Page:Newton's Principia (1846).djvu/168

 a body describing a circle about the centre B, at the distance BC; in the subduplicate ratio of $$\frac{BQ^{2}\times AC\times SP}{AO\times BC}$$ to SY², that is (neglecting the ratios of equality of SP to BC, and BQ² to SY²), in the subduplicate ratio of AC to AO, or ½AB. Q.E.D.

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

. 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.


 * 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.

For (by Cor. 7, Prop. XVI) 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.


 * The same things supposed, I say, that the area of the figure DES, described by the indefinite radius SD, is equal to the area which a body with a radius equal to half the latus rectum of the figure DES, by uniformly revolving about the centre S, may describe in the same time.