CUBE, a regular solid with six square faces; that is, a regular hexahedron. Since the volume of a cube is expressed, in terms of an edge, as in arithmetic and algebra the third power of a quantity is called the cube of that quantity. That is, or 27, is the cube of 3; and is the cube of x.
A number of which a given number is the cube is called the cube root of the lat ter number; that is, since 27 is the cube of 3, 3 is the cube root of 27—symboli cally, 3= .I 27. A number that is not a cube is said to have a cube root, the value being expressed approximately; that is, 4 is not a cube, but we speak of the cube root of 4, 4, the approximate value being 1.587, because is approximately 4. In Greek geometry the duplication of the cube was one of the most fa mous of the unsolved problems. It required the construction of a cube that should have twice the volume of a given cube. This has been proved to be impossible by the aid of the straight edge and compasses alone, but the Greeks were able to effect the con struction by the use of higher curves, notably by the cissoid of Diocles (see CURVE). Hippocrates (c. 43o B.c.) showed that the problem reduced to that of finding two mean proportionals between a line segment and its double; that is algebraically, to that of finding x and y in the proportion a:x=x:y=y:2a, from which and hence the cube with x as an edge has twice the volume of one with a as an edge.