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.
Paris: Hieronymus Drouart, 1621. 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 Méziriac, 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 Problèmes plaisants et délectables 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 Sébastien 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 Qustā ibn Lūqā, 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 Viète 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 n(n + 1)/2, the squares, the pentagonals n(3n − 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). Viète 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 André Weil, whose Number Theory: An Approach through History from Hammurapi to Legendre (Birkhäuser, 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 Viète — 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 − 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 Clément-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-Nadīm’s Fihrist of 987/988 records that Qustā ibn Lūqā (c. 900) wrote a commentary on three and a half books of the Arithmetica, and that Abū’l-Wafā’ (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 Yūnus is also attested but has not survived. The most substantial surviving witness to the Arab Diophantus is the algebra of al-Karajī 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 Problèmes 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 Académie française 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 (Birkhäuser, 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 Qustā ibn Lūqā (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 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. Folio (353 × 225 × 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.
Item #6237
Price: $28,500.00
