John Napier's Logarithm and the Bernoulli Numbers

I draw attention to a formal connection between the generating function of the Bernoulli numbers and an algebraic formulation of a technique used by Henry Briggs in his golden rule, which itself flows directly out of John Napier’s arithmetic. That leads us to consider Napier’s use of an average speed to estimate time of travel over an interval and the fact that the accuracy can be improved by root extractions. Those extractions multiplied by k converge to a logarithm that can be different than Napier’s — the paramount example is the natural logarithm. There we shall find the Bernoulli numbers. They are festooned in so many interesting places: in Bernoulli’s formula for summation of powers of the integers, in Euler’s zeta-function entwined with at the positive even integers, and there again at the trivial zeroes of Riemann’s zetafunction.  In each of these cases, they actually make an appearance.  In the case presented here they are hidden in algebra, but they can be exposed by means of the generating function for the Bernoulli numbers. And so we come to a recursion formula for computing natural logarithms and the exponential function. Computer codes are included for online calculations to support the text.