One striking feature of nineteenth century mathematics, as contrasted with that of previous eras, is the higher degree of rigor and precision demanded by its practitioners. This tendency was...

Why be so Critical? Nineteenth Century Mathematics and the Origins of Analysis  Janet Heine Barnett December 22, 2016 One striking feature of nineteenth century mathematics, as contrasted with that of previous eras, is the higher degree of rigor and precision demanded by its practitioners. This tendency was especially noticeable in analysis, a eld of mathematics that essentially began with the \invention" of calculus by Leibniz and Newton in the mid-17th century. Unlike the calculus studied in an undergraduate course today, however, the calculus of Newton, Leibniz and their immediate followers focusedentirelyonthestudyofgeometric curves,usingalgebra(or`analysis')asanaidintheirwork. This situation changed dramatically in the 18th century when the focus of calculus shifted instead to the study of functions, a change due largely to in uence of the Swiss mathematician and physicist Leonhard Euler (1707{1783). In the hands of Euler and his contemporaries, functions became a powerful problem solving and modelling tool in physics, astronomy, related mathematical elds such as di?erential equations and the calculus of variations. Why then, after nearly 200 years of success in the development and application of calculus techniques, did 19th-century mathematicians feel the need to bring a more critical perspective to the study of calculus? This project explores this question through selected excerpts from the writings of the 19th century mathematicians who led the initiative to raise the level of rigor in the eld of analysis. 1 The Problem with Analysis: Bolzano, Cauchy and Dedekind To begin to get a feel for what mathematicians felt was wrong with the state of analysis at the start of the 19th century, we will read excerpts from three well-known analysts of the time: Bernard Bolzano (1781{1848), Augustin Cauchy (1789{1857) and Richard Dedekind (1831{1916). In these excerpts, these mathematicians expressed their concerns about the relation of calculus (analysis)...