51 G ALGEBRA [HISTORY. attempts at their solution. Much skill and ingenuity have been displayed by writers of more or less eminence in the attempt to elaborate a method of solution applicable to equations of the fifth degree, but they have failed ; whether it be that, like the ancient problems of the quadrature of the circle, and the duplication of the cube, an absolute solution is an impossibility, or whether it is reserved for future mathematicians to start in the research in some new path, and reach the goal by avoiding the old tracks which appear to have been thoroughly traversed in vain. It is scarcely necessary to refer to such writers as Hoene de Wronski, who, in 1811, announced a general method of solving all equations, giving formulge without demonstra tion. In 1 8 1 7, the Academy of Sciences of Lisbon proposed as the subject of a prize, the demonstration of Wronski s formulae. The prize was in the following year awarded to M. Torriani for the refutation of them. The reader will find in the fifth volume of the Reports of the British Association, an elaborate report by Sir W. R. Hamilton on a Method of Decomposition, proposed by Mr G. B. Jerrard in his Mathematical Researches, published at Bristol in a work of great beauty and originality, but which Hamilton is compelled to conclude fails to effect the desired object. In fact, the method which is valid when the proposed equation is itself of a sufficiently elevated degree, fails to reduce the solution of the eqiiation of the iifth degree to that of the fourth. But although the absolute solution of equations of higher orders than the fourth remains amongst the things un- effected, and rather to be hoped for than expected, a very great deal has been done towards preparing the way for approximate, if not for absolute solutions. In the first place, equations of the higher orders, when they assume certain forms, have been shown to be capable of solution. An equation of this kind, to all appearance of a very general and comprehensive form, had been solved by De Moivre in the Philosophical Transactions for 1737. Binomial equations had advanced under the skilful hands of Gauss, who, in his Disquisitiones Arithmeticae, which appeared in 1801, added largely to what had been done by Vandernionde in the classification and solution of such equations ; and subsequently, Abel, a mathematician of Norwegian birth, who died too early for science, completed and extended what Gauss had left imperfect. The collected writings of Abel published at Christiania in 1839, contain original and valuable contributions to this and many other branches of mathematics. But it is not in the solution of equations of certain forms that the greatest advance has been made during the present century. The basis of all methods of solution must evidently be found in the previous separation of the roots, and the efforts of mathematicians have been directed to the discovery of methods of effecting this. The object is not so much to classify the roots into positive and negative, real and imaginary, as to determine the situation and number of the real roots of the equation. The first writer on the subject whose methods appeared in print is Budan, whose treatise, entitled Nouvelle methode pour la resolution des equations numeriques, appeared in 1807. But there is evidence that Fourier had delivered lectures on the subject prior to the publication of Sudan s work, and consequently, without detriment to the claims of Budan, we may admit that the most valuable and original contribution to the science is to be found in Fourier s posthumous work, published by Navier in 1831, entitled Analyse des equations determinees. The theorem which Fourier gave for the discovery of the position, within narrow limits, of a root of an equation, is one of two theorems, each of which is known by mathematicians as "Fourier s Theorem." The other is a theorem of integration, and occurs in the author s magnificent woit Thtorie de la Chalcur. During the interval between the publication of Budan s work and that of Fourier, there appeared a paper in the Philosophical Transactions of the Royal Society for 1819, by W. G. Horner, upon a new method of solving arithmetical II :r:ier equations. From its being somewhat obscurely expressed, the great originality of the memoir did not at once appear. Gradually, however, Mr Homer s method came to be appre ciated, and it now ranks as one of the best processes, approaching, in some points, to Fourier s. In the Mcmoirea des savans etrangers for 1835, appears a memoir, which, if it does not absolutely supersede all that had been previously done in assigning the positions of the real roots of equations, yet in simplicity, completeness, and uni versality of application, surpasses them all. The author, M. Sturm, of French extraction, but born at Geneva, has Sturm. in this memoir linked his name to a theorem which is likely to retain its place amongst the permanent extensions of the domain of analysis as long as the study of algebra shall last. It was presented to the Academy in 1829. Determinants. The solution of simultaneous equations Detcrra of the first degree may be presented under the form of a nuts. 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 Exercices d analyse et de physique, mathematique, vol. ii. p. 161, but now replaced by the title determinants, a name first applied to certain forms of them by Gauss. In his (Jours d analyse algebrique, Cauchy terms them alternate func tions. The germ of the theory of determinants is to be found in the writings of Leibnitz, who, indeed, was far- Leibnit seeing enough to anticipate for it some of the power which, about a century after his time, it began to attain. More than half that 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 Crainci Analyse des lignes courbes algebriques, published at Geneva in 1750, gave the rule which establishes the sign of a product as plus or minus, according as the number of dis placements 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 Gauss. translation by M. Poullet-Delisle was given in 1807. Not withstanding 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, and others followed, and applied the results to geometrical problems. In 1826, Jacobi commenced a series of papers on the subject in Jacoln. 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 asso ciated 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 Sylvesti their originality, by their fecundity, by their grasp of all Cayley. the resources of analysis, these two powerful mathematicians have enriched the Transactions of the Royal Society, Crelle s Journal, the Cambridge and Dublin Mathematical Journal, and the Quarterly Journal of Mathematics, with papers on
this and on kindred branches of science of such value aa
