The Clockwork Universe

by Edward Dolnick

  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 orbit, 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 space 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.

  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. God had indeed designed his cosmos with “ye greatest simplicity.”

  Chapter Forty-Six

  A Visit to Cambridge

  Newton’s moon calculation had buttressed 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 God’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.)

  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?—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 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 ambassador. 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, titled “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 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.

  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 sextant but pen and ink. All he needed was the assumption 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 manuscript.

  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.

  “Now I am upon this subject,” he told a colleague early in his investigation of gravity, “I would gladly know ye bottom of it before I publish my papers.” The matter-of-fact tone obscures Newton’s drivenness. “I never knew him take any Recreation or Pastime,” recalled an assistant, “either in Riding out to take ye Air, Walking, Bowling, or any other Exercise whatever, thinking all Hours lost that was not spent in his Studyes.” Newton would forget to leave his rooms for meals until he was reminded and then “would go very carelessly, with Shooes down at Heels, Stockings unty’d . . . & his Head scarcely comb’d.”

  Such stories were in the standard vein of anecdotes about absentminded professors, already a cliché in the 1600s,51 except that in Newton’s case the theme was not otherworldly dreaminess but energy and singleness of vision. Occasionally a thought would strike Newton as he paced the grounds near his rooms. (It was not quite true that he never took a walk to clear his head.) “When he has sometimes taken a Turn or two he has made a sudden Stand, turn’d himself about, run up ye stairs & like another Archimedes, with a Eureka!, fall to write on his Desk standing, without giving himself the Leasure to draw a Chair to sit down in.”

  Even for Newton the assault on gravity demanded a colossal effort. The problem was finding a way to move from the idealized world of mathematics to the messy world of reality. The diagrams in Newton’s “On Motion” essay for Halley depicted points and curves, much as you might see in any geometry book. But those points represented colossal, complicated objects like the sun and the Earth, not abstract circles and triangles. Did the rules that held for textbook examples apply to objects in the real world?

  Newton was exploring the notion that all objects attracted one another and that the strength of that attraction depended on their masses and the distance between them. Simple words, it seemed, but they presented gigantic difficulties. What was the distance between the apple and the Earth? For two objects separated by an enormous distance, like the Earth and the moon, the question seemed easy. In that case, it hardly mattered precisely where you began measuring. For simplicity’s sake, Newton took “distance” to mean the distance between the centers of the two objects. But when it came to the question of the attraction between an apple and the Earth, what did the center of the Earth have to do with anything? An apple in a tree was thousands of miles from the Earth’s center. What about all those parts of the Earth that weren’t at the center? If everything attracted everything else, wouldn’t the pulls from bits of ground near the tree have to be taken into account? How would you tally up all those millions and millions of pulls, and wouldn’t they combine to overcome the pull from a faraway spot like the center of the Earth?

  Mass was just as bad. The Earth certainly wasn’t a point, though Newton had drawn it that way. It wasn’t even a true sphere. Nor was it uniform throughout. Mountains soared here, oceans swelled there, and, deep underground, strange and unknown structures lurked. And that was just on Earth. What of the sun and the other planets, and what about all their simultaneous pulls? “To do this business right,” Newton wrote Halley in the middle of his bout with the Principia, “is a thing of far greater difficulty than I was aware of.”

  But Newton did do the business right, and astonishingly quickly. In April 1686, less than two years after Halley’s first visit, Newton sent Halley his completed manuscript. His nine-page essay had grown into the Principia’s five hundred pages and two-hundred-odd theorems, propositions, and corollaries. Each argument was dense, compact, and austere, containing not a spare word or the slightest note of warning or encouragement to his hard-pressed readers. The modern-day physicist Subrahmanyan Chandrasekhar studied each theorem and proof minutely. Reading Newton so closely left him more astonished, not less. “That all these problems should have been enunciated, solved, and arranged in logical sequence in seventeen months is beyond human comprehension. It can be accepted only because it is a fact.”

  The Principia was made up of an introduction and three parts, known as Books I, II, and III. Newton began his introduction with three propositions now known as Newton’s laws. These were not summaries of thousands of specific facts, like Kepler’s laws, but magisterial pronouncements about the behavior of nature in general. Newton’s third law, for instance, was the famous “to every action, there is an equal and opposite reaction.” Book I dealt essentially with abstract mathematics, focused on topics like orbits and inverse squares. Newton discussed not the crater-speckled moon or the watery Earth but a moving point P attracted toward a fixed point S and moving in the direction AB, and so on.

  In Book II Newton returned to physics and demolished the theories of those scientists, most notably Descartes, who had tried to describe a mechanism that accounted for the motions of the planets and the other heavenly bodies. Descartes pictured space as pervaded by some kind of ethereal fluid. Whirlpools within that fluid formed “vortices” that carried the planets like twigs in a stream. Something similar happened here on Earth; rocks fell because mini-whirlpools dashed them to the ground.

  Some such “mechanistic” explanation had to be true, Descartes insisted, because the alternative was to believe in magic, to believe that objects could spring into motion on their own or could move under the direction of some distant object that never came in contact with them. That couldn’t be. Science had banished spirits. The only way for objects to interact was by making contact with other objects. That contact could be direct, as in a collision between billiard balls, or by way of countless, intermediate collisions with the too-small-to-see particles that fill the universe. (Descartes maintained that there could be no such thing as a vacuum.)

  Much of Newton’s work in Book II was to show that Descartes’ model was incorrect. Whirlpools would eventually fizzle out. Rather than carry a planet on its eternal rounds, any whirlpool would sooner or later be “swallowed up and lost.” In any case, no such picture could be made to fit with Kepler’s laws.
>
  Then came Book III, which was destined to make the Principia immortal.

  Chapter Forty-Eight

  Trouble with Mr. Hooke

  If not for the Principia’s unsung hero, Edmond Halley, the world might never have seen Book III. At the time he was working to coax the Principia from Newton, Halley had no official standing to speak of. He was a minor official at the Royal Society—albeit a brilliant scientist—who had taken on the task of dealing with Newton because nobody else seemed to be paying attention. Despite its illustrious membership, the Royal Society periodically fell into confusion. This was such a period, with no one quite in charge and meetings often canceled.

  So the task of shepherding along what would become one of the most important works in the history of science fell entirely to Halley. It was Halley who had to deal with the printers and help them navigate the impenetrable text and its countless abstruse diagrams, Halley who had to send page proofs to Newton for his approval, Halley who had to negotiate changes and corrections. Above all, it was Halley who had to keep his temperamental author content.

  John Locke once observed that Newton was “a nice man to deal with”—“nice” in the seventeenth-century sense of “finicky”—which was true but considerably understated. Anyone dealing with Newton needed the delicate touch and elaborate caution of a man trying to disarm a bomb. Until he picked up the Principia from the printer and delivered the first copies to Newton, Halley never dared even for a moment to relax his guard.