On the infinity of infinities of orders of the infinitely large and infinitely small

TL;DR

This paper explores the concept of multiple orders of infinity and infinitesimals, analyzing how functions approach infinity or zero at different rates, based on Euler's 1780 discussion.

Contribution

It provides an interpretation of Euler's ideas on the hierarchy of infinities and infinitesimals, highlighting the infinite gradations of their magnitudes.

Findings

Identifies multiple orders of infinity and infinitesimals in Euler's work

Suggests a hierarchy of infinities based on function growth rates

Highlights the need for further study of Euler's original concepts

Abstract

Translation (by J.B.) from the original Latin of Euler's "De infinities infinitis gradibus tam infinite magnorum quam infinite parvorum" (1780). E507 in the Enestr\"om index. Euler discusses orders of infinity in this paper. In other words this paper is about how different functions approach infinity or 0 at different rates. I was not certain about what Euler means by "infinities infiniti". Probably he means that $x,x^2,x^3$, etc. are infinitely many orders of infinity, and $\log x,(\log x)^2, (\log x)^3$, etc. are infinitely many orders of infinity, so the combinations of them are an infinity of infinities of orders of infinity. In fact Euler mentions other orders of infinity in this paper. It would be worthwhile to study this paper more to figure out exactly what Euler means here. Another translation of the title is "On the infinitely infinite orders of the infinitely large and infinitely small". Here's another place Euler uses the phrase "infinities infiniti". The phrase "infinities infiniti" from the title is used by Euler also in section 21 of E302, "De motu vibratio tympanorum". Truesdell translates this phrase on p. 333 of "The rational mechanics of flexible or elastic bodies" as "infinity of infinities". I'd like to thank Martin Mattmueller for clearing up some questions.