ARCHIMEDES (287-212 B.C.). A Greek geometrician and mechanician, the greatest mathematician of antiquity. He was born in the State of Syracuse, in the island of Sicily. Ho studied probably under Conon at the Unniversitv of Alexandria, spending the major part of his life in Sicily. He was killed in the sack of Syracuse. The most important among his extant works include three on plane geometry, three on solid geometry, one on arithmetic, and three on mechanics. In the treatise on the measurement of the circle, the value of 7r is given as a number less than 3 1/7 and greater than 3 10/71. He also gave formulas for the area of the circle and the ellipse, and for the sector of a spiral whose equation is r=cθ. His demonstration that the area of a segment of a parabola is two-thirds that of the inclosing parallelogram is the first real example of the quadrature (q.v.) of a curvilinear surface. His method of exhaustion is suggestive of the modern methods of calculus. In the works on solid geometry are treated the volumes of spheroids and con oids. His arithmetical work, known by its Latin title, Arenarias (‘sand-reckoner’), contains his famous attempt to express the amount of sand required to fill the universe. This work has given rise to the conjecture that Archimedes invented a new and powerful system of notation, all knowledge of which perished with the work itself: Besides his work in pure mathematics, Archimedes also made valuable contributions to applied mathematics, including applications of geometry to the theory of machines, as levers, pulleys, and screws. He also improved the methods of finding centres of gravity. In accordance with a wish of Archimedes, Marcellus raised in his honor a tomb, on which was engraved a sphere inscribed in a cylinder. Cicero, in his Tusculan Disputations, gives a charming account of his discovery of the tomb in 75 B.C. The most noted editions of Archimedes' works are those of J. Torelli (Oxford, 1792), J. L. Heiberg (Leipzig, 1881) , and T. L. Heath (Cambridge, 1897) .
ARCHIMEDES, THE PRINCIPLE OF. One of the most important principles in the science of hydrostatics, so called because the discovery of it is generally ascribed to the Syracusan philosopher. It may be thus stated: A body, when entirely surrounded by a fluid, is buoyed up by a force equal to the weight of the fluid it displaces. This is an immediate consequence of the principles of fluid pressure, which prove also that the line of action of the upward force is vertically through the centre of gravity of the displaced fluid. When bodies lighter than water are wholly immersed in it, they displace an amount of water of greater weight than their own, so that if left free to adjust themselves they rise to the surface and float, only as much of their bulk being submerged as will displace a quantity of water weighing the same as themselves. Accordingly, while bodies heavier than water displace, when put into it, their own volume, bodies lighter than water displace, when allowed to float on the surface, their own weight of the fluid. Bodies of the same density as water, according to the principle of Archimedes, have no tendency to rise or sink in it, for the water displaced by them weighs precisely the same as they do. Similar statements may be made with respect to bodies surrounded by other liquids or by gases, e.g., the, atmospheric air. The buoyancy of baloons is an illustration of the principle of Archimedes as applied to the atmosphere. See HYDROSTATICS.
ARCHIMEDES' SCREW (called also SPIRAL PUMP). A machine. for raising water, said to have been invented by Archimedes, during his stay in Egypt, for draining and irrigating the land. Its simplest form consists of a flexible tube bent spirally round a solid cylinder, the ends of which are furnished with pivots, so as to admit of the whole turning round its axis, as is shown in Fig. 1. The machine is placed in an inclined position, so that the lower mouth of the tube may dip below the surface of the water to be raised. The lowest bend of the tube will be filled with water, and if now the handle be made to turn in the direction of the hands of a watch, the mouth of the spiral tube will be raised above the surface; and the water inclosed in the tube, having no means of escape, will flow within it until after one revolution, it will occupy the second bend. The first bend has meanwhile received a second charge, which, after a second revolution, flows up into the second bend and takes the place of the first charge, which has now moved up to the third bend. Whn, therefore, as many revolutions of the cylinder have been made as there are turns in the spiral tube, each of the lower bends will be filled with water; and in the course of another revolution, there being no higher bend for the water of the first charge to occupy, it will flow out of the tube by its upper mouth. At each succeeding revolution the lowest bend will be charged and the highest discharged. It will be seen that there may be room to dispose a second tube side by side with the first, round the cylinder, in which case the screw would be called double-threaded. In the ordinary construction of these machines the cylinder itself is hollowed out into a double or triple-threaded screw, and inclosed in a watertight case, which turns round with it, the space between the threads supplying the place of tubes. It is sometimes found convenient to fix the exterior envelope and to make the screw work within it, the outer edge of the latter being as close as possible to the former without actual contact, as is shown in Fig. 2. This modification of the Archimedes' Screw receives the name of "water screw," and frequently of "Dutch screw," from its use in Holland for draining low grounds.
The New International Encyclopaedia Vol. II (New York: Dodd, Mead & Co., 1920) 56-57.