French translation

Newton's proof of Theorem 2

Imagine a point A inside the earth. Draw a sphere S, concentric to the surface of Earth, so that the point A is inside.
(You are strongly recommended to DRAW a picture following these directions).
Think of S as a (very) thin spherical shell representing a thin layer of matter "above" the point A. Draw a cone of very small opening with vertex at A. (In your picture this cone is represented by two lines through the point A, with a very small angle between them. This cone will cut two small pieces of S, on two opposite sides of A. Let us call these pieces B and C. (These pieces look approximately like small ellipses on the surface of the sphere S). I claim that the gravity forces acting on A from the pieces B and C are EQUAL, and have opposite directions (so they compensate each other). Indeed, the masses of the pieces B,C are proportional to their volumes, the volumes are proportional to the areas (on the sphere S), because we assume S is very thin and the thickness is constant everywhere. There areas are proportional to the squares of "radii" of these pieces (more precisely, the great axes of the ellipses), that is to the squares of lengths of arcs of circle B and C on your two-dimensional picture. But the two long and narrow triangles with common vertex at A on your picture are similar (they have three equal angles!), so the squares of their bases B and C are proportional to the squares of distances AB and AC. Thus the masses are proportional to the squares of distances AB and AC, and if we divide by the squares of the same distances (as Newton's gravity law requires), we will conclude that the forces are equal. This is Newton's proof.

You can also do of course a "conventional calculus proof" by evaluating a triple integral over the spherical shell of the function const/r^2, where r is the distance from A. If you set this integral you will see how much simpler and more elegant Newton's original proof is. On the other hand, if you analyze Newton's proof carefully you will see that this is also a "calculus" proof, if this word is used in its original sense of Infinitesimal Calculus. The angle of the cone at A is INFINITESIMALLY small (this is necessary for the conclusion that the triangles were similar), and one has to add the contributions from INFINITELY many such cones. If you like Newton's version more than the "modern calculus" version, you may also enjoy reading Newton himself.

Exercise. What happens if we consider a two dimensional problem? What is the gravitational force acting on a point inside a massive disk, assuming that the density depends only on the distance from the center?

It is not zero anymore. Is it directed away from the center or towards the center? It must be clear from the symmetry consideration that it cannot be directed in any other direction.

This question is reduced to the force exercised by a gravitating circle of uniform density.