In Pursuit of the Unknown (13 page)

Ellipses cemented their role in astronomy around 1600, with the work of Kepler. His astronomical interests began in childhood; at the age of six he witnessed the great comet of 1577,
²
and three years later he saw an eclipse of the Moon. At the University of Tübingen, Kepler showed great talent for mathematics and put it to profitable use casting horoscopes. In those days mathematics, astronomy, and astrology often went together. He combined a heady level of mysticism with a level-headed attention to mathematical detail. A typical example is his
Mysterium Cosmographicum
(‘The Cosmographic Mystery'), a spirited defence of the heliocentric system published in 1596. It combines a clear grasp of Copernicus's theory with what to modern eyes is a very strange speculation relating the distances of the known planets from the Sun to the regular solids. For a long time Kepler regarded this discovery as one of his greatest, revealing the Creator's plan for the universe. He saw his later researches, which we now consider to be far more significant, as mere elaborations of this basic plan. At the time, one advantage of the theory was that it explained why there were precisely six planets (Mercury through Saturn). Between these six orbits lie five gaps, one for each regular solid. With the discovery of Uranus and later Neptune and Pluto (until its recent demotion from planetary status) this feature quickly became a fatal flaw.

Kepler's lasting contribution has its roots in his employment by Tycho Brahe. The two first met in 1600. After a two-month stay and a heated argument Kepler negotiated an acceptable salary. Following a spate of problems in his home city of Graz he moved to Prague, assisting Tycho in the analysis of his planetary observations, especially of Mars. When Tycho unexpectedly died in 1601 Kepler took over his employer's position as imperial mathematician to Rudolph II. His primary role was casting imperial horoscopes, but he also had time to continue his analysis of the orbit of Mars. Following traditional epicyclic principles he refined his model to the point at which its errors, compared with observation, were usually a mere two minutes of arc, the typical error in the observations themselves. However, he didn't stop there because sometimes the errors were bigger, up to eight minutes of arc.

His search eventually led him to two laws of planetary motion, published in
Astronomia Nova
(‘A New Astronomy'). For many years he had tried to fit the orbit of Mars to an ovoid – an egg-shaped curve, sharper at one end than the other – without success. Perhaps he expected the orbit to
be more curved closer to the Sun. In 1605 it occurred to Kepler to try an ellipse, equally rounded at both ends, and to his surprise this did a much better job. He concluded that all planetary orbits are ellipses, his first law. His second law described how the planet moves along its orbit, stating that planets sweep out equal areas in equal times. The book appeared in 1609. Kepler then devoted much of his effort to preparing various astronomical tables, but he returned to the regularities of planetary orbits in 1619 in his
Harmonices Mundi
(‘The Harmony of the World'). This book had some ideas we now find strange, for example that the planets emit musical sounds as they roll round the Sun. But it also includes his third law: the squares of the orbital periods are proportional to the cubes of the distances from the Sun.

Kepler's three laws were all but buried amid a mass of mysticism, religious symbolism, and philosophical speculation. But they represented a giant leap forward, leading Newton to one of the greatest scientific discoveries of all time.

Newton derived his law of gravity from Kepler's three laws of planetary motion. It states that every particle in the universe attracts every other particle with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. In symbols,

Here
F
is the attractive force,
d
is the distance, the ms are the two masses, and G is a specific number, the gravitational constant.
³

Who discovered Newton's law of gravity? It sounds like one of those self-answering questions, like ‘whose statue stands on top of Nelson's column?'. But a reasonable answer is the curator of experiments at the Royal Society, Robert Hooke. When Newton published the law in 1687, in his
Principia
, Hooke accused him of plagiarism. However, Newton provided the first mathematical derivation of elliptical orbits from the law, which was vital in establishing its correctness, and Hooke acknowledged this. Moreover, Newton had cited Hooke, along with several others, in the book. Presumably Hooke felt he deserved more credit; he had suffered similar problems several times before and it was a sore point.

The idea that bodies attract each other had been floating around for a while, and so had its likely mathematical expression. In 1645 the French astronomer Ismaël Boulliau (Bullialdus) wrote his
Astronomia Philolaica
(‘Philolaic Astronomy' – Philolaus was a Greek philosopher who thought that a central fire, not the Earth, was the centre of the universe). In it he wrote:

As for the power by which the Sun seizes or holds the planets, and which, being corporeal, functions in the manner of hands, it is emitted in straight lines throughout the whole extent of the world, and like the species of the Sun, it turns with the body of the Sun; now, seeing that it is corporeal, it becomes weaker and attenuated at a greater distance or interval, and the ratio of its decrease in strength is the same as in the case of light, namely, the duplicate proportion, but inversely, of the distances.

This is the famous ‘inverse square' dependency of the force on distance. There are simple, though naive, reasons to expect such a formula, because the surface area of a sphere varies as the square of its radius. If the same amount of gravitational ‘stuff' spreads out over ever-increasing spheres as it departs from the Sun, then the amount of it received at any point must vary in the inverse proportion to the surface area. Exactly this happens with light, and Boulliau assumed, without much evidence, that gravity must be analogous. He also thought that the planets move along their orbits under their own power, so to speak: ‘No kind of motion presses upon the remaining planets, [which] are driven round by individual forms with which they were provided.'

Hooke's contribution dates to 1666, when he presented a paper to the Royal Society with the title ‘On gravity'. Here he sorted out what Boulliau had got wrong, arguing that an attractive force from the Sun could interfere with a planet's natural tendency to move in a straight line (as specified by Newton's third law of motion) and cause it to follow a curve. He also stated that ‘these attractive powers are so much the more powerful in operating, by how much the nearer the body wrought upon is to their own Centers', showing that he thought the force fell off with distance. But he didn't tell anyone else the mathematical form for this decrease until 1679, when he wrote to Newton: ‘The Attraction always is in a duplicate proportion to the Distance from the Center Reciprocall.' In the same letter he said that this implies that the velocity of a planet varies as the reciprocal of its distance from the Sun. Which is wrong.

When Hooke complained that Newton had stolen his law, Newton was having none of it, pointing out that he had discussed the idea with Christopher Wren before Hooke had sent his letter. To demonstrate prior
art, he cited Boulliau, and also Giovanni Borelli, an Italian physiologist and mathematical physicist. Borelli had suggested that three forces combine to create planetary motion: an inward force caused by the planet's desire to approach the Sun, a sideways force caused by sunlight, and an outward force caused by the Sun's rotation. Score one out of three, and that's generous.

Newton's main point, generally considered decisive, is that whatever else Hooke had done, he had not deduced the exact form of orbits from inverse square law attraction. Newton had. In fact, he had deduced all three of Kepler's laws of planetary motion: elliptical orbits, sweeping out equal areas in equal intervals of time, with the square of the period being proportional to the cube of the distance. ‘Without my Demonstrations,' Newton insisted, the inverse square law ‘cannot be believed by a judicious philosopher to be anywhere accurate.' But he did also accept that ‘Mr Hook is yet a stranger' to this proof. A key feature of Newton's argument is that it applies not just to a point particle, but to a sphere. This extension, which is crucial to planetary motion, had caused Newton considerable effort. His geometric proof is a disguised application of integral calculus, and he was justifiably proud of it. There is also documentary evidence that Newton had been thinking about such questions for quite a while.

At any rate, we name the law after Newton, and this does justice to the importance of his contribution.

The most important aspect of Newton's law of gravitation is not the inverse square law as such. It is the assertion that gravitation acts universally.
Any
two bodies, anywhere in the universe, attract each other. Of course you need an accurate force law (inverse square) to get accurate results, but without universality, you don't know how to write down the equations for any system with more than two bodies. Almost all of the interesting systems, such as the Solar System itself, or the fine structure of the motion of the Moon under the influence of (at least) the Sun and the Earth, involve more than two bodies, so Newton's law would have been almost useless if it had applied only to the context in which he first deduced it.

What motivated this vision of universality? In his 1752
Memoirs of Sir Isaac Newton's Life
, William Stukeley reported a tale Newton had told him in 1726:

The notion of gravitation . . . was occasioned 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? Assuredly the reason is, that the Earth draws it. There must be a drawing power in matter. And the sum of the drawing power in the matter of the Earth must be in the Earth's centre, not in any side of the Earth. Therefore does this apple fall perpendicularly or towards the centre? If matter thus draws matter; it must be in proportion of its quantity. Therefore the apple draws the Earth, as well as the Earth draws the apple.

Whether the story is the literal truth, or a convenient fiction that Newton invented to help him explain his ideas later on, is not entirely clear, but it seems reasonable to take the tale at face value because the idea does not end with apples. The apple was important to Newton because it made him realise that the same law of forces can explain both the motion of the apple and that of the Moon. The only difference is that the Moon also moves sideways; this is why it stays up. Actually, it is always falling towards the Earth, but the sideways motion causes the Earth's surface to fall away as well. Newton, being Newton, didn't stop with this qualitative argument. He did the sums, compared them with observations, and was satisfied that his idea must be correct.

If gravity acts on the apple, the Moon, and the Earth, as an inherent feature of matter, then presumably it acts on everything.

It is not possible to verify the universality of gravitational forces directly; you would have to study all pairs of bodies in the entire universe, and find a way to remove the influence of all the other bodies. But that's not how science works. Instead, it employs a mixture of inference and observations. Universality is a hypothesis, capable of being falsified every time it is applied. Every time it survives falsification – a fancy way to say it gives good results – the justification for using it becomes a little stronger. If (as in this case) it survives thousands of such tests, the justification becomes very strong indeed. However, the hypothesis can never be proved
true:
for all we know, the next experiment might produce incompatible results. Perhaps somewhere in a galaxy far, far away there is one speck of matter, one atom, that is not attracted to everything else. If so, we will never find it; equally, it won't upset our calculations. The inverse square law itself is exceedingly difficult to verify directly, that is, by actually measuring the attractive force. Instead, we apply the law to systems that we can measure by using it to predict orbits, and then check whether the predictions agree with observations.

Even granting universality, it is not enough to write down an accurate law of attraction. That just produces an equation describing the motion. In order to find the motion itself, you have to solve the equation. Even for two bodies, this is not straightforward, and even bearing in mind that he knew in advance what answer to expect, Newton's deduction of elliptical orbits is a
tour de force
. It explains why Kepler's three laws provide a very accurate description of each planet's orbit. It also explains why that description is not exact: other bodies in the solar system, other than the Sun and the planet itself, affect the motion. In order to account for these disturbances, you have to solve the equations of motion for three or more bodies. In particular, if you want to predict the motion of the Moon with high precision, you have to include the Sun and the Earth in your equations. The effects of the other planets, especially Jupiter, are not entirely negligible either, but they show up only in the long term. So, fresh from Newton's success with the motion of two bodies under gravity, mathematicians and physicists moved on to the next case: three bodies. Their initial optimism dissipated rapidly: the three-body case turned out to be very different from the two-body case. In fact, it defied solution.