Napier, Torporley, Menelaus, and Ptolemy: Delambre and De Morgan’s Observations on Seventeenth-Century Restructuring of Spherical Trigonometry

Abstract

An effort to reorganize and systematize planar and spherical trigonometry began in the 15th century with the work of Regiomontanus, extended throughout the 16th century with work by Otho, Rheticus, Pitiscus, and Fincke, and continued into the 17th century by Napier, Torporley, Viète, and others. During the 18th and 19th century, publications by Euler, Taylor, Fourier, and Gauss extended the role of trigonometric functions into new areas including power series, and complex functions of complex variables. An analysis of De Morgan’s criticism of Napier’s and Torporley’s efforts in this area sheds light on the challenges to an historian of mathematics of one era attempting to understand the thought process of mathematicians living in earlier times. In particular, we focus on two areas: the historian’s knowledge of future mathematical developments and modes of expression unknown to those living in the earlier period, and secondly an incomplete, inaccurate, or absent knowledge on the part of the historian of definitions, references, or conventions well known to those of the earlier era. These definitions, references, and conventions were often used without comment or explanation, and occasionally used without mention, since the writer could assume them to be common knowledge to the readers of his time, and that their use would be understood by his readers, even if that use was implicit.