A Formal Connection between 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. Napier used an average speed to estimate time of travel over an interval, and the accuracy of that 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, a good reason to call it natural. The Bernoulli numbers are festooned in so many interesting places: Bernoulli’s  formula for summation of powers of the integers, in Euler’s zeta-function entwined with the positive even integers, and at the trivial zeroes of Riemann’s zeta-function. In each of these cases, they actually make an appearance. In the case presented here, they are hidden in a formality. (Computer codes are included for online calculations in support of the text.)