The Clockwork Universe

by Edward Dolnick

  On May 22, 1686, after Newton had already turned in Books I and II of his manuscript, Halley worked up his nerve and sent Newton a letter with unwelcome news. “There is one thing more I ought to informe you of,” he wrote, “viz, that Mr Hook has some pretensions upon the invention of ye rule of the decrease of Gravity. . . . He says you had the notion from him.” Halley tried to soften the blow by emphasizing the limits of Hooke’s claim. Hooke maintained that he had been the one to come up with the idea of an inverse-square law. He conceded that he had not seen the connection between inverse squares and elliptical orbits; that was Newton’s insight, alone. Even so, Halley wrote, “Mr Hook seems to expect you should make some mention of him.”

  Instead, Newton went through the Principia page by page, diligently striking out Hooke’s name virtually every time he found it. “He has done nothing,” Newton snarled to Halley. Newton bemoaned his mistake in revealing his ideas and thereby opening himself up to attack. He should have known better. “Philosophy [i.e., science] is such an impertinently litigious Lady that a man had as good be engaged in Law suits as have to do with her,” he wrote. “I found it so formerly & now I no sooner come near her again but she gives me warning.”

  The more Newton brooded, the angrier he grew. Crossing out Hooke’s name was too weak a response. Newton told Halley that he had decided not to publish Book III. Halley raced to soothe Newton. He could not do without Newton’s insights; the Royal Society could not; the learned world could not.

  * * *

  Newton could have dismissed the controversy with a gracious tip of the hat to Hooke, for Hooke had indeed done him a favor. In 1684, as we have seen, Halley had asked Newton a question about the inverse-square law, and Newton had immediately given him the answer.

  The reason Newton knew the answer is that Hooke had written him a letter four years before that asked the identical question. What orbit would a planet follow if it were governed by an inverse-square law? “I doubt not but that by your excellent method you will easily find out what that Curve must be,” Hooke had written Newton, “and its proprietys [properties], and suggest a physicall Reason of this proportion.”

  Newton had solved the problem then and put it away. He never replied to Hooke’s letter. This was perhaps inevitable, for Hooke and Newton had been feuding for years. Back in 1671, the Royal Society had heard rumors of a new kind of telescope, supposedly invented by a young Cambridge mathematician. The rumors were true. Newton had designed a telescope that measured a mere six inches but was more powerful than a conventional telescope six feet long. The Royal Society asked to see it, Newton sent it along, and the Society oohed and aahed.

  Newton’s reputation was made. This was Newton’s first contact with the Royal Society, which at once invited him to join. He accepted. Only Hooke, until this new development England’s unchallenged authority on optics and lenses, refused to add his voice to the chorus of praise.

  Even a better-natured man than Hooke might have bristled at all the attention paid to a newcomer (Hooke was seven years older than Newton), but Hooke was fully as proud and prickly as Newton himself. In 1671 Hooke was an established scientific figure; Newton was unknown. Hooke had spent a career crafting instruments like the telescopes that Newton’s new design had so dramatically surpassed; Newton’s main interests were in other areas altogether. And more trouble lay just ahead, though Hooke could not have anticipated it. In a letter to the Royal Society thanking them for taking such heed of his telescope, Newton added a tantalizing sentence. In the course of his “poore & solitary endeavours,” he had found something remarkable.

  Within a month, Newton followed up his coup with the telescope by sending the Royal Society his groundbreaking paper on white light. The nature of light was another of Hooke’s particular interests. Once again, the outsider had barged into staked-out territory and put down his own marker. Deservedly proud of what he had found, Newton for once said so openly. His demonstration that white light was made up of all the colors was, Newton wrote, “the oddest, if not the most considerable detection, which has hitherto been made in the operation of nature.”

  The paper, later hailed as one of the all-time landmarks in science, met with considerable resistance at first, from Hooke most of all. He had already done all of the same experiments, Hooke claimed, and, unlike Newton, he had interpreted them correctly. He said so, dismissively, lengthily, and unwisely. (It was at this point that Newton sent a letter to the hunchbacked Hooke with a mock-gracious passage about how Newton stood “on the shoulders of giants.”) Thirty years would pass—until 1704, the year following Hooke’s death—before the world would hear any more about Newton’s experiments on light.

  Now, in 1686, with the first two books of the Principia in Halley’s hands, Hooke had popped up again. For Hooke to venture yet another criticism, this time directed against Newton’s crowning work, was a sin beyond forgiving. In Newton’s eyes Hooke had done nothing to contribute to a theory of gravitation. He had made a blind guess and not known how to follow it up. The challenge was not to suggest that an inverse-square law might be worth looking at, which anyone might have proposed, but to work out what the universe would look like if that law held.

  Hooke had not even known how to get started, but he had airily dismissed Newton’s revelations as if they were no more than the working out of a few details that Hooke had been too busy for. “Now is not this very fine?” Newton snapped. “Mathematicians that find out, settle & do all the business must content themselves with being nothing but dry calculators & drudges & another that does nothing but pretend & grasp at all things must carry away all the invention. . . .”

  Hooke was a true genius, far more than Salieri to Newton’s Mozart, but he did not come up to Newton’s level. Hooke’s misfortune was to share so many interests with a man fated to win every competition. That left both men trapped. Newton could not bear to be criticized, and Hooke could not bear to be outdone. The two men never did make peace. On the rare occasions when they found themselves thrown together, Hooke stalked out of the room. Newton was just as hostile. Even twenty years after Hooke’s death, Newton could not hear his name spoken without losing his temper.

  During the many years when Hooke was a dominant figure at the Royal Society, Newton made a point of staying away. When Hooke finally died, in 1703, Newton immediately accepted the post of Royal Society president. At about the same time, the Royal Society moved to new quarters. In the course of the move the only known portrait of Hooke vanished.

  Chapter Forty-Nine

  The System of the World

  “I must now again beg you,” Halley wrote Newton at the height of the Hooke affair, “not to let your resentments run so high, as to deprive us of your third book.” Halley would have pleaded even more fervently if Newton had told him outright what riches he had reserved for Book III. Newton gave in to Halley’s pleas. Perhaps he had meant to do so all along, although Newton seldom bothered to bark without also going on to bite.

  The key to Book III was one astonishing theorem. Among the mysteries that Newton had to solve, one of the deepest was this: how could he justify the assumption that any object whatsoever, no matter how tiny or gigantic, no matter how odd its shape, no matter how complicated its makeup, could be treated mathematically as if it were a single point? Newton hadn’t had a choice about simplifying things in that way, because otherwise he could not have gotten started, but it seemed an unlikely fiction.

  Then, in Book III, Newton delivered an extraordinarily subtle, calculus-based proof that a complicated object could legitimately be treated as a single point. In reality the Earth was eight thousand miles in diameter and weighed thousands of billions of tons; mathematically it could be treated as a point with that same unimaginable mass. Make a calculation based on that simplifying assumption—what was the shape of the moon’s orbit, say?—and the result would match snugly with reality.

  Everything depended on the inverse-square law. If the universe had been governed by a diff
erent law, Newton showed, then his argument about treating objects as points would not have held, nor would the planets have fallen into stable orbits. For Newton, this was yet more evidence that God had designed the universe mathematically.

  The Principia seemed to proclaim that message. What, after all, was the meaning of Newton’s demonstration that real-life objects could be treated as idealized, abstract points? It meant that all of the mathematical arguments that Newton had made in Book I turned out to describe the actual workings of the world. Like the world’s most fantastic pop-up book, the geometry text of Book I rose to life as the real-world map of Book III. Newton introduced his key findings with a trumpet flourish. “I now demonstrate the frame of the System of the World,” he wrote, which was to say, “I will now lay out the structure of the universe.”

  And so he did. Starting with his three laws and a small number of propositions, Newton deduced all three of Kepler’s laws, which dealt with the motions of the planets around the sun; he deduced Galileo’s law about objects in free fall, which dealt with the motion of objects here on Earth; he explained the motion of the moon; he explained the path of comets; he explained the tides; he deduced the precise shape of the Earth.

  The heart of the Principia was a breathtaking generalization. Galileo had made a leap from objects sliding down a ramp to objects falling through the air. Newton leaped from the Earth’s pulling an apple to every pair of objects in the universe pulling one another. “There is a power of gravity,” Newton wrote, “pertaining to all bodies, proportional to the several quantities of matter which they contain.” All bodies, everywhere.

  This was the theory of “universal gravitation,” a single force and a single law that extended to the farthest reaches of the universe. Everything pulled on everything else, instantly and across billions of miles of empty space, the entire universe bound together in one vast, abstract web. The sun pulled the Earth, an ant tugged on the moon, stars so far away from Earth that their light takes thousands of years to reach us pull us, and we pull them. “Pick a flower on Earth,” said the physicist Paul Dirac, “and you move the farthest star.”

  With a wave of Newton’s wand, the world fell into place. The law of gravitation—one law—explained the path of a paperweight knocked off a desk, the arc of a cannonball shot across a battlefield, the orbit of a planet circling the sun or a comet on a journey that extended far, far beyond the solar system. An apple that fell a few feet to the ground, in a matter of seconds, obeyed the law of gravitation. So did a comet that traveled hundreds of millions of miles and neared the Earth only once every seventy-five years.

  And Newton had done more than explain the workings of the heavens and the Earth. He had explained everything using the most familiar, literally the most down-to-earth force of all. All babies know, before they learn to talk, that a dropped rattle falls to the ground. Newton proved that if you looked at that observation with enough insight, you could deduce the workings of the cosmos.

  The Principia made its first appearance, in a handsome, leatherbound volume, on July 5, 1687. The scientific world searched for superlatives worthy of Newton’s achievement. “Nearer the gods no mortal may approach,” Halley wrote, in an adulatory poem published with the Principia. A century later the reverence had scarcely died down. Newton was not only the greatest of all scientists but the most fortunate, the French astronomer Lagrange declared, for there was only one universe to find, and he had found it.

  Halley watched over the Principia all the way to the end, and past it. The Royal Society had only ventured into publishing once before. In 1685 it had published a lavish volume called The History of Fishes and lost money. Now the Society instructed Halley to print the Principia at his own expense, since he was the one who had committed it to publication in the first place. Halley agreed, though he was far from rich. The work appeared, to vast acclaim, but the Society’s finances fell further into disarray. It began paying Halley his salary in unsold copies of The History of Fishes.

  Chapter Fifty

  Only Three People

  From the beginning, the Principia had a reputation for difficulty. When Newton brushed by some students on the street one day, he heard one of them mutter, “There goes the man that writt a book that neither he nor any body else understands.” It was almost true. When the Principia first appeared, it baffled all but the ablest scientists and mathematicians. (The first print run was tiny, between three and four hundred.) “It is doubtful,” wrote the historian Charles C. Gillispie, “whether any work of comparable influence can ever have been read by so few persons.”

  The historian A. Rupert Hall fleshed out Gillispie’s remark. Perhaps half a dozen scientists, Hall reckoned, fully grasped Newton’s message on first reading it. Their astonished praise, coupled with efforts at recasting Newton’s arguments, quickly drew new admirers. In time popular books would help spread Newton’s message. Voltaire wrote one of the most successful, Elements of Newton’s Philosophy, much as Bertrand Russell would later write ABC of Relativity. An Italian writer produced Newtonianism for Ladies, and an English author using the pen name Tom Telescope wrote a hit children’s book.

  But in physics a mystique of impenetrability only adds to a theory’s allure. In 1919, when the New York Times ran a story on Einstein and relativity, a subheadline declared, “A Book for 12 Wise Men.” A smaller headline added, “No More in All the World Could Comprehend It.” A few years later a journalist asked the astronomer Arthur Eddington if it was true that only three people in the world understood general relativity. Eddington thought a moment and then replied, “I’m trying to think who the third person is.”

  Two features, beyond the difficulty of its mathematical arguments, made the Principia so hard to grasp. The first reflected Newton’s hybrid status as part medieval genius, part modern scientist. Through the whole vast book Newton relies on concepts from calculus—infinitesimals, limits, straight lines homing in ever closer to curves—that he had invented two decades before. But he rarely mentions calculus explicitly or explains the strategy behind his arguments, and he makes only indirect use of calculus’s labor-saving machinery.

  Instead he makes modern arguments using old-fashioned tools. What looks at a glance like classical geometry turns out to be a more exotic beast, a kind of mathematical centaur. Euclid would have been baffled. “An ancient and venerable mathematical science had been pressed into service in a subject area for which it seems inappropriate,” writes one modern physicist. “Newton’s geometry seems to shriek and groan under the strain, but it works perfectly.”

  There are almost no other historical examples of so strange a performance as this use/nonuse of calculus. To get something of its flavor, we have to imagine far-fetched scenarios. Think, for instance, of a genius who grew up using Roman numerals but then invented Arabic numerals. And then imagine that he conceived an incredibly complex theory that relied heavily on the special properties of Arabic numerals—the way they make calculations easy, for instance. Finally, imagine that when he presented that theory to the world he used no Arabic numerals at all, but only Roman numerals manipulated in obscure and never-explained ways.

  Decades after the Principia, Newton offered an explanation. In his own investigations, he said, he had used calculus. Then, out of respect for tradition and so that others could follow his reasoning, he had translated his findings into classical, geometric language. “By the help of the new Analysis [i.e., calculus] Mr. Newton found out most of the Propositions in his Principia Philosophiae,” he wrote, referring to himself in the third person, but then he recast his mathematical arguments so that “the System of the Heavens might be founded upon good Geometry.”

  Newton’s account made sense, and for centuries scholars took it at face value. He knew he was presenting a revolutionary theory. To declare that he had reached startling conclusions by way of a strange, new technique that he had himself invented would have been to invite trouble and doubt. One revolution at a time.

  But it no
w turns out that Newton did not use calculus’s shortcuts in private and then reframe them. “There is no letter,” declared one of the most eminent Newtonians, I. Bernard Cohen, “no draft of a proposition, no text of any kind—not even a lone scrap of paper—that would indicate a private mode of composition other than the public one we know in the Principia.” The reason that Newton claimed otherwise was evidently to score points against Leibniz. “He wanted,” wrote Cohen, “to show that he understood and was using the calculus long before Leibniz.”

  This is curious, for Newton had understood calculus long before Leibniz, and so it would have made perfect sense for him to have drawn on its hugely powerful techniques. But he did not. The reason, evidently, was that he was such a geometric virtuoso that he felt no impulse to deploy the powerful new arsenal that he himself had built. “As we read the Principia,” the nineteenth-century scientist William Whewell would write, “we feel as when we are in an ancient armoury where the weapons are of gigantic size; and as we look at them, we marvel what manner of men they were who could use as weapons what we can scarcely lift as a burden.”

  Chapter Fifty-One

  Just Crazy Enough

  The second reason that the Principia was so baffling is more easily stated—the theory made no sense. This is not to deny that the theory of gravitation “works.” It works astonishingly well. When NASA sent a man to the moon, every calculation along the way turned out precisely as Newton would have forecast centuries before. Nor does the model break down when applied to the farthest corners of the universe or the largest structures in nature. A theory that Newton devised by pondering the solar system and its one sun turns out to apply to galaxies made up of billions upon billions of suns, galaxies whose existence was unknown in Newton’s day.