Page:The Construction of the Wonderful Canon of Logarithms.djvu/44

20 CONSTRUCTION OF THE CANON.



Thus, the preceding figure being repeated, and S T being produced beyond T to 0, so that 0 S is to T S as T S to d S. I say that b c, the logarithm of the sine d S, is greater than T d and less than 0 T. For in the same time that g is borne from 0 to T, g is borne from T to d, because (by 24) 0 T is such a part of 0 S as T d is of T S, and in the same time (by the definition of a logarithm) is a borne from b to c; so that o T, T d, and b c are distances traversed in equal times. But since g when moving between T and 0 is swifter than at T, and between T and d slower, but at T is equally swift with a (by 26); it follows that 0 T the distance traversed by g moving swiftly is greater, and T d the distance traversed by g moving slowly is less, than b c the distance traversed by the point a with its medium motion, in just the same moments of time; the latter is, consequently, a certain mean between the two former.

Therefore 0 T is called the greater limit, and T d