How did Newton use the Moon’s motion to compare gravity in space with gravity near Earth?

To compare gravity in space with gravity near Earth, Isaac Newton analyzed the Moon’s orbital motion using his laws of motion.12

Newton recognized that the Moon requires a constant inward centripetal force to maintain its circular orbit rather than moving in a straight line, as dictated by his law of inertia. He compared the distance the Moon is diverted from a tangential path in one second with the distance a body falls from rest near Earth’s surface in one second.12

Because the Moon’s orbital radius is approximately 60 times the radius of Earth, Newton calculated that the Moon’s inward acceleration is $1/{60}^2$, or 1/3,600, of the acceleration experienced by a falling object at Earth’s surface. This correlation demonstrated that the same gravitational force governs both terrestrial and celestial bodies, acting as an inverse square law where the force diminishes as the square of the distance between the centers of two bodies increases.123