while away in England, and with at least one letter to his father confined to reminders to keep up the house and not let the gutter run over, Ramanujan wrote his mother about the titanic struggle unleashed in Europe with the onset of the Great War, down to details of the number of men fighting, the width of the battle fronts, the use of airplanes in combat, and the contribution of Indian rajas to the British war effort.

The genius of Hinduism, then, was that it left room for everyone. It was a profoundly tolerant religion. It denied no other faiths. It set out no single path. It prescribed no one canon of worship and belief. It embraced everything and everyone. Whatever your personality there was a god or goddess, an incarnation, a figure, a deity, with which to identify, from which to draw comfort, to rouse you to a higher or deeper spirituality. There were gods for every purpose, to suit any frame of mind, any mood, any psyche, any stage or station of life. In taking on different forms, God became formless; in different names, nameless.

In the preface to the Synopsis he suggested that “abler hands than mine” might have done a better job with it, but that “abler hands might also, perhaps, be more usefully employed”—

his mind was free, or, shall we say, was the slave of his genius.”

A determination to succeed and to sacrifice everything in the attempt.

He was also a young man who hung around the house, who had flunked out of two colleges, who had no job, who indulged in mystical disquisitions that few understood, and in mathematics that no one did.

The word leisure has undergone a shift since the time Ramachandra Rao used it in this context. Today, in phrases like leisure activity or leisure suit, it implies recreation or play. But the word actually goes back to the Middle English leisour, meaning freedom or opportunity. And as the Oxford English Dictionary makes clear, it’s freedom not from but “to do something specified or implied” [emphasis added]. Thus, E. T. Bell writes of a famous seventeenth-century French mathematician, Pierre de Fermat, that he found in the King’s service “plenty of leisure”—leisure, that is, for mathematics.

The European “gentleman of leisure,” free from the need to earn a livelihood, presumably channeled his time and energy into higher moral and intellectual realms. Ramanujan did not belong to such an aristocracy of birth, but he claimed membership in an aristocracy of the intellect. In seeking “leisure,” he sought nothing more than what thousands

The European “gentleman of leisure,” free from the need to earn a livelihood, presumably channeled his time and energy into higher moral and intellectual realms. Ramanujan did not belong to such an aristocracy of birth, but he claimed membership in an aristocracy of the intellect. In seeking “leisure,” he sought nothing more than what thousands born to elite status around the world took as their due.

he proposed the formation of a mathematical society. Behind the idea lay simple want. Just as Ramanujan had so depended on whatever few mathematical books had come his way, so did Indian mathematicians generally suffer a lack of books and journals from Europe and America. The society, in Ramaswami’s conception, would subscribe to journals and buy books, then circulate them to members. Twenty-five rupees per year from even half a dozen members would be enough to get the society off the ground.

“Paper, The Great Immortalizer,” its one-page ad was headed. “Good Paper,” it went on, “has helped preserve and propagate the great thoughts of Man.”

years later, asked as part of a survey what most struck them about England, students from former Asian and African colonies invariably mentioned the sight of white men doing manual labor.)

Hardy was a cricket aficionado of almost pathological proportion. He played it, watched it, studied it, lived it. He analyzed its tactics, rated its champions. He included cricket metaphors in his math papers. “The problem is most easily grasped in the language of cricket,” he would write in a Swedish mathematical journal; foreigners failed to grasp it at all.

Hardy judged God, and found Him wanting.

C. P. Snow once reported that the longer you spent in Einstein’s company, the more extraordinary he seemed;

He does not look relaxed; in no photograph of Hardy does he ever look relaxed. Always there’s that haunted look in his eyes, like “a slightly

He does not look relaxed; in no photograph of Hardy does he ever look relaxed.

Over the years, the school graduated more than its share of those whom one school historian could aptly class as “gentlemanly rebels and intellectual reformers.” And twenty years later, it would be fair to describe Hardy as one of them. Winchester didn’t try to, of course; its more usual products were reserved, patrician, conservative social and political leaders. But those who did rebel often became distinguished rebels.

History of the English-Speaking Peoples.

“I shall never forget,” Hardy later wrote of Jordan’s book, whose second, much-improved edition had just appeared in 1896, “the astonishment with which I read that remarkable work, the first inspiration for so many mathematicians of my generation, and learnt for the first time as I read it what mathematics really meant.”

At the time Hardy joined, the Apostles had not yet reached the point where, as Duncan Grant would put it, “even the womanisers pretend to be sods, lest they shouldn’t be thought respectable.”

That Hardy himself was at least of homosexual disposition is scarcely in doubt. No woman, aside from his mother and sister, played the slightest substantive role in his life. And he had numerous male friends of whom he was passionately fond. In 1903, for example, he shared a double suite of rooms at Trinity with R. K. Gaye,

Littlewood, who worked with him for almost forty years, called him “a non-practicing homosexual.”

one senses him at the edge of their world, not its center. Being a mathematician, and a pure mathematician at that, may have isolated him; within the Shakespeare Society, for example, he was gently ribbed for putting “his knowledge of higher mathematics” to use in calculating the tab for a recent dinner at five shillings, one penny—an “alarming sum.”

among Havelock Ellis’s thousand or so British “geniuses,” 26 percent never married. In the academic and intellectual circles of which Hardy was a part, such a monastic sort of life actually represented one pole of common practice.

About the only time Hardy and other fellows encountered women was among the bedmakers who tidied up college rooms—and they were said to be selected for their plainness, age, and safely married status, presumably so as to minimize the distraction they represented to students and fellows of the colleges.

Despite suggestive evidence, then, one cannot conclude that Hardy was a practicing homosexual. And yet, in one sense, it doesn’t matter.

Cambridge didn’t offer the doctorate, a German innovation, until after World War I, hoping to lure Americans otherwise drawn to Germany.) In 1906, be became a

Cambridge didn’t offer the doctorate, a German innovation, until after World War I, hoping to lure Americans otherwise drawn to Germany.)

Around this time, a pupil of E. W. Barnes, director of mathematical studies at Trinity, sought Barnes’s advice about what lectures to attend. Go to Hardy’s, he recommended. The pupil hesitated. “Well,” replied Barnes, “you need not go to Hardy’s lectures if you don’t want, but you will regret it—as indeed,” recalled the pupil many years later, “I have.”

Thought, Hardy used to say, was for him impossible without words. The very act of writing out his lecture notes and mathematical papers gave him pleasure, merged his aesthetic and purely intellectual sides. Why, if you didn’t know math was supposed to be dry and cold, and had only a page from one of his manuscripts to go on, you might think you’d stumbled on a specimen of some new art form beholden to Chinese calligraphy. Here were inequality symbols that slashed across the page, sweeping integral signs an inch and a quarter high, sigmas that resonated like the key signatures on a musical staff. There was a spaciousness about how he wrote out mathematics, a lightness, as if rejecting

Thought, Hardy used to say, was for him impossible without words. The very act of writing out his lecture notes and mathematical papers gave him pleasure, merged his aesthetic and purely intellectual sides. Why, if you didn’t know math was supposed to be dry and cold, and had only a page from one of his manuscripts to go on, you might think you’d stumbled on a specimen of some new art form beholden to Chinese calligraphy. Here were inequality symbols that slashed across the page, sweeping integral signs an inch and a quarter high, sigmas that resonated like the key signatures on a musical staff. There was a spaciousness about how he wrote out mathematics, a lightness, as if rejecting the cramped, ungenerous formalities of the printed notation.

Like everything else Hardy ever wrote, his textbook was readable. This was not simply page after gray page of formula. His were real explanations of difficult ideas presented in clear, cogent English prose.

Nowadays,” somebody said later, “there are only three really great English mathematicians: Hardy, Littlewood, and Hardy-Littlewood.”

some—at least in jest—doubted he existed at all. But exist he did, and in 1913, when Ramanujan’s letter arrived, it was natural

Because Littlewood disdained bright, sparkling company and stayed away from mathematics conferences, some—at least in jest—doubted he existed at all. But exist he did,

it soon became obvious that Ramanujan must possess much more general theorems and was keeping a great deal up his sleeve.”

“They must be true because, if they were not true, no one would have the imagination to invent them.”

It wasn’t the first time a letter had launched the career of a famous mathematician. Indeed, as the mathematician Louis J. Mordell would later insist, “It is really an easy matter for anyone who has done brilliant mathematical work to bring himself to the attention of the mathematical world, no matter how obscure or unknown he is or how insignificant a position he occupies. All he need do is to send an account of his results to a leading authority,”

many who claim the mantle of “new and original” are indeed new, and original—but not better.

But otherwise, contact between them was slight. Janaki was barely fourteen. He worked in the heady realms of pure mathematics, whereas she, beyond simple Tamil, had no education at all. Occasionally, when he took a break, he would ask her to later jog his memory with language like, “the one you were working on downstairs,” or “the one you were working on before eating.” But intellectually, they could share nothing. Nor did he try to force-feed her. She didn’t ask him to, and he didn’t volunteer.

this time Ramanujan furnished proofs for many of his assertions, more analytical mathematicians would later shoot them full of holes. Yet the results themselves—the theorems Ramanujan offered as true—were true. Bruce Berndt, an American Ramanujan scholar, would see in that curious split a message for mathematicians today: “We might allow our thoughts to occasionally escape from the chains of rigor,” he advised, “and, in their freedom, to discover new pathways through the forest.”

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.

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?”

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.

Mahalonobis was astounded. How, he asked Ramanujan, had he done it? “Immediately I heard the problem it was clear that the solution should obviously be a continued fraction; I then thought, Which continued fraction? And the answer came to my mind.”

“There are regions of mathematics in which the precepts of modern rigour may be disregarded with comparative safety,” Hardy would write, “but the Analytic Theory of Numbers is not one of them.”

Hardy, whose insistence on rigor had sent him off almost single-handedly to reform English mathematics and to write his classic text on pure mathematics; who had told Bertrand Russell two years before that he would be happy to prove, really prove, anything: “If I could prove by logic that you would die in five minutes, I should be sorry you were going to die, but my sorrow would be very much mitigated by pleasure in the proof.”

Ramanujan, Intuition Incarnate, had run smack into Hardy, the Apostle of Proof.

the proof was made to seem the culmination of a hundred closely reasoned steps ranging over a dozen pages. There, mathematics could seem no more than a neat lockstep march to certainty, B following directly from A, C from B, . . . Z from Y. But no mathematician actually worked that way; logic like that reflected the demands of formal proof but hinted little at the insights leading to Z. Rather, as Hardy himself would write, “a mathematician usually discovers a theorem by an effort of intuition; the conclusion strikes him as plausible, and he sets to work to manufacture a proof.”

The theorem itself was apt to emerge just as other creative products do—in a flash of insight, or through a succession of small insights, preceded by countless hours of slogging through the problem. You might, early on, try a few special cases to informally “prove” the result to your own satisfaction. Then later, you might go back and, with a full arsenal of mathematical weapons, supply the kind of fine-textured proof Hardy championed. But all that came later—after you had something to prove.

Rigor, Littlewood would observe, “is not of first-rate importance in analysis beyond the undergraduate state, and can be supplied, given a real idea, by any competent professional.” Given a real idea—that was the rare commodity.

systematically and publish it,” he wrote Narayana Iyer in November

After years as a mathematician known only to himself, then only to Madras, Ramanujan plainly relished the prospect of appearing in prestigious English mathematics journals. To appear in print was the only tangible sign of recognition you could hold up to family and friends, the only way the world would know what you’d done.

Pollard wrestled manfully with the argument and was rewarded by a severe headache. I gave up the struggle earlier.”

“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.

Hardy had been among several to declare in 1909 that Berlin pudding and Hirsch-horn fritters ought never appear again on the menu. Later, he and Littlewood defended plum pudding with wine and camperdown sauce against those who would see it blacklisted; over the days and weeks, alliances were built around the issue, compromises forged. At another point, Hardy was embroiled in a battle over whether fruit pies should be served hot, subscribing to the view that, as another fellow put it, “a man who will eat a hot fruit pie is unfit for decent society.”

“One evening, when dining at a club, I tried in my innocence to open a conversation across the table, and I admired the skill with which the intrusion was fended off without the slightest suggestion of discourtesy.” Fended off, though, it was.

As usual, it was Euler who made the first real dent in the problem.

Once, when the mathematician Louis J. Mordell wrote him, complaining that his papers were getting picked apart by the editors of a mathematical journal because of minor stylistic faults, Hardy refused to indulge him. “I know I have spent over three hours over the journal proofs of a note of yours,” he wrote back, “and have made over thirty corrections on a page. All ‘trivialities’—so trivial that you have never noticed them, or at any rate commented on them: but a morning’s work gone west.” The remainder of Hardy’s eleven-page letter showed similar irritation. Don’t be so easy on yourself, it said.

The work awaits you.

Ramanujan, sick and in the hospital, was apologizing to Hardy for having failed to do more mathematics!

After three years in Cambridge, his life was Hardy, the four walls of his room, and work. For thirty hours at a stretch he’d sometimes work, then sleep for twenty.

Back in India, Janaki later recalled, Ramanujan would sometimes abruptly stop eating, or else hurry through it, to pursue a mathematical thought; meals were something to be dispensed with.

Ramanujan sometimes cooked only once a day, or sometimes only once every other day, and then at weird hours in the early morning.

“I am not in need of anything as I have gained a perfect control over my taste and can live on mere rice with a little salt and lemon juice for an indefinite time.”

Ramanujan, in the language of the Polish emigré mathematician Mark Kac, was a “magician,” rather than an “ordinary genius.” An ordinary genius is a fellow that you and I would be just as good as, if we were only many times better. There is no mystery as to how his mind works. Once we understand what he has done, we feel certain that we, too, could have done it. It is different with the magicians. They are, to use mathematical jargon, in the orthogonal complement of where we are and the working of their minds is for all intents and purposes incomprehensible. Even after we understand what they have done, the process by which they have done it is completely dark.

Despite his emphasis on rigor, G. H. Hardy was not blind to the virtues of vague, intuitive mental processes in mathematics. Bromwich, he would write, for example, “would have had a happier life, and been a greater mathematician, if his mind had worked with less precision.

Almost the only time he did write about mathematical creativity came many years later, near the end of his life, in a review of a book by Jacques Hadamard, The Psychology of Invention in the Mathematical Field. What philosophers or poets could say about the creative process in mathematics, Hardy felt, was next to nothing. But Hadamard was a mathematician, and a great one. He had run up the highest score ever recorded on the entrance examination to the Ecole Polytechnique, France’s premier school of science. He had, with the Belgian Charles J. de la Vallée-Poussin, proved the prime number theorem. What he had to say about “invention in the mathematical field” was worth listening to. Hardy agreed with Hadamard that unconscious activity often plays a decisive part in discovery; that periods of ineffective effort are often followed, after intervals of rest or distraction, by moments of sudden illumination; that these flashes of inspiration are explicable only as the result of activities of which the agent has been unaware—the evidence for all this seems overwhelming.

I have often been asked whether Ramanujan had any special secret; whether his methods differed in kind from those of other mathematicians; whether there was anything really abnormal in his mode of thought. I cannot answer these questions with any confidence or conviction; but I do not believe it. My belief is that all mathematicians think, at bottom, in the same kind of way, and that Ramanujan was no exception.

Narasimhan; the two of them had set out for Bombay on the twenty-first. But not his wife. Why fret over Janaki? sniped Komalatammal.

intellectual cousin,

Ramanujan’s work during the year before he died could be seen to support an old nostrum of the tuberculosis literature—that the tuberculous patient, as he succumbed, was driven to an ever-higher creative pitch; that approaching death inspired a final flurry of creativity impossible during normal times.

If I could attain every scientific ambition of my life, the frontiers of the Empire would not be advanced, not even a black man would be blown to pieces, no one’s fortunes would be made, and least of all my own. A pure mathematician must leave to happier colleagues the great task of alleviating the sufferings of humanity.

Ramanujan was a man for whom, as Littlewood put it, “the clear-cut idea of what is meant by proof . . . he perhaps did not possess at all”; once he had become satisfied of a theorem’s truth, he had scant interest in proving it to others. The word proof, here, applies in its mathematical sense. And yet, construed more loosely, Ramanujan truly had nothing to prove. He was his own man. He made himself.

“I did not invent him,” Hardy once said of Ramanujan. “Like other great men he invented himself.” He was svayambhu.