Eisenstein's Misunderstood Geometric Proof of the Quadratic Reciprocity Theorem

1 Introduction The Quadratic Reciprocity Theorem has played a central role in the development of number theory, and formed the rst deep law governing prime numbers. Its numerous proofs from many distinct points of view testify to its position at the heart of the subject. The theorem was discovered by Eu-ler, and restated by Legendre in terms of the symbol now bearing his name, but was rst proven by Gauss. The eight diierent proofs Gauss published in the early 1800s, for what he called the Fundamental Theorem, were followed by dozens more before the century was over, including four given by Gotthold Eisenstein in the years 1844{45. Our aim is to take a new look at Eisenstein's geometric proof, in which he presents a particularly beautiful and economical adaptation of Gauss' third proof, and to draw attention to all the advantages of his proof over Gauss', most of which have apparently heretofore been overlooked. 0 The authors would like to thank Keith Dennis for his assistance in locating sources, Tom Hoeksema and Carol Walker for their enthusiastic support of the Honors courses which led to this work, and Joe Zund for telling us where to look.