Broadly, this was what Ramanujan was doing in the theorem on page 3 of his letter to Hardy and all through the section labeled “IV. Theorems on Integrals.”
This particular integral, he was saying, could be represented in terms of gamma functions. (The gamma function is like the more familiar “factorial”—4!, read “four factorial,” = 4 × 3 × 2 × 1—except that it extends the idea to numbers other than integers.) Hardy figured he could prove this theorem. Later he tried, and succeeded, though it proved harder than he thought. None of Ramanujan’s other integrals were trifling exercises, either, and all would wind up, years later, the object of papers devoted to them. Still, Hardy judged, these were among the least impressive of Ramanujan’s results.
More so were the infinite series, two of which were:
and
The first wasn’t new to Hardy, who recognized it as going back to a mathematician named Bauer. The second seemed little different. To a layman, in fact, it and kindred ones in Ramanujan’s letter might seem scarcely intimidating at all; save for pi and the gamma function they were nothing but ordinary numbers. But Hardy and others would show how these series were derived from a class of functions called hypergeometric series first explored by Leonhard Euler and Carl Friedrich Gauss and as algebraically formidable as anybody could want.
Sometime before 1910, Hardy learned later, Ramanujan had come up with a general formula, later to be known as the Dougall-Ramanujan Identity, which under the right conditions could be made to fairly spew out infinite series. Just as an ordinary beer can is made in a huge factory, the ordinary numbers in Ramanujan’s series were the deceptively simple end product of complex mathematical machinery. Of course, on the day he got Ramanujan’s letter, Hardy knew nothing of this. He knew only that these series formulas weren’t what they seemed. Compared to the integrals, they struck him as “much more intriguing, and it soon became obvious that Ramanujan must possess much more general theorems and was keeping a great deal up his sleeve.”
Some theorems in Ramanujan’s letter, of course, did look comfortably familiar. For example,
If αβ = π2, then
Hardy had proved theorems like it, had even offered a similar one as a mathematical question in the Education Times fourteen years before. Some of Ramanujan’s formulas actually went back to the days of Laplace and Jacobi a century before. Of course, it was quite something that this Indian had rediscovered them.
But now, then, what was Hardy to make of this one, which he found on the last page of Ramanujan’s letter?
then
This was a relationship between continued fractions, in which the compressed notation for, say, the function u actually means this:
The publication of this result some years hence would set off a flurry of work by English mathematicians. Rogers would furnish one ten-page proof for it in 1921. Darling would explore it, too. In 1929, Watson would approach it from a different angle, trying to steer clear of the tricky mathematical terrain of theta functions. But in 1913, Hardy could make nothing of it, classing it among a group of Ramanujan’s theorems which, he would write, “defeated me completely; I had never seen anything in the least like them before. A single look at them is enough to show that they could only be written down by a mathematician of the highest class.” And then, in a classic Hardy flourish, he added: “They must be true because, if they were not true, no one would have the imagination to invent them.”
As Hardy and Littlewood probed the theorems before them, trying to make out what they said, where they fit into the mathematical canon, and how they might be proved or disproved, they began to reach a judgment. That the Indian’s mathematics was strange and individual had been evident from the start. But now they were coming to see his work as something more. It was not “individual” in the way a rebellious teenager tries to be, camouflaging his ordinariness behind bizarre dress or hair. It was much more. “There is always more in one of Ramanujan’s formulae than meets the eye, as anyone who sets to work to verify those which look the easiest will soon discover,” Hardy would write later. “In some the interest lies very deep, in others comparatively near the surface; but there is not one which is not curious and entertaining.”
The more they looked, the more dazzled they became. “Of the theorems sent without demonstration, by this clerk of whom we had never heard,” one of their Trinity colleagues, E. H. Neville, would later write, “not one could have been set in the most advanced mathematical examination in the world.” Hardy would rank Ramanujan’s letter as “certainly the most remarkable I have ever received,” its author “a mathematician of the highest quality, a man of altogether exceptional originality and power.”
And so, before midnight, Hardy and Littlewood began to appreciate that for the past three hours they had been rummaging through the papers of a mathematical genius.
• • •
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,” as Jacobi had in writing Legendre on elliptic functions, or as Hermite had in writing Jacobi on number theory.
And yet, if Mordell was right—if “it is really an easy matter”—why had Gauss spurned Abel? Carl Friedrich Gauss was the premier mathematician of his time, and, perhaps, of all time. The Norwegian Niels Henrik Abel, just twenty-two at the time he wrote Gauss, had proved that some equations of the fifth degree (like x5 + 3x4 + … = 0) could never be solved algebraically. That was a real coup, especially since leading mathematicians had for years sought a general solution that, Abel now showed, didn’t exist. Yet when he sent his proof to Gauss, the man history records as “the Prince of Mathematics” tossed it aside without reading it. “Here,” one account has him saying, dismissing Abel’s paper as the work of a crank, “is another of those monstrosities.”
Then, too, if “it is really an easy matter,” why had Ramanujan’s brilliance failed to cast an equal spell on Baker and Hobson, the other two Cambridge mathematicians to whom he had written?
Certainly Henry Frederick Baker, forty-eight at the time he heard from Ramanujan, qualified as the kind of “leading authority” Mordell had in mind. He held a special Cayley lectureship. He had been elected a Fellow of the Royal Society, at the age of thirty-two, in 1898. He had received the Sylvester Medal in 1910. He had been president of the London Mathematical Society until the year before.
But as one biographer noted after his death, Baker “was little affected by the revolution brought about amongst the Cambridge mathematical analysts by G. H. Hardy in the first decade of the twentieth century. During this period Baker’s position was essentially that of one of the leaders of the older generation.” That he was immune to Hardy, of course, did not itself explain his indifference to Ramanujan; it did, however, suggest some reticence about embracing new ideas. Indeed, Baker was said to so revere the great mathematicians of the past that it choked his own originality. His upcoming second marriage, which took place in 1913, may also have left him less open to the importunings of an unknown Indian clerk.
The other Cambridge mathematician, a Senior Wrangler, was E. W. Hobson, who was in his late fifties when he heard from Ramanujan and more eminent even than Baker. His high forehead, prominent mustache, and striking eyes helped make him, in Hardy’s words, “a distinguished and conspicuous figure” around Cambridge.
But he was remembered, too, as a dull lecturer, and after he died his most important book was described in words like “systematic,” “exhaustive,” and “comprehensive,” never in language suggesting great imagination or flair. “An old stick-in-the-mud,” someone once called him. For some years, he was a Tripos coach (one student was John Maynard Keynes), largely ignoring math
ematical research. He would take a conventional stand on the coming war, and vehemently opposed granting degrees to women. These were sensibilities, then, hardly primed for unfamiliar theorems coming from an unorthodox source.
Of course, Ramanujan’s fate had always hung on a knife edge, and it had never taken more than the slightest want of imagination, the briefest hesitancy, to tip the balance against him. Only the most stubborn persistence on the part of his friend Rajagopalachari had gained him the sympathy of Ramachandra Rao. And Hardy himself was put off by Ramanujan’s letter before he was won over by it. The cards are stacked, against any original mind, and perhaps properly so. After all, many who claim the mantle of “new and original” are indeed new, and original—but not better. So, in a sense, it should be neither surprising nor reason for any but the mildest rebuke that Hobson and Baker said no.
Nor should it be surprising that no one in India had made much of Ramanujan’s work. Hardy was perhaps England’s premier mathematician, the beneficiary of the finest education, in touch with the latest mathematical thought and, to boot, an expert in several fields Ramanujan plowed… . And yet a day with Ramanujan’s theorems had left him bewildered. I had never seen anything in the least like them before. Like the Indians, Hardy did not know what to make of Ramanujan’s work. Like them, he doubted his own judgment of it. Indeed, it is not just that he discerned genius in Ramanujan that redounds to his credit today; it is that he battered down his own wall of skepticism to do so.
That Ramanujan was Indian probably didn’t taint him in Hardy’s eyes. True, Hardy’s knowledge of India may have been as mired in imperial stereotypes as that of most other Englishmen; when he was ten, back in Cranleigh, the school magazine had featured a day at “An Indian Bazaar,” rife with bejeweled maidens, dagger-bearing Ghurkas, and filthy fakirs cursing “English dogs” under their breaths. But by 1913, Hardy had already made mathematical contact with several Indians. A professor of mathematics at Allahabad, Umes Chandra Ghosh, had, in 1899, given the solution to one of his earliest questions in the Educational Times. And in 1908, another of Hardy’s questions, on infinite series, had drawn a response by V. Ramaswami Iyer, who two years before had founded the Indian Mathematical Society and two years later would befriend Ramanujan. (Then, too, the Indian cricket sensation Ranjitsinjhi had been in his prime when Hardy was an undergraduate, perhaps also helping to overturn any lingering prejudices.)
Growing up along Horseshoe Lane and attending Cranleigh School may have made Hardy readier than other Cambridge dons to see merit dressed in exotic garb. All his life, certainly, he was sympathetic to the underdog. Mary Cartwright, who met him a few years after his Ramanujan period, recalled that, as a woman mathematician, “I was a depressed class”—and so enjoyed Hardy’s favor. Snow wrote that Hardy preferred the downtrodden of all types “to the people whom he called the large bottomed: the description was more psychological than physiological… . [They] were the confident, booming, imperialist bourgeois English. The designation included most bishops, headmasters, judges, and … politicians.” When Hardy attended a cricket match and knew none of the competitors, he would tap his own favorites on the spot. These “had to be the under-privileged, young men from obscure schools, Indians, the unlucky and diffident. He wished for their success and, alternatively, for the downfall of their opposites.”
Further tipping the balance in Ramanujan’s favor was Hardy’s willingness to stray from safe, familiar paths. Hobson and Baker were both from an earlier generation, more settled, perhaps at a time in their lives when they were less eager to take on something new. Hardy, on the other hand, was a generation younger and had a penchant for the unorthodox and the unexpected. He had left familiar Cranleigh for Winchester. He had allowed a sixth-rate novelist, “Alan St. Aubyn,” to deflect him toward Trinity. He had weighed leaving mathematics; then, he had embraced Professor Love’s suggestion that he dip into Camille Jordan’s Cours d’analyse. He had joined the Apostles. He had broken precedent to take the Tripos after his second year instead of his third.
Each time Hardy had opened himself, he had come away enriched. Now, something wildly new and alien had presented itself to him in the form of a long, mathematics-dense letter from India. Once again, he opened his heart and mind to it. Once again, he would be the better for it.
2. “I HAVE FOUND IN YOU A FRIEND …”
“No one who was in the mathematical circles in Cambridge at that time can forget the sensation” caused by [Ramanujan’s] letter, wrote E. H. Neville years later. Hardy showed it to everyone, sent parts of it to experts in particular fields. (Midst all the excitement, Ramanujan’s original cover letter, along with one page of formulas, got lost.) Meanwhile, Hardy had sprung into action, advising the India Office in London of his interest in Ramanujan and of his wish to bring him to Cambridge.
It was not until a windy Saturday, February eighth, the day following his birthday, that Hardy sat down to deliver to Ramanujan the verdict on his gifts that Cambridge already knew. “Trinity College, Cambridge,” he wrote at the top, and the date, then began: “Dear Sir, I was exceedingly interested by your letter and by the theorems …” With an opening like that, Ramanujan would, at least, have to read on.
But in the very next sentence, Hardy threw out his first caveat: “You will however understand that before I can judge properly of the value of what you have done, it is essential that I should see proofs of some of your assertions.”
Proof. It wasn’t the first time the word had come up in Ramanujan’s mathematical life. But it had never before borne such weight and eminence. Carr’s Synopsis, Ramanujan’s model for presenting mathematical results, had set out no proofs, at least none more involved than a word or two in outline. That had been enough for Carr, and enough for Ramanujan. Now, Hardy was saying, it was not enough. The mere assertion of a result, however true it might seem to be, did not suffice. And all through his letter to Ramanujan he would sound the same insistent theme:
I want particularly to see your proofs of your assertions here. You will understand that, in this theory, everything depends on rigorous exactitude of proof.
And again:
assuming your proofs to be rigorous …
And:
Of course in all these questions everything depends on absolute rigour.
On the whole, Hardy’s letter was lavish with encouragement. True, some of Ramanujan’s theorems were already well known, or were simple extensions of known theorems. But even these, Hardy allowed, represented an achievement. “I need not say that if what you say about your lack of training is to be taken literally, the fact that you should have rediscovered such interesting results is all to your credit.”
Then, too, Hardy hazarded, some of Ramanujan’s results, while themselves of little note, were perhaps examples of general methods and thus more important than they seemed. “You always state your results in such particular forms that it is difficult to be sure about this.”
Littlewood was also intrigued by Ramanujan’s work, Hardy mentioned, even adding as a sort of appendix, “Further Notes Suggested by Mr. Littlewood.” Most of these dealt with Ramanujan’s work on prime numbers, a subject in which Littlewood had recently made a stunning, if not yet published, advance. So Hardy’s message from Littlewood carried an especially fervent plea: “Please send the formula for the no. of primes & …”—here it was again—“as much proof as possible quickly.”
Hardy’s whole letter was like that—shot through with urgency, with a barely contained excitement, that Ramanujan would have been dull indeed not to sense. At the bottom of page 6, Hardy wrote, “I hope very much that you will send me as quickly as possible”—he underlined it with a veritable slash across the page—“a few of your proofs, and follow this more at your leisure by a more detailed account of your work on primes and divergent series.”
And he went on: “It seems to me quite likely that you have done a good deal of work worth publication; and if you can produce satisfactory demonstration, I sh
ould be very glad to do what I can to secure it.”
• • •
Hardy’s letter probably arrived late in the third week of February. But his endorsement of Ramanujan had reached Madras earlier. Almost a week before writing Ramanujan, Hardy had contacted the India Office and, by February 3, a certain Mr. Mallet had already written Arthur Davies, secretary to the Advisory Committee for Indian Students in Madras. Later in the month, Davies met with Ramanujan and, at Sir Francis Spring’s behest, Narayana Iyer, apprising him of Hardy’s wish that Ramanujan come to Cambridge.
But as Hardy soon learned, Ramanujan wasn’t coming. Religious scruples, or a cultural resistance that verged on it, got in the way; Brahmins and other observant Hindus were enjoined not to cross the seas. And that, it seemed for a long time, was that.
Meanwhile, in Madras, the balance delicately poised since Ramanujan’s meeting with Ramachandra Rao in late 1910, teetering inconclusively between success and failure, now came down firmly on Ramanujan’s side. All that had been wanting was for a mathematician of unimpeachable credentials to weigh in with a verdict. Now Hardy had delivered it.
On February 25, Gilbert Walker was shown Ramanujan’s work. Walker was a former Senior Wrangler, a former fellow and mathematical lecturer at Trinity, and was now, at the age of forty-five, head of the Indian Meteorological Department in Simla. At the time of his appointment, he had no meteorological background whatever. But so crucial was predicting the onset of the monsoon in India, and so lively the press furor in the wake of several years of bad predictions, that it was thought a professional mathematician of Walker’s standing—he had just been named a Fellow of the Royal Society—might help defuse the situation.
Now, as Walker was passing through Madras, Sir Francis prevailed on him to look through Ramanujan’s notebooks. The next day, Walker wrote Madras University, asking it to support Ramanujan as a research student. “The character of the work that I saw,” he wrote the university registrar,
The Man Who Knew Infinity Page 22
