Page:The Construction of the Wonderful Canon of Logarithms.djvu/66

42 CONSTRUCTION OF THE CANON. 45 degrees, to e i, now radius. Consequently (by 37) double the logarithm of the sine of 45 degrees is equal to the logarithms of the extremes, namely radius and its half. But the sum of the logarithms of both these is the logarithm of half radius only, because (by 27) the logarithm of radius is nothing. Necessarily, therefore, the double of the logarithm of an arc of 45 degrees is the logarithm of half radius.

Since (by 55) half radius is to the sine of half the given arc as the sine of the complement of that half arc is to the sine of the whole arc, therefore (by 38) the sum of the logarithms of the two extremes, namely half radius and the whole arc, will be equal to the sum of the logarithms of the means, namely the half arc and the complement of the half arc. Whence, also (by 38), if you add the logarithm of half radius, found by 51 or 56, to the given logarithm of the whole arc, and subtract the given logarithm of the complement of the half arc, there will remain the required logarithm of the half arc.

ET there be given the logarithm of half radius (by 51) 6931469; also the arc 69 degrees 20 minutes, and its logarithm 665143. The half are is 34 degrees 40 minutes, whose logarithm