Chapter 13

God and the world.

61. In the passage of theMetaphysicsjust referred to, Aristotle speaks of Xenophanes as “the first partisan of the One,”[306]and the context shows that he means to suggest he was the first of the Eleatics. We have seen already that the certain facts of his life make it very unlikely that he settled at Elea and founded a school there, and it is probable that, as usual in such cases, Aristotle is simply reproducingcertain statements of Plato. At any rate, Plato had spoken of the Eleatics as the “partisans of the Whole,”[307]and he had also spoken of the school as “starting with Xenophanes and even earlier.”[308]The last words, however, show clearly enough what he meant. Just as he called the Herakleiteans “followers of Homer and still more ancient teachers,”[309]so he attached the Eleatic school to Xenophanes and still earlier authorities. We have seen in other instances how these playful and ironical remarks of Plato were taken seriously by his successors, and we need not let this fresh instance of the same thing influence our general view of Xenophanes unduly.

Aristotle goes on to tell us that Xenophanes, “referring to the whole world,[310]said the One was god.” This clearly alludes to frs.23-26, where all human attributes are denied of a god who is said to be one and “the greatest among gods and men.” It may be added that these verses gain very much in point if we may think of them as closely connected with frs.11-16, instead of referring the one set of verses to the Satires and the other to a cosmological poem. It was probably in the same context that Xenophanes called the world or god “equal every way”[311]and denied that it breathed.[312]The statement that, there is no mastership among the gods[313]also goes very well with fr.26. A god has no wants, nor is it fitting for one god to be the servant of others, like Iris and Hermes in Homer.

Monotheism or polytheism.

62. That this “god” is just the world, Aristotle tells us, and the use of the word θεός is quite in accordance with Anaximander’s. Xenophanes regarded it as sentient, though without any special organs of sense, and it sways all things by the thought of its mind. He also calls it “one god,” and, if that is monotheism, then Xenophanes was a monotheist, though this is surely not how the word is generally understood. The fact is that the expression “one god” wakens all sorts of associations in our mind which did not exist at all for the Greeks of this time. His contemporaries would have been more likely to call Xenophanes an atheist than anything else. As Eduard Meyer excellently says: “In Greece the question of one god or gods many hardly plays any part. Whether the divine power is thought of as a unity or a plurality, is irrelevant in comparison with the question whether it exists at all, and how its nature and its relation to the world is to be understood.”[314]

On the other hand, it is wrong to say with Freudenthal that Xenophanes was in any sense a polytheist.[315]That he should use the language of polytheism in his elegies is only what we should expect, and the other references to “gods” can be best explained as incidental to his attack on the anthropomorphic gods of Homer and Hesiod. In one case, Freudenthal has pressed a proverbial way of speaking too hard.[316]Least of all can we admit that Xenophanes allowed the existence of subordinate or departmental gods; for it was just the existence of such that he was chiefly concerned to deny. At the same time, I cannot help thinking that Freudenthal was more nearly right than Wilamowitz, who says that Xenophanes “upheld the only real monotheism that has ever existed upon earth.”[317]Diels, I fancy, comes nearer the mark, when he calls it a “somewhat narrow pantheism.”[318]But all these views would have surprised Xenophanes himself about equally. He was really Goethe’sWeltkind, with prophets to right and left of him, and he would have smiled if he had known that one day he was to be regarded as a theologian.

169. For the theological views of Anaximander and Anaximenes, see§ 22and30.

170. Cf. Herod. i. 170 (advice of Bias); vi. 22 sqq. (Kale Akte).

171. On all this, see Rohde,Psyche, pp. 327 sqq. It is probable that he exaggerated the degree to which these ideas were already developed among the Thracians, but the essential connexion of the new view of the soul with Northern worships is confirmed by the tradition over and over again.

172. See Meyer,Gesch. des Alterth.ii. § 461. The exaggerated rôle often attributed to priesthoods is a survival of French eighteenth century thinking.

173. See E. Meyer,Gesch. des Alterth.ii. §§ 453-460, who rightly emphasises the fact that the Orphic theogony is the continuation of Hesiod’s work. As we have seen, some of it is even older than Hesiod.

174. For the gold plates of Thourioi and Petelia, see the Appendix to Miss Harrison’sProlegomena to the Study of Greek Religion, where the text of them is discussed and a translation given by Professor Gilbert Murray.

175. This was the oldest name for these “mysteries,” and it simply means “sacraments” (cf. ἔοργα).Orgiaare not necessarily “orgiastic.” That association of ideas merely comes from the fact that they belonged to the worship of Dionysos.

176. Herodotos mentions that Isagoras and those of his γένος worshipped the Karian Zeus (v. 66), and it is probable that theOrgeonesattached by Kleisthenes to the Atticphratriaiwere associations of this kind. See Foucart,Les associations religieuses chez les Grecs.

177. A striking parallel is afforded to all this by what we are told in Robertson Smith’sReligion of the Semites, p. 339. “The leading feature that distinguished them” (the Semitic mysteries of the seventh centuryB.C.) “from the old public cults with which they came into competition, is that they were not based on the principle of nationality, but sought recruits from men of every race who were willing to accept initiation through the mystic sacraments.”

178. ThePhaedois dedicated, as it were, to Echekrates and the Pythagorean society at Phleious, and it is evident that Plato in his youth was impressed by the religious side of Pythagoreanism, though the influence of Pythagorean science is not clearly marked till a later period. Note specially the ἄτραπος ofPhd.66 b 4. InRep.x. 600 b 1, Plato speaks of Pythagoras as the originator of a private ὁδός τις βίου.

179. Cf. especially the point of view of theAuction of Lives(Βίων πρᾶσις).

180. For the Προτρεπτικός of Aristotle, see Bywater inJ. Phil.ii. p. 55; Diels inArch.i. p. 477; and the notes onEthics, i. 5, in my edition.

181. Plato,Rep.520 c 1, καταβατέον οὖν ἐν μέρει. The allegory of the Cave seems to be Orphic, and I believe Professor Stewart’s suggestion (Myths of Plato, p. 252, n. 2), that Plato had the κατάβασις εἰς Ἅιδου in mind, to be quite justified. The idea of rescuing the “spirits in prison” is thoroughly Orphic.

182. For Empedokles, see§ 119; for the Pythagoreans, see§ 149.

183. Cf.Phd.69 c 2, καὶ κινδυνεύουσι καὶ οἱ τὰς τελετὰς ἡμῖν οὗτοι καταστήσαντες οὐ φαῦλοί τινες εἶναι, ἀλλὰ τῷ ὄντι πάλαι αἰνίττεσθαι κ.τ.λ. The gentle irony of this and similar passages ought to be unmistakable.

184. Arist. fr. 45, 1483 a 19, τοὺς τελουμένους οὐ μαθεῖν τι δεῖν, ἀλλὰ παθεῖν καὶ διατεθῆναι.

185. See E. Rohde’s admirable papers,“Die Quellen des Iamblichus in seiner Biographie des Pythagoras”(Rh. Mus.xxvi., xxvii.).

186. Iamblichos was a disciple of Porphyry, and contemporary with Constantine. TheLife of Pythagorashas been edited by Nauck (1884). Nikomachos belongs to the beginning of the second centuryA.D.There is no evidence that he added anything to the authorities he followed, but these were already vitiated by Neopythagorean fables. Still, it is to him we chiefly owe the preservation of the valuable evidence of Aristoxenos.

187. Porphyry’sLife of Pythagorasis the only considerable extract from hisHistory of Philosophy, in four books, that has survived. The romance of Antonius is the original parodied by Lucian in hisVera Historia.

188. The importance of the life in Laertios Diogenes lies in the fact that it gives us the story current at Alexandria before the rise of Neopythagoreanism and the promulgation of the gospel according to Apollonios of Tyana.

189. Andron of Ephesos wrote a work on the Seven Wise Men, calledThe Tripod, in allusion to the well-known story. The feats ascribed to Pythagoras in the Aristotelian treatise remind us of an ecclesiastical legend. For example, he kills a deadly snake by biting it; he was seen at Kroton and Metapontion at the same time; he exhibited his golden thigh at Olympia, and was addressed by a voice from heaven when crossing the river Kasas. The same authority stated that he was identified by the Krotoniates with Apollo Hyperboreios (Arist. fr. 186).

190. Herod. iv. 95.

191. Cf. Herod. iv. 95, and Herakleitos, fr.17(R. P. 31 a). Herodotos represents him as living at Samos. On the other hand, Aristoxenos said that he came from one of the islands which the Athenians occupied after expelling the Tyrrhenians (Diog. viii. 1). This suggests Lemnos, from which the Tyrrhenian “Pelasgians” were expelled by Miltiades (Herod. vi. 140), or possibly some other island which was occupied at the same time. There were also Tyrrhenians at Imbros. This explains the story that he was an Etrurian or a Tyrian. Other accounts bring him into connexion with Phleious, but that is perhaps a pious invention of the Pythagorean society which flourished there at the beginning of the fourth centuryB.C.Pausanias (ii. 13, 1) gives it as a Phleiasian tradition that Hippasos, the great-grandfather of Pythagoras, had emigrated from Phleious to Samos.

192. Eratosthenes identified Pythagoras with the Olympic victor of Ol. XLVIII. 1 (588/7B.C.), but Apollodoros gave hisfloruitas 532/1, the era of Polykrates. He doubtless based this on the statement of Aristoxenos quoted by Porphyry (V. Pyth.9), that Pythagoras left Samos from dislike to the tyranny of Polykrates (R. P. 53 a). For a full discussion, see Jacoby, pp. 215 sqq.

193. Herakl. fr.16,17(R. P. 31, 31 a).

194. It occurs first in theBousirisof Isokrates, § 28 (R. P. 52).

195. Herod. ii. 81 (R. P. 52 a). The comma at Αἰγυπτίοισι is clearly right. Herodotos believed that the worship of Dionysos was introduced from Egypt by Melampous (ii. 49), and he means to suggest that the Orphics got these practices from the worshippers of Bakchos, while the Pythagoreans got them from the Orphics.

196. Herod. ii. 123 (R. P.ib.). The words “whose names I know, but do not write” cannot refer to Pythagoras; for it is only of contemporaries that Herodotos speaks in this way (cf. i. 51; iv. 48). Stein’s suggestion that he meant Empedokles seems to me convincing. Herodotos may have met him at Thourioi. Nor is there any reason to suppose that οἱ μὲν πρότερον refers specially to the Pythagoreans. If Herodotos had ever heard of Pythagoras visiting Egypt, he would surely have said so in one or other of these passages. There was no occasion for reserve, as Pythagoras must have died before Herodotos was born.

197. Porph.V. Pyth.9 (R. P. 53 a).

198. From what Herodotos tells us of Demokedes (iii. 131) we can see that the medical school of Kroton was founded before the time of Pythagoras. Cf. Wachtler,De Alcmaeone Crotoniata, p. 91.

199. It may be taken as certain that Pythagoras spent his last days at Metapontion; Aristoxenos said so (ap.Iambl.V. Pyth.249), and Cicero (De Fin.v. 4) speaks of the honours which continued to be paid to his memory in that city (R. P. 57 c). Cf. also Andron, fr. 6 (F.H.G.ii. 347).

200. For these distinctions, see Porphyry (V. Pyth.37) and Iamblichos (V. Pyth.80), quoted R. P. 56 and 56 b. The name ἀκουσματικοί is clearly related to the ἀκούσματα, with which we shall have to deal shortly (§ 44).

201. For the “mystic silence,” see Aristoxenos,ap.Diog. viii. 15 (R. P. 55 a). Tannery,“Sur le secret dans l’école de Pythagore”(Arch.i. pp. 28 sqq.), thinks that the mathematical doctrines were the secrets of the school, and that these were divulged by Hippasos; but the most reasonable view is that there were no secrets at all except of a ritual kind.

202. Plato,Rep.x. 600 a, implies that Pythagoras held no public office. The view that the Pythagorean sect was a political league, maintained in modern times by Krische (De societatis a Pythagora conditae scopo politico, 1830), goes back, as Rohde has shown (loc. cit.), to Dikaiarchos, the champion of the “Practical Life,” just as the view that it was primarily a scientific society goes back to the mathematician and musician Aristoxenos. The former antedated Archytas, just as the latter antedated Philolaos (see Chap. VII.§ 138). Grote’s good sense enabled him to see this quite clearly (vol. iv. pp. 329 sqq.).

203. Meyer,Gesch. des Alterth.ii. § 502, Anm. It is still necessary to insist upon this, as the idea that the Pythagoreans represented the “Dorian ideal” dies very hard. In hisKulturhistorische Beiträge(Heft i. p. 59), Max C. P. Schmidt imagines that later writers call the founder of the sect Pythagoras instead of Pythagores, as he is called by Herakleitos and Demokritos, because he had become “a Dorian of the Dorians.” The fact is simply that Πυθαγόρας is the Attic form of Πυθαγόρης, and that the writers in question wrote Attic. Similarly, Plato calls Archytas, who did belong to a Dorian state, Archytes, though Aristoxenos and others retained the Dorian form of his name.

204. Kylon, the chief opponent of the Pythagoreans, is described by Aristoxenos (Iambl.V. Pyth.248) as γένει καὶ δόξῃ καὶ πλούτῳ πρωτεύων τῶν πολιτῶν. Taras, later the chief seat of the Pythagoreans, was a democracy. The truth is that, at this time, the new religion appealed to the people rather than the aristocracies, which were apt to be “free-thinking” (Meyer,Gesch. des Alt.iii. § 252). Xenophanes, not Pythagoras, is their man.

205. We have the authority of Aristotle, fr. 186, 1510 b 20, for the identification of Pythagoras with Apollo Hyperboreios. The names of Abaris and Aristeas stand for a mystical movement parallel to the Orphic, but based on the worship of Apollo. The later tradition makes them predecessors of Pythagoras; and that this has some historical basis, appears from Herod. iv. 13 sqq., and above all from the statement that Aristeas had a statue at Metapontion, where Pythagoras died. The connexion of Pythagoras with Zamolxis belongs to the same order of ideas. As the legend of the Hyperboreans is Delian, we see that the religion taught by Pythagoras was genuinely Ionian in its origin.

206. See Rohde,Rh. Mus.xxvi. p. 565, n. 1. The narrative in the text (Iambl.V. Pyth.250; R. P. 59 b) goes back to Aristoxenos and Dikaiarchos (R. P. 59 a). There is no reason to suppose that their view of Pythagoras has vitiated their account of what must have been a perfectly well-known piece of history. According to the later story, Pythagoras himself was burned to death in the house of Milo, along with his disciples. This is merely a dramatic compression of the whole series of events into a single scene; we have seen that Pythagoras died at Metapontion before the final catastrophe. The valuable reference in Polybios ii. 39 (R. P. 59) to the burning of Pythagorean συνέδρια certainly implies that the disturbances went on for a very considerable time.

207. Plato,Phd.61 d 7, e 7.

208. When discussing the Pythagorean system, Aristotle always refers it to “the Pythagoreans,” not to Pythagoras himself. That this was intentional seems to be proved by the phrase οἱ καλούμενοι Πυθαγόρειοι, which occurs more than once (e.g.Met.Α, 5. 985 b 23;de Caelo, Β, 13. 293 a 20). Pythagoras himself is only thrice mentioned in the whole Aristotelian corpus, and in only one of these places (M. Mor.1182 a 11) is any philosophical doctrine ascribed to him. We are told there that he was the first to discuss the subject of goodness, and that he made the mistake of identifying its various forms with numbers. But this is just one of the things which prove the late date of theMagna Moralia. Aristotle himself is quite clear that what he knew as the Pythagorean system belonged in the main to the days of Empedokles, Anaxagoras, and Leukippos; for, after mentioning these, he goes on to describe the Pythagoreans as “contemporary with and earlier than them” (ἐν δὲ τούτοις καὶ πρὸ τούτων,Met.Α, 5. 985 b 23).

209. The fragments of the Πυθαγορικαὶ ἀποφάσεις of Aristoxenos are given by Diels,Vors.pp. 282 sqq.

210.V. Pyth.19 (R. P. 55).

211. See Diels,Dox.p. 150; and“Ein gefälschtes Pythagorasbuch”(Arch.iii. pp. 451 sqq.). Cf. also Bernays,Die Heraklitischen Briefe, n. 1.

212. The proper Greek term for this is παλιγγενεσία, and the inaccurate μετεμψύχωσις only occurs in late writers. Hippolytos and Clement of Alexandria say μετενσωμάτωσις, which is accurate but cumbrous. See Rohde,Psyche, p. 428, n. 2.

213. On the significance of this, see above, p. 93.

214. Dieterich,“Mutter Erde”(Archiv für Religionswissenschaft, viii. pp. 29 and 47).

215. Aristoxenosap.Diog. viii. 20, πάντα μὲν τὰ ἄλλα συγχωρεῖν αὐτὸν ἐσθίειν ἔμψυχα, μόνον δ’ ἀπέχεσθαι βοὸς ἀροτῆρος καὶ κριοῦ.

216. Aristoxenosap.Gell. iv. 11, 5, Πυθαγόρας δὲ τῶν ὀσπρίων μάλιστα τὸν κύαμον ἐδοκίμασεν· λειαντικόν τε γὰρ εἶναι καὶ διαχωρητικόν· διὸ καὶ μάλιστα κέχρηται αὐτῷ;ib.6,“porculis quoque minusculis et haedis tenerioribus victitasse, idem Aristoxenus refert.”It is, of course, possible that Aristoxenos may be right about the taboo on beans. We know that it was Orphic, and it may have been transferred to the Pythagoreans by mistake. That, however, would not affect the general conclusion that at least some Pythagoreans practised abstinence from various kinds of food, which is all that is required.

217. The sect of the “Akousmatics” was said to descend from Hippasos (Iambl.V. Pyth.81; R. P. 56). Now Hippasos was the author of a μυστικὸς λόγος (Diog. viii. 7; R. P. 56 c), that is to say, of a superstitious ceremonial or ritual handbook, probably containing Akousmata like those we are about to consider; for we are told that it was written ἐπὶ διαβολῇ Πυθαγόρου.

218. Diels has collected these fragments in a convenient form (Vors.pp. 291 sqq.). For our purpose the most important passages are Antiphanes, fr. 135, Kock, ὥσπερ Πυθαγορίζων ἐσθίει | ἔμψυχον οὐδέν; Alexis, fr. 220, οἱ Πυθαγορίζοντες γάρ, ὡς ἀκούομεν, | οὔτ’ ὄψον ἐσθίουσιν οὔτ’ ἄλλ’ οὐδὲ ἓν | ἔμψυχον; fr. 196 (from the Πυθαγορίζουσα), ἡ δ’ ἑστίασις ἰσχάδες καὶ στέμφυλα | καὶ τυρὸς ἔσται· ταῦτα γὰρ θύειν νόμος | τοῖς Πυθαγορείοις; Aristophon, fr. 9 (from the Πυθαγοριστής), πρὸς τῶν θεῶν οἰόμεθα τοὺς πάλαι ποτέ, | τοὺς Πυθαγοριστὰς γενομένους ὄντως ῥυπᾶν | ἑκόντας ἢ φορεῖν τριβῶνας ἡδέως; Mnesimachos, fr. 1, ὡς Πυθαγοριστὶ θύομεν τῷ Λοξίᾳ | ἔμψυχον οὐδὲν ἐσθίοντες παντελῶς. See also Theokritos, xiv. 5, τοιοῦτος καὶ πρᾶν τις ἀφίκετο Πυθαγορικτάς, | ὠχρὸς κἀνυποδητός· Ἀθηναῖος δ’ ἔφατ’ ἦμεν.

219. See Bernays,Theophrastos’ Schrift über Frömmigkeit. Porphyry’s tract, Περὶ ἀποχῆς ἐμψύχων, was doubtless saved from the general destruction of his writings by its conformity to the ascetic tendencies of the age. Even St. Jerome made constant use of it in his polemic against Iovianus, though he is careful not to mention Porphyry’s name (Theophr. Schr.n. 2). The tract is addressed to Castricius Firmus, the disciple and friend of Plotinos, who had fallen away from the strict vegetarianism of the Pythagoreans.

220. The passage occursDe Abst.p. 58, 25 Nauck: ἱστοροῦσι δέ τινες καὶ αὐτοὺς ἅπτεσθαι τῶν ἐμψύχων τοὺς Πυθαγορείους, ὅτε θύοιεν θεοῖς. The part of the work from which this is taken comes from one Clodius, on whom see Bernay,Theophr. Schr.p. 11. He was probably the rhetorician Sextus Clodius, and a contemporary of Cicero. Bernays has shown that he made use of the work of Herakleides of Pontos (ib.n. 19). On “mystic sacrifice” generally, see Robertson Smith,Rel. Sem.i. p. 276.

221. Porphyry (V. Pyth.c 15) has preserved a tradition to the effect that Pythagoras recommended a flesh diet for athletes (Milo?). This story must have originated at the same time as those related by Aristoxenos, and in a similar way. In fact, Bernays has shown that it comes from Herakleides of Pontos (Theophr. Schr.n. 8). Iamblichos (V. Pyth.5. 25) and others (Diog. viii. 13, 47) got out of this by supposing it referred to a gymnast of the same name. We see here very distinctly how the Neoplatonists for their own ends endeavoured to go back to the original form of the Pythagorean legend, and to explain away the fourth century reconstruction.

222. For these see Diels,Vors.pp. 282 sqq.

223. There is an excellent collection of Ἀκούσματα καὶ σύμβολα in Diels,Vors.pp. 279 sqq., where the authorities will be found. It is impossible to discuss these in detail here, but students of folklore will see at once to what order of ideas they belong.

224. Herakl. fr.17(R. P. 31 a). The word ἱστορίη is in itself quite general. What it chiefly means here we see from a valuable notice preserved by Iamblichos,V. Pyth.89, ἐκαλεῖτο δὲ ἡ γεωμετρία πρὸς Πυθαγόρου ἱστορία. Tannery’s interpretation of this statement is based on a misunderstanding, and need not be discussed here.

225. Herod. iv. 95.

226. Arist. Περὶ τῶν Πυθαγορείων, fr. 186, 1510 a 39, Πυθαγόρας Μνησάρχου υἱὸς τὸ μὲν πρῶτον διεπονεῖτο περὶ τὰ μαθήματα καὶ τοὺς ἀριθμούς, ὕστερον δέ ποτε καὶ τῆς Φερεκύδου τερατοποιΐας οὐκ ἀπέστη.

227. Its immediate source is to be found in Plato,Laws, 790 d 2 sqq., where the Korybantic rites are adduced as an instance. For a full account see Rohde,Psyche, p. 336, n. 2.

228. Plato gives this as the Pythagorean view inPhd.62 b, for the interpretation of which cf. Espinas inArch.viii. pp. 449 sqq. Plato distinctly implies that it was not merely the theory of Philolaos, but something older.

229. See Döring inArch.v. pp. 505 sqq. There seems to be a reference to the theory of the “three lives” in Herakleitos, fr.111. It was apparently taught in the Pythagorean Society of Phleious; for Herakleides made Pythagoras expound it in a conversation with the tyrant of Phleious (Cic.Tusc.v. 3; Diog. pr. 12, viii. 8), and it is developed by Plato in a dialogue which is, as it were, dedicated to Echekrates. If it should be thought that this is interpreting Pythagoras too much in the light of Schopenhauer, it may be answered that even the Orphics came very near such a theory. The soul must not drink of Lethe, but go past it and drink of the water of Memory, before it can claim to become one of the heroes. This has obvious points of contact with Plato’s ἀνάμνησις, and the only question is how much of thePhaedowe are to ascribe to Pythagorean sources. A great deal, I suspect. See Prof. Stewart’sMyths of Plato, pp. 152 sqq.

230. Stob. i. p. 20, 1, ἐκ τῶν Ἀριστοξένου περὶ ἀριθμητικῆς, Τὴν δὲ περὶ τοὺς ἀριθμοὺς πραγματείαν μάλιστα πάντων τιμῆσαι δοκεῖ Πυθαγόρας καὶ προαγαγεῖν ἐπὶ τὸ πρόσθεν ἀπαγαγὼν ἀπὸ τῆς τῶν ἐμπόρων χρείας.

231. Apart from the story in Iamblichos (V. Pyth.148) that Eurytos heard the voice of Philolaos from the grave after he had been many years dead, it is to be noticed that he is mentioned after him in the statement of Aristoxenos referred to (Diog. viii. 46; R. P. 62).

232. Arist.Met.Ν, 5. 1092 b 8 (R. P. 76 a). Aristotle does not quote the authority of Archytas here, but the source of his statement is made quite clear by Theophr.Met.p. vi. a 19 (Usener), τοῦτο γὰρ (sc. τὸ μὴ μέχρι του προελθόντα παύεσθαι) τελέου καὶ φρονοῦντος, ὅπερ Ἀρχύτας ποτ’ ἔφη ποιεῖν Εὔρυτον διατιθέντα τινὰς ψήφους· λέγειν γὰρ ὡς ὅδε μὲν ἀνθρώπου ὁ ἀριθμός, ὅδε δὲ ἵππου, ὅδε δ’ ἄλλου τινὸς τυγχάνει.

233. Arithmetic is older than geometry, and was much more advanced in Egypt, though still in the form which the Greeks called λογιστική rather than as ἀριθμητική proper. Even Plato puts Arithmetic before Geometry in theRepublicin deference to the tradition. His own theory of number, however, suggested the inversion of this order which we find carried out in Euclid.

234. Nikomachos of Gerasa,Introd. Arithm.p. 83, 12, Hoche, Πρότερον δὲ ἐπιγνωστέον ὅτι ἕκαστον γράμμα ᾧ σημειούμεθα ἀριθμόν, οἷον τὸ ι, ᾧ τὸ δέκα, τὸ κ, ᾧ τὰ εἴκοσι, τὸ ω, ᾧ τὰ ὀκτακόσια, νόμῳ καὶ συνθήματι ἀνθρωπίνῳ, ἀλλ’ οὐ φύσει σημαντικόν, ἐστι τοῦ ἀριθμοῦ, κ.τ.λ. The same symbolism is used by Theo,Expositio, pp. 31 sqq. Cf. also Iambl.Introd.p. 56, 27, Pistelli, ἰστέον γὰρ ὡς τὸ παλαιὸν φυσικώτερον οἱ πρόσθεν ἐσημαίνοντο τὰς τοῦ ἀριθμοῦ ποσότητας, ἀλλ’ οὐχ ὥσπερ οἱ νῦν συμβολικῶς.

235. Cf. the formula Οὐ μὰ τὸν ἁμετέρᾳ γενεᾷ παραδόντα τετρακτύν, which is all the more likely to be old that it is put into the mouth of Pythagoras by the forger of the Χρυσᾶ ἔπη, thus making him swear by himself! See Diels,Arch.iii. p. 457. The Doric dialect shows, however, that it belongs to the later generations of the school.

236. Speusippos wrote a work on the Pythagorean numbers, based chiefly on Philolaos, and a considerable fragment of it is preserved in theTheologumena Arithmetica. It will be found in Diels,Vorsokratiker, p. 235, 15, and is discussed by Tannery,Science hellène, pp. 374 sqq.

237. For these see Theon,Expositio, pp. 93 sqq. Hiller. The τετρακτύς used by Plato in theTimaeusis the second described by Theon (Exp.p. 94, 10 sqq.). It is no doubt Pythagorean, but hardly as old as Pythagoras.

238. Cf. Milhaud,Philosophes géomètres, pp. 115 sqq. Aristotle puts the matter thus (Phys.Γ, 4. 203 a 13): περιτιθεμένων γὰρ τῶν γνωμόνων περὶ τὸ ἓν καὶ χωρὶς ὁτὲ μὲν ἄλλο ἀεὶ γίγνεσθαι τὸ εἶδος, ὁτὲ δὲ ἕν. This is more clearly stated by Ps.-Plut. (Stob. i. p. 22, 16), Ἔτι δὲ τῇ μονάδι τῶν ἐφεξῆς περισσῶν περιτιθεμένων ὁ γινόμενος ἀεὶ τετράγωνός ἐστι· τῶν δὲ ἀρτίων ὁμοίως περιτιθεμένων ἑτερομήκεις καὶ ἄνισοι πάντες ἀποβαίνουσιν, ἴσως δὲ ἰσάκις οὐδείς. I cannot feel satisfied with any of the explanations which have been given of the words καὶ χωρίς in the Aristotelian passage (see Zeller, p. 351, n. 2), and I would therefore suggest ταῖς χώραις comparing Boutheros (Stob. i. p. 19, 9), who says, according to the MS. reading, Καὶ ὁ μὲν (ὁ περισσός), ὁπόταν γεννῶνται ἀνὰ λόγον καὶ πρὸς μονάδας, ταῖς αὑτοῦ χώραις καταλαμβάνει τοὺς ταῖς γραμμαῖς περιεχομένους (sc. ἀριθμούς).

239. In the fragment referred to above (p. 113,n.236), Speusippos speaks of four as the first pyramidal number; but this is taken from Philolaos, so we cannot safely ascribe it to Pythagoras.

240. We have ὅροι of a series (ἔκθεσις), then of a proportion, and in later times of a syllogism. The signs :, ::, and ∴ are a survival of the original use. The term χώρα is often used by the later Pythagoreans, though Attic usage required χωρίον for a rectangle. The spaces between the γραμμαί of theabacusand the chess-board were also called χῶραι.

241. In his commentary on Euclid i. 44, Proclus tells us on the authority of Eudemos that the παραβολή, ἔλλειψις, and ὑπερβολή of χωρία were Pythagorean inventions. For an account of these and the subsequent application of the terms in Conic Sections, see Milhaud,Philosophes géomètres, pp. 81 sqq.

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