Superficies; Loci ad Superficiem ; four books on Conic Sections; and treatises on other branches of the Mathematics.

Archimedes, one of the greatest geometricians of antiquity, was the first who approximated to the ratio of the cir

P Archimedes was born at Syracuse, and related to Hiero, King of Sicily: he was remarkable for his extraordinary application to mathematical studies, but more so for his skill and surprising inventions in Mechanics. He excelled likewise in Hydrostatics, Astronomy, Optics, and almost every other science; he exhibited the motions of the heavenly bodies in a pleasing and instructive manner, within a sphere of glass of his own contrivance and workmanship; he likewise contrived curious and powerful machines and engines for raising weights, hurling stones, darts, &c. launching ships, and for exhausting the water out of them, draining marshes, &c. When the Roman Consul, Marcellus, besieged Syracuse, the machines of Archimedes were employed: these showered upon the enemy a cloud of destructive darts, and stones of rast weight and in great quantities; their ships were lifted into the air by his cranes, levers, hooks, &c. and dashed against the rocks, or precipitated to the bottom of the sea; nor could they find safety in retreat: his powerful burning glasses reflected the condensed rays of the sun upon them with such effect, that many of them were burned. Syracuse was however at last taken by storm, and Archimedes, too deeply engaged in some geometrical speculations to be conscious of what had happened, was slain by a Roman soldier. Marcellus was grieved at his death, which happened A. C. 210, and took care of his funeral. Cicero, when he was Questor of Sicily, discovered the tomb of Archimedes overgrown with bushes and weeds, having the sphere and cylinder engraved on it, with an inscription which time had rendered illegible.

His reply to Hiero, who was one day admiring and praising his machines, can be regarded only as an empty boast. "Give me," said the exulting philosopher," a place to stand on, and I will lift the earth.” (Aos μoi #8 5w, xai any yn nivnow.) This however may be easily proved to be impossible; for, granting him a place, with the simplest machine, it would require a man to move swifter than a cannon shot during the space of 100 years, to lift the earth only one inch in all that time.-Hiero ordered a golden crown to be made, but suspecting that the artists had purloined some of the gold and substituted base metal in its stead, he employed our philosopher to detect the cheat; Archimedes tried for some time in vain, but one day as he went into the bath, he observed that his body excluded just as much water as was equal to its bulk: the thought immediately struck him that this discovery had furnished ample data for solving his difficulty; upon which he leaped out of the bath, and ran through the streets homewards, crying out, signxa! svo̟nxa! I have found it! I have found it!-The best edition of his works is that of Torelli, edited at the Clarendon Press, Oxford, fol. 1792, by Dr. Robertson, Savilian Professor of Astronomy.

cumference of a circle to its diameter, A. C. 250: this he effected by circumscribing about, and inscribing in the circle regular polygons of 96 sides, and making a numerical calculation of their perimeters; by means of this process he made the ratio as 22 to 7, which is a determination near enough the truth for common practical operations, where great exactness is not required, and has the advantage of being expressed by small numbers. He was the next after Hippocrates, who squared a curvilineal space; he applied himself with ardour to the investigation of the measures, proportions, and properties of the conic sections, spirals, cylinders, cones, spheres, conoids, spheroids, &c. On these subjects the following works of his are still extant, viz. two books on the Sphere and Cylinder; and treatises on the Dimensions of the Circle; on Spirals; on Conoids and Spheroids; and on the Centres of Gravity.

The next geometer of note after Archimedes, was Apollonius Pergæus, A. C. 230: this great man studied for a long time in the schools of Alexandria under the disciples of Euclid, and was the author of several valuable works on Geometry, which were so much esteemed, that they procured him the honourable title of the great Geometrician. His principal work, and the most perfect of the kind among the ancients, is his treatise on the Conic Sections, in eight books; seven only of these have been preserved, the four first in the original Greek, and the 5th, 6th, and 7th in an Arabic version ".

According to Pappus and Eutocius, the following works were likewise written by Apollonius, viz. 1. The Section of a Spaće. 2. The Section of a Ratio. 3. The Determinate Section. 4. The Inclinations. 5. The Tangencies, and 6. The Plane Loci; each of these treatises consisting of two books. Pappus has left us some particulars of the above works, which are all concerning them that now remain; but from these scanty materials, many restorations have been made, viz. by Vieta, Snellius, Ghetaldus, Fermat, Schooten, Alex. Anderson, Halley, Simson, Horsley, Lawson, Wales, and Burrow. The best edition of the Conics of Apollonius is that by Dr. Halley, fol. Oxon. 1710.

The age of Archimedes and Apollonius has with justice been stiled, THE GOLDEN AGE OF ANCIENT GEOMETRY, as the science never acquired so great a degree of brilliancy at other period of the Grecian history.

any

The duplication of the cube, quadrature of the circle, trisection of an angle, &c. were problems of which the ancients never lost sight; many of the propositions in the Elements, particularly prop. 27, 28, and 29 of the sixth book, are intimately connected with the solution, and probably originated in the attempts to obtain it. The application of the conic sections to this purpose by Menechmus, has been already noticed about the same time Dinostratus invented the quadratrix, a mechanical curve, possessing the triple advantage of trisecting and multiplying an angle, and squaring the circle. The conchoid of Nicomedes, who flourished A. C. 250, has been applied by both ancient and modern geometers equally to the trisection, finding two mean proportionals, and the construction of other solid problems; for which purposes this curve has been preferred by Archimedes, Pappus, and Newton, to any other. (See Newton's Arithmetica Universalis, p. 288, 289.) The cissoid, another curve, being an improvement on the conchoid, was invented by Diocles about 150 years before Christ.

Hero, Dositheus, Eratosthenes, and Hypsicles, who flourished in the, second century before Christ, and Geminius who flourished in the first, were all eminent for their skill in Geometry: indeed the science continued to be cultivated with ardour by a numerous list of geometricians produced by the Alexandrian school, until that famous seat of learning fell a prey to the blind and merciless bigotry of the Arabs. The first who wrote on the sphere and its circles to any consi

Here the eighth book is supplied, and likewise is added, Serenus' treatise on the Section of the Cylinder and Cone, printed from the original Greek, with a Latin translation,

derable extent, at least whose works have been preserved, was Theodosius, A. C. 60: this work, in which the propositions are demonstrated with equal strictness and elegance, forms the basis of spherical Trigonometry, as practised by the moderns. About the same time, or shortly after, Menelaus wrote his treatise on Chords, which is lost; but his work on Spherical Triangles, containing the construction and trigonometrical method of resolving them, according to the ancient practice, is still extant. We are particularly indebted to Pappus, A. D. 380, and Proclus, A. D. 430, for their laborious researches; many particulars relating to the sciences of the Greeks would have been lost to posterity, but for their writings: the former was an eminent mathematician of Alexandria, and author of several learned and useful works, particularly eight books of Mathematical Collections, of which the first and part of the second are wanting. These books contain a great variety of useful information relating to Geometry, Arithmetic, Mechanics, &c. with the solution of problems of different sorts. Proclus likewise studied at Alexandria, and afterwards presided over the Platonic school at Athens; he wrote, besides many other works, Commentaries on the first book of Euclid, on the Mathematics, on Philosophy; also a treatise De Sphæra, which was published by Dr. Bainbridge, Savilian Professor of Geometry at Oxford, in 1620. The writings of the Greek geometricians were translated and commented on by several learned Arabians, but the improvements they introduced were chiefly of the practi eal kind; among these may be mentioned the fundamental propositions of Trigonometry, in which, by the substitution of sines instead of the chords, and other convenient abridgements, they greatly simplified the theory and solutions of plane and spherical triangles. These improvements are aseribed to Mahomet Ebn Musa, a geometer of whom there still exists a work on Plane and Spherical Figures. We like

wise possess a work on Surveying, written by Mahomet of Baghdad, which some modern authors have ascribed to Euclid.

A few learned men, famous for their skill in Geometry, flourished in the West during the fifteenth century. Of these the chief were the Cardinals Bessarion and Cusa, Purbach, Nicholas Oresme, Bianchini, George of Trabezonde, Lucas de Burgo, Schonerus, Walther, and Regiomontanus; the latter wrote a treatise on Plane and Spherical Trigonometry, A. D. 1464; in which, among other improvements, he introduced the use of the tangents, and applied Algebra to the solution of geometrical problems; this is the more surprising, as it occurred several years before the publication of any of the works of De Burgo, who is generally supposed to have been the introducer of Algebra into Europe..

On the revival of learning in Europe about the beginning of the sixteenth century, the study of Geometry began to be cultivated with great attention; the works of the Greek geometricians were eagerly sought after and translated into Latin or Italian, and served as guides to those who had a taste for that correct reasoning, for which the ancient Geometry is so justly famed, or were desirous of availing themselves of the knowledge of its application and use, as connected with the necessary business of life. As early as 1522, John Werner, a celebrated astronomer of Nuremberg, published some tracts on the Conic Sections, and on other geometrical subjects. Tartalea composed a treatise on Arithmetic, Algebra, Geometry, Mensuration, &c. entitled, Trattato di Numeri et Misure, 1556, being the first modern work

Nicolas De Cusa was born of poor parents, A. D. 1401; his application to learning and his personal merit, however, raised him to the rank of bishop and cardinal: his claim to the honour of having squared the circle was ably refuted by Regiomontanus; nevertheless he was a man of very extraordinary parts, and excelled in the knowledge of law, divinity, natural philosophy, and geometry, on which subjects he is said to have written some excellent treatises. He died in 1464.