Why is force squared in the derivation of Newton's gravitational law?

[ecastro]

The derivation of the law have been put up in the forums but I have a question regarding its derivation.

I understood everything from the assumptions to the application of Newton's Third Law, but I got stocked at this step:

$$\frac{m}{k} = \frac{M}{k'}$$.

This is similar to

$$\frac{C}{M} = \frac{c}{m} = \frac{k}{4 \pi^2}$$

at this site, http://www.relativitycalculator.com/Newton_Universal_Gravity_Law.shtml.

According to the same site, the next step requires the force to be squared. Why is this so? Is it merely to acquire the force $F$ between the two bodies? Aren't there any other ways to calculate the force other than multiplication?

 

[Nugatory]

According to the same site, the next step requires the force to be squared. Why is this so? Is it merely to acquire the force $F$ between the two bodies? Aren't there any other ways to calculate the force other than multiplication?

It's just a convenient algebra trick to get both $m$ and $M$ into the equation for $f$. We have $f=f'$ so we can multiply both sides of that equation by $f$ to get one equation that can be solved for $f$ in terms of $k$, $m$, and $M$.

 

[ecastro]

Alright. Thank you for your help. :D