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I have often heared that Newton compared the centripetal acceleration of an (imaginary) cannonball revolving around the earth just above the surface with the centripetal acceleration of the moon and discovered that a 60 times larger distance results in a $\frac{1}{3600} = \frac{1}{60^2}$ fraction of the centripetal acceleration. From this he concluded the hypothesis of the inverse square law for gravity.

What is true about this story? Do you have references to Newtons writings where he describes this idea?

Did Newton also have other "derivations" to the inverse square law? If so, which idea was most important from his point of view?

Is it also true that he abandoned the idea at first time because he didn't have an accurate value for the earth diameter and then got back to it over 15 years later when a better value was available?

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