Part 4 (1/2)
_Their Cla.s.sification._ In the infancy of algebra, these equations were cla.s.sed according to the number of their terms. But this cla.s.sification 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 cla.s.sification 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 quant.i.ty. 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 a.n.a.lysts 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 a.n.a.lysts 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 surpa.s.ses the scope of our intelligence. We must besides observe that the formula which would express the _root_ of an equation of the _m^{th}_ degree would necessarily include radicals of the _m^{th}_ 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 cla.s.s of a.n.a.lytical 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.
It is, besides, important to observe that, even supposing we had obtained the resolution of _algebraic_ equations of any degree whatever, we would still have treated only a very small part of _algebra_, properly so called, that is, of the calculus of direct functions, including the resolution of all the equations which can be formed by the known a.n.a.lytical functions.
Finally, we must remember that, by an undeniable law of human nature, our means for conceiving new questions being much more powerful than our resources for resolving them, or, in other words, the human mind being much more ready to inquire than to reason, we shall necessarily always remain _below_ the difficulty, no matter to what degree of development our intellectual labour may arrive. Thus, even though we should some day discover the complete resolution of all the a.n.a.lytical equations at present known, chimerical as the supposition is, there can be no doubt that, before attaining this end, and probably even as a subsidiary means, we would have already overcome the difficulty (a much smaller one, though still very great) of conceiving new a.n.a.lytical elements, the introduction of which would give rise to cla.s.ses of equations of which, at present, we are completely ignorant; so that a similar imperfection in algebraic science would be continually reproduced, in spite of the real and very important increase of the absolute ma.s.s of our knowledge.
_What we know in Algebra._ In the present condition of algebra, the complete resolution of the equations of the first four degrees, of any binomial equations, of certain particular equations of the higher degrees, and of a very small number of exponential, logarithmic, or circular equations, const.i.tute the fundamental methods which are presented by the calculus of direct functions for the solution of mathematical problems. But, limited as these elements are, geometers have nevertheless succeeded in treating, in a truly admirable manner, a very great number of important questions, as we shall find in the course of the volume. The general improvements introduced within a century into the total system of mathematical a.n.a.lysis, have had for their princ.i.p.al object to make immeasurably useful this little knowledge which we have, instead of tending to increase it. This result has been so fully obtained, that most frequently this calculus has no real share in the complete solution of the question, except by its most simple parts; those which have reference to equations of the two first degrees, with one or more variables.
NUMERICAL RESOLUTION OF EQUATIONS.
The extreme imperfection of algebra, with respect to the resolution of equations, has led a.n.a.lysts to occupy themselves with a new cla.s.s of questions, whose true character should be here noted. They have busied themselves in filling up the immense gap in the resolution of algebraic equations of the higher degrees, by what they have named the _numerical resolution_ of equations. Not being able to obtain, in general, the _formula_ which expresses what explicit function of the given quant.i.ties the unknown one is, they have sought (in the absence of this kind of resolution, the only one really _algebraic_) to determine, independently of that formula, at least the _value_ of each unknown quant.i.ty, for various designated systems of particular values attributed to the given quant.i.ties. By the successive labours of a.n.a.lysts, this incomplete and illegitimate operation, which presents an intimate mixture of truly algebraic questions with others which are purely arithmetical, has been rendered possible in all cases for equations of any degree and even of any form. The methods for this which we now possess are sufficiently general, although the calculations to which they lead are often so complicated as to render it almost impossible to execute them. We have nothing else to do, then, in this part of algebra, but to simplify the methods sufficiently to render them regularly applicable, which we may hope hereafter to effect. In this condition of the calculus of direct functions, we endeavour, in its application, so to dispose the proposed questions as finally to require only this numerical resolution of the equations.
_Its limited Usefulness._ Valuable as is such a resource in the absence of the veritable solution, it is essential not to misconceive the true character of these methods, which a.n.a.lysts rightly regard as a very imperfect algebra. In fact, we are far from being always able to reduce our mathematical questions to depend finally upon only the _numerical_ resolution of equations; that can be done only for questions quite isolated or truly final, that is, for the smallest number. Most questions, in fact, are only preparatory, and intended to serve as an indispensable preparation for the solution of other questions. Now, for such an object, it is evident that it is not the actual _value_ of the unknown quant.i.ty which it is important to discover, but the _formula_, which shows how it is derived from the other quant.i.ties under consideration. It is this which happens, for example, in a very extensive cla.s.s of cases, whenever a certain question includes at the same time several unknown quant.i.ties. We have then, first of all, to separate them. By suitably employing the simple and general method so happily invented by a.n.a.lysts, and which consists in referring all the other unknown quant.i.ties to one of them, the difficulty would always disappear if we knew how to obtain the algebraic resolution of the equations under consideration, while the _numerical_ solution would then be perfectly useless. It is only for want of knowing the _algebraic_ resolution of equations with a single unknown quant.i.ty, that we are obliged to treat _Elimination_ as a distinct question, which forms one of the greatest special difficulties of common algebra. Laborious as are the methods by the aid of which we overcome this difficulty, they are not even applicable, in an entirely general manner, to the elimination of one unknown quant.i.ty between two equations of any form whatever.
In the most simple questions, and when we have really to resolve only a single equation with a single unknown quant.i.ty, this _numerical_ resolution is none the less a very imperfect method, even when it is strictly sufficient. It presents, in fact, this serious inconvenience of obliging us to repeat the whole series of operations for the slightest change which may take place in a single one of the quant.i.ties considered, although their relations to one another remain unchanged; the calculations made for one case not enabling us to dispense with any of those which relate to a case very slightly different. This happens because of our inability to abstract and treat separately that purely algebraic part of the question which is common to all the cases which result from the mere variation of the given numbers.
According to the preceding considerations, the calculus of direct functions, viewed in its present state, divides into two very distinct branches, according as its subject is the _algebraic_ resolution of equations or their _numerical_ resolution. The first department, the only one truly satisfactory, is unhappily very limited, and will probably always remain so; the second, too often insufficient, has, at least, the advantage of a much greater generality. The necessity of clearly distinguis.h.i.+ng these two parts is evident, because of the essentially different object proposed in each, and consequently the peculiar point of view under which quant.i.ties are therein considered.
_Different Divisions of the two Methods of Resolution._ If, moreover, we consider these parts with reference to the different methods of which each is composed, we find in their logical distribution an entirely different arrangement. In fact, the first part must be divided according to the nature of the equations which we are able to resolve, and independently of every consideration relative to the _values_ of the unknown quant.i.ties. In the second part, on the contrary, it is not according to the _degrees_ of the equations that the methods are naturally distinguished, since they are applicable to equations of any degree whatever; it is according to the numerical character of the _values_ of the unknown quant.i.ties; for, in calculating these numbers directly, without deducing them from general formulas, different means would evidently be employed when the numbers are not susceptible of having their values determined otherwise than by a series of approximations, always incomplete, or when they can be obtained with entire exactness. This distinction of _incommensurable_ and of _commensurable_ roots, which require quite different principles for their determination, important as it is in the numerical resolution of equations, is entirely insignificant in the algebraic resolution, in which the _rational_ or _irrational_ nature of the numbers which are obtained is a mere accident of the calculation, which cannot exercise any influence over the methods employed; it is, in a word, a simple arithmetical consideration. We may say as much, though in a less degree, of the division of the commensurable roots themselves into _entire_ and _fractional_. In fine, the case is the same, in a still greater degree, with the most general cla.s.sification of roots, as _real_ and _imaginary_. All these different considerations, which are preponderant as to the numerical resolution of equations, and which are of no importance in their algebraic resolution, render more and more sensible the essentially distinct nature of these two princ.i.p.al parts of algebra.
THE THEORY OF EQUATIONS.
These two departments, which const.i.tute the immediate object of the calculus of direct functions, are subordinate to a third one, purely speculative, from which both of them borrow their most powerful resources, and which has been very exactly designated by the general name of _Theory of Equations_, although it as yet relates only to _Algebraic_ equations. The numerical resolution of equations, because of its generality, has special need of this rational foundation.