That of Lagrange.This perfect unity of analysis, and this purely abstract character of its fundamental notions, are found in the highest degree in the conception of Lagrange, and are found there alone; it is, for this reason, the most rational and the most philosophical of all. Carefully removing every heterogeneous consideration, Lagrange has reduced the transcendental analysis to its true peculiar character, that of presenting a very extensive class of analytical transformations, which facilitate in a remarkable degree the expression of the conditions of various problems. At the same time, this analysis is thus necessarily presented as a simple extension of ordinary analysis; it is only a higher algebra. All the different parts of abstract mathematics, previously so incoherent, have from that moment admitted of being conceived as forming a single system.
Unhappily, this conception, which possesses such fundamental properties, independently of its so simple and so lucid notation, and which is undoubtedly destined to become the final theory of transcendental analysis, because of its high philosophical superiority over all the other methods proposed, presents in its present state toomany difficulties in its applications, as compared with the conception of Newton, and still more with that of Leibnitz, to be as yet exclusively adopted. Lagrange himself has succeeded only with great difficulty in rediscovering, by his method, the principal results already obtained by the infinitesimal method for the solution of the general questions of geometry and mechanics; we may judge from that what obstacles would be found in treating in the same manner questions which were truly new and important. It is true that Lagrange, on several occasions, has shown that difficulties call forth, from men of genius, superior efforts, capable of leading to the greatest results. It was thus that, in trying to adapt his method to the examination of the curvature of lines, which seemed so far from admitting its application, he arrived at that beautiful theory of contacts which has so greatly perfected that important part of geometry. But, in spite of such happy exceptions, the conception of Lagrange has nevertheless remained, as a whole, essentially unsuited to applications.
The final result of the general comparison which I have too briefly sketched, is, then, as already suggested, that, in order to really understand the transcendental analysis, we should not only consider it in its principles according to the three fundamental conceptions of Leibnitz, of Newton, and of Lagrange, but should besides accustom ourselves to carry out almost indifferently, according to these three principal methods, and especially according to the first and the last, the solution of all important questions, whether of the pure calculus of indirect functions or of its applications. This is a course which I could not too strongly recommend to all those who desireto judge philosophically of this admirable creation of the human mind, as well as to those who wish to learn to make use of this powerful instrument with success and with facility. In all the other parts of mathematical science, the consideration of different methods for a single class of questions may be useful, even independently of its historical interest, but it is not indispensable; here, on the contrary, it is strictly necessary.
Having determined with precision, in this chapter, the philosophical character of the calculus of indirect functions, according to the principal fundamental conceptions of which it admits, we have next to consider, in the following chapter, the logical division and the general composition of this calculus.
Thecalculus of indirect functions, in accordance with the considerations explained in the preceding chapter, is necessarily divided into two parts (or, more properly, is decomposed into two differentcalculientirely distinct, although intimately connected by their nature), according as it is proposed to find the relations between the auxiliary magnitudes (the introduction of which constitutes the general spirit of this calculus) by means of the relations between the corresponding primitive magnitudes; or, conversely, to try to discover these direct equations by means of the indirect equations originally established. Such is, in fact, constantly the double object of the transcendental analysis.
These two systems have received different names, according to the point of view under which this analysis has been regarded. The infinitesimal method, properly so called, having been the most generally employed for the reasons which have been given, almost all geometers employ habitually the denominations ofDifferential Calculusand ofIntegral Calculus, established by Leibnitz, and which are, in fact, very rational consequences of his conception. Newton, in accordance with his method, named the first theCalculus of Fluxions, and the second theCalculus of Fluents, expressions which were commonly employed in England. Finally, followingthe eminently philosophical theory founded by Lagrange, one would be called theCalculus of Derived Functions, and the other theCalculus of Primitive Functions. I will continue to make use of the terms of Leibnitz, as being more convenient for the formation of secondary expressions, although I ought, in accordance with the suggestions made in the preceding chapter, to employ concurrently all the different conceptions, approaching as nearly as possible to that of Lagrange.
The differential calculus is evidently the logical basis of the integral calculus; for we do not and cannot know how to integrate directly any other differential expressions than those produced by the differentiation of the ten simple functions which constitute the general elements of our analysis. The art of integration consists, then, essentially in bringing all the other cases, as far as is possible, to finally depend on only this small number of fundamental integrations.
In considering the whole body of the transcendental analysis, as I have characterized it in the preceding chapter, it is not at first apparent what can be the peculiar utility of the differential calculus, independently of this necessary relation with the integral calculus, which seems as if it must be, by itself, the only one directly indispensable. In fact, the elimination of theinfinitesimalsor of thederivatives, introduced as auxiliaries to facilitate the establishment of equations, constituting, as we have seen, the final and invariable object of the calculus of indirect functions, it is natural to think that the calculus which teaches how to deduce from the equations betweenthese auxiliary magnitudes, those which exist between the primitive magnitudes themselves, ought strictly to suffice for the general wants of the transcendental analysis without our perceiving, at the first glance, what special and constant part the solution of the inverse question can have in such an analysis. It would be a real error, though a common one, to assign to the differential calculus, in order to explain its peculiar, direct, and necessary influence, the destination of forming the differential equations, from which the integral calculus then enables us to arrive at the finite equations; for the primitive formation of differential equations is not and cannot be, properly speaking, the object of any calculus, since, on the contrary, it forms by its nature the indispensable starting point of any calculus whatever. How, in particular, could the differential calculus, which in itself is reduced to teaching the means ofdifferentiatingthe different equations, be a general procedure for establishing them? That which in every application of the transcendental analysis really facilitates the formation of equations, is the infinitesimalmethod, and not the infinitesimalcalculus, which is perfectly distinct from it, although it is its indispensable complement. Such a consideration would, then, give a false idea of the special destination which characterizes the differential calculus in the general system of the transcendental analysis.
But we should nevertheless very imperfectly conceive the real peculiar importance of this first branch of the calculus of indirect functions, if we saw in it only a simple preliminary labour, having no other general and essential object than to prepare indispensable foundations for the integral calculus. As the ideas on this matterare generally confused, I think that I ought here to explain in a summary manner this important relation as I view it, and to show that in every application of the transcendental analysis a primary, direct, and necessary part is constantly assigned to the differential calculus.
1.Use of the Differential Calculus as preparatory to that of the Integral.In forming the differential equations of any phenomenon whatever, it is very seldom that we limit ourselves to introduce differentially only those magnitudes whose relations are sought. To impose that condition would be to uselessly diminish the resources presented by the transcendental analysis for the expression of the mathematical laws of phenomena. Most frequently we introduce into the primitive equations, through their differentials, other magnitudes whose relations are already known or supposed to be so, and without the consideration of which it would be frequently impossible to establish equations. Thus, for example, in the general problem of the rectification of curves, the differential equation,
ds2=dy2+dx2, ords2=dx2+dy2+dz2,
is not only established between the desired function s and the independent variablex, to which it is referred, but, at the same time, there have been introduced, as indispensable intermediaries, the differentials of one or two other functions,yandz, which are among the data of the problem; it would not have been possible to form directly the equation betweendsanddx, which would, besides, be peculiar to each curve considered. It is the same for most questions. Now in these cases it is evident that the differential equation is not immediately suitable for integration. It is previously necessary that the differentialsof the functions supposed to be known, which have been employed as intermediaries, should be entirely eliminated, in order that equations may be obtained between the differentials of the functions which alone are sought and those of the really independent variables, after which the question depends on only the integral calculus. Now this preparatory elimination of certain differentials, in order to reduce the infinitesimals to the smallest number possible, belongs simply to the differential calculus; for it must evidently be done by determining, by means of the equations between the functions supposed to be known, taken as intermediaries, the relations of their differentials, which is merely a question of differentiation. Thus, for example, in the case of rectifications, it will be first necessary to calculatedy, ordyanddz, by differentiating the equation or the equations of each curve proposed; after eliminating these expressions, the general differential formula above enunciated will then contain onlydsanddx; having arrived at this point, the elimination of the infinitesimals can be completed only by the integral calculus.
Such is, then, the general office necessarily belonging to the differential calculus in the complete solution of the questions which exact the employment of the transcendental analysis; to produce, as far as is possible, the elimination of the infinitesimals, that is, to reduce in each case the primitive differential equations so that they shall contain only the differentials of the really independent variables, and those of the functions sought, by causing to disappear, by elimination, the differentials of all the other known functions which may have been taken as intermediaries at the time of the formation of the differentialequations of the problem which is under consideration.
2.Employment of the Differential Calculus alone.For certain questions, which, although few in number, have none the less, as we shall see hereafter, a very great importance, the magnitudes which are sought enter directly, and not by their differentials, into the primitive differential equations, which then contain differentially only the different known functions employed as intermediaries, in accordance with the preceding explanation. These cases are the most favourable of all; for it is evident that the differential calculus is then entirely sufficient for the complete elimination of the infinitesimals, without the question giving rise to any integration. This is what occurs, for example, in the problem oftangentsin geometry; in that ofvelocitiesin mechanics, &c.
3.Employment of the Integral Calculus alone.Finally, some other questions, the number of which is also very small, but the importance of which is no less great, present a second exceptional case, which is in its nature exactly the converse of the preceding. They are those in which the differential equations are found to be immediately ready for integration, because they contain, at their first formation, only the infinitesimals which relate to the functions sought, or to the really independent variables, without its being necessary to introduce, differentially, other functions as intermediaries. If in these new cases we introduce these last functions, since, by hypothesis, they will enter directly and not by their differentials, ordinary algebra will suffice to eliminate them, and to bring the question to depend on only the integral calculus. The differential calculus will then have nospecial part in the complete solution of the problem, which will depend entirely upon the integral calculus. The general question ofquadraturesoffers an important example of this, for the differential equation being thendA = ydx, will become immediately fit for integration as soon as we shall have eliminated, by means of the equation of the proposed curve, the intermediary functiony, which does not enter into it differentially. The same circumstances exist in the problem ofcubatures, and in some others equally important.
Three classes of Questions hence resulting.As a general result of the previous considerations, it is then necessary to divide into three classes the mathematical questions which require the use of the transcendental analysis; thefirstclass comprises the problems susceptible of being entirely resolved by means of the differential calculus alone, without any need of the integral calculus; thesecond, those which are, on the contrary, entirely dependent upon the integral calculus, without the differential calculus having any part in their solution; lastly, in thethirdand the most extensive, which constitutes the normal case, the two others being only exceptional, the differential and the integral calculus have each in their turn a distinct and necessary part in the complete solution of the problem, the former making the primitive differential equations undergo a preparation which is indispensable for the application of the latter. Such are exactly their general relations, of which too indefinite and inexact ideas are generally formed.
Let us now take a general survey of the logical composition of each calculus, beginning with the differential.
In the exposition of the transcendental analysis, it is customary to intermingle with the purely analytical part (which reduces itself to the treatment of the abstract principles of differentiation and integration) the study of its different principal applications, especially those which concern geometry. This confusion of ideas, which is a consequence of the actual manner in which the science has been developed, presents, in the dogmatic point of view, serious inconveniences in this respect, that it makes it difficult properly to conceive either analysis or geometry. Having to consider here the most rational co-ordination which is possible, I shall include, in the following sketch, only the calculus of indirect functions properly so called, reserving for the portion of this volume which relates to the philosophical study ofconcretemathematics the general examination of its great geometrical and mechanical applications.
Two Cases: explicit and implicit Functions.The fundamental division of the differential calculus, or of the general subject of differentiation, consists in distinguishing two cases, according as the analytical functions which are to be differentiated areexplicitorimplicit; from which flow two parts ordinarily designated by the names of differentiationof formulasand differentiationof equations. It is easy to understand,à priori, the importance of this classification. In fact, such a distinction would be illusory if the ordinary analysis was perfect; that is, if we knew how to resolve all equations algebraically, for then it would be possible to render everyimplicitfunctionexplicit; and, by differentiatingit in that state alone, the second part of the differential calculus would be immediately comprised in the first, without giving rise to any new difficulty. But the algebraical resolution of equations being, as we have seen, still almost in its infancy, and as yet impossible for most cases, it is plain that the case is very different, since we have, properly speaking, to differentiate a function without knowing it, although it is determinate. The differentiation of implicit functions constitutes then, by its nature, a question truly distinct from that presented by explicit functions, and necessarily more complicated. It is thus evident that we must commence with the differentiation of formulas, and reduce the differentiation of equations to this primary case by certain invariable analytical considerations, which need not be here mentioned.
These two general cases of differentiation are also distinct in another point of view equally necessary, and too important to be left unnoticed. The relation which is obtained between the differentials is constantly more indirect, in comparison with that of the finite quantities, in the differentiation of implicit functions than in that of explicit functions. We know, in fact, from the considerations presented by Lagrange on the general formation of differential equations, that, on the one hand, the same primitive equation may give rise to a greater or less number of derived equations of very different forms, although at bottom equivalent, depending upon which of the arbitrary constants is eliminated, which is not the case in the differentiation of explicit formulas; and that, on the other hand, the unlimited system of the different primitive equations, which correspond to thesame derived equation, presents a much more profound analytical variety than that of the different functions, which admit of one same explicit differential, and which are distinguished from each other only by a constant term. Implicit functions must therefore be regarded as being in reality still more modified by differentiation than explicit functions. We shall again meet with this consideration relatively to the integral calculus, where it acquires a preponderant importance.
Two Sub-cases: A single Variable or several Variables.Each of the two fundamental parts of the Differential Calculus is subdivided into two very distinct theories, according as we are required to differentiate functions of a single variable or functions of several independent variables. This second case is, by its nature, quite distinct from the first, and evidently presents more complication, even in considering only explicit functions, and still more those which are implicit. As to the rest, one of these cases is deduced from the other in a general manner, by the aid of an invariable and very simple principle, which consists in regarding the total differential of a function which is produced by the simultaneous increments of the different independent variables which it contains, as the sum of the partial differentials which would be produced by the separate increment of each variable in turn, if all the others were constant. It is necessary, besides, carefully to remark, in connection with this subject, a new idea which is introduced by the distinction of functions into those of one variable and of several; it is the consideration of these different special derived functions, relating to each variable separately, and the number of which increases more andmore in proportion as the order of the derivation becomes higher, and also when the variables become more numerous. It results from this that the differential relations belonging to functions of several variables are, by their nature, both much more indirect, and especially much more indeterminate, than those relating to functions of a single variable. This is most apparent in the case of implicit functions, in which, in the place of the simple arbitrary constants which elimination causes to disappear when we form the proper differential equations for functions of a single variable, it is the arbitrary functions of the proposed variables which are then eliminated; whence must result special difficulties when these equations come to be integrated.
Finally, to complete this summary sketch of the different essential parts of the differential calculus proper, I should add, that in the differentiation of implicit functions, whether of a single variable or of several, it is necessary to make another distinction; that of the case in which it is required to differentiate at once different functions of this kind,combinedin certain primitive equations, from that in which all these functions areseparate.
The functions are evidently, in fact, still more implicit in the first case than in the second, if we consider that the same imperfection of ordinary analysis, which forbids our converting every implicit function into an equivalent explicit function, in like manner renders us unable to separate the functions which enter simultaneously into any system of equations. It is then necessary to differentiate, not only without knowing how to resolve the primitive equations, but even without beingable to effect the proper eliminations among them, thus producing a new difficulty.
Reduction of the whole to the Differentiation of the ten elementary Functions.Such, then, are the natural connection and the logical distribution of the different principal theories which compose the general system of differentiation. Since the differentiation of implicit functions is deduced from that of explicit functions by a single constant principle, and the differentiation of functions of several variables is reduced by another fixed principle to that of functions of a single variable, the whole of the differential calculus is finally found to rest upon the differentiation of explicit functions with a single variable, the only one which is ever executed directly. Now it is easy to understand that this first theory, the necessary basis of the entire system, consists simply in the differentiation of the ten simple functions, which are the uniform elements of all our analytical combinations, and the list of which has been given in the first chapter, on page 51; for the differentiation of compound functions is evidently deduced, in an immediate and necessary manner, from that of the simple functions which compose them. It is, then, to the knowledge of these ten fundamental differentials, and to that of the two general principles just mentioned, which bring under it all the other possible cases, that the whole system of differentiation is properly reduced. We see, by the combination of these different considerations, how simple and how perfect is the entire system of the differential calculus. It certainly constitutes, in its logical relations, the most interesting spectacle which mathematical analysis can present to our understanding.
Transformation of derived Functions for new Variables.The general sketch which I have just summarily drawn would nevertheless present an important deficiency, if I did not here distinctly indicate a final theory, which forms, by its nature, the indispensable complement of the system of differentiation. It is that which has for its object the constant transformation of derived functions, as a result of determinate changes in the independent variables, whence results the possibility of referring to new variables all the general differential formulas primitively established for others. This question is now resolved in the most complete and the most simple manner, as are all those of which the differential calculus is composed. It is easy to conceive the general importance which it must have in any of the applications of the transcendental analysis, the fundamental resources of which it may be considered as augmenting, by permitting us to choose (in order to form the differential equations, in the first place, with more ease) that system of independent variables which may appear to be the most advantageous, although it is not to be finally retained. It is thus, for example, that most of the principal questions of geometry are resolved much more easily by referring the lines and surfaces torectilinearco-ordinates, and that we may, nevertheless, have occasion to express these lines, etc., analytically by the aid ofpolarco-ordinates, or in any other manner. We will then be able to commence the differential solution of the problem by employing the rectilinear system, but only as an intermediate step, from which, by the general theory here referred to, we can pass to the final system, which sometimes could not have been considered directly.
Different Orders of Differentiation.In the logical classification of the differential calculus which has just been given, some may be inclined to suggest a serious omission, since I have not subdivided each of its four essential parts according to another general consideration, which seems at first view very important; namely, that of the higher or lower order of differentiation. But it is easy to understand that this distinction has no real influence in the differential calculus, inasmuch as it does not give rise to any new difficulty. If, indeed, the differential calculus was not rigorously complete, that is, if we did not know how to differentiate at will any function whatever, the differentiation to the second or higher order of each determinate function might engender special difficulties. But the perfect universality of the differential calculus plainly gives us the assurance of being able to differentiate, to any order whatever, all known functions whatever, the question reducing itself to a constantly repeated differentiation of the first order. This distinction, unimportant as it is for the differential calculus, acquires, however, a very great importance in the integral calculus, on account of the extreme imperfection of the latter.
Analytical Applications.Finally, though this is not the place to consider the various applications of the differential calculus, yet an exception may be made for those which consist in the solution of questions which are purely analytical, which ought, indeed, to be logically treated in continuation of a system of differentiation, because of the evident homogeneity of the considerations involved. These questions may be reduced to three essential ones.
Firstly, thedevelopment into seriesof functions of one or more variables, or, more generally, the transformation of functions, which constitutes the most beautiful and the most important application of the differential calculus to general analysis, and which comprises, besides the fundamental series discovered by Taylor, the remarkable series discovered by Maclaurin, John Bernouilli, Lagrange, &c.:
Secondly, the generaltheory of maxima and minimavalues for any functions whatever, of one or more variables; one of the most interesting problems which analysis can present, however elementary it may now have become, and to the complete solution of which the differential calculus naturally applies:
Thirdly, the general determination of the true value of functions which present themselves under anindeterminateappearance for certain hypotheses made on the values of the corresponding variables; which is the least extensive and the least important of the three.
The first question is certainly the principal one in all points of view; it is also the most susceptible of receiving a new extension hereafter, especially by conceiving, in a broader manner than has yet been done, the employment of the differential calculus in the transformation of functions, on which subject Lagrange has left some valuable hints.
Having thus summarily, though perhaps too briefly, considered the chief points in the differential calculus, I now proceed to an equally rapid exposition of a systematic outline of the Integral Calculus, properly so called, that is, the abstract subject of integration.
Its Fundamental Division.The fundamental division of the Integral Calculus is founded on the same principle as that of the Differential Calculus, in distinguishing the integration ofexplicitdifferential formulas, and the integration ofimplicitdifferentials or of differential equations. The separation of these two cases is even much more profound in relation to integration than to differentiation. In the differential calculus, in fact, this distinction rests, as we have seen, only on the extreme imperfection of ordinary analysis. But, on the other hand, it is easy to see that, even though all equations could be algebraically resolved, differential equations would none the less constitute a case of integration quite distinct from that presented by the explicit differential formulas; for, limiting ourselves, for the sake of simplicity, to the first order, and to a single functionyof a single variablex, if we suppose any differential equation betweenx,y, anddy/dx, to be resolved with reference tody/dx, the expression of the derived function being then generally found to contain the primitive function itself, which is the object of the inquiry, the question of integration will not have at all changed its nature, and the solution will not really have made any other progress than that of having brought the proposed differential equation to be of only the first degree relatively to the derived function, which is in itself of little importance. The differential would not then be determined in a manner much lessimplicitthan before, as regards the integration, which would continue to present essentially the same characteristic difficulty.The algebraic resolution of equations could not make the case which we are considering come within the simple integration of explicit differentials, except in the special cases in which the proposed differential equation did not contain the primitive function itself, which would consequently permit us, by resolving it, to finddy/dxin terms ofxonly, and thus to reduce the question to the class of quadratures. Still greater difficulties would evidently be found in differential equations of higher orders, or containing simultaneously different functions of several independent variables.
The integration of differential equations is then necessarily more complicated than that of explicit differentials, by the elaboration of which last the integral calculus has been created, and upon which the others have been made to depend as far as it has been possible. All the various analytical methods which have been proposed for integrating differential equations, whether it be the separation of the variables, the method of multipliers, &c., have in fact for their object to reduce these integrations to those of differential formulas, the only one which, by its nature, can be undertaken directly. Unfortunately, imperfect as is still this necessary base of the whole integral calculus, the art of reducing to it the integration of differential equations is still less advanced.
Subdivisions: one variable or several.Each of these two fundamental branches of the integral calculus is next subdivided into two others (as in the differential calculus, and for precisely analogous reasons), according as we consider functions with asingle variable, or functions withseveral independent variables.
This distinction is, like the preceding one, still more important for integration than for differentiation. This is especially remarkable in reference to differential equations. Indeed, those which depend on several independent variables may evidently present this characteristic and much more serious difficulty, that the desired function may be differentially defined by a simple relation between its different special derivatives relative to the different variables taken separately. Hence results the most difficult and also the most extensive branch of the integral calculus, which is commonly named theIntegral Calculus of partial differences, created by D'Alembert, and in which, according to the just appreciation of Lagrange, geometers ought to have seen a really new calculus, the philosophical character of which has not yet been determined with sufficient exactness. A very striking difference between this case and that of equations with a single independent variable consists, as has been already observed, in the arbitrary functions which take the place of the simple arbitrary constants, in order to give to the corresponding integrals all the proper generality.
It is scarcely necessary to say that this higher branch of transcendental analysis is still entirely in its infancy, since, even in the most simple case, that of an equation of the first order between the partial derivatives of a single function with two independent variables, we are not yet completely able to reduce the integration to that of the ordinary differential equations. The integration of functions of several variables is much farther advanced in the case (infinitely more simple indeed) in which it has to do with only explicit differential formulas. We can then, in fact, when these formulas fulfil the necessaryconditions of integrability, always reduce their integration to quadratures.
Other Subdivisions: different Orders of Differentiation.A new general distinction, applicable as a subdivision to the integration of explicit or implicit differentials, with one variable or several, is drawn from thehigher or lower order of the differentials: a distinction which, as we have above remarked, does not give rise to any special question in the differential calculus.
Relatively toexplicit differentials, whether of one variable or of several, the necessity of distinguishing their different orders belongs only to the extreme imperfection of the integral calculus. In fact, if we could always integrate every differential formula of the first order, the integration of a formula of the second order, or of any other, would evidently not form a new question, since, by integrating it at first in the first degree, we would arrive at the differential expression of the immediately preceding order, from which, by a suitable series of analogous integrations, we would be certain of finally arriving at the primitive function, the final object of these operations. But the little knowledge which we possess on integration of even the first order causes quite another state of affairs, so that a higher order of differentials produces new difficulties; for, having differential formulas of any order above the first, it may happen that we may be able to integrate them, either once, or several times in succession, and that we may still be unable to go back to the primitive functions, if these preliminary labours have produced, for the differentials of a lower order, expressions whose integrals are not known. This circumstance must occur so much the oftener (the number of knownintegrals being still very small), seeing that these successive integrals are generally very different functions from the derivatives which have produced them.
With reference toimplicit differentials, the distinction of orders is still more important; for, besides the preceding reason, the influence of which is evidently analogous in this case, and is even greater, it is easy to perceive that the higher order of the differential equations necessarily gives rise to questions of a new nature. In fact, even if we could integrate every equation of the first order relating to a single function, that would not be sufficient for obtaining the final integral of an equation of any order whatever, inasmuch as every differential equation is not reducible to that of an immediately inferior order. Thus, for example, if we have given any relation betweenx,y,dx/dy, andd2y/dx2, to determine a functionyof a variablex, we shall not be able to deduce from it at once, after effecting a first integration, the corresponding differential relation betweenx,y, anddy/dx, from which, by a second integration, we could ascend to the primitive equations. This would not necessarily take place, at least without introducing new auxiliary functions, unless the proposed equation of the second order did not contain the required functiony, together with its derivatives. As a general principle, differential equations will have to be regarded as presenting cases which are more and moreimplicit, as they are of a higher order, and which cannot be made to depend on one another except by special methods, the investigation of which consequently forms a new class of questions, with respectto which we as yet know scarcely any thing, even for functions of a single variable.[10]
Another equivalent distinction.Still farther, when we examine more profoundly this distinction of different orders of differential equations, we find that it can be always made to come under a final general distinction, relative to differential equations, which remains to be noticed. Differential equations with one or more independent variables may contain simply a single function, or (in a case evidently more complicated and more implicit, which corresponds to the differentiation of simultaneous implicit functions) we may have to determine at the same time several functions from the differential equations in which they are found united, together with their different derivatives. It is clear that such a state of the question necessarily presents a new special difficulty, that of separating the different functions desired, by forming for each, from the proposed differential equations, an isolated differential equation which does not contain the other functions or their derivatives. This preliminary labour, which is analogous to the elimination of algebra, is evidently indispensable before attempting any direct integration, since we cannot undertake generally (except by special artifices which are very rarely applicable) to determine directly several distinct functions at once.
Now it is easy to establish the exact and necessary coincidence of this new distinction with the precedingone respecting the order of differential equations. We know, in fact, that the general method for isolating functions in simultaneous differential equations consists essentially in forming differential equations, separately in relation to each function, and of an order equal to the sum of all those of the different proposed equations. This transformation can always be effected. On the other hand, every differential equation of any order in relation to a single function might evidently always be reduced to the first order, by introducing a suitable number of auxiliary differential equations, containing at the same time the different anterior derivatives regarded as new functions to be determined. This method has, indeed, sometimes been actually employed with success, though it is not the natural one.
Here, then, are two necessarily equivalent orders of conditions in the general theory of differential equations; the simultaneousness of a greater or smaller number of functions, and the higher or lower order of differentiation of a single function. By augmenting the order of the differential equations, we can isolate all the functions; and, by artificially multiplying the number of the functions, we can reduce all the equations to the first order. There is, consequently, in both cases, only one and the same difficulty from two different points of sight. But, however we may conceive it, this new difficulty is none the less real, and constitutes none the less, by its nature, a marked separation between the integration of equations of the first order and that of equations of a higher order. I prefer to indicate the distinction under this last form as being more simple, more general, and more logical.
Quadratures.From the different considerations which have been indicated respecting the logical dependence of the various principal parts of the integral calculus, we see that the integration of explicit differential formulas of the first order and of a single variable is the necessary basis of all other integrations, which we never succeed in effecting but so far as we reduce them to this elementary case, evidently the only one which, by its nature, is capable of being treated directly. This simple fundamental integration is often designated by the convenient expression ofquadratures, seeing that every integral of this kind, Sf(x)dx, may, in fact, be regarded as representing the area of a curve, the equation of which in rectilinear co-ordinates would bey=f(x). Such a class of questions corresponds, in the differential calculus, to the elementary case of the differentiation of explicit functions of a single variable. But the integral question is, by its nature, very differently complicated, and especially much more extensive than the differential question. This latter is, in fact, necessarily reduced, as we have seen, to the differentiation of the ten simple functions, the elements of all which are considered in analysis. On the other hand, the integration of compound functions does not necessarily follow from that of the simple functions, each combination of which may present special difficulties with respect to the integral calculus. Hence results the naturally indefinite extent, and the so varied complication of the question ofquadratures, upon which, in spite of all the efforts of analysts, we still possess so little complete knowledge.
In decomposing this question, as is natural, according to the different forms which may be assumed by thederivative function, we distinguish the case ofalgebraicfunctions and that oftranscendentalfunctions.
Integration of Transcendental Functions.The truly analytical integration of transcendental functions is as yet very little advanced, whether forexponential, or forlogarithmic, or forcircularfunctions. But a very small number of cases of these three different kinds have as yet been treated, and those chosen from among the simplest; and still the necessary calculations are in most cases extremely laborious. A circumstance which we ought particularly to remark in its philosophical connection is, that the different procedures of quadrature have no relation to any general view of integration, and consist of simple artifices very incoherent with each other, and very numerous, because of the very limited extent of each.
One of these artifices should, however, here be noticed, which, without being really a method of integration, is nevertheless remarkable for its generality; it is the procedure invented by John Bernouilli, and known under the name ofintegration by parts, by means of which every integral may be reduced to another which is sometimes found to be more easy to be obtained. This ingenious relation deserves to be noticed for another reason, as having suggested the first idea of that transformation of integrals yet unknown, which has lately received a greater extension, and of which M. Fourier especially has made so new and important a use in the analytical questions produced by the theory of heat.
Integration of Algebraic Functions.As to the integration of algebraic functions, it is farther advanced. However, we know scarcely any thing in relation to irrationalfunctions, the integrals of which have been obtained only in extremely limited cases, and particularly by rendering them rational. The integration of rational functions is thus far the only theory of the integral calculus which has admitted of being treated in a truly complete manner; in a logical point of view, it forms, then, its most satisfactory part, but perhaps also the least important. It is even essential to remark, in order to have a just idea of the extreme imperfection of the integral calculus, that this case, limited as it is, is not entirely resolved except for what properly concerns integration viewed in an abstract manner; for, in the execution, the theory finds its progress most frequently quite stopped, independently of the complication of the calculations, by the imperfection of ordinary analysis, seeing that it makes the integration finally depend upon the algebraic resolution of equations, which greatly limits its use.
To grasp in a general manner the spirit of the different procedures which are employed in quadratures, we must observe that, by their nature, they can be primitively founded only on the differentiation of the ten simple functions. The results of this, conversely considered, establish as many direct theorems of the integral calculus, the only ones which can be directly known. All the art of integration afterwards consists, as has been said in the beginning of this chapter, in reducing all the other quadratures, so far as is possible, to this small number of elementary ones, which unhappily we are in most cases unable to effect.
Singular Solutions.In this systematic enumeration of the various essential parts of the integral calculus, considered in their logical relations, I have designedly neglected(in order not to break the chain of sequence) to consider a very important theory, which forms implicitly a portion of the general theory of the integration of differential equations, but which I ought here to notice separately, as being, so to speak, outside of the integral calculus, and being nevertheless of the greatest interest, both by its logical perfection and by the extent of its applications. I refer to what are calledSingular Solutionsof differential equations, called sometimes, but improperly,particularsolutions, which have been the subject of very remarkable investigations by Euler and Laplace, and of which Lagrange especially has presented such a beautiful and simple general theory. Clairaut, who first had occasion to remark their existence, saw in them a paradox of the integral calculus, since these solutions have the peculiarity of satisfying the differential equations without being comprised in the corresponding general integrals. Lagrange has since explained this paradox in the most ingenious and most satisfactory manner, by showing how such solutions are always derived from the general integral by the variation of the arbitrary constants. He was also the first to suitably appreciate the importance of this theory, and it is with good reason that he devoted to it so full a development in his "Calculus of Functions." In a logical point of view, this theory deserves all our attention by the character of perfect generality which it admits of, since Lagrange has given invariable and very simple procedures for finding thesingularsolution of any differential equation which is susceptible of it; and, what is no less remarkable, these procedures require no integration, consisting only of differentiations, and are therefore alwaysapplicable. Differentiation has thus become, by a happy artifice, a means of compensating, in certain circumstances, for the imperfection of the integral calculus. Indeed, certain problems especially require, by their nature, the knowledge of thesesingularsolutions; such, for example, in geometry, are all the questions in which a curve is to be determined from any property of its tangent or its osculating circle. In all cases of this kind, after having expressed this property by a differential equation, it will be, in its analytical relations, thesingularequation which will form the most important object of the inquiry, since it alone will represent the required curve; the general integral, which thenceforth it becomes unnecessary to know, designating only the system of the tangents, or of the osculating circles of this curve. We may hence easily understand all the importance of this theory, which seems to me to be not as yet sufficiently appreciated by most geometers.
Definite Integrals.Finally, to complete our review of the vast collection of analytical researches of which is composed the integral calculus, properly so called, there remains to be mentioned one theory, very important in all the applications of the transcendental analysis, which I have had to leave outside of the system, as not being really destined for veritable integration, and proposing, on the contrary, to supply the place of the knowledge of truly analytical integrals, which are most generally unknown. I refer to the determination ofdefinite integrals.
The expression, always possible, of integrals in infinite series, may at first be viewed as a happy general means of compensating for the extreme imperfection of the integral calculus. But the employment of such series,because of their complication, and of the difficulty of discovering the law of their terms, is commonly of only moderate utility in the algebraic point of view, although sometimes very essential relations have been thence deduced. It is particularly in the arithmetical point of view that this procedure acquires a great importance, as a means of calculating what are calleddefinite integrals, that is, the values of the required functions for certain determinate values of the corresponding variables.
An inquiry of this nature exactly corresponds, in transcendental analysis, to the numerical resolution of equations in ordinary analysis. Being generally unable to obtain the veritable integral—named by opposition thegeneralorindefiniteintegral; that is, the function which, differentiated, has produced the proposed differential formula—analysts have been obliged to employ themselves in determining at least, without knowing this function, the particular numerical values which it would take on assigning certain designated values to the variables. This is evidently resolving the arithmetical question without having previously resolved the corresponding algebraic one, which most generally is the most important one. Such an analysis is, then, by its nature, as imperfect as we have seen the numerical resolution of equations to be. It presents, like this last, a vicious confusion of arithmetical and algebraic considerations, whence result analogous inconveniences both in the purely logical point of view and in the applications. We need not here repeat the considerations suggested in our third chapter. But it will be understood that, unable as we almost always are to obtain the true integrals, it is of the highest importance to have been ableto obtain this solution, incomplete and necessarily insufficient as it is. Now this has been fortunately attained at the present day for all cases, the determination of the value of definite integrals having been reduced to entirely general methods, which leave nothing to desire, in a great number of cases, but less complication in the calculations, an object towards which are at present directed all the special transformations of analysts. Regarding now this sort oftranscendental arithmeticas perfect, the difficulty in the applications is essentially reduced to making the proposed research depend, finally, on a simple determination of definite integrals, which evidently cannot always be possible, whatever analytical skill may be employed in effecting such a transformation.
Prospects of the Integral Calculus.From the considerations indicated in this chapter, we see that, while the differential calculus constitutes by its nature a limited and perfect system, to which nothing essential remains to be added, the integral calculus, or the simple system of integration, presents necessarily an inexhaustible field for the activity of the human mind, independently of the indefinite applications of which the transcendental analysis is evidently susceptible. The general argument by which I have endeavoured, in the second chapter, to make apparent the impossibility of ever discovering the algebraic solution of equations of any degree and form whatsoever, has undoubtedly infinitely more force with regard to the search for a single method of integration, invariably applicable to all cases. "It is," says Lagrange, "one of those problems whose general solution we cannot hope for." The more we meditate onthis subject, the more we will be convinced that such a research is utterly chimerical, as being far above the feeble reach of our intelligence; although the labours of geometers must certainly augment hereafter the amount of our knowledge respecting integration, and thus create methods of greater generality. The transcendental analysis is still too near its origin—there is especially too little time since it has been conceived in a truly rational manner—for us now to be able to have a correct idea of what it will hereafter become. But, whatever should be our legitimate hopes, let us not forget to consider, before all, the limits which are imposed by our intellectual constitution, and which, though not susceptible of a precise determination, have none the less an incontestable reality.
I am induced to think that, when geometers shall have exhausted the most important applications of our present transcendental analysis, instead of striving to impress upon it, as now conceived, a chimerical perfection, they will rather create new resources by changing the mode of derivation of the auxiliary quantities introduced in order to facilitate the establishment of equations, and the formation of which might follow an infinity of other laws besides the very simple relation which has been chosen, according to the conception suggested in the first chapter. The resources of this nature appear to me susceptible of a much greater fecundity than those which would consist of merely pushing farther our present calculus of indirect functions. It is a suggestion which I submit to the geometers who have turned their thoughts towards the general philosophy of analysis.
Finally, although, in the summary exposition which was the object of this chapter, I have had to exhibit thecondition of extreme imperfection which still belongs to the integral calculus, the student would have a false idea of the general resources of the transcendental analysis if he gave that consideration too great an importance. It is with it, indeed, as with ordinary analysis, in which a very small amount of fundamental knowledge respecting the resolution of equations has been employed with an immense degree of utility. Little advanced as geometers really are as yet in the science of integrations, they have nevertheless obtained, from their scanty abstract conceptions, the solution of a multitude of questions of the first importance in geometry, in mechanics, in thermology, &c. The philosophical explanation of this double general fact results from the necessarily preponderating importance and grasp ofabstractbranches of knowledge, the least of which is naturally found to correspond to a crowd ofconcreteresearches, man having no other resource for the successive extension of his intellectual means than in the consideration of ideas more and more abstract, and still positive.
In order to finish the complete exposition of the philosophical character of the transcendental analysis, there remains to be considered a final conception, by which the immortal Lagrange has rendered this analysis still better adapted to facilitate the establishment of equations in the most difficult problems, by considering a class of equations still moreindirectthan the ordinary differential equations. It is theCalculus, or, rather, theMethod of Variations; the general appreciation of which will be our next subject.
In order to grasp with more ease the philosophical character of theMethod of Variations, it will be well to begin by considering in a summary manner the special nature of the problems, the general resolution of which has rendered necessary the formation of this hyper-transcendental analysis. It is still too near its origin, and its applications have been too few, to allow us to obtain a sufficiently clear general idea of it from a purely abstract exposition of its fundamental theory.
The mathematical questions which have given birth to theCalculus of Variationsconsist generally in the investigation of themaximaandminimaof certain indeterminate integral formulas, which express the analytical law of such or such a phenomenon of geometry or mechanics, considered independently of any particular subject. Geometers for a long time designated all the questions of this character by the common name ofIsoperimetrical Problems, which, however, is really suitable to only the smallest number of them.
Ordinary Questions of Maxima and Minima.In the common theory ofmaximaandminima, it is proposed to discover, with reference to a given function of one or more variables, what particular values must be assigned to these variables, in order that the correspondingvalue of the proposed function may be amaximumor aminimumwith respect to those values which immediately precede and follow it; that is, properly speaking, we seek to know at what instant the function ceases to increase and commences to decrease, or reciprocally. The differential calculus is perfectly sufficient, as we know, for the general resolution of this class of questions, by showing that the values of the different variables, which suit either the maximum or minimum, must always reduce to zero the different first derivatives of the given function, taken separately with reference to each independent variable, and by indicating, moreover, a suitable characteristic for distinguishing the maximum from the minimum; consisting, in the case of a function of a single variable, for example, in the derived function of the second order taking a negative value for the maximum, and a positive value for the minimum. Such are the well-known fundamental conditions belonging to the greatest number of cases.
A new Class of Questions.The construction of this general theory having necessarily destroyed the chief interest which questions of this kind had for geometers, they almost immediately rose to the consideration of a new order of problems, at once much more important and of much greater difficulty—those ofisoperimeters. It is, then, no longerthe values of the variablesbelonging to the maximum or the minimum of a given function that it is required to determine. It isthe form of the function itselfwhich is required to be discovered, from the condition of the maximum or of the minimum of a certain definite integral, merely indicated, which depends upon that function.
Solid of least Resistance.The oldest question of this nature is that ofthe solid of least resistance, treated by Newton in the second book of the Principia, in which he determines what ought to be the meridian curve of a solid of revolution, in order that the resistance experienced by that body in the direction of its axis may be the least possible. But the course pursued by Newton, from the nature of his special method of transcendental analysis, had not a character sufficiently simple, sufficiently general, and especially sufficiently analytical, to attract geometers to this new order of problems. To effect this, the application of the infinitesimal method was needed; and this was done, in 1695, by John Bernouilli, in proposing the celebrated problem of theBrachystochrone.
This problem, which afterwards suggested such a long series of analogous questions, consists in determining the curve which a heavy body must follow in order to descend from one point to another in the shortest possible time. Limiting the conditions to the simple fall in a vacuum, the only case which was at first considered, it is easily found that the required curve must be a reversed cycloid with a horizontal base, and with its origin at the highest point. But the question may become singularly complicated, either by taking into account the resistance of the medium, or the change in the intensity of gravity.
Isoperimeters.Although this new class of problems was in the first place furnished by mechanics, it is in geometry that the principal investigations of this character were subsequently made. Thus it was proposed to discover which, among all the curves of the same contourtraced between two given points, is that whose area is a maximum or minimum, whence has come the name ofProblem of Isoperimeters; or it was required that the maximum or minimum should belong to the surface produced by the revolution of the required curve about an axis, or to the corresponding volume; in other cases, it was the vertical height of the center of gravity of the unknown curve, or of the surface and of the volume which it might generate, which was to become a maximum or minimum, &c. Finally, these problems were varied and complicated almost to infinity by the Bernouillis, by Taylor, and especially by Euler, before Lagrange reduced their solution to an abstract and entirely general method, the discovery of which has put a stop to the enthusiasm of geometers for such an order of inquiries. This is not the place for tracing the history of this subject. I have only enumerated some of the simplest principal questions, in order to render apparent the original general object of the method of variations.
Analytical Nature of these Problems.We see that all these problems, considered in an analytical point of view, consist, by their nature, in determining what form a certain unknown function of one or more variables ought to have, in order that such or such an integral, dependent upon that function, shall have, within assigned limits, a value which is a maximum or a minimum with respect to all those which it would take if the required function had any other form whatever.
Thus, for example, in the problem of thebrachystochrone, it is well known that ify=f(z),x= π(z), are the rectilinear equations of the required curve, supposing the axes ofxand ofyto be horizontal, and the axis ofzto be vertical, the time of the fall of a heavy body in that curve from the point whose ordinate isz1, to that whose ordinate isz2, is expressed in general terms by the definite integral
∫_{z_{2}}z_{1}√(1 + (f'(z))2+ (π'(z))2/(2gz))dz.
It is, then, necessary to find what the two unknown functionsfand π must be, in order that this integral may be a minimum.
In the same way, to demand what is the curve among all plane isoperimetrical curves, which includes the greatest area, is the same thing as to propose to find, among all the functionsf(x)which can give a certain constant value to the integral
∫dx√(1 + (f'(x))2),
that one which renders the integral ∫f(x)dx, taken between the same limits, a maximum. It is evidently always so in other questions of this class.
Methods of the older Geometers.In the solutions which geometers before Lagrange gave of these problems, they proposed, in substance, to reduce them to the ordinary theory of maxima and minima. But the means employed to effect this transformation consisted in special simple artifices peculiar to each case, and the discovery of which did not admit of invariable and certain rules, so that every really new question constantly reproduced analogous difficulties, without the solutions previously obtained being really of any essential aid, otherwise than by their discipline and training of the mind. In a word, this branch of mathematics presented, then, the necessary imperfection which always exists when the part common to all questions of the same class has notyet been distinctly grasped in order to be treated in an abstract and thenceforth general manner.
Lagrange, in endeavouring to bring all the different problems of isoperimeters to depend upon a common analysis, organized into a distinct calculus, was led to conceive a new kind of differentiation, to which he has applied the characteristic δ, reserving the characteristicdfor the common differentials. These differentials of a new species, which he has designated under the name ofVariations, consist of the infinitely small increments which the integrals receive, not by virtue of analogous increments on the part of the corresponding variables, as in the ordinary transcendental analysis, but by supposing that theformof the function placed under the sign of integration undergoes an infinitely small change. This distinction is easily conceived with reference to curves, in which we see the ordinate, or any other variable of the curve, admit of two sorts of differentials, evidently very different, according as we pass from one point to another infinitely near it on the same curve, or to the corresponding point of the infinitely near curve produced by a certain determinate modification of the first curve.[11]It is moreover clear, that the relativevariationsof different magnitudes connected with each other by any laws whatever are calculated, all but the characteristic, almost exactly in the same manner as the differentials. Finally,from the general notion ofvariationsare in like manner deduced the fundamental principles of the algorithm proper to this method, consisting simply in the evidently permissible liberty of transposing at will the characteristics specially appropriated to variations, before or after those which correspond to the ordinary differentials.
This abstract conception having been once formed, Lagrange was able to reduce with ease, and in the most general manner, all the problems ofIsoperimetersto the simple ordinary theory ofmaximaandminima. To obtain a clear idea of this great and happy transformation, we must previously consider an essential distinction which arises in the different questions of isoperimeters.
Two Classes of Questions.These investigations must, in fact, be divided into two general classes, according as the maxima and minima demanded areabsoluteorrelative, to employ the abridged expressions of geometers.
Questions of the first Class.Thefirst caseis that in which the indeterminate definite integrals, the maximum or minimum of which is sought, are not subjected, by the nature of the problem, to any condition; as happens, for example, in the problem of thebrachystochrone, in which the choice is to be made between all imaginable curves. Thesecondcase takes place when, on the contrary, the variable integrals can vary only according to certain conditions, which usually consist in other definite integrals (which depend, in like manner, upon the required functions) always retaining the same given value; as, for example, in all the geometrical questions relating to realisoperimetricalfigures, and in which, by the nature of the problem, the integral relating to thelength of the curve, or to the area of the surface, must remain constant during the variation of that integral which is the object of the proposed investigation.
TheCalculus of Variationsgives immediately the general solution of questions of the former class; for it evidently follows, from the ordinary theory of maxima and minima, that the required relation must reduce to zero thevariationof the proposed integral with reference to each independent variable; which gives the condition common to both the maximum and the minimum: and, as a characteristic for distinguishing the one from the other, that the variation of the second order of the same integral must be negative for the maximum and positive for the minimum. Thus, for example, in the problem of the brachystochrone, we will have, in order to determine the nature of the curve sought, the equation of condition
δ∫_{z_{2}}z_{1}√([1 + (f'(z))2+ (π'(z))2]/(2gz))dz= 0,
which, being decomposed into two, with respect to the two unknown functionsfand π, which are independent of each other, will completely express the analytical definition of the required curve. The only difficulty peculiar to this new analysis consists in the elimination of the characteristic δ, for which the calculus of variations furnishes invariable and complete rules, founded, in general, on the method of "integration by parts," from which Lagrange has thus derived immense advantage. The constant object of this first analytical elaboration (which this is not the place for treating in detail) is to arrive at real differential equations, which can always be done; and thereby the question comes under the ordinarytranscendental analysis, which furnishes the solution, at least so far as to reduce it to pure algebra if the integration can be effected. The general object of the method of variations is to effect this transformation, for which Lagrange has established rules, which are simple, invariable, and certain of success.
Equations of Limits.Among the greatest special advantages of the method of variations, compared with the previous isolated solutions of isoperimetrical problems, is the important consideration of what Lagrange callsEquations of Limits, which were entirely neglected before him, though without them the greater part of the particular solutions remained necessarily incomplete. When the limits of the proposed integrals are to be fixed, their variations being zero, there is no occasion for noticing them. But it is no longer so when these limits, instead of being rigorously invariable, are only subjected to certain conditions; as, for example, if the two points between which the required curve is to be traced are not fixed, and have only to remain upon given lines or surfaces. Then it is necessary to pay attention to the variation of their co-ordinates, and to establish between them the relations which correspond to the equations of these lines or of these surfaces.