Euler's De Serie Lambertina (E532) English Translation

The original De Serie Lambertina (sometimes cited as De Serie Lambertine) is available from this link. It is Enestrom number E532. This paper was written around 1776 and published around 1779 (Acta Academiae Scientiarum Imperialis Petropolitanae, Volume 1779: II, pp. 29-51), or 1783, depending on the reference.

Euler reviews the results of Johann Heinrich Lambert's 1758 work Observationes Variae in Mathesin Puram (Various observation on pure mathematics) in Acta Helvetica Physico-Mathematico-Botanico-Medica (specifically, Acta Helvetica Vol III pp. 128-168, available here from Biodiversity Heritage Library; I extracted the article as a PDF here on Google Drive). I'll be reviewing the mathematics involved in this in a future post, but some details are available from this link, an answer of mine on History of Math and Science StackExchange.

The equation considered by Euler is a trinomial symmetric in two parameters, α and β. (This is in contrast to Lambert's equation, a general polynomial). The reason for this is explained in the translation given here. 

Commentary on this paper, and the results within it, can be found in:

Wang, F. (2017). Proof of a series solution for Euler’s trinomial equation. ACM Communications in Computer Algebra, 50(4), 136–144. doi:10.1145/3055282.3055284 

Which is available on Sci-Hub here.

The original, rewritten using LaTeX, is shown below, and the English translation is below that. These are both rough drafts, so there are probably errors and typos (particularly in the Latin text). The original LaTeX source files are available here (Latin) and here (English). 



Comments

  1. victormdAugust 6, 2023 at 9:39 PM

    Hey, i first became interested in this particular subject by thinking about how Lambert might have found out the infinite series to the trinomial he is famous for. By doing some research i came to this blog and read your papers above which were really helpful. I'd like, though, to read lambert's original paper and the only version i could find has the pages on the method missing ( i got it from this link http://www.kuttaka.org/~JHL/L1758c.html ).
    Im wondering if you'd have a more complete version available to share? Thank you anyway for yhe post

    ReplyDelete
    Replies
    1. Sam GallagherAugust 7, 2023 at 5:46 AM

      I don't have any translation to share at the moment, although I could make a start on one. I added a link to Lambert's original paper to this page, and as you might've read already, the first endnote of On Lambert Series contains the details of how his trinomial equation is derived. This method was based on prior work by Daniel Bernoulli, ("Observationes de seriebus recurrentibus", Commentarii Academiae Scientiarum Imperialis Petropolitanae, Tome III pp. 85-100 (1728)).
      If you have specific sections you'd like translated, I'll start on those and can be done sooner. If you want quick turnaround and don't mind spending a bit of money, I might recommend Quintus' Latin Translation (https://www.thelatintranslator.com/).

      Delete

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