## Rerum arithmeticarum libri sex, quoru[m] primi duo adiecta habent scholia, maximi (ut conjectura est) Planudis. Item liber de numeris polygonis seu multiangulis. Opus incomparabile, veræ arithmeticæ logisticæ perfectionem continens, paucis adhuc visum. A Guil[lelmo] Xylandro Augustano incredibili labore Latinè redditum, & commentariis explanatum, inq[ue] lucem editum. Ad illustriss. principe[m] Ludovicum Vuirtembergensem.

Basel: Eusebius & Heirs of Nikolaus Episcopius, 1575.

First edition, very rare, of the first systematic treatise on algebra, which “inspired the rebirth of number theory” (Britannica), translated from the Greek by Wilhelm Holtzmann [Xylander] – the original Greek text was not published until 1621. “The appearance of [this] translation had an immediate and enormous influence on the development and shaping of Algebra’” (Heath). “The work marks the high point of Alexandrian Greek algebra: Diophantus introduced symbolism into algebra, dealt with powers as high as six ... and delved extensively into the solution of indeterminate equations, founding the branch of algebra now known as Diophantine analysis” (Norman).

.First edition, very rare, of the first systematic treatise on algebra (Smith, *Rara Arithmetica*, p. 348), which “inspired the rebirth of number theory” (Britannica), translated from the Greek by Wilhelm Holtzmann [Xylander] – the original Greek text was not published until 1621. “The appearance of [this] translation had an immediate and enormous influence on the development and shaping of Algebra’” (Heath, *Diophantus of Alexandria* (1910), p. 26). “The work marks the high point of Alexandrian Greek algebra: Diophantus introduced symbolism into algebra, dealt with powers as high as six (in contrast to classical Greek mathematicians, who did not consider powers higher than three), and delved extensively into the solution of indeterminate equations, founding the branch of algebra now known as Diophantine analysis” (Norman). “Xylander was an enthusiast for Diophantus, and his preface and notes are often delightful reading. Unfortunately the book is now very rare” (Heath, *History of Greek Mathematics *(1921), Vol. 2, p. 454). “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 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” (Watkins, *Number Theory*: *A Historical Approach* (2013), pp. 91-2). Xylander’s translation also includes a fragment of the only other work of Diophantus that has come down to us, his treatise on ‘polygonal numbers’ (or ‘figurate numbers’) – the numbers of dots that can be arranged in the shape of a regular polygon. It was famously in his copy of the 1621 reprint of Xylander’s translation that Pierre de Fermat made his marginal annotations, including his statement of ‘Fermat’s last theorem.’ ABPC/RBH list just two other copies since Honeyman, both in modern bindings. OCLC lists 8 copies in North America.

“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 translation by Qusta ibn Luqa. 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., *x*^{2}), 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 4*n* + 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 8*n* + 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.

“The contents of the three missing books of the *Arithmetica* can be surmised from the introduction, where, after saying that the reduction of a problem should “if possible” conclude with a binomial equation, Diophantus adds that he will “later on” treat the case of a trinomial equation—a promise not fulfilled in the extant part” (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*(3*n* – 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 *x*^{2} + *n* = *u*^{2}, *x*^{2} - *n* = *v*^{2}; 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) … 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” (Weil, *Number Theory*: *An Approach Through History from Hammurapi to Legendre*, (1984), p. 31).

“Xylander’s achievement has been, as a rule, quite inadequately appreciated. Very few writers on Diophantus seem to have studied the book itself: a fact which may be partly accounted for by its rarity. Even Nesselmann, whose book [*Die Algebra der Greichen*] appeared in 1842, says that he has never been able to find a copy. Nesselmann, however, seems to have come nearest to a proper appreciation of the value of the work: he says, ‘Xylander’s work remains, in spite of the various defects which are unavoidable in a first edition of so difficult an author, especially when based on only one MS and that full of errors, a highly meritorious achievement … The mathematical public was put in possession of Diophantus’ work, and the appearance of the translation had an immediate and enormous influence on the development and shaping of Algebra’” (Heath, *Diophantus*, pp. 25-6).

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.

Adams D-652; DSB IV, 110-19; Honeyman 890; Norman 641; Macclesfield 636 (this copy). For a detailed analysis of the history, contents, and influence of the Arithmetica, see Chap. 4 of Katz & Parshall, *Taming the Unknown*, 2014. For what is known (and conjectured) about Diophantus and the history of the *Arithmetica*, see: Schappacher, ‘Diophantus of Alexandria: a text and its history’ (http://irma.math.unistra.fr/~schappa/NSch/Publications_files/1998cBis_Dioph.pdf).

Folio (307 x 200 mm), pp. [xii], 152, printers' device with inscription ‘Episcop’ on title, woodcut initials, printed marginal notes. Nineteenth-century half-morocco by Hatton, South Library bookplate on front paste-down (small sliver cut from centre of top of title page, first and last leaves a bit soiled, light foxing).

Item #4710

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Price:
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