by geometrical principles, the distance of vessels from the shore. Thales introduced the study of this science into Greece. He has the credit of making several discoveries of properties of the circle, and others respecting the comparison of triangles. The only discovery known to be his, however, is the proposition that all the angles in a semicircle are right angles. It is said, that, on the occasion of this last discovery, he offered in gratitude a sacrifice to the Muses. Thales founded a school in Greece, called the Ionian School. The first elementary treatise of geometry is said to have been composed by Anaximander, a disciple of Thales, and his successor in his school. Anaximander was succeeded by Anaxagoras, who is the first recorded to have attempted the Quadrature of the Circle. Pythagoras was another disciple of Thales. He travelled into Egypt and India in pursuit of knowledge. He settled in Tarentum, in Italy, where he founded the celebrated Pythagorean School (550 B.c.) He discovered the important proposition respecting the squares described on the sides of a right-angled triangle. On this occasion he is said to have sacrificed to the Muses a hundred oxen; a statement which is very improbable, both on account of the limited fortune of philosophers, and his philosophical creed respecting the transmigration of souls, which prohibited not only the destruction of animals, but even the use of animal food. He. was also the first that observed that relation of lines called their incommensurability, and that invented the regular solids, afterwards called the Platonic bodies. Hippocrates of Chios discovered the Quadrature of the Lune which has been called his lune, and he first proved that the solution of the problem of the Duplication of the Cube depends upon the finding of two mean proportionals; a question which arose from the request of the oracle at Delos to double the cubic altar of Apollo. He was the second who composed a treatise of geometry, which has been lost. The origin of the Platonic School (400 B.C.) is considered to be one of the most important epochs in the history of the science. Its founder, Plato, is the reputed inventor of the method of Geometrical Analysis ; and to this school we are indebted for the theory of the Conic Sections and of Geometrical Loci. These discoveries were probably the result of researches instituted for a very different purpose, namely, the duplication of the cube and the trisection of an angle-problems that were at this time of great celebrity; no less than eleven of the ancient geometers gave a solution of the former problem. The partiality of Plato for this science is evinced by his placing an inscription over the door of his school, forbidding those to enter who were ignorant of geometry. Archytus of Tarentum, who belonged to the Pythagorean School, was the eighth successor of its founder, and the contemporary of Plato, whom he frequently visited. He solved the problem of two mean proportionals, and made several other discoveries by means of the geometrical analysis, which he learned from Plato. He was also well acquainted with the principlesof mechanics. His flying pigeon is a proof of his reputation for mechanical invention, and he is said to be the inventor of the pulley and the screw. Leon of Neoclis, a philosopher, who was one of Plato's disciples, remodelled and extended the elements of geometry, to adapt them to the advanced state of the science, which had received considerable improvements from the time of Hippocrates. Eudoxus of Cnidus, a friend of Plato, and who is stated by Cicero to have been bred in the Egyptian School, is said by Archimedes to have discovered the contents of the pyramid and cone; and some consider him as the inventor of curve lines in general, and of the theory of proportion in Euclid's Elements of Geometry. Menæchmus improved the Conic Sections, and gave two solutions of the problem of two mean proportionals. Dinostratus discovered a property of the Quadratrix, a curve which was no doubt invented for the purpose of solving the problem of the trisection of an angle, and the invention of which is ascribed to Hippias of Elis. Xenocrates, one of Plato's successors, composed works on arithmetic and geometry. Aristæus composed a treatise on Solid Loci, and another on Conic Sections. He had thus acquired great proficiency in what was called the sublime geometry; and he is said to have been the instructor of Euclid. The Peripatetic School, founded by Aristotle (about 350 B. C.), was not so devoted as the Platonic to the study of mathematics. But Theophrastus, the successor of Aristotle, composed several mathematical works, and a complete history of the science, which have unfortunately been lost. Geometry was cultivated with much attention in the Alexandrian school, which was established by Ptolemy Lagus (300 B. c.), who was a great patron of learned men. The celebrated geometrician Euclid, whose native place is unknown, but who had first studied geometry at Athens, went from Greece to Alexandria, where he settled during the time of the first Ptolemy. He composed treatises on various branches of mathematics, but his most eminent work is his Elements of Geometry. He was no doubt acquainted with the treatises of Hippocrates and Leon, but the fact of his treatise entirely supplanting theirs among the ancients, is a sufficient proof of their inferiority. Euclid has shown so much judgment in the composition of this treatise, that, notwithstanding the great additions made to geometry since his time, it has continued for upwards of two thousand years to sustain the highest reputation as an elementary treatise. He also composed a work called Data, another on Loci ad Superficiem, relating to curves of double curvature, and one on Porisms. The two latter have been lost; but, from the account given by Pappus, it appears that the last was a profound work relating to the analysis of the most abstruse and general problems. Archimedes (born 287 B. c.) was the most celebrated of all the ancient mathematicians. He enjoyed the most extensive, and also the most popular reputation ; for to his abstract researches he added several striking mechanical inventions ; and he happened also to be placed in circumstances that made their value be appreciated. He began his career of geometrical discovery at that point at which ordinary minds are disposed to stop. He cultivated all the branches of mathematical science, but particularly the most difficult branches of geometry, relating to the areas of curves and the sections of curve surfaces. He employed, and with great address, in most of his discoveries, that subtle invention of the ancients, the Method of Exhaustions. He discovered that beautiful relation between a sphere and its circumscribing cylinder, that the capacity and surface of the former are respectively two-thirds of those of the latter; and also, that the surfaces of any two of their corresponding zones, formed by planes perpendicular to the axis of the cylinder, are equal. He also proved that the area of a circle is equal to half the rectangle contained by its circumference and radius, and he showed how to approximate as near as may be required to the quadrature of the circle, and found that of the parabola, which is memorable as the first example of the complete quadrature of a curve. He discovered several properties of conoids and spheroids, and of the spiral invented by Conon, but which, in honour of him who de monstrated its properties, is called the spiral of Archimedes. The writings of Archimedes prove to what an extent the sagacity of a mind like his could carry the ancient method of investigation ; but they also prove the fact that there is a limit to its application. From the number and importance of his discoveries, though not to be compared to those of Newton, Archimedes has been called the Newton of antiquity. Eratosthenes, a native of Cyrene, was called to the Alexandrian School by Ptolemy Euergetes, who made him his librarian on account of his extensive and general attainments in learning. He has given a construction for the duplication of the cube. His knowledge of geography and astronomy enabled him to discover, for the first time on record, a method for measuring the circumference of the earth. Apollonius of Perga, in Pamphylia, was nearly a contemporary of Archimedes. He studied in the Alexandrian School in the time of Ptolemy Philopater. He greatly improved the conic sections, and composed several geometrical treatises, of which only one has reached us entire, namely, that on the Section of a Ratio ; it is in Arabic, and was translated by Dr Halley. His other treatises were, on the Section of Space; on Determinate Section ; on Tangencies; on Inclinations; on Plane Loci. He was an inventive and profound geometer, and his discoveries were so highly esteemed, that he was called the Great Geometer. The dispositions of Apollonius and Euclid were very different; the former was remarkable for vanity, jealousy of the merits B of others, and detraction; the latter, for mildness, modesty, ; and benevolence. Passing over the names of several geometers of little note, who lived about this period, we may merely mention Theodosius (50 B. c.), who composed an excellent work on Spherics, which is considered to be one of the most valuable remains of the ancient geometry; and Dionysiodorus, who found a solution of the problem of Archimedes, to cut a hemisphere in a given ratio by a plane parallel to its base, a problem that required considerable skill. Menalaus (second century) was the author of a treatise on trigonometry, and another on spherics. Pappus and Theon of Alexandria (380 A.D.) were celebrated for their commentaries and annotations on the ancient geometry. To Pappus we are indebted for an excellent work, his Mathematical Collections, which consisted of eight books, of which the first and half of the second have perished. It is chiefly the elementary works of the ancient geometers that have been preserved; their more abstruse writings have either perished, or are only known by the accounts or abridgements of them that are given by Pappus. In this work he collected several detached original researches ; and the preface to his seventh book is particularly valuable, as it has preserved several works.on geometrical analysis, which would otherwise have been unknown. Hypatia, the accomplished and learned daughter of Theon, was so great a proficient in geometrical science, that she was considered qualified to be her father's successor in the Alexandrian School. But with all her merits, her fate was tragical, for she fell a victim to an infuriated and fanatical mob. The philosopher Proclus, who was the head of the Platonic school, wrote a prolix commentary on the first book of Euclid, containing many curious observations on the metaphysics, and many interesting facts in the history of the science. Diocles invented the cissoid for the purpose of finding two mean proportionals, and Sporus gave a solution of this problem. Eutocius also gives the former the credit of solving the problem of Archimedes respecting the section of a hemisphere. The age of important discovery was long past before the |