Introduction to the Analysis of the Infinite, volume 1
The Euler Archive has a PDF of an 1885 German translation.
In the Introductio in analysin infinitorum (this volume, together with E102), Euler lays the foundations of modern mathematical analysis. He summarizes his numerous discoveries in infinite series, infinite products, and continued fractions, including the summation of the series 1/1k + 1/2k + 1/3k + ... for all even values of k from 2 to 26. Perhaps more importantly, the Introductio makes the function the central concept of analysis; Euler introduces the notation f(x) for a function and uses it for implicit as well as explicit functions, and for both continuous and discontinuous functions. In addition, he calls attention to the central role of e and ex in analysis. At the end of Chapter 7, he uses infinitesimal analysis to develop the definitions ex = (1 + x/i)i and ln(1+x) = i((1+x)1/i – 1), where i represents an infintely small quantity. These formulations put ex and ln(x) on an equal basis for the first time. Additionally, Euler proves that every rational number can be written as a finite continued fraction and that the continued fraction of an irrational number is infinite. He also shows how infinite series correspond to infinite continued fractions; in particular, Euler derives continued fraction expansions for e and √e.
Original Source Citation
Lausanne: Marcum-Michaelem Bousquet, Volume 1, pp. 1-320.
Opera Omnia Citation
Series 1, Volume 8, pp.1-392.