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of motion. But it was Kepler who deduced from observations the first mathematical (albeit only phenomenological) laws of planetary motion. Kepler used a huge body of data left by the astronomer Tycho Brahe to determine the orbit of Mars. He referred to the ensuing hundreds of sheets of calculations as “my warfare with Mars.” Except for two discrepancies, a circular orbit matched all the observations. Still, Kepler was not satisfied with this solution, and he later described his thought process: “If I had believed that we could ignore these eight minutes [of arc; about a quarter of the diameter of a full moon], I would have patched up my hypothesis…accordingly. Now, since it was not permissible to disregard, those eight minutes alone pointed the path to a complete reformation in astronomy.” The consequences of this meticulousness were dramatic. Kepler inferred that the orbits of the planets are not circular but elliptical, and he formulated two additional, quantitative laws that applied to all the planets. When these laws were coupled with Newton’s laws of motion, they served as the basis for Newton’s law of universal gravitation. Recall, however, that along the way Descartes proposed his theory of vortices, in which planets were carried around the Sun by vortices of circularly moving particles. This theory could not get very far, even before Newton showed it to be inconsistent, because Descartes never developed a systematic mathematical treatment of his vortices.

What do we learn from this concise history? There can be no doubt that Newton’s law of gravitation was the work of a genius. But this genius was not operating in a vacuum. Some of the foundations had been painstakingly laid down by previous scientists. As I noted in chapter 4, even much lesser mathematicians than Newton, such as the architect Christopher Wren and the physicist Robert Hooke, independently suggested the inverse square law of attraction. Newton’s greatness showed in his unique ability to put it all together in the form of a unifying theory, and in his insistence on providing a mathematical proof of the consequences of his theory. Why was this formalism as accurate as it was? Partly because it treated the most fundamental problem—the forces between two gravitating bodies and the resulting motion. No other complicating factors were involved. It was for this problem and this problem alone that Newton obtained a complete solution. Hence, the fundamental theory was extremely accurate, but its implications had to undergo continuous refinement. The solar system is composed of more than two bodies. When the effects of the other planets are included (still according to the inverse square law), the orbits are no longer simple ellipses. For instance, the Earth’s orbit is found to slowly change its orientation in space, in a motion known as precession, similar to that exhibited by the axis of a rotating top. In fact, modern studies have shown that, contrary to Laplace’s expectations, the orbits of the planets may eventually even become chaotic. Newton’s fundamental theory itself, of course, was later subsumed by Einstein’s general relativity. And the emergence of that theory also followed a series of false starts and near misses. So the accuracy of a theory cannot be anticipated. The proof of the pudding is in the eating—modifications and amendments continue to be made until the desired accuracy is obtained. Those few cases in which a superior accuracy is achieved in a single step have the appearance of miracles.

There is, clearly, one crucial fact in the background that makes the search for fundamental laws worthwhile. This is the fact that nature has been kind to us by being governed by universal laws, rather than by mere parochial bylaws. A hydrogen atom on Earth, at the other edge of the Milky Way galaxy, or even in a galaxy that is ten billion light-years away, behaves in precisely the same manner. And this is true in any direction we look and at any time. Mathematicians and physicists have invented a mathematical term to refer to such properties; they