The simplification of the arithmetical processes of involution and evolution

Abstract

It is a self-evident statement that addition and subtraction are the most simple arithmetical processes, so that any other process may be said to be completely simplified when it is replaced by either of these two. The invention of logarithms completely simplified multiplication and division, but only reduced involution and evolution to multiplication and division. In modern applied science there are many laws and empirical formulæ in which fractional indices occur, and the calculations sometimes become troublesome on this account. Although the obvious course is to perform the necessary multiplication or division of the logarithms, in such cases, by the addition or subtraction of their logarithms, no one seems to have considered it worth while to construct a Table giving the logarithms of the logarithms of numbers directly. The only step in this direction is the invention of the "log-log" slide-rule, which is very limited in its range, and the accuracy of its results mat be open to question. The prospect of abolishing even multiplication and division from all ordinary calculations, and so making addition and subtraction the only necessary arithmetical processes, was sufficient inducement to the author to construct such a Table. Before describing the difficulties that had to be overcome, a few words on the proposed nomenclature are necessary. In the first place, the word "logarithm" is unpronounceable and too long, which most people realise and avoid by calling it "log "; secondly, the inverse function has been very clumsily termed "anti-logarithm"; and, thirdly, the logarithm of the lagarithm has been given the cacophonous name " log-log." It, therefore, seems permissible to devise more convenient names, constructed on some system. It is proposed to replace logarithm by "log," and to call the inverse function the "illog." The name "log-log" would sound better if reduced to "lolog," and its inverse function could then be systematically named the "illolog."