# Euclid: Transmission of the Elements.

# Euclid: Transmission of the Elements.

Any attempt to plot the course of Euclid’s *Elements* from the third century b.c. through the subsequent history of mathematics and science is an extraordinarily difficult task. No other work—scientific, philosophical, or literary—has, in making its way from antiquity to the present, fallen under an editor’s pen with anything like an equal frequency. And with good reason: it served, for almost 2,000 years, as the standard text of the core of basic mathematics. As such, the editorial attention it constantly received was to be expected as a matter of course. The complexity of the history of this attention is, moreover, not simply one of a multiplicity of translations; it includes an amazing variety of redactions, emendations, abbreviations, commentaries, scholia, and special versions for special purposes.

**The Elements in Greek Antiquity.** The history of the *Elements* properly begins within later Greek mathematics itself. Comments on Euclid’s major work were evidently far from uncommon. Indeed, Proclus (410–485), the author of the major extant Greek commentary on the *Elements*, several times refers to similar efforts by his predecessors in a way that makes it clear that the production of works or glosses on or about Euclid was a frequent—even all too frequent and not particularly valuable—activity. It would seem that Proclus had in mind an already considerable body of scholia and remarks (largely, perhaps, in various separate philosophical and scientific works) on the *Elements*, as well as other commentaries specifically devoted to it. We know that at least four such commentaries, or at least partial commentaries, existed. The earliest was written by Hero of Alexandria, but we know of its contents only through the few references in Proclus himself and through fragments preserved in the Arabic commentary of al-Nayrīzī (*d. ca*. 922). Far more important and extensive was the commentary of Pappus of Alexandria, a work whose Greek text is also lost but of which we possess an Arabic translation of the comments on book X. Proclus also mentions the Neoplatonist Porphyry (*ca*. 232–304), although it is doubtful that his work on Euclid would have been as systematic and penetrating as those of Hero and Pappus. Finally, although he did not compose a commentary specifically on the *Elements* itself, mention should be made of Geminus of Rhodes, whose lost work on the order or doctrine of mathematics (its exact title is uncertain) so often served Proclus with valuable source material. In the period following Proclus, it should be noted that Simplicius, in addition to his well-known commentaries on a number of works of Aristotle, also wrote a *Commentary on the Premises* [or *the Proemium] of the Book of Euclid*. Again we are indebted to al-Nayrīzī, who preserved fragments of the work.

To these more formal works on the *Elements*, one should add the substantial number of Greek scholia. Many derive from the commentaries of Proclus and Pappus, the latter being especially significant when they derive from the lost books of his work. Others are of a much later date, to say nothing of an inferior quality, and reach all the way to the fourteenth century (where the arithmetical comments of the monk Barlaam to book II of the *Elements* stand as the most extensive so-called scholium of all).

The event, however, that had the most enduring effect within the Greek phase of the transmission of the *Elements* was the edition and slight emendation it underwent at the hands of Theon of Alexandria (fourth century; not to be confused with the secondcentury Neoplatonist, Theon of Smyrna). The result of Theon’s efforts furnished the text for every Greek edition of Euclid until the nineteenth century. Fortunately, in his commentary to Ptolemy’s *Almagest*, Theon indicates that he was responsible for an addendum to the final proposition of book VI in his “edition (έκδоσιѕ) of the *Elements”;* for it was this confession that furnished scholars with their first clue in unraveling the problem of the pre-Theonine, “pristine” Euclid. In 1808 Frangois Peyrard noted that a Vatican manuscript (Vat. graec. 190) which Napoleon had appropriated for Paris did not contain the addition Theon had referred to. This, coupled with other notable differences from the usual Theonine editions of the *Elements*, led Peyrard to conclude that he had before him a more ancient version of Euclid’s text. Accordingly, he employed the Vatican codex, as well as several others, in correcting the text presented by the *editio princeps* of Simon Grynaeus (Basel, 1533). Others, utilizing occasional additional (but always Theonine) manuscripts or earlier editions, continued to improve Peyrard’s text, but it was not until J. L. Heiberg began the reconstruction of the text anew on the basis of the Vatican and almost all other known manuscripts that a critical edition of the *Elements* was finally (1883–1888) established. Heiberg not only in great measure succeeded in getting behind the numerous Theonine alterations and additions, but also was able to sift out a considerable number of pre-Theonine interpolations. In addition to the authority of the non-Theonine Vatican manuscript, he culled papyri fragments, scholia, and every known ancient quotation of, or reference to, the *Elements* for evidence in his construction of the “original” Euclid. The result still stands.

**The Medieval Arabic Euclid.** A most appropriate introduction to the dissemination of the *Elements* throughout the Islamic world can be had by quoting the entry on Euclid in the *Fihrist* (“Index”) of the tenth-century biobibliographer Muḥammad ibn Isḥāq ibn Abī Yaʿqūb al-Nadīm:

A geometer, he was the son of Naucrates, who was in turn the son of B[a]r[a]niq[e]s. He taught geometry and is found as an author in this field earlier than Archimedes and others; he belonged among those called mathematical philosophers. On his book

Of the Elements of Geometry:Its title is στοιΧεῑα, which means “elements of geometry.” It was twice translated by al-Ḥajjāj ibn Yūsuf ibn Maṭar: one translation, the first, is known under the name of Hārūnian, while the other carries the label Maʾmūnian and is the one to be relied and depended upon. Furthermore, Isḥāq ibn Ḥunayn also translated the work, a translation in turn revised by Thābit ibn Qurra al-Ḥarrānī. Moreover, Abū ʿUthman al-Dimashgī translated several books of this same work; I have seen the tenth in Mosul, in the library of ʿAlī ibn Aḥmad al-ʿImrānī (one of whose pupils was Abu’l-Ṣagr al-Qabīṣī who in turn in our time lectures on theAlmagest). Hero commented upon this book [i.e. theElements) and resolved its difficulties. Al-Nayrīzī also commented upon it, as did al-Karābīsī, of whom further mention will be made later. Further, al-Jawharī (who will also be treated below) wrote a commentary on the whole work from beginning to end. Another commentary on book V was done by al-Māhānī. I am also informed by the physician Naẓīf that he saw the Greek of book X of Euclid and that it contained forty more propositions than that which we have (109 propositions) and that he had decided to translate it into Arabic. It is also reported by Yūḥannā al-Qass [i.e., the priest] that he saw the proposition which Thābit claimed to belong to book I, maintaining that it was in the Greek version; and Naẓīf said that he had shown it to him [Yūḥannā?). Furthermore, Abū Jaʿfar al-Khāzin al-Khurāsānī (who will be mentioned again below) composed a commentary on Euclid’s book, as did Abu’l-Wafā’, although the latter did not finish his. Then a man by the name of Ibn Rāhiwayh al-Arrajānī commented on book X, while Abū’l-Qāsim al-Anṭāgī commented on the whole work and this has come out [been published?]. Further, a commentary was made by Sanad ibn ʿAlī (nine books of which, and a part of the tenth, were seen by Abū ʿAli) and book X was commented upon by Abū Yūsuf al-Rāzī at the instance of Ibn al-ʿAmīd. In his treatiseOn the Aims of Euclid’s Bookai-Kindī mentioned that this book had been composed by a man by the name of Apollonius the Carpenter and that he drafted it in fifteen parts. Now, at the time when this composition had already become obsolete and in need of revision, one of the kings of Alexandria became interested in the study of geometry. Euclid was alive at this time and the king commissioned him to rework the book and comment upon it; this Euclid did and thus it came about that it was ascribed to him [as author]. Later, Hypsicles, a pupil of Euclid, discovered two further books, the fourteenth and the fifteenth; he brought them to the king and they were added to the others. And all this took place in Alexandria. Among Euclid’s other writings belong: The bookOn Appearances[i.e., thePhaenomena]. The bookOn the Difference of Images[i.e., the Optical. The bookOn Given Magnitudes[i.e., theData]. The bookOn Tones, known under the titleOn Music(spurious). The bookOn Division, revised by Thābit. The bookOn Practical Applications[i.e., thePorisma] (spurious). The bookOn the Canon. The bookOn the Heavy and the Light. The bookOn Composition(spurious). The bookOn Resolution(spurious).

Al-Nadīm’s report immediately reveals the extensive attention Euclid had already received by the end of the tenth century: two complete translations, each in turn revised, perhaps two partial translations, and an amazing variety of commentaries. What is more, this flurry of activity over the *Elements* was to continue for at least 300 years more. But before recounting the more salient aspects of this later history, it will be necessary to expand certain facets of al-Nadīm’s account of the earlier efforts to work Euclid into the mainstream of Islamic mathematics. By way of introduction it may be worth indicating that the totally fanciful account reported from al-Kindī of how Euclid came to compose the *Elements* may well have derived, as Thomas Heath has maintained, from a confusing misinterpretation of the Greek preface to book XIV by Hypsicles. More important than Islamic beliefs as to the origin of the *Elements*, however, is the history of how and when this work was introduced to the Arabic-speaking world. Here al-Nadīm is more reliably informed. The first translation by al-Ḥajjāj (*fl. ca*. 786–833) to which he refers was made, as the label he assigns it indicates, under the ʿAbbāsid caliphate of Hārmūn al-Rashīd (786–809), at the instance of his vizier Yaḥyā ibn Khālid ibn Barmak. We also know that a manuscript of the *Elements* was obtained from the Byzantine emperor by an earlier caliph, ai-Manṣmūr (754–775), although apparently without then occasioning its translation into Arabic. And this patronage of science by the ʿAbbāsid caliphs is even more in evidence in Ḥajjāj’s realization that a second, shorter recension of his translation would be likely to gain the favor of Maʾmmūn (813–833). It is this version alone which appears to be extant (books I-VI, XI-XIII only). The first six books exist in a Leiden manuscript conjoined with al-Nayrīzī’s commentary, and from the prefatory remarks of this work we learn that, in preparing his second version of the *Elements*, Ḥajjāj “left out the superfluities, filled up the gaps, corrected or removed the errors, until he had perfected the book and made it more certain, and had summarized it, as it is found in the present version. This was done for specialists, without changing any of its substance, while he left the first version as it was for the vulgar.” Although what we have of Ḥajjāj’s second version has not yet undergone a thorough analysis, that it was composed with something of the notion of a school text in mind seems evident. For, to cite several instances, the tendency to distinguish separate cases of a proposition and the use of numerical examples to illustrate various proofs point toward a preoccupation with pedagogical concerns that was to become fairly characteristic of the Arabic Euclid and of the medieval Latin versions that derived from it.

The second, largely new translation of Euclid was accomplished, as al-Nadīm tells us, by Isḥāq ibn Ḥunayn, son of Ḥunayn ibn Isḥāq, the most illustrious of all translators of Greek works into Arabic. Again a second recension was prepared, in this instance by a scholar who in his own right holds a major position within the history of Islamic mathematics, Thabīt ibn Qurra. Although no copies of Ishaq’s initial version appear to have survived, we do possess a number of manuscripts of the Isḥāq–Thābit recension. Further study of these manuscripts is needed to say much in detail of the character of this translation, but we do know that Thābit utilized Greek manuscripts in whatever reworking he did of the text (as stated in a marginal note to a Hebrew translation of the *Elements* and confirmed by Thābit’s own reference to a Greek text). Whether Isḥāq (or even Thābit) relied to any great extent on one of the Ḥajjāj versions for any sort of guidance is problematic. For, in a comparison of a single manuscript of what are presumably books XI-XIII of Ḥajjāj with their corresponding parts in the Isḥāq–Thābit redaction, Martin Klamroth, the first scholar to examine the two translations in depth, confessed that the difference was slight. But perhaps, assuming the ascription of Klamroth’s manuscript of XI-XIII to Ḥajjāj correct, this lack of variation occurs only in the later books.

It is at this point perhaps noteworthy that Klamroth was of the opinion that the Arabic tradition as a whole is closer, as we have it, to the original Euclid than the text presented by extant Greek manuscripts. Heiberg, however, marshaled a considerable amount of evidence against Klamroth’s contention and clearly confirmed the superior reliability of the Greek tradition. At the same time, he established the filiation of the Isḥāq–Thābit version and a particular divergent Greek manuscript.

To complete al-Nadīm’s account of translations, mention should be made that Abū ʿUthman al-Dimashqī (*fl. ca*, 908–932) not only translated parts of the *Elements* but also the commentary of Pappus to book X (the latter alone being extant). Furthermore, al-Nadīm’s report of the intention of Naẓīf ibn Yumn (*d. ca*. 990) to translate book X appears to be reflected in various additions and modifications deriving from the Greek that are extant in Arabic under Naẓīf’s name. Finally, although it escaped al-Nadīm’s notice, the spurious books XIV and XV of the *Elements* were translated by the Baghdad mathematician and astronomer Qusṭā ibn Lūqā.

The full roster of Arabic translations of Euclid’s major work only begins to sketch the program of activity concerning the *Elements* within Islamic mathematics and science. The numerous commentaries mentioned by al-Nadīm are adequate testimony to that. But even before one turns to these, attention should be drawn to yet other forms that found expression among Arabic treatments of Euclid. Quite distinct from translations proper (*naql*) there are a number of epitomes or summaries (*ikhtiṣār* or *mukhtaṣar*), recensions (*taḥrīr*), and emendations (*iṣlaḥ*) of the *Elements*.

Undoubtedly the most famous of the epitomes is that included by the Persian philosopher Avicenna (Ibn Sīnā) in the section on geometry in his voluminous philosophical encyclopedia, the *Kitāb al-Shifāʾ*. All fifteen books of the *Elements* are present, but with abbreviated proofs. Nor was Avicenna alone in his attempt to distill Euclid into a more compact dosage; we have already seen that Ḥajjāj considered one of the primary virtues of his second version of the *Elements* to be its shorter length, and other summaries were composed by Muẓaffar al-Asfuzārī (*d*. before 1122), a colleague of Omar Khayyām (al-Khayyāmi), and also, if we can believe a report by the fourteenthcentury historian Ibn Khaldūn (*Muqaddima*, VI, 20), by one Ibn al-Ṣalt (presumably the Hispano-Muslim physician, astronomer, and logician Abūʾl-Ṣalt [1067/1068–1134]).

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-Ṭuūsī. 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 thirteenbook *Taḥrīr*. However, one of these codices explicitly asserts that the work was completed on 10 Muḥarram 1298. Since al-Ṭuūsī 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.

One other *Tahrīr* of the *Elements* bears specific mention: that of Muḥyi ’l-Dīn al-Maghribī (*fl*. thirteenth century), a mathematician and astronomer who worked in both Syria and Marāgha and to whom we owe editions (literally “purifications,” *tahdhīb*) of Greek works on spherical trigonometry (Theodosius and Menelaeus) and of Apollonius’ *Conies*, and a similar work entitled *The Essence* (*Khulāṣa*) *of the Almagest*. His *Taḥrīr* may have been written shortly after the genuine fifteen-book *Taḥrīr* of al-Ṭūsī, since it is found in a manuscript dated 1260/1261. It contains, on the other hand, a preface that is similar in many ways to that found in the later (1298) *Taḥrīr*, wrongly, it appears, ascribed to al-Ṭūsī. It also complains of the faults in previous attempts to treat Euclid, but it is more specific in assigning at least some of the blame to Avicenna, a certain al-Nīsābūri, and Abū Jaʿfar al-Khāzin (cited in al-Nadīm’s chronicle). Al-Maghribī’s work sets out to remedy these faults and especially to explain all of the puzzles (*shmūkuk*) occasioned by Euclid and to supply the added lemmas (*muqaddamāt*) necessary for various proofs. In sum, one can say that al-Maghribī’s *Taḥrīr*, as well as the others we have mentioned above, began from the existing translations of the *Elements* and, through the incorporation (albeit in revised form) of presumably a good many of the notions contained in earlier commentaries as well as through the creation of much original material, proceeded to the preparation of an improved Euclid that may well have been ultimately intended to serve more adequately than Euclid himself as a school text. Exactly what this improved *Elements* contains as its most salient characteristics will be revealed only after a great deal more analysis of the relevant texts. And the same must be said for the translations of Ḥajjāj and Isḥāq–Thābit.

The third type of redaction of the *Elements* mentioned above, those labeled “Emendations” (*Iṣlāḥ*), is difficult to characterize beyond what is revealed by the title, since no known copies have survived of those to which reference is made by Islamic scholars. We are told, for example, that al-Kindī composed an Iṣlāḥ of the *Elements* in addition to his work *On the Aims (Aghrāḍ) of Euclid’s* Book. Similarly, *Iṣlāḥ’s* were written by the astronomer al-Jawharī and the Persian philosopher and scientist Athīr al-Dīn al-Abharī (*d*. 1265), but we know of them only through fragmentary quotations in other works. Further, in a way related to the emending of Euclid, it should be mentioned that the contribution of at least a few Islamic mathematicians to the transmission of the *Elements* appeared in the form of specific additions (*ziyādāt*), often merely to particular propositions within the text.

There remain the substantial number of Arabic commentaries, alternatively entitled *tafsīr* or *shurūḥ*, on Euclid. One can, indeed, extend their sequence considerably beyond that revealed in the *Fihrist* In another passage of that work al-Nadīm notes what would be, were the reference correct, the very first such commentary on the *Elements:* one ascribed to the central figure of Arabic alchemy, Jābir ibn Ḥayyān. But this clearly seems to be an error, introduced by a later scribal addition, for the thirteenth-century astronomer Jābir ibn Aflaḥ. When we turn, however, to the list of genuine commentaries in al-Nadīm and supplement it with information drawn from later sources, the number becomes so considerable (nearly fifty, of which more than half are extant in some form or another) that only the most notable can be mentioned here. Among the most significant recorded by al-Nadīm is that by the Persian mathematician and astronomer al-Māhānī, who commented on book X and on book V, and that by the somewhat later al-Nayrīzī. The latter, which is often a source for comments from lost Greek works on Euclid, was translated into Latin in the twelfth century by Gerard of Cremona. When one pushes beyond the Euclid entry of the *Fihrist*, note should be made of the particularly astute commentary on book V written by the Andalusian mathematician Ibn Muʿādh al-Jayyānī. It contains, apart from Greek mathematics itself, the first known comprehension of the brilliant definition of the equality of ratios formulated by Eudoxus. In fact, apart from several brief glosses in the medieval Latin Euclid, this definition was seldom properly understood in the West before Isaac Barrow in the seventeenth century. Finally, some note should be made of the fact that figures in Islam who derived appreciable eminence from other pursuits also saw fit to expend time in commenting on the *Elements*. Thus, one might cite the philosophers al-Kindī and al-Fārābī, who commented on books I and V. And similar attention should be drawn to the treatises on Euclid written by Alhazen (Ibn al-Haytham), author of the extremely significant textbook on optics, *Kitāb at-Manāẓir*, and to the commentary dealing with the problems of parallels, ratios, and proportion by the even more famous Persian mathematician and poet Omar Khayyām.

A somewhat more informative outline of the commentaries can be had if one turns from their authors to the questions and subjects they treat. Although so few have been edited, to say nothing of studied, that only the most tentative attempt can be made to assay the contents of these commentaries, it is nevertheless possible to see at least some of the areas of major concern. To begin with, it should be made clear that the commentaries were more often than not on parts, and not the whole, of the *Elements*. Thus, as one expects within almost any body of Euclidean commentarial literature, considerable effort was spent in mulling over premises, i.e., definitions, postulates, and axioms (for example, in the treatises of al-Karābīsī [see al-Nadīm’s report], al-Fārābī, Ibn al-Haytham, and Omar Khayyām referred to above). As a subclass of this genre of concern, emphasis should be placed upon special tracts, or passages in more general commentaries, that carried on the series of attempts already made in Greek mathematics to prove the parallels postulate (thus, to cite but a portion of the literature, we have two separate treatises on this topic written by Thābit ibn Qurra, a separate work dealing with it by al-Nayrīzī, and treatments of it in the *Taḥrīr* of both al-Maghrībi and al-Ṭūsī [both the genuine and the spurious *Taḥrīr* of the latter]).

Moving beyond the concern expressed over premises, one is immediately struck by the unusually high proportion of commentaries on books V and X. Although further investigation is needed to establish all of the motives behind the larger share of attention received by these books, a preliminary conjecture can easily be made. On the one hand, the extreme complexity of the treatment of irrational magnitudes in book X undoubtedly required more exposition and explanation to assure comprehension. On the other hand, the central role played by the theory of proportion contained in book V throughout all geometry probably caused Islamic mathematicians, rightly, to view this book as more fundamental than others. This, coupled with the consideration that some trouble was had in appreciating the Eudoxean definition of equal ratios that is included in book V, most likely gave it a position of some priority in the eyes of potential commentators.

One feature of the series of Arabic commentaries on the *Elements* should be recorded: Although the greater number of such commentators were mathematicians, astronomers, or physicians (or some combination thereof), a minority were not that, but rather philosophers. Of course, a philosopher of the mark of al-Kindī was as much concerned with things scientific as he was with things philosophical. But others, such as al-Fārābī and Avicenna, did not have his scientific interests or acumen. Yet they too commented on, or epitomized, the *Elements*. We are also informed that the philosopher and Shāfiʿite theologian Fakhr al-Dīn al-Rāzī (1149–1210) wrote on Euclid’s premises and that the Cordovan philosopher, physician, and Aristotelian commentator par excellence Averroës (Ibn Rushd) wrote a treatise on what was needed from Euclid for the study of Ptolemy’s *Almagest*. It is likely, to be sure, that such works on the *Elements* written by philosophers (most, unfortunately, are lost) were less penetrating and exacting than the more mathematical product of other commentators; they are, nonetheless, still significant as a measure of the extent to which the importance of Euclid had penetrated Islamic thought. In sum, the Arabic phase of the *Elements’* history may well prove to be not merely the most manifold but, even mathematically, the most creative of all.

Other medieval Near Eastern translations of the *Elements* all seem to have been based on one or another of the Arabic versions already mentioned. This is certainly the case with the Persian translation (completed in 1282–1283) of al-Ṭuūsī’s fifteen-book *Taḥrīr*, ostensibly made by his pupil Quṭb al-Dīn al-Shīrāzī (1236–1311). Similarly, although there are a fair number of medieval Hebrew compendia and special recensions of the *Elements*, the basic thirteenth-century Hebrew translation (or translations) appears to derive from the revised Isḥāq–Thābit version but contains marginal reference to the Ḥajjāj translation as well. It is still problematic whether we have here two distinct Hebrew translations or the collaborative effort over a number of years (*ca*, 1255–1270) of the two scholars involved: Moses ibn Tibbon and Jacob ben Maḥir ibn Tibbon.

Even more debatable is the issue of the Syriac version of Euclid. It was frequently the case that Arabic translations of Greek works were executed via a Syriac intermediary. It is, however, rather doubtful that this was true with the *Elements*. We do possess fragments of a Syriac redaction in a fifteenth- or sixteenth-century manuscript, and comparison of these fragments with the Arabic tradition clearly indicates a filiation, although without any absolute evidence of the direction in which the parentage must have run. If one asks how early the Syriac edition must be dated, present evidence necessitates moving it back to the eleventh or twelfth century. For instance, we know that the Syriac polymath Abu’l-Faraj (Bar Hebraeus, 1226–1286) lectured on Euclid at Maragha in 1268. Furthermore, reference to a Syriac version of the *Elements* is made in the 1298 pseudoal-Ṭūsī *Taḥrīr* and in mathematical opuscula of Ibn al-Sarī (*d*. 1153). Finally, note should be made of fragments of an Armenian version of the *Elements*, for it too appears to be related to the Arabic (Isḥāq–Thābit) tradition. It seems most probable that this Armenian Euclid was the work of Gregory Magistros (*d*. 1058), in one of whose letters we find the announcement that he had begun a translation of the *Elements*. If to this we add the fact, as one scholar has urged, that Gregory knew only Greek and Syriac, but no Arabic, it would appear that he based his translation in some way or another on the Syriac version under discussion. This gives us a terminus ante quem of the first half of the eleventh century for this version, but there is no other evidence on the basis of which we can, with any certainty, assign it an earlier date. One can merely indicate that the editor of these Syriac fragments, G. Furlani, judged them to have a very close relation to the Arabic text of Ḥajjāj and that they were, in his view, in some way derived from this text. He dismissed the apparent contrary evidence one might derive from the Syriac transcription of Greek terms, since this often occurs in Syriac works that we know were based on Arabic originals. However, the second scholar to examine the fragments, Mlle. Claire Baudoux, claimed a definite link with the Isḥāq–Thābit translation (not investigated by Furlani) and concluded that the Syriac redaction preceded Isḥāq and served as an intermediary between it and the Greek original. Nevertheless, it would seem that the issue must stand unresolved until a fresh comparison is made with both Arabic translations and all relevant evidence is presented in detail. Until then, it would seem more plausible to hold the tentative conclusion that the Syriac version had an Arabic source, and not vice versa.

As the article we have quoted above from the *Fihrist* already indicates, Euclid’s other works also existed in Arabic, although al-Nadīm has omitted the names of their translators. Indeed, we are still not able to identify translators in all instances. Thus, although the original translator of *On the Division of Figures* remains unknown, we do have information that Thābit ibn Qurra revised the translation, and it is, as a matter of fact, on the basis of this revision that, together with other Latin material drawn from the work of the thirteenth-century mathematician Leonardo Fibonacci, we have been able to reconstruct the contents of this Euclidean treatise. Similarly, we know that Thābit also corrected the translation of the work *On the Heavy and the Light*. There is also a treatise extant in Arabic called *The Book of Euclid on the Balance*, but there is no further information concerning its provenance.

Three other minor works, the *Data*, the *Phaenomena*, and the *Optica* (the Arabs were not aware of the pseudo-Euclidean *Catoptrica*), have a similar Islamic history. All three were part of that collection of shorter works known as the “middle books” *(mutawassiṭāt)*, which functioned as appropriate texts for the segment of mathematics falling between the *Elements* and Ptolemy’s *Almagest*. Both the *Data* and the *Optica* underwent Isḥāq–Thābit translation-revisions and later *Taḥrīr* at the hands of al-Ṭūsī. Of the *Phaenomena* we are reliably informed only of the recension done by al-Ṭūsī.

**The Medieval Latin Euclid: The Greek–Latin Phase.** The first known Latin reference to Euclid is found in Cicero (*De oratore*, III, 132)—surely a good number of years before any attempt was made to translate the *Elements*. This latter aspect of the Latin history of Euclid begins, as far as extant sources tell us, with a fragment attributed to the third-century astrologer Censorinus. What we have in this fragment that gives excerpts from the *Elements* might also be reflected in the Euclid passages in the *De nuptiis* of Martianus Capella, although some historians feel that Martianus may have been utilizing a Greek source as well as some Latin adaptation of (or at least of parts of) the *Elements*.

The second piece of evidence in the history of Latin renditions of Euclid is found in a fifth-century palimpsest in the Biblioteca Capitolare at Verona. Treated with chemicals in the nineteenth century, it is now all but impossible to decipher. We can establish, however, that it contains fragments of a translation from books XII-XIII of the *Elements*. Very little else can be said with any surety of the translation, although its most recent editor, M. Geymonat, has urged that the palimpsest be dated slightly later and has suggested that Boethius was the author of the translation of the fragments that it contains.

Whether or not this suggestion is correct, it is to the problem of the Boethian Euclid that we must now turn. We do know that Boethius made such a translation because Cassiodorus refers to it in his *Institutiones* (II, 6, 3: “ex quibus Euclidem translatum Romanae linguae idem vir magnificus Boethius edidit”) and also preserved a letter from Theodoric to Boethius himself (*Variae*, I, 45, 4) in which the existence of the translation is again attested. However, we are far less well informed of the extent and nature of this translation, for the “Boethian” geometries—or better, geometrical materials—that have come down to us are in a late fragmentary form. Basically, the excerpts we possess of Boethius’ translation derive from four sources, each considerably later than the date of his actual translating efforts: (1) excerpts in the third recension of book II of Cassiodorus’ *Institutiones* (eighth or ninth century); (2) excerpts in a number of manuscripts of a later redaction of the *Agrimensores*, a collection (made *ca,* 450) of materials concerned with surveying, land division, mapmaking, the rules of land tenure, etc. (ca. ninth century); (3) excerpts within the so-called five-book “Boethian” geometry (eighth century); (4) excerpts within the so-called two-book “Boethian” geometry (eleventh century). Special note might be taken of the full content of the last two sources, inasmuch as they appear in the literature under Boethius’ name. The earlier of the two compilations, in five books, consists of gromatic material in book I and in part of book V, of excerpts from Boethius’ *Arithmetica* in book II, and of excerpts from his translation of Euclid in books III-IV and in the initial section of book V. The twobook version of the “Boethian” geometry seems to have been compiled by a Lotharingian scholar without especially acute mathematical ability and contains its excerpts from Boethius’ translation in book I, as well as a brief preface and a concluding section on the abacus, while book II consists largely of *Agrimensores* material. If one combines the extracts of Boethius *Elements* from these two works with the extracts found in the Cassiodorus and *Agrimensores* sources listed above, the total schedule, as it were, of translated Euclid amounts to (a) almost all the definitions, postulates, and axioms of books I-V of the *Elements;* (b) the enunciations of almost all the propositions of books I-IV; and (c) the proofs for book I, propositions 1–3. The above four sources containing these extracts often overlap in the items they include, but it is notable that a sequence of the enunciations of propositions from book III (i.e., 7–22) is found only in the five-book “Boethian” geometry, while the definitions of book V appear only in the recension of Cassiodorus. (The relation of the four sources can be seen in the chart below.)

The ninth- through fifteenth-century manuscripts in which these sources (especially the last three) of Euclidean excerpts appear are, for the most part, collections containing other material pertinent to the quadrivium. But even when, with new and more complete translations of the *Elements* in the twelfth century, this kind of collection began to lose the dominant position it once held in medieval Latin mathematics, traces of the Boethian Euclid linger on through occasional conjunction with the newly translated material. Thus, we know of at least two different melanges of parts of the Boethian excerpts with one of the translations of the *Elements* from the Arabic by Adelard of Bath (that labeled Adelard II below). One of these melanges dates from about 1200 and seems to have been compiled by a North German scholar with appreciably more mathematical wit than, for example, the author of the two-book “Boethius” discussed above. It is preserved in a single thirteenth-century manuscript: Lüneburg, Ratsbücherei MS miscell. D 4°48. The second mélange occurs in four manuscripts, three of them of the twelfth century, but little has been done to attempt to determine the provenance of its author. Further, cognizance should be taken of the fact that the Boethian “source” of both mélanges seems to have been the two-book *Geometry*. Finally, although we do have these attempts to combine the Greek–Latin Boethian extracts of the *Elements* with the Arabic-Latin tradition deriving from Adelard, it should be made clear that they constituted but a minor part of the medieval Euclid in the West; the Adelardian-based tradition was soon to hold all in sway.

However, before we move to this tradition and to the Arabic-Latin Euclid in general, two other Greek–Latin medieval versions must be mentioned. Of the first we have but a fragment (I, 37–38 and II, 8–9). It exists in a single tenth-century manuscript in Munich. Although extremely literal, its translator, an Italian, knew little of what he was doing, since he translated as numbers the letters designating geometrical figures.

The second Greek–Latin Euclid we must discuss constitutes the most exact translation ever made of the *Elements*, being a *de verbo ad verbum* rendering in which the order of words and occasionally the syntax itself are often more Greek than Latin. Based solely on a Theonine text, the translation is known from two extant manuscripts and covers books I-XIII and XV. Neither manuscript names the translator, but a stylistic analysis of the text has established that he is identical with the anonymous twelfth-century translator of Ptolemy’s *Almagest* from the Greek. A preface fortunately attached to the latter translation informs us that our nameless author was a one-time medical student at Salerno who, learning of the existence in Palermo (*ca*. 1160) of a Greek codex of Ptolemy, journeyed to Sicily in order to see this treasure and, after a period of further scientific preparation, set himself to putting it in Latin. Presumably our translator did the same for the *Elements* shortly thereafter (since no mention of such an effort is made in his description of other of his activities in his preface to his version of Ptolemy). When one turns to the translation itself, it is immediately evident that its author was extremely acute, both as an editor and as a mathematician. Not only does he give an extraordinarily exact rendering of the Greek, but on occasion he also employs brackets to indicate several passages in an alternate Greek manuscript he was using. What is more, several times he employs these same brackets to improve the logic of a proof. Unfortunately, the superb Latin Euclid he produced exerted very little, if any, influence upon his medieval successors. (It might also be indicated that one manuscript of this translation contains a pastiche of books XIV-XV in place of the missing book XIV; it too derives from Greek sources and even castigates translators from the Arabic for being insufficiently careful.)

The Greek–Latin phase of the medieval Euclid is also, perhaps, the most appropriate point at which mention should be made of the minor Euclidean works during this period. For, contrary to what proved to be true for the *Elements*, these shorter works have a medieval Latin history that derives predominantly—in all instances through anonymous translators—from the Greek. Thus, in place of the apparently lost version of the *Data* from the Arabic by Gerard of Cremona (who also translated the *Elements*), we possess several codices of an accurate rendering made in the twelfth century directly from the Greek. Similarly, although there do exist copies of Gerard’s Arabic-Latin *Optica*, they are over whelmingly outnumbered by manuscripts containing Greek–Latin translations. Indeed, there appear to be two distinct versions of the *Optica* from the Greek, some manuscripts of which are so variant as to lead one to expect an even more complicated history. There also seem to be several versions of the pseudo-Euclid *Catoptrica* made from the Greek. What is more, there is a totally separate *De speculis* translated from the Arabic (by Gerard?) and ascribed to Euclid. We know of no Greek or Arabic original from which it may have derived, although it does exist in Hebrew in several manuscripts. The *Sectio canonis* had several propositions from it transmitted through the medium of Boethius’ *De institutione musica*. The *Phaenomena*, on the other hand, was not put into Latin before the Renaissance.

**The Medieval Latin Euclid: The Arabic-Latin Phase.** Once integral translations of the *Elements* from the Arabic were available to the medieval scholar, all Greek–Latin fragments and versions receded into the background. The new, dominant tradition was, however, twofold; one wing derived from the Ḥajjāj Euclid, the other from that of Isḥāq–Thābit, the recensions of al-Ṭūsī and al-Maghribī coming too late, of course, to enter into the competition of translating activity in the twelfth century.

The Latin *Elements* based upon the Isḥāq–Thābit text was the accomplishment of Gerard of Cremona, the most industrious of all translators of scientific, philosophical, and medical works from the Arabic. We know that he translated the *Elements* from its citation in his *Vita*, written by one of his pupils and appended to one of his translations of Galen. Identified among extant manuscripts in 1901, Gerard’s Euclid was soon realized to be the closest to the Greek tradition of all Arabic–Latin versions. It alone contains Greek material—for example, the preface to the spurious book XIV—absent from the other versions. Ironically, however, it clearly seems to have been less used, and less influential, than the somewhat more inaccurate (Adelardian-based) editions. It derives its more faithful reflection of the Greek original from the fact that the Arabic of Isḥāq–Thābit, upon which it was based, is itself a more exact reproduction of the Greek. We have no explicit ascription stating that Gerard worked from this particular Arabic translation, but even the most preliminary examination of Gerard’s text reveals that this was in all probability the case. For instance, the phrase “Thebit dixit” occurs frequently throughout the body of the translation. At least some of these occurrences—perhaps almost all of them—are not due to Gerard’s reflecting on the text he was rendering, but to a direct translation of that text itself, since several citations that have been published by Klamroth from the Arabic and are reproduced in Gerard indicate that Thābit is named therein as well. (Third-person references by an author to himself are, of course, quite common.) While awaiting evidence that will issue from a direct comparison of the Isḥāq–Thābit and Gerard texts, note should be made of the fact that Gerard has the two propositions added after VIII, 25, and the corollaries to VIII, 14–15, which are characteristic of Isḥāq–Thābit. Yet this is not the only Arabic version Gerard had before him, at least not in its pure form. For he includes VIII, 16, which is not, according to Klamroth, in the Isḥāq–Thābit manuscripts examined. Furthermore, after having followed these manuscripts by reproducing VIII, 11–12, as two separate propositions, at the conclusion of book VIII Gerard has an addendum claiming that these two propositions were found as one in *alio libra;* the addendum continues by reproducing this combined version, proof and all. Exactly who found this combined version of VIII. 11–12, in another book is problematic; use of the first person in this passage in Gerard is not conclusive, since it could derive directly from his Arabic text. We do know, however, that the Adelardian tradition ostensibly based on the Ḥajjāj Euclid (of which book VIII is not extant) does conflate the two propositions in question. Therefore, either Gerard utilized texts of both Isḥāq–Thābit and Ḥajjāj in making his translation or, which seems more likely, he based his labors on an Isḥāq–Thābit text that contained material drawn from one or another of the Ḥajjāj versions.

Gerard also contributed to the literature of Euclides Latinus by translating the commentary of al-Nayrīzī on the *Elements*, the commentary of Muḥammad ibn ʿAbd al-Bāqī (*fl. ca*. 1100) on book X, and at least part of Dimashqīʾs translation of Pappus’ commentary on book X.

By far the most important share of the medieval Arabic–Latin Euclid belongs to the English translator, mathematician, and philosopher Adelard of Bath. Not only was Adelard himself the author of at least three versions of the *Elements*, but he served as the point of departure for numerous offspring redactions and revisions as well. Taken together as the Adelardian tradition, they soon gained a virtual monopoly when it came to using and quoting Euclid in the Latin Middle Ages.

The first version due to Adelard himself (hereafter specified by the Roman numeral I) is the only one within the whole tradition that is, properly speaking, a translation. As such, there is clear indication that it was based on the Arabic of Ḥajjāj. Thus, the proofs in Adelard I appear to correspond quite well with what we have of Ḥajjāj. Further, Adelard I carries the same three added definitions (at least two, however, going back in some way to the Greek) to book III found in Ḥajjāj and agrees with him in reproducing the maximum of six separate cases in the proof of III, 35. But it also seems clear that Adelard I did not utilize Ḥajjāj as we have him today. The fact that he does not reproduce the arithmetical examples in books II and VI, or the added propositions in book V, that are present in the extant Ḥajjāj is not the point at issue, since these features are most likely from the commentary of al-Nayrīzī to which our Ḥajjāj text is attached. What is significant is the fact that Adelard I does contain the corollaries to II, 4, and VI, 8, which are not present in our Ḥajjāj and, even more suggestive, does not include VI, 12, which is contained in our Ḥajjāj. If one couples the latter fact with the statement in the pseudo-Ṭūsī *Taḥrīr* that Ḥajjāj did not include VI, 12, it then becomes most probable that there existed another Ḥajjāj, slightly variant from the text we possess, and that it was this Arabic version which Adelard I employed. One could argue that it would have been the first version Ḥajjāj prepared under Hārūn al-Rashīd, but we know that this earlier redaction was somewhat longer than the second version—which, presumably, is the one we possess. And Adelard I in other respects (particularly in the length and detail of the proofs) appears to correspond well enough with the second extant version to make derivation from a longer Arabic original unlikely. A variant of Ḥajjāj’s second redaction seems a more plausible source.

Adelard II, on the other hand, does not give rise to similar problems, since it is not a translation but an abbreviated edition or, as Adelard himself calls it (in his version III), a *commentum*. Although the briefest of Adelard’s efforts in putting Euclid into Latin, it was unquestionably the most popular. This is clear not merely on grounds of the far greater number of extant copies of Adelard II, but also because the translations given here of the “enunciations” (of definitions, postulates, axioms, and propositions) were subsequently appropriated for a good many other versions by editors other than Adelard. Indeed, the diversity thus growing out of Adelard II appears to be present within it as well, since there is considerable variation among the almost fifty manuscript copies thus far identified. The earliest extant codex (Oxford, Trinity College 47) presents, for example, a text that is more concise than any consistently presented by other manuscripts.

The characteristic feature of Adelard II lies in its proofs, which are not truly proofs at all, but *commenta* furnishing relevant directions in the event one should wish to carry out a proof. One is constantly reminded, for instance, of just which proposition or definition or axiom one is building upon, or whether the argument—should it be carried out—is direct or indirect. The *commenta* talk about the proposition and its potential proof; the language is, in our terms, metamathematical. Moreover, this talk about the proof often puts greater emphasis upon the constructions to be utilized than on the proof proper.

Adelard III, referred to by Roger Bacon as an *editio specialis*, continues this fondness for the metamathematical remark, but now embeds them within and throughout full proofs as such. In the bargain, one often finds that such reflections veer from the proposition and the proof at hand to external mathematical matters.

Adelard II and III have much in common besides their author. Both contain Arabisms; both contain Grecisms; and III quotes II. More important, however, both make use of original Latin material: they employ notions drawn from Boethian arithmetic, use classical expressions, and even (Adelard III) allude to Ovid.

A fourth major constituent of the Adelard tradition is the version of the *Elements* prepared by Campanus of Novara. It too takes over the Adelard II enunciations and, through the formulation of proofs that seem largely independent of Adelard, fashions what is, from the mathematical standpoint, the most adequate Arabic–Latin Euclid of all (its earliest dated extant copy being that of 1259). The *additiones* Campanus made to his basic Euclidean text are particularly notable. With an eye to making the *Elements* as self-contained as possible, he devoted considerable care to the elucidation and discussion of what he felt to be obscure and debatable points. He also attempted to work Euclid more into the current of thirteenth-century mathematics by relating the *Elements* to, and even supplementing it with, material drawn from the *Arithmetica* of Jordanus de Nemore.

Furthermore, Campanus and the three versions of Adelard (especially II and III) served as sources for an amazing multiplication of other versions of the *Elements*, Although the extent to which they diverge from one or another of these sources may not be great or marked with much originality, and although they frequently seem to concentrate on selected books of Euclid, one can still discern among extant manuscripts some fifteen or more additional “editions” belonging to the Adelardian tradition. They range from the thirteenth through the fifteenth centuries and none, as far as a cursory investigation has shown, bears the name of an author or a compiler.

As a whole the Adelardian tradition formed the dominant medieval Euclid. Further, although by far the greatest share of this tradition was not a strict translation from its Arabic source, the divergence from the original Greek thus occasioned caused little difficulty. The few missing propositions were easily remedied, and changes in the order of theorems gave rise to no mathematical qualms at all. Misunderstanding of what Euclid intended also seems to have been quite infrequent, save for the always problematic criterion of Eudoxus for the equality of ratios, which was ensconced in book V, definition 5. Here the most influential medieval interpretation—that of Campanus—curiously seems to have been conditioned by a strange quirk in transmission. For in place of the genuine V, definition 4, Campanus (and all other constituents of the Adelardian tradition) has another, mathematically useless, definition; and in his attempt to make sense of this, Campanus formulated a mechanics of explanation that he in turn extended to his discussion, and consequent misunderstanding, of the “Eudoxean” V, definition 5. Thus, we are witness to a unique instance in which the existence of a spurious fragment within the textual tradition seriously affected interpretation of something genuine.

The most impressive characteristic of the Adelardian–Campanus *Elements* is not, however, to be found in missing or misunderstood fragments of the text, but rather in the frequent additions made to it, additions which often take the form of supplementary propositions or premises but also occur as reflective remarks within the proofs to standard Euclidean theorems. It is not possible to tabulate even a small fraction of these additions, but it is important to realize something of the basis of their concern. To begin with, the motive behind many, indeed most, of them was to render the whole of their Euclid more didactic in tone. The trend toward a “textbook” *Elements*, noted in its Arabic history (and even, in a way, in Theon of Alexandria’s new Greek redaction), was being extended. The reflections mentioned above about the structure of proofs are surely part of this increased didacticism. The labeling of the divisions within a proof, express directions as to how to carry out required constructions and how to draw threedimensional figures, indications of what “sister” propositions can be found elsewhere in the *Elements*, and even clarifying references to notions from astronomy and music are all evidence of the same. We are also witness to the erosion (again pedagogically helpful) of the strict barrier fixed by the Greeks between number and magnitude. For the care not to employ the general propositions of book V in the arithmetical books VII-IX has been pushed out of sight, and one can find admittedly insufficient numerical proofs in propositions (especially in books V and X) dealing with general magnitude.

Of even greater interest is an ever-present preoccupation with premises and with what is fundamental. Axioms are everywhere added (even in the middle of proofs) to cover all possible gaps in the chain of reasoning, and considerable attention is paid to the logic of what is going on. Once again one sees a fit with didactic aims. But this emphasis on basic notions and assumptions was directed not only toward making the geometry of the *Elements* more accessible to those toiling in the medieval faculty of arts; it was also keyed to the bearing of issues within this geometry upon external, largely philosophical problems. Most notable in this regard is the time spent in worrying over the conceptions of incommensurability, of the so-called horn angle (between the circumference of a circle and a tangent to it), and of the divisibility of magnitudes. For these conceptions all relate, at bottom, to the problems of infinity and continuity that so often exercised the wits of medieval philosophers. The Adelardian tradition furnished, as it were, a schoolbook Euclid that admirably fit Scholastic interests both within and beyond the bounds of medieval mathematics.

There is one other medieval Latin version of the *Elements* that is connected, more tenuously to be sure, with the Adelard versions. Its connection derives from its appropriation of most of the Adelard II enunciations, although frequently with substantial change. It exists in anonymous form in a single manuscript (Paris, BN lat. 16646). We know, however, that this codex was willed to the Sorbonne in 1271 by Gérard d’Abbeville and is in all probability identical with a manuscript described in the *Biblionomia* (*ca*. 1246) of Richard de Fournival. Richard, however, identifies the manuscript as “Euclidis geometria, arismetrica et stereometria ex commentario Hermanni secundi.” This version is, therefore, presumably the work of Hermann of Carinthia (*fl. ca*. 1140–1150), well-known translator of astronomical and astrological texts from the Arabic. Its proofs differ from those of the Adelardian tradition, and the occurrence of Arabisms not in this tradition has been viewed as evidence for Hermann’s use of another Arabic text in compiling his redaction. The recent suggestion that this text was the Isḥāq–Thābit translation seems doubtful, however, for unlike Gerard of Cremona’s version, ostensibly based on that Arabic translation, Hermann not only lacks the references to “Thebit” but also does not have the additions in book VIII (see above) characteristic of Isḥāq–Thābit. On the other hand, it is true that Hermann does show variations from the text of Ḥajjāj as we have it; but all of these, it appears, are also in Adelard I, which clearly derives, as we have seen, from some Ḥajjāj text or another. Hermann repeats the differences noted above in discussing Adelard I and with him alone, among all Latin versions, carries the full six separate cases for III, 35. One other piece of evidence might be noted: In a series of propositions (V, 20–23) dealing with proportion, the Greek Euclid specifies only one (V, 22) for “any number of magnitudes whatever,” the others being stated merely for three magnitudes. Now all versions save Adelard I and Hermann, including our Ḥajjāj and, to judge from Gerard’s translation, Isḥāq–Thābit, adopt the policy of stating all four propositions in general form. Hermann and Adelard I, on the other hand, retain the “three magnitudes” version of the Greek for V, 20–21, 23 but also substitute *tres* for *quotlibet* in V, 22. The filiation of these two is, therefore, quite close. One can conclude, then, that Hermann used at least both Adelard II (from which he derives many enunciations) and presumably the same version of Ḥajjāj used by Adelard I (and possibly also Adelard I itself). Finally, it has been noticed that Hermann contains the Arabicism *aelman geme* (corresponding to *ʿilm jāmiʿ*) to refer to axioms or common notions, while both Ḥajjāj and, to judge from Klamroth, Isḥāq–Thābit employ *al-ʿulūm al-mutaʿārafa*. Yet here Hermann could still be following some Ḥajjāj text, since *ʿilm jāmiʿ* (also used, incidentally, in Avicenna’s epitome of Euclid) occurs as a marginal alternative in our manuscript of Ḥajjāj.

Occasional claims have also been made for two other identifiable medieval Latin versions of the *Elements*. One, purported to be by Alfred the Great, has been shown to derive from erroneous marginal ascriptions (in a single manuscript) of the Adelard II version to “Alfredus.” The second, a seventeenthcentury manuscript catalogue, refers to a version of the *Elements* as “ex Arab, in Lat. versa per Joan. Ocreatum.” We do know of passages at the beginning of book V and in book X (props. 9, 23, 24) in certain manuscripts of Adelard II that do appeal to “Ocrea Johannis,” but in such a way that this may be but a reference to separate (marginal?) comments on Euclid or to some other mathematical treatise, rather than to a distinct translation or version of the *Elements*.

If, in conclusion, one compares the very substantial amount of material constituting the Arabic–Latin Euclid with the equally extensive history the *Elements* had in Islam, several rather striking differences are apparent, even at the present, extremely preliminary stage of investigation of these two traditions. In both one finds an overwhelming number of versions, editions, and variants of Euclid proper. Although here one has a common trait, there is, on the other hand, no doubt that the number of commentaries composed in Arabic far exceeds those in Latin. Indeed, there seems to be but one “original” commentary proper in Latin, questionably ascribed to the thirteenthcentury Dominican philosopher Albertus Magnus and in any event greatly dependent upon earlier translated material (notably the commentary of al-Nayrīzī). When we do find *Questiones super Euclidem*, they treat more of general problems within geometry, mathematics, or natural philosophy than they do of issues specifically tied to particular Euclidean premises or theorems. In point of fact, this bears upon a second element of contrast between the Arabic and Latin traditions—that the latter seems to have moved more rapidly toward serving the interests of philosophy, while the former remained more strictly mathematical in its concerns. One does not find, for example, anything like the Arabic debate about the parallels postulate in the Latin texts. But then Islamic mathematics itself was more lively and creative than that of the medieval Latin West.

**The Renaissance and Modern Euclid.** Four events seem to have been the most outstanding in determining the course of the *Elements* in the sixteenth and succeeding centuries: (1) the publication of the medieval version of Campanus of Novara, initially as the first printed Euclid at Venice (1482) by Erhard Ratdolt, and at many other places and dates in the ensuing 100 years; (2) a new Latin translation from the Greek by Bartolomeo Zamberti in 1505; (3) the *editio princeps* of the Greek text by Simon Grynaeus at Basel in 1533; (4) another Greek–Latin translation made in 1572 by Federico Commandino. The publications resulting from these four versions show their effect in almost all later translations and versions, be they Latin or vernacular.

Of Campanus we have spoken above. The printed Euclid following his was, ignoring the publication in various forms of the “Boethian” fragments, not the influential one of Zamberti but only portions of the *Elements* included in the gigantic encyclopedia *De expetendis et fugiendis rebus* (Venice, 1501) of Georgius Valla (*d*. 1499). This was not, to be sure, an easily accessible Euclid; for in addition to being an extraordinarily cumbersome book to use, the selections from the *Elements* are scattered among materials translated from other Greek mathematical and scientific texts. The first publication of a Greek-based Latin *Elements* as an integral whole was that at Venice in 1505 prepared by Bartolomeo Zamberti (*b. ca*. 1473). His translation derived from a strictly Theonine Greek text, a factor which has Zamberti attributing the proofs to this Alexandrian redactor *(cum expositione Theonis insignis mathematici)*. The work also contains translations of the minor Euclidean works (which were also, in part, in Valla’s encyclopedia).

Zamberti was most conscious of the advantages he believed to accrue from his working from a Greek text. This enabled him, he claimed, to add things hitherto missing and properly to arrange and prove again much found in the version of Campanus. Indeed, his animus against his medieval predecessor is far from gentle: his Euclid was, Zamberti complains, replete with “wondrous ghosts, dreams and fantasies” *(miris larvis, somniis et phantasmatibus)*. Campanus himself he labels *interpres barbarissimus*.

The attack thus launched by Zamberti was almost immediately answered by new editions of Campanus, the most notable of them being that prepared at Venice in 1509 by the Franciscan Luca Pacioli. Pacioli regarded himself as a corrector (*castigator*) who freed Campanus from the errors of copyists, especially in the matter of incorrectly drawn figures. In direct reply to Zamberti, Campanus was now presented as *interpres fidissimus*.

A kind of detente was subsequently reached between the Campanus and Zamberti camps, for there was soon a series of published *Elements* reproducing the editions of both *in toto*, the first appearing at Paris in 1516. Each theorem and proof first occurs *ex Campano* and is immediately followed by its mate and proof *Theon ex Zamberto*. The *additiones* due to Campanus appear in place but are appropriately set off and indicated as such.

The end of the first third of the sixteenth century brought with it the first publication of the Greek text of the *Elements*. The German theologian Simon Grynaeus (*d*. 1541) accomplished this, working from two manuscripts, with an occasional reference to Zamberti’s Latin. His edition, which included the text of Proclus’ *Commentary* as well, was the only complete one of the Greek before the eighteenth century. The other Greek Euclids of the Renaissance were all partial, most frequently offering only the enunciations of the propositions in Greek (usually with accompanying translation). The most significant of such “piecemeal” *Elements* is unquestionably that of the Swiss mathematician and clockmaker Conrad Dasypodius, or Rauchfuss (1532–1600). Dasypodius makes it abundantly clear that his edition, issued in three parts at Strasbourg in 1564, was intended as a school-text Euclid. For this reason he believed it more convenient to give merely the enunciations of books III-XIII to accompany the full text of books I—II, all in Greek with Latin translation. The pedagogical design of his publication is also seen from the fact that in spite of his exclusion of a great deal of genuine Euclid, he nevertheless saw fit to include the text of the more readily comprehensible arithmetical version of book II that was composed by the Basilian monk Barlaam (*d. ca*. 1350).

In any event, the printing of the complete Greek text in 1533, plus the earlier appearance of both Campanus and Zamberti, provided the raw material, as it were, for the first, pre-Commandino, phase of the Renaissance Euclid. The irony is that Campanus and Zamberti, and not the Greek *editio princeps*, played the dominant role. Some note of the most significant of the many early Renaissance printed editions of the *Elements* will make this clear. (The substantial, but totally uninvestigated, manuscript material of this period is here excluded from consideration.)

If one focuses, to begin with, upon Latin Euclids, the first important “new” version (Paris, 1536) is that of the French mathematician Oronce Fine (1494–1555). Yet his contribution seems to have been to insert the Greek text of the enunciations for the *libri sex priores* in the appropriate places in Zamberti’s Latin translation of the whole of these books. Similarly, his compatriot Jacques Peletier du Mans brought out another six-book Latin version (Lyons, 1557), this time based, as Commandino noted, more on Campanus’ Arabic–Latin edition than upon anything Greek. (It should be recorded, however, that Peletier supplemented what he took from Campanus’ *additiones* with some interesting ones of his own.) At Paris in 1566 yet a third French scholar, Franciscus Flussatus Candalla (François de Foix, Comte de Candale, 1502–1594) produced a Latin *Elements*. Covering all fifteen books, and appending three more on the inscription and circumscription of solids, the appeal is once again not to the Greek text as such, but to Zamberti and Campanus. And when there is something not derived from these two, it seems as often as not to have been Candalla’s own invention.

A contemporary summary view of the status of Euclid scholarship was revealed when, a few years before Candalla’s expanded *Elements*, Johannes Buteo published his *De quadratura circuli* at Lyons in 1559. This work contained as an appendix Buteo’s *Annotationum opuscula in errores Campani, Zamberti, Orontii, Peletarii... interpretum Euclidis*. Campanus was, he felt, the best of these editors, for his errors derive from his Arabic source and not from an ineptitude in mathematics. Zamberti, on the other hand, although he worked directly from the Greek, showed less acumen in geometry. Even less adequate, in Buteo’s judgment, were the works of Fine and Peletier, the latter taking the greatest liberties with the text and ineptly adding or omitting as he saw fit.

We have thus far spoken merely of sixteenth-century Latin translations, but the same pattern reflecting the central impact of Campanus and Zamberti can also be discerned in the most notable vernacular renderings. The earliest of these to be printed was the Italian translation by the mathematician, mechanician, and natural philosopher Niccolò Tartaglia. Its first edition appeared at Venice in 1543. When Tartaglia submits that his redaction was made *secondo le due tradittioni*, there is no question that Campanus—who appears to be heavily favored—and Zamberti are meant. When Campanus has added propositions or premises, Tartaglia has appropriately translated them and noted their absence *nella seconda tradittione*, while things omitted by Campanus but included by Zamberti receive the reverse treatment.

The next languages to receive the privilege of displaying Euclid among their goods were French (by Pierre Forcadel at Paris in 1564) and German (at the hands of Johann Scheubel and Wilhelm Holtzmann in 1558 and 1562). We are better informed, however, of the circumstances surrounding the production of the more elaborate, first complete English edition. Yet before we describe this, it will be well to note an even earlier intrusion of Euclidean materials into English. This is found in *The Pathway to Knowledg* (London, 1551) of the Tudor mathematical practitioner Robert Recorde. Recorde fully recognized the ground he was breaking, for in anticipation of the dismay even Euclid’s opening definitions would likely cause in the “simple ignorant” who were to be his readers, he cautioned: “For nother is there anie matter more straunge in the englishe tungue, then this whereof never booke was written before now, in that tungue.” Recorde’s purpose was distinctly practical, and he expressly mentioned the significance of geometry for surveying, land measure, and building. The *Pathway* contains the enunciations of books I-IV of Euclid, reworked and reordered to serve his practical aims.

The first proper English translation was the work of Sir Henry Billingsley, later lord mayor of London (*d*. 1606), and appeared at London in 1570 with a preface by John Dee, patron and sometime practitioner of the mathematical arts. A truly monumental folio volume, Billingsley’s translation contains “manifolde additions, Scholies, Annotations and Inventions... gathered out of the most famous and chiefe Mathematiciens, both of old time and in our age” and even includes pasted flaps of paper that can be folded up to produce three-dimensional models for the propositions of book XI. Each book begins with a summary statement that includes considerable commentary and often an assessment of the views of Billingsley’s predecessors, most notably those of Campanus and Zamberti. The role these two scholars played in Billingsley’s labors is confirmed in yet another way. There exists in the Princeton University Library a copy of the 1533 *editio princeps* of the Greek text of the *Elements* bound together with a 1558 Basel “combined” edition of Campanus and Zamberti. It is not known how these volumes came into Princeton’s possession, but both contain manuscript notes in Billingsley’s hand. The fact that these notes are found on only five pages of the Greek text, but on well over 200 of both Campanus and Zamberti, is clearly suggestive of Billingsley’s major source. Once again, the two basic Latin versions, one medieval and one Renaissance, have exhibited the considerable extent of their influence.

However, the better part of this influence was interrupted suddenly and decisively by the fourth major version listed above: the publication at Pesaro in 1572 of the Latin translation by Federico Commandino of Urbino. Commandino—who, in addition to the place he holds in the history of physics deriving from his *Liber de centro gravitatis* (Bologna, 1565), prepared exacting Latin versions of many other Greek mathematical works—was clearly the most competent mathematician of all Renaissance editors of Euclid. He was also most astute in his scholarship, for we know that in addition to the 1533 *editio princeps*, he employed at least one other Greek manuscript in establishing the text for his translation. For the first time, save for the anonymous translation in the twelfth century, we now have a version (no matter what language) of the *Elements* that is solidly based on a tolerably critical Greek original. It even includes, also for the first time, a rendering of numerous Greek scholia. Aware, but critical, of the efforts of his predecessors, Commandino leaves no doubt of the advantage of staying closer to the Greek sources so many of them had minimized, if not ignored. The result of his labors may prove to be of less fascination than other versions, since it so closely follows the Greek we already know, but the importance it held for the subsequent modern history of the *Elements* is immeasurable. It came to serve, in sum, as the base of almost all other proper translations before Peyrard’s discovery of the “pristine” Euclid in the early nineteenth century. Thus, to cite only the most notable cases in point, Greek texts of the *Elements* with accompanying Latin translation frequently based the latter on Commandino: for example, Henry Briggs’s *Elementorum Euclidis libri VI priores* (London, 1620) and even David Gregory’s 1703 Oxford edition of Euclid’s *Opera omnia* (which was the standard, pre-nineteenth-century source for the Greek text). Commandino was also followed in later strictly Latin versions: that of Robert Simson, simultaneously issued in English at Glasgow in 1756; and even that of Samuel Horsley, appearing at London in 1802. Vernacular translations often followed a similar course, beginning with the Italian translation, revised by Commandino himself, appearing at Urbino in 1575 and extending to and beyond the English version by John Keill, Savilian professor of astronomy at Oxford, in 1708.

In all translations based heavily on Commandino, one naturally remained close to the (Theonine) Greek tradition; but there were also other efforts after, as well as before, Commandino that did not stay so nearly on course. These were the numerous commentaries on Euclid, the various schoolbook *Elements*, and, in a class by itself, the edition of Christopher Calvius.

The commentaries of the sixteenth through eighteenth centuries were almost always limited to specific books or parts of the *Elements*. We have already noted the 1559 *Annotationum opuscula* of Buteo. but a considerable amount of related commentarial literature began to flourish around the same time. Giovanni Battista Benedetti (1530–1590) brought out his *Resolutio omnium Euclidis problematum... una tantummodo circini data apertura* at Venice in 1553 in response to a controversy that had recently arisen out of some reflection by several Italian scholars on Euclid. Petrus Ramus, who had previously produced a Latin version of the *Elements* in 1545, published at Frankfurt in 1559 his *Scholae mathematicae*, in which he scrutinized the structure of Euclid from the standpoint of logic. Along related lines, mention might be made of the curious *Euclideae demonstrationes in syllogismos resolutae* (Strasbourg, 1564) of Conrad Dasypodius and Christianus Herlinus. Such works as these were in a way extensions, perhaps fanciful ones, of the medieval Scholastic concern with the logic of the *Elements*. Yet another development can be seen in the various attempts to reduce the *Elements* to practice. We have already noticed this standpoint in Robert Recorde, and to this one could add the first German translation of books I-VI—published by Wilhelm Holtzmann (Xylander) in 1562—which was written with the likes of painters, goldsmiths, and builders in mind; and the Italian version (1613–1625) of Antonio Cataldi, which expressly declared itself to be an *Elementi ridotti alla practica*.

On a more specific plane, commentaries on book V, and particularly upon the Eudoxean definition of equal ratios that we have already seen to be problematic, continued in the sixteenth and seventeenth centuries. Beginning with the almost totally unknown works of Giambattista Politi, *Super definitiones et propositiones* quae supponuntur ab Euclide in quinto *Elementorum* eius (Siena, 1529), and of Elia Vineto Santone, *Definitiones elementi quincti et sexti Euclidis* (Bordeaux, 1575), the issue was also broached by Galileo in the added “Fifth Day” of his *Discorsi... a due nuove scienze* (an addendum first published at Florence in 1674 in Vincenzo Viviani’s *Quinto libro degli Elementi d’Euclide*). Finally, note should be made of two of the most impressive early modern commentaries on selected aspects of Euclid. The first is Henry Savile’s lectures *Praelectiones tresdecim in principium Elementorum Euclidis Oxoniae habitae MDCXX* (Oxford, 1621), which cover only the premises and first eight propositions of book I but do so in an extraordinarily penetrating, and still valuable, way. The last work to be mentioned is so famous that one often forgets that it formed part of the commentarial literature on the *Elements*—the *Euclides ab omni naevo vindicatus* (Milan, 1733) of Girolamo Saccheri, in which this Jesuit mathematician and logician fashioned the attempt to prove Euclid’s parallels postulate that has won him so prominent a place in the histories of non-Euclidean geometry.

Closely connected with the commentarial literature we have sampled is the magisterial Latin version of the *Elements* composed by another, much earlier Jesuit scholar, Christopher Clavius (1537–1612). The first edition of his *Euclidis Elementorum libri XV* appeared at Rome in 1574. Not, properly speaking, a translation, as Clavius himself admitted, but a personal redaction compiled from such earlier authors as Campanus, Zamberti, and Commandino, the work is chiefly notable, to say nothing of immensely valuable, for the great amount of auxiliary material it contains. Separate *praxeis* are specified for the constructions involved in the problems, long *excursus* appear on such debatable issues as the horn angle, and virtually self-contained treatises on such topics as composite ratios, mean proportionals, the species of proportionality not treated in Euclid, and the quadratrix are inserted at appropriate places. Indeed, by Clavius’ own count, to the 486 propositions he calculated in his Greek-based Euclid, he admits to adding 671 others of his own; “in universum ergo 1234 propositiones in nostro Euclide demonstrantur,” he concludes. And the value of what he has compiled matches, especially for the historian, its mass.

The final segment of the modern history of Euclid that requires description is what might most appropriately be called the handbook tradition, both Latin and vernacular, of the *Elements*. Many of the briefer Renaissance versions already mentioned are properly part of this tradition, and if one sets no limit on size, editions like that of Clavius would also qualify. In point of fact, the undercurrent of didacticism we have seen to be present in the medieval Arabic and Latin versions of the *Elements* can justly be regarded as the beginning of this handbook, or school text, tradition.

In the seventeenth century, however, the tradition takes on a more definite form. Numerous examples could be cited from this period, but all of them show the tendency to shorten proofs, to leave out propositions—and even whole books—of little use, and to introduce symbols wherever feasible to facilitate comprehension. This did not mean, to be sure, the disappearance of the sorts of supplementary material characteristic of so many translations and redactions of the *Elements*. On the contrary, such material was often rearranged and retained, and even created anew, when it seemed to be fruitful from the instructional point of view. For instance, one of the most popular (some twenty editions through the first few years of the nineteenth century) handbooks, the *Elementa geometriae planae et solidae* (Antwerp, 1654) of André Tacquet (1612–1660), covers books I-VI and XI-XII, with added material from Archimedes. Its proofs are compendious, but it makes up for its gain in this regard through the addition of a substantial number of pedagogically useful corollaries and scholia. On the other hand, the *Euclidis Elementorum libri XV breviter demonstrate* (Cambridge, 1655) of Isaac Barrow (1630–1677) stubbornly holds to its status as an epitome. Producer of perhaps the shortest handbook of all of books I-XV, Barrow achieved this maximum of condensation by appropriating the symbolism of William Oughtred (1574–1660) that the latter employed in a *declaratio* of book X of Euclid in his *Clavis Mathematicae* (Oxford, 1648 ff.). In his preface, Barrow claimed that his goal was “to conjoin the greatest Compendiousness of Demonstration with as much perspicuity as the quality of the subject would admit.” Although his success struck some (for example, John Keill) as producing a somewhat obscure compendium, this did not prevent the appearance of numerous (some ten) editions, several of them in English. Vernacular handbooks appeared in other languages as well, perhaps the most notable being *Les Elémens d’Euclide* (Lyons, 1672) of the French Jesuit Claude-François Milliet de Chales (1621–1678). Appearing earlier in Latin (Lyons, 1660), this handbook, covering, like Tacquet’s, books I-VI and XI-XII, went through some twenty-four subsequent editions, including translations into English and Italian.

The next stage in the handbook tradition belongs to the nineteenth century, where there occurred a veritable avalanche of Euclid primers, frequently radically divergent from any imaginable text of the *Elements*. Quite separate from these attempts to make Euclid proper for the grammar schools, lycées, and Gymnasia of the 1800’s, the rise of classical philology carried with it the efforts to establish a sound and critical text of the *Elements*. These efforts, in turn, gave rise to the annotated translations of the present century, with an audience primarily the historian and the classicist, rather than the mathematican.

Only a paltry few of the almost innumerable versions of the *Elements* dating from the Renaissance to the present have even been mentioned above—most are merely listed in bibliographies and remain totally unexamined. Even the few titles of this period that have been cited have received little more than fleeting attention—often limited to their preface— from historians. Further study will, one feels certain, reveal much more of the significance this mass of Euclidean material holds for the history of mathematics and science as a whole.

## BIBLIOGRAPHY

Abbreviations of frequently cited works:

Clagett, *Medieval Euclid* = Marshall Clagett, “The Medieval Latin Translations From the Arabic of the Elements of Euclid, With Special Emphasis on the Versions of Adelard of Bath,” in *Isis*, **44** (1953), 16–42.

Curtze, *Supplementum* = *Anaritii in decem libros priores Elementorum Euclidis ex interpretatione Gherardi Cremonensis*, Maximilian Curtze, ed. (Leipzig, 1899), supplement to *Euclidis Opera omnia*, Heiberg and Menge, eds.

Heath, *Euclid* = Thomas L. Heath, *The Thirteen Books of Euclid’s Elements Translated From the Text of Heiberg With Introduction and Commentary*, 3 vols. (2nd ed., Cambridge, 1925; repr. New York, 1956).

Heiberg, *Euclides* = *Euclidis Opera omnia*, J. L. Heiberg and H. Menge, eds., 8 vols. (Leipzig, 1883–1916).

Heiberg, *Litt. Stud*. = J. L. Heiberg, *Litterärgeschichtliche Studien über Euklid* (Leipzig, 1882).

Heiberg, *Paralipomena* = J. L. Heiberg, “Paralipomena zu Euklid,” in *Hermes*, **38** (1903), 46–74, 161–201, 321–356.

Klamroth, *Arab. Euklid* = Martin Klamroth, “Ueber den arabischen Euklid,” in *Zeitschrift der Deutschen morgenländischen Gesellschaft*, **35** (1881), 270–326, 788.

Sabra, *Simplicius* = A. I. Sabra, “Simplicius’s Proof of Euclid’s Parallels Postulate,” in *Journal of the Warburg and Courtauld Institutes*, **32** (1969), 1–24.

Sabra, *Thābit* = A. I. Sabra, “Thābit ibn Qurra on Euclid’s Parallels Postulate,” in *Journal of the Warburg and Courtauld Institutes*, **31** (1968), 12–32.

General Euclidean Bibliographies

The most complete bibliography of Euclid is still that of Pietro Riccardi, *Saggio di una bibliografia Euclidea*, in 5 pts., (Bologna, 1887–1893); this work also appeared in the *Memorie della Reale Accademia delle Scienze dell” Istituto di Bologna*, 4th ser., **8** (1887), 401–532; **9** (1888), 321–343; 5th ser., **1** (1890), 27–84; **3** (1892), 639–694. More complete bibliographic information on pre-1600 eds. of the *Elements*, and works dealing with Euclid, can be found in Charles Thomas-Sanford, *Early Editions of Euclid’s Elements* (London, 1926). Other bibliographies are listed in the bibliography to pt. I of the present article.

The Elements in Greek Antiquity

1. *Establishment of the* “*pristine*” *Greek text*. A history of the text in capsule form was first given in Heiberg, *Litt, Stud.*, pp. 176–186. Heiberg, *Euclides*, V (Leipzig, 1888), xxiii-1xxvi gives a more complete analysis of the Theonine and pre-Theonine texts, together with an outline of the criteria and methods used in establishing the latter. Further material relevant to the textual problem is found in Heiberg, *Paralipomena*, pp. 47–53, 59–74, 161–201. Heath, *Euclid* I, 46–63, gives a summary of all of the Heiberg material above.

2. *Greek commentaries*. The ed. of the Greek text of Proclus by G. Friedlein is noted in pt. I of the present article, together with several trans. To this one should now add the English trans. of Glenn Morrow (Princeton, 1970). Of the literature on Proclus, the most useful to cite is J. G. van Pesch, *De Procli fontibus* (Leiden, 1900). For the commentary of Pappus, extant only in Arabic, see the following section. Heath, *Euclid*, I, 19–45, gives a convenient summary of Greek commentarial literature. To this one should add Sabra, *Simplicius*, for material on this commentary, extant only in Arabic fragments (and Latin trans. thereof). Finally, Heiberg has treated the commentaries as well as the citations of Euclid in all other later Greek authors (notably commentators on Aristotle); this material is assembled in Heiberg, *Litt. Stud.*, pp. 154–175, 186–224; and *Paralipomena*, pp. 353–354.

3. *Greek Scholia* Most of these are published in Heiberg, *Euclides*, V (Leipzig, 1888), 71–738, supplemented by Heiberg, *Paralipomena*, pp. 321–352. Scholia to the minor Euclidean works are published in vols. VI-VIII of Heiberg, *Euclides*. The most complete discussion of the scholia is J. L. Heiberg, “Om Scholierne til Euklids Elementer” (with a Franch résumé), in *Kongelige Danske Videnskabernes Selskabs Skrifter*, Hist-philosofisk afdeling II, **3** (1888), 227–304. Once again there is a summary of this Danish article in Heath, *Euclid*, I, 64–74.

The Medieval Arabic Euclid

1. *General works*. Serious study of the Arabic Euclid began with J. C. Gartz, *De interpretibus et explanatoribus Euclidis arabicis schediasma historicum* (Halle, 1823) and was continued in J. G. Wenrich, *De auctorum graecorum versionibus et commentariis syriacis, arabicis, armeniacis persicisque commentatio* (Leipzig, 1842), pp. 176–189. The problem of the reports of Euclid in Arabic literature was broached in Heiberg, *Litt. Stud.*, pp. 1–21; but the major step was taken in Klamroth, *Arab. Euklid*. Klamroth’s contentions concerning the superiority of the Arabic tradition were answered by Heiberg in “Die arabische Tradition der Elemente Euklid’s in *Zeitschrift für Mathematik und Physik*, **29** (1884), 1–22. This was followed by the summary article, which included material on Arabic commentators, of Moritz Steinschneider, “Euklid bei den Arabern: Eine bibliographische Studie”, in *Zeitschrift für Mathematik und Physik*, Hist.-lit. Abt., **31** (1886), 81–110. Cf. Steinschneider’s *Die arabischen Uebersetzungen aus dem Griechischen*(Graz, 1960), pp. 156–164 (originally published in *Centralblatt für Bibliothekswesen, supp*. **5,** 1889). See also the article by A. G. Kapp in section 6 below. A summary view of our knowledge (as of the beginning of the present century) of the *Elements* in Islam can be found in Heath, *Euclid*, I, 75–90. M. Klamroth has also published a translation of some of the summaries of Greek works by the ninth-century historian al-Yaʿqūbī which includes a résumé of the *Elements:* “Ueber die Auszüge aus griechischen Schriftstellern bei al-Jaʿqūbī in *Zeitschrift der Deutschen morgenländischen Gesellschaft*, **42** (1888), 3–9. The standard bibliography of Arabic mathematics and mathematicians is Heinrich Suter, *Die Mathe-matiker und Astronomen der Araber und ihre Werke, Abhandlungen zur Geschichte der mathematischen Wissenschaften*, X (Leipzig, 1900), with *Nachträge und Berichti-gungen, op, cit.*, XIV (Leipzig, 1902), 155–185. The most recent history of Islamic mathematics, with appended bibliography, is contained in A. P. Juschkewitsch [Youschkevitch], *Geschichte der Mathematik im Mittelalter* (Basel, 1964; original Russian ed., Moscow, 1961).

2. *The translation ofal-Hajjaj*. We know of but a single MS containing (presumably) the second Ḥajjāj version together with Nayrīzī’s commentary for books I-VI (and a few lines of VII) alone: Leiden, 399, 1. This has been ed. with a modern Latin trans, by J. L. Heiberg, R. O. Besthorn, *et al., Codex Leidensis 399, 1: Euclidis Elementa ex interpretatione al-Hadschdschadschii cum commentariis al-Narizii*, in 3 pts. (Copenhagen, 1893–1932). Confirmation is needed of the report of two further MSS containing a Ḥajjāj version of books XI-XII: MSS Copenhagen LXXXI and Istanbul, Fātih 3439. There is no secondary literature specifically devoted to the Ḥajjāj *Elements*, but information is contained in Klamroth, *Arab. Eukld*.

3. *The translation of Isḥāq–Thābit*. The most frequently cited MS of this version is Oxford, Bodleian Libr., MS Thurston 11 (279 in Nicoll’s catalogue), dated 1238. This was one of the two basic codices employed in Klamroth, *Arab. Euklid*. The literature also makes continual reference to MS Bodl. Or. 448 (280 in Nicoll) as an Isḥāq–Thābit text; it is not this, but rather a copy of al-Maghribī’s *Taḥrīr* (the error derives from a marginal misascription to Thābit that was reported by Nicoll in his catalogue). There are, however, a number of other extant copies of Isḥāq–Thābit. Intention to edit the Isḥāq–Thābit trans. was announced (but apparently abandoned) by Claire Baudoux, “Une édition polyglotte orientale des Eléments d’Euclide: La version arabe d’Ishāq et ses derivées”, in *Archeion*, **19** (1937), 70–71. Of the literature on the reviser of this trans., Thābit ibn Qurra, see Eilhard Wiedemann, “Ueber Thābit, sein Leben und Wirken,” in *Sitzungsberichte der physikalisch-medizinischen Sozietät zu Erlangen*, **52** (1922), 189–219; A. Sayili, “Thābit ibn Qurra’s Generalization of the Pythagorean Theorem,” in *Isis***51** (1960), 35–37; and section 7 below. An integral ed. and Russian trans. of Thābit’s mathematical works is in preparation.

4. *The epitomes of Avicenna and others*. A. I. Sabra has edited Avicenna’s compendium of the *Elements*, and it will appear in the Cairo ed. of Avicenna’s *Kitāb al-Shifāʾ*. A brief description of this compendium was published by Karl Lokotsch, *Avicenna als Mathematiker, besonders die planimetrischen Bücher seiner Euklidübersetzung* (Erfurt, 1912). A copy of a poem praising Euclid, ascribed to Avicenna in a MS found in the Topkapi Museum at Istanbul, is the subject of A. S. Unver, “Avicenna’s Praise of Euclid,” in *Journal of the History of Medicine*, **2** (1947), 198–200 (other occurrences of the poem, however, disagree with this ascription). The only other Euclid compendium treated in the literature is that of al-Asfuzārī, in L. A. Sédillot, “Notice de plusieurs opuscules mathématiques: V. Quatorzième livre de I’épitome de I’lmam Muzhaffar-al-Isferledi sur les Elements d’Euclide,” in *Notices et extraits des manuscrits de la Bibliothèque du Roi*, **13** (1838), 146–148.

5. *The Taḥrīr of al-Tūsī, pseudo-Tūsī, and al-Mghribī*. The genuine, fifteen-book *Taḥrīr* of al-Tūsī exists in an overwhelming number of MSS and has also been frequently printed (Istanbul, 1801; Calcutta, 1824; Lucknow, 1873–1874; Delhi, 1873–1874; Tehran, 1881). Indication of the spurious nature of the thirteen-book *Taḥrīr* usually ascribed to al-Ṭūsī is established by Sabra, *Thābit*, n. 11, and *Simplicius*, postscript, p. 18; doubt is also raised in B. A. Rozenfeld, A. K. Kubesov, and G. S. Sobirov, “Kto by avtorom rimskogo izdania ‘Izlozhenia Evklida Nasir ad-Dina at-Tusi’” (“Who Was the Author of the Rome Edition ‘Recension of Euclid by Naṣīr al-Dīn al-Tmūsī’?”), in *Voprosy istorii estestvoznaniya i tekhniki*, **20** (1966), 51–53. The spurious *Taḥrīr* was printed at Rome in 1594, and we know of only two extant MSS: Bibl. Laur. Or. 2. and Or. 51; the latter carries the 1298 date, causing, among other factors, the problems with al-Ṭūsī’s authorship. Almost all of the literature dealing with al-Ṭūsī and Euclid treats of the spurious *Taḥrīr:* H. Suter. “Einiges von Nasīr el-Dīn’s Euklid-Ausgabe,” in *Bibliotheca mathematica* 2nd ser., **6** (1892), 3–6; E. Wiedemann, “Zu der Redaktion von Euklids Elementen durch *Naṣīr al Din al Ṭūsī” in Sitzungsberichte der physikalisch-medizinischen Sozietät zu Ertangen*, **58/59** (1926/1927), 228–236; C. Thaer, “Die Eukid-Überlieferung durch al-Ṭūsī,” in *Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik*, Abt. *B*, Studien, 3 (1936), 116–121. More general works on al-Ṭūsī as a mathematician include an ed. of the Arabic text of the Rasā ʿil al-Ṭūsī, 2 vols. (Hyderabad, 1939–1940); E. Wiedemann, “Naṣīr al Din al Ṭūsī,” in *Sitzungsberichte der physikalisch-medizinischen Sozietät zu Erlangen*, **60** (1928), 289–316; and B. A. Rozenfeld, “O matematicheskikh rabotakh Nasireddina Ṭūsī” (“On the Mathematical Works of Naṣīr al-Dīn al-Ṭūsī,”), in *Istoriko-matematicheskie issledovaniya*, **4** (1951), 489–512. See also section 7 below. The Taḥrīr of al-Maghribī is found in the thirteenth-century MS Bodl. Or. 448, as well as two later codices in Istanbul. It is identified and discussed, together with the ed. and trans, of a fragment from it in Sabra, *Simplicius*, pp. 13–18, 21–24.

6. *Commentaries. The Arabic translation of the commentary of Pappus on book X*. Extracts of the Arabic text, together with a French trans., were first published by Franz Woepcke in “Essai d’une restitution de travaux perdus d’Apollonius sur les quantités irrationnelles.” in *Mémoires présentés par divers savants à l’Académic des sciences*, **14** (1856), 658–720 (also published separately). Woepcke also published the full text of the commentary without date or place of publication (Paris, 1855[?]). This was in turn trans. into German with comments by H. Suter, “Der Kommentar des Pappus zum X Buche des Euklides aus der arabischen Übersetzung des Abū ʿOthmān al-DimashlḲī ins Deutsche übertragen,” in *Abhandlungen zur Geschichte der Naturwissenschaften und der Medizin*, **4** (1922), 9–78. A new ed. of the Arabic text with notes and English trans. was published by William Thomson and Gustav Junge, *The Commentary of Pappus on Book X of Euclid’s Elements* (Cambridge, Mass., 1930). Critical remarks on this text were published by G. Bergstrasser, “Pappos Kommentar zum Zehnten Buch von Euklid’s Elementen,” in *Der Islam*, **21** (1933), 195–222. A fragment of Gerard of Cremona’s trans. of this Pappus text is printed in G. Junge, “Das Fragment der lateinischen Übersetzung des Pappus-Kommentars zum 10. Buche Euklids,” in *Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik*, Abt. **B, 3** , Studien (1936), 1–17.

*Arabic commentaries*. Very few of the great number of these have been published or studied. A list, quite complete in terms of present knowledge, giving brief indications of author, subject, and relevant bibliography, can be found in E. B. Plooij, *Euclid’s Conception of Ratio and his Definition of Proportional Magnitudes as Criticized by Arabian Commentators* (Rotterdam, n.d.), pp. 3–13. More elaborate is A. G. Kapp, “Arabische Übersetzer und Kommentatoren Euklids, sowie deren math. naturwiss. Werke auf Grund des Ta’rikh al-Ḥukamā’ des Ibn al-Qifṭī,” in *Isis*, **22** (1934), 150–172; **23** (1935), 54–99; **24** (1935), 37-79; as the title indicates, this extensive article contains much material trans. from the biobibliographical work of Qifṭī (*ca*. 1172–1248). Arabic commentaries on book X are treated in G. P. Matvievskaya, *Uchenie o chisle na srednevekovom blizhnem i srednem vostoke* (“Studies on Number in the Medieval Near and Middle East”; Tashkent, 1967), pp. 191–229; The following commentaries (listed in approximate chronological order) have been ed. or trans. (if only partially) and analyzed: (1) al-Nayrīzī: books I–VI in Arabic in the Heiberg-Besthorn ed. of Ḥajjāj cited in section 2 above; books I–X (incomplete[?]) in the Latin trans. of Gerard of Cremona in Curtze, *Supplementum* pp. 1–252 only (the remainder of this volume containing the commentary not of al-Nayrīzī, but of ʿAbd al-Baqi; see below). Determination through examination of al-Nayrīzī of various interpolations in the text of the *Elements* was done in Heiberg, *Paralipomena*, pp. 54–59. (2) Al-Fārābī on books I and V has been trans. into Russian on the basis of its two Hebrew copies (MSS Munich 36 and 290, not edited): M. F. Bokshteyn and B. A. Rozenfeld, “Kommentarii Abu Nasra al-Farabi k trudnostyam vo vvedeniakh k pervoy i pyatoy knigam Evklida” (“The Commentary of Abū Naṣr al-Fārābī on the Difficulties in the Introduction to Books I and V of Euclid”), in *Akademiya nauk SSR, Problemy vostokovedeniya*, no. 4 (1959), 93–104. The Arabic text of this brief work of al-Fārābī has now also apparently been discovered: Escorial MS Arab. 612, 109r–11Iv. A fragment of this, or of another Euclidean opusculum by al-Fārābī, is Tehran, Faculty of Theology, MS 123-D, 80v–82r. See also A. Kubesov and B. A. Rozenfeld, “On the Geometrical Treatise of al-Fārābī,” in *Archives internationales d’histoire des sciences*, **22** (1969), 50.(3) Ibn al-Haytham, *On the Premises of Euclid*, has also received a (partial) Russian trans. by B. A. Rozenfeld as “Kniga kommentariev k vvedeniam knigi Evklida ‘Naehala’” (“Book of Commentaries to Introductions to Euclid’s *Elements*”), in *Istoriko-matematicheskie issle-dovaniya*, **11** (1958), 743–762. (4) Ibn Muʿādh al-Jayyānī on book V has been reproduced in facsimile with accompanying English trans, in the book of E. B. Plooij cited above. This book also contains a trans. of passages relevant to book V from the commentaries of al-Māhānī, al-Nayrīzī, Ibn al-Haytham, and Omar Khayyām. (5) Omar Khayyām’s work on Euclid has received the most attention of all. A. I. Sabra has published a critical Arabic text (without trans.) as *Explanation of the Difficulties in Euclid’s Postulates* (Alexandria, 1961). There is an earlier ed., on the basis of a single MS, by T. Erani (Tehran, 1936). A Russian trans., with commentary, has been published by B. A. Rozenfeld and A. P. Youschkevitch in *Istoriko-matematicheskie issledovaniya***6** (1953), 67–107, 143–168; repr. with a MS facsimile in Omar Khayyām, *Traktaty* (Moscow,1961). The English trans. by Amir-Móez in *Scripta mathematica*, **24** (1959), 272–303, must be used with great care. (6) ‘Abdal-Bāqī’s commentary on book X in Gerard of Cremona’s Latin trans. is printed in Curtze, *Supplementum*, pp. 252–368. H. Suter has given corrections to Curtze’s text in “Uberden Kommentar des Muḥ b. ʿAbdelbāgi zum 10 Buche des Euklides,” in *Bibliotheca mathematica*, 3rd ser., 7 (1907), 234–251. See the following section for literature on yet other commentarial material.

7. *On the parallels postulate*. The importance of this postulate in the history of mathematics is reflected not merely in the frequency of its discussion by Islamic authors, but also by the attention it has received from modern historians. As an introduction, see B. A. Rozenfeld and A.P. Youschkevitch, *The Prehistory of Non-Euclidean Geometry in the Middle East, XXV International Congress of Orientalists, Papers Presented by the USSR Delegation* (Moscow, 1960). Compare B. A. Rozenfeld, “The Theory of Parallel Lines in the Medieval East,” in *Actes du XI ^{e} Congrès International d’Histoire des Sciences, Varsovie-Cracovie* 1965,

**3**(Warsaw. etc., 1968), 175–178. More specifically, two treatments of Thābit ibn Qurra are trans. and analyzed in Sabra,

*Thābit*. The problem is also the subject of Sabra,

*Simplicius*. In fact, these two articles contain a mine of information pertinent to the issue throughout Islamic mathematics. The two Thābit treatise have also been analyzed and trans. into Russian by B. A. Rozenfeld and A. P. Youschkevitch in

*Istoriko-matematicheskie issledovaniya*,

**14**(1961), 587–597; and

**15**(1963), 363–380. Extracts from the treatments of the postulate by al-Jawharī, Qayṣar ibn Abi ’al-Qāsim, and al-Maghribī are found in the two Sabra articles. The greatest amount of attention has been paid to al-Ṭūsī’s struggles with the problem, beginning with a trans. into Latin by Edward Pocock of the proof of the postulate in the pseudo-Ṭūsī Taḥrīr; this was printed in John Wallis,

*Opera mathematica*, II (Oxford, 1693), 669–673. Both this proof and that in the genuine fifteen-book al-Ṭūsī Taḥrīr were published and analyzed in Arabic by A. I. Sabra, “Burhān Naṣīr al-Dīn al-Tūsī ’alā muṣādarat Uqlīdis al-khāmisa,” in

*Bulletin of the Faculty of Arts of the University of Alexandria*,

**13**(1959), 133–170. Russian treatment again occurs in G. D. Mamedbeili,

*Mukhammad Nasureddin Tusi o teorii parallelnykh liny i teorii otnosheny*(“Muḥammad Naṣīr al-Dsīn al-Ṭūsī on the Theory of Parallel Lines and the Theory of Proporation” Baku, 1959), and in

*Istoriko-mathematicheskie issledovaniya*,

**13**(1960), 475–532. Finally, the article of H. Dilgan, “Demonstration du V

^{e}postulat d’Euclide par Schams-ed-Din Samarkandi, Traduction de l’ouvrage Aschkal-ut-tessis de Samarkandi,” in

*Revue d’histoire des sciences*,

**13**(1960), 191–196, does not contain a proof by Samarqandī, but rather one by Athīr al-Dīn al-Abharī that was reproduced in a commentary to Samarqandī’s work.

8. *Translations into other Near Eastern languages*. The most adequate account of Hebrew versions and commentaries is in Moritz Steinschneider, *Die hebräischen Übersetsungen des Milltelalters und die Juden als Dolmetscher* (Berlin, 1893; repr. Graz, 1956), 503–513. The fragments of the Syriac version were published and trans. by G. Furlani, “Bruchstuck eine syrischen Paraphrase der ‘Elemente’ Elemente” des Eukleides,” in *Zeitschrift für Semitistik und uerwante Gebiete*, **3** (1924), 27–52, 212–235. Furlani held that the paraphrase was derived from the Arabic of Ḥajjāj. This was questioned, and the opposing view placing it before, and as a source of, the Isḥāq trans., by C. Baudoux, “La version syriaque des ‘Eléments.’ d’Euclide,” in II^{e}*Congrès national des sciences* (Brussels, 1935), pp. 73–75. The fragments of the early Armenian version were published and trans. (into Latin) by Maurice Leroy, “La traduction arménienne d’Euclide,” in *Annuaire de l’Institut de philologie et d’histoire orientales et slaves (Mélanges Franz Cumont)*, **4** (1936), 785–816. The letter of Gregory Magistros announcing his translating activity with respect to Euclid was published and analyzed by Leroy in the same *Annuaire*, **3** (1935), 263–294. Additional material on Armenian Euclids can be found in the article (not presently examined) by T. G. Tumanyai, “‘Nachala’s Evklida po drevnearmyanskim istochnikam” (“Euclid’s *Elements* in Ancient Armenian Sources”), in *Istoriko-matematicheskie issledovaniya*, **6** (1953), 659–671, and in G. B. Petrosian and A. G. Abramyan, “A Newly Discovered Armenian Text of Euclid’s Geometry,” in *Proceedings of the Tenth International Congress of the History of Science, Ithaca*, 1962, II (Paris, 1964), 651–654.

9. *Euclid’s minor works*. The Arabic trans. of the Euclidean opuscula, together with a discussion of their role as “middle books” in Islamic mathematics and astronomy, was first examined by M. Steinschneider, “Die ‘mittleren’ Bücher der Araber und ihre Bearbeiter,” in *Zeitschrift für Mathematik und Physik*, **10** (1865), 456–498. There is little material dealing specifically with these shorter works, but in addition to the general literature in section 1 above, see Clemens Thaer, “Euklids Data in arabischer Fassung,” in *Hermes*, **77** (1942), 197–205. The prolegomena in vols. VI–VIII of Heiberg, *Euclides*, also contains information on the Arabic phase of these opuscula. For literature dealing with the Islamic role in the work *On the Division of Figures* and the works on mechanics, see pt. I of the present article.

The Medieval Latin Euclid: The Greek–Latin Phase

1. *Euclidean material in Roman authors*. The fragmets in Censorinus are appended in F. Hultsch’s ed. (Leipzig, 1867) of the *De die natali*, pp. 60–63. For Euclid in Martianus Capella, see the *De nuptiis philologiae et mercurii*, VI, 708 ff.

2. *The Verona palimpsest*. The Eiclid framents have recently been edited, with facsimile and notes, by Mario Geymonat, *Euclidis latine facti fragmenta Veronensia* (Milan, 1964). This work contains references to all other previous literature on the palimpset, both of paleographers and historians of mathematics.

3. *The Boethian Euclid excerpts*. The best account of all of the variables involved is the absolutely fundamental work of Menso Folkerts, “*Boethius” Geometrie II: Ein mathematisches Lehrbuch des Mittelalters* (Wiesbaden, 1970). This contains a critical ed. of (a) the two-book “Boethian” *Geometry;* (b) the Euclid excerpts preserved in all four earlier medieval sources. The Boethian–Adelard mélanges in Ratsbücherei Lüneburg MS miscell. D4° 48 have been treated and ed. by Folkerts in *Ein neuer Text des Euclides Latinus: Faksimiledruck der Handschrift Lüneburg D 4° 48, f. 13r-17v* (Hildesheim, 1970), and, together with a consideration of the mélanges in the Paris and Munich MSS (see following section), in “Anonyme lateinische Euklidbearbeitungen aus dem 12. Jahrhundert,” in *Denkschriften der Österreichischen Akademie der Wissenschaften*, Math-naturwiss. Klasse (1970), 5–42. *See also* Folkerts’ earlier article, “Das Problem der pseudo-boethischen Geometric,” in *Sudhoff’s Archiv für Geschichte der Medizin und der Naturwissenschaften*, **52** (1968), 152–161. An earlier work that also attempted, as a tangential problem, to sort out the threads of the “Boethian” Euclid is Nicolaus Bubnov, *Gerberti Opera mathematica* (Berlin 1899, repr. Hildesheim, 1963). Before Folkerts the standard ed. of the two-book geometry was that of Gottfried Friedlein in his text of Boethius’ De *institution arithmetica... de institutione musica... accedit geometria quae fertur Boetii* (Leipzig, 1867), pp. 372–428. The five-book “Boethian” geometry still does not exist in a critical ed. The first two books have appeared, however, among Boethius’ works in J. P. Migne, *Patrologia Latina*, vol. LXIII, cols. 1352–1364 (cols. 1307–1352 contain the two-book geometry now in Folkerts). Books I, III, IV, and part of V are in F. Blume, K. Lachmann, and A. Rudoff, *Die Schriften der römischen Feldmesser*, I (Berlin, 1848), 377–412. The remaining section of book V is unedited. Although the five-book geometry has therefore not received adequate editing, a most exacting analysis of its MS sources and history has been made by C. Thulin, *Zur Überlieferungsgeschichte des Corpus Agrimensorum. Exzerptenhandschriften und Kompendien* (Göteborg, 1911). The Euclid excerpts found in Cassiodorus have been edited by R. A. B. Mynors in his text of the *Institutiones* (Oxford, 1937), pp. 169–172. A recent important article that treats of the role of “Boethian” geometry in the earlier Middle Ages is B. L. Ullman, “Geometry in the Medieval Quadrivium,” in *Studi di bibliografia e di storia in onore di Tammaro de Marinis*, IV (Verona, 1964), 263–285. Among the earlier literature on the problems of Boethius and the *Elements* are H. Weissenborn, “Die Boetius-Frage,” in *Abhandlungen zur Geschichte der Mathematik***2** (1879), 185–240; J. L. Heiberg, “Beiträge zur Geschichte der Mathematik im Mittelalter, II. Euklid’s Elemente im Mittelalter,” in *Zeitschrift für Mathematik und Physik* Hist-lit. Abt., **35** (1890), 48–58, 81–100; Georg Ernst, *De geometricis illis quae sub Boëthii nomine nobis tradita sunt, quaestiones* (Bayreuth, 1903); M. Manitius, “Collation aus eienem geometricschen Tractat,” in *Hermes*, **39** (1904), 291–300, and “Collationen aus der Ars geometrica,” *ibid*., **41** (1906), 278–292; and several pieces by Paul Tannery, now included in his *Mémoires scientifiques*, V (Paris, 1922), 79–102, 211–228, 246–250.

4. *Boethian-Adelaridan mélanges*. The Lüneburg MS mélange has been ed. by Folkerts (see above). A second mélange exists in the four MSS Paris, BN lat. 10257; Oxford, Bodl. Digby 98; Munich, CLM 1302, and CLM 23511. The Paris MS has been ed. in the unpublished dissertation of George D. Goldat, “The Early Medieval Traditions of Euclid’s (Madison, Wisc., 1956).

5. *Munich manuscript fragment*. This has been ed., from MS Univ. Munich 2° 757, by Curtze, *Supplementum*, pp. xvi-xxvi. Corrections to Curtze’s text can be found in Heiberg, *Paralipomena*, pp. 354–356, and *Bibliotheca mathematica*, 3rd ser.,**2** (1901), 365–366. A new edition of the text has been prepared by Mario Geymonat in “Nuovi frammenti della geometric ‘Boeziana’ in un codice del IX secolo?” in *Scriptorium***21** (1967), 3–16. Geymonat dates the fragment as of the ninth century rather than the tenth; whether this be correct or not, the question that he poses of Boethius’ authorship for this fragment should in all probability be answered negatively.

6.*Twelfth-century Greek–Latin translation*. This is found in only two extant MSS: Paris, BN lat. 7373, and Florence, Bib. Naz. Centr. Fondo Conventi Soppressi C I 448. The trans. has been analyzed in full in John Murdoch, “Euclides Graeco-Latinus. A Hitherto Unknown Medieval Latin Translation of the *Elements* Made Directly from the Greek,” in *Harvard Studies in Classical Philology*, **71** (1966), 249–302. The Greek–Latin version of Ptolemy’s *Almagest* made by the same translator is discussed, and its preface edited, in C. H. Haskins, *Studies in the History of Mediaeval Science* (Cambridge, Mass., 1924), ch. 9.

7. *Medieval Latin versions of the Euclidean opuscula*. General information on the trans. of these minor works can be found in Heiberg, *Eurclides* prolegomena to vols. VI-VIII. In *fact*, the text of one of the Greek–Latin renderings of the *Optica* has been ed. in Heiberg, *Euclides*, VII (1895), 3–121. The Greek–Latin version of the *Data* was first noted by A. A. Björnbo, “Die mittelalterlichten lateinischen Übersetzungen aus dem Griechischen auf dem Gebiete der mathematischen Wissenschaften,” in *Festschrift Moritz Cantor* (Leipzig, 1909), p. 98. It has since been edited in the unpublished dissertation of Shuntaro Ito, “The Medieval Latin Translation of the Data of Euclid” (Madison, Wisc., 1964). The pseudo-Euclidean *De speculis*—to be distinguished from the equally spurious *Catoptria*—has been edited in A. A. Björnbo and S. Vogl, *Alkindi, Tideus und Pseudo-Euklid. Drei optische Werke, Abhandlungen zur Geschichte der mathematischen Wissenschaften*, vol. XXVI, pt. 3 (Leipzig, 1911); cf. S. Vogl in *Festschrift Moritz Cantor* (Leipzig, 1909), pp. 127–143.

The Medieval Latin Euclid: The Arabic–Latin Phase

1. *General*. A brief resumé of our earlier knowledge of this wing of the medieval Latin Euclid can be found in Heath, *Euclid* I, 93–96. The fundamental comprehensive description is now Clagett, *Medieval Euclid* which includes appendices that present sample texts from all of the basic twelfth-century versions constituting the Arabic–Latin *Elements*.

2. *The translation of Gerard of Cremona*. The most complete discussion, including a listing of MSS, is Clagett, *Medieval Euclid*, pp. 27–28, 38–41. *See also* A.A. Björnbo, “Gerhard von Cremonas Uebersetzung von Alkwarizmis Algebra und von Euklids Elementen,” in *Bibliotheca mathematica*, 3rd ser., **6** (1905), 239–248. Still useful for Gerard’s life and career is B. Boncompagni, “Della vitae delle opere di Gherardo cremonense,” in *Atti dell” Accademia pontificia de’ Nuovi Lincei*, 1st ser., **4** (1851), 387–493. A more critical text of Gerard’s *vita et libri translati* appended in a number of MSS to his trans. of Galen’s *Ars parva* has been given, with annotations to the list of works trans., by F. Wüstenfeld, *Die Übersetzungen arabischer Werke in das Lateinische seit dem XI Jahrhundert. Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen* (Göttingen, 1877), pp. 57–81. For Gerard’s trans. from the Arabic of commentaries on the *Elements*, see section 6 of the Arabic Euclid bibliography above.

3. *The Adelardian tradition*. The first extensive article treating of Adelard’s role in the transmission of Euclid was that of Herman Weissenborn, “Die Übersetzung des Euklid aus dem Arabischen in das Lateinische durch Adelhard von Bath...,” in *Zeitschrift für Mathematik und Physik*, Hist.-lit. Abt., **25** (1880), 143–166. It was Clagett, *Medieval Euclid*, pp. 18–25, who first distinguished the three separate versions to be ascribed to Adelard. This article also lists a good portion of extant MSS of the three recensions. A more detailed analysis of the nature of these three versions, together with that of Campanus of Novara, is given in J. Murdoch, “The Medieval Euclid: Salient Aspects of the Translations of the *Elements* by Adelard of Bath and Campanus of Novara,” in XII^{e} Congrès International d’Histoire des Sciences, Colloques, in *Revue de synthèse*, **89** (1968), 67–94. For the misinterpretation within the Adelard tradition and within medieval mathematics in general of the Eudoxean definition of equal ratios, see J. Murdoch, “The Medieval Language of Proportions: Elements of the Interaction With Greek Foundations and the Development of New Mathematical Techniques,” in A. C. Crombie, ed., *Scientific Change* (London, 1963), pp. 237–271, 334–343. The erroneous ascription of an Adelard version to Alfred the Great was set forth in Edgar Jorg, *Des Boetius und des Alfredus Magnus Kommentar zu den Elementen des Euklid (Nach dem Codex [Z. L. CCCXXXII] B. der Bibliotheca Nazionale di S. Marco zu Venedig), Zweities Buch* (Bottrop, 1935); that this particular MS contains merely both an Adelard II (Boethius) and an Adelard III (Alfred) version was established by M. Clagett, “King Alfred and the *Elements* of Euclid,” in *Isis*, **45** (1954), 269–277. 269–277. Works on Adelard himself are C. H. Haskins, *Studies in the History of Mediaeval Science* (Cambridge, Mass., 1924), ch. 2, and the frequently overenthusiastic book of Franz Bliemetzrieder, *Adelard von Bath* (Munich, 1935). On the version of Campanus, in addition to the article of Murdoch cited above, see Hermann Weissenborn, *Die Uebersetzungen des Euklid durch Campano und Zamberti* (Halle, 1882). Further biobibliographical information on Campanus is contained in the text and trans. of his *Theorica planetarum*, as edited by Francis S. Benjamin, Jr., and G. J. Toomer (in press).

4. *The translation of Hermann of Carinthia*. See Clagett, *Medieval Euclid*, pp. 26–27, 38–42, and the ed. of books I-VI by H. L. L. Busard, “The Translation of the *Elements* of Euclid From the Arabic Into Latin by Hermann of Carinthia (?),” in *Janus*, **54** (1967), I-142.

5. *Other translations and commentaries*. The supposed reference to a pre-Adelardian *Elements* in England “Yn tyme of good kyng Adelstones day” (as stated by a fourteenth-century verse) has been shown to apply to masonry, and not geometry, by F. A. Yeldham, “The Alleged Early English Version of Euclid,” in *Isis*, **9** (1927), 234–238. For the problem of references to a trans. by Johannes Ocreat, see Clagett, *Medieval Euclid*, pp. 21–22. A commentary on books I-IV of the *Elements* that exists in a single MS (Vienna, Dominik. 80/45) and is there ascribed to Albertus Magnus is discussed by J. E. Hoffmann, “Ueber eine Euklid-Bearbeitung, die dem Albertus Magnus zuschrieben wird,” in *Proceedings of the International Congress of Mathematicians, Cambridge, 1958*, pp. 554–566; and by B. Geyer, “Die mathematischen Schriften des Albertus Magnus,” in *Angelicum*, **35** (1958), 159–175. An example of later medieval *questiones super Euclidem* are those of Nicole Oresme, recently edited (Leiden, 1961) by H. L. L. Busard; cf. J. E. Murdoch, in *Scripta mathematica*, **27** (1964), 67–91.

The Renaissance and Modern Euclid

1. *General*. There exists very little literature dealing with the transmission of Euclid from 1500 to the present; even bibliographies have not received much attention since the nineteenth century. And there is absolutely no work covering the fairly extensive body of MS materials from the sixteenth and seventeenth centuries. For the printed materials, the most adequate general works are the bibliographies cited above of Riccardi and Sanford, together with the brief survey of the principal eds. of the *Elements* in Heath, *Euclid*, I, 97–113. Dates and places of the versions of the *Elements* that are mentioned above have been given in the body of the text and will not be repeated here.

2. *Latin and Greek editions in the Renaissance and early modern period*. An outline of the major eds. is given in Heiberg, *Euclides*, V (1888), ci-cxiii. Heiberg has also treated of the significance for Euclid and Greek mathematics of Giorgio Valla and his encyclopedic *De expetendis et fugiendis rebus* in “Philologischen Studien zu griechischen Mathematikern: III. Die Handschriften George Vallas von griechischen Mathematikern,” in *Jahrbuch für classische Philologie*, **12,** supp. (1881), 337–402; and *Beiträge zur Geschichte Georg Valla’s und seiner Bibliothek, Centralblatt für Bibliothekswesen*, Beiheft 16 (Leipzig, 1896). On Zamberti see the monograph of Weissenborn on Campanus and this author that is cited in the section on the Adelardian tradition, above. There is no adequate work on Commandino or Clavius, especially concerning their role in the trans. and dissemination of Greek mathematics. Note has been taken, however, that in the seventeenth century the Jesuit Ricci, a student of Clavius, was instrumental in effecting a Chinese version of the latter’s *Elements:* see L. Vanhee, “Euclide en chinois et mandchou,” in *Isis*, **30** (1939), 84–88. The possibility of an earlier, thirteenth-century translation of Euclid into Chinese has been briefly discussed by Joseph Needham and Wang Ling, *Science and Civilisation in China*, III (Cambridge, 1959), 105.

Girolamo Saccheri’s *Euclidis ab omni naevo vindicatus*has received a modern ed. and English trans. by G. B. Halsted (Chicago, 1920). But this contains only book I of Saccheri’s treatise. For book II (dealing with the theory of proportion) see Linda Allegri, “Book II of Girolamo Saccheri’s *Euclides ab omni naevo vindicatus,” in Proceedings of the Tenth International Congress of the History of Science, Ithaca, 1962*, II (Paris, 1964), 663–665; an English trans. is to be found in the same author’s unpublished dissertation, “The Mathematical Works of Girolamo Saccheri, S. J. (1667–1733)” (Columbia University, 1960).

3. *Vernacular translations*. A recent detailed treatment of the appearance of the *Elements* in England up to *ca*. 1700 (in both English and Latin) is Diana M. Simpkins, “Early Editions of Euclid in England,” in *Annals of Science*, **22** (1966), 225–249. Robert Recorde and his inclusion of Euclidean material in *The Pathway to Knowledg* is the subject of Joy B. Easton, “A Tudor Euclid,” in *Scripta mathematica*, **27** (1964), 339–355. Most extensive attention has been paid to the 1570 English trans. by Sir Henry Billingsley. In addition to the work of Simpkins, above, see G. B. Halsted, “Note on the First English Euclid,” in *American Journal of Mathematics*, **2** (1879), 46–48 (which contains the first notice of the volumes at Princeton with marginalia in Billingsley’s hand); W. F. Shenton, “The First English Euclid,” in *American Mathematical Monthly*, **35** (1928), 505–512; R. C. Archibald, “The First Translation of Euclid’s *Elements* Into English and Its Source,” in *American Mathematical Monthly*, **57** (1950), 443–452. See also Edward Rosen, “John Dee and Commandino,” in *Scripta mathematica*, **28** (1970), 325. An annotated bibliography of French trans. of Euclid is Marie Lacoarret, “Les traductions françaises des oeuvres d’Euclide,” in *Revue d’histoire des sciences*, **10** (1957), 38–58. I. J. Depman has written of unnoticed Russian eds. of the *Elements in Istorikomatematicheskie issledovaniya*, **3** (1950), 467–473.

4. *The Nineteenth and twentieth centuries*. The most complete survey of the great number of nineteenth-century school-text Euclids is still to be found in the Riccardi bibliography (introductory section above). The most notable twentieth-century trans. that contain considerable historical and analytic annotation are, in English, Heath, *Euclid;* in Italian, Federigo Enriques, *et al*., eds., *Gli Elementi d’Euclide e la critica antica e moderna*, 4 vols. (Rome-Bologna, 1925–1935); in Dutch, E. J. Dijksterhuis, *De Elementen van Euclides*, 2 vols. (Groningen, 1929–1930); in Russian, D. D. Morduchai-Boltovskogo, *Nachala Evklida*, 3 vols. (Moscow–Leningrad, 1948–1950); in French, of books VII-IX only, Jean Itard, *Les livres arithmetiques d’Euclide* (Paris, 1961).

AdCampTrad | = Variant versions deriving from those of Adelard and Campanus |

Adel I | = First translation of Adelard of Bath |

Adel II | = Commentum of Adelard of Bath |

Adel III | = Editio specialis of Adelard of Bath |

Anr | = Commentary of al-Naurīzī |

ArCm | = Arabic commentaries |

Arm | = Armenian version presumably made by Gregory Magistros |

Avic | = Epitome of Avicenna |

B | = Translation of Boethius |

Bill | = English translation of Henry Billingsley |

Camp | = Version of Campanus of Novara |

Cand | = Versionof Franciscus Flussatus Candalla |

Cen | = Fragments in Censorinus |

Clav | = Edition of Christopher Clavius |

Comm | = Translation of Federico Com madino |

DPap | = Translation by al-Dimashqī of book X of Pappus’ Commentary |

EdPr | = Editio Princeps of Greek text by Simon Grynaeus |

G | = Original Greek text |

Gl | = Pre-Thenine Greek text |

Ga | = Greek text employed for Arabic translations |

GC | = Translation by Gerard of Cremona of Isḥāq–Thābit text |

GCAnr | = Gerard of Cremona’s ranslation of al-Nayrīzī’s commentary |

Gcomm | = Greek commentaries |

GCPap | = Gerard of Cremona’s translation of (Part of) Dimasqī’s translation of Pappus on book X |

Gh | = Greek text utilized by Hero of Alexandria |

Gt | = Redaction of Greek text by Theon of Alexandria |

HC | = Version of Hermann of Carinthia |

Hero | = Hero of Alexandria’s commentary |

Hj 1 | = First version of Ḥajjāj |

Hj 2 | = Second version of Ḥajjāj |

Hj Trad | = Ḥajjāj Arabic tradition |

Ish | = Translation of Isḥāq ibn Ḥunayn |

I-T | = Translation of Isḥāq as revised by Thābit ibn Qurra |

I-T Trad | = Isḥāq-Thābit tradition |

Ma | = Boethian excerpts preserved in Cassiodorus |

Magh | = Taḥrīr of al-Maghribī |

Mb | = Boethian excerpts preserved in Agrimensores material |

Mc | = Boethian excerts in five-book geometry of “Boethius” |

Md | = Boethian excerts in two-book geometry of “Boethius” |

Mel I | = Boethian-Adeardian mélanges in MS Lüneburg D 4° 48 |

Mel II | = Boethian–Adelardian mélanges in Paris and Munich MSS |

Mu | = Frangments of translation in MS Univ. Munich 2° 757 |

Pap | = Greek commentary of Pappus of Alexandria |

Par | = Anonumous Greek–Latin translation of twelfth century |

Proc | = Greek cmmentary of Parclus on book I |

Ps-T | = Thirteen-book Taḥrīr erroneously ascribed to al-Tūsī |

Simp | = Commentary of Simplicius on the premises |

Syr | = Syriac redaction |

Tart | = Italian translation by Niccolò Trataglia |

Ṭūsī | = Genuine fifteen-book Taḥrīr by al-Ṭūsī |

Ver | = Verona palimpset of fifth century |

X | = Ancestors not further specifield here |

Zamb | = Translation of Bartolomeo Zam berti of Theonine text. |

John Murdoch

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# Euclid: Transmission of the Elements.

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**Euclid: Transmission of the Elements.**