SEC. II.] OF NATURAL PHILOSOPHY. 105 times, in spaces void of resistance, are completed into the parallelograms ABCV, DEFZ : the forces in B and E are one to the other in the ulti mate ratio of the diagonals BV, EZ, when those arcs are diminished in infinitum. For the motions BC and EF of the body (by Cor. 1 of the Laws) are compounded of the motions Be, BV, and E/", EZ : but BV and EZ, which are equal to Cc and F/, in the demonstration of this Proposi tion, were generated by the impulses of the centripetal force in B and E; and are therefore proportional to those impulses. COR. 4. The forces by which bodies, in spaces void of resistance, are drawn back from rectilinear motions, and turned into curvilinear orbits, are one to another as the versed sines of arcs described in equal times ; which versed sines tend to the centre of force, and bisect the chords when those arcs are diminished to infinity. For such versed sines are the halves of the diagonals mentioned in Cor. 3. COR. 5. And therefore those forces are to the force of gravity as the said versed sines to the versed sines perpendicular to the horizon of those para bolic arcs which projectiles describe in the same time. COR. 6. And the same things do all hold good (by Cor. 5 of the Laws), when the planes in which the bodies are moved, together with the centres of force which are placed in those planes, are not at rest, but move uni formly forward in right lines. PROPOSITION II. THEOREM II. Every body that moves in any curve line described in a plane, and by a radius, drawn to a point either immovable, or moving forward with an uniform rectilinear motion, describes about that point areas propor tional to the times, is urged by a centripetal force directed to that point CASE. 1. For every body that moves in a curve line, is (by Law 1) turned aside from its rectilinear course by the action of some force that impels it. And that force by which the body is turned off from its rectilinear course, and is made to describe, in equal times, the equal least triangles SAB, SBC, SCD, &c., about the immovable point S (by Prop. XL. Book 1, Elem. and Law II), acts in the place B, according to the direction of a line par