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EUB a house in Coney Street, York, to which he removed in the summer of 1848, but his repose there was not long; he died of congestion of the lungs on the 13th of November, 1849, and was honoured by his native town with a public funeral. He was never married, and having led a very quiet life, he left a considerable fortune.—(See the "Autobiography" quoted, and the Life of Wm. Etty, R.A. By A. Gilchrist, 2 vols. 8vo. London, 1855.)—R. N. W.  EUBULIDES, a philosopher of Miletus (384-322 .), was the contemporary of Aristotle, whom he assailed with animosity. He was a successor of Euclides of Megara. A doubtful tradition makes him the teacher of Demosthenes. He left no writings; and his name has been preserved chiefly in connection with the Mentiens, the Electra, the Cornuta, and other logical puzzles of the same sort, which were much used by the Megarean school in confuting the adversaries of their Eleatic doctrines. The Cornuta was this: "Whatever you have not lost you have: you have not lost horns: you have horns." The right of Eubulides to the sophisms which pass by his name is dubious, as several of them are quoted by earlier writers.—G. R. L.  EUBULUS, an Athenian poet, flourished in the earlier part of the fourth century. His name is one of the most distinguished among the poets of the middle comedy, whose period almost coincides with his career. He took his subjects chiefly from mythology. Sometimes he introduced parodies of the tragic poets, particularly Euripides. Later writers pillaged largely from him. Suidas says that Eubulus was the author of one hundred and four plays. There are still extant the names of fifty of them. His language is described as graceful and unaffected, and on the whole pure and correct.—G. R. L.  EUCHERIUS, archbishop of Lyons in the fifth century. Like Nilus, he left his family (he took his two sons with him) to practice asceticism in the desert. The fame of his virtues soon got abroad; whereupon he was offered the archbishopric of Lyons, which he unwillingly accepted. He was present at the first council of Orange in 441, and died about 454. His writings are mostly intended to set forth the excellencies of the ascetic life.—R. M., A.  EUCLID of Alexandria, a celebrated Greek writer on mathematical subjects, best known as the author of the "Elements." Of his history little is known with certainty. According to the Arabian editors, he was of Greek extraction, son of Naucrates and grandson of Zenarchus, born at Tyre, and resident at Damascus; but not much credit is attached to this statement. More authentic information is derived from the Greek commentator Proclus. He states that Euclid lived in the time of Ptolemy I., son of Lagus (323-283 .), and gave to that prince the celebrated reply that there is no royal road to geometry; that he was therefore younger than Plato's immediate disciples, but elder than Archimedes; and that he was of the Platonic sect. From this statement it is inferred that Euclid either founded the celebrated mathematical school of Alexandria, or was one of the earliest teachers there. At all events, the supposition (at one time common) that the author of the "Elements" was Euclid of Megara, the friend of Socrates, who lived nearly a century before the time of Ptolemy and was not a mathematician, is quite inconsistent with the account of Proclus. David Gregory, in the preface to his edition of Euclid's works (Oxon. 1703), quotes a remark of Sir Henry Savile, that as we know from external sources so little of Euclid's biography, we must fall back upon those great monuments of his genius, the writings which have descended to us; and all succeeding biographers have been forced to adopt the same course. The works of Euclid mentioned by Proclus are the following—"On Optics and Catoptrics" (treatises ascribed to Euclid with these names are extant, but it is doubted if they are by the author of the "Elements"); "Elements of Music" (two treatises noticed below); a treatise "On Divisions," not now extant; the "Elements of Geometry," the well-known work, identified with the name of Euclid; a treatise "On Fallacies." This which was preliminary to the Elements has unfortunately been lost. "Porisms" also lost. Pappus of Alexandria, who flourished in 370, gives an account of the three books of Euclid's "Porisms," which, from corruptions of the text, is nearly unintelligible, and has given rise to much speculation. It was at one time a favourite exercise of geometers to attempt the restoration of the "Porisms" from the description of Pappus. The most successful attempt is that of Dr. Robert Simson, professor of mathematics in the university of Glasgow from 1711 to 1758. The restoration of the "Porisms," preceded by the text of Pappus in Latin, is to be found in Simson's posthumous works printed at the expense of Earl Stanhope by Foulis, Glasgow, 1776.—(See also Lord Brougham's Life of Simson.)—Besides these. Pappus mentions the following works—Four books of "Conic Sections," not now extant; "Phænomena," a treatise on geometrical astronomy, which has come down to us; two books of "Loci ad Superficiem," not extant, and the nature of which is not known (Lord Brougham, in his Life of Simson, asserts that this was a treatise on curves of double curvature, but assigns no reason); the book of "Data," which we still have; and two books of "Plani Loci."—None of the writings of Euclid which are extant, and probably none of those which are lost, is nearly as important as the "Elements," which, as a work of mathematical art, is still almost unrivalled. It is, however, certain that, though immense credit must be given to Euclid for the arrangement and method of the work, the whole is by no means original. Proclus, after giving a list of Euclid's predecessors who had done much to advance the science of geometry, assigns to him the merit of "bringing together the elements, arranging many propositions of Eudoxus, completing many of Theætetus, and giving undeniable proofs of propositions previously inaccurately demonstrated." How just are the praises he afterwards bestows on his marvellous accuracy, judicious method, and freedom from false reasoning, must not only be felt by all thoughtful students, but appears from the impression which many of the modern editors seem to have received, that any imperfections that may be detected must be due, not to Euclid, but to the want of skill of the ancient editor. Dr. Robert Simson, whose translation is the basis of almost all the English editions, never speaks of correcting, but always of restoring the text of Euclid. It is suggested by Professor De Morgan in his learned article on Euclid in Dr. Smith's Dictionary of Biography and Mythology, that the "Elements" had not received the author's final corrections. This has occurred also to the present writer in reference to the state of the earlier part of the third book; and it is, perhaps, a more plausible way than Simson's of accounting for the imperfections of the text, while it is an equal testimony to the high standard of the work in general.

The ultimate object which Euclid proposed to himself in the "Elements" is somewhat doubtful. According to Proclus, the discussion of the five regular solids was the main aim of the whole, and this is confirmed by the opinion of Kepler. But by a modern reader, whose estimate, however, of the relative importance of propositions is probably very different, it might be conjectured that Euclid's aim is the comparison of the areas of plane figures, and of the contents of solids; the method of comparison being by the construction of squares equal in area to plane figures, and cubes equal in volume to solids. From this point of view, the main object of the first book is the description of a rectangle of given breadth equal in area to any rectilinear figure, which is extended in the second book to the description of a square equal to any given rectilinear figure; the forty-seventh proposition of the first book being incidentally introduced, partly for its own importance, and partly to show how a square may be constructed equal to the sum of several figures. Having reached this point, Euclid proceeds in the third book to discuss the properties of the circle; and in the fourth the construction of regular polygons described in and about circles. In the fifth book the theory of proportion is discussed, as applicable to magnitudes of all kinds. "There is nothing," says Dr. Isaac Barrow, "in the whole body of the 'Elements' of a more subtile invention, nothing more solidly established, and more accurately handled than the doctrine of proportionals"—an opinion which will be adopted by all who have detected the ingenuity with which the definition of the equality of ratios is made to overcome the difficulty of applying proportion to incommensurable lines, and felt the impossibility of substituting any simpler and equally exact method. In the ninth book the theory of proportion is applied to plane rectilinear figures, and to sectors and arcs of equal circles. The next three books treat of arithmetic. The tenth is on incommensurable quantities; and it has been plausibly suggested, that this investigation was undertaken in the hope of solving the problem of drawing a straight line equal to the circumference of a circle of which the diameter is given. The elements of solid geometry are the subject of the eleventh book; and in the twelfth, besides comparing the contents of certain solids, Euclid 