Eighteenth-Century Advances in Understanding π

Overview

Mankind has been fascinated with π for millennia and attempts to calculate it exactly have taken place for nearly as long as mathematics has existed. In the eighteenth century, rapid advances in mathematics led to a deeper understanding of numbers in general and of π in specific. This enhanced understanding captured the public's attention, discouraged those trying to solve the ancient problem of "squaring the circle," and helped to advance mathematics enormously.

Background

Earlier than 2000 b.c. Babylonian and Egyptian mathematicians realized that the relationship between a circle's diameter and its circumference (which is what π is) was unchanging, regardless of the circle's size. They also quickly realized that this number fell somewhere between the values of 3 1/7 and 3 1/8, values that were to crop up repeatedly for the next three millennia.

One of the first formal mathematical attempts to calculate a value for π was conducted by Archimedes (287-212 b.c.) in the third century b.c. in his attempt to square the circle (described later in this essay). Archimedes realized that he could draw a polygon on the outside of a circle with the center of each side just touching the outer edge of the circle as it was drawn. He could repeat this process with a polygon on the circle's interior, in this case, with the "points" just touching the interior of the circle. The length of the sides of each polygon was easy to determine, and the length of the circle's circumference had to fall between the length of sides of the two polygons. Archimedes then went one step further and understood that, as the number of sides of these polygons increased, they came to resemble circles more and more closely until, with an infinite number of sides, they would be circles. Therefore, by adding sides repeatedly, Archimedes proposed to calculate an exact value for π, if only enough sides could be added. This approach is amazing because it anticipated the development of calculus by 2,000 years.

Unfortunately, Archimedes was killed by an invading soldier and was never able to finish his calculations, but he came up with the most accurate value for π that would be calculated for the next 2,000 years. In fact, Archimedes's method for calculating π was not improved upon for many centuries: a tribute to his genius. In fact, through the reign of the Roman Empire and the Dark Ages, very little original work was done on the problem of the nature of π or in calculating it more precisely than previous estimates.

Later efforts to determine the value of π precisely were no more successful than Archimedes and, in many cases, simply used his techniques or copied his numbers. One of the motivating factors behind these efforts to calculate π was the challenge of "squaring the circle," that is, using only an unmarked straight edge and a compass, to construct a square with exactly the same area as a circle. This remained one of the outstanding problems in mathematics from the time of the ancient Greeks until the late nineteenth century, when Ferdinand Lindemann (1852-1939) proved π to be a transcendental number. A transcendental number is one that cannot form the root for an algebraic equation. For example, the equation x² - 2x + 1 = 0 has solutions (or roots) that are equal to 1 and -1 because either of these numbers, in place of x, will make this equation correct. The solution to the equation x² - 2 = 0 is √2, an irrational number because it never terminates or repeats. However, this is an algebraic number, not a transcendental number because it forms the root of an algebraic equation. Since all transcendental numbers are also irrational (meaning they neither terminate nor repeat digits, unlike 1/2, 1/3, and so forth), this finding meant that π would never be calculated exactly. However, for the prospective circle-squarers, π's transcendence was of more fundamental importance; if π could never be calculated precisely and if it could not be forced to be the root of an algebraic equation, then it would be forever impossible to square the circle using only simple tools.

The next refinement in attempting to calculate π came in the late sixteenth century when François Viète (1540-1603), a French mathematician, became the first known person to attack the problem using trigonometric techniques. This resulted in an infinite series, that is, a never-ending series of terms, each related to the previous one algebraically and each smaller than the last. As each new term is calculated and added to the previous term, the series is said to "converge" on the correct answer to the problem, although an exact solution may never be reached. Using Viète's methods, and other infinite series, π was calculated to 140 places by the end of the eighteenth century.

The eighteenth century saw a number of significant advances in understanding π as a number. John Machin became the first person to calculate π to 100 decimal places in 1706. That same year saw the first use of the Greek letter π to describe the ratio of a circle's circumference to its diameter; prior to that time there was no formal name for this ratio. In 1719, the Frenchman Thomas de Lagny (1660-1734) calculated π to 127 decimal places, and in the middle of the century, the brilliant Swiss mathematician Leonhard Euler (1707-1783) developed an algorithm for calculating π that was superior to anything that preceded it.

At this same time, too, mathematicians were beginning to better appreciate some of the properties that made π so interesting and so intractable. In 1766, Johann Lambert (1728-1777) proved π to be an irrational number, that is, one that could not be expressed simply as the ratio of two integers. By proving this, Lambert also proved, as noted above, that any attempt to find an exact value for π was fruitless. A few years later, in 1775, Euler suggested that π might be transcendental, but at that time, transcendental numbers had not yet been proven to exist. This would not happen for another 65 years when Joseph Liouville (1809-1882), the great French mathematician, showed their existence. All of these events were important and all were destined to have an impact.

Impact

First, we should discuss the phenomenon of squaring the circle a little more. This problem was first posed by the Greeks, sometime before 400 b.c. It gained some degree of notoriety because it remained unsolved for so long a time. In fact, proposed solutions to this ancient problem had become so notorious that, in 1775, the Academy at Paris passed a resolution that solutions to this problem would no longer be officially considered by the Academy. However, in 1931 the American Carl Heisel self-published a book claiming to have successfully squared the circle, and claimants still come forward to this day. Precisely because of its longevity, this problem continues to fascinate, even in the face of clear mathematical proof of the impossibility of a solution.

The progress made in understanding π has helped to better understand irrational and transcendental numbers in general, numbers that are of great importance to mathematics, physics, engineering, and other fields of inquiry. As one example, the base for the Napieran logarithms, e, is a transcendental number that is also of exceptional importance in a wide variety of fields, including calculations as routine as determining the interest on a loan.

In the case of π, it turns out to be of fundamental importance to our understanding of many vital properties of physics and engineering. One reason for this is that so many objects are circular or spherical, and π is used to calculate the surface area and (if appropriate) volumes of such objects. As one example, in 1998 the radiation dose was measured from a gamma ray burst at a great distance from Earth. (As the burst expanded in space, one can think of it as filling the surface area of a sphere because the light from the burst expands equally in all directions.) Using the equation for calculating the surface area of a sphere, and knowing how much of the sphere is taken up by the instrument, one can then find out exactly how much energy was released by this gamma ray burst (as it turns out, about 10⁵³ ergs of energy, making it the most energetic event ever witnessed). The solution to this calculation is utterly dependent on knowing the value of π to several decimal places.

Pi is also used to describe cyclic motion, such as the oscillations of a pendulum, a clock spring, a rotating CD, or the pistons in a car engine. Physicists in the eighteenth century realized that, for example, a swinging pendulum repeated the same positions at regular intervals, exactly as a spinning disk did. This meant that they could use the same equations to describe both rotating motion and regular oscillations. Because π is, by definition, the ratio of a circle's circumference to its diameter, any problems that deal with rotation or oscillations can only be solved by using accurate values of π.

Finally, in the present day, the fact of π's transcendence and irrationality make it an ideal candidate for proving the power of ever-faster computers. There seems to be some fascination on the part of the public to learn that the newest computer has just added another million places to the value of π, even though five or six places suffice for most serious computations. The fact that π never ends just makes it that much more appealing a target. This suggests, too, that in the last 2,000 years our fascination with π has not changed appreciably.

P. ANDREW KARAM