The Indiana Legislature Redefines Pi

It has all the hallmarks of an urban legend. A Midwestern state (which one varies with the telling) was so unsophisticated that its legislature once passed a law declaring the value of the mathematical constant pi to be 4 (or 3, or 3.2, or some other simple, exact number) instead of, as every urban sophisticate knows (not!), 3.14159. Unlike some urban legends, there is a kernel of truth to this tale—wrapped in a whole lot of condescension.

Pi is a very simple concept, the ratio of the circumference of a circle to its diameter. It is crucial to determining most mathematical relationships involving circles and other curves. It is also crucial to solving the ancient problem of “squaring the circle,” constructing with only a compass and a straightedge a square whose area is exactly equal to that of a given circle. What is fascinating about pi is its propensity for suddenly showing up in the solution to problems that have nothing (apparently) to do with curves, such as problems in statistics and probability.

What is bedeviling about it is that pi is an irrational number. That means that it cannot be exactly expressed as either a fraction or a decimal (3.14159 is pi only to five decimal places; there is no limit to how many places to which the number can be carried—the current record is 1.9 trillion places—and it still won’t be expressed exactly).

Determining pi as closely as possible has been the occupation of mathematicians, among them the greatest who ever lived, for as long as the concept of pi has been around, which is well over three thousand years. By simply wrapping a piece of string around something circular and seeing how many diameters are contained within that length, it is clear that pi is slightly over three.

In 1650 B.C., an Egyptian scribe wrote, “Cut off 1/9 of a diameter and construct a square upon the remainder; this has the same area as the circle.” This squaring of the circle implies a value for pi of 3.16049, which isn’t bad. It is better, for instance, than the Biblical verses (1 Kings 7:23 and 2 Chronicles 4:2) that imply a value for pi of exactly 3.

Archimedes (287–212 B.C.) did better still, using a typically brilliant new method. He even wrote a book about it, called The Measurement of the Circle. In it he wrote that “the ratio of the circumference of any circle to its diameter is less than 3 1/7 and more than 3 10/71.” Averaging those two extremes gives a value for pi of 3.1419, which is accurate to within three ten thousandths of the true value. He, or perhaps a younger colleague of his, Apollonius, later refined the value to 3.1416.

There wouldn’t be a closer approximation in the West for more than 1,700 years, although a Chinese mathematician would do slightly better in the fifth century after Christ. More sophisticated mathematics, the Arabic numeral system, and the all-important zero, which finally came to Europe in the twelfth century, allowed ever more precise approximations of pi. In 1593 it was calculated to 15 places; by the end of the seventeenth century an Englishman named Abraham Sharp calculated it to 72 places.

This was an astonishing feat of mathematics, but it had no practical applications. Engineers rarely need more than four decimal places, and the Romans, the greatest engineers of the ancient world, usually employed the easy-to-use value of 3 1/8 for pi, even though they knew perfectly well that 3 1/7 was closer. Even physicists rarely need more than seven places.

Instead people with the time and the patience did it for the sheer joy of breaking the record. At the end of the eighteenth century the record was 140 decimal places; by 1853 it was 440. Then in 1874 an Englishman named William Shanks calculated pi to 707 places, a record that stood until the dawn of the computer age. William Shanks would go to his grave not knowing that he had made a mistake after the 527th decimal place and the next 180 places of his calculation were wrong, an error discovered only in 1945.

While serious mathematicians were amusing themselves with calculating pi, far greater numbers of crackpots were trying to square the circle, or produce a square with the same area as a given circle. Mathematicians knew in their bones that that was impossible, but they couldn’t prove it. The German mathematician Ferdinand von Lindemann, in 1882, did exactly that, by proving that pi is not only irrational but transcendental as well. A transcendental number, without getting too technical, is one that cannot be expressed exactly as an algebraic equation. No line whose length is transcendental can be constructed using only a compass and straightedge, and therefore the circle cannot be squared.

Mathematicians were agog at von Lindemann’s accomplishment, but crackpots kept right on trying to do the impossible. In 1897 one of them, a physician named Edwin J. Goodwin, from the wonderfully named town of Solitude, Indiana, approached his state representative, Taylor Record, with a proposed bill. In exchange for passing the bill, the state of Indiana would have free use of the mathematical truths contained therein, while other jurisdictions, presumably, would have to pay a royalty. (Mathematical truths, of course, can be neither patented nor copyrighted.)

The “truths” buried in the nearly incomprehensible bill were no fewer than three different implied values of pi: 3, 3.2, and 4. Representative Record, who later claimed not to have the slightest idea what was in the bill, submitted it, and it eventually was approved by the committee on education and then passed unanimously by the House of Representatives, 110 years ago today, 67-0. It is a safe bet that the other 66 representatives were as mystified by the bill’s contents as was Representative Record. They presumably just figured that if those contents proved useful, the state would be sitting pretty. The next day, the Indianapolis Journal wrote that it was the strangest bill ever to pass the House.

Out-of-state newspapers began to pick up on the story and local ones to run sarcastic editorials. On February 12, 1897, the state senate ran for cover and voted to table the measure.

While a silly bill, it was not the silliest one ever entertained by an American legislature by a long shot. And it never came close to becoming law.