[FERMAT, Pierre de (1601-1665)]. DIOPHANTUS of Alexandria (fl. A.D. 250). Arithmeticorum libri sex, et de numeris multangulis liber unus. Latin translation by Guilelmus Xylander (Wilhelm Holtzmann, 1532-1576). Edited by Claude Bachet de Mziriac (1581-1638). Commentary by Pierre de Fermat. Toulouse: Bernard Bosc, 1670.
2o (338 x 230 mm). Text in Greek and Latin in double columns, commentary in Latin, single column. A few woodcut text diagrams, engraved title vignette by Rabault, 2 engraved head-pieces, one engraved initial, woodcut initials and tailpieces. (Marginal dampstain to sheet X1.4, sheets X2.3 and Nn2.3 misbound in reverse order, marginal paper flaw to Hh4.) Contemporary vellum over pasteboard (covers soiled). Provenance: Cashel Library (19th-century inscription with shelfmark).
FIRST EDITION OF FERMAT'S THEOREMS RELATING TO NUMBER THEORY. A Toulouse parlementaire with a passion for numbers, Fermat was one of the most brilliant mathematicians of his or any time. He made important discoveries in analytic geometry and algebra, and was the first European to make extensive contributions to the theory of numbers, which he restricted in principle to the domain of integers, establishing it as an independent branch of mathematics. Since most of his work in the realm of number theory remained unpublished in his lifetime and of limited circulation later, "it was neither understood nor appreciated until Euler revived it and initiated the line of continuous research that culminated in the work of Gauss and Kummer in the early nineteenth century. Indeed, many of Fermat's results are basic elements of number theory today... The importance of Fermat's work in the theory of numbers lay less in any contribution to contemporary developments in mathematics than in their stimulative effect on later generations" (DSB). His work paved the way notably for the development of the differential calculus.
Fermat showed not the slightest interest in publishing his work, which remained confined to his correspondence, personal notes, and to marginal jottings in a copy of the 1621 editio princeps, edited by Claude Bachet, of Diophantus' Arithmetica. Fermat's marginalia included not only arguments against some of Bachet's conclusions, but also new problems inspired by Diophantus. After his death, Fermat's eldest son Clment-Samuel published his father's marginalia in this new edition. Most famous of the 48 observations by Fermat included here is the tantalizing note that appears on fol. H3r, stating the still unproved theorem "regarding the impossibility of finding a positive integer n > 2 for which the equation xn + yn = zn holds true for the positive integers x, y, and z" (Norman). Fermat noted that he had discovered a "truly marvelous demonstration" of this proposition, but that the margin was too narrow to transcribe it. The proposition, so simple in form, became known as the single most difficult problem in mathematics, and for over 300 years no mathematician succeeded either in disproving it or in finding Fermat's mysterious proof. In 1995 Andrew Wiles, professor of mathematics at Princeton, who had been obsessed with Fermat's proposition since the age of 10, completed a 130-page proof of the theorem (first presented in 1993, with a flaw that required revision), using the most advanced techniques of modern mathematics. His achievement was described by fellow mathematicians as the mathematical equivalent "of splitting the atom or finding the structure of DNA" (Singh, p. 279). Nonetheless, Wiles had had to resort to sophisticated 20th-century techniques not available to Fermat. The exact form of Fermat's proof, if indeed he had a genuine one, thus remains one of the great unsolved puzzles of mathematics.
Simon Singh, Fermat's Enigma (New York, 1997); Smith, p. 348; Norman 777.