On Euler's contributions to the four-squares theorem

Abstract

A passage in Euler's notebook 132 (written between 1740 and 1744) concerning the four-squares theorem is interpreted in a new way. The proof of the four-squares theorem is based on three lemmas. Euler proved two of them from which he deduced the theorem on the representation as a sum of four squares of fractions. But Euler was not able to prove the third lemma though it is formulated in notebook 132