I. Pythagoras of Samos

Character of the tradition.

37. It is no easy task to give an account of Pythagoras that can claim to be regarded as history. Our principal sources of information[[185]] are the Lives composed by Iamblichos, Porphyry, and Laertios Diogenes. That of Iamblichos is a wretched compilation, based chiefly on the work of the arithmetician Nikomachos of Gerasa in Judaea, and the romance of Apollonios of Tyana, who regarded himself as a second Pythagoras, and accordingly took great liberties with his materials.[[186]] Porphyry stands, as a writer, on a far higher level than Iamblichos; but his authorities do not inspire us with more confidence. He, too, made use of Nikomachos, and of a certain novelist called Antonius Diogenes, author of a work entitled Marvels from beyond Thule.[[187]] Diogenes quotes, as usual, a considerable number of authorities, and the statements he makes must be estimated according to the nature of the sources from which they were drawn.[[188]] So far, it must be confessed, our material does not seem promising. Further examination shows, however, that a good many fragments of two much older authorities, Aristoxenos and Dikaiarchos, are embedded in the mass. These writers were both disciples of Aristotle; they were natives of Southern Italy, and contemporary with the last generation of the Pythagorean school. Both wrote accounts of Pythagoras; and Aristoxenos, who was personally intimate with the last representatives of scientific Pythagoreanism, also made a collection of the sayings of his friends. Now the Neopythagorean story, as we have it in Iamblichos, is a tissue of incredible and fantastic myths; but, if we sift out the statements which go back to Aristoxenos and Dikaiarchos, we can easily construct a rational narrative, in which Pythagoras appears not as a miracle-monger and religious innovator, but simply as a moralist and statesman. We might then be tempted to suppose that this is the genuine tradition; but that would be altogether a mistake. There is, in fact, a third and still earlier stratum in the Lives, and this agrees with the latest accounts in representing Pythagoras as a wonder-worker and a religious reformer.

Some of the most striking miracles of Pythagoras are related on the authority of Andron’s Tripod, and of Aristotle’s work on the Pythagoreans.[[189]] Both these treatises belong to the fourth century B.C., and are therefore untouched by Neopythagorean fancies. Further, it is only by assuming the still earlier existence of this view that we can explain the allusions of Herodotos. The Hellespontine Greeks told him that Salmoxis or Zamolxis had been a slave of Pythagoras,[[190]] and Salmoxis is a figure of the same class as Abaris and Aristeas.

It seems, then, that both the oldest and the latest accounts agree in representing Pythagoras as a man of the class to which Epimenides and Onomakritos belonged—in fact, as a sort of “medicine-man”; but, for some reason, there was an attempt to save his memory from this imputation, and that attempt belonged to the fourth century B.C. The significance of this will appear in the sequel.

Life of Pythagoras.

38. We may be said to know for certain that Pythagoras passed his early manhood at Samos, and was the son of Mnesarchos;[[191]] and he “flourished,” we are told, in the reign of Polykrates.[[192]] This date cannot be far wrong; for Herakleitos already speaks of him in the past tense.[[193]]

The extensive travels attributed to Pythagoras by late writers are, of course, apocryphal. Even the statement that he visited Egypt, though far from improbable if we consider the close relations between Polykrates of Samos and Amasis, rests on no sufficient authority.[[194]] Herodotos, it is true, observes that the Egyptians agreed in certain practices with the rules called Orphic and Bacchic, which are really Egyptian, and with the Pythagoreans;[[195]] but this does not imply that the Pythagoreans derived these directly from Egypt. He says also in another place that the belief in transmigration came from Egypt, though certain Greeks, both at an earlier and a later date, had passed it off as their own. He refuses, however, to give their names, so he can hardly be referring to Pythagoras.[[196]] Nor does it matter; for the Egyptians did not believe in transmigration at all, and Herodotos was simply deceived by the priests or the symbolism of the monuments.

Aristoxenos said that Pythagoras left Samos in order to escape from the tyranny of Polykrates.[[197]] It was at Kroton, a city already famous for its medical school,[[198]] that he founded his society. How long he remained there we do not know; he died at Metapontion, whither he had retired on the first signal of revolt against his influence.[[199]]

The Order.

39. There is no reason to believe that the detailed statements which have been handed down with regard to the organisation of the Pythagorean Order rest upon any historical basis, and in the case of many of them we can still see how they came to be made. The distinction of grades within the Order, variously called Mathematicians and Akousmatics, Esoterics and Exoterics, Pythagoreans and Pythagorists,[[200]] is an invention designed to explain how there came to be two widely different sets of people, each calling themselves disciples of Pythagoras, in the fourth century B.C. So, too, the statement that the Pythagoreans were bound to inviolable secrecy, which goes back to Aristoxenos,[[201]] is intended to explain why there is no trace of the Pythagorean philosophy proper before Philolaos.

The Pythagorean Order was simply, in its origin, a religious fraternity of the type described above, and not, as has sometimes been maintained, a political league.[[202]] Nor had it anything to do with the “Dorian aristocratic ideal.” Pythagoras was an Ionian, and the Order was originally confined to Achaian states.[[203]] Nor is there the slightest evidence that the Pythagoreans favoured the aristocratic rather than the democratic party.[[204]] The main purpose of the Order was to secure for its own members a more adequate satisfaction of the religious instinct than that supplied by the State religion. It was, in fact, an institution for the cultivation of holiness. In this respect it resembled an Orphic society, though it seems that Apollo, rather than Dionysos, was the chief Pythagorean god. That is doubtless why the Krotoniates identified Pythagoras with Apollo Hyperboreios.[[205]] From the nature of the case, however, an independent society within a Greek state was apt to be brought into conflict with the larger body. The only way in which it could then assert its right to exist was by identifying the State with itself, that is, by securing the control of the sovereign power. The history of the Pythagorean Order, so far as it can be traced, is, accordingly, the history of an attempt to supersede the State; and its political action is to be explained as a mere incident of that attempt.

Downfall of the Order.

40. For a time the new Order seems actually to have succeeded in securing the supreme power, but reaction came at last. Under the leadership of Kylon, a wealthy noble, Kroton was able to assert itself victoriously against the Pythagorean domination. This, we may well believe, had been galling enough. The “rule of the saints” would be nothing to it; and we can still imagine and sympathise with the irritation felt by the plain man of those days at having his legislation done for him by a set of incomprehensible pedants, who made a point of abstaining from beans, and would not let him beat his own dog because they recognised in its howls the voice of a departed friend (Xenophanes, fr. 7). This feeling would be aggravated by the private religious worship of the Society. Greek states could never pardon the introduction of new gods. Their objection to this was not, however, that the gods in question were false gods. If they had been, it would not have mattered so much. What they could not tolerate was that any one should establish a private means of communication between himself and the unseen powers. That introduced an unknown and incalculable element into the arrangements of the State, which might very likely be hostile to those citizens who had no means of propitiating the intruding divinity.

Aristoxenos’s version of the events which led to the downfall of the Pythagorean Order is given at length by Iamblichos. According to this, Pythagoras had refused to receive Kylon into his Society, and he therefore became a bitter foe of the Order. From this cause Pythagoras removed from Kroton to Metapontion, where he died. The Pythagoreans, however, still retained possession of the government of Kroton, till at last the partisans of Kylon set fire to Milo’s house, where they were assembled. Of those in the house only two, Archippos and Lysis, escaped. Archippos retired to Taras; Lysis, first to Achaia and then to Thebes, where he became later on the teacher of Epameinondas. The Pythagoreans who remained concentrated themselves at Rhegion; but, as things went from bad to worse, they all left Italy except Archippos.[[206]]

This account has all the air of being historical. The mention of Lysis proves, however, that those events were spread over more than one generation. The coup d’état of Kroton can hardly have occurred before 450 B.C., if the teacher of Epameinondas escaped from it, and it may well have been even later. But it must have been before 410 B.C. that the Pythagoreans left Rhegion for Hellas; Philolaos was certainly at Thebes about that time.[[207]]

The political power of the Pythagoreans as an Order was now gone for ever, though we shall see that some of them returned to Italy at a later date. In exile they seem to have dropped the merely magical and superstitious parts of their system, and this enabled them to take their place as one of the scientific schools of Hellas.

Want of evidence as to the teaching of Pythagoras.

41. Of the opinions of Pythagoras we know even less than of his life. Aristotle clearly knew nothing for certain of ethical or physical doctrines going back to the founder of the Society himself.[[208]] Aristoxenos only gave a string of moral precepts.[[209]] Dikaiarchos is quoted by Porphyry as asserting that hardly anything of what Pythagoras taught his disciples was known except the doctrine of transmigration, the periodic cycle, and the kinship of all living creatures.[[210]] The fact is, that, like all teachers who introduce a new way of living rather than a new view of the world, Pythagoras preferred oral instruction to the dissemination of his opinions by writing, and it was not till Alexandrian times that any one ventured to forge books in his name. The writings ascribed to the earliest Pythagoreans were also forgeries of the same period.[[211]] The early history of Pythagoreanism is, therefore, wholly conjectural; but we may still make an attempt to understand, in a very general way, what the position of Pythagoras in the history of Greek thought must have been.

Transmigration.

42. In the first place, then, there can be no doubt that he really taught the doctrine of transmigration.[[212]] The story told by the Greeks of the Hellespont and Pontos as to his relations with Salmoxis could never have gained currency by the time of Herodotos if he had not been known as a man who taught strange views of the life after death.[[213]] Now the doctrine of transmigration is most easily to be explained as a development of the savage belief in the kinship of men and beasts, as all alike children of the Earth,[[214]] a view which Dikaiarchos said Pythagoras certainly held. Further, among savages, this belief is commonly associated with a system of taboos on certain kinds of food, and the Pythagorean rule is best known for its prescription of similar forms of abstinence. This in itself goes far to show that it originated in the same ideas, and we have seen that the revival of these would be quite natural in connexion with the foundation of a new religious society. There is a further consideration which tells strongly in the same direction. In India we have a precisely similar doctrine, and yet it is not possible to assume any actual borrowing of Indian ideas at this date. The only explanation which will account for the facts is that the two systems were independently evolved from the same primitive ideas. These are found in many parts of the world; but it seems to have been only in India and in Greece that they were developed into an elaborate doctrine.

Abstinence.

43. It has indeed been doubted whether we have a right to accept what we are told by such late writers as Porphyry on the subject of Pythagorean abstinence. Aristoxenos, whom we have admitted to be one of our earliest witnesses, may be cited to prove that the original Pythagoreans knew nothing of these restrictions on the use of animal flesh and beans. He undoubtedly said that Pythagoras did not abstain from animal flesh in general, but only from that of the ploughing ox and the ram.[[215]] He also said that Pythagoras preferred beans to every other vegetable, as being the most laxative, and that he was partial to sucking-pigs and tender kids.[[216]] Aristoxenos, however, is a witness who very often breaks down under cross-examination, and the palpable exaggeration of these statements shows that he is endeavouring to combat a belief which existed in his own day. We are therefore able to show, out of his own mouth, that the tradition which made the Pythagoreans abstain from animal flesh and beans goes back to a time long before there were any Neopythagoreans interested in upholding it. Still, it may be asked what motive Aristoxenos could have had for denying the common belief? The answer is simple and instructive. He had been the friend of the last of the Pythagoreans; and, in their time, the merely superstitious part of Pythagoreanism had been dropped, except by some zealots whom the heads of the Society refused to acknowledge. That is why he represents Pythagoras himself in so different a light from both the older and the later traditions; it is because he gives us the view of the more enlightened sect of the Order. Those who clung faithfully to the old practices were now regarded as heretics, and all manner of theories were set on foot to account for their existence. It was related, for instance, that they descended from one of the “Akousmatics,” who had never been initiated into the deeper mysteries of the “Mathematicians.”[[217]] All this, however, is pure invention. The satire of the poets of the Middle Comedy proves clearly enough that, even though the friends of Aristoxenos did not practise abstinence, there were plenty of people in the fourth century, calling themselves followers of Pythagoras, who did.[[218]] History has not been kind to the Akousmatics, but they never wholly died out. The names of Diodoros of Aspendos and Nigidius Figulus help to bridge the gulf between them and Apollonios of Tyana.

We know, then, that Pythagoras taught the kinship of beasts and men, and we infer that his rule of abstinence from flesh was based, not upon humanitarian or ascetic grounds, but on taboo. This is strikingly confirmed by a fact which we are told in Porphyry’s Defence of Abstinence. The statement in question does not indeed go back to Theophrastos, as so much of Porphyry’s tract certainly does;[[219]] but it is, in all probability, due to Herakleides of Pontos, and is to the effect that, though the Pythagoreans did as a rule abstain from flesh, they nevertheless ate it when they sacrificed to the gods.[[220]] Now, among savage peoples, we often find that the sacred animal is slain and eaten sacramentally by its kinsmen on certain solemn occasions, though in ordinary circumstances this would be the greatest of all impieties. Here, again, we have to do with a very primitive belief; and we need not therefore attach any weight to the denials of Aristoxenos.[[221]]

Akousmata.

44. We shall now know what to think of the various Pythagorean rules and precepts which have come down to us. These are of two kinds, and have very different sources. Some of them, derived from the collection of Aristoxenos, and for the most part preserved by Iamblichos, are mere precepts of morality. They do not pretend to go back to Pythagoras himself; they are only the sayings which the last generation of “Mathematicians” heard from their predecessors.[[222]] The second class is of a very different nature, and the sayings which belong to it are called Akousmata,[[223]] which points to their being the property of that sect of Pythagoreans which had faithfully preserved the old customs. Later writers interpret them as “symbols” of moral truth; but their interpretations are extremely far-fetched, and it does not require a very practised eye to see that they are genuine taboos of a thoroughly primitive type. I give a few examples in order that the reader may judge what the famous Pythagorean rule of life was really like.

It would be easy to multiply proofs of the close connexion between Pythagoreanism and primitive modes of thought, but what has been said is really sufficient for our purpose. The kinship of men and beasts, the abstinence from flesh, and the doctrine of transmigration all hang together and form a perfectly intelligible whole from the point of view which has been indicated.

Pythagoras as a man of science.

45. Were this all, we should be tempted to delete the name of Pythagoras from the history of philosophy altogether, and relegate him to the class of “medicine-men” (γόητες) along with Epimenides and Onomakritos. This, however, would be quite wrong. As we shall see, the Pythagorean Society became one of the chief scientific schools of Hellas, and it is certain that Pythagorean science as well as Pythagorean religion originated with the master himself. Herakleitos, who is not partial to him, says that Pythagoras had pursued scientific investigation further than other men, though he also says that he turned his much learning into an art of mischief.[[224]] Herodotos called Pythagoras “by no means the weakest sophist of the Hellenes,” a title which at this date does not imply the slightest disparagement.[[225]] Aristotle even said that Pythagoras first busied himself with mathematics and numbers, and that it was later on he attached himself to the miracle-mongering of Pherekydes.[[226]] Is it possible for us to trace any connexion between these two sides of his activity?

We have seen that the aim of the Orphic and other Orgia was to obtain release from the “wheel of birth” by means of “purifications,” which were generally of a very primitive type. The new thing in the Society founded by Pythagoras seems to have been that, while it admitted all these half-savage customs, it at the same time suggested a more exalted idea of what “purification” really was. Aristoxenos tells us that the Pythagoreans employed music to purge the soul as they used medicine to purge the body, and it is abundantly clear that Aristotle’s famous theory of κάθαρσις is derived from Pythagorean sources.[[227]] Such methods of purifying the soul were familiar in the Orgia of the Korybantes, and will serve to explain the Pythagorean interest in Harmonics. But there is more than this. If we can trust Herakleides so far, it was Pythagoras who first distinguished the “three lives,” the Theoretic, the Practical, and the Apolaustic, which Aristotle made use of in the Ethics. The general theory of these lives is clear, and it is impossible to doubt that in substance it belongs to the very beginning of the school. It is to this effect. We are strangers in this world, and the body is the tomb of the soul, and yet we must not seek to escape by self-murder; for we are the chattels of God who is our herdsman, and without his command we have no right to make our escape.[[228]] In this life, there are three kinds of men, just as there are three sorts of people who come to the Olympic Games. The lowest class is made up of those who come to buy and sell, and next above them are those who come to compete. Best of all, however, are those who come simply to look on (θεωρεῖν). The greatest purification of all is, therefore, disinterested science, and it is the man who devotes himself to that, the true philosopher, who has most effectually released himself from the “wheel of birth.” It would be rash to say that Pythagoras expressed himself exactly in this manner; but all these ideas are genuinely Pythagorean, and it is only in some such way that we can bridge the gulf which separates Pythagoras the man of science from Pythagoras the religious teacher.[[229]] We must now endeavour to discover how much of the later Pythagorean science may reasonably be ascribed to Pythagoras himself.

Arithmetic.

46. In his treatise on Arithmetic, Aristoxenos said that Pythagoras was the first to carry that study beyond the needs of commerce,[[230]] and his statement is confirmed by everything we otherwise know. By the end of the fifth century B.C., we find that there is a widespread interest in such subjects and that these are studied for their own sake. Now this new interest cannot have been wholly the work of a school; it must have originated with some great man, and there is no one but Pythagoras to whom we can refer it. As, however, he wrote nothing, we have no sure means of distinguishing his own teaching from that of his followers in the next generation or two. All we can safely say is that, the more primitive any Pythagorean doctrine appears, the more likely it is to be that of Pythagoras himself, and all the more so if it can be shown to have points of contact with views which we know to have been held in his own time or shortly before it. In particular, when we find the later Pythagoreans teaching things that were already something of an anachronism in their own day, we may be reasonably sure that we are dealing with survivals which only the authority of the master’s name could have preserved. Some of these must be mentioned at once, though the developed system belongs to a later part of our story. It is only by separating its earliest form from its later that the true place of Pythagoreanism in Greek thought can be made clear, though we must always remember that no one can now pretend to draw the line between its successive stages with any certainty.

The figures.

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]]

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.

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.

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.

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]]

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]]

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]]

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.

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.

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.

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]]

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]]

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.