observe in Fig. 1 that the versed sine A E, of the angle A O C,
represents in a general way the distance that the body A will be
deflected from the tangent A D toward the center O while describing the
arc A C.

The same law of deflection is shown, in smaller angles, in Fig. 2. In
this figure, also, you observe in each of the angles A O B and A O C
that the deflection, from the tangential direction toward the center, of
a body moving in the arc A C is represented by the versed sine of the
angle. The tangent to the arc at A, from which this deflection is
measured, is omitted in this figure to avoid confusion. It is shown
sufficiently in Fig. 1. The angles in Fig. 2 are still pretty large
angles, being 12° and 24° respectively. These large angles are used for
convenience of illustration; but it should be explained that this law
does not really hold in them, as is evident, because the arc is longer
than the tangent to which it would be connected by a line parallel with
the versed sine. The law is absolutely true only when the tangent and
arc coincide, and approximately so for exceedingly small angles.

[Illustration: Fig. 2]

In reality, however, we have only to do with the case in which the arc
and the tangent do coincide, and in which the law that the deflection is
_equal to_ the versed sine of the angle is absolutely true. Here, in
observing this most familiar thing, we are, at a single step, taken to
that which is utterly beyond our comprehension. The angles we have to
consider disappear, not only from our sight, but even from our
conception. As in every other case when we push a physical investigation
to its limit, so here also, we find our power of thought transcended,
and ourselves in the presence of the infinite.