Title: A History of Pi
Author: Petr Beckmann
Publisher: St.Martin’s Press 1971
ISBN: 978-0-312-38185-1
Pages: 189
Petr Beckmann was born in Czechoslovakia, in 1924 and worked there as a research scientist at the Czechoslovak Academy of Sciences till 1963. At that time, he traveled to the University of Colorado as a visiting professor, and decided to stay on to escape from the repressive communist government at home. In this remarkable, but not unique book on the history of pi (p) – mathematics’ most renowned number – Beckmann looks at the steady human progress made from prehistoric to post modern times in reconciling the curious scientific concept with the organisation of knowledge. Though an electrical engineer by profession, his interest and grasp of history is amazing and delightfully expressed in numerous asides which help to light up a point of doubtful clarity. In the survey which spans over four millennia, Beckmann offers bouquets and brickbats alternatively to one dynasty or nation to the other. He is all praise for the Greek, but mortally opposed to the Romans. He repudiates the Soviets under communism in the most harsh terms, which is expected from a person who had to save his own skin by emigrating to a foreign country from the bloodthirsty communist regime in Czechoslovakia.
The ratio between the circum ference and diameter of a circle was noted by the ancients. But, calculating the value was a herculean task, considering that they had no scales, decimal system or long division. Babylonian and Egyptian priest-mathematicians found the value of pi nearly correct to two decimal places. The value obtained was 3.125 in Babylonia and 3.16 in Egypt. This is far sophisticated than the value given in the Bible (the integer 3!). The eastern societies performed superbly, the value obtained for pi being 3.1416 in India (by Aryabhata and Bhaskara) and 3.1415926 in China. Archimedes developed an analytical method to find the value, but was killed by a Roman soldier. It is said that the only representative of the Romans who finds a place in the history of science is the soldier who killed Archimedes. The author goes into histrionics in condemning the Roman empire. Though conceptually appealing to the author’s arguments, calling the Romans pests and Julius Caesar a thug is a little off the mark.
Alexandria was the light house of knowledge in the ancient world for about 600 years spanning the birth of common era. The Alexandrian Library was destroyed many times in history – first, in 48 BCE by Julius Caesar, in 272 CE by emperor Aurelianus, in 295 CE by emperor Diocletian, in 391 CE by Bishop Theophilus (Rome had Christianized by then), and in 415 by Bishop Cyril who also ordered the hacking to death of the famous woman mathematician, Hypatia as a devotee of pagan learning. The end came in 646 when the Arab Muslim general Amr ibn-al-As completely wiped it off, which is traditionally marked as the beginning of the middle ages when knowledge was eclipsed and religion cast it wide wings of ignorance and superstition over much of Asia and Europe. The middle ages ended in the 15th century, but the transition was not swift or marked. Even after a century thereafter, high crimes by priests and the Church continued unabated. Cardinal Torquemada condemned Spanish mathematician Valmes to be burnt at the stake in 1486 for finding the solution to the quartic equation. The cardinal held that it Valmes’ action was against the will of God that such a solution was inaccessible to human understanding. Giordano Bruno was burnt alive in 1600 for postulating that the earth moves round the sun. Galileo Galilee was arrested and punished in 1633 who later recanted. However the entire middle ages was not a period of unmitigated darkness. Sparks flew here and there, as Fibonacci (Leonardo of Pisa) learnt and practiced the Indian numeric system from Arab merchants ad calculated pi as 3.1418. Francois Viete invented an analytical method to calculate pi to any precision, as the sum of an infinite series.
By the end of the 16th century, pi was known to 30 decimal places, by 18th century, it rose to 140 decimal places, by the end of 19th, it reached 707 and in 1967, computers found 500,000 decimal places for pi. The current record is 2.7 trillion places, established in 2010. Even though mathematicians delved deep into the numeric thick forest, no periodicity was found. Johann Lambert (1767) and Adrien-Marie Legendre (1794) proved that pi is an irrational number (which cannot be represented as the fraction of two integers). Legendre also proved that p2 was also irrational, demolishing hopes that p might be the square root of a rational number. F Lindemann established in 1882 that it is also a transcendental number. Various mnemonic devices are available to memorise the digits of pi to many places. The number of letters in each word of the following sentence represents the successive digits of p, “How I Want A Drink, Alcoholic Of Course, After The Heavy Lectures Involving Quantum Mechanics” (3.14159265329). In French and German, there are poems that help to memorize up to 29 decimal places. The 32nd digit of p is zero, limiting attempts beyond this point!
William Jones first used the letter ‘p', for pi in the current sense of the term in 1706. He intended it as a short form for ‘periphery’. However, Jones was not a prominent person in the mathematics of his era, and his notation was not copied by anyone. Things suddenly changed when Leonhard Euler, one of the great mathematicians of all time, used the symbol in 1737, replacing his earlier notations of ‘c’ or ‘p’, for the same quantity. Newton and Euler devised methods to easily calculate p to any significant number of decimal places. Laplace obtained a mechanical method, called Monte Carlo Method, to find p, taking into account the probability of a randomly thrown needle to intersect the ruled lines on a horizontal surface.
The book is eminently readable and quite compact to even finish it in one go. The language is simple, but clear and crisp. The reader is invited and led by hand to the concepts which the author illustrates in graphic detail. Being written by an engineer, it reflects the wisdom of the world and highly practical ways of lesser known concepts of mathematics. Instead of sticking solely to trace p, free transgressions from the thread are repeated many times over, but the reader finds each one to be quite amusing and informative. Every student of science and technology must read this little marvel of a book. The author is sarcastic and forceful on fanaticism of any kind, whether it be professed by the church or the communists. He says about the destruction of books in Mexico by the missionaries as, “In the 1560’s, Diego de Landa, Bishop of Yucatan, burned the literature of the Maya on the grounds that “they contained nothing in which there were not to be seen superstition and lies of the devil”. What remained was burnt by the natives who had been converted to the Bishop’s religion of love and tolerance” (p.35). We can also discern striking generalizations such as “Then there is Roman engineering: the Roman roads, aqueducts, the Colosseum. Warfare, alas, has always been beneficial to engineering. In a healthy society, engineering design gets smarter and smarter; in gangster states, it gets bigger and bigger” (p.56). The book also spells out what is a transcendental number, which is a term not understood properly even by students who had even undergone college mathematics. Transcendental numbers are not only irrational, but couldn’t even be roots of an algebraic equation.
On the down side, Beckmann’s cavalier approach to the eastern societies which contributed to the progress of science when Europe was besmirched with superstition is to be deplored. Touching on the point of accidental discovery of the Ahmes papyrus, he says, “Histories like these (accidental location of ancient papyrii) make one wonder how many such priceless documents have been used up by the Arabs as toilet papyrii” (p.23), as if the Arabs were good at nothing else! Also Beckmann attributes the discovery of zero to the Chinese, whereas the honour is conferred on India by general consensus, as shown by, “The Chinese discovered the equivalent of the digit zero. Like the Babylonians, they wrote numbers by digits multiplying powers of the base (10 in China), just like we do” (p.29). Frontal attacks on the Romans, also would seem to be disproportionate considering the times in which those emperors ruled. Politics often seeps out of the background information as accusations against the Russian communists and admonitions against the Egyptians, against whom Beckmann claims that Archimedean screw pumps were still in use in Egypt whose rulers think it more important to destroy Israel than to provide their people with modern irrigation (p.63). The book is also riddled with a little too many equations and inscrutable diagrams. Such things are not helped in any way by the author’s proclivity to provide mathematical proof of formulas or theorems.
Even with all these shortcomings, the book is a delight to read. The cover, designed frugally with a black background with a large p in the middle, is so appealing. The book was published in 1970, and the details on computer simulations of p need updating in the present context of superior computing power affordable even to amateurs. The book is also important because pi has been a special number for all the people. Every year, March 14, (3/14) is celebrated as the PI Day!
The book is highly recommended.
Rating: 3 Star
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