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

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Euler, Leonhard

Bell, Jordan

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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.

Comment: 13 pages; E507 in the Enestroem index. Corrected a few typos

Comment: 13 pages; E507 in the Enestroem index. Corrected a few typos

##### Keywords

Mathematics - History and Overview, Mathematics - Classical Analysis and ODEs, 01A50, 26A12