The agony and the ecstasy – the development of logarithms by Henry Briggs

Abstract

Two recent papers published elsewhere set out the original development of logarithms by John Napier: in spreadsheets are used, and the computation of these logarithms is well within the grasp of a student looking for an unusual PC based project. It is an interesting exercise to continue the computer aided investigation of the subsequent development of tables of logarithms in the early 17th century by that other major participant, Henry Briggs (1560–1631): only now can previously unknown flaws in Briggs’ extensive numerical work be found, without the trauma of spending years of arithmetical drudgery. This work is also suitable for a student project using a PC maths package, making use of variable place arithmetic. The story that unfolds is remarkable: some parts, though well-documented, are possibly not so well-known as they deserve - e.g. D. T. Whiteside has shown that Briggs discovered the binomial series expansion for (1 + α)1/2 for small α, while devising a finite difference algorithm for extracting square roots. Florian Cajori pieced together the analytical aspects of the development of logarithms in the early years of the 20th century, but largely ignored the numerical aspects of the work, especially that of Briggs (and Mercator). The aspects of Briggs’ work investigated here is guided by his usually good intuition, while of course there are pitfalls into which he occasionally stumbles, as one might expect from someone engaging in numerical work ab initio: with the analytical methods of the calculus still being largely over the mathematical horizon, Briggs presents his methods without formal proof, but uses numerical examples instead.