The Clockwork Universe Part 8

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The crucial belief of Isaac Newton and his fellow scientists was that G.o.d had designed the world on mathematical lines. All nature followed precise laws. The belief derived from the Greeks, who had been amazed to find that music and mathematics were deeply intertwined. Here they explore the relation between the weight of a bell, or the volume of a gla.s.s, and its pitch.Pythagoras, shown below, is credited with being the first to find this connection, "one of the truly momentous discoveries in the history of mankind."

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Newton's theory of gravity propelled him to instant fame. He liked to tell the story, depicted in this j.a.panese print, that the crucial insight came from watching an apple fall. The story is quite likely a myth. Before anyone knew of his mathematical genius, Newton had dazzled the Royal Society with this compact yet powerful telescope.

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Edmond Halley (known today for Halley's Comet) was a brilliant astronomer and, just as surprisingly, a man so congenial that he could get along with Isaac Newton. Halley took on the task of coaxing the reluctant, secretive Newton into publis.h.i.+ng his masterpiece, Principia Mathematica. Principia Mathematica. The 500-page book, in Latin and dense with mathematics, might never have appeared without Halley's labors. The 500-page book, in Latin and dense with mathematics, might never have appeared without Halley's labors.

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Newton's tomb, in Westminster Abbey. From the moment he unveiled the theory of gravity Newton was hailed as almost superhuman. Voltaire observed Newton's funeral and was stunned to see dukes and earls carrying the casket. "I have seen a professor of mathematics, simply because he was great in his vocation, buried like a king who had been good to his subjects."

Chapter Thirty-Five.

Barricaded Against the Beast The closer mathematicians looked at infinity, the stranger it seemed. Take one of the simplest drawings imaginable, a straight line one inch long. That line is made up of points, and there are infinitely many of them. Now draw a line two inches long. The longer line must have twice as many points as the shorter one (what else could make it twice as long?). But a matching technique, much like Galileo had used with numbers, shows that there are precisely the same number of points on both lines.

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The proof is pictorial. Make a dot as in the drawing and draw a straight line from it through the two lines. Any such line pairs up a point on the short line with a point on the longer line. Like a perfectly orderly dance, everyone has a partner. No point on either line is left out, and no point has to share a partner with anyone else. How can that be?

Worse was to come. Exactly the same argument shows that a line ten inches long is made of precisely as many points as a line one inch long. So is a line ten miles long, or ten thousand. Could anything send a clearer message that infinity was a topic best left to philosophers and mathematicians, and completely unsuited to hardheaded scientists?

Infinity is built into mathematics from the beginning, because numbers go on forever. If someone made a claim about all the human beings on Earth-no person alive today is nine feet tall-in principle you could test it by gathering everyone into a line and working your way along from first person to last. But no such test can work for numbers, because the line never ends. For every number there is another, bigger number (and also another half as big).

But it was by no means clear that infinity had anything to do with the real world. That was fine. Seventeenth-century scientists, like all their predecessors, would happily have left the paradoxes of infinity to those who enjoyed such things. These practical men of science glanced at infinity, saw that it could not be tamed, and booted it out the door so that they could concentrate on the real-life questions that preoccupied them.

No sooner had they set to work than they heard a clawing at the window.

The most basic challenge in seventeenth-century science was to describe how objects move. To move is to change position. Infinity kept fighting its way into the picture because change comes in two forms. One is easy. The other would challenge and tantalize some of the most powerful thinkers the world had ever seen.

The easy form is steady change, as when a car rolls down the highway with the cruise control set at sixty miles an hour. The car is changing position, but one moment looks much like another. Now think of a rock falling off a cliff. The rock is changing position, like the car, but it is changing speed at every instant, too. That kind of changing changing change happens all around us. We see it when a population grows, or a bullet tears through the air, or an epidemic sweeps through a city. Something is changing, and the rate at which it is changing is changing, too. change happens all around us. We see it when a population grows, or a bullet tears through the air, or an epidemic sweeps through a city. Something is changing, and the rate at which it is changing is changing, too.

Look again at the falling rock. Galileo showed that, as time pa.s.sed, the rock fell faster and faster. At every instant its speed was different. But what did it mean to talk about speed at a given instant? As it turned out, that was where infinity came in. To answer even the most mundane question-how fast is a rock moving?-these seventeenth-century scientists would have to grapple with the most abstract, highfalutin question imaginable: what is the nature of infinity?

It was easy to talk about average average speed, which posed no abstruse riddles. If a traveler in a hackney coach covered a distance of ten miles in an hour, then his average speed was plainly ten miles per hour. But what about speed not over a long interval but at a specific moment? That was trouble. What if the horses pulling the coach labored up a steep hill and then sped down the far side and then stumbled and slowed for a moment and then regained their footing and sped back up? With the coach's speed varying unpredictably, how could you possibly know its speed at a precise instant, at, for instance, the moment it pa.s.sed in front of the Fox and Hounds Tavern? speed, which posed no abstruse riddles. If a traveler in a hackney coach covered a distance of ten miles in an hour, then his average speed was plainly ten miles per hour. But what about speed not over a long interval but at a specific moment? That was trouble. What if the horses pulling the coach labored up a steep hill and then sped down the far side and then stumbled and slowed for a moment and then regained their footing and sped back up? With the coach's speed varying unpredictably, how could you possibly know its speed at a precise instant, at, for instance, the moment it pa.s.sed in front of the Fox and Hounds Tavern?

The point wasn't that anyone needed to know precisely how fast coaches traveled. For any practical question about making a journey from here to there, a rough guess would do. The coach's speed was only important as the key to a larger question: how could you devise a mathematical language that captured the ever-changing world and all its myriad moving parts? How could you see the world as G.o.d saw it?

Before you could tackle the world in general, then, it made sense to try to sort out something as familiar as a horse-drawn coach. For decades mathematicians had all tried to solve the mystery of instantaneous speed in the same way. Speed, they knew, was a measure of how much distance the coach covered in a given time. Suppose the coach happened to pa.s.s the Fox and Hounds at precisely noon. To get a rough guess of its speed at that moment, you might see how far down the road it was an hour later. If the coach had traveled eight miles between noon and one o'clock, its speed at noon was likely somewhere near eight miles per hour. But maybe not. An hour is a long while, and anything could have happened during that time. The horses might have stopped to graze the gra.s.s. They might have been stung by hornets and broken into a sprint. It would be better to guess the coach's speed at the stroke of noon by looking at how far it traveled in a shorter interval that included noon, such as from noon to 12:30. A shorter interval still, say from noon to 12:15, would be better yet. From noon to 12:01 would be even better, and from noon to one second after noon would be better than that.

Success seemed close enough to touch. To measure speed at the instant the clock struck noon, all you had to do was look at how much distance the coach covered in shorter and shorter intervals beginning at noon.

And then, with victory at hand, it flew out of reach. An instant, by definition, is briefer than the tiniest fraction of a second. How much distance did the coach cover in an instant? No distance at all, because it takes some amount of time to travel even the shortest distance. "Ten miles per hour" is a perfectly sensible speed. What could "zero distance in zero seconds" possibly mean?

Chapter Thirty-Six.

Out of the Whirlpool The answer began with Descartes' graphs. Since steady motion was far easier to deal with than uneven motion, scientists started there. Imagine a man trudging home from work at the end of a long day, dragging himself along at 2 miles per hour. A younger colleague might scoot along at 4 miles per hour. A runner might whiz by at 8 miles per hour.

We could chart their journeys in a table that shows how much distance they covered.

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But a graph, a la Descartes, makes matters clearer. A steady pace corresponds to a straight line, as we see in the drawing below, and the faster the pace the steeper the line's slope. Slope, in other words, is a measure of speed. (Slope is a textbook term with a symbol-laden definition, but the technical meaning is the same as the everyday one. A line's slope is simply a measure of how quickly a situation is changing. A flat slope means no change at all; a steep slope, like a spike in blood pressure, means a fast change.) is a textbook term with a symbol-laden definition, but the technical meaning is the same as the everyday one. A line's slope is simply a measure of how quickly a situation is changing. A flat slope means no change at all; a steep slope, like a spike in blood pressure, means a fast change.) [image]

So we can say, with the aid of our picture, precisely what it means to travel at a steady speed of 2 miles per hour (or 4, or 8). It means that, if we were to make a graph of the traveler's path, the result would be a straight line with a certain slope.

This seems simple enough, and so it is, but there is a subtle point hidden inside our tidy graph. The picture has let us dodge a vital but tricky question: What does it mean to travel at two miles per hour if you're not traveling for a full hour? if you're not traveling for a full hour? Before Descartes came along, such questions had sp.a.w.ned endless confusion. But we have no need to vanish into the philosophical fog. We can nearly do without words and debates and definitions altogether. At least in the case of a traveler moving at a steady speed, we can blithely make statements like, " Before Descartes came along, such questions had sp.a.w.ned endless confusion. But we have no need to vanish into the philosophical fog. We can nearly do without words and debates and definitions altogether. At least in the case of a traveler moving at a steady speed, we can blithely make statements like, "At this precise instant she's traveling at a rate of two miles she's traveling at a rate of two miles per hour per hour." All this with the aid of a graph.

But suppose our task was to look at a more complicated journey than a steady march down the street. What does a graph of a cannonball's flight look like? Galileo knew that. It looks like this, as we have seen before.

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Cannonball's flight Now our trick seems to have let us down. So long as we were dealing with graphs of straight lines, we'd found a way to talk about instantaneous speed. It was easy to talk about the slope of a straight line, because the slope was always the same. But what does it mean to talk about the slope of a curve, which by definition is never never straight? straight?

The question was important for two reasons. First, few changes in real life are as simple as the steady plink plink plink plink plink plink of drops from a leaky faucet. Second, if thinkers could devise a way to deal with complicated change of of drops from a leaky faucet. Second, if thinkers could devise a way to deal with complicated change of one one sort, then presumably they could deal with complicated changes of sort, then presumably they could deal with complicated changes of many many sorts. Mathematics is so powerful-and so difficult for us to learn-because it is a universal tool. We balk at algebra, for instance, because those inscrutable sorts. Mathematics is so powerful-and so difficult for us to learn-because it is a universal tool. We balk at algebra, for instance, because those inscrutable x x's are so off-putting. But algebra is useful precisely because it allows us to fill in the blanks in countless different ways.

A mathematics of change dangled the same promise. Planets and comets speeding across the heavens, populations growing and shrinking, bank accounts swelling, divers plummeting, s...o...b..nks melting, all would yield up their mysteries. Questions that asked when a given change would reach its high point or its low-what angle should a cannon be tilted at to shoot the farthest? when will a growing population level off? what is the ideal shape for the arch of a bridge?-could be answered quickly and definitively.

This was a gleaming prize. But how to win it?

The riddle at the heart of the mystery of motion was the question of speed at a given instant. What did it mean? How could you keep from drowning in the whirlpool of "zero distance in zero seconds"?

Answering that question meant learning how to focus on infinitesimally brief stretches of time. The first step was to see that Zeno was not so unnerving as he had seemed. Take his argument that it would take forever to cross a room because it would take a certain amount of time to cross to the halfway point, and then more time to cross half the remaining distance, and so on.

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Zeno's paradox. If it takes 1 second to walk to the middle of a room, and second more to walk half the rest of the way, and second more to walk halfway again, and so on, then it takes an infinitely long time an infinitely long time to cross the room. to cross the room.

In essence Zeno's argument is a claim about infinity. It seems common sense to say that if you add up numbers forever, and if each number you add is bigger than zero, then eventually the sum is infinite. If you piled up blocks forever forever, wouldn't the stack eventually reach the ceiling, no matter how vast a room you started in?

Well, no, actually, not necessarily.

It all depends on the size of each new block that you added to the stack. If all the blocks were the same size, then the tower would would eventually reach the ceiling, the moon, the stars. And even if each new block were thinner than its predecessor, a tower might still grow forever. eventually reach the ceiling, the moon, the stars. And even if each new block were thinner than its predecessor, a tower might still grow forever.43 But it might not, if you picked the sizes of the blocks just so. But it might not, if you picked the sizes of the blocks just so.

In modern terms, Zeno's paradox amounts to saying that if you add up 1 + + + + 1/16 +... the total is infinite. Zeno never framed it that way. He focused not on a specific chain of fractions like this one, but on a general argument about what had to be true for any any endless list of numbers whatsoever. endless list of numbers whatsoever.

But Zeno was wrong. If the sum were infinite, as he believed, then it would be bigger than any number you can think of-bigger than 100, bigger than 100,000, and so on. But Zeno's sum does not not exceed any number you can name. On the contrary, the sum is the perfectly ordinary number 2. exceed any number you can name. On the contrary, the sum is the perfectly ordinary number 2.

In a minute, we'll see why that is. But think how surprising this result is. Suppose you took a block one inch high, and put a -inch-thick block on top of it, and then a -inch-thick one on top of that, and so on. If you added new blocks forever, one a second, through your lifetime and your children's lifetimes and the universe's lifetime, even so the tower would never reach above a two-year-old's ankles.

Part Three: Into the Light

Chapter Thirty-Seven.

All Men Are Created Equal.

The taming of infinity represents another of those breakthroughs where a once-baffling abstraction, like "zero" or "negative five," comes to seem simple in hindsight. The key was to stay relentlessly down-to-earth and never to venture into such murky territory as "the nature of infinity."

The abstraction that would save the day was the notion of a "limit." The mathematical sense is close to the everyday one. In one of the Lincoln-Douglas debates, Abraham Lincoln asked his listeners why the Declaration of Independence a.s.serted that "all men are created equal." Not because the founders believed that all men had already attained equality, Lincoln said. That That was an "obvious untruth." The founders' point, Lincoln declared, was that equality for all was a goal that should be "constantly looked to, constantly labored for, and even though never perfectly attained, constantly approximated." was an "obvious untruth." The founders' point, Lincoln declared, was that equality for all was a goal that should be "constantly looked to, constantly labored for, and even though never perfectly attained, constantly approximated."

In the same sense, a mathematical limit is a goal, a target that a sequence of numbers comes ever closer to. The sequence doesn't have to reach reach the limit, but it does have to get nearer and nearer. The limit of the sequence 1, .1, .01, .001, .0001,... is the number 0, even though the sequence never gets there. Similarly, the limit of , , , 7/8, 9/10, 10/11,... is the number 1, also never attained. The sequence 1, 2, 1, 2, 1, 2,... does not have a limit, because it hops back and forth forever and never homes in on a target. the limit, but it does have to get nearer and nearer. The limit of the sequence 1, .1, .01, .001, .0001,... is the number 0, even though the sequence never gets there. Similarly, the limit of , , , 7/8, 9/10, 10/11,... is the number 1, also never attained. The sequence 1, 2, 1, 2, 1, 2,... does not have a limit, because it hops back and forth forever and never homes in on a target.44 Zeno cast his paradox in the form of a story about a journey across a room. In the 1500s and 1600s a few intrepid mathematicians reframed his tale as a statement about numbers. From that perspective, the question was whether or not 1 + + + + 1/16 +... added up to infinity. Zeno's answer was "yes," because the numbers go on forever and each contributes something to the sum. But when mathematicians turned from Zeno's words to their numbers and began adding, they found something odd. They began with 1 + . That made 1 . Nothing dire there. How about 1 + + ? That came to 1 . Still okay. 1 + + + ? That was 1 7/8. They added more and more terms and never ran into trouble. The running total continued to grow, but it became ever clearer that the number 2 represented a kind of boundary. You could draw arbitrarily close to that boundary-within one-thousandth or one-billionth or even closer-but certainly you could never break through in the way that a runner breaks the tape at the finish line.

For the practical-minded scientists of the seventeenth century, this meant the end of Zeno. In the battle with infinity, they declared victory. Zeno had maintained that if it took one second to reach the middle of a room, it would take forever forever to cross to the other side. Not so, said the new mathematicians. It would take two seconds. to cross to the other side. Not so, said the new mathematicians. It would take two seconds.

Why did that strike them as so momentous? Because when they had taken up the question they truly wanted to answer-what does instantaneous speed instantaneous speed mean?-they had run head-on into Zeno's paradox. They had wanted to know a hackney coach's speed at the instant of noon and had found themselves ensnarled in an infinite regress of questions of the form, mean?-they had run head-on into Zeno's paradox. They had wanted to know a hackney coach's speed at the instant of noon and had found themselves ensnarled in an infinite regress of questions of the form, what was the coach's speed between 12 what was the coach's speed between 12:00 and one minute after? between 12:00 and 30 seconds after? between 12:00 and 15 seconds after? between 12:00 and... ?

This was the seventeenth-century's counterpart of phone-menu h.e.l.l ("if your call is about billing, press 1"), and early scientists groaned in despair because the questions continued endlessly, and escape seemed impossible. But now their victory over Zeno gave them hope. Yes, the questions about the coach did did go on forever. But suppose you looked at the coach's speed in briefer and briefer intervals and found that that sequence of speeds homed in on a limit? go on forever. But suppose you looked at the coach's speed in briefer and briefer intervals and found that that sequence of speeds homed in on a limit?

Then your troubles would be over. That limit would be a number-a definite, perfectly ordinary number. That That was what "instantaneous speed" meant. Nothing to it. But the greatest mathematicians of antiquity, and all their descendants for another fifteen centuries, had failed to see it. was what "instantaneous speed" meant. Nothing to it. But the greatest mathematicians of antiquity, and all their descendants for another fifteen centuries, had failed to see it.

This was not quite calculus, but it was a giant step toward it. In essence, calculus would be a mathematical microscope, a tool that let you pin motion down and scrutinize it tip to toe. Some moments were more important than others-the arrow's height at the instant it reached its peak, the cannonball's speed at the instant it smashed into a city's wall, a comet's speed when it rounded the sun-and, with calculus's help, you could fix those particular moments to a slide and study them close-up.

Or so the newly optimistic mathematicians presumed. But when they grabbed the microscope, they found that no matter how they twisted and tweaked its k.n.o.bs they simply could not bring the image into focus. The problem, they soon saw, was that everything hinged on the notion of limits, and limits weren't as straightforward as they had thought.

As with all other abstractions, the problem was trying to wrestle with a phantom. What did it mean, precisely and quant.i.tatively, for a sequence of numbers to come very close to a limit? "The planet Mars comes close to the Earth when it is 50 million miles away," one modern mathematician observes. "On the other hand, a bullet comes close to a person if it gets within a few inches of him." How close is close?

Even Isaac Newton and Gottfried Leibniz, the boldest thinkers of their age and the leaders of the a.s.sault on infinity, found themselves tangled up in confusion and contradiction. For one thing, infinity seemed to come in a disarming variety of forms. In ordinary usage, infinity infinity conjured up thoughts of boundless immensity. Now, though, in all this talk of speed at a given instant, it seemed vital to sort out the meaning of "infinitely small" lengths and "infinitely brief" stretches of time, as well. conjured up thoughts of boundless immensity. Now, though, in all this talk of speed at a given instant, it seemed vital to sort out the meaning of "infinitely small" lengths and "infinitely brief" stretches of time, as well.

Worse still, the tiny distances and the tiny intervals of time were all mingled together. Speed means distance divided by time. That was not a problem when you were dealing with large, familiar units like miles and hours. But how could you keep your eyes from blurring when it came to dividing ever-shorter distances by ever-briefer time spans?

No one could think how to cla.s.sify these vanis.h.i.+ngly small times and lengths. Leibniz talked of "infinitesimals," which were by definition "the smallest possible numbers," but that definition raised as many questions as it answered. How could a number be smaller than every every fraction? Perhaps infinitesimals were real but too small to see, like the microscopic creatures Leeuwenhoek had recently discovered? As tiny as they were, infinitesimals were bigger than 0. Except sometimes, when they weren't. fraction? Perhaps infinitesimals were real but too small to see, like the microscopic creatures Leeuwenhoek had recently discovered? As tiny as they were, infinitesimals were bigger than 0. Except sometimes, when they weren't.

Leibniz tried to explain, but he only made matters worse. "By... infinitely small, we understand something... indefinitely small, so that each conducts itself as a sort of cla.s.s, and not merely as the last thing of a cla.s.s. If anyone wishes to understand these [the infinitely small] as the ultimate things... it can be done." This was, two of Leibniz's disciples acknowledged, "an enigma rather than an explication." Newton spoke instead of "the ultimate ratio of evanescent quant.i.ties," which was perhaps clear to him but baffling to almost everyone else. "In mathematics the minutest errors are not to be neglected," he insisted in one breath, and in the next he pointed out that these tiny crumbs of numbers were so close to 0 that they could safely be ignored.

Amazingly, things mostly worked out, much as earlier generations had found that things mostly worked out when they manipulated what were then newfangled and still mysterious negative numbers. In the case of calculus, a seemingly mystical abracadabra yielded utterly down-to-earth, hardheaded results about such questions as how far cannonb.a.l.l.s would travel and how much damage they would do when they landed. The very name calculus calculus served as a testimonial to the practical value of this new art; served as a testimonial to the practical value of this new art; calculus calculus is the Latin word for "pebble," a reference to the heaps of stones once used as a calculating aid in addition and multiplication. is the Latin word for "pebble," a reference to the heaps of stones once used as a calculating aid in addition and multiplication.

Skeptics contended that any correct results must have been due to happy accidents in which multiple errors canceled themselves out. ("For science it cannot be called," one critic later charged, "when you proceed blindfold and arrive at the Truth not knowing how or by what means.") But so long as the slapdash new techniques kept churning out answers to questions that had always lain out of reach, no one spent much time worrying about rigor. Leibniz, boundlessly optimistic in personality as well as in his philosophical views, argued explicitly that this gift horse should be saddled and ridden, not inspected. It would all work out.

The muddle would last until the 1800s. Only then would a new generation of mathematicians find a way to replace vague intuitions with clear definitions. (The breakthrough was finding a way to define "limits" while banis.h.i.+ng all talk of infinitely small numbers.) In all the intervening years mathematicians and scientists had rejoiced in a bounty they did not understand. Instead they followed the advice of Jean d'Alembert, a French mathematician who lived a century after Newton and Leibniz but during the era when the underpinnings of calculus were still cloaked in mystery.

"Persist," d'Alembert advised, "and faith will come to you."

Chapter Thirty-Eight.

The Miracle Years Both Isaac Newton and Gottfried Leibniz had egos as colossal as their intellects. In the hunt for calculus, each man saw himself as a lone adventurer in unexplored territory. And then, unbeknownst to one another, each gained the prize he sought. Each saw his triumph not as that of a runner bursting past a pack of rivals but as that of a solo mountain climber. They had won their way to a summit, moreover, that no one else even knew existed. Or so they both believed.

Imagine, then, the exultation that each man felt when he planted his flag in the ice and gazed out at the panorama before him, a landscape that extended as far as the eye could see. Picture, too, the pride and satisfaction that came with lone owners.h.i.+p of this vast and beckoning domain. And then imagine the morning when that proprietary delight gave way to shock and horror. Imagine the first glimpse of a puff of smoke in the distance-fog, surely surely, for how could anyone have built a fire in this emptiness? for how could anyone have built a fire in this emptiness?-and then, soon after, the unmistakable sight of someone else's footprints in the snow.

Newton had been the first to learn how to pin down the mysterious infinitesimals that held the key to explaining motion. He kept his discoveries secret from all but a tiny circle for three decades. Victimized by his own temperament-Newton was always torn between indignation at seeing anyone else get credit for work he had done first and fury at the thought of announcing his findings and thereby exposing himself to critics-he might have hesitated forever. As it was, his delay in staking his claim led to one of the bitterest feuds in the history of science.

Newton made his mathematical breakthroughs (and others just as important) in a fever of creativity that historians would later call the "miracle years." He spent eighteen months between 1665 and 1667 at his mother's farm, hiding from the plague that had shut Cambridge down. Newton was twenty-two when he returned home, undistinguished, unknown, and alone.

Intellectually, though, he was not entirely on his own. Calculus was in the air, and such eminent mathematicians as Fermat, Pascal, and Descartes had made considerable advances toward it. Newton had attended mathematical lectures at Cambridge; he had bought and borrowed a few textbooks; he had studied Descartes' newfangled geometry with diligence.

What sparked his mathematical interest in the first place he never said. We can, however, pin down the time and place. Every August, Cambridge played host to an enormous outdoor market called Stourbridge Fair. In row upon row of tents and booths, merchants and hawkers sold clothes, dishes, toys, furniture, books, jewelry, beer, ale, and, in the horrified words of John Bunyan, "l.u.s.ts, pleasures, and delights of all sorts." Newton steered well clear of the swarms of wh.o.r.es, jugglers, and con men. (He had given a good deal of thought to temptation, s.e.xual temptation above all, and had fas.h.i.+oned a strategy. "The way to chast.i.ty is not to struggle directly with incontinent thoughts," he wrote in an essay on monasteries and the early church, "but to avert ye thoughts by some employment, or by reading, or meditating on other things.") Newton made two purchases. They seemed innocuous, but they would revolutionize the intellectual world. "In '63 [Newton] being at Stourbridge fair bought a book of astrology to see what there was in it," according to a young admirer who had the story from Newton himself. Perhaps in the same year-scholars have not settled the matter-he bought a trinket, a gla.s.s prism. Children liked to play with prisms because it was pretty to see how they caught the light.

The astrology book had no significance in itself, but it helped change history. Newton "read it 'til he came to a figure of the heavens which he could not understand for want of being acquainted with trigonometry," he recalled many years later. "Bought a book of trigonometry, but was not able to understand the demonstrations. Got Euclid to fit himself for understanding the ground of trigonometry."

At that point Newton's backtracking came to an end. To his relief, he found that Euclid was no challenge. "Read only the t.i.tles of the propositions," he would recall, "which he found so easy to understand that he wondered how anybody would amuse themselves to write any demonstrations of them."

Newton turned from Euclid's cla.s.sical geometry to Descartes' recent recasting of the entire subject. This was not so easy. He made it through two or three pages of Descartes but then lost his way. He started over and this time managed to understand three or four pages. He slogged along in this fas.h.i.+on, inching his way forward until he lost his bearings and then doubling back to the beginning "& continued so doing till he made himself Master of the whole without having the least light or instruction from any body." Every aspiring mathematician knows the frustration of spending entire days staring at a single page in a textbook, or even a single line, waiting for insight to dawn. It is heartening to see one of the greatest of all mathematicians in almost the same plight.

Newton's pride in finally mastering Descartes' Geometry Geometry had two aspects, and both were typical of him. He had accomplished a great deal, and he had done it without a word of guidance "from any body." And he had only begun. To this point he had studied work that others had already done. From here on, he would be advancing into unexplored territory. In early 1665, less than two years from the day he had picked up the astrology booklet, he recorded his first mathematical discovery. He proved what is now called the binomial theorem, to this day one of the essential results in all of mathematics. had two aspects, and both were typical of him. He had accomplished a great deal, and he had done it without a word of guidance "from any body." And he had only begun. To this point he had studied work that others had already done. From here on, he would be advancing into unexplored territory. In early 1665, less than two years from the day he had picked up the astrology booklet, he recorded his first mathematical discovery. He proved what is now called the binomial theorem, to this day one of the essential results in all of mathematics.45 This was the opening salvo of the "miracle years." This was the opening salvo of the "miracle years."

Newton's summary of what came next remains startling three and a half centuries later. Even those unfamiliar with the vocabulary cannot miss the rat-tat-tat pacing of discoveries that spilled out almost too quickly to list. "The same year in May I found the method of Tangents... & in November had the direct method of fluxions & the next year in January had the Theory of Colours & in May following I had entrance into ye inverse method of fluxions. And the same year I began to think of gravity extending to ye orb of the Moon...."

Over the course of eighteen months, that is, Newton first invented a great chunk of calculus, everything to do with what is now called differentiation. Then he briefly put mathematics aside and turned to physics. Taking up his Stourbridge Fair prisms (he had bought a second one) and shutting up his room except for a pinhole that admitted a shaft of sunlight, he discovered the nature of light. Then he turned back to calculus. The subject falls naturally into two halves, although that is by no means evident early on. In early 1665 Newton had invented and then investigated the first half; now he knocked off the other half, this time inventing the techniques now known as integration. Then he proved that the two halves, which looked completely different, were in fact intimately related and could be used in tandem in hugely powerful ways. Then he began thinking about the nature of gravity. "All this," he wrote, "was in the two plague years of 16651666. For in those days I was in the prime of my age for invention & minded Mathematicks & Philosophy more than at any time since."

Newton was indeed in his prime at twenty-three, for mathematics and physics are games for the young. Einstein was twenty-six when he came up with the special theory of relativity, Heisenberg twenty-five when he formulated the uncertainty principle, Niels Bohr twenty-eight when he proposed a revolutionary model of the atom. "If you haven't done outstanding work in mathematics by 30, you never will," says Ronald Graham, one of today's best-regarded mathematicians.

The greats flare up early, like athletes, and they burn out just as quickly. Paul Dirac, a physicist who won his n.o.bel Prize for work he did at twenty-six, made the point with wry bleakness, in verse. (He wrote his poem while still in his twenties.) Age is, of course, a fever chillthat every physicist must fear.He's better dead than living stillwhen once he's past his thirtieth year.