I am studying Zangwill's Modern Electrodynamics but I'm having trouble following an argument he makes about solid angles in preparation for deriving the integral form of Gauss's law.
He defines the solid angle $\Omega_{S}$ subtended by a surface $S$ at a point $O$ as
\begin{equation*} \Omega_{S} = \int_{S}{\frac{\mathbf{r}_{S}}{\left|\mathbf{r}_{S}\right|^{3}}\cdot\hat{\mathbf{n}}\,\mathrm{d}S} \end{equation*}
where $\mathbf{r}_{S}$ is the vector from $O$ to the area element $\mathrm{d}S$ and $\hat{\boldsymbol{n}}$ is the outwardly directed normal vector to $S$ at $\mathrm{d}S$.
He asks the reader to consider a closed surface $S$ that encloses a volume $V$ and puts two cases: one in which the point $O$ lies inside of $V$ and one in which the point $O$ lies outside of $V$. He provides the following figure.
Here is his subsequent explanation:
"The left side of [the figure] shows that every ray from $O$ intersects $S$ an even number of times if $O$ is outside of $V$. Adjacent intersections where the ray enters and exists $V$ make the same projection onto a unit sphere centered at $O$ but cancel one another in [the integral] because $\hat{\mathbf{n}}\cdot\hat{\mathbf{r}}_{S}$ changes sign. Conversely, if $O$ is inside of $V$ (right side of [the figure]), every ray makes an odd number of intersections and the set of intersections closest to $O$ project onto the complete unit sphere with surface area $4\pi$."
This argument relying on the parity of the number of intersections of certain rays doesn't make sense to me. I would be grateful for any pointers or clarification.
