circle of which A B C is an arc, and with a velocity which will carry it
from A to B in one second of time. Then in this time the body is
deflected from the tangential direction a distance equal to A D, the
versed sine of the angle A O B. Now let us suppose the velocity of this
body to be doubled in the same circle. In one second of time it moves
from A to C, and is deflected from the tangential direction of motion a
distance equal to A E, the versed sine of the angle, A O C. But A E is
four times A D. Here we see in a given circle of revolution the
deflection varying as the square of the speed. The slight error already
pointed out in these large angles is disregarded.

The following table will show, by comparison of the versed sines of very
small angles, the deflection in a given circle varying as the square of
the speed, when we penetrate to them, so nearly that the error is not
disclosed at the fifteenth place of decimals.

The versed sine of 1" is 0.000,000,000,011,752
" " " " 2" is 0.000,000,000,047,008
" " " " 3" is 0.000,000,000,105,768
" " " " 4" is 0.000,000,000,188,032
" " " " 5" is 0.000,000,000,293,805
" " " " 6" is 0.000,000,000,423,072
" " " " 7" is 0.000,000,000,575,848
" " " " 8" is 0.000,000,000,752,128
" " " " 9" is 0.000,000,000,951,912
" " " " 10" is 0.000,000,001,175,222
" " " " 100" is 0.000,000,117,522,250

You observe the deflection for 10" of arc is 100 times as great, and for
100" of arc is 10,000 times as great as it is for 1" of arc. So far as