Page:Newton's Principia (1846).djvu/352

$\scriptstyle {\frac {N-1}{N+1}}$H, and the height described will be $\scriptstyle {\frac {2PF}{G}}$ - 1,3862943611F + 4,605170186LF. If the fluid be of a sufficient depth, we may neglect the term 4,605170186LF; and $\scriptstyle {\frac {2PF}{G}}$ - 1,3862943611F will be the altitude described, nearly. These things appear by Prop. IX, Book II, and its Corollaries, and are true upon this supposition, that the globe meets with no other resistance but that which arises from the inactivity of matter. Now if it really meet with any resistance of another kind, the descent will be slower, and from the quantity of that retardation will be known the quantity of this new resistance.

That the velocity and descent of a body falling in a fluid might more easily be known, I have composed the following table; the first column of which denotes the times of descent; the second shews the velocities acquired in falling, the greatest velocity being 100000000: the third exhibits the spaces described by falling in those times, 2F being the space which the body describes in the time G with the greatest velocity; and the fourth gives the spaces described with the greatest velocity in the same times. The numbers in the fourth column are $\scriptstyle {\frac {2P}{G}}$, and by subducting the number 1,3862944 - 4,6051702L, are found the numbers in the third column; and these numbers must be multiplied by the space F to obtain the spaces described in falling. A fifth column is added to all these, containing the spaces described in the same times by a body falling in vacuo with the force of B its comparative weight.

The Times P. || Velocities of the body falling in the fluid. || The spaces described in falling in the fluid. || The spaces described with the greatest motion. || The spaces described by falling in vacuo.
0,001G 0,01G 0,1G 0,2G 0,3G 0,4G 0,5G 0,6G 0,7G 0,8G 0,9G 1G 2G 3G 4G 5G 6G 7G 8G 9G 10G	99999$\scriptstyle {\frac {29}{30}}$ 999967 9966799 19737532 29131261 37994896 46211716 53704957 60436778 66403677 71629787 76159416 96402758 99505475 99932930 99990920 99998771 99999834 99999980 99999997 99999999$\scriptstyle {\frac {3}{5}}$	0,000001F 0,0001F 0,0099834F 0,0397361F 0,0886815F 0,1559070F 0,2402290F 0,3402706F 0,4545405F 0,5815071F 0,7196609F 0,8675617F 2,6500055F 4,6186570F 6,6143765F 8,6137964F 10,6137179F 12,6137073F 14,6137059F 16,6137057F 18,6137056F	0,002F 0,02F 0,2F 0,4F 0,6F 0,8F 1,0F 1,2F 1,4F 1,6F 1,8F 2F 4F 6F 8F 10F 12F 14F 16F 18F 20F	0,000001F 0,0001F 0,01F 0,04F 0,09F 0,16F 0,25F 0,36F 0,49F 0,64F 0,81F 1F 4F 9F 16F 25F 36F 49F 64F 81F 100F