Isaac Newton - James Gleick (2004)

Chapter 12. Every Body Perseveres

IT WAS ORDERED, that a letter of thanks be written to Mr NEWTON,” recorded Halley, as clerk of the Royal Society, on April 28, 1686, “… and that in the meantime the book be put into the hands of Mr HALLEY.”1

Only Halley knew what was in “the book”—a first sheaf of manuscript pages, copied in Cambridge by Newton’s amanuensis2 and dispatched to London with the grand title Philosophiæ Naturalis Principia Mathematica. Halley had been forewarning the Royal Society: “a mathematical demonstration of the Copernican hypothesis”; “makes out all the phenomena of the celestial motions by the only supposition of a gravitation towards the centre of the sun decreasing as the squares of the distances therefrom reciprocally.”3 Hooke heard him.

It was Halley, three weeks later, who undertook the letter of thanks: “Your Incomparable treatise,” etc. He had persuaded the members, none of whom could have read the manuscript, to have it printed, in a large quarto, with woodcuts for the diagrams. There was just one thing more he felt obliged to tell Newton: “viz, that Mr Hook has some pretensions upon the invention of the rule of the decrease of Gravity.… He sais you had the notion from him [and] seems to expect you should make some mention of him, in the preface.…”4

What Newton had delivered was Book I of the Principia. He had completed much of Book II, and Book III lay not far behind. He interrupted himself to feed his fury, search through old manuscripts, and pour forth a thunderous rant, mostly for the benefit of Halley. He railed that Hooke was a bungler and a pretender:

This carriage towards me is very strange & undeserved, so that I cannot forbeare in stating that point of justice to tell you further … he should rather have excused himself by reason of his inability. For tis plain by his words he knew not how to go about it. Now is this not very fine? 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.…

Mr Hook has erred in the invention he pretends to & his error is the cause of all the stirr he makes.…

He imagins he obliged me by telling me his Theory, but I thought my self disobliged by being upon his own mistake corrected magisterially & taught a Theory which every body knew & I had a truer notion of it then himself. Should a man who thinks himself knowing, & loves to shew it in correction & instructing others, come to you when you are busy, & notwithstanding your excuse, press discourses upon you & through his own mistakes correct you & multiply discourses & then make this use of it, to boast that he taught you all he spake & oblige you to acknowledge it & cry out injury & injustice if you do not, I beleive you would think him a man of a strange unsociable temper.5

In his drafts of Book II, Newton had mentioned the most illustrious Hooke—“Cl[arissimus] Hookius”6—but now he struck all mention of Hooke and threatened to give up on Book III. “Philosophy is such an impertinently litigious Lady that a man had as good be engaged in Law suits as have to do with her. I found it so formerly & now I no sooner come near her again but she gives me warning.”7 Hooke had not been the first to propose the inverse-square law of attraction; anyway, for him it was a guess. It stood in isolation, like countless other guesses at the nature of the world. For Newton, it was embedded, linked, inevitable. Each part of Newton’s growing system reinforced the others. In its mutual dependency lay its strength.

Halley, meanwhile, found himself entangled in the business of publishing. The Royal Society had never actually agreed to print the book. Indeed, it had only underwritten the publication of one book before, a lavish and disastrously unsuccessful two-volume History of Fishes.8 After much discussion the council members did vote to order the Principia printed—but by Halley, at his own expense. They offered him leftover copies of History of Fishesin place of his salary. No matter. The young Halley was a believer, and he embraced his burden: the proof sheets mangled and lost, the complex abstruse woodcuts, the clearing of errata, and above all the nourishing of his author by cajolement and flattery. “You will do your self the honour of perfecting scientifically what all past ages have but blindly groped after.”9 The flattery was sincere, at least.

Halley sent sixty copies of Philosophiæ Naturalis Principia Mathematica on a wagon from London to Cambridge in July 1687. He implored Newton to hand out twenty to university colleagues and carry forty around to booksellers, for sale at five or six shillings apiece.10 The book opened with a florid ode of praise to its author, composed by Halley. When an adulatory anonymous review appeared in the Philosophical Transactions, this, too, was by Halley.11

Without further ado, having defined his terms, Newton announced the laws of motion.

Law 1. 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. A cannonball would fly in a straight line forever, were it not for air resistance and the downward force of gravity. The first law stated, without naming, the principle of inertia, Galileo’s principle, refined. Two states—being at rest and moving uniformly—are to be treated as the same. If a flying cannonball embodies a force, so does the cannonball at rest.

Law 2. A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed. Force generates motion, and these are quantities, to be added and multiplied according to mathematical rules.

Law 3. 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. If a finger presses a stone, the stone presses back against the finger. If a horse pulls a stone, the stone pulls the horse. Actions are interactions—no preference of vantage point to be assigned. If the earth tugs at the moon, the moon tugs back.12

He presented these as axioms, to serve as the foundation for an edifice of reasoning and proof. “Law”—lex—was a strong and peculiar choice of words.13 Bacon had spoken of laws, fundamental and universal. It was no coincidence that Descartes, in his own book called Principles of Philosophy, had attempted a set of three laws, regulæ quædam sive leges naturæ, specifically concerning motion, including a law of inertia. For Newton, the laws formed the bedrock on which a whole system would lie.

A law is not a cause, yet it is more than a description. A law is a rule of conduct: here, God’s law, for every piece of creation. A law is to be obeyed, by inanimate particles as well as sentient creatures. Newton chose to speak not so much of God as of nature. “Nature is exceedingly simple and conformable to herself. Whatever reasoning holds for greater motions, should hold for lesser ones as well.”14

Newton formed his argument in classic Greek geometrical style: axioms, lemmas, corollaries; Q.E.D. As the best model available for perfection in knowledge, it gave his physical program the stamp of certainty. He proved facts about triangles and tangents, chords and parallelograms, and from there, by a long chain of argument, proved facts about the moon and the tides. On his own path to these discoveries, he had invented a new mathematics, the integral and differential calculus. The calculus and the discoveries were of a piece. But he severed the connection now. Rather than offer his readers an esoteric new mathematics as the basis for his claims, he grounded them in orthodox geometry—orthodox, yet still new, because he had to incorporate infinities and infinitesimals. Static though his diagrams looked, they depicted processes of dynamic change. His lemmas spoke of quantities that constantly tend to equality or diminish indefinitely; of areas that simultaneously approach and ultimately vanishof momentary increments and ultimate ratios and curvilinear limits. He drew lines and triangles that looked finite but were meant to be on the point of vanishing. He cloaked modern analysis in antique disguise.15 He tried to prepare his readers for paradoxes.

It may be objected that there is no such thing as an ultimate proportion of vanishing quantities, inasmuch as before vanishing the proportion is not ultimate, and after vanishing it does not exist at all.… But the answer is easy.… the ultimate ratio of vanishing quantities is to be understood not as the ratio of quantities before they vanish or after they have vanished, but the ratio with which they vanish.16

The diagrams appeared to represent space, but time kept creeping in: “Let the time be divided into equal parts.… If the areas are very nearly proportional to the times …”

When he and Hooke had debated the paths of comets and falling objects, they had dodged one crucial problem. All the earth’s substance is not concentrated at its center but spread across the volume of a great sphere—countless parts, all responsible for the earth’s attractive power. If the earth as a whole exerts a gravitational force, that force must be calculated as the sum of all the forces exerted by those parts. For an object near the earth’s surface, some of that mass would be down below and some would be off to the side. In later terms this would be a problem of integral calculus; in the Principia he solved it geometrically, proving that a perfect spherical shell would attract objects outside it exactly as by a force inversely proportional to the square of the distance to the center.17

Meanwhile, he had to solve the path of a projectile attracted to this center, not with constant force, but with a force that varies continually because it depends on the distance. He had to create a dynamics for velocities changing from moment to moment, both in magnitude and in direction, in three dimensions. No philosopher had ever conceived such a thing, much less produced it.

A handful of mathematicians and astronomers on earth could hope to follow the argument. The Principia’s reputation for unreadability spread faster than the book itself. A Cambridge student was said to have remarked, as the figure of its author passed by, “There goes the man that writt a book that neither he nor anybody else understands.”18 Newton himself said that he had considered composing a “popular” version but chose instead to narrow his readership, to avoid disputations—or, as he put it privately, “to avoid being baited by little smatterers in mathematicks.”19

Yet as the chain of proof proceeded, it shifted subtly toward the practical. The propositions took on a quality of how to. Given a focus, find the elliptical orbit. Given three points, draw three slanted straight lines to a fourth point. Find the velocity of waves. Find the resistance of a sphere moving through a fluid. Find orbits when neither focus is given. Q.E.D. gave way to Q.E.F. and Q.E.I.: that which was to be done and that which was to be found out. Given a parabolic trajectory, find a body’s position at an assigned time.

There was meat for observant astronomers.

On the way, Newton paused to obliterate the Cartesian cosmology, with its celestial vortices. Descartes, with his own Principia Philosophiæ, was his chief forebear; Descartes had given him the essential principle of inertia; it was Descartes, more than any other, whom he now wished to bury. Newton banished the vortices by taking them seriously: he did the mathematics. He created methods to compute the rotation of bodies in a fluid medium; he calculated relentlessly and imaginatively, until he demonstrated that such vortices could not persist. The motion would be lost; the rotation would cease. The observed orbits of Mars and Venus could not be reconciled with the motion of the earth. “The hypothesis of vortices … serves less to clarify the celestial motions than to obscure them,” he concluded.20 It was enough to say that the moon and planets and comets glide in free space, obeying the laws of motion, under the influence of gravity.

Book III gave The System of the World. It gathered together the phenomena of the cosmos. It did this flaunting an exactitude unlike anything in the history of philosophy. Phenomenon 1: the four known satellites of Jupiter. Newton had four sets of observations to combine. He produced some numbers: their orbital periods in days, hours, minutes, and seconds, and their greatest distance from the planet, to the nearest thousandth of Jupiter’s radius. He did the same for the five planets, Mercury, Venus, Mars, Jupiter, and Saturn. And for the moon.

From the propositions established in Book I, he now proved that all these satellites are pulled away from straight lines and into orbits by a force toward a center—of Jupiter, the sun, or the earth—and that this force varies inversely as the square of the distance. He used the word gravitate. “The moon gravitates toward the earth and by the force of gravity is always drawn back from rectilinear motion and kept in its orbit.”21 He performed an apple-moon computation with data he had lacked in Woolsthorpe twenty years before. The moon’s orbit takes 27 days, 7 hours, 43 minutes. The earth measures 123,249,600 Paris feet around. If the same force that keeps the moon in orbit draws a falling body “in our regions,” then a body should fall, in one second, 15 feet, 1 inch, and 17/9 lines (twelfths of an inch). “And heavy bodies do actually descend to the earth with this very force.” No one could time a falling body with such precision, but Newton had some numbers from beating pendulums, and, performing the arithmetic, he slyly exaggerated the accuracy.22 He said he had tested gold, silver, lead, glass, sand, salt, wood, water, and wheat—suspending them in a pair of identical wooden boxes from eleven-foot cords and timing these pendulums so precisely that he could detect a difference of one part in a thousand.23

Furthermore, he proposed, the heavenly bodies must perturb one another: Jupiter influencing Saturn’s motion, the sun influencing the earth, and the sun and moon both perturbing the sea. “All the planets are heavy toward one another.”24 He pronounced:

It is now established that this force is gravity, and therefore we shall call it gravity from now on.

One flash of inspiration had not brought Newton here. The path to universal gravitation had led through a sequence of claims, each stranger than the last. A force draws bodies toward the center of the earth. This force extends all the way to the moon, pulling the moon exactly as it pulls an apple. An identical force—but toward the center of the sun—keeps the earth in orbit. Planets each have their own gravity; Jupiter is to its satellites as the sun is to the planets. And they all attract one another, in proportion to their mass. As the earth pulls the moon, the moon pulls back, adding its gravity to the sun’s, sweeping the oceans around the globe in a daily flood. The force points toward the centers of bodies, not because of anything special in the centers, but as a mathematical consequence of this final claim: that every particle of matter in the universe attracts every other particle. From this generalization all the rest followed. Gravity is universal.

Newton worked out measurements for weights on the different planets. He calculated the densities of the planets, suggesting that the earth was four times denser than either Jupiter or the sun. He proposed that the planets had been set at different distances so that they might enjoy more or less of the sun’s heat; if the earth were as distant as Saturn, he said, our water would freeze.25

He calculated the shape of the earth—not an exact sphere, but oblate, bulging at the equator because of its rotation. He calculated that a given mass would weigh differently at different altitudes; indeed, “our fellow countryman Halley, sailing in about the year 1677 to the island of St. Helena, found that his pendulum clock went more slowly there than in London, but he did not record the difference.”26

He explained the slow precession of the earth’s rotation axis, the third and most mysterious of its known motions. Like a top slightly off balance, the earth changes the orientation of its axis against the background of the stars, by about one degree every seventy-two years. No one had even guessed at a reason before. Newton computed the precession as the complex result of the gravitational pull of the sun and moon on the earth’s equatorial bulge.

Into this tapestry he wove a theory of comets. If gravity was truly universal, it must apply to these seemingly random visitors as well. They behaved as distant, eccentric satellites of the sun, orbiting in elongated ellipses, crossing the plane of the planets, or even ellipses extended to infinity—parabolas and hyperbolas, in which case the comet never would return.

These elements meshed and turned together like the parts of a machine, the work of a perfect mechanic, like an intricate clock, a metaphor that occurred to many as news of the Principia spread. Yet Newton himself never succumbed to this fantasy of pure order and perfect determinism. Continuing to calculate where calculation was impossible, he saw ahead to the chaos that could emerge in the interactions of many bodies, rather than just two or three. The center of the planetary system, he saw, is not exactly the sun, but rather the oscillating common center of gravity. Planetary orbits were not exact ellipses after all, and certainly not the same ellipse repeated. “Each time a planet revolves it traces a fresh orbit, as happens also with the motion of the Moon, and each orbit is dependent upon the combined motions of all the planets, not to mention their actions upon each other,” he wrote. “Unless I am much mistaken, it would exceed the force of human wit to consider so many causes of motion at the same time, and to define the motions by exact laws which would allow of an easy calculation.”27

The comet of 1680—“as observed by Flamsteed” and “corrected by Dr. Halley.” Newton also collated sightings by Ponthio in Rome, Gallet in Avignon, Ango at La Fleche, “a young man” at Cambridge, and Mr. Arthur Staorer near Hunting Creek, in Maryland, in the confines of Virginia. “Thinking it would not be improper, I have given … a true representation of the orbit which this comet described, and of the tail which it emitted in several places.” He concludes that the tails of comets always rise away from the sun and “must be derived from some reflecting matter”—smoke, or vapor(illustration credit 12.1)

Yet he solved another messy, bewildering phenomenon, the tides. He had assembled data, crude and scattered though they were. Samuel Sturmy had recorded observations from the mouth of the River Avon, three miles below Bristol. Samuel Colepress had measured the ebb and flow in Plymouth Harbor. Newton considered the Pacific Ocean and the Ethiopic Sea, bays in Normandy and at Pegu in the East Indies.28 Halley himself had analyzed observations by sailors in Batsha Harbor in the port of Tunking in China. None of these lent themselves to a rigorous chain of calculation, but the pattern of two high tides per twenty-five hours was clear and global. Newton marshaled the data and made his theoretical claim. The moon and sun both pull the seas; their combined gravity creates the tides by raising a symmetrical pair of bulges on opposite sides of the earth.

Kepler had suggested a lunar influence on the seas. Galileo had mocked him for it:

That concept is completely repugnant to my mind.… I cannot bring myself to give credence to such causes as lights, warm temperatures, predominances of occult qualities, and similar idle imaginings.…

I am more astonished at Kepler than at any other.… Though he has at his fingertips the motions attributed to the Earth, he has nevertheless lent his ear and his assent to the moon’s dominion over the waters, to occult properties, and to such puerilities.29

Now Newton, too, resorted to invisible action at a distance. Such arcana had to offend the new philosophers.

Before confronting the phenomena, Newton stated “Rules of Philosophizing”—rules for science, even more fundamental in their way than the laws of motion.

No more causes of natural things should be admitted than are both true and sufficient to explain their phenomena. Do not multiply explanations when one will suffice.

The causes assigned to natural effects of the same kind must be, so far as possible, the same. Assume that humans and animals breathe for the same reason; that stones fall in America as they do in Europe; that light is reflected the same way by the earth and the planets.30

But the mechanical philosophy already had rules, and Newton was flouting one of them in spectacular fashion. Physical causes were supposed to be direct: matter striking or pressing on matter, not emitting invisible forces to act from afar. Action at a distance, across the void, smacked of magic. Occult explanations were supposed to be forbidden. In eliminating Descartes’s vortices he had pulled away a much-needed crutch. He had nothing mechanical to offer instead. Indeed, Huygens, when he first heard about Newton’s system of the world, replied, “I don’t care that he’s not a Cartesian as long as he doesn’t serve us up conjectures such as attractions.”31 As a strategy for forestalling the inevitable criticism, Newton danced a two-step, confessional and defiant.

I have explained the phenomena of the heavens and of our sea by the force of gravity, but I have not yet assigned a cause to gravity.… I have not as yet been able to deduce … the reasons for these properties of gravity, and I do not feign hypotheses. For whatever is not deduced from the phenomena must be called a hypothesis; and hypotheses, whether metaphysical or physical, or based on occult qualities, or mechanical, have no place in experimental philosophy.…32

So gravity was not mechanical, not occult, not a hypothesis. He had proved it by mathematics. “It is enough,” he said, “that gravity really exists and acts according to the laws that we have set forth and is sufficient to explain all the motions of the heavenly bodies and of our sea.”33 It could not be denied, even if its essence could not be understood.

He had declared at the outset that his mission was to discover the forces of nature. He deduced forces from celestial bodies’ motion, as observed and recorded. He made a great claim—the System of the World—and yet declared his program incomplete. In fact, incompleteness was its greatest virtue. He bequeathed to science, that institution in its throes of birth, a research program, practical and open-ended. There was work to do, predictions to be computed and then verified.

“If only we could derive the other phenomena of nature from mechanical principles by the same kind of reasoning!” he wrote. “For many things lead me to have a suspicion that all phenomena may depend on certain forces by which the particles of bodies, by causes not yet known, either are impelled toward one another and cohere in regular figures, or are repelled from one another and recede.”34 Unknown forces—as mysterious still as the forces he sought through his decades-long investigation of alchemy. His suspicion foresaw the program of modern physics: certain forces, attraction and repulsion, final causes not yet known.