The Quadratic Reciprocity Theorem has played a central role in the development of number theory, and formed the first 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 Euler, and restated by Legendre in terms of the symbol now bearing his name, but was first proven by Gauss. The eight different 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.

It is hard to imagine today the sensation caused by Eisenstein when
he burst upon the mathematical world. In the autumn of 1843, at age twenty,
this self-taught mathematician had barely received his high school
certificate and entered the Friedrich-Wilhelms University of Berlin, when he
produced a flood of publications, instantly making him
one of the leading mathematicians of the early nineteenth century.
On July 14, 1844, Gauss wrote to C. Gerling, saying ``I have recently made the
aquaintance of a young mathematician, Eisenstein from Berlin, who came here with
a letter of recommendation from Humboldt. This man, who is still very young,
exhibits * very* excellent talent, and will certainly achieve great things''
[4]. In 1844 Eisenstein contributed no less
than 16 of the 27 mathematical articles in Volume 27 of Crelle's Journal,
and by his third semester as a student he had
received an honorary doctorate from Breslau [9]. Both Gauss and
the great scientist and explorer
Alexander von Humboldt made great efforts, for the most part in vain,
to obtain recognition
and financial security for the impoverished Eisenstein. Gauss wrote
to Humboldt that Eisenstein's talent was ``that which
nature bestows upon only a few in each century'' [3]. He did obtain a
position as a Privatdozent (unsalaried lecturer)
at the University in Berlin, and was eventually admitted to the
Berlin Academy of Sciences in early 1852. But by then his lifelong poor
health had seriously deteriorated, and later that same year he died, aged
29, of tuberculosis. Gotthold Eisenstein stands with Abel and Galois as
another nineteenth century mathematical genius with a tragic and short
life [3,9].

Eisenstein's geometric proof appeared in Crelle's Journal under the title
* Geometrischer Beweis des Fundamentaltheorems für die quadratischen
Reste* [5].
It is intimately connected to Gauss' third proof (published in
[6] and translated in [10]). Many expositions of
Eisenstein's proof, beginning with [1,2], have
observed only one of its three geometric aspects, and
have overlooked the other important differences between the two proofs.
The result has been a failure to recognize and fully appreciate all the
ways in which Eisenstein greatly streamlines and illuminates Gauss'
proof, and thereby reveals its essence. For instance, Gauss' third
proof is based on a result known as Gauss' Lemma, and Eisenstein was
particularly pleased with a shortcut he found to avoid the
technicalities involved in applying it:

``I did not rest until I freed my geometric proof ... from the Lemma on which it still depended, and it is now so simple that it can be communicated in a couple of lines.'' [7, pp. 173--4,]

We believe that the elegance of Eisenstein's proof deserves wide appreciation, and we present it here along with a comparison to Gauss' third proof.

Tue Feb 9 17:35:23 MST 1999