Newton claimed he had proved using calculus that objects at or near the surface of a gravitating body such as the Earth do not experience any bias in the gravitational pull resulting from the particles of mass directly beneath the observer’s feet being that much closer than the particles at the exact centre of the gravitating body.

The equation GM/r² dictates that the gravitational pull is inversely proportional to the distance squared, so any particle of mass directly beneath the observer’s feet would have a disproportionally greater effect than the same sized particle at the exact centre of the Earth or on the other side of the planet. But Newton stated that calculus revealed that any effect such as this is completely cancelled out.

However trying to picture this effect in my mind’s eye I can’t see how this effect could be cancelled out.

It is experimentally provable that g = 9.8 m/sec² at the Earth’s surface. Also the equation GM/r² completely balances at the Earth’s surface and when the appropriate values and units are used, produces the figure of 9.8 m/sec² for g. However if any gravitational bias were to occur as a result of the observation outlined above, it ruins everything. The escape velocity would change, the orbits of Earth bound satellites would also change and it could even affect the orbits of the planets.

Has anyone ever seen Newton's proof by calculus on this question and how does it work exactly? I have not been able to trace any mention of it on the internet to date.