ALGEBRA, according to present English usage that branch of mathematics which considers primarily the representation of numbers by means of letters. Speaking more precisely, that part of mathematics which considers the relations and properties of numbers by the aid of general symbols, usually letters (a,b,c, . . . x,y,z) and signs of operation (+,—,X, • . .) and relation (=,>, <, . . .). For example, the statement that the area of a rect angle is equal to the product of the number of units of length in the base multiplied by the number of units in the height, is alge braically expressed by the symbols A = bh, where A stands for the number of square units of area; b, for the number of linear units in the base ; and h, for the number of linear units in the height. Similarly, the expression
states that there is a number (represented by x) such that its square, plus five times itself, is equal to 14. Algebraic symbolism and operations enter into nearly all branches of science, including the various subdivisions of mathematics. In certain cases, however, as in vector analysis (q.v.), the letters are not restricted to the representation of numbers. In its broader sense algebra treats of equations, poly nomials, continued fractions, series, number sequences, forms, determinants, and new types of numbers. These topics will be found discussed in separate articles. It considers the fundamental theorem that every integral equation f(x) with real coefficients has at least one real or complex root (see EQUATIONS), indeterminate equations (see DIOPHANTINE EQUATIONS), general algebraic equa tions of the third and fourth degrees, numerical higher equations (see EQUATIONS), and it enters into such important branches as the calculus, trigonometry and the theory of functions (qq.v.). As a preparation for all this work, however, there is the elementary part of the subject. It is the purpose of this present article to discuss a number of topics which will show to the general reader the nature of algebra in its initial stage, and to show the teacher why algebra is taught in the schools. This discussion will be sup plemented by references to treatises which will enable the student to find the underlying philosophical principles of the science.
Ifby the word algebra we mean that branch of mathematics by which we learn how to solve the equa tion
written in this way, the science begins in the 17th century. If we allow the equation to be written with other and less convenient symbols, it may be considered as beginning at least as early as the 3rd century. If we permit it to be stated in words and solved, for simple cases of positive roots, by the aid of geometric figures, the science was known to Euclid and others of the Alex andrian school as early as 30o B.C. If we permit of more or less scientific guessing in achieving a solution, algebra may be said to have been known nearly 2000 years B.c., and it had probably at tracted the attention of the intellectual class much earlier. The scope of all early algebra was limited to a study of equations or to the solution of problems which at present would be solved by their aid. In the 16th century, of ter the advent of the printed book in Europe, the field was enlarged through the efforts of men like Christoff Rudolf (Die Coss, 15 2 5 ; 2nd ed. by Michael Stifel, Konigsberg,
Robert Recorde (Whetstone of Witte, London, 1557), Rafael Bombelli (L'Algebra parte maggiore dell' arimetica, Bologna, 1572) and Christopher Clavius (Algebra, Rome, 16o8), becoming more of a generalized arithmetic, the fundamental operations with numbers being duplicated with rather crude algebraic symbols. The perfecting of symbolism in the 17th century greatly extended the domain of algebra and rendered pos sible the development of the higher branches of the subject.
Thename "algebra" is quite fortuitous. When Mohammed ibn Musa al-K howarizmi (Mohammed, son of Moses, the Khowarezmite), a native of Khowarezm (Khwarezm), wrote in Baghdad (c. 825), he gave to one of his works the name Al-jebr w'al-muqabalah. The title is sometimes translated as "res toration and equation," but the meaning was not clear even to the later Arab writers. Of late it has been thought that al-jebr is Arabic, while muqabalah is from the Persian, and that each meant or referred to an equation. At any rate, al-Khowarizmi's work was the first to bear the title "algebra," and the treatise was so important as to cause the name to be adopted, often with strange variations in spelling, by later writers. Various other names have been given to the science, such as arithmetica (see ARITH METIC), Bija Ganita (Brahmagupta's Hindu treatise, c. 628, the term meaning calculation with primitive elements), T'ien-yuen (Chinese, "celestial element"), Kigen seih5 (one of the Japanese names, referring to "revealing unknowns"), Fakhri (al-Karkhi, c. 1020, who gave this name to his algebra in honour of his patron, Fakhr al-Mulk), Regola de la cosa (Rule of the cosa, Lat. causa, Fr. chose, the unknown quantity), Ars magna ("great art," used by Cardan in 1545), the German Die Coss and English Cossike arte (both in the i6th century) . Most of these names referred to the science of the equation, and this is the meaning assigned to elementary algebra in certain European languages at present, the fundamental operations with literal expressions being then included under the term "arithmetic." Summary of the Changes in Meaning. We may, therefore, summarize the leading steps in the growth of algebra as follows: (I) The period of the puzzle problem relating to numbers, with little or no symbolism, C. 1800 B.C. to A.D. 275; (2) The inclusion of the geometric problem of completing the square, thus leading to the finding of a line that would be represented by x in a modern quadratic equation ; (3) The development of a systematic although a crude symbolism, applied to the theory of numbers, as in the Arithmetica of Diophantus (c. 275), this theory including certain indeterminate equations; (4) A more critical study of equations with some approach to scientific treatment, as in the period of Muslim ascendancy, c. Boo- c. I2oo; (5) The rise of the theory of equations, beginning with the solution of the cubic and the biquad ratic in the i6th century (see CARDAN ; TARTAGLIA); (6) The development of a convenient symbolism, chiefly in the century 1550-1650, changing algebra from a crude theory of equations to an analytic subject concerned with algebraic numbers and poly nomials; and (7) The modern development of higher algebra.
