Abstract
Translated from the Latin original "Novae demonstrationes circa resolutionem
numerorum in quadrata" (1774). E445 in the Enestrom index. See Chapter III,
section XI of Weil's "Number theory: an approach through history". Also, a very
clear proof of the four squares theorem based on Euler's is Theorem 370 in
Hardy and Wright, Än introduction to the theory of numbers", fifth ed. It uses
Theorem 87 in Hardy and Wright, but otherwise does not assume anything else
from their book. I translated most of the paper and checked those details a few
months ago, but only finished last few parts now. If anything isn't clear
please email me.
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