What Euler could further write, or the unnoticed "big bang" of the fractional calculus

Abstract

In this short communication, an attempt is made to continue beyond paragraph 29 of Euler's famous paper in Vol. 5 of Comment. Acad. Sci. Petropol. (1738), using his style of storytelling to extrapolate the audacity of his approach from fractional differentiation to fractional integration. To add the authenticity and the amusement to the imitation, the emulated paragraphs 30-32 are first presented in Latin version followed by the English translation. This reconstruction aims to demonstrate that Euler could consider not only fractional differentiation, but also fractional-order integration and its inverse relationship with differentiation of the same fractional order. © 2013 Versita Warsaw and Springer-Verlag Wien. MSC 2010: 26A33*33B15* 01A99*01A50.