Page:The Construction of the Wonderful Canon of Logarithms.djvu/43

CONSTRUCTION OF THE CANON. 19

26.The logarithm of a given sine ts that number which has increased avithmetically with the same velocity throughout as that with which radius began to decrease geometrically, and in the same time as radius has decreased to the given sine.

Let the line T S be radius, and d S a given sine in the same line; let g move geometrically from T to d in certain determinate moments of time. Again, let bi be another line, infinite towards i, along which, from b, let a move arithmetically with the same velocity as g had at first when at T; and from the fixed point b in the direction of i let a advance in just the same moments of time up to the point c. The number measuring the line b c is called the logarithm of the given sine d S.

27.Whence nothing is the logarithm of radius.

For, referring to the figure, when g is at T making its distance from S radius, the arithmetical point d beginning at b has never proceeded thence. Whence by the definition of distance nothing will be the logarithm of radius.

28. Whence

1 S, 2 S, 3 S, 4 S, &c, for when quantities are continued proportionally, their differences are also continued in the same proportion. Now the distances are by hypothesis a proportionally, and the spaces traversed are their differences, wherefore it is proved that the spaces traversed are continued in the same ratio as the distances.