Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/175

 Rh will be alo placed in a conic ection, paing through the ame five points B, C, A, p, P. when the point M is perpetually placed in a right line. Wherefore the two conic ections will both pas through the ame given points, againt corol. 3. lem. 20. It is therefore aburd to uppoe that the point M is placed in a curve line. Q. E. D.

To decribe a trajectory that hall pas through five given given points. Pl. 9 Fig. 5.

Let the five given points be A, B, C, P, D. From any one of them as A, to any other two as B, C, which may be called the poles, draw the right lines AB, AO, and parallel to thoe the lines TPS, PRQ through the fourth point P. Then from the two poles B, C, draw through the fifth point D two indefinite lines BDE, CRD, meeting with the lat drawn lines TPS, PRQ (the 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 egments Pt, Pr, proportional to PT, PR; and if through their extremities t, r, and the poles B, C, the right lines Bt, Cr are drawn, meeting in d, that point d will be placed in the trajectory required. For (by lem. 20.) that point d is placed in a conic ection paing through the four points A, B, C, P; and the lines Rr, Tt vanihing, the point d comes to coincide with the point D. Wherefore the conic ection paes through the five points A, B, C, P, D. Q. E. D. Rh