Did Indiana Once Try To Make A Law That Said Pi Was Rational?

There are so many awesome things about math, but one of the most reassuring is its certainty. If something can be proven – not always a given, we grant you – then it’s proven forever; no going back a few centuries later and realizing you were wrong all along because nipples exist.

Take pi, for example. It’s famously irrational, and we aren’t just saying that because we’ve calculated 100 trillion digits of it so far and the end is still nowhere in sight. The fact – and it is a fact – that the constant can’t be written as a fraction of two whole numbers has been known for at least some 360 years at this point: the first known proof is credited to Johann Heinrich Lambert in 1761, and at least five or six others have been developed since.

That didn’t seem to matter, though, to 19th-century physician and amateur mathematician – emphasis on the “amateur”, there – Edward J Goodwin. It was he who, in 1894, wrote a paper that claimed to prove that pi was rational – and who, three years later, tried to make that result the law of the land.

The Indiana Pi Bill

And so, in 1897, House Bill No. 246 was brought before the Indiana State Legislature.

Be it enacted by the General Assembly of the State of Indiana: 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 on one side. The diameter employed as the linear unit according to the present rule in computing the circle’s area is entirely wrong, as it represents the circle’s area one and one-fifth times the area of a square whose perimeter is equal to the circumference of the circle. This is because one fifth of the diameter fails to be represented four times in the circle’s circumference. For example: if we multiply the perimeter of a square by one-fourth of any line one-fifth greater than one side, we can in like manner make the square’s area to appear one-fifth greater than the fact, as is done by taking the diameter for the linear unit instead of the quadrant of the circle’s circumference.

It goes on like this for a while – it is, you’ll be relieved (though not surprised) to hear, fucking nonsense from start to finish: as Petr Beckmann noted in his A History of Pi, it “not only contradict[s] elementary geometry, but also appear[s] to contradict [itself]” – before concluding the proposed act with a few spurious boasts about Goodwin’s mathematical credentials. Had it passed, the value of pi would – in Indiana, at least – have been legally defined as a rational number.

And the craziest thing of all? It might have actually gotten through – had it not been for the chance intervention of a math professor who knew better.

Squaring the circle

It may have gone down in history as the “Indiana Pi Bill”, but the text of the proposed law actually never mentions the constant in question. Goodwin’s claim was slightly more roundabout than that: he purported to have found a method of “squaring the circle”.

But what does that mean, exactly? Well, technically, “squaring the circle” is an Ancient Greek problem, but really, it goes back to the very dawn of mathematics – back when all we had going for us as a species was a geometry set and a give ‘em hell attitude. The question is this: given a circle of a particular area, how do you draw a square with the exact same area?

It might not sound too difficult when set out like that, and indeed, the Greeks were pretty good at this “squaring” process when it came to other shapes. As early as 300 BCE, Euclid had already set out an algorithm for converting a polygon with any number of sides into a square with the same area – perhaps, therefore, squaring a circle would be a natural development of that process. After all, as at least one ancient philosopher argued, what is a circle if not the ultimate limit of a series of n-sided regular convex polygons?

Sadly for the ancients, though, squaring the circle is impossible. How can we be sure? Well, it turns out that finding a way to solve this problem is equivalent to proving that pi is rational – that, if you were wondering why we haven’t mentioned it yet, is where the constant in question comes into all of this. You can run through it yourself pretty easily, actually: the area of a square of side length s is s²; the area of a circle of radius r is πr²; for the two to be equal, therefore, π would have to be rational.

But pi isn’t just not rational, it’s transcendental – kind of like the veganism to irrationality’s vegetarianism. It would take a couple of millennia for that to be proven once and for all – it was eventually closed as a problem by Ferdinand von Lindemann in 1882 – but by the time of Goodwin’s quote-unquote “result”, it was a well-established fact. 

So what was Goodwin up to?

Goodwin’s pi

Possibly the weirdest thing of all about the “Indiana Pi Bill” is that, despite being infamous as the bill that tried to legislate a rational value for pi, nobody has ever managed to figure out what that value was meant to be.

There are some clues: towards the end of the second of three, the bill claims that “the ratio of the diameter and circumference is as five-fourths to four”, which is equivalent to saying that pi equals 3.2. That’s… not bad, as approximations to pi go – that is to say, it’s quite a bit less accurate than the Babylonians managed in the 17th century BCE, but it’s way better than the value implied in section one of the bill, which at 9.2376 was “probably […] the biggest overestimate of π in the history of mathematics,” Beckmann noted.

In fact, had the bill passed, Goodwin’s own work would have been outlawed in the state. Not that he ever seems to have done anything so passe as to use the correct irrational value of the constant, but a perusal of his various writings turns up at least nine different values of pi.

What was he thinking? Well, as pointed out by Gizmodo, Goodwin was an… interesting character. He didn’t rely on frivolous things like “evidence” or “logic” to support his theorems – he had something better. He knew the true value of pi, he explained in an 1897 interview with The Indianapolis Sun, because God had revealed it to him in March 1888. 

Presumably, precisely which of the many values God pinpointed it as has been lost to time. 

An unlikely hero

When the Indiana Pi Bill hit the state assembly, nobody was quite sure what to make of it. That’s not totally surprising, since the bill is essentially 50 percent jargon and 50 percent nonsense – but unfortunately, that seems to have been enough to win over some of the local lawmakers. After being bounced from the House Committee on Canals – no, we don’t know why either – to the Committee on Education, the bill was passed unanimously.

Indiana was one step closer to having pi legally defined as a rational number discovered by a man who claimed God had told him the value. It just had to pass the State Senate – which is when the magnificently named Clarence Abiathar Waldo stepped in.

There was no particular reason Waldo should have known about the Pi Bill – he was only in the statehouse that day to lobby for a bigger budget for Purdue University, where he taught math. Instead, he found himself witnessing the State Assembly debating whether to redefine pi by statute.

Alarmed, he took it upon himself to “coach” – his words – the senators on the facts of, you know, basic geometry. Thanks to his intervention, when the bill finally reached the Senate floor, it was met with derision.

“The Senators made bad puns about it, ridiculed it and laughed over it,” reported Will E Edington of DePauw University some years later. “The fun lasted half an hour.”

In the end, the Indiana Pi Bill got exactly the respect it deserved: It was indefinitely postponed, as not being a “subject for legislation,” Edington confirmed.

“Senator Hubbell characterized the bill as utter folly,” he wrote. “The Senate might as well try to legislate water to run uphill as to establish mathematical truth by law.”

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