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I am trying to confirm the initial publication of Lagrange's Four Squares Theorem.

Most of my sources give that it was proved by him in $1770$. However, the generally very good Penguin Dictionary of Mathematics (4th ed.) edited by David Nelson, has this date as $1772$.

I am pretty sure that this date is inaccurate, as everywhere else (as I say) has this date as $1770$.

$1770$ was, of course, the date when Lagrange published his Réflexions sur la Résolution Algébrique des Equations, in which can be found a complete restudy of all the known methods of solving the cubic and quartic equations.

My assumption is that the proof of the Four Squares Theorem can be found somewhere in that book, but I have been unable to access a copy of this to confirm it.

Is anyone able to confirm that Lagrange's Four Squares Theorem can be found, complete with proof, in Réflexions sur la Résolution Algébrique des Equations?

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The proof was submitted to the Berlin Academy in 1770, but published only in 1772. It does not appear in Réflexions, the citation is Lagrange, J.-L.: Démonstration d’un théorème d’arithmétique. Nouveaux mémoires de l’Académie royale des sciences et belles-lettres de Berlin, Année 1770, pp. 123–133 (1772). This memoir is reproduced in Œuvres de Lagrange, ed. J.-A. Serret, t. 3, Gauthier-Villars, Paris, pp. 189–201 (1869), and the reproduction is accessible on Gallica.

The history of the theorem involves the work of Bachet de Méziriac, Fermat and Euler prior to Lagrange. Boucard gives a detailed history in Lagrange and the four-square theorem:

"Lagrange’s proof of the four-square theorem appeared in 1772. As he himself remarks in his historical introduction, the ‘‘theorem by Mr Bachet’’ [17, p. 190] has been the subject of previous research. Claude-Gaspard Bachet de Méziriac is well known for his Problémes plaisans et délectables qui se font par les nombres, published in 1612 and then again in 1624... It is from this work that, some years later, Fermat learned about the four-square theorem. He made mention of this problem in his correspondence starting in the 1630s... No proof by Fermat has been found, but some clues can be found in a letter to Pierre de Carcavi written in 1659, where Fermat claims there that the four-square theorem belongs to a series of a priori unconnected arithmetic questions that can be proved using the ‘‘method of infinite descent’’.

Goldbach sent Euler two letters, in December 1729 and May 1730, detailing some conjectures by Fermat and a few of his own results; Euler replied on 4 June 1730, confirming his interest in number theory and in particular in the ‘‘not inelegant theorem’’ stating that every number can be written as a sum of four squares... Euler presented several memoirs about sums of squares, including a proof of a weak version of the four-square theorem [23]: every integer number is a sum of at most four integer or rational squares.

Lagrange finally gives a complete proof of the four-square theorem in a dozen-page memoir [17]. Unlike Euler, he has not published other texts about sums of squares. At the time, the two scholars were engaged in a regular correspondence, in which they often discussed arithmetical questions, but never the four-square theorem... In his memoir titled Novae demonstrationes circa resolutionem numerorum in quadrata submitted to the Academy of St. Petersburg in 1772, but published in the Acta eruditorum only in 1780, Euler happily announces that he has finally managed to find a complete proof of the four-square theorem, very different from Lagrange’s and, above all, less laboured."

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