  Although Newton does not so expressly single out other results of Book 1, a few deserve comment here. The key that opened the way to Newton's theory of motion under centripetal forces was his discovery of how to generalize to non-circular trajectories the solution that he and Huygens had obtained for the central force in uniform circular motion. Figure 2 shows Newton's diagram for this generalization from the first edition. Suppose first that the trajectory APQ is part of a circle of radius SP along which the body at P is moving uniformly. Both Newton and Huygens had reasoned that the displacement QR from the tangent is proportional to the product of the force retaining the body in its circular orbit and the square of the time t for the body to