We can discuss very small angles.  We talk familiarly about the angle which is subtended by 1” of arc.  On Fig. 2, a short line is drawn near to the radius O A’.  The distance between O A’ and this short line is 1 deg. of the arc A’ B’.  If we divide this distance by 3,600, we get 1” of arc.  The upper line of the Table of versed sines given below is the versed sine of 1” of arc.  It takes 1,296,000 of these angles to fill a circular space.  These are a great many angles, but they do not make a circle.  They make a polygon.  If the radius 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