Mathematicians Develop New Ways to Calculate π

Overview

During the Renaissance, mathematicians continued their centuries-old fascination with π. As part of this, they calculated π to ever-greater precision while developing new formulae to add digits more quickly. Part of this fascination with π was the equally old quest to "square the circle," and part was simply human curiosity. In spite of their lack of success in quadrature (the term for squaring the circle), these efforts did yield greater insights into the nature of π, some aspects of geometry, and other areas of mathematics.

Background

The ancient Egyptians and Babylonians may have been the first to notice that the ratio of a circle's circumference to its diameter was not an even number. By about 2000 b.c., in fact, the Egyptians had determined this ratio was about 31/8, while the Babylonians put it at about 4 × (8/9)². These come out to be equal to 3.125 and about 3.161, respectively; not far from the modern value of 3.1416.... These relationships remained largely unchanged until the time of the Greeks, nearly 1,600 years later.

Among the ancient Greeks, Archimedes stands out as a mathematician who made particularly important advances in determining the value of pi, among his many other accomplishments. Using a method that remained largely unchanged for nearly two millennia, Archimedes determined that π had a value of between 3.14084 and 3.14258; remarkably close to the actual value. In reaching this value, he also developed mathematical techniques that seemed to anticipate the development of some parts of differential calculus.

One of the driving forces behind calculating π to greater and greater numbers of digits was a fascination, in some cases an obsession, with the quadrature problem. This problem simply asked a person, using only an unmarked straight edge and a compass, to create a square with exactly the same area as a circle of a given diameter. The quadrature problem seems to have originated with the ancient Greeks, and occupied an inordinate amount of attention for nearly 2,000 years. Part of the reason for the attention is that success seemed, in many cases, tantalizingly close, while remaining just out of grasp. Somewhere along the way, mathematicians and other "circle-squarers" realized that, if only they could determine the exact value of π, they would have a much better chance of developing an algorithm that would let them achieve their goal. So work continued.

Archimedes' method for calculating π relied on the fact that a polygon drawn on the outside of a circle would always have a greater circumference than the circle itself, while a polygon drawn on the inside of a circle would always have a smaller circumference. By adding more sides to each of these polygons, they would gradually approach the same circumference, and if they had infinitely many sides, they would have exactly the same circumference as the circle itself. In effect, the circumferences of the two polygons would gradually approach that of the circle that was being "squeezed" between the interior and exterior shapes.

The next advance in calculating the value of π came in the 1500s when François Viète (1540-1603) developed an elaboration on Archimedes' method. While his basic method would have been both understood and appreciated by the Greek, Viète became the first to use an analytically derived infinite series to aid in the calculations. This, in and of itself, was a major accomplishment for not only π-calculators, but for all of mathematics.

The first attacks on Archimedes' method itself were not launched until 1621, when Dutch physicist and mathematician, Willebrord Snell (1580-1626) developed a superior method of calculation. This was verified by another Dutchman, Christiaan Huygens (1629-1695). His method would yield as many accurate digits for π in just a few iterations as could be garnered by Archimedes' method using a polygon of over 400,000 sides.

The next significant advance took only another 44 years, when English mathematician John Wallis (1616-1703), like Viète, used an infinite series to calculate the value of π. Unlike Viète, Wallis's formula did not rely on a series of square roots, which are always difficult to calculate by hand. Instead, Wallis developed a method that used ratios of whole numbers and trigonometric operations, giving a much faster and more accurate way to calculate π, and allowing many more decimal places to be added to its known value. Wallis's contribution is significant from another standpoint; it was the last important advance to be made that did not use calculus and the methods of analytical geometry, recently invented by Isaac Newton (1642-1727) and Gottfried Leibniz (1646-1716).

Impact

The impacts of these advances range from significant to almost trivial. On the trivial side, it became ever easier to calculate π to an ever-longer string of digits. More important than simply knowing a string of digits for π, however, was the mathematical understanding that came from this process. Finally, the analytical process that began with Viète led to a much deeper understanding of the nature of π, eventually proving that the exact value of π could never be known because it is a transcendental number; a number that neither repeats itself nor terminates. This discovery, in turn, was to forever quash attempts to square the circle, something that had remained in the public imagination for many centuries.

From one standpoint, adding calculated decimal places to the known value of π was akin to mounting the heads of dead animals on the wall; it accomplished little except to give some fleeting fame to a mathematician. However, from another standpoint, even so apparently mundane a task could still carry some degree of significance. For example, Viète was able to show his method generated digits much more quickly than did Archimedes'. By reproducing the digits of those who came before, each method was validated, as was the math that went into developing that method. And, by generating an increasingly long string of digits for π, mathematicians began to suspect that it was not so simple a number as it at first appeared. These suspicions gained ground, even as circle-squarers appeared, each convinced he or she had accomplished a millennia-old task. Finally, a very small subdiscipline began to take shape that looked for patterns in the digits of π. It was thought that, by studying the digits closely enough, one might be able to find a method to predict the next digit without having to go through the laborious calculations already developed. Unfortunately, such efforts were in vain, and many properties of π remain inscrutable to this day. However, it is also of interest to note that, even today, stories appear in the media on a regular basis noting that someone (or, increasingly, a new computer) has broken the record for calculating π by generating another few million digits. So, even today, the digit hunters continue to be noticed by the general population.

There was, however, a great deal of mathematical understanding that came from these attempts to determine the value of π once and for all. For example, in addition to being the best way to calculate π for nearly 2,000 years, Archimedes' method for calculating π was one stepping stone on the way to developing differential calculus. He, and many centuries of mathematicians who followed him, was limited by the lack of decimal notation (i.e. writing 3.25 instead of 3¹/₄), but he was squarely on the track of methods that would later be used to develop the concepts of calculus. Further along in history, Viète's use of infinite series to help calculate π was revolutionary, and was, in fact, the first known use of analytically derived infinite series in any part of mathematics. The improvements made by Snell and Wallis helped to greatly extend the utility of infinite series, which are today used in a very large number of scientific, mathematical, and engineering applications. These mathematical tools would likely have been developed regardless of their use in attacking this one problem, but it is very likely that continued frustration in attempts to solve the quadrature problem served to accelerate progress in these, and other related areas.

Finally, over time the quadrature problem had attained a notoriety approaching that of Fermat's Last Theorem, finally solved in 1993. Although the quadrature problem was not resolved during this period (indeed, it was later found to be impossible to do), it did capture the public attention. Or, more appropriately, it captured the attention of those who were literate, educated, and had some amount of free time in which to pursue such idle interests; a very small fraction of the total population. However, this relatively small number of people was very important in the society of the times because they were the ones upon whom much of society rested. The educated elite helped set public policy, wrote the works by which their age is known, advised the government, ran businesses, taught, and did the myriad other things that helped establish, preserve, and spread their culture and government throughout the world. Those nations with a relatively large middle and upper class, such as the Dutch and English, dominated world affairs more than their relatively small size would suggest; while larger nations, such as Russia, remained relative backwaters both culturally and politically for many centuries.

The nations that became dominant—especially France, Holland, and England—were the same nations in which the quadrature problem seemed to gain the most notoriety, because it was these nations that had relatively large groups of intellectuals who could appreciate and try to solve it. Their interest and near misses, in turn, encouraged their contemporaries to try their hands at the problem, or to at least encourage further attempts. This culminated in a decree by the French Academy that they would no longer accept for publication any papers claiming to have solved this problem. The final proof, in the nineteenth century, of π's status as a transcendental number, drove the last nail into this particular coffin, forever dashing mathematical hopes of a solution. However, it should be noted that no small number of mathematically uninformed continue to try to repeal rigorous mathematical proof, including the government of one state in the United States. This fact alone demonstrates the continuing hold of π on the imagination of the general public, and the continuing impact of this number on the general public, as well as on mathematicians and scientists.

P. ANDREW KARAM