H. Grassmann, Die lineale Ausdehnungslehre (Leipzig, 1844), Die Ausdehnungslehre (Berlin, 1862) (these are reprinted with valuable emendations and notes in his Gesammelte math. u. phys. Werke, vol. i., Leipzig (2 parts), 1894, 1896), and papers in Grunert’s Arch. vi., Crelle, xlix.-lxxxiv., Math. Ann. vii. xii.; B. and C. S. Peirce, “Linear Associative Algebra,” Amer. Journ. Math. iv. (privately circulated, 1871); A. Cayley, on Matrices, Phil. Trans. cxlviii., on Multiple Algebra, Quart. M. Journ. xxii.; J. J. Sylvester, on Universal Algebra (i.e. Matrices), Amer. Journ. Math. vi.; H. J. S. Smith, on Linear Indeterminate Equations, Phil. Trans. cli.; R. S. Ball, Theory of Screws (Dublin, 1876); and papers in Phil. Trans. clxiv., and Trans. R. Ir. Ac. xxv.; W. K. Clifford, on Biquaternions, Proc. L. M. S. iv.; A. Buchheim, on Extensive Calculus and its Applications, Proc. L. M. S. xv.-xvii.; H. Taber, on Matrices, Amer. J. M. xii.; K. Weierstrass, “Zur Theorie der aus n Haupteinheiten gebildeten complexen Grössen,” Götting. Nachr. (1884); G. Frobenius, on Bilinear Forms, Crelle, lxxxiv., and Berl. Ber. (1896); L. Kronecker, on Complex Numbers and Modular Systems, Berl. Ber. (1888); G. Scheffers, “Complexe Zahlensysteme,” Math. Ann. xxxix. (this contains a bibliography up to 1890); S. Lie, Vorlesungen über continuirliche Gruppen (Leipzig, 1893), ch. xxi.; A. M‘Aulay, “Algebra after Hamilton, or Multenions,” Proc. R. S. E., 1908, 28. p. 503. For a more complete account see H. Hankel Theorie der complexen Zahlensysteme (Leipzig, 1867); O. Stolz, Vorlesungen über allgemeine Arithmetik (ibid., 1883); A. N. Whitehead, A Treatise on Universal Algebra, with Applications (vol. i., Cambridge, 1898) (a very comprehensive work, to which the writer of this article is in many ways indebted); and the Encyclopädie d. math. Wissenschaften (vol. i., Leipzig, 1898), &c., §§ A 1 (H. Schubert), A 4 (E. Study), and B 1 c (G. Landsberg). For the history of the development of ordinary algebra M. Cantor’s Vorlesungen über Geschichte der Mathematik is the standard authority. (G. B. M.)
C. History
Various derivations of the word “algebra,” which is of Arabian origin, have been given by different writers. The first mention of the word is to be found in the title of a work by Mahommed ben Musa al-Khwarizmi (Hovarezmi), who flourished about the beginning of the 9th century. Etymology.The full title is ilm al-jebr wa’l-muqābala, which contains the ideas of restitution and comparison, or opposition and comparison, or resolution and equation, jebr being derived from the verb jabara, to reunite, and muqābala, from gabala, to make equal. (The root jabara is also met with in the word algebrista, which means a “bone-setter,” and is still in common use in Spain.) The same derivation is given by Lucas Paciolus (Luca Pacioli), who reproduces the phrase in the transliterated form alghebra e almucabala, and ascribes the invention of the art to the Arabians.
Other writers have derived the word from the Arabic particle al (the definite article), and geber, meaning “man.” Since, however, Geber happened to be the name of a celebrated Moorish philosopher who flourished in about the 11th or 12th century, it has been supposed that he was the founder of algebra, which has since perpetuated his name. The evidence of Peter Ramus (1515–1572) on this point is interesting, but he gives no authority for his singular statements. In the preface to his Arithmeticae libri duo et totidem Algebrae (1560) he says: “The name Algebra is Syriac, signifying the art or doctrine of an excellent man. For Geber, in Syriac, is a name applied to men, and is sometimes a term of honour, as master or doctor among us. There was a certain learned mathematician who sent his algebra, written in the Syriac language, to Alexander the Great, and he named it almucabala, that is, the book of dark or mysterious things, which others would rather call the doctrine of algebra. To this day the same book is in great estimation among the learned in the oriental nations, and by the Indians, who cultivate this art, it is called aljabra and alboret; though the name of the author himself is not known.” The uncertain authority of these statements, and the plausibility of the preceding explanation, have caused philologists to accept the derivation from al and jabara. Robert Recorde in his Whetstone of Witte (1557) uses the variant algeber, while John Dee (1527–1608) affirms that algiebar, and not algebra, is the correct form, and appeals to the authority of the Arabian Avicenna.
Although the term “algebra” is now in universal use, various other appellations were used by the Italian mathematicians during the Renaissance. Thus we find Paciolus calling it l ’Arte Magiore; ditta dal vulgo la Regula de la Cosa over Alghebra e Almucabala. The name l ’arte magiore, the greater art, is designed to distinguish it from l ’arte minore, the lesser art, a term which he applied to the modern arithmetic. His second variant, la regula de la cosa, the rule of the thing or unknown quantity, appears to have been in common use in Italy, and the word cosa was preserved for several centuries in the forms coss or algebra, cossic or algebraic, cossist or algebraist, &c. Other Italian writers termed it the Regula rei et census, the rule of the thing and the product, or the root and the square. The principle underlying this expression is probably to be found in the fact that it measured the limits of their attainments in algebra, for they were unable to solve equations of a higher degree than the quadratic or square.
Franciscus Vieta (François Viète) named it Specious Arithmetic, on account of the species of the quantities involved, which he represented symbolically by the various letters of the alphabet. Sir Isaac Newton introduced the term Universal Arithmetic, since it is concerned with the doctrine of operations, not affected on numbers, but on general symbols.
Notwithstanding these and other idiosyncratic appellations, European mathematicians have adhered to the older name, by which the subject is now universally known.
It is difficult to assign the invention of any art or science definitely to any particular age or race. The few fragmentary records, which have come down to us from past civilizations, must not be regarded as representing the totality of their knowledge, and the omission of a science or art does not necessarily imply that the science or art was unknown. It was formerly the custom to assign the invention of algebra to the Greeks, but since the decipherment of the Rhind papyrus by Eisenlohr this view has changed, for in this work there are distinct signs of an algebraic analysis. The particular problem—a heap (hau) and its seventh makes 19—is solved as we should now solve a simple equation; but Ahmes varies his methods in other similar problems. This discovery carries the invention of algebra back to about 1700 B.C., if not earlier.
It is probable that the algebra of the Egyptians was of a most rudimentary nature, for otherwise we should expect to find traces of it in the works of the Greek geometers, of whom Thales of Miletus (640–546 B.C.) was the first. Notwithstanding the prolixity of writers and the Greek algebra.number of the writings, all attempts at extracting an algebraic analysis from their geometrical theorems and problems have been fruitless, and it is generally conceded that their analysis was geometrical and had little or no affinity to algebra. The first extant work which approaches to a treatise on algebra is by Diophantus (q.v.), an Alexandrian mathematician, who flourished about A.D. 350. The original, which consisted of a preface and thirteen books, is now lost, but we have a Latin translation of the first six books and a fragment of another on polygonal numbers by Xylander of Augsburg (1575), and Latin and Greek translations by Gaspar Bachet de Merizac (1621–1670). Other editions have been published, of which we may mention Pierre Fermat’s (1670), T. L. Heath’s (1885) and P. Tannery’s (1893–1895). In the preface to this work, which is dedicated to one Dionysus, Diophantus explains his notation, naming the square, cube and fourth powers, dynamis, cubus, dynamodinimus, and so on, according to the sum in the indices. The unknown he terms arithmos, the number, and in solutions he marks it by the final ς; he explains the generation of powers, the rules for multiplication and division of simple quantities, but he does not treat of the addition, subtraction, multiplication and division of compound quantities. He then proceeds to discuss various artifices for the simplification of equations, giving methods which are still in common use. In the body of the work he displays considerable ingenuity in reducing his problems to simple equations, which admit either of direct solution, or fall into the class known as indeterminate equations. This latter class he discussed so assiduously that they are often known as Diophantine problems, and the methods of resolving them as the Diophantine analysis (see Equation, Indeterminate). It is
