 The “phenomena” with which Book 3 of the Principia begins record the observed relative motions of the principal satellites in the solar system with respect to their primary bodies (the planets with respect to the sun, the moons of Jupiter and Saturn with respect to their planets, the earth’s moon with respect to the earth). All of these satellites obey Kepler’s laws (at the time often called “rules”) of orbital motion; and, appealing to his first law of motion (the law of inertia), Newton is able to derive purely mathematically that each of the satellites in question experiences an inverse-square acceleration directed towards its respective primary body. Moreover, the so-called “moon test” (developed in Proposition 4 of Book 3) shows that the inverse-square acceleration governing the moon’s orbit is, when the distance in question approaches the surface of the earth, numerically equal to the constant acceleration of terrestrial gravity figuring in Galileo’s law of fall. Newton concludes (by the first and second of his Rules for the Study of Natural Philosophy) that the (centripetal) force holding the moon in its orbit is the same force as terrestrial gravity.