and perpendicular on the leſſer; and the contrary will happen if thoſe legs meet in the remoteſt point L. Whence if the centre of the trajectory is given, the axes will be given; and thoſe being given, the foci will be readily found.
But the ſquares of the axes are one to the other as KH to LH, and thence it is eaſy to deſcribe a trajectory given in kind through four given points. For if two of the given points are made the poles C, B, the third will give the moveable angles PCK, PBK; but thoſe being given, the circle BGKC may be deſcribed. Then, becauſe the trajectory is given in kind, the ratio of OH to OK, and therefore OH it ſelf will be given. About the centre O, with the interval OH, deſcribe another circle. and the right line that touches this circle and paſſes through the concourſe of the legs CK, BK, when the firſt legs CP, BP, meet in the fourth given point, will be the ruler MN, by means of which the trajectory may be deſcribed. Whence alſo on the other hand a trapezium given in kind (excepting a few caſes that are impoſſible) may be inſcribed in a given conic ſection.
There are alſo other lemma's by the help of which trajectories given in kind may be deſcribed through given points, and touching given lines. Of ſuch a fort is this, that if a right line is drawn through any point given by poſition. that may cut a given conic ſection in two points, and the diſtance of the interſections is biſected, the point of biſection will touch another conic ſection of the ſame kind with the former, and having its axes parallel to the axes of the former. But I haſten to things of greater uſe.
