This paper examines the content, and subsequent reception, of Riemann's early, and little-noted, memoir Versuch einer allgemeinen Auffassung der Integration und Differentiation (An attempt at a general conception of integration and differentiation), in which he propounded his concept of a fractional calculus. Taking his cue from Lagrange's contribution to the treatement of the calculus, and bringing in divergent series, he adopted a stance directly at odds with Cauchy's. A century further on, Hardy showed that, in some respects, Riemann's arguments may be reinterpreted on the lines of Borel's theory of divergent series. It is still an open question, however, whether this calculus is altogether amenable to such a reading – most notably regarding the matter of complementary functions. Appended to this reappraisal is the first-ever translation into French of Riemann's paper.