A new look at E.G.!Bjorling and the Cauchy sum theorem

We give a new account of Bjorling'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 Bjorling's 1846 proof of the theorem. Our intention is also to discuss Bjorling's convergence conditions in view of Grattan-Guinness' distinction between history and heritage. In connection to Bjorling's convergence theory we discuss the interpretation of Cauchy's infinitesimals.