Question

Proof of the equation F = Gm₁m₂/r^{2
Where F is the force of gravitation and G is the gravitational constant}

Answer 1

There's something called inductive reasoning, wherein we divide a huge system in a smaller bits and evaluate the nature of all those bits.

​​​​​What Newton did was to study the behavior of tiny particles and he postulated that behavior into the equation stated above. He observed that the particles were attracted to each other - the greater the masses - the greater was the attraction. So the obvious conclusion was a proportionality of m1.m2.

Then, the dependence on distance was noted. The dependence proceed out to be inverse and showed signs of possible exponential dependence. However such wide difference as observed in an exponential variation was not possible in a physical system of celestial masses because the system is largely uniform in energy distribution - so only possible answer could be an inverse square dependence.

The actual proof is complicated and beyond your understanding, if you still insist on knowing it, here it is: