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ALGEBRA. 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 .) 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).

AL'GEBRA'IC CURVE. See.

ALGECIRAS, Til'jj-sl'ras, Sp. pron. al'Ha- the'rag (From Ar. al-jazirah, the island, penin- sula). A seaport of Spain, in the province of Cadiz, six miles west of Gibraltar (Map: Spain, C 4). Its harbor is good, and protected by a' fort. It is a well-built city, with fine churches and monasteries. It maintains a brisk and prof- itable coasting trade. Pop., 1900, 13,131. This was the Pontus Novus of Roman times, and later was the first place seized by the Moors from Africa (711). They called it Algeciras, the green island, from the islet at the har- bor's mouth, still known as Isla Verde. Alfonso XI. besieged it for twenty months, and took it after his victory at Rio Salado. 1344. The Moors are said to have used gunpowder for the first time at this siege, at which all Christendom was represented by the knights and crusaders who mustered under Alfonso's banner. Consult: Roure, "Bahía de Algeciras," in Memorias de ingenieros del ejército, Volume XVI. (Madrid, 1899).

AL'GER, (1834-99). An American writer of juvenile literature. He was born at Revere, Mass., graduated at Harvard in 1852, and afterward at the Harvard Divinity School. He became pastor of the Unitarian church at Brewster, Mass.. in 1864, but two years later went to New York, where he labored for the improvement of the condition of street boys. He wrote much for newspapers and periodicals, and published about seventy books, of which nearly 800,000 copies have been sold. These writings include the popular Ragged Dick, Tattered Tom, and Luck and Pluck series.

ALGER, Russell Alexander (1836—). An American soldier and politician, born in Lafay- ette, O. Orphaned at eleven, he worked on a farm, attended Richfield Academy (Ohio) for several winters, studied law at Akron, O., was admitted to the bar in 1859. and began practice at Cleveland. He removed to Michigan in Janu- ary, 1860, and in September, 1861, enlisted in the Union Army as a volunteer, serving succes- sively as captain, major, and lieutenant-colonel, and being brevetted first as brigadier-general and then as major-general "for gallant and meritorious services." He returned to Michigan in 1864, and engaged in the lumber business, in which he soon amassed a large fortune. He took an active interest in both local and national politics, and was Governor of Michigan from 1885 to 1886. He was chosen Commander-in- Chief of the G. A. R. in 1889, and in 1897 became Secretary of War in President McKinley's cab- inet. His administration of the department dur- ing the Spanish-American War met with the most vigorous criticism. He was charged with being directly or indirectly responsible for the unsan- itary condition of camps, the overcrowding and unfitness of transports, the insufficiency of phy- sicians and medicines, the bad quality of food, and the incompetence of subordinate officers. An investigating committee, appointed by the President, in the main exonerated Alger. He resigned August 1, 1899. He wrote The Spanish-American War (New York, 1901).

ALGER, (1822—). An