Indivisibles, Infinitesimals and a Tale of Seventeenth-Century Mathematics

Abstract

SummaryIn this article, we describe clever arguments by Torricelli and Roberval that employ indivisibles to find the volume of Gabriel's trumpet and the area under the cycloid. We detail 17th-century objections to these non-rigorous but highly intuitive techniques, as well as the controversy surrounding indivisibles. After reviewing the fundamentals of infinitesimal calculus and its rigorous footing provided by Robinson in the 1960s, we are able to revisit the 17th-century solutions. In changing from indivisible to infinitesimal-based arguments, we salvage the beautiful intuition found in these works.