Newton and the Counterfeiter: The Unknown Detective Career of the World's Greatest Scientist

by Thomas Levenson

  Mysterious, but calculable. To do so, he needed to take one last, great step and create a mathematical expression to describe the intensity of whatever it was connecting the earth and the moon with the distance between the two bodies. He found inspiration in Kepler's third law of planetary motion, which relates the time it takes for a planet to complete its orbit with its distance from the sun. By analyzing that law, Newton concluded (as he later put it) that "the forces which keep the planets in their orbs must [be] reciprocally as the squares of their distances from the centers around which they revolve." That is, the force of gravity falls off in proportion to the square of the distance between any two objects.

  With that, it was just a matter of plugging in the numbers to calculate the moon's orbit. Here he ran into trouble. From his pendulum experiments, he had a fairly precise measurement of one crucial term, the strength of gravity at the earth's surface. But he still needed to know the distance between the moon and the earth, a calculation that turned on knowledge of the earth's size. This was a number Newton could not determine for himself, so he used the common mariner's guess that one degree of the earth's circumference was equal to "sixty measured Miles only." That was wrong, well off the accurate figure of slightly more than sixty-nine miles. The error propagated throughout his calculation, and nothing Newton could do would make the moon's path work out. He had some guesses as to what might be happening, but these were loose thoughts, and as yet he knew no way to reduce them to the discipline of mathematics.

  The setback was enough to provoke Newton to move on. New ideas were crowding in. Optics came next, a series of inquiries into the nature of light that would bring him a first, ambivalent brush with fame in the early 1670s. Thus engaged, Newton let the matter of the moon rest.

  But if his miracle years, as they have come to be known, did not produce the finished Newtonian system, still by the end of his enforced seclusion Newton understood that any new physical system could succeed only by "subjecting motion to number." His attempt to analyze the gravitational interaction of the earth and the moon provided the model: any claim of a relationship, any proposed connection between phenomena, had to be tested against the rigor of a mathematical description.

  Many of the central ideas that would form the essential content of his physics were there too, though an enormous amount of labor remained to get from those first drafts to the finished construction of the system. Newton would have to redefine what he and his contemporaries thought they knew about the most basic concepts of matter and motion just to arrive at a set of definitions that he could turn to account. For example, he was still groping for a way to express the crucial conception of force that would allow him to bring the full force of mathematics to bear. By 1666, he had got this far: "Tis known by ye light of nature ... yt equall forces shall effect an equall change in equall bodys ... for in loosing or ... getting ye same quantity of motion a body suffers the ye same quantity of mutation in its state."

  The core of the idea is there: that a change in the motion of a body is proportional to the amount of force impressed on it. But to turn that conception into the detailed, rich form it would take as Newton's second law of motion would require long, long hours of deep thought. The same would prove to be true for all his efforts over the next twenty years as they evolved into the finished edifice of his great work, Philosophiœ naturalis principia mathematica—The Mathematical Principles of Natural Philosophy—better known as the Principia. For all his raw intelligence, Newton's ultimate achievement turned on his genius for perseverance. His one close college friend, John Wickens, marveled at his ability to forget all else in the rapt observation of the comet of 1664. Two decades later, Humphrey Newton, Isaac Newton's assistant and copyist (and no relation), saw the same. "When he has sometimes taken a turn or two [outdoors] has made a sudden stand, turn'd himself about and run up ye stairs, like another Archimedes, with an Eureka, fall to write on his Desk standing, without giving himself the Leasure to draw a Chair to sit down in." If something mattered to him, the man pursued it relentlessly.

  Equally crucial to his ultimate success, Newton was never a purely abstract thinker. He gained his central insight into the concept of force from evidence "known by ye light of nature." He tested his ideas about gravity and the motion of the moon with data drawn from his own painstaking experiments and the imperfect observations of others. When it came time to analyze the physics of the tides, the landlocked Newton sought out data from travelers the world over; barely straying from his desk in the room next to Trinity College's Great Gate, he gathered evidence from Plymouth and Chepstow, from the Strait of Magellan, from the South China Sea. He stabbed his own eye, built his own furnaces, constructed his own optical instruments (most famously the first reflecting telescope); he weighed, measured, tested, smelled, worked—hard—with his own hands, to discover the answer to whatever had sparked his curiosity.

  Newton labored through the summer. That September, the Great Fire of London came. It lasted five days, finally exhausting itself on September 7. Almost all of the city within the walls was destroyed, and some beyond, 436 acres in all. More than thirteen thousand houses burned, eighty-seven churches, and old St. Paul's Cathedral. The sixty tons of lead in the cathedral roof melted; a river of molten metal flowed into the Thames. Just six people are known to have died, though it seems almost certain that the true number was much greater.

  But once the fire destroyed the dense and deadly slums that cosseted infection, the plague finally burned itself out. That winter, reports of cases dropped, then vanished, until by spring it became clear that the epidemic was truly done.

  In April 1667, Newton returned to his rooms at Trinity College. He had left two years earlier with the ink barely dry on his bachelor of arts degree. In the interval, he had become the greatest mathematician in the world, and the equal of any natural philosopher then living. No one knew. He had published nothing, communicated his results to no one. So the situation would remain, in essence, for two decades.

  3. "I Have Calculated It"

  ISAAC NEWTON CLAMBERED up the academic pyramid as rapidly as his abilities warranted. In 1669, when Newton was twenty-six, his former teacher Isaac Barrow resigned the Lucasian Professorship of Mathematics in his favor, and from that point on he was set. The chair was his for as long as he chose to keep it. It provided him with room, board, and about one hundred pounds a year—plenty for an unmarried man with virtually no living expenses. In return, all he had to do was deliver one course of lectures every three terms. Even that duty did not impinge much on his time. Humphrey Newton reported that the professor would speak for as much as half an hour if anyone actually showed up, but that "oftimes he did in a manner, for want of Hearers, read to ye Walls."

  Aside from such minimal nods toward the instruction of the young, Newton did as he pleased. He loathed distractions, had little gift for casual talk, and entertained few visitors. He gave virtually all his waking hours to his research. Humphrey Newton again: "I never knew him [to] take any Recreation or Pastime, either in riding out to take air, Walking, bowling, or any other Exercise whatever, Thinking all Hours lost, that was not spent in his Studyes." He seemed offended by the demands of his body. Humphrey reported that Newton "grudg'd that short Time he spent in eating & sleeping"; that his housekeeper would find "both Dinner & Supper scarcely tasted of"; that "He very seldom sat by the fire in his Chamber, excepting that long frosty winter, which made him creep to it against his will." His one diversion was his garden, a small plot on Trinity's grounds, "which was never out of Order, in which he would, at some seldom Times, take a short Walk or two, not enduring to see a weed in it." That was it—a life wholly committed to his studies, except for a very occasional conversation with a handful of acquaintances and a few stolen minutes pulling weeds.

  But work to what end? Year after year, he published next to nothing, and he had almost no discernible impact on his contemporaries. As Richard Westfall put it: "Had Newton died in 1684 and his papers surviv
ed, we would know from them that a genius had lived. Instead of hailing him as a figure who had shaped the modern intellect, however, we would at most...[lament] his failure to reach fulfillment."

  And then, one August day in 1684, Edmond Halley stopped by. Halley was one of that handful of acquaintances who could always gain admittance to Newton's rooms in Trinity. The pair had met two years earlier, just after Halley's return from France, where he'd meticulously observed the comet that would later be named for him. Newton had made his own sketches of the comet, and he welcomed a fellow enthusiast into the circle of those whose letters he would answer, whose conversation he welcomed.

  Today Halley brought no pressing scientific news. He had come down from London to the countryside near Cambridge on family business, and his visit to Newton was merely social. But in the course of their conversation, Halley recalled a technical point he had been meaning to take up with his friend.

  Halley's request had seemed trivial enough. Would Isaac Newton please settle a bet? The previous January, Halley, Robert Hooke, and the architect Sir Christopher Wren had talked on after a meeting of the Royal Society. Wren wondered if it was true that the motion of the planets obeyed an inverse square law of gravity—the same inverse square relationship that Newton had investigated during the plague years. Halley readily confessed that he could not solve the problem, but Hooke had boasted that he had already proved that the inverse square law held true, and "that upon that principle all the Laws of the celestiall motions were to be demonstrated."

  When pressed, though, Hooke refused to reveal his results, and Wren openly doubted his claim. Wren knew how tricky the question was. Seven years before, Isaac Newton had visited him in his London home, where the two men discussed the complexity of the problem of discovering "heavenly motions upon philosophical principles." Accordingly, Wren would not take a claim of a solution on faith. Instead, he offered a prize, a book worth forty shillings, to the man who could solve the problem within two months. Hooke puffed up, declaring that he would hold his work back so that "others triing and failing, might know how to value it." But two months passed, and then several weeks more, and Hooke revealed nothing. Halley, diplomatically, did not write that Hooke had failed, but that "I do not yet find that in that particular he has been as good as his word."

  There the matter rested, until Halley put Wren's question to Newton: "what he thought the Curve would be that would be described by the Planets supposing the force of attraction towards the Sun to be reciprocal to the square of their distance from it." Newton immediately replied that it would be an ellipse. Halley, "struck with joy & amazement," asked how he could be so sure, and Newton replied, "Why ... I have calculated it."

  Halley asked at once to see the calculation, but, according to the story he later told, Newton could not find it when he rummaged through his papers. Giving up for the moment, he promised Halley that he would "renew it & send it to him."

  While Halley waited in London, Newton tried to re-create his old work—and failed. He had made an error in one of his diagrams in the prior attempt, and his elegant geometric argument collapsed with the mistake. He labored on, however, and by November he had worked it out.

  In his new calculation, Newton analyzed the motions of the planets using a branch of geometry concerned with conic sections. Conic sections are the curves made when a plane slices through a cone. Depending on the angle and location of the cut, you get a circle (if the plane intersects either cone at a ninety-degree angle), an ellipse (if the plane bisects one cone at an angle other than ninety degrees), a parabola (if the curve cuts through the side of the cones but does not slice all the way through its circumference), or the symmetrical double curve called a hyperbola (produced only if there are two identical cones laid tip to tip).

  As he calculated, Newton was able to show that for an object in a system of two bodies bound by an inverse square attraction, the only closed path available is an ellipse, with the more massive body at one focus. Depending on the distance, the speed, and the ratio of masses of the two bodies, such ellipses can be very nearly circular—as is the case for the earth, whose orbit deviates by less than two percent from a perfect circle. As the force acting on two bodies weakens with distance, more elongated ellipses and open-ended trajectories (parabolas or hyperbolas) become valid solutions for the equations of motion that describe the path of a body moving under the influence of an inverse square force. To the practical matter at hand, Newton had proved that in the case of two bodies, one orbiting the other, an inverse square relationship for the attraction of gravity produces an orbit that traces a conic section, which becomes the closed path of an ellipse in the case of our sun's planets.

  QED.

  Newton wrote up the work in a nine-page manuscript titled De motu corporum in gyrum—"On the Motion of Bodies in Orbit." He let Halley know the work was done, and then presumably settled back into his usual routine.

  That peace could not last, not if Halley had anything to do with it. He grasped the significance of De Motu immediately. This was no mere set-piece response to an after-dinner challenge. Rather, it was the foundation of a revolution of the entire science of motion. He raced back to Cambridge in November, copied Newton's paper in his own hand, and in December was able to tell the Royal Society that he had permission to publish the work in the register of the Royal Society as soon as Newton revised it.

  And then ... nothing.

  Halley had not expected anything more than a quick revision of the brief paper he had already seen. The final, corrected version of De Motu was supposed to follow soon after his second meeting with Newton. When it failed to arrive on schedule, Halley took the precaution of registering his preliminary copy with the Royal Society, establishing its priority. Then he resumed his vigil, waiting for more to come from Cambridge. Still nothing, not in what remained of 1684, and not through the first part of 1685.

  Newton, for all of his periodic public silences, wrote constantly. He committed millions of words to paper over his long life, often recopying three or more near-identical drafts of the same document. He was a conscientious letter writer too. His correspondence fills seven folio volumes. While that is not an extraordinary total for a time when the learned of Europe (and America) communicated with each other by letter, it represents a formidable stream of prose. But between December 1684 and the summer of 1686, when he delivered to Halley the final versions of the first two parts of his promised, and now greatly expanded, treatise, he is known to have written just seven letters. Two of them are mere notes. The remaining five were all to John Flamsteed, the Astronomer Royal, asking him for his observations of the planets, of Jupiter's moons, and of comets, all to help him in a series of calculations whose true nature he did not choose to share.

  Much later, Newton admitted what had happened. "After I began to work on the inequalities of the motions of the moon, and then also began to explore other aspects of the laws and measures of gravity and of other forces," he wrote, "I thought that publication should be put off to another time, so that I might investigate these other things and publish all my results together." He was trying to create a new science, one he called "rational mechanics." This new discipline would be comprehensive, able to gather in the whole of nature. It would be, he wrote, "the science, expressed in exact propositions and demonstrations, of the motions that result from any forces whatever and of the forces that are required for any motions whatever."

  Newton writes here of a science advanced by a method that would be exact in its laws and analyses. Fully developed, it would yield an absolute, precise account of cause and effect, true for all encounters between matter and force, whatever they may be. This was his aim in writing what was about to become the Principia, at once the blueprint and the manifesto for such a science. He began with three simple statements that could cut through the confusion and muddled thought that had tangled all previous attempts to account for motion in nature. First came his ultimate understanding of what he dubbed inertia: "Every
body perseveres in its state of being at rest or of moving uniformly straight forward except insofar as it is compelled to change its state by forces impressed."

  His second axiom stated the precise relationship between force and motion: "A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed." Last he addressed the question of what happens when forces and objects interact: " To any action there is always an opposite and equal reaction; in other words, the actions of two bodies upon each other are always equal and always opposite in direction" (italics in the original).

  Thus, the famous three laws of motion, stated not as propositions to be demonstrated but as pillars of reality. This was, Newton recognized, an extraordinary moment, and he composed his text accordingly, in an echo of the literature he knew best. He began with a revelation, a bald statement of fundamental truths, then followed with five hundred pages of exegesis that showed what could be done from this seemingly simple point of origin.

  Books One and Two—both titled "The Motion of Bodies"—demonstrated how much his three laws could explain. After some preliminaries, Newton reworked the material he had shown Halley to derive the properties of the different orbits produced by an inverse square law of gravity. He analyzed mathematically how objects governed by the three laws collide and rebound. He calculated what happens when objects travel through different media—water instead of air, for example. He pondered the issues of density and compression, and created the mathematical tools to describe what happens to fluids under pressure. He analyzed the motion of a pendulum. He inserted some older mathematical work on conic sections, apparently simply because he had it lying around. He attempted an analysis of wave dynamics and the propagation of sound. On and on, through every phenomenon that could be conceived as matter in motion.