E445 -- Novae demonstrationes circa resolutionem numerorum in quadrata
(Proof that every Integer is a Sum of Four Squares)
Euler wrote this in response to a paper by Lagrange.
According to the records it was submitted on September 21, 1772, then taken
back in 1773 and on March 24, 1774, it was presented to the St. Petersburg Academy for the
The following details have been related by Ed Sandifer:
On November 19, 1773, the Academy had its meeting to pick the articles to
publish in the Mathematics section of the next volume of the Novi
Commentarii. They picked eight articles by Euler, E-445 and E-448 to
E-454. Three days later, they received two articles from Daniel Bernoulli
in Basel, one of which was very closely related to one of Euler's articles,
not on the list (E-447) about sums of sines of angles that are in geometric
progression. They decided to publish that one right next to Euler's
article, but that made the Mathematics section of that volume too big. So,
Euler volunteered to withdraw one of his articles. He picked E-445 and sent
it to the Acta Eruditorum instead.
- Originally published in Acta Eruditorum 1777, 1780, pp. 48-69
Opera Omnia: Series 1, Volume 3, pp. 218 - 238
- Reprinted in Acta. acad. sc. Petrop. 1777: II, printed 1780, pp. 48-69 [E445a]
- Reprinted in Commentat. arithm. 1, 1849, pp. 538-548 [E445b]
- A partial translation of what he considers to be the interesting parts of the paper
was done by Erik Temple Bell, and appears on pp. 91-94 of Smith's Source Book in
Mathematics (1929, McGraw-Hill).
- Original Publication: E445 (Note: this original publication is largely unreadable. A scan of the Opera Omnia's version of this paper (now with expired copyright) has been prepared by Christian Aebi: E445
- English Translation (Jordan Bell): E445
- The Euler Archive attempts to monitor current scholarship for articles and books that may be of interest to Euler Scholars. Selected references we have found that discuss or cite E445 include:
- Pieper H., “On Euler's contributions to the 4-squares theorem.” Historia Mathematica, 20 (1), pp. 12-18 (Feb 1993).
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