Whether bored by the long voyage or inspired by the prospect of soon breaking into the Mediterranean, Ramanujan grew expansive. From Suez, at the entrance to the canal, and Port Said at its other end, exactly two weeks out of Madras, he posted at least four letters back to India. One, to Viswanatha Sastri, bore stamps showing the pyramids. One went to R. Krishna Rao, nephew of Ramachandra Rao. “I do not know whether I have to go to Cambridge directly or stay at London and then go,” he wrote after telling of the voyage thus far. “I shall write to you after I reach England and everything is settled. My best compliments to your brother and respects and warmest thanks to your uncle.”
The next day, the Nevasa sailed from Port Said and steamed into the Mediterranean. On April 7, after a stop in Genoa, from which Ramanujan posted another letter home, she left Marseilles.
Then it was through the Strait of Gibraltar and up along the Spanish coast through the Bay of Biscay to England. The Nevasa docked first at Plymouth, then steamed up the English Channel and arrived at the mouth of the Thames on April 14.
• • •
It was a bright, lovely day, a little warmer than usual, without so much as a trace of overcast—more of the run of fine weather that, on Easter Sunday, two days before, had brought Londoners out to the parks and streets of the city by the hundreds of thousands. Now, waiting for Ramanujan at the dock was Neville and his older brother, who had arrived there by car. They drove to 21 Cromwell Road, in the South Kensington district of London, a reception center for Indian students just arrived in England.
London was a city of five million people, spilling over its ancient borders into the villages and hamlets of Surrey and Middlesex. In population, it was ten times larger than Madras. Madras was the capital of South India? Then London was the capital of the world, the nerve center from which the empire was directed. The clopping of horses’ hooves, the jingle of harnesses, and the clatter of hansom cabs over cobblestoned streets could still be heard in London, but these had begun to give way to the roar and smoke of Studebaker Cabriolets, of Wolsley “Torpedo Phaetons,” of double-decker buses bearing signs advertising Nestlé’s chocolate. London was fast. Even Hardy’s friend Leonard Woolf, who had returned to London in 1911 after seven years in Ceylon, found London aquiver, marching to “a tempo clearly faster and noisier than what I was accustomed to.”
Back in Madras, the Englishmen Ramanujan had known were mostly educated and upper-class, with accents to match, never seen to stoop to manual labor. Now, on the streets of London, he heard the nasal Cockney twang of rag merchants. He saw lamplighters who patrolled the streets at dusk with their long poles. He saw knife grinders manning little two-wheeled carts, men selling muffins who heralded their wares by ringing a bell. Here, there was every sort of Englishman—men in bowlers and flat workingmen’s caps, women in finery and in rags.
Just off the boat and England was already a strange new world for any Indian. Cromwell Road, to which Neville now took Ramanujan, was supposed to ease the transition. Of course it didn’t. The National India Association had offices there. Several rooms in the stately Georgian-styled corner building were available to students passing through. And across the street, the imposing edifice of the Natural History Museum, fairly glowing in its two-toned stone and adorned with sculpted griffins, lent an appropriately imperial luster to the immigrant’s first days in England. But Cromwell Road typically failed in its mission; a study a few years later would chide the reception center for invariably making things worse for newcomers, not better.
But unlike most Indians in England, Ramanujan had by his side, in Neville, a Cambridge don to help smooth the way. He also met A. S. Ramalingam, a twenty-three-year-old engineer from Cuddalore, south of Madras, who had been in England for four years and who also tried to help him feel at home. In any case, Ramanujan survived whatever rigors Cromwell Road could inflict and on April 18 went with Neville to Cambridge. Soon he was settled in Neville’s house on Chestertown Road, in a little suburb of Cambridge just across the River Cam from the town itself.
Chestertown Road was a street of fine townhouses, some turreted and cupolaed, their front yards typically set off from the street by wrought-iron fences. The bay-windowed Neville house sat one in from the end of the block on a gentle arc of street beside the river, occupying a peculiarly shaped plot just sixteen feet wide on the street side but fanning out to more than fifty in the back. Neville and his new wife, Alice, had moved in the year before and now, for two months in the spring of 1914, it was Ramanujan’s introduction to the English home.
It had been a modestly scaled two-story affair when first built around the middle of the previous century. Early on, though, it had been enlarged. A long, fingerlike projection, built of the same tan brick, jutted out the back of the house and followed the oddly angled property line. An added third story, with three more rooms, gave the house a height that made it like an exclamation point to the little group of houses at the end of the block.
In any case, the Neville house was by now quite spacious, and wherever they settled him in the sprawling place, Ramanujan was bound to have a measure of privacy he had never enjoyed before. The back of the house faced a large garden, which had once been a pear orchard. From the second-floor sitting room, Ramanujan could look out over the River Cam and Victoria Bridge and at the broad expanse of Midsummer Common with its crisscross of pedestrian paths leading to the old stone courts and cloisters of the colleges.
There was, of course, business to attend to—fees to pay, paperwork to work through. Hardy and Neville took care of most of it. A printed list of first-year students, prepared after Michaelmas Term (pronounced Mih-kel-miss, and starting in mid-October) of the previous year, listed each student alphabetically. Now, squeezing it in between Pugh, F. H. and Rawlins, J. D., someone dipped his pen in black ink and wrote in Ramanujan’s name by hand.
Those early weeks were rich with new promise, graced by a wondrous spring. Day after day the weather was lovely and warm. May flowers bloomed in April. Tracts of open countryside were transformed into great seas of bluebells. At the end of the month, King George visited Cambridge, where he was greeted by thousands of schoolchildren, waving tiny Union Jacks, trying to gain a glimpse of the royal Daimler.
Meanwhile, Ramanujan had already set to work with Hardy and Littlewood; Littlewood, for one, saw him about once a week, Hardy much more often. Ramanujan was productive, working hard, happy. “Mr. Hardy, Mr. Neville and others here are very unassuming, kind and obliging,” he wrote home in June.
Ramanujan had not come to Cambridge to go to school. But arriving in time for the Easter term, which began in late April, he did attend a few lectures. Some were Hardy’s. Others, on elliptic integrals, were given by Arthur Berry, a King’s College mathematician in his early fifties. One morning early in the term, Berry stood at the blackboard working out some formulas and at one point looked over to Ramanujan, whose face fairly glowed with excitement. Was he, Berry inquired, following the lecture? Ramanujan nodded. Did he care to add anything? At that, Ramanujan stood, went to the blackboard, took the chalk, and wrote down results Berry had not yet proved and which, Berry concluded later, he could not have known before.
Soon the word was getting out about Ramanujan. W. N. Bailey, then an undergraduate, heard “strange rumors that he had been unable to pass examinations, and that he had run away from such terrors. But apart from these rumors we only knew that his name was Ramanujan, and even this was pronounced wrongly,” probably Rah-ma-noo-jn. People didn’t often see him; he was usually busy in his rooms. But when they did, they noticed him—remembered his squat, solitary figure as, in the words of one, he “waddled” across Trinity’s Great Court, his feet in slippers, unable yet to wear Western shoes.
It was Hardy’s rooms in New Court to which Ramanujan was apt to be bound. This smaller quadrangle was “new” only by Cambridge standards, of course, having been built in 1823, two centuries or so after most of the rest of the college. Hardy lived on the second floor
of Staircase A, just over the portal leading out to the Avenue, a double row of two-hundred-year-old lime trees parading across the Backs. It was a long haul from the Nevilles’ to Hardy’s rooms—across the bridge at the far end of town to Midsummer Common, along one of several footpaths crossing it, and onto Park Parade; then by one or another old cobblestoned street to the Great Gate, and only then into the college itself. All in all, perhaps a twenty-minute hike to New Court at the far southwestern edge of the college.
That, apparently, was too far. In early June, after about six weeks on Chestertown Road, Ramanujan moved into rooms on Staircase P in Whewell’s Court. It would be “inconvenient for the professors and myself if I stay outside the college,” he wrote to a friend.
Probably, he was sad to leave the Nevilles. Neville was the first Englishman to win Ramanujan’s confidence and, from the moment Ramanujan disembarked from the Nevasa, had done much to ease his adjustment to English life. Then, too, if later accounts are any guide, he and his wife Alice were the consummate hosts, their hospitality a legend. They were young, liberal-minded, and by now had an emotional stake in Ramanujan. In all likelihood, they doted on him.
In Whewell’s Court, only about five minutes from Hardy’s rooms, Ramanujan could look out his window across to where Hardy had lived as an undergraduate twenty years before. But Ramanujan had more than twenty years’ worth of mathematical catching up to do. His education had ended, in a sense, when George Shoobridge Carr put the finishing touches on his Synopsis in 1886. And Carr’s mathematics was old when it was new, mostly barren of anything developed past about 1850.
Ramanujan, then, had much to learn. But, then again, so did Hardy.
2. TOGETHER
Together now in Cambridge, there was no longer the need for those long, awkward letters, across a gulf of culture and geography, with all their chance for misunderstanding. Now, as he would for the next few years, Ramanujan saw Hardy nearly every day and could show him the methods he had developed in India that he’d been loath to describe by international post. Meanwhile, Hardy had the notebooks themselves before him and, with their author by his side, could study them as much as he wished.
Many of the 120 theorems Ramanujan had sent him in those first two letters, Hardy could see now, had been plucked intact from the notebooks. Here, in chapter 5, section 30 of the second notebook, was what Ramanujan had written in the first letter about a class of numbers built up from “an odd number of dissimilar prime divisors.” From chapter 5 also came much of the work going into Ramanujan’s first published paper on Bernoulli numbers. In chapter 6 was that bizarre stuff from the first letter about divergent series that, Ramanujan had feared, might persuade Hardy he was destined for the lunatic asylum—the one where 1 + 2 + 3 + 4 + … unaccountably added up to −1/12. On its face, that was ridiculous; yet it sought to give meaning to divergent series—which at first glance added up to nothing more revealing or precise than infinity. But now Hardy found something like Ramanujan’s reasoning behind it, which involved a “constant” that, as Ramanujan wrote, “is like the center of gravity of a body”—a concept borrowed from, of all places, elementary physics.
A few of Ramanujan’s results were, Hardy could see, wrong. Some were not as profound as Ramanujan liked to think. Some were independent rediscoveries of what Western mathematicians had found fifty years before, or a hundred, or two. But many—perhaps a third, Hardy would reckon, perhaps two-thirds, later mathematicians would estimate—were breathtakingly new. Ramanujan’s fat, mathematics-rich letters, Hardy now saw, represented but the thinnest sampling, the barest tip of the iceberg, of what had accumulated over the past decade in his notebooks. There were thousands of theorems, corollaries, and examples. Maybe three thousand, maybe four. For page after page, they stretched on, rarely watered down by proof or explanation, almost aphoristic in their compression, all their mathematical truths boiled down to a line or two.
The notebooks would frustrate whole generations of mathematicians, who were forever underestimating the sheer density of mathematical riches they contained. In 1921, after having for seven years been exposed to them, Hardy would note that “a mass of unpublished material” still awaited analysis. Two years later, having devoted a paper to Ramanujan’s work in chapters 12 and 13 of his first notebook, on hypergeometric series, he had to report that those were, in fact, “the only two chapters which, up to the present, I have been able to subject to a really searching analysis.”
Around that time, Hardy was visited by the Hungarian mathematician George Polya, who borrowed from him his copy of Ramanujan’s notebooks, not yet then published. A few days later, Polya, in something like a panic, fairly threw them back at Hardy. No, he didn’t want them. Because, he said, once caught in the web of Ramanujan’s bewitching theorems, he would spend the rest of his life trying to prove them and never discover anything of his own.
In 1929, G. N. Watson, professor of pure mathematics at the University of Birmingham and a former Trinity fellow, and B. M. Wilson, who had known Ramanujan in Cambridge and was then at Liverpool University, set out on a mathematical odyssey through Ramanujan’s notebooks. Two years later, recounting their progress, Watson admitted the task was “not a light one.” A single pair of modular equations, for example, had taken him a month to prove. Yet Ramanujan was so rewarding, he wrote, that he and Wilson thought it “worth while to spend a fairly substantial fraction of our lives in editing the Note Books and making Ramanujan’s earlier discoveries accessible.” He estimated the job might take another five years. In fact, before his energy waned in the late 1930s, Watson had devoted most of a decade to the job, producing more than two dozen papers and masses of notes never published. (Wilson could give the project only fours years more; he died, after routine surgery, in 1935.)
In 1977, the American mathematician Bruce Berndt took up where Watson and Wilson left off. After thirteen years of work, having published three volumes devoted solely to the notebooks, he is still at it today, the task unfinished.
Plainly, then, in the months after Ramanujan arrived in England, Hardy and Littlewood could hardly have more than skimmed the surface of the notebooks, dipping into them at points, lingering over particularly intriguing results, trying to prove this one or simply understand that one. But this first glance was enough to reinforce the impression left by the letters. After the second letter, Littlewood had written Hardy, “I can believe that he’s at least a Jacobi.” Hardy was to weigh in with a tribute more lavish yet. “It was his insight into algebraical formulae, transformation of infinite series, and so forth, that was most amazing,” he would write. In these areas, “I have never met his equal, and I can compare him only with Euler or Jacobi.”
• • •
Euler and Jacobi were both towering figures on the stage of mathematical history. Leonhard Euler has been called “the most productive mathematician of the eighteenth century,” the author of almost eight hundred books and papers, many of them after he was blind, in every field of mathematics known in his day. It was Euler who, in his 1748 book, Introductio in analysin infinitorum, gave the trigonometric functions the form they have today. Today’s mathematics texts fairly spill over with Euler’s constant, and Euler’s criterion, and the Euler-Maclaurin formula, and Eulerian integrals, and Euler numbers. Born twenty years after Euler’s death, Karl Gustav Jacob Jacobi was nearly his equal in genius. The son of a Berlin banker, he pioneered elliptic functions and applied them to number theory. His name, too, is enshrined in mathematics, in Jacobi’s theorem and Jacobi’s polynomials.
But Euler and Jacobi were not just generic “great mathematicians”; it was not capriciously that Hardy and Littlewood had compared Ramanujan to them. Rather, these two men represented a particular mathematical tradition of which Ramanujan, too, was part—that of “formalism.” Formal, here, carries no suggestion of “stiff” or “stodgy.” Euler, Jacobi, and Ramanujan had (along with deep insight) a knack for manipulating formulas, a delight in mathematical form for its own s
ake. A “formal result” suggests one fairly bubbling up from the formulas themselves, almost irrespective of what those formulas mean. Computers today manipulate three-dimensional contours regardless of whether they represent economic forecasts or car bumpers. Some painters care as much for form, line, or texture as they do subject matter. The mind of the mathematical formalist works along similar lines.
All mathematicians, of course, manipulate formulas. But formalists were almost magicians at it, uncannily selecting just the tricks and techniques needed to obtain intriguing new results. They would replace one variable in an equation by another, thus reducing it to simpler form. They would know when to integrate a function, when to differentiate it, when to construct a new function, when to worry about rigor, when to ignore it.
But already by 1914, a faint odor of derision clung to them. For one thing, mathematicians of more finicky tastes clucked at how formalists sometimes steamrolled over certain mathematical niceties. By this light, they were holdovers from a prerigorous past whose ingenious formulas sometimes failed to stand the test of close reasoning.
But more, formalists were seen to inhabit a mathematical backwater. Useful formulas tended to be found early in the development of a new mathematical field, pointing the way to future work. But as a field matured and these early formulas were applied and extended, they often grew too complicated to be useful. By Ramanujan’s time, something like this had happened in branch after branch of mathematics.
The Man Who Knew Infinity Page 26