# 1911 Encyclopædia Britannica/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*, 2*a*, the lines,
we have *a* : *x* :: *x* : *y* :: *y* : 2*a*, from which it follows that *x*^{3} = 2*a*^{3}.
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* (= *a*^{3}) is its cube. Similarly the “cube root” of a quantity is another quantity which when multiplied by itself twice gives the original quantity; thus *a*1/3 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).