Chapter II: Science and Religion (2)

47. Now one of the most remarkable statements that we have about Pythagoreanism is what we are told of Eurytos on the unimpeachable authority of Archytas. Eurytos was the disciple of Philolaos, and Aristoxenos expressly mentioned him along with Philolaos as having taught the last of the Pythagoreans, the men with whom he himself was personally acquainted. He therefore belongs to the beginning of the fourth century B.C., by which time the Pythagorean system was fully developed, and he was no eccentric enthusiast, but one of the foremost men in the school.[231] We are told of him, then, that he used to give the number of all sorts of things, such as horses and men, and that he demonstrated these by arranging pebbles in a certain way. It is to be noted further that Aristotle compares his procedure to that of those who bring numbers into figures like the triangle and the square.[232]

Footnote 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).

Footnote 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.

Now these statements, and especially the remark of Aristotle last quoted, seem to imply the existence at this date, and earlier, of a numerical symbolism quite distinct from the alphabetical notation on the one hand and from the Euclidean representation of numbers by lines on the other. The former was inconvenient for arithmetical purposes, just because the zero was one of the few things the Greeks did not invent, and they were therefore unable to develop a really serviceable numerical symbolism based on position. The latter, as will appear shortly, is intimately bound up with that absorption of arithmetic by geometry, which is at least as old as Plato, but cannot be primitive.[233] It seems rather that numbers were represented by dots arranged in symmetrical and easily recognised patterns, of which the marking of dice or dominoes gives us the best idea. And these markings are, in fact, the best proof that this is a genuinely primitive method of indicating numbers; for they are of unknown antiquity, and go back to the time when men could only count by arranging numbers in such patterns, each of which became, as it were, a fresh unit. This way of counting may well be as old as reckoning with the fingers, or even older.

Footnote 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 the _Republic_ in deference to the tradition. His own
theory of number, however, suggested the inversion of this order which
we find carried out in Euclid.

It is, therefore, very significant that we do not find any adequate account of what Aristotle can have meant by “those who bring numbers into figures like the triangle and the square” till we come to certain late writers who called themselves Pythagoreans, and revived the study of arithmetic as a science independent of geometry. These men not only abandoned the linear symbolism of Euclid, but also regarded the alphabetical notation, which they did use, as something conventional, and inadequate to represent the true nature of number. Nikomachos of Gerasa says expressly that the letters used to represent numbers are only significant by human usage and convention. The most natural way would be to represent linear or prime numbers by a row of units, polygonal numbers by units arranged so as to mark out the various plane figures, and solid numbers by units disposed in pyramids and so forth.[234] He therefore gives us figures like this:—

α α α α
α α α ααα
α α α α α α α α
α α α α ααα
α α α α α

Now it ought to be obvious that this is no innovation, but, like so many things in Neopythagoreanism, a reversion to primitive usage. Of course the employment of the letter _alpha_ to represent the units is derived from the conventional notation; but otherwise we are clearly in presence of something which belongs to the very earliest stage of the science—something, in fact, which gives the only possible clue to the meaning of Aristotle’s remark, and to what we are told of the method of Eurytos.

Footnote 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.

[Sidenote: Triangular, square, and oblong numbers.]

48. This is still further confirmed by the tradition which represents the great revelation made by Pythagoras to mankind as having been precisely a figure of this kind, namely the _tetraktys_, by which the Pythagoreans used to swear,[235] and we have no less an authority than Speusippos for holding that the whole theory which it implies was genuinely Pythagorean.[236] In later days there were many kinds of _tetraktys_,[237] but the original one, that by which the Pythagoreans swore, was the “tetraktys of the dekad.” It was a figure like this—

•
• •
• • •
• • • •

and represented the number ten as the triangle of four. In other words, it showed at a glance that 1 + 2 + 3 + 4 = 10. Speusippos tells us of several properties which the Pythagoreans discovered in the dekad. It is, for instance, the first number that has in it an equal number of prime and composite numbers. How much of this goes back to Pythagoras himself, we cannot tell; but we are probably justified in referring to him the conclusion that it is “according to nature” that all Hellenes and barbarians count up to ten and then begin over again.

Footnote 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.

Footnote 236:

Speusippos wrote a work on the Pythagorean numbers, based chiefly on
Philolaos, and a considerable fragment of it is preserved in the
_Theologumena Arithmetica_. It will be found in Diels,
_Vorsokratiker_, p. 235, 15, and is discussed by Tannery, _Science
hellène_, pp. 374 sqq.

Footnote 237:

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

It is obvious that the _tetraktys_ may be indefinitely extended so as to exhibit the sums of the series of successive numbers in a graphic form, and these sums are accordingly called “triangular numbers.”

For similar reasons, the sums of the series of successive odd numbers are called “square numbers,” and those of successive even numbers “oblong.” If odd numbers are added to the unit in the form of _gnomons_, the result is always a similar figure, namely a square, while, if even numbers are added, we get a series of rectangles,[238] as shown by the figure:—

Square Numbers. Oblong Numbers.
─────────────┐ ───────────────┐
• • •  │ • • • • │
───────┐ │ ───────────┐ │
• • │ • │ • • • │ • │
──┐ │ │ ──────┐ │ │
• │ • │ • │ • • │ • │ • │

It is clear, then, that we are entitled to refer the study of sums of series to Pythagoras himself; but whether he went beyond the oblong, and studied pyramidal or cubic numbers, we cannot say.[239]

Footnote 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).

Footnote 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.

[Sidenote: Geometry and harmonics.]

49. It is easy to see how this way of representing numbers would suggest problems of a geometrical nature. The dots which stand for the pebbles are regularly called “boundary-stones” (ὅροι, _termini_, “terms”), and the area which they occupy, or rather mark out, is the “field” (χώρα).[240] This is evidently a very early way of speaking, and may therefore be referred to Pythagoras himself. Now it must have struck him that “fields” could be compared as well as numbers,[241] and it is even likely that he knew the rough methods of doing this which were traditional in Egypt, though certainly these would fail to satisfy him. Once more the tradition is singularly helpful in suggesting the direction that his thoughts must have taken. He knew, of course, the use of the triangle 3, 4, 5 in constructing right angles. We have seen (p. 24) that it was familiar in the East from a very early date, and that Thales introduced it to the Hellenes, if they did not know it already. In later writers it is actually called the “Pythagorean triangle.” Now the Pythagorean proposition _par excellence_ is just that, in a right-angled triangle, the square on the hypotenuse is equal to the squares on the other two sides, and the so-called Pythagorean triangle is the application of its converse to a particular case. The very name “hypotenuse” affords strong confirmation of the intimate connexion between the two things. It means literally “the cord stretching over against,” and this is surely just the rope of the “harpedonapt.”[242] An early tradition says that Pythagoras sacrificed an ox when he discovered the proof of this proposition, and indeed it was the real foundation of scientific mathematics.[243]

Footnote 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 the _abacus_ and the chess-board were also called
χῶραι.

Footnote 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.

Footnote 242:

The verb ὑποτείνειν is, of course, used intransitively. The
explanation suggested in the text seems to me much simpler than that
of Max C. P. Schmidt (_Kulturhistorische Beiträge_, Heft i. pp. 64
sqq.). He explains the hypotenuse as the longest string in a
triangular harp; but my view seems more in accordance with analogy. So
ἡ κάθετος is, literally, a plumb-line.

Footnote 243:

The statement comes from Eudemos; for it is found in Proclus’s
commentary on Euclid i. 47. Whether historical or not, it is no
Neopythagorean fancy.

[Sidenote: Incommensurability.]

50. One great disappointment, however, awaited Pythagoras. It follows at once from the Pythagorean proposition that the square on the diagonal of a square is double the square on its side, and this ought surely to be capable of numerical expression. As a matter of fact, however, there is no square number which can be divided into two equal square numbers, and so the problem cannot be solved. In this sense, it is doubtless true that Pythagoras discovered the incommensurability of the diagonal and the side of a square, and the proof mentioned by Aristotle, namely, that, if they were commensurable, we should have to say that an even number was equal to an odd number, is distinctly Pythagorean in character.[244] However that may be, it is certain that Pythagoras did not care to pursue the subject any further. He had, as it were, stumbled on the fact that the square root of two is a surd, but we know that it was left for Plato’s friends, Theodoros of Kyrene and Theaitetos, to give a complete theory of the matter.[245] The fact is that the discovery of the Pythagorean proposition, by giving birth to geometry, had really superseded the old view of quantity as a sum of units; but it was not till Plato’s time that the full consequences of this were seen.[246] For the present, the incommensurability of the diagonal and the square remained, as has been said, a “scandalous exception.” Our tradition says that Hippasos of Metapontion was drowned at sea for revealing this skeleton in the cupboard.[247]

Footnote 244:

Arist. _An. Pr._ Α, 23. 41 a 26, ὅτι ἀσύμμετρος ἡ διάμετρος διὰ τὸ
γίγνεσθαι τὰ περιττὰ ἴσα τοῖς ἀρτίοις συμμέτρου τεθείσης. The proofs
given at the end of Euclid’s Tenth Book (vol. iii. pp. 408 sqq.,
Heiberg) turn on this very point. They are not Euclidean, and may be
substantially Pythagorean. Cf. Milhaud, _Philosophes géomètres_, p.
94.

Footnote 245:

Plato, _Theaet._ 147 d 3 sqq.

Footnote 246:

How novel these consequences were, is shown by the fact that in
_Laws_, 819 d 5, the Athenian Stranger says that he had only realised
them late in life.

Footnote 247:

This version of the tradition is mentioned in Iamblichos, _V. Pyth._
247, and looks older than the other, which we shall come to later (§
148). Hippasos is the _enfant terrible_ of Pythagoreanism, and the
traditions about him are full of instruction.

[Sidenote: Proportion and harmony.]

51. These last considerations show that, while it is quite safe to attribute the substance of the First Book of Euclid to Pythagoras, the arithmetic of Books VII.-IX., and the “geometrical algebra” of Book II. are certainly not his. They operate with lines or with areas instead of with units, and the relations which they establish therefore hold good whether they are capable of numerical expression or not. That is doubtless why arithmetic is not treated in Euclid till after plane geometry, a complete inversion of the original order. For the same reason, the doctrine of proportion which we find in Euclid cannot be Pythagorean, and is indeed the work of Eudoxos. Yet it is clear that the early Pythagoreans, and probably Pythagoras himself, studied proportion in their own way, and that the three “medieties” in particular go back to the founder, especially as the most complicated of them, the “harmonic,” stands in close relation to his discovery of the octave. If we take the harmonic proportion 12 : 8 : 6,[248] we find that 12 : 6 is the octave, 12 : 8 the fifth, and 8 : 6 the fourth, and it can hardly be doubted that it was Pythagoras himself who discovered these intervals. The stories which have come down to us about his observing the harmonic intervals in a smithy, and then weighing the hammers that produced them, or of his suspending weights corresponding to those of the hammers to equal strings, are, indeed, impossible and absurd; but it is sheer waste of time to rationalise them.[249] For our purpose their absurdity is their chief merit. They are not stories which any Greek mathematician or musician could possibly have invented, but genuine popular tales bearing witness to the existence of a real tradition that Pythagoras was the author of this momentous discovery.

Footnote 248:

Plato (_Tim._ 36 a 3) defines the harmonic mean as τὴν ... ταὐτῷ μέρει
τῶν ἄκρων αὐτῶν ὑπερέχουσαν καὶ ὑπερεχομένην. The harmonic mean of 12
and 6 is therefore 8; for 8 = 12 - 12/3 = 6 + 6/3.

Footnote 249:

For these stories and a criticism of them, see Max C. P. Schmidt,
_Kulturhistorische Beiträge_, i. pp. 78 sqq. The smith’s hammers
belong to the region of _Märchen_, and it is not true either that the
notes would be determined by the weight of the hammers, or that, if
they were, the weights hung to equal strings would produce the notes.
These inaccuracies were pointed out by Montucla (Martin, _Études sur
le Timée_, i. p. 391).

[Sidenote: Things are numbers.]

52. It was this too, no doubt, that led Pythagoras to say all things were numbers. We shall see that, at a later date, the Pythagoreans identified these numbers with geometrical figures; but the mere fact that they called them “numbers,” when taken in connexion with what we are told about the method of Eurytos, is sufficient to show this was not the original sense of the doctrine. It is enough to suppose that Pythagoras reasoned somewhat as follows. If musical sounds can be reduced to numbers, why should not everything else? There are many likenesses to number in things, and it may well be that a lucky experiment, like that by which the octave was discovered, will reveal their true numerical nature. The Neopythagorean writers, going back in this as in other matters to the earliest tradition of the school, indulge their fancy in tracing out analogies between things and numbers in endless variety; but we are fortunately dispensed from following them in these vagaries. Aristotle tells us distinctly that the Pythagoreans explained only a few things by means of numbers,[250] which means that Pythagoras himself left no developed doctrine on the subject, while the Pythagoreans of the fifth century did not care to add anything of the sort to the school tradition. Aristotle does imply, however, that, according to them the “right time” (καιρός) was seven, justice was four, and marriage three. These identifications, with a few others like them, we may safely refer to Pythagoras or his immediate successors; but we must not attach much importance to them. They are mere sports of the analogical fancy. If we wish to understand the cosmology of Pythagoras, we must start, not from them, but from any statements we can find that present points of contact with the teaching of the Milesian school. These, we may fairly infer, belong to the system in its most primitive form.

Footnote 250:

Arist. _Met._ Μ, 4. 1078 b 21 (R. P. 78); Zeller, p. 390, n. 2. The
_Theologumena Arithmetica_, wrongly attributed to Nikomachos of
Gerasa, is full of fanciful doctrine on this subject (R. P. 78 a).
Alexander _in Met._ p. 38, 8, gives a few definitions which may be old
(R. P. 78 c).

[Sidenote: Cosmology.]

53. Now the most striking statement of this kind is one of Aristotle’s. The Pythagoreans held, he tells us, that there was “boundless breath” outside the heavens, and that it was inhaled by the world.[251] In substance, this is the doctrine of Anaximenes, and it becomes practically certain that it was that of Pythagoras, when we find that Xenophanes denied it.[252] We may infer, then, that the further development of the idea is also due to Pythagoras himself. We are told that, after the first unit had been formed—however that may have taken place—the nearest part of the Boundless was first drawn in and limited;[253] and further, that it is just the Boundless thus inhaled that keeps the units separate from each other.[254] It represents the interval between them. This is a very primitive way of describing the nature of discrete quantity.

Footnote 251:

Arist. _Phys._ Δ, 6. 213 b 22 (R. P. 75).

Footnote 252:

Diog. ix. 19 (R. P. 103 c). It is true that Diogenes is here drawing
from a biographical rather than a doxographical source (_Dox._ p.
168), but this touch can hardly be an invention.

Footnote 253:

Arist. _Met._ Μ, 3. 1091 a 13 (R. P. 74).

Footnote 254:

Arist. _Phys._ Δ, 6. 213 b 23 (R. P. 75 a). The words διορίζει τὰς
φύσεις have caused unnecessary difficulty, because they have been
supposed to attribute the function of limiting to the ἄπειρον.
Aristotle makes it quite clear that his meaning is that stated in the
text. Cf. especially the words χωρισμοῦ τινος τῶν ἐφεξῆς καὶ
διορίσεως. The term διωρισμένον is the proper antithesis to συνεχές.
In his work on the Pythagorean philosophy, Aristotle used instead the
phrase διορίζει τὰς χώρας (Stob. i. p. 156, 8; R. P. 75), which is
also quite intelligible if we remember what the Pythagoreans meant by
χώρα (cf. p. 115, _n._ 240).

In the passages of Aristotle just referred to, the Boundless is also spoken of as the void or empty. This identification of air and the void is a confusion which we have already met with in Anaximenes, and it need not surprise us to find it here too.[255] We find also, as we might expect, distinct traces of the other confusion, that of air and vapour. It seems certain, in fact, that Pythagoras identified the Limit with fire, and the Boundless with darkness. We are told by Aristotle that Hippasos made Fire the first principle,[256] and we shall see that Parmenides, in discussing the opinions of his contemporaries, attributes to them the view that there were two primary “forms,” Fire and Night.[257] We also find that Light and Darkness appear in the Pythagorean table of opposites under the heads of the Limit and the Unlimited respectively.[258] The identification of breath with darkness here implied is a strong proof of the primitive character of the doctrine; for in the sixth century darkness was supposed to be a sort of vapour, while in the fifth, its true nature was well known. Plato, with his usual historical tact, makes the Pythagorean Timaios describe mist and darkness as condensed air.[259] We must think, then, of a “field” of darkness or breath marked out by luminous units, an imagination which the starry heavens would naturally suggest. It is even probable that we should ascribe to Pythagoras the Milesian view of a plurality of worlds, though it would not have been natural for him to speak of an infinite number. We know, at least, that Petron, one of the early Pythagoreans, said there were just a hundred and eighty-three worlds arranged in a triangle;[260] and Plato makes Timaios admit, when laying down that there is only one world, that something might be urged in favour of the view that there are five, as there are five regular solids.[261]

Footnote 255:

Cf. Arist. _Phys._ Δ, 6. 213 a 27, οἱ δ’ ἄνθρωποι ... φασὶν ἐν ᾦ ὅλως
μηδέν ἐστι, τοῦτ’ εἶναι κενόν, διὸ τὸ πλῆρες ἀέρος κενὸν εἶναι; _de
Part. An._ Β, 10. 656 b 15, τὸ γὰρ κενὸν καλούμενον ἀέρος πλῆρές ἐστι;
_de An._ Β, 10 419 b 34, δοκεῖ γὰρ εἶναι κενὸν ὁ ἀήρ.

Footnote 256:

Arist. _Met._ Α, 3. 984 a 7 (R. P. 56 c).

Footnote 257:

See Chap. IV. § 91.

Footnote 258:

Arist. _Met._ Α, 5. 986 a 25 (R. P. 66).

Footnote 259:

Plato, _Tim._ 58 d 2.

Footnote 260:

This is quoted by Plutarch, _de def. orac._ 422 b, d, from Phanias of
Eresos, who gave it on the authority of Hippys of Rhegion. If we may
follow Wilamowitz (_Hermes_, xix. p. 444) in supposing that this
really means Hippasos of Metapontion (and it was in Rhegion that the
Pythagoreans took refuge), this is a very valuable piece of evidence.

Footnote 261:

Plato, _Tim._ 55 c 7 sqq.

[Sidenote: The heavenly bodies.]

54. Anaximander had regarded the heavenly bodies as wheels of “air” filled with fire which escapes through certain openings (§ 19), and there is evidence that Pythagoras adopted the same view.[262] We have seen that Anaximander only assumed the existence of three such wheels, and held that the wheel of the sun was the lowest. It is extremely probable that Pythagoras identified the intervals between these rings with the three musical intervals which he had discovered, the fourth, the fifth, and the octave. That would be the most natural beginning for the later doctrine of the “harmony of the spheres,” though that expression would be doubly misleading if applied to any theory we can properly ascribe to Pythagoras himself. The word ἁρμονία does not mean harmony, and the “spheres” are an anachronism. We are still at the stage when wheels or rings were considered sufficient to account for the motions of the heavenly bodies. It is also to be observed that sun, moon, planets, and fixed stars must all be regarded as moving in the same direction from east to west. Pythagoras certainly did not ascribe to the planets an orbital motion of their own from west to east. The old idea was rather that they were left behind more or less every day. As compared with the fixed stars, Saturn is left behind least of all, and the Moon most; so, instead of saying that the Moon took a shorter time than Saturn to complete its path through the signs of the Zodiac, men said Saturn travelled quicker than the Moon, because it more nearly succeeds in keeping up with the signs. Instead of holding that Saturn takes thirty years to complete its revolution, they said it took the fixed stars thirty years to pass Saturn, and only twenty-nine days and a half to pass the Moon. This is one of the most important points to bear in mind regarding the planetary systems of the Greeks, and we shall return to it again.[263]

Footnote 262:

This will be found in Chap. IV. § 93.

Footnote 263:

For a clear statement of this view (which was still that of
Demokritos), see Lucretius, v. 621 sqq. The view that the planets had
an orbital motion from west to east is attributed by Aetios, ii. 16,
3, to Alkmaion (§ 96), which certainly implies that Pythagoras did not
hold it. As we shall see (§ 152), it is far from clear that any of the
Pythagoreans did. It seems rather to be Plato’s discovery.

The account just given of the views of Pythagoras is, no doubt, conjectural and incomplete. We have simply assigned to him those portions of the Pythagorean system which appear to be the oldest, and it has not even been possible at this stage to cite fully the evidence on which our discussion is based. It will only appear in its true light when we have examined the second part of the poem of Parmenides and the system of the later Pythagoreans.[264] For reasons which will then be apparent, I do not venture to ascribe to Pythagoras himself the theory of the earth’s revolution round the central fire. It seems safest to suppose that he still adhered to the geocentric hypothesis of Anaximander. In spite of this, however, it will be clear that he opened a new period in the development of Greek science, and it was certainly to his school that its greatest discoveries were directly or indirectly due. When Plato deliberately attributes some of his own most important discoveries to the Pythagoreans, he was acknowledging in a characteristic way the debt he owed them.

Footnote 264:

See Chap. IV. §§ 92-93, and Chap. VII. §§ 150-152.

II. XENOPHANES OF KOLOPHON

[Sidenote: Life.]

55. We have seen how Pythagoras identified himself with the religious movement of his time; we have now to consider a very different manifestation of the reaction against that view of the gods which the poets had made familiar to every one. Xenophanes denied the anthropomorphic gods altogether, but was quite unaffected by the revival of more primitive ideas that was going on all round him. We still have a fragment of an elegy in which he ridiculed Pythagoras and the doctrine of transmigration. “Once, they say, he was passing by when a dog was being ill-treated. ‘Stop!’ he said, ‘don’t hit it! It is the soul of a friend! I knew it when I heard its voice.’”[265] We are also told that he opposed the views of Thales and Pythagoras, and attacked Epimenides, which is likely enough, though no fragments of the kind have come down to us.[266] His chief importance lies in the fact that he was the author of the quarrel between philosophy and poetry which culminated in Plato’s _Republic_.

Footnote 265:

See fr. 7 (= 18 Karst.), _ap._ Diog. viii. 36 (R. P. 88).

Footnote 266:

Diog. ix. 18 (R. P. 97). We know that Xenophanes referred to the
prediction of an eclipse by Thales (Chap. I. p. 41, _n._ 62). We shall
see that his own view of the sun was hardly consistent with the
possibility of such a prediction, so it may have been in connexion
with this that he opposed him.

It is not easy to determine the date of Xenophanes. Timaios said he was a contemporary of Hieron and Epicharmos, and he certainly seems to have played a part in the anecdotical romance of Hieron’s court which amused the Greeks of the fourth century much as that of Croesus and the Seven Wise Men amused those of the fifth.[267] As Hieron reigned from 478 to 467 B.C., that would make it impossible to date the birth of Xenophanes much earlier than 570 B.C., even if we suppose him to have lived till the age of a hundred. On the other hand, both Sextus and Clement say that Apollodoros gave Ol. XL. (620-616 B.C.) as the date of his birth, and the former adds that his days were prolonged till the time of Dareios and Cyrus.[268] Again, Diogenes, whose information on such matters mostly comes from Apollodoros, says that he flourished in Ol. LX. (540-537 B.C.), and Diels holds that Apollodoros really said so.[269] However that may be, it is evident that the date 540 B.C. is based on the assumption that he went to Elea in the year of its foundation, and is, therefore, a mere combination.[270]

Footnote 267:

Timaios _ap._ Clem. _Strom._ i. p. 533 (R. P. 95). There is only one
anecdote which actually represents Xenophanes in conversation with
Hieron (Plut. _Reg. apophth._ 175 e), but it is natural to understand
Arist. _Met._ Γ, 5. 1010 a 4 as an allusion to a remark made by
Epicharmos to him. Aristotle has more than one anecdote about
Xenophanes, and it seems most likely that he derived them from the
romance of which Xenophon’s _Strom._ is an echo.

Footnote 268:

Clem., _loc. cit._; Sext. _Strom._ i. 257. The mention of Cyrus is
confirmed by Hipp. _Strom._ i. 94. Diels thinks that Dareios was
mentioned first for metrical reasons; but no one has satisfactorily
explained why Cyrus should be mentioned at all, unless the early date
was intended. On the whole subject, see Jacoby, pp. 204 sqq., who is
certainly wrong in supposing that ἄχρι τῶν Δαρείου καὶ Κύρου χρόνων
can mean “during the times of Dareios and Cyrus.”

Footnote 269:

_Strom._ xxxi. p. 22. He assumes an early corruption of N into M. As
Apollodoros gave the Athenian archon, and not the Olympiad, we might
with more probability suppose a confusion due to two archons having
the same name.

Footnote 270:

As Elea was founded by the Phokaians six years after they left Phokaia
(Herod. i. 164 sqq.) its date is just 540-39 B.C. Cf. the way in which
Apollodoros dated Empedokles by the era of Thourioi (§ 98).

What we do know for certain is that Xenophanes had led a wandering life from the age of twenty-five, and that he was still alive and making poetry at the age of ninety-two. He says himself (fr. 8 = 24 Karst.; R. P. 97):—

There are by this time threescore years and seven that have tossed my
careworn soul[271] up and down the land of Hellas; and there were then
five-and-twenty years from my birth, if I can say aught truly about
these matters.

Footnote 271:

Bergk (_Litteraturgesch._ ii. p. 418, n. 23) took φροντίς here to mean
the literary work of Xenophanes, but it is surely an anachronism to
suppose that at this date it could be used like the Latin _cura_.

It is tempting to suppose that in this passage Xenophanes was referring to the conquest of Ionia by Harpagos, and that he is, in fact, answering the question asked in another poem[272] (fr. 22 = 17 Karst.; R. P. 95 a):—

This is the sort of thing we should say by the fireside in the
winter-time, as we lie on soft couches after a good meal, drinking
sweet wine and crunching chickpeas: “Of what country are you, and how
old are you, good sir? And how old were you when the Mede appeared?”

Footnote 272:

It was certainly another poem; for it is in hexameters while the
preceding fragment is in elegiacs.

We cannot, however, be sure of this, and we must be content with what is, after all, for our purpose the main fact, namely, that he refers to Pythagoras in the past tense, and is in turn so referred to by Herakleitos.[273]

Footnote 273:

Xenophanes, fr. 7 (above, p. 124, _n._ 265); Herakleitos, frs. 16, 17
(below, p. 147).

Theophrastos said that Xenophanes had “heard” Anaximander,[274] and we shall see that he was certainly acquainted with the Ionian cosmology. When driven from his native city, he lived in Sicily, chiefly, we are told, at Zankle and Katana.[275] Like Archilochos before him, he unburdened his soul in elegies and satires, which he recited at the banquets where, we may suppose, the refugees tried to keep up the usages of good Ionian society. The statement that he was a rhapsode has no foundation at all.[276] The singer of elegies was no professional like the rhapsode, but the social equal of his listeners. In his ninety-second year he was still, we have seen, leading a wandering life, which is hardly consistent with the statement that he settled at Elea and founded a school there, especially if we are to think of him as spending his last days at Hieron’s court. It is quite probable that he visited Elea, and it is just possible that he wrote a poem of two thousand hexameters on the foundation of that city, which was naturally a subject of interest to all the Ionic _émigrés_.[277] But it is very remarkable that no ancient writer expressly says that he ever was at Elea, and the only thing besides the doubtful poem referred to which connects him with it is a single anecdote of Aristotle’s as to the answer he gave the Eleates when they asked whether they should sacrifice to Leukothea and lament her or not. “If you think her a goddess,” he said, “do not lament her; if not, do not sacrifice to her.” That is absolutely all, and it is only an apophthegm.[278] It is strange there should be no more if Xenophanes had really found a home at last in the Phokaian colony.

Footnote 274:

Diog. ix. 21 (R. P. 96 a).

Footnote 275:

Diog. ix. 18 (R. P. 96). The use of the old name Zankle, instead of
the later Messene, points to an early source for this
statement—probably the elegies of Xenophanes himself.

Footnote 276:

Diog. ix. 18 (R. P. 97) says αὐτὸς ἐρραψῴδει τὰ ἑαυτοῦ, which is a
very different thing. Nothing is said anywhere of his reciting Homer,
and the word ῥαψῳδεῖν is used quite loosely for “to recite.” Gomperz’s
imaginative picture (_Greek Thinkers_, vol. i. p. 155) has no further
support than this single word. Nor is there any trace of Homeric
influence in the fragments. They are in the usual elegiac style.

Footnote 277:

The statement is justly suspected by Hiller (_Rh. Mus._ xxxiii. p.
529) to come from Lobon of Argos, who provided the Seven Wise Men,
Epimenides, etc., with stichometric notices, all duly recorded in
Diogenes. Even if true, however, it proves nothing.

Footnote 278:

Arist. _Rhet._ Β, 26. 1400 b 5 (R. P. 98 a). Anecdotes like this are
really anonymous. Plutarch transfers the story to Egypt (_P. Ph. Fr._
p. 22, § 13), and others tell it of Herakleitos. It is hardly safe to
build on such a foundation.

[Sidenote: Poems.]

56. According to a notice preserved in Diogenes, Xenophanes wrote in hexameters and also composed elegies and iambics against Homer and Hesiod.[279] No good authority says anything about his having written a philosophical poem.[280] Simplicius tells us he had never met with the verses about the earth stretching infinitely downwards (fr. 28),[281] and this means that the Academy possessed no copy of such a poem, which would be very strange if it had ever existed. Simplicius was able to find the complete works of much smaller men. Nor does internal evidence lend any support to the view that he wrote a philosophical poem. Diels refers about twenty-eight lines to it, but they would all come in quite as naturally in his attacks on Homer and Hesiod, as I have endeavoured to show. It is also significant that a considerable number of them are derived from commentators on Homer.[282] It seems probable, then, that Xenophanes expressed his theological and philosophical views incidentally in his satires. That would be quite in the manner of the time, as we can see from the remains of Epicharmos.

Footnote 279:

Diog. ix. 18 (R. P. 97). The word ἐπικόπτων is a reminiscence of
Timon, fr. 60; Diels, Ξεινοφάνης ὑπάτυφος Ὁμηραπάτης ἐπικόπτης.

Footnote 280:

The oldest reference to a poem Περὶ φύσεως is in the Geneva scholium
on _Il._ xxi. 196 (quoting fr. 30), and this goes back to Krates of
Mallos. We must remember, however, that such titles are of later date
than Xenophanes, and he had been given a place among philosophers long
before the time of Krates. All we can say, therefore, is that the
Pergamene librarians gave the title Περὶ φύσεως to some poem of
Xenophanes.

Footnote 281:

Simpl. _de Caelo_, p. 522, 7 (R. P. 97 b). It is true that two of our
fragments (25 and 26) are preserved by Simplicius, but he got them
from Alexander. Probably they were quoted by Theophrastos; for it is
plain that Alexander had no first-hand knowledge of Xenophanes either.
If he had, he would not have been taken in by _M.X.G._ (See p. 138,
_n._ 305.)

Footnote 282:

Three fragments (27, 31, 33) come from the _Homeric Allegories_, two
(30, 32) are from Homeric scholia.

The satires themselves are called _Silloi_ by late writers, and this name may go back to Xenophanes himself. It is also possible, however, that it originates in the fact that Timon of Phleious, the “sillographer” (_c._ 259 B.C.), put much of his satire upon philosophers into the mouth of Xenophanes. Only one iambic line has been preserved, and that is immediately followed by a hexameter (fr. 14 = 5 Karst.). This suggests that Xenophanes inserted iambic lines among his hexameters in the manner of the _Margites_, which would be a very natural thing for him to do.[283]

Footnote 283:

Cf. Wilamowitz, Progr. Gryphiswald. 1880.

[Sidenote: The fragments.]

57. I give all the fragments of any importance according to the text and arrangement of Diels.

ELEGIES

(1)

Now is the floor clean, and the hands and cups of all; one sets
twisted garlands on our heads, another hands us fragrant ointment on a
salver. The mixing bowls stand ready, full of gladness, and there is
more wine at hand that promises never to leave us in the lurch, soft
and smelling of flowers in the jars. In the midst the frankincense
sends up its holy smoke, and there is cold water, sweet and clean.
Brown loaves are set before us and a lordly table laden with cheese
and rich honey. The altar in the midst is clustered round with
flowers; song and revel fill the halls.

But first it is meet that men should hymn the god with joyful song,
with holy tales and pure words; then after libation and prayer made
that we may have strength to do right—for that is in truth the better
way—no sin is it to drink as much as a man can take and get home
without an attendant, so he be not stricken in years. And above all
men is he to be praised who after drinking gives goodly proof of
himself in the trial of skill, as memory and voice will serve him. Let
him not sing of Titans and Giants—those fictions of the men of old—nor
of turbulent civil broils in which is no good thing at all; but ever
give heedful reverence to the gods.

(2)

What if a man win victory in swiftness of foot, or in the
_pentathlon_, at Olympia, where is the precinct of Zeus by Pisa’s
springs, or in wrestling,—what if by cruel boxing or that fearful
sport men call _pankration_ he become more glorious in the citizens’
eyes, and win a place of honour in the sight of all at the games, his
food at the public cost from the State, and a gift to be an heirloom
for him,—what if he conquer in the chariot-race,—he will not deserve
all this for his portion so much as I do. Far better is our art than
the strength of men and of horses! These are but thoughtless
judgments, nor is it fitting to set strength before our art. Even if
there arise a mighty boxer among a people, or one great in the
_pentathlon_ or at wrestling, or one excelling in swiftness of
foot—and that stands in honour before all tasks of men at the
games—the city would be none the better governed for that. It is but
little joy a city gets of it if a man conquer at the games by Pisa’s
banks; it is not this that makes fat the store-houses of a city.

(3)

They learnt dainty and unprofitable ways from the Lydians, so long as
they were free from hateful tyranny; they went to the market-place
with cloaks of purple dye, not less than a thousand of them all told,
vainglorious and proud of their comely tresses, reeking with fragrance
from cunning salves.

SATIRES

(10)

Since all at first have learnt according to Homer....

(11)

Homer and Hesiod have ascribed to the gods all things that are a shame
and a disgrace among mortals, stealings and adulteries and deceivings
of one another. R. P. 99.

(12)

They have uttered many, many lawless deeds of the gods, stealings and
adulteries and deceivings of one another. R. P. _ib._

(14)

But mortals deem that the gods are begotten as they are, and have
clothes[284] like theirs, and voice and form. R. P. 100.

(15)

Yes, and if oxen and horses or lions had hands, and could paint with
their hands, and produce works of art as men do, horses would paint
the forms of the gods like horses, and oxen like oxen, and make their
bodies in the image of their several kinds. R. P. _ib._

(16)

The Ethiopians make their gods black and snub-nosed; the Thracians say
theirs have blue eyes and red hair. R. P. 100 b.

(18)

The gods have not revealed all things to men from the beginning, but
by seeking they find in time what is better. R. P. 104 b.

(23)

One god, the greatest among gods and men, neither in form like unto
mortals nor in thought.... R. P. 100.

(24)

He sees all over, thinks all over, and hears all over. R. P. 102.

(25)

But without toil he swayeth all things by the thought of his mind. R.
P. 108 b.

(26)

And he abideth ever in the selfsame place, moving not at all; nor doth
it befit him to go about now hither now thither. R. P. 110 a.

(27)

All things come from the earth, and in earth all things end. R. P. 103
a.

(28)

This limit of the earth above is seen at our feet in contact with the
air;[285] below it reaches down without a limit. R. P. 103.

(29)

All things are earth and water that come into being and grow. R. P.
103.

(30)

The sea is the source of water and the source of wind; for neither in
the clouds (would there be any blasts of wind blowing forth) from
within without the mighty sea, nor rivers’ streams nor rain-water from
the sky. The mighty sea is father of clouds and of winds and of
rivers.[286] R. P. 103.

(31)

The sun swinging over[287] the earth and warming it....

(32)

She that they call Iris is a cloud likewise, purple, scarlet and green
to behold. R. P. 103.

(33)

For we all are born of earth and water. R. P. _ib._

(34)

There never was nor will be a man who has certain knowledge about the
gods and about all the things I speak of. Even if he should chance to
say the complete truth, yet he himself knows not that it is so. But
all may have their fancy. R. P. 104.

(35)

Let these be taken as fancies[288] something like the truth. R. P. 104
a.

(36)

All of them[289] that are visible for mortals to behold.

(37)

And in some caves water drips....

(38)

If god had not made brown honey, men would think figs far sweeter than
they do.

Footnote 284:

I formerly, with Zeller, preferred Theodoret’s reading αἴσθησιν, but
both Clement and Eusebios have ἐσθῆτα, and Theodoret is entirely
dependent on them.

Footnote 285:

Reading ἠέρι for καὶ ῥεῖ with Diels.

Footnote 286:

This fragment has been recovered in its entirety from the Geneva
scholia on Homer (see _Arch._ iv. p. 652). The words in brackets are
added by Diels. See also Praechter, “Zu Xenophanes” (_Philol._ xviii.
p. 308).

Footnote 287:

The word is ὑπεριέμενος. This is quoted from the _Allegories_ as an
explanation of the name Hyperion, and doubtless Xenophanes so meant
it.

Footnote 288:

Reading δεδοξάσθω with Wilamowitz.

Footnote 289:

As Diels suggests, this probably refers to the stars, which Xenophanes
held to be clouds.

[Sidenote: The heavenly bodies.]

58. The intention of one of these fragments (fr. 32) is perfectly clear. “Iris too” is a cloud, and we may infer that the same thing had just been said of the sun, moon, and stars; for the doxographers tell us that these were all explained as “clouds ignited by motion.”[290] To the same context clearly belongs the explanation of the St. Elmo’s fire which Aetios has preserved. “The things like stars which appear on ships,” we are told, “which some call the Dioskouroi, are little clouds made luminous by motion.”[291] In the doxographers this explanation is repeated with trifling variations under the head of moon, stars, comets, lightning, shooting stars, and so forth, which gives the appearance of a systematic cosmology.[292] But the system is due to the arrangement of the work of Theophrastos, and not to Xenophanes; for it is obvious that a very few hexameters added to those we possess would amply account for the whole doxography.

Footnote 290:

Cf. Diels _ad loc._ (_P. Ph. Fr._ p. 44), “ut Sol et cetera astra,
quae cum in nebulas evanescerent, deorum simul opinio casura erat.”
Cf. _Arch._ x. p. 533.

Footnote 291:

Aet. ii. 18, 1 (_Dox._ p. 347), Ξενοφάνης τοὺς ἐπὶ τῶν πλοίων
φαινομένους οἷον ἀστέρας, οὓς καὶ Διοσκούρους καλοῦσί τινες, νεφέλια
εἶναι κατὰ τὴν ποιὰν κίνησιν παραλάμποντα.

What we hear of the sun presents some difficulties. We are told, on the one hand, that it too was an ignited cloud; but this can hardly be right. The evaporation of the sea from which clouds arise is distinctly said to be due to the sun’s heat. Theophrastos stated that the sun, according to Xenophanes, was a collection of sparks from the moist exhalation; but even this leaves the exhalation itself unexplained.[293] That, however, matters little, if the chief aim of Xenophanes was to discredit the anthropomorphic gods, rather than to give a scientific theory of the heavenly bodies. The important thing is that Helios too is a temporary phenomenon. The sun does not go round the earth, as Anaximander taught, but straight on, and the appearance of a circular path is solely due to its increasing distance. So it is not the same sun that rises next morning, but a new one altogether; while the old one “tumbles into a hole” when it comes to certain uninhabited regions of the earth. Besides that, there are many suns and moons, one of each for every region of the earth.[294] It is obvious that things of that kind cannot be gods.

Footnote 292:

The passages from Aetios are collected in _P. Ph. Fr._ pp. 32 sqq.
(_Vors._ p. 42).

Footnote 293:

Aet. ii. 20, 3 (_Dox._ p. 348), Ξενοφάνης ἐκ νεφῶν πεπυρωμένων εἶναι
τὸν ἥλιον. Θεόφραστος ἐν τοῖς Φυσικοῖς γέγραφεν ἐκ πυριδίων μὲν τῶν
συναθροιζομένων ἐκ τῆς ὑγρᾶς ἀναθυμιάσεως, συναθροιζόντων δὲ τὸν
ἥλιον.

Footnote 294:

Aet. ii. 24, 9 (_Dox._ p. 355). πολλοὺς εἶναι ἡλίους καὶ σελήνας κατὰ
κλίματα τῆς γῆς καὶ ἀποτομὰς καὶ ζώνας, κατὰ δέ τινα καιρὸν ἐμπίπτειν
τὸν δίσκον εἴς τινα ἀποτομὴν τῆς γῆς οὐκ οἰκουμένην ὑφ’ ἡμῶν καὶ οὕτως
ὥσπερ κενεμβατοῦντα ἔκλειψιν ὑποφαίνειν· ὁ δ’ αὐτὸς τὸν ἥλιον εἰς
ἄπειρον μὲν προιέναι, δοκεῖν δὲ κυκλεῖσθαι διὰ τὴν ἀπόστασιν. It is
clear that in this notice ἔκλειψινἕκλειψιν has been erroneously
substituted for δύσιν, as it has also in Aet. ii. 24, 4 (_Dox._ p.
354).

The vigorous expression “tumbling into a hole”[295] seems clearly to come from the verses of Xenophanes himself, and there are others of a similar kind, which we must suppose were quoted by Theophrastos. The stars go out in the daytime, but glow again at night “like charcoal embers.”[296] The sun is of some use in producing the world and the living creatures in it, but the moon “does no work in the boat.”[297] Such expressions can only be meant to make the heavenly bodies appear ridiculous, and it will therefore be well to ask whether the other supposed cosmological fragments can be interpreted on the same principle.

Footnote 295:

That this is the meaning of ὥσπερ κενεμβατοῦντα appears sufficiently
from the passages referred to in Liddell and Scott.

Footnote 296:

Aet. ii. 13, 14 (_Dox._ p. 343), ἀναζωπυρεῖν νύκτωρ καθάπερ τοὺς
ἄνθρακας.

Footnote 297:

Aet. ii. 30, 8 (_Dox._ p. 362), τὸν μὲν ἥλιον χρήσιμον εἶναι πρὸς τὴν
τοῦ κόσμου καὶ τὴν τῶν ἐν αὐτῷ ζῴων γένεσίν τε καὶ διοίκησιν, τὴν δὲ
σελήνην παρέλκειν, The verb παρέλκειν means “to cork.” Cf.
Aristophanes, _Pax_, 1306.

[Sidenote: Earth and water.]

59. In fr. 29 Xenophanes says that “all things are earth and water,” and Hippolytos has preserved the account given by Theophrastos of the context in which this occurred. It was as follows:—

Xenophanes said that a mixture of the earth with the sea is taking
place, and that it is being gradually dissolved by the moisture. He
says that he has the following proofs of this. Shells are found in
midland districts and on hills, and he says that in the quarries at
Syracuse has been found the imprint of a fish and of seaweed, at Paros
the form of an anchovy in the depth of the stone, and at Malta flat
impressions of all marine animals. These, he says, were produced when
all things were formerly mud, and the outlines were dried in the mud.
All human beings are destroyed when the earth has been carried down
into the sea and turned to mud. This change takes place for all the
worlds.—Hipp. _Ref._ i. 14 (R. P. 103 a).