How did Isaac Newton use Kepler’s third law in developing universal gravitation?

Isaac Newton utilized Kepler’s third law as a critical mathematical tool to determine the nature of the force governing planetary motion. By analyzing uniform circular motion, Newton derived a formula for the centripetal force required to keep an object in a circular path based on its velocity and mass.12

When he substituted the relationship defined by Kepler’s third law—which states that the squares of the orbital periods of planets are proportional to the cubes of their mean distances from the Sun—into his centripetal force formula, he discovered that the force must vary inversely with the square of the distance ($1/r^2$) from the Sun.12

This realization allowed Newton to conclude that the same inverse-square force attracting the planets to the Sun also acted upon the moons of Jupiter and the Moon relative to Earth. By correlating these observations, he established the law of universal gravitation, demonstrating that a single, consistent force governs the motion of all massive bodies in the universe.12