will either paſs through the two points P, p, or touch the two right lines TR, tr. Q. E. F.
Proposition XIX. Problem XI.
About a given focus, to deſcribe parabolic trajectory, which ſhall paſs through given points, and touch right lines given by poſition. Pl. 7. Fig. 3.
Let S the the focus, P a point, and TR a tangent of the trajectory to be deſcribed. About P as a centre, with the interval PS, deſcribe the circle FG; From the focus let fall ST perpendicular on the tangent, and produce the ſame to V, ſo as TV may be equal to ST. After the ſame manner another circle fg is to be deſcribed, if another point p is given; or another point v is to be found, if another tangent tr is given; then draw the right line IF, which ſhall touch the two circles FG, fg, if two points P, p are given, or paſs through the two points V, v, if two tangents TR, tr are given, or touch the circle FG and paſs through the point V, if the point P and the tangent TR are given. On FI let fall the perpendicular SL and bisect the ſame in K; and with the axis SK, and principal vertex K deſcribe a parabola. I ſay the thing is done. For this parabola (becauſe SK is equal to IK, and SP to FP) will paſs through the point P; and (by cor. 3. lem. 14.) becauſe ST is equal to TV and STR; a right angle, it will touch the right line TR. Q. E. F.
