A Brief Guide to the Great Equations (9 page)

DESCRIPTION
: Gravity exists in all bodies universally, and its strength between two bodies depends on their masses and inversely as the square of the distance between their centres.

DISCOVERER
: Isaac Newton

DATE
: 1684–87

The high point of the Scientific Revolution was Isaac Newton’s discovery of the law of universal gravitation. All objects attract each other with a force directly proportional to the product of their masses and inversely proportional to the square of their separation. By subsuming under a single mathematical law the chief physical phenomena of the observable universe Newton demonstrated that terrestrial physics and celestial physics are one and the same.

Just as surely as people know that if you push an object it moves, they also know that, if you drop things like apples, they fall to the ground. No one had to discover this behaviour. But Newton’s equation – first published not in the form of the familiar equation
F
_g
=
Gm
₁
m
₂
/
r
²
but as a verbal description –
was
a discovery. And it did more than quantify falling behaviour, stating the key quantities involved and how they relate.

The appearance of this relation – in Newton’s
Principia
, the same book in which he published his second law – was the culminating moment of the Scientific Revolution, as Cohen said, for it knit together heavens and earth as part of the same world and obeying the same laws. But the impact of this equation extended yet further. It helped enshrine Newton as a symbol, not only of scientist, explorer, and genius, but – strangely enough, given that the Aristotelian scientific picture was being ushered off the horizon – also of humanity’s quest for actualization and perfection:
This
is what we can accomplish when our minds are fully engaged. Indeed, Newton’s discovery of universal gravitation seemed a close encounter with divinity:
This
is as close to God as we humans can ever hope to get.

It is thus not a coincidence that Newton’s discovery is firmly connected with a story involving an apple, recollecting that other famous apple story – the biblical story of the Garden of Eden, and the first fruit of the Tree of Knowledge to be grasped by humans.

In Aristotle’s cosmic ecosystem, falling was a special behaviour that only certain kinds of things did, and only in certain places in the universe. Falling was one among many different kinds of motion and change, and had nothing to do with the tides, nor with the circular motions of the planets and other heavenly objects. It was a natural motion by which a thing made of some proportion of earth returned to its natural place via its own internal power. The causes of falling therefore included the composition of the object, its natural place in the earth, and the object’s tendency to return to that place. For a long time, under Aristotle’s influence, the downward falling of objects to the earth was viewed as but one of several different types of ‘attractions’ and motions in the universe. So was his view that
the quickness of fall depends on the heaviness of the object – which is, after all, confirmed by our everyday experience. As the character Rosencrantz, holding up a ball and feather at one point in the movie of Tom Stoppard’s play
Rosencrantz and Guildenstern Are Dead
, says, ‘You would think this would fall faster than this [drops them, ball hits the ground first]. And you would be absolutely right.’

But some ancient authors broke with Aristotle in proposing the existence of various kinds of connections between phenomena on earth and in the heavens, the most conspicuous being that between the moon and the tides. Aristotle had struggled to produce a mechanical explanation for tidal motions – involving the wind – but others thought the connection somehow more direct. The Greek scholar Posidonius (ca. 135–51
BC
), along with several other ancient authors, produced the un-Aristotelian notion of forces permeating the cosmos that were not based in any one substance (substances being the only things that truly existed for Aristotle), but which linked substances together. These cosmic forces were called ‘sympathies’, after the Greek for ‘feeling together.’
¹

In the ancient and medieval world, the exploration of physical influences among heavenly bodies, and between the heavenly bodies and objects on earth, was generally called ‘astrology.’ But we must not confuse this with the current socially acceptable form of bigotry that seems to entitle the human beings who believe in it to prejudge the character of others based solely on their dates of birth. Ancient and medieval astrology indeed had its share of charlatans who did that sort of thing. But astrology also had a serious side, springing from the quite reasonable assumption that physical influences existed in the universe that linked some things to other faraway things, and the scholarly conviction that it was possible to investigate and describe these influences. As science historian David C. Lindberg says, ‘Almost any ancient philosopher would have considered it extraordinarily foolish to deny the existence of such connections.’
²
The work of astrologers, at the beginning, had an enormous positive result in developing notions of long-range forces.
³

Yet the problem of explaining these connections, including why bodies fell, remained puzzling. Was the force something external or internal to the falling object, or something else? In 1504, Nicoletto Vernias, writing on free fall, declared, ‘This question is the most difficult of all questions in physics.’
⁴

The question was transformed in 1543, when Nicolaus Copernicus (1473–1543) published
On the Revolutions of the Heavenly Spheres
, a book proposing that the sun, not the earth, was the centre of the solar system. This book – according to legend the author received the first published copy on his deathbed – assumed that gravity was a volition implanted by God into things. Yet it profoundly influenced those who investigated cosmic forces, inasmuch as it implied that the gravity or heaviness of bodies on earth was not cosmically unique but presumably experienced on the other bodies orbiting the sun – and perhaps by the moon and even the sun.
⁵
Every body had its own gravity.

Another milestone in thinking about cosmic forces was
De Magnete
(1600), William Gilbert’s treatise on magnetism. Magnetism clearly sprang from the mutual interaction between the earth and various substances, and Gilbert noted that its strength varied with distance. He also suspected that magnetic force was active even when the bodies it affected were at rest. Gilbert ridiculed the notion that bodies at rest were unaffected by this force as like thinking that houses are governed by walls, roof, and floor rather than the families inhabiting them.

The work of Johannes Kepler (1571–1630), who sought a mathematical description of the cosmic force binding the sun and planets, built on that of Copernicus and Gilbert. Kepler’s early studies in theology were unexpectedly interrupted when he landed a job as a mathematician, and he developed an ambivalent relationship with astrology. Like the astrologers, he was passionately devoted to the idea that harmonies pervaded the universe, anchored in an overarching harmony established by God. Yet Kepler scorned the methods of astrologers, for they were firmly committed to everyday language in their studies, and knew nothing of the precise language of mathematics used by professional astronomers. Without mathematics, Kepler thought, astrologers could not detect the cosmic harmonies, and would be ignorant of the structure of the world.

The strength of a force that extends out in a plane will weaken directly as the distance from the source, while that which radiates in all directions will weaken with the square of the distance: the inverse square law.

Kepler was among the first, for instance, to realize that the intensity of light varies according to an inverse square law. An inverse square law states that some property weakens with the square of the distance. In the case of the intensity of light, which radiates out in all directions from a source, it is simple geometry. If you double the distance from a source, for instance, the area over which the same light must be distributed increases (and its intensity weakens) by four; if you triple the distance, the area increases by nine.

Kepler’s astronomical work borrowed elements from both Copernicus
and Gilbert. From Copernicus Kepler took the heliocentric picture of the solar system, and the notion that gravity is an attractive force; Kepler, in fact, wrote a popular seven-volume textbook on Copernican astronomy called the
Epitome
[Introduction]
to Copernican Astronomy
(1618–21). From Gilbert Kepler took the notion that this force involves a ‘mutual’ attraction. The stone moves toward the earth even as the earth moves toward the stone – and two stones, if placed in distant space somewhere, would attract each other. Moreover, the attraction weakens with distance; the further a planet is from the sun, the weaker the attraction and the slower it moves. But Kepler concluded that the force by which the sun held the planets did not radiate in all directions, but only stretched out to the planets in the plane of their orbit. Why should it radiate in all directions, given that its ‘purpose’ was only to grip the planets? Kepler therefore concluded that the force varied only inversely with the distance from the sun, not with the inverse square of the distance.

When Kepler tried to figure out the mathematical relationships governing the planetary motions, though, he encountered a puzzle. According to Copernicus, the planets revolve about the sun, and in circular orbits, for all celestial motions had been considered circular since the time of Aristotle. But before the use of telescopes in astronomy began in 1609, the best data of the day had been taken by the Danish astronomer Tycho Brahe (1546–1601), whom Kepler knew and trusted – and Kepler found that these data could not
quite
be fitted to a circular orbit model. The discrepancy was tiny, almost insignificant, a mere 8 minutes of arc, or just barely more than the naked eye was able to discriminate. Kepler spent six years trying to incorporate those 8 minutes of arc into the Copernican system. He could not.

Others might have written off the discrepancy as due to observational error or to some unknown factor. Yet Kepler trusted
both
Copernicus’s heliocentric model
and
Brahe’s data. Because he did, he was led to consider a radically new idea: that the planets move, not circularly, but in elliptical orbits with the sun at one focus. Furthermore,
he concluded that, regardless of whether the planets move quickly when near the sun, or more slowly when more distant, an imaginary line stretching from the sun to a planet sweeps out equal areas in equal times. These conclusions were the first two of Kepler’s famous three laws, and the third was another mathematical relationship: that the squares of the times of revolution of any two planets are proportional to the cubes of their distances from the sun.
⁶

Kepler found these laws beautiful and harmonious. He also claimed that this beauty and harmony was what had caused God to use these laws to construct the universe in the first place. ‘This notion of causality’, notes philosopher E. A. Burtt, ‘is substantially the Aristotelian formal cause reinterpreted in terms of exact mathematics.’
⁷
And Kepler saw the force binding sun and planets as a secular version of an animate force. ‘If for the word ‘soul’ you substitute the word ‘force’, ‘ he wrote, ‘you have the very same principle on which [my] Celestial Physics [is based]... For once I believed that the cause which moved the planets was precisely a soul... But when I pondered that this moving cause grows weaker with distance...I concluded that this force is something corporeal.’
⁸
This almost seamless transition in Kepler’s thinking between the sun gripping the planets like a spirit and gripping them with a corporeal force is a classic illustration of what French philosopher Auguste Comte called the transition between theological and metaphysical thinking.