Euler, infinitesimals and limits

This paper examines the Eulerian notion of infinitesimal or evanescent quantity and compares it with the modern notion of limit and non-standard analysis concepts. The Eulerian infinitesimal, when interpreted using the conceptual instruments available to modern mathematics, seems to be a fluid mixture of different elements, a continuous leap from a vague idea of limit to a confused notion of infinitesimal. However, while the modern notions of limit, hyperreal numbers etc. derive from precise definitions or constructions and are developed within non-equivocal contexts, Euler did not define the notion of evanescent quantity, either explicitly or implicitly: it is substantially a 'common notion', directly derived from the experience of physical and geometric quantities, which can diminish until they vanish. Infinitesimals are fictions, useful instruments for dealing with finite quantities, and thus Euler could state that the algorithm of the calculus did not concern differentials but functions. The vague and imprecise Eulerian notions can be reformulated in modern terms only by introducing extraneous elements into Euler's work and changing the logical and conceptual frame-work of his mathematics