This latter observation ‘set Newton on the right track’, though Newton would spend the rest of his life denying Hooke’s contribution.15 In the early 1680s, Newton did not have the law of universal gravitation yet; for one thing, he still treated comets as aliens to the solar system. But he used Hooke’s method of analyzing curved motion by decomposing it into a straight-line centripetal force and straight-line inertial motion to great effect. It opened the door to think of everything – falling bodies, planets – as governed by one centre-seeking force. Newton also employed Hooke’s method, plus the inverse square law, to establish the fundamental connection of Kepler’s laws of motion. A body, attracted by another body by an inverse square force, travels around it in an elliptical path with the central body at one focus, and a line drawn between the central and orbiting bodies sweeps out equal areas in equal times.16
Then Halley dropped by to visit Newton in Cambridge in August 1684. A contemporary described the visit:
After they had been some time together, Dr [Halley] asked him 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. Sir Isaac replied immediately that it would be an ellipsis. The doctor, struck with joy & amazement, asked him how he knew it. Why, saith he, I have calculated it. Whereupon Dr. Halley asked him for his calculation without any further delay. Sir Isaac looked among his papers but could not find it, but he promised him to renew it and then to send it to him.17
Was this Newton’s paranoia and secretiveness, or had he really misplaced the calculation? We can’t say. In any case, Newton set out to rework the calculation for Halley, and by early December this transmuted into the first draft of a short, nine-page work entitled De Motu (Concerning Motion). In it, Newton treated the sun as fixed and immobile: as a body that attracted everything else in the solar system but remained unaffected by the planets swirling about it. This work brought Newton to the threshold of universal gravitation, but it lacked a key idea. According to Newton’s third law of motion – for any action there is an equal and opposite reaction – if the sun tugged on a planet it meant that the planet also tugged back on the sun, affecting its motion. This seems to have occurred to Newton only after completing the first draft of De Motu.
Newton therefore set about revising the work, which he finished by the end of December 1684. This is the first document to embody the key insight of universal gravitation – that all bodies act on each other – using the phrase ‘eorum omnium actiones in se invicem’, or ‘the actions of all these on each other.’ If the sun had one planet orbiting about it, for instance, the two bodies would revolve about a common centre of gravity. But the solar system contains many planets, each of which tugs on the sun and on one another. No planet therefore moves in a perfect ellipse, nor ever follows the same path twice. Indeed, Newton wrote, to calculate the complex net result of all the tugs ‘exceeds, unless I am mistaken, the reach of the entire human intellect.’18
Newton had not only achieved a deeper insight into the solar system but had also transformed scientific procedure. He had transformed Galileo’s thought experiment of an infinite plane without resistances into a complete world-stage, on which masses appear and do nothing but move under the influence of forces. Scientists create models on this world-stage – such as Kepler’s laws of motion – and compare these models to observations of the real world. But these models are only approximations, and have to be constantly refined. Newton’s early work had been motivated by Kepler’s laws, which he had assumed to be accurate descriptions – and these had led him to conclude that Kepler’s laws were wrong, and to predict deviations from them.19
Newton gave De Motu to Halley in December 1684. Halley asked Newton if he could publish it, but Newton refused. Instead, he set about expanding De Motu, weaving together his new insights into the structure of the solar system with other insights, including Hooke’s idea of analyzing circular motion into two components.
The result, which appeared after 18 months of labour in 1686, was the Principia, the single most influential piece of writing in science. Near the beginning of Book I, Newton skillfully uses Hooke’s method of analyzing curved motions by breaking them down into centripetal forces plus inertia, to derive Kepler’s laws, among others. In a part of Book II, Newton demonstrates that Descartes’ vortices could not explain the motions of planets, and promises an adequate explanation. In Book III, the ‘System of the World’, Newton follows through on that promise. He carries out the ‘moon test’, measuring the force tugging objects on the earth’s surface, and shows that it has the same strength as the force with which the earth tugs the moon; furthermore, this force has the same strength as that between the sun and the planets, and the other planets and their satellites. Until now, Newton announces dramatically a few paragraphs later, we have called these all ‘centripetal’ forces – but now that we are sure it is the same force, we can call it by one name: gravity. Gravity ‘exists in all bodies universally’, and its strength between two bodies depends on their masses and ‘will be inversely as the square of the distance between the centres.’ As we write it now, Fg = Gm1m2/r2.
Hooke later claimed priority for the discovery of this law, and we can see why. But we can also see why Newton (and many historians) rejected this claim. Newton clearly profited from Hooke’s work, but when Newton famously said that he saw farther than others because he stood on the shoulders of giants, the statement owed its truth to its irony. Newton was alluding sarcastically to Hooke’s diminutive stature; that is, the boost from him was more like that of a footstool than a tower. Hooke had suggested the inverse square law chiefly with respect to one body, or at most with respect to celestial bodies, while Newton made its universality explicit. Hooke’s most important boost to Newton had been in showing Newton how to analyse curved orbital motions. But the priority issue is further obscured, both factually and morally, by Newton’s mendacious practice in memoirs and conversations of backdating key events in his work on universal gravitation – including the moon test – to clinch priority over Hooke. Still, what makes Newton stand out as its discoverer – above Boulliau, Hooke, and others – is his clear statement that gravity is not just a force by which certain bodies grip or are gripped by certain other bodies – not bodies falling and the bodies to which they fall, nor heavenly bodies to one another – but all bodies to all other bodies.
There were some odd features of Newton’s account. Why, for instance, was the mass of a body involved in the gravitational force the same as the mass in the push-pull force described by F = ma? It didn’t have to be. Was this an accident? If so, it was an awfully strange accident. The answer to this puzzle would play a role in the development of general relativity over 200 years later. But in Newton’s time, one would have to think carefully to see it as a puzzle, so daring and dazzling was the sweep of Newton’s vision.
It was a deeply democratic vision. Gravity is a universal force, and it does not matter what a body looks like, nor where it lives in the universe, but solely how much mass it has. Galileo had universalized things, and achieved insights, by turning all chandeliers into pendulums. Newton now universalized even more ambitiously by turning all bodies into attractors. Gravitation is all bodies, all the time, everywhere.
The Law That Explained Law
Newton’s equation of universal gravitation was hailed as the capstone of one of the most profound transformations of Western science. It led to Newton becoming the ‘gold standard’ against which scholars in other sciences compared the superstars in their fields. James Clerk Maxwell, for example, hailed Ampère as the Newton of electricity, while Alfred R. Wallace, Thomas Huxley, and others called Darwin the Newton of biology.
Diagram of Newton’s cannonball thought experiment illustrating the idea of an orbit. What would happen if one shot a cannonball horizontally from a peak that poked above the atmosphere? The more forcefully the cannonball is shot, the farther around the
earth it will travel. With enough force, it returns to the peak, and follows the same path over and over again.
Moreover, Newton’s law of gravitation was often cited as the kind of law that a mature science required. François Magendie, in his classic textbook Elementary Sketch of Physiology (1817), lamented the absence from his field of ‘an intellect of the first order to come and discover the laws of the vital force in the same way Newton made known the laws of attraction.’
But the influence of Newton’s equation extended well beyond science – to education, philosophy, theology, and other areas of human culture. It also helped to change the very notion of ‘law’ itself.
In modern times, the concept of a scientific law has a specific meaning; it is something descriptive of nature and its behaviour. For example, in his book The Software of the Universe: An Introduction to the History and Philosophy of Laws of Nature, the philosopher Mauro Dorato from the University of Rome = calls a scientific law ‘a mathematical relationship between properties of physical systems.’
But it was not always that way. For the ancient Greeks, laws were not descriptive but normative, from the Greek nomos, the custom or behaviour of human beings. A law was an order that a ruler gave to subjects, who could then choose to obey or not obey. (For the nonhuman parts of the world, what we understand in terms of laws was then expressed in the idea of the thing’s characteristic nature.) Even as late as the seventeenth century, many scientists refused to apply the term ‘law’ to regularities in nature, insisting that it was no more than a metaphorical extension of social language to the natural world. But the growing appreciation for the clockworklike structure of the cosmos inclined others, such as Descartes, to describe creation as a juridical act by a supreme lawgiver. The difference between the human and nonhuman order is that the latter obey God unconsciously, while the former obey (or disobey) consciously.
Newton viewed the world in this manner. He saw himself as describing a universal principle that pervades the entire universe and affects everything in it, something whose influence is direct, immediate, and authoritative. The very universality of this principle, and the care with which Newton states that gravitation is not a property in matter, was part and parcel of his view that he was describing the actions of a supreme lawgiver.20 Newton’s matter is lifeless; it moves only when touched by a force. This got ‘out of God’s way’, to guarantee that the Creator had a free hand.21 Newton’s mechanical view of the universe, as full of objects passively responding to forces from without, thus was not only consistent with a supreme lawgiver, but required it. How could there be laws and no lawgiver? As he wrote, ‘This most Elegant System of the Planets and Comets could not be produced but by and under the Contrivance and Domination of an Intelligent and Powerful Being.’ (That Sir Isaac Newton could have thought this regarding the origin of the solar system, which is now easily accounted for by the action of simple principles over time, makes us marvel at the outrageous hubris of those much smaller intellects today who are so confident that their inability to explain something’s origin means that this thing must have been the act of a god.)
Newton’s equation of gravitation gave an enormous boost to the inclination to view laws descriptively rather than normatively. The influence was reversed: natural language was now extended to the social world.
One of Newton’s assistants, the Royal Society member John Theophilus Desaguliers, composed a poem entitled ‘The Newtonian System of the World, the Best Model of Government.’ Desaguliers found in the Newtonian system, consisting of ‘the most regular Attraction of universal Gravity, (or attraction) whose Power is diffus’d from the Sun to the very Centres of all the Planets and Comets’ to be a ‘lively image of our System’ of government (the British), namely, ‘The limited Monarchy, whereby our Liberties, Rights and Privileges are so well secured.’ Thanks to this, he concluded, ‘the Happiness that we enjoy under His present MAJESTY’s Government’ is a sign ‘that A-T-T-R-A-C-T-I-O-N is now as universal in the Political, as the Philosophical World.’22
But political theorists also began to use Newtonian language – so much so that it sharply influenced the modern conception of democracy, as Cohen detailed in his 1995 book Science and the Founding Fathers: Science in the Political Thought of Thomas Jefferson, Benjamin Franklin, John Adams, and James Madison. All of the U.S. founding fathers read Newton, Cohen pointed out. Jefferson, whom Cohen describes as ‘surely the only president of the United States who ever read Newton’s Principia’, had several copies of the Principia in his library and Newton’s portrait on the wall; Franklin was so deeply impressed by Newton as a young man that he tried to meet him in London; Adams once cited Newton’s laws of motion in a political debate; and Madison wrote an essay comparing nature and human affairs.
Even the birth of socialism is tied up with Newton’s law. For the political thinker Henri de Saint-Simon (1760–1825), who was one of the founders of socialism, Newton’s law was not only the purest example of scientific thinking but also provided the model for creating a science of human social life, based on universal fraternity and collective organization. Saint-Simon once had a vision in which God disclosed to him that Newton sat at his right hand and decreed that the world should be governed by a committee called the Council of Newton. Its primary task, besides improving humanity – Saint-Simon is quoting God now – was to discover ‘a new law of gravitation applicable to social bodies.’ Newton’s equation was not just a key fact, but the key fact, unifying science and provoking the search for a law of social order that would work not merely between individuals and groups but also nations. Saint-Simon even criticized Newton for failing to turn gravity into an all-encompassing philosophical system.23 The sooner humanity found this law and reorganized society accordingly, the sooner it would be liberated.
To be sure, Saint-Simon was a flamboyant character, and the kind of megalomaniac aristocrat – idealist, bad writer, idiot, and eccentric – with which early nineteenth-century socialism was amply stocked. But he was not alone. Other political thinkers, including Pierre Cabanis (1757–1808), Charles Fourier (1772–1837), and Giovanni Morelli (1816–1891), tried to apply the notion of gravitational attraction to human life in holding that free, subjective, conscious individuals were nonetheless compelled by universal, deterministic scientific law – a notion that also influenced Karl Marx (1818–1883).
Newton’s equation of universal gravitation did more than quantify the attraction between objects, be they pebbles, spacecraft, or planets. Among other things, it inspired scholars in other fields – even political theory – to seek descriptive, mathematical, and universal laws. If the Pythagorean theorem was a proof that exhibited Proof, Newton’s equation of universal gravitation was a law that exhibited Law. In so doing, it not only altered our understanding of nature, but also our conception of science and human life.
The equation remains a symbol of the achievement of knowledge and rationality. In George Orwell’s novel 1984, the final sign that protagonist Winston Smith (after accepting that 2 + 2 = 5) had fully capitulated to the thought police – had been thoroughly broken and ceased to think – is that he denies the law of gravity.
Interlude
THAT APPLE
Then ye who now on heavenly nectar fare, Come celebrate with me in song the name Of Newton, to the Muses dear; for he Unlocked the hidden treasuries of Truth: So richly through his mind had Phoebus cast The radiance of his own divinity. Nearer the gods no mortal may approach.
– Edmond Halley, Ode to Newton
What of the apple?
The story that Newton discovered universal gravitation after seeing an apple fall is one of the oldest and most familiar legends of science.1 The incident is said to have taken place some time late in 1665 or 1666 at his mother’s orchard in Woolsthorpe, Lincolnshire, where Newton had retreated from studies at Cambridge to escape the plague. The story has long been dismissed as fiction, for several reasons. First, it seems just too theatrical to be true. Second, a cranky but
influential early biographer named David Brewster doubted the story. Third, and most importantly, the story is just not how great revolutions happen. The causal force implied by the story – that seeing an apple fall created the law of universal gravitation in Newton’s mind, without much further analysis and reflection – has to be false. As Newton’s biographer Richard Westfall observes, ‘The story vulgarizes universal gravitation by treating it as a bright idea.’2
Yet biographers have found abundant evidence that the ultimate source is Sir Isaac himself, who told the story to several different people – including his niece (who passed it on to Voltaire) and to his friend William Stukeley (1687–1765). Here’s the version from Stukeley’s memoirs:
The weather being warm [Kensington, England, April 15, 1726], we went into the garden and drank tea, under shade of some apple-trees, only he and myself. Amidst other discourses, he told me, he was just in the same situation, as when formerly, the notion of gravitation came into his mind. It was occasion’d by the fall of an apple, as he sat in contemplative mood. Why should that apple always descend perpendicularly to the ground, thought he to himself. Why should it not go sideways or upwards, but constantly to the earth’s centre?3
But we should still be skeptical. Why should this notoriously shy, secretive, and possessive person suddenly become garrulous, expansive, and giving about the origins of his greatest discovery? It doesn’t sound like the genuine Newton. Many writers and historians, in fact, suspect that it was not – that he was being devious, in an attempt to attack Hooke. For Hooke had claimed to be the first to come up with the inverse square law for gravitation, and had even once written a letter to Newton seeking Newton’s approval of the claim that he, Hooke, had come up with the law first. In telling this story, Newton was predating his discovery of gravitation to the 1660s, and thus removing the ground from Hooke’s claim. Such a deception sounds more like the genuine Newton, even if it is not the truthful Newton.
A Brief Guide to the Great Equations Page 8
