Algebra

[It. and Sp.; Fr. algèbre, probably from the Ar. Al jebr e al mokabalah, restoration and reduction.] ( Math.) The method of calculating indeterminate quantities. It is a sort of universal arithmetic, founded on the same principles as common arithmetic, and proceeding by rules and operations precisely similar. But it is not confined merely to questions relating to numbers, being applied generally to investigate the relations that subsist among quantities of all kinds, whether arithmetical or geometrical. The reasoning is carried on by general symbols, which are of two kinds, those which denote quantity, and those which denote the affections or relations, or properties of quantities, and operations to be performed on them. For representing quantities or magnitudes, the letters of the alphabet are employed. Thus, in the solution of an arithmetical problem, a number may be represented by the letter a; in geometry, a may represent a line or an angle; in mechanics, a force. The relations of quantities are expre ed by other conventional symbols. The relation of equality is expre ed by the sign; thus, to expre that the quantity represented by a is equal to the quantity represented by b, we write ab. The symbol > or b signifies that a is greater than b, and a mined. But in algebra the symbols that are employed are perfectly general, and may represent any numbers whatever; and the expre ions which result from combining them according to the conditions of the problem, indicate the solution not of a particular question, but of all questions whatever, in which numbers are subjected to the same series of operations. In this manner the general properties of numbers are discovered. For ex ample, the expre ion ( a + b ) ( a - b ), which signifies that the sum of the two numbers a and b is to be multiplied by their difference, becomes, on performing the multiplication, a2-b2; whence we infer this general or universal truth, namely, that the product of the sum and the difference of any two numbers is equal to the difference of the squares of those numbers. Arithmetic could only prove the property to be true in respect of particular numbers. The only nece ary preliminary to the study of A. is a good knowledge of the four rules of arithmetic, and of common and decimal fractions. Algebra'ic, Algebraical, a. Relating to algebra. Algebraical Curve . A curve of which the relation between the absci a and the ordinates is expre ed by an equation which contains only algebraic quantities. The term algebraic is here used in contradistinction to transcendental, under which is comprehended infinite series and quantities of the following kind: log. z , az sin. z , cos. x , tan. z , . Algebraic Equations . An equation of which the terms contain only algebraic quantities. - See EQUATION. Algebra'ically, adv . By algebraic proce . Algebraist, n. One who is versed in algebra. Algebraize, v . a . To perform by algebra, or reduce to algebraical form. Algeciras, a seaport-town of Spain, And alus i a, prov. of Cadiz, on the W. side of the Bay of Gibraltar, opposite to and 6 m. W. of Gibraltar. Pop . about 12,000. It was the scene of a naval engagement between the English and French, 1801. Al'gemesi, or ALGEMESIA, a town of Spain, prov. and 18. m. S. of Valencia. Pop. 5,000.