Newton and the Counterfeiter (4 page)

I
SAAC 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 y
^e
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 survived, 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.

He wrote on through the fall and winter of 1685, stating propositions and theorems, presenting proofs, extracting corollaries from concepts already established, page after page, proof after proof, until the sheer mass overwhelmed all challenges. Throughout that time, Newton's always impressive appetite for work became total. "He very rarely went to Bed till 2 or 3 of the clock, sometimes not till 5 or 6, lying about 4 or 5 hours," observed Humphrey Newton. On rising, "his earnest & indefatigable Studyes retain'd Him, so that He scarcely knewe the Hour of Prayer."

It took Newton almost two years to finish Book Two. Its last theorem completes his demolition of Descartes' vortices—those whirlpools in some strange medium that were supposed to drive the motion of the planets and stars. Newton showed no pity, concluding dismissively that his predecessor's work served "less to clarify the celestial motions than to obscure them."

With that bit of old business settled, Newton turned to his ultimate aim. In the preface to the
Principia,
Newton wrote, "The whole difficulty of philosophy seems to be to discover the forces of nature from the phenomena of motions and then to demonstrate the other phenomena from these forces." Books One and Two had covered only the first half of that territory, presenting "the laws and conditions of motions"; but as Newton wrote, those laws were "not, however, philosophical but strictly mathematical." Now, he declared, it was time to put such abstraction to the test of experience. "It still remains for us," he wrote, "to exhibit the system of the world from these same principles."

At first reading, Book Three, which he in fact titled "The System of the World," falls short. No mere forty-two propositions could possibly comprehend all of experience. But, as usual, Newton said what he meant. In a mere hundred pages or so of mathematical reasoning, he did not promise to capture all that moved in the observable universe. Rather, he offered a system with which to do so—the method that, as it has turned out, his successors have employed to explore all of material reality through the enterprise we call science.