# The New International Encyclopædia/Algebra

Edition of 1905. See also Algebra on Wikipedia; and the disclaimer.
​AL'GEBRA. A branch of pure mathematics that materially simplifies the solution of arithmetical problems, especially through the use of equations. It also forms the introduction to all of the higher branches of mathematical science, except pure geometry.
The name is derived from the title of the Arabic work by AM-Khuwarizmi (q.v.), Ilm al-jabr wa'l muqubalah, meaning "the science of redintegration and equation;" that is, the science that relates to the reduction of equations to integral form and to the transposition of terms. The title appeared thereafter in various forms, as ludus algebræ almugrabalæque, and algiebar and almachabel, but the abbreviation algebra was finally adopted. The science also went under various other names in the fifteenth and sixteenth centuries, as the ars magna (Cardan, 1545), the arte maggiore, the regola de la cosa (because the unknown quantity was denominated cosa, the "thing"), and hence in early English the cossike art, and in German the Coss.
The exact limitations of algebra are not generally agreed upon by mathematicians, and hence various definitions have been proposed for the science. It has been proposed to limit it to the theory of equations, as the etymology of the word would suggest; but this has become a separate branch of mathematics. Perhaps the most satisfactory definition, especially as it brings out the distinction between algebra and arithmetic, is that of Comte: "Algebra is the calculus of functions, and arithmetic is the calculus of values." This distinction would include some arithmetic in ordinary school algebra (e. g., the study of surds), and some algebra in common arithmetic (e.g., the formula for square root).
The oldest known manuscript in which algebra is treated is that of Ahmes, the Egyptian scribe, who, about 1700 B.C., copied a treatise dating perhaps from 2500 B.C. In this appears the simple equation in the form, "Hau (literally heap), its seventh, its whole, it makes 19," which, put in modern symbols, means
-|-aj=19. In Euclid's Elements (about 300
B.C.) a knowledge of certain quadratic equations is shown. It was Diophantus of Alexandria (q.v.), however, who made the first attempt (fourth ​century A.D.) to work out the science. In the following century Aryabhatta (q.v.) made some contributions to the subject. Little was then done until about 800 A.D., when Al-Khuwarizmi wrote. His efforts were followed by another period of comparative repose, until the Italian algebraists of the sixteenth century undertook the solution of the cubic equation. (See Equation.) In this, building upon the efforts of Ferreo and Tartaglia, Cardan was successful (1545), although there is reason to believe that the real honor belongs to Tartaglia. Soon after, Ferrari and Bombelli (1579) gave the solution of the biquadratic equation.
The principal improvements in the succeeding century related to symbolism. It took a long time, however, to pass from the radical sign of Chuquet (1484), IJ'. 10 through various forms, as -/j^ 10, to our common symbol I'TO and to the more refined IO14. Similarly it was only by slow stops that progress was made from Cardan's cubus p 6 rebus æqualis 20, for x3 + 6x = 20, through Vieta's
IC — SQ + IG.V cequ. 40, for a;' — 8a;= + 16a^ = 40
and Descartes'
or cc ax — 66, for or ^ ax — 6",
and Hudde's
X oc qx. r, for x^ = qx + r.
to the modern notation. To the Frenchman Vieta, whose first book on algebra, In Artem Analyticam Isagoge, appeared in 1591, credit is due for the introduction of the use of letters to represent known as well as unknown quantities.
The next step led to the recognition of the nature of the various number systems of algebra. The meaning of the negative number began to be really appreciated through the application of algebra to geometry by Descartes (1637), and the meaning of the so-called "imaginary," when Wes- sel (1797) published his memoir on complex numbers, or, more strictly, when Gauss (q.v.) brought the matter to the attention of mathema- ticians (1832).
The effort to solve the quintic equation, seriously begun in the sixteenth century, had met with failure. It was only after the opening of the nineteenth century that Abel, by the use of the theory of groups discovered by Galois, gave the first satisfactory proof of the fact, anticipated by Gauss and announced by Rulfini, that it is impossible to express the roots of a general equation as algebraic functions of the coefficients when the degree exceeds the fourth.
Among the later additions to the science of algebra may be mentioned the subjects of Deter- minants (q.v.). Complex Number (q.v.). Sub- stitutions and Groups (q.v.). Form, and the modern treatment of Equation (q.v.). Under these heads may be found historical sketches dealing with the recent developments of algebra.
Bibliography. For the modern history, con- sult: M. Merriman and Woodward, Higher Math- ematics (New York, 1896), and Fink, History of Mathematics (Chicago, 1900). For elementary theory, Smith, Teaching of Elementary Mathe- matics (New York, 1900). For modern higher algebra, Netto, Vorlesungen über Algebra (Leip- zig, 1898-1900); Biermann, Elemente der höheren Mathematik; H. Weber, Lehrbuch der Algebra (Leipzig, 1895), and Salmon, Modern Higher Algebra (Dublin, 1885); for a com-
pendium, Pund, Algebra mit Einschluss der elementaren Zahlentheorie (Leipzig, 1899); Pierce, Linear Associative Algebra (New York 1882).