(7) On Floating Bodies (Περὶ ὀχουμένων) is a treatise in two books, the first of which establishes the general principles of hydrostatics, and the second discusses with the greatest completeness the positions of rest and stability of a right segment of a paraboloid of revolution floating in a fluid.
(8) The Psammites (Ψαμμίτης, Lat. Arenarius, or sand reckoner), a small treatise, addressed to Gelo, the eldest son of Hiero, expounding, as applied to reckoning the number of grains of sand that could be contained in a sphere of the size of our “universe,” a system of naming large numbers according to “orders” and “periods” which would enable any number to be expressed up to that which we should write with 1 followed by 80,000 ciphers!
(9) A Collection of Lemmas, consisting of fifteen propositions in plane geometry. This has come down to us through a Latin version of an Arabic manuscript; it cannot, however, have been written by Archimedes in its present form, as his name is quoted in it more than once.
Lastly, Archimedes is credited with the famous Cattle-Problem, enunciated in the epigram edited by G.E. Lessing in 1773, which purports to have been sent by Archimedes to the mathematicians at Alexandria in a letter to Eratosthenes. Of lost works by Archimedes we can identify the following: (1) investigations on polyhedra mentioned by Pappus; (2) Άρχαί, Principles, a book addressed to Zeuxippus and dealing with the naming of numbers on the system explained in the Sand Reckoner; (3) Περὶ ζυγῶν, On balances or levers; (4) Κεντροβαρικά, On centres of gravity; (5) Κατοπτρικά, an optical work from which Theon of Alexandria quotes a remark about refraction; (6) Έφόδιον, a Method, mentioned by Suidas; (7) Περὶ σφαιροποιἶας, On Sphere-making, in which Archimedes explained the construction of the sphere which he made to imitate the motions of the sun, the moon and the five planets in the heavens. Cicero actually saw this contrivance and describes it (De Rep. i. c. 14, §§ 21-22).
Bibliography.—The editio princeps of the works of Archimedes, with the commentary of Eutocius, is that printed at Basel, in 1544, in Greek and Latin, by Hervagius. D. Rivault’s edition (Paris, 1615) gave the enunciations in Greek and the proofs in Latin somewhat retouched. A Latin version of them was published by Isaac Barrow in 1675 (London, 4to); Nicolas Tartaglia published in Latin the treatises on Centres of Gravity, on the Quadrature of the Parabola, on the Measurement of the Circle, and on Floating Bodies, i. (Venice, 1543); Trojanus Curtius published the two books on Floating Bodies in 1565 after Tartaglia’s death; Frederic Commandine edited the Aldine edition of 1558, 4to, which contains Circuli Dimensio, De Lineis Spiralibus, Quadratura Paraboles, De Conoidibus et Spheroidibus, and De numero Arenae; and in 1565 the same mathematician published the two books De iis quae vehuntur in aqua. J. Torelli’s monumental edition of the works with the commentaries of Eutocius, published at Oxford in 1792, folio, remained the best Greek text until the definitive text edited, with Eutocius’ commentaries, Latin translation, &c., by J.L. Heiberg (Leipzig, 1880-1881) superseded it. The Arenarius and Dimensio Circuli, with Eutocius’ commentary on the latter, were edited by Wallis with Latin translation and notes in 1678 (Oxford), and the Arenarius was also published in English by George Anderson (London, 1784), with useful notes and illustrations. The first modern translation of the works is the French edition published by F. Peyrard (Paris, 1808, 2 vols. 8vo.). A valuable German translation with notes, by E. Nizze, was published at Stralsund in 1824. There is a complete edition in modern notation by T.L. Heath (The Works of Archimedes, Cambridge, 1897). On Archimedes himself, see Plutarch’s Life of Marcellus.
(T. L. H.)
ARCHIMEDES, SCREW OF, a machine for raising water, said to have been invented by Archimedes, for the purpose of removing water from the hold of a large ship that had been built by King Hiero II. of Syracuse. It consists of a water-tight cylinder, enclosing a chamber walled off by spiral divisions running from end to end, inclined to the horizon, with its lower open end placed in the water to be raised. The water, while occupying the lowest portion in each successive division of the spiral chamber, is lifted mechanically by the turning of the machine. Other forms have the spiral revolving free in a fixed cylinder, or consist simply of a tube wound spirally about a cylindrical axis. The same principle is sometimes used in machines for handling wheat, &c. (see [Conveyors]).
ARCHIPELAGO, a name now applied to any island-studded sea, but originally the distinctive designation of what is now generally known as the Aegean Sea (Αἰγαῖον πέλαγος), its ancient name having been revived. Several etymologies have been proposed: e.g. (1) it is a corruption of the ancient name, Egeopelago; (2) it is from the modern Greek, Άγιο πέλαγο, the Holy Sea; (3) it arose at the time of the Latin empire, and means the Sea of the Kingdom (Archi); (4) it is a translation of the Turkish name, Ak Denghiz, Argon Pelagos, the White Sea; (5) it is simply Archipelagus, Italian, arcipelago, the chief sea. For the Grecian Archipelago see [Aegean Sea]. Other archipelagoes are described in their respective places.