Newton was fascinated by the fact “which is very remarkable” that the two “principal cases of attractions” (i.e. the inverse square and direct distance laws) are distinguished from others in two independent ways:

1. (i) the attraction of a spherically symmetric body at a point outside “preserves the same law of increase or decrease” as that of a point mass (call this the spherical property);

2. (ii) a particle can move in an elliptical orbit, though with different centres of force in the two cases (call this the elliptical property).