Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/230

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SP to BC, and to ) in the ſubduplicate ratio of AC to AO or . Q. E. D.

Cor. 1. When the points B and S come to coincide, TC will become to TS, as AC to AO.

Cor. 2. A body revolving in any circle at a given diſtance from the centre, by its motion converted upwards will ascend to double its diſtance from the centre.


Proposition XXXUV. Theorem X.

Plate 15, Figure 5
Plate 15, Figure 5

If the figure BED is a parabola, I ſay that the velocity of a falling body in any place C is equal to the velocity by which a body may uniformly deſcribe a circle about the centre B at half the interval BC. Pl. 15. fig. 5.

For (by cor. 7. prop. 16.) the velocity of a body deſcribing a parabola RPB about the centre S, in any place P, is equal to the velocity of a body uniformly deſcribing a circle about the same centre S at half the interval SP. Let the breadth CP of the parabola be diminiſhed in infinitum, so as the parabolic arc PfB may come to coincide with the right line CB, the centre S with the vertex B, and the interval SP with the interval BC, and the proposition will be manifeſt. Q. E. D.