Pythagoreanism

"If someone associates with a true Pythagorean, what will he will get from him, and in what quantity? I would say: statesmanship, geometry, astronomy, arithmetic, harmonics, music, medicine, complete and god-given prophecy, and also the higher rewards — greatness of mind, of soul, and of manner, steadiness, piety, knowledge of the gods and not just supposition, familiarity with blessed spirits and not just faith, friendship with both gods and spirits, self-sufficiency, persistence, frugality, reduction of essential needs, ease of perception, of movement, and of breath, good color, health, cheerfulness, and immortality."

"It seems to me that they do well to study mathematics, and it is not at all strange that they have correct knowledge about each thing, what it is. For if they knew rightly the nature of the whole, they were also likely to see well what is the nature of the parts. About geometry, indeed, and arithmetic and astronomy, they have handed us down a clear understanding, and not least also about music. For these seem to be sister sciences; for they deal with sister subjects, the first two forms of being."

"They [the Pythagoreans] say the things themselves are Numbers and do not place the objects of mathematics between forms and sensible things. ...Since again, they saw that the modifications and the ratios of the musical scales were expressible in numbers—since, then, all other things seemed in their whole nature to be modelled on numbers, and numbers seemed to be the first things in the whole of nature, they supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number... and the whole arrangement of the heavens they collected and fitted into their scheme; and if there was a gap anywhere, they readily made additions so as to make their whole theory coherent."

"These thinkers seem to consider that number is the principle both as matter for things and as constituting their attributes and permanent states."

"They thought they found in numbers, more than in fire, earth, or water, many resemblances to things which are and become; thus such and such an attribute of numbers is justice, another is soul and mind, another is opportunity, and so on; and again they saw in numbers the attributes and ratios of the musical scales. Since, then, all other things seemed in their whole nature to be assimilated to numbers, while numbers seemed to be the first things in the whole of nature, they supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number."

"It has fallen to the lot of one people, the ancient Greeks, to endow human thought with two outlooks on the universe neither of which has blurred appreciably in more than two thousand years. ...The first was the explicit recognition that proof by deductive reasoning offers a foundation for the structure of number and form. The second was the daring conjecture that nature can be understood by human beings through mathematics, and that mathematics is the language most adequate for idealizing the complexity of nature into appreciable simplicity. Both are attributed by persistent Greek tradition to Pythagoras in the sixth century before Christ. ...there is an equally persistent tradition that it was Thales... who first proved a theorem in geometry. But there seems to be no claim that Thales... proposed the inerrant tactic of definitions, postulates, deductive proof, theorem as a universal method in mathematics. ...in attributing any specific advance to Pythagoras himself, it must be remembered that the Pythagorean brotherhood was one of the world's earliest unpriestly cooperative scientific societies, if not the first, and that its members assigned the common work of all by mutual consent to their master."

"None of Pythagoras' own work has survived, but the ideas fathered on him by his followers would be the most potent in modern history. Pure knowledge, the Pythagoreans argued, was the purification (catharsis) of the soul... rising above the data of the human senses. The pure essential reality... was found only in the realm of numbers. The simple, wonderful proportion if numbers would explain the harmonies of music... [T]hey introduced the musical terminology of the octave, the fifth, the fourth, expressed as 2:1, 3:1, and 4:3. ..."

"In Copernicus' time Pythagoreans still believed that the only way to truth was by mathematics."

"The Pythagorean mathematical concepts, abstracted from sense impressions of nature, were... projected into nature and considered to be the structural elements of the universe. [Pythagoreans] attempted to construct the whole heaven out of numbers, the stars being... material points. ...they identified the regular geometric solids... with the different sorts of substances in nature. ...This confusion of the abstract and the concrete, of rational conception and empirical description, which was characteristic of the whole Pythagorean school and of much later thought, will be found to bear significantly on the development of the concepts of calculus. It has often been inexactly described as mysticism, but such stigmatization appears to be somewhat unfair. Pythagorean deduction a priori having met with remarkable success in its field, an attempt (unwarranted...) was made to apply it to the description of the world of events, in which the Ionian hylozoistic interpretations a posteriori had made very little headway. This attack on the problem was highly rational and not entirely unsuccessful, even though it was an inversion of the scientific procedure, in that it made induction secondary to deduction."

"Ionian philosophers... had sought to identify a first principle for all things. Thales had thought to find this in water, but others preferred to think of air or fire as the basic element. The Pythagoreans had taken a more abstract direction, postulating that number... was the basic stuff behind phenomena; this numerical atomism... had come under attack by the followers of Parmenides of Elea... The fundamental tenet of the was the unity and permanence of being... contrasted with the Pythagorean ideas of multiplicity and change. Of Parmenides' disciples the best known was Zeno the Eleatic... who propounded arguments to prove the inconsistency in the concepts of multiplicity and divisibility."

"We may... go to our... statement from Aristotle's treatise on the Pythagoreans, that according to them the universe draws in from the Unlimited time and breath and the void. The cosmic nucleus starts from the unit-seed, which generates mathematically the number-series and physically the distinct forms of matter. ...it feeds on the Unlimited outside and imposes form or limit on it. Physically speaking this Unlimited is [potential or] unformed matter... mathematically it is extension not yet delimited by number or figure. ...As apeiron in the full sense, it was... duration without beginning, end, or internal division—not time, in Plutarch's words, but only the shapeless and unformed raw material of time... As soon... as it had been drawn or breathed in by the unit, or limiting principle, number is imposed on it and at once it is time in the proper sense. ...the Limit, that is the growing cosmos, breathed in... imposed form on sheer extension, and by developing the heavenly bodies to swing in regular, repetitive circular motion... it took in the raw material of time and turned it into time itself."

"It is certain that the Theory of Numbers originated in the school of Pythagoras."

"Those who dwelt in the common auditorium adopted this oath: "I swear by the discoverer of the Tetraktys, which is the spring of all our wisdom; The perennial fount and root of Nature.""

"The tetrad was called by the Pythagoreans every number, because it comprehends in itself all the numbers as far as to the decad, and the decad itself; for the sum of 1, 2, 3, and 4, is 10. Hence both the decad and the tetrad were said by them to be every number; the decad indeed in energy, but the tetrad in capacity. The sum likewise of these four numbers was said by them to constitute the tetractys, in which all harmonic ratios are included. For 4 to 1, which is a quadruple ratio, forms the symphony bisdiapason; the ratio of 3 to 2, which is sesquialter forms the symphony diapente; 4 to 3, which is sesquitertian, the symphony diatessaron; and 2 to 1, which is a duple ratio, forms the diapason."

"Nicomachus... mentions the customary Pythagorean divisions of quantum and the science that deals with each. Quantum is either discrete or continuous. Discrete quantum in itself considered, is the subject of Arithmetic; if in relation, the subject of Music. Continuous quantum, if immovable, is the subject of Geometry; if movable, of Spheric (Astronomy). These four sciences formed the of the Pythagoreans. With the (which Nicomachus does not mention) of Grammar, Logic, and Rhetoric, they composed the seven liberal arts taught in the schools of the Roman Empire."

"The Neo-Pythagoreans treated all the divisions of philosophy. In Metaphysics they held that the Unit and the (indeterminate) Two are the basis of all things. the Unit being the form, and the Two the matter. ...The Unit being the prior principle may be identified with Deity, and, as such, was thought of either as the former [creator] of indefinite matter into individual things, or, as in Neo-Platonism, as the transcendent origin of the derivative Unit and Two. Another mode of conception was to identify the numbers with the Platonic Ideas and then to think of the Unit as comprehending them in the same manner as the mind comprehends its thoughts and gives them form. In Logic the Neo-Pythagoreans were for the most part imitators of Aristotle. Their Physics was Aristotelian and Stoic. Their Anthropology was Platonic. In Ethics and Politics they merely reechoed the Academy and the Lyceum with Stoic additions. In all this Neo-Pythagoreanism has little originality."

"Why was the Tetraktys so revered? Because to the eyes of the sixth century BC Pythagoreans, it seemed to outline the entire nature of the universe. In geometry — the springboard to the Greeks' epochal revolution in thought — the number 1 represented a point... 2 represented a line... 3 represented a surface... and 4 represented a three-dimensional tetrahedral solid... The Tetraktys, therefore appeared to encompass all the perceived dimensions of space."

"On the question whether mathematics was discovered or invented, Pythagoras and the Pythagoreans had no doubt — mathematics was real, immutable, omnipresent, and more sublime than anything that could conceivably emerge from the human mind. The Pythagoreans literally embedded the universe into mathematics. In fact, to the Pythagoreans, God was not a mathematician — mathematics was God! ...By setting the stage, and to some extent the agenda, for the next generation of philosophers — Plato in particular — the Pythagoreans established a commanding position in Western thought."

"As a moral philosopher, many of his precepts relating to the conduct of life will be found in the verses which bear the name of the Golden Verses of Pythagoras. It is probable they were composed by some one of his school, and contain the substance of his moral teaching. The speculations of the early philosophers did not end in the investigation of the properties of number and space. The Pythagoreans attempted to find, and dreamed they had found, in the forms of geometrical figures and in certain numbers, the principles of all science and knowledge, whether physical or moral. The figures of Geometry were regarded as having reference to other truths besides the mere abstract properties of space. They regarded the unit, as the point; the duad, as the line; the triad, as the surface; and the tetractys, as the geometrical volume. They assumed the pentad as the physical body with its physical qualities. They seem to have been the first who reckoned the elements to be five in number, on the supposition of their derivation from the five regular solids. They made the cube, earth; the pyramid, fire; the octohedron, air; the icosahedron, water; and the dodecahedron, aether. The analogy of the five senses and the five elements was another favourite notion of the Pythagoreans."

"Almost all the theories, religious philosophical and mathematical, taught by the Pythagoreans, were known in India in the sixth century BCE, and the Pythagoreans, like the Jains and the Buddhists, refrained from the destruction of life and eating meat."

"While most s emphasized the reality of change — in particular, the Atomists, followers of and Democritus — the Pythagoreans stressed the study of the unchangeable elements in nature and society. In their search for the eternal laws of the universe they studied geometry, arithmetic, astronomy, and music (the '). Their most outstanding leader was Archytas of Tarentum...and to whose school, if we follow... E. [Eva] Frank, much of the Pythagorean brand of mathematics may be ascribed. ...Numbers were divided into classes: odd, even, even-times-even, odd-times-odd, prime and composite, perfect, friendly, triangular, square, pentagonal, etc. ...Of particular importance was the ratio of numbers (logos, Lat. ratio). Equality of ratio formed a proportion. They discriminated between an arithmetical (2b = a + c), geometrical (b^2 = ac), and a harmonical (\frac{2}{b} = \frac{1}{a} + \frac{1}{c}) proportion that they interpreted philosophically and socially."

"The Pythagoreans knew some properties of s... how a plane can be filled by... regular triangles, squares, or regular hexagons, and space by cubes... [They] may also have known the regular oktahedron and dodekahedron—the latter figure because pyrite, found in Italy, crystallizes in dodekahedra, and models... date to Etruscan times."

"[T]he most striking result of the Greeks' faith that the world could be understood in terms of rational principles was the invention of abstract mathematics. The most grandiose ambition they conceived was to explain all the properties of Nature in arithmetical terms alone. This was the aim of the Pythagoreans... [T]hey... knew that the phenomena of the Heavens recurred in a cyclical manner; and... discovered ...that the sound of a vibrating string ...is simply related to the length ...and its 'harmonics' always go with simple fractional lengths. ...[S]ince the Pythagoreans were a religious brotherhood... they thought that this search would lead to more than explanations alone. If one discovered the mathematical harmonies in things, one should... discover how to put oneself in harmony with Nature. ...[T]hey had ...positive grounds for thinking that both astronomy and acoustics were at the bottom arithmetical; and the study of simple fractions was called 'music' right down until the late Middle Ages."

"It is usually maintained that the Platonic or Socratic philosophy, like the rest of Greek speculation, was original, indigenous, owing very little to any outside influence. But the quest and life and faith of Socrates were as un-Greek as anything could possibly be: that was one of the reasons why the Greeks killed him: the essence of his life belonged to a world unknown to them, and therefore dangerous in their eyes […] There is only one “philosopher” whose doctrines, both practical and theoretical, appear to have resembled Plato’s in spirit and aim as well as in substance; and that one is Pythagoras. It is noteworthy that Pythagoras is the only great thinker of Greece whom Plato never criticises, but of whom he speaks with the greatest deference and respect […] instancing him as the great example of a teacher whose teaching had in it living truth enough to inspire a band of devoted disciples, and to transform their lives as well as their beliefs. And every one of those doctrines, which we know formed the “gospel” of Pythagoras and of the Pythagorean brotherhood at Crotona, was an almost exact reproduction of the cardinal doctrines of the Indian Vidya and the Indian Yoga—so much so that Indian Vedantists today do not hesitate to claim Pythagoras as one of themselves, one of their great expounders, whose very name was only the Greek form of the Indian title Pitta Guru, or Father-Teacher.’"

"It has been no easy task to revise this volume in such a way as to make it more worthy of the favour with which it has been received. Most of it has had to be rewritten in the light of certain discoveries made since the publication of the first edition, above all, that of the extracts from Menon’s Iατρικά, which have furnished, as I believe, a clue to the history of Pythagoreanism."

"[T]he authority of Anaximenes was so great that both Leukippos and Demokritos adhered to his theory of a disc-like earth. ...This, in spite of the fact that the spherical form of the earth was already a commonplace in circles affected by Pythagoreanism."

"The main purpose of the Orgia was to "purify" the believer’s soul, and so enable it to escape from the "wheel of birth," and it was for... this end that the Orphics were organised in communities. Religious associations must have been known to the Greeks from a fairly early date; but the oldest of these were based... 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. This was, in fact, the establishment of churches, though there is no evidence that these were connected... such... that we could rightly speak of them as a single church. The Pythagoreans came nearer to realising that."

"[T]he religious revival... 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’s Phaedo, which was written under the influence of Pythagorean ideas."

"The Phaedo is dedicated... to Echekrates and the Pythagorean society at Phleious, and it is evident that Plato in his youth was impressed by the religious side of Pythagoreanism, though the influence of Pythagorean science is not clearly marked till a later period."

"[A] good many fragments of... Aristoxenos and Dikaiarchos are embedded in the mass. These writers were both disciples of Aristotle; they were natives of Southern Italy, and contemporary with the last generation of the Pythagorean school. Both wrote accounts of Pythagoras; and Aristoxenos, who was personally intimate with the last representatives of scientific Pythagoreanism, also made a collection of the sayings of his friends."

"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... The distinction of grades within the Order, variously called Mathematicians and Akousmatics, Esoterics and Exoterics, Pythagoreans and Pythagorists, is an invention designed to explain how there came to be two widely different sets of people, each calling themselves disciples of Pythagoras, in the fourth century B.C. So, too, the statement that the Pythagoreans were bound to inviolable secrecy, which goes back to Aristoxenos, 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... and not, as has sometimes been maintained, a political league. Nor had it anything to do with the "Dorian aristocratic ideal." Pythagoras was an Ionian, and the Order was originally confined to Achaian states. Nor is there the slightest evidence that the Pythagoreans favoured the aristocratic rather than the democratic party. The main purpose... was to secure for... members a more adequate satisfaction of the religious instinct than... the State religion. It was... an institution for the cultivation of holiness. ...[I]t 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. ...[H]owever, an independent society within a Greek state was apt to be brought into conflict with the larger body. The only way in which it could then assert its right to exist was... by securing the control of the sovereign power. The history of the Pythagorean Order... is, accordingly, the history of an attempt to supersede the State..."

"When discussing the Pythagorean system, Aristotle always refers it to "the Pythagoreans," not to Pythagoras himself. ...[T]his was intentional ...Pythagoras himself is only thrice mentioned in the whole Aristotelian corpus, and in only one... is any philosophical doctrine ascribed to him. ...Aristotle ...is quite clear that what he knew as the Pythagorean system belonged in the main to the days of Empedokles, Anaxagoras, and Leukippos; for ...he goes on to describe the Pythagoreans as "contemporary with and earlier than them.""

"The Pythagoreans held, [Aristotle] tells us that there was "boundless breath" outside the heavens, and that it was inhaled by the world. In substance, this is the doctrine of Anaximenes, and... it was that of Pythagoras... Xenophanes denied it. ...[F]urther development of the idea is ...due to Pythagoras ...We are told that, after the first unit had been formed ...the nearest part of the Boundless was first drawn in and limited; and... the Boundless thus inhaled... keeps the units separate from each other. It represents the interval between them. This is a... primitive way of describing... discrete quantity."

"In... Aristotle... the Boundless is also... the void or empty. This identification of air and the void is a confusion... in Anaximenes... too. We find also... the other confusion... air and vapour. ...Pythagoras identified the Limit with fire, and the Boundless with darkness. We are told by Aristotle that Hippasos made Fire the first principle... Parmenides... attributes... two primary "forms," Fire and Night. ...Light and Darkness appear in the Pythagorean table of opposites under the heads of the Limit and the Unlimited respectively."

"The identification of breath with darkness ...is a strong proof of the primitive character of the doctrine; for in the sixth century darkness was supposed ...a sort of vapour, while in the fifth, its true nature was ...known. Plato... makes the Pythagorean Timaios describe mist and darkness as condensed air."

"[T]hink, then, of a "field" of darkness or breath marked out by luminous units ...which the starry heavens would naturally suggest."

"It is... probable that we should ascribe to Pythagoras the Milesian view of a plurality of worlds, though... not... infinite ...Petron, one of the early Pythagoreans, said there were ...a hundred and eighty-three worlds arranged in a triangle; and Plato makes Timaios admit, when laying down ...only one world, that something might be urged in favour of ...five, as there are five regular solids."

"Simplicius, with the poem of Parmenides before him, corrects Aristotle by substituting Light and Darkness for Fire and Earth... Parmenides... calls one "form" Light, Flame, and Fire, and the other Night, and we... consider whether these can be identified with the Pythagorean Limit and Unlimited. We have... reason to believe that... the world breathing belonged to the earliest form of Pythagoreanism, and... identifying this "boundless breath" with Darkness, which stands... for the Unlimited. "Air" or mist was always regarded as the dark element. And that which gives definiteness to the vague darkness is... light or fire, and this may account for the prominence given to that element by Hippasos. We may probably conclude... that the Pythagorean distinction between the Limit and the Unlimited... made its first appearance in this crude form. If... we identify darkness with the Limit, and light with the Unlimited, as most critics do, we get into insuperable difficulties."

"In the fourth century, the chief seat of the school is at Taras, and we find the Pythagoreans heading the opposition to Dionysios of Syracuse. ...[In] this period... Archytas... the friend of Plato... almost realised, if he did not suggest, the ideal of the . ...He was also the inventor of mathematical mechanics."

"At the same time, Pythagoreanism had taken root in Hellas. Lysis... remained at Thebes, where Simmias and Kebes had heard Philolaos, and there was an important community of Pythagoreans at Phleious. Aristoxenos was personally acquainted with the last generation of the school, and mentioned by name Xenophilos the Chalkidian from , with Phanton, Echekrates, Diokles, and Polymnestos of Phleious. They were all, he said, disciples of Philolaos and Eurytos. Plato was on friendly terms with these men, and dedicated the Phaedo to them. Xenophilos was the teacher of Aristoxenos..."

"It seems natural to suppose... the Pythagorean elements of Plato’s Phaedo and Gorgias come mainly from Philolaos. Plato makes Sokrates express surprise that Simmias and Kebes had not learnt from him why it is unlawful for a man to take his [own] life, and it seems to be implied that the Pythagoreans at Thebes used the word "philosopher" in the... sense of... seeking to find... release from the burden of this life? It is... probable that Philolaos spoke of the body... as the tomb... of the soul. ...[H]e taught the old Pythagorean religious doctrine in some form, and... likely... laid stress upon knowledge as a means of release. ...Plato ...is by far the best authority ...on the subject."

"We know... Philolaos wrote on "numbers"; for Speusippos followed him in the account he gave of the Pythagorean theories on that subject. It is probable... he busied himself... with arithmetic, and... his geometry was... primitive... Eurytos was his disciple, and... his views were... crude."

"Philolaos wrote on medicine, and... while... influenced by the... Sicilian school, he opposed them from the Pythagorean standpoint. ...[H]e said... our bodies were composed only of the warm... [O]nly after birth... the cold was introduced by respiration. The connexion... with the old Pythagorean theory is obvious. Just as the Fire in the macrocosm draws in and limits the cold dark breath which surrounds the world... so do our bodies inhale cold breath... Philolaos made , blood, and the causes of disease..."

"Philolaos... is a sufficiently remarkable figure... and has... been spoken of as a "precursor of Copernicus.""

"Plato was intimate with these men and was deeply impressed by their religious teaching, though... he did not adopt it... He was still more attracted by the scientific side of Pythagoreanism, and... this exercised a great influence on him. His own system in its final form had many points of contact with it, as he is careful to mark in the ' But... he is apt to develop Pythagoreanism on lines of his own, which may or may not have commended themselves to Archytas, but are no guide to the views of Philolaos and Eurytos. He is not careful... to claim the authorship of his own improvements in the system. He did not believe that cosmology could be an exact science, and he... therefore... credit[s] Timaios the Lokrian, or "ancient sages"... with theories which... had their birth in the Academy."

"Plato had many enemies and detractors, and this literary device enabled them to bring against him the charge of plagiarism. Aristoxenos... made the extraordinary statement that most of the Republic was... found in a work by Protagoras. ...He seems also... the... source of the story that Plato bought "three Pythagorean books" from Philolaos and copied the Timaeus out of them. ...[A]ccounts... imply... Plato bought... either a book by Pythagoras, or... notes of his teaching..."

"We know nothing of Timaios except what Plato tells us... and he may... be a fictitious character like the Eleatic Stranger."

"We are told that the other book which passed under the name of Pythagoras was really by Lysis."

"[W]e have... testimony that the five "Platonic figures,"... were discovered in the Academy. In the to Euclid... Pythagoreans only knew the , the pyramid (), and the , while the and the were discovered by Theaitetos."

"This sufficiently justifies... regarding the "fragments of Philolaos" with... more than suspicion."

"[W]e cannot safely take Plato as our guide to the original meaning of the Pythagorean theory, though... from him alone... we can learn to regard it sympathetically."

"Aristotle... was... out of sympathy with Pythagorean ways of thinking, but took... great... pains to understand them. This was... because they played so great a part in the philosophy of Plato and his successors, and he had to make the relation of the two doctrines as clear as he could to... his disciples."

"[W]e have to... interpret what Aristotle tells us in the spirit of Plato, and... consider how the doctrine... is related to the systems which had preceded it. ...[This] delicate operation... has been made... safer by recent discoveries in the early history of mathematics and medicine."

"Platonic elements which have crept into later accounts... are of two kinds. First... genuine Academic formulae... as... identification of the Limit and the Unlimited with the One and the Indeterminate Dyad; ...secondly ...the Neoplatonic doctrine which represents it as an opposition between God and Matter. ...[N]o one will any longer attribute these doctrines to the Pythagoreans of the fifth century."

"[T]he problem... is still extremely difficult."

"According to Aristotle, the Pythagoreans said Things are numbers, though that does not appear to be the doctrine of the fragments of "Philolaos." According to them, things have number, which make them knowable, while their real essence is... unknowable. ...[B]ut ...things are numbers seems meaningless. We have seen reason for believing that it is due to Pythagoras..., though we did not feel able to say... clearly what he meant..."

"There is no such doubt... [in] his school. Aristotle says they used the formula in a cosmological sense. The world... was made of numbers in the same sense as others had said it was made of "four roots" or "innumerable seeds." It will not do to dismiss this as mysticism."

"Whatever we may think of Pythagoras, the Pythagoreans of the fifth century were scientific men, and they must have meant something... definite. ...[T]hey used the words Things are numbers in a ...non-natural sense, but there is no difficulty in such a supposition."

"The Pythagoreans had... a great veneration for the... words of the Master... but... veneration is often accompanied by a singular licence of interpretation."

"Aristotle is... decided in his opinion that Pythagoreanism was intended to be a cosmological system like the others. "Though the Pythagoreans... made use of less obvious s and elements than the rest, seeing that they did not derive them from sensible objects, yet all their discussions and studies had reference to nature alone. They describe the origin of the heavens, and they observe the phenomena of its parts, all that happens to it and all it does." They apply their first principles entirely to these things, "agreeing... with the other natural philosophers in holding that reality was just what could be perceived by the senses, and is contained within the compass of the heavens," though "the first principles and causes of which they made use were... adequate to explain realities of a higher order than the sensible.""

"The doctrine is more precisely stated by Aristotle to be that the elements of numbers are the elements of things, and... therefore things are numbers. He is equally positive that these "things" are sensible things, and... are... the bodies of which the world is constructed. This construction... out of numbers was a real process in time, which the Pythagoreans described in detail."

"[T]he numbers were intended to be mathematical... though... not separated from... things of sense. ...[T]hey were not mere predicates of something else, but had an independent reality... "They did not hold that the limited and the unlimited and the one were... substances, such as fire, water... [etc.,] but... the unlimited itself and the one itself were the reality of the things of which they are predicated, and that is why they said that number was the reality of everything.""

"Accordingly the numbers are, in Aristotle’s own language, not only the formal, but also the material, cause of things. According to the Pythagoreans, things are made of numbers in the same sense as they were made of fire, air, or water in the theories of their predecessors."

"Aristotle notes that the point in which the Pythagoreans agreed with Plato was in giving numbers an independent reality of their own; while Plato differed from the Pythagoreans in holding that this reality was distinguishable from that of sensible things."

"Aristotle speaks of certain "elements"... of numbers, which were also the elements of things. ...Primarily, the "elements of number" are the Odd and the Even... identified in a somewhat violent way with the Limit and the Unlimited... the original principles of the Pythagorean cosmology. Aristotle tells us... the Even... gives things their unlimited character when... contained in them and limited by the Odd... [C]ommentators... understand... this to mean... the Even is... the cause of infinite divisibility. They get into great difficulties, however..."

"Simplicius... preserved an explanation, in all probability Alexander’s... that they called the even number unlimited "because every even is divided into equal parts, and what is divided into equal parts is unlimited in respect of bipartition; for division into equals and halves goes on '. But, when the odd is added, it limits it; for it prevents its division into equal parts.""

"[W]e must not impute to the Pythagoreans... that even numbers can be halved indefinitely. They had... studied the properties of the decad, and... must have known that... 6 and 10 do not admit of this."

"In this way, then, the Odd and the Even were identified with the Limit and the Unlimited, and it is possible... Pythagoras... had taken this step... by... Unlimited he meant something spatially extended, and... identified... with air, night, or the void, so we are prepared to find... his followers also thought of the Unlimited as extended."

"Aristotle... argues... if the Unlimited is... a reality, and not merely the predicate of some other reality, then every part of it must be unlimited... just as every part of air is air. The same thing is implied in his statement that the Pythagorean Unlimited was outside the heavens. Further than this, it is hardly safe to go."

"Philolaos and his followers cannot have regarded the Unlimited in the old Pythagorean way as Air; for... they adopted the theory of Empedokles as to that "element," and accounted for it otherwise. ...[T]hey can hardly have regarded it as an absolute void; for that conception was introduced by the Atomists. ...[T]hey meant by the Unlimited the ', without analysing that... further."

"As the Unlimited is spatial, the Limit must be spatial too, and we should... expect... the point... line, and... surface were regarded as... forms of the Limit. That was the later doctrine; but the characteristic feature of Pythagoreanism is... that the point was not... a limit, but... the first product of the Limit and the Unlimited, and was identified with the arithmetical unit. According[ly]... the point has one dimension, the line two, the surface three, and the solid four... [i.e.,] Pythagorean points have magnitude... lines breadth, and... surfaces thickness. The whole theory... turns on the definition of the point as a unit “having position." ...[O]ut of such elements ...it seemed possible to construct a world."

"[T]his way of regarding the point... line, and... surface is closely bound... with... representing numbers by dots... in symmetrical patterns... attribut[ed]... to the Pythagoreans."

"The science of geometry had... made considerable advances, but the old view of quantity as a sum of units had not been revised... so... [ such a] doctrine... was inevitable."

"Aristotle is... decided as to Pythagorean points having magnitude. "They construct the whole world out of numbers... but they suppose the units have magnitude. As to how the first unit with magnitude arose, they appear to be at a loss.""

"Aristotle criticises the Pythagoreans. They held, he says, that in one part of the world Opinion prevailed, while a little above it or below it were to be found Injustice or Separation or Mixture, each... a number. But in the very same regions of the heavens were... things having magnitude which were also numbers. How can this be, since Justice has no magnitude? This means... the Pythagoreans... failed to give... clear account of the relation between these... fanciful analogies and their quasi-geometrical construction of the universe."

"[W]hat distinguished the Pythagoreanism of this period from its earlier form was that it sought to adapt... to the new theory of "elements." ...[T]his ...makes it necessary ...to take up ...consideration of the system ...in connexion with the pluralists."

"When the Pythagoreans returned to Southern Italy, they must have found views... there which... demanded a partial reconstruction of their own system. ...Empedokles founded a philosophical society, but ...influence[d] ...the medical school of these regions; and ...Philolaos played a part in the history of medicine."

"The tradition is that the Pythagoreans explained the elements as built up of geometrical figures, a theory... in the more developed form... attained in Plato’s Timaeus. If they were to retain their position as... leaders of medical study... they were bound to account for the elements."

"[T]he Pythagorean construction of the elements was... that... in Plato’s Timaeus. ...[T]here is good reason for believing they only knew three of the regular solids, the , the pyramid (), and the . Plato starts from fire and earth, and... the construction οf the elements proceeds... such... that the and the can easily be transformed into pyramids, while the cube and the dodecahedron cannot. ...[I]t follows that, while air and water pass readily into fire, earth cannot... and the dodecaedron is reserved for another purpose... This would... suit the Pythagorean system; for it would leave room for a dualism... outlined in the Second Part of the poem of Parmenides."

"Hippasos made Fire the first principle, and... from the Timaeus... it would be possible to represent air and water as forms of fire. The other element is... earth, not air, as... it was in early Pythagoreanism. That would be a... result of the discovery of atmospheric air by Empedokles and of his general theory of the elements. It would... explain the... fact... that Aristotle identifies the two "forms" spoken of by Parmenides with Fire and Earth."

"The most interesting point in the theory is... the use... of the ... identified... with the "sphere of the universe," or... in the Philolaic fragment, with the "hull of the sphere." ...[I]t must be taken in close connexion with the word "" applied to the central fire. The structure of the world was compared to the building of a ship..."

"In the Phaedo we read that the "true earth,"... looked at from above, is "many-coloured like the balls that are made of twelve pieces of leather." In the Timaeus... "Further, as there is still one construction left, the fifth, God made use of it for the universe when he painted it." ...[T]he approaches more nearly to the than any other of the regular solids. The twelve pieces of leather used to make a ball would... be s; and, if the material were not flexible like leather, we should have a dodecahedron instead of a sphere. This points to the Pythagoreans having had at least the rudiments of the "" formulated later by Eudoxos."

"They must have studied the properties of circles by means of inscribed polygons and those of spheres by means of inscribed solids. That gives us a high idea of their mathematical attainments; but that it is not too high, is shown by the fact that the famous lunules of Hippokrates date from the middle of the fifth century. The inclusion of straight and curved in the "table of opposites" under the head of Limit and Unlimited points in the same direction."

"The tradition confirms... the importance of the in the Pythagorean system. According to one account, Hippasos was drowned at sea for revealing its construction and claiming the discovery as his οwn."

"[T]he Pythagoreans adopted the pentagram or pentalpha as their symbol. The use... in later magic is well known; and Paracelsus... employed it as a symbol of health, which is... what the Pythagoreans called it."

"The view that the soul is a "harmony," or... attunement, is intimately connected with the theory of the four elements. It cannot have belonged to the earliest... Pythagoreanism; for... in Plato’s Phaedo, it is... inconsistent with the idea that the soul can exist independently of the body. It is... opposite of the belief that "any soul can enter any body." ...[F]rom the Phaedo... it was accepted by Simmias and Kebes, who had heard Philolaos at Thebes, and by Echekrates of Phleious, who was the disciple of Philolaos and Eurytos."

"The account of the doctrine given by Plato is... in accordance with the view that it was of medical origin. Simmias says: "Our body being... strung and held together by the warm and the cold, the dry and the moist... [etc.,] our soul is a sort of temperament and attunement of these, when... mingled... well and in due proportion. If, then, our soul is an attunement,... when the body has been relaxed or strung up out of measure by diseases and other ills, the soul must... perish at once." This is... an application of the theory of Alkmaion, and is in accordance with... the Sicilian school of medicine. It completes the evidence that the Pythagoreanism of the end of the fifth century was an adaptation of the old doctrine to the new principles introduced by Empedokles."

"The planetary system which Aristotle attributes to "the Pythagoreans" and Aetios to Philolaos is... remarkable. The earth is no longer in the middle of the world; its place... taken by a central fire, which is not... the sun. Round this fire revolve ten bodies. First comes the Antichthon or , and next the earth, which thus becomes one of the planets. After the earth comes the moon, then the sun, the five planets, and the heaven of the fixed stars. We do not see the central fire and the antichthon because... [our] side of the earth... is always turned away from them.., explained by the analogy of the moon. ...[M]en living on the other side of it would never see the earth. ...[A]ll these bodies rotate on their axes in the same time as they revolve round the central fire."

"Plato gives a description of the earth and its position... entirely opposed to... [antichthon theory], but is accepted... by Simmias the disciple of Philolaos. It is undoubtedly... Pythagorean... and marks... advance on the Ionian views then current at Athens. ...Plato states it as ...a novelty that the earth does not require ...support ...to keep it in its place. ...Anaxagoras had not been able to shake himself free of that idea, and Demokritos still held it."

"The... inference from the Phaedo would... be that the theory of a spherical earth, kept in the middle of the world by its equilibrium, was that of Philolaos... If so, the doctrine of the central fire would belong to a somewhat later generation of the school, and Plato may have learnt it from Archytas and his friends after he had written the Phaedo."

"[I]t is... incredible that the heaven of the fixed stars should have been regarded as stationary. That would have been the most startling paradox that any scientific man had yet propounded, and we should have expected the comic poets and popular literature generally to raise the cry of atheism... [W]e should have expected Aristotle to say something... He made the circular motion of the heavens the... keystone of his system, and would have regarded... a stationary heaven as blasphemous. ...[H]e argues against those who, like the Pythagoreans and Plato, regarded the earth as in motion; but he does not attribute the view that the heavens are stationary to any one. There is no necessary connexion between the two ideas. All the heavenly bodies may be moving as rapidly as we please, provided that their relative motions are such as to account for the phenomena."

"It seems probable that the... earth’s revolution round the central fire... originated in the account... by Empedokles of the sun's light. The two... are brought into... connexion by Aetios, who says... Empedokles believed in two suns, while Philolaos believed in two or... three. The theory of Empedokles... gives two inconsistent explanations of night."

"The central fire received a number of mythological names. ...[W]e are dealing with a real scientific hypothesis. It was a great thing... that the phenomena could best be "saved" by a central luminary, and that the earth must... be a revolving sphere like the planets. [W]e are almost tempted to say that the identification of the central fire with the sun... suggested for the first time in the Academy, is a mere detail in comparison. The great thing was that the earth should... take its place among the planets... once... done.., we can... search for the true "hearth" of the planetary system... It is probable... that... this theory... made it possible for Herakleides of Pontos and Aristarchos of Samos to reach the heliocentric hypothesis, and it was... Aristotle’s reversion to the geocentric theory which made it necessary for Copernicus to discover the truth afresh. We have his own word for it that the Pythagorean theory put him on the right track."

"The existence of the antichthon was... a hypothesis intended to account for... eclipses. ...Aristotle says that the Pythagoreans invented it... to bring the number of revolving bodies up to ten; but that is a... sally... Aristotle... knew better. In his work on the Pythagoreans... he said... eclipses of the moon were caused sometimes by.... the earth and sometimes by... the antichthon... the same statement was made by Philip of Opous..."

"Aristotle shows... how the theory originated... that some thought there might be a considerable number of bodies revolving round the centre, though invisible because of the intervention of the earth, and... they accounted... for there being more eclipses of the moon than of the sun. ...Aristotle regarded the two hypotheses as of the same nature."

"Anaximenes... assumed... existence of dark planets to account for the frequency of s, and Anaxagoras... revived that view. Certain Pythagoreans had placed these dark planets between the earth and the central fire... to account for their invisibility, and the next stage was to reduce them to a single body. ...[A]gain ...the Pythagoreans tried to simplify the hypotheses of ...predecessors."

"We must not assume ...Pythagoreans made the sun, moon, and planets, including the earth, revolve in the opposite direction to the heaven of the fixed stars. ...Alkmaion is said to have agreed with "some of the mathematicians" in holding this view, but it is never ascribed to Pythagoras or even to Philolaos."

"The old theory was that all... heavenly bodies revolved... from east to west, but that the planets revolved more slowly the further they were removed from the heavens, so... those... nearest the earth are "overtaken" by those that are further away. This view was... maintained by Demokritos, and that it was... Pythagorean... follow[s] from... the "harmony of the spheres." [W]e cannot attribute this theory in... later form to the Pythagoreans of the fifth century, but we have... testimony of Aristotle... that those Pythagoreans whose doctrine he knew believed... heavenly bodies produced s in their courses. ...[V]elocities of these bodies depended on the distances between... [which] corresponded to the intervals of the . He... implies that the heaven of the fixed stars takes part in the concert; for... "the sun, the moon, and the stars, so great in magnitude and in number as they are..." ...[T]he slower bodies give out a deep note and the swifter a high note."

"[P]revailing tradition gives the high note of the octave to the heaven of the fixed stars... [I]t follows that all the heavenly bodies revolve in the same direction, and... their velocity increases in proportion to their distance from the centre."

"The theory that the proper motion of the sun, moon, and planets is from west to east, and that they also share in the motion from east to west of the heaven of the fixed stars, makes its first appearance in the in Plato’s Republic, and is fully worked out in the Timaeus. In the Republic it is still associated with the "harmony of the spheres,"..."

"In the Timaeus... the slowest of the heavenly bodies appear the fastest and vice versa; and, as this... is... a Pythagorean [speaking], we might suppose the theory of a composite movement to have been anticipated by some... [in] that school."

"Pythagoreans were... open to new ideas."

"[T]he theory is... emphatically expressed by the Athenian Stranger in the Laws, who is... Plato... expounding a novel theory."

"[A] view... Aristotle sometimes attributes to the Pythagoreans... things were "like numbers." He does not appear to regard this as inconsistent with the doctrine that things are numbers..."

"Aristoxenos represented the Pythagoreans as teaching that things were like numbers, and there are other traces of an attempt to make... this... the original doctrine. A letter... purporting to be... Theano... wife of Pythagoras... says... she hears many of the Hellenes think Pythagoras said things were made of number, whereas he... said they were made according to number. ...[T]his fourth-century theory had to be explained away... later... and Iamblichos... tells... that it was Hippasos who said number was the of things."

"Aristotle seems to find only a verbal difference between Plato and the Pythagoreans. The metaphor of [numbers'] "participation" was merely substituted for that of [numbers'] "imitation." ...Aristotle’s ascription of the doctrine of "imitation" to the Pythagoreans is... justified by the Phaedo."

"The arguments for immortality ...come from various sources. Those derived from the doctrine of Reminiscence... sometimes... supposed... Pythagorean, are only known to the Pythagoreans by hearsay, and Simmias requires to have the whole psychology of the subject explained... When... we come to the question what it is that our sensations remind us of, his attitude changes. The view that the equal itself is alone real, and that what we call... things are imperfect imitations of it, is... familiar to him. He requires no proof... and is... convinced of the immortality of the soul... because Sokrates makes him see that the theory of forms implies it."

"Sokrates does not introduce the theory as a novelty. The reality of the "ideas" is the... reality "we are always talking about," and they are explained in a peculiar vocabulary... of a school."

"Whose theory is it? It is usually supposed... Plato’s... though nowadays it is... his "early theory of ideas,"... that he modified... profoundly in later life. But there are serious difficulties in this view."

"Plato... was not present at the conversation... in the Phaedo. Did any philosopher ever propound a new theory of his own by representing it as already familiar to... distinguished living contemporaries? It would be rash... to ascribe the theory to Sokrates, and there seems nothing... but to suppose that the doctrine of “forms” originally took shape in Pythagorean circles, perhaps under Sokratic influence. ... Simmias and Kebes were not only Pythagoreans but disciples of Sokrates; for... Xenophon has included them in his list of true Sokratics."

"We have... ground for believing... the Megarians had adopted a like theory under similar influences, and Plato states... that Eukleides and of were present at the conversation recorded in the Phaedo. ...[U]se of the words εἴδη and ἰδέαι to express ultimate realities is pre-Platonic, and it seems most natural to regard it as of Pythagorean origin."

"Parmenides had already called the original Pythagorean "elements" μορφαί, and Philistion called the "elements" of Empedokles ἰδέαι. If the ascription of this terminology to the Pythagoreans is correct, we may say that the Pythagorean "forms" developed into the atoms of Leukippos and Demokritos on the one hand, and into the "ideas" of Plato on the other."

"We... exceeded the limits... by tracing the history of Pythagoreanism... to... where it becomes practically indistinguishable from the earliest form of ; but it was necessary... to put the statements of our authorities in their true light."

"Aristoxenos is not likely... mistaken with regard to the opinions of the men he had known personally, and Aristotle’s statements must have had some foundation."

"We must assume... a later form of Pythagoreanism... was closely akin to early . [T]he fifth-century doctrine was of the more primitive type..."

"Whether or not we accept the hypothesis of direct influence from Persia on the Ionian Greeks in the sixth century, any student of Orphic and Pythagorean thought cannot fail to see that the similarities between it and Persian religion are so close as to warrant out regarding them as expressions of the same view of life, and using the one system to interpret the other. The characteristic preoccupation of Pythagoreanism with astronomy and the contemplation of the heavens becomes transparently clear, when we see it in the light of notions like , , and ."

"The School of Pythagoras, in our opinion, represents the main current of that mystical tradition which we have set in contrast with the scientific tendency. The terms 'mystical' and 'scientific,' ...are ...not to be understood as if ...all the philosophers we class as mystic were unscientific. The fact that we regard Parmenides, the discoverer of Logic, as an offshoot of Pythagoreanism, and Plato... as finding in the Italian philosophy the chief source of his inspiration, will be enough to refute such a misunderstanding. Moreover, the Pythagorean School... developed a scientific doctrine closely resembling the Milesian Atomism; and Empedocles, again, attempted to combine the two types of philosophy."

"Behind the School of Pythagoras, we can discern, in the socalled Orphic revival, one of these reformations of Dionysiac religion. ...[T]the Pythagorean philosophy... is always passing from mysticism to science, as its religion had passed from Dionysus to Apollo. Yet, philosophy and religion alike do not cease to be mystical at the root; and the attempt to hold the two ends together involves religion in certain contradictions, and leads philosophy to corresponding dilemmas..."

"[T]hroughout the mystical systems inspired by Orphism, we... find the fundamental contrast between... principles of Light and Darkness, identified with Good and Evil. This cosmic dualism is the counterpart of the dualism in the... soul; for... physis and soul... are... identical in substance. The soul in its pure state consists of fire, like the divine stars from which it falls; in its impure state, throughout... reincarnation, it... is infected with the baser elements, and weighed down... In the cosmologies... the manifold world of sense will be viewed as a degradation from the purity of real being. Such systems will tend to be other-worldly, putting all value in the unseen unity of God, and condemning the visible world as false and illusive, a turbid medium... obscured in mist and darkness. These characteristics are common to all the systems which came out of the Pythagorean movement—Pythagoreanism proper, and the philosophies of Parmenides, Empedocles, and Plato."

"The doctrines of mysticism are secret, because they are not cold, abstract beliefs, or articles in a creed, which can be taught and explained by intellectual processes... The 'truth' which mysticism guards is... only... learnt by being experienced (παθεῖν μαθεῖν); it is... not an intellectual, but an emotional experience—that invasive, flooding sense of oneness, of reunion and communion with... the life of the world... Being an emotional, non-rational state, it is indescribable, and incommunicable save by suggestion. To induce that state, by the stimulus of collective excitement and all the pageantry of dramatic ceremonial, is the aim of mystic ritual. The 'truth' can only come to those who submit themselves to these... because it is... to be immediately felt, not conveyed by dogmatic instruction. For that reason only... 'mysteries' are reserved to the initiate, who have undergone 'purification,' ...a state of mind which fits them for the consummate experience. Pythagoreanism presents... an attempt to intellectualise... Orphism, while preserving its social form, and... spirit... Orphism ceases to be a cult, and becomes a Way of life. As a revival, Pythagoreanism means a return to an earlier simplicity... simple enough to adapt itself to a new movement of the spirit. Pythagoreanism is... a complex phenomenon, containing the germs of several tendencies... philosophies that emerged from the school... separating towards divergent issues, or intertwined in ingenious reconciliations. Our analysis must take account of three strata, superimposed... Dionysus, Orpheus, Pythagoras. From Dionysus come the unity of all life, in the cycle of death and rebirth, and the conception of the or collective soul, immanent in the group as a whole, and yet something more than any or all... To Orpheus is due the shift of focus from earth to heaven, the substitution for the vivid, emotional experience of the renewal of life in nature, of the worship of a distant and passionless perfection in the region of light, from which the soul, now immortal, is fallen into the body of this death, and which it aspires to regain by the formal observances of asceticism. But the Orphic still clung to the emotional... reunion and... ritual that induced it, and... to the passionate spectacle (theoria) of the suffering God. Pythagoras gave a new meaning to theoria... as the passionless contemplation of rational, unchanging truth... a 'pursuit of wisdom' (philosophia). The way of life is still also a way of death; but now... death to the emotions and lusts... and a release of the intellect to soar into the untroubled of theory... by which the soul can 'follow God' (ἕπεσθαι θεῷ)... beyond the stars. Orgiastic ritual... drives a... nail into the coffin of the soul, and binds it... to its earthly prison-house. ...[O]only certain ascetic prescriptions of the Orphic askesis are retained, to symbolise a turning away from lower desires, that might enthral... reason."

"To this society men and women were admitted without distinction; they had all possessions in common, and a 'common fellowship and mode of life.' ...[N]o individual... was allowed to claim the credit of any discovery... It was vulgarly supposed that the school must have wished to keep its knowledge to itself as a 'mysterious' doctrine, as if there were any conceivable reason for hiding a theorem in geometry or harmonics. ...What is to be gathered from the story of Hippasos is that the pious Pythagoreans believed that the Master’s spirit dwelt continually within his church, and was the source of all its inspiration. ...The impiety lay, not in divulging a discovery in mathematics, but in claiming to have invented what could only have come from... a group-soul... living on after [Pythagoras'] death as the Logos of his disciples."

"[T]he Pythagorean One, or Monad, splits into two principles, male and female, the Even and the Odd, which are the elements of all numbers and so of the universe. ...One is not simply a numerical unit, which gives rise to other numbers by ...addition. That conception belongs to the later atomistic number-doctrine ...In the earlier Pythagoreanism, we must think of the One (which is not itself a number at all) as analogous to Anaximander’s ἄπειρον. It is the primary, undifferentiated group-soul, or physis, of the universe, and numbers must arise from it by a process of differentiation or 'separating out' (ἀπόκρισις). Similarly, each of these numbers is not a collection of units, built up by addition, but itself a sort of minor group-soul—a distinct 'nature,' with various mystical properties. In the same way, it is by dividing up the whole interval of the octave that the harmonic proportions are determined."

"Pythagorean science... will inevitably reproduce the later and inconsistent conception of the atomic, indestructible, individual soul. This... was... present in Orphic religion, fallen from its first Dionysiac faith in the one continuous life in all things, towards the Olympian conception of athanasia. The later Pythagoreans of the fifth century 'construct the whole world out of numbers, but they suppose the units to have magnitude. As to how the first unit with magnitude arose, they appear to be at a loss.' ...at a loss, because they could not realise that this physical doctrine was ...a reflection of the belief in a plurality of immortal souls, which contradicted their older faith that Soul was a Harmony—a bond linking all things in one. This Soul had formerly been the One God manifest in the logos; now it is broken up into a multitude of individual atoms, each claiming an immortal and separate persistence. And the material world suffers a corresponding change. In place of the doctrine of procession from the Monad, bodies are built up out of numbers, now conceived as collections of ultimate units, having position and magnitude. Thus, Pythagoreanism is led... from a temporal monism to a spatial pluralism—a doctrine of number-atoms hardly distinguishable from the atoms of Leukippus and Democritus, who, as Aristotle says, like these Pythagoreans, 'in a sense make all things to be numbers and to consist of numbers.' But the development of this number-atomism was predestined by religious representations of the nature of soul older than Pythagoreanism itself, and already contained in the blend of Dionysiac and Olympian conceptions inherited by Pythagoras from Orphism."

"The tendency which impelled Pythagorean science towards a materialistic atomism is only the recoil of that same tendency which exalted Pythagoras, from his position as the indwelling daemon of his church, to the distant heaven of the immortals. It is the tendency to dualism. When God ceases to be the immanent Soul of the world, living and dying in its ceaseless round of change, and ascends to the region of immutable perfection, it is because man has acquired a soul of his own, a little indestructible atom of immortality, a self-subsistent individual. 'Nature' likewise loses her unity, continuity, and indwelling life, and is remodelled as an aggregate of little indestructible atoms of matter. But note the consequence: she, too, is now self-subsistent. The world of matter becomes the undisputed dominion of Destiny, or Chance, or Necessity—of Moira, ', . There is no place in it for the God who has vanished beyond the stars."

"Not one of the philosophical ideas in Part I of the commentary is peculiarly Neoplatonic. The doctrine of the Threeness of things... is found in Aristotle and goes back to the early Pythagoreans or to Homer even; paragraph 8 is mathematical in content rather than philosophical... although there is an allusion in it to the Monad as the principle of finitudes, again a very early Pythagorean doctrine; and these two paragraphs are the source of [Heinrich] Sitter's suggestion of the authorship of Proclus. As a matter of fact, the philosophical notions in Part I have been borrowed for the most part directly from Plato, with two or three exceptions that are Aristotelian... Plato's Theaetetus, Parmenides, and the Laws, are specifically mentioned. The Timaeus forms the background of much of the thought. And the Platonism of a mathematician of the turn of the third century A. D. need not surprise us, if we but recall Aristotle's accusation that the Academy tended to turn philosophy into mathematics."

"§1. The aim of Book X of Euclid's treatise on the Elements is to investigate the commensurable and incommensurable, the rational and irrational continuous quantities. This science (or knowledge) had its origin in the sect (or school) of Pythagoras, but underwent an important development at the hands of the Athenian, Theaetetus, who had a natural aptitude for this as for other branches of mathematics most worthy of admiration."

"§2. Since this treatise (i. e. Book X of Euclid.) has the aforesaid aim and object, it will not be unprofitable for us to consolidate the good which it contains. Indeed the sect (or school) of Pythagoras was so affected by its reverence for these things that a saying became current in it, namely, that he who first disclosed the knowledge of surds or irrationals and spread it abroad among the common herd, perished by drowning: which is most probably a parable by which they sought to express their conviction that firstly, it is better to conceal (or veil) every surd, or irrational, or inconceivable in the universe, and, secondly, that the soul which by error or heedlessness discovers or reveals anything of this nature which is in it or in this world, wanders [thereafter] hither and thither on the sea of nonidentity (i. e. lacking all similarity of quality or accident), immersed in the stream of the coming-to-be and the passing-away, where there is no standard of measurement. This was the consideration which Pythagoreans and the Athenian Stranger held to be an incentive to particular care and concern for these things and to imply of necessity the grossest foolishness in him who imagined these things to be of no account."