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 stress, and have
observed the increasing angular velocity, the two obviously increasing
in the same ratio. That is the operation or action which the fourth law
of centrifugal force expresses. An examination of this same figure (Fig.
2) will show you at once the reason for it in the increasing deflection
which the body suffers, as its circle of revolution is contracted. If we
take the velocity A' B', double the velocity A B, and transfer it to the