  He had other reasons as well. The theory of gravity entails that gravity below the surface of a uniformly dense sphere varies linearly with the distance from the center, and hence, at least to a first approximation, this is how gravity varies below the surface of the Earth. Centripetal forces that vary as 1/r3 hold if and only if the trajectory is a  spiral;[31] and, given any stationary orbit governed by centripetal forces, superposition of a 1/r3 centripetal force will cause that orbit to precess, as in the case of the lunar  orbit.[32] Still, Newton's main reason appears to have been the one given in the quotation.