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 same things as three in the Optics, is one of the reasons given by Gregory for deeming that work spurious. Several other reasons will be found in Gregory’s preface to his edition of Euclid’s works.

In some editions of Euclid’s works there is given a book on the Divisions of Superficies, which consists of a few propositions, showing how a straight line may be drawn to divide in a given ratio triangles, quadrilaterals and pentagons. This was supposed by John Dee of London, who transcribed or translated it, and entrusted it for publication to his friend Federico Commandino of Urbino, to be the treatise of Euclid referred to by Proclus as . Dee mentions that, in the copy from which he wrote, the book was ascribed to Machomet of Bagdad, and adduces two or three reasons for thinking it to be Euclid’s. This opinion, however, he does not seem to have held very strongly, nor does it appear that it was adopted by Commandino. The book does not exist in Greek.

The fragment, in Latin, De levi et ponderoso, which is of no value, and was printed at the end of Gregory’s edition only in order that nothing might be left out, is mentioned neither by Pappus nor Proclus, and occurs first in Bartholomew Zamberti’s edition of 1537. There is no reason for supposing it to be genuine.

The following works attributed to Euclid are not now extant:—

1. Three books on Porisms ( ) are mentioned both by Pappus and Proclus, and the former gives an abstract of them, with the lemmas assumed. (See .)

2. Two books are mentioned, named  , which is rendered Locorum ad superficiem by Commandino and subsequent geometers. These books were subservient to the analysis of loci, but the four lemmas which refer to them and which occur at the end of the seventh book of the Mathematical Collection, throw very little light on their contents. R. Simson’s opinion was that they treated of curves of double curvature, and he intended at one time to write a treatise on the subject. (See Trail’s Life of Dr Simson).

3. Pappus says that Euclid wrote four books on the Conic Sections ( ), which Apollonius amplified, and to which he added four more. It is known that, in the time of Euclid, the parabola was considered as the section of a right-angled cone, the ellipse that of an acute-angled cone, the hyperbola that of an obtuse-angled cone, and that Apollonius was the first who showed that the three sections could be obtained from any cone. There is good ground therefore for supposing that the first four books of Apollonius’s Conics, which are still extant, resemble Euclid’s Conics even less than Euclid’s Elements do those of Eudoxus and Theaetetus.

4. A book on Fallacies ( ) is mentioned by Proclus, who says that Euclid wrote it for the purpose of exercising beginners in the detection of errors in reasoning.

This notice of Euclid would be incomplete without some account of the earliest and the most important editions of his works. Passing over the commentators of the Alexandrian school, the first European translator of any part of Euclid is Boëtius (500), author of the De consolatione philosophiae. His Euclidis Megarensis geometriae libri duo contain nearly all the definitions of the first three books of the Elements, the postulates, and most of the axioms. The enunciations, with diagrams but no proofs, are given of most of the propositions in the first, second and fourth books, and a few from the third. Some centuries afterwards, Euclid was translated into Arabic, but the only printed version in that language is the one made of the thirteen books of the Elements by Nasir Al-Dīn Al-Tūsī (13th century), which appeared at Rome in 1594.

The first printed edition of Euclid was a translation of the fifteen books of the Elements from the Arabic, made, it is supposed, by Adelard of Bath (12th century), with the comments of Campanus of Novara. It appeared at Venice in 1482, printed by Erhardus Ratdolt, and dedicated to the doge Giovanni Mocenigo. This edition represents Euclid very inadequately; the comments are often foolish, propositions are sometimes omitted, sometimes joined together, useless cases are interpolated, and now and then Euclid’s order changed.

The first printed translation from the Greek is that of Bartholomew Zamberti, which appeared at Venice in 1505. Its contents will be seen from the title: Euclidis megarēsis philosophi platonici Mathematicaru disciplinarū Janitoris: Habent in hoc volumine quicūq ad mathematicā substantiā aspirāt: elemētorum libros xiii ''cū expositione Theonis insignis mathematici ... Quibus ... adjuncta.'' Deputatum scilicet Euclidi volumē xiiii cū expositiōe Hypsi. ''Alex. Itidēq Phaeno. Specu. Perspe. cum expositione Theonis ac'' mirandus ille liber Datorum cum expostiōe Pappi Mechanici una ''cū Marini dialectici protheoria. Bar. Zāber. Vene. Interpte.''

The first printed Greek text was published at Basel, in 1533, with the title . It was edited by Simon Grynaeus from two MSS. sent to him, the one from Venice by Lazarus Bayfius, and the other from Paris by John Ruellius. The four books of Proclus’s commentary are given at the end from an Oxford MS. supplied by John Claymundus.

The English edition, the only one which contains all the extant works attributed to Euclid, is that of Dr David Gregory, published at Oxford in 1703, with the title, . Euclidis quae supersunt omnia. The text is that of the Basel edition, corrected from the MSS. bequeathed by Sir Henry Savile, and from Savile’s annotations on his own copy. The Latin translation, which accompanies the Greek on the same page, is for the most part that of Commandino. The French edition has the title, Les Œuvres d’Euclide, traduites en Latin et en Français, d’après un manuscrit ''très-ancien qui était resté inconnu jusqu’à nos jours. Par F. Peyrard,'' Traducteur des œuvres d’Archimède. It was published at Paris in three volumes, the first of which appeared in 1814, the second in 1816 and the third in 1818. It contains the Elements and the Data, which are, says the editor, certainly the only works which remain to us of this ever-celebrated geometer. The texts of the Basel and Oxford editions were collated with 23 MSS., one of which belonged to the library of the Vatican, but had been sent to Paris by the comte de Peluse (Monge). The Vatican MS. was supposed to date from the 9th century; and to its readings Peyrard gave the greatest weight. What may be called the German edition has the title . Euclidis Elementa ex optimis libris in usum Tironum Graece edita ab Ernesto Ferdinando August. It was published at Berlin in two parts, the first of which appeared in 1826 and the second in 1829. The above mentioned texts were collated with three other MSS. Modern standard editions are by Dr Heiberg of Copenhagen, Euclidis Elementa, edidit et Latine interpretatus est J. L. Heiberg. vols. i.-v. (Lipsiae, 1883–1888), and by T. L. Heath, The Thirteen Books of Euclid’s Elements, vols. i.-iii. (Cambridge, 1908).

Of translations of the Elements into modern languages the number is very large. The first English translation, published at London in 1570, has the title, The Elements of Geometrie of the most auncient ''Philosopher Euclide of Megara. Faithfully (now first) translated into'' ''the Englishe toung, by H. Billingsley, Citizen of London. Whereunto'' are annexed certaine Scholies, Annotations and Inventions, of the best Mathematiciens, both of time past and in this our age. The first French translation of the whole of the Elements has the title, Les ''Quinze Livres des Elements d’Euclide. Traduicts de Latin en François.'' Par D. Henrion, Mathematicien. The first edition of it was published at Paris in 1615, and a second, corrected and augmented, in 1623. Pierre Forcadel de Beziés had published at Paris in 1564 a translation of the first six books of the Elements, and in 1565 of the seventh, eighth and ninth books. An Italian translation, with the title, Euclide Megarense acutissimo philosopho solo introduttore delle ''Scientie Mathematice. Diligentemente rassettato, et alla integrità'' ridotto, per il degno professore di tal Scientie Nicolò Tartalea Brisciano, was published at Venice in 1569, and Federico Commandino’s translation appeared at Urbino in 1575; a Spanish version, Los ''Seis Libros primeros de la geometria de Euclides. Traduzidos en'' lēgua Española por Rodrigo Camorano, Astrologo y Mathematico, at Seville in 1576; and a Turkish one, translated from the edition of J. Bonnycastle by Husaīn Rifkī, at Bulak in 1825. Dr Robert Simson’s editions of the first six and the eleventh and twelfth books of the Elements, and of the Data.

.—The authors and editions above referred to; Fabricius, Bibliotheca Graeca, vol. iv.; Murhard’s Litteratur der mathematischen Wissenschaften; Heilbronner’s Historia matheseos universae; De Morgan’s article “Eucleides” in Smith’s Dictionary of Biography and Mythology; Moritz Cantor’s Geschichte der Mathematik, vol. i.

 EUCRATIDES, king of Bactria (c. 175–129 ), came to the throne by a rebellion against the dynasty of Euthydemus, whose son Demetrius had conquered western India. His authority was challenged by a great many other pretenders and Greek dynasts in Sogdiana, Aria (Herat), Drangiana (Sijistan), &c., whose names—Pantaleon, Agathocles, Antimachus, Antalcidas “the victorious” ( ), Plato, whose unique coin is dated from the year 147 of the Seleucid era (= 166 ), and others—are known only from coins with Greek and Indian legends. In the west the Parthian king Mithradates I. began to enlarge his kingdom and attacked Eucratides; he succeeded in conquering two provinces between Bactria and Parthia, called by Strabo “the country of Aspiones and Turiua,” two Iranian names. But the principal opponent of Eucratides was (q.v.) of India, who attacked him with a large army “of 300,000 men”; Eucratides fled with 300 men into a fortress and was besieged. But at last he beat