Overview

Harmonices Mundi, which was the culmination of Kepler's revolutionary contribution to science and contained his third law of planetary motion, failed to account for the masses of the planets. Indeed, when we refer to Keplerian orbits, we are implicitly assuming that these masses are truly negligible, and that Kepler's so-called "laws" are exact. In fact, however, with the exception of two-body motion, the problems of celestial mechanics are, generally, incapable of exact mathematical solution. In many ways, this was fortunate for the development of science and engineering. (Indeed, if even the solution of Kepler's equation in the two-body problem had been simple to obtain in closed form, the history of mathematics might have been considerably altered.)

Celestial mechanics became the driving force which spurred the great mathematicians to incredible efforts to find useful methods of analyzing planetary motion. The elegant tools which they invented for this purpose had astonishing applicability in many diverse fields.

Sir Isaac Newton was the first to consider the attraction exerted by spheres and spheroids of uniform and varying density on a particle. In the Principia, Proposition 74, he showed that the attraction of a homogeneous sphere on a particle is the same as if the mass of the sphere were concentrated at its center. This was not an easy problem even for Newton. In 1684, almost 20 years after he began to apply his law of gravitation to planetary motion, his friend Edmund Halley urged him to publish his results. However, Newton...