How Eighteenth-Century Mathematics Was Transformed into Nineteenth-Century School Curricula

Abstract

Felix Klein’s phrase of a “double discontinuity” between mathematics at schools and universities had an implicit connotation, which is forgotten today, but should be known to understand him adequately. As Klein said, the traditional school mathematics of his time was determined by Euler’s views, whereas he intended to introduce into schools the ideas of contemporary modern mathematics. In order to explain Klein’s argument, the chapter discusses the mathematical curriculum of German gymnasia during the nineteenth century. The roots of contents and underlying vision in Eulerian mathematics, especially in Euler’s Introductio in analysin infinitorum, are shown as well as the transformations of Euler’s concepts during the century. Klein’s reform movement amounted to a real paradigm shift between two different visions. There were numerous ways of compromise between the two paradigms. This is exemplified by studying the teaching at a concrete gymnasium, the Ratsgymnasium in Bielefeld.