How did Newton derive his law of gravity?

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For some reason I have been extremely curious of how Newton derived his mathematical law for gravitation. You know

$$F= \frac{GMm}{r^2}$$

So the first thing I did before i posted this was search Google and I foundhttp://www.relativitycalculator.com/Newton_Universal_Gravity_Law.shtml" . It talks about a possible derivation of how Newton could have derived his law of gravity. It looks pretty good and I understand most of it BUT there is one part I don't understand. There is one part in the derivation that looks like this.

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

In this situation k is the gravitational constant and 4pi is... 4pi. My question is WHERE did this k/(4pi) come from? Why set these two equations equal to k/(4pi)? Can someone tell me where this k/(4pi) came from??

Thanks for your responce :D

 

[alxm]

Well, get rid of it and see what happens!

$$\frac{C}{M}=\frac{c}{m}=k$$
then
$$C = kM, c=km$$
$$f^2 = f*f' = m\frac{4\pi^2C}{r^2}M\frac{4\pi^2c}{r^2}$$
$$= m\frac{4\pi^2kM}{r^2}M\frac{4\pi^2km}{r^2}$$
$$= 16\pi^4k^2\frac{M^2m^2}{r^4}$$

$$f = 4\pi^2k\frac{Mm}{r^2}$$

So you get the same result but with the unnecessary constant of $$4\pi^2$$ in front, so they defined k in a way that eliminates that constant..

Of course, if you want to, you could also just say $$G = 4\pi^2k$$.