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 uppoed to be the points of contact ituated any where ele in the tangents, and through any of the points H, I, K, L, as I, ituated 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 ections) be to $$\scriptstyle LP^2$$, as the rectangle CID is to the rectangle CLD, that is (by the contruction) 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, ince the tangents meet in G, the rectangle XIY or $$\scriptstyle IZ^2$$ will (by the properties of the conic ections) be to $$\scriptstyle A^2$$ as $$\scriptstyle GP^2$$ is to $$\scriptstyle GA^2$$, and conequently 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 ame 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 thee points are found the trajectory may be decribed as in the firt cae of the preceding problem. Q. E. F.

In this propoition, and cae 2. of the foregoing, the contructions are the ame, whether the right line XY cut the trajectory in X and Y or not; neither do they depend upon that ection. But the contructions being demontrated where that right line does cut the trajectory, the contructions, where it does not,