Comte - The positive philosophy. Vol. 1

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to use till after the formation of the infinitesimal analysis.

This analysis has another property, besides that of facilitating the study of the mathematical laws of all phenomena, and perhaps not less important than that. The differential formulas exhibit an extreme generality, expressing in 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 teas introduced 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 positive approximations between classes of wholly different phenomena, through the 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, 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 invariable rules.

This beautiful method is, however, imperfect in its logical basis. At first, geometers 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 difficult problems. It became necessary, however, to recur to the basis of the new analysis, to establish the rigorous exactness of the processes 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 operations 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 especially, which was unquestionably exact. This af-

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forded 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 Carnot who furnished this at last, by showing that the method was founded on the principle 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 Liebnitz, than is even vet commonly supposed. His reasoning 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 infinitely small errors in all the equations; and when the relations of finite quantities are reached, these relations must be rigorously exact, since the only errors then possible must be finite ones, which cannot have entered: and thus the final equations become perfect. Carnot’s theory is doubtless more subtle than solid; but it has no other radical logical vice than that of the infinitesimal method itself, of which it is, as it seems to me, the natural develop meet. 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 exhibited to allow of my giving only the principal idea of the other two methods.

Newton offered his conception under several different forms in succession. That which is now most commonly adopted, at least on the continent, was called by himself, sometimes the Method of prime and ultimate Ratios, sometimes the Method of Limits, by which last term it is now usually known.

Under this Method, the auxiliaries introduced are the limits of the ratios of the simultaneous increments of the primitive quantities; or, in other words, the final ratios of these increments, 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

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ratios of those increments, and from our being therefore able always to substitute for each increment any other magnitude more east to treat, provided their final ratio is the ratio of equality or, in other words, that the limit of their ratio is unity. It is clear, in fact, that the calculus of limits can be in no way affected by this substitution. Starting from this principle. We find nearly the equivalent of the facilities offered 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 fill and arrange Newton’s ideas on the subject.

Newton had another view, however, which ought to be presented here, because it is still 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 transcendental analysis. I mean the Calculus of fluxions and of fluents, 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 lank whatever. The different quantities presented by the curve, the abscissa, the ordinate, the arc, the area, etc., will be regarded as simultaneously 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 fluent. Henceforth, the transcendental analysis will, according to this conception, 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 guests themselves. What has just been stated respecting 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 method of prime and ultimate ratios, which alone is reducible to a calculus. It therefore necessarily admits of the same general

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advantages in the various principal applications of the transcendental analysis, without its being requisite for us to offer special proofs of this.

Lagrange’s conception consists, in its admirable simplicity, in considering the transcendental analysis to be a great algebraic artifice, by which, to facilitate the establishment of equations, we must introduce, in the place of or with the primitive 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 between the primitive magnitudes. The transcendental analysis is then only a simple, but very considerable extension of ordinary analysis. It has long been a common practice with geometers to introduce, in analytical investigations, in the place of the magnitudes in question, their different powers, or their logarithms, or their sines, etc., in order to simplify the equations, and even to obtain them more easily. Successive derivation is a general artifice of the same nature, only of greater extent, 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 angst be so under this method of derivation, rather than any other transformation. This is the weak side of Lagrange’s great idea. We have not yet become able to lay hold of its precise advantages, in an abstract manner, and without recurrence to the other conceptions of the transcendental analysis. These advantages can be established only in the separate consideration of each principal question; and this verification becomes laborious; in the treatment of a complex problem.

Other theories have been proposed, such as Euler’s Calculus of vanishing quantities: but they are merely modifications of the three just exhibited. We must next compare and estimate these methods; and in the first place observe their perfect and necessary conformity.

Considering the three methods in regard to their destination, independently of preliminary ideas, It is clear that they all consist in the same general logical artifice; that is, the introduction of a certain system of auxiliary magnitudes uniformly correlative with those under investigation; the auxiliaries being substituted for the express object of facili-

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tating the analytical expression of the mathematical laws of phenomena, though they must be finally eliminated be the help of a special calculus. It was this which determined 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 will 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 preliminary 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 lies for its object the establishment of equations, which are the indispensable point of departure for all analytical investigations. But this preliminary elaboration has been remarkably simplified by the creation of the transcendental 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 operation to finding the equations between auxiliary magnitudes, whence the Calculus leads to equations directly relating to the proposed magnitudes, which had formerly to be established directly. Whether these indirect equations are differential equations, according to Leibnitz, or equations of limits, according to Newton, or derived equations, according 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 identical, 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 properties of the derived function of Lagrange. The transcendental analysis, then, examined abstractly and in its principle, is always the same, whatever conception is adopted, and the processes of the Calculus of indirect functions are necessarily identical in these different methods which must therefore, under any application whatever, lead to rigorously uniform results.

If we endeavour to estimate their comparative value, we shall find in each of the three conceptions advantages and inconveniences which are peculiar to it, and which prevent geometers from adhering to any

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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 magnitudes. We owe to its use the high perfection attained by all the general theories of geometry and mechanics. Whatever may be the speculative opinions of geometers as to the infinitesimal method, they all employ it in the treatment of any new question. Lagrange himself, after having reconstructed the 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 false 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 ingenious principle of the compensation of errors, we may, as we have already seen, explain the necessary exactness of processes which compose the method; but it is a radical inconvenience to be obliged to indicate, in Mathematics, two classes of reasonings so unlike, as that the one order are perfectly rigorous, while by the others we designedly commit errors which have to be afterwards compensated. 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 irreproachable. 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 such a transformation. Finally, the infinitesimal method exhibits the very serious defect of breaking the unity of abstract mathematics by creating 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 independent, tends to prevent the formation of general analytical 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 Leitnitz. The notion of limits is in fact remarkable for its distinctness and precision. The equations are, in this case, regarded as

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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 infinitesimal 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 remarked, the idea of limits, though clear and exact, is not the less a foreign idea, on which analytical theories ought not to be dependent.

This perfect unity of analysis, and a purely abstract character in the fundamental ideas, are found in the conception of Lagrange, and there alone. It is therefore the most philosophical of all. Discarding every heterogeneous consideration, Lagrange reduced the transcendental analysis to its proper character,—that of presenting a very extensive class transformations, which facilitate in a remarkable decree 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 philosophical superiority marks it for adoption as the final theory of transcendental analysis; but it presents too many difficulties in its application, in comparison with the others, to admit of its exclusive preference at present. Lagrange himself had great difficulty 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 important. Though Lagrange, stimulated by difficulty, obtained 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 transcendental analysis thoroughly, we should not only study it in its principles according to all these conceptions, but should accustom ourselves to employ them all (and especially the first and last) almost indifferently, in the solution of all important questions, whether of the calculus of indirect functions in itself, or of its applications. In all the other departments of mathematical science, the consideration of different methods for a single class of questions may be useful, apart from the historical interest which it presents; but it is not indispensable. Here, on the contrary, it is strictly

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indispensable. Without it there can be no philosophical judgment of this admirable creation of the human mind; nor any success and facility in the use of this powerful instrument.

The Differential and Integral Calculus.

The Calculus of Indirect functions is 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 relations between the corresponding primitive magnitudes; by the other we seek, conversely, these direct equations by means of the indirect equations first established. this is the double 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 infinitesimal method, properly so called, being most in use, almost all geometers employ the terms Differential Calculus and Integral Calculus 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 expressions, though we must, as has been shown, employ all the conceptions concurrently, approaching as nearly as may be to that of Lagrange.

The differential calculus is obviously the rational basis of the integral. We have seen that ten simple functions constitute the elements of our analysis. We cannot know how to integrate directly any other differential expressions than those produced by the differentiation of those ten functions. 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 indispensable one; in fact, the elimination of the infinitesimals or the derivatives, introduced as auxiliaries, being the final object of the calculus of indirect functions, it is natural to think that the calculus which teaches us to deduce the equations between the primitive magni-

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tudes from those between the auxiliary magnitudes must meet all the general 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 departure of any calculus whatever. The very use of the differential calculus is enabling us to differentiate the various equations; and it cannot therefore be the process for establishing them. This common error arises from confounding the infinitesimal calls with the infinitesimal 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 functions if we saw in it only a preliminary process, designed merely to prepare an indispensable basis for the integral calculus. A few words will show that a primary direct and necessary office is always assigned to the differential calculus. In forming differential equations, we 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 functions, the relations of their differentials, and this is merely a question of differentiation. This is the way in which the differential calculus not only prepares a basis for the gral, but manes it available in a multitude of cases which could not otherwise be treated. There are some questions, few, but highly important, which admit of the employment of the differential calculus alone. They are those in which the magnitudes sought enter directly, and not by their differentials, into the primitive differential equations, which then contain differentially only the various known functions employed, as we saw just now, as intermediaries. This calculus is here entirely sufficient for the elimination of the infinitesimals, without the question giving rise to any integration. There are also questions, few, but highly important, which are the converse of the last, requiring

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the employment of the integral calculus alone. In these, 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 the introduction, differentially, of any intermediaries being required. If intermediary functions are introduced, they will, by the hypothesis, enter directly, and not by their differentials; and then, ordinary algebra will serve for their elimination, and to bring the question to depend on the integral calculus only. The differential calculus is, in such cases, not essential to the solution of the problem, which will depend entirely on the integral calculus. Thus, all questions to which the analysis is applicable are contained in three classes. The first class comprehends the problems which may be resolved by the differential calculus alone. The second, those which may be resolved by the integral calculus alone. These are only exceptional; the third constituting 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 either calculus, which are quite a different study from that of the abstract principles of differentiation and integration. The consequence 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 Mathematics that the applications should be studied.

The first division of the differential calculus is grounded on the condition whether the functions to be differentiated are explicit or implicit; the one giving rise to the differentiation of formulas, and the other to the differentiation of equations. This classified lion is rendered necessary by the imperfection of ordinary 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 number of cases; and we have to differen-