CHAPTER VIII.

PRINCIPAL TRANSLATIONS AND EDITIONS OF THE ELEMENTS.

CICERO is the first Latin author to mention Euclid1; but it is not likely that in Cicero's time Euclid had been translated into Latin or was studied to any considerable extent by the Romans; for, as Cicero says in another place, while geometry was held in high honour among the Greeks, so that nothing was more brilliant than their mathematicians, the Romans limited its scope by having regard only to its utility for measurements and calculations. How very little theoretical geometry satisfied the Roman agrimensores is evidenced by the work of Balbus de mensuris3, where some of the definitions of Eucl. Book I. are given. Again, the extracts from the Elements found in the fragment attributed to Censorinus (fl. 238 A.D.) are confined to the definitions, postulates, and common notions. But by degrees the Elements passed even among the Romans into the curriculum of a liberal education; for Martianus Capella speaks of the effect of the enunciation of the proposition "how to construct an equilateral triangle on a given straight line" among a company of philosophers, who, recognising the first proposition of the Elements, straightway break out into encomiums on Euclid. But the Elements were then (c. 470 A.D.) doubtless read in Greek; for what Martianus Capella gives was drawn from a Greek source, as is shown by the occurrence of Greek words and by the wrong translation of I. def. 1 ("punctum vero est cuius pars nihil est "). Martianus may, it is true, have quoted, not from Euclid himself, but from Heron or some other ancient

source.

But it is clear from a certain palimpsest at Verona that some scholar had already attempted to translate the Elements into Latin. This palimpsest' has part of the "Moral reflections on the Book of Job" by Pope Gregory the Great written in a hand of the 9th c. above certain fragments which in the opinion of the best judges date from the 4th c. Among these are fragments of Vergil and of Livy, as well as a geometrical fragment which purports to be taken from the 14th and 15th Books of Euclid. As a matter of fact it is from Books XII. and XIII. and is of the nature of a free rendering, or rather a new

arrangement, of Euclid with the propositions in different order1. The MS. was evidently the translator's own copy, because some words are struck out and replaced by synonyms. We do not know whether the translator completed the translation of the whole, or in what relation his version stood to our other sources.

Magnus Aurelius Cassiodorius (b. about 475 A.D.) in the geometrical part of his encyclopaedia De artibus ac disciplinis liberalium literarum says that geometry was represented among the Greeks by Euclid, Apollonius, Archimedes, and others, "of whom Euclid was given us translated into the Latin language by the same great man Boethius"; also in his collection of letters is a letter from Theodoric to Boethius containing the words, "for in your translations ... Nicomachus the arithmetician, and Euclid the geometer, are heard in the Ausonian tongue." The so-called Geometry of Boethius which has come down to us by no means constitutes a translation of Euclid. The MSS. variously give five, four, three or two Books, but they represent only two distinct compilations, one normally in five Books and the other in two. Even the latter, which was edited by Friedlein, is not genuine', but appears to have been put together in the 11th c., from various sources. It begins with the definitions of Eucl. I., and in these are traces of perfectly correct readings which are not found even in the MSS. of the 10th c., but which can be traced in Proclus and other ancient sources; then come the Postulates (five only), the Axioms (three only), and after these some definitions of Eucl. II., III., IV. Next come the enunciations of Eucl. I., of ten propositions of Book II., and of some from Books III., IV., but always without proofs; there follows an extraordinary passage which indicates that the author will now give something of his own in elucidation of Euclid, though what follows is a literal translation of the proofs of Eucl. 1. 1-3. This latter passage, although it affords a strong argument against the genuineness of this part of the work, shows that the Pseudoboethius had a Latin translation of Euclid from which he extracted the three propositions.

Curtze has reproduced, in the preface to his edition of the translation by Gherard of Cremona of an-Nairīzī's Arabic commentary on Euclid, some interesting fragments of a translation of Euclid taken from a Munich MS. of the 10th c. They are on two leaves used for the cover of the MS. (Bibliothecae Regiae Universitatis Monacensis 2o 757) and consist of portions of Eucl. 1. 37, 38 and II. 8, translated literally word for word from the Greek text. The translator seems to have been an Italian (cf. the words "capitolo nono" used for the ninth prop. of Book II.) who knew very little Greek and had morcover little mathematical knowledge. For example, he translates the capital letters denoting points in figures as if they were numerals: thus rà ABг,

1 The fragment was deciphered by W. Studemund, who communicated his results to Cantor.

2 Cassiodorius, Variae, I. 45, p. 40, 12 ed. Mommsen.

See especially Weissenborn in Abhandlungen zur Gesch. d. Math. 11. p. 185 sq.; Heiberg in Philologus, XLIII. p. 507 sq.; Cantor, I3, p. 580 sq.

1

AEZ is translated "que primo secundo et tertio quarto quinto et septimo," T becomes "tricentissimo" and so on. The Greek MS. which he used was evidently written in uncials, for AEZO becomes in one place "quod autem septimo nono," showing that he mistook AE for the particle dé, and rai ó ETU is rendered "sicut tricentissimo et quadringentissimo," showing that the letters must have been written KAIOCTU.

The date of the Englishman Athelhard (Æthelhard) is approximately fixed by some remarks in his work Perdifficiles Quaestiones Naturales which, on the ground of the personal allusions they contain, must be assigned to the first thirty years of the 12th c. He wrote a number of philosophical works. Little is known about his life. He is said to have studied at Tours and Laon, and to have lectured at the latter school. He travelled to Spain, Greece, Asia Minor and Egypt, and acquired a knowledge of Arabic, which enabled him to translate from the Arabic into Latin, among other works, the Elements of Euclid. The date of this translation must be put at about 1120. MSS. purporting to contain Athelhard's version are extant in the British Museum (Harleian No. 5404 and others), Oxford (Trin. Coll. 47 and Ball. Coll. 257 of 12th c.), Nürnberg (Johannes Regiomontanus' copy) and Erfurt.

Among the very numerous works of Gherard of Cremona (11141187) are mentioned translations of "15 Books of Euclid" and of the Data. Till recently this translation of the Elements was supposed to be lost; but Axel Anthon Björnbo has succeeded (1904) in discovering a translation from the Arabic which is different from the two others known to us (those by Athelhard and Campanus respectively), and which he, on grounds apparently convincing, holds to be Gherard's. Already in 1901 Björnbo had found Books X.-XV. of this translation in a MS. at Rome (Codex Reginensis lat. 1268 of 14th c.)3; but three years later he had traced three MSS. containing the whole of the same translation at Paris (Cod. Paris. 7216, 15th c.), Boulogne-sur-Mer (Cod. Bononiens. 196, 14th c.), and Bruges (Cod. Brugens. 521, 14th c.), and another at Oxford (Cod. Digby 174, end of 12th c.) containing a fragment, XI. 2 to XIV. The occurrence of Greek words in this translation such as rombus, romboides (where Athelhard keeps the Arabic terms), ambligonius, orthogonius, gnomo, pyramis etc., show that the translation is independent of Athelhard's. Gherard appears to have had before him an old translation of Euclid from the Greek which Athelhard also often followed, especially in his terminology, using it however in a very different manner. Again, there are some Arabic terms, e.g. meguar for axis of rotation, which Athelhard did not use, but which is found in almost all the translations that are with certainty attributed to Gherard of Cremona; there occurs also the 1 Cantor, Gesch. d. Math. 1, p. 906.

2 Boncompagni, Della vita e delle opere di Gherardo Cremonese, Rome, 1851, p. 5.

* Described in an appendix to Studien über Menelaos' Sphärik (Abhandlungen zur Geschichte der mathematischen Wissenschaften, XIV., 1902).

See Bibliotheca Mathematica, VI, 1905-6, pp. 242-8.

expression "superficies equidistantium laterum et rectorum angulorum," found also in Gherard's translation of an-Nairīzī, where Athelhard says "parallelogrammum rectangulum." The translation is much clearer than Athelhard's: it is neither abbreviated nor "edited" as Athelhard's appears to have been; it is a word-for-word translation of an Arabic MS. containing a revised and critical edition of Thabit's version. It contains several notes quoted from Thabit himself (Thebit dixit), e.g. about alternative proofs etc. which Thabit found "in another Greek MS.," and is therefore a further testimony to Thabit's critical treatment of the text after Greek MSS. The new editor also added critical remarks of his own, e.g. on other proofs which he found in other Arabic versions, but not in the Greek: whence it is clear that he compared the Thabit version before him with other versions as carefully as Thabit collated the Greek MSS. Lastly, the new editor speaks of "Thebit qui transtulit hunc librum in arabicam linguam" and of "translatio Thebit," which may tend to confirm the statement of al-Qifti who credited Thabit with an independent translation, and not (as the Fihrist does) with a mere improvement of the version of Isḥāq b. Hunain.

Gherard's translation of the Arabic commentary of an-Nairizi on the first ten Books of the Elements was discovered by Maximilian Curtze in a MS. at Cracow and published as a supplementary volume to Heiberg and Menge's Euclid': it will often be referred to in this work.

Next in chronological order comes Johannes Campanus (Campano) of Novara. He is mentioned by Roger Bacon (1214-1294) as a prominent mathematician of his time, and this indication of his date is confirmed by the fact that he was chaplain to Pope Urban IV, who was Pope from 1261 to 1281. His most important achievement was his edition of the Elements including the two Books XIV. and XV. which are not Euclid's. The sources of Athelhard's and Campanus' translations, and the relation between them, have been the subject of much discussion, which does not seem to have led as yet to any definite conclusion. Cantor (II, p. 91) gives references and some particulars. It appears that there is a MS. at Munich (Cod. lat. Mon. 13021) written by Sigboto in the 12th c. at Prüfning near Regensburg, and denoted by Curtze by the letter R, which contains the enunciations of part of Euclid. The Munich MSS. of Athelhard and Campanus' translations have many enunciations textually identical with those in R, so that the source of all three must, for these enunciations, have

1 Anaritii in decem libros priores Elementorum Euclidis Commentarii ex interpretatione Gherardi Cremonensis in codice Cracoviensi 569 servata edidit Maximilianus Curtze, Leipzig (Teubner), 1899.

2 Cantor, II, p. 88.

Tiraboschi, Storia della letteratura italiana, IV. 145-160.

♦ H. Weissenborn in Zeitschrift für Math. u. Physik, xxv., Supplement, pp. 143-166, and in his monograph, Die Übersetzungen des Euklid durch Campano und Zamberti (1882); Max. Curtze in Philologische Rundschau (1881), 1. pp. 943-950, and in Jahresbericht über die Fortschritte der classischen Alterthumswissenschaft, XL. (1884, 111.) pp. 19-22; Heiberg in Zeitschrift für Math. u. Physik, xxxv., hist.-litt. Abth., pp. 48—58 and pp. 81—6.

been the same; in others Athelhard and Campanus diverge completely from R, which in these places follows the Greek text and is therefore genuine and authoritative. In the 32nd definition occurs the word "elinuam," the Arabic term for "rhombus," and throughout the translation are a number of Arabic figures. But R was not translated from the Arabic, as is shown by (among other things) its close resemblance to the translation from Euclid given on pp. 377 sqq. of the Gromatici Veteres and to the so-called geometry of Boethius. The explanation of the Arabic figures and the word "elinuam" in Def. 32 appears to be that R was a late copy of an earlier original with corruptions introduced in many places; thus in Def. 32 a part of the text was completely lost and was supplied by some intelligent copyist who inserted the word "elinuam," which was known to him, and also the Arabic figures. Thus Athelhard certainly was not the first to translate Euclid into Latin; there must have been in existence before the 11th c. a Latin translation which was the common source of R, the passage in the Gromatici, and "Boethius." As in the two latter there occur the proofs as well as the enunciations of I. 1-3, it is possible that this translation originally contained the proofs also. Athelhard must have had before him this translation of the enunciations, as well as the Arabic source from which he obtained his proofs. That some sort of translation, or at least fragments of one, were available before Athelhard's time even in England is indicated by some old English verses':

"The clerk Euclide on this wyse hit fonde

Thys craft of gemetry yn Egypte londe

Yn Egypte he tawghte hyt ful wyde,

In dyvers londe on every syde.

Mony erys afterwarde y understonde

Yer that the craft com ynto thys londe.

Thys craft com into England, as y yow say,

Yn tyme of good kyng Adelstone's day,"

which would put the introduction of Euclid into England as far back as 924-940 A.D.

We now come to the relation between Athelhard and Campanus. That their translations were not independent, as Weissenborn would have us believe, is clear from the fact that in all MSS. and editions, apart from orthographical differences and such small differences as are bound to arise when MSS. are copied by persons with some knowledge of the subject-matter, the definitions, postulates, axioms, and the 364 enunciations are word for word identical in Athelhard and Campanus; and this is the case not only where both have the same text as R but where they diverge from it. Hence it would seem that Campanus used Athelhard's translation and only developed the proofs by means of another redaction of the Arabian Euclid. It is true that the difference between the proofs of the propositions in the two translations is considerable; Athelhard's are short and com1 Quoted by Halliwell in Rara Mathematica (p. 56 note) from мs. Bib. Reg. Mus. Brit. 17 A. 1. f. 2-3.