The form of Universal Gravitation

The first thing to note is that the force of gravity must be independent of the type of material since all objects fall with the same acceleration, g. Therefore, the gravitational force must depend on mass alone. The dependence appears to be linear since Fgrav. = mg for an object of mass m. Consider two point masses, m1 and m2, as shown below.
The force acting on m1 should be F12 = m1k1, where k1 depends, at the very least, on the distance between the point masses. We have already established that k1 must be proportional to 1/r122, the distance between the point masses and that the force should be directed along a line connecting the two and pointing toward m2 (the latter comes from the fact that we assumed that the force of attraction for a circular orbit was a central one, i.e. the force connects the centers-of-mass). By the same reasoning, the force acting on m2 should be F21 = m2k2. But, by Newton's Third Law, we must have F21 = F12 and oppositely directed along the line between the point masses and pointing toward m1. The simplest consistent mathematical form is to have
k1 = Gm2/r 122
k2 = Gm1/r 122

where G is the universal constant of gravitation and must have units of Nt*m2/kg2. Therefore, Newton asserted that the gravitational force between two point masses is proportional to the product of the masses and inversely proportional to the square of the distance between the masses.