There's an old urban legend about a state legislature that passed a law redefining pi so that it equaled 3. This story is a myth - but the true story that inspired it is actually even more ridiculous and bizarre.


The mythical version goes something like this: a state legislature is threatening to pass a law that will declare pi to not equal 3.14159... but instead just 3. Which legislature varies in the telling - I first heard it as Kansas, while the most common version of this hoax circulating today seems to focus on Alabama. But the upshot is always the same, as this redefinition is meant to bring pi in line with biblical standards, as well as making kids feel better about themselves by not making them deal with something so frustratingly inexact. As a satire of religious ignorance, it's not exactly Jonathan Swift, but it's remained quite a popular urban legend.

The fact is, there's only one known case in American history where someone actually tried to get his state legislature to redefine the value of pi. It happened in Indiana in 1897, and it had nothing to do with biblical values or building children's self-esteem through exactitude. Indeed, pi isn't even mentioned in the bill in question. This is the story of an amateur crank named Dr. Edwin J. Goodwin and his foolhardy lunge for mathematical immortality.


Redefining Pi

Dr. Goodwin only wanted to redefine pi insofar as it would allow him to solve a problem that had stumped masters of geometry for over two thousand years. The question was this: using only a standard compass and straightedge, how do you construct a square that has exactly the same area as a given circle? Professional mathematicians had long since given up on this as impossible - in fact, the French Academy passed a resolution in 1775 saying they would no longer even bother to examine any more proposed solutions for squaring the circle, so sure were they that it was impossible.

There are rigorous mathematical proofs of this assertion, but it's not hard to see why it's impossible to square a circle. A circle has an area of πr2, where r is the radius of the circle. A square has an area of s2, where s is the length of each side. To square a circle, πr2 must equal s2. Since this challenge begins with a circle, we'll say its radius is equal to 1, which means π(12) = s2, or π = s2. That means that each side of the square we're constructing equals the square root of pi, or √π.

The question then, is whether it's possible to construct a line that is √π times the size of the radius of the circle using just a compass and a straightedge. It's easy enough to construct a line that is twice the length of a line - you just use the compass to measure the length of the original line, then swing it around and extend out the line using the straightedge. Using this line of thinking, it's possible to create any rational number - in other words, any number that can be expressed as the ratio of two integers. It would take a long time to create a line that is, say, 195/137 the length of another line, but you could theoretically do it.


In fact, it's also possible to create lines whose lengths are irrational numbers. If you construct a right triangle whose two shorter sides are the same length, then the Pythagorean theorem tells us the hypotenuse will equal the square root of 12 + 12, or √2. If pi was simply an irrational number, it would still be constructible, assuming you could express it as the root of some combination of rational numbers. (That's admittedly a rather imprecise way to describe it, but you get the idea.)

But pi - and, by extension, the square root of pi - is transcendental, meaning there's no algebraic equation that describes it. There's no equation you can set up that will spit out pi, a fact that had been suspected by mathematicians for some time before it was rigorously proven by Ferdinand von Lindemann in 1882. This is what made squaring the circle impossible, and why this task became the exclusive domain of deluded cranks.


Goodwin Squares the Circle

Not much is known about Edwin J. Goodwin - indeed, even his name is a matter of some dispute, as contemporary sources list it variously as Edwin and Edward. According to his obituary in 1902, he was born in 1825 and originally came from Virginia, where he had "received an excellent education", one we can only assume did not focus on basic mathematics. By the 1890s, he was a country doctor living in Solitude, Indiana, a tiny community on the very southwestern tip of the state, a region that is sometimes called Indiana's Pocket.


In 1894, Dr. Goodwin published his paper on how to square the circle in the seventh issue of the inaugural volume of American Mathematical Monthly, which remains an active and respected publication to this day. Perhaps "remains" is the wrong word - after all, if their first volume featured Dr. Goodwin's obviously false theories, it's hard to see how respectable they could be. Here's a sample of his piece, "Quadrature of the Circle":

This new measure of the circle has happily brought to light the ratio of the chord and arc of 90°, which is as 7:8; and also the ratio of the diagonal and one side of a square, which is as 10:7. These two ratios show the-numerical relation of diameter to circumference to be as 5/4:4. Authorities will please note that while the finite ratio (5/4:4) represents the area of the circle to be more than the orthodox ratio, yet the ratio (3.1416) represents the area of a circle whose circumference equals 4 two % greater than the finite ratio (5/4:4), as will be seen by comparing the terms of their respective proportions, stated as follows: 1:3.20:: 1.25:4, 1:.3.1416::1.2732:4.


It's gibberish, of course, and it's hard to say whether an advanced understanding of mathematics would actually make that more or less difficult to understand. Some masochistic mathematicians have tried in vain to figure out just what Goodwin thought the value of pi actually was. There are at least nine different values of pi found in his various writings, including 4, 3.2325..., and even 9.2376..., which Petr Beckman in his A History of Pi declares probably "the biggest overestimate of π in the history of mathematics." The upshot of all this, apparently, was that pi was now some constructible number - which one in particular is hard to determine, but I think it's 3.2, maybe - which meant that it was possible to square the circle. You can see a valiant attempt to make sense of Goodwin's circle on the left.


What The π Were They Thinking?

It's easy here to dismiss Goodwin as a loon...and it's probably only going to get easier when I tell you about the larger implications of his ideas. In an 1897 interview with the Indianapolis Sun, Goodwin explained he had only started working on this problem in 1888 as an outgrowth of his own homegrown ideology, a bizarre hodgepodge of science, philosophy, and religion that he claimed God had revealed to him in the 1880s.


His ideas all flowed from his One Law of the Universe, which he explained in his self-published opus A New Physical Truth: Universal Inequality of the Law of all Creation, "All change depends on an inequality in the adjustment of forces whereby particles and aggregates compress to and repel from centres while acting in lines least resisting." Goodwin claimed that God had given him the true value of pi in March of 1888, and his mathematical work flowed from there.

This still leaves open the question of what could have possibly possessed the publishers of American Mathematical Monthly to publish this transparent drivel, particularly in their first ever volume. Later publishers admitted that quality control wasn't great on the "rustic" early issues, and founding editor Benjamin Fickel seemed to have an ax to grind with Euclidean geometry, which led him to give a lot of time to slightly more respectable attacks on orthodox geometry. It's possible Goodwin's work got swept up in this revolutionary fervor.


Two rather more mundane reasons also go a long way to solving this mystery: first, the fact that the paper was published in the miscellaneous section "Notes and Queries", where the general public could send in their own submissions, and second, that his paper carries the disclaimer "Published by the request of the author." Basically, American Mathematical Monthly was running short on actual material, so they padded out an issue of their first volume by humoring a crank.


Mr. Goodwin Goes to Indianapolis

If his July 1894 appearance in American Mathematical Monthly was the opening shot in Goodwin's great war on geometry, then January 18, 1897 was to be the day of his great triumph. The septuagenarian Goodwin had convinced his state representative, the rather wonderfully named Taylor I. Record, to introduce House Bill 246, which would establish his method for squaring the circle as a part of Indiana law.


The first two sections of the bill were taken up with the same old pseudomathematical gobbledygook - although, if anything, these were somehow even less clear than what Goodwin had written in American Mathematical Monthly - but the preamble and final section reveal some fascinating insights into just what Goodwin hoped to achieve with the bill:

A Bill for an act introducing a new mathematical truth and offered as a contribution to education to be used only by the State of Indiana free of cost by paying any royalties whatever on the same, provided it is accepted and adopted by the official action of the Legislature of 1897... In further proof of the value of the author's proposed contribution to education and offered as a gift to the State of Indiana, is the fact of his solutions of the trisection of the angle, duplication of the cube and quadrature of the circle having been already accepted as contributions to science by the American Mathematical Monthly, the leading exponent of mathematical thought in this country. And be it remembered that these noted problems had been long since given up by scientific bodies as insolvable mysteries and above man's ability to comprehend.


Of course, Goodwin was eliding over how tenuous his appearance in American Mathematical Monthly really was. But why should such quibbles matter? After all, Dr. Goodwin was offering a tremendous gift to his home state and countless future generations of Indiana schoolchildren, all of whom could learn from his genius without worrying about paying him royalties. As Goodwin was keen to stress, he had copyrighted this pearl of mathematical wisdom way back in 1889. Never mind that you probably can't actually copyright a mathematical formula - Goodwin was on a roll.

For his part, Representative Record was upfront about the fact that he didn't actually understand what the bill was saying or whether it was worthwhile, and it was fairly obvious that he had had nothing to do with actually writing it. Record was simply introducing the bill as a favor to Goodwin, who had worked as a presumably decently well-respected physician in his constituency for the last two decades.


The Pi Bill's Progress

House Bill 246 hit the Indiana General Assembly with a dull thud. Nobody really knew what to make of it, and the bill was referred to, of all places, the House Committee on Canals. Considering this committee's nickname was the Committee on Swamp Lands, it's thought that this rather bizarre decision was some representative's idea of a joke. In any event, Canals Committee Chairman M.B. Butler took the bill just seriously enough to recognize he had no business considering it, and referred it again to the Committee on Education, which I might argue was an even more inappropriate place for it.


The Indianapolis newspapers had picked up the story of this strange bill, with one offering a spirited and devastatingly accurate debunking of the whole thing, tracing the history of pi from ancient times to Ferdinand von Lindemann. The only problem was that this particular paper, Der Tagliche Telegraph, was only published in German, so nobody took much notice. The English-speaking newspapers were embarrassingly far more credulous, with a writer for the Indianapolis Sentinel saying he was lobbying for the bill and noting its impressive credentials:

[Goodwin] and State Superintendent of Public Instruction Geeting believe that it is the long-sought solution of the problem, and they are seeking to have it adopted by the legislature. Dr. Goodwin, the author, is a mathematician of note.


That was bad. The Indianpolis Journal was worse:

It may not be the function of a Legislature to endorse such discoveries, but the average editor will not gain much by trying to make fun of a discovery that has been endorsed by the American Mathematical Journal; approved by the professors of the National Astronomical Observatory of Washington, including Professor Hall, who discovered the moons of Mars; declared absolutely perfect by professors at Ann Arbor and Johns Hopkins Universities; and copyrighted as original in seven countries of Europe. The average editor is hardly well enough versed in high mathematics to attempt to down such an array of authorities as that. Dr. Goodwin's discovery is as genuine as that of Newton or Galileo, and it will endure, whether the Legislature endorses it or not.


It's hard to know where to begin with all this, except to say the source of this writer's information was undoubtedly Dr. Goodwin himself. We do know that Goodwin wrote letters to professors at the National Astronomical Observatory - who, it should be pointed out, were astronomers, not mathematicians - and he certainly felt he had persuaded them, whatever they might have actually thought.

Whatever the case, Goodwin had picked up some support from the press, and on February 2 Representative S.E. Nicholson, the chair of the Education Committee, recommended that the full House pass the bill. On February 5, they did just that, by a vote of 67 to 0. Whether any of those 67 representatives had even glanced at the bill is an open question. The bill was now headed to the senate, and Dr. Goodwin's legislative vindication seemed close at hand.


Professor Waldo Steps In

With the notable exception of some German-speaking journalists, House Bill 246 had apparently completely escaped the attention of anyone with even basic mathematical knowledge. That changed with the arrival of the head professor of mathematics at Purdue University, the even more wonderfully named Clarence Abiathar Waldo. Born in upstate New York in 1852, Waldo had been teaching mathematics at various universities since 1877, and in 1897 was serving as the president of the Indiana Academy of Science.


In 1917, twenty years after his run-in with House Bill 246, Professor Waldo wrote his own account in Proceedings of the Indiana Academy of Science. He got some details wrong - he remembered it as happening in 1899, not 1897 - but the thing is positively drenched in withering sarcasm:

In the early spring of [1897] vague rumors reached Indiana University and Purdue that some sort of mathematical legislation was pending at Indianapolis. It was evident, however, that the state solons [lawmakers] there assembled thought themselves well equipped to attack the problems, whatever they might be, with wisdom and justice for they made no appeal for help to their two state supported fountains of erudition.


It was just that question of state support that brought Waldo to the statehouse, as he was there to lobby for Purdue during the ongoing budget talks. He recalls entering floor of the House to hear Goodwin's bill being read, and an ex-teacher turned legislator raving about "a new and correct value of π" that Goodwin would allow Indiana to use without paying him a royalty. Professor Waldo made no secret of his reaction to it all with this third-person account:

The roll was then called and the bill passed its third and final reading in the lower house. A member then showed the writer a copy of the bill just passed and asked him if he would like an introduction to the learned doctor, its author. He declined the courtesy with thanks remarking that he was acquainted with as many crazy people as he cared to know.


Professor Waldo had seen enough. One way or another, he was going to stop this bill.

The Death of the Pi Bill

The bill reached the Senate on February 10, and the next day it was given its latest head-scratching referral, this time to the Committee on Temperance. (If anything, you'd need a stiff drink to have any chance of fathoming what Dr. Goodwin was on about.) On February 12, Senator Harry S. New, the committee chairman, returned the bill with a recommendation that it pass.


By his own account, Professor Waldo had "properly coached" some senators so that they could speak out against the bill - or, as he rather wonderfully put it, "they threw out with much merriment the epoch making discovery of the Wise Man from the Pocket." Waldo undoubtedly had some effect on the bill's failure, but by now news of the bill had spread to Chicago and the larger cities back East, and the general consensus of the national newspapers was that the bill, its creator, and the legislature considering it were all completely ridiculous. According to an account in The Indianapolis News, Senator Orrin Hubbell led the charge against the bill when it reached the floor:

Representative Record's mathematical bill legalizing a formula for squaring the circle was brought up and made fun of. The Senators made bad puns about it, ridiculed it and laughed over it. The fun lasted half an hour. Senator Hubbell said that it was not meet for the Senate, which was costing the State $250 a day, to waste its time in such frivolity. He said that in reading the leading newspapers of Chicago and the East, he found that the Indiana State Legislature had laid itself open to ridicule by the action already taken on the bill. He thought consideration of such a proposition was not dignified or worthy of the Senate.


The Indianapolis Journal reported that Hubbell had declared the bill "utter folly", and that the legislature "might as well try to legislate water to run up hill as to establish mathematical truth by law." The Indianapolis Journal captured a moment that reflected the senate's general attitude toward mathematics:

Senator Drummond did not want the rule suspended until he had some information as to the purpose of the bill. "It may be I am densely ignorant on this question of Mathematics," he said. "Consent! Consent!" said Senator Ellison. There was loud laughter at this sally.


It's an amusing zinger, but the fact is that nobody in the senate seemed particularly concerned with their complete ignorance of the mathematics involved. The Journal reports the rather underwhelming reason why the senate as a whole ultimately rejected the bill:

Although the bill was not acted on favorably no one who spoke against it intimated that there was anything wrong with the theories it advances. All of the senators who spoke on the bill admitted that they were ignorant of the merits of the proposition. It was simply regarded as not being a subject for legislation.


What It All Meant

And, at last, the Indiana Pi Bill, as it had come to be known, was defeated. Twenty years later, a triumphant Professor Waldo claimed "it was probably the Indiana Academy of Science alone which prevented [this monstrosity]" - while writing in said academy's own journal, I should point out - and that "if this deduction is correct then that one act of prevention was worth more to Indiana, jealous of her fair fame as she is, than all she ever contributed or can contribute to the publication of the proceedings of her Academy of Science."


Perhaps. But based on what the senators themselves had to say, this seems just as much a case of the forces of ignorance defeating the forces of craziness. It's a victory, perhaps, but it's an ugly win all the same.

As for Dr. Goodwin, he died just five years later in 1902 at the old age of 77. The local newspaper for New Harmony, Indiana printed this obituary, which was every bit as ridiculous and yet strangely endearing as its subject:

He felt that he had a great invention and wished the world to have the benefit of it. In years to come Dr. Goodwin's plan for measuring the heavens may receive the approbation which was untiringly sought by its originator. As years went on and he saw the child of his genius still unreceived by the scientific world, he became broken with disappointment, although he never lost hope and trusted that before his end came he would see the world awakened to the greatness of his plan and taste for a moment the sweetness of success. He was doomed to disappointment, and in the peaceful confines of village life the tragedy of a fruitless ambition was enacted.


You know, I feel bad for the poor guy. But that doesn't change the fact that pi is equal to 3.141592...

Sources

"What Might Have Been" by C.A. Waldo in Proceedings of the Indiana Academy of Science, 1916-17

"House Bill No. 246, Indiana State Legislature, 1897" by Will E. Edington in Proceedings of the Indiana Academy of Science, 1935

"Indiana's Squared Circle" by Arthur E. Hallerberg in Mathematics Magazine, May 1977

History of Pi by Petr Beckmann

Mathematical Cranks by Underwood Dudley


Image Credits

Pi by Benjamin Haas, via Shutterstock.

Compass by gcpics, via Shutterstock.

Goodwin's circle by Henning Makholm on Wikimedia.

Indiana State Capitol by Daniel Schwen and Massimo Catarinella on Wikimedia.

Professor Waldo from Purdue Yearbook via Wikimedia.