This text is an excerpt from a reprint of the 3rd edition of John Burnet's famous study of Presocratic philosophy, Early Greek Philosophy, originally published in 1920. The spelling has been modernized throughout. To make the text compatible with Internet search engines, Greek names have been Latinized to their common forms and Greek characters have been transliterated to Latin.

Pythagoras of Samos

John Burnet

37. Character of the Tradition

It is not easy to give any account of Pythagoras that can claim to be regarded as historical. The earliest reference to him, indeed, is practically a contemporary one. Some verses are quoted from Xenophanes, in which we are told that Pythagoras once heard a dog howling and appealed to its master not to beat it, as he recognized the voice of a departed friend. From this we know that he taught the doctrine of transmigration. Heraclitus, in the next generation, speaks of his having carried scientific investigation (historię) further than any one, though he made use of it for purposes of imposture. Later, though still within the century, Herodotus speaks of him as "not the weakest scientific man (sophistęs) among the Hellenes," and he says he had been told by the Greeks of the Hellespont that the legendary Scythian Salmoxis had been a slave of Pythagoras at Samos. He does not believe that; for he knew Salmoxis lived many years before Pythagoras. The story, however, is evidence that Pythagoras was well known in the fifth century, both as a scientific man and as a preacher of immortality. That takes us some way.

Plato was deeply interested in Pythagoreanism, but he is curiously reserved about Pythagoras. He only mentions him once by name in all his writings, and all we are told then is that he won the affections of his followers in an unusual degree (diapherontôs ęgapęthę) by teaching them a "way of life," which was still called Pythagorean. Even the Pythagoreans are only once mentioned by name, in the passage where Socrates is made to say that they regard music and astronomy as sister sciences. On the other hand, Plato tells us a good deal about men whom we know from other sources to have been Pythagoreans, but he avoids the name. For all he says, we should only have been able to guess that Echecrates and Philolaus belonged to the school. Usually Pythagorean views are given anonymously, as those of "ingenious persons" (kompsoi tives) or the like, and we are not even told expressly that Timaeus the Locrian, into whose mouth Plato has placed an unmistakably Pythagorean cosmology, belonged to the society. We are left to infer it from the fact that he comes from Italy. Aristotle imitates his master's reserve in this matter. The name of Pythagoras occurs only twice in the genuine works that have come down to us. In one place we are told that Alcmaeon was a young man in the old age of Pythagoras, and the other is a quotation from Alcidamas to the effect that "the men of Italy honored Pythagoras." Aristotle is not so shy of the word "Pythagorean" as Plato, but he uses it in a curious way. He says such things as "the men of Italy who are called Pythagoreans," and he usually refers to particular doctrines as those of "some of the Pythagoreans." It looks as if there was some doubt in the fourth century as to who the genuine Pythagoreans were. We shall see why as we go on.

Aristotle also wrote a special treatise on the Pythagoreans which has not come down to us, but from which quotations are found in later writers. These are of great value, as they have to do with the religious side of Pythagoreanism.

The only other ancient authorities on Pythagoras were Aristoxenus of Taras, Dicaearchus of Messene, and Timaeus of Tauromenium, who all had special opportunities of knowing something about him. The account of the Pythagorean Order in the Life of Pythagoras by Iamblichus is based mainly on Timaeus, who was no doubt an uncritical historian, but who had access to information about Italy and Sicily which makes his testimony very valuable when it can be recovered. Aristoxenus had been personally acquainted with the last generation of the Pythagorean society at Phlius. It is evident, however, that he wished to represent Pythagoras simply as a man of science, and was anxious to refute the idea that he was a religious teacher. In the same way, Dicaearchus tried to make out that Pythagoras was simply a statesman and reformer.

When we come to the Lives of Pythagoras, by Porphyry, Iamblichus, and Diogenes Laertius, we find ourselves once more in the region of the miraculous. They are based on authorities of a very suspicious character, and the result is a mass of incredible fiction. It would be quite wrong, however, to ignore the miraculous elements in the legend of Pythagoras; for some of the most striking miracles are quoted from Aristotle's work on the Pythagoreans and from the Tripod of Andron of Ephesus, both of which belong to the fourth century B.C., and cannot have been influenced by Neopythagorean fancies. The fact is that the oldest and the latest accounts agree in representing Pythagoras as a wonder-worker; but, for some reason, an attempt was made in the fourth century to save his memory from that imputation. This helps to account for the cautious references of Plato and Aristotle, but its full significance will only appear later.

38. Life of Pythagoras

We may be said to know for certain that Pythagoras passed his early manhood at Samos, and was the son of Mnesarchus; and he "flourished," we are told, in the reign of Polycrates (532 B.C.). This date cannot be far wrong; for Heraclitus already speaks of him in the past tense.

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 Polycrates of Samos and Amasis, rests on no sufficient authority. Herodotus, 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; but this does not imply that the Pythagoreans derived these directly from Egypt. He says also 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. Nor does it matter; for the Egyptians did not believe in transmigration at all, and Herodotus was deceived by the priests or the symbolism of the monuments.

Aristoxenus said that Pythagoras left Samos in order to escape from the tyranny of Polycrates. It was at Croton, a city which had long been in friendly relations with Samos and was famed for its athletes and its doctors, that he founded his society. Timaeus appears to have said that he came to Italy in 529 B.C. and remained at Croton for twenty years. He died at Metapontum, whither he had retired when the Crotoniates rose in revolt against his authority.

39. The Order

The Pythagorean Order was simply, in its origin, a religious fraternity, and not, as has been maintained, a political league. Nor had it anything whatever to do with the "Dorian aristocratic ideal." Pythagoras was an Ionian, and the Order was originally confined to Achaean states. Moreover the "Dorian aristocratic ideal" is a fiction based on the Socratic idealization of Sparta and Crete. Corinth, Argos, and Syracuse are quite forgotten. Nor is there any evidence that the Pythagoreans favored the aristocratic party. The main purpose of the Order was the cultivation of holiness. In this respect it resembled an Orphic society, though Apollo, and not Dionysus, was the chief Pythagorean god. That is doubtless due to the connection of Pythagoras with Delos, and explains why the Crotoniates identified him with Apollo Hyperboreus.

40. Downfall of the Order

For a time the new Order succeeded in securing supreme power in the Achaean cities, but reaction soon came. Our accounts of these events are much confused by failure to distinguish between the revolt of Cylon in the lifetime of Pythagoras himself, and the later risings which led to the expulsion of the Pythagoreans from Italy. It is only if we keep these apart that we begin to see our way. Timaeus appears to have connected the rising of Cylon closely with the events which led to the destruction of Sybaris (510 B.C.). We gather that in some way Pythagoras had shown sympathy with the Sybarites, and had urged the people of Croton to receive certain refugees who had been expelled by the tyrant Telys. There is no ground for the assertion that he sympathized with these refugees because they were "aristocrats"; they were victims of a tyrant and suppliants, and it is not hard to understand that the Ionian Pythagoras should have felt a certain kindness for the men of the great but unfortunate Ionian city. Cylon, who is expressly stated by Aristoxenus to have been one of the first men of Croton in wealth and birth, was able to bring about the retirement of Pythagoras to Metapontum, another Achaean city, and it was there that he passed his remaining years.

Disturbances still went on, however, at Croton after the departure of Pythagoras for Metapontum and after his death. At last, we are told, the Cyloneans set fire to the house of the athlete Milo, where the Pythagoreans were assembled. Of those in the house only two, who were young and strong, Archippus and Lysis, escaped. Archippus retired to Taras, a democratic Dorian state: Lysis, first to Achaea and afterwards to Thebes, where he was later the teacher of Epaminondas. It is impossible to date these events accurately, but the mention of Lysis proves that they were spread over more than one generation. The coup d'Etat of Croton can hardly have occurred before 450 B.C., if the teacher of Epaminondas escaped from it, nor can it have been much later or we should have heard of it in connection with the foundation of Thourioi in 444 B.C. In a valuable passage, doubtless derived from Timaeus, Polybius tells us of the burning of the Pythagorean "lodges" (sunedria) in all the Achaean cities, and the way in which he speaks suggests that this went on for a considerable time, till at last peace and order were restored by the Achaeans of Peloponnesus. We shall see that at a later date some of the Pythagoreans were able to return to Italy, and once more acquired great influence there.

41. Want of Evidence as to the Teaching of Pythagoras

Of the opinions of Pythagoras we know even less than of his life. Plato and Aristotle clearly knew nothing for certain of ethical or physical doctrines going back to the founder himself. Aristoxenus gave a string of moral precepts. Dicaearchus said 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. Pythagoras apparently preferred oral instruction to the dissemination of his opinions by writing, and it was not till Alexandrian times that anyone ventured to forge books in his name. The writings ascribed to the first Pythagoreans were also forgeries of the same period. 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.

42. Transmigration

In the first place, as we have seen, he taught the doctrine of transmigration. Now this is most easily to be explained as a development of the primitive belief in the kinship of men and beasts, a view which Dicaearchus said Pythagoras held. Further, 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. It seems certain that Pythagoras brought this with him from Ionia. Timaeus told how at Delos he refused to sacrifice on any but the oldest altar, that of Apollo the Father, where only bloodless sacrifices were allowed.

43. Abstinence

It has indeed been doubted whether we can accept what we are told by such late writers as Porphyry on the subject of Pythagorean abstinence. Aristoxenus undoubtedly said Pythagoras did not abstain from animal flesh in general, but only from that of the ploughing ox and the ram. 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. The palpable exaggeration of these statements shows, however, that he is endeavoring to combat a belief which existed in his own day, so we can 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 the Neopythagoreans. The explanation is that Aristoxenus had been the friend of the last of the Pythagoreans; and, in their time, the strict observance had been relaxed, except by some zealots whom the heads of the Society refused to acknowledge. The "Pythagorists" who clung to the old practices were now regarded as heretics, and it was said that the Akousmatics, as they were called, were really followers of Hippasus, who had been excommunicated for revealing secret doctrines. The genuine followers of Pythagoras were the Mathematicians. The satire of the poets of the Middle Comedy proves, however, that, even though the friends of Aristoxenus did not practice abstinence, there were plenty of people in the fourth century, calling themselves followers of Pythagoras, who did. We know also from Isocrates that they still observed the rule of silence. History has not been kind to the Akousmatics, but they never wholly died out. The names of Diodorus of Aspendus and Nigidius Figulus help to bridge the gulf between them and Apollonius of Tyana.

We have seen that Pythagoras taught the kinship of beasts and men, and we infer that his rule of abstinence from flesh was based, not on humanitarian or ascetic grounds, but on taboo. This is strikingly confirmed by a statement in Porphyry's Defence of Abstinence, to the effect that, though the Pythagoreans did as a rule abstain from flesh, they nevertheless ate it when they sacrificed to the gods. Now, among primitive peoples, we often find that the sacred animal is slain and eaten on certain solemn occasions, though in ordinary circumstances this would be the greatest of all impieties. Here, again, we have a primitive belief; and we need not attach any weight to the denials of Aristoxenus.

44. Akousmata

We shall now know what to think of the Pythagorean rules and precepts that have come down to us. These are of two kinds, and have different sources. Some of them, derived from Aristoxenus, and for the most part preserved by Iamblichus, 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. The second class is of a different nature, and consists of rules called Akousmata, which points to their being the property of the sect which had faithfully preserved the old customs. Later writers interpret them as "symbols" of moral truth; but it does not require a practiced eye to see that they are genuine taboos. I give a few examples to show what the Pythagorean rule was really like.

It would be easy to multiply proofs of the close connection between Pythagoreanism and primitive modes of thought, but what has been said is sufficient for our purpose.

45. Pythagoras as a Man of Science

Now, were this all, we should be tempted to delete the name of Pythagoras from the history of philosophy, and relegate him to the class of "medicine-men" (goętes) along with Epimenides and Onomacritus. That, however, would be quite wrong. The Pythagorean Society became the chief scientific school of Greece, and it is certain that Pythagorean science goes back to the early years of the fifth century, and therefore to the founder. Heraclitus, who is not partial to him, says that Pythagoras had pursued scientific investigation further than other men. Herodotus called Pythagoras "by no means the weakest sophist of the Hellenes," a title which at this date does not imply the slightest disparagement, but does imply scientific studies. Aristotle said that Pythagoras at first busied himself with mathematics and numbers, though he adds that later he did not renounce the miracle-mongering of Pherecydes. Can we trace any connection 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" of a primitive type. The new thing in the society founded by Pythagoras seems to have been that, while it admitted all these old practices, it at the same time suggested a deeper idea of what "purification" really is. Aristoxenus said that the Pythagoreans employed music to purge the soul as they used medicine to purge the body. 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 Heraclides, 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 doctrine 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. 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 to look on (theorein). The greatest purification of all is, therefore, 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. It is easy to understand that most of his followers would rest content with the humbler kinds of purification, and this will account for the sect of the Akousmatics. A few would rise to the higher doctrine, and we have now to ask how much of the later Pythagorean science may be ascribed to Pythagoras himself.

46. Arithmetic

In his treatise on Arithmetic, Aristoxenus said that Pythagoras was the first to carry that study beyond the needs of commerce, 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 pretty sure 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 place of Pythagoreanism in Greek thought can be made clear, though we must remember that no one can now pretend to draw the line between its successive stages with any certainty.

47. The Figures

One of the most remarkable statements we have about Pythagoreanism is what we are told of Eurytus on the unimpeachable authority of Archytas. Eurytus was the disciple of Philolaus, and Aristoxenus mentioned him along with Philolaus as having taught the last of the Pythagoreans, the men with whom he himself was 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. 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. Moreover, Aristotle compares his procedure to that of those who bring numbers into figures (schęmata) like the triangle and the square.

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, because the zero was not yet invented. The representation of numbers by lines was adopted to avoid the difficulties raised by the discovery of irrational quantities, and is of much later date. It seems rather that numbers were originally represented by dots arranged in symmetrical and easily recognized 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.

It is, therefore, significant that we do not find any clue to what Aristotle 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 inadequate to represent the true nature of number. Nicomachus of Gerasa says, expressly that the letters used to represent numbers are purely conventional. The natural thing 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. We therefore find figures like this:


Now it ought to be obvious that this is no innovation. 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. We also gather that the dots were supposed to represent pebbles (psęphoi), and this throws light on early methods of what we still call calculation.

48. Triangular, Square and Oblong Numbers

That Aristotle refers to this seems clear, and is confirmed by the tradition that the great revelation made by Pythagoras to mankind was precisely a figure of this kind, the tetraktys, by which the Pythagoreans used to swear, and we have the authority of Speusippus for holding that the whole theory was Pythagorean. In later days there were many kinds of tetraktys, 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.

It showed at a glance that 1 + 2 + 3 + 4 = 10. Speusippus 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 integers 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 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, as shown by the figure:


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.

49. Geometry and Harmonics

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" (horoi, termini, "terms"), and the area they mark out is the "field" (chôra). This is evidently an early way of speaking, and may be referred to Pythagoras himself. Now it must have struck him that "fields" could be compared as well as numbers, and it is likely that he knew the rough methods of doing this traditional in Egypt, though certainly these would fail to satisfy him. Once more the tradition is helpful in suggesting the direction his thoughts must have taken. He knew, of course, the use of the triangle 3, 4, 5 in constructing right angles. We have seen (§ XI) 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" (hupoteinousa) affords strong confirmation of the intimate connection between the two things. It means literally "the cord stretching over against," and this is surely just the rope of the "arpedonapt." It is, therefore, quite possible that this proposition was really discovered by Pythagoras, though we cannot be sure of that, and though the demonstration of it which Euclid gives is certainly not his.

50. Incommensurability

One great disappointment, however, awaited him. 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 arithmetical 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 may be true that Pythagoras discovered the incommensurability of the diagonal and the side of a square, and the proof mentioned by Aristotle, namely, that, iff they were commensurable, we should have to say that an even number was equal to an odd number, is distinctly Pythagorean in character. However that may be, it is certain that Pythagoras did not care to pursue the subject any further. He may have stumbled on the fact that the square root of two is a surd, but we know that it was left for Plato's friends, Theodorus of Cyrene and Theaetetus, to give a complete theory of irrationals. For the present, the incommensurability of the diagonal and the square remained, as has been said, a "scandalous exception." Our tradition says that Hippasus of Metapontum was drowned at sea for revealing this skeleton in the cupboard.

51. Proportion and Harmony

These last considerations show that, while it is quite safe to attribute the substance of the early books of Euclid to the early Pythagoreans, his arithmetical method is certainly not theirs. It operates with lines instead of with units, and it can therefore be applied to relations which are not capable of being expressed as equations between rational numbers. 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 Eudoxus. Yet it is clear that the early Pythagoreans, and probably Pythagoras himself, studied proportion in their own way, and that the three "medieties" (mesotętes) 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, we find that 12:6 is the octave, 12:8 the fifth, and 8:6 the fourth, and it can hardly be doubted that Pythagoras himself discovered these intervals. The stories about his observing the harmonic intervals in a smithy, and then weighing the hammers that produced them, or suspending weights corresponding to those of the hammers to equal strings, are, indeed, impossible and absurd; but it is sheer waste of time to rationalize them. For our purpose their absurdity is their chief merit. They are not stories which any Greek mathematician could possibly have invented, but popular tales bearing witness to the existence of a real tradition that Pythagoras was the author of this momentous discovery. On the other hand, the statement that he discovered the "consonances" by measuring the lengths corresponding to them on the monochord is quite credible and involves no error in acoustics.

52. Things Are Numbers

It was this, 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," taken in connection with what we are told about the method of Eurytus, 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 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 trading 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, 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 tradition. Aristotle does imply, however, that according to them the "right time" (kairos) 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 too much importance to them. 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.

53. Cosmology

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. In substance, that is the doctrine of Anaximenes, and it becomes practically certain that it was taught by Pythagoras, when we find that Xenophanes denied it. We may infer that the further development of the idea is also due to Pythagoras. 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; and that it is the Boundless thus inhaled that keeps the units separate from each other. It represents the interval between them. This is a primitive way of describing discrete quantity.

In these passages of Aristotle, the "breath" is also spoken of as the void or empty. This is a confusion we have already met with in Anaximenes, and it need not surprise us to find it here. We find also clear traces of the other confusion, that of air and vapor. It seems certain, in fact, that Pythagoras identified the Limit with fire, and the Boundless with darkness. We are told by Aristotle that Hippasus made Fire the first principle, 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. We also find that Light and Darkness appear in the Pythagorean table of opposites under the heads of the Limit and the Unlimited respectively. 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 vapor, while in the fifth its true nature was known. Plato, with his usual historical tact, makes the Pythagorean Timaeus describe mist and darkness as condensed air. We must think, then, of a "field" of darkness or breath marked out by luminous units, an imagination 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.

54. The Heavenly Bodies

Anaximander had regarded the heavenly bodies as wheels of "air" filled with fire which escapes through certain orifices (§ 21), and there is evidence that Pythagoras adopted the same view. We have seen that Anaximander only assumed the existence of three such wheels, and it is extremely probable that Pythagoras identified the intervals between these with the three musical intervals he had discovered, the fourth, the fifth, and the octave. That would be the most natural beginning for the doctrine of the "harmony of the spheres," though the expression would be doubly misleading if applied to any theory we can properly ascribe to Pythagoras himself. The word harmonia does not mean harmony, but octave, and the "spheres" are an anachronism. We are still at the stage when wheels or rings were considered sufficient to account for the heavenly bodies.

The distinction between the diurnal revolution of the heavens from east to west, and the slower revolutions of the sun, moon, and planets from west to east, may also be referred to the early days of the school, and probably to Pythagoras himself. It obviously involves a complete break with the theory of a vortex, and suggests that the heavens are spherical. That, however, was the only way to get out of the difficulties of Anaximander's system. If it is to be taken seriously, we must suppose that the motions of the sun, moon, and planets are composite. On the one hand, they have their own revolutions with varying angular velocities from west to east, but they are also carried along by the diurnal revolution from east to west. Apparently this was expressed by saying that the motions of the planetary orbits, which are oblique to the celestial equator, are mastered (krateitai) by the diurnal revolution. The Ionians, down to the time of Democritus, never accepted this view. They clung to the theory of the vortex, which made it necessary to hold that all the heavenly bodies revolved in the same direction, so that those which, on the Pythagorean system, have the greatest angular velocity have the least on theirs. On the Pythagorean view, Saturn, for instance, takes about thirty years to complete its revolution; on the Ionian view it is "left behind" far less than any other planet, that is, it more nearly keeps pace with the signs of the Zodiac.

For reasons which will appear later, we may confidently attribute to Pythagoras himself the discovery of the sphericity of the earth, which the Ionians, even Anaxagoras and Democritus, refused to accept. It is probable, however, that he still adhered to the geocentric system, and that the discovery that the earth was a planet belongs to a later generation (§ 150).

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. It is clear at any rate that the great contribution of Pythagoras to science was his discovery that the concordant intervals could be expressed by simple numerical ratios. In principle, at least, that suggests an entirely new view of the relation between the traditional "opposites." If a perfect attunement (harmonia) of the high and the low can be attained by observing these ratios, it is clear that other opposites may be similarly harmonized. The hot and the cold, the wet and the dry, may be united in a just blend (krasis), an idea to which our word "temperature" still bears witness. The medical doctrine of the "temperaments" is derived from the same source. Moreover, the famous doctrine of the Mean is only an application of the same idea to the problem of conduct. It is not too much to say that Greek philosophy was henceforward to be dominated by the notion of the perfectly tuned string.