The philosophy of mathematics · Auguste Comte
NUMERICAL RESOLUTION OF EQUATIONS.
The extreme imperfection of algebra, with respect to the resolution of equations, has led analysts to occupy themselves with a new class 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 quantities 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 quantity, for various designated systems of particular values attributed to the given quantities. By the successive labours of analysts, 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 analysts 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 quantity which it is important to discover, but the _formula_, which shows how it is derived from the other quantities under consideration. It is this which happens, for example, in a very extensive class of cases, whenever a certain question includes at the same time several unknown quantities. We have then, first of all, to separate them. By suitably employing the simple and general method so happily invented by analysts, and which consists in referring all the other unknown quantities 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 quantity, 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 quantity 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 quantity, 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 quantities 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 distinguishing these two parts is evident, because of the essentially different object proposed in each, and consequently the peculiar point of view under which quantities 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 quantities. 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 quantities; 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 classification 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 principal parts of algebra.
Text from Project Gutenberg, public domain in the United States.