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

 SP to BC, and $$\scriptstyle BQ^2$$ to $$\scriptstyle SY^2$$) in the ubduplicate ratio of AC to AO or $$\scriptstyle \frac 12AB$$. Q. E. D.

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

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

If the figure BED is a parabola, I ay that the velocity of a falling body in any place C is equal to the velocity by which a body may uniformly decribe 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 decribing a parabola RPB about the centre S, in any place P, is equal to the velocity of a body uniformly decribing a circle about the same centre S at half the interval SP. Let the breadth CP of the parabola be diminihed 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 manifet. Q. E. D.