## [Arabic title: Kitāb Taḥrīr Uṣūl li-Ūqlīdis min ta'līf khwajah Naṣīr al-Dīn al-Ṭūsī]. Euclidis Elementorum geometricorum libri tredecim ex traditione doctissimi Nasiridini Tusini nunc primum Arabicè impressi.

Rome: in Typographia Medicea, 1594.

First edition of Euclid in Arabic, “possibly the most remarkable of all printed editions of Euclid” (Thomas-Stanford). Euclid’s *Elements,* the “oldest mathematical textbook still in common use today” (*PMM*), “has exercised an influence upon the human mind greater than that of any other work except the Bible” (*DSB*). It is the only work of classical antiquity to have remained continuously in print, and to be used continuously as a textbook from the pre-Christian era to the 20th century. It is the foundation work not only for geometry but also for number theory. Euclid was first re-introduced to medieval Europe through Adelard of Bath’s Latin translation of an Arabic manuscript of the *Elements* – a testament to the enduring importance of intellectual exchange between Europe and the Islamic world. This first printed Arabic edition (the only Arabic edition printed in Europe until the 19^{th} century) is a *taḥrīr* of the *Elements* – not a translation but a redaction, a re-working of older Arabic translations, with additions and commentary. The most significant of the additions is an attempted proof of the ‘parallel postulate’ (pp. 28-33) – that through any point not on a given straight line passes a unique line that does not intersect the given line. This attempted proof exerted a considerable influence on the historical development of non-Euclidean geometry. A Latin translation of it was published by the great Oxford mathematician John Wallis in his *Opera mathematica* (1693) (based upon a lecture he had delivered 30 years earlier), and this in turn “became the starting point for the work of Saccheri and ultimately for the discovery of non-Euclidean geometry” (Katz, p. 271). This redaction of the *Elements *has traditionally been ascribed to the great Persian polymath Naṣīr al-Dīn al-Ṭūsī (1201-74) (and is so ascribed on the title page), but this is no longer accepted by scholars. The actual author, usually designated ‘Pseudo-Ṭūsī’, is unknown, although some believe it may have been al-Ṭūsī’s son, Sadr al-Din. This beautiful edition of the *Elements *was printed at the press established by Ferdinando de’ Medici under Pope Gregory XIII to disseminate works in oriental languages. Some copies (including the present one) contain the first 12 books of the *Elements*, others also contain the 13^{th} book. The reason for the omission of the 13^{th} book in some copies is unknown, but copies of the ‘shorter’ version are held in many great libraries (Oxford, Christ Church and Trinity; one, if not more, of the copies in Cambridge University Library, plus other Cambridge copies (see Adams); Göttingen; Munich; the Vitry copy in a contemporary German binding (Sotheby’s 10-11 April 2002, lot 352); and others). In addition, while in the present copy the title page is partly in Arabic and partly in Latin, some copies have a title page entirely in Arabic (and the title verso is also reset – see Thomas Stanford).

*Provenance*: Tsar of Russia (ink stamps on title). Adam Litawor Chreptowicz (1768-1844), philanthropist and patron of science, Knight of Malta, Knight of Honor and Devotion of the Grand Catholic Priory in Russia (inscription at foot of title page, marginal annotations). He augmented the library of the Counts of Chreptowicz in Szczorsy (now in Belarus) which contained more than 10,000 volumes. The library was founded by his father Joachim Litawor Chreptowicz (1729-1812), Polish-Lithuanian nobleman, writer, poet, politician of the Grand Duchy of Lithuania, marshal of the Lithuanian Tribunal, and the last Grand Chancellor of Lithuania.

Born ca. 300 BC in Alexandria, Egypt, “Euclid compiled his *Elements* from a number of works of earlier men. Among these are Hippocrates of Chios (flourished *c.* 440 BC), not to be confused with the physician Hippocrates of Cos (*c.* 460–375 BC). The latest compiler before Euclid was Theudius, whose textbook was used in the Academy and was probably the one used by Aristotle (384–322 BC). The older elements were at once superseded by Euclid’s and then forgotten. For his subject matter Euclid doubtless drew upon all his predecessors, but it is clear that the whole design of his work was his own …

“Euclid understood that building a logical and rigorous geometry depends on the foundation—a foundation that Euclid began in Book I with 23 definitions (such as “a point is that which has no part” and “a line is a length without breadth”), five unproved assumptions that Euclid called postulates (now known as axioms), and five further unproved assumptions that he called common notions. Book I then proves elementary theorems about triangles and parallelograms and ends with the Pythagorean theorem …

“The subject of Book II has been called geometric algebra because it states algebraic identities as theorems about equivalent geometric figures. Book II contains a construction of “the section,” the division of a line into two parts such that the ratio of the larger to the smaller segment is equal to the ratio of the original line to the larger segment. (This division was renamed the golden section in the Renaissance after artists and architects rediscovered its pleasing proportions.) Book II also generalizes the Pythagorean theorem to arbitrary triangles, a result that is equivalent to the law of cosines. Book III deals with properties of circles and Book IV with the construction of regular polygons, in particular the pentagon.

“Book V shifts from plane geometry to expound a general theory of ratios and proportions that is attributed by Proclus (along with Book XII) to Eudoxus of Cnidus (*c.* 395/390–342/337 BC). While Book V can be read independently of the rest of the *Elements*, its solution to the problem of incommensurables (irrational numbers) is essential to later books. In addition, it formed the foundation for a geometric theory of numbers until an analytic theory developed in the late 19th century. Book VI applies this theory of ratios to plane geometry, mainly triangles and parallelograms, culminating in the “application of areas,” a procedure for solving quadratic problems by geometric means.

“Books VII–IX contain elements of number theory, where number (*arithmos*) means positive integers greater than 1. Beginning with 22 new definitions—such as unity, even, odd, and prime—these books develop various properties of the positive integers. For instance, Book VII describes a method, *antanaresis* (now known as the Euclidean algorithm), for finding the greatest common divisor of two or more numbers; Book VIII examines numbers in continued proportions, now known as geometric sequences (such as *ax*, *ax*^{2}, *ax*^{3}, *ax*^{4}, …); and Book IX proves that there are an infinite number of primes …

“Book X, which comprises roughly one-fourth of the *Elements*, seems disproportionate to the importance of its classification of incommensurable lines and areas (although study of this book would inspire Johannes Kepler [1571–1630] in his search for a cosmological model) … Book XI concerns the intersections of planes, lines, and parallelepipeds (solids with parallel parallelograms as opposite faces). Book XII applies Eudoxus’s method of exhaustion to prove that the areas of circles are to one another as the squares of their diameters and that the volumes of spheres are to one another as the cubes of their diameters” (*Britannica*).

Euclid’s *Elements *first became known in Europe through Latin translations of Arabic manuscripts translated from the Greek. The most important of these are two by al-Ḥajjāj ibn Yūsuf ibn Maṭar – first for the ʿAbbāsid caliph Hārūn al-Rashīd (ruled 786–809) and again for the caliph al-Maʾmūn (ruled 813–833) – and a third by Isḥāq ibn Ḥunayn (died 910), son of Ḥunayn ibn Isḥāq (808–873), which was revised by Thābit ibn Qurrah (c. 836–901).

“More significant within the history of Islamic mathematics are the various recensions or *Taḥrīr* of the *Elements*. The best known is that of the Persian philosopher and scientist Nāṣir al-Dīn al-Tūsī, who composed similar editions of many other Greek mathematical, astronomical, and optical works. We know that at least one *Taḥrīr Uṣmūl Uqlīdis* (‘Recension of Euclid’s Elements’) was completed by al-Ṭūsī in 1248. It covered all fifteen books and made use of both the Ḥajjāj and Isḥāq–Thābit translations. There is, however, yet another *Taḥrir* of the *Elements* that is traditionally ascribed to al-Ṭūsī. Although it covers only books I–XIII, it is considerably more detailed than the more frequently appearing 1248 version. Printed in Rome in 1594, we know of only two extant manuscripts (both at the Biblioteca Medicea-Laurenziana in Florence) of this thirteen-book *Taḥrīr*. However, one of these codices explicitly asserts that the work was completed on 10 Muḥarram 1298. Since al-Ṭūsī died in 1274, this gives grounds (and there appear to be other reasons as well) for seriously doubting the ascription to him.

“Yet whatever conclusion may finally be reached concerning its authorship, the preface to this *Taḥrīr* is particularly instructive with respect to the reason for composing such redactions of the *Elements* and with regard to the kind of added material they would be likely to contain. Beginning with a few remarks specifying the place of geometry within the classification of the sciences and several fanciful statements about Euclid’s biography, this preface makes special note of the two previously executed translations by Ḥajjāj and (revised by) Thābit and then launches into a more elaborate description of all else Islamic scholars had done with, and to, the *Elements*. This interim ‘history’ of *Euclides Arabus* tells us that much effort had been spent in removing all difficulties from the text and in clarifying its numerous obscurities. Examples were inserted to make complex things more obvious and, moving in the opposite direction, some things that were too obvious were left out. Some related propositions were combined and treated as one, implicit assumptions were made explicit, and care was taken to specify (at least by number) just which previous theorems were being utilized in a particular proof. And all of this was done, our preface continues, not just in the body of the text of these versions of the *Elements*, but everywhere in the margins and even between lines. The varieties of information produced in such a fashion are now, the author of the present *Taḥrīr* submits, sorely in need of proper arrangement and clarification, and he goes on to reveal his intention of satisfying this need through the presentation as a unified whole of the original text, together with relevant commentary. His resulting *Taḥrīr* needs much closer scrutiny in order to set forth the complete spectrum of all of the types of added material it contains, but it is clear from the preface we have been summarizing that it presumably includes, in addition to its own original contributions, many features similar to those its author has just recounted among the works of his predecessors” (DSB).

The most significant of this added material is Pseudo-Ṭūsī’s attempted proof of Euclid’s fifth postulate, or parallel postulate. This was originally stated in the following form:

‘That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles.’

Euclid was dissatisfied with the fifth postulate and he tried to avoid its use as long as possible – in fact the first 28 propositions of *The Elements* are proved without using it. In his commentary on the* Elements*, Proclus (410-485) noted several incorrect attempts to deduce the fifth postulate from the other four, including one by Ptolemy, and gave a false proof of his own. However, his work was important for stating the following postulate, which is equivalent to the fifth postulate (in the presence of the other postulates):

‘Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line.’

Several mathematicians in the mediaeval Islamic world attempted to prove the parallel postulate, or to establish equivalent formulations of it, including Ibn al-Haytham (Alhazen) (965–1039), Omar Khayyám (1050–1123), Thābit ibn Qurrah, and al-Ṭūsī himself in his *Taḥrīr* of 1248. But their work, important as it was, did not influence the subsequent development of non-Euclidean geometry, as it remained unpublished until after the impossibility of deducing the parallel postulate from the other postulates of Euclidean geometry had been definitively established by János Bolyai and Nikolai Lobachevsky in the early 19^{th} century. The the proof given by Pseudo-Ṭūsī in the Arabic Euclid, the only attempt by Islamic scholars to be published in the early modern period, exerted a very substantial influence on subsequent developments.

“As the only printed Arabic version of Euclid’s work available to the European scholarly community, the Rome edition was studied far more than any Arabic manuscript. It was not so often studied in its entirety, but usually only with regard to specific mathematical points. From the beginning, the focus of attention has been on its attempted demonstration of Euclid’s parallel postulate …

“The Arabic text of the Pseudo-Ṭūsī attempt to demonstrate the parallel lines postulate was eventually translated into Latin by Edward Pococke (1604-91). This translation must have been made before 1663 when John Wallis (1616-1703) discussed the attempt to demonstrate the parallel postulate in his lecture as Savilian Professor of Geometry at Oxford (11 July 1663). But Wallis was apparently not keen to publish his thoughts, so it was thirty years later that this quotation from Pococke became widely available in printed form (Wallis, *Opera *(1693), Vol. II, pp. 665-678). Through Wallis, knowledge of this Arabic discussion of Euclid’s postulate entered the mainstream of European geometrical discussion. Wallis’s discussion of the postulate directly influenced Saccheri (1667-1733) in his famous attack on the parallels postulate in his *Euclides ab omni naevo vindicatus* (1733). It was his discussion and criticism of the approach of Pseudo-Ṭūsī to the parallels postulate (in Scholion III of proposition XXI) that helped to lay the foundation for the modern study of non-Euclidean geometry. And thus the Pseudo-Ṭūsī attempt to demonstrate the parallel lines postulate entered the mainstream of European discussion under a mistaken identity” (De Young, pp. 291-292).

The Medici Oriental Press (Typographia Medicea) “was set up by Ferdinand de Medici (1549-1609), Grand Duke of Tuscany, with Giovan Battista Raimondi (ca. 1536-1614) as its director … Raimondi constructed a very ambitious publication plan that would have included some 80 titles. Unfortunately, the press of immediate activities and duties gradually drew Ferdinand’s interests away from the press and his financial contributions became less and less. Eventually, Raimondi himself purchased the press from Ferdinand, but without financial resources he was unable to complete more than a tiny fraction of what he had once intended.

“The publishing program represented one of the earliest attempts to print Arabic in Europe. That Euclid should have been chosen as one of the Arabic texts chosen to be printed by the press only serves to remind us of the remarkable role that the *Elements *has played in the intellectual history of Mediterranean and European cultures. By the time this Arabic version was issued by the press the *Elements* had already been printed numerous times in Europe. This was the only Arabic print edition to be published prior to the nineteenth century and the only printed Arabic edition to be produced in Europe. Inevitably it became the Arabic treatise that was most widely studied by European mathematicians and historians of mathematics over the past centuries …

“The initial print run of the Arabic *Elements* was an astonishing 3000 copies. Clearly, Raimondi and the press administrators must have expected a considerable market from the Arabic speaking Middle East for this first venture into printing Arabic mathematics. The records of the press also attest that these expectations were not met – only a little more than 1000 copies had been sold when the press ceased operations” (De Young, pp. 266-268).

The reasons for the two different title pages (Arabic only and Arabic-Latin), and for the numerous examples with 12 books, have not been determined. Cassinet argues that the 13-book version has priority over that with 12-books – he claims, without providing evidence, that not enough copies of the 13^{th} book were printed, so copies without it were issued when supplies of the 13^{th} book were exhausted. However, it seems to us that, in the absence of further evidence, one can just as easily argue that the 12-book version has priority because the 13^{th} book was printed last and some copies were issued before it was ready. Cassinet himself points out that at least one copy is known (at Geneva) with only the first six books, which Cassinet attributes to the desire of a scholar to see the work before the remaining books were printed.

Adams E-990; Brunet II: 1087; Honeyman 1014; Macclesfield 715 (lacking title page); Mortimer Italian 175; Thomas-Stanford 46. Cassinet, ‘L’aventure de l’édition des Eléments d’Euclide en arabe par la Société Typographique Médicis vers 1594,’ *Révue française d’Histoire du Livre* 78-79 (1993), pp. 5-51. De Young, ‘Further adventures of the Rome 1594 Arabic redaction of Euclid’s *Elements*,’ *Archive for History of Exact Sciences* 66 (2012), pp. 265-294. Katz, *History of Mathematics*, 1998. For an account of Pseudo-Ṭūsī’s attempted proof of the parallel postulate, see Bonola, *Non-Euclidean Geometry* (1912), pp. 10-12.

Folio (315 x 212 mm), pp. [1], 2-400 (numbered in Arabic at the top of each page and, except for pp. 3-6 & 9-24, in Roman at the bottom), title page in both Arabic and Latin, Arabic type in two sizes by Robert Granjon. diagrams and mathematical figures throughout, all pages within double-rule border, woodcut headpieces (some browning and staining). Contemporary calf, double gilt rules to boards, spine gilt in compartments with gilt lettering, gilt supralibros on each board (joints cracked, extremities worn, coat of arms on gilt supralibros excised). Custom long-grained black morrocco box with gilt spine lettering.

Item #6098

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Price:
$88,500.00
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