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D B F H K C G

Chapter II: Euclid’s Other Works

IN giving a list of the Euclidean treatises other than the Elements, I shall be brief: for fuller accounts of them, or speculations with regard to them, reference should be made to the standard histories of mathematics.

Chapter III: Greek Commentators Other Than Proclus

THAT there was no lack of commentaries on the Elements before the time of Proclus is evident from the terms in which Proclus refers to them; and he leaves us in equally little doubt as to the value which, in his opinion, the generality of them possessed. Thus he says in one place (at the end of his second prologue):

Chapter IV: Proclus and His Sources

IT is well known that the commentary of Proclus on Eucl. Book I. is one of the two main sources of information as to the history of Greek geometry which we possess, the other being the Collection of Pappus. They are the more precious because the original works of the forerunners of Euclid, Archimedes and Apollonius are lost, having probably been discarded and forgotten almost immediately after the appearance of the masterpieces of that great trio.

Chapter V: The Text

IT is well known that the title of Simson’s edition of Euclid (first brought out in Latin and English in 1756) claims that, in it, “the errors by which Theon, or others, have long ago vitiated these books are corrected, and some of Euclid’s demonstrations are restored”; and readers of Simson’s notes are familiar with the phrases used, where anything in the text does not seem to him satisfactory, to the effect that the demonstration has been spoiled, or things have been interpolated or omitted, by Theon “or some other unskilful editor.” Now most of the MSS. of the Greek text prove by their titles that they proceed from the recension of the Elements by Theon; they purport to be either “from the edition of Theon” (ἐκ τῆς Θέωνος ἐκδόσεως) or “from the lectures of Theon” (ἀπὸ συνουσιῶν τοῦ Θέωνος). This was Theon of Alexandria (4th c. A.D.) who also wrote a commentary on Ptolemy, in which there occurs a passage of the greatest importance in this connexion: “But that sectors in equal circles are to one another as the angles on which they stand has been proved by me in my edition of the Elements at the end of the sixth book.” Thus Theon himself says that he edited the Elements and also that the second part of VI. 33, found in nearly all the MSS., is his addition.

Chapter VI: The Scholia

HEIBERG has collected scholia, to the number of about 1500, in Vol. V. of his edition of Euclid, and has also discussed and classified them in a separate short treatise, in which he added a few others.

Chapter VII: Euclid in Arabia

WE are told by Ḥāji Khalfa that the Caliph al-Manṣūr (754–775) sent a mission to the Byzantine Emperor as the result of which he obtained from him a copy of Euclid among other Greek books, and again that the Caliph al-Ma’mūn (813–833) obtained manuscripts of Euclid, among others, from the Byzantines. The version of the Elements by al-Ḥajjāj b. Yūsuf b. Maṭar is, if not the very first, at least one of the first books translated from the Greek into Arabic. According to the Fihrist it was translated by al-Ḥajjāj twice; the first translation was known as “Hārūnī” (“for Hārūn”), the second bore the name “Ma’mūnī” (“for al-Ma’mūn”) and was the more trustworthy. Six Books of the second of these versions survive in a Leiden MS. (Codex Leidensis 399, 1) now in part published by Besthorn and Heiberg. In the preface to this MS. it is stated that, in the reign of Hārūn ar-Rashīd (786–809), al-Ḥajjāj was commanded by Yaḥyā b. Khālid b. Barmak to translate the book into Arabic. Then, when al-Ma’mūn became Caliph, as he was devoted to learning, al-Ḥajjāj saw that he would secure the favour of al-Ma’mūn “if he illustrated and expounded this book and reduced it to smaller dimensions. He accordingly left out the superfluities, filled up the gaps, corrected or removed the errors, until he had gone through the book and reduced it, when corrected and explained, to smaller dimensions, as in this copy, but without altering the substance, for the use of men endowed with ability and devoted to learning, the earlier edition being left in the hands of readers.”

Chapter VIII: Principal Translations and Editions

CICERO is the first Latin author to mention Euclid; 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 mensuris, 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.

F D E A C B

Chapter IX: The Nature of Elements

IT would not be easy to find a more lucid explanation of the terms element and elementary, and of the distinction between them, than is found in Proclus, who is doubtless, here as so often, quoting from Geminus. There are, says Proclus, in the whole of geometry certain leading theorems, bearing to those which follow the relation of a principle, all-pervading, and furnishing proofs of many properties. Such theorems are called by the name of elements; and their function may be compared to that of the letters of the alphabet in relation to language, letters being indeed called by the same name in Greek (στοιχεῖα).