Newton's Gravitational Law?

Question

Using the Newton's laws of motion along with Kepler's laws, how is Newton's inverse square law of gravity deduced?

Answer 1

Kepler's law R^3/r^3 = w^2/W^2 = 1/T^2//1/t^2; where R and r are two orbital distances from a common mass M (e.g., the Sun), w and W are the respective angular velocities, and T and t are the periods where 1/T = w and 1/t = W and w < W and r < R.

We can find R^2/r^2 = w^2r/W^2R, then GmM/r^2//GnM/R^2 = GmMw^2//GnMW^2R = F/f; where F = GmM/r^2 and f = GnM/R^2 for planets of mass m and n at respective orbits r and R. And there you have it if you assume the orbits are circular, which they were thought to be some years ago.