## Arithmeticorum libri sex, et de numeris multangulis liber unus. Nunc primum Graece & Latine editi, atque commentariis illustrati. Auctore C. G. Bacheto.

Paris: H. Drouart, 1621.

Editio princeps of the first systematic treatise on algebra (Smith, Rara Arithmetica, p. 348), comprising the first printing of the Greek text, accompanied by Xylander’s Latin translation (1575) and Bachet’s commentary. “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 is a collection of problems – the known Greek books contain 189 problems – and even though the solutions presented by Diophantus are always quite specific, his solutions do tend to suggest general methods. As a result, Diophantus has often been called the father of algebra, in part because of these methods, but also because of the systematic use of notation and terminology that he introduced in this work. For example, even though he did not have the notation we now use for exponents, he nonetheless had his own effective symbolic way of representing polynomials. But the spirit of the Arithmetica has far more in common with modern number theory than with today’s practice of algebra … By 1575, the Arithmetica had been translated into Latin, and subsequently into French, and Rafael Bombelli had substantially revised his Algebra before its publication in 1572 as a result of reading a manuscript of Diophantus in the Vatican library and even included 143 problems taken directly from the Arithmetica. But the most famous translation – and certainly the most important, at least until a definitive translation appeared in the late nineteenth century – was the 1621 edition by Claude Gaspard Bachet de Méziriac” (Watkins, Number Theory: A Historical Approach (2013), pp. 91-2). “It contains a good deal of matter upon the Greek theory of numbers, notably the ‘Clavdii Gasparis Bacheti Sebusiani, in Diophantus Porismatvm, Liber Primus,’ ‘Liber Secundus,’ and ‘Liber Tertius’” (Smith, p. 368). It was famously in his copy of this 1621 edition of Diophantus that Pierre de Fermat made his marginal annotations, including his statement of ‘Fermat’s last theorem.’ Bachet’s edition was reprinted in 1670 by Fermat’s son Clément-Samuel, including his father’s annotations.

“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 Méziriac, 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” (Weil, pp. 31-34).

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.

There are two imprints of this edition; one by Drouart and one by Cramoisy, which are identical except for the title vignette (no priority established).

Honeyman 891. Weil, Number Theory: An Approach Through History from Hammurapi to Legendre, 1984.

Folio (338 x 217 mm), pp. [12], 32, 451, [1, blank], 58, [2, errata]. Title printed in red and black, woodcut vignette (old owner’s signature erased from the lower right corner of title). Contemporary calf with raised bands, old repair to front board, richly gilt spine, red sprinkled edges, very light dampstain to the upper right corner, and slight browning throughout, altogether a very genuine, attractive copy with wide margins.

Item #2441

Price: \$12,500.00