  This is one of the few places in the Principia where Newton singles out a result in an aside in this way. That an attracting sphere can be treated as if the mass were concentrated at its center in the case of attractive forces that vary linearly with the distance was not so notable, for as Newton shows in Section 13, in this case of attractive forces an attracting body always can be treated as if the mass were located at its center of gravity, regardless of shape. The truly notable finding is that it is also true of spheres in the case of inverse-square forces. The subsequent results in Sections 12 and 13 indicate that, in the case of all other kinds of centripetal force, the attraction toward a sphere is not the same as attraction toward all its mass concentrated in the center; and even in the inverse-square case, the result does not hold for other shapes or for spheres that do not have spherically symmetric density.