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The Indivisibles of the Continuum

This chapter considers some significant developments in seventeenth-century mathematics which are part of the pre-history of the infinitesimal calculus. In particular, I examine the “method of indivisibles” proposed by Bonaventura Cavalieri and various developments of this method by Evangelista Torricelli, Gilles Personne de Roberval, and John Wallis. From the beginning, the method of indivisibles faced objections that aimed to show that it was either conceptually ill-founded (in supposing that the continuum could be composed of dimensionless points) or that its application would lead to error. I show that Cavalieri’s original formulation of the method attempted to sidestep the question of whether a continuous magnitude could be composed of indivisibles, while Torricelli proposed to avoid paradox by taking indivisibles to have both non-zero (yet infinitesimal) magnitude and internal structure. In contrast, Roberval and Wallis showed significantly less interest in addressing foundational issues and were content to maintain that the method could (at least in principle) be reduced to Archimedean exhaustion techniques.