Why does Newton’s law of gravitation use the inverse square of distance?

Newton’s law of gravitation uses the inverse square of distance because this relationship is required to explain the observed motions of planets and their moons, which were described by Johannes Kepler’s laws of planetary motion.12

Specifically, Newton demonstrated that if planets follow elliptical orbits, the central force pulling them toward the Sun must vary as the inverse square of their distance ($1/r^2$). By applying his laws of motion, Newton showed that if objects in circular motion have periods that vary as $r^{3/2}$—as established by Kepler’s third law—the required centripetal acceleration, and thus the force, must decrease as the square of the distance.32

Newton further confirmed this by comparing the acceleration of the Moon in its orbit to the acceleration of objects falling freely on Earth. He found that the Moon’s inward acceleration was approximately $(1/60{)}^2$ of the acceleration at Earth’s surface, matching the ratio of the square of Earth’s radius to the square of the Moon’s orbital distance. This consistency allowed him to formulate the law of universal gravitation, which posits that all massive bodies attract one another with a force proportional to the product of their masses and inversely proportional to the square of their separation.45