and then, ordinary algebra will serve for their elimination, and to bring the question to depend on the integral calcu- lus only, 'ihe differential calculus is, in such cases, not essential to the solution of the problem, which will depend entirely on the integral calcidus. Thus, all questions to which the analysis is applicable are contained io three classes. The first class comprehends the problems which may be resolved by the differential calculus alone. The second, tho-e which may be resolved by the integral cal- culus alone. 'J hese are only exceptional; the third con- stituting the normal case; that in which the differential and integral calculus have each a distinct and necessary part in the solution of problems.
The Differential Calculus.
The entire system of the differential calculus is simple and perfect, while the integral calculus remains extremely imperfect.
We have nothing to do here with the applications of 78 POSITIVE PHILOSOPHY.
either calculus, which are quite a different tial Calculus study from that of the abstract principles of differentiation and integration. The con- sequence of the common practice of confounding these principles with their application, especially in geometry, is that it becomes difficult to conceive of either analysis or geometry. It is in the department of Concrete Mathe- matics that the applications should he studied.
The first division of the differential calculus is grounded on the condition whether the functions to be differentiated rp ^..., are explicit or implicit; the one giving rise to the differentiation of formulas, and the other to the differentiation of equations. This classifica- tion is rendered necessary by the imperfection of oi'dinary analysis; for if we knew how to resolve all equations algebraically, it would be possible to render every implicit function explicit; and, by differentiating it only in that state, the second part of the differential calculus would be immediately included in the first, without giving rise to any new difficulty. But the algebraic resolution of equations is, as we know, still scarcely past its infancy, and unknown for the greater numV>er of cases; and we have to differentiate a function without knowing it, though it is determinate. Thus we have two classes of questions, the differentiation of implicit functions being a distinct ease from that of explicit functions, and much more complicated. We have to begin by the differentiation of formiilas, and we may then refer to this first case the differentiation of equations, by certain analytical considera- tions which we are not concerned with here. There is another view in which the two general cases of differentia- tion are distinct. The relation obtained between the differentials is always more indirect, in comparison with that of the finite quantities, in the differentiation of im- plicit, than in that of explicit functions. We shall meet with this consideration in the case of the integral calculus, where it acquires a preponderaut im]iortance.
Q TV • • Each of these parts of the diff'erential calculus IS again divided: and this sub- division exhibits two very distinct theories, according as we have to differentiate functions of a single variable, or SUBDIVISIONS OF THE DIFFERENTIAL CALCULUS. 79 functions of several independent variables, — the second branch being of far greater complexity than the first, in the case of explicit functions, and. much more in that of imi^iicit. One more distinction remains, to complete this brief sketch of the parts of the differential calculus. The case in which it is required to differentiate at once different implicit functions combined in certain primitive equations must be distinguished from that in which all these functions are separate. The same imperfection of ordinary analysis which prevents our converting every implicit function into an equivalent explicit oue, renders us unable to separate the functions which enter simultaneously into auv system of equations; and the functions are evidently still more implicit in the case of combined, than of separate functions: and. in differentiating, we are not only unable to resolve the primitive equations, but even to effect the j^roper elimination among them.
We have now seen the different parts of this calculus in their natural connection and tl\e elenients rational distribution. The whole calculus is finally found to rest upon the differentiation of explicit functions with a single varialile, — the only one which is ever executed dii'ectly. Now, it is easy to understand that this first theory, this necessary basis of the whole system, simply consists of the differentiation of the elementary functions, ten in number, which C(nnpose all our analytical combinations; for the differentiation of compound functions is evidently deduced, immediately and necessarily, from that of their constituent simple functions. We find, then, the whole system of differentiation reduced, to the know- ledge of the ten fundnmental differentials, and to that of the two general princi]>les, by one of which the differentia- tion of impHcit functions is deduced from that of explicit, and by the other, the differentiation of functions of several variables is reduced to that of functions of a single variable. Such is the simplicity and perfection of the system of the differential calculus.
The transformation of derived Functions for Transfornia- new variables is a theory which must be just tionof dt'iived mentioned, to avoid the omission of an indis- functions tor pensable complement of the system of diffe- "'^^ variables.
80 POSITIVE PHILOSOPHY.
rentiatiou. It is as finished and perfect as tlie other parts of this calculus; and its great importance is in its increasing our resources by permitting us to choose, to facilitate the formation of differential equations, that system of indepen- dent variables which may appear to be most advantageous, though it may afterwards be relinquished, as an inter- mediate stiq>, by which, through this theory, we may pass to the final system, which sometimes could not have been considered directly...Though we cannot hei'e consider the conapnhcatioas crete applications of this calculus, we must glance at those which are analytii-al, because they a.re of the same nature with the theory, and should be looked at in connection with it. These questions are reducible to three essential ones. First, the development into series of functions of one or more variables; or, more generally, the transformation of functions, which constitutes the mo>t beautiful and the most important application of the differential calculus to general analysis, and which comprises, besides the fundamental ser es discovered by Taylor, the remarkable series discovered by Maclaurin, John Beruouilli, Lagrange and others. Feeondly, the general theory of maxima and minima values for any functions whatever of one or more variables: one of the most iutex-esting problems that analysis can present, how- ever elementary it has become. The th rd is the least im- portant of the three: — it is the determination of the true value of functions which present themselves under an in- determinate ap])earance, for certain hypotheses made on the values of the corresponding variables. In every view, the first question is the most eminent; it is also the most susce[)tible of future extension, especially by ccmceiving, in a larger manner than hitherto, of the employment of tlie differential calculus for the transformation of functions, about which Lagrange left some valuable suggestions which have been neither generalized nor followed up.
It is with regret that I confine myself to the generalities which aie the proper subjects of this work; so extensive and so interesting are the developments wliicli might other- wise be offered. Insufficient and summary as are the views of the Differential Calculus just offered, we must be no THE INTEGRAL CALCULUS. 81 less rapid in our survey of the Integral Calculus, properly so called; that is, the abstract subject of integration.
The Integral Calculus.
The division of the Integral Calculus, like that of the Differential, 'proceeds on the Sjcu"us°''^' principle of distinguishing the integration of explicit differential formulas from the integration of itnplicit differentials,or of differential en nations. The t^ t •.
separation ot these two cases is even more radical in the case of integration than in the other. In the differential calculus this distinction rests, as we have seen, only on the extreme imperfection of ordinary analysis. But, on the other hand, it is clear that even if all equations could be algebraically resolved, differential equations would nevertheless constitute a case of integration altogether dis- tinct from that presented by explicit differential formulas. Their integration is necessarily more complicated than that of explicit differentials, by the elaboration of which the integral calculus was originated, and on which the others have been made to depend, as far as possible. All the various analytical processes hitherto proposed for the inte- gration of differential equations, whether by the separation of variables, or the method of multipliers, or other means, have been designed to reduce these integrations to those of differential formulas, the only object which can be directly undertaken. Unhappily, imperfect as is this neces- sary basis of the whole integral calculus, the art of reducing to it the integration of differential equations is even much less advanced.
As in the case of the differential calculus, „,,...T £ T 1 i! J.1 i. subdivisions, and tor analogous I'easons, each ot these two branches of the integral calculus is divided again, accord- ing as we consider functions with a single ^.,. variable or functions with several indepen- ^^ several ^' dent variables. This distinction is, like the preceding, even more important for integration than for differentiation This is especially remarkable with respect to differential equations. In fact, those which relate to several independent variables may evidently present this I. G 82 POSITIVE PHILOSOPHY.
characteristic and higher difficulty — that the function sought may be differentially defined by a simple relation between its various special derivatives with regard to the different variables taken separately. Thence results the most difficult, and also the most extended branch of the integral calculus, which is commonly called the Integral Calculus of partial differences, created by D'Alembert, in which, as Lagrange truly perceived, geometers should have recognized a new calculus, the philosophical character of which has not yet been precisely decided. This higher branch of transcendental analysis is still entirely in its infancy. In the very simplest case, we cannot completely reduce the integration to that of the ordinary differential equations.
A new distinction, highly important here, differentiation tl'oi^igli ^lot in the differential calculus, where it is a mistake to insist upon it, is drawn from the higher or lower order of the differentials. We may regard this distinction as a subdivision in the integra- tion of explicit or implicit differentials. With regard to explicit differentials, whether of one variable or of several, the necessity of distinguishing their different orders is occasioned merely by the extreme imperfection of the in- tegral calculus; and, with reference to implicit differentials, the distinction of orders is more important still. In the first case, we know so little of integration of even the first order of differential formulas, that differential formulas of a high order produce new difficulties in arriving at the primitive function which is our object. And in the second case, there is the additional difficulty that the higher order of the differential equations necessarily gives rise to ques- tions of a new kind. The higher the order of differential equations, the more implicit are the cases which they pre- sent; and they can be made to depend on each other only by special methods, the investigation of which, in conse- quence, forms a new class of questions, with regard to the simplest cases of which we as yet know next to nothing.
The necessary basis of all other integrations is, as we see from the foregoing considerations, that of explicit diffe- rential formulas of the first order and of a single variable; and we cannot succeed in effecting other integrations but ALGEBRAIC AND TRANSCENDENTAL FUNCTIONS. 83 by reducing them to this elementary case, Avhich is the only one capable of being treated directly. This,. l- t • simple fundamental integration, often con- veniently called quadratures, corresponds in the differential calculus to the elementary case of the differentiation of ex- plicit functions of a single variable. But the integral ques- tion is, by its nature, quite otherwise comjilicated, 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 si:)ecial difficulties with respect to the integral calculus. Hence the indefinite extent and varied complication of the question of quadra- tures, of which we know scarcely anything completel} after all the efforts of analysts.
The question is divided into the two cases ai • i. • of algebraic functions and transcenclental func- functions tions. The algebraic class is the more ad- vanced 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 satisfac- tory 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 integra- tion 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 inte- ^,., gration or transcendental functions is quite functions in its infancy as yet, as regards either ex- ponential, 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.
84 POSITIVE PHILOSOPHY.
The theory of Singular Sohdions (some- lonT '"'''' ^^'^^^ ^^^^^^ Particular Solutions), fully de- veloped 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 im- plicitly a portion of the general theory of the integration of differential equations; but I have left it till now, be- cause 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.
^,.,. One more theorv remains to be noticed, to tecrals complete our review ot that collection or analytical researches which constitutes the integral calciilus. 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 de- finite 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 equa- tions 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, differen- tiated, has produced the proj^osed differential formula, have DEFINITE INTEGRALS. 85 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 with- out 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, uuder our inability to obtain the true integrals, it is of the utmost importance to have been able to obtain this solution, incomplete and in- sufiicient 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 ti'ansformations. This kind of transcendental arithmetic being considered perfect, the difiiculty in its applications is reduced to making the pro- posed inquiry finally depend only on a simple determina- tion of definite integrals; a thing which evidently cannot be always possible, whatever analytical skill may be em- ployed in effecting so forced a transformation.
We have now seen that while the differen- Prospects of tial calculus constitutes by its nature a limited the Integral and perfect system, the integral calculus, or Calculus, 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 inte- gration 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 86 POSITIVE PHILOSOPHY.
a widei' generality. The transcendental analysis is yet too near its origin, it has too recently been regarded in a truly rational manner, for ns to have any idea what it may here- after 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 deriva- tion 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 re- sources than in ixrging further our present calculus of in- direct functions; and I am persuaded that when geometers have exhausted the most imjjortant applications of our present transcendental analysis, they will turn their atten- tion 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 enter- taining a false idea of the general resources of the trans- cendental analysis to attach too much importance to this consideration. As in ordinary analysis, we find here that a very small amount of fundamental Icuowledge respecting the resolution of equations is of inestimable use. However little advanced geometers are as yet in the science of inte-' grations, they have nevertheless derived from their few absti'act 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 corresjDonds to a multitude of concrete researches, Man having no other resoui'ce 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 Ms Calculus or Method of Variations, Lagrange im- proved 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.
These problems are those which were long Problems known by the name of Isoperimetrical Pro- giving rise to Hems; a name which is truly applicable to *"*** Calculus, only a very small number of them. They consist in the investigation of the maxima and minima of certain inde- tex'minate integral formulas which express the analytical law of such or such a geometrical or mechanical pheno- menon, 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 sepai'ately 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 maximvim and a positive for the minimum. Such are the fundamental conditions belonging 88 POSITIVE PHILOSOPHY.
to the majority of cases; and where modifications take place, they are equally subject to invariable, though more complicated abstract rules.
The construction of this general theory having destroyed the chief interests of geometers in this kind of questions, they rose almost immediately to the consideration of a new order of problems, at once more important and more diffi- cult,— those of isoperimeters. It was then no longer the values of the variables proper to the maximum or the mini- mum of a given function that had to be determined. It was the form of the function itself that had to be dis- covered, accoi'ding to the condition of the maximum or minimum of a certain definite integral, merely indicated, which depended on that fvmction. We cannot here follow the history of these problems, the oldest of which is that of the solid of least resistance, ti'eated by Newton in the second book of the ' Principia,' in which he determines what must 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. Mechanics first furnished this new class of problems; but it was from geometry that the subjects of the principal investi- gations were afterwards derived. They were varied and complicated almost infinitely by the labours of the best geometers, when Lagrange reduced their solution to an