322 MATHEMATICS. 



CHAPTER II. 



ALGEBRA. 



ALGEBRA is that branch of Mathematics in which the relations of 

 quantities are expressed, and problems resolved, by means of letters 

 and other symbols. The name is derived from the Arabic phrase, 

 J?l gebr u al mocabela, signifying the reduction of equations : and 

 from the generality of its results, it has also been called Universal 

 Arithmetic. It presupposes a knowledge of Arithmetic, or at least 

 of the elementary rules, on the general principles of which it also 

 depends ; but in representing unknown or variable quantities by 

 letters, and expressing their relations by means of other symbols, it 

 reaches a wide range of useful and curious problems, and theorems, 

 which common Arithmetic could never grasp. 



The first germs of Algebra are found in the writings of Diophantus 

 of Alexandria ; who flourished A. D. 350, and is the reputed inventor 

 of the indeterminate analysis. His works, however, are merely a 

 collection of difficult questions concerning squares and cubes, and the 

 general properties of numbers. Here ends the history of Algebra 

 among the ancients : and, accordingly, its invention is ascribed by 

 some writers to the Hindoos; and by others to the Arabians; to 

 whom we are indebted, as has already been mentioned, for its intro- 

 duction into Europe. The earliest mentioned Hindoo writer on 

 Algebra, is said to have been the astronomer, Aryabhatta, probably 

 as early as the fifth century of our era. Some of the Arabians admit 

 that they received their Algebra from India ; but others attribute its 

 invention to their countryman, Mahomed Ben Musa, about A. D. 800; 

 and, in either case, it was doubtless improved by their mathematical 

 knowledge derived from Greek authors. 



The first printed treatise on Algebra, entitled Summa de Jlrith- 

 metica, was published in Italy, in 1494, by Lucas Paccioli de Borgo; 

 but it only extended to quadratic equations. The first resolution of 

 cubic equations, is claimed by Tartaglia, (or Tartalea), about 1535; 

 and that of biquadratic equations is ascribed to Ferrari, by Cardan 

 of Pavia, in his book De Arte Magna, published in 1545. Cardan 

 used letters to represent unknown quantities : but Vieta of France, 

 who died in 1603, first applied them to known quantities ; and thus 

 generalized the solutions. Vieta also improved the modes of resolv- 

 ing equations ; particularly by approximation. Harriot, of England, 

 who died in 1621, first discovered that every algebraic equation is 

 composed of as many factors of the first degree, as are indicated by 

 the degree of the equation. Descartes first introduced the use of 

 exponents ; and explained the nature of the negative roots of an 

 equation : and he also made the application of indeterminate co- 

 efficients, to resolve equations into their several factors. Newton 

 enriched Algebra, not only by farther discoveries concerning equa- 

 tions, but by the invention of the binomial theorem, for problems 

 of involution and evolution. The later discoveries of Maclaurin, 



