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EUC proves, by the famous method of exhaustions, that circles are to one another as the squares of their diameters, and spheres as the cubes of their diameters. The thirteenth then treats of the length of the sides of regular polygons, and the description of the regular solids. The fourteenth and fifteenth books are by Hypsicles, and not by Euclid. If the view here taken of the aim of this remarkable work be correct, it will be seen, that though it is completely successful as regards rectilineal plane figures, and though the areas of circles and the volumes of spheres are satisfactorily compared with one another, yet the end, which it is conjectured Euclid hoped to attain, of constructing a square equal in area to any given circle, and a cube equal in volume to a given sphere, is not reached. Of the other extant works of Euclid, the book of "Data" is next in value to the "Elements"—it has, however, been completely superseded by modern analysis. The principal edition of the whole extant works of Euclid in Greek, with a Latin translation, is the Oxford folio of 1703, edited by David Gregory. The most accessible edition of the "Elements" in Greek, is that of E. F. August, Berlin, 1826. The Oxford edition, by James Williamson, B.D., in 2 vols. 4to, 1781, contains a literal translation into English of the thirteen books of the "Elements," and Simson's edition (8vo., Foulis; Glasgow, 1762) contains, besides the first six books, and the eleventh and twelfth, the book of "Data," all translated and "restored." This edition is the foundation of nearly all the modern English editions.—H. B.  [Euclid is supposed to have written two tracts on music, the "Introduction to Harmony," and the "Section of the Canon," but his authorship of these is questioned, on the ground that two of the most perfect manuscripts of them, variously bear the names of Pappus and Cleonidas. Meibomius, who includes the two treatises in his collection of the writings on music of seven ancient Greek authors, assumes that these are the names of persons who corrected each the transcript to which his is affixed, and that Euclid was the writer. On the other hand. Dr. Wallis suggests that the two tracts are the production of different authors, arguing from the internal evidence of the one contradicting the division of the intervals of Aristoxenes, and the other following the principles of this famous philosopher; and M. Peyrard rejects both the tracts, as positively forming no part of the works of Euclid. The two tracts were first printed under the name of Cleonidas—with no reference to Euclid or his other writings—translated into Latin by G. Valla, at Venice in 1498. Another Latin translation by J. Pena, printed at Paris in 1557, ascribes them to Euclid, and this view is supported by the authority of Proclus, Diadochus, and Pappus of Alexandria, early commentators on the great mathematician, who both allude to these as his compositions. Whoever was their author, the tracts are eminently valuable as showing more clearly than any previous writings the principles of Greek music. The "Section of the Canon," in particular, expressly defines the scale, with the distinctions of the minor and major semitones, according to the system of Pythagoras. It appears to have been the authority for all subsequent writers on the subject; for, though Aristides Quintilianus added the numerical calculations of the proportions of the several intervals, these but illustrate Euclid's "Section," departing from it in no respect whatever.]—G. A. M.  EUCLID, born, according to one account, in the Sicilian Gela, removed at an early period to the Greek city, which gave the name to that sect of which he was the founder. A disciple of Socrates, he remained, while his master lived, a devoted student of his philosophy, and one of the most zealous of his personal friends. On the death of Socrates, 399,. Euclid returned to Megara, where he opened an asylum for his brother disciples, and established a school of his own. Euclid was partly an eclectic; partly he partook of the spirit of the sophists. The philosophers of Megara approached the Eleatics in the fundamental principle of their metaphysics, while in the subtilties of their logical exposition they anticipated the epoch of the schoolmen. Reversing the dictum of Socrates, "Virtue is knowledge," they asserted that knowledge was virtue, and their one-sided view, which made speculation the end of life, is controverted by Plato in the Philebus. Asserting that the universe, as a whole, is identical with the supreme Good, they denied the separate existence of Evil. The various forms of virtue are merely phases of this universal good, and the apparent manifestations of vice are merely degrees of its privation. Laertius preserves an anecdote, in which Socrates is made to accuse his pupil of knowing how to debate with sophists and not with men, which, along with Timon's reference to the "wrangling Euclid, who infected all Megara with a mania for disputation," illustrates the reputation which his sect enjoyed for arguments more intricate in their form than profound in their matter. The details of those arguments are for the most part unknown to us; we have only a few isolated specimens to indicate their tenor We are told that Euclid was wont to assail not the premises, but the conclusions of his opponent, and that he rejected all reasoning from analogy. If the objects compared were unlike, he said the analogy was necessarily fallacious; if like, we must know them to be so from a knowledge of both, and had best examine the objects themselves—an argument which well represents the tendency to subordinate the discovery of truth to the elaboration of ingenious dilemmas which characterized throughout the shallow and brilliant dialectic of the Megarean school.—J. N.  EUCRATIDES, King of Bactria, contemporary with Mithridates I., whose encroachments he long and valiantly resisted. Eucratides made extensive conquests in northern India, and he is called by Strabo lord of a thousand cities. In spite of the incessant hostility with which he was pursued by Mithridates, he appears to have retained possession of these conquests till his death. He was assassinated by his son. The commencement of his reign is with some degree of probability assigned to the year 160 .—J. S., G.  EUDÆMON,, a learned jesuit, born in Crete in 1560; died in 1625. He was said to have been descended from the Palæologi. Coming to Rome he entered with the jesuits, and afterwards taught philosophy and theology at Padua. Eudæmon was looked upon as the author of the celebrated "Admonitio ad Regem Ludovicum XIII.," a work that was universally condemned by the French nation, as striking at the power of the king in ecclesiastical matters. He is also remembered for having entered the lists with several English writers against popery—such as Prideaux, Abbot, and Collins.—R. M., A.  EUDEMUS of Rhodes, a Greek philosopher, contemporary and disciple of Aristotle. The honour of tilling the place of Aristotle after his death, was divided by the Peripatetics between him and Theophrastus. The interest which attaches to his name is that which he acquired as editor of and commentator upon the Aristotelian writings. For the form in which we have the greater part of the works of Aristotle we are indebted to the care of Eudemus, to whom the manuscript of the Metaphysics was committed by the Stagyrite himself. No particulars of the life of this philosopher have come down to us.—J. S., G.  EUDES, Count of Paris, eldest son of Duke Robert the Brave, inherited his father's martial spirit, and distinguished himself against the Norman rovers in the reign of the emperor, Charles le Gros. When that monarch was deposed in 887, his son and heir being of tender age, the French nobles turned their eyes on Eudes, and at their invitation he assumed the sovereignty of France, professing, however, to hold it for the young Carlovingian prince. The opposition subsequently raised by the partisans of the latter, and his flight to claim the aid of the German emperor, led to a partition of the kingdom in a council held at Worms. Charles the Simple received the northern half, and Eudes retained the southern till his death in 898.—W. B.  EUDES, Duke of Aquitaine. See. <section end="316H" /> <section begin="316I" />EUDES I., Duke of Burgundy, inherited that large fief of the French crown from his brother, Hugh I., in 1078. He fought in the wars of Philip I. against the Norman conqueror of England, and took part in the expedition of 1087, which carried a French force to the assistance of Alfonso VI. of Castile against the Moors. Ten years later he joined the first crusade, and died in Cilicia in 1103.—W. B. <section end="316I" /> <section begin="316J" />EUDES II., Duke of Burgundy, grandson of the preceding, succeeded his father, Hugh II., in 1142. He held the dukedom for twenty years, during which he had some disputes with his neighbours, the count of Champagne and the bishop of Langres, respecting his feudal rights. He was succeeded in 1162 by his son, Hugh III.—W. B. <section end="316J" /> <section begin="316Knop" />EUDES III., Duke of Burgundy, was the son of Hugh III., from whom he inherited the dukedom in 1193. He took part in the crusade against Raymond of Toulouse and the Albigenses, held a high command in the army with which Philip Augustus humbled the count of Flanders, and was about to accompany the expedition against the Saracen power in Egypt, when he died in 1218.—W. B. <section end="316Knop" />