The notion of continuity in Euler and its development in the works of Arbogast, Charles and Fourier

Abstract In this paper, we aim to examine key episodes in the historical development of the concept of continuity, tracing its development from Euler’s early analytical works to the beginning of the nineteenth century. Particular attention is given to Euler’s formulation of an analytical criterion of continuity and to a critical, yet often overlooked, aspect of his approach: his treatment of certain curves in the second volume of the Introductio in analysin infinitorum . Our analysis shows that Euler’s criterion encounters a fundamental difficulty when applied, a difficulty that, while seemingly recognized by Euler himself, could not be adequately resolved within the constraints of his theoretical framework. By considering the contributions of later mathematicians such as Arbogast, Charles, and Fourier, we show how Euler’s notion of continuity was progressively found inadequate in light of new mathematical developments. This historical trajectory not only highlights the limitations of Euler’s notion of continuity but also shows that the recognition of these limitations constituted a necessary step in laying the conceptual groundwork for the emergence of the modern, analytically grounded definition of continuity.