ant lines. The sine and the
cosine I shall have occasion to employ in the latter part of my lecture.
Now you know what the versed sine of an angle is, and are able to
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 deg. and 24 deg. 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.

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