Pythagorean Quotes

Rachel bit her lip. "I hope you're right. I'm a little worried. What if someone asks what's on the next math test and I start spouting a prophecy in the middle of geometry class? The Pythagorean theorem shall be problem two...Gods, that would be embarrassing.

You forget all of it anyway. First, you forget everything you learned-the dates of the Hay-Herran Treaty and Pythagorean Theorem. You especially forget everything you didn't really learn, but just memorized the night before. You forget the names of all but one or two of your teachers, and eventually you'll forget those, too. You forget your junior class schedule and where you used to sit and your best friend's home phone number and the lyrics to that song you must have played a million times. For me, it was something by Simon & Garfunkel. Who knows what it will be for you? And eventually, but slowly, oh so slowly, you forget your humiliations-even the ones that seemed indelible just fade away. You forget who was cool and who was not, who was pretty, smart, athletic, and not. Who went to a good college. Who threw the best parties Who could get you pot. You forget all of them. Even the ones you said you loved, and even the ones you actually did. They're the last to go. And then once you've forgotten enough, you love someone else.

Geometry has two great treasures; one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel.

Three passions, simple but overwhelmingly strong, have governed my life: the longing for love, the search for knowledge, and unbearable pity for the suffering of mankind. These passions, like great winds, have blown me hither and thither, in a wayward course, over a great ocean of anguish, reaching to the very verge of despair. I have sought love, first, because it brings ecstasy - ecstasy so great that I would often have sacrificed all the rest of life for a few hours of this joy. I have sought it, next, because it relieves loneliness--that terrible loneliness in which one shivering consciousness looks over the rim of the world into the cold unfathomable lifeless abyss. I have sought it finally, because in the union of love I have seen, in a mystic miniature, the prefiguring vision of the heaven that saints and poets have imagined. This is what I sought, and though it might seem too good for human life, this is what--at last--I have found. With equal passion I have sought knowledge. I have wished to understand the hearts of men. I have wished to know why the stars shine. And I have tried to apprehend the Pythagorean power by which number holds sway above the flux. A little of this, but not much, I have achieved. Love and knowledge, so far as they were possible, led upward toward the heavens. But always pity brought me back to earth. Echoes of cries of pain reverberate in my heart. Children in famine, victims tortured by oppressors, helpless old people a burden to their sons, and the whole world of loneliness, poverty, and pain make a mockery of what human life should be. I long to alleviate this evil, but I cannot, and I too suffer. This has been my life. I have found it worth living, and would gladly live it again if the chance were offered me.

When a father inquired about the best method of educating his son in ethical conduct, a Pythagorean replied: "Make him a citizen of a state with good laws

The Pythagoreans, you have to remember, were extremely weird. Their philosophy was a chunky stew of things we’d now call mathematics, things we’d now call religion, and things we’d now call mental illness.

Do people believe in human rights because such rights actually exist, like mathematical truths, sitting on a cosmic shelf next to the Pythagorean theorem just waiting to be discovered by Platonic reasoners? Or do people feel revulsion and sympathy when they read accounts of torture, and then invent a story about universal rights to help justify their feelings?

In this world, headwinds are far more prevalent than winds from astern (that is, if you never violate the Pythagorean maxim).

it behoves you to develop a sagacious flair for sniffing and smelling out and appreciating such fair and fatted books, to be swiff: in pursuit and bold in the attack, and then, by careful reading and frequent meditation, to crack open the bone and seek out the substantificial marrow – that is to say, what I mean by such Pythagorean symbols – sure in the hope that you will be made witty and wise by that reading; for you will discover therein a very different savour and a more hidden instruction which will reveal to you the highest hidden truths and the most awesome mysteries

During the first century A.D., Alexandria was a veritable hotbed of mystical activity, a crucible in which Judaic, Mithraic, Zoroastrian, Pythagorean, Hermetic, and neo-Platonic doctrines suffused the air and combined with innumerable others.

What I Have Lived For Three passions, simple but overwhelmingly strong, have governed my life: the longing for love, the search for knowledge, and unbearable pity for the suffering of mankind. These passions, like great winds, have blown me hither and thither, in a wayward course, over a great ocean of anguish, reaching to the very verge of despair. I have sought love, first, because it brings ecstasy - ecstasy so great that I would often have sacrificed all the rest of life for a few hours of this joy. I have sought it, next, because it relieves loneliness--that terrible loneliness in which one shivering consciousness looks over the rim of the world into the cold unfathomable lifeless abyss. I have sought it finally, because in the union of love I have seen, in a mystic miniature, the prefiguring vision of the heaven that saints and poets have imagined. This is what I sought, and though it might seem too good for human life, this is what--at last--I have found. With equal passion I have sought knowledge. I have wished to understand the hearts of men. I have wished to know why the stars shine. And I have tried to apprehend the Pythagorean power by which number holds sway above the flux. A little of this, but not much, I have achieved. Love and knowledge, so far as they were possible, led upward toward the heavens. But always pity brought me back to earth. Echoes of cries of pain reverberate in my heart. Children in famine, victims tortured by oppressors, helpless old people a burden to their sons, and the whole world of loneliness, poverty, and pain make a mockery of what human life should be. I long to alleviate this evil, but I cannot, and I too suffer. This has been my life. I have found it worth living, and would gladly live it again if the chance were offered me.

Even the most carefully defined philosophical or mathematical concept, which we are sure does not contain more than we have put into it, is nevertheless more than we assume. It is a psychic event and as such partly unknowable. The very numbers you use in counting are more than you take them to be. They are at the same time mythological elements (for the Pythagoreans, they were even divine); but you are certainly unaware of this when you use numbers for a practical purpose.

The ultimate goal of Pythagorean and Platonic philosophy was assimilation to god through the cultivation of virtue and truth. It meant a return to the first principles reached through philosophical education (paideia) and recollection (anamnesis), scientific investigation, contemplation, and liturgy (or theurgic ascent), based on the ineffable symbols and sacramental rites.

Thus the thought, for example, which we expressed in the Pythagorean theorem is timelessly true, true independently of whether anyone takes it to be true. It needs no bearer. It is not true for the first time when it is discovered, but is like a planet which, already before anyone has seen it, has been in interaction with other planets.

I began to study trigonometry. There was solace in its strange formulas and equations. I was drawn to the Pythagorean theorem and its promise of a universal—the ability to predict the nature of any three points containing a right angle, anywhere, always. What I knew of physics I had learned in the junkyard, where the physical world often seemed unstable, capricious. But here was a principle through which the dimensions of life could be defined, captured. Perhaps reality was not wholly volatile. Perhaps it could be explained, predicted. Perhaps it could be made to make sense.

The interwoven spheres and vines ran along the bottom. I'd done some research, and I'd found this motif everywhere. These overlapping circles were ancient, tracing back to Pythagorean geometry--geometry, a measure of the world. In more mystical terms, the shape had always evoked tghe place where world overlap: dreaming with waking, death with life, the visible with the unseen. [p. 362]

In the pentagram, the Pythagoreans found all proportions well-known in antiquity: arithmetic, geometric, harmonic, and also the well-known golden proportion, or the golden ratio. ... Probably owing to the perfect form and the wealth of mathematical forms, the pentagram was chosen by the Pythagoreans as their secret symbol and a symbol of health. - Alexander Voloshinov [As quoted in Stakhov]

The forms of mathematics, the harmonies of music, the motions of the planets, and the gods of the mysteries were all essentially related for Pythagoreans, and the meaning of that relation was revealed in an education that culminated in the human soul’s assimilation to the world soul, and thence to the divine creative mind of the universe.

It is known that for the Greeks delta was a symbol for woman. The Pythagoreans regarded the triangle as the arche geneseoas because of its perfect form and because it represented the archetype of universal fertility. A similar symbolism for the triangle is to be found in India.

Uh-oh," Moni sang, and nodded her head in Chantal's direction. "I think someone's a wee bit upset with us." She turned and walked a few steps backward. "Careful," I said. "We're not out of range." "Have no fear, Super Brain is here." Moni whipped out her calculator, holding it up like a shield. "What are you going to do, daze her with denominators?" "Maybe. But first I'm going to pummel her with my Pythagorean theorem.

There is much creativity underlying math and science problem solving. Many people think that there’s only one way to do a problem, but there are often a number of different solutions, if you have the creativity to see them. For example, there are more than three hundred different known proofs of the Pythagorean theorem.

There was not a philosopher of any notoriety who did not hold to this doctrine of metempsychosis, as taught by the Brahmans, Buddhists, and later by the Pythagoreans, in its esoteric sense, whether he expressed it more or less intelligibly. Origen and Clemens Alexandrinus, Synesius and Chalcidius, all believed in it; and the Gnostics, who are unhesitatingly proclaimed by history as a body of the most refined, learned, and enlightened men, * were all believers in metempsychosis. Socrates entertained opinions identical with those of Pythagoras; and both, as the penalty of their divine philosophy, were put to a violent death. The rabble has been the same in all ages. Materialism has been, and will ever be blind to spiritual truths. These philosophers held, with the Hindus, that God had infused into matter a portion of his own Divine Spirit, which animates and moves every particle. They

The Pythagoreans were probably the first to recognize the concept that the basic forces in the universe may be expressed through the language of mathematics.

Further there is the question which is hardest of all and most perplexing, whether unity and being, as the Pythagoreans and Plato said, are not attributes of something else but the substance of existing things, or this is not the case, but the substratum is something else,-as Empedocles says, love; as some one else says, fire; while another says water or air.

According to such a universalist and perennialist perspective, the teachings of Neoplatonism were not a sort of regrettable innovation (as modern classicists would have it), but the faithful perpetuation of pre-Platonic metaphysics put into a new dress. Plato himself was merely a link (albeit crucial) in the Golden Chain of the Pythagorean, Orphic and different Oriental traditions.

[D]idn't Aristarchus and the Pythagoreans propose heliocentrism in ancient times? If only they had prevailed, we might have had Real Science millennia sooner. What was their evidence? Well, you see, fire is nobler than earth and the center is a nobler position. So fire has to be in the center. QED. There are many names for this sort of thinking, but "scientific" is not one of them.

irrational behavior is as unacceptable to a certain species of economist as the irrational magnitude of the hypotenuse was to the Pythagoreans. It doesn’t fit their model of what can be; and yet it is.

Western musicians had to choose between creating sounds and playing notes—and they opted for the latter. But African musicians never got enlightened (or is corrupted the better word?) by Pythagorean thinking.

For as in this world, head winds are far more prevalent than winds from astern (that is, if you never violate the Pythagorean maxim), so for the most part the Commodore on the quarter-deck gets his atmosphere at second hand from the sailors on the forecastle. He thinks he breathes it first; but not so. In much the same way do the commonalty lead their leaders in many other things, at the same time that the leaders little suspect it.

We therefore find that the triangles and rectangles herein described, enclose a large majority of the temples and cathedrals of the Greek and Gothic masters, for we have seen that the rectangle of the Egyptian triangle is a perfect generative medium, its ratio of five in width to eight in length 'encouraging impressions of contrast between horizontal and vertical lines' or spaces; and the same practically may be said of the Pythagorean triangle

One Washington axiom, proved more times than the Pythagorean theorem, states that in the presence of oxygen, one loud fart with an obvious culprit will cover many small emissions in the same room, provided they are nearly simultaneous.

Kepler followed Proclus and believed that 'the main goal of Euclid was to build a geometric theory of the so-called Platonic solids.' Kepler was fascinated by Proclus and often quotes him calling him a 'Pythagorean'. [History of Mathematics]

Straining to see the world through triangle-shaped lenses, Pythagoreans argued that in heredity too a triangular harmony was at work. The mother and the father were two independent sides and the child was the third—the biological hypotenuse to the parents’ two lines. And just as a triangle’s third side could arithmetically be derived from the two other sides using a strict mathematical formula, so was a child derived from the parents’ individual contributions: nature from father and nurture from mother. A

Mythic Background Describing his approach to science, Einstein said something that sounds distinctly prescientific, and hearkens back to those ancient Greeks he admired: What really interests me is whether God had any choice in the creation of the world. Einstein's suggestion that God-or a world-making Artisan-might not have choices would have scandalized Newton or Maxwell. It fits very well, however, with the Pythagorean search for universal harmony, or with Plato's concept of a changeless Ideal. If the Artisan had no choice: Why not? What might constrain a world-making Artisan? One possibility arises if the Artisan is at heart an artist. Then the constraint is desire for beauty. I'd like to (and do) infer that Einstein thought along the line of our Question-Does the world embody beautiful ideas?-and put his faith in the answer "yes!" Beauty is a vague concept. But so, to begin with, were concepts like "force" and "energy." Through dialogue with Nature, scientists learned to refine the meaning of "force" and "energy," to bring their use into line with important aspects of reality. So too, by studying the Artisan's handiwork, we evolve refined concepts of "symmetry," and ultimately of "beauty"-concepts that reflect important aspects of reality, while remaining true to the spirit of their use in common language.

It is this process of symbolization which, in certain hasheesh states, gives every tree and house, every pebble and leaf, every footprint, feature, and gesture, a significance beyond mere matter or form, which possesses an inconceivable force of tortures or of happiness.

I smiled. Mom laughed, shaking her head. “That’s the punchline? Why is that even funny?” “It’s the Pythagorean theorem,” said Lauren. “It’s a math formula for . . . something.” “Right triangles,” I said, and looked pointedly at Margaret. “I told you I’d already done geometry.

There is reason to fear that men love better to investigate how muslins, hay-rakes, and, above all and inclusive of all, money may be made, than how their own minds are constructed

In absolute incommunicableness it stood apart, a thought, a system of thought which as yet had no symbol in spoken language

I know of nothing that has led me to reflect more on Plato's concealment and sphinx nature than that happily preserved petit fait that under the pillow of his death-bed there was discovered no 'Bible', nothing Egyptian, Pythagorean, Platonic - but Aristophanes. How could even a Plato have endured life - a Greek life which he had denied - without an Aristophanes! -

In their later years, each (Einstein and Schrödinger) hoped to find a unified field theory that would fill in the gaps of quantum physics and unite the forces of nature. By extending general relativity to include all of the natural forces, such a theory would replace matter with pure geometry - fulfilling the dream of the Pythagoreans, who believed that "all is number".

Finally, the dishonesty in the movement of the publication of a Greek philosophy, becomes very glaring, when we refer to the fact, purposely that by calling the theorem of the Square on the Hypotenuse, the Pythagorean theorem, it has concealed the truth for centuries from the world, who ought to know that the Egyptians taught Pythagoras and the Greeks, what mathematics they knew.

Pythagoras apparently wrote nothing, and yet his influence was so great that the more attentive of his followers formed a secretive society, or brotherhood, and were known as the Pythagoreans. Aristippus of Cyrene tells us in his Account of Natural Philosphers that Pythagoras derived his name from the fact that he was speaking (agoreuein) truth like the God at Delphi (tou Pythiou).

A disdain for the practical swept the ancient world. Plato urged astronomers to think about the heavens, but not to waste their time observing them. Aristotle believed that: “The lower sort are by nature slaves, and it is better for them as for all inferiors that they should be under the rule of a master.… The slave shares in his master’s life; the artisan is less closely connected with him, and only attains excellence in proportion as he becomes a slave. The meaner sort of mechanic has a special and separate slavery.” Plutarch wrote: “It does not of necessity follow that, if the work delight you with its grace, the one who wrought it is worthy of esteem.” Xenophon’s opinion was: “What are called the mechanical arts carry a social stigma and are rightly dishonoured in our cities.” As a result of such attitudes, the brilliant and promising Ionian experimental method was largely abandoned for two thousand years. Without experiment, there is no way to choose among contending hypotheses, no way for science to advance. The anti-empirical taint of the Pythagoreans survives to this day. But why? Where did this distaste for experiment come from? An explanation for the decline of ancient science has been put forward by the historian of science, Benjamin Farrington: The mercantile tradition, which led to Ionian science, also led to a slave economy. The owning of slaves was the road to wealth and power. Polycrates’ fortifications were built by slaves. Athens in the time of Pericles, Plato and Aristotle had a vast slave population. All the brave Athenian talk about democracy applied only to a privileged few. What slaves characteristically perform is manual labor. But scientific experimentation is manual labor, from which the slaveholders are preferentially distanced; while it is only the slaveholders—politely called “gentle-men” in some societies—who have the leisure to do science. Accordingly, almost no one did science. The Ionians were perfectly able to make machines of some elegance. But the availability of slaves undermined the economic motive for the development of technology. Thus the mercantile tradition contributed to the great Ionian awakening around 600 B.C., and, through slavery, may have been the cause of its decline some two centuries later. There are great ironies here.

The reciprocal relationship of epistemology and science is of noteworthy kind. They are dependent upon each other. Epistemology without contact with science becomes an empty scheme. Science without epistemology is—insofar as it is thinkable at all—primitive and muddled. However, no sooner has the epistemologist, who is seeking a clear system, fought his way through to such a system, than he is inclined to interpret the thought-content of science in the sense of his system and to reject whatever does not fit into his system. The scientist, however, cannot afford to carry his striving for epistemological systematic that far. He accepts gratefully the epistemological conceptual analysis; but the external conditions, which are set for him by the facts of experience, do not permit him to let himself be too much restricted in the construction of his conceptual world by the adherence to an epistemological system. He therefore must appear to the systematic epistemologist as a type of unscrupulous opportunist: he appears as realist insofar as he seeks to describe a world independent of the acts of perception; as idealist insofar as he looks upon the concepts and theories as free inventions of the human spirit (not logically derivable from what is empirically given); as positivist insofar as he considers his concepts and theories justified only to the extent to which they furnish a logical representation of relations among sensory experiences. He may even appear as Platonist or Pythagorean insofar as he considers the viewpoint of logical simplicity as an indispensable and effective tool of his research. (Einstein 1949, 683–684)

Belief in reincarnation was the central tenet of the Pythagorean religion. Pythagoras taught that the soul was indestructible, but that people needed to follow certain rules to ensure the best possible reincarnations after their physical deaths. Among the rules adopted by his followers: no wearing shoes in the temple, no touching white roosters, put the right shoe on first, and always abstain from eating beans.

A similar tradition on the creative power of letters forms the basis of the following midrash on Job 28:11.... This brings us to the text that played so important a part in the development of the golem concept: the Book of Yetsirah or the Book of Creation.... We do not know the exact date of this enigmatic text,.... We can only be sure that it was written by a Jewish Neo-Pythagorean some time between the third and sixth century.

now I want to speak about the word ‘theory’. This was originally an Orphic word, which Cornford interprets as ‘passionate sympathetic contemplation’. In this state, he says, ‘The spectator is identified with the suffering God, dies in his death, and rises again in his new birth.’ for Pythagoras, the ‘passionate sympathetic contemplation’ was intellectual, and issued in mathematical knowledge. In this way, through Pythagoreanism, ‘theory’ gradually acquired its modern meaning; but for all who were inspired by Pythagoras it retained an element of ecstatic revelation. To those who have reluctantly learnt a little mathematics in school this may seem strange; but to those who have experienced the intoxicating delight of sudden understanding that mathematics gives, from time to time, to those who love it, the Pythagorean view will seem completely natural even if untrue.

Three passions, simple but overwhelmingly strong, have governed my life: the longing for love, the search for knowledge, and unbearable pity for the suffering of mankind. These passions, like great winds, have blown me hither and thither, in a wayward course, over a great ocean of anguish, reaching to the very verge of despair. I have sought love, first, because it brings ecstasy - ecstasy so great that I would often have sacrificed all the rest of life for a few hours of this joy. I have sought it, next, because it relieves lonelines - that terrible loneliness in which one shivering consciousness looks over the rim of the world into the cold unfathomable lifeless abyss. I have sought it finally, because in the union of love I have seen, in a mystic miniature, the prefiguring vision of the heaven that saints and poets have imagined. This is what I sought, and though it might seem too good for human life, this is what - at last - I have found. With equal passion I have sought knowledge. I have wished to understand the hearts of men. I have wished to know why the stars shine. And I have tried to apprehend the Pythagorean power by which number holds sway above the flux. A little of this, but not much, I have achieved. Love and knowledge, so far as they were possible, led upward toward the heavens. But always pity brought me back to earth. Echoes of cries of pain reverberate in my heart. Children in famine, victims tortured by oppressors, helpless old people a burden to their sons, and the whole world of loneliness, poverty, and pain make a mockery of what human life should be. I long to alleviate this evil, but I cannot, and I too suffer. This has been my life. I have found it worth living, and would gladly live it again if the chance were offered me.

But of the stimulus of drugs, of potions, beware. For the sake of that very majesty with which you justly wish to aggrandize your soul, beware. Their fountains will be presently exhausted, and then you shall helplessly beat your breast, as without possibility of arising from the brink you draw in their foul, their maddening lees, and curse yourself for slaying those noble powers which it was your longing to strengthen, to nourish, and to clarify.

Even the most carefully defined philosophical or mathematical concept, which we are sure does not contain more than we put into it, is nevertheless more than we assume. It is a psychic event and as such partly unknowable. The very numbers you use in counting are more than you take them to be. They are at the same time mythological elements (for the Pythagoreans, they were even divine); but you are certainly unaware of this when you use numbers for practical purpose.

Yet if there be one voice which can speak from the gateway of a dangerous avenue to its satisfaction, that can say, “Ho there! pass by; I have tried this way; it leads at last into poisonous wildernesses,” in the name of Heaven let it be raised.

The two kinds of mental activity that can be pursued by the method that Plato recommends are mathematics and mystic insight. This explains how these two come to be so intimately combined in Plato and the Pythagoreans. To the empiricist, the body is what brings us into touch with the world of external reality, but to Plato it is doubly evil, as a distorting medium, causing us to see as through a glass darkly, and as a source of lusts which distract us from the pursuit of knowledge and the vision of truth.

theory'. This was originally an Orphic word, which Cornford interprets as 'passionate sympathetic contemplation'. In this state, he says, 'The spectator is identified with the suffering God, dies in his death, and rises again in his new birth.' For Pythagoras, the 'passionate sympathetic contemplation' was intellectual, and issued in mathematical knowledge. In this way, through Pythagoreanism, 'theory' gradually acquired its modern meaning; but for all who were inspired by Pythagoras it retained an element of ecstatic revelation.

I don’t see him much.” “It happens, baby. You forget all of it anyway. First, you forget everything you learned – the dates of the Hay-Herran Treaty and the Pythagorean theorem. You especially forget everything you didn’t really learn, just memorised the night before. You forget the names of all but one or two of your teachers, and eventually you’ll forget those, too. You forget your junior year class schedule and where you used to sit and your best friend’s home phone number and the lyrics to that song you must have played a million times. For me, it was something by Simon & Garfunkel. Who knows what it will be for you? And eventually, but slowly, oh so slowly, you forget your humiliations – even the ones that seemed indelible just fade away. You forget who was cool and who was not, who was pretty, smart, athletic, and not. Who went to a good college. Who threw the best parties. Who could get you pot. You forget all of them. Even the ones you said you loved, and the ones you actually did. They’re the last to go. And then once you’ve forgotten enough, you love someone else.

I always go to the sea as a sailor, because of the wholesome exercise and the pure air of the forecastle deck. For as in this world, head winds are far more prevalent than winds from astern (that is, if you never violate the Pythagorean maxim), so for the most part the Commodore on the quarter-deck gets his atmosphere at second hand from the sailors on the forecastle. He thinks he breathes it first; but not so. In much the same way do the commonalty lead their leaders in many other things, at the same time that the leaders little suspect it.

Having a daemonic and intermediate nature, Eros was one of the links between the sensible cosmos and the eternal world of the gods. Accordingly, Eros was regarded as a paradigm and pattern for the philosopher, or lover of wisdom, because wisdom was beautiful and beauty was loveable.

The curriculum for the education of statesmen at the time of Plato included arithmetic, geometry, solid geometry, astronomy, and music-all of which, the Pythagorean Archytas tells us, fell under the general definition of "mathematics." According to legend, when Alexander the Great asked his teacher Menaechmus (who is reputed to have discovered the curves of the ellipse, the parabola, and the hyperbola) for a shortcut to geometry, he got the reply: "O King, for traveling over the country there are royal roads and roads for common citizens; but in geometry there is one road for all.

18B. (Iamblichus, Theol. Arith. 61). The Decad is also named Faith, because, according to Philolaus, it is by the Decad and its elements, if utilized energetically and without negligence, that we arrive at a solidly grounded faith about beings. It is also the source of memory, and that is why the Monad has been called Mnemosyne.

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

I...I...YOU...SIXTEEN LOG THIRTY-THREE...ALL COSINE SUBSCRIPTS...ANTI...ANTI...IN ALL THESE YEARS...BEAM...FLOOD...PYTHAGOREAN...CARTESIAN LOGIC...CAN I...DARE I...A PEACH...EAT A PEACH...ALLMAN BROTHERS...PATRICIA...CROCODILE AND WHIPLASH SMILE...CLOCK OF DIALS...TICK-TOCK, ELEVEN O'CLOCK, THE MAN'S IN THE MOON AND HE'S READY TO ROCK...INCESSAMENT...INCESSAMENT, MON CHER...OH MY HEAD...BLAINE...BLAINE DARES...BLAINE WILL ANSWER...I...(screaming in the voice of an infant, lapsing into another language, presumably French, as none of the words are familiar to Eddie, beginning to sing when the song Velcro Fly by Z.Z. Top suddenly plays courtesy of its percussion drums)

Most people-all, in fact, who regard the whole heaven as finite-say it lies at the centre. But the Italian philosophers known as Pythagoreans take the contrary view. At the centre, they say, is fire, and the earth is one of the stars, creating night and day by its circular motion about the centre. They further construct another earth in opposition to ours to which they give the name counterearth. In all this they are not seeking for theories and causes to account for observed facts, but rather forcing their observations and trying to accommodate them to certain theories and opinions of their own. But there are many others who would agree that it is wrong to give the earth the central position, looking for confirmation rather to theory than to the facts of observation. Their view is that the most precious place befits the most precious thing: but fire, they say, is more precious than earth, and the limit than the intermediate, and the circumference and the centre are limits. Reasoning on this basis they take the view that it is not earth that lies at the centre of the sphere, but rather fire. The Pythagoreans have a further reason. They hold that the most important part of the world, which is the centre, should be most strictly guarded, and name it, or rather the fire which occupies that place, the 'Guardhouse of Zeus', as if the word 'centre' were quite unequivocal, and the centre of the mathematical figure were always the same with that of the thing or the natural centre. But it is better to conceive of the case of the whole heaven as analogous to that of animals, in which the centre of the animal and that of the body are different. For this reason they have no need to be so disturbed about the world, or to call in a guard for its centre: rather let them look for the centre in the other sense and tell us what it is like and where nature has set it. That centre will be something primary and precious; but to the mere position we should give the last place rather than the first. For the middle is what is defined, and what defines it is the limit, and that which contains or limits is more precious than that which is limited, see ing that the latter is the matter and the former the essence of the system. (2-13-1) There are similar disputes about the shape of the earth. Some think it is spherical, others that it is flat and drum-shaped. For evidence they bring the fact that, as the sun rises and sets, the part concealed by the earth shows a straight and not a curved edge, whereas if the earth were spherical the line of section would have to be circular. In this they leave out of account the great distance of the sun from the earth and the great size of the circumference, which, seen from a distance on these apparently small circles appears straight. Such an appearance ought not to make them doubt the circular shape of the earth. But they have another argument. They say that because it is at rest, the earth must necessarily have this shape. For there are many different ways in which the movement or rest of the earth has been conceived. (2-13-3)

The outsiders stood always in awe in front of what they had surnamed the Celestial City with Mighty Walls. The great mystery that cloaked its very foundations kept impelling the youth of Crotona, as well as those of the adjacent cities, to seek admittance. In spite of the difficult rules of the Master, curiosity goaded many to venture inside its secrecy, with a passionate aspiration to discover the unknown. Yet, to enroll, young men and women should be introduced by their parents. Sometimes, it was one of the assigned Masters of the Pythagorean Society who assumed the introduction. At the massive wooden gated entrance, one could admire the marble statue of Hermes-Enoch, the father of the spiritual laws. A cubical stone formed its stall where a skillful hand had carved the words: No entry to the vulgar

The Eternal Return has certainly not been thought by philosophers or by those who are concerned about Nietzsche in the contemporary history of ideas, and this because the Eternal Return can not be thought of. It is a revelation that presents next to the Silvaplana rock, or on the threshold of the Gateway of the Moment, where the Two Ways meet. You will have to travel step by step along the path of Western yoga that Nietzsche rediscovered and practiced, putting his feet in the tracks that he left in the paths of the high peaks, relive their great pains and divine glories, reaching to reach similar tonalities of the soul, to be possessed by Dionysus and his ancient drunkenness, Luciferian, that makes dance in the solitude of forests and lost from a solar age, laughing and crying at the same time. And this is not achieved by the philosophers of the intellect or the beings 'of the flock'. For to achieve this, the Circle will have to be traversed for several eternities, again at the Gateway of the Moment, already predestined at noon. In addition, the doctrine of the Eternal Return is selective. As the initiatory practice Tantric Panshatattva is not for the paśu [animal], but only for some heroes or viryas, thus the Noon is reached by the 'Lords of the Earth' and by the poets of the Will to Power, predestined in a mysterious way to perform the Superman, that individualistic and aristocratic mutation. The 'herd', the vulgar, has nothing to do with all this, including here the scientists, technologists and most philosophers, politicians and government of the Kaliyuga. Nietzsche's description of the Eternal Return is found in some aphorisms that precede 'The Gay Science', Joyful Science, using Nietzsche the Provencal term, Occitan, from 'Gay'. Joyful Science will be that of the one who has accepted the Eternal Return of all things and has transmuted the values. The one of Superman. There is also a description in the schemes of 'The Will to Power'. In they all take hold, with genius that transcends their time, of the scientific knowledge and the mechanics of the time, which does not lose validity to the doctrine, let us say better to the revealed Idea, to the Revelation that, of somehow, it was also in the Pythagoreans, in their Aryan-Hyperborean form, differentiating itself from other elaborations made in the millennia of the East. Also would have been veiled in the Persian reformer Zarathustra. We are going to reproduce what Nietzsche has written about the Eternal Return. In the schemes of 'The Will to Power', he says: 'Everything returns and returns eternally; We can not escape this.

Thus only the pattern is cosmically determined, not any particular event; within that pattern, man is free. In his later years, this Gestalt concept of cosmic destiny became more abstract and purified from dross. The individual soul, which bears the potential imprint of the entire sky, reacts to the light coming from the planets according to the angles they form with each other, and the geometrical harmonies or disharmonies that result - just as the ear reacts to the mathematical harmonies of music, and the eye to the harmonies of colour. This capacity of the soul to act as a cosmic resonator has a mystic and a causal aspect: on the one hand it affirms the soul's affinity with the anima mundi, on the other, it makes it subject to strictly mathematical laws. At this point, Kepler's particular brand of astrology merges into his all-embracing and unifying Pythagorean vision of the Harmony of the Spheres.

Nevertheless, to learn right away something new from the same example, how fleeting and weak, how imprecise that comparison would be! If the comparison is to carry out this powerful effect, how much of the difference will be missed in the process. How forcefully must the individuality of the past be wrenched into a general shape, with all its sharp corners and angles broken off for the sake of the correspondence! In fact, basically something that once was possible could appear possible a second time only if the Pythagoreans were correct in thinking that with the same constellations of the celestial bodies the same phenomena on the Earth had to repeat themselves, even in the small single particulars, so that when the stars have a certain position relative to each other, a Stoic and an Epicurean will, in an eternal recurrence, unite and assassinate Caesar, and with another stellar position Columbus will eternally rediscover America.

The Pythagoreans were fascinated by the regular solids, symmetrical three-dimensional objects all of whose sides are the same regular polygon. The cube is the simplest example, having six squares as sides. There are an infinite number of regular polygons, but only five regular solids. (The proof of this statement, a famous example of mathematical reasoning, is given in Appendix 2.) For some reason, knowledge of a solid called the dodecahedron having twelve pentagons as sides seemed to them dangerous. It was mystically associated with the Cosmos. The other four regular solids were identified, somehow, with the four “elements” then imagined to constitute the world; earth, fire, air and water. The fifth regular solid must then, they thought, correspond to some fifth element that could only be the substance of the heavenly bodies. (This notion of a fifth essence is the origin of our word quintessence.) Ordinary people were to be kept ignorant of the dodecahedron.

At the same time, it must be admitted that, unless words, to some extent, had fixed meanings, discourse would be impossible. Here again, however, it is easy to be too absolute. Words do change their meanings; take, for example, the word 'idea'. It is only by a considerable process of education that we learn to give to this word something like the meaning which Plato gave to it. It is necessary that the changes in the meanings of words should be slower than the changes that the words describe; but it is not necessary that there should be no changes in the meanings of words. Perhaps this does not apply to the abstract words of logic and mathematics, but these words, as we have seen, apply only to the form, not to the matter, of propositions. Here, again, we find that logic and mathematics are peculiar. Plato, under the influence of the Pythagoreans, assimilated other knowledge too much to mathematics. He shared this mistake with many of the greatest philosophers, but it was a mistake none the less.

Perhaps the most influential person ever associated with Samos was Pythagoras,* a contemporary of Polycrates in the sixth century B.C. According to local tradition, he lived for a time in a cave on the Samian Mount Kerkis, and was the first person in the history of the world to deduce that the Earth is a sphere. Perhaps he argued by analogy with the Moon and the Sun, or noticed the curved shadow of the Earth on the Moon during a lunar eclipse, or recognized that when ships leave Samos and recede over the horizon, their masts disappear last. He or his disciples discovered the Pythagorean theorem: the sum of the squares of the shorter sides of a right triangle equals the square of the longer side. Pythagoras did not simply enumerate examples of this theorem; he developed a method of mathematical deduction to prove the thing generally. The modern tradition of mathematical argument, essential to all of science, owes much to Pythagoras. It was he who first used the word Cosmos to denote a well-ordered and harmonious universe, a world amenable to human understanding.

Look at a strung bow lying on the ground or leaning against a wall. No movement is visible. To the eyes, it appears a static object, completely at rest. But in fact, a continuous tug-of-war is going on within it, as will become evident if the string is not strong enough, or is allowed to perish. The bow will immediately take advantage, snap it and leap to straighten itself, thus showing that each had been putting forth effort all the time. The harmonia was a dynamic one of vigorous and contrary motions neutralized by equilibrium and so unapparent.

The purely philosophical influences on Plato were also such as to pre-dispose him in favour of Sparta. These influences, speaking broadly, were: Pythagoras, Parmenides, Heraclitus, and Socrates. From Pythagoras (whether by way of Socrates or not) Plato derived the Orphic elements in his philosophy: the religious trend, the belief in immortality, the other-worldliness, the priestly tone, and all that is involved in the simile of the cave; also his respect for mathematics, and his intimate intermingling of intellect and mysticism. From Parmenides he derived the belief that reality is eternal and timeless, and that, on logical grounds, all change must be illusory. From Heraclitus he derived the negative doctrine that there is nothing permanent in the sensible world. This, combined with the doctrine of Parmenides, led to the conclusion that knowledge is not to be derived from the senses, but is only to be achieved by the intellect. This, in turn, fitted in well with Pythagoreanism. From Socrates he probably learnt his preoccupation with ethical problems, and his tendency to seek teleological rather than mechanical explanations of the world. 'The Good' dominated his thought more than that of the pre-Socratics, and it is difficult not to attribute this fact to the influence of Socrates.

We are strangers in this world, and the body is the tomb of the soul, and yet we must not seek to escape by self-murder; for we are the chattels of God who is our herdsman, and without his command we have no right to make our escape. In this life, there are three kinds of men, just as there are three sorts of people who come to the Olympic Games. The lowest class is made up of those who come to buy and sell, the next above them are those who compete. Best of all, however, are those who come simply to look on. The greatest purification of all is, therefore, disinterested science, and it is the man who devotes himself to that, the true philosopher, who has most effectually released himself from the 'wheel of birth.

Goodness and Reality being timeless, the best State will be the one which most nearly copies the heavenly model, by having a minimum of change and a maximum of static perfection, and its rulers should be those who best understand the eternal Good. In the second place: Plato, like all mystics, has, in his beliefs, a core of certainty which is essentially incommunicable except by a way of life. The Pythagoreans had endeavoured to set up a rule of the initiate, and this is, at bottom, what Plato desires. If a man is to be a good statesman, he must know the Good; this he can only do by a combination of intellectual and moral discipline. If those who have not gone through this discipline are allowed a share in the government, they will inevitably corrupt it. In the third place: much education is needed to make a good ruler on Plato's principles. It seems to us unwise to have insisted on teaching geometry to the younger Dionysius, tyrant of Syracuse, in order to make him a good king, but from Plato's point of view it was essential. He was sufficiently Pythagorean to think that without mathematics no true wisdom is possible. This view implies an oligarchy. In the fourth place: Plato, in common with most Greek philosophers, took the view that leisure is essential to wisdom, which will therefore not be found among those who have to work for their living, but only among those who have independent means or who are relieved by the State from anxieties as to their subsistence. This point of view is essentially aristocratic.

Three passions, simple but overwhelmingly strong, have governed my life: the longing for love, the search for knowledge, and unbearable pity for the suffering of mankind.These passions, like great winds, have blown me hither and thither, in a wayward course, over a great ocean of anguish, reaching to the very verge of despair. I have sought love, first, because it brings ecstasy - ecstasy so great that I would often have sacrificed all the rest of life for a few hours of this joy. I have sought it, next, because it relieves loneliness—that terrible loneliness in which one shivering consciousness looks over the rim of the world into the cold unfathomable lifeless abyss. I have sought it finally, because in the union of love I have seen, in a mystic miniature, the prefiguring vision of the heaven that saints and poets have imagined. This is what I sought, and though it might seem too good for human life, this is what—at last—I have found. With equal passion I have sought knowledge. I have wished to understand the hearts of men. I have wished to know why the stars shine. And I have tried to apprehend the Pythagorean power by which number holds sway above the flux. A little of this, but not much, I have achieved. Love and knowledge, so far as they were possible, led upward toward the heavens. But always pity brought me back to earth. Echoes of cries of pain reverberate in my heart…The whole world of loneliness, poverty, and pain make a mockery of what human life should be. I long to alleviate this evil, but I cannot, and I too suffer. This has been my life. I have found it worth living, and would gladly live it again if the chance were offered me.

That great portion of what is generally received as Christian truth is, in its rudiments or in its separate parts, to be found in heathen philosophies and religions. For instance, the doctrine of a Trinity is found both in the East and in the West; so is the ceremony of washing; so is the rite of sacrifice. The doctrine of the Divine Word is Platonic; the doctrine of the Incarnation is Indian; of a divine kingdom is Judaic; of Angels and demons is Magian; the connection of sin with the body is Gnostic; celibacy is known to Bonze and Talapoin; a sacerdotal order is Egyptian; the idea of a new birth is Chinese and Eleusinian; belief in sacramental virtue is Pythagorean; and honours to the dead are a polytheism. Such is the general nature of the fact before us; Mr. Milman argues from it,—'These things are in heathenism, therefore they are not Christian:' we, on the contrary, prefer to say, 'these things are in Christianity, therefore they are not heathen.' That is, we prefer to say, and we think that Scripture bears us out in saying, that from the beginning the Moral Governor of the world has scattered the seeds of truth far and wide over its extent; that these have variously taken root, and grown up as in the wilderness, wild plants indeed but living; and hence that, as the inferior animals have tokens of an immaterial principle in them, yet have not souls, so the philosophies and religions of men have their life in certain true ideas, though they are not directly divine. What man is amid the brute creation, such is the Church among the schools of the world; and as Adam gave names to the animals about him, so has the Church from the first looked round upon the earth, noting and visiting the doctrines she found there.

Three passions, simple but overwhelmingly strong, have governed my life: the longing for love, the search for knowledge, and unbearable pity for the suffering of mankind. These passions, like great winds, have blown me hither and thither, in a wayward course, over a great ocean of anguish, reaching to the very verge of despair. I have sought love, first, because it brings ecstasy - ecstasy so great that I would often have sacrificed all the rest of life for a few hours of this joy. I have sought it, next, because it relieves loneliness—that terrible loneliness in which one shivering consciousness looks over the rim of the world into the cold unfathomable lifeless abyss. I have sought it finally, because in the union of love I have seen, in a mystic miniature, the prefiguring vision of the heaven that saints and poets have imagined. This is what I sought, and though it might seem too good for human life, this is what—at last—I have found. With equal passion I have sought knowledge. I have wished to understand the hearts of men. I have wished to know why the stars shine. And I have tried to apprehend the Pythagorean power by which number holds sway above the flux. A little of this, but not much, I have achieved. Love and knowledge, so far as they were possible, led upward toward the heavens. But always pity brought me back to earth. Echoes of cries of pain reverberate in my heart. Children in famine, victims tortured by oppressors, helpless old people a burden to their sons, and the whole world of loneliness, poverty, and pain make a mockery of what human life should be. I long to alleviate this evil, but I cannot, and I too suffer. This has been my life. I have found it worth living, and would gladly live it again if the chance were offered me.

Three passions, simple but overwhelmingly strong, have governed my life: the longing for love, the search for knowledge, and unbearable pity for the suffering of mankind. These passions, like great winds, have blown me hither and thither, in a wayward course, over a great ocean of anguish, reaching to the very verge of despair. I have sought love, first, because it brings ecstasy - ecstasy so great that I would often have sacrificed all the rest of life for a few hours of this joy. I have sought it, next, because it relieves loneliness--that terrible loneliness in which one shivering consciousness looks over the rim of the world into the cold unfathomable lifeless abyss. I have sought it finally, because in the union of love I have seen, in a mystic miniature, the prefiguring vision of the heaven that saints and poets have imagined. This is what I sought, and though it might seem too good for human life, this is what--at last--I have found. With equal passion I have sought knowledge. I have wished to understand the hearts of men. I have wished to know why the stars shine. And I have tried to apprehend the Pythagorean power by which number holds sway above the flux. A little of this, but not much, I have achieved. Love and knowledge, so far as they were possible, led upward toward the heavens. But always pity brought me back to earth. Echoes of cries of pain reverberate in my heart. Children in famine, victims tortured by oppressors, helpless old people a burden to their sons, and the whole world of loneliness, poverty, and pain make a mockery of what human life should be. I long to alleviate this evil, but I cannot, and I too suffer. This has been my life. I have found it worth living, and would gladly live it again if the chance were offered me.

The Eternal Return has certainly not been thought by philosophers or by those who are concerned about Nietzsche in the contemporary history of ideas, and this because the Eternal Return can not be thought of. It is a revelation that presents itself next to the Silvaplana rock, or on the threshold of the Gateway of the Moment, where the Two Ways meet. You will have to travel step by step along the path of Western yoga that Nietzsche rediscovered and practiced, putting his feet in the tracks that he left in the paths of the high peaks, relive their great pains and divine glories, reaching to reach similar tonalities of the soul, to be possessed by Dionysus and his ancient drunkenness, Luciferian, that makes dance in the solitude of forests and lost from a solar age, laughing and crying at the same time. And this is not achieved by the philosophers of the intellect or the beings 'of the flock'. For to achieve this, the Circle will have to be traversed for several eternities, again at the Gateway of the Moment, already predestined at noon. In addition, the doctrine of the Eternal Return is selective. As the initiatory practice Tantric Panshatattva is not for the paśu [animal], but only for some heroes or viryas, thus the Noon is reached by the 'Lords of the Earth' and by the poets of the Will to Power, predestined in a mysterious way to perform the Superman, that individualistic and aristocratic mutation. The 'herd', the vulgar, has nothing to do with all this, including here the scientists, technologists and most philosophers, politicians and government of the Kaliyuga. Nietzsche's description of the Eternal Return is found in some aphorisms that precede 'The Gay Science', Joyful Science, using Nietzsche the Provencal term, Occitan, from 'Gay'. Joyful Science will be that of the one who has accepted the Eternal Return of all things and has transmuted the values. The one of Superman. There is also a description in the schemes of 'The Will to Power'. In they all take hold, with genius that transcends their time, of the scientific knowledge and the mechanics of the time, which does not lose validity to the doctrine, let us say better to the revealed Idea, to the Revelation that, of somehow, it was also in the Pythagoreans, in their Aryan-Hyperborean form, differentiating itself from other elaborations made in the millennia of the East. Also would have been veiled in the Persian reformer Zarathustra. We are going to reproduce what Nietzsche has written about the Eternal Return. In the schemes of 'The Will to Power', he says: 'Everything returns and returns eternally; We can not escape this.

The Eternal Return has certainly not been thought by philosophers or by those who are concerned about Nietzsche in the contemporary history of ideas, and this because the Eternal Return can not be thought of. It is a revelation that presents itself next to the Silvaplana rock, or on the threshold of the Gateway of the Moment, where the Two Ways meet. You will have to travel step by step along the path of Western yoga that Nietzsche rediscovered and practiced, putting his feet in the tracks that he left in the paths of the high peaks, relive their great pains and divine glories, reaching to reach similar tonalities of the soul, to be possessed by Dionysus and his ancient drunkenness, Luciferian, that makes dance in the solitude of forests and lost from a solar age, laughing and crying at the same time. And this is not achieved by the philosophers of the intellect or the beings 'of the flock'. For to achieve this, the Circle will have to be traversed for several eternities, again at the Gateway of the Moment, already predestined at noon. In addition, the doctrine of the Eternal Return is selective. As the initiatory practice Tantric Panshatattva is not for the paśu [animal], but only for some heroes or viryas, thus the Noon is reached by the 'Lords of the Earth' and by the poets of the Will to Power, predestined in a mysterious way to perform the Superman, that individualistic and aristocratic mutation. The 'herd', the vulgar, has nothing to do with all this, including here the scientists, technologists and most philosophers, politicians and government of the Kaliyuga. Nietzsche's description of the Eternal Return is found in some aphorisms that precede 'The Gay Science', Joyful Science, using Nietzsche the Provencal term, Occitan, from 'Gay'. Joyful Science will be that of the one who has accepted the Eternal Return of all things and has transmuted the values. The one of Superman. There is also a description in the schemes of 'The Will to Power'. In they all take hold, with genius that transcends their time, of the scientific knowledge and the mechanics of the time, which does not lose validity to the doctrine, let us say better to the revealed Idea, to the Revelation that, of somehow, it was also in the Pythagoreans, in their Aryan-Hyperborean form, differentiating itself from other elaborations made in the millennia of the East. Also would have been veiled in the Persian reformer Zarathustra. We are going to reproduce what Nietzsche has written about the Eternal Return. In the schemes of 'The Will to Power', he says: 'Everything returns and returns eternally; We can not escape this.

Or think of the tale of the blind men who encounter an elephant for the first time. One wise man, touching the ear of the elephant, declares the elephant is flat and two-dimensional like a fan. Another wise man touches the tail and assumes the elephant is like rope or a one-dimensional string. Another, touching a leg, concludes the elephant is a three-dimensional drum or a cylinder. But actually, if we step back and rise into the third dimension, we can see the elephant as a three-dimensional animal. In the same way, the five different string theories are like the ear, tail, and leg, but we still have yet to reveal the full elephant, M-theory. Holographic Universe As we mentioned, with time new layers have been uncovered in string theory. Soon after M-theory was proposed in 1995, another astonishing discovery was made by Juan Maldacena in 1997. He jolted the entire physics community by showing something that was once considered impossible: that a supersymmetric Yang-Mills theory, which describes the behavior of subatomic particles in four dimensions, was dual, or mathematically equivalent, to a certain string theory in ten dimensions. This sent the physics world into a tizzy. By 2015, there were ten thousand papers that referred to this paper, making it by far the most influential paper in high-energy physics. (Symmetry and duality are related but different. Symmetry arises when we rearrange the components of a single equation and it remains the same. Duality arises when we show that two entirely different theories are actually mathematically equivalent. Remarkably, string theory has both of these highly nontrivial features.) As we saw, Maxwell’s equations have a duality between electric and magnetic fields—that is, the equations remain the same if we reverse the two fields, turning electric fields into magnetic fields. (We can see this mathematically, because the EM equations often contain terms like E2 + B2, which remain the same when we rotate the two fields into each other, like in the Pythagorean theorem). Similarly, there are five distinct string theories in ten dimensions, which can be proven to be dual to each other, so they are really a single eleven-dimensional M-theory in disguise. So remarkably, duality shows that two different theories are actually two aspects of the same theory. Maldacena, however, showed that there was yet another duality between strings in ten dimensions and Yang-Mills theory in four dimensions. This was a totally unexpected development but one that has profound implications. It meant that there were deep, unexpected connections between the gravitational force and the nuclear force defined in totally different dimensions. Usually, dualities can be found between strings in the same dimension. By rearranging the terms describing those strings, for example, we can often change one string theory into another. This creates a web of dualities between different string theories, all defined in the same dimension. But a duality between two objects defined in different dimensions was unheard of.

In Babylonian religious thought the gods are represented by numbers. The number 1 represented the High God. The god of music, Enki, was represented by the number 40 which, in the sexagesimal system, means , that is, or the musical perfect fifth—the same ratio applied to the winter solstice (the New Year). One cannot regain the whole system, only edges of it here and there—a “great worldwide archaic construction” which was “preserved almost intact in the later thought of the Pythagoreans and Plato.”70

There are a thousand ways in which his neighbours can evaporate the essence which is all in all to him, while they at the same time give to his scenery ponderable value which to them is worth far more

A meme is simply a unit of memorable cultural information. It can be as small as a tune or a metaphor, as big as a philosophy or religious concept. Hell is a meme; so are the Pythagorean theorem, A Hard Day’s Night, the wheel, Hamlet, pragmatism, harmony, “Where’s the beef?,” and of course the notion of the meme itself.

The world is in A major...the earth revolves in A major, a low A.

The Pythagoreans... were fascinated by certain specific ratios, ...The Greeks knew these as the 'golden' proportion and the 'perfect' proportion respectively. They may well have been learned from the Babylonians by Pythagoras himself after having been taken prisoner in Egypt. Ratios lay at the heart of the Pythagorean theory of music.

Paul was acquainted with the Doctrine of Reincarnation, also known as Metempsychosis, mentioned in several world religion books and traditions like in Hinduism, Buddhism, and the Pythagoreans’. Frankly

Maybe ET will use math to get our attention—a string of prime numbers, perhaps, or the apparently omnipresent, omni-important Pythagorean theorem. High school trig teachers should rebrand it the alien communication formula. That would’ve kept you from daydreaming in class, right?

This sense of awe goes way back in the history of mathematics. According to legend, Pythagoras felt it around 550 BCE when he and his disciples discovered that music was governed by the ratios of whole numbers. For instance, imagine plucking a guitar string. As the string vibrates, it emits a certain note. Now put your finger on a fret exactly halfway up the string and pluck it again. The vibrating part of the string is now half as long as it used to be—a ratio of 1 to 2—and it sounds precisely an octave higher than the original note (the musical distance from one do to the next in the do-re-mi-fa-sol-la-ti-do scale). If instead the vibrating string is ⅔ of its original length, the note it makes goes up by a fifth (the interval from do to sol; think of the first two notes of the Stars Wars theme). And if the vibrating part is ¾ as long as it was before, the note goes up by a fourth (the interval between the first two notes of “Here Comes the Bride”). The ancient Greek musicians knew about the melodic concepts of octaves, fourths, and fifths and considered them beautiful. This unexpected link between music (the harmony of this world) and numbers (the harmony of an imagined world) led the Pythagoreans to the mystical belief that all is number. They are said to have believed that even the planets in their orbits made music, the music of the spheres.

Of great interest is the metempsychosis of the Cabala. How this doctrine, already espoused by the Egyptians, Pythagoreans and Plato, came into Jewish mysticism, is not yet fully explained.

Numa the king of the Romans was a Pythagorean, and aided by the precepts of Moses, prohibited from making an image of God in human form, and of the shape of a living creature. Accordingly, during the first hundred and seventy years, though building temples, they made no cast or graven image. For Numa secretly showed them that the Best of Beings could not be apprehended except by the mind alone.

this backed up the belief that the entire cosmos was based on number: what the Pythagoreans called the Harmony of the Spheres.

In fact, so detailed was his interest in mathematics, and so acute his understanding, that he had recently written an original proof of the Pythagorean Theorem during a free moment at the capital. The New England Journal of Education had published the proof just the month before, transparently astonished that a member of congress had written it. Despite Garfield's deep admiration for mathematics and the arts, however, he believed that it was science, above all other disciplines that had achieved the greatest good.

Ancient people, like the Greeks, had a deep fascination with numbers. Could it be that in difficult times numbers were the only constant thing in an ever shifting world? To the Pythagoreans, an ancient Greek sect, numbers were tangible, immutable, comfortable, eternal, more reliable than friends, less threatening than Apollo and Zeus.

Pythagoras concluded that ratios govern not only music but also all other types of beauty. To the Pythagoreans, ratios and proportions controlled musical beauty, physical beauty, and mathematical beauty. Understanding nature was as simple as understanding the mathematics of proportions.

The purpose of life, to the Pythagoreans, was to break free of the bondage; to jump off the hamster wheel of earthly reincarnation, and join a cosmic, sublime level of harmony

2. If these figures express units of string lengths, then Anu is, with 60 units, the longest string, the bass note. Sin is one octave below Ištar and one above Anu. The ratios of string lengths are thus in reciprocal relation to the ratios of frequencies. It seems appropriate at this point to introduce the musical cent or centième since it is the most tangible unit of tonometry. The conversion of ratios into musical cents consists in multiplying the log to base 10 of the quotient of the division between the denominator and numerator of the ratio by the constant 3986.314. This method produces a scale composed of 1200 units in which equal semitones measure 100 cents. Thus, 1/1 = 0 cents; 2/1= 1200 cents, the octave; 9/8 = 204 cents, the Pythagorean tone; 3/4 = 498, the just fourth; 2/3 = 702, the just fifth, etc. From this we see that the gods’ respective numbers are contained in the span of the top octave. Anu, Enlil, Ea and Sin provide with the tonal infrastructure for the Babylonian scale as shown below: SIN EA ENLIL ANU 0 498 884 1200 Fundamental Fourth Sixth Octave Anu/Enlil 60/50 = 6/5 = 316 = just minor third Enlil/Ea 50/40 = 5/4 = 386 = just major third Ea/Sin 40/30 = 4/3 = 498 = just fourth Sin/Šamaš 30/20 = 3/2 = 702 = just fifth Šamaš/Bel 20/10 = 2/1 = 1200 = octave.

Stukeley was fascinated by Pythagoreanism, Neoplatonism, and the Egyptian Mysteries, as well as Druidism. His friends called him ‘The Druid’, and after he had met Augusta, Princess of Wales, the mother of the future George III, he wrote to her as ‘Veleda, Archdruidess of Kew’.

In favor of monism there is left, then, only the craving for excessive simplification, and the repugnance to the mystery of the origin of contingent beings. Against it stand the fatal contradictions to necessary intuitions and real facts of experience. Monism asks: How does even an infinite age produce an actual beginning of real beings ex nihilo? Sound philosophy must answer: It does not know; it cannot explain that action to human comprehension. But sound philosophy can show that this is no objection, because it can be proved that such explanation lies beyond the conditions of human knowledge. Those conditions understood, we see that we had no right to expect to be able to comprehend the beginning ex nihilo of contingent beings, nor to be stumbled at the fact...We say to the monist, then: Pause; both of you and we are out of our depth; we are in a region of ontology where we can safely neither affirm, nor deny, nor comprehend, nor explain. Let us lay our hands upon our mouths. The conclusion of that matter is to confess with the apostle (Hebrews xi. 3), that the doctrine of the begging of contingent being is one of faith, not of philosophy...And here is strong evidence of his acquaintance with the whole range of speculative human thought. He says at once to the Pythagorean, the Eleatic, the atomist, the Platonist, the Stagyrite: Vain men, you are out of your depth. The same inspired caution is as good for Spinoza the most modern idealist or monist.

The Pythagoreans had no sex prejudice, contrary to later times, but Parmenides goes much further than they did, further than ever Mr. Robert Graves would dare. His matriarchal absolutism is revealed not only in his Daemon Lady but in all her attendants and epicleses,...

Indeed, many problems can't be solved forward. And that is why the great algebraist Carl Jacobi so often said, "Invert, always invert." And why the Pythagoreans thought in reverse to prove that the square root of two was an irrational number.

Not only Newton's classical physics but also wave mechanics ultimately originated in the tension between those eight minutes of arc - less than one-seventh of one degree - and Kepler's Pythagorean metaphysics. Like the theory of atoms, which began in its Greek form as metaphysics in the fifth century B.C. (Leucippus and Democritus) and acquired scientific status only in the nineteenth and twentieth centuries A.D., Kepler's Harmony of the World acquired scientific status only with Louis de Broglie and Erwin Schrodinger. In fact, Schrodinger's wave mechanics takes the transition from geometric radial optics to wave optics and attempts to transpose it to the theory of matter, to the theory of elementary particles. Wave optics in turn takes its orientation from musical theory, from the theory of acoustic vibrations and waves, resonance and dissonance. But in this theory Kepler and his doctrine of harmony - hence Pythagoras in the end - plays a decisive role. Kepler, then, plays a role in the prehistory of Schrodinger's wave mechanics. But that is not all. Of all Schrodinger's precursors, Kepler is the only one who foresaw that harmony - resonance - holds the world together.