fundamental in trigonometry. C E is the sine, O E is the cosine, and E A
is the versed sine of the angle A O C. Respecting these three lines
there are many things to be observed. I will call your attention to the
following only:

_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