DIOPHANTUS of Alexandria.

Rerum Arithmeticarum Libri sex, quoru[m] primi duo adiecta habent scholia, maximi (ut coniectura est) Planudis. Item liber de numeris polygonis seu multiangulis. Opus incomparabile, verae arithmeticae logisticae perfectionem continens, paucis adhuc uisum. A Guil[lelmo] Xylandro Augustano incredibili labore Latine redditum, & commentariis explanatum, inque lucem editum. Ad illustriss. principem Ludovicum Vuirtembergensem.

Basel: Eusebius Episcopius & Heirs of Nikolaus Episcopius, 1575. / Hardcover.

First edition of Diophantus — the first printing of the Arithmetica in any language in any form — owned, annotated, and signed by Giovanni Camillo Gloriosi (1572–1643), Galileo’s successor in the chair of mathematics at the University of Padua, and subsequently acquired from Gloriosi’s estate for five hundred gold coins by Ramiro de Guzmán, Duke of Medina de las Torres and Viceroy of Naples, who bound the book in red morocco with the combined armorial of himself and his wife Anna Carafa de Stigliano on the covers. When Antonio Favaro undertook his 1904 study of Gloriosi in the series on Galileo’s acquaintances and correspondents, he posed what seemed a straightforward question — what had become of the mathematician’s books and papers? — and reported that every effort he had made to trace them had been entirely in vain. The present copy, bound in the viceroy’s unmistakable red morocco and carrying Gloriosi’s signature on the last leaf with extensive mathematical annotations in his hand, is distinguished among the now-identified survivors of that library by the density of the annotations and by their demonstrable bearing on Gloriosi’s own published mathematical work.

Gloriosi was a Neapolitan, Jesuit-trained, and came to mathematics through the algebraic tradition rather than through natural philosophy. In 1604 a friar asked Galileo to write on his behalf for a lectureship in mathematics; the appointment did not materialise, but the correspondence opened a cordial, if occasionally pointed, relationship between the two men. By 1606 Gloriosi was in Venice, moving in the circle around Paolo Sarpi and Giovanfrancesco Sagredo, where he also met Antonio Santini and Marino Ghetaldi, who introduced him to the algebra of François Viète — the immediate and decisive influence on his reading of Diophantus. In October 1613, with Galileo’s recommendation and on the strength of his first publication, Ad theorema geometricum (Venice: Tommaso Baglioni, 1613), Gloriosi was nominated to the chair Galileo had just vacated at Padua. He held the post until 1622, returning thereafter to Naples, where he lived as a private gentleman, maintained correspondence with the mathematical communities at Padua, Venice, Bologna, and the Roman College, and continued to exchange letters with Galileo until at least 1635. He died in January 1643, leaving four surviving letters to Galileo as the record of a thirty-year intellectual acquaintance.

Gloriosi’s annotations transform the copy from a bibliographical rarity into a document of working mathematical scholarship. Dating from 1611 and 1612 — the two years immediately preceding the Ad theorema geometricum and the Padua appointment — they fall into two kinds. Many are brief marginal identifiers, a single Latin word (most often Theorema) placed beside the statement of a particular proposition to fix its status in the flow of Diophantus’s argument. The majority, however, are substantive mathematical calculations in Gloriosi’s fine italic hand, revising, extending, and in a number of cases correcting the work of Diophantus and of the book’s Latin translator and commentator, Wilhelm Holtzmann of Augsburg (Xylander). They are concentrated in Book II and in the later books in which the more intricate indeterminate problems occur, and they are dense enough in places to fill margins on both sides of the printed page. Where Xylander’s own calculation has gone astray, Gloriosi writes out the corrected arithmetic to several orders of fractional precision; where a proposition requires a generalisation Diophantus had not offered, he supplies it.

The most consequential of these annotations occupies the lower margin of page 59, against Proposition 19 of Book II — the problem that asks a given number to be divided into three parts such that each part, on donating a specified fraction of itself plus a fixed number of units to the next, yields three equal results. For the number 80, with the fractions one-fifth, one-sixth, and one-seventh and the added units 6, 7, and 8 respectively, Xylander’s solution failed: the parts summed to 80, but the distribution did not in fact satisfy the equations. Gloriosi noted the failure precisely — aequatio facta est ad 16 2/3 cum fieri debebant ad 26 2/3, the equation had been set at 16 2/3 when it should have been set at 26 2/3 — and then, in the same annotation, recorded the three correct fractional parts (1N 9530/363, 10200/363, and 9310/363) obtained from the corrected equation. He rejected the possibility that Diophantus himself had erred, insisting that the Alexandrian would not have proposed a problem without knowing its solution, and argued instead that the Greek text had been mutilated in transmission. The task he set himself was to recover a solution using, as Diophantus’s method required, a single hypothesis. This is exactly the reconstruction carried out on pages 28–29 of the Ad theorema geometricum of 1613 — the annotation and the printed text correspond point by point in the equation, in the numerator 9530 over the common denominator 363, and in the subsequent reasoning — and the book prints no other source for the reconstruction. The margin of the present copy is the working source.

The causal chain is tight. The marginal notes of 1611–1612 fed into the printed argument of 1613, the printed argument of 1613 supplied the published credential on which Galileo’s recommendation built, and the appointment to the Padua chair followed in October of the same year. The book at hand is therefore not merely an annotated copy of a famous mathematical work but the physical support for the single published achievement that elevated Gloriosi into the most visible Italian chair of mathematics. That it emerged from a library thought lost — and emerged intact, bound by a seventeenth-century viceroy, with the annotations complete and legible — is among the more unusual recoveries of the last generation of rare-science-book scholarship.

Favaro’s search had run up against a problem he could not solve from the evidence then available. Gloriosi had died a private gentleman in Naples; his nephew, in Tomasini’s words in the 1644 Elogia, was a stranger to the study of letters who disposed of the entire library at a single stroke for five hundred gold coins to the viceroy. The viceroy then transferred the books to Spain, and after his own death in 1668 the collection dispersed into the Madrid book trade, where the Gloriosi association was no longer visible to anyone not already looking for it. Fernando Bouza, working from Tomasini’s text and from the catalogue records of Spanish libraries, reconstructed the trajectory in a 2024 article in Galilæana and catalogued thirty-five printed books once belonging to Gloriosi, almost all of them in Madrid — at the Biblioteca Nacional de España, the Biblioteca Histórica of the Universidad Complutense, the Biblioteca Francisco de Zabálburu, and the Real Academia de Bellas Artes de San Fernando — together with a further two outside Madrid (a 1521 Alfonsine Tables rebound in the eighteenth century, and a 1545 Cardano Ars Magna sold at the 1861 Guglielmo Libri auction at Sotheby’s). In each of the thirty-five Madrid books, Gloriosi is identified by his characteristic signature alone, set on the title verso or after the colophon; Bouza does not describe any of them as carrying substantive mathematical annotation. The present copy stands apart from that group in two respects: the density of Gloriosi’s 1611–1612 marginalia, and the specific point-for-point correspondence between the note on page 59 and pages 28–29 of the printed Ad theorema geometricum, which ties this particular volume to a particular publication in a way no other survivor has yet been shown to do.

Ramiro Núñez de Guzmán (1600–1668), second Duke of Medina de las Torres by marriage to Anna Carafa and son-in-law of the Count-Duke of Olivares, served as Viceroy of Naples from 1637 to 1644 and was one of the most powerful Spanish grandees of his generation. His library-building followed the pattern common to seventeenth-century Italian viceregal courts, in which the acquisition of a scholar’s entire legacy was an act of cultural prestige as well as of intellectual collecting. The red morocco binding with his armorial stamps — exclusive bindings known to collectors as medines, combining the viceroy’s quartered arms on one cover with those of Anna Carafa (REVOLUTA FOECUNDANT, the Carafa stars and crescent) on the other — was presumably executed in Naples or in Madrid after the purchase. Of the small number of these bindings that survive, the one on the present volume is identical in tool, format, and armorial layout to that on the Bodleian sammelband Rigaud.e.148, which contains Gloriosi’s own copies of Galileo’s Sidereus nuncius (1610) and Il Saggiatore (1623), Giulio Cesare La Galla’s 1612 De phoenomenis in orbe lunae, Francesco Sizzi’s 1611 Dianoia astronomica, and Mario Guiducci’s 1620 Lettera al padre Tarquinio Galluzzi. Taken together, the sammelband and the present copy demonstrate that the viceroy bound the mathematical and the Galilean-astronomical portions of Gloriosi’s library in a single uniform style, and that Gloriosi himself had studied Galileo’s principal works at first hand.

After Medina de las Torres’s death in 1668 the library began to disperse. The principal buyer was William Godolphin (c. 1634–1696), the English diplomat and Catholic convert then resident at the court in Madrid, whose prominent ownership inscriptions identify a substantial block of former Guzmán books. The present Diophantus was not among them. A second buyer, identified only as ‘Álvarez’, signed his name on the title page; the inscription survives, covered by a contemporary paper slip that has been preserved in place. The same ‘Álvarez’ signature appears on three other former Guzmán books currently traceable in the Spanish antiquarian market, and in one of them — a 1600 Brescia edition of Alessandro Manerba’s Moralis sylva — Godolphin’s own title-page inscription overlaps Álvarez’s, showing that the two were contemporaries and that Álvarez transferred part of his collection on to Godolphin. The absence of Godolphin’s characteristic title-page or colophon signature from the present copy indicates that Álvarez acquired the book directly from the viceroy’s dispersal and retained it, and that it never entered Godolphin’s library. Álvarez is therefore the third known owner, standing between Guzmán and the Earls of Macclesfield, from whose library at Shirburn Castle the book came to the market as lot 636 in the 2005 sale.

The edition Gloriosi annotated was, in 1611–1612, the only printed Diophantus in existence. Wilhelm Holtzmann of Augsburg (1532–1576), who Hellenised his name as Xylander, was a classical philologist and professor of Greek at Heidelberg, and his Latin Arithmetica of 1575 was the first complete European rendering of the text. The Greek editio princeps would not appear for another forty-six years, when Bachet de Méziriac printed it in Paris in 1621 (item 20 in this catalogue, Bachet’s own large-paper copy). Xylander worked from a single Byzantine manuscript derived, like every surviving Greek witness, from a single lost archetype, and the manuscript was, in André Weil’s phrase, marred throughout by the numerical errors of professional scribes who had not been mathematicians. He laboured for several years under these conditions, supplied the text with a running Latin commentary — the Xylandri sections set beneath each Diophantine proposition in the present volume — and dedicated the book to his pupil Prince Ludwig of Württemberg. Thomas Heath, writing in 1910, observed that Xylander’s achievement had been inadequately appreciated by later commentators largely because the book itself was so rare: Nesselmann, preparing his 1842 Algebra der Griechen, was unable to find a copy at all. The translation’s immediate and enormous influence on the shaping of European algebra, as Heath put it, ran through Bombelli, Stevin, Viète, and — through Bachet’s 1621 reprinting with improvements — Pierre de Fermat. Xylander himself did not live to see that influence take hold: he died the year after publication.

The Arithmetica itself, composed at Alexandria in approximately AD 250, is the first systematic treatise on algebra and the founding text of the tradition now called Diophantine analysis: the search for rational or integer solutions to polynomial equations in several unknowns. Diophantus introduced the earliest sustained symbolism in Greek mathematics — a character for the unknown, for its powers up to the sixth, and for the operations of addition and subtraction — and treated roughly two hundred and sixty problems whose solutions, though always given in specific numerical terms, tend to suggest general methods. The work was originally in thirteen books. Six survived in Greek, transmitted by Byzantine scholars from Michael Psellus through Maximus Planudes (whose scholia on the first two books Xylander prints alongside the text) to the codex Cardinal Bessarion rescued before the fall of Constantinople and that Regiomontanus discovered at Venice; four further books surfaced in 1968 in a ninth-century Arabic translation by Qustā ibn Lūqā, dispersing the suspicion that the ancient numbering had corresponded straightforwardly to the surviving Greek sequence. Three books remain lost. The Arab reception had in fact been considerable: al-Nadīm’s index of the sciences (987–988) lists commentaries by Qustā ibn Lūqā and by Abū’l-Wafā’, and a substantial fraction of the problems in al-Karajī’s algebra are drawn directly from Diophantus’s first three books.

Xylander’s volume prints, at the end, a fragment of the only other surviving work by Diophantus — a treatise on polygonal numbers, which is differentiated from the Arithmetica by its use of geometric proofs and which breaks off in the middle of its investigation of the number of ways in which a given number can be expressed as a polygonal. The full transmission history of both texts, from Bessarion and Regiomontanus through Bombelli’s partial assimilation in his 1572 Algebra (271 problems, of which 147 were taken directly from Diophantus) to Viète’s Zetetica of 1593 and on to Bachet’s definitive 1621 edition, runs entirely through this 1575 volume. It was in the margins of a copy of Bachet’s 1621 reprint that Pierre de Fermat, in the mid-1630s, wrote the forty-eight observations that founded modern number theory — among them, on page 85, against Problem II.8 on the decomposition of a square into two squares, the proposition now known as Fermat’s Last Theorem, whose proof by Andrew Wiles in 1995 closed a gap that had stood for three hundred and fifty-eight years. The 1670 reprint of Bachet’s edition with Fermat’s observations printed in the margins, the book that carried the Last Theorem into circulation, is item 19 in this catalogue.

Auction records since Honeyman list only three other copies of the 1575 Xylander; each is in a nineteenth- or twentieth-century binding and none has significant provenance. OCLC records eight copies in North American libraries. Copies in contemporary armorial bindings with identifiable early mathematical ownership are essentially unrecorded in commerce of the last century, and the present volume — the Gloriosi copy in the Medina de las Torres binding, standing as the material support for the 1613 Ad theorema geometricum and for the Padua appointment that followed it — is without known parallel.

Almost nothing is known of the life of Diophantus. He quotes Hypsicles, and so must have worked after roughly 150 BC; he is quoted in turn by Theon of Alexandria, and so must have worked before AD 364. The conventional placement around AD 250 rests on a single passage in an eleventh-century Byzantine letter and on the absence of Diophantus’s name from the commentaries of Pappus. His place of birth is unknown, his teachers unknown, and the fourteen-line Greek epigram in the Palatine Anthology that purports to record his age at death is generally regarded as a mathematical exercise rather than a biographical document.

References: Adams D-652 — DSB IV, 110–19 — Honeyman 890 — Norman 641 — Macclesfield 636 (this copy) — Bouza, ‘The mathematician and the viceroy’, Galilæana XXI, 1 (2024), pp. 201–220 — Favaro, Amici e corrispondenti di Galileo Galilei. IX. Giovanni Camillo Gloriosi, 1904 — Tomasini, Elogia virorum literis et sapientia illustrium, 1644 — Heath, Diophantus of Alexandria: A Study in the History of Greek Algebra, 2nd ed., 1910 — Heath, A History of Greek Mathematics, 1921, vol. II, pp. 448–517 — Weil, Number Theory: An Approach Through History from Hammurapi to Legendre, 1984 — Katz & Parshall, Taming the Unknown, 2014, ch. 4 — Schappacher, ‘Diophantus of Alexandria: a text and its history’ (IRMA, Strasbourg) — Smith, Rara Arithmetica, p. 348.

Folio (307 × 200 mm), pp. [xii], 152. Printer’s device on title (legend Episcop), woodcut initials, printed marginal notes. Occasional foxing, light damp stains to blank corners of some leaves. Mid-seventeenth-century red morocco with gilt arms of the Duke of Medina de las Torres and his wife on the covers, elaborate gilt borders and corner fleurons, spine gilt; damage to upper edge of front board affecting gilt border.

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Item #6346

Price: $125,000.00