First Principles
IN THE NEXT YEAR a comet came. In England it arose faint in the early morning sky for a few weeks in November till it approached the sun and faded in the dawn. Few saw it.
A more dramatic spectacle appeared in the nights of December. Newton saw it with naked eye on December 12: a comet whose great tail, broader than the moon, stretched over the full length of King’s College Chapel. He tracked it almost nightly through the first months of 1681.1 A young astronomer traveling to France, Edmond Halley, a new Fellow of the Royal Society, was amazed at its brilliance.2 Robert Hooke observed it several times in London. Across the Atlantic Ocean, where a handful of colonists were struggling to survive on a newfound continent, Increase Mather delivered a sermon, “Heaven’s Alarm to the World,” to warn Puritans of God’s displeasure.3
Halley served as a sometime assistant to a new officeholder, the Astronomer Royal. This was John Flamsteed, a clergyman and self-taught skywatcher appointed by the King in 1675, responsible for creating and equipping an observatory on a hilltop across the River Thames at Greenwich. The Astronomer’s chief mission was to perfect star charts for the Navy’s navigators. Flamsteed did this assiduously, recording star places with his telescope and sextant night after night, more than a thousand observations each year. Yet he had not seen the November comet. Now letters from England and Europe alerted him to it.4
Whatever comets were, omens or freaks, their singularity was taken for granted: each glowing visitor arrived, crossed the sky in a straight path, and departed, never to be seen again. Kepler had said this authoritatively, and what else could a culture of short collective memory believe?
But this year European astronomers recorded two: a faint predawn comet that came and went in November 1680, and a great giant that appeared a month later and dominated the skies till March. Flamsteed thought comets might behave like planets.5 Immersed as he was in the geometry of the sky, charting the changes in celestial perspective as the earth orbited the sun, he predicted that the comet he had missed in November might yet return. He watched the sky for it. His intuition was rewarded; he spied a tail on December 10, and the tail and head together, near Mercury, two days later. He had a friend at Cambridge, James Crompton, and he sent notes of his observations, hoping Crompton could pass them on to Newton. A fortnight later he wrote again, speculating, “If we suppose it a consumeing substant ’tis much decayed and the Fuell spent which nourishes the blaze but I have much to say against this hypothesis however you may consider of it and Pray let me have your opinion.”6 Newton read this and remained silent.
A month later Flamsteed tried again. “It may seem that the exteriour coat of the Comet may be composed of a liquid.… It was never well defined nor shewed any perfect limb but like a wisp of hay.”7 He was persuaded more than ever that the two comets were one. After all, he had predicted the reappearance. He struggled to explain the peculiar motion he had recorded. Suppose, he said, the sun attracts the planets and other bodies that come within its “Vortex”—perhaps by some form of magnetism. Then the comet would approach the sun in a straight line, and this path could be bent into a curve by the pressure of the ethereal vortex.8 How to explain its return? Flamsteed suggested a corresponding force of repulsion; he likened the sun to a magnet with two poles, one attracting and one repelling.
Finally Newton replied. He objected to the notion of magnetism in the sun for a simple reason: “because the 
is a vehemently hot body & magnetick bodies when made red hot lose their virtue.” He was not persuaded that the two comets were one and the same, because his exquisitely careful measurements of their transit, and all the others he could collect—6 degrees a day, 36 minutes a day, 3½ degrees a day—seemed to show acceleration suddenly alternating with retardation.9 “It is very irregular.” Even so, he diagrammed Flamsteed’s proposal, the comet nearing the sun, swerving just short of it, and veering away. This he declared unlikely. Instead he suggested that the comet could have gone all the way around the sun and then returned.10 He diagrammed this alternative, too. And he conceded a crucial point to Flamsteed’s intuition: “I can easily allow an attractive power in the whereby the Planets are kept in their courses about him from going away in tangent lines.”
He had never before said this so plainly. In the gestation of the calculus, in 1666, he had relied on tangents to curves—the straight lines from which curves veer, through the accumulation of infinitesimal changes. In laying the groundwork for laws of motion, he had relied on the tendency of all bodies to continue in straight lines. But he had also persisted in thinking of planetary orbits as a matter of balance between two forces: one pulling inward and the other, “centrifugal,” flinging outward. Now he spoke of just one force, pulling a planet away from what would otherwise be a straight trajectory.
This very conception had arrived at his desk not long before in a letter from his old antagonist Hooke. Now Secretary to the Royal Society, in charge of the Philosophical Transactions, Hooke wrote imploring Newton to return to the fold. He made glancing mention of their previous misunderstandings: “Difference in opinion if such there be me thinks shoud not be the occasion of Enmity.”11 And he asked for a particular favor: would Newton share any objections he might have to his idea, published five years before, that the motions of planets could be simply a compound of a straight-line tangent and “an attractive motion towards the centrall body.” A straight line plus a continuous deflection equals an orbit.
Newton, just back in Cambridge after settling his mother’s affairs, lost no time in composing his reply. He emphasized how remote he was from philosophical matters:
heartily sorry I am that I am at present unfurnished with matter answerable to your expectations. For I have been this last half year in Lincolnshire cumbred with concerns.… I have had no time to entertein Philosophical meditations.… And before that, I had for some years past been endeavouring to bend my self from Philosophy … which makes me almost wholy unacquainted with what Philosophers at London or abroad have lately been employed about.… I am almost as little concerned about it as one tradesman uses to be about another man’s trade or a country man about learning.12
Hooke’s essay offered a “System of the World.”13 It paralleled much of Newton’s undisclosed thinking about gravity and orbits in 1666, though Hooke’s system lacked a mathematical foundation. All celestial bodies, Hooke supposed, have “an attraction or gravitating power towards their own centers.” They attract their own substance and also other bodies that come “within the sphere of their activity.” All bodies travel in a straight line until their course is deflected, perhaps into a circle or an ellipse, by “some other effectual powers.” And the power of this attraction depends on distance.
Newton professed to know nothing of Hooke’s idea. “Perhaps you will incline the more to beleive me when I tell you that I did not before the receipt of your last letter, so much as heare (that I remember) of your Hypotheses.”14 He threw Hooke a sop, however: an outline of an experiment to demonstrate the earth’s daily spin by dropping a ball from a height. “The vulgar” believed that, as the earth turns eastward under the ball, the ball would land slightly to the west of its starting point, having been left behind during its fall. On the contrary, Newton proposed that the ball should land to the east. At its initial height, it would be rotating eastward with a slightly greater velocity than objects down on the surface; thus it should “outrun” the perpendicular and “shoot forward to the east side.” For a trial, he suggested a pistol bullet on a silk line, outdoors on a very calm day, or in a high church, with its windows well stopped to block the wind.
He drew a diagram to illustrate the point. In it he allowed his imaginary ball to continue in a spiral to the center of the earth.15 This was an error, and Hooke pounced. Having promised days earlier to keep their correspondence private, he now read Newton’s letter aloud to the Royal Society and publicly contradicted it.16 An object falling through the earth would act like an orbiting planet, he said. It would not descend in a spiral—“nothing att all akin to a spirall”—but rather, “my theory of circular motion makes me suppose,” continue to fall and rise in a sort of orbit, perhaps an ellipse or “Elleptueid.”17
How a body falls to the center of the earth: Newton and Hooke’s debate of 1679. (illustration credit 11.1)
a. Newton: A body dropped from a height at A should be carried forward by its motion and land to the east of the perpendicular, “quite contrary to the opinion of the vulgar.” (But he continues the path—erroneously—in a spiral to the center.)
b. Hooke: “But as to the curve Line which you seem to suppose it to Desend by … Vizt a kind of spirall … my theory of circular motion makes me suppose it would be very differing and nothing att all akin to a spirall but rather a kind Elleptueid.”
c. Newton: The true path, supposing a hollow earth and no resistance, would be even more complex—“an alternating ascent & descent.”
Once again Hooke had managed to drive Newton into a rage.18 Newton replied once more and retreated to silence. Yet in their brief exchange the two men engaged as never before on the turf of this peculiar, un-physical, ill-defined thought experiment. It was “a Speculation of noe Use yet,” Hooke agreed. After all, the earth was solid, not void. They exchanged dueling diagrams.
They goaded each other into defining the terms of a profound problem. Hooke drew an ellipse.19 Newton replied with a diagram based on the supposition that the attractive force would remain constant but also considered the case where gravity was—to an unspecified degree—greater nearer the center. He also let Hooke know that he was bringing potent mathematics to bear: “The innumerable & infinitly little motions (for I here consider motion according to the method of indivisibles) …” Both men were thinking in terms of a celestial attractive force, binding planets to the sun and moons to the planets. They were writing about gravity as though they believed in it. Both now considered it as a force that pulls heavy objects down to the earth. But what could be said about the power of this force? First Hooke had said that it depended on a body’s distance from the center of the earth. He had been trying in vain to measure this, with brass wires and weights atop St. Paul’s steeple and Westminster Abbey. Meanwhile the intrepid Halley, an eager seagoing traveler, had carried a pendulum up a 2,500-foot hill on St. Helena, south of the equator, and judged that it swung more slowly there.
Hooke and Newton had both jettisoned the Cartesian notion of vortices. They were explaining the planet’s motion with no resort to ethereal pressure (or, for that matter, resistance). They had both come to believe in a body’s inherent force—its tendency to remain at rest or in motion—a concept for which they had no name. They were dancing around a pair of questions, one the mirror of the other:
What curve will be traced by a body orbiting another in an inverse-square gravitational field? (An ellipse.)
What gravitational force law can be inferred from a body orbiting another in a perfect ellipse? (An inverse-square law.)
Hooke finally did put this to Newton: “My supposition is that the Attraction always is in a duplicate proportion to the Distance from the Center Reciprocall”—that is, inversely as the square of distance.20 He got no reply. He tried again:
It now remaines to know the proprietys of a curve Line … made by a centrall attractive power … in a Duplicate proportion to the Distances reciprocally taken. I doubt not but that by your excellent method you will easily find out what that Curve must be, and its proprietys, and suggest a physicall Reason of this proportion.21
Hooke had finally formulated the problem exactly. He acknowledged Newton’s superior powers. He set forth a procedure: find the mathematical curve, suggest a physical reason. But he never received a reply.
Four years later Edmond Halley made a pilgrimage to Cambridge. Halley had been discussing planetary motion in coffee-houses with Hooke and the architect Christopher Wren. Some boasting ensued. Halley himself had worked out (as Newton had in 1666) a connection between an inverse-square law and Kepler’s rule of periods—that the cube of a planet’s distance from the sun varies as the square of its orbital year. Wren claimed that he himself had guessed at the inverse-square law years before Hooke, but could not quite work out the mathematics. Hooke asserted that he could show how to base all celestial motion on the inverse-square law and that he was keeping the details secret for now, until more people had tried and failed; only then would they appreciate his work.22 Halley doubted that Hooke knew as much as he claimed.
Halley put the question to Newton directly in August 1684: supposing an inverse-square law of attraction toward the sun, what sort of curve would a planet make? Newton told him: an ellipse. He said he had calculated this long before. He would not give Halley the proof—he said he could not lay his hands on it—but promised to redo it and send it along.
Months passed. He began with definitions. He wrote only in Latin now, the words less sullied by everyday use. Quantitas materiæ—quantity of matter. What did this mean exactly? He tried: “that which arises from its density and bulk conjointly.” Twice the density and twice the space would mean four times the amount of matter. Like weight, but weight would not do; he could see ahead to traps of circular reasoning. Weight would depend on gravity, and gravity could not be presupposed. So, quantity of matter: “This quantity I designate under the name of body or mass.”23 Then, quantity of motion: the product of velocity and mass. And force—innate, or impressed, or “centripetal”—a coinage, to mean action toward a center. Centripetal force could be absolute, accelerative, or motive. For the reasoning to come, he needed a foundation of words that did not exist in any language.
He could not, or would not, give Halley a simple answer. First he sent a treatise of nine pages, “On the Motion of Bodies in Orbit.”24 It firmly tied a centripetal force, inversely proportional to the square of distance, not only to the specific geometry of the ellipse but to all Kepler’s observations of orbital motion. Halley rushed back to Cambridge. His one copy had become an object of desire in London. Flamsteed complained: “I beleive I shall not get a sight of [it] till our common freind Mr Hooke & the rest of the towne have been first satisfied.”25 Halley begged to publish the treatise, and he begged for more pages, but Newton was not finished.
As he wrote, computed, and wrote more, he saw the pins of a cosmic lock tumbling into place, one by one. He pondered comets again: if they obeyed the same laws as planets, they must be an extreme case, with vastly elongated orbits. He wrote Flamsteed asking for more data.26 He first asked about two particular stars, but Flamsteed guessed immediately that his quarry was the comet. “Now I am upon this subject,” Newton said, “I would gladly know the bottom of it before I publish my papers.” He needed numbers for the moons of Jupiter, too. Even stranger: he wanted tables of the tides. If celestial laws were to be established, all the phenomena must obey them.
The birth of universal gravitation: Newton proves by geometry that if a body Q orbits in an ellipse, the implied force toward the focus S (not the center C) varies inversely with the square of distance. (illustration credit 11.2)
The alchemical furnaces went cold; the theological manuscripts were shelved. A fever possessed him, like none since the plague years. He ate mainly in his room, a few bites standing up. He wrote standing at his desk. When he did venture outside, he would seem lost, walk erratically, turn and stop for no apparent reason, and disappear inside once again.27 Thousands of sheets of manuscript lay all around, here and at Woolsthorpe, ink fading on parchment, the jots and scribbles of four decades, undated and disorganized. He had never written like this: with a great purpose, and meaning his words to be read.
Though he had dropped alchemy for now, Newton had learned from it. He embraced invisible forces. He knew he was going to have to allow planets to influence one another from a distance. He was writing the principles of philosophy. But not just that: the mathematical principles of natural philosophy. “For the whole difficulty of philosophy,” he wrote, “seems to be to discover the forces of nature from the phenomena of motions and then to demonstrate the other phenomena from these forces.”28 The planets, the comets, the moon, and the sea. He promised a mechanical program—no occult qualities. He promised proof. Yet there was mystery in his forces still.
First principles. “Time, space, place, and motion”—he wished to blot out everyday knowledge of these words. He gave them new meanings, or, as he saw it, redeemed their true and sacred meanings.29 He had no authority to rely on—this unsocial, unpublished professor—so it was a sort of bluff, but he made good on it. He established time as independent of our sensations; he established space as independent of matter. Thenceforth time and space were special words, specially understood and owned by the virtuosi—the scientists.
Absolute, true, and mathematical time, in and of itself, and of its own nature, without reference to anything external, flows uniformly.…
Absolute space, of its own true nature without reference to anything external, always remains homogeneous and immovable.…30
Our eyes perceive only relative motion: a sailor’s progress along his ship, or the ship’s progress on the earth. But the earth, too, moves, in reference to space—itself immovable because it is purely mathematical, abstracted from our senses. Of time and space he made a frame for the universe and a credo for a new age.
