A Brief Guide to the Great Equations (10 page)

But what kind of thing was this corporeal force? This question would be debated for most of the rest of the seventeenth century. Some agreed with Kepler that it was a corporeal force. Others, such as Descartes, thought that it was purely mechanical and a product of tiny motions, called vortices, in a fluidlike substance called the ether in which the solar system was submerged.
⁹
Galileo, taking what Comte would have called a step into scientific thinking, preferred to stop discussing the nature of gravity altogether, and focus on measuring its quantitative effects. Just give us the numbers, please.

In 1645, French astronomer Ismael Boulliau (1605–1694) unwittingly stumbled across, and rejected, the right formula for the
strength of this force. Boulliau is a fascinating figure in the history of science, known for his accurate astronomical tables and quirky intellectual commitments. He was one of the first astronomers to accept Kepler’s idea that the planets move in elliptical orbits – but also one of the last astronomers to take astrology seriously, which led him to attack Kepler and his use of mathematics. On astrological grounds, Boulliau vehemently rejected Kepler’s conclusion that planetary motion was governed by an impersonal force from the sun whose strength weakened with distance. If there were such a force, Boulliau proclaimed, laughing at Kepler’s ridiculous idea, it would have to spread out in all directions, like light, meaning that it would weaken as the
square
of the distance. But this is absurd! Boulliau could not believe that God would act in such a way.
¹⁰

Several other scientists, however, realized that the force between planets and sun might indeed radiate in all directions, meaning that something like an ‘inverse square’ relationship was not absurd, and indeed probably involved in whatever force operated between sun and planets – but they thought that this relationship was the outcome of a tug of war between a centre-fleeing, or centrifugal, force and a centre-seeking force, with the inverse square behaviour as the outcome. These scientists also suspected that Kepler’s laws could be derived from an inverse square relationship.

One was Robert Hooke (1605–1703), the curator of experiments at the Royal Society in London. In 1674, Hooke proposed that the earth and all other celestial bodies possess ‘an attraction or gravitating power towards their own Centres’, which attracts not only parts of that body but all other bodies ‘within the sphere of their activity’, with the strength of the force depending on the distance between the bodies.
¹¹
Yet Hooke did not have the mathematical ability to use this surmise to calculate planetary motions. In 1679, still seeking an answer, he wrote a letter to the ablest mathematician around, Isaac Newton. What, Hooke asked Newton, do you think of my ideas about ‘an attractive motion towards the central body’?
¹²
And in January 1680, after exchanging letters with Newton, Hooke mentioned
his idea that the attractive force varied according to an inverse square law. If this were so, he asked Newton, would the planetary paths work out?

In 1680, as it happened, several events sparked interest in the motions of celestial objects, and curiosity about their behaviour. One was the appearance of a large and dramatic comet in the heavens, which was examined with interest by British astronomer Edmond Halley (1656–1742). That comet was followed by another comet in 1682 – this one is now known as ‘Halley’s comet’ – and yet another in 1684. Until this time, comets were assumed to be alien, randomly appearing objects in the solar systems, not governed by its laws. This opinion would soon change.

In January 1684, in a London coffee house, Halley, Hooke, and the scientist and architect Sir Christopher Wren (1632–1723) discussed the nature of planetary paths, and whether they could be accounted for by an inverse square law. Halley said he had tried and failed to calculate the paths based on such a law. Hooke boastfully said that he had done it but refused to produce a demonstration. Wren, both skeptical and impatient, challenged them to produce a demonstration within 2 months, saying that he would reward the one who did so with a book worth 40 shillings. The 2-month period expired, but the next time Halley wound up in Cambridge, that August, he broached the subject with Newton. That visit was the single most transformative event in Newton’s life. And it begat one of the most important events in Western science and culture – the birth of the
Principia
– in which the law of universal gravitation was a by-product.
¹³

Laws are like sausages, runs the old saw: the less you know how they are made, the more you respect the product. This remark is more clever than true. For what do you expect? If you truly understand
human creativity, you have no difficulty with knowing how tasty sausages and just regulations are made. But the remark does highlight a curious conundrum of creativity: that something momentous can arise from base origins. Few triumphs of the human mind illustrate this conundrum as sharply as Newton’s path to the law of universal gravitation. That path was marked by raging ambition, empty posturing, obsessive secrecy, seething jealousy, and transparent lies, but the product was breathtakingly brilliant. It was, Richard Feynman once said, ‘one of the most far-reaching generalizations of the human mind.’
¹⁴

Newton’s path to universal gravitation evolved during the same time period as his path to force discussed in the previous chapter. It began when he was a student at Trinity College, Cambridge, and jotted down numerous remarks about gravity in his notebooks. In some, he treats gravity as if it were an impetus-like ability internal to things that caused their motion; in others, dealing with celestial motions, he considers Descartes’ explanation that gravity arose from the pressures of particles created by vortices. For a long time, he accepted the notion of a centrifugal force, one that pushed away from a body, as a swinging stone tugs on the end of the tethered string to which it is attached.

Then, about 1680, Newton’s thinking about gravitation was profoundly altered by two key events, one philosophical and the other mathematical. The philosophical event was conversion away from an impetus-like idea of force as something that impelled a body to move from
within
, to the view that motion is caused by a force that acts on a body from
without
. This was accompanied by a dawning realization of the distinction (noticed before him, with varying degrees of clarity, by Robert Boyle, Galileo, and Kepler) between weight and mass, which is necessitated by the idea of forces that vary with distance. Weight varies with the distance from the earth’s surface; a body has a different weight at different altitudes. But the mass of a body, which is a key element of how the body moves, stays the same.

The other key event to profoundly alter Newton’s thinking about
gravitation was the correspondence with Newton’s nemesis Hooke that began in the fall of 1679.

Newton loathed Hooke. In 1673, Hooke had told his Royal Society colleagues – mistakenly but pompously – that Newton’s recent, pathbreaking work on light was wrong, making Newton so annoyed that he threatened to give up science altogether. The correspondence that Hooke initiated in the fall of 1679 and continued for 2 months began equally inauspiciously. Newton made an embarrassing error in his first reply, and again Hooke bruited about Newton’s blunder to his Royal Society cohorts. But Newton was challenged by Hooke’s question about the inverse square law and planetary motions. He was also intrigued by Hooke’s remark that the planets travel in curved paths, not because of the combined action of centrifugal and centripetal forces acting on them, but because of the combined action of a centripetal force and the bodies’ own inertia.

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,
F
_g
=
Gm
₁
m
₂
/
r
²
.

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.