Newton's achievement: Show that … equals volume/

Problems, on the attraction of a sphere, use Figure 14.18 and the law of cosines $q^2=D^2-2\rho D\cos \varphi +{\rho }^2=u$.

Newton's achievement: Show that $\iiint (\cos \varphi )dV/q^2$ equals volume/$D^2$. One hint only: Find $\cos \varphi$ from a second law of cosines ${\rho }^2=D^2-2qD\cos \alpha +q^2$. The $\varphi$ integral should involve $1/q$ and $1/q^3$. Equation (2) integrates $1/q$, leaving $\iiint dV/q^3$ still to do.

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §14.4 Cylindrical and Spherical Coordinates, #26