Proposition XX. Problem XII.
About a given focus to deſcribe any trajectory given in ſpecie, which ſhall paſs thro' given points and touch right lines given by poſition.
Case 1. About the focus (Pl. 7. Fig. 4.) it is required to deſcribe trajectory ABC, paſſing thro' two points B, C. Becauſe the trajectory is given in ſpecie, the ratio of the principal axe to the diſtance of the foci will be given. In that ratio take KB to BS and LC to CS. About the centres B, C, with the intervals BK. CL deſcribe two circles, and on the right line KL, that touches the ſame in K and L, let fall the perpendicular SG; which cut in A and a, ſo that GA may be to AS, and Ga to aS, as KB to BS; and with the axe Aa, and vertices A, a, deſcribe a trajectory. I ſay the thing is done. For let H be the other focus of the deſcribed figure, and ſeeing GA is to AS as Ga to aS, then by diviſion we ſhall have Ga - GA or Aa to aS - AS or SH in the ſame ratio, and therefore in the ratio which the principal axe of the figure to be deſcribed has to the diſtance of its foci; and therefore the deſcribed figure is of the ſame ſpecies with the figure which was to be deſcribed. And ſince KB to BS, and LC to CS are in the ſame ratio, this figure will paſs thro' the points B, C, as is manifeſt from the conic ſections
Case 2. About the focus (Pl. 7. Fig. 5.) it is required to deſcribe a trajectory, which ſhall ſomewhere touch two right lines TR, tr. From the focus On thoſe tangents let fall the perpendiculars ST, St,
