Hardy held to a regular routine. He read the London Times over breakfast, especially the cricket scores. He worked for four hours or so in the morning, then had a light lunch in Hall, perhaps played a little tennis in the afternoon. His career was well in place, his life comfortable, his future secure.
Then the letter came from India.
CHAPTER FIVE
“I Beg to Introduce Myself …”
[1913 to 1914]
1. THE LETTER
The letter, borne in a large envelope covered with Indian stamps, was dated “Madras, 16th January 1913,” and began:
Dear Sir,
I beg to introduce myself to you as a clerk in the Accounts Department of the Port Trust Office at Madras on a salary of only £20 per annum. I am now about 23 years of age. I have had no University education but I have undergone the ordinary school course. After leaving school I have been employing the spare time at my disposal to work at Mathematics. I have not trodden through the conventional regular course which is followed in a University course, but I am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as “startling.”
Some insignificant clerk in some backwater of an office five thousand miles away apparently sought to incite both pity and wonder. There was a nerviness about him: I have not trodden through the conventional regular course. By the second paragraph he was insisting he could give meaning to negative values of the gamma function. By the third he was disputing an assertion in a mathematical pamphlet Hardy had written three years before, part of a series called the Cambridge Tracts in Mathematics and Mathematical Physics.
It was called Orders of Infinity: The ‘Infinitarcalcul’ of Paul Du Bois-Reymond, and in it Hardy dealt with how mathematical functions can grow toward infinity more or less rapidly. For example, f(x) = x3 approaches infinity faster than g(x) = 3x. Both functions, as x grows larger, grow without bound; both, it can be crudely said, “reach” infinity. But the first does so more quickly than the second. By the time x = 100, for example, the first function has exploded to 1,000,000, while the second is still mired at 300. At one point, Hardy had cited a familiar mathematical expression from the theory of prime numbers. This expression consisted of, first, a term involving logarithms and, second, an error term, ρ(x), that simply represented how far wrong the first term was. On page 36, Hardy had asserted that “the precise order of ρ(x) has not been determined.”
Well, Ramanujan now wrote Hardy, it had been determined; he had determined it. “I have found an expression [for the number of prime numbers] which very nearly approximates to the real result, the error being negligible.” He was saying that the prime number theorem, as it was known in the mathematical world, and as it had first been given form by Legendre and then more precisely by Gauss, was inadequate and incomplete, and that he, an unknown Indian clerk, had something better.
This was the hook with which Ramanujan set out to snare Hardy’s attention. He concluded:
I would request you to go through the enclosed papers. Being poor, if you are convinced that there is anything of value I would like to have my theorems published. I have not given the actual investigations nor the expressions that I get but I have indicated the lines on which I proceed. Being inexperienced I would very highly value any advice you give me. Requesting to be excused for the trouble I give you.
I remain,
Dear Sir,
Yours truly,
S. Ramanujan
This was not, of course, the end of the matter, but the beginning. The “enclosed papers” to which Ramanujan referred went on for nine pages (and also probably included a copy of his published paper on Bernoulli numbers). The first page or so read like an inventor’s patent claim, and with the same almost rhythmic ring of brash certainty:
I have found a function which exactly represents the no. of prime nos. less than x, “exactly” in the sense that the difference between the function and the actual no. of primes is generally 0 or some small finite value even when x becomes infinite. I have got the function in the form of infinite series and have expressed it in two ways.
I have also got expressions to find the actual no. of prime nos. of the form An + B, which are less than any given number however large.
I have found out expressions for finding not only irregularly increasing functions but also irregular functions without increase (e.g. the no. of divisors of natural nos.) not merely the order but the exact form. The following are a few examples from my theorems… .
Now ordinary English virtually disappeared, giving way to the language of algebra, trigonometry, and calculus. There were theorems in number theory, theorems devoted to evaluating definite integrals, theorems on summing infinite series, theorems on transforming series and integrals, theorems offering intriguing approximations to series and integrals—perhaps fifty of them in all.
The whole letter, all ten pages of it, was written out in large, legible, rounded schoolboy script distinguished only by crossed t’s that didn’t cross. His handwriting had always been neat; but here, if possible, it was neater still, as if he realized the gulf of skepticism that divided him from Hardy and dared not let an illegible scrawl widen it.
It was a wise precaution; the gulf was indeed great. For Hardy, Ramanujan’s pages of theorems were like an alien forest whose trees were familiar enough to call trees, yet so strange they seemed to have come from another planet; it was the strangeness of Ramanujan’s theorems that struck him first, not their brilliance. The Indian, he supposed, was just another crank. He was forever getting bizarre manuscripts from strangers that, as his friend Snow later put it, “pretended to prove the prophetic wisdom of the Great Pyramid, the revelations of the Elders of Zion, or the cryptograms that Bacon had inserted in the plays of the so-called Shakespeare.”
And so, after a perfunctory glance, he put the manuscript aside and soon lost himself in the day’s London Times which, in late January 1913, told of opium abuse in China, port hands in Lisbon gone on strike, the French battling Arab rebels near Mogador, the House of Lords debating home rule for Ireland… . He may have skipped over the account of the Buxton divorce trial, where Mrs. Buxton was accused of adultery with Henry Arthur Mornington Wellesley, Lord Crowley. But he likely didn’t miss the news of England’s one-goal rugby victory over the French before twelve thousand spectators at Twickenham.
Around nine that morning, he set to work on mathematics, kept at it until about one, then ambled over to Hall for lunch. Then it was off to the university courts on Grange Road for a game of “real” tennis (which is what the English called the indoor variant that antedated the lawn tennis more popular today). But that day, in the corner of his mind normally left serene by vigorous athletics, something was wrong. The Indian manuscript scraped and tugged at his composure with, as Snow wrote, its “wild theorems. Theorems such as he had never seen before, nor imagined.”
Were they wild and unimaginable because they were silly, or trivial, or just plain wrong, with nothing to support them? Or because they were the work of some rare flower of exotic genius?
Or maybe they were merely well-known theorems the Indian had found in some book and cleverly disguised by expressing in slightly different form—making it just a matter of time before Hardy found them out?
Or perhaps it was all a practical joke? Hoaxes, after all, were much in vogue just then. Many Englishmen holding high positions in the Indian Civil Service had endured the mathematical Tripos or were otherwise versed in mathematics—well versed enough, perhaps, to pull off such a stunt. And how best to dupe your old Cambridge friend Hardy? Why, you’d garb familiar “theorems” in unfamiliar attire, purposely twist them into weird shapes. But who in India was adept enough to do it? Maybe the hoax had originated in Europe. But would the perpetrator have gone to the trouble of securing a genuine Madras postmark … ?
Vagrant, fragmentary thoughts like these bubbled through Hardy’
s head as, returning from tennis, he walked back across one of the Cam bridges, then over the expanse of lawn that was the Backs, and through the gateway into New Court. Back in his second-floor suite of rooms, which were built over one of the gateways, he again sat down with the letter from India. Outside the Gothic mullioned windows of his room, the winter light began to fade.
Years later, most of the formulas in Ramanujan’s letter would become the subjects of papers in the Journal of the London Mathematical Society and other mathematical journals. In them, their authors, including Hardy himself, would take two, or five, or ten pages to formally prove those not already known. But now, proving them wasn’t Hardy’s aim. Now he was content to see if there was anything to them at all. And even that was not apparent—in part because, as Hardy wrote later, “some curious specialization of a constant or a parameter made the real meaning of a formula difficult to grasp.” Roughly speaking, it was as if, instead of stating the aphorism “penny-wise, pound-foolish,” Ramanujan had for his own reasons expressed it as “two pennies wise, seven-and-a-half pounds foolish”—leaving the listener, distracted by the particulars, harder pressed to extract its meaning. In any case, it only compounded Hardy’s perplexity.
Darkness fell. It was almost time for dinner. The formulas grew no more straightforward, the quality of the man who had written them no clearer. Genius or fraud? You couldn’t idly riffle through these pages and tell. Yes, Hardy decided, Littlewood would have to see them, too.
• • •
John Edensor Littlewood was just two years older than Ramanujan, but while Ramanujan foundered in India, he had been mathematically schooled by England’s best.
He came from old English yeoman stock; Littlewood archers, it was said, fought at the Battle of Agincourt in 1415. More recently, his ancestors had been robustly middle-class professionals—ministers, schoolmasters, publishers, doctors and the like. Both his grandfather and father had studied mathematics at Cambridge; his grandfather became a theologian, his father headmaster of a school in South Africa, where John lived for eight years. Back in England when he was fourteen, he attended St. Paul’s School. There he caught the mathematics bug.
In his memoir, A Mathematician’s Miscellany, Littlewood would assert, without artifice or conceit, that he was a “prodigy.” Prodigy or not, he profited from an education that was everything Ramanujan’s was not. About the time Ramanujan was studying S. L. Loney’s Trigonometry, so was Littlewood. But whereas Ramanujan’s formal exposure to mathematical ideas ended there, Littlewood’s had just begun. Over the next three years he made his way through Macaulay’s Geometrical Conics, Smith’s Analytical Conics, Edwards’s Differential Calculus, Williamson’s Integral Calculus, Casey’s Sequel to Euclid, Hobson’s Trigonometry, Routh’s Dynamics of a Particle, Murray’s Differential Equations, Smith’s Solid Geometry, Burnside and Panton’s Theory of Equations, and more—all before so much as sitting for the Trinity College entrance scholarship exam in December 1902. By this time, Ramanujan had not even encountered Carr.
Two years later, in 1905, Littlewood was Senior Wrangler. But the fellowship normally his almost by right mysteriously went to someone else. For three years, he left to assume a lectureship at the University of Manchester. There, in “exile,” he endured an oppressive load of lecturing, conferences, and paperwork—the normal lot of faculty members at provincial universities. After that, beginning in 1910, he was back at Trinity for good.
In the words of two of his biographers, he was “a rough-hewn earthy person with a charm of his own.” He was strong, virile, vigorous. He had been a crack gymnast at school, had played cricket, would become an accomplished rock climber and skier and, even into his eighties, could be seen hiking through the East Anglian countryside around Cambridge. He wasn’t especially tall, but a photograph of him lecturing in academic robes suggests an enormous, hulking masculinity. Another of him and Hardy together shows Littlewood dominating the picture, hat planted firmly atop his head, slope-shouldered, feet spread as if ready for a fight; the smaller Hardy seems to recede into the background.
Like Hardy, Littlewood remained a bachelor. But unlike Hardy, he thoroughly enjoyed the company of women. While at Manchester, he was the best dancer in his group. He would write of his grandmother that she was “a remarkable woman from an able family, but unfortunately very saintly.” He was not. His long-term relationship with a married woman, with whose family he long shared a house in Cornwall, and with whom he had a daughter, would become well known in Cambridge.
As a mathematician, Littlewood was “the man most likely to storm and smash a really deep and formidable problem: there is no one else who can command such a combination of insight, technique, and power.” So said Hardy himself. The two men first met, at least intellectually, in 1906 when an early Littlewood paper to the London Mathematical Society sparked disagreement as to its merits and it went to Hardy as referee. When Littlewood returned to Trinity in 1910, he worked closely enough with him for Hardy to acknowledge his help in the preface to Orders of Infinity, the book that had caught Ramanujan’s eye in India.
Their actual collaboration began inauspiciously, when a proof they submitted to the London Mathematical Society in June 1911 turned out to be flawed. But then, the first of their more than one hundred papers appeared in 1912, and after that they were mathematically inseparable. “Nowadays,” somebody said later, “there are only three really great English mathematicians: Hardy, Littlewood, and Hardy-Littlewood.” 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, and in 1913, when Ramanujan’s letter arrived, it was natural that Hardy thought to show it to him.
Littlewood had recently moved into rooms on D Staircase of Nevile’s Court. Pausing in the arched doorway at the staircase’s base, he could sight through the portico to the arches across the court framed within it. It was an arresting view—perhaps one reason he remained there for sixty-five years, until his death in 1977. From Hardy’s rooms, it was just nineteen steps down the winding stone staircase, then forty paces through the gate into Nevile’s Court and around to D Staircase. Yet normally the two men communicated by mail or college messenger and did not, in any case, routinely run off to confer with one another in person.
And so, that winter evening in 1913, to let Littlewood know he wished to meet with him after Hall, Hardy sent word by messenger.
• • •
About nine o’clock, as Snow reconstructed the day’s events, they met, probably in Littlewood’s rooms, and soon the manuscript lay stretched out before them. Some of the formulas were familiar while others, Hardy would write, “seemed scarcely possible to believe.” Twenty years later, in a talk at Harvard University, he would invite his audience into the day that had so enriched his life. “I should like you to begin,” he said, “by trying to reconstruct the immediate reactions of an ordinary professional mathematician who receives a letter like this from an unknown Hindu clerk.” It was a mathematical audience, so Hardy introduced them to some of Ramanujan’s theorems. Like this one, on the bottom of page three:
The elongated S-like symbol appearing on the left-hand side of this equation, and in many other equations all through the letter, was an integral sign, a notation originating with Newton’s competitor Leibniz. An integral—the idea goes back to the Greeks—is essentially an addition, a sum, but one of a peculiar, precise, and, at first glance, infuriating kind.
Imagine cutting a hot dog into disclike slices. You could wind up with ten sections half an inch thick or a thousand paper-thin slices. But however thin you sliced it, you could, presumably, reassemble the pieces back into a hot dog. Integral calculus, as this branch of mathematics is called, adopts the strategy of taking an infinite number of infinitesimally thin slices and generating mathematical expressions for putting them back together again—for making them whole, or “integral.” This powerful additive process can be used to determine th
e drag force buffeting a wing as it slices through the air, or the gravitational effects of the earth on a man-made satellite, or indeed to solve any problem where the object is to piece together the contributions of many small influences.
You don’t need integral calculus to determine the area of a neat rectangular plot of farmland; you just multiply length times width. But you could use it. And you could use the same additive methods applicable to wings and satellites to calculate the area of an irregularly shaped plot where length-times-width won’t work. Furnish the function that mathematically defines its shape, and in principle you can get its area by “integrating” it—that is, by performing the additive process in a particular, precisely defined way.
Calculus books come littered with hundreds of ways to integrate functions. And yet, pick a function at random and chances are it can’t be integrated—at least not straightforwardly. With “definite integrals” like those Ramanujan offered in his letter to Hardy, however, you’re offered a back-door route to a solution.
A definite integral is “definite” in that you seek to integrate the function over a definite numerical range; the little numbers at top and bottom of the elongated S—the ∞ and 0 in Ramanujan’s equation—tell what it is. (In other words, you mark off a piece of the farm plot whose area you want reckoned.) When you evaluate a definite integral, you don’t wind up with a general algebraic formula (as you do with indefinite integrals) but, in principle, an actual number. And sometimes, by applying the right mathematical tools, you can determine this number without integrating the function first—indeed, without being able to integrate it at all.
The Man Who Knew Infinity Page 21
