Page:The Construction of the Wonderful Canon of Logarithms.djvu/62

38 CONSTRUCTION OF THE CANON.

For since the ratio of every sine to its half is the same as that of radius to 5000000, therefore (by 36) the difference of the logarithms of any sine and of its half is the same as the difference of the logarithms of radius and of its half 5000000. But (by 34) the difference of the logarithms of radius and of the sine 5000000 is the same as the logarithm itself of the sine 5000000, and this logarithm (by 50) will be 6931469.22. Therefore, also, 6931469.22 will be the difference of all logarithms whose sines are in the proportion of two to one. Consequently the double of it, namely 13862938.44, will be the difference of all logarithms whose sines are in the ratio of four to one; and the triple of it, namely 20794407.66, will be the difference of all logarithms whose sines are in the ratio of eight to one.

For (by 50) the sine 8000000 will have. for its logarithm 2231434.68; and (by 51) the difference between the logarithms of the sine 8000000 and of its eighth part 1000000, will be 20794407.66; whence by addition will be produced 23025842.34 for the logarithm of the sine 1000000. And since radius is ten times this, all sines in the ratio of ten to one will have the same difference, 23025842.34, between their logarithms, for the reason and cause already stated (in 51) in reference to the proportion of two to one. And consequently the double of this logarithm, namely 46051684.68, will, as regards the difference of the logarithms, correspond to