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.
21. But here we have supposed, that the centripetal power is of equal strength every where at the same distance from the center. And this is the case of that centripetal power, which will hereafter be shewn to be the cause, that keeps the planets in their courses. But a body may be kept on in a perpetual circuit round a center, although the centripetal power have not this property. Indeed a body may by a centripetal force be kept moving in any curve line whatever, that shall have its concavity turned every where towards the center of the force.
22. To make this evident I shall first propose the case of a body moving through the incurvated figure A B C D E (in fig. 81.) which is composed of the straight lines A B, B C, C D, D E, and E A; the motion being carried on in the following manner. Let the body first move in the line A B with any uniform velocity. When it is arrived at the point B, let it receive an impulse directed toward any point F taken within the figure; and let the impulse be of that strength as to turn the body out of the line A B into the line B C. The body after this impulse, while left to itself, will continue moving in the line B C. At C let the body receive another impulse directed towards the same point F, of such strength, as to turn the body from the line B C into the line C D. At D let the body by another impulse, directed likewise to the point F, be turned out of the line C D into D E. And at E let another impulse, directed toward the point F, turn the body from the line D E into E A. Thus we see how a body may be carried through the figure A B C D E by certain impulses directed always toward the same center, only by their acting on the body at proper intervals, and with due degrees of strength.
23. But farther, when the body is come to the point A, if it there receive another impulse directed like the rest toward the point F, and of such a degree of strength as to turn the body into the line A B, wherein it first moved; I say that the body shall return into this line with the same velocity, as it had at first.
24. Let A B be prolonged beyond B at pleasure, suppose to G; and from G let G H be drawn, which if produced should always continue equidistant from B F, or, according to the more usual phrase, let G H be drawn parallel to B F. Then it appears, from what has been said upon the second law of motion[90], that in the time, wherein the body would have moved from B to G, had it not received a new impulse in B, by the means of that impulse it will have acquired a velocity, which will carry it from B to H. After the same manner, if C I be taken equal to B H, and I K be drawn equidistant from or parallel to C F; the body will have moved from C to K with the velocity, which it has in the line C D, in the same time, as it would have employed in moving from C to I with the velocity, it had in the line B C. Therefore since C I and B H are equal, the body will move through C K in the same time, as it would have taken up in moving from B to G with the original velocity, wherewith it moved through the line A B. Again, D L being taken equal to C K and L M drawn parallel to D F; for the same reason as before the body will move through D M with the velocity, which it has in the line D E, in the same time, as it would imploy in moving through B G with its original velocity. In the last place, if E N be taken equal to D M, and N O be drawn parallel to E F; likewise if A P be taken equal to E O, and P Q be drawn parallel to A F: then the body with the velocity, wherewith it returns into the line A B, will pass through A Q in the same time, as it would have imployed in passing through B G with its original velocity. Now as all this follows directly from what has above been delivered, concerning the effect of oblique impulses impressed upon bodies in motion; so we must here observe farther, that it can be proved by geometry, that A Q will always be equal to E G. The proof of this I am obliged, from the nature of my present design, to omit; but this geometrical proportion being granted, it follows, that the body has returned into the line A B with the velocity, which it had, when it first moved in that line; for the velocity, with which it returns into the line A B, will carry it over the line A Q in the same time, as would have been taken up in its passing over an equal line B G with the original velocity.
25. Thus we have found, how a body may be carried round the figure A B C D E by the action of certain impulses upon it which should all be pointed toward one center. And we likewise see, that when the body is brought back again to the point, whence it first set out; if it there meet with an impulse sufficient to turn it again into the line, wherein it moved at first, its original velocity will be again restored; and by the repetition of the same impulses, the body will be carried again in the same round. Therefore if these impulses, which act on the body at the points B, C, D, E, and A, continue always the same, the body will make round this figure innumerable revolutions.