ALGEBRAIC AND TRANSCENDENTAL FUNCTIONS. 83 by reducing them to this elementary case, which is the only one capable of being treated directly. This Quadratures. simple fundamental integration, often conveniently called quadratures, 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, quite otherwise complicated, and much more extensive than the differential question. We have seen that the latter is reduced to the differentiation of ten simple functions, which furnish the elements of analysis; but 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 the indefinite extent and varied complication of the question of quadratures, of which we know scarcely anything completely after all the efforts of analysts. Algebraic functions. The question is divided into the two cases of algebraic functions and transcendental functions. The algebraic class is the more advanced of the two. In relation to irrational functions, it is true, we know scarcely anything, the integrals of them having been obtained only in very restricted cases, and particularly by rendering them rational. The integration of rational functions is thus far the only theory of this calculus which has admitted of complete treatment; and thus it forms, in a logical point of view, its most satisfactory part, though it is perhaps the least important. Even here, the imperfection of ordinary analysis usually comes in to stop the working of the theory, by which the integration finally depends on the algebraic solution of equations; and thus it is only in what concerns integration viewed in an abstract manner that even this limited case is resolved. And this gives us an idea of the extreme imperfection of the integral calculus. The case of the inteTranscendental gration of transcendental functions is quite functions. in its infancy as yet, as regards either exponential, logarithmic, or circular functions. Very few cases of these kinds have been treated; and though the simplest have been chosen, the necessary calculations are extremely laborious. Singular solutions. The theory of Singular Solutions (sometimes called Particular Solutions), fully developed by Lagrange in his Calculus of Functions, but not yet duly appreciated by geometers, must be noticed here, on account of its logical perfection and the extent of its applications. This theory forms implicitly a portion of the general theory of the integration of differential equations; but I have left it till now, because it is, as it were, outside of the integral calculus, and I wished to preserve the sequence of its parts. Clairaut first observed the existence of these solutions, and he saw in them a paradox of the integral calculus, since they have the property of satisfying the differential equations without being comprehended in the corresponding general integrals. Lagrange explained this paradox by showing how such solutions are always derived from the general integral by the variation of the arbitrary constants. This theory has a character of perfect generality; for Lagrange has given invariable and very simple processes for finding the singular solution of any differential equation which admits of it and, what is very remarkable, these processes require no integration, consisting only of differentiations, and being therefore always applicable. Thus has differentiation become, by a happy artifice, a means of compensating, in certain circumstances, for the imperfection of the integral calculus. Definite integrals. t; One more theory remains to be noticed, to complete our review of that collection of analytical researches which constitutes the integral calculus. It takes its place outside of the system, because, instead of being destined for true integration, it proposes to supply the defect of our ignorance of really analytical integrals. I refer to the determination of definite integrals. These definite integrals are the values of the required functions for certain determinate values of the corresponding variables. The use of these in transcendental analysis corresponds to the numerical resolution of equations in ordinary analysis. Analysts being usually unable to obtain the real integral (called in opposition the general or indefinite integral), that is, the function which, differentiated, has produced the proposed differential formula, have been driven to determining, at least, without knowing this function, the particular numerical values which it would take on assigning certain declared values to the variables. This is evidently resolving the arithmetical question without having first resolved the corresponding algebraic one, which is generally the most important; and such an analysis is, by its nature, as imperfect as that of the numerical resolution of equations. Inconveniences, logical and practical, result from such a confusion of arithmetical and algebraic considerations. But, under our inability to obtain the true integrals, it is of the utmost importance to have been able to obtain this solution, incomplete and insufficient as it is. This has now been attained for all cases, the determination of the value of definite integrals having been reduced to entirely general methods, which leave nothing to be desired, in many cases, but less complexity in the calculations; an object to which analysts are now directing all their special transformations. This kind of transcendental arithmetic being considered perfect, the difficulty in its applications is reduced to making the proposed inquiry finally depend only on a simple determination of definite integrals; a thing which evidently cannot be always possible, whatever analytical skill may be employed in effecting so forced a transformation. Prospects of the Integral Calculus. We have now seen that while the differential calculus constitutes by its nature a limited and perfect system, the integral calculus, or the simple subject of integration, offers inexhaustible scope for the activity of the human mind, independently of the indefinite applications of which transcendental analysis is evidently capable. The reasons which convince us of the impossibility of ever achieving the general resolution of algebraic equations of any degree whatever, are yet more decisive against our attainment of a single method of integration applicable to all cases. "It is," said Lagrange, one of those problems whose general solution we cannot hope for." The more we meditate on the subject, the more convinced we shall be that such a research is wholly chimerical, as transcending the scope of our understanding, though the labours of geometers must certainly add in time to our knowledge of integration, and create procedures of 66 a wider generality. The transcendental analysis is yet too near its origin, it has too recently been regarded in a truly rational manner, for us to have any idea what it may hereafter become. But, whatever may be our legitimate hopes, we must ever, in the first place, consider the limits imposed by our intellectual constitution, which are not the less real because we cannot precisely assign them. I have hinted that a future augmentation of our resources may probably arise from a change in the mode of derivation of the auxiliary quantities introduced to facilitate the establishment of equations. Their formation might follow a multitude of other laws besides the very simple relation which has been selected. I discern here far greater resources than in urging further our present calculus of indirect functions; and I am persuaded that when geometers have exhausted the most important applications of our present transcendental analysis, they will turn their attention in this direction, instead of straining after perfection where it cannot be found. I submit this view to geometers whose meditations are fixed on the general philosophy of analysis. As for the rest, though I was bound to exhibit in my summary exposition the state of extreme imperfection in which the integral calculus still remains, it would be entertaining a false idea of the general resources of the transcendental analysis to attach too much importance to this consideration. As in ordinary analysis, we find here that a very small amount of fundamental knowledge respecting the resolution of equations is of inestimable use. However little advanced geometers are as yet in the science of integrations, they have nevertheless derived from their few abstract notions the solution of a multitude of questions of the highest importance in geometry, mechanics, thermology, etc. The philosophical explanation of this double general fact is found in the preponderating importance and scope of abstract science, the smallest portion of which naturally corresponds to a multitude of concrete researches, Man having no other resource for the successive extension of his intellectual means than in the contemplation of ideas more and more abstract, and nevertheless positive. LAGRANGE'S METHOD OF VARIATIONS. 87 Calculus of Variations. By his Calculus or Method of Variations, Lagrange improved the capacity of the transcendental analysis for the establishment of equations in the most difficult problems, by considering a class of equations still more indirect than differential equations properly so called. It is still too near its origin, and its applications have been too few, to admit of its being understood by a purely abstract account of its theory; and it is therefore necessary to indicate briefly the special nature of the problems which have given rise to this hyper-transcendental analysis. Problems giving rise to this Calculus. These problems are those which were long known by the name of Isoperimetrical Problems; a name which is truly applicable to only a very small number of them. They consist in the investigation of the maxima and minima of certain indeterminate integral formulas which express the analytical law of such or such a geometrical or mechanical phenomenon, considered independently of any particular subject. In the ordinary theory of maxima and minima, we seek, with regard to a given function of one or more variables, what particular values must be assigned to these variables, in order that the corresponding value of the proposed function may be a maximum or a minimum with respect to those values which immediately precede and follow it:-that is, we inquire, properly speaking, at what instant the function ceases to increase in order to begin to decrease, or the reverse. The differential calculus fully suffices, 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 render null the different derivatives of the first order of the given function, taken separately with relation to each independent variable; and by indicating moreover a character suitable for distinguishing the maximum from the minimum, which consists, 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 for the minimum. Such are the fundamental conditions belonging |