Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/176



The ame otherwie. Pl. 9. Fig. 6.

Of the given points join any three as A, B, C; and about two of them B, C, as poles, making the angles ABC, ACB of a given magnitude to revolve, apply the legs BA, CA, firt to the point D, then to the point P, and mark the points M, N, in which the other legs BL, CL intersect each the other in both caes. Draw the indefinite right line MN and let thoe moveable angles revolve about their poles B, C, in uch manner that the interection, which is now uppoed to be m, of the legs BL, CL, or BM, CM and always fall in that indefinite right line MN; and, the interection which is now uppoed to be d, of the legs BA, CA, or BD, CD, will decribe the trajectory required PADdB. For, (by lem. 21.) the point d will be placed in a conic ection paing through the points B, C; and when the point m comes to coincide with the points L, M, N, the point d will (by contruction) come to coincide with the points A, D, P. Wherefore a conic ection will be decribed that hall pas through the five points A, B, C, P, D. Q. E. F.

Hence a right line may be readily drawn which hall be a tangent to the trajectory in any given point B. Let the point d come to coincide with the point B, and the right line Bd will become the tangent required.

Hence alo may be found the centres, diameters, and latera recta of the trajectories, as in cor. 2. lem. 19.