Free To Explain the World: The Discovery of Modern Science by Steven Weinberg Page A

the answer to this question led to Newton’s discovery of the inverse square law of gravitation.
    Next, by “Kepler’s rule of the periodical times of the Planets being in sesquialterate proportion of their distances from the center of their Orbs” Newton meant what we now call Kepler’s third law: that the square of the periods of the planets in their orbits is proportional to the cubes of the mean radii of their orbits, or in other words, the periods are proportional to the 3 / 2 power (the “sesquialterate proportion”) of the mean radii. * The period of a body moving with speed v around a circle of radius r is the circumference 2π r divided by the speed v , so for circular orbits Kepler’s third law tells us that r 2 / v 2 is proportional to r 3 , and therefore their inverses are proportional: v 2 / r 2 is proportional to 1/ r 3 . It follows that the force keeping the planets in orbit, which is proportional to v 2 / r , must be proportional to 1/ r 2 . This is the inverse square law of gravity.
    This in itself might be regarded as just a way of restating Kepler’s third law. Nothing in Newton’s consideration of the planets makes any connection between the force holding the planets in their orbits and the commonly experienced phenomena associated with gravity on the Earth’s surface. This connection was provided by Newton’s consideration of the Moon. Newton’s statement that he “compared the Moon in her Orb with the force of gravity at the surface of the Earth & found them answer pretty nearly” indicates that he had calculated the centripetal acceleration of the Moon, and found that it was less than the acceleration of falling bodies on the surface of the Earth by just the ratio one would expect if these accelerations were inversely proportional to the square of the distance from the center of the Earth.
    To be specific, Newton took the radius of the Moon’s orbit (well known from observations of the Moon’s diurnal parallax) to be 60 Earth radii; it is actually about 60.2 Earth radii. He used a crude estimate of the Earth’s radius, * which gave a crude value for the radius of the Moon’s orbit, and knowing that the sidereal period of the Moon’s revolution around the Earth is 27.3 days, he could estimate the Moon’s velocity and from that its centripetal acceleration. This acceleration turned out to be less than the acceleration of falling bodies on the surface of the Earth by a factor roughly (very roughly) equal to 1/(60) 2 , as expected if the force holding the Moon in its orbit is the same that attracts bodies on the Earth’s surface, but reduced in accordance with the inverse square law. (See Technical Note 33 .) This is what Newton meant when he said that he found that the forces “answer pretty nearly.”
    This was the climactic step in the unification of the celestial and terrestrial in science. Copernicus had placed the Earth among the planets, Tycho had shown that there is change in the heavens, and Galileo had seen that the Moon’s surface is rough,like the Earth’s, but none of this related the motion of planets to forces that could be observed on Earth. Descartes had tried to understand the motions of the solar system as the result of vortices in the ether, not unlike vortices in a pool of water on Earth, but his theory had no success. Now Newton had shown that the force that keeps the Moon in its orbit around the Earth and the planets in their orbits around the Sun is the same as the force of gravity that causes an apple to fall to the ground in Lincolnshire, all governed by the same quantitative laws. After this the distinction between the celestial and terrestrial, which had constrained physical speculation from Aristotle on, had to be forever abandoned. But this was still far short of a principle of universal gravitation, which would assert that every body in the universe, not just the Earth and Sun, attracts every other body with a force that decreases as the inverse square