Lagrange's Method of Solving Polynomial Roots

 Today, by chance, I decided to check out Joseph-Louis Lagrange's works, Oeuvres Lagrange, in particular tome 3. It's a remarkable coincidence, because I have only downloaded one book (tome) of Oeuvres Lagrange, book 3, and for no reason in particular I decided to browse it, finding that the first article relates to the solution of general algebraic and transcendental equations! This is a coincidence because the papers from Euler and Lambert that I've been working on (see Welcome article for now) relate to this very topic, and Lagrange is essentially picking up where Lambert left off, using series to represent the roots of polynomials. 

    Lagrange's paper, Nouvelle méthode pour résoudre les équations littérales par le moyen des séries (Mémoires de l'Académie royale des Sciences et Belles-Lettres de Berlin, t. XXIV, 1770) does a far superior job to Lambert's series solution of the same problem, because Lambert's series could only determine any power of the largest (or sometimes the smallest) real root of a polynomial. Lagrange, on the other hand, describes a method for calculating any power of any root by method of series. The improvement is dramatic, and it's certainly a step forward in the history of the theory of equations. And while it's not particularly groundbreaking, I'm still thrilled to find something so relevant, and in a language I can translate! So I took a few hours, and translated the document in beautiful LaTex typesetting, to closely match the original. See it down below. 


About the translation: 

This translation was made by Sam Gallagher, 13 Nov 2020. It covers the first section of Lagrange's paper (cited above) translated from French (as it appeared in the Oeuvres Langrage t. 3) into English. The goal of the translation was to preserve Lagrange's writing style, which is actually unusually simplistic and straightforward. This may be because Lagrange was native Italian, and only later naturalized to French, but I don't know what his French education and experience was like at this point. I do know that he had been the director of mathematics at the Prussian Academy of Sciences in Berlin, Prussia (see: his Wikipedia page) and it wasn't until 1787 (17 years after this paper was published) that he moved to France. I had a lot of fun with this translation, and maybe I'll continue further. I do not know for sure if this article has been translated before, so before I invest any more time I'll do more background and see if a translation in English exists.

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