Euler's output was split fairly evenly between pure and applied mathematics, the latter including many topics that we would today classify as physics. Most of his papers fall decisively into one category or the other, but it wasn't at all rare for one of his works of applied mathematics to include new techniques or results in analysis. This frequently happened in the Paris Prize competition, for example, where the questions were generally of a practical nature. This month, we'll look at an astronomical paper [E401] that proposes numerical techniques for approximating a body's velocity and acceleration. Remarkably, one of the results in E401 was probably the first step in the development of the calculus of operations and seems to have influenced Lagrange's foundational program for the calculus.

Euler read E401, “A New Method for Comparing Observations of the Moon to the Theory,” to the Berlin Academy on February 6, 1766, just a few months before his return to St. Petersburg. Because he quoted astronomical data from the summer of 1765, it's nearly certain that his results date that year. Nevertheless, the paper appeared in the Berlin Academy's volume for 1763, which was only published in 1770. Euler probably had it included in this volume, despite its later composition date, because it's a follow-up to E398, “A New Method for Determining the Perturbations in the Motion of Heavenly Bodies Caused by their Mutual Attraction,” which he read to the Academy on July 8, 1762. Because of the long publication delay in the 1763 volume, Euler was able to arrange matters so that E398 was immediately followed by three papers that build upon it results: E399, read on December 18, 1763, which applies the methods E398 to the moon, E400, read on December 4, 1765, which considers the general three body problem, and E401.