Did Euler Know Quadratic Reciprocity?: New Insights from a Forgotten Work - The Judgment of History

Paul Bialek (Trinity International University) and Dominic W. Klyve (Central Washington University)

How do we judge if a mathematician from a different time and place “knew” a statement or theory from modern mathematics?  Not only the language and notation, but even the standards of proof and description have changed dramatically over the centuries.  Euler's Claim 3.1 was presented by Euler as a “theorem,” as were dozens of other statements in E164, despite the fact that he provided no proofs.  Some people claim that Gauss knew group theory, since the elements of so much of modern group theory can be seen in his Disquisitiones.  Others claim that Fermat (or even Archimedes!) knew a body of theory equivalent to basic integral calculus.  We must not fall into the trap of claiming that somebody didn't know a modern result just because their vocabulary or standard of proof was different, but it is also dangerous to give credit where none is due, by anachronistically reading modern understanding into the words of historical figures (see the discussion in [12]).

What then shall we do?  With historical sensitivity, we must try to understand the work of the players involved.  We must be willing to translate their work into modern language (difficult to avoid when we are comparing their conceptions with ours), while being careful not to put words (or symbols) into the mouths (or pens) of the original writers.  We must be willing to read between the lines of what they wrote, but not to insert whole new lines of thought.  With this is mind, we turn to our big question: “Did Euler know quadratic reciprocity?”