 The actual existence of quasi-inertial frames, corresponding to the abstract cases of Proposition LXV, was a crucial part of Newton’s argument for universal gravitation—more precisely, that the force holding the planets and their satellites in their respective orbits is, in fact, the same force as gravity. One crucial ground for the identification was the fact that the interplanetary force shares the most striking feature of terrestrial gravity, namely, that it imparts the same acceleration to all terrestrial bodies. This principle was discovered by Galileo, of course, but Newton tested it more severely, and with a greater variety of test bodies. He constructed pendulums of identical wooden boxes suspended from strings of equal length, which he filled with different materials; he found that these differences made no difference to the speed of falling over many oscillations of the pendulums. By this means he showed that Galileo’s principle holds to a much higher degree of accuracy than Galileo was able to show, and inferred that a body’s weight toward the earth is generally proportional to its mass. (1687b, Book III, Proposition VI). But Newton extended this principle beyond terrestrial gravity, to the accelerative forces acting on the planets and their satellites. Proposition IV, Corollary VI, from Book I, showed that an orbiting body that obeys Kepler’s third law is urged toward the center by an inverse-square force. Newton could then show that the centripetal forces acting on Jupiter’s moons depend only on the inverse-square of the distance towards Jupiter’s center: 