Chapter VII: The Pythagoreans (1)

[Sidenote: The Pythagorean school.]

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.[732] 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.[733] 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.[734] Plato was on friendly terms with these men, and dedicated the _Phaedo_ to them.[735] Xenophilos was the teacher of Aristoxenos, and lived in perfect health at Athens till the age of a hundred and five.[736]

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._

[Sidenote: Philolaos.]

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.”[737] 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,[738] 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.[739] It is extremely probable that Philolaos spoke of the body (σῶμα) as the tomb (σῆμα) of the soul.[740] 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.[741] 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,[742] 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.[743] 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._

[Sidenote: Plato and the Pythagoreans.]

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_.[744] 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.[745] 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_.[746] 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_.”[747] 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_.[748] 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;[749] 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.[750] 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.[751]

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”;[752] 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.[753] In the first century B.C., Demetrios Magnes was able to quote the opening words of the work published by Philolaos.[754] 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.[755] Boeckh has shown that the work ascribed to Philolaos probably consisted of three books also, and Proclus referred to it as the _Bakchai_,[756] 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.

[Sidenote: The “Fragments of Philolaos.”]

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.[757] 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.[758] 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.[759] 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.[760] Alkmaion of Kroton seems to have written in Ionic.[761] Diels says, it is true, that Philolaos and then Archytas were the first Pythagoreans to use the dialect of their homes;[762] but Philolaos can hardly be said to have had a home,[763] 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,[764] 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.[765] 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,[766] 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.[767] 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).

[Sidenote: The Problem.]

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;[768] and secondly, there is the Neoplatonic doctrine which represents it as an opposition between God and Matter.[769] 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.[770] 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.

[Sidenote: Aristotle on the Numbers.]

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.”[771] 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,”[772] though “the first principles and causes of which they made use were really adequate to explain realities of a higher order than the sensible.”[773]

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.[774] He is equally positive that these “things” are sensible things,[775] and indeed that they are bodies,[776] the bodies of which the world is constructed.[777] This construction of the world out of numbers was a real process in time, which the Pythagoreans described in detail.[778]

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.[779] 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.”[780] Accordingly the numbers are, in Aristotle’s own language, not only the formal, but also the material, cause of things.[781] 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.[782] 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.

[Sidenote: The elements of numbers.]

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,[783] 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.”[784] 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.”[785] 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.”[786] 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.”[787] 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:

[Plut.] _ap._ Stob. i. p. 22, 19, καὶ μὴν εἰς δύο διαιρουμένων ἴσα τοῦ
μὲν περισσοῦ μονὰς ἐν μέσῳ περιέστι, τοῦ δὲ ἀρτίου κενὴ λείπεται χώρα
καὶ ἀδέσποτος καὶ ἀνάριθμος, ὡς ἂν ἐνδεοῦς καὶ ἀτελοῦς ὄντος.

Footnote 787:

Plut. _de E apud Delphos_, 388 a, ταῖς γὰρ εἰς ἴσα τομαῖς τῶν ἀριθμῶν,
ὁ μὲν ἄρτιος πάντῃ διϊστάμενος ὑπολείπει τινὰ δεκτικὴν ἀρχὴν οἷον ἐν
ἑαυτῷ καὶ χώραν, ἐν δὲ τῷ περιττῷ ταὐτὸ παθόντι μέσον ἀεὶ περίεστι τῆς
νεμήσεως γόνιμον. The words which I have omitted in translating refer
to the further identification of Odd and Even with Male and Female.
The passages quoted by Heidel might be added to. Cf., for instance,
what Nikomachos says (p. 13, 10, Hoche), ἔστι δὲ ἄρτιον μὲν ὃ οἷόν τε
εἰς δύο ἴσα διαιρεθῆναι μονάδος μέσον μὴ παρεμπιπτούσης, περιττὸν δὲ
τὸ μὴ δυνάμενον εἰς δύο ἴσα μερισθῆναι διὰ τὴν προειρημένην τῆς
μονάδος μεσιτείαν. He significantly adds that this definition is ἐκ
τῆς δημώδους ὑπολήψεως.

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.[788] The same thing is implied in his statement that the Pythagorean Unlimited was outside the heavens.[789] 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.[790] 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.”[791] 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.

[Sidenote: The numbers as magnitudes.]

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.”[792] Zeller holds that this is only an inference of Aristotle’s,[793] 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.[794]

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.[795] 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?[796] 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,[797] and he has also shown that the cosmology was the whole of the system.[798] 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.

[Sidenote: The numbers and the elements.]

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.[799] 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_.[800] 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.[801] Now it is very significant that Plato starts from fire and earth,[802] 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,[803] 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.[804] 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.

[Sidenote: The dodecahedron.]

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.”[805] 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.[806] The structure of the world was compared to the building of a ship, an idea of which there are other traces.[807] 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.”[808] 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.”[809] 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.[810] 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.[811]

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.[812] 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.[813]

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.

[Sidenote: The Soul a “Harmony.”]

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.”[814] 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.[815] 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.”[816] 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.

[Sidenote: The central fire.]