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 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:water," 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 mensuration of the
See also:cube, and its relations to other geometrical solids are treated in the article POLYHEDRON; in the same article are treated the Archimedean solids, the truncated and snub-cube; reference should be made to the article 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, 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: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:Apollo .
An altar was built having an edge double the length of the
See also:original; but the plague was unabated, the oracles not having been obeyed . The error was discovered, and the Delians applied to Plato for his advice, and Plato referred them to
See also:Eudoxus . This
See also:story is 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 language of algebraic geometry, is to solve geometrically a cubic equation . According to
See also:Proclus, a man named Hippias, probably Hippias of Ells (c . 46o B.c.), trisected an angle with a
See also:curve, named the quadratrix (q.v.) .
See also:Archytas of Tarentum (c . 430 B.c.) solved the problems by means of sections of a
See also:half cylinder; according to Eutocius, Menaechmus solved them by means of the inter-sections of conic sections; and Eudoxus also gave a 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. thelocus 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 conchoid (q.v.); Diodes the cissoid (q.v.); Dinostratus studied the quadratrix invented by Hippias; all these curves furnished solutions, as is also the case with the
See also:trisectrix, a
See also:special form of Pascal's limagon (q.v.) . These problems were also attacked by the Arabian mathematicians;
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 ARITHMETIC and ALGEBRA) . A " cubic equation " is one in which the highest 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 compound term being added together . In mensuration, " cubature " is sometimes used to denote the volume of a solid; the word is parallel with " quadrature, ", to de termine the area of a
See also:surface (see MENSURATION; INFINITESIMAL CALCULUS) .
CUBA (the aboriginal name)
CUBEBS (Arab. kababah)
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