First.—­Their length is always less than the radius.  The radius is expressed by 1, or unity.  So, these lines being less than unity, their length is always expressed by decimals, which mean equal to such a proportion of the radius.

Second.—­The cosine and the versed sine are together equal to the radius, so that the versed sine is always 1, less the cosine.

Third.—­If I diminish the angle A O C, by moving the radius O C toward O A, the sine C E diminishes rapidly, and the versed sine E A also diminishes, but more slowly, while the cosine O E increases.  This you will see represented in the smaller angles shown in Fig. 2.  If, finally, I make O C to coincide with O A, the angle is obliterated, the sine and the versed sine have both disappeared, and the cosine has become the radius.

Fourth.—­If, on the contrary, I enlarge the angle A O C by moving the radius O C toward O B, then the sine and the versed sine both increase, and the cosine diminishes; and if, finally, I make O C coincide with O B, then the cosine has disappeared, the sine has become the radius O B, and the versed sine has become the radius O A, thus forming the two sides inclosing the right angle A O B. The study of this explanation will make you familiar with these important 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.