Alexandrie

DIOPHANTUS of Alexandria; BACHET DE MEZIRIAC, Claude GaspardBiblio.com

Diophanti Alexandrini Arithmeticorum libri sex et De numeris multangulis liber unus. Nunc primùm Graecè & Latinè editi atque absolutissimis Commentariis illustrati. Auctore Claudio Gaspare Bacheto Meziriaco Sebusiano V.C.

16216237Paris: Hieronymus Drouart 1621. First edition. First edition of the Greek text extremely rare large-paper copy of the foundational work of algebra - the edition that Pierre de Fermat acquired in the mid-1630s and annotated with the forty-eight marginal observations that are the founding documents of modern number theory. Fermat's copy is lost; these are the sheets from the same Paris setting that he had in front of him. Claude Gaspard Bachet de MÃ©ziriac a country gentleman of Bas-Quercy whose entry into number theory had come through his 1612 ProblÃ¨mes plaisants spent several years establishing a corrected Greek text improving Xylander's 1575 Latin translation filling the lacunae correcting errors and generalising Diophantus's procedures; his edition remained standard until Tannery's Teubner text of 1893-1895. Against Problem II.8 at page 85 - on the decomposition of a square into two squares - Fermat wrote the proposition now known as Fermat's Last Theorem closed by Andrew Wiles in 1995 three hundred and fifty-eight years after the margin.. Editio Princeps of the First Systematic Treatise on Algebra. First edition of the Greek text extremely rare large-paper copy of the foundational work of algebra and the book in whose margins Pierre de Fermat wrote the most celebrated annotations in the history of mathematics. Before this volume appeared from the Paris press of Hieronymus Drouart in 1621 the Arithmetica of Diophantus of Alexandria existed in print only in Latin: in Wilhelm Holzmann's pioneering but rough 1575 Basel translation - its translator who hellenized his name as Xylander was a humanist who had taken up algebra as a hobby - and in the partial adaptations Rafael Bombelli had incorporated into his 1572 Algebra after reading a Greek manuscript in the Vatican Library. Claude Gaspard Bachet de MÃ©ziriac a country gentleman of Bas-Quercy extraction with classical tastes and no professional mathematical training had come to number theory through mathematical recreations - the puzzles of the Greek Anthology and the Renaissance tradition of mathematical amusement that he had collected in his ProblÃ¨mes plaisants et dÃ©lectables qui se font par les nombres Lyon 1612 - and from there to Diophantus. He spent several years before 1621 establishing a corrected Greek text improving Xylander's Latin translation where Xylander had failed to understand his source filling the lacunae of the defective archetype from which all surviving manuscripts descend identifying and correcting numerical errors generalising Diophantus's procedures and appending three books of his own Porismata. The result is both the editio princeps of the Greek text and the standard scholarly edition unsurpassed until Paul Tannery's Teubner Diophantus of 1893-1895 and the single most consequential textual achievement in the early-modern reception of Greek mathematics. The copy offered here preserves the edition in its most ambitious form. Ordinary copies of the 1621 Bachet measure approximately 337 by 218 millimetres; this copy measures 353 by 225 placing it squarely within the small large-paper issue that Drouart ran alongside the ordinary impression. It is bound in its original yapped vellum the overhanging edges still intact and flexible; the edges of the text block are sprinkled red in the French manner of the early seventeenth century; the spine carries a handwritten manuscript title in Greek and Latin - Diophanti Alexandreos Arithmetikon kai peri polygonon arithmon bibl. followed by cum Commentariis Cl. Gasp. Bacheti - in a plainly contemporary hand the vellum itself serving as the label. The sheets are fresh crisp and unpressed. The title page printed in red and black carries a fine engraved vignette of a flowering thistle within an oval frame a cherub at either side of the cartouche and two satyrs at the base among fruits and foliage surrounded by the dividing mottoes si frote patere aut and ne tan abstine. Two imprints of the 1621 edition were issued simultaneously one under the name of Hieronymus Drouart sub Scuto Solari and one under the name of SÃ©bastien Cramoisy identical in every other respect; the sheets are from the same setting and no priority between the two has been established. The present copy is the Drouart imprint. The Arithmetica itself is the foundational work of Greek algebra. Of the thirteen books that Diophantus's introduction - addressed to one Dionysius arguably Saint Dionysius of Alexandria - promises the Byzantine tradition preserved six in Greek and those are the books printed here. A ninth-century Arabic translation by QustÄ ibn LÅ«qÄ supplying four further books was discovered by Jacques Sesiano in the Astan-i Quds library at Meshed Iran in 1968 - one of the great manuscript finds of the twentieth century. The consensus that has emerged from Sesiano's edition is that the Arabic books correspond to Diophantus's original Books IV through VII and that the six preserved Greek books must be renumbered as Books I through III and after a lacuna Books VIII through X; Books XI through XIII are irretrievably lost. What Bachet had before him in 1621 and what Fermat in his turn would read and annotate was therefore the Byzantine six-book corpus: 189 problems in indeterminate analysis closing with a fragment on the theory of polygonal numbers cast in the older geometrical idiom. The introduction explains Diophantus's symbolism which is the first and only occurrence of algebraic notation anywhere in surviving Greek mathematics. He uses abbreviated signs for the unknown quantity corresponding to the modern x and for its powers up to the sixth for subtraction and for equality. The symbols are scribal abbreviations rather than arbitrary conventions but they function as an effective algebraic language intermediate between the purely verbal mathematics of his Greek predecessors and the fully symbolic algebra that would emerge in the sixteenth century with ViÃ¨te and in the seventeenth with Descartes. Diophantus teaches the multiplication of positive and negative terms and the reduction of an equation to one with only positive terms - the standard form preferred in antiquity which treats negative coefficients as impermissible answers to be moved across the equals sign rather than as legitimate quantities in their own right. Throughout the work he uses the word arithmos rendered by Bachet as numerus to mean what would now be called a positive rational; negative and irrational solutions are never acknowledged. The Arithmetica is a collection of approximately two hundred and sixty problems in what is now called indeterminate analysis - the search for rational solutions to polynomial equations that have more unknowns than equations and that therefore admit in general infinitely many solutions to be found by ingenuity rather than algorithm. The problems of Book I are mostly simple illustrations of the algebraic reckoning that Diophantus has just established. The distinctive features of his method emerge in Books II and III which became the seedbed of modern number theory. In three problems of Book II - the first of them the celebrated Problem II.8 on dividing a given square into two squares whose rational solutions are the Pythagorean triples - Diophantus shows how to represent any given square as a sum of two rational squares; any given non-square that is itself the sum of two known squares as a sum of two other squares; and any given rational number as the difference of two squares. The second of these problems presupposes knowledge of one decomposition hinting that not every integer admits such a decomposition - a question Diophantus later addresses by giving the correct necessary condition: the number must not contain a prime factor of the form 4n 3 raised to an odd power. Diophantus states this condition without proof. It was taken up by Fermat proved by Euler and generalised by Gauss into the representation theory that occupies more than half of the Disquisitiones Arithmeticae. The Arabic books although not available to Bachet confirm the architecture of Diophantus's project as Bachet had intuited it. Their prefaces state that their purpose is to provide the reader with experience and skill and they extend the basic methods of Books I through III to problems of higher degree reducible to binomial equations. The former Greek Books IV and V now Books VIII and IX solve more demanding problems: one decomposes a given integer into two squares arbitrarily close to each other; another decomposes an integer into three squares excluding the impossible case of integers of the form 8n 7 - a result Diophantus asserts but does not prove and that would not be proved until the eighteenth century. The former Book VI now Book X treats right-angled triangles with rational sides subject to various further conditions. The work closes with a fragment of a separate treatise on polygonal numbers - those that can be arranged as regular polygons of dots: triangular numbers nn 1/2 the squares the pentagonals n3n âˆ’ 1/2 and so on - differentiated from the Arithmetica proper by its use of geometrical proofs rather than algebraic methods and breaking off in the middle of an investigation of how many ways a given integer can be a polygonal number. The textual transmission of the Arithmetica is itself a chapter in the history of the book. In Byzantium where the Greek archetype was preserved Michael Psellus in the eleventh century saw what was perhaps the only surviving copy; Georgius Pachymeres 1240-1310 wrote a paraphrase of Book I; Maximus Planudes c. 1255-1310 wrote a commentary on Books I and II. Cardinal Bessarion rescued the manuscript from Constantinople before its fall in 1453 and Regiomontanus discovered it at Venice about 1463 proposing to make a Latin translation that he never produced. For a century thereafter nothing further was heard of Diophantus. He was rediscovered by Rafael Bombelli 1526-1572 the engineer from Bologna whose day job was draining the Chiana marshes and who read a Greek manuscript of the Arithmetica in the Vatican Library about 1570 translated most of it into Italian for his own use and incorporated one hundred and forty-seven of its problems eighty-one with the same numerical values into his Algebra of 1572 - the book that introduced complex numbers into European mathematics and that Bombelli substantially revised before publication as a consequence of his encounter with Diophantus. Three years later the first complete Latin translation by Wilhelm Xylander appeared at Basel; it was the basis for a free French rendering of the first four books by Simon Stevin 1585. ViÃ¨te drew thirty-four problems from Diophantus for his Zetetica of 1593 restricting himself to those that did not violate his principle of homogeneity of dimension. Bachet's 1621 edition superseded every one of these earlier engagements. He studied the text with a thoroughness that no mere philologist could have equalled: Xylander had all too often failed to make sense of corrupt passages where Bachet succeeded because Bachet could read the mathematics as well as the Greek. He filled lacunae identified and corrected the numerical errors that earlier scribes and Xylander after them had introduced generalised the procedures and devised new problems continuing Diophantus's programme. His great critic and admirer AndrÃ© Weil whose Number Theory: An Approach through History from Hammurapi to Legendre BirkhÃ¤user 1984 remains the standard scholarly account observed that Samuel Fermat's praise of Bachet in the preface to the 1670 reprint was by no means excessive and that Bachet's apparent disadvantage - his imperfect grasp of the new symbolic algebra of ViÃ¨te - may actually have benefited number theory in the end. Because he could not readily translate Diophantus into the algebraic language then emerging Bachet laid emphasis on those aspects of the text that were most properly arithmetical and prominently among these on questions regarding the decomposition of integers into sums of squares. It was Bachet who asked for the conditions under which an integer is a sum of two or of three squares and who extracted from Diophantus the conjecture that every integer is a sum of four squares and asked for a proof. The questions passed directly to Fermat and through Fermat to Euler Lagrange and Gauss. Fermat acquired his copy of the Bachet Diophantus in the mid-1630s probably through the circle of Carcavi and Mersenne in Paris. Over the course of his long judicial career at Toulouse and Castres he annotated it with forty-eight marginal observations each written against a particular problem in Bachet's text that together constitute the founding documents of modern number theory. They include the two-square theorem which states that every prime congruent to one modulo four is the sum of two squares in essentially one way; the conjecture later proved by Lagrange that every integer is the sum of four squares; Fermat's Little Theorem that for any prime p and any integer a not divisible by p the quantity a raised to the power p âˆ’ 1 is congruent to one modulo p; the method of infinite descent as a rigorous technique for negative existence proofs; and - written in the margin of Problem II.8 at page 85 of this edition adjacent to Diophantus's treatment of the decomposition of a square into two squares - the claim that no cube can be decomposed into two cubes no fourth power into two fourth powers and in general no power higher than the second into two powers of the same kind together with the famous remark that he had discovered a truly marvellous proof of this proposition which the narrowness of the margin could not contain. The proposition is Fermat's Last Theorem. It resisted proof for three hundred and fifty-eight years until Andrew Wiles closed it in a one-hundred-page paper in the Annals of Mathematics in 1995 using techniques - the theory of modular forms the arithmetic of semistable elliptic curves the Galois representations emerging from the Langlands programme - that Fermat could not have envisaged. Fermat's annotated copy the physical object on which he wrote those forty-eight notes has been lost for over three centuries; its contents survive because ClÃ©ment-Samuel de Fermat working from his father's papers in the years after the elder Fermat's death in 1665 transcribed the observations and printed each at the appropriate point in Bachet's text in the 1670 Toulouse reprint the companion item in the present catalogue. Every surviving copy of the 1621 Bachet therefore bears in a material sense the weight of Fermat's missing copy: these are the sheets from the same setting of the same edition in the same Greek and Latin and with the same commentary against which Fermat was reading and writing. The decisive turn in the reception of Fermat's programme came a century after his death when Leonhard Euler produced the proof of the Last Theorem for exponent three in 1770 invoking the method of descent Fermat had developed for other purposes. Sophie Germain in the first decade of the nineteenth century opened a substantial class of exponents; Dirichlet and Legendre settled exponent five in 1825; Kummer's introduction of ideal numbers in the 1840s addressed to the failure of unique factorisation in the relevant cyclotomic integers founded algebraic number theory and proved Fermat's proposition for all regular primes. By the late nineteenth century the Last Theorem had become the most celebrated unsolved problem in mathematics and the whole of twentieth-century algebraic number theory and arithmetic geometry can be read as a cumulative response to the challenge Fermat set down in the margin of a page that is printed in this very edition. The two-square theorem the four-square theorem the little theorem and the sum of three triangular numbers all passed through similar cycles of Fermat's claim delayed demonstration and subsequent theoretical elaboration - and each of them like the Last Theorem itself began its public life as a marginal note in Fermat's copy of Bachet's 1621 Diophantus. The Arab reception of Diophantus although largely lost is documented by bibliographical sources that Bachet could not have known. Al-NadÄ«m's Fihrist of 987/988 records that QustÄ ibn LÅ«qÄ c. 900 wrote a commentary on three and a half books of the Arithmetica and that AbÅ«'l-WafÄ' 940-998 wrote both a commentary and a book of proofs of the propositions Diophantus had used; a commentary by Ibn al-Haytham with marginal notations by Ibn YÅ«nus is also attested but has not survived. The most substantial surviving witness to the Arab Diophantus is the algebra of al-KarajÄ« in early eleventh-century Baghdad which absorbed a third of the problems from Diophantus's Book I all the problems from Book II beginning with II.8 and almost all of Book III. In the Latin West problems of Diophantine type first appeared in Leonardo of Pisa's Liber Abbaci of 1202 transmitted from Arabic sources during Leonardo's journeys around the Mediterranean. The long chain from the lost Greek archetype through Byzantium Venice Rome Basel and Paris to the copies of Bachet's 1621 edition that passed into the working libraries of seventeenth-century mathematicians is reconstructed here in a single volume. Bachet himself having seen the Arithmetica through the press in 1621 retired to his country estate married and apparently gave up all mathematical activity beyond a second edition of his ProblÃ¨mes plaisants that incorporated material originally intended for a treatise on arithmetic which he never wrote. He died in 1638 three years before his election to the AcadÃ©mie franÃ§aise and a generation before his work would reach its full consequence through Fermat's marginalia. Almost nothing is known about the life of Diophantus himself. He quotes Hypsicles fl. c. 150 BC and is quoted by Theon of Alexandria c. AD 364 and a date around AD 250 is generally accepted; that would place him in late Hellenistic Alexandria at the chronological edge of the tradition of Greek mathematical creativity that had begun with Thales. The Byzantine epigram that gives his age at death as eighty-four extrapolated from a series of Diophantine conditions on the years of his boyhood youth marriage and the birth and death of his son is textually reliable but biographically useless. What survives of him is the mathematics. The book's working life in the seventeenth and early eighteenth centuries can be measured by the hands through which it passed. Isaac Newton owned a copy Harrison 524 now at Trinity College Cambridge and referred to Diophantus in his mathematical notebooks. Leibniz studied the work. Euler began his revival of Fermat's number-theoretic programme in the 1730s by working through the Observationes in the 1670 reprint of precisely this edition. The continuous research that culminated in Gauss's Disquisitiones Arithmeticae of 1801 and from there in the whole nineteenth-century elaboration of algebraic and analytic number theory runs back through Euler and Lagrange to Fermat's marginalia and through Fermat's marginalia to Bachet's 1621 printing of the Greek text. Large-paper copies such as the one offered here were produced in small numbers - probably as presentation copies for Bachet's dedicatees and for purchase by serious mathematicians who wanted the generous margins for their own annotations. The sheets of this copy are unpressed and the state of preservation exceptional; the contemporary yapped vellum binding with its handwritten Greek and Latin spine title in a seventeenth-century hand is the same working binding in which the volume left the Paris trade in 1621. References: Honeyman 891 - Smith Rara Arithmetica pp. 348 and 368 - Brunet II 702 - Weil Number Theory: An Approach through History from Hammurapi to Legendre BirkhÃ¤user 1984 chapters I-III - Heath Diophantus of Alexandria: A Study in the History of Greek Algebra Cambridge second edition 1910; Dover reprint 1964 - Sesiano Books IV to VII of Diophantus' Arithmetica in the Arabic Translation attributed to QustÄ ibn LÅ«qÄ Springer 1982 - Bashmakova Diophantus and Diophantine Equations Mathematical Association of America 1997 - Mahoney The Mathematical Career of Pierre de Fermat Princeton University Press second edition 1994 - Goldstein Un thÃ©orÃ¨me 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. Folio 353 Ã— 225 Ã— 40 mm pp. 12 32 451 1 blank 58 2 errata. Title printed in red and black with large engraved allegorical vignette flowering thistle within oval cartouche cherubs above and satyrs below mottoes si frote patere aut and ne tan abstine. Greek and Latin in parallel columns throughout the Arithmetica. Woodcut historiated initials. Book headings in large capitals Greek capitals for the Greek heads. Separate signatures and pagination for Bachet's Porismatum libri tres pp. 1-32 and for the polygonal numbers fragment pp. 1-58 at the end followed by two leaves of errata. Contemporary yapped vellum the overhanging edges intact edges of the text block sprinkled red handwritten manuscript title on the spine in Greek and Latin in a period hand directly on the vellum. A fine crisp fresh unpressed copy with the full generous margins of the large-paper issue. The Drouart imprint; the Cramoisy imprint issued simultaneously from the same sheets differs only in the bookseller's name on the title page and no priority between the two has been established. . Hieronymus Drouart unknown

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FERMAT, Pierre de; BACHET DE MEZIRIAC, Claude-Gaspard; DIOPHANTUS OF ALEXANDRIABiblio.com

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

16706238Toulouse: Bernard Bosc 1670. First edition. First edition large-paper issue with the rare engraved portrait of Pierre de Fermat by François Poilly - rare in this edition - and with the editor's presentation inscription on the title page: de Molieres ex dono authoris placing this copy with Louis de Molières Pierre de Fermat's brother-in-law and trésorier de France at Montauban in the year of publication. Prepared by Clément-Samuel de Fermat from his father's marginalia on Bachet's Diophantus the volume prints Fermat's forty-eight number-theoretic observations among them at page 61 the editio princeps of Fermat's Last Theorem - the marginal claim that no power above the second decomposes into two like powers and that a marvellous proof exists which the margin cannot contain. The theorem held for three hundred and fifty-eight years generating algebraic number theory the arithmetic of elliptic curves and modular forms in the course of the search for its proof until Wiles closed it in 1995. A contemporary hand has attempted corrections to Fermat's observation on cube differences at page 135.. Box: 378 x 264 x45 mm. The First Printing of Fermat's Last Theorem. 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. . Bernard Bosc unknown

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