The first section is on the method of prime and ultimate ratios, by the use of which Newton was able, in effect, to integrate. He applied this to the curvature and the areas of curves, and proved that, at the very beginning of the motion of a body from rest under any force, the space described is proportional to the force and the square of the time. The second section is concerned with the motion of a particle under a central force. It contains the well-known propositions that if the force is central the area swept out by the vector to the centre is proportional to the time, and conversely that if such area is proportional to the time the particle is acted on by a central force. Newton further discussed particular cases of circular, elliptic, and spiral motion. In the third section he dealt with motion in a conic under a central force to the focus, showed that in this case the force must vary inversely as the square of the distance, and conversely that if a particle be projected from any point in any direction with any velocity under such a force it must describe a conic about the centre of force as a focus, and that in such elliptic orbits the periodic times are in the sesquiplicate ratio of the major axes of the ellipses. He also explained how to treat the problem if disturbing forces are introduced. These two sections solved the problem of planetary motion if the planets could be treated as particles and did not disturb one another's motions. The fourth and fifth sections are given up to the proof of certain geometrical propositions in conics required for subsequent discussions: in particular the construction of a conic when a focus and three other conditions or when five points on it or five tangents to it are given. In the sixth section Newton returned to the problem of the motion of a particle in an ellipse under a central force to a focus, and discussed how to determine the position of the particle at any given time. (Kepler's Problem.) The seventh and eighth sections are devoted to the motion of a particle under a central force which is any function of the distance. The geometrical treatment of these problems is ingenious, but necessarily more involved than when modern analysis is used. In the ninth section Newton dealt with the motion of particles in orbits which are revolving about the centre of force, and on the motion of the apses of such orbits: this introduced the theory of disturbing forces. The tenth section is concerned with constrained motion, and particularly with the motion of pendulums. The eleventh section deals with the motion of particles under their mutual attractions and incidentally with the problem of three bodies. These three sections afford a notable illustration of Newton's analytical powers.