stitute tlie quantities whose relations we are seeking. There are relations between these differentials which are simpler and more discoverable than those of the [ivimitive quantities; and hy these we maj afterwards (through a special calculus employed to eliminate these auxdiary in- finitesimals) recur to the equations sought, which it would usually have been im])Ossible to obtain directly. This indirect analysis may have various degrees of indirectness; for, when there is too miich difiiculty in forming the equa- tion between the diiferentials of the magnitudes under notice, a second application of the method is required, the differentials being now treated as new primitive quantities. and a relation being sought between their infinitely small elements, or second differentials, and so on; the same transformation beiug repeated any numlier of times, pro- vided the whole number of auxiliaries be fiually eliminated. It may be asked by novices in these studies, how these auxiliary quantities can be of use while they are of the same species with the magnitudes to be treated, seeing that the greater or less value of any quantity cannot affect any inquiry which has nothing to do with value at all. The explanation is this. We must begin by distinguishing the different orders of infinitely small quantities, obtaining a precise idea of this by considering them as l»eing either the successive powers of the same primitive infinitely small quantity, or as being quantities which may be regarded as having finite ratios \v'itli these powers; so that, for instance, the second or third or other differentials of the same variable are classed as infinitely small quantities of the second, third or other order, because it is easy to exhibit in them finite multiples of the second, third, or other powers of a certain first differential. These preliminary ideas being laid down, the spirit of the infinitesimal analysis consists in constantly neglecting the infinitely small quantities in comparison with finite quantities; and generally, the infinitely small quantities of any order whatever in comparison with all those of an inferior order. We see at once how such a power must facilitate the formation of equations between the differentials of quantities, since we can substitute for these differentials such other elements as we may choose, and as will be more simjjle to treat, only observing the con- I. F 66 POSITIVE PHILOSOPHY.
dition that the new elements shall differ from the precediniij only by quantities infinitely small in relation to them. It is thus that it becomes possible in geometry to treat curved lines as composed of an infinity of rectilinear elements, and curved surfaces as formed of plane elements; and, in mechanics, varied motions as an infinite series of uniform motion'', succeeding each other at infinitely small intervals of time. Such a mere hint as this of the varied application of this method may give some idea of the vast scope of the conception of transcendental analysis, as formed by Leibnitz. It is, beyond all question, the loftiest idea ever yet attained by the human mind.
It is clear that this conception was necessary to complete the basis of mathematical science, by enabling us to estab- lish, in a broad and practical manner, the relation of the concrete to the abstract. In this respect, we must regard it as the necessary complement of the great fundamental idea of Descartes on the general analytical representation of natural phenomena; an idea which could not be duly estimated or put to use till after the formation of the infinitesimal analysis.
This analysis has another property, besides that of facili- tating the study of the mathematical laws of all phenomena, and perhaps not less important than that. The differential ^,.. formulas exhibit an extreme generality, exthe formulas pressing m a single equation each determinate phenomenon, however varied may be the subjects to which it belongs. Thus, one such equation gives the tangents of all curves, another their rectifications, a third their quadratures; and, in the same way, one invariable formula expresses the mathematical law of all variable motion; and one single equation represents the distribution of heat in any body, and for any case. This remarkable generality is the basis of the loftiest views of the geometers. Thus this analysis has not only furnished a general method for forming equations indirectly which could not have been directly discovered, but it has intro- duced a new order of more natural laws for our use in the mathematical study of natural phenomena, enabling us to rise at times to a perception of ])ositi\ e approximations between classes of wholly different phenomena, through the JUSTIFICATION OF THE LEIBNITZIAN METHOD. 67 analogies presented by the differential expressions of their mathematical laws. In virtue of this second property of the analysis, the entire system of an immense science, like geometry or mechanics, has submitted to a condensation into a small number of analytical formulas, from which the solution of all particular problems can be deduced, by in- variable rules.
This beautiful method is, however, iniper-.,.
feet in its logical basis. At first, geometers ^.j^^ Method were naturally more intent upon extending the discovery and multiplying its applications than upon establishing the logical foundation of its processes. It was enough for some time to be able to produce, in answer to objections, unhoped-for solutions of the most difiicult problems. It became necessary, however, to recur to the basis of the new analysis, to establish the rigorous exact- ness of the jirocesses employed, notwithstanding their apparent breaches of the ordinary laws of reasoning, Leibnitz himself failed to justify his conception, giving, when urged, an answer which represented it as a mere approximative calculus, the successive opei'ations of which might, it is evident, admit an augmenting amount of error. Some of his successors were satisfied with showing that its results accorded with those obtained by ordinary algebra, or the geometry of the ancients, reproducing by these last some solutions which could be at first obtained only by the new method. Some, again, demonstrated the conformity of the new conception with others; that of Newton espe- cially, which was unquestionably exact. This afforded a practical justification: but, in a case of such unequalled importance, a logical justification is also required, — a direct proof of the necessary rationality of the infinitesimal method. It was Caruot who furnished this at last, by showing that the method was founded on the pi'inciple of the necessary compensation of errors. We cannot say that all the logical scaffolding of the infinitesimal method may not have a merely provisional existence, vicious as it is in its nature: but, in the present state of our knowledge, Carnot's principle of the necessary compensation of errors is of more importance, in legitimating the analysis of Leibnitz, than is even yet commonly supposed. His reason- 68 POSITIVE PHILOSOPHY.
ing is founded on the conception of infinitesimal quantities indefinitely decreasing, while those from which they are derived are fixed. The infinitely small errors introduced with the auxiliaries cannot have occasioned other than infi- nitely small errors in all the equations; and when the relations of finite c^uantities are reached, these relations must be rigorously exact, since the only errors then pos- sible must be finite ones, which cannot have entered: and thus the final equations become perfect. Caruot's theory is doubtless more subtle than solid; but it has no other radical logical vice than that of the infiaitesimal method itself, of which it is, as it seems to me, the natural develop- ment and general explanation; so that it must be adopted as long as that method is directly employed.
The philosophical character of the transcendental analysis has now been sufficiently exhil>ited to allow of my giving only the principal idea of tlie other two methods.
^,^ Newton offered his conception under several Method different forms in succession. That which is now most commonly adopted, at least on the continent, was called by himself, sometimes the Mdhod of prime and tdtimate Ratios, sometimes the Method of Limits, by which last term it is now usually known.
Under this Method, the auxiliaries introsimultaneous increments of the primitive quantities; or, in other words, the final ratios of these in- crements; limits or final ratios which we can easily show to have a determinate and finite value. A special calculus, which is the equivalent of the infinitesimal calculus, is afterwards employed, to rise from the equations between these limits to the corresponding equations between the primitive quantities themselves.
The power of easy expression of the mathematical laws of phenomena given by this analysis arises from the calculus applying, not to the increments themselves of the proposed quantities, but to the limits of the ratios of those incre- ments; and from our being therefore able always to sub- stitute for each increment any other magnitude more easy to treat, provided their final ratio is the ratio of equality; or, in other words, that the limit of their ratio is unity. It Newton's method of limits. 69 is clear, in fact, that the calculus of limits can he in no way affected by this substitution. Starting from this principle, we find nearly the equivalent of the facilities offei'ed by the analysis of Leibnitz, which are merely considered from another point of view. Thus, curves will be regarded as the limits of a series of rectilinear polygons, and variable motions as the limits of an aggregate of uniform motions of continually nearer approximation, etc., etc. Such is, in substance, Newton's conception; or rather, that which Maclaurin and d'Alembert have offered as the most rational basis of the transcendental analysis, in the endeavour to fix and arrange Newton's ideas on the subject.
Newton had another view, however, which ought to be presented here, because it is still fj^p^Ig"^ ^"' the special form of the calculus of indirect functions commonly adopted by English geometers; and also, on account of its ingenious clearness in some cases, and of its having furnished the notation best adapted to this manner of regarding the ti'anscendental analysis. I mean the Calculus oi fluxions and oifltients, founded on the general notion of velocities.
To facilitate the conception of the fundamental idea, let us conceive of every curve as generated by a point affected by a motion varying according to any law whatever. The different quantities presented by the (turve, the abscissa, the ordinate, the arc, tlie area, etc., will be regarded as simiil- taneously produced by successive degrees during this motion. The velocity with which each one will have been described will be called the fluxion of that quantity, which inversely would have been called its fluted Henceforth, the transcendental analysis will, accoi'ding to this concep- tion, consist in forming directly the equations between the fluxions of the proposed quantities, to deduce from them afterwards, by a special Calculus, the equations between the fluents themselves. Wliat has just been stated respect- ing curves may evidently be transferred to any magnitudes whatever, regarded, by the help of a suitable image, as some being produced by the motion of others. This method is evidently the same with that of limits complicated with the foreign idea of motion. It is, in fact, only a way of representing, by a comparison derived from mechanics, the 70 POSITIVE PHILOSOPHY.
luetliod of prime aud ultimate ratios, which alone is redu- cible to a calculus. It therefore necessarily admits of the same genei'al advantages in the various principal applica- tions of the transcendental analysis, without its being requisite for us to offer special proofs of this.
Lagrange's conception consists, in its ad- Method mirable simplicity, in considering the transcendental analysis to be a great algebraic artifice, by which, to facilitate the establishment of equa- tions, we must introduce, in the place of or with the primi- tive functions, their derived functions; that is, according to the definition of Lagrange, the coefficient of the first term of the increment of each function, arranged according to the ascending powers of the increment of its variable. The Calculus of indirect functions, properly so called, is destined here, as well as in the conceptions of Leibnitz and Newton, to eliminate these derivatives, employed as auxiliaries, to deduce from their relations the corresponding equations 1>etween the primitive magnitudes. The transcendental analysis is then only a simple, but very considerable exten- sion of ordinary analysis. It has long been a common practice with geometers to introduce, in analytical investi- gations, in the place of the magnitudes in question, their different powers, or their logarithms, or their sines, etc., in order to simjjlify the equations, and even to obtain them more easily. Successive derivation is a general artifice of the same nature, only of greater exteutT and commanding, in consequence, much more important resources for this common object.
But, though we may easily conceive, a priori, that the auxiliary use of these derivatives may facilitate the study of equations, it is not easy to explain why it must be so under this method of derivation, rather than any other transformation. This is the weak side of Lagrange's great idea. We liave not yet become able to lay hold of its pre- cise advantages, in an abstract manner, and without recur- rence to the other conceptions of the transcendental analysis. These advantages can be established only in the separate consideration of each principal question; and this verifica- tion becomes laborious, in the treatment of a complex problem.
COxMPAEISON OF THE THREE METHODS. 71 Other theories have been proposed, such as Euler's Cal- culus of vanishing quantities: but they are merely modifi- cations of the thi'ee just exhibited. We must next com- pare and estimate these methods; and in the first j)lace observe their perfect and necessary conformity.
Considering the three methods in regard. •,, r, j to their destination, independently of pre- three m'etliod^ liminary ideas, it is clear that they all con- sist in the same general logical artifice; that is, the in- troduction of a certain system of auxiliary magnitudes uniformly correlative Avith those under investigation; the auxiliaries being substituted for the express object of facilitating the analytical expression of the mathematical laws of phenomena, though they must be finally eliminated by the help of a special calculus. It was this which deter- mined me to define the transcendental analysis as the Calculus of indirect functions, in order to mark its true philosophical character, while excluding all discussion about the best manner of conceiving and applying it. Whatever may be the method employed, the general effect of this analysis is to bring every mathematical question more speedily into the domain of the calculus, and thus to lessen considerably the grand difficulty of the passage from the concrete to the abstract. We cannot hope that the Calculus Avill ever lay hold of all questions of natural philosophy — geometrical, mechanical, thermological, etc. — from their birth. That would be a contradiction. In every problem there must be a certain ])reliminai"y operation before the calculus can be of any use, and one which could not by its nature be subjected to abstract and invariable rules: — it is that which has for its object the establishment of equations, which are the indispensable point of departure for all ana- lytical investigations. But this preliminary elaboration has been remarkably simplified by the creation of the transcen- dental analysis, which has thus hastened the moment at which general and abstract processes may be uniformly and exactly applied to the solution, by reducing the opera- tion to fiudiug the equations between auxiliary magnitudes, whence the Calculus leads to equations directly relating to the proj^osed magnitudes, which had formerly to be estab- lished directly. Whether these indirect equations are 72 POSITIVE PHILOSOPHY.
differential equations, according to Leibnitz, or equations of ZimiVs, according to Newton, ov derived equations, accord- ing to Lagrange, the general procedure is evidently always the same. The coincidence is not only in the result but in the process; for the auxiliaries introduced are really iden- tical, being only regarded from different points of view. The conceptions of Leibnitz and of Newton consist in making known in any case two general necessary proper- ties of the derived function of Lagrange. The transcen- dental analysis, then, examined abstractly and in its principle, is always the same, whatever conception is adopted; and the processes of the Calculus of indirect func- tions ai^e necessarily identical in these different methods, which must therefore, under any aj^plication whatever, lead to rigorously uniform results.
„.. If we endeavour to estiinate their comparai\l^ r.r.1,',1^ '' tive value, we shall find in each of the three live Veil lit;,,, _ conceptions advantages and inconveniences which are peculiar to it, and which prevent geometers from adhering to any one of them, as exclusive and final.
The method of Leibnitz has eminently the advantage in the rapidity and ease with which it effects the formation of equations between auxiliary magnitiides. We owe to its use the high perfection attained by all the general theories of geometry and mechanics. Whatever may be the specu- lative opinions of geometers as to the infinitesimal method, they all employ it in the treatment of any new question. Lagrange himself, after having reconstructed tht^ analysis on a new basis, rendered a candid and decisive homage to the conception of Leibnitz, by employing it exclusively in the whole system of his "Analytical Mechanics." Such a fact needs no comment. Yet are we obliged to admit, with Lagrange, that the conception of Leibnitz is radically vicious in its logical relations. He himself declared the notion of infinitely small quantities to be a faJse idea: and it is in fact impossible to conceive of them clearly, though we may sometimes fancy that we do. This false idea bears, to my mind, the characteristic impress of the metaphysical age of its birth and tendencies of its originator. By the in- genious princij^le of the compensation of errors, we may, as we have already seen, explain the necessary exactness of COMPARISON OF THE THREE METHODS. 78 the processes which compose the method; but it is a radical inconvenience to be obhged to indicate, in Mathe- matics, two clashes of reasonings so unlike, as tliat the one order are perfectly rigoi'ons, while by the others we de- signedly commit errors which have to be afterwards com- pensated. There is nothing very logical in this; nor is anything obtained by pleading, as some do, that this method can be made to enter into that of limits, which is logically irrepr<.)achable. This is eluding the difficulty, and not resolving it; and besides, the advantages of this method, its ease and rapidity, are almost entirely lost under sucli a transformation. Finally, the infinitesimal method exhibits the very serious defect of breaking the unity of abstract mathematics by ci'eating a transcendental analysis founded upon principles widely different from those which serve as a basis to ordinary analysis. This division of analysis into two systems, almost wholly inde- pendent, tends to prevent the formation of general analy- tical conceptions. To estimate the consequences duly, we must recur in thought to the state of the science before Lagrange had established a general and complete harmony between these two great sections.
Newton's conception is free from the logical objections imputable to that of Leibnitz. The notion of limits is in fact remarkable for its distinctness and precision. The equations are, in this case, regarded as exact from their origin; and the general rules of reasoning are as constantly observed as in ordinary analysis. But it is weak in resources, and embarrassing in operation, compared with the infini- tesimal method. In its applications, the relative inferiority of this theory is very strongly marked. It also separates the ordinary and transcendental analysis, though not so conspicuously as the theory of Leibnitz. As Lagrange re- marked, the idea of liiidts, though clear and exact, is not the less a foreign idea, on which analytical theories ought not to be dependent.
This jierfect unity of analysis, and a ])\irely abstract character in the fundamental ideas, are found in the con- ception of Lagrange, and there alone. It is therefore the most philosophical of all. Discarding every heterogeneous consideration, Lagrange reduced the transcendental analysis 74 POSITIVE PHILOSOPHY.
to its proper character, — that of pi'esenting a very extensive class of analytical transformations, which facilitate in a re- markable degree the expression of the conditions of the various problems. This exhibits the conception as a simple extension of ordinary analysis. It is a superior algebra. All the different parts of abstract mathematics, till then so incoherent, might be from that moment conceived of as forming a single system. This jjhilosophical superiority marks it for adoption as the final theory of transcendental analysis; but it presents too many difficulties in its appli- cation, in comj^arison with the others, to admit of its exclusive preference at present. Lagrange himself had great didiculty in rediscovering, by his own method, the principal results already obtained by the infinitesimal method, on general questions in geometry and mechanics; and we may judge by that what obstacles would occur in treating in the same way questions really new and im- portant. Though Lagrange, stimulated by difficulty, ob- tained results in some cases which other men would have despaired of, it is not the less true that his conception has thus far remained, as a whole, essentially unsuited to applications.
The result of such a comparison of these three methods is the conviction that, in order to understand the tran- scendental analysis thoroughly, we should not only study it in its principles according to all these conceptions, but should accustom ourselves to emj^loy them all (and es- peci.dly the first and last) almost indiffei'ently, in the solution of all important questions, whether of the calculus of indirect functions in itself, or of its applications. In all the other dej^artments of mathematical science, the con- sideratii)n of different methods for a single class of ques- tions may be useful, apart from the historical interest which it presents; but it is not indispensable. Here, on the contrary, it is strictly indispensable. Without it there can be no philosophical judgment of this admirable creation of the human mind; nor any success and facilit}" in the use of this powerful instrument.
THE TWO CALCULI OF INDIRECT FUNCTIONS.
THE DIFFERENTIAL AND INTEGRAL CALCULUS.
The Calculus of Indirect functions is j,.^ parts necessarily divided into two parts; or rather, it is composed of two distinct calculi, having the relation of converse action. By the one we seek the relations between the auxiliary magnitudes, by means of the rela- tions between the corresj^onding primitive magnitudes; by the other we seek, conversely, these direct equations by means of the indirect equations first established. This is the doable object of the transcendental analysis.
Different names have been given to the two systems, according to the point of view from which the entire analysis has been regarded. The intiuitesimal method, properly so called, being most in use, almost all geometers employ the terms Differential Calculus and Integral Cal- culus established by Leibnitz. Newton, in accordance with his method, called the first the Calculus of Fluxions, and the second the Calculus of Fluents, terms which were till lately commonly adopted in England. According to the theory of Lagrange, the one would be called the Calculus of Derived Functions, and the other the Calculus of Primitive Functions. I shall make use of the terms of Leibnitz, as the fittest for the formation of secondary ex- pressions, though we must, as has been shown, employ all the conceptions concurrently, apj)roaching as nearly as may be to that of Lagrange.
The dilferential calculus is obviously the rr,i •. x i rational basis of the integral. We have seen velations. that ten simple functions constitute the elements of our analysis. We cannot know how to in- tegrate directly any other differential expressions than those produced by the differentiation of tliose ten func- tions. The art of integration consists therefore in bringing all the other cases, as far as possible, to depend wholly on this small number of simple functions.
It may not be apparent to all minds what can be the proper utility of the differential calculus, independently of this necessary connection with the integral calculus, which seems as if it must be in itself the only directly indis- pensable one; in fact, the elimination of the infinitesimals 76 POSITIVE PHILOSOPHY.
or the derivatives, introduced as auxiliaries, beinc^ the final object of the calculus of indirect functions, it is natural to think that the calculus which teaches us to deduce the equations betAveen the primitive magnitudes from those between the auxiliary magnitudes must meet all the gene- ral needs of the transcendental analysis, without our seeing at first what special and constant part the solution of the inverse question can have in such an analysis. A common answer is assigning to the differential calculus the office of forming the differential equations; but this is clearly an error; for the primitive formation of differential equations is not the business of any calculus, for it is, on the contrary, the point of departin*e of any calculus whatever. The very use of the differential calculus is enabling us to differentiate the var'ous equations; and it cannot therefore be the process for establishing them. This common error arises from confounding the infinitesimal calculus with the in- finitesimal method, which last facilitates the formation of equations, in every application of the transcendental analysis. The calculus is the indispensable complement of the method; but it is perfectly distinct from it. But again, we should much misconceive the peculiar importance of this first branch of the calculus of indirect func- tions if we saw in it only a preliminary process, designed merely to pi-epare an indisi-tensable ba^is for the integral calculus. A few words will show that a jirimary direct and necessary office is always assigned to the differential of the two '"'*^ rarely restrict ourselves to introducing differentially only those magnitudes whose relations are sought. It would often be impossible to establish, equations without introducing other magnitudes whose relations are, or are supposed to be, known. Now in such cases it is necessary that the differentials of these intermediaries should be eliminated before the equations are fit for integration. This elimination belongs to the differential calculus; for it must be done by determining, by means of the equations between the intermediary func- tions, the relations of their differentials; and Ihis is merely a question of differentiation. This is the way in which the differential calculus not only prepares a basis for the EXAMPLES OF THE TWO CALCULI. 11 integral, Liit makes it available in a multitude of cases which could not otherwise be treated. There Cases of the are some questions, few, but highly imjjor- Differential tant, wliicli admit of the emjjloymeut of the calculus alone, differential calculus alone. They are those in which the magnitudes sought enter directly, and not by their diffe- rentials, into the jirimitive differential equations, which then contain differentially only the various known func- tions employed, as we saw just now, as intermediaries. This calculus is here entirely sufficient for the elimination of the intinitesimals, without the question giving rise to any integration. There are also questions, few, but highly important, which are the converse of the last, requiring the employment of the integral calculus alone. Cases of the In these, the differential equations are found Integral cal- to be immediately ready for integration, cuius alone, because they contain, at their first formation, only the in- finitesimals which relate to the functions sought, or to the really independent variables, without the introduction, differential y, of any intermediaries being required. If intermediary functions are introduced, they wi 1, by the hypothesis, enter directly, and not by their differentials;