to bring these two things together to arrive at the interpretation given above.

Footnote 795:

Zeller, p. 382.

Footnote 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.

Footnote 797:

Zeller, p. 404.

Footnote 798:

Ibid. pp. 467 sqq.

147. When we come to details, we seem to see that what distinguished the Pythagoreanism of this period from its earlier form was that it sought to adapt itself to the new theory of “elements.” It is just this which makes it necessary for us to take up the consideration of the system once more in connexion with the pluralists. When the Pythagoreans returned to Southern Italy, they must have found views prevalent there which imperatively demanded a partial reconstruction of their own system. We do not know that Empedokles founded a philosophical society, but there can be no doubt of his influence on the medical school of these regions; and we also know now that Philolaos played a part in the history of medicine. This discovery gives us the clue to the historical connexion, which formerly seemed obscure. The tradition is that the Pythagoreans explained the elements as built up of geometrical figures, a theory which we can study for ourselves in the more developed form which it attained in Plato’s Timaeus. If they were to retain their position as the leaders of medical study in Italy, they were bound to account for the elements.

Footnote 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.

Footnote 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.

We must not take it for granted, however, that the Pythagorean construction of the elements was exactly the same as that which we find in Plato’s Timaeus. It has been mentioned already that there is good reason for believing they only knew three of the regular solids, the cube, the pyramid (tetrahedron), and the dodecahedron. Now it is very significant that Plato starts from fire and earth, and in the construction of the elements proceeds in such a way that the octahedron and the icosahedron can easily be transformed into pyramids, while the cube and the dodecahedron cannot. From this it follows that, while air and water pass readily into fire, earth cannot do so, and the dodecahedron is reserved for another purpose, which we shall consider presently. This would exactly suit the Pythagorean system; for it would leave room for a dualism of the kind outlined in the Second Part of the poem of Parmenides. We know that Hippasos made Fire the first principle, and we see from the Timaeus how it would be possible to represent air and water as forms of fire. The other element is, however, earth, not air, as we have seen reason to believe that it was in early Pythagoreanism. That would be a natural result of the discovery of atmospheric air by Empedokles and of his general theory of the elements. It would also explain the puzzling fact, which we had to leave unexplained above, that Aristotle identifies the two “forms” spoken of by Parmenides with Fire and Earth. All this is, of course, problematical; but it will not be found easy to account otherwise for the facts.

Footnote 801:

See above, p. 329, n. 767.

Footnote 802:

Plato, Tim. 31 b 5.

Footnote 803:

Plato, Tim. 54 c 4. It is to be observed that in Tim. 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.

Footnote 804:

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

148. The most interesting point in the theory is, perhaps, the use made of the dodecahedron. It was identified, we are told, with the “sphere of the universe,” or, as it is put in the Philolaic fragment, with the “hull of the sphere.” Whatever we may think of the authenticity of the fragments, there is no reason to doubt that this is a genuine Pythagorean expression, and it must be taken in close connexion with the word “keel” applied to the central fire. The structure of the world was compared to the building of a ship, an idea of which there are other traces. The key to what we are told of the dodecahedron is given by Plato. In the Phaedo we read that the “true earth,” if looked at from above, is “many-coloured like the balls that are made of twelve pieces of leather.” In the Timaeus the same thing is referred to in these words: “Further, as there is still one construction left, the fifth, God made use of it for the universe when he painted it.” The point is that the dodecahedron approaches more nearly to the sphere than any other of the regular solids. The twelve pieces of leather used to make a ball would all be regular pentagons; and, if the material were not flexible like leather, we should have a dodecahedron instead of a sphere. This points to the Pythagoreans having had at least the rudiments of the “method of exhaustion” formulated later by Eudoxos. They must have studied the properties of circles by means of inscribed polygons and those of spheres by means of inscribed solids. That gives us a high idea of their mathematical attainments; but that it is not too high, is shown by the fact that the famous lunules of Hippokrates date from the middle of the fifth century. The inclusion of straight and curved in the “table of opposites” under the head of Limit and Unlimited points in the same direction.

Footnote 805:

Aet. ii. 6, 5 (R. P. 80); “Philolaos,” fr. 12 (= 20 M.; R. P. 79). On the ὁλκάς, see Gundermann in Rhein. 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.

Footnote 806:

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

Footnote 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.

Footnote 808:

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

Footnote 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.

Footnote 810:

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

Footnote 811:

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

The tradition confirms in an interesting way the importance of the dodecahedron in the Pythagorean system. According to one account, Hippasos was drowned at sea for revealing its construction and claiming the discovery as his own. What that construction was, we may partially infer from the fact that the Pythagoreans adopted the pentagram or pentalpha as their symbol. The use of this figure in later magic is well known; and Paracelsus still employed it as a symbol of health, which is exactly what the Pythagoreans called it.

Footnote 812:

Iambl. V. Pyth. 247. Cf. above, Chap. II. p. 117, n. 247.

Footnote 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.

149. The view that the soul is a “harmony,” or rather an attunement, is intimately connected with the theory of the four elements. It cannot have belonged to the earliest form of Pythagoreanism; for, as shown in Plato’s Phaedo, it is quite inconsistent with the idea that the soul can exist independently of the body. It is the very opposite of the belief that “any soul can enter any body.” On the other hand, we know also from the Phaedo that it was accepted by Simmias and Kebes, who had heard Philolaos at Thebes, and by Echekrates of Phleious, who was the disciple of Philolaos and Eurytos. The account of the doctrine given by Plato is quite in accordance with the view that it was of medical origin. Simmias says: “Our body being, as it were, strung and held together by the warm and the cold, the dry and the moist, and things of that sort, our soul is a sort of temperament and attunement of these, when they are mingled with one another well and in due proportion. If, then, our soul is an attunement, it is clear that, when the body has been relaxed or strung up out of measure by diseases and other ills, the soul must necessarily perish at once.” This is clearly an application of the theory of Alkmaion (§ 96), and is in accordance with the views of the Sicilian school of medicine. It completes the evidence that the Pythagoreanism of the end of the fifth century was an adaptation of the old doctrine to the new principles introduced by Empedokles.

Footnote 814:

Arist. de An. Α, 3. 407 b 20 (R. P. 86 c).

Footnote 815:

Plato, Phd. 85 e sqq.; and for Echekrates, ib. 88 d.

Footnote 816:

Plato, Phd. 86 b 7-c 5.

150. The planetary system which Aristotle attributes to “the Pythagoreans” and Aetios to Philolaos is sufficiently remarkable. 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 the Antichthon or 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 the antichthon because the side of the earth on which we live is always turned away from them. 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.

Footnote 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.

Footnote 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.

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 the Phaedo Plato 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. 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. The natural inference from the Phaedo would 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 may have learnt it from Archytas and his friends after he had written the Phaedo. However that may be, it is of such importance that it cannot be omitted here.

Footnote 819:

Plato, Phd. 108 e 4 sqq. Simmias assents to this doctrine in the emphatic words Καὶ ὀρθῶς γε.

Footnote 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.

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.” That remark, however, would prove too much; for in the Timaeus Plato 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. 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.

Footnote 821:

Arist. de Caelo, Β, 13. 293 a 18 sqq. (R. P. 83).

Footnote 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).

In any case, it is wholly incredible that the heaven of 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; 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.

Footnote 823:

Arist. de Caelo, Β, 13. 293 b 15 sqq.

Footnote 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.”

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. 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. 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.

Footnote 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.

Footnote 826:

Chap. VI. § 113.

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.” 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 possible for Herakleides of Pontos and Aristarchos of Samos to reach the heliocentric hypothesis, 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.

Footnote 827:

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

Footnote 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).

Footnote 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.

151. The existence of the antichthon was 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; 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 the antichthon; and the same statement was made by Philip of Opous, a very competent authority on the matter. 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 invisible to 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. This is mentioned in close connexion with the antichthon, 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 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.

Footnote 830:

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

Footnote 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).

Footnote 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.

Footnote 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).

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” 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 removed from 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. Further, we are told that the slower bodies give out a deep note and the swifter a high note. Now the prevailing tradition gives the high note of the octave to the heaven of the fixed stars, from which it follows that all the heavenly bodies revolve in the same direction, and that their velocity increases in proportion to their distance from the centre.

Footnote 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.

Footnote 835:

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

Footnote 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 the antichthon as προσλαμβανόμενος.

Footnote 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.

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’s Republic, and is fully worked out in the Timaeus. In the Republic it is still associated with the “harmony of the spheres,” though we are not told how it is reconciled with that theory in detail. In the Timaeus we read that the slowest of the heavenly bodies appear the fastest and vice 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. That is, of course, possible; for the Pythagoreans 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 the Laws, 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.

Footnote 838:

The difficulty appears clearly in Adam’s note on Republic, 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.

Footnote 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”).

Footnote 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.

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 things are numbers, though it is hard to see how he could reconcile the two. There is no doubt, however, that Aristoxenos represented the Pythagoreans as teaching that things were like numbers, 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 made of number, whereas he really said they were made according to number. 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.

Footnote 841:

Cf. especially Met. 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.

Footnote 842:

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

Footnote 843:

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

Footnote 844:

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

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 the Phaedo. 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. 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. 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.

Footnote 845:

Plato, Phd. 73 a sqq.

Footnote 846:

Ibid. 74 a sqq.

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.” 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 the Phaedo. 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. 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 of Megara were present at the conversation recorded in the Phaedo. There were, no doubt, more “friends of the ideas” 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.

Footnote 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.”

Footnote 848:

Xen. Mem. i. 2, 48.

Footnote 849:

Plato, Soph. 248 a 4.

Footnote 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.

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.