Ancient History & Civilisation

PYTHAGORAS AND FIFTH-CENTURY PYTHAGOREANISM
(PYTHAGORAS OF SAMOS, PHILOLAUS OF CROTON, PETRON OF HIMERA, EURYTUS OF CROTON)

We have no extant fragments of Pythagoras himself—he probably wrote nothing—and the historical record is indelibly confused by his great fame, since this meant that later generations attributed all kinds of ideas and mathematical theorems to their illustrious founder, with no regard for the modern concept of historical truth. The difficulty of recovering pre-Platonic Pythagorean thought is increased by the fact that many of Plato’s ideas are Pythagorean in inspiration, and he was such a famous philosopher that subsequent writings about Pythagoreanism are tainted, as some scholars see it, with Platonic views. There is a large number of such writings, and they need judicious mining for nuggets of genuine early Pythagorean thought. However, others regard Pythagoreanism more as a continuing and stable tradition, from which Plato borrowed; if this is the case, post-Platonic evidence about Pythagoreanism may be just as informative about the tradition as any other.

Pythagoras soon became well known as a sage: he lived around the end of the sixth century, and T1–5 were all written within about fifty years of his death. Heraclitus grumpily accuses Pythagoras of plagiarism (probably from Orphic texts) and lack of insight, but Herodotus, Ion, and Empedocles see him as a great teacher. Since later tradition credits Pythagoras with teaching reincarnation,1 it is likely that T6, along with Xenophanes F20 (p. 30), are also early references to Pythagoras. T7 (from Aristotle) and T8 (from a contemporary of Aristotle) confirm that metempsychosis was central to early Pythagorean thought.2 The religious flavour of early Pythagoreanism is also clear in T9–11 (certain practices were forbidden to members of the sect), T12 (miracles were ascribed to Pythagoras), and T14 and T18 (some Pythagoreans took a vow of silence). The connection between Pythagoreanism and the Orphic religion (hinted at in T9) is hard to unravel, but the following elements of Orphism are almost certainly relevant: the soul is imprisoned in the body until it has paid the penalty for past misdeeds; a life of ritual purity is required to cleanse our souls; ascetic prescriptions for purity include abstention from blood sacrifice, and from eating meat and most fish. For the Pythagoreans, vegetarianism was a natural consequence of their belief in the transmigration of the soul: today’s dinner may be your dead grandmother. But since they also believed that plants had souls of a kind, it is not known how far down the food chain they took their proscription, or even whether only certain kinds of meat were prohibited, rather than all meat. Moreover, since other testimonia commend sacrifice (see T15, T18, T22), it is not clear to what extent, if at all, they undertook the radical step of abstaining from sacrifice. On the imprisonment of the soul, see the fragment F1 of the fifth-century Pythagorean Philolaus, a contemporary of Socrates. The idea that the soul is independent of the body, and in some sense represents one’s true self, has of course been of immense significance in Western thought. Boosted by Plato and Christianity, until recently it was taken more or less for granted.

The meaning of many of the Pythagorean prohibitions, such as those listed in T10 and T11, is obscure. They were a particularly famous feature of the Pythagorean way of life, and were known as akousmata (‘things heard’, or passed down by word of mouth) orsumbola (‘tokens’ or ‘passwords’). At any rate, it is clear that anyone attempting to obey these injunctions would have to remain alert, rather than succumbing to the semi-sleep state that constitutes normal consciousness. T13, although late testimony, is probably based on the fourth-century writer Aristoxenus, and well sums up the mystical thrust of Pythagorean practices, one of the consequences of which, given by Plato at Phaedo 61e-62c, is a prohibition of suicide; if I am the gods’ subordinate, I do not have the right to take my own life. Even their mathematical teaching was subordinate to the aim of harmonizing one’s life with god’s wishes. In short, it is likely that Pythagoras was a teacher of perennial wisdom, rather than a Presocratic philosopher in the Milesian mould. Some of his followers later developed his views into a more scientific form (or, just possibly, revealed them where they had previously been considered secret). We hear of many individual Pythagoreans (over 200), but few of them are more than just a name: we rarely know enough to be able to attribute particular doctrines to them. And all we can safely say about the doctrines of Pythagoras himself is summed up in T14 (which probably stems from Dicaearchus), with the possible addition of T15–16.

The biographical tradition concerning Pythagoras is often contradictory, but it is reasonably safe to say that though he was born on the eastern Greek island of Samos, at the time of its greatest prosperity, he fled from there during the reign of the tyrant Polycrates (535–522) and settled in southern Italy, first in Croton, and then later in Metapontum, where he died. The move to Metapontum may have been made necessary by hostility towards Pythagorean political influence in Croton; there were two waves of attacks on Pythagoreans in southern Italy, one c.510 and the other c.450. Pythagoras’ activities in southern Italy included setting up communes (T17), run on religious and mystical principles (T13), which also gained political power in a number of communities in southern Italy (T19). For the first time, women were admitted into these schools. Others, however, downplay the religious side of these communes and try to see them purely as political pressure groups.

Plato’s view of the Pythagoreans covers both their quasi-monastic way of life and their interest in mathematics and science (T20–1); the idea that astronomy and harmonics are sister sciences was probably traditionally Pythagorean, but was certainly expressed by Plato’s contemporary, Archytas of Tarentum. These are Plato’s only two explicit references to Pythagoras or Pythagoreans, but they do not reveal the extent of his debt to them in certain passages of Gorgias, Phaedo, and Republic, in Philebus and Timaeus as a whole, and in his famous ‘unwritten doctrines’. Not all akousmata were commandments, and the essence of Pythagorean arithmology is expressed in the centrality of the tetraktys to their system (T22, T23). The tetraktys is the decad considered as the sum of the first four numbers, and is usually portrayed as a triangular number:

Image

It could be, and was, used to express the arithmetical, geometric, and harmonic relations between the first ten numbers, in a number of complex ways.3 Some Pythagorean arithmology has survived today, although we may have shed the geometrical conception of mathematics the Pythagoreans perpetuated in favour of abstract notation; they were the first to define different kinds of numbers as, for instance, odd and even, square and cube, prime and composite; and we are still impressed by the fact that, for instance, successive odd numbers always add up to successive square numbers. T23 is only the tip of the iceberg of uses to which the Pythagoreans put the tetraktys. The musical use, prominent in Sextus, is certainly early (as T24 shows), and many would attribute the discovery of the mathematics of the primary musical intervals to Pythagoras himself (e.g. T16, also an early piece of evidence). But how much further the first Pythagoreans went in mathematical musicology is complex and unclear; in T21 Plato complains that they did not pay enough attention to pure mathematics.

But, if T25–8 are to be trusted, the connection the Pythagoreans saw between number and the universe lay not just in the kinds of correspondences the tetraktys could display. Aristotle tells us that they saw number as somehow the principle of all things. This view is likely to be confusing, until we appreciate that the Pythagoreans were not Milesians: they were not interested in the material nature of things so much as their organization. Thus as T25 suggests, and T29 and T30 show at greater length, even abstract concepts such as justice could be accommodated. Note also that the Pythagorean attribution of properties to numbers was not stable; T29 calls either 4 or 9 ‘justice’, while T30 is an extended reflection on how 5 (the pentad) can be seen as justice. It is clear that Aristotle talks correctly of numbers being ‘analogues’ or ‘resemblances’ of things (T25); so when elsewhere he talks as if the Pythagoreans identified things with numbers (T26–8, and see also Alexander at the end of T29) and suggested that things were literally made out of numbers, he is trying too hard to incorporate Pythagorean views into his own theory of the material cause. At any rate, on Aristotle’s evidence, according to the Pythagoreans things are numbers,4 things are like numbers, and the elements of number, the limit and the unlimited, or the even and the odd, are the elements of all things.

The famous table of opposites with which Aristotle concludes T25 is puzzling, because it seems to combine different kinds of opposites. However, in each of the ten pairs, the first one should be seen as a limiter and the second as something unlimited (see Philolaus in F3, below). This begins to suggest a way in which limit and the unlimited are the elements of things, or account in some way for the properties of things. Every property of every object can be seen to be either a limiter or unlimited. What is particularly important about this is that (at any rate, by the later fifth century) the Pythagoreans had clearly moved beyond the Milesian conception of the opposites as concrete stuffs to the realization that they were abstract qualities. But in any case, although the primary pairs, limit-unlimited and odd-even, are early the full table of opposites may stem from fourth-century Pythagoreanism, later than the time-frame of this book. With the words ‘Other members of the same school’, Aristotle distinguishes its authors from the fifth-century Pythagoreans he had previously been discussing.

Pythagorean interest in number led them to investigate its properties widely, and there is no doubt that they made significant advances in mathematics (though nowhere near as many as later tradition credits them with), as well as in the pseudo-science of arithmology. Here it is especially hard to know which, if any, of the theorems derive from Pythagoras himself. T31–5 give a few important theorems which we may date with some but not total confidence to early in the history of Pythagorean mathematics (see alsoT15 on ‘Pythagoras’ theorem’—but that, in any case, may have been learnt from Babylon, where knowledge of the Pythagorean triangle goes back to about 1700BCE). T36 shows that mathematics (or arithmology) was considered esoteric. But T37 suggests that the arithmologists, representing the mystical side of Pythagoreanism, considered themselves the only true Pythagoreans; and it is true that our sources do show a tendency to label any mathematician a ‘Pythagorean’, solely because he worked on mathematics.

The Pythagoreans (and especially Philolaus and Archytas) greatly enhanced our knowledge of astronomy. Although it is probably going too far to suggest that T38 shows that they saw the earth as simply one of the planets (since they had no conception of a heliocentric universe), and although it is not clear that they discovered the correct order of the planets, let alone explained the irregularities of their motions, they did distinguish the planets from the sun, moon, and fixed stars (T39–40), and they recognized that the heavenly bodies were of an enormous size (T41). T42 is an ingenious accommodation of the fieriness of the sun with the teaching about heavenly fire contained in T40. Aristotle’s notorious accusation in T25 that the Pythagoreans invented heavenly bodies for arithmological purposes is clearly the last resort of an intellectual failing to understand a system constructed more for its resonance with the inner psyche of people than for its correspondence with observable facts. The mystical or mind-expanding aspect of Pythagoreanism is never far from the surface: T41 introduces us to the famous Pythagorean doctrine of the Harmony of the Spheres, the beauty of which has cast a spell on all subsequent generations.5 Anaximander’s proportionate universe is here given majestic elaboration, but note that in its earliest manifestation, as reported here by Aristotle, it is not clear how many notes make up the harmony—that is, it is not clear that the early Pythagoreans distinguished the five visible planets and assigned them each a different sound. T43, sounding like something from H. G. Wells, again reminds us that we are in the domain of shamanistic visions, not science.

Pythagorean cosmogony is difficult to reconstruct, and our sources are full of obvious contradictions, or at least alternative views. As T25 and F2 show, the opposites, limit and the unlimited, are primary. The imposition of limit on the unlimited creates the universe, the One, which is both even and odd simultaneously. The other numbers, which are somehow identical with things, proceed from the One. Aristotle was severely critical of this view, both because it involved the generation of numbers (which he considered eternal: Metaphysics 1091a12–22) and because it constructed the material universe out of immaterial entities, numbers (Metaphysics 1090a30–35). It is clear (and Aristotle, Physics 203a confirms it) that the Pythagoreans thought of the universe as spherical and as being surrounded by ‘the unlimited’ (the same word as Anaximander’s ‘boundless’). Some kind of drawing in takes place, perhaps like an inbreath (T44—5); this introduces void, which distinguishes one thing, one number, from another. The first thing to be distinguished in this way is the central hearth of the universe (F6), and then the rest of the major features of the universe—the planets and so on (F7). It is legitimate to connect fragments of Philolaus with Aristotle’s testimonia about ‘the Pythagoreans’, because it is likely that Philolaus is actually the Pythagorean Aristotle most commonly has in mind. But at the same time it is clear that to be a Pythagorean meant, primarily, to practise a certain way of life, not to adhere to a particular cosmology in all its details, and so we do hear of significant theoretical differences between thinkers classified as ‘Pythagoreans’.

Philolaus’ cosmogony is the most sophisticated extant. His thinking reflects the symmetry of Anaximander’s universe, which balances up and down in an era before knowledge of gravity (F7). He based his cosmogony entirely on a primary pair of opposites, limitation and unlimitedness (which are most profitably thought of as that which provides structure and that which becomes structured, or quasi-Aristotelian form and matter), thus continuing both the Milesian reduction of the first principles of the universe to as few as possible and their emphasis on opposites, but in response to Parmenides’ strictures made the ‘being’ of these things eternal (F5). Since it was standard Pythagorean teaching that odd numbers limit, while even numbers are unlimited,6 it is likely that he was an orthodox Pythagorean at least to the extent that his cosmogony was arithmological. Harmony, or mathematically conformable adjustment, relates the odd and even numbers, limiters and unlimiteds. The harmony or structure of the world is always uppermost in Philolaus’ mind as the chief thing he needs to explain.

Also noteworthy are his comments—almost asides—on the limitations of human knowledge. The true essence of things is accessible only to the gods, or perhaps to a man with divine knowledge; and in the nature of things we cannot know the infinite (F4, F5). In part, Philolaus is here criticizing Milesian or similar attempts to divine an ultimate reality behind the things of this world. He is suggesting that this is impossible, and that the best one can do, instead, is to try to say what the necessary preconditions are for the world we are faced with to exist. Those necessary preconditions are, he suggests, the existence of things that limit and things that are unlimited, and of harmonia to bind them together; and he suggests that these are easily identifiable features of our world. One can, then, analyse any event or entity into something unlimited which has been limited in a harmonious fashion. As a Pythagorean, Philolaus would probably argue that ultimately the limiter and the unlimited are numerical, but in the first instance this is not necessary: this book, for instance, is simply unlimited vegetable matter which has been limited by something (human will?) in a harmonious fashion.

It is not clear in detail how Philolaus or other Pythagoreans explained the creation of the various minutiae of life on earth (T46 is tantalizing, but its attribution to Philolaus is controversial), but they did speculate about the nature of the human soul (T47 andT48). In T48, which is probably the theory of Philolaus, it is likely that Aristotle has been unduly influenced by Plato’s elaboration of this theory in his dialogue Phaedo (especially 86b-c), and that originally Philolaus said that the soul was a numerical ratio rather than a blending of opposites. It is more likely to be authentically Pythagorean that the soul is or has its own harmony. Apart from anything else, if the soul is the harmony of the bodily elements, it is hard to see how the soul could survive the dissolution of the body, and yet transmigration of the soul was standard Pythagorean doctrine. As T49 and F8 show, Philolaus also speculated about the nature of the body. He might have added that warm bodies grow cold on death. As we have found with the Milesians, Philolaus here adumbrates an analogy between macrocosm and microcosm. Just as the universe is formed first out of central fire, and then draws in void from the unlimited (T44–5), so a new-born human is hot and draws in air from outside.

I conclude with almost all we know about two other fifth-century Pythagoreans, Eurytus of Croton (a pupil of Philolaus) and Petron of Himera.7 T50 shows how bizarre and amazing early Pythagorean cosmological speculation could be. The testimonia about Eurytus (T51–2) are more interesting: they demonstrate how, in Aristotle’s terms, Pythagoreans could think that everything was made out of numbers. An unkind interpretation has Eurytus playing silly games—blocking out a pre-drawn figure of a human being with 250 pebbles and then saying, ‘Eureka! 250 is the number of a human being!’ More charitably, his reasoning was probably that if 3 is the minimum number required to define a triangle, and 4 a pyramid, then there may be a minimum number required to define thespecific form of a human being. On this view, Eurytus may be seen as moving towards the kind of science we have nowadays, which is based on mathematics.

T1 (DK 22B129; KRS 256) There was no more diligent investigator than Pythagoras the son of Mnesarchus; he made a selection from these writings and created a wisdom of his own, a thing of wide learning and fraudulent artifice. (Heraclitus [fr. 129 Diels/Kranz] in Diogenes Laertius, Lives of the Eminent Philosophers 8.6.3–5 Long)

T2 (DK 22B40; KRS 255) Wide learning does not teach insight; otherwise it would have taught Hesiod and Pythagoras, not to mention Xenophanes and Hecataeus. (Heraclitus [fr. 40 Diels/Kranz] in Diogenes Laertius, Lives of Eminent Philosophers 9.1.5–7 Long)

T3 (DK 14A2; KRS 257) [Herodotus records a story, which he himself does not believe, that the Thracian deity Salmoxis had once been a slave of Pythagoras, and duped the Thracian tribe, the Getae, into a belief in personal immortality by hiding away for three years and then reappearing. In the course of telling the story he says:] Now, Salmoxis had experienced life in Ionia and was familiar with Ionian customs, which are more profound than those of the Thracians, who are an uncivilized and rather naïve people; after all, he had associated with Greeks, and in particular with Pythagoras, who was hardly the weakest intellect in Greece. (Herodotus, Histories 4.95.2.4–7 Hude)

T4 (DK 36B4; KRS 258) Ion of Chios says about Pherecydes:

Well furnished, then, with manly vigour and dignity,

Even when dead he has a pleasant life for his soul,

If Pythagoras really knew what he was talking about,

And he excelled in knowing and studying men’s views.

(Ion of Chios [fr. 5 Diehl] in Diogenes Laertius, Lives of Eminent Philosophers 1.120.5–8 Long)

T5 (DK 31B129; KRS 259) Empedocles too testifies to this when he says about Pythagoras:

There was among them a certain man of rare knowledge,

Master especially of all kinds of wise deeds,

Who had acquired the greatest wealth of mind:

For whenever he reached out with his entire mind

He easily saw each and every individual thing 5

In ten and twenty lifetimes of men.

(Empedocles [fr. 129 Diels/Kranz] in Porphyry, Life of Pythagoras 30.7–14 Nauck)

T6 (DK 14A1; KRS 261) The Egyptians were also the first to claim that the soul of a human being is immortal, and that each time the body dies the soul enters another creature just as it is being born. They also say that when the soul has made the round of every creature on land, in the sea, and in the air, it once more clothes itself in the body of a human being just as it is being born, and that a complete cycle takes three thousand years. This theory has been adopted by certain Greeks too—some from a long time ago, some more recently—who presented it as if it were their own. I know their names, but I will not write them down. (Herodotus, Histories 2.123.2–3 Hude)

T7 (DK 58B39) They [Aristotle’s predecessors] try only to describe the soul, but they fail to go into any kind of detail about the body which is to receive the soul, as if it were possible (as it is in the Pythagorean tales) for just any old soul to be clothed in just any old body. (Aristotle, On the Soul 407b20–3 Ross)

T8 (DK 14A8) Heraclides of Pontus says that Pythagoras used to say about himself that he had once been born as Aethalides and was regarded as a son of Hermes. Hermes told him that he could choose anything he wanted except immortality, and he asked to be able to retain, both alive and dead, the memory of things that had happened. He therefore remembered everything during his lifetimes, and when dead he still preserved the same memories. Later he entered into Euphorbus and was wounded by Menelaus. Euphorbus used to say that he had formerly been born as Aethalides and had received the gift from Hermes, and used to tell of the journeying of his soul and all its migrations, recount all the plants and creatures to which it had belonged, and describe everything he had experienced in Hades and the experiences undergone by the rest of the souls there. When Euphorbus died, his soul moved into Hermotimus, who also wanted to prove the point, so he went to Branchidae, entered the sanctuary of Apollo, and pointed out the shield which Menelaus had dedicated there … When Hermotimus died, he became Pyrrhus, the fisherman from Delos, and again remembered everything, how he had formerly been Aethalides, then Euphorbus, then Hermotimus, and then Pyrrhus. And when Pyrrhus died, he became Pythagoras and remembered everything that has just been mentioned. (Heraclides of Pontus [fr. 89 Wehrli] in Diogenes Laertius, Lives of Eminent Philosophers 8.4–5 Long)

F1 (DK 44B14) The ancient theologians and prophets testify to the fact that the soul has been yoked to the body as a punishment of some kind and that it has been buried in the body as in a tomb. (Philolaus [fr. 14 Diels/Kranz] in Clement, Miscellanies 2.203.11 Stählin/Früchtel)

T9 (DK 14A1; KRS 263) It is against religious law for the Egyptians to take anything woollen into their sanctuaries or to be buried along with any woollen items. This custom of theirs accords with Orphic and Bacchic rites, as they are called (though they are actually Egyptian and Pythagorean), because no initiate of these rites either is allowed to be buried in woollen clothing. (Herodotus, Histories 2.81.1–2 Hude)

T10 (DK 58C3; KRS 275) In On the Pythagoreans Aristotle explains the Pythagorean injunction to abstain from beans as being due either to the fact that they resemble the genitals in shape, or because they resemble the gates of Hades (since it is the only plant which has no joints), or because they ruin the constitution, or because they resemble the nature of the universe, or because they are oligarchic, in the sense that they are used in the election of magistrates by lot.* And the injunction not to pick up things that have fallen he explains as being an attempt to accustom them not to eat in immoderate quantities, or due to the fact that it signals someone’s death … The injunction not to touch a white cock is due to the fact that the creature is sacred to the New Month and is a suppliant … The injunction not to touch any sacred fish is due to the fact that the same food should not be served to gods and men, just as free men and slaves should have different food too. The injunction not to break a loaf is due to the fact that in olden days friends used to meet over a single loaf. (Aristotle [fr. 195 Rose] in Diogenes Laertius, Lives of Eminent Philosophers 8.34.1–35.2 Long)

T11 (DK 58C6; KRS 276) There was another kind of token, such as do not step over a balance (i.e. do not desire more than your share), and do not poke a fire with a sword (i.e. avoid irritating with sharp words anyone who is seething with anger), and do not pluck leaves from a garland (i.e. do not maltreat the laws, which are the garlands of communities). Then again there were other similar tokens, such as do not eat heart (i.e. do not upset yourself with regrets), and do not sit on a bushel (i.e. do not live an idle life), anddo not turn back from a journey (i.e. do not cling to this life when you are dying), and do not walk on the highways (a recommendation not to follow the opinions of the many, but the views of those few people who are educated), and do not let swallows in your house(i.e. do not take in as lodgers chatterboxes with no control over their tongues) … (Aristotle [fr. 197 Rose] in Porphyry, Life of Pythagoras 42.1–15 Nauck)

T12 (DK 14A7; KRS 273) He was once seen in Croton and Metapontum at the same time of the same day. (Aristotle [fr. 191 Rose] in Apollonius, Enquiry into Miracles 6.2e Giannini)

T13 (DK 58D2; KRS 456) The aim of all the Pythagorean precision about what should and should not be done is association with the divine. This is their starting-point, and their way of life has been wholly organized with a view to following God. The thinking behind their philosophy is that people behave in an absurd fashion if they try to find any source for the good other than the gods … Since there is a god, since he has supreme authority, since it goes without saying that one should ask for the good from whoever has authority [rather than from a subordinate], and since everyone gives good things to those whom they love and who please them, and the opposite to those who do the opposite of pleasing them, it obviously follows that we should act in ways which please God. (Iamblichus, Pythagorean Life 137 Deubner)

T14 (DK 14A8a; KRS 285) But no one can tell for certain what Pythagoras used to say to his companions, because of the extraordinary silence they practised. However, certain of his teachings became particularly well known throughout the world: first, his claim that the soul is immortal; second, that it changes into other species of living things; third, that past events happen again in specific cycles, and that nothing is simply new; and fourth, that we should regard all ensouled creatures as akin. (Porphyry, Life of Pythagoras19.6–13 Nauck)

T15 (KRS 434) Anticleides says that Pythagoras was particularly interested in the arithmetical aspect of geometry, and discovered the properties of the monochord. Nor did he neglect medicine either. Apollodorus the mathematician says that Pythagoras sacrificed a hecatomb when he discovered that the square on the hypotenuse of the right-angled triangle is equal to the squares on the sides which encompass the right angle. (Anticleides [fr. 1 Jacoby] in Diogenes Laertius, Lives of Eminent Philosophers 8.11.10–12.5 Long)

T16 In his Introduction to Music Heraclides says that, according to Xenocrates, it was Pythagoras who discovered that the musical intervals also come about inevitably because of number, in the sense that they consist in a comparison of one quantity with another, and that he also looked into the question of what makes the intervals concordant or discordant, and in general what factors are responsible for harmony and disharmony (Xenocrates [fr. 9 Heinze] in Porphyry, Commentary on Ptolemy’s ‘Harmonics’30.1–6 During]

T17 (KRS 271) At any rate, in his ninth book Timaeus says, ‘When the younger men came to him and expressed their desire to associate with him, he did not immediately accede to their request, but said that their property would also have to be held in common with other members.’ (Timaeus [fr. 13a Jacoby] in a scholiast on Plato, Phaedrus 279c, Greene p. 88)

T18 (DK 14A4) Pythagoras of Samos visited Egypt and studied with the Egyptians. He was the first to import philosophy in general into Greece, and he was especially concerned, more conspicuously than anyone else, with sacrifice and ritual purification in sanctuaries, since he thought that even if, as a result of these practices, no advantage accrued to him from the gods, they would at least gain him a particularly fine reputation among men. And this is exactly what happened. He became so much more famous than anyone else that all the young men wanted to become his disciples, while the older men preferred to see their sons associating with him than looking after their own affairs. And it is impossible to mistrust their opinion, because even now those who claim to be his followers are more impressive in their silence than those with the greatest reputation for eloquence. (Isocrates, Busiris 28.5–29.9 van Hook)

T19 (DK 14A16; KRS 267) Cylon of Croton was one of the leading men of his community, thanks to his birth, reputation, and wealth, but in other respects he was a cruel, brutal, disruptive, and tyrannical man. He expressed a heart-felt desire to join in the Pythagorean way of life and met with Pythagoras himself, who was then an old man, but was rejected because of the character flaws I have already mentioned. As a result of this he and his friends declared unrelenting war on Pythagoras and his companions … Nevertheless, for a while the true goodness of the Pythagoreans prevailed, along with the desire of the communities themselves to have their political affairs administered by them. But eventually the Cylonians’ intrigues against the men reached such a pitch that when the Pythagoreans convened in Milo’s house in Croton to discuss political business, the Cylonians set fire to the house and burnt to death all the men inside, except for the two youngest and strongest, Archippus and Lysis, who managed to break out. But the Italian communities ignored what had happened, and so the Pythagoreans abandoned their involvement in politics … The remaining Pythagoreans gathered in Rhegium and continued to associate with one another there, but as time went on and the political situation deteriorated they left Italy, with the exception of Archytas of Tarentum. (Aristoxenus [fr. 11 Müller] in Iamblichus, Pythagorean Life 248.8–251.3 Deubner)

T20 (DK 14A10; KRS 252) So there is no evidence of Homer’s having been a public benefactor, but what about in private? Is there any evidence that, during his lifetime, he was a mentor to people, and that they used to value him for his teaching and then handed down to their successors a particular Homeric way of life? This is what happened to Pythagoras: he wasn’t only held in extremely high regard for his teaching during his lifetime, but his successors even now call their way of life Pythagorean and somehow seem to stand out from all other people. (Plato, Republic 600a9-b5 Burnet)

T21 (DK 47B1; KRS 253) The eyes are made for astronomy, and by the same token the ears are presumably made for the type of movement that constitutes music. If so, these branches of knowledge are allied to each other. This is what the Pythagoreans claim, and we should agree, Glaucon, don’t you think? Music is a difficult subject, so we’ll consult the Pythagoreans to find out their views … [Socrates and Glaucon go on to criticize the kind of musicologists who ‘laboriously measure the interrelations between audible concords and sounds’] But I wasn’t thinking of those people, but the ones we were saying just now would explain music to us, because they act in the same way that astronomers do. They limit their research to the numbers they can find within audible concords, but they fail to come up with general matters for elucidation, such as which numbers form concords together and which don’t, and why some do and some don’t. (Plato, Republic 530d6-e2, 531b7-c4 Burnet)

T22 (DK 58C4; KRS 277) The philosophy of the acousmatics consists in unproved and unjustified akousmata, to the effect that one should act in such-and-such a way, and they try to preserve everything else which is said to stem from Pythagoras as divine dogma. They claim that they say nothing of their own accord and that it would be not be right for them to do so, and even go so far as to account those of their number the most advanced in terms of wisdom who have grasped the most akousmata. There are three categories of these so-called akousmata: some of them indicate what a thing is, some of them indicate superlatives, and some of them indicate what one should or should not do. For example, among those that indicate what a thing is are: What are the Isles of the Blessed? The sun and moon.* Or again: What is the Delphic oracle? The tetraktys, which is the harmony in which the Sirens sing. Examples of those that indicate superlatives are: What is most moral? To sacrifice. Or: What is wisest? Number. (Iamblichus,Pythagorean Life82.1–15 Deubner)

T23 (KRS 279) In order to indicate this [the importance of number in things] the Pythagoreans are accustomed on occasion to say that ‘There is a resemblance to number in all things’, and also on occasion to swear their most characteristic oath: ‘No, by him who handed down to our company the tetraktys, the fount which holds the roots of ever-flowing nature.’ By ‘him who handed down’ they mean Pythagoras, whom they regarded as divine, and by the ’tetraktys’ they mean a certain number which, being composed out of the first four numbers, produces the most perfect number—that is, ten (for 1 + 2 + 3 + 4 = 10). This number is the first tetraktys and it is called ‘the fount of ever-flowing nature’ because it is their view that the whole universe is organized on harmonic principles, and harmony is a system of three concords (the fourth, the fifth, and the octave), and the ratios of these three concords are found in the four numbers I have already mentioned—that is, in 1, 2, 3, and 4. For the fourth is constituted by 4: 3, the fifth by 3:2, and the octave by 2:1. (Sextus Empiricus, Against the Professors 7.94–6 Bury)

T24 (DK 18A12) A certain Hippasus prepared four bronze discs in such a way that, although their diameters were equal, the thickness of the first was in the ratio 4:3 to that of the second, in the ratio 3:2 of that to the third, and in the ratio 2:1 to that of the fourth. When struck, they produced a concord. (Aristoxenus [fr. 77 Müller] in a scholiast on Plato, Phaedo 108d, Greene p. 15)

T25 (DK 58B4, B5; KRS 430) At the same time [as Leucippus and Democritus] and earlier than them were the so-called Pythagoreans, who were interested in mathematics. They were the first to make mathematics prominent, and because this discipline constituted their education they thought that its principles were the principles of all things. Now, in the nature of things, numbers are the primary mathematical principles; they also imagined that they could perceive in numbers many analogues to things that are and that come into being (more analogues than fire and earth and water reveal)—such-and-such an attribute of numbers being justice, such-and-such an attribute being soul and mind, due season another, and so on for pretty well everything else; moreover, they saw that the attributes and ratios of harmonies depend on numbers. Since, then, the whole natural world seemed basically to be an analogue of numbers, and numbers seemed to be the primary facet of the natural world, they concluded that the elements of numbers are the elements of all things, and that the whole universe is harmony and number. They collected together all the properties of numbers and harmonies which were arguably conformable to the attributes and parts of the universe, and to its organization as a whole, and fitted them into place; and the existence of any gaps only made them long for the whole thing to form a connected system. Here is an example of what I mean: ten was, to their way of thinking, a perfect number, and one which encompassed the nature of numbers in general, and they said that there were ten bodies moving through the heavens; but since there are only nine visible heavenly bodies, they came up with a tenth, the counter-earth …

They hold that the elements of number are the even and the odd, of which the even is unlimited and the odd limited; one is formed from both even and odd, since it is both even and odd; number is formed from one and, as I have said, numbers constitute the whole universe. Other members of the same school say that there are ten principles, which they arrange in co-ordinate pairs: limit and unlimited; odd and even; unity and multiplicity; right and left; male and female; still and moving; straight and bent; light and darkness; good and bad; square and oblong. (Aristotle, Metaphysics 985b23–986a26 Ross)

T26 (DK 58A8) The Pythagoreans spoke of two causes in the same way, but added, as an idiosyncratic feature, that the limited and the unlimited and the one were not separate natures, on a par with fire or earth or something, but the unlimited itself and the one itself were taken to be the substance of the things of which they are predicated. This is why they said that number was the substance of everything. (Aristotle, Metaphysics 987a13–19 Ross)

T27 The Pythagoreans, as a result of observing that many properties of numbers exist in perceptible bodies, came up with the idea that existing things are numbers, but not separate numbers: they said that existing things consist of numbers. Why? Because the properties of numbers exist in musical harmony, in the heavens, and in many other cases. (Aristotle, Metaphysics 1090a20–5 Ross)

T28 (DK 58B9; KRS 431) The Pythagoreans recognize only one kind of number, mathematical number, but they say that it is not separate, but that perceptible things are made up of it. For they construct the whole universe out of numbers—and not numbers made up of abstract units, but they take their numerical units to have spatial magnitude. But they apparently have no way to explain how the first spatially extended unit was put together. (Aristotle, Metaphysics 1080b16–21 Ross)

T29 Aristotle has shown the kinds of analogues the Pythagoreans said existed between numbers and the things that are and that come into being. On the assumption that reciprocity or equality is a property of justice, and finding that equality is also a property of numbers, they said that justice is the first square number, on the grounds that the first of a series of things with the same definition is, in each case, most truly what it is said to be. Some said that the number of justice was 4, because, being the first square number, it is divided into equal parts and is itself equal (since it is 2 × 2), but others said that it was 9, since it is the first square number produced by multiplying an odd number—3—by itself. Again, they said that 7 was due season, since natural things seem to have their perfect seasons of birth and completion in terms of sevens … Since the sun is responsible for the seasons, they thought, according to Aristotle, that it was located in the place of the seventh number, which they call ‘due season’; for the sun, they said, occupied the seventh rank among the ten bodies which move around the centre and the hearth. First come the sphere of the fixed stars and the five spheres of the planets, and then the sun; after the sun, the moon occupies the eighth place, the earth the ninth, and then the counter-earth.* Since 7 neither generates any other number within the decad nor is generated by any of them, they called it ‘Athena’ … Marriage, they said, was 5, because it is the union of male and female and they thought that the odd was male and the even female; and 5 is the first number formed from the first even number, 2, and the first odd number, 3; for, as I said, they thought that the odd was male and the even female. Reason (which was what they called soul) and substance they identified with 1. Because it is unchanging, everywhere alike, and a ruling principle, they called reason a monad, or 1; but they also applied these names to substance, because it is primary. Opinion they identified with 2 because it can move in two directions; they also called it movement and addition. Picking out such analogues between things and numbers, they assumed numbers to be the first principles of things, and said that all things are made up of numbers. (Aristotle [fr. 203 Rose] in Alexander of Aphrodisias, Commentary on Aristotle’s ‘Metaphysics’, CAG 1, 38.8–39.19 Hayduck)

T30 In the first place, we must set out in a row the sequence of numbers from the monad up to nine: 1, 2, 3, 4, 5, 6, 7, 8, 9. Then we must add up the amount of all of them together, and since the row contains nine terms, we must look for the ninth part of the total, to see if it is already naturally present among the numbers in the row; and we will find that the property of being the ninth belongs only to the mean itself. So the pentad is another thing which has neither excess nor defectiveness in it, and it will turn out to provide this property for the rest of the numbers, so that it is a kind of justice, on the analogy of a weighing instrument. For if we suppose that the row of numbers is some such weighing instrument, and the mean number 5 is the hole of the balance, then all the parts towards the ennead, starting with the hexad, will sink down because of their quantity, and those towards the monad, starting with the tetrad, will rise up because of their fewness, and the ones which have the advantage will altogether be triple the total of the ones over which they have the advantage, but 5 itself, as the hole in the beam, partakes of neither advantage nor disadvantage, but it alone has equality and sameness.

The parts adjacent to it gradually decrease in advantage or disadvantage the closer they get to it, just like the parts which move away little by little from the scales on the beam towards the balance. The ennead and the monad are at the furthest distance, whence the ennead has the greatest advantage, the monad the greatest disadvantage, each by a full tetrad. A little further in from these are the ogdoad and the dyad, whence the ogdoad has a little less excess, the dyad a little less defectiveness; in each case the excess or defectiveness is a triad. Then, next to these, are the hebdomad and the triad, whence the triad is defective and the hebdomad excessive by the next amount—they are a dyad away from the centre. Further in from these and next to the pentad, as it were to the balance, are the tetrad and the hexad, which has the least excess, for no smaller number than this can be thought of.

When the beam is suspended, the parts with excess make excessive both the angle at the scales and the angle at the balance, while the parts with defectiveness make the angle defective in both cases, and the obtuse angle is the excessive one, since a right angle has the principle of maximum equality.

Since in a case of injustice those who are wronged and those who do wrong are equivalent, just as in a case of inequality the greater and the lesser parts are equivalent, but nevertheless those who do wrong are more unjust than those who suffer wrong (for the onegroup requires punishment, the other compensation and help), therefore the parts which are at a distance on the side of the obtuse angle, where the weighing instrument is concerned and in the terms of our mathematical illustration (i.e. the parts with advantage), are progressively further away from the mean, which is justice; but the parts on the side of the acute angle will increasingly approach and come near, and as it were through continually suffering wrong in being at a disadvantage, while the others will travel downwards and into corruption and immersion in evil, they will rise up and take refuge in God through their need for retribution and compensation.

At any rate, if it is necessary, taking the beam as a whole, for equality to be in this mathematical illustration, then again such a thing will be contrived thanks to the pentad’s participation as it were in a kind of justice. For one possibility is that if all the parts which are arranged at a fifth remove from the excessive parts are subtracted from them and added to the disadvantaged parts, then what is being sought will be the result.* Alternatively, thanks to the pentad’s being a point of distinction and reciprocal separation, if the disadvantaged one which is closest to the balance on that side is subtracted from the one which is furthest from the balance on the excessive side and added to the one which is furthest from the balance on the other side (i.e. 1)—if, to effect equalization, 4 is subtracted from 9 and added to 1; and from 8, 3 is subtracted, which will be the addition to 2; and from 7, 2 is subtracted, and added to 3; and from 6, 1 is subtracted, which is the addition to 4 to effect equalization, then all of them equally, both the ones which have been punished, as excessive, and the ones which have been set right, as wronged, will be assimilated to the mean of justice. For all of them will be 5 each, and 5 alone remains unsubtracted and unadded, so that it is neither more nor less, but it alone encompasses by nature what is fitting and appropriate. (Ps.-Iamblichus, The Theology of Arithmetic 37.4–39.24 de Falco)

T31 (DK 18A15) In the old days, in the time of Pythagoras and the mathematicians of his ilk, there were only three means, the arithmetic, the geometric, and the third in the list, the one which used to be called the subcontrary mean, but which was renamed the harmonic by the circle of Archytas and Hippasus, because it seemed to encompass the ratios relevant to what is harmonized and (Iamblichus, Commentary on Nicomachus’ ‘Introduction to Arithmetic’ 100.19–25 Pistelli)

T32 (DK 58B21; KRS 436) Eudemus the Peripatetic attributes to the Pythagoreans the discovery of the theorem that the internal angles of every triangle are equal to two right angles. He says that they proved the theorem in question as follows.

Image

Let ABC be a triangle, and through A let the line DE be drawn parallel to BC. Since BC and DE are parallel, and the alternate angles are equal, then the angle DAB is equal to the angle ABC, and EAC is equal to ACB. Let BAC be added to both. Then the angles DAB, BAC, and CAE, that is, the angles DAB and BAE, that is, two right angles, are equal to the three angles of the triangle. Therefore the three angles of the triangle are equal to two right angles. (Eudemus [fr. 88 Spengel] in Proclus, Commentary on Euclid379.2–16 Friedlein)

T33 (DK 58B20; KRS 435) These things are ancient, according to Eudemus, and are discoveries of the Muse of the Pythagoreans—I mean, the application of areas, and their exceeding and falling short.* (Eudemus [fr. 89 Spengel] in Proclus, Commentary on Euclid419.15–17 Friedlein)

T34 (DK 58B1) Pythagoras … discovered the construction of the cosmic figures.* (Proclus, Commentary on Euclid 65.19 Friedlein)

T35 The Pythagoreans proposed the following elegant theorem about diameter and side numbers. When to a diameter there is added the side of which it is the diameter, it becomes a side, while the side, when added to itself and receiving its own diameter in addition as well, becomes a diameter. This is proved with the aid of a diagram by Euclid in the second book of the Elements. If a straight line is bisected and a straight line is added to it, the square on the whole line (that is, including the added line) plus the square on the added line by itself are together double the square on the half and of the square on the straight line made up of the half and the added line.* (Proclus, Commentary on Plato’s ‘Republic’ 2.27.11–22 Kroll)

T36 (DK 18A4) Concerning Hippasus, they say that he was a Pythagorean, and that because he was the first to publish and construct the sphere of twelve pentagons [the dodecahedron], he died at sea for this act of impiety.* They add that although he gained the reputation for this discovery, it really belongs, as does everything else, to ‘the master’. This is how they refer to Pythagoras, since they never call him by name. (Iamblichus, Pythagorean Life 88.13–19 Deubner)

T37 (KRS 280) Of those who practised Pythagorean philosophy, the acousmatics are admitted to be Pythagoreans by the others, but they withhold the title from the mathematicians, saying that their branch of study stems from Hippasus rather than Pythagoras … Those of the Pythagoreans who are concerned with mathematics, however, recognize the others as Pythagoreans, but claim that they are more deserving of the title. (Iamblichus, On General Mathematical Knowledge 76.19–77.2 Festa)

T38 (DK 58B37; KRS 446) Most of those who maintain that the universe is finite say that the earth lies at the centre, but with this the Pythagoreans, as they are known, from Italy, disagree. They say that there is fire in the centre, that the earth is one of the heavenly bodies, and that it is its motion around the centre that creates night and day. Moreover, they invent another earth, opposite to ours, which they call the ‘counter-earth’. (Aristotle, On the Heavens 293a18–24 Allan)

T39 Those who deny that the earth lies at the centre claim that it moves in a circle around the centre, and that it is not just the earth that does this, but also the counter-earth, as I have already mentioned. Some even think that there might be several such bodies in motion around the centre, which are invisible to us because the earth is in the way. This allows them to explain the greater frequency of lunar over solar eclipses: they say that each of these invisible bodies, and not just the earth, blocks the moon. (Aristotle, On the Heavens 293b18–25 Allan)

T40 (DK 44A16; KRS 447) Philolaus says that there is fire in the middle, around the centre, and he calls it the ‘hearth of the universe’ and the ‘house of Zeus’, ‘mother of the gods’, ‘altar, bond, and measure of nature’. Then again, he says, there is another fire surrounding the universe at the periphery. But he says that the centre is naturally primary, and that around the centre dance ten divine bodies—heaven, planets, and then the sun, and then under the sun the moon, and then under the moon the earth, and then under the earth the counter-earth, and last in this whole sequence the hearth-fire which is located around the centre. (Aëtius, Opinions 2.7.7 Diels)

T41 (DK 58B35; KRS 449) It is clear from what has been said that the notion that the movement of the heavenly bodies produces a harmony, because the sounds they make are concordant, is untrue, despite having been ingeniously and brilliantly expressed by its authors. The idea was that bodies that are large are bound to make a sound, since here on earth bodies far inferior in size and speed of movement make sounds. So given that the sun and moon and stars, in all their quantity and enormity of size, are moving at such a great speed, it is impossible, they claimed, for them not to produce an incredibly loud noise. Having made this assumption, and having also supposed that the speeds of the heavenly bodies, as judged by their distances, are in the same ratios as musical concordances, they claim that the sound produced by the circular motion of the heavenly bodies is harmonic. And they explain the apparent absurdity of our inability to hear this sound by claiming that the sound is present to us right from the moment of our birth, with the result that it is never distinguished by comparison with a contrasting silence. (Aristotle, On the Heavens 290b12–27 Allan)

T42 (DK 44A19; KRS 448) Philolaus the Pythagorean says that the sun is glass-like, so that it receives the direct light of the fire in the universe and filters its light and heat to us.* This means that in a sense there are two suns, the fiery one in the heavens and the one which is dependent on it and is fiery in a mirror-like way—unless one were to say that there is also a third, which is the light that is spread from the mirror to us by reflection. For this light too we call a sun, and it is, so to speak, the image of an image. (Aëtius,Opinions 2.20.12 Diels)

T43 (DK 44A20) Some of the Pythagoreans, including Philolaus, say that the moon looks like the earth because it is inhabited, just like our earth, but by creatures and plants which are taller and more beautiful; for creatures there are fifteen times as strong as those here, and never excrete anything, and their day is fifteen times longer than ours here. (Aëtius, Opinions 2.30.1 Diels)

T44 (DK 58B30; KRS 443) The Pythagoreans also claim that there is such a thing as void. According to them, it enters the universe from the infinite breath because the universe breathes in void as well as breath. What void does, they say, is differentiate things; they think of void as being a kind of separation and distinction when one thing comes after another. This happens first among the numbers, because on their view it is the void that distinguishes one number from another.* (Aristotle, Physics 213b22–7 Ross)

T45 (DK 58B30; KRS 444) In the first book of his work on Pythagorean philosophy Aristotle writes that the universe is one, and that time and breath and the void, which differentiates the places of all individual things, are drawn into the universe from the unlimited. (Aristotle [fr. 201 Rose] in John of Stobi, Anthology 1.18.1c Wachsmuth/Hense)

F2 (DK 44B1; KRS 424) Nature in the universe was harmonized out of both things which are unlimited and things which limit; this applies to the universe as a whole and to all its components. (Philolaus [fr. 1 Diels/Kranz] in Diogenes Laertius, Lives of Eminent Philosophers 8.85.13–14 Long)

F3 (DK 44B2; KRS 425) All the things that exist must be either limiting or unlimited, or both limiting and unlimited. But they cannot be only unlimited. So since they evidently arise neither from things that are all limiters nor from things that are all unlimited, it clearly follows that the universe and its components were harmonized out of both things which limit and things which are unlimited. And the facts of things also make this clear, since some things arise from limiters and are limiters, while others arise from both limiters and unlimiteds and both limit and fail to impose limit, and others arise from unlimiteds and are plainly unlimited.* (Philolaus [fr. 2 Diels/Kranz] in John of Stobi, Anthology 1.21.7a Wachsmuth/Hense)

F4 (DK 44B4; KRS 427) And everything which is known has number, because otherwise it is impossible for anything to be the object of thought or knowledge. (Philolaus [fr. 4 Diels/Kranz] in John of Stobi, Anthology 1.21.7b Wachsmuth/Hense)

F5 (DK 44B6; KRS 429) On the subject of nature and harmony, this is how things stand: the being of things, qua eternal, and nature itself are accessible only to divine and not human knowledge—except that it is impossible for any of the things that exist and are known by us to have arisen without the prior existence of the being of the things out of which the universe is composed, namely limiters and unlimiteds. Now, since these sources existed in all their dissimilarity and incompatibility, it would have been impossible for them to have been made into an orderly universe unless harmony had been present in some form or other. Things that were similar and compatible had no need of harmony, but things that were dissimilar and incompatible and incommensurate had to be connected by this kind of harmony, if they are to persist in an ordered universe. (Philolaus [fr. 6 Diels/Kranz] in John of Stobi, Anthology 1.21.7d Wachsmuth/Hense)

F6 (DK 44B7; KRS 441) The first thing to be harmonized, the one, in the centre of the sphere, is called the hearth. (Philolaus [fr. 7 Diels/Kranz] in John of Stobi, Anthology 1.21.8 Wachsmuth/Hense)

F7 (DK 44B17) The universe is single. It originally arose from the centre, and from the centre upwards and downwards in the same way. For what is above the centre is the opposite in disposition to what is below, in the sense that to lower things the lowest part is like the highest part, and the same goes for the upper things too. For the relation to the centre is the same in either case, except that their positions are reversed.* (Philolaus [fr. 17 Diels/Kranz] in John of Stobi, Anthology 1.15.7 Wachsmuth/Hense)

T46 (DK 44A12) Philolaus says that after mathematical magnitude has become three-dimensional thanks to the tetrad [i.e. has progressed to solidity from the primary point (1), line (2), and plane figure (triangle, 3)], there is the quality and ‘colour’ of visible nature in the pentad, and ensoulment in the hexad, and intelligence and health and what he calls ‘light’ in the hebdomad, and then next, with the ogdoad, things come by love and friendship and wisdom and creative thought. (Ps.-Iamblichus, The Theology of Arithmetic74.10–15 de Falco)

T47 (DK 58B40; KRS 450) The doctrine handed down by the Pythagoreans seems to have the same purport [that respiration is the prerequisite for life], since some of them identified the soul with the motes in the air, while others said that the soul was what caused these motes to move. The reason for the importance of these motes in their theory is that they are apparently in continuous motion, even when there is not the slightest breath of wind. (Aristotle, On the Soul 404a16–20 Ross)

T48 (DK 44A23; KRS 451) There is another theory about the soul that has come down to us, which many people find the most plausible one around … They say that the soul is a kind of attunement (harmonia), on the grounds that attunement is a mixture and compound of opposites, and the body is made up of opposites. (Aristotle, On the Soul 407b27–32 Ross)

T49 (DK 44A27; KRS 445) Philolaus of Croton says that our bodies are composed of heat and have no share in cold. The evidence he adduces for this is as follows. Semen is warm, and it is semen that is constitutive of a living creature; and the place where semen is deposited—that is, the womb—is warmer. The womb resembles semen, and anything that is like anything else has the same property as that which it resembles. Since the constitutive agent has no share in cold and the place where it is deposited has no share in cold, it obviously follows that the living creature which is constituted will be of the same kind. With regard to its constitution he refers to the following facts. Immediately after birth a living creature inhales the external air, which is cold, and then expels it again, as if it were discharging a debt. Also, the reason why it has an instinctive appetite for the external air is to enable our bodies, which are too hot, by drawing in the air from outside, to be cooled by it. This is the way in which he describes the composition of our bodies.

As for diseases, he says that they arise as a result of bile, blood and phlegm, which are the sources of diseases. He says that blood is thickened when the flesh is compressed internally, and thinned when the vessels in the flesh are dilated. He says that phlegm is composed of the waters of the body. He says that bile is a discharge from flesh … While most people claim that phlegm is cold, he supposes that it is hot by nature, and derives the word ‘phlegm’ from phlegein, to burn. So, he says, it is because they have a share in phlegm that inflammatory agents cause inflammation. These are the sources of diseases, according to him. Secondary causes, he says, are either excess or lack of warmth, food, cold, and so on. (Meno in Anonymus Londinensis, 18.20–19.21 Jones)

F8 (DK 44B13) There are four sources of a rational creature (as Philolaus also says in On Nature)—brain, heart, navel, and genitals: ‘Head for thought, heart for soul and for feeling, navel for the embryo to take root and to grow, genitals for the emission of seed and for birth. The brain provides the source for man, the heart for animals, the navel for plants, the genitals for them all; for they all both sprout and grow from seed.’ (Philolaus [fr. 13 Diels/Kranz] in Ps.-Iamblichus, The Theology of Arithmetic 25.17–26.3 de Falco)

T50 (DK 16A1) He [a non-Greek sage met by Cleombrotus, one of the participants in this dialogue of Plutarch] said that the number of worlds is not infinite, nor one, nor five, but 183, arranged in a triangle of which each side has sixty worlds. Each of the three remaining worlds is situated at an angle. The worlds that are next to one another are contiguous and revolve gently, as in a dance. The interior of the triangle is the common hearth of all the worlds, and is called the plain of truth, in which lie unchanging the essences, forms, and patterns of things past and future. Around them time is communicated to the worlds like an effluence from eternity. Human souls may see and contemplate these things once in 10,000 years, provided they have lived well. The best mystery rites on earth are only a shadow of that initiation and rite. If our philosophical discussions are not conducted with a view to recollecting the beauties there, they are in vain … But he is convicted by the number of his worlds, which is not Egyptian or Indian, but Dorian, from Sicily, the idea of a man from Himera called Petron. Now, I have not read his work and I do not know if it has been preserved, but Hippys of Rhegium, according to Phanias of Eresus, reports that this was the opinion and teaching of Petron, that there are 183 worlds in contact with one another according to element. But what ‘contact according to element’ means he does not make clear, nor does he add any proof. (Plutarch, On the Decline of Oracles 422b3-e6 Babbit)

T51 (DK 45A3; KRS 433) Nothing at all clear has been said about how numbers are the causes of substantial things and of being. Is it that they are limits, as points are the limits of magnitudes? This is how Eurytus used to arrange things, to see what was the number of what—that such-and-such is the number of a human being, and such-and-such the number of a horse. In the way that people adduce numbers to explain the shapes of a triangle or a square, he used to make likenesses of the forms of creatures and plants with his pebbles. (Aristotle, Metaphysics 1092b8–13 Ross)

T52 (DK 45A3) Suppose, for the sake of argument, that 250 is the definition of a human being and 360 of a plant. On this assumption he used to take 250 pebbles (green, black, red, and all sorts of colours), smear the wall with plaster, draw an outline of a man (or a plant), and then fix some of the pebbles on the outline of the face, others on the hands, others elsewhere, and he would fill in the outline of the whole imitation human being with pebbles equal in number to the units which he said defined a human being. (Ps.-Alexander, Commentary on Aristotle’s ‘Metaphysics’, CAG 1, 827.9–17 Hayduck)

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R. S. Brumbaugh and J. Schwartz, ‘Pythagoras and Beans: A Medical Explanation’, Classical World, 73 (1980), 421–3.

W. Burkert, Lore and Science in Ancient Pythagoreanism (Cambridge, Mass.: Harvard University Press, 1972).

A. Burns, ‘The Fragments of Philolaus and Aristotle’s Account of Pythagorean Theories in Metaphysics A’, Classica et Mediaevalia, 25 (1964), 93–128.

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