The philosophy of mathematics Part 4

_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.

This last and important branch of algebra is naturally divided into two orders of questions, viz., those which refer to the _composition_ of equations, and those which concern their _transformation_; these latter having for their object to modify the roots of an equation without knowing them, in accordance with any given law, providing that this law is uniform in relation to all the parts.[9]

[Footnote 9: The fundamental principle on which reposes the theory of equations, and which is so frequently applied in all mathematical a.n.a.lysis--the decomposition of algebraic, rational, and entire functions, of any degree whatever, into factors of the first degree--is never employed except for functions of a single variable, without any one having examined if it ought to be extended to functions of several variables. The general impossibility of such a decomposition is demonstrated by the author in detail, but more properly belongs to a special treatise.]

THE METHOD OF INDETERMINATE COEFFICIENTS.

To complete this rapid general enumeration of the different essential parts of the calculus of direct functions, I must, lastly, mention expressly one of the most fruitful and important theories of algebra proper, that relating to the transformation of functions into series by the aid of what is called the _Method of indeterminate Coefficients_.

This method, so eminently a.n.a.lytical, and which must be regarded as one of the most remarkable discoveries of Descartes, has undoubtedly lost some of its importance since the invention and the development of the infinitesimal calculus, the place of which it might so happily take in some particular respects. But the increasing extension of the transcendental a.n.a.lysis, although it has rendered this method much less necessary, has, on the other hand, multiplied its applications and enlarged its resources; so that by the useful combination between the two theories, which has finally been effected, the use of the method of indeterminate coefficients has become at present much more extensive than it was even before the formation of the calculus of indirect functions.

Having thus sketched the general outlines of algebra proper, I have now to offer some considerations on several leading points in the calculus of direct functions, our ideas of which may be advantageously made more clear by a philosophical examination.

IMAGINARY QUANt.i.tIES.

The difficulties connected with several peculiar symbols to which algebraic calculations sometimes lead, and especially to the expressions called _imaginary_, have been, I think, much exaggerated through purely metaphysical considerations, which have been forced upon them, in the place of regarding these abnormal results in their true point of view as simple a.n.a.lytical facts. Viewing them thus, we readily see that, since the spirit of mathematical a.n.a.lysis consists in considering magnitudes in reference to their relations only, and without any regard to their determinate value, a.n.a.lysts are obliged to admit indifferently every kind of expression which can be engendered by algebraic combinations.

The interdiction of even one expression because of its apparent singularity would destroy the generality of their conceptions. The common embarra.s.sment on this subject seems to me to proceed essentially from an unconscious confusion between the idea of _function_ and the idea of _value_, or, what comes to the same thing, between the _algebraic_ and the _arithmetical_ point of view. A thorough examination would show mathematical a.n.a.lysis to be much more clear in its nature than even mathematicians commonly suppose.

NEGATIVE QUANt.i.tIES.

As to negative quant.i.ties, which have given rise to so many misplaced discussions, as irrational as useless, we must distinguish between their _abstract_ signification and their _concrete_ interpretation, which have been almost always confounded up to the present day. Under the first point of view, the theory of negative quant.i.ties can be established in a complete manner by a single algebraical consideration. The necessity of admitting such expressions is the same as for imaginary quant.i.ties, as above indicated; and their employment as an a.n.a.lytical artifice, to render the formulas more comprehensive, is a mechanism of calculation which cannot really give rise to any serious difficulty. We may therefore regard the abstract theory of negative quant.i.ties as leaving nothing essential to desire; it presents no obstacles but those inappropriately introduced by sophistical considerations.

It is far from being so, however, with their concrete theory. This consists essentially in that admirable property of the signs + and-, of representing a.n.a.lytically the oppositions of directions of which certain magnitudes are susceptible. This _general theorem_ on the relation of the concrete to the abstract in mathematics is one of the most beautiful discoveries which we owe to the genius of Descartes, who obtained it as a simple result of properly directed philosophical observation. A great number of geometers have since striven to establish directly its general demonstration, but thus far their efforts have been illusory. Their vain metaphysical considerations and heterogeneous minglings of the abstract and the concrete have so confused the subject, that it becomes necessary to here distinctly enunciate the general fact. It consists in this: if, in any equation whatever, expressing the relation of certain quant.i.ties which are susceptible of opposition of directions, one or more of those quant.i.ties come to be reckoned in a direction contrary to that which belonged to them when the equation was first established, it will not be necessary to form directly a new equation for this second state of the phenomena; it will suffice to change, in the first equation, the sign of each of the quant.i.ties which shall have changed its direction; and the equation, thus modified, will always rigorously coincide with that which we would have arrived at in recommencing to investigate, for this new case, the a.n.a.lytical law of the phenomenon. The general theorem consists in this constant and necessary coincidence. Now, as yet, no one has succeeded in directly proving this; we have a.s.sured ourselves of it only by a great number of geometrical and mechanical verifications, which are, it is true, sufficiently multiplied, and especially sufficiently varied, to prevent any clear mind from having the least doubt of the exact.i.tude and the generality of this essential property, but which, in a philosophical point of view, do not at all dispense with the research for so important an explanation. The extreme extent of the theorem must make us comprehend both the fundamental difficulties of this research and the high utility for the perfecting of mathematical science which would belong to the general conception of this great truth. This imperfection of theory, however, has not prevented geometers from making the most extensive and the most important use of this property in all parts of concrete mathematics.

It follows from the above general enunciation of the fact, independently of any demonstration, that the property of which we speak must never be applied to magnitudes whose directions are continually varying, without giving rise to a simple opposition of direction; in that case, the sign with which every result of calculation is necessarily affected is not susceptible of any concrete interpretation, and the attempts sometimes made to establish one are erroneous. This circ.u.mstance occurs, among other occasions, in the case of a radius vector in geometry, and diverging forces in mechanics.

PRINCIPLE OF h.o.m.oGENEITY.

A second general theorem on the relation of the concrete to the abstract is that which is ordinarily designated under the name of _Principle of h.o.m.ogeneity_. It is undoubtedly much less important in its applications than the preceding, but it particularly merits our attention as having, by its nature, a still greater extent, since it is applicable to all phenomena without distinction, and because of the real utility which it often possesses for the verification of their a.n.a.lytical laws. I can, moreover, exhibit a direct and general demonstration of it which seems to me very simple. It is founded on this single observation, which is self-evident, that the exact.i.tude of every relation between any concrete magnitudes whatsoever is independent of the value of the _units_ to which they are referred for the purpose of expressing them in numbers.

For example, the relation which exists between the three sides of a right-angled triangle is the same, whether they are measured by yards, or by miles, or by inches.

It follows from this general consideration, that every equation which expresses the a.n.a.lytical law of any phenomenon must possess this property of being in no way altered, when all the quant.i.ties which are found in it are made to undergo simultaneously the change corresponding to that which their respective units would experience. Now this change evidently consists in all the quant.i.ties of each sort becoming at once _m_ times smaller, if the unit which corresponds to them becomes _m_ times greater, or reciprocally. Thus every equation which represents any concrete relation whatever must possess this characteristic of remaining the same, when we make _m_ times greater all the quant.i.ties which it contains, and which express the magnitudes between which the relation exists; excepting always the numbers which designate simply the mutual _ratios_ of these different magnitudes, and which therefore remain invariable during the change of the units. It is this property which const.i.tutes the law of h.o.m.ogeneity in its most extended signification, that is, of whatever a.n.a.lytical functions the equations may be composed.