[Sidenote: The writings and works of Archimedes.]
Great as is the fame of Euclid, it is eclipsed by that of Archimedes the Syracusan, born B.C. 287, whose connection with Egyptian science is not alone testified by tradition, but also by such facts as his acknowledged friendship with Conon of Alexandria, and his invention of the screw still bearing his name, intended for raising the waters of the Nile. Among his mathematical works, the most interesting, perhaps, in his own estimation, as we may judge from the incident that he directed the diagram thereof to be engraved on his tombstone, was his demonstration that the solid content of a sphere is two-thirds that of its circumscribing cylinder. It was by this mark that Cicero, when Quæstor of Sicily, discovered the tomb of Archimedes grown over with weeds. This theorem was, however, only one of a large number of a like kind, which he treated of in his two books on the sphere and cylinder in an equally masterly manner, and with equal success. His position as a geometer is perhaps better understood from the assertion made respecting him by a modern mathematician, that he came as near to the discovery of the Differential Calculus as can be done without the aid of algebraic transformations. Among the special problems he treated of may be mentioned the quadrature of the circle, his determination of the ratio of the circumference to the diameter being between: 3·1428 and 3·1408, the true value, as is now known, being 3·1416 nearly. He also wrote on Conoids and Spheroids, and upon that spiral still passing under his name, the genesis of which had been suggested to him by Conon. In his work entitled "Psammites" he alludes to the astronomical system subsequently established by Copernicus, whose name has been given to it. He also mentions the attempts which had been made to measure the size of the earth; the chief object of the work being, however, to prove not only that the sands upon the sea-shore can be numbered, but even those required to fill the entire space within the sphere of the fixed stars; the result being, according to our system of arithmetic, a less number than is expressed by unity followed by 63 ciphers. Such a book is the sport of a geometrical giant wantonly amusing himself with his strength. Among his mathematical investigations must not be omitted the quadrature of the parabola. His fame depends, however, not so much on his mathematical triumphs as upon his brilliant discoveries in physics and his mechanical inventions. How he laid the foundation of Hydrostatics is familiar to everyone, through the story of Hiero's crown. A certain artisan having adulterated the gold given him by King Hiero to form a crown, Archimedes discovered while he was accidentally stepping into a bath, that the falsification might be detected, and thereby invented the method for the determination of specific gravity. From these investigations he was naturally led to the consideration of the equilibrium of floating bodies; but his grand achievement in the mechanical direction was his discovery of the true theory of the lever: his surprising merit in these respects is demonstrated by the fact that no advance was made in theoretical mechanics during the eighteen centuries intervening between him and Leonardo da Vinci. Of minor matters not fewer than forty mechanical inventions have been attributed to him. Among these are the endless screw, the screw pump, a hydraulic organ, and burning mirrors. His genius is well indicated by the saying popularly attributed to him, "Give me whereon to stand, and I will move the earth," and by the anecdotes told of his exertions against Marcellus during the siege of Syracuse; his invention of catapults and other engines for throwing projectiles, as darts and heavy stones, claws which, reaching over the walls, lifted up into the air ships and their crews, and then suddenly dropped them into the sea; burning mirrors, by which, at a great distance, the Roman fleet was set on fire. It is related that Marcellus, honouring his intellect, gave the strictest orders that no harm should be done to him at the taking of the town, and that he was killed, unfortunately, by an ignorant soldier--unfortunately, for Europe was not able to produce his equal for nearly two thousand years.