Euler, Goldbach, and "Fermat's Theorem"

While preparing the correspondence between Leonhard Euler and Christian Goldbach for publication, Martin Mattmueller asked whether the lemma given in the postscript of Euler's letter dated July 26, 1749, was enough for completing the proof of Fermat's Four Squares Theorem. In this article we will show that Euler's result can in fact be used for proving this result via induction.