In the most abstract fields-music, mathematics, physics, even chess-the young thrive. Child prodigies are not quite common, but they turn up regularly. Perhaps it makes sense that if a Mozart or a Bobby Fischer were to appear anywhere, it would be in a self-contained field that does not require insight into the quirks of human psychology. We are unlikely ever to meet a twelve-year-old Tolstoy.

But that is only part of the story. Penetrating to the heart of abstract fields seems to demand a degree of intellectual firepower, an intensity of focus and stamina, that only the young can muster. For the greats, these truly are miracle years. "I know that when I was in my late teens and early twenties the world was just a Roman candle-rockets all the time," recalled I. I. Rabi, another n.o.bel Prizewinning physicist. "You lose that sort of thing as time goes on.... Physics is an otherworld thing. It requires a taste for things unseen, even unheard of-a high degree of abstraction.... These faculties die off somehow when you grow up."

Nerve and brashness are as vital as brainpower. A novice sets out to change the world, confident that he can find what has eluded every other seeker. The expert knows all the reasons why the quest is impossible. The result is that the young make the breakthroughs. The pattern is different in the arts. "Look at a composer or a writer-one can divide his work into early, middle, and late, and the late work is always better, more mature," observed Subrahmanyan Chandrasekhar, the astrophysicist who won a n.o.bel Prize for his work on black holes (and worked into his eighties). Even so, he declared in his old age, "For scientists, the early work is always better."

At age thirty-five or forty, when a politician would still count as a fresh face, when doctors in certain specialties might only recently have completed their training, mathematicians and physicists know they have probably pa.s.sed their peak. In the arts, talent often crests at around forty. Michelangelo completed the ceiling of the Sistine Chapel at thirty-seven; Beethoven finished his Fifth Symphony at thirty-seven; Tolstoy published War and Peace War and Peace at forty-one; Shakespeare wrote at forty-one; Shakespeare wrote King Lear King Lear at forty-two. But the list of artists who continued to produce masterpieces decades later than that-Monet, Cervantes, t.i.tian, Pica.s.so, Verdi-is long. at forty-two. But the list of artists who continued to produce masterpieces decades later than that-Monet, Cervantes, t.i.tian, Pica.s.so, Verdi-is long.

Science and mathematics have no such roster. In the end, the work simply becomes too difficult. Newton would make great advances in mathematics after his miracle years, but he would never again match the creative fervor of that first outburst. Looking back at his career in his old age, he remarked that "no old Men (excepting Dr. Wallis)"-this was Newton's eminent contemporary John Wallis-"love Mathematicks."

From his earliest youth, Newton had seen himself as different from others, set apart and meant for special things. He read great significance into his birth on Christmas Day, his lack of a father, and his seemingly miraculous survival in infancy. The depth and sincerity of his religious faith are beyond question, and so was his belief that G.o.d had set him apart and whispered His secrets into his ear. Others had studied the prophecies in the Bible just as he had, Newton noted, but they had met only "difficulty & ill success." He was unsurprised. Understanding was reserved for "a remnant, a few scattered persons which G.o.d hath chosen." Guess who.

He took the Latin form of his name, Isaacus Nevtonus, and found in it an anagram, Ieova sanctus unus Ieova sanctus unus, or the one holy Jehovah the one holy Jehovah. He drew attention to the pa.s.sage in Isaiah where G.o.d promises the righteous that "I will give thee the treasures of darkness, and hidden riches of secret places."

By the end of the miracle years, Newton found himself awash in hidden riches. He knew more mathematics than anyone else in the world (and therefore more than anyone who had ever lived). No one even suspected. "The fact that he was unknown does not alter the other fact that the young man not yet twenty-four, without benefit of formal instruction, had become the leading mathematician of Europe," wrote Richard Westfall, Newton's preeminent biographer. "And the only one who really mattered, Newton himself, understood his position clearly enough. He had studied the acknowledged masters. He knew the limits they could not surpa.s.s. He had outstripped them all, and by far."

Newton had always felt felt himself isolated from others. Now at twenty-three, wrote Westfall, he finally had objective proof that he was not like other men. "In 1665, as he realized the full extent of his achievement in mathematics, Newton must have felt the burden of genius settle upon him, the terrible burden which he would have to carry in the isolation it imposed for more than sixty years." himself isolated from others. Now at twenty-three, wrote Westfall, he finally had objective proof that he was not like other men. "In 1665, as he realized the full extent of his achievement in mathematics, Newton must have felt the burden of genius settle upon him, the terrible burden which he would have to carry in the isolation it imposed for more than sixty years."

Chapter Thirty-Nine.

All Mystery Banished Isaac Newton believed that he had been tapped by G.o.d to decipher the workings of the universe. Gottfried Leibniz thought that Newton had set his sights too low. Leibniz shared Newton's yearning to find nature's mathematical structure, which in their era meant almost inevitably that both men would mount an a.s.sault on calculus, but in Leibniz's view mathematics was only one piece in a much larger puzzle.

Leibniz was perhaps the last man who thought it was possible to know everything. The universe was perfectly rational, he believed, and its every feature had a purpose. With enough attention you could explain it all, just as you could deduce the function of every spoke and spring in a carriage.

For Leibniz, one of the greatest philosophers of the age, this was more than a demonstration of almost pathological optimism (though it was that, too). More important, Leibniz's faith was a matter of philosophical conviction. The universe had had to make perfect sense because it had been created by an infinitely wise, infinitely rational G.o.d. To a powerful enough intellect, every true observation about the world would be self-evident, just as every true statement in geometry would immediately be obvious. In all such cases, the conclusion was built in from the start, as in the statement "all bachelors are unmarried." We humans might not be clever enough to see through the undergrowth that obscures the world, but to G.o.d every truth shines bright and clear. to make perfect sense because it had been created by an infinitely wise, infinitely rational G.o.d. To a powerful enough intellect, every true observation about the world would be self-evident, just as every true statement in geometry would immediately be obvious. In all such cases, the conclusion was built in from the start, as in the statement "all bachelors are unmarried." We humans might not be clever enough to see through the undergrowth that obscures the world, but to G.o.d every truth shines bright and clear.

In fact, though, Leibniz felt certain that G.o.d had designed the world so that we can can understand it. Newton took a more cautious stand. Humans could read the mind of G.o.d, he believed, but perhaps not all of it. "I don't know what I may seem to the world," Newton famously declared in his old age, though he knew perfectly well, "but, as to myself, I seem to have been only like a boy playing on the seash.o.r.e, and diverting myself in now and then finding a smoother pebble or a prettier sh.e.l.l than ordinary, whilst the great ocean of truth lay all undiscovered before me." understand it. Newton took a more cautious stand. Humans could read the mind of G.o.d, he believed, but perhaps not all of it. "I don't know what I may seem to the world," Newton famously declared in his old age, though he knew perfectly well, "but, as to myself, I seem to have been only like a boy playing on the seash.o.r.e, and diverting myself in now and then finding a smoother pebble or a prettier sh.e.l.l than ordinary, whilst the great ocean of truth lay all undiscovered before me."

Newton's point was not simply that some questions had yet to be answered. Some questions might not have have answers, or at least not answers we can grasp. Why had G.o.d chosen to create something rather than nothing? Why had He made the sun just the size it is? Newton believed that such mysteries might lie beyond human comprehension. Certainly they were outside the range of scientific inquiry. "As a blind man has no idea of colors," Newton wrote, "so have we no idea of the manner by which the all-wise G.o.d perceives and understands all things." answers, or at least not answers we can grasp. Why had G.o.d chosen to create something rather than nothing? Why had He made the sun just the size it is? Newton believed that such mysteries might lie beyond human comprehension. Certainly they were outside the range of scientific inquiry. "As a blind man has no idea of colors," Newton wrote, "so have we no idea of the manner by which the all-wise G.o.d perceives and understands all things."

Leibniz accepted no such bounds. G.o.d, he famously declared, had created the best of all possible worlds. This was not an a.s.sumption, in Leibniz's view, but a deduction. G.o.d was by definition all-powerful and all-knowing, so it followed at once that the world could not have been better designed. (Even for one of the ablest of all philosophers, this made for an impossible tangle. If logic compelled G.o.d to create the very world we find ourselves in, didn't that mean that He had no choice in the matter? But surely to be G.o.d meant to have infinite infinite choice?) choice?) Voltaire would later take endless delight, in Candide Candide, in pummeling Leibniz. On Candide Candide's very first page, we meet Leibniz's stand-in, Dr. Pangloss, the greatest philosopher in the world. Pangloss's specialty is "metaphysico-theologo-cosmolonigology." The world, Pangloss explains contentedly, has been made expressly for our benefit. "The nose is formed for spectacles, therefore we wear spectacles.... Pigs were made to be eaten, therefore we eat pork all the year round."

Pangloss and the hero of the novel, a naive young man named Candide, spend the book beset by calamity-Voltaire cheerily throws in an earthquake, a bout of syphilis, a stint as a galley slave, for starters. Bloodied and battered though both men may be, Pangloss pops up from every crisis as undaunted as a jack-in-the-box, pointing out once more that this is the best of all possible worlds.

This was great fun-Voltaire was an immensely popular writer, and Candide Candide was his most popular work-but it was a bit misleading. Leibniz knew perfectly well that the world abounded in horrors. (He had been born during the Thirty Years' War.) His point was not that all was sunshine, but that no better alternative was possible. G.o.d had considered every conceivable universe before settling on this one. Other universes might have been good, but ours is better. G.o.d could, for instance, have made humans only as intelligent as dogs. That might have made for a happier world, but happiness is not the only virtue. In a world of poodles and Great Danes, who would paint pictures and write symphonies? was his most popular work-but it was a bit misleading. Leibniz knew perfectly well that the world abounded in horrors. (He had been born during the Thirty Years' War.) His point was not that all was sunshine, but that no better alternative was possible. G.o.d had considered every conceivable universe before settling on this one. Other universes might have been good, but ours is better. G.o.d could, for instance, have made humans only as intelligent as dogs. That might have made for a happier world, but happiness is not the only virtue. In a world of poodles and Great Danes, who would paint pictures and write symphonies?

Or G.o.d might have built us so that we always chose to do good rather than evil. In such a world, we would all be kind, but we would all be automatons. In His wisdom, G.o.d had decided against it. A world with sin was better than a world without choice. Not perfect, in other words, but better than any possible alternative. It was this complacency that infuriated Voltaire. He raged against Leibniz not because Leibniz was blind to the world's miseries but because he so easily reconciled himself to them.

But Leibniz's G.o.d was as rational as he was. For every conceivable world, He totted up the pros and cons and then subtracted the one from the other to compute a final grade. (It is perhaps no surprise that Leibniz invented calculus; in searching for the world that would receive the highest possible score, G.o.d was essentially solving a calculus problem.) Since G.o.d had necessarily created the best of all possible worlds, Leibniz went on, we can deduce its properties by pure thought. The best possible world was the one that placed the highest value on the pursuit of intellectual pleasure-here the philosopher showed his hand-and the greatest of all intellectual pleasures was finding order in apparent disorder. It was certain, therefore, that G.o.d meant for us to solve all the world's riddles. Leibniz was "perhaps the most resolute champion of rationalism who ever appeared in the history of philosophy," in the words of the philosopher Ernst Ca.s.sirer. "For Leibniz there... is nothing in heaven or on earth, no mystery in religion, no secret in nature, which can defy the power and effort of reason."

Surely, then, Leibniz could solve the problem of describing the natural world in the language of mathematics.

Chapter Forty.

Talking Dogs and Unsuspected Powers Leibniz gave the impression that he intended to pursue every one of nature's secrets himself. "In the century of Kepler, Galileo, Descartes, Pascal, and Newton," one historian wrote, "the most versatile genius of all was Gottfried Wilhelm Leibniz." The grandest topics intrigued him, and so did the humblest. Leibniz invented a new kind of nail, with ridged sides to keep it from working free. He traveled to see a talking dog and reported to the French Academy that it had "an apt.i.tude that was hard to find in another dog." (The wondrous beast could p.r.o.nounce the French words for tea, coffee, and chocolate, and some two dozen more.) He drew up detailed plans for "a museum of everything that could be imagined," roughly a cross between a science exhibition and a Ripley's Believe It or Not museum. It would feature clowns and fireworks, races between mechanical horses, rope dancers, fire eaters, musical instruments that played by themselves, gambling halls (to bring in money), inventions, an anatomical theater, transfusions, telescopes, demonstrations of how the human voice could shatter a drinking gla.s.s or how light reflected from a mirror could ignite a fire.

Leibniz's energy and curiosity never flagged, but he could scarcely keep up with all the ideas careening around his head. "I have so much that is new in mathematics, so many thoughts in philosophy, so numerous literary observations of other kinds, which I do not wish to lose, that I am often at a loss what to do first," he lamented.

Many of these ventures consumed years, partly because they were so ambitious, partly because Leibniz tackled everything at once. He continued to work on his calculating machine, for example, and on devising a symbolic language that would allow disputes in ethics and philosophy to be solved like problems in algebra. "If controversies were to arise, there would be no more need of disputing between two philosophers than between two accountants. For it would suffice to take their pencils in their hands, to sit down to their slates, and to say to each other (with a friend as witness, if they liked): 'Let us calculate.' "

Leibniz wrote endlessly, at high speed, often while b.u.mping along the road in a coach. Today a diligent team of editors is laboring to turn well over one hundred thousand ma.n.u.script pages into a Collected Works, but they do not expect to complete the project in their lifetimes. Volume 4, to choose an example at random, comes under the heading of "Philosophical Writings," and consists of three "books." Each book contains over a thousand pages. The editors envision sixty such volumes.

Thinkers who take on the whole world, as Leibniz did, are out of fashion today. Even in his own era, he was a hard man to get the measure of. Astonishingly brilliant, jaw-droppingly vain, charming, overbearing, a visionary one minute and a self-deluded dreamer the next, he was plainly a lot of work. Not everyone was inclined to make the effort. Still, in Bertrand Russell's words, "Leibniz was one of the supreme intellects of all time." If anything, his reputation among scientists and mathematicians has grown through the centuries, as ideas of his that once seemed simply baffling have come into focus.

More than three hundred years ago, for instance, Leibniz envisioned the digital computer. He had discovered the binary language of 0s and 1s now familiar to every computer programmer,46 and, more remarkably, he had imagined how this two-letter alphabet could be used to write instructions for an all-purpose reasoning machine. and, more remarkably, he had imagined how this two-letter alphabet could be used to write instructions for an all-purpose reasoning machine.

The computer that Leibniz had in mind relied not on electrical signals-this was almost a century before Benjamin Franklin would stand outdoors with a kite in a lightning storm-but on marbles tumbling down chutes in a kind of pinball machine. "A container shall be provided with holes in such a way that they can be opened and closed," Leibniz wrote. "They are to be open at those places that correspond to a 1 and remain closed at those that correspond to a 0. Through the opened gates small cubes or marbles are to fall into tracks, through the others nothing."

Leibniz was born in Germany, but he spent his glory years in the glittering Paris of Louis XIV, when the Sun King had just begun building Versailles and emptying the royal treasury. Leibniz arrived in Paris in 1672, at age twenty-six, a dapper young diplomat sporting a long wig. The dark curls and silk stockings were standard fare, but the torrent of words that spilled forth from the new arrival dazed his listeners. Leibniz had come to Paris with characteristically bold plans. Germany dreaded an invasion by the French, who had grand territorial ambitions. Leibniz's mission was to convince Louis XIV that an incursion into Germany would do him little good. What he ought to do instead, what would prove a triumph worthy of so ill.u.s.trious a monarch, was to conquer Egypt.

In four years Leibniz never managed to win an audience with the king. (France, as everyone had feared, spent the next several decades embroiling Europe in one war after another.) Leibniz spent his time productively nonetheless, somehow combining an endless series of visits with one count or duke or bishop after another with the deepest investigations into science and mathematics.

Leibniz's conquest of mathematics came as a surprise. Unlike nearly all the other great figures in the field, he came to it late. Leibniz's academic training had centered on law and diplomacy. In those fields, as well as philosophy and history and a dozen others, he knew everything. But at twenty-six, one historian writes, Leibniz's knowledge of mathematics was "deplorable."

He would remedy that. In Paris he set to work under the guidance of some leading mathematicians, notably the brilliant Dutch scientist Christiaan Huygens. For the most part, though, he taught himself. He took up cla.s.sic works, like Euclid, and recent ones, like Pascal and Descartes, and dipped in and out at random like a library patron flipping through the books on the "new arrivals" shelf. Even Newton had found that newfangled doctrines like Descartes' geometry slowed him to a crawl. Not Leibniz. "I read [mathematics] almost as one reads tales of romance," he boasted.

He read voraciously and compet.i.tively. These were difficult, compact works by brilliant men writing for a tiny audience of peers, not textbooks meant for students, and Leibniz measured himself against the top figures in this new field. "It seemed to me," he wrote shortly after beginning his crash course, "I do not know by what rash confidence in my own ability, that I might become the equal of these if I so desired." The time had come to stop reading about what other people had done and to make discoveries of his own.

By now it was 1675. Leibniz was thirty but still, at that advanced mathematical age, at the peak of his powers. The riddle that taunted every mathematician was the infinitesimal, the key to understanding motion at a given instant. Nearly a decade before, Newton had solved the mystery and invented what is now called calculus. He had told almost no one, preferring to wrap that secret knowledge around himself like a warm cloak. Now, unaware of what Newton had already done, Leibniz set out after the same prize.

In the course of one astonishing year-a miracle year of his own-he found it. Newton had kept his discovery to himself, because of his hatred of controversy and because the security of his professorship at Cambridge meant he did not have to scramble for recognition. Leibniz did not publish an account of his discovery of calculus for nine years, but his silence is harder to explain. Leibniz never had a safe position like Newton's. Throughout his long career, he was dependent on the whims of his royal patrons, forever trapped in the role of an intellectual court jester. That might have made him more more eager to publish, anything to make his status less precarious, but it did not. eager to publish, anything to make his status less precarious, but it did not.

The reasons for his delay have disappeared into a biographical black hole. Leibniz wrote endlessly on every conceivable topic-his correspondence alone consisted of fifteen thousand letters, many of them more essays than notes-but he remained silent on the question of his long hesitation. Scholars can only fill the void with guesses.

Perhaps he was gun-shy as a result of a fiasco at the very beginning of his mathematical career. On his first trip to England, in 1672, Leibniz had met several prominent mathematicians (but not Newton) and happily rattled on about his discoveries. The bragging was innocent, but Leibniz was such a mathematical novice that he talked himself into trouble. At an elegant dinner party in London, presided over by Robert Boyle, Leibniz claimed as his own a result (involving the sum of a certain infinitely long sequence of fractions) that was in fact well-known. Another guest set him straight. In time the episode blew over. Still, Leibniz may have decided to make sure that he stood on firm ground before he announced far bolder mathematical claims.

Or perhaps he decided that formal publication was beside the point because the audience he needed to reach had already learned of his achievement through informal channels-rumors and letters. Or the task of developing a full-fledged theory, as opposed to a collection of techniques for special cases, may have proved unexpectedly difficult. Or Leibniz may have judged that he needed to make a bigger splash-from an impossible-to-miss invention like the telescope or from some diplomatic coup-than any mathematical discovery could provide.

Eventually, in 1684, Leibniz told the world what he had discovered. By then he and Newton had exchanged friendly but guarded letters discussing mathematics in detail but tiptoeing around the whole subject of calculus. (Rather than tell Leibniz directly what he had found, Newton concealed his most important discoveries in two encrypted messages. One read, "6accdae13eff7i319n4o4qrr4s8t12ux.") In the published article announcing his discovery of calculus, Leibniz made no mention of Newton or any of his other predecessors.

In the case of Newton, at least, that oversight was all but inevitable, since Leibniz had no way of knowing what Newton had found. A perfect alibi, one might think, but it proved anything but. Leibniz's "oversight" was destined to poison the last decades of his life.

Chapter Forty-One.

The World in Close-Up Newton and Leibniz framed their discoveries in different vocabulary, but both had found the same thing. The challenge confronting both men was finding a way to stop time in its tracks. Their solution, hundreds of years before the birth of photography, was essentially to imagine the movie camera. They pictured the world not as the continuous, flowing panorama we see but as a series of still photos, each one barely different from those before and after it, and all the frames flashing before the eye too quickly to register as static images.

But how could you be sure that, no matter what moment you wanted to scrutinize, there happened to be a sharply focused image on hand? It seemed clear that the briefer the interval between successive still photos, the better. The problem was finding a stopping place-if sixty-four frames a second was good, wouldn't 128 be better? Or 1,000, or 100,000?

Think of Galileo near the top of the Leaning Tower, a bit winded from the long climb. He extends an arm out into s.p.a.ce, opens his fingers, and releases the rock he has lugged up all this way. It falls faster and faster-in each successive second, that is, it covers more distance than it did the second before-as the figures in this table show. (As we have seen, it was no easy matter to make such measurements without clocks or cameras, which was why Galileo ended up working with ramps rather than towers.) [image]

Galileo found that rocks fall according to a precise rule that can be expressed in symbols. Scientists write the rule as d d = 16 = 16 t t, where d d stands for distance and stands for distance and t t stands for time. In one second, a rock falls a distance of 16 1 feet, or 16 feet. In two seconds, it falls a distance of 16 4 feet, or 64 feet; in three seconds, 16 9 feet, or 144 feet. stands for time. In one second, a rock falls a distance of 16 1 feet, or 16 feet. In two seconds, it falls a distance of 16 4 feet, or 64 feet; in three seconds, 16 9 feet, or 144 feet.

[image]

The graph shows how far a rock dropped from a height falls in t seconds. The rock obeys the rule d = 16 t2.

The table can be converted to a graph, and, as usual, a picture helps reveal what the numbers only imply. (And a picture, unlike a table, shows the rock's position at every every moment rather than at a select few.) The horizontal axis depicts time, the vertical axis distance. The drawing shows the distance the rock has fallen at a given time. At the moment that Galileo uncurled his fingers and released the rock (in other words, at moment rather than at a select few.) The horizontal axis depicts time, the vertical axis distance. The drawing shows the distance the rock has fallen at a given time. At the moment that Galileo uncurled his fingers and released the rock (in other words, at t t = 0), the rock has fallen 0 feet. At 1 second it has fallen 16 feet; in 2 seconds, 64 feet; and so on. = 0), the rock has fallen 0 feet. At 1 second it has fallen 16 feet; in 2 seconds, 64 feet; and so on.

The transition from rock to table to graph is one of increasing abstraction, and early mathematicians had a hard time keeping their bearings. Few things could be more tangible than a rock. When Galileo let it go, anyone who happened to be pa.s.sing by could see it. The table took that ordinary event-a stone whooshing toward the ground-and transformed it into a list of numbers. The graph represented still another move away from everyday reality. It shows a curve that represents the rock's distance from Galileo's hand, but the curve in the graph only matches the real-life descent of the falling rock in a subtle way. The actual rock dropped in a straight line. The graph shows a curve, a parabola. Worse yet, the rock fell down down, while the parabola headed up up. To "see" the rock falling in the way that the graph depicts required a far more laborious and roundabout process than merely looking at a rock.

And yet it was this second way of looking at a rock's fall, this unnatural way, that held the key to nature's secrets. For it was this curve that showed Newton and Leibniz how to seize time in their fists and hold it still. We saw in chapter 36 that in a graph that shows distance measured against time, a straight line corresponds to a steady speed. (The greater the speed, the steeper the slope of that line, because a steeper slope indicates more distance covered in a given amount of time.) But in the graph of a falling rock, we have a curve, not a straight line. How can we talk about the rock's speed? In particular, how can we find its speed at a specific instant, at, for instance, precisely one second into its fall?

We could do it, Newton and Leibniz explained, if we could find the slope of the curve at precisely the one-second point. Which they proceeded to do. The idea was to look at the curve in extraordinary close-up. If you look closely enough, a curve looks like a straight line. (A jogger on a huge circular track would feel as if she were running in a straight line. Only a bird's-eye view would reveal the track's true shape.) And although curves are hard to work with, straight lines are easy.

First they froze time by selecting a single frame from nature's ongoing movie. (Newton and Leibniz worked in parallel, unaware of one another, as we have seen, but they independently hit on the same strategy.) Second, they tunneled into that frame, as if it were a slide under a microscope.

In the case of a falling rock, they began by freezing the picture at the instant t t = 1 second. They wanted to know the rock's speed at that moment, but the only information they had to work with was a graph depicting time and distance. Even so, they were nearly done. = 1 second. They wanted to know the rock's speed at that moment, but the only information they had to work with was a graph depicting time and distance. Even so, they were nearly done.

All they had to do was focus their conceptual microscope. Speed is a measure of distance traveled in a given time. Sixty miles per hour Sixty miles per hour. Three inches per second Three inches per second. To solve the problem they cared about, they began by solving an easier problem, in the hope that the solution to the easy problem would point to the solution they truly wanted.

The rock's speed at a given instant was hard to find because that speed was constantly changing. But the rock's average average speed over any particular span of time was easy to find. ( Just divide the distance the rock fell by the length of the time span.) With that in mind, Newton and Leibniz did something clever. They put the actual rock to one side for a moment and concentrated instead on an easier-to-deal-with imaginary rock. The great virtue of this imaginary rock was that, in contrast with a real rock, it fell at a constant speed. What speed to pick? speed over any particular span of time was easy to find. ( Just divide the distance the rock fell by the length of the time span.) With that in mind, Newton and Leibniz did something clever. They put the actual rock to one side for a moment and concentrated instead on an easier-to-deal-with imaginary rock. The great virtue of this imaginary rock was that, in contrast with a real rock, it fell at a constant speed. What speed to pick?

The answer, Newton and Leibniz decided, was that the imaginary rock should fall at a steady speed that exactly matched the average speed of the actual rock in the interval between t t = 1 and = 1 and t t = 2. This roundabout procedure seems like a detour, but in fact it brought them closer to their goal. = 2. This roundabout procedure seems like a detour, but in fact it brought them closer to their goal.

Look at the graph below. The dotted line depicts the imaginary rock, the curve the real rock. At the one-second mark (in other words, at t t = 1) the imaginary rock and the real one have both fallen 16 feet. At = 1) the imaginary rock and the real one have both fallen 16 feet. At t t = 2, both the imaginary rock and the real one have fallen 64 feet. = 2, both the imaginary rock and the real one have fallen 64 feet.

[image]

The dotted line represents the fall of an imaginary rock traveling at constant speed. The slope of the dotted line gives the imaginary rock's speed in the one-second interval between t = 1 and t = 2.

The dotted line is straight. That's crucial. Why? Because it means we can talk about its slope, which is a number-a regular, run-of-the-mill number, not an infinitesimal or any other colorful beast. That number is the speed of the imaginary rock. (It is easy to compute. Slope is a measure of steepness, which means that it is a ratio of vertical change to horizontal change. In this case, the vertical change was from 16 feet to 64 feet, and the horizontal change was from 1 second to 2 seconds, so the slope was [6416] feet [21] seconds, or 48 feet per second.) Now Newton and Leibniz made their big move. Forty-eight feet per second was the imaginary rock's speed over a one-second span. That gave a fair approximation to what they really wanted to know about, an actual rock's speed at the precise instant t t = 1. = 1.

How could you get a better approximation? By zooming in for a closer look at the graph. And the way to do that was once again to focus your attention at t t = 1 but this time to look at a shorter time interval than one second. As usual, pictures came to the rescue. = 1 but this time to look at a shorter time interval than one second. As usual, pictures came to the rescue.

Look at the diagram below. The new, dashed line represents the path of a new imaginary rock. This This imaginary rock, too, is falling at a constant speed. What speed? Not the same speed as the first imaginary rock. This new imaginary rock is falling at a speed exactly equal to the actual rock's average speed in a newer, shorter time interval, the interval between imaginary rock, too, is falling at a constant speed. What speed? Not the same speed as the first imaginary rock. This new imaginary rock is falling at a speed exactly equal to the actual rock's average speed in a newer, shorter time interval, the interval between t t = 1 and = 1 and t t = 1 = 1 . The point is that the speed of this new imaginary rock gives us a better estimate of the actual rock's speed at the instant t t = 1. = 1.

[image]

The dashed line represents the fall of a new imaginary rock. The slope of the line gives the imaginary rock's speed, which is constant, in the one-half second interval between t = 1 and t = 1 .

If we zoomed in on an even shorter interval starting at t t = 1, we could draw still another straight line. We might, for instance, focus on the interval between = 1, we could draw still another straight line. We might, for instance, focus on the interval between t t = 1 second and = 1 second and t t = 1 = 1 seconds. The new line, too, would have a slope that we could compute. We could repeat the procedure still another time, this time focusing on a yet-shorter interval, say between seconds. The new line, too, would have a slope that we could compute. We could repeat the procedure still another time, this time focusing on a yet-shorter interval, say between t t = 1 second and = 1 second and t t = 1 seconds. And so on. = 1 seconds. And so on.

Newton and Leibniz saw that you could continue drawing new straight lines forever. forever. Pictorially, you would be drawing straight lines that pa.s.sed through two dots on the curve. One dot was fixed in place at Pictorially, you would be drawing straight lines that pa.s.sed through two dots on the curve. One dot was fixed in place at t t = 1, and the other moved down the curve, like a bead on a wire, approaching ever nearer to the fixed dot. = 1, and the other moved down the curve, like a bead on a wire, approaching ever nearer to the fixed dot.

Those lines would approach ever nearer to one particular straight line. That "target" line was unique, in a natural way-it was the line that just grazed the curve at a single place, the point corresponding to t t = 1. The target line-in mathematical jargon the tangent line-was the prize that all the fuss was about. (In the diagram below, the tangent line is the straight line made up of short dashes.) Until this moment, mathematicians had never managed to close their fingers around the notion of instantaneous speed. Now they had it. = 1. The target line-in mathematical jargon the tangent line-was the prize that all the fuss was about. (In the diagram below, the tangent line is the straight line made up of short dashes.) Until this moment, mathematicians had never managed to close their fingers around the notion of instantaneous speed. Now they had it.

This was an enormous breakthrough, and perhaps a recap is in order to make sure we see just what Newton and Leibniz had done. They had found a way to define a moving object's speed at a given instant. Instantaneous speed was the number that average speeds approached Instantaneous speed was the number that average speeds approached, as you looked at shorter and shorter time intervals. as you looked at shorter and shorter time intervals.

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The slope of the tangent line (short dashes) represents the speed of a falling rock at the instant t = 1 second.

Instantaneous speed wasn't a paradoxical idea or an arcane one. You could grab it and examine it at your leisure. The speed of a moving object at a given instant was just an ordinary number, the slope of the tangent line at that point. How did you compute that slope? By looking at the slopes of the straight lines that approached the tangent line and seeing if those numbers approached a limit. That limit was the number we were after, the grail in this long quest.

In the case of Galileo's rock, Newton and Leibniz found that the rock's speed at the instant when it had been falling for 1 second was precisely 32 feet per second. They discovered an array of tricks that made such calculations easy whenever you had an equation to work with. Nearly always, you did. (I will skip the procedure, but it is a hint at how neatly things work out that the number they ended up with in the Galileo example-32-can be written as 16 2, and there were both a 16 and a 2 lurking in the equation of the curve they started with, d d = 16 = 16 t t.) Better yet, the same calculation that revealed the rock's speed at a single instant also told its speed at every every instant. Without bothering to lift a finger or draw another straight line (let alone an infinite sequence of straight lines homing in on a target line), this once-and-for-all calculation showed that the rock's speed at any time instant. Without bothering to lift a finger or draw another straight line (let alone an infinite sequence of straight lines homing in on a target line), this once-and-for-all calculation showed that the rock's speed at any time t t was precisely 32 was precisely 32t. The speed was always changing, but a single formula captured all the changes. When the rock had been falling for 2 seconds, its speed was 64 feet per second (32 2). At 2 seconds, its speed was 80 feet per second (32 2 ); at three seconds, 96 feet per second, and so on.

This new tool for describing the moving, changing world was called calculus. With its discovery, every scientist in the world suddenly held in his hands a magical machine. Pose a question that asked how far? how fast? how high? how far? how fast? how high? and then press a b.u.t.ton, and the machine spit out the answer. Calculus made it easy to take a snapshot-to freeze the action at any given instant-and then to examine, at leisure, an arrow momentarily motionless against the sky or an athlete hovering in midleap. and then press a b.u.t.ton, and the machine spit out the answer. Calculus made it easy to take a snapshot-to freeze the action at any given instant-and then to examine, at leisure, an arrow momentarily motionless against the sky or an athlete hovering in midleap.

Questions that had been out of reach forever now took only a moment. How fast is a high diver traveling when she hits the water? If you shoot a rifle with the barrel at a given angle, how far will the bullet travel? What will its speed be when it reaches its target? If a drunken reveler shoots a pistol in the air to celebrate, how high will the bullet rise? More to the point, how fast will it be traveling when it returns to the ground?

Calculus was "the philosopher's stone that changed everything it touched to gold," one historian wrote, and he seemed almost resentful of the new tool's power. "Difficulties that would have baffled Archimedes were easily overcome by men not worthy to strew the sand in which he traced his diagrams."

Chapter Forty-Two.

When the Cable Snaps If infinity had not always inspired terror, like some mythological dragon blocking access to a castle, someone would have discovered calculus long before Newton and Leibniz. They did not slay the dragon-the crucial concepts in calculus all hinge on infinity-but they did manage to capture and tame it. Their successors harnessed it to a plow and set it to work. For the next two centuries, science would consist largely of finding ways to exploit the new power that calculus provided. Patterns once invisible to the naked eye now showed up in vivid color. Galileo had expended huge amounts of effort to come up with the law of falling bodies, for example, but his d d = 16 = 16 t t equation contained far more information than he ever knew. Without calculus he could not see it. With calculus, there was no missing it. equation contained far more information than he ever knew. Without calculus he could not see it. With calculus, there was no missing it.

Galileo knew that his law described position; he didn't know that it contained within itself a hidden law that described speed. Better yet, the law describing position was complicated, the law describing speed far simpler. In words, Galileo's position law says that after t t seconds have pa.s.sed, an object's distance from its starting point is 16 seconds have pa.s.sed, an object's distance from its starting point is 16 t t feet. It is the feet. It is the t t in that equation, rather than a simple in that equation, rather than a simple t t, that makes life complicated. As we have seen, calculus takes that law and, with only the briefest of calculations, extracts from it a new law, this one for the speed of a falling object. In words, when an object has been falling for t t seconds, its speed is exactly 32 seconds, its speed is exactly 32t feet per second. In symbols (using feet per second. In symbols (using v v for velocity), for velocity), v v = 32 = 32t.

That tidy speed equation contains three surprises. First, it's simple. There's no longer any need to worry about messy numbers like t t. Plain old t t will do. Second, it holds for will do. Second, it holds for every every falling object, pebbles and meteorites alike. Third, this single equation tells you a falling object's speed at falling object, pebbles and meteorites alike. Third, this single equation tells you a falling object's speed at every every instant instant t t, whether t t represents 1 second or 5.3 seconds or 50. There's never any need to switch to a new equation or to modify this one. For a complete description of falling objects, this is the only equation you'll ever need. represents 1 second or 5.3 seconds or 50. There's never any need to switch to a new equation or to modify this one. For a complete description of falling objects, this is the only equation you'll ever need.

We began with a law that described the position of a falling body and saw that it concealed within itself a simpler law describing speed. Scientists looked at that speed law and saw that it, too, concealed within itself a simpler law. And that that one, that gem inside a gem, is truly a fundamental insight into the way the world works. one, that gem inside a gem, is truly a fundamental insight into the way the world works.

What is speed? It's a measure of how fast you're changing position. It is, to put it in a slightly more general way, a rate of change. (To be barreling down the highway at 80 miles per hour means that you're changing position at a rate of 80 miles every hour.) If we repeat the same process, by starting with speed and looking at its its rate of change-in other words, if we compute the falling rock's acceleration-what do we find? rate of change-in other words, if we compute the falling rock's acceleration-what do we find?

We find good news. Calculus tells us, literally at a glance, that a falling rock's acceleration never changes. Unlike position, that is, which depends on time in a complicated way, and unlike speed, which depends on time in a simpler way, acceleration doesn't depend on time at all. Whether a rock has been falling for one second or ten, its acceleration is always the same. It is always 32 feet per second per second. Every second that a rock continues to fall, in other words, its speed increases by another 32 feet per second. This This is nature's doubly concealed secret. is nature's doubly concealed secret.

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When a rock falls, its position changes in a complicated way, its velocity in a simpler way, and its acceleration in the simplest possible way.

There is a pattern in the position column, but it hardly blazes forth. The pattern in the speed column is less obscure. The pattern in the acceleration column is transparent. What do all falling objects have in common? Not their weight or color or size. Not the height they fall from or the time they take to reach the ground or their speed on impact or their greatest speed. What is true of all falling objects-an elevator snapping its cables, an egg slipping through a cook's fingers, Icarus with the wax melting from his wings-is that they all accelerate at precisely the same rate.

Acceleration is a familiar word ("the acceleration in my old car was just pitiful"), but it is a remarkably abstract notion. "It is not a fundamental quant.i.ty, such as length or ma.s.s," writes the mathematician Ian Stewart. "It is a rate of change. In fact, it is a 'second order' rate of change-that is, a rate of change of a rate of change." is a familiar word ("the acceleration in my old car was just pitiful"), but it is a remarkably abstract notion. "It is not a fundamental quant.i.ty, such as length or ma.s.s," writes the mathematician Ian Stewart. "It is a rate of change. In fact, it is a 'second order' rate of change-that is, a rate of change of a rate of change."