Mathematicians From Greece

"Pythagoras... assumed as first principles the numbers and symmetries existing among them, which he calls harmonies, and the elements compounded of both, that are called geometrical. ...he says that the nature of Number is the Decad."

"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. Paracelsus was a target for mockery and malignance. Thomas Vaughan seemed to be an outcast, and few wished to meet with him. Thus was the reign of darkness manifested."

"If one examines the reasons for the persecution of the best minds of different nations, and compares the reasons for the persecution and banishment of Pythagoras, Anaxagoras, Socrates, Plato, and others, one can observe that in each case the accusations and reasons for banishment were almost identical and unfounded. But in the following centuries full exoneration came, as if there had never been any defamation. It would be correct to conclude that such workers were too exalted for the consciousness of their contemporaries, and the sword of the executioner was ever ready to cut off a head held high. Pericles was recognized in his time only after people had reduced him to a sorry state. Only in that state could his fellow citizens accept him as an equal! A book should be written about the causes of the persecution of great individuals. By comparing the causes is it possible to trace the evil will."

"Pythagoras, it seems, did not only call the supreme Deity a monad, but also a tetrad, or tetractys... It is, in the golden verses, said to be the fountain of the eternal nature; and by Hierocles, the maker of all things, the intelligent god, the cause of the heavenly and sensible god, that is, of the animated world or heaven. The later Pythogoreans endeavour to give reasons why God should be called Tetractys, from certain mysteries in the number four; but... much more probable... this name was really nothing else but the tetragrammaton, or that proper name of the supreme God amongst the Hebrews, consisting of four letters; nor is it strange Pythagoras should be so well acquainted with the name Jehovah, since, besides travelling into other parts of the East, he is by Josephus, Porphyry, and others, to have conversed with the Hebrews also."

"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."

"Whenever he heard a person who was making use of his symbols, he immediately took him into his circle, and made him a friend."

"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 school of the Pythagoras and those of the Neo-Platonists kept up the tradition for Greece, and we know that Pythagoras gained some of his learning in India, while Plato studied, and was initiated in the schools of Egypt. More precise information has been published of the Grecian schools than of others; the Pythagorean had pledged disciples as well as an outer discipline, the inner circle passing through three degrees during five years of probation."

"It is further interesting to remark that the finest characters among women with which ancient Greece presents us were formed in the school of Pythagoras, and the same is true of the men."

"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 repeats our archaic doctrine when stating that the Ego (Nous) is eternal with Deity; that the soul only passed through various stages to arrive at divine excellence; while thumos returned to the earth, and even the phren, the lower Manas, was eliminated."

"I have often admired the mystical way of Pythagoras, and the secret Magic of numbers."

"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."

"The apparently ancient reports of the importance of Pythagoras and his pupils in laying the foundations of mathematics crumble on touch, and what we can get hold of is not authentic testimony by the efforts latecomers to paper over a crack, which they obviously found surprising, by the use of various kinds of reconstruction and reinterpretation. On the other hand, there are ancient and unassailable indications of a Greek mathematics antedating Pythagoras and quite outside his sphere of influence."

"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."

"Pythagoras did not possess a proof of the theorem which bears his name... he was temperamentally uninterested in proofs of this nature, as may be gleaned from... his numerological deductions. ...the Pythagorean theorem was known to Thales. ...the hypotenuse theorem is a direct consequence of the principle of similitude, and... Thales was fully conversant with the theory of similar triangles. On the other hand, there is no doubt that Pythagoras fully appreciated the metaphysical implications. ...this relation ...was to Pythagoras and the Pythagoreans a basic law of nature, and... a brilliant confirmation of their number philosophy."

"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."

"Pythagoras is the founder of European culture in the Western Mediterranean sphere."

"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, s 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."

"After his father's death, though he was still but a youth, his aspect was so venerable, and his habits so temperate that he was honored and even reverenced by elderly men, attracting the attention of all who saw and heard him speak, creating the most profound impression. That is the reason that many plausibly asserted that he was a child of the divinity. Enjoying the privilege of such a renown, of an education so thorough from infancy, and of so impressive a natural appearance he showed that he deserved all these advantages by deserving them, by the adornment of piety and discipline, by exquisite habits, by firmness of soul, and by a body duly subjected to the mandates of reason. An inimitable quiet and serenity marked all his words and actions, soaring above all laughter, emulation, contention, or any other irregularity or eccentricity; his influence at Samos was that of some beneficent divinity. His great renown, while yet a youth, reached not only men as illustrious for their wisdom as Thales at Miletus, and Bias at Prione, but also extended to the neighboring cities. He was celebrated everywhere as the "long-haired Samian," and by the multitude was given credit for being under divine inspiration."

"Pythagoras conceived that the first attention that should be given to men should be addressed to the senses, as when one perceives beautiful figures and forms, or hears beautiful rhythms and melodies. Consequently he laid down that the first erudition was that which subsists through music's melodies and rhythms, and from these he obtained remedies of human manners and passions, and restored the pristine harmony of the faculties of the soul."

"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 is impossible to decide whether a particular detail of the Pythagorean universe was the work of the master, or filled in by a pupil—a remark which equally applies to Leonardo or Michelangelo. But there can be no doubt that the basic features were conceived by a single mind; that Pythagoras of Samos was both the founder of a new religious philosophy, and the founder of Science, as the word is understood today."

"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."

"It was through philosophy, he said, that he had come to be surprised at nothing."

"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."

"Such things taught he, though advising above all things to speak the truth, for this alone deifies men. For as he had learned from the Magi, who call God Oremasdes, God's body is light, and his soul is truth. He taught much else, which he claimed to have learned from Aristoclea at Delphi."

"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."