Cube

Cube (Gr. κύβος, 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 thus formed was cubical, having six square planes for its surfaces.” In his cosmology Plato assigned this solid to “earth,” for “‘earth’ is the least mobile of the four (elements—‘fire,’ ‘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 stable.” The mensuration of the 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 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 Euergetes he narrates the history of the problem. The Delians, suffering a dire pestilence, consulted their oracles, and were ordered to double the volume of the altar to their tutelary god, Apollo. An altar was built having an edge double the length of the 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 Eudoxus. This story is mere fable, for the problem is far older than Plato.

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 = 2a³. 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 Elis (c. 460 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 intersections 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.); Diocles the cissoid (q.v.); Dinostratus studied the quadratrix invented by Hippias; all these curves furnished solutions, as is also the case with the trisectrix, a special form of Pascal’s limaçon (q.v.). These problems were also attacked by the Arabian mathematicians; Tobit ben Korra (836-901) is credited with a solution, while Abul Gud solved it by means of a parabola and an equilateral hyperbola.

In algebra, the “cube” of a quantity is the quantity multiplied by itself twice, i.e. if a be the quantity a × a × a (= a³) is its cube. Similarly the “cube root” of a quantity is another quantity which when multiplied by itself twice gives the original quantity; thus a1/3 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 term of a power higher than the third, the 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 determine the area of a surface (see Mensuration; Infinitesimal Calculus).

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