Their Classification. In the infancy of algebra, these equations were classed according to the number of their terms. But this classification was evidently faulty, since it separated cases which were really similar, and brought together others which had nothing in common besides this unimportant characteristic. It has been retained only for equations with two terms, which are, in fact, capable of being resolved in a manner peculiar to themselves.
The classification of equations by what is called their degrees, is, on the other hand, eminently natural, for this distinction rigorously determines the greater or less difficulty of their resolution. This gradation is apparent in the cases of all the equations which can be resolved; but it may be indicated in a general manner independently of the fact of the resolution. We need only consider that the most general equation of each degree necessarily comprehends all those of the different inferior degrees, as must also the formula which determines the unknown quantity. Consequently, however slight we may suppose the difficulty peculiar to the degree which we are considering, since it is inevitably complicated in the execution with those presented by all the preceding degrees, the resolution really offers more and more obstacles, in proportion as the degree of the equation is elevated.
ALGEBRAIC RESOLUTION OF EQUATIONS.
Its Limits. The resolution of algebraic equations is as yet known to us only in the four first degrees, such is the increase of difficulty noticed above. In this respect, algebra has made no considerable progress since the labours of Descartes and the Italian analysts of the sixteenth century, although in the last two centuries there has been perhaps scarcely a single geometer who has not busied himself in trying to advance the resolution of equations. The general equation of the fifth degree itself has thus far resisted all attacks.
The constantly increasing complication which the formulas for resolving equations must necessarily present, in proportion as the degree increases (the difficulty of using the formula of the fourth degree rendering it almost inapplicable), has determined analysts to renounce, by a tacit agreement, the pursuit of such researches, although they are far from regarding it as impossible to obtain the resolution of equations of the fifth degree, and of several other higher ones.
General Solution. The only question of this kind which would be really of great importance, at least in its logical relations, would be the general resolution of algebraic equations of any degree whatsoever. Now, the more we meditate on this subject, the more we are led to think, with Lagrange, that it really surpasses the scope of our intelligence. We must besides observe that the formula which would express the root of an equation of the mth degree would necessarily include radicals of the mth order (or functions of an equivalent multiplicity), because of the m determinations which it must admit. Since we have seen, besides, that this formula must also embrace, as a particular case, that formula which corresponds to every lower degree, it follows that it would inevitably also contain radicals of the next lower degree, the next lower to that, &c., so that, even if it were possible to discover it, it would almost always present too great a complication to be capable of being usefully employed, unless we could succeed in simplifying it, at the same time retaining all its generality, by the introduction of a new class of analytical elements of which we yet have no idea. We have, then, reason to believe that, without having already here arrived at the limits imposed by the feeble extent of our intelligence, we should not be long in reaching them if we actively and earnestly prolonged this series of investigations.