AL'GEBRA is a branch of pure mathematics. • The name is derived from the Arabs, who call the science at gebr teal supplementing and equalizing—in refer ence to the transposition and reduction of the terms of an equation. Among the Italians in Carly times it was called ante maggiore, as having to do with the higher kinds of calculation, and still oftener regola de la coca, because the unknown quantity was denom inated coca, the " thing ;" hence the name of cessike art, given to it by early English writers.
The term algebraical is generally used somewhat vaguely, to denote any expression or calculation in which signs are used to denote the operations, and letters or other symbols are put instead of numbers. But it is perhaps better to restrict the name A. to the doc trine of equations (q.v.). Literal arithmetic, then, or multiplying, dividing, etc., with letters instead of Arabic ciphers, is properly only a preparation for A. ; while analysis (q.v.), in the widest sense, would embrace A. as its first part. A. itself is divided into two chief branches. The first treats of equations involving unknown quantities having a determinate value; in the other, called the diophantine or indeterminate analysis, the unknown quantities have no exactly fixed values, but depend in some degree upon assumption.
The oldest work in the west on A. is that of Diophantus of Alexandria, in the 4th c. after Christ. It consisted originally of 13 books, and contained arithmetical problems ; only 6 books are now extant. They are written in Greek, and evince no little acute ness. The modern Europeans got their first acquaintance with A.. not directly from the Greeks, but, like most other knowledge, through the Arabs, who derived it, again, from the Hindoos. The chief European source was the work of Mohammed Ben Musa, who lived in the time of caliph Al Mamun (81:3-833) ; it has been translated into English by Dr. Rosen (Load. 1831). An Italian merchant, Leonardo Bonaccio, of Pisa, traveling
in the east about 1200, acquired a knowledge of the science, and introduced it among his countrymen on his return ; he has left a work on A. not yet printed. The first work on A. after the revival of learning is that of the Minorite friar Paciolo or Luca Borgo (Yen. 1494). Scipio Ferreo in Bologna, discovered, in 1505, the solution of one case of cubic equations. Tartaglia of Brescia (d. 1557) carried cubic equations still further, and imparted his discoveries to Cardan of Milan, as a secret. Cardan extended the discovery himself, and published, in 1545, the solution known as " Cardan's rule." Ludovico Fer rari and .13ombelli (1579) gave the solution of biquadratic equations. A. was first culti vated in Germany by Christian Rudolf, in a work printed in 1524 ; Stifel followed with his Arithmetica Integra (Nttrnb. 1544). Robert Rccorde, in England, and Pelletier, in France, wrote about 1550. Vieta, a Frenchman (d. 1603) first made the grand step of using letters to denote the known quantities as well as the unknown. Harriot, in Eng land (1631), and Girard, in Holland. (1633), still further improved on the advances made by Vieta. The Geometrie (1637) of Descartes makes an epoch in A. ; it is rich in new investigations. Descartes applied A. to geometry, and was the first to represent the nature of curves by means of equations. Fermat also contributed much to the science ; and so did the Arithmetica Universalic of Newton. To these names may be added Mac laurin, Moivre, Taylor, and Fontaine. Among the chief promoters of A., in more recent times, are Euler, Lagrange, Gauss, Abel, Fourier, Peacock, De Morgan, etc.