Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/187

through which the conic ſection ought to paſs in this new figure; and compleating the parallelogram hikl, let the right lines hi, ik, kl be ſo cut in c, d, e, that be may be to the ſquare root of the rectangle ahb, ic to id, and ke to kd, as the ſum of the right lines hi and kl is to the ſum of the three lines. the firſt whereof is the right line ik, and the other two are the ſquare roots of the rectangles ahb and alb; and c, d, e, will be the points of contact. For by the properties of the conic ſections $\scriptstyle hc^{2}$ to the rectangle ahb, and $\scriptstyle ic^{2}$ to $\scriptstyle id^{2}$, and $\scriptstyle ke^{2}$ to $\scriptstyle kd^{2}$ and $\scriptstyle cl^{2}$ to the rectangle alb, are all in the ſame ratio; and therefore hc to the ſquare root of ahb, ic to id, kc to kd, and el to the ſquare root of alb, are in the ſubduplicate of that ratio; and by compoſition in the given ratio of the ſum of all the antecedents hu + kl, to the ſum of all the conſequents $\scriptstyle {\sqrt {ahb}}+ik+{\sqrt {alb}}$. Wherefore from that given ratio we have the points of contact c, d, c, in the new figure. By the inverted operations of the laſt, lemma, let thoſe points be tranſferred into the firſt figure, and the trajectory will be there deſcribed by prob. 14. Q. E. F. But according as the points a, b, fall between the points h, l, or without them, the points c, d, e, muſt be taken either between the points b, i, k, l, or without them. If one of the points h, l, falls between the points h, l, and the other without the points h, l, the problem is impoſſible.