Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/182

or between the points K and H, I and L, or without them; then draw RS cutting the tangents in A and P, and A and P will be the points of contact. For if A and P are ſuppoſed to be the points of contact ſituated any where elſe in the tangents, and through any of the points H, I, K, L, as I, ſituated in either tangent HI a right line IT is drawn, parallel to the other tangent KL, and meeting the curve in X and I and in that right line there be taken IZ equal to a mean proportional between IX and IT; the rectangle XIT or $\scriptstyle IZ^{2}$, will (by the properties of the conic ſections) be to $\scriptstyle LP^{2}$, as the rectangle CID is to the rectangle CLD, that is (by the conſtruction) as $\scriptstyle I^{2}$ is to $\scriptstyle SL^{2}$, and therefore IZ is to LP; as SI to SL. Wherefore the points S, P, Z, are in one right line. Moreover, ſince the tangents meet in G, the rectangle XIY or $\scriptstyle IZ^{2}$ will (by the properties of the conic ſections) be to $\scriptstyle A^{2}$ as $\scriptstyle GP^{2}$ is to $\scriptstyle GA^{2}$, and conſequently IZ will be to IA, as GP to GA. Wherefore the points P, Z, A, lie in one right line, and therefore the points S, P, and A are in one right line. And the ſame argument will prove that the points R, P, and A are in one right line. Wherefore the points of contact A and P lie in the right line RS. But after theſe points are found the trajectory may be deſcribed as in the firſt caſe of the preceding problem. Q. E. F.

In this propoſition, and caſe 2. of the foregoing, the conſtructions are the ſame, whether the right line XY cut the trajectory in X and Y or not; neither do they depend upon that ſection. But the conſtructions being demonſtrated where that right line does cut the trajectory, the conſtructions, where it does not,