In uniform circular motion, an equal angle or fraction of the arc of a circle is covered in equal times. So, for example, it takes twice as long to go two-thirds of the way around a circle as it does to go one-third of the way around. Kepler found something different for elliptical orbits: As the planet moves along its orbit, it sweeps out a little wedge-shaped area within the ellipse. When it is close to the Sun, in a given period of time it traces out a large arc in its orbit, but the area represented by that arc is not very large because the planet is then near the Sun. When the planet is far from the Sun, it covers a much smaller arc in the same period of time, but that arc corresponds to a bigger area because the Sun is now more distant. Kepler found that these two areas were precisely the same no matter how elliptical the orbit: the long skinny area, corresponding to the planet far from the Sun, and the shorter, squatter area, when the planet is close to the Sun, are exactly equal. This was Kepler’s second law of planetary motion: Planets sweep out equal areas in equal times.
Kepler’s first two laws may seem a little remote and abstract: planets move in ellipses, and sweep out equal areas in equal times. Well, so what? Circular motion is easier to grasp. We might have a tendency to dismiss these laws as mere mathematical tinkering, something removed from everyday life. But these are the laws our planet obeys as we ourselves, glued by gravity to the surface of the Earth, hurtle through interplanetary space. We move in accord with laws of nature that Kepler first discovered. When we send spacecraft to the planets, when we observe double stars, when we examine the motion of distant galaxies; we find that throughout the universe Kepler’s laws are obeyed.
Many years later, Kepler came upon his third and last law of planetary motion, a law that relates the motion of various planets to one another, that lays out correctly the clockwork of the solar system. He described it in a book called The Harmonies of the World. Kepler understood many things by the word harmony: the order and beauty of planetary motion, the existence of mathematical laws explaining that motion – an idea that goes back to Pythagoras – and even harmony in the musical sense, the ‘harmony of the spheres.’ Unlike the orbits of Mercury and Mars, the orbits of other planets depart so little from circularity that we cannot make out their true shapes even in an extremely accurate diagram. The Earth is our moving platform from which we observe the motion of the other planets against the backdrop of distant constellations. The inner planets move rapidly in their orbits – that is why Mercury has the name it does: Mercury was the messenger of the gods. Venus, Earth and Mars move progressively less rapidly about the Sun. The outer planets, such as Jupiter and Saturn, move stately and slow, as befits the kings of the gods.
Kepler’s third or harmonic law states that the squares of the periods of the planets (the times for them to complete one orbit) are proportional to the cubes of their average distance from the Sun; the more distant the planet, the more slowly it moves, but according to a precise mathematical law: P2 = a3, where P represents the period of revolution of the planet about the Sun, measured in years, and a the distance of the planet from the Sun measured in ‘astronomical units.’ An astronomical unit is the distance of the Earth from the Sun. Jupiter, for example, is five astronomical units from the Sun, and a3 = 5 x 5 x 5 =125. What number times itself equals 125? Why, 11, close enough. And 11 years is the period for Jupiter to go once around the Sun. A similar argument applies for every planet and asteroid and comet. Not content merely to have extracted from Nature the laws of planetary motion, Kepler endeavored to find some still more fundamental underlying cause, some influence of the Sun on the kinematics of worlds. The planets sped up on approaching the Sun and slowed down on retreating from it. Somehow the distant planets sensed the Sun’s presence. Magnetism also was an influence felt at a distance, and in a stunning anticipation of the idea of universal gravitation, Kepler suggested that the underlying cause was akin to magnetism:
My aim in this is to show that the celestial machine is to be likened not to a divine organism but rather to a clockwork . . . , insofar as nearly all the manifold movements are carried out by means of a single, quite simple magnetic force, as in the case of a clockwork [where] all motions [are caused] by a simple weight.
Magnetism is, of course, not the same as gravity, but Kepler’s fundamental innovation here is nothing short of breathtaking: he proposed that quantitative physical laws that apply to the Earth are also the underpinnings of quantitative physical laws that govern the heavens. It was the first nonmystical explanation of motion in the heavens; it made the Earth a province of the Cosmos. ‘Astronomy,’ he said ‘is part of physics.’ Kepler stood at a cusp in history; the last scientific astrologer was the first astrophysicist.
Not given to quiet understatement, Kepler assessed his discoveries in these words:
With this symphony of voices man can play through the eternity of time in less than an hour, and can taste in small measure the delight of God, the Supreme Artist . . . I yield freely to the sacred frenzy . . . the die is cast, and I am writing the book – to be read either now or by posterity, it matters not. It can wait a century for a reader, as God Himself has waited 6,000 years for a witness.
Within the ‘symphony of voices,’ Kepler believed that the speed of each planet corresponds to certain notes in the Latinate musical scale popular in his day – do, re, mi, fa, sol, la, ti, do. He claimed that in the harmony of the spheres, the tones of Earth are fa and mi, that the Earth is forever humming fa and mi, and that they stand in a straightforward way for the Latin word for famine. He argued, not unsuccessfully, that the Earth was best described by that single doleful word.
Exactly eight days after Kepler’s discovery of his third law, the incident that unleashed the Thirty Years’ War transpired in Prague. The war’s convulsions shattered the lives of millions, Kepler among them. He lost his wife and son to an epidemic carried by the soldiery, his royal patron was deposed, and he was excommunicated by the Lutheran Church for his uncompromising individualism on matters of doctrine. Kepler was a refugee once again. The conflict, portrayed by both the Catholics and the Protestants as a holy war, was more an exploitation of religious fanaticism by those hungry for land and power. In the past, wars had tended to be resolved when the belligerent princes had exhausted their resources. But now organized pillage was introduced as a means of keeping armies in the field. The savaged population of Europe stood helpless as plowshares and pruning hooks were literally beaten into swords and spears.*
* Some examples are still to be seen in the Graz armory.
Waves of rumor and paranoia swept through the countryside, enveloping especially the powerless. Among the many scapegoats chosen were elderly women living alone, who were charged with witchcraft. Kepler’s mother was carried away in the middle of the night in a laundry chest. In Kepler’s little hometown of Weil der Stadt, roughly three women were tortured and killed as witches every year between 1615 and 1629. And Katharina Kepler was a cantankerous old woman. She engaged in disputes that annoyed the local nobility, and she sold soporific and perhaps hallucinogenic drugs as do contemporary Mexican curanderas. Poor Kepler believed that he himself had contributed to her arrest.
It came about because Kepler wrote one of the first works of science fiction, intended to explain and popularize science. It was called the Somnium, ‘The Dream.’ He imagined a journey to the Moon, the space travelers standing on the lunar surface and observing the lovely planet Earth rotating slowly in the sky above them. By changing our perspective we can figure out how worlds work. In Kepler’s time one of the chief objections to the idea that the Earth turns was the fact that people do not feel the motion. In the Somnium he tried to make the rotation of the Earth plausible, dramatic, comprehensible: ‘As long as the multitude does not err …. I want to be on the side of the many. Therefore, I take great pains to explain to as many people as possible.’ (On another occasion he wrote in a letter, ‘Do not sentence me completely to the treadmill of mathematical calculations – leave me time for philosophical speculations, my sole delight.’*)