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 [by J. de Billy] ex varijs ... Fermat epistolis [ed. Samuel de Fermat].

Toulouse: Bernard Bosc, 1670.

First edition of Fermat’s annotated edition of Diophantus’ Arithmetica. This is the first printing of Fermat’s contributions to the theory of numbers, of which he is the undisputed founder, including his famous statement of ‘Fermat’s last theorem.’ Since most of Fermat’s work in number theory remained unpublished in his lifetime, “it was neither understood nor appreciated until Euler revived it and initiated the line of continuous research that culminated in the work of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” (DSB). Fermat showed little interest in publishing his work, which remained confined to his correspondence, personal notes, and to marginal jottings in his 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 Clement-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 “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 marvellous demonstration” of this proposition, but that the margin was too narrow to transcribe it. This simple statement became known as the single most difficult problem in mathematics, and for over 300 years no mathematician succeeded in either proving or disproving it. In 1995 Andrew Wiles, professor of mathematics at Princeton, who had been obsessed with Fermat’s last theorem since the age of 10, completed a 130-page proof (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, Fermat’s Enigma (1997), p. 279). Although Fermat’s marginal jottings in Diophantus hold a special place in the history of mathematics, much of what we know of Fermat’s methods of proof is found in his letters to the French Jesuit Jacques de Billy, a pupil of Bachet, printed for the first time in the present work as Doctrinae Analyticae Inventum Novum.

Almost nothing is known about the life of Diophantus. His place of birth is unknown and his arrival in Alexandria could have been at any time within a five-century period. In his writings Diophantus quotes Hyspicles and therefore must have lived after 150 BC; on the other hand his own work is quoted by Theon of Alexandria and therefore he must have lived before AD 364. A date around AD 250 is generally accepted. “The Arithmetica is essentially a logistical work, but with the difference that Diophantus’ problems are purely numerical with the single exception of problem V, 30. In his solutions Diophantus showed himself a master in the field of indeterminate analysis, and apart from Pappus he was the only great mathematician during the decline of Hellenism” (DSB).

“The Arithmetica begins with an introduction addressed to Dionysius—arguably St. Dionysius of Alexandria. After some generalities about numbers, Diophantus explains his symbolism—he uses symbols for the unknown (corresponding to our x) and its powers, positive or negative, as well as for some arithmetic operations—most of these symbols are clearly scribal abbreviations. This is the first and only occurrence of algebraic symbolism before the 15th century. After teaching multiplication of the powers of the unknown, Diophantus explains the multiplication of positive and negative terms and then how to reduce an equation to one with only positive terms (the standard form preferred in antiquity). With these preliminaries out of the way, Diophantus proceeds to the problems. Indeed, the Arithmetica is essentially a collection of problems with solutions, about 260 in the part still extant.

The introduction also states that the work is divided into 13 books. Six of these books were known in Europe in the late 15th century, transmitted in Greek by Byzantine scholars and numbered from I to VI; four other books were discovered in 1968 in a 9th-century Arabic translation by Qusṭā ibn Lūqā. However, the Arabic text lacks mathematical symbolism, and it appears to be based on a later Greek commentary—perhaps that of Hypatia (c. 370–415)—that diluted Diophantus’s exposition. We now know that the numbering of the Greek books must be modified: Arithmetica thus consists of Books I to III in Greek, Books IV to VII in Arabic, and, presumably, Books VIII to X in Greek (the former Greek Books IV to VI). Further renumbering is unlikely; it is fairly certain that the Byzantines only knew the six books they transmitted and the Arabs no more than Books I to VII in the commented version.

“The problems of Book I are not characteristic, being mostly simple problems used to illustrate algebraic reckoning. The distinctive features of Diophantus’s problems appear in the later books: they are indeterminate (having more than one solution), are of the second degree or are reducible to the second degree (the highest power on variable terms is 2, i.e., x2), and end with the determination of a positive rational value for the unknown that will make a given algebraic expression a numerical square or sometimes a cube. (Throughout his book Diophantus uses “number” to refer to what are now called positive, rational numbers; thus, a square number is the square of some positive, rational number.) Books II and III also teach general methods. In three problems of Book II it is explained how to represent: (1) any given square number as a sum of the squares of two rational numbers; (2) any given non-square number, which is the sum of two known squares, as a sum of two other squares; and (3) any given rational number as the difference of two squares. While the first and third problems are stated generally, the assumed knowledge of one solution in the second problem suggests that not every rational number is the sum of two squares. Diophantus later gives the condition for an integer: the given number must not contain any prime factor of the form 4n + 3 raised to an odd power, where n is a non-negative integer. Such examples motivated the rebirth of number theory. Although Diophantus is typically satisfied to obtain one solution to a problem, he occasionally mentions in problems that an infinite number of solutions exists.

“In Books IV to VII Diophantus extends basic methods such as those outlined above to problems of higher degrees that can be reduced to a binomial equation of the first- or second-degree. The prefaces to these books state that their purpose is to provide the reader with “experience and skill.” While this recent discovery does not increase knowledge of Diophantus’s mathematics, it does alter the appraisal of his pedagogical ability. Books VIII and IX (presumably Greek Books IV and V) solve more difficult problems, even if the basic methods remain the same. For instance, one problem involves decomposing a given integer into the sum of two squares that are arbitrarily close to one another. A similar problem involves decomposing a given integer into the sum of three squares; in it, Diophantus excludes the impossible case of integers of the form 8n + 7 (again, n is a non-negative integer). Book X (presumably Greek Book VI) deals with right-angled triangles with rational sides and subject to various further conditions” (Britannica).

The work closes with a fragment of a treatise on ‘polygonal numbers’, which were numbers were of special interest to the Pythagoreans. The polygonal numbers are the numbers of dots which can be arranged in the shape of a regular polygon: for example, the triangular numbers are the numbers n(n + 1)/2 for n = 1, 2, 3, …; the square numbers are (of course) the squares; the pentagonal numbers are n(3n – 1)/2, etc. This text “is immediately differentiated from the Arithmetica by its use of geometric proofs. The first section treats several lemmas on polygonal numbers, a subject already long known to the Greeks. The definition of these numbers is new; it is equivalent to that given by Hypsicles, which Diophantus cites … The work breaks off during the investigation of how many ways a number p can be a polygonal number” (DSB).

“In their endeavor to acquire the knowledge of the Greeks, the Arabs—relatively late, it is true—became acquainted with the Arithmetica. Al Nadīm (987/988) reports in his index of the sciences that Qusṭā ibn Lūqā (ca. 900) wrote a Commentary on Three and One Half Books of Diophantus’ Work on Arithmetical Problems and that Abū’l-Wafāʾ (940–988) likewise wrote A Commentary on Diophantus’ Algebra, as well as a Book on the Proofs of the Propositions Used by Diophantus and of Those That He Himself [Abu’l-Wafā’] Has Presented in His Commentary. These writings, as well as a commentary by Ibn al-Haytham on the Arithmetica (with marginal notations by Ibn Yūnus), have not been preserved. On the other hand, Arab texts do exist that exhibit a concern for indeterminate problems. An anonymous manuscript (written before 972) treats the problem x2 + n = u2, x2 - n = v2; a manuscript of the same period contains a treatise by al-Ḥusain (second half of tenth century) that is concerned with the theory of rational right triangles. But most especially, one recognizes the influence of Diophantus on al-Karajī. In his algebra he took over from Diophantus’ treatise a third of the exercises of book I; all those in book II beginning with II, 8; and almost all of book III …

“Problems of the type found in the Arithmetica first appeared in the West in the Liber abbaci of Leonardo of Pisa (1202); he undoubtedly became acquainted with them from Arabic sources during his journeys in the Mediterranean area. A Greek text of Diophantus was available only in Byzantium, where Michael Psellus saw what was perhaps the only copy still in existence. Georgius Pachymeres (1240–1310) wrote a paraphrase with extracts from the first book, and later Maximus Planudes (ca. 1255–1310) wrote a commentary to the first two books. Among the manuscripts that Cardinal Bessarion rescued before the fall of Byzantium was that of Diophantus, which Regiomontanus discovered in Venice. His intention to produce a Latin translation was not realized. Then for a century nothing was heard about Diophantus. He was rediscovered by Bombelli, who in his Algebra of 1572, which contained 271 problems, took no fewer than 147 from Diophantus, including eighty-one with the same numerical values. Three years later the first Latin translation, by Xylander, appeared in Basel; it was the basis for a free French rendering of the first four books by Simon Stevin (1585). Viète also took thirty-four problems from Diophantus (including thirteen with the same numerical values) for his Zetetica (1593); he restricted himself to problems that did not contradict the principle of dimension. Finally, in 1621 the Greek text was prepared for printing by Bachet de Méziriac, who added Xylander’s Latin translation, which he was able to improve in many respects. Bachet studied the contents carefully, filled in the lacunae, ascertained and corrected the errors, generalized the solutions, and devised new problems. He, and especially Fermat, who took issue with Bachet’s statements, thus became the founders of modern number theory, which then—through Euler, Gauss, and many others—experienced an unexpected development” (DSB).

“For a mathematician of the sixteenth century, Diophantus was no easy text to decipher, and no small share of the credit for its re-discovery must go to Rafael Bombelli (1526-72) and to Xylander (1532-76). Bombelli read and translated most of it in Rome, for his own use, about 1570, and then incorporated this into his Italian Algebra of 1572; Xylander was the first to attempt a complete translation, and published the fruit of his efforts in Basel in 1575. Of course the substantial difficulties of such undertakings were greatly increased by the comparatively poor condition of the available manuscript texts. As has been the case with virtually all Greek classical authors, these were all derived from a single codex (the so-called “archetype”, now lost) marred by copying mistakes and omissions. Worst of all, in the case of Diophantus, were the numerical errors. Undoubtedly the copying had been done by professional scribes, not by mathematicians; but perhaps this was just as well; would-be mathematicians might have made things even worse.

“The first translator, [Wilhelm] Holzmann, who hellenized his name as Xylander, was a humanist and a classical scholar who took up algebra as a hobby. Bombelli was a busy engineer, designing canals, desiccating marshes, and he dedicated to mathematics little more than those periods of leisure which were granted to him by his munificent employer and patron, the bishop of Melfi … Thus it was still a huge task that awaited the future editor, translator and commentator of Diophantus, even after all the work done by his predecessors. As Fermat’s son Samuel expressed it in his preface to the Diophantus of 1670 (echoing, no doubt, his father’s sentiments, perhaps his very words): “Bombelli, in his Algebra, was not acting as a translator for Diophantus, since he mixed his own problems with those of the Greek author; neither was Viete, who, as he was opening up new roads for algebra, was concerned with bringing his own inventions into the limelight rather than with serving as a torch-bearer for those of Diophantus. Thus it took Xylander's unremitting labors and Bachet's admirable acumen to supply us with the translation and interpretation of Diophantus's great work”.

“Claude Gaspar Bachet, sieur de Meziriac, was a country gentleman of independent means, with classical tastes, and no mathematician. Somehow he developed an interest for mathematical recreations and puzzles of the kind found in many epigrams of the Greek Anthology as well as in medieval and Renaissance mathematical texts, or nowadays in the puzzle columns of our newspapers and magazines. In 1612 he published in Lyon a collection of such puzzles under the title Problèmes plaisants et délectable quise font par les nombres. As this indicates, he was thus led to number theory, and so to Diophantus. The latter must have occupied him for several years prior to its publication in 1621; after seeing it through the press he retired to his country estate, got married, and apparently gave up all mathematical activity, except that he prepared a second edition of his Problèmes of 1612, incorporating into it some of the materials he had intended for a treatise on arithmetic which never saw the light of day.

“Samuel Fermat’s praise of Bachet was by no means excessive. No mere philologist would have made sense out of ever so many corrupt passages in the manuscripts; Xylander had all too often failed to do so. Bachet never tires of drawing attention to the defects of Xylander’s translation and comments, while naively extolling his own merits. He even ventures to speak disparagingly of Viète’s algebraic methods, which he neither appreciated nor understood; this did not stop him from lifting two Porisms and some important problems about cubes out of Viète’s Zetetica without a word of acknowledgment. Nevertheless, his is the merit of having provided his successors and notably Fermat with a reliable text of Diophantus along with a mathematically sound translation and commentary. Even his lack of understanding for the new algebra may be said to have benefited number theory in the end … he invariably laid the emphasis on those aspects of the text which were more properly arithmetical, and, prominently among these, on all questions regarding the decomposition of integers into sums of squares. He asked for the conditions for an integer to be a sum of two or of three squares; he extracted from Diophantus the conjecture that every integer is a sum of four squares, and asked for a proof. The curtain could rise; the stage was set. Fermat could make his entrance” (Weil, Number Theory, pp. 31-34).

Fermat’s marginal notes in his copy of Bachet’s Diophantus probably date from the late 1630s. We mention only four. “In a note in his copy of Diophantus and in a letter to Mersenne, Fermat generalized on the well-known 3, 4, 5 right triangle relationship by asserting the following theorems: A prime of the form 4n + 1 is the hypotenuse of one and only one right triangle with integral arms; the square of 4n + 1 is the hypotenuse of two and only two such right triangles; its cube, of three; its biquadrate, of 4; and so on, ad infinitum. As an example, consider the case of n = 1. Then 4n + 1 = 5 and 3, 4, 5 are the sides of the one and only right triangle with 5 as hypotenuse. However, 52 is the hypotenuse of two, and only two, right triangles 15, 20, 25 and 7, 24, 25. Also 53 is the hypotenuse of three, and only three, right triangles 75, 100, 125; 35, 120, 125; and 44, 117, 125” (Kline, Mathematical Thought from Ancient to Modern Times, p. 275).

A further theorem on right-angled triangles with integer sides stated by Fermat is that the area of such a triangle cannot be a square. Just in this one case Fermat did find room in the margin for the proof, next to the very last proposition of Diophantus (pp. 338-9). The proof, which is by a method Fermat called “infinite descent,” is explained, in modern terminology, in Weil, Number Theory, p. 77.

A second set of problems concerns polygonal numbers, the numbers of dots which can be arranged in the shape of a regular polygon. For example, the triangular numbers are the numbers n(n + 1)/2 for n = 1, 2, 3, …; the square numbers are (of course) the squares; the pentagonal numbers are n(3n – 1)/2, etc. “On polygonal numbers Fermat stated in his copy of Diophantus the important theorem that every positive integer is itself triangular or the sum of 2 or 3 triangular numbers; every positive integer is itself square or a sum of 2, 3, or 4 squares; every positive integer is either pentagonal or a sum of 2, 3, 4, or 5 pentagonal numbers; and so on for higher polygonal numbers” (Kline, p. 277). Weil is doubtful that Fermat could have proved these assertion: “no suggestion can be offered at present as to how Fermat could possibly have proved that every integer is a sum of three triangular numbers, and one cannot help thinking that on this point he may have deceived himself” (Weil, Review, p. 1148).

The most famous marginal note is, of course, that which accompanies Diophantus’ Proposition II, 8 (p. 61), “To divide a given square number into two squares,” for which Diophantus gives the answer (in our notation)

[a(m2 + 1)]2 = (2am)2 + [a(m2 - 1)]2.

Fermat adds: “In contrast, it is impossible to divide a cube into two cubes, or a fourth power into two fourth powers, or in general any power beyond the square into powers of the same degree; of this I have discovered a very wonderful demonstration. This margin is too narrow to contain it.”

In other words, the equation xn + yn = zn has no solution with x, y, z being positive integers and n an integer greater than 2. Fermat gave the proof for n = 4 using infinite descent: in fact, it is an easy consequence of his result about the area of right-angled triangles with integer sides described above. To prove the result in general it is now sufficient to prove it when n is an odd prime number (if x, y, z is a solution with n = pq, then xp, yp, zp is a solution with n = q). The case n = 3 was proved by Euler by infinite descent, and Weil suggests that Euler’s proof may have been within Fermat’s compass. Fermat had in fact posed these two cases as a challenge to the English mathematicians (notably Brouncker and Wallis) in 1657, so it may be reasonable to suppose that he had proofs at that time.

Over the next two centuries (1637–1839), the conjecture was proved for only the primes 3, 5, and 7, although Sophie Germain devised an approach that was relevant to an entire class of primes. In the mid-19th century, Ernst Kummer extended this and proved the theorem for all ‘regular’ primes, leaving irregular primes to be analyzed individually. Building on Kummer’s work and using sophisticated computer studies, other mathematicians were able to extend the proof to cover all prime exponents up to four million.

In 1908, the German industrialist and amateur mathematician Paul Wolfskehl bequeathed 100,000 marks to the Göttingen Academy of Sciences to be offered as a prize for a complete proof of Fermat's Last Theorem. On 27 June 1908, the Academy published nine rules for awarding the prize. Among other things, these rules required that the proof be published in a peer-reviewed journal; the prize would not be awarded until two years after the publication; and that no prize would be given after 13 September 2007, roughly a century after the competition was begun. In the first year alone (1907–1908), 621 attempted proofs were submitted, although by the 1970s, the rate of submission had decreased to roughly 3–4 attempted proofs per month. According to F. Schlichting, a Wolfskehl reviewer, most of the proofs were based on elementary methods taught in schools, and often submitted by “people with a technical education but a failed career.” In the words of mathematical historian Howard Eves, “Fermat's Last Theorem has the peculiar distinction of being the mathematical problem for which the greatest number of incorrect proofs have been published.”

Around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama suspected a link might exist between elliptic curves and modular forms, two completely different areas of mathematics. Known at the time as the Taniyama–Shimura-Weil conjecture, and (eventually) as the modularity theorem. In 1984, Gerhard Frey noticed an apparent link between the modularity theorem and Fermat’s Last Theorem. This potential link was confirmed two years later by Ken Ribet, who gave a conditional proof of Fermat’s Last Theorem that depended on the modularity theorem. On hearing this, English mathematician Andrew Wiles decided to try to prove the modularity theorem as a way to prove Fermat’s Last Theorem. In 1993, after six years working secretly on the problem, Wiles succeeded in proving enough of the modularity theorem to prove Fermat’s Last Theorem for odd prime exponents. A flaw was discovered in one part of his original paper during peer review and required a further year and collaboration with a past student, Richard Taylor, to resolve. As a result, the final proof in 1995 was accompanied by a second smaller joint paper. Wiles collected the Wolfskehl prize money, then worth \$50,000, on 27 June 1997.

In his dedication to Colbert (complete with handsome engraved head-piece and initial), Fermat’s son states that his father’s observations on Diophantus may be “mole quidem exiguas, se pondere, ni fallor, maiores.” The great Paul Tannery in his Teubner text (Diophanti Alexandrini Opera omnia: cum graecis commentariis, vol. I, p. v, 1895) is extremely dismissive: “Negligenda erat Fermatiana [editio] (Tolosae, 1670) quae textum eundem [as Bachet] mendose repetivit.” However, Bachet’s edition was not easily available, and this edition was eagerly sought at the time. In a letter, John Collins remarks how difficult it was to obtain a copy of this work, which, he tells us, was only sent to Paris in a small number of copies. However, by August 1672, it was available in London (Macclesfield sale catalogue).

Honeyman 893; Hoffman II, p.109; Macclesfield 638 (large paper copy); Norman 777; Smith, Rara arithmetica, p. 348. Weil, Review of Mahoney (ibid.), Bulletin of the American Mathematical Society 79 (1973), pp. 1138-49. Weil, Number Theory: An Approach Through History from Hammurapi to Legendre, 1984.

Folio (354 x 235 mm), pp. [xii], 341; 48, with engraved portrait frontispiece and numerous woodcut diagrams (occasional browning as usual with this book). Contemporary vellum with (later?) leather spine label (upper corners worn, small split to upper joint, label chipped). A very large, unpressed copy.

Item #4489

Price: \$50,000.00