The name of Euclid marks another epoch in the history of geometry, and the chief interest of the vague sketch above given of the labors of his predecessors lies in its demonstrating the great mass of materials from which he constructed his E'lements—the variety of treatises which prepared the way for that great work whose pre-eminence has now for over 2,000 years been undisputed. In the Elements, Euclid collected all the theorems which had been invented by his predecessors in Egypt and Greece, and digested them into fifteen books, demonstrating and arranging the whole in a very accu rate and perfect manner. Sec EUCLID. Next to Euclid, of the ancient writers whose works are extant, must be named Apollonius Pergneus, who flourished about 230 r.c., and about 100 years later than Euclid, and was called "the great geometrician," on account of his work on the conics, and other ingenious geometrical writings. Much about the same time with Apollonius flourished Archimedes, not less celebrated for his geometrical than for his mechanical inventions. See ARCHIMEDES, and APOLIONIUS or PERGA. It may be mentioned that Apollonius first gave the names of ellipse and hyper bola to two of the conic sections, the third of which had previously been called the 'parabola by Archimedes.
For a long period after the time of Archimedes, we find few names of note in con nection with geometry. We but mention Nicomedes, Hipparchus, and Theodosius of Tripoli. The Greeks, however, never intermitted their attention to the science; they continued it even after their subjugation by the Romans, and we find them producing many excellent geometers after the translation of the Roman empire, and within our era: Ptolemy (q.v.), who died 147 A.D.; Pappus (q.v.), who lived in the time of Theo dosius (379-395 A.D.); Proclus, who lived in the 5th, and Eutocius, in the Gth century. The works of all these writers are still extant. Meantime, the Romans, the dominant race, even in the most flourishing time of the republic, were so ignorant of the science, that, according to Tacitus, they gave the name of mathematicians (q.v.) to those who practiced divination and judicial astrology. As may be supposed, their domination was not favorable to the science, and only one Roman name can be mentioned—viz., Boetldus, who lived towards the close of the 5th c., who attained eminence in geometry; and of his writings, it must be said, as of the Roman literature generally, that they were but compilations and reflections of Greek thought. But if the Roman empire was unfavor able, its downfall, and the consequent inundation of ignorance and barbarism, were still more so. The rise of the Mohammedan power in the 7th c., and the rapid and desolat ing consequences which followed, further hastened the extinction of the Greek sciences. The time now came when those. who devoted themselves to science were
everywhere branded as magicians, and exposed to popular fury. It was in these times that, fortunately for civilization, an asylum was found for the spirit of inquiry in Arabia. An acquaintance with the science of the Hindus prepared the Arabians for the reception of the writings of the Greek astronomers and mathematicians; and the dispersion of the scientific coteries of Alexandria gave to Bagdad many preceptors in the learning of the west. In little more than a century after it took place, the Arabians were the most zealous patrons and cultivators of Greek science; froni the 0th to the 14th centuries, they produced many astronomers, geometricians, etc. ; and through them the mathematical sciences were again restored to Europe towards the close of the 14th c., being first received in Spain and Italy. The revival of ancient literature in Europe, and the dis covery of the art of printing about the middle of the 16th c.,•concurred to diffuse a knowledge of the science of the Greeks, which came into notice with their general literature; and from this date, many names occur of eminent geometricians. During the 16th c., Euclid was held in such estimation, that no attemPts were made to advance the science beyond the point at which he left it. Commentaries and translations of the Rauents,. of Euclid were rife; but till the time of Kepler, DO attempts were made to improve or extend the methods of geometry. Kepler (q.v.) introduced the principle of infinity into geometry. Next, Descartes, seizing the results of Vieta's discoveries in the use of sylibols, invented the new or the analytical algebraical geometry, which vastly extended the domains of the science. It then required but the invention of the calculus to give the science that grand sweep and power which it now possesses. Fora notice of some of the more recent improvements iu geometrical methods, see TRANSVERSALS, POLARS, PROJECTIONS. The reader will also find a very excellent view of the growth of the science in the introduction to Mr. Pott's Euclid (Loam), 1845); also under I lie various names of those mentioned in this article, will be. found fuller notices of their contribu tions to the science. No full,list can be given of the contributors, but it would be unjust not to refer here to Johann .11.111er (called Regiomontanus), Copernicus; Tarfaglia, Vieta, Galileo, Fermat., Roberval, Pascal, Huygbens, Barrow, Newton, the Gregories, Lagrange, Clairaut, Euler, Robert Simson—whose translation of Euclid may be regarded as the standard text in English—Mathew Stewart, Brook Taylor, Maclaurin, Monge, Poncelet, Carnot,.ChaSles, and sir William Hamilton of Dublin. See also QTJATERNIONS.