being the Historia física y política, and also the earlier work on which
they are based, Historia económica-política y estadística de ... Cuba
(Havana, 1831); treatises on administrative law in Cuba by
J. M. Morilla (Havana, 1847; 2nd ed., 1865, 2 vols.) and A. Govin
(3 vols., Havana, 1882–1883); A. S. Rowan and M. M. Ramsay,
The Island of Cuba (New York, 1896); Coleccion de reales ordenes,
decretos y disposiciones (Havana, serial, 1857–1898); Spanish Rule
in Cuba. Laws Governing the Island. Reviews Published by the
Colonial Office in Madrid ... (New York, for the Spanish legation,
1896); and compilations of Spanish colonial laws listed under
article Indies, Laws of the. On the new Republican régime:
Gaceta Oficial (Havana, 1903– ); reports of departments of
government; M. Romero Palafox, Agenda de la republica de Cuba
(Havana, 1905). See also the Civil Reports of the United States
military governors, J. R. Brooke (2 vols., 1899; Havana and
Washington, 1900), L. Wood (33 vols., 1900–1902; Washington,
1901–1902).
History.—The works (see above) of Sagra, Humboldt and Arango are indispensable; also those of Francisco Calcagno, Diccionario biográfico Cubano (ostensibly, New York, 1878); Vidal Morales y Morales, Iniciadores y primeros mártires de la revolución Cubana (Havana, 1901); José Ahumada y Centurión, Memoria histórica política de ... Cuba (Havana, 1874); Jacobo de la Pezuela, Diccionario geográfico-estadístico-histórico de ... Cuba (4 tom., Madrid, 1863–1866); Historia de ... Cuba, (4 tom., Madrid, 1868–1878; supplanting his Ensayo histórico de ... Cuba, Madrid and New York, 1842); and José Antonio Saco, Obras (2 vols., New York, 1853), Papeles (3 tom., Paris, 1858–1859), and Coleccion postuma de Papeles (Havana, 1881). Also: Rodriguez Ferrer, op. cit. above, vol. 2 (Madrid, 1888); P. G. Guitéras, Historia de . . . Cuba (2 vols., New York, 1865–1866). Of great value is J. Zaragoza, Las Insurrecciones en Cuba. Apuntes para la historia política (2 tom., Madrid, 1872–1873); also J. I. Rodriguez, Vida de ... Félix Varela (New York, 1878), and Vida de D. José de la Luz (New York, 1874; 2nd ed., 1879). On early history see Coleccion de documentos inéditos relativos al descubrimiento ... de ultramar (series 2, vols. 1, 4, 6, Madrid, 1885–1890). On archaeology, N. Fort y Roldan, Cuba indigena (Madrid, 1881); M. Rodriguez Ferrer (see above); and especially A. Bachiller y Morales, Cuba primitiva (Havana, 1883). For the history of the Cuban international problem consult José Ignacio Rodriguez, Idea de la anexion de la isla de Cuba à los Estados Unidos de America (Havana, 1900), and J. M. Callahan, Cuba and International Relations (Johns Hopkins University, Baltimore, 1898), which supplement each other. On the domestic reform problem there is an enormous literature, from which may be selected (see general histories above and works cited under § Administration of this bibliography): M. Torrente, Bosquejo económico-político (2 tom., Madrid-Havana, 1852–1853); D. A. Galiano, Cuba en 1858 (Madrid, 1859); José de la Concha, twice Captain-General of Cuba, Memorias sobre el estado político, gobierno y administración de ... Cuba (Madrid, 1853); A. Lopez de Letona, Isla de Cuba, reflexiones (Madrid, 1856); F. A. Conte, Aspiraciones del partido liberal de Cuba (Havana, 1892); P. Valiente, Réformes dans les îles de Cuba et de Porto Rico (Paris, 1869); C. de Sedano, Cuba: Estudios políticos (Madrid, 1872); H. H. S. Aimes, History of Slavery in Cuba, 1511–1868 (New York, 1907); F. Armas y Cèspedes, De la esclavitud en Cuba (Madrid, 1866), and Régimen político de las Antillas Españolas (Palma, 1882); R. Cabrera, Cuba y sus Jueces (Havana, 1887; 9th ed., Philadelphia, 1895; 8th ed., in English, Cuba and the Cubans, Philadelphia, 1896); P. de Alzola y Minondo, El Problema Cubano (Bilbao, 1898); various works by R. M. de Labra, including La Cuestion social en las Antillas Españolas (Madrid, 1874), Sistemas coloniales (Madrid, 1874), &c.; R. Montoro, Discursos ... 1878–1893 (Philadelphia, 1894); Labra et al., El Problema colonial contemporánea (2 vols., Madrid, 1894); articles by Em. Castelar et al., in Spanish reviews (1895–1898). On the period since 1899 the best two books in English are C. M. Pepper, To-morrow in Cuba (New York, 1899); A. G. Robinson, Cuba and the Intervention (New York, 1905). (F. S. P.)
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).
