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 with the dimensions of spheres, cones, “solid rhombi” and cylinders, all demonstrated in a strictly geometrical method. The first book contains forty-four propositions, and those in which the most important results are finally obtained are: 13 (surface of right cylinder), 14, 15 (surface of right cone), 33 (surface of sphere), 34 (volume of sphere and its relation to that of circumscribing cylinder), 42, 43 (surface of segment of sphere), 44 (volume of sector of sphere). The second book is in nine propositions, eight of which deal with segments of spheres and include the problems of cutting a given sphere by a plane so that (a) the surfaces, (b) the volumes, of the segments are in a given ratio (Props. 3, 4), and of constructing a segment of a sphere similar to one given segment and having (a) its volume, (b) its surface, equal to that of another (5, 6).

(2) The Measurement of the Circle ( ) is a short book of three propositions, the main result being obtained in Prop. 2, which shows that the circumference of a circle is less than 3$1⁄7$ and greater than 3$10⁄71$ times its diameter. Inscribing in and circumscribing about a circle two polygons, each of ninety-six sides, and assuming that the perimeter of the circle lay between those of the polygons, he obtained the limits he has assigned by sheer calculation, starting from two close approximations to the value of √3, which he assumes as known (265/153 < √3 < 1351/780).

(3) On Conoids and Spheroids ( ) is a treatise in thirty-two propositions, on the solids generated by the revolution of the conic sections about their axes, the main results being the comparisons of the volume of any segment cut off by a plane with that of a cone having the same base and axis (Props. 21, 22 for the paraboloid, 25, 26 for the hyperboloid, and 27-32 for the spheroid).

(4) On Spirals ( ) is a book of twenty-eight propositions. Propositions 1-11 are preliminary, 13-20 contain tangential properties of the curve now known as the spiral of Archimedes, and 21-28 show how to express the area included between any portion of the curve and the radii vectores to its extremities.

(5) On the Equilibrium of Planes or Centres of Gravity of Planes ( ). This consists of two books, and may be called the foundation of theoretical mechanics, for the previous contributions of Aristotle were comparatively vague and unscientific. In the first book there are fifteen propositions, with seven postulates; and demonstrations are given, much the same as those still employed, of the centres of gravity (1) of any two weights, (2) of any parallelogram, (3) of any triangle, (4) of any trapezium. The second book in ten propositions is devoted to the finding the centres of gravity (1) of a parabolic segment, (2) of the area included between any two parallel chords and the portions of the curve intercepted by them.

(6) The Quadrature of the Parabola ( ) is a book in twenty-four propositions, containing two demonstrations that the area of any segment of a parabola is 4/3 of the triangle which has the same base as the segment and equal height. The first (a mechanical proof) begins, after some preliminary propositions on the parabola, in Prop. 6, ending with an integration in Prop. 16. The second (a geometrical proof) is expounded in Props. 17-24.

(7) On Floating Bodies ( ) 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 (, 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) , Principles, a book addressed to Zeuxippus and dealing with the naming of numbers on the system explained in the Sand Reckoner; (3)  , On balances or levers; (4)  , On centres of gravity; (5)  , an optical work from which Theon of Alexandria quotes a remark about refraction; (6)  , a Method, mentioned by Suidas; (7)  , 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).

.—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.

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 ). 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 ( ), 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, , the Holy Sea; (3) it arose at the time of the Latin empire, and means the Sea of the Kingdom (Archè); (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. Other archipelagoes are described in their respective places. ARCHIPPUS, an Athenian poet of the Old Comedy, who flourished towards the end of the 5th century His most famous play was the Fishes, in which he satirized the fondness of the Athenian epicures for fish. The Alexandrian critics attributed to him the authorship of four plays previously assigned to Aristophanes. Archippus was ridiculed by his contemporaries for his fondness for playing upon words (Schol. on Aristophanes, Wasps, 481).

Titles and fragments of six plays are preserved, for which see T. Kock, Comicorum Atticorum Fragmenta, i. (1880); or A. Meineke, Poetarum Comicorum Graecorum Fragmenta (1855).

ARCHITECTURE (Lat. architectura, from the Gr. <span title="architéktōn">, a master-builder), the art of building in such a way as to accord with principles determined, not merely by the ends the edifice is intended to serve, but by high considerations of beauty and harmony (see ). It cannot be defined as the art of building simply, or even of building well. So far as mere excellence of construction is concerned, see and its allied articles. The end of building as such is convenience, use, irrespective of appearance; and the employment of materials to this end is regulated by the mechanical principles of the constructive art. The end of architecture as an art, on the other hand, is so to arrange the plan, masses and enrichments of a structure as to impart to it interest, beauty, grandeur, unity, power. Architecture thus necessitates the possession by the builder of gifts of imagination as well as of technical skill, and