 In each of these cases, that is, Newton found that the centripetal acceleration behaves like gravitational acceleration, and so the bodies’ forces toward their respective centers are, essentially, their weights toward those centers. Moreover, the orbits of Jupiter’s moons provided a completely novel test of Galileo’s principle, on an extremely large scales of mass and distance. For he showed that Jupiter and its moons—within the limits of observational accuracy—undergo the same accelerations toward the sun (cf. 1687b, Book I, Proposition 65; Book III, Proposition VI). Any non-negligible difference in these accelerations would produce corresponding irregularities in the satellites’ orbits. 