Number theory. An approach through history. From Hammurapi to Legendre. (English) Zbl 0531.10001

Weil, André

Boston-Basel-Stuttgart: Birkhäuser. XXI, 375 p. DM 74.00 (1984).

In his latest book the author presents the history of the pre-Gaussian period of the theory of numbers. Two names dominate that period - Fermat and Euler, and so it is not surprising that three quarters of the book are devoted to the analysis of their research. The author states in the introduction that ”it is the author’s fond hope that some readers at least will find it possible to get their initiation into number theory by following the itinerary retraced in this volume”. In the reviewer’s opinion he did succeed in this task. The book makes a fascinating reading, permitting to perceive the birth of new ideas, and to understand why they should have been born. One sees here, how the theory of quadratic forms started with primitive Babylonian tables of Pythagorean triangles, then went through the early, one would like to say, experimental stage in the hands of Diophantus and Fermat, how it has been expanded by Euler and then culminated in the general theory, created by Lagrange, which gave later way to the modern development begun by Gauss (which however lies already outside the scope of this book).
There are four chapters: ”Protohistory”, ”Fermat and his correspondents”, ”Euler” and ”An age of transition: Lagrange and Legendre”, and also several appendices, which introduce a modern point of view and provide proofs for many mentioned results.
The book is strongly recommended to anybody interested in the history of mathematics and should be on the shelf of every number-theorist.

Reviewer: W.Narkiewicz

MSC:

11-03	History of number theory
01A05	General histories, source books
01A45	History of mathematics in the 17th century
01A50	History of mathematics in the 18th century

Keywords:

Fermat; Euler; Legendre; Lagrange; quadratic forms; diophantine analysis; history of preGaussian period of theory of numbers