| In this study, the historical development of the foundations of the mathematics of curve length is described. In particular, the development of definitions for curve length, the investigation of the implications of such definitions, and the development of general procedures or formulas for determining or expressing the length of a curve are considered. This exposition focuses on certain key stages in the evolution of the mathematics of curve length. Among these important stages are the following: Archimedes' introduction of assumptions to justify the comparison of the lengths of certain curves and his estimation of (pi) by polygonal approximation; the development of the usual integral formula for curve length in the last half of the seventeenth century; Duhamel's (1841) statement of an explicit definition for curve length as the limit of the lengths of a sequence of inscribed polygons whose maximal side-lengths approach zero and the use of this definition to justify the Archimedean assumptions and the curve length integral formula; the comprehensive investigations by Scheefer (1884) and Jordan (1887) of the validity of Duhamel's definition and variations thereof and of conditions guaranteeing that a curve has finite length; and research into the extent of the applicability of the integral formula that ultimately relied on the use of Lebesgue integration. Twentieth-century investigations of curve length are also surveyed. Relationships between the concept of curve length and other mathematical topics (e.g., surface area and functions of bounded variation) are discussed as there are encountered throughout the course of this study. In concluding remarks, illustrations are given of how the contents of this study may be used to augment the teaching of curve length and of how the topic of curve length can be used as a recurring, unifying theme in post-secondary mathematics education. |
