SEC. IILJ OF NATURAL PHILOSOPHY. 279 SECTION III. Of the motions of bodies which are resisted partly in the ratio of the ve locities, and partly in the duplicate of the same ratio. PROPOSITION XI. THEOREM VIII. If a body be resisted partly in the ratio and partly in the duplicate ratio of its velocity, and moves in a similar medium by its innate force only; and the times be taken in arithmetical progression; then quantities reciprocally proportional to the velocities, increased by a cer tain given quantity, will be in geometrical progression. With the centre C, and the rectangular asymptotes ^ OADd and CH, describe an hyperbola BEe, and let