162 THE MATHEMATICAL PRINCIPLES [BOOK 1 a body describing a circle about the centre B, at the distance BC, in thr BQ2 X AC X SP subduplicate ratio of - -r-^ - ^ - to SY2, that is (neglecting the raX Jo tios of equality of SP to BC, and BQ,2 to SY2), in the subduplicate ratio of AC to AO, or iAB. 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 XXXIV. THEOREM X. 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 cen tre S, in any place P, is equal to the velocity of a body uniformly describing a circle about the c same centre S at half the interval SP. Let the breadth CP of the parabola be diminished in itifiiiitirni, so as the parabolic arc P/B may come to coincide with the right line CB, the centre S s with the vertex B, and the interval SP with the interval BC, and the proposition will be manifest. Q.E.D. PROPOSITION XXXV. THEOREM XL The same things supposed, I say, that the area of the figure DES, de scribed by the indefinite radius SD, is equal to the area which a body with a radius equal to h'df the latus rectum of the figure DES, by uniformly revolving about the centre S, may describe in the same tijiw. 1 JD/ AJ