Euclid's Elements, compiled over two thousand, three hundred years ago, is a textbook on geometry and number theory that of any book, apart from the Bible, has had the most readers over the centuries. Until the late twentieth century its theorems formed the basis of school geometry. Euclid aimed to derive as many conclusions as possible starting from the fewest number of assumptions or postulates. He started with ten 'common notions' and 'postulates' and derived four hundred and sixty-five theorems as logical consequences in thirteen 'books'. The subjects dealt with are mainly geometry: lines, angles, similar and congruent triangles, areas, the theorem inappropriately associated with Pythagoras, circles, polygons, volumes of parallelepipeds, prisms, pyramids, the sphere, but also number theory, including prime numbers and irrational numbers.
In the Renaissance of learning that took place in Baghdad in the eighth to tenth centuries, the Elements were translated from Greek into Arabic and commented upon. The constructive criticism of Euclid and other Greek authors led to a school of Islamic mathematics that flourished over several centuries. We may mention, for example, Muslim efforts to 'prove' Euclid's famous 'Fifth Postulate', which is equivalent to stating that two lines will be parallel if the interior angles formed by a transversal add up to 180°.
The great scholar and polymath Nasir al-Din al-Tusi (1201-74 AD) was responsible for editions of most of the Greek astronomical and mathematical works that had been translated into Arabic between the eighth and tenth centuries. His enormous output in such editions or recensions was almost matched by his own independent works on those subjects.
THE TRADITION OF TRANSLATING EUCLID'S ELEMENTS
All discussion and development of Euclidean ideas in Medieval Arabic science were based on two early translations from the Greek original text. Thanks to the record of Abu'l-Faraj Muhammad ibn Ishaq al-Nadim, who lived in Damascus in the tenth century AD, we have a record of these translations in his Kitab al-fihrist (Sayyid 2009). Abu-l-Faraj recorded that the earliest translation of Euclid’s text was made by al-Hajjaj ibn Yusuf ibn Maṭar during the reign of Caliph Harun al-Rashid (r.786-809 AD), which he later revised under the reign of Caliph al-Ma’mun (r.813-833 AD). A second translation is reported to have been made by Ishaq ibn Hunayn (d. circa 910 AD), which was subsequently revised by the mathematician Thabit ibn Qurra (d.901 AD).
The surviving manuscripts have undergone repeated editing and it has not been possible to recover the original translations or their reported revisions. Several sources have preserved either short quotations ascribed to al-Hajjaj, or reports describing how his work differed from the revision of Thabit. There are two textual strands, sometimes denoted as Group A and Group B, represented in surviving Arabic translation manuscripts. They differ in technical vocabulary and sometimes in the ordering of definitions and propositions (De Young 2004). The Group B manuscripts sometimes seem to preserve features that a few Arabic commentators ascribe to the work of al-Hajjaj.
It was in this complicated transmission from the Greek that a remarkable edifice of comment and discussion developed in Arabic over the ensuing centuries. There were commentaries, some on the entire Elements, others on selected books, along with summaries, extracts, editions and corrections, created within this burgeoning intellectual and mathematical enterprise. The present treatise represents one of these secondary elaborations that grew on the foundation of the original Arabic translations.
THE PRESENT MANUSCRIPT
Because of the editing evident in this version, it is not easy to situate it in relation to the various textual families of the surviving manuscripts. The convention of diagram labelling seems to suggest a connection to Group A manuscripts. At least two features, however, suggest a possible influence from the Group B primary transmission version.
Propositions VIII, 20 and 21 (numbered in this manuscript as 18 and 19 because of a error in numbering) are given a shortened demonstration that is typical of some Group B manuscripts. The attribution of the longer demonstrations to Ishaq by an anonymous commentator implies that the shorter version derives from the transmission attributed to al-Hajjaj (De Young 2003, pp.152-3). Moreover, these two propositions are normally followed in the Ishaq transmission by two propositions demonstrating the converse and ascribed to Thabit. These propositions are missing from this version of the Elements (De Young 2003, pp.153-14). An additional piece of evidence that suggests possible ties to the Hajjaj transmission is the inversion of the order of propositions IX, 11 and 12 compared to their order in the transmission ascribed to Thabit (De Young 2003, p.154). Thus, at least some of the sources used to create this edition appear to have preserved some traces from the early translation attributed to al-Hajjaj.
This translation is interesting as it seems to combine different versions which were circulating at the time. Based mainly on al-Maghribi, this volume presents itself as a selection of al-Maghribi’s definition and also contains few lemmas (muqaddima) which are not present in al-Maghribi’s version.
Several of the definitions listed in this volume seem to repeat almost exactly the diction used in the Tahrir kitab usul, by Muhyi al-Din al-Maghribi (d. circa 1281-91 AD). Some have been deliberately omitted (for example the one on the circle and its components, or the chord of an angle, both in book I) or changed (see for example the definition of parallel lined which differs from al-Maghribi’s version and follows instead the formulation found in group A). We face a comparable situation when examining the propositions. The enunciation (muqaddima), setting out (mithal), and specification (sharita) which are separate statements in al-Maghribi’s text (and also in the primary Arabic transmission) are combined into a single statement in this edition.
This technique for condensing the Euclidean text is not unknown in Arabic transmission: the epitome of the Elements included by Abu 'Ali ibn Sina (d.428 AH/1037 AD) and his philosophical compendium, Kitab al-shifa' both present these enunciations condensed in one statement.
The present edition includes a few lemmas (muqaddima) not present in al-Maghribi’s version, an addition that leads us to think this volume is a product of more than one tradition. For example, in preceding proposition I and 22, the author states: let line AB be divided into three parts at [points] G, D and [let] the sum of any two of its parts be longer than the remaining [part]. Then if there is drawn at G a circle with distance GA and likewise [there is drawn] at D [a circle] with distance BD, the two of them intersect. This lemma is not unique. That Euclid had omitted to demonstrate that the two circles constructed in this demonstration would meet was already pointed out by Proclus (Morrow 1970, pp.257-60). A similar discussion can be found in Arabic in Ibn al-Haytham’s Hall shukuk kitab Uqlidis, and also in the Tahrir of Nasir al-Din al-Tusi. This lemma is intended to rectify this omission by Euclid.
Because of the editing evident in this version, it is difficult to place it precisely in relation to the various textual families of the surviving manuscripts. As noted earlier, the conventions of diagram labelling seems to suggest a connection to Group A manuscripts, although at least two features suggest a possible influence from the Group B primary transmission. This text, although incomplete, is crucially important as it presents itself as a combination of these two groups.
We are grateful to Dr. Gregg De Young for his assistance in cataloguing this lot.
Please note that this lot is accompanied by a Carbon date test stating that the paper is dated to the thirteenth century.
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