The Math Department

About the History of the Calculation of π

 

The number π, defined as the ratio of the circumference of a circle to its diameter, has been an object of curiosity and study to mathematicians for thousands of years. Although it rises from one of the simplest and most symmetric shapes, it presents many mathematical mysteries: it is irrational, and indeed, transcendental. It is one of the so-called “fundamental constants” – it appears in important roles in geometry, trigonometry, and even in unexpected fields like statistics. It seems to have a mystique that has endured over the centuries. People memorize its digits to the thousandth decimal place; they search for esoteric patterns in its decimal expansion; they develop computer algorithms to find its billionth digit and beyond.

 

Nobody knows who the person was that first defined π, or that first estimated it. As long as people have built permanent structures or measured things, there has likely been some awareness of the number. There is evidence that approximations to π were used by the ancient Mesopotamians at the beginning of recorded history. In early written fragments from the ancient Egyptians and Babylonians, we find accounts of measuring circles.

 

The early approximations were doubtlessly discovered simply by measurement – rough estimates for π can be made using as little as a rope and a large circular object. Although not very accurate, the ancient Babylonian value of 25/8 and Egyptian value of 256/81 were good enough for most architectural purposes.

 

The great mathematician Archimedes, around 250 BCE, used a much more mathematically sophisticated method to find estimates for π. He first inscribed a polygon in a unit circle, and then circumscribed another along its outside. The perimeters of the polygons then provided lower and upper bounds for the value of π.

 

Suppose the polygons have 2ⁿ sides. If the central angle of the polygon is 2θ, then the length of each side of the inner polygon can be shown to be the sine of the angle θ, and the length of each side of the outer polygon is the tangent of the angle θ. Using the half-angle identities for the sine and tangent, it is not hard to derive iterative formulas for the polygon perimeters. It turns out that, given the inner perimeter I_n and the outer perimeter O_n, I_n+1 is the geometric mean of I_n and O_n, and O_n+1 is the harmonic mean of I_n and O_n.

 

This was stunning mathematics; Archimedes did not have a very usable number system to work with and did not have the tools of modern trigonometry. He also used a limiting process that would find echoes centuries later in the work of Leibniz and Newton on the integral calculus.

 

Archimedes’ method was the most effective known for centuries, until the time of the Renaissance. By this method of exhaustion, Archimedes found about five correct decimal places for π; the next significant improvement did not come until fifteen hundred years later, when Al-Kashi in Samarkand used a similar method to find fourteen correct digits.

 

The sixteenth century mathematician Ludolph van Ceulen was obsessed with computing the digits of π, and did find the best approximation for π yet. For his effort, his value of π was engraved on his tombstone.  In old texts, the number π is occasionally refered to as the Ludolphine constant in his honor.

 

It took decades for him to do, but he managed to compute 35 correct digits. His method was also the same method of exhaustion used by Archimedes, though instead of using polygons with only a few hundred sides, van Ceulen had used a 2⁶²-gon for his approximation. It would take the development of the calculus before more efficient methods would be discovered.

 

The prodigious Leonhard Euler was the first mathematician to regularly use the Greek letter π to represent the number. He wrote hundreds of papers and, as the leading mathematician of the day, was widely read, so his notation stuck.

 

Hundreds of methods have been found to approximate π, discovered by both famous and obscure mathematicians. Wallis, Gregory, Euler, and Ramanujan all derived well-known series for π. John Machin’s arctangent formula, published in 1704,  π = 16 arctan(1/5) – 4 arctan(1/239), is computationally rather simple and, using series approximations for the arctangent, has very rapid convergence. For the next two hundred fifty years, nearly every new record π computation was accomplished using his formula.

 

The pinnacle of hand computation of π was achieved in the nineteenth century. William Shanks was a mathematician who spent a great amount of time compiling logarithm tables, prime number tables, and the like; in the days before calculating machines, large tables of such values were essential for work in engineering or physics.

 

He did try his hand at π, using Machin’s formula, and in 1873, after many years of labor, presented 707 digits of the number.

 

But π can be a cruel and fickle muse. It was later discovered that Shanks had made a mistake, and that after the 527^th digit, his calculations were wrong. Nevertheless, it was the best value of π available until the advent of high-speed digital computers.

 

Lambert proved a very important result about π in 1794. He established that π is an irrational number. Not until 1874 was it proved by Hermite that π is in fact transcendental. This laid to rest the ancient problem of squaring the circle. It also killed anyone’s hopes of actually finding the “true” value of π – since π is irrational, it cannot be written in a finite number of digits, and since it is transcendental, it cannot be written using a finite number of radical signs either.

 

Despite these discoveries, a physician, Dr. Edwin Goodwin, introduced a bill before the Indiana House of Representatives in 1897 suggesting that the state officially peg the value of π at 3 (or maybe 3.2, or maybe 10, or maybe 9. The text of the bill itself is a little confused on this seemingly important matter.) The good doctor graciously offered to allow the Indiana schools to teach this marvelous mathematical “fact” without paying him any royalties – but schools outside the state, of course, would have to pay.

 

The bill passed the House unanimously; however, mathematical sense prevailed in the Indiana Senate. The bill was met there with ridicule, and tossed in the dustbin. One cannot help but wonder, though, what advances in mathematics might have been made had the schoolchildren of Indiana been taught that π was really and truly 3 (or 3.2, or 6, or Dr. Goodwin’s birthday, or whatever rubbish he was trying to say.)

 

This constant π, intimately tied to a plain geometric concept, which might seem at first only slightly useful shorthand, is found in the mathematics of geometry, of probability, of analysis – it is very familiar. To many of us, it is one of the friendliest and most wonderful numbers.

 

Very few people, after all, have spent their entire lives computing the value of any other number; very few numbers have had nearly as much written about them or said about them; very few other mathematical ideas spark the recognition or interest of the general public like π does. It has its own history, its own legends and its own folklore. It was used in the work of the ancient star mappers, and it is used in the work of particle physicists today - like the circle, it just seems to keep coming around and around. That is perhaps part of the reason why it still intrigues people today.