The Man Who Knew Infinity

by Robert Kanigel

  And so the smart money, as it were, had largely abandoned the search for formulas. The formalist came to be seen as one who stopped short of deep thinking and churned out narrow results, through mere conjurer’s tricks, that failed to turn over important new ground. His was a style of mathematics not so much profound as clever; that smacked not of High Science but Low Art—or Black Magic.

  Ramanujan’s mathematics, if it fit any category, fit this one. And yet, Hardy could see that if Ramanujan possessed conjurer’s tricks, they were ones of almost Mephistophelean potency. Ramanujan was a formalist who undermined the stereotypes. “It is possible that the great days of formulae are finished and that Ramanujan ought to have been born 100 years ago,” Hardy would write. But, he acknowledged, “He was by far the greatest formalist of his time,” one whose mathematical sleight of hand no one could match, and whose theorems, however he got them, later generations of mathematicians would esteem as elegant, unexpected, and deep.

  It was with some sense, then, of mingled mystery and awe that Hardy and Littlewood came away from their first long look at Ramanujan’s notebooks. “The beauty and singularity of his results is entirely uncanny,” Littlewood would comment in a review of Ramanujan’s papers published later. “Are they odder than one would expect things selected for oddity to be? The moral seems to be that we never expect enough; the reader at any rate experiences shocks of delighted surprise.”

  For Hardy’s part, confronting the mystery of Ramanujan’s mind would constitute, as his friend Snow had it, “the most singular experience of his life: what did modern mathematics look like to someone who had the deepest insight, but who had literally never heard of most of it?”

  Ramanujan “combined a power of generalization, a feeling for form, and a capacity for rapid modification of his hypotheses, that were often really startling, and made him, in his own peculiar field, without a rival in his day,” Hardy would conclude. As for his ultimate influence, Hardy couldn’t, at the time he wrote, say; in a sense, its very peculiarity undercut it. “It would be greater,” he suggested, “if it were less strange.”

  But, he added, “One gift it has which no one can deny—profound and invincible originality.”

  • • •

  Having gone so out on a limb to bring Ramanujan to Cambridge, Hardy, after familiarizing himself with the notebooks, probably felt a little relieved, too. And proud. “Ramanujan was,” he would write, “my discovery. I did not invent him—like other great men, he invented himself—but I was the first really competent person who had the chance to see some of his work, and I can still remember with satisfaction that I could recognize at once what a treasure I had found.”

  It didn’t take long to see that much in Ramanujan’s notebooks was well worth publishing. Editorial work was needed, of course; his results needed to be shaped, cast into lucid English, their notation made more familiar. Hardy, ever the mathematical journalist, now proceeded to do just that. “All of Ramanujan’s manuscripts passed through my hands,” he wrote, “and I edited them very carefully for publication. The earlier ones I rewrote completely.” (But, he added, he contributed nothing to the mathematics itself—an assertion made entirely credible by his readiness to take credit, in their jointly bylined papers, whenever he did contribute. “Ramanujan,” he wrote, “was almost absurdly scrupulous in his desire to acknowledge the slightest help.”) Now, in any event, Ramanujan’s bare notebook entries began to take new form, as mathematical papers fit to be seen and read by the world. By June, he and Hardy had the beginnings of two papers, one of which was ready enough to show around.

  The second Thursday of each month normally saw Hardy board the 2:15 P.M. train out of Cambridge to attend that evening’s London Mathematical Society meeting. When on one particular second Thursday—June 11, 1914—he greeted his friends at the society’s meeting room near Piccadilly, he bore a manuscript with theorems by Ramanujan. Among those there to hear them presented was Hobson, one of those to whom Ramanujan had written a year and a half before. Bromwich was there, too; his book on infinite series was the one Ramanujan had been so sternly advised to read. So was Professor Love, Hardy’s advisor from his undergraduate days. So was Littlewood.

  Ramanujan himself was not.

  • • •

  The following year, 1915, would see a flood of papers published by Ramanujan, including the one Hardy presented that June evening to the London Mathematical Society. But 1914, the year of his arrival in England, saw only one. Comprising mostly Indian work, it appeared in the Quarterly Journal of Mathematics under the title “Modular Equations and Approximations to Pi.”

  Every schoolchild knows that pi gives the ratio of a circle’s circumference to its diameter, about 3.14. Why waste time pursuing new ways to approximate it? Surely not for the sake of fixing it more precisely; even by the midnineteenth century, mathematicians had determined pi to five hundred decimal places, far in excess of any practical need. (Two Canadian mathematicians, the brothers Jonathan M. Borwein and Peter B. Borwein, have noted that thirty-nine decimal places will fix the circumference of a circle around the known universe to within the radius of a hydrogen atom.)

  But pi is not merely the ubiquitous actor in high school geometry problems; it is stitched across the whole tapestry of mathematics, not just geometry’s little corner of it. Since mathematicians find it more convenient to express angles in pi-based “radians” than in everyday degrees, pi occupies a key place in trigonometry, too. It is intimately related to that other transcendental number, e, and to “imaginary” numbers through Euler’s elegant relationship,

  eiπ = − 1

  which in a single, strange, beautiful statement of mathematical truth ties trigonometry and geometry to natural logarithms and thence to the whole world of “imaginary” numbers. Pi even shows up in the mathematics of probability. Drop a needle onto a table finely scored by parallel lines each separated by the length of the needle and the chance of its intersecting a line is 2/pi. Again and again pi pops up. So finding new ways to express it can reveal hidden links between seemingly disparate mathematical realms.

  Ancient societies were usually content to figure pi at, simply, 3. The seventh-century Indian mathematician Brahmagupta put it as the square root of 10, which is about 3.16. In the West, early efforts to define pi were pursued geometrically; you circumscribe circles, drop perpendiculars, bisect angles, draw parallels and wind up with pi as the length of some line segment. “Squaring the circle,” as this classic problem is called, turned out to be impossible. But Archimedes took another geometric approach and came up with a value of pi equal to between 310/70 and 310/71.

  In the midseventeenth century, the powerful tools of the calculus were brought to bear, leading to a variety of infinite series that converged to pi. Newton himself came up with one that gave pi to fifteen decimal places. “I am ashamed to tell you,” he confessed to a colleague, “to how many figures I carried these calculations, having no other business at the time.”

  Series yielding pi, or approximations to it, can be of surpassing grace, like this one, attributed variously to Leibniz, or the Scottish mathematician James Gregory, or to mathematicians in Kerala:

  John Wallis came up with this infinite product at about the same time:

  Another pretty one is:

  Thus pi, about as unruly a number as you can imagine (no pattern in its digits, even out to millions of decimal places, has ever been found), can be represented by series of the most appealing simplicity.

  Ramanujan’s early letters to Hardy included several such series approximations. Now, his twenty-three-page paper was filled with other routes to pi. Many rested on modular equations, a subject, going back to the work of the French mathematician Legendre in 1825, that he had exhaustively surveyed in his notebooks. Roughly, a modular equation relates a function of a variable, x, with the same function expressed in terms of x raised to an integral power (x3 or x4, for instance, but not x3.2). The trick, of course, is to find a function, f(x),
to satisfy it. Not surprisingly, they are rare. But when they do show up, it turns out, they often display special properties mathematicians can exploit. Ramanujan found that certain such functions, satisfying certain modular equations, gave solutions that, under certain circumstances, could be used to closely approximate pi.

  Some of his results, it turned out, had been anticipated by European mathematicians, like Kronecker, Hermite, and Weber. Still, Hardy would write later, Ramanujan’s paper was “of the greatest interest and contains a large number of new results.” If nothing else, it was astounding how rapidly some of his series converged to pi. Leibniz’s series, on page 209, is lovely—but almost worthless for getting pi; three-decimal-point accuracy demands no fewer than five hundred terms. Some of Ramanujan’s series, on the other hand, converged with astonishing rapidity. In one, the very first term gave pi to eight decimal places. Many years later, Ramanujan’s work would provide the basis for the fastest-known algorithm, or step-by-step method, for determining pi by computer.

  • • •

  During most of the ten years since he’d encountered Carr’s Synopsis, Ramanujan had inhabited an intellectual wilderness. In India, he’d been surrounded by family, friends, familiar faces. He was a South Indian among other South Indians, a Tamil-speaker among other Tamil-speakers, a Brahmin among other Brahmins. And yet, he was also a mathematical genius of perhaps once-in-a-century standing cut off from the mathematics of his time. He roused wonder and admiration among those, like Narayana Iyer and Seshu Iyer, who could glimpse into his theorems. Yet no one had been able to truly appreciate his work. He had been alone. He had had no peers.

  And now, in Littlewood and Hardy, he did have them. In Cambridge he had at last found an intellectual home, a community of mathematicians who saw in his work all that he saw in it. And, at least in the beginning, that more than made up for being a stranger in an alien land.

  The Cambridge into which he stepped that day in April 1914 was a Cambridge where the Playhouse, on Mill Road, played The Fatal Legacy, billed as “a grand and absorbing drama, with an exciting foxhunt.” A barbed-wire fence was going up on the Old Chestertown Recreation Ground to protect its green for the playing of bowls. And the big story in the paper was that Caesar, the late King Edward’s favorite dog, the Irish terrier who had accompanied his Royal Master everywhere, had died the previous Saturday.

  To a South Indian, the knives and forks the English used and in which Ramachandra Rao had tried to school Ramanujan back in India seemed like an invasion—hard metallic things penetrating the mouth. Feet long unconstrained by anything more than sandals felt pinched by shoes; it took months to get used to them. English names all ran together in a blur. And whether oval or square, topped with brown hair or blond, their faces seemed so alike in their essential whiteness; you could talk for hours with an Englishman yet fail to recognize him later on the street. Then, of course, a South Indian found that when he gave that little undulating jiggle of the head that back home meant something between a simple acknowledgment and a yes, the English were apt to take it as a no.

  Still, in the glow of Ramanujan’s arrival and his first few months, any feelings of homesickness, loneliness, or frustration must have been fairly swept under the emotional rug. Yes, wrote Neville, who witnessed his adjustment up close, “He felt the petty miseries of life in a strange civilization, the vegetables that were unpalatable because they were unfamiliar, the shoes that tormented feet that had been unconfined for twenty-six years. But he was a happy man, reveling in the mathematical society which he was entering and idolized by the Indian students.” He had all the money his tastes required. He had perfect leisure to pursue his work. In this old town of cobbled walks, grassy courts, and medieval chapels, whole universes away from Madras, Ramanujan had found a kind of intellectual nirvana.

  But then, the first cannon sounded.

  3. THE FLAMES OF LOUVAIN

  The Great War was both expected and unexpected.

  Everyone knew it was coming. All during 1913 and 1914, Europe seethed. The assassination of the Austrian crown prince Franz Ferdinand on June 28, 1914 set in motion a chain of events which ultimately proved irresistible. Germany and France, locked into alliances, declared war. When German armies bound for France swept through Belgium, violating its neutrality, England declared war, too. That came at 11 A.M. on August 4, 1914. All Europe was soon engulfed, Germany and the Central Powers marshaled against France, Britain, and the other Allied Powers.

  To many, the war was something hotly sought, a chance to burn off tensions built up over forty years of peace. Europe marched into war, flags flying, to the sound of martial music. The uniforms were fresh, the ranks and files still intact. The enemy would be taught its lesson and the war itself would be over in a month or two. The troops, somebody said, would be “home before the leaves fall.”

  That was the unexpected part, the war’s terrible surprise—that it was no brief but glorious orgasm of arms but rather ground along, month after month, year after cruel year. During August, German armies roared through Belgium, in obedience to the Schlieffen Plan with which they hoped to humble France. The French, with their Plan Seventeen, took a similarly offensive stance, aiming straight for Berlin. But hopes for rapid victory on both sides were dashed within the war’s first six weeks. The great armies met, fought, bled. Plans went wrong. And after the Battle of the Marne in September, trench warfare replaced great sweeping battlefield maneuvers. “Running from Switzerland to the Channel like a gangrenous wound across French and Belgian territory,” Barbara Tuchman wrote in The Guns of August, “the trenches determined the war of position and attrition, the brutal, mud-filled, murderous insanity known as the Western Front that was to last for four more years.”

  On September 11, while Germany and France grappled at the Marne, Ramanujan wrote his mother, reassuring her that “there is no war in this country. War is going on only in the neighboring country. That is to say, war is waged in a country that is as far as Rangoon is away from the city [Madras].”

  That wasn’t true; the war was much closer. In fact, Cambridge had already felt the war’s impact. In its first week, back in early August, the Sixth Division, from Ireland, converged on Cambridge and set up tents on Midsummer Common, just across from Neville’s house in Chestertown. Corpus Christi College became temporary headquarters for the Officers Training Corps; professors, undergraduates, and fellows all volunteered to help. At Trinity, the columned area under Wren’s great library was made into an open-air hospital; wooden boards laid on the uneven stone floor, to keep the beds level, now muffled the echoes that had been a fixture of those stone precincts for centuries. In the court’s northwest corner, near the winding staircase that climbed up to the library, bathrooms were installed. The south cloister, near where Littlewood lived, had lights strung from the ceiling and became an operating theater. Meanwhile, in Hardy’s New Court, rooms were made into offices.

  On August 14, an ambulance with a great red cross on its side bore the first wounded patient to what was, officially, the First Eastern General Hospital—the open-air hospital at Trinity. Later in the month, the Germans burned Louvain, and two British divisions retreated from positions they had defended along the Mons Canal in Belgium. In the nine hours of battle before the retreat began, the fighting cost sixteen hundred British casualties. Of the wounded, many were soon in Nevile’s Court, in double rows of beds under the library, where Henry Jackson, the college’s vice-master, a venerable classicist, would see them on his way to Hall.

  “We have a new Cambridge,” wrote Jackson, “with 1700 men in statu pupillari instead of 3600… . Medical Colonels and Majors and Captains dine in hall in khaki.”

  In early September, the Sixth Division left for France. But Cambridge was still crowded with men under arms and would remain the final training station for a whole succession of army divisions bound for the front. When it rained, horse-drawn gun carriages and other army vehicles churned up inches-deep mud on unasphalted roads.
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  On September 20, the Chapel Choir met on the lawn in the middle of Nevile’s Court to sing hymns to the wounded soldiers in the surrounding cloisters.

  In the early days of the war, jingoism had not yet been buried by cynicism. “The depravation of Germany—its gospel of iniquity and selfishness—is appalling,” wrote Jackson. “For though I never thought the Prussians gentlemen, I had a profound respect for their industry and efficiency, and I attributed to them domestic virtues. As it is, their good qualities subserve what is evil.” Henry Butler, master of Trinity, had no trouble believing “that the German infantry can neither shoot nor stand the bayonet. As to the last, they turn and run and get stuck in the back. Many of [the Nevile’s Court wounded] assure me that they have seen women and children driven in front of the enemy when they charge.”

  Feeling against Germany swelled. Even Ramanujan was caught up in it, writing his mother about the German advance across Belgium. “Germans set fire to many a city, slaughter and throw away all the people, the children, the women and the old.”

  • • •

  The popular English magazine Strand had long carried a page, entitled “Perplexities,” devoted to intriguing puzzles, numbered and charmingly titled, like “The Fly and the Honey,” or “The Tessellated Tiles,” the answers being furnished the following month. Each Christmas, though, “Perplexities” expanded, the author fitting his puzzles into a short story. Now, in December 1914, “Puzzles at a Village Inn” took readers to the imaginary town of Little Wurzelfold, where the main topic of interest was what had just happened in Louvain.

  In late August, pursuing an explicit policy of brutalization against civilian populations, German troops began burning the medieval Belgian city of Louvain, on the road between Liege and Brussels. House by house and street by street they set Louvain to the torch, destroying its great library, with its quarter million books and medieval manuscripts, and killing many civilians. The burning of Louvain horrified the world, galvanized public opinion against Germany, and united France, Russia, and England more irrevocably yet. “The March of the Hun,” English newspapers declared. “Treason to Civilization.” It was an early turning point of the war, doing much to set its tone. Louvain came to symbolize the breakdown of civilization. And now it reached even the “Perplexities” page of Strand.