# Is it true that Newton used the gedankenexperiment of a cannonball revolving around the earth to find the inverse square law of gravity?

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?

• He did use the thought experiment, but it was not about inferring the inverse square law (he inferred it from Kepler's laws already, like others before him, see Who was first to explain intuitively the inverse square law of gravity?). It was to illustrate how manifestations of gravity on Earth can be continuously related to orbital motions of planets and moons, and argue that the law is universal, see Pesnell, The flight of Newton's cannonball. Commented Nov 10, 2020 at 20:31
• He used Moon instead of a cannon ball. Commented Nov 10, 2020 at 21:13
• See A Treatise of the System of the World (published posthumously in 1728), page 6 for the picture. Commented Nov 12, 2020 at 9:43
• It was not the way Newton discovered the law. It is used to explain it (page 5): "That by means of centripetal forces, the planets may be retained in certain orbits, we may easily understand, if we consider the motions of projectiles." Commented Nov 12, 2020 at 9:45