stoutfellow

I rank Isaac Newton as the second-greatest mathematician of all time, behind only Archimedes. Newton was responsible for a considerable amount of truly innovative work; a notable example is the discovery of the generalized binomial theorem, which was inspired by but not derived from work of John Wallis. However, his two most important achievements were of a different character. They were works of unification, in which Newton recognized that discoveries of his predecessors could be seen as aspects of a single overarching theory. In this and the next Ramble, I'll discuss these achievements.

It was a commonplace of medieval philosophy that the universe could be divided into two quite distinct spheres, the mundane and the celestial, with the boundary being the orbit of the Moon. In the celestial sphere, though there was change, it was periodic, reversible. The stars might wheel about the Pole Star, but they return to their original positions. The Sun treads its measure through the Zodiac each year, always the same. In the mundane sphere, though some changes were periodic - the seasons, the tides - the dominant trend involved the irreversible. Old men die; they do not return to the womb. A house burns down; it does not rise again from the ashes.

The first major crack in this mind-set came with Tycho Brahe's observation of the supernova of 1572. Here, a star had blossomed where none had ever been seen before; after a time, it winked out, never to reappear. The periodicity of the celestial sphere failed, in this instance, and the medieval conception was from that day on doomed.

Still, something of that mindset remained, with curious effects. In the generation after Tycho, his student Johannes Kepler established a mathematical model describing the motion of the planets about the Sun. At almost the same time, Galileo Galilei was studying the motion of objects on Earth: pendulums, ballistic projectiles, and the like. He too developed mathematical laws to describe this kind of motion - a law of inertia, the constant acceleration of falling bodies, the parabolic motion of projectiles. All this, of course, pertained to objects in the immediate vicinity of the Earth.

Part of the genius of Newton's formulation of the Law of Universal Gravitation and the laws of motion was that it unified the celestial dynamics of Kepler with the mundane dynamics of Galileo. For the first time, a single mathematical theory was seen to govern both spheres, above and below the orbit of the Moon, and it was a relatively simple theory. (Kepler's laws, in particular, though much simpler than, say, the constructions of the later Ptolemaists, still had an ad hoc flavor to them. Why should orbits satisfy a 3:2 power law? Why should the radius vector sweep out equal areas in equal times? Newton showed that the seeming arbitrariness of these laws - and those of Galileo - were simply the working-out of Newton's simpler laws in particular contexts.) It is this unification and simplification that makes Newton's theory of gravitation the great achievement that it is.

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