cone, and that Apolionius 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 Conies, which are still extant, resemble Euclid's Conies even less than Euclid's Elements do those of Eudoxus and Thesetetus.
4. A book on Fallacies (Uepl if/fvSapiuv) 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 Boetius (500 A.D.), author of the De Consolatione Philosophic?. His Euclidis Megarensis Geometrice 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-Din Al-Tiisi (13fch century), which appeared at Rome in 1594. Judging from the unusual number of diagrams in this edition, the translation of Euclid's text is probably rather free.
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 ISTovara. 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: Eudidis megaresis philosophi platonid Mathematical u% disciplinary Janitoris: Ilabent in hoc volumine quicilq^ ad mathematical substantia aspirat: elemetorum libros xiii cu expositione Theonis insignis mathematics Quibus .... adjuncta. Deputatum scilicet Euclidi volume xiiii cu expositioe Hypsi. Alex. Itideq-& Phaeno. Specu. Perspe. cum expositione Theonis. ac mirandus ille liber Datorum cum expositioe Pappi Mechanici ima cu Marini dialectici protheoria. Bar. Zdher. Vene. Interpte.
The first printed Greek text was published at Basel, in 1533, with the title Ev K f18ov SrotxeiW /5i/3A- ie CK rdv e wvos o-wovo-iwv. It was edited by Simon Grynams from two MSB. 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, Ewv-A.e(. Scw TO. o-wo>o a. Euclidis quce supersunt omnia. A Jo Q text is tliat of tbe Basel edition, corrected from the . 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 Commandine.
The French edition has the title, Les Oeuvres d Eudide, tradaites en Latin et en Francis, d apres un manuscrit tres-anaen qui etait reste inconnu jmqu a, nos jours. Par J>. 1 eyrard, Traducteur des oeuvres d Archimede. It was isned at Paris in three volumes, the first of which ap peared 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 Eu/cAei Sov 2roixei- Euclidis Elementa ex optimis libris in usum Tironum Greece 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. All the abovementioned texts were collated with three other MSS.
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 Geometric of the most auncient Philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by II. Billingsley, Citizen of London. Whereunto are annexed certaine Scholies, Annotations, and Inventions, of the best Jfatkematiciens, 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 Eudide. Traduicts de Latin en Francois. Par D. Henrion, Mathcmaticien. The first edition of it was printed in 1614, and a second, corrected and augmented, was published at Paris in 1623. An Italian translation, with the title, Euclide Megarense aciitissimo jihilosoplio solo introduttore delle Scientie Mathematice. Diligentemente rassettato, et alia integritH ridotto, per il degno professore di tal Scientie Nicolo Tartalea Brisciano, was published at Venice in 1569; a Spanish version, Los Seis Libros primeros de la geometria de Eudides. Traduzidos en legua Espaiiola por Rodrigo Qamorano, Astrologo y Mathematico, at Seville in 1576; and a Turkish one 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, which form the basis of all the modern school texts of Euclid, are so common that it is not considered necessary to describe them.
Authorities. The authors and editions above referred to; Fabricii BibliolhecaGrceca,o. iv.; Murhard's iterator dcr Mathcmatischen Wissenschaftcn; Heilbronner's Historia Mathescos Universes; De Morgan's article "Eucleides" in Smith's Dictionary of Biography and Mythology.(j. s. m.)
