In Newton's superb theorem: An elementary geometric proof (Argument on Page 4, accompanying diagram on Page 11), the author attempts to prove Proposition 70 in Newton's Principia (gravitational force = 0 inside a hollow sphere) with a more streamlined version of Newton's original argument.
But the problem is I don't understand why we need to use infinitesimal cones instead of normal ones, since the area on the sphere formed by the cone can be calculated from $r^2\theta$, where $\theta$ is the solid angle of the cone's vertex, shouldn't the conclusion just fall out?
If, however, I resort to infinitesimal cones, why should the elemental area $dS$ be $\frac{r^2d\Omega}{\cos \alpha}$? Where does $\cos\alpha$ come from, and why does it matter that the angle between the normal to the surface and the cone be equal on both sides?
