Better yet, the same calculation that revealed the rock’s speed at a single instant also told its speed at 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 t 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 ½ seconds, its speed was 80 feet per second (32 × 2 ½); 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? and then press a button, 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 = 16 t2 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 seconds have passed, an object’s distance from its starting point is 16 t2 feet. It is the t2 in that equation, rather than a simple 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 seconds, its speed is exactly 32t feet per second. In symbols (using v for velocity), v = 32t.
That tidy speed equation contains three surprises. First, it’s simple. There’s no longer any need to worry about messy numbers like t2. Plain old t will do. Second, it holds for every falling object, pebbles and meteorites alike. Third, this single equation tells you a falling object’s speed at every instant t, whether 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.
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 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 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 is nature’s doubly concealed secret.
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 quantity, such as length or mass,” 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.”
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 prickly 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 measure it, but only in an indirect and cumbersome 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 cannonballs and arrows and leaping kangaroos all tr
avel 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 analyzing how things change as time passes. 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 quantity 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 analyzed 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 circumstances where something shifts abruptly, where the world jumps from one state to a different one entirely without passing 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 multitalented 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 lust 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 God, who had designed that universe? Almost no one could understand the technical issues, but everyone enjoyed the sight of intellectual titans 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 passed not to Anne’s nearest relative but, so great was the fear of Catholic power, to her nearest Protestant relative. This was a fifty-four-year-old German nobleman 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), 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, 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 participants 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 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 assassins ambushed the duchess’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.
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