How exacly did **Sir Isaac Newton** went from now knowing any formula describing gravity to the conclusion that: F = G (m1 * m2)/r^2
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The version I heard was that he looked at the moon, and wondered why the moon went around the Earth, and if was due to the same force that made things fall on the Earth (the apple was probably made up, but makes for a nice story). He figured out that if you were to shoot a bullet fast enough, it would fall down precisely as fast as the Earth curved away underneath it. By knowing how fast gravity accelerated things on the surface of the Earth, and knowing how large the Earth is, Newton figured out how fast you would have to shoot a bullet – about 7.5 km/s. Once you have that figured out, you can carry out the same exercise with something as far away as the moon, and he figured out that if the force between two object fell off like the square of the distance, the same gravity we feel on Earth was exactly right to keep the moon going around the Earth every 28 days.
That gets you r^2, which is the key part of gravity. For m1 and m2, we see that everything accelerates under gravity at the same rate (credit to Galileo for this), so since F=ma, if a stays the same when m goes up, then F must go up by the same amount. So, F is proportional to mass. You’re basically done now, so F=Gm1m2/r^2 for some unknown constant G. We had to wait for Cavendish to get the value for G.
Incidentally, the same gravity law is there in Kepler’s laws, which Newton would have been well aware of. Kepler’s third law, which states that the time it takes a planet to go around the sun scales like the distance of the planet to the sun to the 3/2 power, only works out if the force falls like the distance squared.
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