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The Mathematical Career of Pierre de Fermat, 1601-1665

Michael S. Mahoney
Princeton University Press
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Álvaro Lozano-Robledo
, on

The Mathematical Career of Pierre de Fermat is an account of the life and mathematical works of one of the greatest minds of the seventeenth century. For this account, the author intertwines the biographical information available about Fermat (including his parliamentary, judiciary and legal duties) with his mathematical work. Most interestingly, his life and work are explained in the social and mathematical context of his period and also compared to his predecessors and followers. As a whole, the book produces a very vivid picture of the person, the mathematician, his mathematics and the time period.

The book is divided as follows. The first chapter begins with a general description of the diversity of mathematics before Fermat's time, followed by a brief description of Fermat's duties in the parliament and as a lawyer. The chapter ends with a first attempt at describing what led "the learned councillor from Toulouse", as Mersenne once referred to him, to become interested in mathematics. In Chapter II, the author argues that the work of  Francois Viète (and his school of thought) provided crucial inspiration to Fermat. As a result, the chapter is a survey of Viete's work and a survey of Fermat's work in that context. Chapter III describes Fermat's efforts in the restoration of Apollonius' Plane Loci and the methods of analytic geometry that streamed from it. Chapter IV is dedicated to Fermat's methods for finding maxima, minima and tangents (and a number of disputes with René Descartes). In Chapter V, the author summarizes the achievements on quadratures during the period of 1643–1657. Finally, Chapter VI describes Fermat's numerous contributions to number theory during his fruitful career. The book concludes with a retrospective essay on the importance of Fermat's work, an appendix dealing with his work on optics, mechanics and probability, and a second appendix with a full chronological bibliography of Fermat's works.

As already mentioned, the book The Mathematical Career…  manages to portray all different aspects of Fermat's life and work (his interests, his style, his acquaintances, his refusal to publish under his own name…) in a very effective and clear way. Even though I did not read the entire book, I very much enjoyed reading several chapters. The text flows nicely and the more mathematical sections of the book are explained both in the notation of the period and modern notation, helping the reader assimilate the proofs and ideas involved and appreciate their importance.

I am not a historian of mathematics so I am not one to judge (nor doubt) the accuracy of the text, or the opinions and theories that attempt to explain certain obscure aspects of Fermat's work. However, as a non-historian, I believe this book is an excellent source of information (and further references abound in the footnotes) about "the learned councillor from Toulouse."

Álvaro Lozano-Robledo is H. C. Wang Assistant Professor at Cornell University.

 BLL* — The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

 Preface (1994) ix
Introduction xi
Acknowledgments xvii
I. The Personal Touch 1
1. Mathematics in 1620
2. Fermat's Life and Career in Parlement
3. Motivation to Mathematics
II. Nullum Non Problema Solvere: Viete's Analytic Program and Its Influence on Fermat 26
1. Algebra, Analysis, and the Analytic Art
2. Following the "Precepts of the Art"
3. Fermat's Style of Work and His Influence on His Contemporaries
III. The Royal Road 72
1. Introduction
2. Fermat's Analytic Geometry, the Ad locos pianos et solidos isagoge
3. The Origins of the Isagoge: Apollonius' Plane Loci and Conics
4. Extensions of the System of the isagoge: The Isagoge ad locos ad superficiern
5. Uses of the System of the Isagoge: Graphic Solution and Classification of Equations
IV. Fashioning One's Own Luck 143
1. Introduction
2. The Roots of an Equation and the Roots of a Method
3. Of Dubious Parentage: The Method of Tangents
4. Looking Under the Bed: Descartes vs. Fermat, 1637-38
5. The Aftermath: Proceeding By Touch
6. Learning New Tricks: The Letter to Brulart
7. Fine Tuning: The Path Toward Quadrature and Rectification
V. Archimedes and The Theory of Equations 214
1. Introduction
2. From Spirals to Conoids
3. The Method of Centers of Gravity
4. The Treatise on Quadrature (ca. 1658)
5. The Treatise on Rectification (1660)
6. Fermat and the Calculus
VI. Between Traditions 283
1. Introduction
2. Numbers, Perfect and Not So Perfect
3. Triangles and Squares
4. Reclaiming the Patrimony: The Challenges of 1657
5. One Final Attempt: The "Relation" to Carcavi (1659) and the Method of Infinite Descent
6. Infinite Descent and the "Last Theorem"
Epilogue: Fermat in Retrospect 361
Appendix I: Sidelights on A Mathematical Career 368
1. Mechanics
2. Optics
3. Probability
Appendix II: Bibliographical Essay and Chronological Conspectus of Fermat's Works 411
Index 425