Algebraic Versus Geometric Thought and Expression in the Early Calculus

Abstract

The language of the early calculus was much more geometrical than the analytic and algebraic style that was pioneered by Euler and still dominates today. For instance, functions such as sin(x) and log(x) were largely absent from the early calculus, with geometric paraphrases used in their place. From a modern standpoint, one may be inclined to assume that the eventual triumph of the more analytic perspective was a straightforward case of progress, and that the geometric aspects of the early calculus were a historical artifact ultimately hampering this development. Interestingly, however, in private notes, the pioneers of the calculus showed a readiness to disregard traditionalism and operate freely in a more proto-modern style than they allowed themselves in their publications. This suggests that the adherence to the geometrical mode in published works was a deliberate choice selected with full awareness of the analytic alternative. Indeed, the geometrical paradigm was no mere blind conservatism or lip service to classical foundations; rather, it arguably had genuine merits, for example, as an intuition-boosting heuristic strategy. This aspect of the early calculus can serve as a case study that illuminates the relation between official expression and informal thought in mathematics more generally. For one thing, it complicates the common historiographic assumption that fidelity to historical thought is best achieved by following the original text’s mode of expression as closely as possible.