FIRST.--I have called your attention to the fact that the direction in which the revolving body is deflected from the 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 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.