Author(s):

Amy Shell-Gellasch (Beloit College)

The works of Napier and Briggs contain many tables of logarithms to various bases to facilitate the calculations. Napier’s original 1614 work was, like most scientific works of the time, written in Latin. Englishman Edward Wright translated Napier’s text into English. This version of the *Descriptio* was printed in two editions, 1616 and 1618. The second edition contains an appendix of tables of logarithms, one entry of which assigns 2.302585 to 10. In modern notation this gives us approximately log_{2.718} 10 = 2.302585 [11]. In general, this appendix contains logs of the form (in modern notation) 10^{6} log_{e}x. Thus this is a table of what we would now call natural logarithms. Thus, buried in the work of Napier is the first implicit use of *e* as a constant. (Keep in mind that when I refer to *e*, it was not given that designation until Euler over a century later.)* *However, it is commonly believed that this appendix was not written by Napier, but was compiled by William Oughtred [11].

William Oughtred (1574-1660) is known for giving us such notation as X for multiplication and :: for proportion [10]. In relation to his work on logarithms, Oughtred capitalized on Napier’s definition of the log in 1622 to create a simple device for performing arithmetical operations. By placing two rulers in logarithmic scale next to each other, he found he could perform multiplications and other operations by sliding these two rulers along one another, creating the first slide rule [6, 9]. Mitchell and Strain [11] conjecture that this contribution to the development of natural logarithms and *e* is largely forgotten due to the fact that Briggs’s revision of Napier’s work was much more efficient, thus leaving Napier’s earlier works and, along with them, the appendix of Oughtred, largely unused.

Amy Shell-Gellasch (Beloit College), "Napier's *e* - Oughtred," *Convergence* (April 2010), DOI:10.4169/loci003209