Clarence Abiathar Waldo could not believe what he was seeing. It was not uncommon for that room full of politicians to applaud nonsense as if there was no tomorrow, the really strange thing was that it was mathematical nonsense. Waldo had arrived there by chance, but what he was dealing with was no stranger to him, after all, he was a math teacher and president of the Indiana Academy of Sciences.
Apparently, the “Indiana House of Representatives” had just unanimously approved a mathematical demonstration. The simple fact of submitting something as objective as mathematics to popular opinion was already suspicious, but what the supposed demonstration claimed was even more alarming. From the complex phrases that filled the bill, it was derived that the whole history of mathematics was wrong, and with it, that physics and engineering were built on rotten foundations. The person responsible for this stir was a doctor, Dr. Edwin J. Goodwin. A person who would go down in history as the man who wanted to decree that π was worth exactly 3.2.
The story goes that Waldo refused to meet Goodwin because “he had already met too many crazy people.” In fact, as soon as he learned of the abomination that they were trying to approve, he decided to alert the Indiana senators so that, when the bill reached them, they knew the type of barbarities it contained. But what did Goodwin say was so heretical? To understand it we have to know what is hidden behind the bombastic concept of “the square of the circle”.
The forms of mathematics
In Ancient Greece “mathematics” was practically a synonym for “geometry.” At that time the algebra was not yet mature and the sums, subtractions and even square roots were calculated with ruler and compass. Geometry was a powerful tool that allowed solving many problems. However, there was one that resisted them: How to draw a square whose area was identical to that of a given circle?
It may seem simple, but it has a trick. We can start with confidence, imagining a circle of radius equal to 1. Luckily we have a formula that allows us to calculate the area of a circle from its radius: π multiplied by its radius to the cube. Since the cube of 1 (its radius) remains 1, our circle will have an area equal to π. This is fantastic, because we know that the area of a square is calculated by multiplying one of its sides by itself, and that, therefore, each side of our square will have to be worth exactly the root of π (so that multiplied are worth the same than the area of the circle). But as I said, there is a trick, how would you do all this using only geometric drawings? How would you draw a measure line π on which you can calculate its square root?
The infinity of decimals in π is not the real problem. What prevented the geometers from solving the quadrature of the circle was not knowing how to obtain a line of length π. What measures did they have to add or subtract to get π? For centuries, amateurs and math professionals tried to solve the problem, attracted by the simplicity of the approach and the legendary air it had taken. However, nobody managed to solve it, in any case they approached him.
Recall that π is the relationship between the perimeter of a circle and its diameter. It is a constant value that arises from dividing what measures the contour of any circle by twice its radius. If we could stretch a circle we would see that its perimeter fits in its diameter three times and a little more. A little bit that mathematicians had a hard time calculating.
The first approach dates back to 3800 years ago in Ancient Egypt and is the work of a scribe named Ahmes. The result was 3,16049 being quite close to the currently known 3.1415926535 … Other geniuses of the past, such as Archimedes or Zu Chongzhi, tried to get even closer, inscribing polygons from increasingly more sides within the circle. First a square whose corners touched the circumference, then a pentagon, then a hexagon and so on to get so many corners that the circumference and the polygon that was enclosed were almost overlapping. Simultaneously, they repeated the same with the circumscribed polygons (surrounding the circumference) and achieved a more than acceptable result. Archimedes used 99-sided polygons, obtaining a value for π equal to 3.1429, while Zu added even more angles closer to the real value, with 3.1415926. The “exhaustive method” was a success.
However, mathematicians had begun to suspect that π could be an irrational number, that is: with infinite decimals that are not repeated periodically. However, it was not easy to prove mathematically, in fact, it was not achieved until 1761, thanks to Johann Heinrich Lambert and almost 20 centuries after the approach of Archimedes. In fact, so far we have calculated the first 31 trillion digits that make up π. All this thanks to the computers and the 170 Terabytes of memory they have needed to do so.
The good doctor
Things did not advance much since then, until one good day in 1888 a doctor without mathematical training named Edwin J. Goodwin said he had solved the impossible riddle, the circle had squared. The good doctor was a tall, mustache man who at that time was already over sixty years old and we could say that he was not in his best age. Not long ago, his wife had died and his clinic had burned to the ground. As if this were not enough, he decided to move to change his air and as he relates, his new community dedicated himself to spreading rumors about the alleged negligence that had caused him to flee his land. In this context his supposed mathematical demonstration was born, almost as a revelation. Goodwin said he never devoted too much time to mathematics and less to this famous problem, he just crossed his path and the answer came to his mind.
The doctor was so excited that he wanted to share his knowledge with the world, though, for a small fee, so he decided to patent the method he had used to prove it. However, there was an exception, Goodwin wanted his land to prosper, so he decided to make a deal with the county authorities. In exchange for offering it to all Indiana residents free of charge, the state should accept it in the 1897 legislature. That is the real reason why the proposal was approved unanimously before reaching the Senate: because a man with an indecipherable jargon but apparently scientific wanted to give a gift to his state.
It would have been a commendable gesture, especially if the demonstration had not been meaningless. It rounded oddly the measurements of his figures, challenging the most fundamental principles of geometry. From his calculations it was derived that the classic Pythagorean formula relating the legs and the hypotenuse of a right triangle was incorrect. And, what is more, in his reasoning it was implied that π was 3.2 which was simply impossible, as Lambert had shown. Either π and Pythagoras were wrong or Goodwin had made a mistake in his demonstration.
Luckily, Waldo was there to detect all these inconsistencies and alert the authorities. However, his intervention unleashed the mocks of the senators and what is worse, of the press. Mathematicians of all kinds fired at the easy target that was Goodwin who, cornered, intensified his defense, even denying that the diameter of a circle had anything to do with π. Goodwin never accepted his mistake and in his last years he left us phrases like the following:
What Goodwin possibly did not know was that not only was his reasoning wrong, but that a few years earlier, in 1882, Carl Louis Ferdinand von Lindemann had shown that the squaring of the circle was in itself impossible. More specifically, what Lidemann showed was that π was a transcendent number, impossible to obtain by operating with rational numbers. Therefore, if π is transcendent, it means that there is no way to obtain it through classical geometry, the same methods with which the challenge of squaring the circle arises.
Goodwin and his strange value of π starred in what is undoubtedly one of the most rocambolesque stories of mathematics. Today it seems absurd that a group of politicians try to approve a theorem democratically, and yet similar situations continue to occur around us. In science and mathematics something is true or false beyond what we want. Things do not change because we want it with all our strength, they simply are. Accepting scientific evidence is the only healthy way to approach knowledge, even when it contradicts us. Science provides knowledge, but it should never be done on demand, even if we pursue something as beautiful and utopian as the quadrature of the circle.
DON’T KEEP IT UP:
- Despite what is often said, the bill did not expressly speak of the value of π, but of the quadrature of the circle. The value of π was one of the inconsistencies that were derived from that test.
- Some sources say that in the state of Alabama a law was passed by which π was worth 3. It was alleged that it would be easier to operate with it and that it would undermine the self-esteem of its students. Actually it is just a hoax created by comedian Mark Boslough.
- The name of π comes from 1706, not from the ancient Greeks. It was coined by William Jones for being the initial of the Greek word to refer to the periphery: περιφέρεια.
- Arthur E. Hallerburg “House Bill No.246 Revisited” Valparaiso University (1974).
- Carl B. Boyer & Uta C. Merzbach “History Mathematics.” Jossey-Bass; Edition: 3rd (2010).
- Randy Schwartz “Pi is Transcendental: Von Lindemann’s Proof Made Accessible to Today’s Undergraduates.” (2015).