SEC. II.] OF NATURAL PHILOSOPHY. 25CJ and the velocity at the beginning of the second time KL by the line K& and the length described in the hrst time by the area AKA*B, all the fol lowing velocities will be expounded by the following lines \J. Mm, .fee. and the lengths described, by the areas K/, I mi. &c. And, by compo sition, if the whole time be expounded by AM, the sum of its parts, the whole length described will be expounded by AM/ftB the sum of its parts. Now conceive the time AM to be divided into the parts AK, KL, LM, (fee so that CA, CK, CL, CM, (fee. may be in a geometrical progression ; and those parts will be in the same progression, and the velocities AB, K/r, L/, Mm, (fee., will be in the same progression inversely, and the spaces de scribed Ak, K/, Lw, (fee., will be equal. Q,.E.D. COR. 1. Hence it appears, that if the time be expounded by any part AD of the asymptote, and the velocity in the beginning of the time by the ordinate AB, the velocity at the end of the time will be expounded by the ordinate DG ; and the whole space described by the adjacent hyperbolic area ABGD ; and the space which any body can describe in the same time AD, with the first velocity AB, in a non-resisting medium, by the rectan gle AB X AD. COR 2. Hence the space described in a resisting medium is given, by taking it to the space described with the uniform velocity AB in a nonresisting medium, as the hyperbolic area ABGD to the rectangle AB X AD. COR. 3. The resistance of the medium is also given, by making it equal, in the very beginning of the motion, to an uniform centripetal force, which could generate, in a body falling through a non-resisting medium, the ve locity AB in the time AC. For if BT be drawn touching the hyperbola in B. and meeting the asymptote in T, the right line AT will be equal to AC, and will express the time in which the first resistance, uniformly con tinned, may take away the whole velocity AB COR. 4. And thence is also given the proportion of this resistance to the force of gravity, or ay other given centripetal force. COR. 5. And, vice versa, if there is given the proportion of the resist; nee to any given centripetal force, the time AC is also given, in which c centripetal force equal to the resistance may generate any velocity as AB ; and thence is given the point B. through which the hyperbola, having CH CD for its asymptotes, is to be described : as also the space ABGD, which a body, by beginning its motion with that velocity AB, can describe in any time AD. in a similar resisting medium. PROPOSITION VI. THEOREM lVrc Homogeneous and equal spherical bodies, opposed hy resistances that are in the duplicate ratio of the velocities, and moving on by their innate force only, will, in times which are reciprocally as the velocities at thr.