We give a new account of Björling’s contribution to uniform convergence in connection with Cauchy’s theorem on the continuity of an infinite series. Moreover, we give a complete translation from Swedish into English of Björling’s 1846 proof of the theorem. Our intention is also to discuss Björling’s convergence conditions in view of Grattan-Guinness’ distinction between history and heritage. In connection to Björling’s convergence theory we discuss the interpretation of Cauchy’s infinitesimals.

In this paper we discuss visualizations in mathematics from a historical and didactical perspective. We consider historical debates from the 17th and 19th centuries regarding the role of intuition and visualizations in mathematics. We also consider the problem of what a visualization in mathematical learning can achieve. In an empirical study we investigate what mathematical conclusions university students made on the basis of a visualization. We emphasize that a visualization in mathematics should always be considered in its proper context.

In this paper we consider the major development of mathematical analysis during the mid-nineteenth century. On the basis of Jahnke’s (Hist Math 20(3):265–284, 1993 ) distinction between considering mathematics as an empirical science based on time and space and considering mathematics as a purely conceptual science we discuss the Swedish nineteenth century mathematician E.G. Björling’s general view of real- and complexvalued functions. We argue that Björling had a tendency to sometimes consider mathematical objects in a naturalistic way. One example is (...) how Björling interprets Cauchy’s definition of the logarithm function with respect to complex variables, which is investigated in the paper. Furthermore, in view of an article written by Björling (Kongl Vetens Akad Förh Stockholm 166–228, 1852 ) we consider Cauchy’s theorem on power series expansions of complex valued functions. We investigate Björling’s, Cauchy’s and the Belgian mathematician Lamarle’s different conditions for expanding a complex function of a complex variable in a power series. We argue that one reason why Cauchy’s theorem was controversial could be the ambiguities of fundamental concepts in analysis that existed during the mid-nineteenth century. This problem is demonstrated with examples from Björling, Cauchy and Lamarle. (shrink)