by Marcus Chown
The story of Newton’s apple, as already mentioned, may be apocryphal. But the point is that, crucially, Newton realised that the force that pulls an apple towards the Earth is the same one that keeps the Moon trapped in orbit around the Earth.
Such a connection between a falling apple and the Moon is not at all obvious. After all, the Moon does not appear to be falling. Newton’s genius was to realise that appearances are deceptive.
Imagine a cannon firing a cannonball horizontally across the ground. After travelling a short distance, the cannonball falls to the earth. Picture a bigger cannon that shoots a cannonball faster. The ball travels further before it hits the ground. Now imagine a truly enormous cannon that fires a cannonball at an enormous speed of 28,080 kilometres an hour. For this cannonball the curvature of the Earth is now of critical importance because as fast as the cannonball falls towards the ground the ground underneath the cannonball curves away from it. The ball, though it is perpetually descending towards the ground, never gets any closer. Instead, it orbits round and round the Earth, falling forever in a circle. ‘The knack of flying,’ as Douglas Adams so pointedly observed, ‘is learning how to throw yourself at the ground and miss.’23
The Moon is falling for ever in a circle. So the apple and the Moon are doing the same thing. It is just not obvious they are because the apple has no speed parallel to the ground and so falls vertically whereas the Moon, like an ultra-high-velocity cannonball, does have a speed parallel to the ground and so falls in a circle.
Today, children still ask: why does the Moon not fall down? Why do artificial satellites not fall down? What is keeping them up? The thing they do not realise is that nothing is keeping them up. They are falling down! A common misconception is that astronauts in space are weightless because there is no gravity. In fact, gravity even at the altitude of the International Space Station is about 89 per cent of that on the Earth’s surface. The astronauts on board are weightless not because they are beyond gravity but because they are falling.
All Newton had to do in order to prove that gravity is a universal force — operating between all masses, whether in the heavens or on Earth – was to compare the effect of gravity exerted by the Earth on the apple with the effect of gravity exerted by the Earth on the Moon. If he was right, the ratio of the two effects should be explicable by a single force which weakens with distance according to an inverse-square law.
Newton turned his attention to a falling apple. He knew -because people like Galileo had measured it – that in its first second of fall an apple descends 490 centimetres (16 feet). The next question Newton needed to answer was: how far does the Moon fall in 1 second?
Newton knew that the Moon is 384,400 kilometres from the centre of the Earth.24 This enabled him to calculate the circumference of the Moon’s orbit. Since he knew that the Moon travels around this orbit once every 27.3 days, he could calculate the speed of the Moon.
The natural motion of the Moon would be to continue at this speed in a perfectly straight line. But, of course the path of the Moon is continually bent away from this straight line and towards the Earth by the force of the Earth’s gravity. It is a matter of geometry to calculate how far in 1 second the Moon falls away from a straight-line path and towards the Earth. When Newton did the calculation, he arrived at a distance of 0.136 centimetres (roughly 1/20 of an inch). So now he knew that the gravity of the Earth at the distance of the Moon is 0.136/490 = ∽1/3,600th of that at the surface of the Earth (~ means ‘approximately’).
The Earth’s surface is 6,370 kilometres from the centre of the Earth whereas the Moon, as already mentioned, is 384,400 kilometres from the centre of the Earth.25 In other words, the Moon is about 60 times further from the centre of the Earth than is the Earth’s surface. Notice that 602 = 3,600 – the exact amount by which the gravity at the distance of the Moon is weaker than at the surface of the Earth. Newton had proved that a single force that weakens with the square of distance tugs on both terrestrial apples and celestial bodies. Gravity is indeed a universal force.
It is worth pausing for a moment to consider what this means. It means there is a force between every chunk of matter in the Universe and every other chunk of matter. In other words, there is a force of gravity between you and a person walking past you on the street; between you and the mobile phone in your pocket; even between your left earlobe and the big toe of your right foot. In all these everyday circumstances, the force of gravity is far too weak to have any noticeable effect. But gravity is stronger the more stuff there is. It is cumulative. This is why the gravity of the Earth, with a mass of 5.98 million million million million tonnes, is noticeable, and why it pins our feet to the ground.
Because gravity is a universal force, it tries to pull together a collection of massive particles into the most compact form possible, which is a sphere. This can happen only if the matter can flow like treacle, which in practice requires the body to be squeezed very hard by its own gravity. Since ice is easier to squeeze than rock, the threshold mass is different for bodies made of ice than for bodies made of rock. In our Solar System, all icy bodies bigger than about 600 kilometres across are round whereas all bodies smaller than this are potato-shaped. For bodies made of rock, the threshold is about 400 kilometres.
Ultimately, the shape of a celestial body is determined by the strength of gravity, which crushes matter, and the strength of the electromagnetic force, which makes matter stiff so it can oppose gravity. The electromagnetic force between a proton and an electron in hydrogen, the lightest atom, is about 1040 – or 1 followed by 40 zeroes – times bigger than the force of gravity between them. So, for the force of gravity to overwhelm the force of electromagnetism, a huge number of atoms need to be in one location, which is why gravity triumphs only for bodies bigger than 400 to 600 kilometres across.
There is a subtlety here. Gravity certainly grows stronger the more matter there is. And this definitely explains why our feet are pinned to the ground by the mass of the Earth. But gravity is not merely a force that big masses exert on smaller ones. It is a mutual force which massive bodies exert on each other. The Earth exerts a gravitational force on our bodies and our bodies exert an equal gravitational force on the Earth. Despite this, we all of course know that we fall towards the Earth and the Earth does not noticeably fall towards us. But this has nothing to do with gravity and everything to do with inertia, the inherent resistance of massive bodies to any changes in their motion.
Bigger masses have more resistance to being budged – in fact, this is the very definition of mass – and the Earth is enormously more massive than a person so enormously more difficult to budge. The British comedy writer Andy Hamilton nailed a profound truth when he quipped: ‘Is that why I am attracted to big women and big women are not attracted to me?’26 Actually, big women are attracted to Hamilton but, because of their larger mass, Hamilton’s gravitational effect on them is smaller than their effect on him. Similarly, the Earth does fall towards you or an apple but by an imperceptibly small amount. ‘When Newton sat in his garden,’ says philosopher A. C. Grayling, ‘[he] saw what no one had seen before: that an apple draws the world to itself, and the Earth the apple, through a mutual force of nature that holds all things, from the planets to the stars, in unifying embrace.’27
‘Millions saw the apple fall,’ said the American financier Bernard Baruch, ‘but Newton was the one who asked why.’28
Faith in simplicity
It was an extraordinary leap of the imagination to see that the Moon, though it does not appear to be, is falling, and that, furthermore, it is falling because of the very same force that causes an apple to fall from a tree: the gravitational pull of the Earth. The heavens at the time were widely considered to be the domain of angels and of God himself. The Greeks had even imagined them made of an ethereal fifth essence, entirely distinct from the everyday ‘elements’ of earth, fire, air and water. But Newton saw no distinction between up there and down here. In a world still dominated by re
ligious dogma, he had the courage to bring the heavens down to earth. The same laws that govern the behaviour of bodies on Earth govern the behaviour of bodies in the Universe. There exist universal laws – ones that apply in all places and at all times. And Newton, a man living at the very dawn of science, whose father, unable to write, had endorsed his will with an X, had penetrated to the heart of nature and seen one such universal law.
It was the first of the great unifications of science. Later, Charles Darwin would unify the world of humans with the rest of the animal kingdom; James Clerk Maxwell would unify electricity, magnetism and light; and Albert Einstein would unify space, time and gravity. Today’s physicists seek the ultimate unification – or what they imagine to be the ultimate unification – of gravity and ‘quantum theory’, the theory of the microscopic world of atoms and their constituents.
But Newton’s law of gravity was not only universal, it was simple. ‘Truth is ever to be found in simplicity, and not in the multiplicity and confusion of things,’ wrote Newton.29 Had the law of gravity not been simple, of course, it would never have been possible for a man of the seventeenth century – even a man of Newton’s genius — to have found it. Think how lucky this is. The Universe at a fundamental level could easily be governed by complex laws, utterly opaque and impenetrable to the three-pound brain of a jumped-up ape not long descended from the trees onto an East African plain. But it isn’t. The Universe is orchestrated by simple laws.
Following Newton’s lead, others have sought and found yet more simple universal laws. In fact, the belief that such laws exist is the unacknowledged faith behind physics, the light that guides physicists struggling to penetrate the darkness at the frontier of their field. No one knows why the Universe at a fundamental level is simple just as no one knows why it is mathematical. But it was Newton, 350 years ago, who first showed it is both of these things.30
Newton’s universal law describes the gravitational force between particles of matter. In fact, as Newton was first to realise, ultimately, this is all there is to the Universe: particles and forces. ‘The attractions of gravity, magnetism, and electricity, reach to very sensible distances, and so have been observed,’ wrote Newton. ‘But there may be others which reach to so small distances as to hitherto escape observations . . . some force, which in immediate contact is exceeding strong, at small distances performs the chemical operations above-mentioned, and reaches not far from the particles with any sensible effect.’31 We now know that the electromagnetic force is responsible for Newton’s ‘chemical operations’ and that there are indeed two other fundamental forces of nature which had ‘escaped observations’ and are exceedingly strong only at small distances.
The job of physicists, as Newton so presciently recognised, is twofold. First, to find the fundamental forces of nature. And, second, to discover how those fundamental forces, working in concert, have conspired to assemble the fundamental particles of nature into the fantastically rich Universe we see around us, complete with galaxies and stars, planets and moons, trees and people.
Twenty-two years of silence
Newton found his universal law of gravity in 1666. But he did not announce it to the world for twenty-two years. No one knows why, though there are several possibilities. One is that when Newton compared the effect of the Earth’s gravity at the Moon’s distance with the effect of gravity on the ground, it did not confirm the inverse-square law. His seventeenth-century estimate for the distance between the Earth and the Moon was wrong. By the time he realised and discovered the correct value, he had already moved on to other scientific puzzles.
Another plausible reason why Newton did not publish his law of gravity straight away is that he tacitly assumed that the gravitational pull of the Earth is the same as if all of its mass is concentrated at its centre. Recall that, in deducing the inverse-square law, Newton compared the distance of the Moon from the centre of the Earth with the distance of an apple from the centre of the Earth.
The essence of Newton’s theory of universal gravity is that it is a force acting between every piece of matter and every other piece of matter. That means that the gravitational force exerted on the Moon by the Earth is in fact the gravitational force exerted on the Moon by Mount Everest plus the gravitational force exerted on the Moon by the core of the Earth plus the gravitational force exerted on the Moon by every last sand grain on every beach bordering every continent on Earth . . . In fact, the gravitational pull on the Moon is the sum total of the pull exerted by all the untold zillions upon zillions of particles of matter that make up the Earth.
Newton believed that that pull is exactly the same as if all the matter of the Earth is concentrated at a single point at the centre. Almost certainly he could not prove it. But, in the words of the twentieth-century physicist Richard Feynman: ‘You can know more than you can ever prove.’32 And Newton always knew more than he could prove.
Newton’s powers of intuition were formidable. After hours or days or weeks of concentrating on a problem, he would see the solution before him – its inevitability, its obviousness, its rightness. But it is not enough to know the truth. It is necessary to convince others as well. And that meant sitting at a desk with a quill and parchment and dressing up gut instinct with a plodding step-by-step explanation in the toddler language of mortals: mathematics.
One thing was obvious to Newton. The world is a ball with one identical hemisphere on either side of the line joining the Moon to the centre of the Earth. Because of the symmetry of this situation, the gravitational forces exerted by all the chunks of matter in one hemisphere on all the chunks of matter in the second hemisphere will be exactly countered by the gravitational forces exerted by all the chunks of matter in the second hemisphere on all the chunks of matter in the first hemisphere. They will cancel each other out. Consequently, the force of gravity of the Earth on the Moon will act entirely along the line joining the centre of the Earth to the Moon. This is a start. But it is still quite a way from saying that that pull will act along the line joining the centre of the Earth to the Moon as if the entire mass of the Earth is concentrated at a point at the centre of the Earth. This is the thing Newton saw so clearly in his mind’s eye in 1666 but could not prove.
Or maybe he could prove it. But just not in the way anyone else living on Earth in 1666 could possibly understand.
In May 1666, Newton invented ‘integral calculus’. He called it his ‘inverse method of fluxions’. It is a piece of mathematical magic with which he could add up the contributions from an infinite number of infinitesimally small masses (or an infinite number of infinitesimally small any things). It was the perfect instrument to prove that the gravity exerted by the Earth is the same as if all of its mass were concentrated at a point at its centre. But, since Newton had only just invented integral calculus and told not a soul about it, a calculus-based proof was one that Newton, and Newton alone, would understand.33 Telling the world ‘I’ve got a brilliant proof but, before you can appreciate it, I need to teach you an entire field of abstruse mathematics I’ve just invented’ was unlikely to impress anyone.
But Newton was a complex, and contradictory, beast. In addition to the scientific reasons for not announcing his universal law of gravity in 1666 there may well have been powerful psychological reasons. For a start, he was ridiculously, insanely secretive. At school at Grantham, he was tormented, perhaps for his differentness, by the school bully. According to Newton’s own account, after the boy kicked him in the stomach, Newton dragged him to the church by his ears and rubbed his nose against the church wall.34 Despite his victory, the traumatic experience made Newton paranoid about exposing any part of himself – even the abstract intellectual constructs of his mind – to the remotest possibility of attack. Possessed of one skin too few, Newton failed to see the robust scepticism of others as an essential part of healthy scientific discourse but instead regarded it as a personal assault by scientific pygmies on ideas he did not need to defend because he knew they were true.
Newton was a prickly, bad-tempered and, at times, vindictive man, who during his lifetime engaged in long-running, bitter and often demeaning feuds with other scientists. There is a strong element of the pot calling the kettle black in Newton’s observation: ‘We build too many walls and not enough bridges.’ There is a certain irony in his statement: ‘I can calculate the motion of heavenly bodies, but not the madness of people.’
‘Tact is the art of making a point without making an enemy,’ said Newton. Unfortunately, he was never able to practise what he preached. He had insight but little ability to act on that insight.
But, then, no person is devoid of all contradictions. The twentieth-century physicist George Gamow told a story about Newton which may or may not be true.35 Newton loved his cat, said Gamow. And, to let his cat in and out of his study, he cut a hole in his study door – a kind of seventeenth-century cat flap without the flap. But then his cat had kittens. So what did Newton do? According to Gamow, he cut a whole row of little holes in the door . . . one for each of the kittens. He was the greatest super-genius of all time and yet he did not realise that the kittens could all go through the big hole.
Newton’s obsessive secretiveness may have stemmed from something even deeper. Despite being born a premature weakling, he lived to the grand old age of eighty-four, retaining perfect eyesight and all but one of his adult teeth.36 When he died, he left behind a box of papers for posterity. So explosive was its content that a bishop who opened it and scanned the documents promptly slammed the lid shut in horror.37 Among other things, the box contained Newton’s writings on religion. He was a deeply religious man who believed in one and only one God. He totally rejected the religious orthodoxy of the ‘Trinity’ of God, the Son, Jesus, and the Holy Ghost. His own investigations had revealed that the ideology of the ‘three persons in one’ Godhead was foisted on the Church by devious means at the First Council of Nicaea, convened in AD 325 by the Roman Emperor Constantine I.