Page:The New International Encyclopædia 1st ed. v. 05.djvu/753

* CUBE. 649 CUBICULUM. if it is a power of a commensurable (q.v.) num- ber, it is called a perfect power. Roots of per- fect powers are often readily obtained by factor- ing; e.g. to find the cube root of 210; 210 = 6 • 6 • 0, therefore is the cube root of 210. If the root is incommensurable, the binomial for- mula, logarithms, or the equation (q.v.) is available. Every number which satisfies the equation a-" = 1, or a^ — 1 = is a cube root of i. Butar"- — 1 = is the same as (x — 1) (x' -- a; -|- 1 ) =0, and equating each factor to and solving, X = 1, — i + i vf — 3>— i— 2 i/ — 3. the three cube roots of unity. (See Complex Number. ) The three cube roots of 8 are 2, 2(-| + 1/^^^), 2{-i-iV^^S). Thus any number has three cube roots, one real and two imaginary. In extensive calculations, tables of roots and of logaritlims are employed. Duplication of the cube or the Delian problem, according to tradition, originated with the oracle of Delos, which declared to the Athenians that a pestilence prevailing among them would cease if they doubled the altar of Apollo — i.e. replaced his cubical altar by another of twice its con- tents. The problem reduces to the solution of ■ the continued proportion a: x ^ x: y ^ y: 2a. or to the solution of x' = 2a^. This was effected geometrically by Hippocrates, Plato, Mentech- nius, Archytas, and others, but not by elementary geometry. This is one of the three gi'eaf prob- lems whose appearance has been of wonderful significance in the development of mathematics. Consult: Gow, History of Orcek Mathematics (Cambridge, 1884) ; Klein, Tortriige uher aiisgeifiihlte Fragen der Elementargeometrie (Leipzig, 1895) ; Famous Pruhlems of Elemen- tary Geometry, trans, by Beman and Smith (Boston, 1897). CTT'BEBS, or CUBEB PEPTER ( Fr. culehe, from Ar. kababa). The dried unripe berries of* Piper officinalis, a species of climbing shrub of the natural order Piperaceae, very closely allied to the true peppers. Piper officinalis is a native of Penang, Java, New Guinea, etc., and is said to be extensively cultivated in some parts of Java. Its spikes are solitary, opposite to the leaves, and usually produce about fifty berries, which are globular, and, when dried, have much re- semblance to black pepper, except in their lighter color and the stalk with which they are furnished. Piper canina, a native of the Sunda and Molucca islands, is supposed also to yield part of the cuhebs of connncrce, and the berries of Piper ribesioides possess similar properties. Cubebs are less pungent and more pleasantly aromatic than black pepper: they are used in the East as a condiment, but in Europe chiefly for medicinal purposes. They act as a stinuilant, and are sometimes found useful in ca.ses of indigestion; also in chronic catarrh and in many all'cctions of the mucous membrane, particuhirly those of the urino-genital system. The chief constituents of cubebs are a volatile oil, resin, cubebic acid, cubebin, and wax. Cubebs are administered in many ways. For illustraf ion, see Plate of Cypress, etc, CUBE ROOT. See Cube; Involltio.x. CUBIC EQUATION. A rational integral equation of the third degree is called a cubic equation. It is called binary, ternary, or qua- ternary according as it is homogeneous of the third degree in two, three, or four unknowns. The general form of a cubic equation of one un- known is «a;' + 6a;' + ex -- d ^ 0. It is shown in algebra that this equation can be reduced to one of the form oe' -- px -{- q =^ 0. Every cubic equation of this form has three roots, of whidi one is real and the others real or imaginary. The roots will all be real when p is negative, and ~= i^4— This is known as the irreducible case 27 4 • in solving the equation. Only one root is real when p is positive, or when it is negative and — <C^' If » is negative and ~=~, two of 27 ^ 4 "^ ° 2(4 the roots are equal. The cubic equation may be solved by the following fornuila, due to Tartaglia and Ferro, Italian mathematicians of the six- teenth century, but known as Cardan's formula : Besides Ferro, Tartaglia, and Cardan, Vieta, Euler, and others contributed to the early the ory of cubic equations. In case the roots of a cubic equation are all real their values are more readily calculated by means of trigonometric formulas — e.g. assume x = « cosa, and the equa- tion a;' + pa; + 5 = may be expressed by cos'o + 9 1 — 5 COSO + -J cos'o — % cosa — =: 0. But from trigonometry cos 3a „, , — J — ^0; therefore, equat- ing corresponding coefficients of coso, and solv- ing the equations, n = ■y — ~7v and cos3a = — 4q I JL 1 • Hence x may now be computed from I 4:pJ 3 4j) a; := » • cosa; n •& + a 2ff For history and methods, consult Matthicssen, Qrundzuge der antiken und modernen Algebra der litteralen Glcichungen (Leipzig, 1896). See also Cardan, .Jerome. CUBICULUM (Lat., bedroom, from cubare, to lie down). A term used to designate a small room or cell in a Roman house, containing a bed or couch, and opening off the c(nirt; also a recess or .alcove, a box at the theatre or circus; and lastly, a final resting-place or burial-recess for one or more bodies in the early Christian catacombs.