Throwing Some Light on the Vast Darkness that is Analysis: Niels Henrik Abel's Critical Revision and the Concept of Absolute Convergence

During the first half of the nineteenth century, mathematical analysis underwent a transition from a predominantly formula-centred practice to a more concept-centred one. Central to this development was the reorientation of analysis originating in A ugustin- L ouis C auchy's (1789–1857) treatment of infinite series in his Cours d'analyse . In this work, C auchy set out to rigorize analysis, thereby critically examining and reproving central analytical results. One of C auchy's first and most ardent followers was the Norwegian N iels H enrik A bel (1802–1829) who vowed to shed some light on the vast darkness in analysis.
This paper investigates some important aspects of A bel's contribution to the reorientation in analysis. In particular, it stresses the role for critical revision in the process of rigorization. By critically examining past practice, the new practice sought to explain the relative success of the previous—now outdated—approach. This is illustrated by discussing a number of issues related to A bel's new proof of the binomial theorem (1826) including his reactions to the exception that he encountered to one of the central theorems of C auchy's theory.
Following this discussion, the formation of new concepts as the result of critical revisions is illustrated by analysing the early history of the concept of absolute convergence. Thereby, it is shown how a new concept was distilled, investigated, put to use and eventually superseded.