[Footnote 7: Simple as may seem, for example, the equation _a^x_ + _b^x_ = _c^x_, we do not yet know how to resolve it, which may give some idea of the extreme imperfection of this part of algebra.] ALGEBRAIC EQUATIONS. Considering now only these _Algebraic_ equations, we must observe, in the first place, that although they may often contain _irrational_ functions of the unknown quantities as well as _rational_ functions, we can always, by more or less easy transformations, make the first case come under the second, so that it is with this last that analysts have had to occupy themselves exclusively in order to resolve all sorts of _algebraic_ equations. _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.[8] It has been retained only for equations with two terms, which are, in fact, capable of being resolved in a manner peculiar to themselves. [Footnote 8: The same error was afterward committed, in the infancy of the infinitesimal calculus, in relation to the integration of differential equations.] 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.