A History of Π (Pi)

by Petr Becmann (1971)

How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics

Jean Henri Fabre - "History records the names of royal bastards, but cannot tell us the origin of wheat."

p. 12 2,000 B.C., the dawn of the document history of mathematics. From the docs of this time it is evident that by then the Babylonians and the Egyptians (at least) were aware of the existence and significance of the constant π; π = C/D

But the Babylonians and the Egyptians knew more about pi than its mere existence. They had also found its approximate value., Babylonians had found π = 3 1/8, Egyptians 4(8/9)²

p. 23 the Ahmes Papyrus contains 84 problems and their solutions (but often no hint on how the solution was found). [show all work!]

p. 26 early Hindu knowledge was summarized by Aryabhata in the Aryabhatiya, written in 499 A.D. This gives the solutions to many problems, but usually without a hint of how they were found [again, show all work!]

p. 27 The Chinese were singular among the ancient peoples in that they used the decimal system from the very beginning

p. 29 until the infidel digit 0 was imported, few men in EUrope had mastered the art of multiplication and division, let alone the extraction of square roots which was needed to calculate π in the Archimedean way.

p. 30 Astronomy for calendar making was one of the earliest activities involving mathematics in all ancient societies. In Egypt and Babylon, as well as in Maya society, the priests had a monopoly of learning. In all three societies the priests were the astronomers, calendar makers and time keepers.

p. 32 The UN, a grotesque assembly of propaganda-bent hacks, has found itself unable to condemn international terrorism by criminals, much less to reform the calendar.

p. 33 The Maya were incomparably better equipped for numerical calculations than the Egyptians were; they had discovered the zero digit and the positional notation that had escaped the genius of Archimedes, and that held up European arithmetic for a thousand years after the Maya were familiar with it.

p. 35 In the 1560's, Diego de Landa, Bishop of Yucutan, burned the literature of the Maya on the grounds that 'they contained nothing in which there were not to be seen superstition and the devil.'" What remained was burned by the natives who had been converted to the Bishop's religion of love and tolerance."

p. 36 "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." (Menaechmus, 4th century B.C., when his pupil Alexander the Great asked for a shortcut to geometry.)

p. 36 "The thugs always win, but the thinkers always outlast them."

p. 37 in his theory of the Sun, which denied that the Sun was a deity, Anaxagoras of Clazomenae had gone too far, and for some time he was imprisoned in Athens for impiety.

the Sophis philosopher Antiphon enunciates his "principle of exhaustion", which was to have profound influence on mathematicians in their quest for the value of π until the invention of the calculus in the 17th cent.

p. 39 (reductio ad absurdum, proof by contradiction or indirect proof) if we want to prove that a statement is true, we first assume that it is not true an use this assumption to achieve an absurd result (or a result that contradicts the assumption); since the result is absurd, the premise must have been false; if a statement is either true or false (tertium non datur), the statement must be true.

p. 48 Euclid is not the father of geometry; he is the father of mathematical rigor.

p. 49 "The stumble block was that the Greeks could not conceive of an infinite sum adding up to a finite number (Zeno's paradox, Achilles and the tortoise)

p. 55 "What the Romans excelled in was bullying, bludgeoning, butchering and blood baths. Like the Soviet EMpire, the Roman EMpire enslaved peoples whose cultural level was far above their own. . . . looted them, stealing their art treasures, abducting their scientists and copying their technical know-how, which the Romans' barren society was rarely able to improve on. . . . the light of culture which Rome is supposed to have emanated was a borrowed light

p. 56 Roman engineering: roads, aquaducts, Colosseum. Warfare has always been beneficial to engineering. Yet there are unmistakable trends in the engineering of the gangster states. In a healthy society, engineering design gets smarter and smarter; in gangster states, it gets bigger and bigger. In WWII, the democracies produced radar and split the atom; German basic research was far behind in these fields and devoted its efforts to projecs like big lenses so big they could burn Britain, and bells so big that their sound would be lethal. Roman engineering, too, was void of all subtlety. Roads ran absolutely straight, when they came to a mountain they ran over the top as pigheadedly as one of STalin's frontal assaults.

p. 58 The Roman's contribution to science was mostly limited to butchering antiquity's greatest mathematician, burning the Library of ALexandria, and slowly stifling the sciences that flourished in the colonies of their EMpire.

p. 60 No matter what was being discussed in the senate, Marcus Porcius Cato (the Elder)'s speeches would always end with the words Ceterum censeo Carthaginem esse delendam, a sentence that has been copied in innumerable variations by people to whom vicious bigots like Cato were presented as examples of the noble Roman spirit.

Such is the background of the Punic Wars, which lead us back to the story of π. Buring the Second Punic War, the Romans sent an expeditionary force under Claudies Marcellus to Sicily in 214 B.C. the king of Syracus had renewed his alliance with Carthage; The Syracuseans had been taught the secret of the lever and of the pultiple pulley, and they put it to use in their artillery and marine defenses. In the end the invincible Roman legions became so filled with fear that they would run as soon as they saw a piece of rope or wood projecting over the wall. God is on the side of the big battallions and Syracuse final falls to the Romans. Inside the city was the 75-year old thinker who had grasped the secret of the lever, the pulley and the princple of mechanical advantage. Plutarch tells us that the thinker was alone, examining a diagram closely; he did not see that the Roman siege had finally proved victorious; Suddenly a Roman soldier came up to the thinker and ordered him to follow him to Marcellus. The thinker would not get up and go until he had finished his problem and worked out the proof. "Don't touch my circles!"" shouted the thinker. The soldier became enraged, drew his sword and killed the thinker. The thinker was Archimedes.

"unimpeachable logic"

p. 72 The mediaeval zealots did not always, like the Bishop of Yucatan or the Crusaders at Constantinople, burn scientific books as work of the devil. Sometimes they would only wash off the text for the sake of the parchment, so that they might besmirch it with their superstitious garbage. (palimpsest) this is how Archimedes' The Method was recovered in 1906 in Constantinople.

p. 73 48 B.C. , Caesar, and the unrest following his departure, results in most of the Library of Alxeandria's books being burned and lost forever.

p. 74 the library is burned again in 272 A.D. after the Roman Emperor Aurelianus quells an Egyptian uprisning, and again in 295 when Diocletian suppresses another revolt.. In 391 a Christian mob led by Bishop Thephilus destroys the Temple of Serapis, where some books were held. Another Bishop, Cyril, led Christian mob against astronomer and mathematician Hypatia, who is hacked to death in 415, marking the end of ALexandria as a center of mathematical learning.

p. 78 In the early days of civilization, science and religion were in the same hands, in the hands of the priests, who had a monopoly on learning. In Egypt, Babylon, the Yucatan, and everywhere else in the Belt, they were the timekeepers and surveyors, the stronomers and geometers. They soon learned that knowledge is power: power over the gullible and uneducated. . . . they would impress the uneducated with much mumbo-jumbo about the imminet floods or recessions of the Sacred River (Egyptian preists using nilometers)

p. 79 the Pythagorean theorem had been known to every rope-stretcher (surveyor) from the Nile to the Yang-Tse Kiang for over a thousand years before Pythagoras' witchcraft.

p. 80 mediaeval Christianity, in many ways the most terrible of the three catastrophies, persecuted science with the torture chamber and the stake, and almost succeeded in extinguishing it completely.

in 1204, the fourth crusade captured Constantinople and sacked it with horrors unparalled even in the bloody age of the crusades; the classical works that had survived until then were put to the torch by the crusaders in what is generally considered the biggest single loss to classical literature.

p. 81 1600, Giordano Bruna is burned alive in Rome for claiming that the earth moves round the sun. In 1633, 70-year old Galileo Galilei went through the torture chambers of the Inquisition until he was willing to sign a recantation.

p. 87 E pur si muove! Giordano's last cry from the burning stake? incorrectly attributed to Galileo??

p. 87 The classification of history into ancient, mediaeval and modern is but a sympton of the white man's arrogance; for while Europe was suffering the imbecility of the Dark Ages, the rest of the world went on living.

like Tertullian, Muslim philosopher al-Ghazzali (1058-1111) writes that scientific studies shake men's faith in God and undermine religion, and that they lead to loss of belief in the origin of the world and the Creator.

p. 88 But in China too, emperor Tsin Shi Hwang-di (3rd century B.C.) is warned of idling scholars whose influence was founded on books, , orders all literature books in China burned.

p. 104 savant problem solvers: Johann Dase

I have made this letter longer than usual because I lacked the time to make it short. (Blaise Pascal)

p. 122 Chines version of the Pascal triangle, published in 1303, 320 years before Pascal

p. 140 Newton sends solution to problem to Jean Bernoulli without signature, but Bernoulli instantly recognized the author exclaiming tanquam ex ungue leonem (as the lion is known by its claws)

p. Lisez Euler, lisez Euler, c'est notre maitre a tous (Laplace's advice to his students, Read Euler, he is our master in everything)

The Nothing That Is: A Natural History of Zero

by Robert Kaplan (Oxford University Press, 1999)

Still It Moves

When do Roman numeral abbreviations begin?  When does IIII become IV?  When does a position system come into play with Roman numerals?

p. 20 "Why didn't the Greeks pursue this way [using "o" symbol for angular degrees] -- for the zero hardly appears outside their astronomical writings? And why after all, hadn't such an inventive people come up long since with writing numbers positionally, and the zero this entails? Why, at the peak of their power, did they step . . . yet further away from what would have aided thought?"

p. 26 [In ancient Greece, singers learning the Illiad and Odyssey by heart] "Memory was often equated with knowledge, knowledge with wisdom--so that the external memory of texts (that repository of our culture, binding us to generations gone) must have been for them something like musical scores: you feel a bit down when a concert has to perform with on in front of him."

p. 34 "What facilitates thought impoverishes imagination."

p. 57 "not whether the Indians came up with the dot or circle for zero -- but more significantly, how they thought about this zero once they had it. Remarkably enough, the dot was used by them not only for zero but for the unknown, the way we use x."

p. 59 "Nouns name things, [Willard van Orman Quine} wrote, and a thing cannot be both red and not red, for example. But 'something is red' and 'somehting is not red' are both true (the story is told of Quine that when a pianist playing Mozart apologized for striking a wrong note, Quine assured him that he had just played something else perfectly)."

"What can be nothing one moment and smoething the next . . . ? The answer lies in our always having mistranslated this word 'void' or 'empty'. For the Hindus there is no unqualified nothingness. In the same spirit as our Law of the Conservation of Matter, substance for them cannot disappear but can only change is form of nature."

"Sunya' isn't so much vacancy as receptivity, a womb-like hollow ready to swell."

"This is the zero of the counting board: a column already there, but with no counters in it yet. This is the zero of the place-holder notation, having no value itself but giving value by its presence to other numerals."

p. 65 "the concerns of philosophers in ancient Greece, at a time just prior to the spread of their thoughts to India, were such as made an appropriate setting for considering zero and the unknown in similar ways, so that it shouldn't surprise us that the symbols for the two intermingled."

p. 73 infinity isn't a number (not even a stupid number as those who mistranslate the Latin phrase think: Infinitus est numerus stultorum. So what then should we make of a/0?

p. 75 "dangerous Saracen magic" (using algebra and higher order mathematics)

p. 102 "These technical difficulties, combined with the slow spread of knowledge before books were printed and writing in the vernacular was common, added to the reputation that the Arabic numerals already had for being dangerous Saracen magic. Even when they began to appear as dates on coin and monuments, banks were still reluctant to sue them, and for good reason: zero was the villain again, since it could be turned into a 6 or a 9 by the unscrupulous, who could also slip in a digit or two before it. So in Florence the City Council passed an ordinance in 1299 making it illegal to use numbers when entering amounts of money in account books: sums had to be written out in words. [non per cifras sed par literas claros - not in figures [cipher] but in clear letters]. Even as late as 1549, a canon in Antwerp warned merchants not to use numerals in contracts or drafts. We laugh at those who can't count--but in the thirteenth century they laughed at those who could, making 'cipher' and 'the zero of algorismus' terms of derision."

p. 103 millennialists had to reckon then--as they do now--with the difficulty that years ending in zero were the last of their decade, century or millennium, not the first of the next (so for us, January 1st of 2001 begins the third millennium, and the festivities of the year before celebrate only private rites of passage)."

p. 107 Filius Bonacci, or Fibonacci - Leonardo of Pisa, publishes Liber Abaci in 1202


p. 108 "the rising tide of commerce swelled the demand for careful calculations and records of transactions. The Arabic numerals were taught but distrusted. The counting board was still the champion device.

p. 130 our word algebra comes from the title of Al-Khowarizmi's book al-jabr wa'l muqabalah, translated as 'restoration and reduction" or "completion and comparison'. [x2 -39 + 8x = -2x] you first restored the equation (al-jabr_ by moving the negative terms to the other side making them positive. [x2 + 8x +2x = 39] Then you reduced the equation, that is you combined like terms. [x2 + 10x = 39] then you would use your nogging to figure that x = 3, -13.

p. 130 John Napier (Baron of Merchiston near Edinburgh, 1500s) would show that by setting equations like this to zero you could solve for x with a single general technique. He also invented logarithms.

p. 139 Don't we need to find even more fundamental truths to derive these last two 'laws' from? And if we do, won't they require antecedents, and so down this eternal spiral to where their fires are not quenched? For a truth even deeper is that the kind of certainty demanded by deductive thought is unattainable because of the nature of deductive thought. To stop the infinite regress we have to say at some point: 'We hold these truths to be self-evident.'"

p. 141 "mathematicians eventually looked away from what addition and multiplication were and sat down to codify how they acted."

p. 144 "Only selective forgetting of the past lets us move on, taking what was once dubious as the most banal of certainties, what was gained through struggle as our birthright. So with zero. The sermons it spoke in place-holding shrank to a letter of our thinking's alphabet, its volumes on solving equations to a sentence in mathematical primers."

p. 152 [Johann Bernoulli, 1691] "a quantity which is diminished or increased by an infinitely small quantity is neither increased nor decreased" [the birth of differential calculus] just go ahead and faith will follow -- Allez avant et la foi vous viendra.

p. 175 "the light of a billion stars, the background hum from the Big Bang and ever fainter echoes of countless pasts are expanding through one another everywhere, making the hollow night claustrophobic."

p. 176 "In that corner of the library where no young and hardly any old codgers go are stored the books of yesterday's certainties, the delicate charm of their fustiness ruffled only by the ever modern authority of their tone."

p. 190 "The fashion for Existentialism passed, preserved now only in successive waves of adolescene, before the getting and spending set in."

p. 195 "Those for whom all this nothingness is oppresively real take comfort at least in this, that extinction means the end of oppressive reality."