138 THE MATHEMATICAL PRINCIPLES IBOOK L former with the former, and the latter with the latter) in T and R. Then drawing the right line tr parallel to TR, cutting off from the right lines PT, PR, any segments Pt, Pr, proportional to PT, PR ; and if through their extremities, t, r, and the poles B, C, the right lines lit, Cr are drawn, meeting in d, that point d will be placed in the trajectory required. For (by Lena. XX) that point d is placed in a conic section passing through the four points A, B, C, P ; and the lines R/', TV vanishing, the point d comes to coincide with the point D. Wherefore the conic section passes through the five points A, B, C, P, D. Q.E.D. The same otherwise. Of the given points join any three, as A, B, C ; and about two of them 15, C, as poles, making the angles ABC, ACB of a given magnitude to revolve, apply the legs BA, CA, first to the point D, then to the point P, and mark the points M, N, in which the other legs BL, CL intersect each other in both cases. C' Draw the indefinite right line MN, and let those moveable angles revolve about their poles B, C, in such manner that the intersection, which is now supposed to be ???, of the legs BL, CL; or BM7 CM, may always fall in that indefinite right line MN ; and the intersection, which is now supposed to be d, of the legs BA ^A, or BD; CD, will describe the trajectory required, PADc/B. For (by Lem. XXI) the point d will be placed in a conic section passingthrough the points B, C ; and when the point m comes to coincide with the points L, M, N, the point d will (by construction) come to coin cide with the points A, D, P. Wherefore a conic section will be described that shall pass through the five points A, B. C, P, D. Q,.E.F. COR. 1. Hence a right line may be readily drawn which shall be a tan gent to the trajectory in any given point B. Let the point d come to co incide with the point B, arid the right line Bt/ Avill become the tangent required. COR. 2. Hence also may be found the centres, diameters, and latera recta of the trajectories, as in Cor. 2, Lem. XIX. SCHOLIUM. The former of these constructions will be- c come something more simple by joining , and in that line, produced, if need be, aking Bp to BP as PR is to PT ; and t rough p draw the indefinite right inc j0e parallel to S PT, and in that line pe taking always pe equal to Pi , and draw the right lines Be, Cr