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°
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