The calculation of extensive tables of logarithms could not fail to give rise to the method of interpolation; and such indeed was its origin. After Napier had made his memorable discovery, and he and Briggs had recognized the advantages that would result from the adoption of 10 as the base of a system of logarithms, it was Briggs who courageously undertook the immense labour required for calculating those famous tables, published in London in 1624, which have not been surpassed in extent and accuracy by any subsequent work. It is well known, however, that Briggs left a great hiatus in those tables, and that the logarithms of numbers between 20,000 and 90,000 are not to be found in them; but scarcely were the first tables printed, when Briggs undertook with fresh energy the computation of the Logarithms of the Trigonometrical Lines, and he was on the point of terminating this vast enterprise when arrested by death. It was his friend Gellibrand who completed the work, and gave it to the world in 1633.

Having thus found the required formula for the value of , viz.:—

we are now in a position to processed with the transformation of the expression found in the Number of this Journal