Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/178



The ame otherwie. Pl. 10. Fig. 2.

Let the angle CBH of a given magnitude revolve about the pole B, as alo the rectilinear radius DC both ways produced. about the pole C. Mark the points M, N, on which the leg BC of the angle cuts that radius when BH the other leg thereof meets the ame radius in the points P and D. Then drawing the indefinite line MN, let that radius CP or CD and the leg BC of the angle perpetually meet in this line; and the point of concoure of the other leg BH with the radius will delineate the trajectory required.

For if in the contructions of the preceding problem the point A comes to a coincidence with the point B the lines CA and CB will coincide, and the line AB, in its lat ituation, will become the tangent BH; and therefore the contructions there et down will become the ame with the contructions here decribed. Wherefore the concoure of the leg BH with the radius will decribe a conic ection paing through the points C, D, P, and touching the line BH in the point B. Q. E. F.

Suppoe the four points B, C, D, P, (Pl. 10. Fig. 3.) given, being ituated without the tangent HI join each two by the lines BD, CP, meeting in G, and cutting the tangent in H and I. Cut the tangent in A in uch manner that HA may be to IA, as the rectangle under mean proportional between CG and GP, and a mean proportional between BH and HD, is to a rectangle under a mean proportional between GD and GB, and a mean pr0portional between PI and IC; and A will be the point of contact. For if HX, a parallel to the right