A History of ? (Pi), by Petr Beckmann, 4th ed. Boulder, Colo.: Golem Press, 1977, 202 pp., $9.95.
It is perhaps surprising that one number has been studied by generations of mathematicians, both professional and amateur, to a far greater extent than any other. That number is the ratio of the circumference of a circle to its diameter, denoted by the Greek letter pi (?). Beckmann's book is a lucid and entertaining account of these investigations. It is written in a lively and witty, if irreverent, style, with the mathematical details clearly worked out.
Though not evident from the title, this book has much in it of interest to libertarians. The author is a professor of electrical engineering and a champion of liberty and the free market. He discusses many of those aspects of political and social history that have influenced investigators of ?, pulling no punches in his attacks on authoritarianism, militarism, and mysticism.
The story of ? begins around 2000 B.C. with the realization that if one divides the circumference of a circle—any circle—by its diameter, one always gets the same number. By a straightforward measurement one can deduce, as did the Egyptians, that ? lies between 3.1 and 3.2. Much of the work on ? during the next 4,000 years consisted of finding better and better approximations to its value, even though, for any imaginable application, accuracy to 25 decimal places suffices. (The book's end sheets reproduce a computer printout of the first 10,000 decimal places of ?.) But there is more to the story of ? than mere digit hunting.
Though they solved many difficult problems of geometry, the Greeks could not determine whether it was possible, using only straight-edge and compass, to construct a square equal in area to a given circle—the famous problem of squaring the circle. Since the area of a circle is ? times the square of the radius, this problem is closely related to the nature of the number ?. After being studied for over 2,000 years by many of the world's most able mathematicians, the problem was finally solved. Lindemann, in 1882, showed that it is not possible to square the circle using straightedge and compass.
His rigorous proof of impossibility, however, did little to quell those amateur geometers known as circle-squarers. Diehards surface sporadically, claiming to have found a miraculous construction that succeeds in squaring the circle. Most often this involves a claim that ? is rational—a quotient of two whole numbers—even though it was shown in 1761 that this is not the case. For instance, E. Goodwin, of Solitude, Indiana, claimed, at the same time, that ? is equal to 4, 160/49, and 16/5. Goodwin differed from most circle-squarers, though, in that he attempted in 1897 to have the Indiana legislature pass a bill stating and approving his new mathematical truths. Beckmann's thorough account of these events includes a reproduction of the original bill, which passed the Indiana House; he also describes the exploits of other circle-squarers.
The book is enhanced by many fine illustrations, and the fourth edition has cleared up misprints found in the third. This very readable book is highly recommended to anyone interested in a fresh and unconventional view of the history of science.