Page:The Construction of the Wonderful Canon of Logarithms.djvu/51

CONSTRUCTION OF THE CANON. 27 sines, Then the difference of the logarithms of the sines d S and e S lies between the limits V T the greater and T c the less. For by hypothesis, e S is to d e as T S to T V, and d S is to d e as T S to T c; therefore, from the nature of proportionals, two conclusions follow:—

Firstly, that V S is to TS as T S to c S.

Secondly, that the ratio of T S to c S is the same as that of d S to e S. And therefore (by 36) the difference of the logarithms of the sines d S and e S is equal to the difference of the logarithms of the radius T S and the sine c S. But (by 34) this difference is the logarithm of the sine c S itself; and (by 28) this logarithm is included between the limits V T the greater and T c the less, because by the first conclusion above stated, V S greater than radius is to T S radius as T S is to c S. Whence, necessarily, the difference of the logarithms of the sines d S and e S lies between the limits V T the greater and T c the less, which was to be proved.

Since (by 39) the less sine is to the difference of the sines as radius to the greater limit of the difference of the logarithms; and the greater sine is to the difference of the sines as radius to the less limit of the difference of the logarithms; it follows, from the nature of proportionals, that radius being multiplied by the difference of the given sines and the product being divided by the less sine, the greater limit will be produced; and the product being divided by the greater sine, the less limit will be produced. Example