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Encyclopedia Britannica - Main :: CRE-DAH |
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CUBE (Gr. K46os, a cube) , in geometry, a solid bounded by six equal squares, so placed that the angle between any pair of adjacent faces is a right angle. This solid played an all-important part in the geometry and cosmology of the Greeks. Plato (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 meeting at the centre, form a single square. Six of these squares joined together formed eight solid angles, each produced by three plane right angles: and the shape of the body
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 volume of a given cube, was attacked with 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 historical importance with the problems of " trisecting an angle " and " squaring the circle." The origin of the problem is open to conjecture. The Pythagorean discovery of " squaring a square," i.e. constructing a square of twice the area of a given square (which follows as a corollary to the Pythagorean 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 letter to Ptolemy
double
double
original
Hippocrates of 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 Proclus, a man named Hippias, probably Hippias of Ells (c. 46o B.c.), trisected an angle with a mechanical curve, named the quadratrix (q.v.). Archytas of Tarentum (c. 430 B.c.) solved the problems by means of sections of a 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. the locus of a point carried on a rod which is caused to move according to a definite 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 trisectrix
special
parabola
In 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 root " of a quantity is another quantity which when multiplied by itself twice gives the original
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 surface (see MENSURATION; INFINITESIMAL CALCULUS). End of Article: CUBE (Gr. K46os, a cube) If you wish, you can link directly to this article.
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