Bernoulli on Arc Length

Abstract

The academic life of the Bernoulli family was always surrounded by controversy. The disputes between Johann (John) and his older brother and former teacher Jacob and with his son Daniel are famous and well documented. An interesting discussion of this remarkable family is found in Section 12.6 of [3]. After the death of L’Hopital, John claimed the authorship of his classical analysis book. In the controversy between Leibniz and Newton about the creation of calculus, he stood on Leibniz’ side. His controversial positions were not restricted to mathematics: he was even accused of denying the possibility of the resurrection of Christ. In the course of our study of the history of elliptic integrals, we found a paper by Johann Bernoulli [1] which, in our opinion, both illuminates the calculation of arc lengths of smooth curves, a topic covered in most undergraduate calculus programs around the world, and provides an additional tool for producing new and interesting examples of rectifiable curves. According to Bernoulli, these are curves whose arc length can be expressed as elementary functions of their end points. The paper contains a main theorem that is perfectly valid even today, and admits a nice interpretation in terms of the notion of radius of curvature. Furthermore, we discovered in it a colorful antecedent of Landen integral transformations [2]. Let y = y(x) be a differentiable function defined on [a, b]. Then its arc length is defined by