10. THIS I prove as follows. Let D E and F E be continued beyond E. In D E thus continued take at pleasure the length E H, and let H I be so drawn, as to be equidistant from the line G E. Then, by what has been written upon the second law of motion[84], it follows, that after the impulse on the body in E it will move through E I in the same time, as it would have imployed in moving from E to H, with the velocity which it had in the line D E. In F E prolonged take E K equal to E I, and draw K L equidistant from G E. Then, because the body is thrown back in the line F E with the same velocity as that wherewith it went forward in that line; if, when the body was returned to E, it were permitted to go straight on, it would pass through E K in the same time, as it took up in passing through E I, when it went forward in the line E F. But, if at the body’s return to the point E, such an impulse directed toward the point D were to be given it, whereby it should be turned into the line D E; I say, that the impulse necessary to produce this effect must be equal to that, which turned the body out of the line D E into E F; and that the velocity, with which the body will return into the line E D, is the same, as that wherewith it before moved through this line from D to E. Because E K is equal to E I, and K L and H I, being each equidistant from G E, are by consequence equidistant from each other; it follows, that the two triangular figures I E H and K E L are altogether like and equal to each other. If I were writing to mathematicians, I might refer them to some proportions in the elements of EUCLID for the proof of this[85] but as I do not here address my self to such, so I think this assertion will be evident enough without a proof in form; at least I must desire my readers to receive it as a proposition true in geometry. But these two triangular figures being altogether like each other and equal; as E K is equal to E I, so E L is equal to E H, and K L equal to H I. Now the body after its return to E being turned out of the line F E into E D by an impulse acting upon it in E, after the manner above expressed; the body will receive such a velocity by this impulse, as will carry it through E L in the same time, as it would have imployed in passing through E K, if it had gone on in that line undisturbed. And it has already been observed, that the time, in which the body would pass over E K with the velocity wherewith it returns, is equal to the time it took up in going forward from E to I; that is, equal to the time, in which it would have gone through E H with the velocity, wherewith it moved from D to E. Therefore the time, in which the body will pass through E L after its return into the line E D, is the same, as would have been taken up by the body in passing through E H with the velocity, wherewith the body first moved in the line D E. Since therefore E L and E H are equal, the body returns into