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Essai sur La Théorie des Nombres

Adrien-Marie Legendre
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Library Collection
[Reviewed by
Fernando Q. Gouvêa
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Modern technology has created new publishing possibilities. Visit and type in the name of a historically significant author, such as Adrien-Marie Legendre. You will find an abundance of books on offer, from such publishers as Nabu and Kessinger. These often come with a disclaimer, something like this:

This is a reproduction of a book published before 1923. This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book.

What is going on is that older books in the public domain have been scanned and made available in electronic form. They can often be downloaded for free from libraries; one of the most extensive is the Gallica collection from the Bibliothèque Nationale de France. What the publishers are doing is producing print-on-demand editions of these texts for those who still want paper. It is a valuable service, but the risk of “imperfections” is a real one: I bought a copy of Peirce’s Linear Associative Algebra and discovered that several pages were illegible.

What Cambridge University Press offers in their Cambridge Library Collection is the same service, but with CUP standards for quality, both with regard to the books selected for reprinting and with regard to the actual production of the books. The book under review, a reprint of the second edition of Legendre’s famous book on number theory, demonstrates these features nicely: it is an important book, nicely and clearly printed and well bound.

Legendre’s Essai is one of the founding documents in number theory; many of the issues it discusses set the agenda for the next few years of research. Unfortunately, the first edition contained some mistakes, which were pointed out and corrected by Gauss in the other fouding document, the Disquisitiones Mathematicae. (You will soon be able get this from Cambridge as well: it’s volume one of Gauss’s Werke, to appear in November 2011.) The 1808 edition of the Essai, reprinted here, includes Legendre’s reaction to Gauss. In his preface, he claims to have corrected the mistakes and enlarged the book considerably. Some mistakes remained, alas, and eventually there was a third edition as well.

One particularly charming part of this book is Legendre’s “proof” that any arithmetic progression a + nd with gcd(a,d) = 1 contains infinitely many primes. This is Dirichlet’s Theorem on primes in arithmetic progressions, and the fact that it’s not known as “Legendre’s theorem” is a clue that the proof is incorrect. Asking students in a course on elementary number theory to find the mistake would make a truly fantastic exercise.

I hope mathematicians will welcome the Cambridge Library Collection and that libraries will take full advantage of the renewed availability of such rare volumes. For now, I’m delighted to have a copy of Legendre to study.

Fernando Q. Gouvêa loves old books, teaches elementary number theory, and sometimes asks students to do difficult things. He is Carter Professor of Mathematics at Colby College in Waterville, ME.

Part I. Exposition de diverses méthodes et propositions relatives à l'analyse indéterminée
Part II. Propriétés générales des nombres
Part III. Théorie des nombres considérés comme décomposables en trois carrés
Part IV. Méthodes et recherches diverses
Part V. Usage de l'analyse indéterminée dans la résolution de l'équation