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I've worked out the inverse square law using centripetal acceleration and Kepler's law. But I can't quite see how Newton worked out for sure that the force is directly proportional to both masses. I know it's pretty intuitive, but I thought maybe there was a mathematical way of confirming that assumption. The closest I've gotten is by accounting for the fact that two bodies would orbit about a common center of mass if the force acts on both of them. This gave me F[itex]\propto[/itex]mM/(m+M)R^2. So that's pretty close, but now I've got extra stuff in the bottom. I know that if I use Newton's modification of Kepler's Law, the (m+M) in the denominator goes away. And I can see why this modification works, since for the sun and any planet in solar masses (m+M) is essentially 1 and doesn't change the values. But I don't see how he would know for sure what modifications Kepler's Law needed without just assuming his law of gravity and then modifying Kepler's law to fit.
Any suggestions? Did he just assume his law and go from there, or is there a way to derive it?
Any suggestions? Did he just assume his law and go from there, or is there a way to derive it?