CHAPTER IISCIENCE AND RELIGION
Migrations to the West.
32. So far we have not met with any trace of direct antagonism between science and popular beliefs, though the views of the Milesian cosmologists were really as inconsistent with the religions of the people as with the mythology of the anthropomorphic poets.[169]Two things hastened the conflict—the shifting of the scene to the West, and the religious revival which swept over Hellas in the sixth centuryB.C.
The chief figures in the philosophical history of the period were Pythagoras of Samos and Xenophanes of Kolophon. Both were Ionians by birth, and yet both spent the greater part of their lives in the West. We see from Herodotos how the Persian advance in Asia Minor occasioned a series of migrations to Sicily and Southern Italy;[170]and this, of course, made a great difference to philosophy as well as to religion. The new views had probably grown up so naturally and gradually in Ionia that the shock of conflict and reaction was avoided; but that could no longer be so, when they were transplanted to a region where men were wholly unprepared to receive them.
Another, though a somewhat later, effect of these migrations was to bring Science into contact with Rhetoric, one of the most characteristic products of Western Hellas. Already in Parmenides we may note the presence of that dialectical and controversial spirit which was destined to have so great an influence on Greek thought, and it was just this fusion of the art of arguing for victory with the search for truth that before long gave birth to Logic.
The religious revival.
33. Most important of all in its influence on philosophy was the religious revival which culminated about this time. The religion of continental Hellas had developed in a very different way from that of Ionia. In particular, the worship of Dionysos, which came from Thrace, and is barely mentioned in Homer, contained in germ a wholly new way of looking at man’s relation to the world. It would certainly be wrong to credit the Thracians themselves with any very exalted views; but there can be no doubt that, to the Greeks, the phenomenon of ecstasy suggested that the soul was something more than a feeble double of the self, and that it was only when “out of the body” it could show its true nature.[171]To a less extent, such ideas were also suggested by the worship of Demeter, whose mysteries were celebrated at Eleusis; though, in later days, these came to take the leading place in men’s minds. That was because they were incorporated in the public religion of Athens.
Before the time with which we are dealing, tradition shows us dimly an age of inspired prophets—Bakidesand Sibyls—followed by one of strange medicine-men like Abaris and Aristeas of Prokonnesos. With Epimenides of Crete, we touch the fringe of history, while Pherekydes of Syros is the contemporary of the early cosmologists, and we still have some fragments of his discourse. It looked as if Greek religion were about to enter upon the same stage as that already reached by the religions of the East; and, but for the rise of science, it is hard to see what could have checked this tendency. It is usual to say that the Greeks were saved from a religion of the Oriental type by their having no priesthood; but this is to mistake the effect for the cause. Priesthoods do not make dogmas, though they preserve them once they are made; and in the earlier stages of their development, the Oriental peoples had no priesthoods either in the sense intended.[172]It was not so much the absence of a priesthood as the existence of the scientific schools that saved Greece.
The Orphic religion.
34. The new religion—for in one sense it was new, though in another as old as mankind—reached its highest point of development with the foundation of the Orphic communities. So far as we can see, the original home of these was Attika; but they spread with extraordinary rapidity, especially in Southern Italy and Sicily.[173]They were first of all associations for the worship of Dionysos; but they were distinguished by two features which were new among the Hellenes. They looked to a revelation as the sourceof religious authority, and they were organised as artificial communities. The poems which contained their theology were ascribed to the Thracian Orpheus, who had himself descended into Hades, and was therefore a safe guide through the perils which beset the disembodied soul in the next world. We have considerable remains of this literature, but they are mostly of late date, and cannot safely be used as evidence for the beliefs of the sixth century. We do know, however, that the leading ideas of Orphicism were quite early. A number of thin gold plates with Orphic verses inscribed on them have been discovered in Southern Italy;[174]and though these are somewhat later in date than the period with which we are dealing, they belong to the time when Orphicism was a living creed and not a fantastic revival. What can be made out from them as to the doctrine has a startling resemblance to the beliefs which were prevalent in India about the same time, though it seems impossible that there should have been any actual contact between India and Greece at this date. The main purpose of theOrgia[175]was to “purify” the believer’s soul, and so enable it to escape from the “wheel of birth,” and it was for the better attainment of this end that the Orphics were organised in communities. Religious associations must have been known to the Greeks from a fairly early date;[176]but the oldest ofthese were based, at least in theory, on the tie of kindred blood. What was new was the institution of communities to which any one might be admitted by initiation.[177]This was, in fact, the establishment of churches, though there is no evidence that these were connected with each other in such a way that we could rightly speak of them as a single church. The Pythagoreans came nearer to realising that.
Philosophy as a Way of Life.
35. We have to take account of the religious revival here, chiefly because it suggested the view that philosophy was above all a “way of life.” Science too was a “purification,” a means of escape from the “wheel.” This is the view expressed so strongly in Plato’sPhaedo, which was written under the influence of Pythagorean ideas.[178]Sokrates became to his followers the ideal “wise man,” and it was to this side of his personality the Cynics mainly attached themselves. From them proceeded the Stoic sage and the Christian saint, and also the whole brood of impostors whom Lucian has pilloried for our edification.[179]Saints and sages are apt to appear in questionable shapes, andApollonios of Tyana showed in the end where this view may lead. It was not wholly absent from any Greek philosophy after the days of Pythagoras. Aristotle is as much possessed by it as any one, as we may see from the Tenth Book of theEthics, and as we should see still more distinctly if we possessed such works as theProtreptikosin their entirety.[180]Plato, indeed, tried to make the ideal wise man of service to the state and mankind by his doctrine of the philosopher king. It was he alone, so far as we know, that insisted on philosophers descending by turns into the cave from which they had been released and coming to the help of their former fellow-prisoners.[181]That was not, however, the view that prevailed, and the “wise man” became more and more detached from the world. Apollonios of Tyana was quite entitled to regard himself as the spiritual heir of Pythagoras; for the theurgy and thaumaturgy of the late Greek schools was but the fruit of the seed sown in the generation before the Persian Wars.
No doctrine in the “Mysteries.”
36. On the other hand, it would be wrong to suppose that Orphicism or the Mysteries suggested any definite doctrines to philosophers, at least during the period which we are about to consider. We have admitted that they really implied a new view of the soul, and we might therefore have expected to find that they profoundly modified men’s theory of the world and their relation to it. The striking thing isthat this did not happen. Even those philosophers who were most closely in touch with the religious movement, like Empedokles and the Pythagoreans, held views about the soul which really contradicted the theory implied by their religious practices.[182]There is no room for an immortal soul in any philosophy of this period. Up to Plato’s time immortality was never treated in a scientific way, but merely assumed in the Orphic rites, to which Plato half seriously turns for confirmation of his own teaching.[183]
All this is easily accounted for. With us a religious revival generally means the vivid realisation of a new or forgotten doctrine, while ancient religion has properly no doctrine at all. “The initiated,” Aristotle said, “were not expected to learn anything, but merely to be affected in a certain way and put into a certain frame of mind.”[184]Nothing was required but that the ritual should be correctly performed, and the worshipper was free to give any explanation of it he pleased. It might be as exalted as that of Pindar and Sophokles, or as material as that of the itinerant mystery-mongers described by Plato in theRepublic. The essential thing was that he should duly sacrifice his pig.
Character of the tradition.
37. It is no easy task to give an account of Pythagoras that can claim to be regarded as history. Ourprincipal 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 entitledMarvels 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. Bothwrote 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’sTripod, and of Aristotle’s work on the Pythagoreans.[189]Both these treatises belong to the fourth centuryB.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 centuryB.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 theEgyptians 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 calledMathematiciansandAkousmatics,EsotericsandExoterics,PythagoreansandPythagorists,[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 centuryB.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 aristocraticideal.” 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. Theonly 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 thedownfall 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. Thecoup d’étatof Kroton can hardly have occurred before 450B.C., if the teacher of Epameinondas escaped from it, and it may well have been even later. But it must have been before 410B.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 hisname. 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 considerationwhich 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 beliefwhich 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 theAkousmatics, 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’sDefence 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 eatensacramentally 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 calledAkousmata,[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 Pythagoreanscience 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 otherOrgiawas 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]Suchmethods of purifying the soul were familiar in theOrgiaof 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 theEthics. 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 sciencefrom 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 centuryB.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 weknow 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 centuryB.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 thosewho 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 αα α α αααα α α α α α α αα α α α αααα α α α α
α α α αα α α αααα α α α α α α αα α α α αααα α α α α
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 letteralphato 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.