7. BUT we may observe farther, that if the centripetal power, while the body increases its distance from the center, retain sufficient strength to make the lines drawn from the center to the body to become at length less oblique to the curve; then if this diminution of the obliquity continue, till at last the line drawn from the center to the body shall cease to be obliquely inclined to the curve, and shall become perpendicular thereto; from this instant the body shall no longer recede from the center, but in its following motion it shall again descend, and shall describe a curve line in all respects like to that, which it has described already; provided the centripetal power, every where at the same distance from the center, acts with the same strength. So we observed in the preceding chapter, that, when the motion of a projectile became parallel to the horizon, the projectile no longer ascended, but forthwith directed its course downwards, descending in a line altogether like that, wherein it had before ascended[83].
8. THIS return of the body may be proved by the following proposition: that if the body in any place, suppose at I, were to be stopt, and be thrown directly backward with the velocity, wherewith it was moving forward in that point I; then the body, by the action of the centripetal force upon it, would move back again over the path I H B, in which it had before advanced forward, and would arrive again at the point B in the same space of time, as was taken up in its passage from B to I; the velocity of the body at its return to the point B being the same, as that wherewith it first set out from that point. To give a full demonstration of this proposition, would require that use of mathematics, which I here purpose to avoid; but, I believe, it will appear in great measure evident from the following considerations.
9. SUPPOSE (in fig. 75.) that a body were carried after the following manner through the bent figure A B C D E F, composed of the straight lines A B, B C, C D, D E, E F. First let it be moving in the line A B, from A towards B, with any uniform velocity. At B let the body receive an impulse directed toward some point, as G, taken within the concavity of the figure. Now whereas this body, when once moving in the straight line A B, will continue to move on in this line, so long as it shall be left to it self; but being disturbed at the point B in its motion by the impulse, which there acts upon it, it will be turned out of this line A B into some other straight line, wherein it will afterwards continue to move, as long as it shall be left to itself. Therefore let this impulse have strength sufficient to turn the body into the line B C. Then let the body move on undisturbed from B to C, but at C let it receive another impulse pointed toward the same point G, and of sufficient strength to turn the body into the line C D. At D let a third impulse, directed like the rest to the point G, turn the body into the line D E. And at E let another impulse, directed likewise to the point G, turn the body into the line E F. Now, I say, if the body while moving in the line E F be stopt, and turned back again in this line with the same velocity, as that wherewith it was moving forward in this line; then by the repetition of the former impulse at E the body will be turned into the line E D, and move in it from E to D with the same velocity as before it moved with from D to E; by the repetition of the impulse at D, when the body shall have returned to that point, it will be turned into the line D C; and by the repetition of the other impulses at C and B the body will be brought back again into the line B A, with the velocity, wherewith it first moved in that line.