Mathematicians From Greece

"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."

"Personal religion is derived from ecstasy, theology from mathematics, and both are to be found in Pythagoras."

"The combination of mathematics and theology, which began with Pythagoras, characterized religious philosophy in Greece, in the Middle Ages, and in modern times down to Kant. Orphism before Pythagoras was analogous to Asiatic mystery religions. But in Plato, Saint Augustine, Thomas Aquinas, Descartes, Spinoza, and Kant there is an intimate blending of religion and reasoning, of moral aspiration with logical admiration of what is timeless, which comes from Pythagoras, and distinguishes the intellectualized theology of Europe from the more straightforward mysticism of Asia. It is only in quite recent times that it has been possible to say clearly that Pythagoras was wrong. I do not know of any other man who has been as influential as he was in the sphere of thought. I say this because what appears as Platonism is, when analyzed, found to be in essence Pythagoreanism. The whole conception of an eternal world, revealed to the intellect but not to the senses, is derived from him. But for him, Christians would not have thought of Christ as the Word; but for him, theologians would not have sought logical proofs of God and immortality."

"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."

"Pythagoras was a teacher of the purest system of morals ever propounded to man."

"Pythagoras was a man; and with all his imperfections on his head, we shall look among the race of men, for his better, in yain, yea, for his equal, or his second, but in vain. Pythagoras was entirely a Deist, a steady maintainer of the unity of God, and of the eternal obligations of moral virtue. No Christian writings, even to this day, can compete in sublimity and grandeur with what this illustrious philosopher has laid down concerning God, and the end of all our actions; and it is likely, says Bayle, that he would have carried his orthodoxy much farther, had he had the courage to expose himself to martyrdom."

"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."

"But this marvellous man in some way, whether from the knowlege he received, or from his own proper discoveries, has secured to his species benefits of a more permanent nature, and which shall outlive the revolutions of ages, and the instability of political institutions. He was a profound geometrician. The two theorems, that the internal angles of every right-line triangle are equal to two right angles, and that the square of the hypothenuse of every right angled triangle is equal to the sum of the squares of the other two sides, are ascribed to him. In memory of the latter of these discoveries he is said to have offered a public sacrifice to the Gods; and the theorem is still known by the name of the Pythagorean theorem."

"To inculcate a pure and a simple mode of subsistence was also an express object of pursuit to Pythagoras. He taught a total abstinence from every thing having had the property of animal life. … He taught temperance in all its branches, and a resolute subjection of the appetites of the body to contemplation and the exercises of the mind; and, by the unremitted discipline and authority he exerted over his followers, he caused his lessons to be constantly observed. There was therefore an edifying and an exemplary simplicity that prevailed as far as the influence of Pythagoras extended, that won golden opinions to his adherents at all times that they appeared, and in all places."

"One revolution that Pythagoras worked, was that, whereas, immediately before, those who were most conspicuous among the Greeks as instructors of mankind in understanding and virtue, styled themselves sophists, professors of wisdom, this illustrious man desired to be known only by the appellation of a philosopher, a lover of wisdom. The sophists had previously brought their denomination into discredit and reproach, by the arrogance of their pretensions, and the imperious way in which they attempted to lay down the law to the world. The modesty of this appellation however did not altogether suit with the deep designs of Pythagoras, the ascendancy he resolved to acquire, and the oracular subjection in which he deemed it necessary to hold those who placed themselves under his instruction. This wonderful man set out with making himself a model of the passive and unscrupulous docility which he afterwards required from others. He did not begin to teach till he was forty years of age, and from eighteen to that period he studied in foreign countries, with the resolution to submit to all his teachers enjoined, and to make himself master of their least communicated and most secret wisdom.<!-- In Egypt in particular, we are told that, though he brought a letter of recommendation from Polycrates, his native sovereign, to Amasis, king of that country, who fully concurred with the views of the writer, the priests, jealous of admitting a foreigner into their secrets, baffled him as long as they could, referring him from one college to another, and prescribing to him the most rigorous preparatives, not excluding the rite of circumcision. But Pythagoras endured and underwent every thing, till at length their unwillingness was conquered, and his perseverance received its suitable reward."

"When in the end Pythagoras thought himself fully qualified for the task he had all along had in view, he was no less strict in prescribing ample preliminaries to his own scholars. At the time that a pupil was proposed to him, the master, we are told, examined him with multiplied questions as to his principles, his habits and intentions, observed minutely his voice and manner of speaking, his walk and his gestures, the lines of his countenance, and the expression and management of his eye, and, when he was satisfied with these, then and not till then admitted him as a probationer. It is to be supposed that all this must have been personal. As soon however as this was over, the master was withdrawn from the sight of the pupil; and a noviciate of three and five, in all eight years, was prescribed to the scholar, during which time he was only to hear his instructor from behind a curtain, and the strictest silence was enjoined him through the whole period. As the instructions Pythagoras received in Egypt and the East admitted of no dispute, so in his turn he required an unreserved submission from those who heard him: autos iphae "the master has said it," was deemed a sufficient solution to all doubt and uncertainty."

"To give the greater authority and effect to his communications Pythagoras hid himself during the day at least from the great body of his pupils, and was only seen by them at night. Indeed there is no reason to suppose that any one was admitted into his entire familiarity. When he came forth, he appeared in a long garment of the purest white, with a flowing beard, and a garland upon his head. He is said to have been of the finest symmetrical form, with a majestic carriage, and a grave and awful countenance. He suffered his followers to believe that he was one of the Gods, the Hyperborean Apollo, and is said to have told Abaris that he assumed the human form, that he might the better invite men to an easiness of approach and to confidence in him. -->"

"What however seems to be agreed in by all his biographers, is that he professed to have already in different ages appeared in the likeness of man: first as Aethalides, the son of Mercury; and, when his father expressed himself ready to invest him with any gift short of immortality, he prayed that, as the human soul is destined successively to dwell in various forms, he might have the privilege in each to remember his former state of being, which was granted him. From Aethalides he became Euphorbus, who slew Patroclus at the siege of Troy. He then appeared as Hermotimus, then Pyrrhus, a fisherman of Delos, and finally Pythagoras. He said that a period of time was interposed between each transmigration, during which he visited the seat of departed souls; and he professed to relate a part of the wonders he had seen. He is said to have eaten sparingly and in secret, and in all respects to have given himself out for a being not subject to the ordinary laws of nature. Pythagoras therefore pretended to miraculous endowments. Happening to be on the sea-shore when certain fishermen drew to land an enormous multitude of fishes, he desired them to allow him to dispose of the capture, which they consented to, provided he would name the precise number they had caught. He did so, and required that they should throw their prize into the sea again, at the same time paying them the value of the fish."

"The close of the life of Pythagoras was, according to every statement, in the midst of misfortune and violence. Some particulars are related by Iamblichus, which, though he is not an authority beyond all exception, are so characteristic as seem to entitle them to the being transcribed. … Cylon, the richest man, or, as he is in one place styled, the prince, of Crotona, had manifested the greatest partiality to Pythagoras. He was at the same time a man of rude, impatient and boisterous character. He, together with Perialus of Thurium, submitted to all the severities of the Pythagorean school. They passed the three years of probation, and the five years of silence. They were received into the familiarity of the master. They were then initiated, and delivered all their wealth into the common stock. They were however ultimately pronounced deficient in intellectual power, or for some other reason were not judged worthy to continue among the confidential pupils of Pythagoras. They were expelled. The double of the property they had contributed was paid back to them. A monument was set up in memory of what they had been; and they were pronounced dead to the school. … Cylon, from feelings of the deepest reverence and awe for Pythagoras, which he had cherished for years, was filled even to bursting with inextinguishable hatred and revenge. The unparalleled merits, the venerable age of the master whom he had so long followed, had no power to control his violence. His paramount influence in the city insured him the command of a great body of followers. He excited them to a frame of turbulence and riot. He represented to them how intolerable was the despotism of this pretended philosopher. They surrounded the school in which the pupils were accustomed to assemble, and set it on fire. Forty persons perished in the flames. According to some accounts Pythagoras was absent at the time. According to others he and two of his pupils escaped. He retired from Crotona to Metapontum. But the hostility which had broken out in the former city, followed him there. He took refuge in the Temple of the Muses. But he was held so closely besieged that no provisions could be conveyed to him; and he finally perished with hunger, after, according to Laertius, forty days' abstinence."

"It is difficult to imagine any thing more instructive, and more pregnant with matter for salutary reflection, than the contrast presented to us by the character and system of action of Pythagoras on the one hand, and those of the great enquirers of the last two centuries, for example, Bacon, Newton and Locke, on the other. Pythagoras probably does not yield to any one of these in the evidences of true intellectual greatness. In his school, in the followers he trained resembling himself, and in the salutary effects he produced on the institutions of the various republics of Magna Graecia and Sicily, he must be allowed greatly to have excelled them. His discoveries of various propositions in geometry, of the earth as a planet, and of the solar system as now universally recognised, clearly stamp him a genius of the highest order. Yet this man, thus enlightened and philanthropical, established his system of proceeding upon narrow and exclusive principles, and conducted it by methods of artifice, quackery and delusion. One of his leading maxims was, that the great and fundamental truths to the establishment of which he devoted himself, were studiously to be concealed from the vulgar, and only to be imparted to a select few, and after years of the severest noviciate and trial. He learned his earliest lessons of wisdom in Egypt after this method, and he conformed through life to the example which had thus been delivered to him. The severe examination that he made of the candidates previously to their being admitted into his school, and the years of silence that were then prescribed to them, testify this. He instructed them by symbols, obscure and enigmatical propositions, which they were first to exercise their ingenuity to expound. The authority and dogmatical assertions of the master were to remain unquestioned; and the pupils were to fashion themselves to obsequious and implicit submission, and were the furthest in the world from being encouraged to the independent exercise of their own understandings. There was nothing that Pythagoras was more fixed to discountenance, than the communication of the truths upon which he placed the highest value, to the uninitiated. It is not probable therefore that he wrote any thing: all was communicated orally, by such gradations, and with such discretion, as he might think fit to adopt and to exercise."

"All was oracular and dogmatic in the school of Pythagoras. He prized and justly prized the greatness of his attainments and discoveries, and had no conception that any thing could go beyond them. He did not encourage, nay, he resolutely opposed, all true independence of mind, and that undaunted spirit of enterprise which is the atmosphere in which the sublimest thoughts are most naturally generated. He therefore did not throw open the gates of science and wisdom, and invite every comer; but on the contrary narrowed the entrance, and carefully reduced the number of aspirants. He thought not of the most likely methods to give strength and permanence and an extensive sphere to the progress of the human mind. For these reasons he wrote nothing; but consigned all to the frail and uncertain custody of tradition. And distant posterity has amply avenged itself upon the narrowness of his policy; and the name of Pythagoras, which would otherwise have been ranked with the first luminaries of mankind, and consigned to everlasting gratitude, has in consequence of a few radical and fatal mistakes, been often loaded with obloquy, and the hero who bore it been indiscriminately classed among the votaries of imposture and artifice."

"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 the Pythagorean metempsychosis should be thoroughly explained and compared with the modern theory of evolution, it would be found to supply every "missing link" in the chain of the latter. But who of our scientists would consent to lose his precious time over the vagaries of the ancients. Notwithstanding proofs to the contrary, they not only deny that the nations of the archaic periods, but even the ancient philosophers had any positive knowledge of the Heliocentric system. The "Venerable Bedes," the Augustines and Lactantii appear to have smothered, with their dogmatic ignorance, all faith in the more ancient theologists of the pre-Christian centuries. But now philology and a closer acquaintance with Sanskrit literature have partially enabled us to vindicate them from these unmerited imputations. In the Vedas, for instance, we find positive proof that so long ago as 2000 B.C., the Hindu sages and scholars must have been acquainted with the rotundity of our globe and the Heliocentric system. Hence, Pythagoras and Plato knew well this astronomical truth; for Pythagoras obtained his knowledge in India, or from men who had been there, and Plato faithfully echoed his teachings."

"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)"

"Zoroaster, Pythagoras, Epicharmus, Empedocles, Kebes, Euripides, Plato, Euclid, Philo, Boethius, Virgil, Marcus Cicero, Plotinus, Iamblichus, Proclus, Psellus, Synesius, Origen, and, finally, Aristotle himself, far 'from denying our immortality, support it most emphatically. (p. 251)"

"The laws of Manu are the doctrines of Plato, Philo, Zoroaster, Pythagoras, and of the Kabala. The esoterism of every religion may be solved by the latter. The kabalistic doctrine of the allegorical Father and Son, or Xlarrjp and Aoyos is identical with the groundwork of Buddhism. Moses could not reveal to the multitude the sublime secrets of religious speculation, nor the cosmogony of the universe ; the whole resting upon the Hindu Illusion, a clever mask veiling the Sanctum Sanctorum, and which has misled so many theological commentators. (p. 271)"

"Scholastic skeptics, as well as ignorant materialists, have greatly amused themselves for the last two centuries over the absurdities attributed to Pythagoras by his biographer, Iamblichus. The Samian philosopher is said to have persuaded a she-bear to give up eating human flesh; to have forced a white eagle to descend to him from the clouds, and to have subdued him by stroking him gently with the hand, and by talking to him. On another occasion, Pythagoras actually persuaded an ox to renounce eating beans, by merely whispering in the animal's ear! (Iamblichus: "De Vita Pythag.") Oh, ignorance and superstition of our forefathers, how ridiculous they appear in the eyes of our enlightened generations! Let us, however, analyze this absurdity. Every day we see unlettered men, proprietors of strolling menageries, taming and completely subduing the most ferocious animals, merely by the power of their irresistible will... Every one has either witnessed or heard of the seemingly magical power of some mesmerizers and psychologists. They are able to subjugate their patients for any length of time. Regazzoni, the mesmerist who excited such wonder in France and London, has achieved far more extraordinary feats than what is above attributed to Pythagoras. Why, then, accuse the ancient biographers of such men as Pythagoras and Apollonius of Tyana of either wilful misrepresentation or absurd superstition?"

"Pythagoras stands at the fountainhead of our culture. The ideas he set in motion were, according to Daniel Boorstin, "among the most potent in modern history," resulting directly in many of the pillars upon which the modern world is built. In particular, the very existence of science becomes possible only when it is realized that inner, purely subjective, mathematical forms have a resonance with the form and behavior of the external world — a Pythagorean perception. And a world at peace — that is to say, in a nuclear age, the survival of our planet — is predicated upon ideas of universal brotherhood to which Pythagoras, while not the sole author, made an enormous contribution. Even the seeming remoteness of Pythagorean teaching helps one to realize that the current world view, while it seems destined to dominate the planet, is fleeting and temporary and, like others before it, will pass."

"Pythagoras' teachings have enormous relevance in understanding both the sources of our culture and, perhaps more importantly, where it may be heading or may need to head. But to appreciate this we have to understand him in modern terms."

"At the dawn of our century, scientists were proclaiming that our understanding of the world was almost complete. Only one or two small problems in physics remained to be solved. One of these problems had to do with black body radiation and was solved by Max Planck. His solution, however, formed the foundation for quantum mechanics which was to sweep aside almost the whole edifice of fundamental assumptions in physics, and with it our understanding of the world. A hundred years later we are faced with a similar situation. The mechanistic viewpoint that began to dominate our world view in the seventeenth century has almost completed its hegemony. This paradigm, as historian Hugh Kearney points out, stems from only one of three main systems of thought that flowed from Greek thought into the modern world, each of which has dominated our world view at different points in our history. … 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."

"The concept of a harmonious universe ordered according to "the Great Chain of Being" — a chain that connects the continuum of matter, body, mind, soul and spirit — stands as one of the most fundamental ideas of western thought. … It continues to be a profound influence upon the deepest strata of our thought. And yet a major rift has appeared in the consciousness of our time because the theme of harmonia has not been translated into the realm of human conduct. The challenge of our time may be to revive it, and make divine harmony "the great theme" of the next millennium. Any success we have in accomplishing this will be based, in large part, on the achievements of Pythagoras."

"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 know that he (Pythagoras) went to India to be instructed; but the capacity of the learner determines his degree of proficiency, and if Pythagoras on his return had so little knowledge in geometry as to consider the forty-seventh of Euclid as a great discovery, he certainly was entirely incapable of acquiring the Indian method of calculation, through his deficiency of preparatory knowledge… each teacher, or head of sect that drew his knowledge from Indian sources, might conceal his instructions to be reckoned an inventor."

"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."

"Among the sages of this description, to whose useful labors the world is so much indebted, none held a more deservedly conspicuous rank than Pythagoras […] Quitting the land of his nativity […] his zeal for the acquisition of knowledge led him first to Egypt […] Having at this celebrated fountain of learning exhausted the supply without diminishing his thirst, he sought the further means of slaking it, in the then almost unexplored peninsula of India, whence he returned, bringing back with him the doctrine of Metempsychosis, the prejudices against animal diet, the mysterious notions respecting the powers of numbers, and other visionary and fanciful tenets of the East."

"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."