16. IF the centripetal power so change with the change of distance, that the body, after its motion has become oblique to the line drawn from itself to the center, shall again become perpendicular thereto; which we have shewn to be possible in both the cases treated of above; then the body shall in its subsequent motion return again to the distance of A B, and from that distance take a course similar to the former: and thus, if the body move in a space free from all resistance, which has been here all along supposed; it shall continue in a perpetual motion about the center, descending and ascending alternately therefrom. If the body setting out from B (in fig. 78.) in the line B C perpendicular to A B, describe the line B D E, which in D shall be oblique to the line A D, but in E shall again become erect to A E drawn from the body in E to the center A; then from this point E the body shall describe the line E F G altogether like to the line B D E, and at G shall be at the same distance from A, as it was at B. But likewise the line A G shall be erect to the body’s motion. Therefore the body shall proceed to describe from G the line G H I altogether similar to the line G F E, and at I have the same distance from the center, as it had at E; and also have the line A I erect to its motion: so that its following motion must be in the line I K L similar to I H G, and the distance A L equal to A G. Thus the body will go on in a perpetual round without ceasing, alternately inlarging and contracting its distance from the center. [Illustration] 17. IF it so happen, that the point E fall upon the line B A continued beyond A; then the point G will fall on B, I on E, and L also on B; so that the body will describe in this case a simple curve line round the center A, like the line B D E F in fig. 79, in which it will continually revolve from B to E and from E to B without end. 18. IF A E in fig. 78 should happen to be perpendicular to A B, in this case also a simple line will be described; for the point G will fall on the line B A prolonged beyond A, the point I on the line A E prolonged beyond A, and the point L on B: so that the body will describe a line like the curve line B E G I in fig. 80, in which the opposite points B and G are equally distant from A, and the opposite points E and I are also equally distant from the same point A. 19. IN other cases the line described will have a more complex figure. 20. THUS we have endeavoured to shew how a body, while it is constantly attracted towards a center, may notwithstanding by its progressive motion keep it self from falling down to that center; but describe about it an endless circuit, sometimes approaching toward that center, and at other times as much receding from the same.