Historically Speaking—: Identification of Napier's Inequalities

Abstract

The discovery of logarithms by John Napier (1550-1617) is a well known facet in the history of mathematics. His singular accomplishment in defining the logarith mic function of a real variable by providing a numerical description of it, over a wide range of its argument, at small intervals and to several (decimal) places, antedated by many yeara the development of funda mental concepts which the modern stu dent regards as necessary to achieve even the same limited goals. Napier success fully bridged, solely in regard to this function, these lacunae in the mathematical knowledge of bis day. It has long been of interest to identify the concepts which he intuitively invoked. This is not done, it should be clearly said, with any idea of assigning to him some kind of priority for them, but merely in the interests of a elearer appreciation of the ingenuity he displayed and the power of his methods. Two inequalities that he obtained are the key to his numerical resolution of the problem and his consequent table of logarithms. The analytical identification of these inequalities appears to have been overlooked. Before exhibiting this identi-fication we shall speak of the fundamental role that these inequalities played. In the interests of intelligibility we first recollect a few familiar facts regarding Napier's formulation of the problem.