Algebra is a branch of pure mathematics. The name is derived from the Arabs, who call the science Al jebṛ wa'l muqābalah ('redintegration and equation'). The term algebra is generally used to denote a method of calculating by means of letters which are employed to represent the numbers, and signs which are employed to represent their relations. Literal arithmetic, then, or multiplying, dividing, &c. with letters instead of Arabic ciphers, is properly only a preparation for algebra; while Analysis (q.v.), in the widest sense, would embrace algebra as its first part. Algebra itself is divided into two chief branches. The first treats of Equations (q.v.) 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 algebra is that of Diophantus of Alexandria, in the 4th century after Christ. It consisted originally of 13 books, and contained arithmetical problems; only six books are now extant. They are written in Greek, and evince no little acuteness. The modern Europeans got their first acquaintance with algebra, not directly from the Greeks, but, like most other knowledge, through the Arabs, who derived it, again, from the Hindus. The chief European source was the work of Mohammed Ben Musa, who lived in the time of Caliph Al Mamun (813-833); it was translated into English by Rosen (1831). An Italian merchant, Leonardo Bonaccio, of Pisa, travelling in the East about 1200, acquired a knowledge of the science, and introduced it among his countrymen on his return; he left a MS. work on algebra. The first work on algebra after the revival of learning is that of the Minorite friar Paciolo or Luca Borgo (Ven. 1494). Scipio Ferreo in Bologna discovered, in 1505, the solution of one case of cubic equations. Tartalea of Brescia (died 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 Ferrari and Bombelli (1572) gave the solution of biquadratic equations. Algebra was first cultivated in Germany by Christian Rudolf, in a work printed in 1524; Stifel followed with his Arithmetica Integra (Nürnberg, 1544). Robert Recorde in England, and Pelletier in France, wrote about 1550. Vieta, a Frenchman (died 1603), first made the grand step of using letters to denote the known quantities as well as the unknown. Harriot, in England (1631), and Girard, in Holland (1629), still further improved on the advances made by Vieta. The Géométrie (1637) of Descartes makes an epoch in algebra; it is rich in new investigations. Descartes applied algebra 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 Universalis of Newton. To these names may be added Maclaurin, Moivre, Taylor, Fontaine; and later, Euler, Lagrange, Gauss, Abel, Fourier, Peacock, De Morgan, Sylvester, and Cayley. See articles on ANALYSIS, BINOMIAL CALCULUS, DETERMINANTS, DIFFERENCE, DIOPHANTUS, EQUATIONS, FRACTIONS, FUNCTION, GEOMETRY, INVOLUTION, NUMBERS (THEORY OF), PERMUTATIONS AND COMBINATIONS, PROBABILITIES, PROBLEMS, SERIES, &c.