Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/236

 162 S at the given interval SG, take GA to $$\scriptstyle{\frac{1}{2}AS}$$. If that ratio is the ame as of the number 2 to 1 the point A is infinitely remote; in which cae a parabola is to be decribed with any latus rectum to the vertex S, and axis SG; as appears by prop. 34. But if that ratio is les or greater than ratio of 2 to 1, in the former cae a circle, in the latter a rectangular hyperbola, is to be decribed on the diameter SA; as appears by prop. 33. Then about the centre S. with an interval equal to half the latus rectum, decribe the circle HkK, and at the place G of the acending or decending body, and at any other place C, erect the perpendiculars GI, CD; meeting the conic ection or circle in I and D. Then joining SI, SD, let the ectors HSK, HSk be made equal to the egments SEIS, SEDS, and by prop. 35. the body G will decribe the pace GC in the ame time in which the body K may decribe the arc Kk. Q. E. F.

Supposing that the centripetal force is proportional to the altitude or ditance of places from the centre, I ay, that the times and velocities of falling bodies, and the paces which they decribe, are repectively proportional to the arcs, and the right and vered ines of the arcs. Pl. 17. Fig. 1.

Suppoe the body to fall from any place A in the right line AS; and about the centre of force S with