a circle, cut the right line VR produced in H, then with the foci S, H and principal axe equal to VH, deſcribe trajectory. I ſay 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 deſcribed to the diſtance of its foci, (as appears from what we have demonſtrated in Case 1.) and therefore the deſcribed trajectory is of the ſame ſpecies with that which was to be deſcribed; 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 ſections. Q. E. F.
Case 4. About the focus S (Pl. 7. Fig. 7.) it is required to deſcribe a trajectory APB that ſhall touch a right line TR. and paſs thro' any given point P without the tangent, and ſhall be ſimilar to the figure apb, deſcribed with the principal axe ab, and foci s, h. On the tangent TR let fill the perpendicular ST; which produce to V, ſo 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 ſhall be to ab as SP to VS deſcribe a circle cutting the figure apb in p: join sp, and draw SH, ſuch 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 diſtance VH, deſcribe a conic ſection. I ſay the thing is done. For if sv is drawn ſo that it ſhall be to sp as sh is to sq, and ſhall make the angle vsp equal to the angle hsq, and the angle vsh equal to the angle psq, the triangles svh, spq, will be ſimilar, and therefore vh will be to pq, as sh is to sq, that is, (becauſe of the ſimilar triangles VSP, hsq) as VS is to SP or as ab to pq. Wherefore
