fl.l OF NATURAL PHILOSOPHY. PROPOSITION XXXIII. THEOREM IX. The tilings above found being supposed. I say, thai the, velocity of a Jai ling body in any place C is to the velocity of a body, describing a circle about the centre B at the distance BC, in, the subduplicate ratio of AC, the distance of the body from the remoter vertex A of the circle or rectangular hyperbola, to iAB, the principal semi-diameter of the Let AB, the common dia meter of both figures RPB, DEB, be bisected in O; and draw the right line PT that may touch the figure RPB in P, and likewise cut that common diameter AB (pro duced, if need be) in T; and let SY be perpendicular to this line, and BQ to this di ameter, and suppose the latus rectum of the figure RPB to be L. Prom Cor. 9, Prop. XVI, it is manifest that the velocity of a body, moving in the line RPB about the centre S, in any place P, is to the velocity of a body describing a circle about the same centre, at the distance SP, in the subduplicate ratio of the rectangle £L X SP to SY2 Por by the properties of the conic sections ACB is to CP2 as 2AO to L. 2CP5 X AO and therefore — — rrrr; is equal to L. Therefore those, velocities an o-ACB to each other in the subduplicate ratio of CP3 X AO X SP ACB toSY~. More over, by the properties of the conic sections, CO is to BO as BO to TV.? and (by composition or division) as CB to BT. Whence (by division cs composition) BO — or + CO will be to BO as CT to BT, that is, AC CP2 X AO X SP ACB" is equal to will be to AO as CP to BQ; and therefore ~AO X BC * ^°W suPPose GV, tne breadth of the figure RPB, to be diminished in infinitum, so as the point P may come to coincide with the point C, and the point S with the point B. and the line SP with the line BC, and the line SY with the line BQ; and the velocity of the body now descending perpendicularly in the line CB will be to the velocity of 11