Pythagoras Quotes (-570--496)

“Choose always the way that seems the best, however rough it may be. Custom will render it easy and agreeable.”

“In all thou dost first let thy Prayers ascend, And to the Gods thy Labours first commend, From them implore Success, and hope a prosperous End.”

“Envy has been, is, and shall be, the destruction of many. What is there, that Envy hath not defamed, or Malice left undefiled? Truly, no good thing.”

“As long as Man continues to be the ruthless destroyer of lower living beings, he will never know health or peace. For as long as men massacre animals, they will kill each other. Indeed, he who sows the seed of murder and pain cannot reap joy and love.”

“As long as Man continues to be the ruthless destroyer of lower living beings, he will never know health or peace. For as long as men massacre animals, they will kill each other. Indeed, he who sows the seed of murder and pain cannot reap joy and love.”

“It seems probable that the early Greeks were largely indebted to the Phoenicians for their knowledge of practical arithmetic or the art of calculation, and perhaps also learnt from them a few properties of numbers. It may be worthy of note that Pythagoras was a Phoenician; and according to Herodotus, but this is more doubtful, Thales was also of that race.”

“But it is, however, in the teachings of Pythagoras that we find the closest and most frequent identities of teachings and argumentation [about as to the nature and position of the Gods], explained as due to Pythagoras himself having visited India and learned his philosophy there, as tradition asserts. In later centuries we find some peculiarly Sânkhyan and Buddhist ideas playing a prominent part in Gnostic thought.”

“The world is always ungrateful to its great men. Florence has built a statue to Galileo, but hardly even mentions Pythagoras. The former had a ready guide in the treatises of Copernicus, who had been obliged to contend against the universally established Ptolemaic system. But neither Galileo nor modern astronomy discovered the emplacement of the planetary bodies. Thousands of ages before, it was taught by the sages of Middle Asia, and brought thence by Pythagoras, not as a speculation, but as a demonstrated science. "The numerals of Pythagoras," says Porphyry, "were hieroglyphical symbols, by means whereof he explained all ideas concerning the nature of all things." Verily, then, to antiquity alone have we to look for the origin of all things. p. 31”

“Pythagoras called his Gnosis “the knowledge of things that are,” or ἡ γνῶσις τῶν ὄντων, and preserved that knowledge for his pledged disciples only: for those who could digest such mental food and feel satisfied; and he pledged them to silence and secrecy. Occult alphabets and secret ciphers are the development of the old Egyptian hieratic writings, the secret of which was, in the days of old, in the possession only of the Hierogrammatists, or initiated Egyptian priests.”

“Pythagoras was said to have been the first man to call himself philosopher; in fact, the world is indebted to him for the word philosopher. Before that time the wise men called themselves sages, which was interpreted to mean those who know. Pythagoras was more modest. He coined the word philosopher, which he defined as one who is attempting to find out.”

“There remains no firm basis for the belief that Pythagoras was a geometer and in any case no attestation of his having written anything.”

“I wished to show that Pythagoras, the first founder of the vegetable regimen, was at once a very great physicist and a very great physician; that there has been no one of a more cultured and discriminating humanity; that he was a man of wisdom and of experience; that his motive in commending and introducing the new mode of living was derived not from any extravagant superstition, but from the desire to improve the health and the manners of men.”

“Koyré's exaltation of the "Platonic and Pythagorean" elements of the Scientific Revolution... was based on a demonstrably false understanding of how Galileo reached his conclusions. Koyré asserted that Galileo merely used experiments as a check on the theories he devised by mathematical reasoning. But later research has definitively established that Galileo's experiments preceded his attempts to give a mathematical account of their results.”

“Pythagoras could not have been the discoverer of the relation, because... this property was known and used by scholars and artisans of Oriental lands thousands of years before Pythagoras... While deductive geometry is barely more than twenty-five hundred years old, empirical geometry is probably as old as civilization itself.”

“Let us suppose that we have set the problem of finding a solution to the equation scriptstyle x^2=2. , This is a problem for which the Babylonians around 1700 BC found the excellent approximation scriptstyle sqrt{2}. ...This is the identical problem which Pythagoras asserted had no fractional solution and in whose honor he was supposed to have sacrificed a hecatomb of oxen—the problem which caused the existentialist crisis in ancient Greek mathematics. The scriptstyle sqrt{2} exists (as the diagonal of the unit square); yet it does not exist (as a fraction)!”

“A foolish consistency is the hobgoblin of little minds, adored by little statesmen and philosophers and divines. With consistency a great soul has simply nothing to do. He may as well concern himself with his shadow on the wall. Speak what you think now in hard words, and to-morrow speak what to-morrow thinks in hard words again, though it contradict every thing you said to-day. — "Ah, so you shall be sure to be misunderstood." — Is it so bad, then, to be misunderstood? Pythagoras was misunderstood, and Socrates, and Jesus, and Luther, and Copernicus, and Galileo, and Newton, and every pure and wise spirit that ever took flesh. To be great is to be misunderstood.”

“But there was among them a man of prodigious knowledge who acquired the profoundest wealth of understanding and was the greatest master of skilled arts of every kind; for, whenever he willed with his whole heart, he could with ease discern each and every truth in his ten—nay, twenty—men's lives.”

“The games which can be built up from the simple idea of dots and lines... can be a productive source of teaching material. After all, figurate numbers provided the Pythagoreans and neo-Pythagoreans with important theorems about the summing of series.”

“Nor need you question but that Pythagoras a long time be­fore he found the demonstration for which he offered the Heca­tomb, had been certain, that the square of the side subtending the right angle in a rectangle triangle, was equal to the square of the other two sides: and the certainty of the conclusion condu­ced not a little to the investigating of the demonstration, un­derstanding me alwayes to mean in demonstrative Sciences.”

“Some fundamental unity was surely to be discerned either in the matter or the structure of things. The Ionic philosophers chose the former field: Pythagoras took the latter. ...The geometry which he had learnt in Egypt was merely practical. ...It was natural to nascent philosophy to draw, by false analogies, and the use of a brief and deceptive vocabulary,2 enormous conclusions from a very few observed facts: and it is not surprising if Pythagoras, having learnt in Egypt that number was essential to the exact description of forms and of the relations of forms, concluded that number was the cause of form and so of every other quality. Number, he inferred, is quantity and quantity is form and form is quality. Footnote2 Primitive men, on seeing a new thing, look out especially for some resemblance in it to a known thing, so that they may call both by the same name. This developes a habit of pressing small and partial analogies. It also causes many meanings to be at attached to the same word. Hasty and confused theories are the inevitable result.”

“Pythagoras was indeed the first man to call himself a philosopher. Others before had called themselves wise (sophos), but Pythagoras was the first to call himself a philosopher, literally a lover of wisdom. More importantly, for Pythagoras and his followers philosophy was not merely an intellectual pursuit, but a way of life, the aim of which was the assimilation to God.”

“Pythagoras, the son of Mnesarchus, was the most learned of all men of history; and having selected from these writings, he thus formed his own wisdom and extensive learning, and mischievous art.”

“Much learning does not teach wisdom; otherwise it would have taught Hesiod and Pythagoras, and again Xenophanes and Hecataeus.”

“No one will deny that the soul of Pythagoras was sent to mankind from Apollo's domain, having either been one of his attendants, or more intimate associates, which may be inferred both from his birth, and his versatile wisdom.”

“Pythagoras taught, accordingly, that he had himself been originally Euphorbus, and then Callides, thirdly Hermotimus, fourthly Pyrrhus, and lastly Pythagoras; and that those things which had existed, after certain revolutions of time, came into being again; so that nothing in the world should be thought of as new. He said that true philosophy was a meditation on death; that its daily struggle was to draw forth the soul from the prison of the body into liberty: that our learning was recollection, and many other things which Plato works out in his dialogues, especially in the Phaedo and Timæus. For Plato, after having formed the Academy and gained innumerable disciples, felt that his philosophy was deficient on many points, and therefore went to Magna Græcia, and there learned the doctrines of Pythagoras from Archytas of Tarentum and Timæus of Locris: and this system he embodied in the elegant form and style which he had learned from Socrates. The whole of this, as we can prove, Origen carried over into his book Περὶ ᾿Αρχῶν, only changing the name.”

“Nicomachus concludes his first book with a theorem that indicates that mathematics was not yet free from ethical and æsthetic mixture. From Pythagoras onward two ideas were widespread in Greek, especially Platonic, philosophy. These are that the beautiful and the definite are prior to the ugly and the indefinite, and that from them are formed all the parts and classes of the infinite and indefinite. Nicomachus aims to show that in mathematics the same principle holds good in that from equality may be derived all the species of inequality.”

“The Ionians were optimistic, heathenly materialists... Every philosopher of the period seems to have had his own theory regarding the nature of the universe around him. ...The sixth century scene evokes the image of an orchestra expectantly tuning up, each player absorbed in his own instrument only, deaf to the caterwaulings of the others. Then there is a dramatic silence, the conductor enters the stage, raps three times with his baton, and harmony emerges from the chaos. The maestro is Pythagoras of Samos, whose influence on the ideas, and thereby on the destiny, of the human race was probably greater than that of any single man before or after him.”

“It may perhaps help us to realize the human side of our Masters if we remember that many of Them in comparatively recent times have been known as historical characters. The Master K.H. for example, appeared in Europe as the philosopher Pythagoras. Before that He was the Egyptian priest Sarthon, and on yet another occasion chief-priest of a temple at Agade, in Asia Minor, where He was killed in a general massacre of the inhabitants by a host of invading barbarians who swooped down upon them from the hills”

“Ah, Pythagoras' metempsychosis, were that true, This soul should fly from me, and I be changed Unto some brutish beast! All beasts are happy, for when they die, Their souls are soon dissolved in elements; But mine must live still to be plagued in hell.”

“Pythagoras and Plato and Boehme and Paracelsus and Thomas Vaughan were men who bore their lamps amidst their fellowmen in life under a hail of nonunderstanding and abuse. Anyone could approach them, but only a few were able to discern the superearthly radiance behind the earthly face. It is possible to name great Servitors of East and West, North and South. It is possible to peruse their biographies; yet everywhere we feel that the superearthly radiance appears rarely in the course of centuries. One should learn from reality. Let us not link ourselves with the vilifiers of Plato and the persecutors of Confucius. They were oppressed by citizens who were considered the pride of the country. Thus has the world raised its hand against the great Servitors. Be assured that the Brotherhood formed by Pythagoras appeared dangerous in the eyes of the city guard. (175)”

“What appeared here, at the center of the Pythagorean tradition in philosophy, is another view of psyche that seems to owe little or nothing to the pan-vitalism or pan-deism (see theion) that is the legacy of the Milesians.”

“The votaries of Pythagoras of Samos have this story to tell of him, that he was not an Ionian at all, but that, once on a time in Troy, he had been Euphorbus, and that he had come to life after death, but had died as the songs of Homer relate. And they say that he declined to wear apparel made from dead animal products and, to guard his purity, abstained from all flesh diet, and from the offering of animals in sacrifice. For that he would not stain the altars with blood; nay, rather the honey-cake and frankincense and the hymn of praise, these they say were the offerings made to the Gods by this man, who realized that they welcome such tribute more than they do the hecatombs note and the knife laid upon the sacrificial basket. For they say that he had of a certainty social intercourse with the gods, and learnt from them the conditions under which they take pleasure in men or are disgusted, and on this intercourse he based his account of nature.”

“The following became universally known: first, that he maintains that the soul is immortal; second, that it changes into other kinds of living things; third, that events recur in certain cycles and that nothing is ever absolutely new; and fourth, that all living things should be regarded as akin. Pythagoras seems to have been the first to bring these beliefs into Greece.”

“He ordained that his disciples should speak well and think reverently of the Gods, muses and heroes, and likewise of parents and benefactors; that they should obey the laws; that they should not relegate the worship of the Gods to a secondary position, performing it eagerly, even at home; that to the celestial divinities they should sacrifice uncommon offerings; and ordinary ones to the inferior deities. (The world he Divided into) opposite powers; the "one" was a better monad, light, right, equal, stable and straight; while the "other" was an inferior duad, darkness, left, unequal, unstable and movable.”

“According to the account of Proclus (Book II. c. 4 ), Pythagoras was the first who gave to Geometry the form of a deductive science, by shewing the connexion of the geometrical truths then known, and their dependence on certain first principles. ...The traditionary account, that Pythagoras was the founder of scientific mathematics, is in some degree, supported by the statement of Diogenes Laertius, that he was chiefly occupied with the consideration of the properties of number, weight, and extension, besides music and astronomy. The passage of Cicero (De Nat. Deor. III. 36) may be referred to as evidence that later writers were unable to give any precise account of the mathematical discoveries of Pythagoras. To Pythagoras, however, is attributed the discovery of some of the most important elementary properties contained in the first book of Euclid's Elements. The very important truth contained in Prop. 47, Book I. is also ascribed to Pythagoras. ...Proclus attributes to him the discovery of that right-angled triangle, the three sides of which are respectively 3, 4, and 5 units. To Pythagoras also belongs the discovery, that there are only three kinds of regular polygons which can be placed so as to fill up the space round a point; namely, six equilateral triangles, four squares, and three regular hexagons. Proclus attributes to him the doctrine of incommensurables, and the discovery of the five regular solids, which, if not due to Pythagoras, originated in his school. In Astronomy he is reputed to have held, that the Sun is the centre of the system, and that the planets revolve round it. This has been called, from his name, the Pythagorean System, which was revived by Copernicus, A.D.1541, and proved by Newton.”

“Pythagoras transformed the study of geometry into a liberal education, examining the principles of the science from the beginning and probing the theorems in an immaterial and intellectual manner: he it was who discovered the theory of irrationals [or 'proportions'] and the construction of the cosmic figures.”

“Pythagoras, as everyone knows, said that "all things are numbers." This statement, interpreted in a modern way, is logical nonsense, but what he meant was not exactly nonsense. He discovered the importance of numbers in music and the connection which he established between music and arithmetic survives in the mathematical terms "harmonic mean" and "harmonic progression." He thought of numbers as shapes, as they appear on dice or playing cards. We still speak of squares or cubes of numbers, which are terms that we owe to him. He also spoke of oblong numbers, triangular numbers, pyramidal numbers, and so on. These were the numbers of pebbles [or calculi] (or as we would more naturally say, shot) required to make the shapes in question.”

“Inasmuch as I have begun to explain to you how much greater was my impulse to approach philosophy in my youth than to continue it in my old age, I shall not be ashamed to tell you what ardent zeal Pythagoras inspired in me. Sotion used to tell me why Pythagoras abstained from animal food, and why, in later times, Sextius did also. In each case, the reason was different, but it was in each case a noble reason. … Pythagoras … held that all beings were inter-related, and that there was a system of exchange between souls which transmigrated from one bodily shape into another. If one may believe him, no soul perishes or ceases from its functions at all, except for a tiny interval – when it is being poured from one body into another.”

“What seems certain is that Pythagoras developed the idea of mathematical logic... He realized that numbers exist independently of the tangible world and therefore their study was untainted by inaccuracies of perception. This meant he could discover truths which were independent of opinion of prejudice and which were more absolute then any previous knowledge.”

“Number, its kinds; the first kind, intellectual in the divine mind. Number is of two kinds, the Intellectual (or immateriall) and the Scientiall. The intellectuall is that eternal substance of number, which Pythagoras in his discourse concerning the Gods asserted to be the principle most providentiall of all Heaven and Earth, and the nature that is betwixt them. Moreover, it is the root of divine Beings, and of gods, & of Dæmons. This is that which he termed the principle, fountain,and root of all things, and defined it to be that which before all things exists in the divine mind; from which and out of which all things are digested into order, and remain numbred by an indissolube series. For all things which are ordered in the world by nature according to an artificiall course in part and in whole appear to be distinguished and adorn'd by Providence and the All-creating Mind, according to Number; the exemplar being established by applying (as the reason of the principle before the impression of things) the number præxistent in the Intellect of God, maker of the world. This only in intellectual, & wholly immaterial, really a substance according to which as being the most exact artificiall reason, all things are perfected, Time, Heaven, Motion, the Stars and their various revolutions. ...The other kind of number, Scientiall; its principles. Scientiall Number is that which Pythagoras defines the extension and production into act of the seminall reasons which are in the Monad, or a heap of Monads, or a progressian of multitude beginning from Monad, and a regression ending in Monad.”

“In spite of the dominance of mechanistic thought in the contemporary world, a perplexing residue of the magical tradition still survives in the form of several issues, solutions to which do not appear possible within the context of a purely mechanical view of the world.... It is important to recognize that the materialist, scientific paradigm that dominates the late twentieth century world and provides the basis for its dominant institutions, has its basis in the life and work of Pythagoras, one of the most significant representatives of the perennial philosophy and a founder of the magical tradition. This spirit, which gave rise to our world view, is a spirit that must be recaptured if our civilization is to flourish. The choice is a clear one to many, and was summed up in a book title by the late Pythagorean and futurist Buckminster Fuller, Utopia or Oblivion.”

“It was Pythagoras who first called heaven kosmos, because it is perfect, and "adorned" with infinite beauty and living beings.”

“It is very important to note that some 2,500 years ago at the least Pythagoras went from Samos to the Ganges to learn geometry…But he would certainly not have undertaken such a strange journey had the reputation of the Brahmins' science not been long established in Europe…”

“From of old, amid the rage of robbery and blood-lust, it came to wise men's consciousness that the human race was suffering from a malady which necessarily kept it in progressive deterioration. Many a hint from observation of the natural man, as also dim half-legendary memories, had made them guess the primal nature of this man, and that his present state is therefore a degeneration. A mystery enwrapped Pythagoras, the preacher of vegetarianism; no philosopher since him has pondered on the essence of the world, without recurring to his teaching. Silent fellowships were founded, remote from turmoil of the world, to carry out this doctrine as a sanctification from sin and misery.”

“It was a maxim of Pythagoras that the two most excellent things for man were to speak the truth, and to render benefits to each other.”

“The association between religion and mathematically based science has its origins in the mists of history. ...the very dawn of Western culture in sixth-century B.C. Greece. ...[W]hen the Greeks were turning away from the mythological picture immortalized by Homer and Hesiod, the Ionian philosopher Pythagoras of Samos pioneered a worldview in which mathematics was seen as the key to reality. In place of the mythological gods, Pythagoras painted a picture in which the universe was conceived as a great musical instrument resonating with divine mathematical harmonies. ...[inspiring] mystics, theologians, and physicists ever since. ...But to Pythagoras and his followers, mathematics was the key not simply to the physical world, but more importantly to the spiritual world—for they believed that numbers were literally gods. By contemplating numbers and their relationships, the Pythagoreans sought union with the "divine." For them, mathematics was first and foremost a religious activity.”

“Around 600 BCE, Pythagoras observed that the tones of a lyre sound most harmonious when the ratio of string lengths forms a simple whole-number fraction. Inspired by such hints, Pythagoras and his followers made a remarkable intuitive leap. They foresaw the possibility of a different kind of world-model, less dependent on the accident of our senses but more in tune with Nature's hidden harmonies, and ultimately more faithful to reality. This is the meaning of... "All things are number."”

“Pythagoras was a man of the most various accomplishments, and appears to have penetrated in different directions into the depths of human knowledge. He sought wisdom in its retreats of fairest promise, in Egypt and other distant countries. In this investigation he employed the earlier period of his life, probably till he was forty, and devoted the remainder to such modes of proceeding, as appeared to him the most likely to secure the advantage of what he had acquired to a late posterity. He founded a school, and delivered his acquisitions by oral communication to a numerous body of followers. He divided his pupils into two classes, the one neophytes, to whom was explained only the most obvious and general truths, the other who were admitted into the entire confidence of the master. These last he caused to throw their property into a common stock, and to live together in the same place of resort. He appears to have spent the latter half of his life in that part of Italy, called Magna Graecia, so denominated in some degree from the numerous colonies of Grecians by whom it was planted, and partly perhaps from the memory of the illustrious things which Pythagoras achieved there.”

“It is certain that Pythagoras awakened the deepest intellectual sympathy of his age, and that his doctrines exerted a powerful influence upon the mind of Plato. His cardinal idea was that there existed a permanent principle 'of unity beneath the forms, changes, and other phenomena of the universe. Aristotle asserted that he taught that “ numbers are the first principles of all entities.” Ritter has expressed the opinion that the formula of Pythagoras should be taken symbolically, which is doubtless correct. (p. XV Before the Veil)”

“Aristotle was no trustworthy witness. He misrepresented Plato, and he almost caricatured the doctrines of Pythagoras. There is a canon of interpretation, which should guide us in our examinations of every philosophical opinion: The human mind has, under the necessary operation of its own laws, been compelled to entertain the same fundamental ideas, and the human heart to cherish the same feelings in all ages. It is certain that Pythagoras awakened the deepest intellectual sympathy of his age, and that his doctrines exerted a powerful influence upon the mind of Plato. His cardinal idea was that there existed a permanent principle of unity beneath the forms, changes, and other phenomena of the universe. Aristotle asserted that he taught that numbers are the first principles of all entities. Ritter has expressed the opinion that the formula of Pythagoras should be taken symbolically, which is doubtless correct.”

“If we carefully trace the terms nazar, and nazaret, throughout the best known works of ancient writers, we will meet them in connection with “Pagan” as well as Jewish adepts. Thus, Alexander Polyhistor says of Pythagoras that he was a disciple of the Assyrian Nazarrt, whom some suppose to be Ezekiel. Diogenes Laertius states most positively that Pythagoras, after being initiated into all the Mysteries of the Greeks and barbarians, “went into Egypt and afterward visited the Chaldeans and Magi;” and Apuleius maintains that it was Zoroaster who instructed Pythagoras. (p. 140)”

“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.”

“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.”

“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.”

“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.”

“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.”

“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.”

“While most sophists emphasized the reality of change — in particular, the Atomists, followers of Leucippus 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 quadrivium). 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.”

“[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.”