330 THE MATHEMATICAL PRINCIPLES [BOOK I) be destroyed or generated, in the time that the globe describes four third parts of its diameter, as the density of the medium to the density of the ^lobe. Con. 2. The resistance of the globe, cceteris paribus, is in the duplicate ratio of the velocity. CUR. 3. The resistance of the globe, cocteris paribus, is in the duplicate ratio of the diameter. COR. 4. The resistance of the globe is, cceteris paribus, as the density of the medium. COR, 5. The resistance of the globe is in a ratio compounded of the du plicate ratio of the velocity, arid the duplicate ratio of the diameter, and the ratio of the density of the medium. COR. 6. The motion of the globe and its re sistance may be thus expounded Let AB be the time in which the globe may, by its resistance uniformly continued, lose its whole motion. Erect AD, BC perpendicular to AB. J ,et BC be that whole motion, and through the point C, the asymptotes being AD, AB, describe the hyperbola CF. Produce AB to any point E. Erect the perpendicular EF meeting the hyperbola in F. Complete the parallelogram CBEG, and draw AF meeting BC in H. Then if the globe in any time BE, with its first mo tion BC uniformly continued, describes in a non-resisting medium the space CBEG expounded by the area of the parallelogram, the same in a resisting medium will describe the space CBEF expounded by the area of the hvperbola; and its motion at the end of that time will be expounded by EF, the ordinate of the hyperbola, there being lost of its motion the part FG. And its resistance at the end of the same time will be expounded by the length BH, there being lost of its resistance the part CH. All these things appear by Cor. 1 and 3, Prop. V., Book II. COR. 7. Hence if the globe in the time T by the resistance R uniformly continued lose its whole motion M, the same globe in the time t in a resisting medium, wherein the resistance R decreases in a duplicate /M ratio of the velocity, will lose out of its motion M the part ,.i ' the TM part rn . ; remaining ; and will describe a space which is to the space de scribed in the same time t, with the uniform motion M, as the logarithm of T + t the number — ^.— multiplied by the number 2,302585092994 is to the number ^ because the hyperbolic area BCFE is to the rectangle BCGE in that proportion.