The Clockwork Universe Part 10

Acceleration is a measure of how fast velocity is changing, in other words, and that's tricky, because velocity is a measure of how fast position is changing. "You can work out distances with a tape measure," Stewart goes on, "but it is far harder to work out a rate of change of a rate of change of distance. This is why it took humanity a long time, and the genius of a Newton, to discover the law of motion. If the pattern had been an obvious feature of distances, we would have pinned motion down a lot earlier in our history."

Acceleration turns out to be a fundamental feature of the world-unless we understand it, whole areas are off-limits to us-but it does not correspond to anything tangible. We can run a finger across a pineapple's p.r.i.c.kly surface or heft a brick or feel the heat of a cup of coffee even through gloved hands. We could put the brick on a scale or take a ruler and measure it. Acceleration seems different from such things as the pineapple's texture and the brick's weight. We can can measure it, but only in an indirect and c.u.mbersome way, and we cannot quite touch it. measure it, but only in an indirect and c.u.mbersome way, and we cannot quite touch it.

But it is this elusive, abstract property, Newton and Leibniz showed, that tells us how objects fall. Once again, seeing nature's secrets required looking through a mathematical lens.

Calculus had still more riches to offer. It not only revealed that distance, speed, and acceleration were all closely linked, for instance, but also showed how to move from any one of them to any of the others. That was important practically-if you wanted to know about speed, say, but you only had tools to measure time and distance, you could still find all the information you wanted, and you could do it easily-and it was important conceptually. Galileo invested endless hours in showing that if you shoot an arrow or throw a ball it travels in a parabola. Newton and Leibniz reached the same conclusion with hardly any work. All they had to know was that falling objects accelerate at 32 feet per second per second. That single number, decoded with calculus's aid, tells you almost at once that cannonb.a.l.l.s and arrows and leaping kangaroos all travel in parabolas.

Again and again, simple observations or commonplace equations transformed themselves into wondrous insights, the mathematical counterpart of Proust's "little pieces of paper" that "the moment they are immersed in [water] stretch and shape themselves, color and differentiate, become flowers, houses, human figures, firm and recognizable."

Calculus was a device for a.n.a.lyzing how things change as time pa.s.ses. Just what those things were made no difference. How long will it take the world's population to double? How many thousands of years ago was this mummy sealed in his tomb? How soon will the oyster harvest in the Chesapeake Bay fall to zero?

Questions about bests and worsts, when this quant.i.ty was at a maximum or that one at a minimum, could also be readily answered. Of all the roller-coaster tracks that start at a peak here and end at a valley there, which is fastest? Of all the ways to fire a cannon at a fortress high above it on a mountain, which would inflict the heaviest damage? (This was Halley's contribution, written almost as soon as he had heard of calculus. It turns out that he had also found the best angle to shoot a basketball so that it swishes through the hoop.) Of all the shapes of soap bubble one can imagine, which encloses the greatest volume with the least surface? (Nature chooses the ideal solution, a spherical bubble.) Of all the ticket prices a theater could charge, which would bring in the most money?

Not every situation could be a.n.a.lyzed using the techniques of calculus. If in a tiny stretch of time a picture changed only a tiny bit, then calculus worked perfectly. From one millisecond to the next, for instance, a rocket or a sprinter advanced only a tiny distance, and calculus could tell you everything about their paths. But in the strange circ.u.mstances where something shifts abruptly, where the world jumps from one state to a different one entirely without pa.s.sing through any stages in between, then calculus is helpless. (If you're counting the change in your pocket, for example, no coin is smaller than a penny, and so you jump from "twelve cents" to "thirteen cents" to "fourteen cents," with nothing in between.) One of the startling discoveries of twentieth-century science was that the subatomic world works in this herky-jerky way. Electrons jump from here to there, for instance, and in between they are... nowhere. Calculus throws up its hands.

But in the world we can see, most change is smooth and continuous. And whenever anything changes smoothly-when a boat cuts through the water or a bullet slices through the air or a comet speeds across the heavens, when electricity flows or a cup of coffee cools or a river meanders or the high, quavering notes of a violin waft across a room-calculus provides the tools to probe that change.

Scientists wielding the new techniques talked as if they had witnessed sorcery. The old methods compared to the new, one dazed astronomer exclaimed, "as dawn compares to the bright light of noon."

Chapter Forty-Three.

The Best of All Possible Feuds For a long while, Newton and Leibniz spoke of one another in the most flattering terms. Newton wrote Leibniz a friendly letter in 1693, nearly a decade after Leibniz had claimed calculus for himself, hailing Leibniz as "one of the chief geometers of this century, as I have made known on every occasion that presented itself." Surely, Newton went on, there was no need for the two men to squabble. "I value my friends more than mathematical discoveries," the friendless genius declared.

Leibniz was even more effusive. In 1701, at a dinner at the royal palace in Berlin, the queen of Prussia asked Leibniz what Newton had achieved. "Taking Mathematicks from the beginning of the world to the time of Sir Isaac," Leibniz replied, "what he had done was much the better half."

But the kind words were a sham. For years, both rivals had carefully praised one another on the record while slandering each other behind the scenes. Each man composed detailed, malicious attacks on the other and published them anonymously. Each whispered insults and accusations into the ears of colleagues and then professed shock and dismay at hearing his own words parroted back.

The two geniuses had admired one another, more or less, until they realized they were rivals. Newton had long thought of the mult.i.talented Leibniz as a dabbler in mathematics, a brilliant beginner whose genuine interests lay in philosophy and law. Leibniz had no doubts about Newton's mathematical prowess, but he believed that Newton had focused his attention in one specific, limited area. That left Leibniz free to pursue calculus on his own, or so he believed.

By the early 1700s, the clash had erupted into the open. For the next decade and a half, the fighting would grow ever fiercer. Two of the greatest thinkers of the age both clutched the same golden trophy and shouted, "Mine!" Both men were furious, indignant, unrelenting. Each felt sure the other had committed theft and compounded it with slander. Each was convinced his enemy had no motive beyond a blind l.u.s.t for acclaim.

Because calculus was the ideal tool to study the natural world, the debate spilled over from mathematics to science and then from science to theology. What was the nature of the universe? What was the nature of G.o.d, who had designed that universe? Almost no one could understand the technical issues, but everyone enjoyed the sight of intellectual t.i.tans grappling like mud wrestlers. Coffeehouse philosophers weighed in; dinner parties bubbled over with gossip and delicious rumor; aristocrats across Europe chortled over the nastiest insults; in England even the royal family grew deeply involved, reviewing tactics and egging on the combatants. What began as a philosophers' quarrel grew and transmogrified until it became, in the words of the historian Daniel Boorstin, "the spectacle of the century."

Royalty came into the story-and threw an even brighter spotlight on Newton and Leibniz-because of Europe's complicated dynastic politics. When England's Queen Anne died without an heir, in 1714, the throne pa.s.sed not to Anne's nearest relative but, so great was the fear of Catholic power, to her nearest Protestant Protestant relative. This was a fifty-four-year-old German n.o.bleman named Georg Ludwig, Duke of Hanover, a brave, bug-eyed ex-soldier of no particular distinction. In England Georg Ludwig would rule as King George I. relative. This was a fifty-four-year-old German n.o.bleman named Georg Ludwig, Duke of Hanover, a brave, bug-eyed ex-soldier of no particular distinction. In England Georg Ludwig would rule as King George I.

Fond of women and cards but little else, the future king had, according to his mother, "round his brains such a thick crust that I defy any man or woman ever to discover what is in them." No matter, for Georg Ludwig had the next best thing to brains of his own. He had Europe's most renowned intellectual, Gottfried Wilhelm Leibniz, permanently on tap and at the ready.

For nearly forty years, Leibniz had served Georg Ludwig (and his father before him and that father's brother before him him), as historian, adviser, and librarian in charge of cataloging and enlarging the ducal book collection. Among his other tasks, Leibniz had labored to establish the Hanoverian claim to the English throne. Now, with his patron suddenly plucked from the backwaters of Germany and dropped into one of the world's plum jobs, Leibniz saw a chance to return to a world capital. He had visions of accompanying his longtime employer, taking his proper place on a brightly lit stage, and trading ideas with England's greatest thinkers. Georg Ludwig had a different vision.

By the time of King George's coronation, Isaac Newton had long since made his own dazzling ascent. In 1704, he had published his second great work, Opticks Opticks, on the properties of light. In 1705, the onetime farmboy had become Sir Isaac Newton, the first scientist ever knighted. (Queen Anne had performed the ceremony. Anne was no scholar-"When in good humour Queen Anne was meekly stupid, and when in bad humor, was sulkily stupid," the historian Macaulay had observed-but she had savvy counselors who saw political benefit in honoring England's greatest thinker.) By the time of his knighthood, Newton was sixty-two and had largely abandoned scientific research. A few years before, he had left Cambridge in favor of London and accepted a government post as warden of the Mint. At roughly the same time he took on the presidency of the Royal Society, a position he would hold until his death. Old, imposing, intimidating, Newton was universally hailed as the embodiment of genius. English genius, in particular. Many who could not tell a parrot from a parabola gloried in the homage paid to England's greatest son. When dignitaries like Russia's Peter the Great visited London, they made a point of seeing Newton along with the capital's other marvels.

Newton did not become much of a partygoer in his London days, but his new circle of acquaintances did come to include such ornaments as Caroline, Princess of Wales. King George himself kept a close watch on the Newton-Leibniz affair. His motive was not intellectual curiosity-the king's only cultural interests were listening to opera and cutting out paper dolls-but he took malicious delight in having a claim on two of the greatest men of the age. King George seemed an unlikely candidate to preside over a philosophical debate. In Germany his court had been caught up not only in scandal but quite likely in murder.

The problems rose out of a tangled series of romantic liaisons. All the important men at the Hanover court had mistresses, often several at a time, and a diagram of whose bed partner was whose would involve multiple arrows crossing one another and looping back and forth. (Adding to the confusion, nearly all the female partic.i.p.ants in the drama seemed to share the name Sophia or some near variation.) Bed-hopping on the part of the Hanover princes fell well within the bounds of royal privilege. What was not not acceptable was that Georg Ludwig's wife, Sophia Dorothea, had embarked on an affair of her own. Royal spies discovered that the lovers had made plans to run off together. This was unthinkable. A team of hired a.s.sa.s.sins ambushed the d.u.c.h.ess's paramour, stabbed him with a sword, sliced him open with an axe, and left him to bleed to death. Sophia Dorothea was banished to a family castle and forbidden ever to see her children again. She died thirty-two years later, still under house arrest. acceptable was that Georg Ludwig's wife, Sophia Dorothea, had embarked on an affair of her own. Royal spies discovered that the lovers had made plans to run off together. This was unthinkable. A team of hired a.s.sa.s.sins ambushed the d.u.c.h.ess's paramour, stabbed him with a sword, sliced him open with an axe, and left him to bleed to death. Sophia Dorothea was banished to a family castle and forbidden ever to see her children again. She died thirty-two years later, still under house arrest.

Through the years Leibniz's attempts to engage Georg Ludwig had met with about the success one would expect, but the women of the Hanoverian court were as intellectual as the men were crude. While the dukes collected mistresses and plotted murder, their d.u.c.h.esses occupied themselves with philosophy. Georg Ludwig's mother, Sophia, read through Spinoza's controversial writings as soon as they were published and spent long hours questioning Leibniz about the views of the Dutch heretic.

Sophia was only the first of Leibniz's royal devotees. Sophia's daughter Sophia Charlotte (sister to the future King George) had an even closer relationship with Leibniz. And yet a third high-born woman forged a still closer bond. This was Caroline, a twenty-one-year-old princess and friend of Sophia Charlotte. Leibniz became her friend and tutor. Soon after, Caroline married one of Georg Ludwig's brothers. When she was whisked off to England in 1714, Caroline became princess of Wales and in time, as the wife of King George II, queen of England. Leibniz had allies in the highest of circles.

But he was stuck in Germany, and none of his royal friends seemed inclined to send for him. From that outpost, he tried to enlist Caroline on his side in his ongoing war against Newton. Their battle represented not just a confrontation between two men, Leibniz insisted, but between two nations. German pride was at stake. "I dare say," Leibniz wrote to Caroline, "that if the king were at least to make me the equal of Mr. Newton in all things and in all respects, then in these circ.u.mstances it would give honor to Hanover and to Germany in my name."

The appeal to national pride proved ineffective. Newton was all but worshipped in England-as we have noted, Caroline had met him on various grand occasions at court-and the newly arrived king had no desire to challenge English self-regard just to soothe the hurt feelings of his pet philosopher. In any case, King George had his own plans for Leibniz. They did not include science. Leibniz's chief duty, the king reminded him, was to continue his history of the House of Hanover. He had bogged down somewhere around the year 1000.

The wonders of calculus, and the injustice of Newton's theft of it, concerned the king not at all. What was life and death for Leibniz was sport for King George. "The king has joked more than once about my dispute with Mr. Newton," Leibniz lamented.

From his exile in Hanover, Leibniz wrote to Caroline attacking Newton's views on science and theology. Caroline studied the letters intently-they dealt mainly with such questions as whether G.o.d had left the world to run on its own or whether He continued to step in to fine-tune it-and she pa.s.sed them along to a Newton stand-in named Samuel Clarke. On some questions Caroline wrote directly to Newton himself. Clarke composed responses to Leibniz (with Newton's help). The correspondence was soon published, and the so-called Leibniz-Clarke papers became, in one historian's judgment, "perhaps the most famous and influential of all philosophical correspondences."

But to Caroline's exasperation, Leibniz persisted in setting aside deep issues in theology and circling back instead to his priority battle with Newton. The princess scolded her ex-tutor for his "vanity." He and Newton were "the great men of our century," Caroline wrote, "and both of you serve a king who merits you." Why draw out this endless fight? "What difference does it make whether you or Chevalier Newton discovered the calculus?" Caroline demanded.

A good question. The world had the benefit of this splendid new tool, after all, whoever had found it. But to Newton and Leibniz, the answer to Caroline's question was simple. It made all the difference in the world.

Chapter Forty-Four.

Battle's End From its earliest days, science has been a dueling ground. Disputes are guaranteed, because good ideas are "in the air," not dreamed up out of nowhere. Nearly every breakthrough-the telescope, calculus, the theory of evolution, the telephone, the double helix-has multiple parents, all with serious claims. But ownership is all, and scientists turn purple with rage at the thought that someone has won praise for stolen insights. The greats battle as fiercely as the mediocre. Galileo wrote furiously of rivals who claimed that they, not he, had been first to see sunspots. They had, he fumed, "attempted to rob me of that glory which was mine." Even the peaceable Darwin admitted, in a letter to a colleague urging him to write up his work on evolution before he was scooped, that "I certainly should be vexed if anyone were to publish my doctrines before me."

What vexed the mild Darwin sent Newton and Leibniz into apoplectic rages. The reasons had partly to do with mathematics itself. All scientific feuds tend toward the nasty; feuds between mathematicians drip with extra venom. Higher mathematics is a peculiarly frustrating field. So difficult is it that even the best mathematicians often feel that the challenge is just too much, as if a golden retriever had taken on the task of understanding the workings of the internal combustion engine. The rationalizations so helpful elsewhere in science-she had a bigger lab, a larger budget, better colleagues-are no use here. Wealth, connections, charm make no difference. Brainpower is all.

"Almost no one is capable of doing significant mathematics," the American mathematician Alfred W. Adler wrote a few decades ago. "There are no acceptably good mathematicians. Each generation has its few great mathematicians, and mathematics would not even notice the absence of the others. They are useful as teachers, and their research harms no one, but it is of no importance at all. A mathematician is great or he is nothing."

That is a romantic view and probably overstated, but mathematicians take a perverse pride in great-man theories, and they tend to see such doctrines as simple facts. The result is that mathematicians' egos are both strong and brittle, like ceramics. Where they focus their gaze makes all the difference. If someone compares himself with his neighbors, then he might preen himself on his membership in an arcane priesthood. But if he judges himself not by whether he knows more mathematics than most people but by whether he has made any real headway at exploring the immense and dark mathematical woods, then all thoughts of vanity flee, and only puniness remains.

In the case of calculus, the moment of confrontation between Newton and Leibniz was delayed for a time, essentially by incredulity. Neither genius could quite believe that anyone else could have seen as far as he had. Newton enjoyed his discoveries all the more because they were his to savor in solitude, as if he were a reclusive art collector free to commune with his masterpieces behind closed doors. But Newton's retreat from the world was not complete. He could abide adulation but not confrontation, and he had shared some of his mathematical triumphs with a tiny number of appreciative insiders. He ignored their pleas that he tell everyone what he had told them. The notion that his discoveries would speed the advance of science, if only the world knew of them, moved Newton not at all.

For Leibniz, on the other hand, his discoveries had value precisely because they put his merits on display. He never tired of gulping down compliments, but his eagerness for praise had a practical side, too. Each new achievement served as a golden entry on the resume that Leibniz was perpetually thrusting before would-be patrons.

In Newton's view, to unveil a discovery meant to offer the unworthy a chance to paw at it. In Leibniz's view, to proclaim a discovery meant to offer the world a chance to shout its hurrahs.

In history's long view, the battle ended in a stalemate. Historians of mathematics have scoured the private papers of both men and found clear evidence that Newton and Leibniz discovered calculus independently, each man working on his own. Newton was first, in 1666, but he only published decades later, in 1704. Leibniz's discovery followed Newton's by nine years, but he published his findings first, in 1684. And Leibniz, who had a gift for devising useful notations, wrote up his discoveries in a way that other mathematicians found easy to understand and build upon. (Finding the right notation to convey a new concept sounds insignificant, like choosing the right typeface for a book, but in mathematics the choice of symbols can save an idea or doom it. A child can multiply 17 by 19. The greatest scholars in Rome would have struggled with XVII times XIX.)47 The symbols and language that Leibniz devised are still the ones that students learn today. Newton's discovery was identical, at its heart, and in his masterly hands it could be turned to nearly any task. But Newton's calculus is a museum piece today, while a buffed and honed version of Leibniz's remains in universal use. Newton insisted that because he had found calculus before anyone else, there was nothing to debate. Leibniz countered that by casting his ideas in a form that others could follow, and then by telling the world what he had found, he had thrown open a door to a new intellectual kingdom.

So he had, and throughout the 1700s and into the 1800s, European mathematicians inspired by Leibniz ran far in front of their English counterparts. But in their lifetimes, Newton seemed to have won the victory. To stand up to Newton at his peak of fame was nearly hopeless. The awe that Alexander Pope would later encapsulate-"Nature and nature's laws lay hid in night, / G.o.d said 'Let Newton be!' and all was light"-had already become common wisdom.

The battle between the two men smoldered for years before it burst into open flames. In 1711, after about a decade of mutual abuse, Leibniz made a crucial tactical blunder. He sent the Royal Society a letter-both he and Newton were members-complaining of the insults he had endured and asking the Society to sort out the calculus quarrel once and for all. "I throw myself on your sense of justice," he wrote.

He should have chosen a different target. Newton, who was president of the Royal Society, appointed an investigatory committee "numerous and skilful and composed of Gentlemen of several Nations." In fact, the committee was a rubber stamp for Newton himself, who carried out its inquiry single-handedly and then issued his findings in the committee's name. The report came down decisively in Newton's favor. With the Royal Society's imprimatur, the long, d.a.m.ning report was distributed to men of learning across Europe. "We take the Proper Question to be not who Invented this or that Method but who was the first Inventor," Newton declared, for the committee.

The report went further. Years before, it charged, Leibniz had been offered surrept.i.tious peeks at Newton's mathematical papers. There calculus was "Sufficiently Described" to enable "any Intelligent Person" to grasp its secrets. Leibniz had not only lagged years behind Newton in finding calculus, in other words, but he was a sneak and a plagiarist as well.

Next the Philosophical Transactions Philosophical Transactions, the Royal Society's scientific journal, ran a long article reviewing the committee report and repeating its anti-Leibniz charges. The article was unsigned, but Newton was the author. Page after page spelled out the ways in which "Mr. Leibniz" had taken advantage of "Mr. Newton." Naturally Mr. Leibniz had his own version of events, but the anonymous author would have none of it. "Mr. Leibniz cannot be a witness in his own Cause."

Finally the committee report was republished in a new edition accompanied by Newton's anonymous review. The book carried an anonymous preface, "To the Reader." It, too, was written by Newton.

Near the end of his life Newton reminisced to a friend about his long-running feud. "He had," he remarked contentedly, "broke Leibniz' heart."

Chapter Forty-Five.

The Apple and the Moon The greatest scientific triumph of the seventeenth century, Newton's theory of universal gravitation, was in a sense a vehicle for showing off the power and range of the mathematical techniques that Newton and Leibniz had fought to claim. Both men discovered calculus, but it was Newton who provided a stunning demonstration of what it could do.

Until 1687, Isaac Newton had been known mainly, to those who knew him at all, as a brilliant mathematician who worked in self-imposed isolation. No recluse ever broke his silence more audaciously.

Fame came with the publication of the Principia Principia. Newton had been at Cambridge for two decades. University rules required that he teach a cla.s.s or two, but this did not impose much of a burden, either on Newton or anyone else. "So few went to hear Him, & fewer that understood him," one contemporary noted, "that oftimes he did in a manner, for want of Hearers, read to ye Walls."

As Newton told the story, his rise to fame had indeed begun with the fall of an apple. In his old age he occasionally looked back on his career, and eager listeners noted down every word. A worshipful young man named John Conduitt, the husband of Newton's niece, was one of several who heard the apple story firsthand. "In the year 1666 he retired again from Cambridge... to his mother in Lincolnshire," Conduitt wrote, "& whilst he was musing in a garden it came into his thought that the power of gravity (which brought an apple from the tree to the ground) was not limited to a certain distance from the earth but that this power must extend much farther than was usually thought. Why not as high as the moon said he to himself & if so that must influence her motion & perhaps retain her in her orbit, whereupon he fell a calculating...."

The story, which is the one thing everyone knows about Isaac Newton, may well be a myth.48 Despite his craving for privacy, Newton was acutely aware of his own legend, and he was not above adding a bit of gloss here and there. Historians who have scrutinized his private papers believe that his understanding of gravity dawned slowly, over several years, rather than in a flash of insight. He threw in the apple, some suspect, simply for color. Despite his craving for privacy, Newton was acutely aware of his own legend, and he was not above adding a bit of gloss here and there. Historians who have scrutinized his private papers believe that his understanding of gravity dawned slowly, over several years, rather than in a flash of insight. He threw in the apple, some suspect, simply for color.

In any case, it wouldn't have taken an apple to remind Newton that objects fall. Everyone had always known that. The point was to look beyond that fact to the questions it raised. If apples fell to the ground because some force drew them, did that force extend from the tree's branches to its top? And beyond the top to... to where? To the top of a mountain? To the clouds? To the moon? Those questions had seldom been asked. There were many more. What about the apple when it was not not falling? An apple in a tree stays put because it is attached to the branch. No surprise there. But what about the moon? What holds the moon up in the sky? falling? An apple in a tree stays put because it is attached to the branch. No surprise there. But what about the moon? What holds the moon up in the sky?

Before Newton the answer had two parts. The moon stayed in the sky because that was its natural home and because it was made of an ethereal substance that was nothing like the heavy stuffing of bodies here on Earth. But that would no longer do. If the moon was just a big rock, as telescopes seemed to show, why didn't it fall like other rocks?

The answer, Newton came to see, was that it does it does fall fall. The breakthrough was to see how that could be. How could something fall and fall but never arrive? Newton's answer in the case of the moon, a natural satellite, ran much like the argument we have already seen, for an artificial satellite.

We tend to forget the audacity of that explanation, and Newton's plain tone helps us along in our mistake. "I began to think of gravity extending to ye orb of the Moon," he recalled, as if nothing could have been more natural. Newton began to give serious thought, in other words, to asking whether the same force that pulled an apple to the Earth also pulled the moon toward the Earth. But this is to downplay two feats of intellectual daring. Why should anyone have thought that the moon is falling, first of all, when it is plainly hanging placidly in the sky, far beyond our reach or the reach of anything else? And even if we did make the large concession that it is falling, second of all, why should that fall have anything in common with an apple's fall? Why would anyone presume that the same rules governed realms as different as heaven and Earth?

But that is exactly what Newton did presume, for aesthetic and philosophical reasons as much as for scientific ones. Throughout his life Newton believed that G.o.d operated in the simplest, neatest, most efficient way imaginable. That principle served as his starting point whether he was studying the Bible or the natural world. (We have already noted his insistence that "it is ye perfection of G.o.d's works that they are all done with ye greatest simplicity.") The universe had no superfluous parts or forces for exactly the reason that a clock had no superfluous wheels or springs. And so, when Newton's thoughts turned to gravity, it was all but inevitable that he would wonder how much that single force could explain.

Newton's first task was to find a way to turn his intuition about the sweep and simplicity of nature's laws into a specific, testable prediction. Gravity certainly seemed to operate here on Earth; if it did reach all the way to the moon, how would you know it? How would gravity reveal itself? To start with, it seemed clear that if gravity did extend to the moon, its force must diminish over that vast distance. But how much? Newton had two paths to an answer. Fortunately, both gave the same result.

First, he could try intuition and a.n.a.logy. If we see a bright light ten yards off, say, how bright will it be if we move it twice as far away, to twenty yards distance? The answer was well-known. Move a light twice as far away and it will not be half as bright, as you might guess, but only one-fourth as bright. Move it ten times as far away and it will be one-hundredth as bright. (The reason has to do with the way light spreads. Sound works the same way. A piano twenty yards away sounds only one-fourth as loud as a piano ten yards away.) So Newton might have been tempted to guess that the pull of gravity decreases with distance in the same way that the brightness of light does. Physicists today talk about "inverse-square laws," by which they mean that some forces weaken not just in proportion to distance but in proportion to distance squared. (It would later turn out that electricity and magnetism follow inverse-square laws, too.) A second way of looking at gravity's pull gave the same answer. By combining Kepler's third law, which had to do with the size and speed of the planets' orbits, with an observation of his own about objects traveling in a circle, Newton calculated the strength of gravity's pull. Again, he found that gravity obeyed an inverse-square law.

Now came the test. If gravity actually pulled on the moon, how much did it pull? Newton set to work. He knew that the moon orbits the Earth. It travels in a circle, in other words, and not in a straight line. (To be strictly accurate, it travels in an ellipse that is almost but not quite circular, but the distinction does not come into play here.) He knew, as well, what generations of students have since had drummed into them as "Newton's first law"-in modern terms, a body in motion will travel in a straight line at a steady speed unless some force acts on it (and a body at rest will stay at rest unless some force acts on it). a body in motion will travel in a straight line at a steady speed unless some force acts on it (and a body at rest will stay at rest unless some force acts on it).

So some force was acting on the moon, pulling it off a straight-line course. How far off course? That was easy to calculate. To start with, Newton knew the size of the moon's...o...b..t, and he knew that the moon took a month to travel once around that circuit. Taken together, those facts told him the moon's speed. Next came a thought experiment. What would happen to the moon if gravity were magically turned off for a second? Newton's first law gave him the answer-it would shoot off into s.p.a.ce on a straight line, literally going off on a tangent. (If you tied a rock with a piece of string and swung it around your head, the rock would travel in a circle until the string snapped, and then it would fly off in a straight line.) But the moon stays in its circular orbit. Newton knew what that meant. It meant a force was pulling it. Now he needed some numbers. To find out how far the moon was pulled, all he had to do was calculate the distance between where the moon actually is and where it would have been if it had traveled in a straight line. That distance was the fall Newton was looking for-the moon "falls" from a hypothetical straight line to its actual position.

[image]

Newton calculated the distance the moon falls in 1 second, which corresponds to the dashed line in the diagram.

In his quest to compare Earth's pull on the moon and on an apple, Newton was nearly home. He knew how far the moon falls in one second. He had just calculated that. It falls about 1/20 of an inch. He knew how far an apple falls in one second. Galileo had found that out, with his ramps: 16 feet.

All that remained was to look at the ratio of those two falls, the ratio of 1/20 of an inch to 16 feet. The last puzzle piece was the distance from the Earth to the moon. Why did that matter? Because the distance from the Earth to the moon was about 60 times the distance from the center of the Earth to the Earth's surface. Which was to say that the moon was 60 times as far from the center of the Earth as the apple was. If gravity truly did follow an inverse-square law, then the Earth's pull on the moon should be 3,600 times weaker (60 60) than its pull on the apple.

Only the last, crucial calculation remained. The moon fell 1/20 of an inch in one second; an apple fell 16 feet in one second. Was the ratio of 1/20 of an inch to 16 feet the same as the ratio of 1 to 3,600, as Newton had predicted? How did the moon's fall compare with the apple's fall?

Just as Newton had hoped it would, or nearly so. The two ratios almost matched. Newton "compared the force required to keep the Moon in her Orb with the force of gravity," he wrote proudly, "& found them answer pretty nearly." The same calculation carried out today, with far better data than Newton had available, would give even closer agreement. That wasn't necessary. The big message was already clear. Gravity reached from the Earth to the moon. The same force that drew an apple drew the moon. The same law held here and in the heavens. G.o.d had indeed designed his cosmos with "ye greatest simplicity."

Chapter Forty-Six.

A Visit to Cambridge Newton's moon calculation had b.u.t.tressed his faith in simple laws, but he still had an immense distance to cover before he could prove his case. The moon was not the universe. What of Kepler's laws, for instance? The great astronomer had devoted his life to proving that the planets traveled around the sun in ellipses. How did ellipses fit in G.o.d's cosmic architecture?

Stymied by the difficulty of sorting out gravity, or perhaps tempted more by questions in other fields, Newton had put gravity aside after his miracle years. He had made his apple-and-moon calculation when he was in his twenties. For the next twenty years he gave most of his attention to optics, alchemy, and theology instead.

Late on a January afternoon in 1684, Robert Hooke, Christopher Wren, and Edmond Halley left a meeting of the Royal Society and wandered into a coffeehouse to pick up a conversation they had been carrying on all day. Coffee had reached England only a generation before, but coffeehouses had spread everywhere.49 Hooke in particular seemed to thrive in the rowdy atmosphere. In crowded rooms thick with the hubbub of voices and the smells of coffee, chocolate, and tobacco, men sat for hours debating business, politics, and, lately, science. (Rumors and "false news" spread so quickly, as with the Internet today, that the king tried, unsuccessfully, to shut coffeehouses down.) Hooke in particular seemed to thrive in the rowdy atmosphere. In crowded rooms thick with the hubbub of voices and the smells of coffee, chocolate, and tobacco, men sat for hours debating business, politics, and, lately, science. (Rumors and "false news" spread so quickly, as with the Internet today, that the king tried, unsuccessfully, to shut coffeehouses down.) With steaming mugs in hand, the three men resumed talking of astronomy. All three had already guessed, or convinced themselves by the same argument Newton had made using Kepler's third law, that gravity obeyed an inverse-square law. Now they wanted the answer to a related question-if the planets did follow an inverse-square law, what did that tell you about their orbits? This question-in effect, where do Kepler's laws come from?- where do Kepler's laws come from?-was one of the central riddles confronting all the era's scientists.

Halley, a skilled mathematician, admitted to his companions that he had tried to find an answer and failed. Wren, still more skilled, confessed that his his failures had stretched over the course of several years. Hooke, who was sometimes derided as the "universal claimant" for his habit of insisting that every new idea that came along had occurred to him long before, said that he'd solved this problem, too. For the time being, he said coyly, he preferred to keep the answer to himself. "Mr. Hook said that he had it," Halley recalled later, "but that he would conceale it for some time, that others trying and failing might know how to value it, when he should make it publick." failures had stretched over the course of several years. Hooke, who was sometimes derided as the "universal claimant" for his habit of insisting that every new idea that came along had occurred to him long before, said that he'd solved this problem, too. For the time being, he said coyly, he preferred to keep the answer to himself. "Mr. Hook said that he had it," Halley recalled later, "but that he would conceale it for some time, that others trying and failing might know how to value it, when he should make it publick."

Wren, dubious, offered a forty-shilling prize-roughly four hundred dollars today-to anyone who could find an answer within two months. No one did. In August 1684, Halley took the question to Isaac Newton. Halley, one of the few great men of the Royal Society who was charming as well as brilliant, scarcely knew Newton, though he knew his mathematical reputation. But Halley could get along with everyone, and he made a perfect amba.s.sador. Though still only twenty-eight, he had already made his mark in mathematics and astronomy. Just as important, he was game for anything. In years to come he would stumble through London's taverns with Peter the Great, on the czar's visit to London; he would invent a diving bell (in the hope of salvaging treasure from shipwrecks) and would descend deep underwater to test it himself; he would tramp up and down mountains to compare the barometric pressure at the summit and the base; in an era of wooden ships he would survey vast swaths of the world's oceans, from the tropics to "islands of ice."

Now his task was to win over Isaac Newton. "After they had been some time together," as Newton later told the story to a colleague, Halley explained the reason for his visit. He needed Newton's help. The young astronomer spelled out the question that had stumped him, Wren, and Hooke. If the sun attracted the planets with a force that obeyed an inverse-square law, what shape would the planets' orbits be?

"Sir Isaac replied immediately that it would be an Ellipsis." Halley was astonished. "The Doctor struck with joy & amazement asked him how he knew it. Why saith he I have calculated it."

Halley asked if he could see the calculation. Newton rummaged through his papers. Lost. Halley extracted a promise from Newton to work through the mathematics again, and to send him the results.

Chapter Forty-Seven.

Newton Bears Down The paper was not really lost. Newton, the most cautious of men, wanted to reexamine his work before he revealed it to anyone. Looking over his calculations after Halley's visit, Newton did indeed catch a mistake. He corrected it, expanded his notes, and, three months later, sent Halley a formal, nine-page treatise, in Latin, t.i.tled "On the Motion of Bodies in an Orbit." It did far, far more than answer Halley's question.

Kepler's discovery that the planets travel in ellipses, for instance, had never quite made sense. It was a "law" in the sense that it fit the facts, but it seemed dismayingly arbitrary. Why ellipses rather than circles or figure eights? No one knew. Kepler had agonized over the astronomical data for years. Finally, for completely mysterious reasons, ellipses had turned out to be the curves that matched the observations. Now Newton explained where ellipses came from. He showed, using calculus-based arguments, that if a planet travels in an ellipse, then the force that attracts it must must obey an inverse-square law. The flip side was true, too. If a planet orbiting around a fixed point does obey an inverse-square law, then it travels in an ellipse. obey an inverse-square law. The flip side was true, too. If a planet orbiting around a fixed point does obey an inverse-square law, then it travels in an ellipse.50 All this was a matter of strict mathematical fact. Ellipses and inverse-square laws were intimately connected, though it took Newton's genius to see it, just as it had taken a Pythagoras to show that right triangles and certain squares were joined by hidden ties. All this was a matter of strict mathematical fact. Ellipses and inverse-square laws were intimately connected, though it took Newton's genius to see it, just as it had taken a Pythagoras to show that right triangles and certain squares were joined by hidden ties.

Newton had solved the mystery behind Kepler's second law, as well. It, too, summarized countless astronomical observations in one compact, mysterious rule-planets sweep out equal areas in equal times. In his short essay, Newton deduced the second law, as he had deduced the first. His tools were not telescope and s.e.xtant but pen and ink. All he needed was the a.s.sumption that some force draws the planets toward the sun. Starting from that bare statement (without saying anything about the shape of the planets' orbits or whether the sun's pull followed an inverse-square law), Newton demonstrated that Kepler's law had to hold. Mystery gave way to order.

Bowled over, Halley rushed back to Cambridge to talk to Newton again. The world needed to hear what he had found. Remarkably, Newton went along. First, though, he would need to improve his ma.n.u.script.

Thus began one of the most intense investigations in the history of thought. Since his early years at Cambridge, Newton had largely abandoned mathematics. Now his mathematical fever surged up again. For seventeen months Newton focused all his powers on the question of gravity. He worked almost without let-up, with the same ferocious concentration that had marked his miracle years two decades before.

Albert Einstein kept a picture of Newton above his bed, like a teenage boy with a poster of LeBron James. Though he knew better, Einstein talked of how easily Newton made his discoveries. "Nature to him was an open book, whose letters he could read without effort." But the real mark of Newton's style was not ease but power. Newton focused his gaze on whatever problem had newly obsessed him, and then he refused to look away until he had seen to its heart.