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.