The Man Who Knew Infinity

by Robert Kanigel

  And now he got it. In 1915, no fewer than nine of his papers appeared, five of them in English journals, as against six—all but one of them in the Journal of the Indian Mathematical Society—during the previous twenty-seven years.

  The advantages he now enjoyed in England were hardly lost upon him. He had originally promised to return to India after two years. And when he wrote his friend Subramanian in June 1915, he still talked of returning to India the following year. But, in another letter the following month, he suggested his return might be only temporary. “I think it may be necessary,” he wrote, “to stay here a few years more as there is no help nor references in Madras for my work.”

  • • •

  For a while after he came to Cambridge, apparently, Ramanujan’s shyness was read as unfriendliness, and students sometimes taunted him. But by now he was a more popular, even legendary figure, his room accorded the status of a shrine. “It was a thrill to me to discover on reaching Cambridge in July 1915 that I was going to be a contemporary of Ramanujan,” recalled C. D. Deshmukh, until then a student at a college in Bombay. “Ramanujan’s current achievements were common talk amongst us Indian students.” Ananda Rao remembered Ramanujan at tea parties and other social gatherings, mixing freely among both English and Indians. Mahalanobis, with whom he took Sunday morning strolls, recalled him as reserved in large groups, expansive in small ones. In any case, he was the toast of the Indian students—the mathematical genius, the man whom the English had moved heaven and earth to bring to Cambridge. Two surviving photographs from this period show him in a group; in both he occupies the very center of the composition.

  Ramanujan mostly stayed in Cambridge during the long periods between terms. (The academic calendar came to only about twenty-two weeks a year.) But sometimes he’d get to London, where he visited the zoo or the British Museum. One time, he and his friend G. C. Chatterji went to see Charley’s Aunt, a farce about Oxbridge undergraduate life. In it, young Lord Fancourt disguises himself as Charley’s aunt from Brazil, the requisite chaperone for the girl Charley has invited to his rooms for lunch. Since its first performance in 1892, it had become a hardy annual of the London stage, full of wigs and fluttering eyelashes, of tea poured in hats, of silliness and comic confusion. It was, of course, just Ramanujan’s speed. He laughed until he cried.

  By mid-October of 1915, Ramanujan had moved from his rooms in Whewell’s Court to new ones on Staircase D of Bishop’s Hostel, just behind Great Court. This wasn’t the original Bishop’s Hostel built in 1669, whose sharply sloping roof and gabled windows loomed just outside his east window. Two red-brick structures had been built adjacent to the old building in 1878, on the site of what had once been the college stables, and Ramanujan’s rooms were on the second floor of one of them. He had one large sitting room, perhaps fourteen feet wide by twenty long, with a tiny bedroom and a small cooking area—actually the “gyp” room, a combination coal cellar, pantry, and kitchen—just off it.

  The layout was almost identical to his old rooms in Whewell’s Court. But now, Hardy was closer yet. Kumbakonam … Madras … Chestertown Road … Whewell’s Court … it was as if some relentless force drew him ever closer to Hardy. Now, in Bishop’s Hostel, all that divided them was a hundred paces.

  • • •

  Late in 1915, Ramanujan’s big paper on highly composite numbers, his most important body of work during that first year or so in Cambridge, appeared in the Proceedings of the London Mathematical Society. Back the previous June, Hardy had introduced his friends at the Mathematical Society to it. In November, a manuscript was ready. But revisions were required, and it didn’t reach final form until March 1915. Now, at last, it was in print.

  So long, and ranging over so vast a terrain, it was divided into discrete numbered sections and even had its own table of contents page to guide readers through it. Sometime earlier, Ramanujan had invited W. N. Bailey and another mathematician, S. Pollard, to see it. “He started at the beginning,” Bailey would recall, “and quickly turned over the pages as he explained the ideas and the arguments very briefly. Pollard wrestled manfully with the argument and was rewarded by a severe headache. I gave up the struggle earlier.”

  A composite number, recall, is a number that is not prime. The number 21 is composite, the product of 3 and 7. So is 22, whereas 23 is prime, the product only of itself and one. Now for each composite number, you can list all the numbers that divide it. For example, 21 has the divisors 1, 3, 7, and 21. For 22, it’s 1, 2, 11, and 22. Twenty-four can be divided by 1, or 2, or 3, or 4, or 6, or 8, or 12, or 24.

  And this last one, 24, was the kind of number with which Ramanujan’s paper dealt. Its eight divisors numbered more than those of any other composite number less than 24—and made it, in Ramanujan’s terminology, “highly composite.” Twenty-two has four divisors, 21 has four, 20 six. None less than 24 has even as many as seven, much less eight. A highly composite number, then, was in Hardy’s phrase “as unlike a prime as a number can be.” Ramanujan had explored their properties for some time; in the earliest pages of his second notebook he’d listed about a hundred highly composite numbers—the first few are 2, 4, 6, 12, 24, 36, 48, 60, 120—searching for patterns. He found them.

  The prime factors from which any composite number, N, was built could be written in the form,

  N = 2a2 × 3a3 5a5 …

  where a2, a3, a5, and so on are just the powers to which the prime numbers, 2, 3, 5 … , are raised. The highly composite number 24, for example, can be viewed as 23 × 31. In this notation, then, a2 = 3, and a3 = 1. Ramanujan found that, for any highly composite number, a2 was always equal to or larger than a3, a3 was always equal to or larger than a5, and so on. Count as high as you liked, you’d never find a highly composite number like N = 23 × 34 × … . Never. And he showed that, with two exceptions (4 and 36), the last an necessary to construct a highly composite number was always 1. He went on to prove these and other truths through fifty-two pages of reasoning that Hardy would term “of an elementary but highly ingenious character.”

  The problem Ramanujan had addressed, Hardy observed, “is a very peculiar one, standing somewhat apart from the main channels of mathematical research. But there can be no question as to the extraordinary insight and ingenuity which he has shown in treating it, nor any doubt that his memoir is one of the most remarkable published in England for many years.”

  • • •

  At Cambridge, every student had a tutor who looked after him and monitored his progress. Ramanujan’s was E. W. Barnes, who would call Ramanujan perhaps the most brilliant of all the top Trinity students (which included Littlewood) to have come before him. Though later to become a bishop in the Anglican church, Barnes was now a mathematician of some standing; he had been one of Andrew Forsyth’s earliest disciples, had with Hardy been among the leading advocates of Tripos reform, and had made substantial mathematical discoveries of his own. Now, in November 1915, he wrote Francis Dewsbury, registrar of the University of Madras, of Ramanujan’s progress, which he termed “excellent. He is entirely justifying the hopes entertained when he came here.” His two-year scholarship, soon coming to an end, ought to be “extended until, as I confidently expect, he is elected to a Fellowship at the College. Such an election I should expect in October 1917.” Ramanujan, he was saying, was in line to become a Fellow of Trinity.

  A few days later, Hardy also wrote Dewsbury, calling Ramanujan “beyond question the best Indian mathematician of modern times… . He will always be rather eccentric in his choice of subjects and methods of dealing with them… . But of his extraordinary gifts there can be no question; in some ways he is the most remarkable mathematician I have ever known.”

  In Madras, Sir Francis Spring joined the chorus, specifically requesting a two-year extension of Ramanujan’s scholarship. Through the late fall and early winter of 1915–1916, Madras authorities debated whether the scholarship should be extended for one year or two. If just one, Spring wrote Dewsbury, Ramanujan was
inclined not to return to India during the summer of 1916 as planned; for should his scholarship end in the spring of 1917, he’d be back in India just nine months later anyway.

  The university held, however, that, if the Trinity fellowship went through, its own scholarship would overlap. So one year, with the possibility of further extensions, it was.

  The Madras scholarship came to 250 pounds a year, which was supplemented by a 60-pound-per-year “exhibition” from Trinity. In 1914, the average English industrial worker made about 75 pounds per year. The threshold for paying income taxes—reached by less than 7 percent of the working population—was 160 pounds. So even with the 50 pounds he sent to support his family in India, Ramanujan was comfortably fixed—especially given what Hardy would call Ramanujan’s “almost ludicrously simple tastes.”

  Ramanujan had no official college duties. He could do as he pleased. He could immerse himself in mathematics without fretting over financial want, either his own or his family’s. Yet something still nagged at him—his lack of a degree, the tangible public marker of academic achievement. In his case, it was the merest formality. But he wanted it.

  Admission as a research student normally meant you already held a university diploma or certificate. But in his case, the requirement had been waived. And now, in March 1916, he received a B.A. “by research,” on the basis of his long paper on highly composite numbers. He’d put up his five pounds dissertation fee. He’d paid two pounds each to his examiners. And now, a dozen years and two college failures after leaving Town High School, he had his degree.

  In the early afternoon of March 18, Ramanujan posed for a photograph to mark the occasion with a group of students in their academic robes outside the Senate House. The shortest and stockiest of the lot, he stood squarely at attention, like an army recruit in boot camp, his mortarboard sitting flat atop his head. His trouser legs were a couple of inches too short. His suit bulged, its buttons straining.

  Whether because the scholarship had been extended for only a single year or for fear of the U-boats then ravaging British shipping, Ramanujan didn’t return to India in the spring. Over much of the next year, he continued instead to work with Hardy on a problem that would indissolubly link their two names in the annals of mathematics.

  That June, Hardy followed up his letter of the previous year to Dewsbury with an official report, his delight in telling of Ramanujan’s progress marred only by the war:

  In one respect Mr. Ramanujan has been most unfortunate. The war has naturally had disastrous results on the progress of mathematical research. It has distracted three-quarters of the interest that would otherwise have been taken in his work, and has made it almost impossible to bring his results to the notice of the continental mathematicians most certain to appreciate it. It has moreover deprived him of the teaching of Mr. Littlewood, one of the great benefits which his visit to England was intended to secure. All this will pass; and, in spite of it, it is already safe to say that Mr. Ramanujan has justified abundantly all the hopes that were based upon his work in India, and has shown that he possesses powers as remarkable in their way as those of any living mathematician.

  Hardy’s account of Ramanujan’s work was “necessarily fragmentary and incomplete,” he apologized. But

  I have said enough, I hope, to give some idea of its astonishing individuality and power. India has produced many talented mathematicians in recent years, a number of whom have come to Cambridge and attained high academical distinction. They will be the first to recognize that Mr. Ramanujan’s work is of a different category.

  The previous December, British, Australian, and New Zealand troops had suffered a catastrophic defeat at Gallipoli. U-boats continued to take their bloody toll of Allied ships. The machine guns chattered away in France. By the end of the year, lists of sometimes four thousand casualties per day would appear in the newspaper.

  For Ramanujan, in mid-1916, things could hardly have been brighter. But he, too, would ultimately be struck down by the war.

  CHAPTER SEVEN

  The English Chill

  [1916 to 1918]

  1. HIGH TABLE

  Beneath the seemingly unruffled surface of Ramanujan’s life in wartime England lay signs that his nerves were tautly strung, his sensitivities balanced on a hair trigger.

  Sometime probably in early 1916, he learned that his friend Gyanesh Chandra Chatterji, a twenty-one-year-old from the Punjab region who held a Government of India state scholarship to study in Cambridge, planned soon to marry. To help celebrate the good news, he invited Chatterji and his fiancée to dinner.

  Back in India, Ramanujan had probably never cooked in his life, had conceivably never even stepped into a kitchen. But here, with neither wife nor mother to serve him and unwilling to trust to the vegetarian purity of the college kitchen, he’d had to learn. Sometimes, on Sundays, he had Indian friends over for rasam, a thin peppery soup, or other South Indian fare. “Delicious,” a friend later recalled. And once, S. Kasturirangar Iyengar, owner and editor of South India’s preeminent English-language newspaper, the Hindu, visited him in Cambridge and lavished praise on the pongal, a lentil and rice dish, that Ramanujan served him.

  By now he took no little pride in his culinary skills and, to honor Chatterji and his fiancée, set about preparing them a feast.

  On the appointed day, Chatterji showed up at Ramanujan’s rooms in Bishop’s Hostel. With him was his fiancée Ila Rudra, a student at the local teacher’s college; and, probably as chaperone, Mrilani Chattopadhyaha, a woman from Hyderabad in her early thirties then studying ethics at Cambridge’s Newnham College, who would go on to become active in the Indian labor movement and run a school for untouchables.

  With his guests seated and his apartment awash in the cooking smells of South India, Ramanujan served soup. All was well.

  Did his friends wish to have more? asked Ramanujan after a time. They did indeed.

  Then: a third helping? Chatterji apparently accepted. But this time, his fiancée and the other woman declined. The dinner conversation continued… .

  Where is Ramanujan? Abruptly, they looked around and realized they were alone in Ramanujan’s apartment. Their host had vanished.

  For an hour, Chatterji and the others waited. Finally, he walked down the single flight of stairs, out the door, through the gate into Great Court, and across the cobbled courtyard to the porter’s lodge, where he inquired after Ramanujan. Why, yes, Chatterji was told, Mr. Ramanujan had called for a taxi.

  Perplexed, he returned to Ramanujan’s rooms and sat there with the others, waiting. Until ten o’clock they waited, the time by which guests had to leave. Still no Ramanujan.

  And no Ramanujan, either, the next morning, when Chatterji checked on him. For four days, he heard nothing, grew increasingly fearful. Then, on the fifth day he received a telegram from Oxford, about eighty miles away. Could Chatterji wire him five pounds? The amount was today’s equivalent of three or four hundred dollars.

  Next day, Ramanujan was back in Cambridge. “I felt hurt and insulted when the ladies didn’t take the food I served,” he told Chatterji. “I did not want to come [back] in while they were in the house.” He had needed to get as far away, as fast as he could; with the money in his pocket, that was Oxford.

  It was the same impulse, in the face of what he viewed as humiliation, that had driven him to Vizagapatnam ten years earlier. It was the rash, precipitous gesture of a man stretched thin.

  To all appearances, Ramanujan had made a splendid adjustment to a foreign country and an alien life. Mathematically, he had lived up to the fondest hopes of Hardy, Littlewood, Neville, and his other English friends. Socially, too, he had seemed to adapt, at least at first. Neville would tell of his delight in cracking jokes and discussing philosophy and politics, would speak approvingly of his simplicity and “instinctive perfection of manners.”

  But inside, Ramanujan was like a checking account from which funds are only withdrawn, never deposited. Doing mathematics took
vast personal energy. So did adjusting to his new life in England, as anyone will attest who has ever tried to penetrate a foreign culture. Together they drained his physical and emotional reserves. Eventually, the account must run dry.

  Sometime in 1916, perhaps around the time of the Chatterji dinner, it did. A constellation of forces had conspired to stretch thin his nerves, weaken him physically, isolate him socially. Indeed, the meal he prepared for Chatterji and his fiancée may have loomed far larger in his mind than it ever did in theirs. For Ramanujan normally went without mealtime company, unlike Hardy and the other Trinity fellows.

  • • •

  The college fellows in their black academic robes solemnly trooped into the great candlelit Hall. A webbed understructure of wooden arches graced the high ceiling. Dryden and Tennyson, Newton, Thackeray, Francis Bacon, and other Trinity notables glowered down from their portraits on the walls. Silver adorned bare wood tables; the tablecloths of peacetime were gone now for fear the light they reflected might beckon German zeppelins. Once all the fellows were seated, the senior among them, sitting at one end of the long table, recited a Latin grace. And the meal began.

  This was High Table, so called because the tables at which the fellows ate were set on a platform built up about four inches from the floor. At Trinity, as elsewhere in Cambridge, it was the focal point of the college’s social life, and long-standing notions of conversational good form governed it.

  For one thing, you never got too serious. “If one has done a hard day’s thinking one does not want to work at conversation,” Littlewood would say. “Dinner conversation is in fact easy and relaxed. No subject is definitely barred, but we do not talk shop in mixed company, and, Heaven be praised, we abstain from the important and boring subject of politics.” It was a place not of great profundity but of wit, wine, and release—release from the high tension of translating ancient Greek, or writing about the fall of Constantinople, or proving a new theorem in the theory of numbers. Here, the Important receded into distant memory. Here, trifles had their day.