Physical vs. numerical approximation in Isaac Newton’s Principia

The problem with approximation is to find principled grounds for preferring any one over the indefinitely many alternative approximations in equal agreement with observation. From the outset of his efforts on orbital motion Newton’s goal was to show that Kepler’s orbits had a physical standing that the various comparably accurate alternatives lacked. What made this goal difficult was his conclusion, almost from the outset, that the actual motions are too complicated for any representation of them ever to be anything but an approximation. This paper reviews Newton’s efforts, and some of those following his, to turn this very complexity into evidence that Keplerian motion is nonetheless a physical, not merely numerical, approximation. By his and their example those efforts provided a more discriminating way than has generally been recognized to distinguish physical from merely numerical approximations.