CHAPTER TWO
Ranging with Delight
[1903 to 1908]
1. THE BOOK OF CARR
It first came into his hands a few months before he left Town High School, sometime in 1903. Probably, college students staying with Ramanujan’s family showed him the book. In any case, its title bore no hint of the hold it would have on him: A Synopsis of Elementary Results in Pure and Applied Mathematics.
In essence, the book was a compilation of five thousand or so equations, written out one after the other—theorems, formulas, geometric diagrams, and other mathematical facts, marching down the page, tied together by topic, with big, bold-faced numbers beside each for cross-reference. Algebra, trigonometry, calculus, analytic geometry, differential equations—great chunks of mathematics as it was known in the late nineteenth century, ranged not over a whole shelf of textbooks, but compressed within two modest volumes (the second of which Ramanujan may not have seen until later).
“The book is not in any sense a great one,” someone would later say of it, “but Ramanujan has made it famous.”
• • •
The Synopsis was a product of the genius of George Shoobridge Carr. Except that Carr was no genius. He was a mathematician of distinctly middling rank who for years tutored privately in London; the book was a distillation of his coaching notes.
Mathematics students in England during the late nineteenth century were preoccupied to the point of obsession with a notoriously difficult examination, known as the Tripos, one’s ranking on which largely determined one’s career. The Tripos system encouraged what educators today might deride as “teaching to the test,” and soon mathematicians would clamor for its reform. But back in the 1860s, in the period giving rise to the book, its hold went unchallenged. Not surprisingly, given the exam’s importance, armies of private tutors had arisen to coach students for it. Carr was one of them.
Carr himself had a peculiar academic history. Born in 1837 in Teignmouth, near where the Pilgrims sailed for the New World, he attended school in Jersey, a Channel Island off the French coast, and later University College School in London. At least by 1866, and perhaps earlier, he started tutoring. Apparently he took it quite seriously and was forever updating his notes, refining his teaching methods, developing mnemonics to help his pupils cover the vast range of material they were supposed to master.
Then, at thirty-eight, more in the modern style than was common at the time, he decided to go back to school. Admitted to Gonville and Caius College, Cambridge University, he received his B.A. in 1880, and then—four years shy of fifty—his M.A.
He was no star student. In the Tripos, he was classed among the “Senior Optimes,” not the higher-ranking “Wranglers,” and only twelfth among them. He knew he was not the brightest light in the firmament of English mathematics. 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”—presumably in making the original mathematical discoveries to which his intellect or temperament failed to suit him.
But while Carr as a mathematician was no more than normally bright, he had the enthusiasm and love of subject to teach it to those abler than himself. In any case, it was just about the time he was granted his Cambridge B.A. that, on May 23, 1880, from his desk in Hadley, outside London, he put the finishing touches on the first volume—a second appeared in 1886—of the Synopsis which would link his name to Ramanujan’s forever.
• • •
One strength of Carr’s book was a movement, a flow to the formulas seemingly laid down one after the other in artless profusion, that gave the book a sly, seductive logic of its own.
Take, for example, the first statement on the first page:
a2 − b2 = (a − b) (a + b)
This is, first of all, an equation. It says—any equation says—that whatever is on the left-hand side of the equals sign is equivalent to what’s on the right, as in 2 + 2 = 4. Only in this case, it’s not numbers, but symbols—the letters a and b—that figure in the equation. That they are symbols changes nothing. Some equations are true only when their variables take on certain values; the job, then, is to “solve” the equation, to determine those values—x = 3, say, or z = −8.2—that make it valid. But this one, an “identity,” is always true; whatever you make a and b, the statement still holds.
So, try it: Let a = 11, say, and b = 6. What happens?
Well, a + b is just 11 + 6, which is 17.
And a − b is 11 − 6, or 5.
Now, to set off quantities in parentheses, as they are in Carr’s equation, means just to multiply them—(a + b) and (a − b)—together. In this case, (17) (5) is just 17 × 5, or 85. That’s the right-hand side of the equation.
Now for the left. a2, of course, is just a times a, which is 11 × 11, or 121. b2 is 36. a2 − b2, then, is just 121 − 36, or 85. Which is just what the right-hand side of the equation comes to. Sure enough, the two sides match. The equation holds.
You could keep on doing this forever—verifying that the equation holds, with big numbers and little numbers, positive numbers and negative, fractions and decimals. You could do that, but who’d want to?
More sensible is to do what mathematicians do—prove the identity holds generally, for any a and any b. To do that, you dispense with particular numbers and manipulate instead the symbols themselves. You add and subtract the letters a and b, multiply and divide them, just as you would numbers.
In this case, the equation tells us to multiply (a − b) times (a + b). Doing that is about as simple as it looks. If you made $10 an hour, but then got a pay cut of $1 per hour, you could multiply the number of hours you worked by 10, then by 1, and subtract one product from the other. Or you could simply multiply the total hours worked by 9. Same thing. In this case, you can multiply the whole second term, (a + b), by a, then by b, and then subtract one product from the other. Or, symbolically,
(a − b) (a + b) = a(a + b) − b(a + b)
What now? Well, a(a + b) is just a2 + ab. And b(a + b) is just ba + b2. But ba (which means b × a) is just the same as ab. So we get:
(a − b) (a + b) = (a2 + ab) − (ab + b2)
In manipulating their equations, mathematicians often get caught in a clutter of numbers, letters, and symbols. And for the same reasons you do around the house, they periodically take time to tidy up—so they’re not forever stepping over mounds of mathematical debris, and so any attractive qualities of their mathematical habitat are shown off to best advantage. “Grouping like terms” is one form housecleaning takes; you cluster mathematical entities in their appropriate categories. You place dirty clothes in the laundry bin, freshly laundered napkins in the linen drawer, cereal back in the pantry. You put things where they belong.
In this case, we bring everything out from behind the parentheses, add up the a2 terms, and the b2 terms, and the ab terms. And when we do, something interesting happens. The ab terms cancel each other; they “drop out.” The + ab and the − ab add up to a grand total of zero, so that the quantity ab just disappears from the equation. Which leaves us with a2 − b2, which is just what’s on the left-hand side of the original equation—and just what we’re supposed to prove.
What this simple exercise demonstrates is a “proof” of sorts, though a mathematician might shudder at the claim. But at least on casual inspection, it seems that for any a and any b, the two sides of Carr’s equation are the same. We don’t have to check a = 735 and b = .0231. We know it will work because we proved the general case.
So much for Carr’s first equation. His second is this:
a3 − b3 = (a − b) (a2 + ab + b2)
Proving this differs little from proving the first. Working with the symbols, you multiply, add, and subtract, line up like terms—add apples to apples, and oranges to oranges, but never apples and oranges together—hope something cancels out, and soon are left with either side of the equals sign the same. Why, it’s
hardly worth the trouble to go through it… .
And right there, in the normal, natural—and appropriate—impulse to say it’s hardly worth the trouble, we gain a clue to Carr’s pedagogical wisdom (and to how mathematicians, generally, think). The second equation, though different from the first, resembles it, seems an extension or natural progression from it: In following one with the other, Carr was going somewhere. There was a direction, a development, not within the mathematical statements he set down but implicit within the order in which he set them down. The first equation dealt with a and b “raised to the second power,” in the form of a2 − b2; the second with a and b to the third power, as a3 − b3. What, one might now wonder, would be the equation for a4 − b4? Having worked out the first two, you’d suspect you could work it out easily, following the earlier examples. And you’d be right; the answer holds no surprises.
So Carr didn’t set it down at all. That would have been tedious, and trivial. Instead, he generalized:
an − bn = (a − b) (an − 1 + an − 2 b + … bn − 1)
This is the decisive step, for in taking it, the last ordinary number in the equation disappears. It’s not the second power to which a and b are raised this time, or the third, or the eighth, but the nth.
Abruptly, we are in a new world. It’s still simple algebra, but by daring to replace those safe 2s and 3s by the more mysterious n, the equation short-circuits routine mathematical manipulations. Now, you give me a number and I can just write out the equation, merely by substituting for the general n. The ellipsis appearing midway through the equation, the three little dots, just means you continue in the pattern the first two terms establish. O.K., so n = 8? Fine, plug it into Carr’s general equation, and the equation writes itself. Where Carr’s equation says n, you write 8. Where it says n − 1, you write 7, and so on.
As mathematicians might say, the equation with the n’s is more general than the previous two. Or, put another way, the first two equations were merely special cases of the third. Were Ramanujan not already familiar with it—and it’s inconceivable that he wasn’t—he could have confirmed it at a glance. Still, it suggests how he was guided through mathematical realms new to him; it wasn’t just the statements Carr made that counted, but the path he nudged the student along in making them.
And the way in which he set about proving them. Or, rather, not proving them.
In fact, Carr didn’t prove much in his book, certainly not as mathematicians normally do, and not even as we have here. Then, as now, the typical mathematics text methodically worked through a subject, setting out a theorem, then going through the steps of its proof. The student was expected to dutifully follow along behind the author, tracking his logic, perhaps filling in small gaps in his reasoning. “Oh, yes, that follows …” the student thinks. “Yes, I see …”
But mathematics is not best learned passively; you don’t sop it up like a romance novel. You’ve got to go out to it, aggressive and alert, like a chess master pursuing checkmate. And mechanically following a proof laid out by another hardly encourages that, leaves scant opportunity to bring much of yourself to it. Whatever its other merits, the trigonometry text by S. L. Loney that Ramanujan had sailed through a few years before had clung to the mold; it was a text you followed rather than one which demanded you cut your own path.
Carr’s was different.
There was no room for detailed proofs in the Synopsis. Many results were stated without so much as a word of explanation. Sometimes, a little note would be appended to the result. Theorem 245, for example, simply notes, “by (243), (244).” That is, one can arrive at the conclusion of no. 245 by extending the logic of 243 and 244. Theorem 2912 notes: “Proof—By changing x into πx in (2911).” In other words—mathematicians use this trick all the time—by an astute change of variable, the result assumes a clearer, more revealing form. In any case, Carr offered no elaborate demonstrations, no step-by-step proofs, just a gentle pointing of the way.
Scholars would one day probe Carr’s book, searching for the elusive mathematical sophistication that might have inspired Ramanujan. Some would point to how it covered, or failed to cover, this or that mathematical topic. Some would point to its unusually helpful index, others to its broad compass.
But in fact it’s hard to imagine a book more apt to influence a mathematically precocious sixteen-year-old, at least one like Ramanujan. For in baldly stating its results it almost dared you to jump in and prove them for yourself. To Ramanujan, each theorem was its own little research project. Or like a crossword puzzle, with its empty grid begging to be filled in. Or one of those irresistible little quizzes in popular magazines that invite you to rate your creativity or your sex appeal.
Nor was all this just an accident, or a by-product of the concision any compendium might demand; Carr had it in mind all along, and said so in the preface. “I have, in many cases,” he explained,
merely indicated the salient points of a demonstration, or merely referred to the theorems by which the proposition is proved… . The difference in the effect upon the mind between reading a mathematical demonstration, and originating one wholly or partly, is very great. It may be compared to the difference between the pleasure experienced, and interest aroused, when in the one case a traveller is passively conducted through the roads of a novel and unexplored country, and in the other case he discovers the roads for himself with the assistance of a map.
But it wasn’t even a map Carr supplied; rather, advice like, Once out of town, turn left.
A Western mathematician who knew Ramanujan’s work well would later observe that the Synopsis had given him direction, but had “nothing to do with his methods, the most important of which were completely original.” In fact, there were no methods, at least not detailed ones, in Carr’s book. So Ramanujan, charging into the dense mathematical thicket of its five thousand theorems, had largely to fashion his own. That’s what he now abandoned himself to doing. “Through the new world thus opened to him,” two of his Indian biographers later wrote, “Ramanujan went ranging with delight.”
2. THE CAMBRIDGE OF SOUTH INDIA
In 1904, soon after discovering Carr, Ramanujan graduated from high school and entered Kumbakonam’s Government College with a scholarship awarded on the strength of his high school work. He was an F.A. student, for First Arts, a course of study that, by years in school, might today correspond to an associates degree but in India, then, counted for considerably more.
From the center of town, the college was about a twenty-minute walk—along the street that ran by Town High, down to the Cauvery’s edge, then right, along the river to a point opposite the college. The bridge today spanning the river dates only to 1944; before that, a little boat ferried you across. Or else, you’d swim—a feat less daunting in March and April, when the river had dried to a trickle.
Government College was small, its faculty consisting of barely a dozen lecturers. And the best local students had begun to forsake it for larger schools in Madras. Still, for its time and place, it was pretty good—good enough, at any rate, to earn the moniker “the Cambridge of South India.” Its link to the great English university rested in part on the campus’s proximity to the Cauvery, which flowed beside it like the River Cam in Cambridge. But also playing a role was the repute of its graduates and the positions many of them held in South Indian life.
The year 1854 saw the college’s founding on land given by the maharani of Tanjore; you could still see the steps, leading down from the dressing cabin, that royal princesses took down to the river to bathe. Beginning in 1871, existing buildings were repaired and enlarged, new ones built. In the 1880s its last secondary classes were dropped, and it became a full-blown college. Its grounds were enlarged and landscaped. A gymnasium was built. While Ramanujan was there, a hostel for seventy-two students was going up, complete with separate dining facilities for Brahmins.
The college occupied a site of considerable natural beauty. The river streamed by. Birds chirped. Gro
ves of trees afforded shelter from the high, hot sun. Luxuriant vines crawled everywhere, forever threatening to overrun the college buildings. Even with the new construction since the maharani’s time, the college did not dominate its site but rather clung there, at nature’s sufferance. The place was lovely, idyllic, serene.
And the scene of Ramanujan’s first academic debacle.
• • •
One can only guess at the effects of a book like Carr’s Synopsis on a mediocre, or even normally bright student. But in Ramanujan, it had ignited a burst of fiercely singleminded intellectual activity. Until then, he’d kept mathematics in balance with the rest of his life, had been properly attentive to other claims on his energy and time. But now, ensnared by pure mathematics, he lost interest in everything else. He was all math. He couldn’t get enough of it. “College regulations could secure his bodily presence at a lecture on history or physiology,” E. H. Neville, an English mathematician who later befriended Ramanujan, would write, “but his mind was free, or, shall we say, was the slave of his genius.”
As his professor intoned about Roman history, Ramanujan would sit manipulating mathematical formulas. “He was quite unmindful of what was going on around him,” recalled one classmate, N. Hari Rao. “He had no inclination whatsoever for either following the class lessons or taking an interest in any subject other than mathematics.” He showed Hari Rao how to construct “magic squares”—tic-tac-toe grids stuffed with numbers which, in every direction, add up to the same quantity. He worked problems in algebra, trigonometry, calculus. He played with prime numbers, the building blocks of the number system, and explored them for patterns. He got his hands on the few foreign-language math texts in the library and made his way through at least some of them; mathematical symbols, of course, are similar in all languages.
One math professor, P. V. Seshu Iyer, sometimes left him to do as he pleased in class, even encouraging him to tackle problems appearing in mathematics journals like the London Mathematical Gazette. One day Ramanujan showed him his work in an area of mathematics known as infinite series; “ingenious and original,” Seshu Iyer judged it. But attention like that was rare, and Ramanujan’s intellectual eccentricities were, on the whole, little indulged. More typical was the professor from whom Ramanujan borrowed a calculus book who, once he saw how it interfered with Ramanujan’s other schoolwork, demanded its return. Even Seshu Iyer may not have been as solicitious as he later remembered; Ramanujan complained to one friend that he was “indifferent” to him.
The Man Who Knew Infinity Page 6