**(B) HISTORY OF ALGEBRA**

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-- Determinants

-- Gottfried Leibnitz (1646-1716)

-- **Cramer**

-- Carl Friedrich Gauss (1771-1855)

-- Carl Jacobi (1804-51)

-- James Joseph Sylvester (1814-97)

-- Arthur Cayley (1821-95)

**Determinants**. - The solution of simultaneous equations of the first degree may be presented under the form of a set of fractions, the numerators and denominators of which are symmetric products of the coefficients of the unknown quantities in the equations. These products were originally known as *resultants*, a name applied to them by Laplace, and retained as late as 1841 by Cauchy in his *Exercises d'analyse et de physique mathématique*, vol ii p. 161, but now replaced by the title *determinants*, a name first applied to certain forms of them by Gauss. In his *Cours d'analyse algébrique*, Cauchy terms them *alternate* functions.

The germ of the theory of determinants is to be found in the writings of **Leibnitz**, who, indeed, was far seeing enough to anticipate for it some of the power which, about a century after his time, it began to attain. More than half period had indeed elapsed before any trace of its existence can be found in the writings of the mathematicians who succeeded Leibnitz.

The revival of the method is due to **Cramer**, who, in a note to his *Analyse des lignes courbes algébriques*, published at Geneva in 1750, gave the rule which establishes the sign of a product as plus or minus, according as the number of displacements from the typical form has been even or odd. Cramer was followed in the last century by Bezout, Laplace, Lagrange, and Vandermonde.

In 1801 appeared the *Disquisitiones Arithmeticae* of **Gauss**, of which a French translation by M. Poullet-Delisle was given in 1807. Notwithstanding the somewhat obscure form in which this work was presented, its originality gave a new impetus to investigations on this and kindred subjects. To Gauss is due the establishment of the important theorem, that the product of two determinants both of the second and third orders is a determinant. Binet, Cauchy, anfd others followed and applied the results to geometrical problems.

In 1826, **Jacobi** commenced a series of papers on the subject in *Crelle's Journal*. In these papers, which extended over a space of nearly twenty years, the subject was recast and made available for ordinary readers; and at the same time it was enriched by new and important theorems, through which the name of Jacobi is indissolubly associated with this branch of science.

Following the steps of Jacobi, a number of mathematicians of no mean power have entered the field. Pre-eminent above all others are two British names, those of **Sylvester** and **Cayley**. By their originality, by their fecundity, by their grasp of all the resources of analysis, these two powerful mathematicians have enriched the *Transactions of the Royal Society*, *Crelle's Jounral*, the *Cambridge and Dublin Mathematicial Journal*, and the *Quarterly Journal of Mathematics*, with papers on this and on kindred branches of science of such value as to have placed their authors at the head of living mathematicians. The reader will find the subject admirably treated in Baltzer's *Theorie und Anwendung der Determinenten*; and more briefly in Salmon's *Higher Algebra*. Elementary treatises have also been published by Spottiswoode in 1851, by Brioschi in 1854, by Todhunter in his *Theory of Equations* in 1861, and by Dodgson in 1867.

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