In the July 1894 issue of American Mathematical Monthly, amateur mathematician Edward J. Goodwin published a “contribution to science” in the “Queries and Information” section titled “Quadrature of the Circle.” The “contribution” asserted that it was “not mathematically consistent that it should take the side of a square whose perimeter equals that of a greater circle to measure the space contained within the limits of a less circle.”1
Goodwin thought he found a way to “square the circle.” He assumes that a circle with the same perimeter as a square should also have the same area. He then demonstrates that by using the established formula with pi as 3.1416, a circle with the circumference of 16 has the same area as a square with a perimeter of (roughly) 18. Assuming that if the perimeter is equal, the area should be as well, he posits that the solution to the dilemma was to multiply the diameter of the circle by 3.2 — essentially saying that pi has been calculated wrong for millennia.
Today, the American Mathematical Monthly is a well-respected journal published by the Mathematical Association of America, but in 1894, this was an infant, independently published journal, and Goodwin’s claim was published in the seventh issue. Goodwin’s “Quadrature of the Circle” was not even an article — it was a brief explanation of his claim, accompanied with an 1889 copyright notice (apparently, laws of mathematics were subject to copyright law).
But Goodwin’s copyrighted mathematical discovery was fallacious. The assumption that equal perimeters of shapes with a different number of sides should have equal areas is false; as the number of sides increases, so does the area, even as the perimeter is held constant. Thus, a circle will always have the largest area out of any regular polygon. In 1894, this was hardly a new truth, but it would have been new to Goodwin. In an 1897 interview given to the Indianapolis Sun, Goodwin admitted that he was new to mathematics — he earned his living as a country doctor — but he had discovered his solution to the supposed problem through revelation. In this age of skepticism, however, he needed to demonstrate his revelation mathematically for it to gain currency. “If I were to say that the discoveries are revelations to me,” he told the reporter, “they wouldn’t believe it. This is an age of unbelief. Do you know it?”2
In fact, the first exposition of Goodwin’s quadrature of the circle was published in his 1892 book Universal Inequality is the Law of All Creation. In it, Goodwin devotes some attention to the quadrature problem and ends the book with an explanation of how it came to him:
During the first week in March, 1888, the author was supernaturally taught the exact measure of the circle, just as he had been taught three years before, the “Scheme of Universal Creation”. These revelations were due in fulfillment of Scriptural statements and promises. Mathematicians second to none in this country, frankly admit that no authority in the science of numbers can tell how the ratio was discovered. ... To assert that my experience differs from that of any other man, is, to say the least, a declaration of no common import. ... All knowledge is revealed directly or indirectly, and the truths hereby presented are direct revelations and are due in confirmation of scriptural promises.3
Goodwin devoted energy to publicizing his “discovery.” He copyrighted his solution in the United States and several European countries. He brought his results to the scientists at the National Observatory in Washington, DC, where he believed (incorrectly) that he convinced the astronomers there that his revelation was correct. He asked the Smithsonian Institute to offer a substantial monetary award to anybody who could show an exception to his rule. In the 1893 World’s Fair, he tried to give a demonstration of his proof.
Ultimately, the mathematics profession did not adopt Goodwin’s attempt to, almost literally, reinvent the wheel. But the legislature of Goodwin’s home state of Indiana was more interested.
The Indiana Pi Bill
In 1897, the Indiana state legislature introduced House Bill No. 246 — “A Bill for an act introducing a new mathematical truth and offered as a contribution to education to be used on by the State of Indiana free of cost of paying any royalties whatever on the same.”4 In fact, it was Goodwin himself who wrote the bill, offering it as a gift to the legislature.
The Indiana Pi Bill attempted to replace standard mathematical teachings, according to traditional valuations of pi, with Goodwin’s discovery. According to Arthur E. Hallerberg’s history of the bill, “It seems evident that nearly all of the legislators were completely unknowledgeable about the underlying mathematics and the import of the bill.”5
In the bill, Goodwin confuses his own proposition even further. In Section 1, he summarizes his findings: “It has been found that a circular area is to the square on a line equal to the quadrant of the circumference, as the area of an equilateral rectangle is to the square of one side.”6 Hallerberg mathematizes this proposition to show its inconsistency with Goodwin’s original formulation of pi:
Since an “equilateral rectangle” is simply a square, this proportion is Area of circle: Quadrant2 = Area of Square: Side2. Since the right side reduces immediately to 1, we appear to have π2 = (2πr/4)2 or π=4.7
So now, in the Indiana Pi Bill, Goodwin effectively proposes two new values for pi: 3.2 and 4.
The delegate who introduced the bill admitted that he didn’t understand the math involved, but he “introduced it at the request of the country doctor who had been practicing in Posey County for nearly 20 years.”8 The state constitution required that any bill must be read three times before being approved, but the House unanimously passed the bill after only the second reading. The legislators did not understand the math involved, but they assumed that somebody must — after all, Section 3 of the bill claimed that the new concept had “already been accepted as contributions to science by the American Mathematical Monthly, the leading exponent of mathematical thought in this country.”9
The claim that the infant journal was the “leading exponent of mathematical thought” was downright laughable, but this grandiose proclamation was certainly compelling to the politicians who voted to enshrine a mathematical fallacy into fiat law. Even the Indiana Superintendent of Public Instruction endorsed it.
Fortunately for the history of Indiana mathematics, a professor from Purdue University stepped in and explained the fallacies to the state’s senate. Even with this, many senators still voted on the bill, but a lone wise senator spoke up to say “that the Legislature had no power to declare a truth.”10 The Senate did not vote to reject the bill, but it did indefinitely postpone the decision (the bill is still technically awaiting a vote today, as far as I know).
The story of Goodwin’s “Quadrature of the Circle” should be little more than an amusing tale of whacky mathematics, but instead, it serves as one of the starkest reminders of legislative arrogance. The idea that legislators held was essentially that fiat law superseded scientific law, and only by the intervention of a mathematics professor did Indiana’s education system narrowly avoid throwing out millennia-old mathematical formulae in favor of the divinely-revealed fallacy of a country doctor.
- 1(1894) Queries and Information, The American Mathematical Monthly, 1:7, 246-248,DOI: 10.1080/00029890.1894.11997822
- 2Indianapolis Sun, Indianapolis, Indiana, February 6, 1997.
- 3E.J. Goodwin, Universal Inequality is the Law of All Creation (Solitude, IN: 1892), 61-62.
- 4W.E. Edington, House Bill No. 246, Proc. Indiana Acad. Sci., 45 (1935) 206-210.
- 5Arthur E. Hallerberg, “Indiana’s Squared Circle,” Mathematics Magazine, 50. no. 3 (May 1977): 136-140.
- 6House Bill No. 246.
- 7Hallerberg, “Indiana’s Squared Circle.”
- 8Ibid.
- 9House Bill No. 246.
- 10New York Tribune, February 13, 1897.
