Archimedes, the most celebrated of ancient mathematicians, was born at Syracuse about 287 B.C. He is said to have been a kinsman of King Hiero, though he seems to have held no public office, but to have devoted himself entirely to science. In regard to mathematics, we cannot estimate fully the merits of Archimedes without an exact knowledge of the state of the science as he found it; we know, however, that he enriched it with discoveries of the highest importance, on which modern mathematicians have founded their methods of measuring curved surfaces and solids. Euclid considers only a few curved figures in relation one to another, but without comparing them with rectilinear surfaces and solids. The theorems necessary to this transition are laid down by Archimedes in his treatises 'on the Sphere and Cylinder,' 'on Spheroids and Conoids,' and 'on the Measurement of the Circle.' His demonstration that the area of a segment of a parabola is two-thirds of the inclosing parallelogram, is the first real example of the Quadrature (q.v.) of a curvilinear space. In his treatise on spirals, he rises to yet higher investigations, which, however, are not very easily understood, even by masters of the subject.
Archimedes is the only one of the ancients that contributed anything of real value to the theory of mechanics and to hydrostatics. He first established the truth that a body plunged in a fluid loses as much of its weight as is equal to the weight of an equal volume of the fluid. It was by this law that he determined how much alloy the goldsmith, whom Hiero had commissioned to make a crown of pure gold, had fraudulently mixed with the metal. The solution of the problem suggested itself to him as he was entering the bath, and he is reported to have been so overjoyed as to run home naked, exclaiming: 'Eureka! Eureka!' (I have found it! I have found it!) His boast, that if he had a fulcrum or stand-point, he could move the world, betrays the enthusiasm with which the extraordinary effects of his newly invented machines ascribed to him. Among the numerous inventions ascribed to him is that of the endless screw, and the Archimedes Screw (q.v.). During the siege of Syracuse by the Romans, he exerted all his ingenuity in the defence of the city. Polybius, Livy, and Plutarch speak with astonishment of the machines with which he opposed the attacks of the enemy. The improbable story of his having set fire to the ships by means of mirrors, rests on later narratives. When the Romans took the city by surprise (212 B.C.), Archimedes, according to the tradition, was sitting in the public square lost in thought, with all sorts of geometrical figures before him drawn in the sand. As a Roman soldier rushed upon him, he called out to him not to spoil the circle! But the rude warrior cut him down. According to his own direction, a cylinder inclosing a sphere was engraved upon his tombstone, in commemoration of his discovery of the relation between these solids—a discovery on which he set particular value. When Cicero was in Sicily as quaestor (75 B.C.), he found the tomb hid among briars. His extant works, written in Doric Greek, were edited by Torelli (Oxf. 1792), and Heiberg, with a Latin translation (3 vols. Leip. 1880-81).