Newton's mature dynamics and the Principia: A simplified solution to the Kepler problem

Abstract

Central to Newton's solution of the Kepler problem in Proposition 11 of the Principia is the dynamics ratio derived in Proposition 6, which is employed to find the force, given the orbit and the force center. A comparison of the derivation of this critical proposition as it appears in the third edition with the early derivation of it in the first edition reveals the development of Newton's mature dynamics. In particular, there appears in the second edition an alternate version of the dynamics ratio in addition to the original one of the first edition. Revisions of both of these versions of Proposition 6 are carried forth to the third edition and an alternate solution to the Kepler problem of Proposition 11 is developed. I suggest that this alternate version provides a firmer and more fundamental basis for dynamics than does the original version, and I provide a substitute derivation of Proposition 6 and a simplified solution to the Kepler problem of Proposition 11 to defend that suggestion. It is a step that Newton could have taken, but it is one that he chose not to take.