of his residence, or the ara in which he flourished. His chief merit seems to have consisted in reducing to order the scattered parts of the system of ancient mathematics, and in presenting the fundamental principles established by former geometricians in their natural order, adding to them others invented by himself. Professor Playfair, when adverting to the progress made by the Greeks in geometrical science, bears the following honourable testimony to the "author of the Elements." "The elementary truths were connected by Euclid, into one great chain, beginning from the axioms and extending to the five regular solids, the whole digested into such admirable order, and explained with such clearness and precision, that no similar work of superior excellence has appeared, even in the present advanced state of mathematical science."--Playfair's Dissert ap. Encyc. Brit. New Sup. Vol. II. p. 3.

"No book of science," says a modern French writer, "has ever met with success equal to that of Euclid's Elements. They have continued to be taught exclusively in every mathematical school for many centuries; have been translated into almost all languages, and have been commented upon by numerous celebrated geometricians-a sure proof of their transcendent excellence."--(Bossut's History of Mathematics.) Besides this, his principal work, Euclid wrote several other mathematical treatises, some of which are still extant, though others, mentioned by Proclus, are lost.

76. Next in chronological order, but far superior in genius and science, stands the celebrated, the

unrivalled ARCHIMEDES. The scientific discoveries of this great philosopher will more properly claim our attention, when reviewing the history of Physics, or Natural Philosophy, among the Greeks. At present we shall only advert to him, as unquestionably the first of ancient mathematicians. He flourished at Syracuse about A. c. 250, and spent his whole life in the most unwearied and successful endeavours to promote the advancement of the sciences, both abstract and practical. "He assailed," says Playfair, "the most difficult problems in geometry, and by means of the method of exhaustion,* demonstrated many curious and important theorems with regard to the lengths and areas of curves, and the contents of solids." He was the first who discovered the relation between the circumference of a circle to its diameter, not indeed with geometrical strictness, (for this is found to baffle even modern science,) but by a method of approximation admirable in its kind, and which has served as a

* This name was given by the ancients to that process of indirect demonstration, by which a great variety of difficult questions were solved relative to the limitation of figures inscribed on curves, or circumscribed round them, and the proportion of those limits to the areas of the curves with which they are connected. Professor Playfair, after having given a succinct but beautiful explanation of this branch of mathematics, adds, "Few things more ingenious than this method have been devised; and nothing could be more conclusive than the demonstrations resulting from it; but it laboured under two very considerable defects; viz. the long and difficult process by which those demonstrations were obtained, and its indirect form giving no insight into the principle on which the investigation was founded."---Playf. Diss. ut sup. pp. 6, 7.

model for all subsequent investigations of a similar kind. His most celebrated work, (which fully proves the transcendent superiority of his genius to the proudest attainments of ordinary men, and renders him not unworthy of being placed by the side of the illustrious Newton,) is that on the dimensions and quadrature of the circle: besides which, he wrote (on mathematical subjects) treatises on the sphere and cylinder,-on spheroids and conoids,-of spiral lines,-of the quadrature of the parabola, &c. &c., irrespective of his works on natural philosophy, to be noticed hereafter. To the preceding summary of the geometrical labours of Archimedes, it may be added, that he extended and more clearly demonstrated the use of geometrical analysis, the principles of which were first discovered by the disciples of Plato. Eratosthenes and Conon were his contemporaries, either of whom would have acquired, in any other age, great celebrity in this department of science, but their fame was completely eclipsed by the dazzling lustre of the great philosopher of Syracuse. Archimedes requested, in his last moments, that a sphere inscribed in a cylinder might be engraved on his tomb, to perpetuate the memory of what he deemed his most glorious discovery. This was done, but the Sicilians so little honoured his memory, that the place of his sepulture was lost, till discovered by Cicero, when Quæstor of Sicily, by these symbols and a Greek inscription at their base, nearly two hundred years after his death.

77. APOLLONIUS, a native of Perga, next occurs,

after an interval of about 50 years (A. c. 200,) in the series of ancient geometricians. As he is known to have studied and taught mathematics at Alexandria, it is not improbable that he continued the school founded in that city by Euclid. His contemporaries styled him, and not without reason, the Great Geometrician, an appellation confirmed by the suffrages of posterity, who place him next to Archimedes in mathematical attainments. He is known to have composed a great number of works on the higher branches of geometry, all of which have perished, except a few fragments and a Treatise, nearly entire, on the Conic Sections. This celebrated treatise was divided into eight books, the first four of which are still extant in Greek; the three next were only known by means of an Arabic translation and Latin version, effected during the middle ages, until translated into English by Dr. Halley, the astronomer, towards the beginning of the eighteenth century; and the eighth is entirely lost. On this work, Playfair remarks, that "his elaborate and profound researches on that species of curves, which, after the circle, are the most simple and important in geometry, laid the foundation of discoveries which were to illustrate very distant ages." Numerous have been the commentators on this celebrated work, both ancient and modern, of which the principal are, Eutocius of Ascalon, Pappus, and Commandine. Apollonius is believed by many to have been the author of those propositions contained in some editions of Euclid's Elements, and enumerated as the fourteenth and fifteenth books.

The age of Archimedes and Apollonius constituted the most brilliant æra in the history of ancient mathematics. After these great men disappeared, we meet with no geometricians of the first order, and scarcely any that can be considered even second-rate mathematicians. A few, however, may be cursorily mentioned as having contributed in some degree, if not to the advancement, yet at least to the preservation of geometrical science, though it will be seen that they appeared but at distant intervals.

78. Theodosius, who flourished about A. c. 60, is known as the author of a Treatise on Spherics, which may be considered as having prepared the way for the discovery of spherical trigonometry; none of the former geometricians having proceeded beyond plane trigonometry, or the measurement of heights and distances, by the sides and angles of a triangle. The Demonstrations of Theodosius are said to have been expressed with the utmost accuracy and elegance. Menelaus, who flourished about A. D. 55, pursued the same track, and, following up the principles of his predecessor, made considerable progress in spherical trigonometry. His work on Spherical Triangles is said to have been a laborious and learned performance, which contributed to the advancement of astronomical science. After a long interval of more than 300 years, we arrive at the æra of Pappus and Diocles, both of whom flourished about A. D. 385. Pappus is chiefly known as an industrious and valuable commentator on the works of those who had preceded him. "The mathema