Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/155

 a circle, cut the right line VR produced in H, then with the foci S, H and principal axe equal to VH, decribe trajectory. I ay the thing is done. For VH is to SH as VK to SK, and therefore as the principal axe of the trajectory which was to be decribed to the ditance of its foci, (as appears from what we have demontrated in Case 1.) and therefore the decribed trajectory is of the ame pecies with that which was to be decribed; but that right line TR, by which the angle VRS is bisected, touches the trajectory in the point R, is certain from the properties of the conic ections. Q. E. F.

About the focus S (Pl. 7. Fig. 7.) it is required to decribe a trajectory APB that hall touch a right line TR. and pas thro' any given point P without the tangent, and hall be imilar to the figure apb, decribed with the principal axe ab, and foci s, h. On the tangent TR let fill the perpendicular ST; which produce to V, o that TV may be equal to ST. And making the angles hsq, shq equal to the angles VSP, SVP; about q as a centre, and with an interval which hall be to ab as SP to VS decribe a circle cutting the figure apb in p: join sp, and draw SH, uch that it may be to sh, as SP is to sp, and may make the angle PSH equal to the angle psh; and the angle VSH equal to the angle psq. Then with the foci S, H, and principal axe AB equal to the ditance VH, decribe a conic ection. I ay the thing is done. For if sv is drawn o that it hall be to sp as sh is to sq, and hall make the angle vsp equal to the angle hsq, and the angle vsh equal to the angle psq, the triangles svh, spq, will be imilar, and therefore vh will be to pq, as sh is to sq, that is, (becaue of the imilar triangles VSP, hsq) as VS is to SP or as ab to pq. Wherefore