After reading Carl Sagan’s Contact in High School I became curious about pi due to its role in Sagan’s novel. I wanted to learn more about this mysterious constant and picked up Petr Beckmann’s book, A History of Pi (ISBN: 0312381859), shortly thereafter.
I had forgotten all about it until one day, after seeing Darren Aronofsky’s PI, I happened to be cleaning out what used to be my bedroom closet where I came across a stash of books that I have always meant to read. None were lacking an air of pretension. There were tomes about Islam, one called Jesus Before Christianity (ISBN: 0883448327), a pair of ponderous Ayn Rand novels and Beckmann’s book about pi, still dog-eared on the last page I had read.
I had gotten about thirty pages in before I found that I had to put it aside. This was most likely from a combination of already getting my fill of math every morning in Mr. Simmons’ Calculus class, a desire to read something more fun like Michael Moorcock’s Corum books, and the tendency for Beckmann to get into some heavy duty equations that I felt I had to follow in order to get a full appreciation of his writing.
This time I took a different tact in order to complete Beckmann’s book. When the equations came too fast and furious I skipped over them and read what they were accomplishing and the context of their use instead of working out the problems along with their creators (big-brained folks like Newton, Euler, and Lagrange). I found that this was more in line with the author’s original intent.
In Beckmann’s introduction he admits that he is neither a historian nor a mathematician and the lack of these skills makes him eminently qualified to write a history of pi. This is the type of sarcastic, irreverent humor that pervades Beckmann’s writing. Yet, it is also the truth. Beckmann doesn’t put on airs when dealing with history nor does he present the mathematics as either ostensible or over-simplified.
Part of the fun of Beckmann’s book, apart from his well-written chronicles of an elusive constant from ancient Babylon to the computer labs of the early ’70s, is his discarding the illusion of objectivism and unflinchingly displaying his strong opinions. He squares off and knocks the stuffing out of sacred cows like Aristotle ("His ignorance of mathematics and physics, compared to the Greeks of his time, far surpasses the ignorance exhibited by this tireless and tiresome writer in the many subjects that he felt himself called upon to discuss.") and Pythagoras ("It is even doubtful that Pythagoras knew Pythagoras’ Theorem, and if so, whether he could prove it.").
The author traces the evolution of the determination of circle’s area and circumference, exhibiting the advances made throughout the world (the subject of mathematical advancement in the Orient and the Americas often being snubbed). Moreover, Beckmann is careful to show the bumps in the road and the cul-de-sacs of pi’s development. To Beckmann, the periods of least growth in art and science occur during eras of oppression when learning and discovery are curtailed, controlled, or advanced solely for the benefit of the party in charge. Among the guilty parties in the cessation of growth, stagnation and destruction of knowledge that Beckmann frequently targets are Romans, Nazis, Communists, and the Holy Roman Empire. According to the book’s timeline, half of the "discoveries" of the Renaissance were recognized as truths hundred, frequently thousands, of years prior but "lost" due to the eradication of libraries or parsimonious governing of learning materials by the aforementioned totalitarians.
Beckmann writes of science serving war for the Romans and Nazis and of it being outlawed by the Holy Roman Empire (who also banned heretical notions like Arabic numbers and the decimal system) and leaders of the USSR. He refers to these groups in the basest of terms, calling the Romans everything from megalomaniacal thugs to barbarians.
What about the famed aqueducts of Rome with which kinder have long been indoctrinated? "The Roman aqueducts were bigger than those that had been used centuries earlier in the ancient world; but they were administered with extremely poor knowledge of hydraulics. Long after Heron of Alexandria (1st century AD) had designed water clocks, water turbines and two-cylinder water pumps, and had written works on these subjects, the Romans were still describing the performance of their aqueducts in terms of the quinaria, a measure of the cross-section of the flow, as if the volume of the flow did not also depend on its velocity. The same unit was used in charging users of large pipes tapping the aqueduct; the Roman engineers failed to realize that doubling the cross-section would more than double the flow of water. Heron could never have blundered like this."
Undoubtedly, Beckmann’s distaste for dictatorships stems from being born and raised in Prague, Czechoslovakia, from which his family fled Nazi occupation in 1939 at the age of 15. After the war, he left Britain, returning to his homeland, now shrouded by the Iron Curtain. Beckmann received his doctorate in electrical engineering from Prague Technical University amidst an atmosphere of intolerance reminiscent of that which had sealed the fates of mathematicians like Giordano Bruno and Galileo Galilei in the Dark Ages. In regards to his new comrades, Beckmann provides an anecdote about a Prussian reception where Leonhard Euler only spoke in monosyllables. When questioned about his lack of conversation, Euler is said to have remarked "I have arrived from a country where they hang those who talk." Beckmann writes, "How times have changed! In today’s Russia they do them to death in forced labor camps or lunatic asylums."
After defecting in 1961, he resided as a professor of electrical engineering at the University of Colorado. His pen unfettered, he filled his works with social commentary. In contemplating the contributions of figures like Newton and Gauss, Beckmann ponders, "How many little Newtons have died in Viet Nam? How many Ramanujans starve to death in India before they can read or write? How many Lobachevskis languish in Siberian concentration camps?"
Among the impassioned passages and enigmatic equations, Beckmann provides a fascinating account of arithmetical advances. To determine why the focus of the book is on pi as the locus of achievement, begin by contemplating the circle. This shape, constantly encountered in nature, has presented a multitude of problems to mathematicians, chiefly in the lack of an accurate method to measure its area and circumference.
The knowledge of the existence of the constant pi dates back over four thousand years. A crude formula for it can be found in the Old Testament (2 Kings vii.23), "And he made a molten sea, ten cubits from one brim to the other (diameter); it was round all about, and his height was five cubits; and a line of thirty cubits did compass it round about (circumference)." pi equals circumference divided by distance. Thus, by this account, pi = 30/10 which means that pi = 3. In terms of precision, this value is unacceptable.
For centuries, number-crunchers attempted to determine a circle’s circumference by employing a method known as "squaring the circle." That is, creating a polygon within a circle that would provide an area close enough to allow an educated guess of the circle’s circumference. Yet, it is fairly obvious that an accurate estimation of pi can be determined as polygons of infinitely smaller sizes could fill the rounded edges of a circle without ever completely filling the space. Variations of this method were being employed across the globe with varying levels of success. Of all cultures, the Mayans and the Chinese, with benefit of the zero digit, had calculated pi correctly to eight decimal places.
It wasn’t until the seventeenth century that occidental mathematics regained the ground that had been lost during the reign of the Romans and Holy Roman Empire, attaining the same level of expertise that had been present in Alexandria two thousand years prior. And, with the adoption of Arabic Algebra, Geometry and the advent of differential calculus, mathematics could progress. And, even then, it came in fits and starts.
While there is little practical need for pi to be calculated past the first few decimal places, the farther the value strays from 3.14159... the level of inaccuracy increases. Indeed, the precision in the determination of the value of pi allows insight to a culture’s level of achievement as the determination of a circle’s area and circumference is fundamental in matters of math and science including fields such as architecture and engineering.
Thus, the momentous role that pi plays as the cornerstone of learning is revealed. Unquestionably, other transcendental constants of equal or greater importance exist, but pi is prominence in our consciousness originates in the deceptive simplicity of the ubiquitous circle. Yet, pi’s eminence is increased due to the basic philosophical challenges inherent within the infinite. It infringes on our mortality. Pi is also an affront to the aspiration to create order where none is present.
This gives way to the folks Beckmann labels "digit hunters" who are obsessed with the computation of pi to the umpteenth decimal place. He concludes his book by examining the foundations of the quest, which initially began as a search for increased precision of numerical analysis. When the nature of the beast was discovered, hope of finding a pattern in the transcendence increased (as fictionalized in Contact). And, of course, there’s the desire to break records, calculating it farther and faster than the previous person has. In this way, it’s the mathematical equivalent of scaling Mount Everest.
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