SEC. VII.] OF NATURAL PHILOSOPHY. 165 the space GO in the same time in which the body K may describe t*he arc Kk. Q.E.F. PROPOSITION XXXVIII. THEOREM XII. Supposing that the centripetal force is proportional to the altitude or distance of places from the centre, I say, that the times and velocities of falling bodies, and the spaces which they describe, are respectively proportional to the arcs, and the right and versed sines of the arcs. Suppose the body to fall from any place A in the A. right line AS ; and about the centre of force S, with the interval AS, describe the quadrant of a circle AE ; and let CD be the right sine of any arc AD ; and the body A will in the time AD in falling describe the space AC, and in the place C will acquire the ve locity CD. This is demonstrated the same way from Prop. X, as Prop. XXX11 was demonstrated from Prop. XI. COR. 1. Hence the times are equal in which one body falling from the place A arrives at the centre S, and another body revolving describes the quadrantal arc ADE. COR. 2. Wherefore all the times are equal in which bodies falling from whatsoever places arrive at the centre. For all the periodic times of re volving bodies are equal (by Cor. 3; Prop. IV). PROPOSITION XXXIX. PROBLEM XXVIT. Supposing a centripetal force of any kind, and granting the quadratnres of curvilinear figures ; it is required to find the velocity of a bod)/, ascending or descending in a right line, in the several places through which it passes ; as also the time in which it will arrive at any place : and vice versa. Suppose the body E to fall from any place A in the right line AD EC ; and from its place E imagine a perpendicular EG always erected proportional to the centripetal force in that place tending to the centre C ; and let BFG be a curve line, the locus of the point G. And D in the beginning of the motion suppose EG to coincide with the perpendicular AB ; and the velocity of the body in any place E will be as a right line whose square is equal to the cur vilinear area ABGE. Q.E.I. In EG take EM reciprocally proportional to E