Development of the Foundations of Mathematical Analysis from Euler to Riemann

[Opening Paragraph of the Introduction]

Our account of the historical development of mathematical analysis concentrates only on its foundational aspects. We start in the eighteenth century with the achievements and difficulties of the calculus epitomized in the mathematical examination of the motion of the vibrating string, and from there we follow lines of development which led in the early decades of the nineteenth century to the broadening of the subject into what we now know as "mathematical analysis": theories of functions and continuity, limits, and the convergence of series (including Fourier series), as well as the formation and use of the derivative and the integral. The story is divided into six chapters, according both to the selection of problems and also to the historical circumstances of their progress, and an appendix is given over to the introduction of convergence tests. From time to time we meet other branches of mathematics which were related to or which applied the new ideas in analysis, but their detailed history has not been attempted here, on the grounds that they would constitute digressions too extensive from our main theme; instead, either a brief summary has been given in a footnote, or else reference has been made to other sources which do discuss the history of the problem in question. Of especial importance in this context is the fact that this study itself ends with the work of Riemann on the foundations of the calculus, for his other ideas and their consequences led to a new era in analysis (of which set theory and measure theory were the most important components), whose own development would require a separate volume. In addition, we have not attempted to recount the many-sided applications of the techniques in classical analysis itself.