of the circumscribed circle of this
polygon is 1,296,000 feet, which is nearly 213 geographical miles, each
one of its sides will be a straight line, 6.283 feet long. On the
surface of the earth, at the equator, each side of this polygon would be
one-sixtieth of a geographical mile, or 101.46 feet. On the orbit of the
moon, at its mean distance from the earth, each of these straight sides
would be about 6,000 feet long.

The best we are able to do is to conceive of a polygon having an
infinite number of sides, and so an infinite number of angles, the
versed sines of which are infinitely small, and having, also, an
infinite number of tangential directions, in which the body can
successively move. Still, we have not reached the circle. We never can
reach the circle. When you swing a sling around your head, and feel the
uniform stress exerted on your hand through the cord, you are made aware
of an action which is entirely beyond the grasp of our minds and the
reach of our analysis.

So always in practical operation that law is absolutely true which we
observe to be approximated to more and more nearly as we consider
smaller and smaller angles, that the versed sine of the angle is the
measure of its deflection from the straight line of motion, or the
measure of its fall toward the center, which takes place at every point
in the motion of a revolving body.

Then, assuming the absolute truth of this law of deflection, we find
ourselves able to explain all the phenomena of centrifugal force, and to
compute its amount correctly in all cases.

We have now advanced two steps. We have learned _the direction_ and _the
measure_ of the deflection, which a revolving body continually suffers,
and its resistance to which is termed centrifugal force. The direction
is toward the center, and the measure is the versed sine of the angle.

SECOND.--We next come to consider what are known as the laws of
centrifugal force. These laws are four in number. They are, that the
amount of centrifugal force exerted by a revolving body varies in four
ways.

_First_.--Directly as the weight of the body.

_Second_.--In a given circle of revolution, as the square of the speed
or of the number of revolutions per minute; which two expressions in
this case mean the same thing.

_Third_.--With a given number of revolutions per minute, or a given
angular velocity[1] _directly_ as the radius of the circle; and

_Fourth_.--With a given actual velocity,