Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/151

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RS, TV are equal, therefore the right lines SR, VR, as well as the angles TRS, TRV, will be alſo equal. Whence the point R will be in the conic ſection, and the perpendicular TR will touch the ſame: and the contrary. Q. E. D.


Proposition XVIII. Problem X.

From a focus and the principal axes given, to deſcribe elliptic and hyperbolic trajectories, which ſhall paſs through given points, and touch right lines given by poſition. Pl. 7. Fig. 2.

Plate 7, Figure 2
Plate 7, Figure 2

Let S be the common focus of the figures; AB the length of the principal axis of any trajectory; P a point through which the trajectory ſhould paſs; and TR a right line which it ſhould touch. About the centre P, with the interval AB - SP, if the orbit is an ellipſis, or AB + SP if the orbit is an hyperbola, deſcribe the circle HG. On the tangent TR let fall the perpendicular ST and produce the ſame to V, ſo that TV may be equal to ST; and about V as a centre with the interval AB deſcribe the circle FH. In this manner whether two points P, p, are given, or two tangents TR, tr, or a point P and a tangent TR, we are to deſcribe two circles. Let H be their common intersection, and from the foci S, H with the given axis deſcribe the trajectory. I ſay the thing is done. For (becauſe PH + SP in the ellipſis, and PH - SP in the hyperbola is equal to the axis) the deſcribed trajectory will paſs through the point P, and (by the preceding lemma) will touch the right line TR. And by the ſame argument it