Scholium.
The former of theſe conſtructions (Fig. 5.) will become ſomething more ſimple by joining BP, and in that line, produced if need be, taking Bp to BP as PR is to PT; and through p drawing the indefinite right line be parallel to SPT; and in that line pe taking always pe equal to Pr; and drawing the right lines Be, Cr to meet in d. For ſince Pr to Pt, PR to PT, pB to PB, pe to Pt, are all in the ſame ratio, pe and Pr will be always equal. After this manner the points of the trajectory are moſt readily found, unleſs you would rather deſcribe the curve mechanically as in the ſecond conſtruction.
Proposition XXIII. Problem XV.
To deſcribe a trajectory that ſhall paſs through four given points, and touch a right line given by poſition. Pl. 10. Fig. 1.
Case 1. Suppoſe that HB is the given tangent, B the point of contact, and C, D, P, the three other given points. Join BC, and drawing PS parallel to BH, and PQ parallel to BC, compleat the parallelogram BSPQ. Draw BD cutting SP in T and CD cutting PQ in R. Laſtly, drawing any line tr parallel to TR, cutting off from PQ, PS, the ſegments Pr, Pt proportional to PR, PT reſpectively; and drawing Cr, Bt, their point of concourſe will (by lem. 20.) always fall on the trajectory to be deſcribed.
