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When Geometry Entered the Statehouse

For more than 2,000 years, mathematicians tried to “square the circle”: given a circle, construct a square with exactly the same area using only a compass and an unmarked straightedge. Thousands of supposed solutions appeared, but in 1897 one false proof escaped the usual mathematical channels and reached the Indiana legislature.

The proof came from Edward J. Goodwin, a physician and amateur mathematician who believed he had solved the ancient problem in 1894. Three years later he proposed a bill that would let Indiana use his discovery without paying royalties. The premise itself should have raised doubts. Theorems are not normally licensed to states, and legislatures do not establish mathematical truth. More seriously, Goodwin’s work implied that pi was 3.2 rather than approximately 3.14159.

The bill nevertheless survived three readings, bounced from the Committee on Canals to the Committee on Education and passed the Indiana House of Representatives unanimously. Goodwin also appeared to have a respectable credential: his argument had been printed in the American Mathematical Monthly. At the time, however, the journal published submissions marked “by request of the author” without reviewing their validity. A prestigious venue gave the claim a sheen of authority that its mathematics did not earn.

Why the Construction Cannot Work

The impossibility of squaring the circle had already been established. In 1882 Ferdinand von Lindemann proved that pi is transcendental, a result that explains why attempts such as Goodwin’s had to smuggle in a false value.

For a circle of radius 1, the area is pi. A square with the same area would therefore need sides of length sqrt(pi). Compass-and-straightedge geometry can construct only lengths obtainable from integers through a finite sequence of addition, subtraction, multiplication, division and square roots. This permits many complicated-looking numbers, but it excludes others that are easy to describe, such as the cube root of 2.

Pi lies even farther outside that constructible family. Calling a number transcendental means it is not the solution of any nonzero polynomial equation with integer coefficients. Consequently, neither pi nor sqrt(pi) can be produced by the permitted geometric operations. The failure is not a matter of mathematicians needing a more ingenious diagram; the requested construction is impossible under the rules.

Goodwin’s value of 3.2 concealed that barrier. Because 3.2 equals the rational number 16/5, it can be expressed using integers and division. Replacing pi with that tidy fraction made the impossible construction appear manageable, but only by changing the constant that defines every circle.

A Coincidental Rescue

The bill reached the Indiana Senate just as Clarence A. Waldo, the head mathematics professor at Purdue University, happened to be visiting the statehouse to discuss the university’s budget. He overheard the debate, recognized the error and stayed to explain the geometry to state senators. Unflattering newspaper coverage added pressure; the Chicago Tribune mocked the idea that a circle’s dimensions could change when it crossed Indiana’s border.

The Senate never formally rejected the bill. Instead, it voted to postpone consideration indefinitely, leaving the proposal dormant. The episode is funny because the stakes seem absurd, yet it exposes a serious weakness: confused procedures, borrowed prestige and confidence can carry a false claim surprisingly far. Mathematics ultimately stopped the measure not because truth won a vote, but because an expert arrived in time to show that truth was never up for one.