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See also:CUBE (Gr. K46os, a cube) , in See also:geometry, a solid bounded by six equal squares, so placed that the See also:angle between any pair of adjacent faces is a right angle . This solid played an all-important See also:part in the geometry and cosmology of the Greeks . See also:Plato (See also:Timaeus) described the figure in the following terms:—" The isosceles triangle which has its See also:vertical angle a right angle . . combined in sets of four, with the right angles See also:meeting at the centre, See also:form a single square . Six of these squares joined together formed eight solid angles, each produced by three See also:plane right angles: and the shape of the See also:body thus formed was cubical, having six square planes for its surfaces." In his cosmology Plato assigned this solid to " See also:earth," for " ` earth ' is the least See also:mobile of the four (elements—' See also:fire," See also:water," See also:air ' and ` earth ') and most plastic of bodies: and that substance must possess this nature in the highest degree which has its bases most See also:stable." The See also:mensuration of the See also:cube, and its relations to other geometrical solids are treated in the See also:article See also:POLYHEDRON; in the same article are treated the Archimedean solids, the truncated and snub-cube; reference should be made to the article See also:CRYSTALLOGRAPHY for its significance as a crystal form . A famous problem concerning the cube, namely. to construct a cube of twice the See also:volume of a given cube, was attacked with See also:great vigour by the Pythagoreans, See also:Sophists and Platonists . It became known as the " Delian problem " or the " problem of the duplication of the cube," and ranks in See also:historical importance with the problems of " trisecting an angle " and " squaring the circle." The origin of the problem is open to conjecture . The See also:Pythagorean See also:discovery of " squaring a square," i.e. constructing a square of twice the See also:area of a given square (which follows as a corollary to the Pythagorean See also:property of a right-angled triangle, viz. the square of the hypotenuse ,equals the sum of the squares on the sides), may have suggested the strictly analogous problem of doubling a cube . Eratosthenes (c . 200 B.c.), however, gives a picturesque origin to the problem . In a See also:letter to See also:Ptolemy Euergetes he narrates the See also:history of the problem . The Delians, suffering a dire pestilence, consulted their oracles, and were ordered to See also:double the volume of the See also:altar to their tutelary See also:god, See also:Apollo . An altar was built having an edge double the length of the See also:original; but the See also:plague was unabated, the oracles not having been obeyed . The See also:error was discovered, and the Delians applied to Plato for his See also:advice, and Plato referred them to See also:Eudoxus . This See also:story is See also:mere See also:fable, for the problem is far older than Plato . See also:Hippocrates of See also:Chios (c . 430 B.C.), the discoverer of the square of a lune, showed that'the problem reduced to the determination of two mean proportionals between two given lines, one of them being twice the length of the other . Algebraically expressed, if x and y be the required mean proportionals and a, 2a, the lines, we have a: x:: x : y :: y : 2a, from which it follows that x3= 2a'', . Although Hippocrates could not determine the proportionals, his statement of the problem in this form was a great advance, for it was perceived that the problem of trisecting an angle was reducible to a similar form which, in the See also:language of algebraic geometry, is to solve geometrically a cubic See also:equation . According to See also:Proclus, a See also:man named Hippias, probably Hippias of Ells (c . 46o B.c.), trisected an angle with a See also:mechanical See also:curve, named the See also:quadratrix (q.v.) . See also:Archytas of See also:Tarentum (c . 430 B.c.) solved the problems by means of sections of a See also:half See also:cylinder; according to Eutocius, Menaechmus solved them by means of the inter-sections of conic sections; and Eudoxus also gave a See also:solution . All these solutions were condemned by Plato on the ground that they were mechanical and not geometrical, i.e. they were not effected by means of circles and lines . However, no proper geometrical solution, in Plato's sense, was obtained; in fact it is now generally agreed that, with such a restriction, the problem is insoluble . The pursuit of mechanical methods furnished a stimulus to the study of mechanical loci, for example. the See also:locus of a point carried on a See also:rod which is caused to move according to a definite See also:rule . Thus Nicomedes invented the See also:conchoid (q.v.); Diodes the See also:cissoid (q.v.); Dinostratus studied the quadratrix invented by Hippias; all these curves furnished solutions, as is also the See also:case with the See also:trisectrix, a See also:special form of See also:Pascal's limagon (q.v.) . These problems were also attacked by the Arabian mathematicians; See also:Tobit See also:ben Korra (836–901) is credited with a solution, while Abul Gud solved it by means of a See also:parabola and an equilateral See also:hyperbola . In See also:algebra, the " cube " of a quantity is the quantity multiplied by itself twice, i.e. if a be the quantity a X a X a.(= a3) is its cube . Similarly the " cube See also:root " of a quantity is another quantity which when multiplied by itself twice gives the original quantity; thus ai is the cube root of a (see See also:ARITHMETIC and ALGEBRA) . A " cubic equation " is one in which the highest See also:power of the unknown is the cube (see EQUATION); similarly, a " cubic curve " has an equation containing no See also:term of a power higher than the third, the See also:powers of a See also:compound term being added together . In mensuration, " cubature " is sometimes used to denote the volume of a solid; the word is parallel with " See also:quadrature, ", to de termine the area of a See also:surface (see MENSURATION; INFINITESIMAL CALCULUS) . |
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