11. Thus if the body be returned through F E with the velocity, wherewith it moved forward; we have shewn how by the repetition of the impulse, which acted on it at E, the body will return again into the line D E with the velocity, which it had before in that line. By the same process of reasoning it may be proved, that, when the body is returned back to D, the impulse, which before acted on the body at that point, will throw the body into the line D C with the velocity, which it first had in that line; and the other impulses being successively repeated, the body will at length be brought back again into the line B A with the velocity, wherewith it set out in that line.
12. Thus these impulses, by acting over again in an inverted order all their operation on the body, bring it back again through the path, in which it had proceeded forward. And this obtains equally, whatever be the number of the straight lines, whereof this curve figure is composed. Now by a method of reasoning, which Sir Isaac Newton makes great use of, and which he introduced into geometry, thereby greatly inriching that science[87]; we might make a transition from this figure composed of a number of straight lines to a figure of one continued curvature, and from a number of separate impulses repeated at distinct intervals to a continual centripetal force, and shew, that, because what has been here advanced holds universally true, whatever be the number of straight lines, whereof the curve figure A C F is composed, and howsoever frequently the impulses at the angles of this figure are repeated; therefore the same will still remain true, although this figure should be converted into one of a continued curvature, and these distinct impulses should be changed into a continual centripetal force. But as the explaining this method of reasoning is foreign to my present design; so I hope my readers, after what has been said, will find no difficulty in receiving the proposition laid down above: that, if the body, which has moved through the curve line B H I (in fig. 74.) from B to I, when it is come to I, be thrown directly back with the same velocity as that, wherewith it proceeded forward, the centripetal force, by acting over again all its operation on the body, shall bring the body back again in the line I H B: and as the motion of the body in its course from B to I was every where in such a manner oblique to the line drawn from the center to the body, that the centripetal power acted in some degree against the body’s motion, and gradually diminished it; so in the return of the body, the centripetal power will every where draw the body forward, and accelerate its motion by the same degrees, as before it retarded it.
13. This being agreed, suppose the body in K to have the line A K no longer obliquely inclined to its motion. In this case, if the body be turned back, in the manner we have been considering, it must be directed back perpendicularly to A K. But if it had proceeded forward, it would likewise have moved in a direction perpendicular to A K; consequently, whether it move from this point K backward or forward, it must describe the same kind of course. Therefore since by being turned back it will go over again the line K I H B; if it be permitted to go forward, the line K L, which it shall describe, will be altogether similar to the line K H B.
14. In like manner we may determine the nature of the motion, if the line, wherein the body sets out, be inclined (as in fig. 76.) down toward the line B A drawn between the body and the center. If the centripetal power so much increases in strength, as the body approaches, that it can bend the path, in which the body moves, to that degree, as to cause all the lines as A H, A I, A K to remain no less oblique to the motion of the body, than A B is oblique to B C; the body shall continually more and more approach the center. But if the centripetal power increases in so much less a degree, as to permit the line drawn from the center to the body, as it accompanies the body in its motion, at length to become more and more erect to the curve wherein the body moves, and in the end, suppose at K, to become perpendicular thereto; from that time the body shall rise again. This is evident from what has been said above; because for the very same reason here also the body shall proceed from the point K to describe a line altogether similar to the line, in which it has moved from B to K. Thus, as it was observed of the pendulum in the preceding chapter[88], that all the time it approaches towards being perpendicular to the horizon, it more and more descends; but, as soon as it is come into that perpendicular situation, it immediately rises again by the same degrees, as it descended by before: so here the body more and more approaches the center all the time it is moving from B to K; but thence forward it rises from the center again by the same degrees, as it approached by before.
15. If (in fig. 77.) the line B C be perpendicular to A B; then it has been observed above[89], that the centripetal power may be so balanced with the progressive motion of the body, that the body may keep moving round the center A constantly at the same distance; as a body does, when whirled about any point, to which it is tyed by a string. If the centripetal power be too weak to produce this effect, the motion of the body will presently become oblique to the line drawn from itself to the center, after the manner of the first of the two cases, which we have been considering. If the centripetal power be stronger, than what is required to carry the body in a circle, the motion of the body will presently fall in with the second of the cases, we have been considering.
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.
