Catching Proteus

John Wallis sits at his desk in Oxford, England, surrounded by books and mathematical instruments. It is 1655. At the top of a piece of paper he writes the words “Proposition 190” again, hovers over the area underneath, and stops. Out of habit, he crosses the words out. At this point he stands up and sighs, looks out the window, and thinks about his simple, code-breaking youth, before Parliament “honored” him with this Professorship and Chair of Geometry—it really is a terribly drafty workspace. He wonders if these three years of blank pages could have been spent any differently.

At this point, Wallis considers the fact that he has done enough work for the day. He mulls over the idea of walking a few buildings away to talk with his good friend Seth Ward, the Professor of Astronomy, about stars or the rings of Saturn. Nice, healthy, real world stuff. Once again he writes “Proposition 190,” at the top of his sheet of paper. Then he tosses the thing aside to go have a visit across the lawn.

Wallis is writing a book called The Arithmetic of Infinitesimals. Years later, his modern translator will comment that it was “perhaps the one real stroke of genius in Wallis’ long mathematical career,” though she also called it an “often tedious approach through scores of uninspiring Propositions and Corollaries.” When it is done, it will do things like annoy the printer, who will be forced to spend three years fitting Wallis’ odd symbols and drawings into the unwieldy press. It will give Thomas Hobbes, philosopher, fits of cranky rage. It will bridge the gap between geometry and algebra. It will convince a 22-year-old Isaac Newton, after he finishes reading, to invent his calculus. Mostly, it will provide a formula for a funny number called pi, discovering an estimate that is better and cleaner than anything that has come before. After it is done, mathematician Doctor William Oughtred will proclaim Wallis’ “understanding and genius, who have not only gone, but also opened a way into those profoundest mysteries of art, unknown and not thought of by the ancients.”

Right now, however, Wallis is stuck at an uninspired part. In the Comment attached to the end of Proposition 189, he writes, “But here, at last, I am at a loss for words.” He whines at the cruel mathematical fates: “Until now we seem to have carried the thing through happily enough.” He is entirely lost, “For I do not see in what manner I may produce the quantity o.” This value represented by this little square, whose discovery represents the climax of the book, is really pi (If we take 4 and then divide it by o, we get ?).

Pi is a number that appears in every circle: it is the ratio between the circumference and the diameter. In decimal form, it is a number that never ends and never repeats, following no discernible pattern. It is interesting and important because it shows up in all branches of mathematics, from probability to real analysis. The problem is that it’s impossible to define with an exact decimal value, because such a decimal would never end. Indeed, the history of mathematics can in some ways be defined by this side-pursuit, this Holy Grail quest for a good formula for pi. Wallis was one of the first to find one, and his formula laid the groundwork for those that came later.

At Proposition 190, Wallis is still groping for the formula. He doesn’t know all the details yet, though he suspects that he’s approaching something magnificent. Indeed, he calls this elusive o a “slippery Proteus whom we have in hand, both here and above, frequently escaping and disappointing hope.” At times like this the Arithmetica sounds more like a personal journal than a textbook, which is the way Wallis likes to write about math, letting his exuberance for the material pour out unobstructed. When he includes, in the autobiography he wrote near the end of his life, the requisite information about wife and children, he does so only dutifully: “On March 4. 1644/5. I married Susanna daughter of John and Rachel Glyde of Northjam in Sussex; born there about the end of January 1621/2 and baptized Feb. 3 following. By whom I have (beside other children who died young) a Son and two Daughters now surviving.”

But when he talks about his first exposure to mathematics his tone changes completely. “One evening as we were sitting down to supper,” he writes, “a Chaplain of Sir William Waller shewed me an intercepted Letter written in Cipher.” It was a curious little puzzle, and it suited Wallis’ interests from the start. “It was about ten a clock when we rose from Supper,” he writes. “I then withdrew to my chamber to consider of it . . . In about 2 hours time (before I went to bed) I had deciphered it.”

It was the first time he put mathematics to use. Educated at Emmanuel College in Cambridge, he had studied Latin, Greek, Hebrew, Theology, and Logic before undergoing the Holy Orders. As an afterthought, he taught himself rudimentary mathematics from his younger brother’s trade books over the Christmas holiday. With that basic knowledge, Wallis would develop some limited renown for his skill in decoding, working for whatever political group was in power at the moment, by using arithmetic and the laws of numbers to translate letters of utmost state importance.

In 1649 his success led him to a professorship at Oxford, where “Mathematicks, which had before been a pleasing Diversion, was now to be my serious Study.” In 1652 he penned the first words of the Arithmetica, subtitling it, “a New Method of Inquiring into the Quadrature of Curves, and other more difficult mathematical problems.”

What Wallis was doing was moving from geometry to algebra. He was convinced that there was a better way to solve the great mathematical problems than with the circles and conics and outlandish shapes of the Greeks. He wanted to play with numbers. His realm was a pure Platonic one of symbols, ideas. Instead of defining the area of a triangle geometrically he wanted to do it with a series of numbers, added neatly together. That’s how he began, with Proposition 1. Then, in Proposition 3, he ran headlong into the ill-humored Hobbes.

Thomas Hobbes considered himself a terrific mathematician. In a description of his accomplishments written late in life, he wrote, in the third-person, “In mathematics, he solved some most difficult problems, which had been sought in vain by the diligent scrutiny of the greatest geometers since the very beginning of geometry.” In truth he was fairly inept—ridiculed on all sides in the mathematical world—and didn’t quite understand the nuances of more intricate mathematical concepts.

In mathematical philosophy, however, he was annoyingly sharp. It was over such a matter that the Wallis-Hobbes dispute broke out: when, in Proposition 3, Wallis writes, “For the triangle consists, as it were, of an infinite number of parallel lines in arithmetic proportion.” Hobbes had a problem with “as it were.” “’As it were’ is no phrase of a geometrician,” Hobbes scolds. The vague wording, complained Hobbes, belied a greater ill. In simple terms, how could width-less lines, even an infinite number of them, make up a finite shape? Philosophical differences on this small matter of infinity led to an increasingly hostile exchange of letters, whose cheeky titles ranged from, “Due Correction for Mr. Hobbes; or Schoole Discipline, for not saying his Lessons right,” to “Markes of the Absurd Geometry, Rural Language, Scottish Church-Politicks, And Barbarismes of John Wallis Professor of Geometry and Doctor of Divinity.”  

Though the essential mathematics of Hobbes’ challenges were entirely wrong, his qualms about Wallis’s conception of infinity still ring true. An example: if you stand one foot away from a red-brick wall, you are a foot away from it. If you go half the distance to the wall, you are half a foot away. Go half the distance again and you’re even closer. But do this forever and you’ll never hit the wall. That’s infinity.

Hobbes refused to wrap his head around the wall. “Whatsoever we imagine,” he wrote in Leviathan, “is Finite. Therefore there is no Idea, or conception of anything we call Infinite. No man can have in his mind an Image of infinite magnitude.”

Wallis, on the other hand, is perfectly happy to stick an infinite number of lines in a space two inches wide, to have numbers march on in sequence forever. He just doesn’t worry too much about “out there,” where the numbers get really big. He has a gut feeling that it’s not really necessary to be so exact, and ultimately he’s right: this is where we leave Hobbes, who can’t get over his stubborn adherence to principle.

Wallis’s willingness to fudge things, to keep working forward even if he’s not entirely sure what he’s doing, gets him through nearly 200 propositions, concerning a wild range of information on sequences and arithmetical-geometric connections. But now he has hit a dead end, because, around Proposition 167, he’s found this number, o, that keeps popping up in his wonderfully neat and interesting sequences. He doesn’t know what to do with it. It’s a number that defies expectation.

Imagine trying to think of something that doesn’t exist. Not something vaporous or philosophical, or extraterrestrial in its weirdness, but something that can’t be defined by our rules or systems. This is the difficulty with a number like pi: it cannot be described. It’s not a regular number, not an integer, like the amount of rolls you can buy at a bakery. You can’t get it by dividing two such numbers: it’s more irrational than that. You can’t even describe it with square roots or any such exponential notation, which was the extent of numerical inquiry at the time, after centuries of work. It is a strange number: natural, appearing everywhere, but indescribable, transcendental.

(It deserves mentioning here that we’ve skipped over thousands of years of history, from when the Egyptians built the pyramids and the legend that their measurements had pi in them for spiritual purposes. The point being that mathematics has always been tied up with the world around it, a world that Wallis says is full of “Changes and Alterations.” His is a period during which people were murdered for being Catholic and not Protestant. It is a time “when, by our Civil Wars,” Wallis says, “Academical Studies were much interrupted.” This was a little before the time when Wallis proved the existence of the trinity by the three dimensionality of the cube. All these things are swirling around while Wallis sat at his desk, half a lifetime removed from code-breaking and intrigues of state; half-insulated from the real world in this cocoon of a university; momentarily dropping his religious duties so he could grope for this sea prophet, which, when caught, would tell the future of mathematics in its never-ending succession: 3.1415926535897932384626433 . . .)

Again, here is Wallis, at his desk, at Proposition 190, stuck. He does the only thing he can do, which is keep going. He writes, “It will perhaps not be unwelcome to have put forward.” And then he goes into it, writing out his sequences, juggling them, massaging the information out. He knows what he is up against: “I am inclined to believe (what from the beginning I suspected) that this ratio we seek is such that it cannot be forced out in numbers according to any method of notation so far accepted.” He says, growing bolder, that “what arithmeticians usually do in their work, must also be done here; that is, where some impossibility is arrived at, which indeed must be assumed to be done, but nevertheless can not actually be done, they consider some method of representing what is assumed to be done, though it may not be done in reality.”

This is a wonderful, literary, confusing way of saying that he’s going to use what he knows (infinite sequences of bakery-roll-counting numbers) to describe what he doesn’t know (o). It’s an incredible moment when it happens—when he finally gets to “o =” –but somehow the excitement pales here, in words. One wonders if it’s the discovery, the pure math that’s exciting, or the history: the long search for pi, the march towards understanding, the debates over philosophy and the formative influences of politics and religion. Regardless, this is a moment of pure mathematical insight, a beautifully clean solution, and it made a man famous. One imagines that Wallis knows this as he writes down the last numbers. That he has done something, finally, important. Let’s just assume that when he’s done, he shoves some books from his desk, kicks his chair against the wall, and runs across the lawn to tell the Astronomy professor what has happened.