Note N. 76 — SECTION III.

Note 76, p. 20. Sines and cosines. Figure 4 is a circle; n p is the sine, and C p is the cosine of an arc m n. Suppose the radius C m to begin to revolve at m, in the direction m n a; then at the point m the sine is zero, and the cosine is equal to the radius C m. As the line C m revolves and takes the successive positions C n, C a, C b, &c., the sines n p, a q, b r, &c., of the arcs m n, m a, m h, &c., increase, while the corresponding cosines C p, C q, C r, &c., decrease; and when the revolving radius takes the position C d, at right angles to the diameter g m, the sine becomes equal to the radius C d, and the cosine is zero. After passing the point d, the contrary happens; for the sines e K, l V, &c., diminish, and the cosines C K, C V, &c., go on increasing, till at g the sine is zero, and the cosine is equal to the radius C g. The same alternation takes place through the remaining parts g h, h m, of the circle, so that a sine or cosine never can exceed the radius. As the rotation of the earth is invariable, each point of its surface passes through a complete circle, or 360 degrees, in twenty-four hours, at a rate of 15 degrees in an hour. Time, therefore, becomes a measure of angular motion, and vice versâ, the arcs of a circle a measure of time, since these two quantities vary simultaneously and equably; and, as the sines and cosines of the arcs are expressed in terms of the time, they vary with it. Therefore, however long the time may be, and how often soever the radius may revolve round the circle, the sines and cosines never can exceed the radius; and, as the radius is assumed to be equal to unity, their values oscillate between unity and zero.