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 Augustin-Louis Cauchy's (1789–1857) treatment of infinite series in his Cours d’analyse. In this work, Cauchy set out to rigorize analysis, thereby critically examining and reproving central analytical results. One of Cauchy's first and most ardent followers was the Norwegian Niels Henrik Abel (1802–1829) who vowed to shed (...) some light on the vast darkness in analysis.This paper investigates some important aspects of Abel'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 By critically examining past practice, the new practice sought to explain the relative success of the previous—now outdated—approach.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 Abel's new proof of the binomial theorem (1826) including his reactions to the exception that he encountered to one of the central theorems of Cauchy'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 Thereby, it is shown how a new concept was distilled, investigated, put to use and eventually superseded. (shrink)

Euler developed a program which aimed to transform analysis into an autonomous discipline and reorganize the whole of mathematics around it. The implementation of this program presented many difficulties, and the result was not entirely satisfactory. Many of these difficulties concerned the integral calculus. In this paper, we deal with some topics relevant to understand Euler’s conception of analysis and how he developed and implemented his program. In particular, we examine Euler’s contribution to the construction of differential equations and his (...) notion of indefinite integrals and general integrals. We also deal with two remarkable difficulties of Euler’s program. The first concerns singular integrals, which were considered as paradoxical by Euler since they seemed to violate the generality of certain results. The second regards the explicitly use of the geometric representation and meaning of definite integrals, which was gone against his program. We clarify the nature of these difficulties and show that Euler never thought that they undermined his conception of mathematics and that a different foundation was necessary for analysis. (shrink)