Diophanti Alexandrini Arithmeticorum libri sex, et De numeris multangulis liber unus. Cum commentariis C. G. Bacheti V. C. & observationibus D. P. de Fermat Senatoris Tolosani. Accessit Doctrinae Analyticae inventum novum, collectum ex variis eiusdem D. de Fermat Epistolis.
Box: 378 x 264 x45 mm. Toulouse: Bernard Bosc, 1670. First edition, large-paper issue, with the engraved portrait of Pierre de Fermat by François Poilly on the leaf facing the title — rarely found in copies of this edition — and with the editor’s presentation inscription on the title page. The portrait, a fine oval bust set above Fermat’s family arms (on a chevron three eagles, and in base a crescent, for the Fermats of Bas-Quercy), is the work of one of the leading Parisian printmakers of the second half of the seventeenth century; most copies of the 1670 Diophantus lack it, and its presence here, together with the generous margins of the large-paper sheets and the inscription immediately below, marks the volume as one of the small number of copies Clément-Samuel reserved for the inner circle of the Fermat family and their Toulouse connections. Beneath the printed line naming Pierre de Fermat as Senatoris Tolosani, a period hand has added six Latin words: de Molieres ex dono authoris — “to de Molières, from the gift of the author.” The recipient is Louis de Molières (1610–1687), born at Cahors into the noblesse de robe of Bas-Quercy and established as trésorier de France at the Montauban bureau des finances, a post he would hold for forty-two years. His first marriage, in 1646, had been to Louise de Fermat (c. 1613–c. 1650), Pierre de Fermat’s younger sister. Louise had been dead twenty years by the time this copy left the press; Louis had long since remarried a demoiselle de Marqueyret; but the connection between the two robe families of the lower Garonne ran too deep for that to matter. The author named in the inscription is not Pierre — who had died in January 1665 — but Pierre’s eldest son Clément-Samuel de Fermat (c. 1632–1697), the lawyer and conseiller au parlement who had inherited his father’s offices, had spent the five years since his father’s death transcribing the elder Fermat’s mathematical marginalia into publishable form, and who oversaw the volume through the Toulouse press of Bernard Bosc in 1670. By sending a large-paper copy to his late aunt’s widower — the senior surviving link to his father’s family in the generation above his own — Clément-Samuel placed his father’s posthumous monument where it most properly belonged. The volume is the second edition of Bachet de Méziriac’s 1621 Greek-and-Latin Arithmetica of Diophantus of Alexandria, expanded by Clément-Samuel with his father’s forty-eight mathematical observations, and completed by the Doctrinae analyticae inventum novum of the Jesuit Jacques de Billy — a summary account of Fermat’s analytical method drawn from the correspondence Billy had maintained with Fermat in the last years. The Arithmetica itself is the foundational work of Greek algebra and of Diophantine analysis, setting out 189 problems in indeterminate analysis that had occupied mathematicians from Regiomontanus and Bombelli through Viète and Bachet. Fermat had annotated his personal copy of the 1621 Bachet edition — the copy he acquired in 1636 or 1637, probably through the circle of Carcavi and Mersenne — with marginal notes responding to individual Diophantine problems and, in many cases, generalising them into new theorems. That original annotated copy is lost. Its contents survive because Clément-Samuel, working from his father’s papers and almost certainly with the copy itself in hand, transcribed the forty-eight observations and printed each at the appropriate point in the Diophantine text. The result is a conflation of the Bachet edition with Fermat’s marginalia: Greek and Latin in parallel columns for the Diophantus, with Bachet’s commentary and Fermat’s observation intervening at the relevant problems. At its centre — literally and historically — stands the single most consequential marginal note in mathematics. On page 61 of this volume, as a commentary on Diophantus Book II Problem VIII (the problem of dividing a given square into two smaller squares), sits Fermat’s observation in nine lines of italic Latin. Against the proposition that every square decomposes into two squares — the problem whose rational solutions are the Pythagorean triples — Fermat remarks that no cube decomposes into two cubes, no fourth power into two fourth powers, and, in general, no power higher than the second can be decomposed into two powers of the same kind. He has discovered, he adds, a truly marvellous proof of this proposition; the narrowness of the margin cannot contain it. This is Fermat’s Last Theorem. It is printed here for the first time. The original 1621 Bachet that Fermat annotated no longer exists, so the 1670 printing is the sole testimony to how Fermat actually wrote the proposition and the sole source for the evocative remark about the margin. The theorem held. For three hundred and fifty-eight years, Fermat’s claim resisted verification. Leonhard Euler produced the proof for exponent three in 1770, invoking the method of infinite descent that Fermat had set out in other contexts. Sophie Germain in the first decade of the nineteenth century opened a substantial class of exponents — the class of primes now called Sophie Germain primes. Dirichlet and Legendre settled exponent five in 1825. Gabriel Lamé reached exponent seven in 1839 and briefly claimed the full theorem, a claim Liouville corrected within weeks by pointing to a failure of unique factorisation in the relevant cyclotomic integers. Ernst Kummer in 1847, working precisely on that failure, introduced the ideal numbers that would become the foundation of algebraic number theory, and proved Fermat’s proposition for all regular primes. By the late nineteenth century Fermat’s Last Theorem stood as a celebrated challenge, and Paul Wolfskehl’s 1908 bequest of a hundred-thousand-mark prize for a valid demonstration kept thousands of amateur attempts flowing to the University of Göttingen through the First World War and the Weimar collapse. The decisive modern move came in 1986, when Gerhard Frey suggested that any counterexample to the Fermat equation would produce a semistable elliptic curve whose properties must contradict the Taniyama–Shimura–Weil conjecture on modular forms. Kenneth Ribet proved the Frey implication the same year. Andrew Wiles, working almost alone at Princeton, announced a proof of the relevant portion of the modularity conjecture at Cambridge in June 1993; referee Nick Katz identified a subtle error; Wiles and Richard Taylor together closed the gap over fourteen further months; and the finished paper appeared in the Annals of Mathematics in May 1995. Fermat was right. The three and a half centuries between statement and proof generated a disproportionate share of modern number theory. Kummer’s ideal numbers founded algebraic number theory. The theory of cyclotomic fields, the arithmetic of elliptic curves, and the whole modern apparatus of modular forms and Galois representations — together forming the present-day Langlands programme — all derive, directly or by consanguinity, from the long search for Fermat’s proof. Wiles’s demonstration runs past a hundred pages and invokes techniques Fermat could not have envisaged; the opinion of most specialists is that whatever proof Fermat believed he had was probably in error, most likely a descent argument of the kind that works for exponents three and four but cannot be extended. Fermat himself, in a 1659 letter to Carcavi, set out his method of infinite descent in some detail and applied it to prove that the area of a rational right triangle can never be a square number — a proposition that, by a short chain of reasoning, implies his Last Theorem for exponent four. Whether that technique could be stretched to the general case is the question to which the answer, three hundred and thirty-six years later, was Wiles’s hundred pages. Fermat’s engagement with Diophantus ranged far beyond the single marginal note at page 61. Forty-seven further observations thread through the volume, responding to Diophantine problems on rational squares, Pythagorean right triangles, the representation of integers as sums of squares, and the arithmetic of cubes. Several of these observations announce theorems of comparable depth. The two-square theorem — that every prime congruent to one modulo four is the sum of two squares in essentially one way — sits among them, as do the germ of the four-square theorem later proved by Lagrange, the statement that every number is the sum of three triangular numbers, and the generalised Fermat equation x2 − Ay2 = 1 (the Pell equation), which Fermat correctly recognised as always solvable in integers for non-square A. The observation at page 135 — headed OBSERVATIO D.P.F. and placed after Diophantus Book IV Question III — displays Fermat’s characteristic fusion of correction and extension. Bachet had offered a partial treatment of the problem of finding two cubes whose difference equals a given number; Fermat shows that Bachet missed an entire further family of solutions, which follow from his own method by continued iteration in infinitely many cases. Given the two cubes 8 and 1, whose difference is 7, Fermat produces a second pair of rational cubes with the same difference. His printed solution gives the sides 1265/183 and 1256/183, yielding the cubes 2024284625/6128487 and 1981385216/6128487. The verification is clean: the difference of these two new cubes reduces exactly to 7. Across the printed denominators on this page, however, a contemporary hand has drawn firm lines and written substitutions above the print: 61 in place of 183 for the sides, and 226981 in place of 6128487 for the cubes. The substitution is not arbitrary — 183 is three times 61, and 6128487 is twenty-seven times 226981, which is itself 61 cubed. The annotator has evidently noticed that Fermat’s fractions appear to contain a common factor of the cube of three, and has tried to simplify them by cancelling it. But the correction does not preserve the answer. The revised sides 1265/61 and 1256/61 are each three times larger than their printed counterparts; the revised cubes are each twenty-seven times larger; and the difference of the revised cubes becomes 189 rather than 7. The substitution would solve a scaled version of Fermat’s problem — one in which the given cubes were 216 and 27 rather than 8 and 1 — but it does not solve the problem as Fermat poses it on this page. The correction is the work of a contemporary reader who followed Fermat’s argument closely enough to recognise the internal structure of the solution, and who carried enough confidence to intervene in a freshly printed Toulouse folio, but who stopped short of the final verification that would have caught the scaling error. That degree of engagement is itself worth marking. Fermat’s observation on Book IV Question III was considered obscure even among the professional mathematicians of the period; the appearance of contemporary manuscript attention to its numerical detail, in a copy that left the editor’s hands in 1670, places this volume inside the very narrow circle of readers who took Fermat’s more technical observations seriously from the moment of publication. Two further inserted slips of paper, at pages 61 and 197, carry contemporary but more elementary annotations, placing this copy plainly in the hands of a seventeenth-century reader working through the mathematics of the volume rather than merely its production. The slip at page 197 — facing the large printed table of eighty-one integer solutions to a Diophantine problem in four variables from Book V — carries calculations in a reader’s hand involving the quantities eight hundred and ten thousand, a cubic variable, and a squared variable, in a working attempt at the problem treated above. A later eighteenth-century English hand has added a note on the flyleaf, framed as a dismissive verdict on Fermat’s mathematical claims. A discreet twentieth-century dealer’s mark on page 9 identifies the code of Lucien Scheler (1902–1999), the Parisian antiquarian bookseller and poet whose handling of the book places its modern provenance within a narrow compass of known trade hands. The recipient of the 1670 inscription belongs to a world of parliamentary offices and extended family connection that the inscription itself records in six Latin words. Louis de Molières, born at Cahors in 1610, served forty-two years as trésorier de France at the Montauban bureau des finances, one of the senior royal financial posts in lower Languedoc. His first marriage, in 1646, was to Louise de Fermat, daughter of Dominique de Fermat — the consul and leather merchant of Beaumont-de-Lomagne — and therefore sister of Pierre and paternal aunt of Clément-Samuel. Louise died in the late 1640s. Louis remarried a demoiselle de Marqueyret and continued as head of one of the prominent parliamentary families of Bas-Quercy until his death in 1687. His son by the second marriage, Armand de Molières, later served as second président of the Cour des aides at Montauban — the Armand whose name has occasionally been conflated with his father’s in later bibliographic sources, producing the hybrid ‘Louis-Armand’ that appears in some modern descriptions. The present inscription is addressed to Louis senior, Pierre’s brother-in-law and Clément-Samuel’s uncle by marriage, a man whose household at Montauban sat fifty kilometres north of Pierre’s at Toulouse and who by 1670 was the senior family member in the generation linking back to Pierre’s parents at Beaumont. Pierre de Fermat’s reputation does not rest on the Last Theorem alone. A conseiller at the parlement of Toulouse and a magistrate of the Chambre de l’Édit at Castres, he was an amateur mathematician in the technical sense only — an amateur who corresponded with Mersenne, Pascal, Descartes, Huygens, Wallis, Carcavi, and Roberval on terms of complete intellectual equality, and who made fundamental discoveries in four distinct branches of mathematics. In number theory, beyond the Last Theorem, he discovered the theorem now called Fermat’s Little (that for any prime p and integer a not divisible by p, the quantity a raised to the power p minus one is congruent to one modulo p), stated and used the two-square theorem, developed the method of infinite descent as a rigorous technique for negative existence proofs, and extended the theory of amicable numbers well beyond the pair 220 and 284 known since antiquity. In analytic geometry, his Ad locos planos et solidos isagoge — which he sent in manuscript to Carcavi and Mersenne in 1636 — predated Descartes’s Géométrie in composition though not in print. In the calculus of variations, his method of adequality supplied a systematic technique for locating maxima, minima, and tangents that Newton and Leibniz both later acknowledged as precursor. In the summer of 1654, in the correspondence with Pascal that Carcavi preserved, he worked out with Pascal the foundations of the mathematical theory of probability, solving the problem of the division of stakes in interrupted games of chance. In optics he enunciated the principle of least time — Fermat’s principle — which furnished the first variational formulation in physics and served as direct ancestor to the principle of least action and the whole edifice of Lagrangian and Hamiltonian mechanics. Any one of these contributions would secure a reputation; that a sitting magistrate of the Toulouse parlement, pursuing mathematics in stolen evening hours, made all four is the condition Clément-Samuel set himself to commemorate in this volume. Of those four strands, the 1670 Diophantus captures chiefly the number-theoretic Fermat, and within that only the portion he wrote as marginalia on Bachet. His analytic geometry and his general method of maxima et minima appeared in 1679 as Varia Opera Mathematica, again at Toulouse, edited again by Clément-Samuel. His complete correspondence and further manuscripts were assembled definitively only in the late nineteenth century by Paul Tannery and Charles Henry, whose four-volume Œuvres de Fermat (1891–1912, with a supplementary fifth volume by Cornelis de Waard in 1922) remains the standard scholarly edition. But the 1670 edition is the book in which Fermat’s Last Theorem first entered print, the book through which Fermat’s name reached the working mathematicians of the late seventeenth and eighteenth centuries, and the book Euler and Gauss both studied and built on. Its place in the foundational history of number theory is not in dispute. What is less often remarked — and what this particular copy preserves — is the presence in 1670 of readers who took Fermat’s more technical observations seriously enough to attempt corrections in the margins, even when, as at page 135, those corrections did not finally succeed. References: Honeyman 885 — Norman 771 — Smith, Rara Arithmetica, pp. 348–349 — Brunet II, 702 — Roberts & Trent, Bibliotheca Mechanica, p. 108 — Hoffmann 1242 — Weil, Number Theory: An Approach through History from Hammurapi to Legendre (Birkhäuser, 1984), chapters II–IV — Mahoney, The Mathematical Career of Pierre de Fermat (Princeton University Press, second edition 1994) — Goldstein, Un théorème de Fermat et ses lecteurs (Presses Universitaires de Vincennes, 1995) — Singh, Fermat’s Enigma (Fourth Estate, 1997) — Wiles, ‘Modular elliptic curves and Fermat’s Last Theorem’, Annals of Mathematics 141 (1995), pp. 443–551 — Taylor and Wiles, ‘Ring theoretic properties of certain Hecke algebras’, Annals of Mathematics 141 (1995), pp. 553–572. Folio (365 × 246 mm), pp. [xii], 341; 48. Engraved portrait of Pierre de Fermat by François Poilly on the leaf facing the title (Fermat in scholarly dress within an oval frame, his arms below on the plinth) — rarely found in copies of this edition. Engraved allegorical vignette on the title page (Orpheus with the lyre, encircled by the Virgilian motto obloquitur numeris septem discrimina vocum). Numerous woodcut diagrams in the text. Greek and Latin in parallel columns throughout the Diophantus. Separate pagination for the Inventum novum. Light browning. Contemporary calf, gilt fillet on covers, spine richly gilt in compartments with gilt-tooled lettering DIOPHANTI / FERMAT, edges speckled red, binding slightly rubbed. A fine copy.
Item #6238
Price: $165,000.00
