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
is shown by the 15th place of decimals, the versed sine varies as the
square of the angle; or, in a given circle, the deflection, and so the
centrifugal force, of a revolving body varies as the square of the
speed.

The reason for the third law is equally apparent on inspection of Fig.
2. It is obvious, that in the case of bodies making the same number of
revolutions in different circles, the deflection must vary directly as
the diameter of the circle, because for any given angle the versed sine
varies directly as the radius. Thus radius O A' is twice radius O A, and
so the versed sine of the arc A' B' is twice the versed sine of the arc
A B. Here, while the angular velocity is the same, the actual velocity
is doubled by increase in the diameter of the circle, and so the
deflection is doubled. This exhibits the general law, that with a given
angular velocity the centrifugal force varies directly as the radius or
diameter of the circle.

We come now to the reason for the fourth law, that, with a given actual
velocity, the centrifugal force varies _inversely_ as the diameter of
the circle. If any of you ever revolved a weight at the end of a cord
with some velocity, and let the cord wind up, suppose around your hand,
without doing anything to accelerate the motion, then, while the circle
of revolution was growing smaller, the actual velocity continuing nearly
uniform, you have felt the continually increasing stres