e tangential line of motion
is toward the center, on the radial line, which forms a right angle with
the tangent on which the body is moving. The first question that
presents itself is this: What is the measure or amount of this
deflection? The answer is, this measure or amount is the versed sine of
the angle through which the body moves.

Now, I suspect that some of you--some of those whom I am directly
addressing--may not know what the versed sine of an angle is; so I must
tell you. We will refer again to Fig. 1. In this figure, O A is one
radius of the circle in which the body A is revolving. O C is another
radius of this circle. These two radii include between them the angle A
O C. This angle is subtended by the arc A C. If from the point O we let
fall the line C E perpendicular to the radius O A, this line will divide
the radius O A into two parts, O E and E A. Now we have the three
interior lines, or the three lines within the circle, which are
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 import