Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/235

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Proposition XXXVI. Problem XXV.

To determine the times of the deſcent of a body falling from a given place A. Pl. 16. Fig. 3.

Plate 16, Figure 3

Upon the diameter AS, the diſtance of the body from the centre at the beginning. deſcribe the ſemi-circle ADS, as likewiſe the ſemi-circle OKH equal thereto, about the centre S. From any place C of the body, erect the ordinate CD, join SD, and make the ſector OSI equal to the area ASD. It is evident by prop. 35. that the body in falling will deſcribe the ſpace AC in the ſame time in which another body, uniformly revolving about the centre S, may deſcribe the arc OK. Q. E. F.

Proposition XXXVII. Problem XXVI.

To deſcribe the times of the aſcent or deſcent of a body projected upwards or downwards from a given place. Pl. 16. Fig. 4.

Plate 16, Figure 4

Suppoſe the body to go off from the given place G, in the direction of the line GS, with any velocity. In the duplicate ratio of this velocity to the uniform velocity in a circle, with which the body may revolve about the centre