(B) HISTORY OF ALGEBRA

(xiv) General Solution of Equations of the Fifth and Higher Orders, Still a Desideratum

Admitting, then, that every equation has a root, it becomes a question to what extent are we in possession of an analysis by which the root can be ascertained. If the question be put absolutely, we fear the answer must be, that in this matter we are in the same position that we have held for the last three centuries. Cubic and biquadratic equations can be solved, whatever they may be; but equations of higher orders, in which there exists no relation amongst the several coefficients, and no known or assumed connection between the different roots have baffled all 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 formulae without demonstration. In 1817, 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 equation of the fifth degree to that of the fourth.





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