[...] The famous bucket example that follows is offered as illustrating how forces can be distinguished that will then distinguish between true and apparent motion. The final paragraph of the scholium begins and ends as follows:
It is certainly very difficult to find out the true motions of individual bodies and actually to differentiate them from apparent motions, because the parts of that immovable space in which the bodies truly move make no impression on the senses. But the situation is not utterly hopeless…. But in what follows, a fuller explanation will be given of how to determine true motions from their causes, effects, and apparent differences, and, conversely, of how to determine from motions, whether true or apparent, their causes and effects. For this was the purpose for which I composed the following treatise.
What does follow are two books of propositions that provide means for inferring forces from motions and motions from forces and a final book that illustrates how these propositions can be applied to the system of the world first to identify the forces governing motion in our planetary system and then to use them to differentiate between certain true and apparent motions of particular interest. In this respect, the empirical content of the theoretical concepts that Newton has explicated in the section called “Definitions” is inextricably linked with the physical theory presented in the rest of the Principia.
The contention that the empirical reasoning in the Principia does not presuppose an unbridled form of absolute time and space should not be taken as suggesting that Newton's theory is free of fundamental assumptions about time and space that have subsequently proved to be problematic. For example, in the case of space, Newton presupposes that the geometric structure governing which lines are parallel and what the distances are between two points is three-dimensional and Euclidean. In the case of time Newton presupposes that, with suitable corrections for such factors as the speed of light, questions about whether two celestial events happened at the same time can in principle always have a definite answer. And the appeal to forces to distinguish real from apparent non-inertial motions presupposes that free-fall under gravity can always, at least in principle, be distinguished from inertial motion.
Equally, the contention that the empirical reasoning in the Principia does not presuppose an unbridled form of absolute space should not be taken as denying that Newton invoked absolute space as his means for conceptualizing true deviations from inertial motion. Corollary 5 to the Laws of Motion, quoted above, put him in a position to introduce the notion of an inertial frame, but he did not do so, perhaps in part because Corollary 6 showed that even using an inertial frame to define deviations from inertial motion would not suffice. Empirically, nevertheless, the Principia follows astronomical practice in treating celestial motions relative to the fixed stars, and one of its key empirical conclusions (Book 3, Prop. 14, Corol. 1) is that the fixed stars are at rest with respect to the center of gravity of our planetary system.[...]
