Page:The Construction of the Wonderful Canon of Logarithms.djvu/42

18 CONSTRUCTION OF THE CANON. 3 S, 4 S, &c., are said to decrease geometrically, because in equal times they are diminished by unequal spaces similarly proportioned. Let the sine T S be represented in numbers by 10000000, 1 S by 9000000, 2 S by 8100000, 3 S by 7290000, 4 S by 6561000; then these numbers are said to decrease geometrically, being diminished in equal times by a like proportion.

Thus, referring to the preceding figure, I say that when the geometrically moving point G is at T, its velocity is as the distance T S, and when G is at 1 its velocity is as 1 S, and when at 2 its velocity is as 2 S, and so of the others. Hence, whatever be the proportion of the distances T S, 1 S, 2 S, 3 S, 4 S, &c., to each other, that of the velocities of G at the points T, 1, 2, 3, 4, &c., to one another, will be the same.

For we observe that a moving point is declared more or less swift, according as it is seen to be borne over a greater or less space in equal times. Hence the ratio of the spaces traversed is necessarily the same as that of the velocities. But the ratio of the spaces traversed in equal times, T 1, T 2, T 3, T 4, T 5, &c., is that of the distances T S, 1 S, 2 S, 3 S, 4 S, &c Hence it follows that the ratio to one ano, of the distances of G from S, namely T S, 1 S, 2 S, 3 S, 4 S, &c., is the same as that of the velocities of G at the points T, 1, 2, 3, 4, &c., respectively. 1 S,