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 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 globe.

. 2. The resistance of the globe, cæteris paribus, is in the duplicate ratio of the velocity.

. 3. The resistance of the globe, cæteris paribus, is in the duplicate ratio of the diameter.

. 4. The resistance of the globe is, cæteris paribus, as the density of the medium.

. 5. The resistance of the globe is in a ratio compounded of the duplicate ratio of the velocity, and the duplicate ratio of the diameter, and the ratio of the density of the medium.

. 6. The motion of the globe and its resistance 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. Let 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 motion 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 hyperbola; 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.

. 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 ratio of the velocity, will lose out of its motion M the part $$\scriptstyle \frac{tM}{T+t}$$, the part $$\scriptstyle \frac{TM}{T+t}$$ remaining; and will describe a space which is to the space described in the same time t, with the uniform motion M, as the logarithm of the number $$\scriptstyle \frac{T+t}{T}$$ multiplied by the number 2,302585092994 is to the number $$\scriptstyle \frac{t}{T}$$, because the hyperbolic area BCFE is to the rectangle BCGE in that proportion.