Comte - The positive philosophy. Vol. 1

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are many questions treated as incidental in the midst of a body of analytical researches, which, consisting of determinations of values, are truly arithmetical. Of this kind are the construction of a table of logarithms, and the calculation of trigonometrical tables, and some distinct and higher procedures, in short, every operation which has for its object the determination of the values of functions. And we must also include that part of the science of the Calculus which we call the Theory of Numbers, the object of which is to discover the properties inherent in different numbers in virtue of their values, independent of any particular system of numeration. It constitutes a sort of transcendental arithmetic. Though the domain of arithmetic is thus larger than is commonly supposed, this Calculus of values will yet never be more than a point, as it were, in comparison with the calculus of functions, of which mathematical science essentially consists. This is evident, when we look into the real nature of arithmetical questions.

Determinations of values are, in fact, nothing else than real transformations of the functions to be valued. These transformations have a special end; but they are essentially of the same nature as all taught by analysis. In this view, the Calculus of values may be regarded as a supplement, and a particular application of the Calculus of functions, so that arithmetic disappears, as it were, as a distinct section in the body of abstract mathematics. To make this evident, we must observe that when we desire to determine the value of an unknown number whose mode of formation is given, we define nd express that value in merely announcing the arithmetical question, already defined and expressed under a certain form; and that, in determining its value, we merely express it under another determinate form, to which we are in the habit of referring the idea of each particular number by making it re-enter into the regular system of numeration. This is made clear by what happens when the mode of numeration is such that the question is its own answer; is, for instance, when we want to add together seven and thirty, and call the result seven-and-thirty. In adding other numbers, the terms are not so ready, and we transform the question; as when we add together twentythree and fourteen: but not the less is the operation merely one of transformation of a question already defined and expressed. In this view, the calculus of values might be regarded as a particular application of the calculus of functions, arithmetic thereby disappearing, as a distinct section, from the domain of abstract mathematics.—And here we have done with the Calculus of values, and pass to the Calculus of functions, of

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which abstract mathematics is essentially composed.

We have seen that the difficulty of establishing, the relation of the concrete to the abstract is owing to the insufficiency of the very small number of analytical elements that we are in possession of. The obstacle has been surmounted in a great number of important cases: and we will now see how the establishment of the equations of phenomena has been achieved.

The first means of remedying the difficulty of the small number of analytical elements seems to be to create new ones. But a little consideration will show that this resource is illusory. A new analytical element would not serve unless we could immediately determine its value: but how can we determine the value of a function which is simple; that is, which is not formed by a combination of those already known? This appears almost impossible: but the introduction of another elementary abstract function into analysis supposes the simultaneous creation of a new arithmetical operation; which is certainly extremely difficult. If we try to proceed according to the method which procured us the elements we possess, we are left in entire uncertainty; for the artifices thus employed are evidently exhausted. We have thus no idea how to proceed to create new elementary abstract functions. Yet, we must not therefore conclude that we have reached the limit appointed by the powers of our understanding. Special improvements in mathematical analysis have yielded us some partial substitutes, which have increased our resources: but it is clear that the augmentation of these elements cannot proceed but with extreme slowness. It is not in this direction, then, that the human mind has found its means of facilitating the establishment of equations.

This first method being discarded, there remains only one other. As it is impossible to find the equations directly, we must seek for corresponding ones between other auxiliary quantities, connected with the first according to a certain determinate law, and from the relation between which we may ascend to that of the primitive magnitudes. This is the fertile conception which we term the transcendental analysis, and use as our fine/tinstrument for the mathematical exploration of natural phenomena.

This conception has a much larger scope than even profound geometers have hitherto supposed; for the auxiliary quantities resorted to might be derived, according to any law whatever, from the immediate elements of the question. It is well to notice this; because our future

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improved analytical resources may perhaps be found in a new mode of derivation. But, at present, the only auxiliary quantities habitually substituted for the primitive quantities in transcendental analysis are what are called—

1st, infinitely small elements, the differentials of different orders of those quantities, if we conceive of this analysis in the manner of Leitnitz: or

2nd, the fluxions, the limits of the ratios of the simultaneous increments of the primitive quantities, compared with one another; or, more briefly, the prime and ultimate ratios of these increments, if we adopt the conception of Newton: or

3rd, the derivatives, properly so called, of these quantities; that is, the coefficients of the different terms of their respective increments, according to the conception of Lagrange.

These conceptions, and all others that have been proposed, are by their nature identical. The various grounds of preference of each of them will be exhibited hereafter.

We now see that the Calculus of functions, or Algebra, must consist of two distinct branches. The one has for its object the resolution of equations when they are directly established between the magnitudes in question: the other, setting out from equations (generally much more easy to form) between quantities indirectly connected with those of the problem, has to deduce, by invariable analytical procedures, the corresponding equations between the direct magnitudes in question,—bring- ing the problem within the domain of the preceding calculus.—It might seem that the transcendental analysis ought to be studied before the ordinary, as it provides the equations which the other has to resolve. But, though the transcendental is logically independent of the ordinary, it is best to follow the usual method of study, taking the ordinary first; for, the proposed questions always requiring to be completed by ordinary analysis, they must be left in suspense if the instrument of resolution had not been studied beforehand.

To ordinary analysis I propose to give the name of Calculus of Direct Function. To transcendental analysis, (which is known by the names of Infinitesimal Calculus, Calculus of fluxions and of fluents, Calculus of Vanishing quantities, the Differential and Integral Calculus, etc., according to the view in which it has been conceived,) I shall give the title of Calculus of Indirect Functions. I obtain these terms by generalizing and giving precision to the ideas of Lagrange, and employ them to indi-

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cate the exact character of the two forms of analysis.

Section I

Ordinary Analysis, or Calculus of Direct Functions

Algebra is adequate to the solution of mathematical questions which are so simple that we can form directly the equations between the magnitudes considered, without its being necessary to bring into the problem, either in substitution or alliance, any system of auxiliary quantities derived from the primary. It is true, in the majority of important cases, its use requires to be preceded and prepared for by that of the calculus of indirect functions, by which the establishment of equations is facilitated: but though algebra then takes the second place, it is not the less a necessary agent in the solution of the question; so that the Calculus of direct functions must continue to be, by its nature, the basis of mathematical analysis. We must now, then, notice the rational composition of this calculus, and the degree of development it has attained.

Its object being the resolution of equations (that is, the discovery of the mode of formation of unknown quantities by the known, according to the equations which exist between them), it presents as many parts as we can imagine distinct classes of equations; and its extent is therefore rigorously indefinite, because the number of analytical functions susceptible of entering into equations is illimitable, though, as we have seen, composed of a very small number of primitive elements.

The rational classification of equations must evidently be determined by the nature of the analytical elements of which their members are composed. Accordingly, analysts first divide equations with one or more variables into two principal classes, according as they contain functions of only the first three of the ten couples, or as they include also either exponential or circular functions. Though the names of algebraic and transcendental functions given to these principal groups are inapt, the division between the corresponding equations is real enough, insofar as that the resolution of equations containing the transcendental functions is more difficult than that of algebraic equations. Hence the study of the first is extremely imperfect, and our analytical methods relate almost exclusively to the elaboration of the second

Our business now is with these Algebraic equations only. In the first place, we must observe that, though they may often contain irrational functions of the unknown quantities, as well as rational functions, the first case can always be brought under the second, by transforma-

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tions more or less easy so that it is only with the latter that analysts have had to occupy themselves, to resolve all the algebraic equations. As to their classification, the early method of classing them according to the number of their terms has been retained only for equations with two terms, which are, in fact, susceptible of a resolution proper to themselves. The classification by their degrees, long universally established, is eminently natural; for this distinction rigorously determines the greater or less difficulty of their resolution. The gradation can be independently, as well as practically exhibited: for the most general equation of each degree necessarily comprehends all those of the different inferior degrees, as must also the formula which determines the unknown quantity: and therefore, however slight we may, a priori, suppose the difficulty to be of the degree under notice, it must offer more and more obstacles, in proportion to the rank of the degree, because it is complicated in the execution with those of all the preceding degrees.

This increase of difficulty is so great, that the resolution of algebraic equations is as yet known to us only in the four first degrees. In this respect, algebra has advanced but little since the labours of Descartes and the Italian analysts of the sixteenth century; though there has probably not been a single geometer for two centuries past who has not striven to advance the resolution of equations. The general equation of the fifth degree has itself, thus far, resisted all attempts. The formula of the fourth degree is so difficult as to be almost inapplicable; and analysts, while by no means despairing of the resolution of equations of the fifth, and even higher degrees, being obtained, have tacitly agreed to give up such researches.

The only question of this kind which would be of eminent importance, at least in its logical relations, would be the general of algebraic equations of any degree whatever. But the more we ponder this subject, the more we are led to suppose. with Lagrange, that it exceeds the scope of our understandings. Even if the requisite formula could be obtained, it could not be usefully applied unless we could simplify it, without impairing its generality, by the introduction of a new class of analytical elements, of which we have as yet no idea. And, besides, if we had obtained the resolution of algebraic equations of any degree whatever, we should still have treated only a very small part of algebra, properly so called; that is, of the calculus of direct functions, comprehending the resolution of all the equations that can be formed bv the analytical functions known to us at this day. Again, we must remember that by a law of

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our nature, we shall always remain below the difficulty of science, our means of conceiving of new questions being always more powerful than our resources for resolving them; in other words, the human mind being more apt at imagining than at reasoning. Thus, if we had resolved all the analytical equations now known, and if, to do this, we had found new analytical elements, these again would introduce classes of equations of which we now know nothing: and so, however great might be the increase of our knowledge, the imperfection of our algebraic science would be perpetually reproduced.

The methods that we have are, the complete resolution of the equations of the first four degrees; of any binomial equations; of certain special equations of the superior degrees; and of a very small number of exponential, logarithmic, and circular equations. These elements are very limited; but geometers have succeeded in treating with them a great number of important questions in an admirable manner. The improvements introduced within a century into mathematical analysis have contributed more to render the little knowledge that we have immeasurably useful, than to increase it.

To fill up the vast gap in the resolution of algebraic equations of the higher degrees, analysts have had recourse to a new order of questions,— to what they call the numerical resolution of equations. Not being able to obtain the real algebraic formula, they have sought to determine at least the valve. Of each unknown quantity for such or such a designated system of particular values attributed to the given quantities. This operation is a mixture of algebraic with arithmetical questions; and it has been so cultivated as to be rendered possible in all Gases, for equations of any degree and even of any form. The methods for this are now sufficiently (general; and what remains is to simplify them so as to fit them for regular application. While such is the state of algebra, we have to endeavour so to dispose the questions to be worked as to require finally only this numerical resolution of the equations. We must not forget however that this is very imperfect algebra; and it is only isolated, or truly final questions (which are very few), that can be brought finally to depend upon only the numerical resolution of equations. Most questions are only preparatory,—a first stage of the solution of other questions; and in these cases it is evidently not the value of the unknown quantity that we want to discover, but the formula which exhibits its derivation. Even in the most simple questions, when this numerical resolution is strictly sufficient, it is not the less a very imperfect method. Because we

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cannot abstract and treat separately the algebraic part of the question, which is common to all the cases which result from the mere variation of the given numbers, we are obliged to go over again the whole series of operations for the slightest change that may take place in any one of the quantities concerned.

Thus is the calculus of direct functions at present divided into two parts, as it is employed for the algebraic or the numerical resolution of equations. The first, the only satisfactory one, is unfortunately very restricted, and there is little hope that it will ever be otherwise: the second, usually insufficient, has at least the advantage of a much greater generality. They must be carefully Distinguished in our minds, on account of their different objects, and therefore of the different ways in which quantities are considered by. Moreover, there is, in regard to their methods, an entirely different procedure in their rational distribution. In the first part, we have nothing to do with the values of the unknown quantities, and the division must take place according to the nature of the equations which we are able to resolve; whereas in the second, we have nothing to do with the degrees of the equations, as the methods are applicable to equations of any degree whatever; but the concern is with the numerical character of the values of the unknown quantities.

These two parts, which constitute the immediate object of the Calculus of direct functions, are subordinated to a third, purely speculative, from which both derive their most effectual resources, and which has been very exactly designated by the general name of Theory of Equations, though it relates, as yet, only to algebraic equations. The numerical resolution of equations has, on account of its generality, special need of this rational foundation.

Two orders of questions divide this important department of algebra between them; first, those which relate to the composition of equations, and then those that relate to their transformation, the business of these last being to modify the roots of an equation without knowing them, according to any given law, provided this law is uniform in relation to all these roots.

One more theory remains to be noticed, to complete our rapid exhibition of the different essential parts of the calculus of direct functions. This theory, which relates to the transformation of functions into series by the aid of what is called the Method of indeterminate Coefficients, is one of the most fertile and important in algebra. This eminently analytical method is one of the most remarkable discoveries of Descartes. The

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invention and development of the infinitesimal calculus, for which it might be very happily substituted in some respects, has undoubtedly deprived it of some of its importance; but the growing extension of the transcendental analysis has, while lessening its necessity, multiplied its applications and enlarged its resources; so at, by the useful combination of the two theories, the employment of the method of indeterminate coefficients as become much more extensive than it was even before the formation of the calculus of indirect functions.

I have now completed my sketch of the Calculus of Direct Functions. We must next pass on to the more important and extensive branch of our science, the Calculus of Indirect Functions.

Section II

Transcendental Analysis, of Calculus of Indirect Functions

We referred in a former section to the views of the transcendental analysis presented by Leibnitz, Newton, and Lagrange.

We shall see that each conception has advantages of its own, that all are finally equivalent, and that no method has yet been found which unites their respective characteristics. Whenever the combination takes place, it will probably be by some method founded on the conception of Lagrange. The other two will then offer only an historical interest; and meanwhile, the science must be regarded as in a merely provisional state, which requires the use of all the three conceptions at the same time; for it is only by the use of them all that an adequate idea of the analysis and its applications can be formed. The vast extent and difficulty of this part of mathematics, and its recent formation, should prevent our being at all surprised at the existing want of system The conception which will doubtless give a fixed and uniform character to the science has come into the hands of only one new generation of geometers since its creation; and the intellectual habits requisite to perfect it have not been sufficiently formed.

The first germ of the infinitesimal method (which can be conceived of independently of the Calculus) may be recognized in the old Greek Method of Exhaustions, employed to pass from the properties of straight lines to those of curves. The method consisted in substituting for the curve the auxiliary consideration of a polygon, inscribed by means of which the curve itself was reached, the limits of the primitive ratios being suitably taken. There is no doubt of the filiation of ideas in this

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case; but there was in it no equivalent for our modern methods; for the ancients had no logical and general means for the determination of these limits, which was the chief difficulty of the question. The task remaining for modern geometers was to generalize the conception of the ancients, and, considering it in an ate. street manner, to reduce it to a system of calculation, which was impossible to them.

Lagrange justly ascribes to the great geometer Fermat the first idea in this new direction. Fermat may be regarded as having initiated the direct formation of transcendental analysis by his method for the determination of maxima and minima, and for the finding of tangents, in which process he introduced auxiliaries which he afterwards suppressed as null when the equations obtained had underdone certain suitable transformations. After some modifications of the ideas of Fermat in the intermediate time, Leitnitz stripped the process of some complications, and formed the analysis into a general and distinct calculus, having his own notation: and Leibnitz is thus the creator of transcendental analysis, as we employ it now. This pre-eminent discovery was so ripe, as all great conceptions are at the hour of their advent, that Newton had at the same time, or rather earlier, discovered a method exactly equivalent, regarding the analysis from a different point of view, much more logical in itself, but less adapted than that of Leitnitz to give all practicable extent and facility to the fundamental method. Lagrange afterwards, discarding the heterogeneous considerations which had guided Leibnitz and Newton, reduced the analysis to a purely algebraic system, which only wants more aptitude for application.

We will notice the three methods in their order. The method of Leibnitz consists in introducing into the calculus, in order to facilitate the establishment of equations, the infinitely small elements or differentials which are supposed to constitute the quantities whose relations we are seeking. There are relations between these differentials which are simpler and more discoverable than those of the primitive quantities; and by these we may afterwards (through a special calculus employed to eliminate these auxiliary infinitesimals) recur to the equations sought, which it would usually have been impossible to obtain directly. This indirect analysis may have various decrees of indirectness for, when there is too much difficulty in forming the equation between the differentials 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,

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or second differentials, and so on; the same transformation being repeated any number of times, provided the whole number of auxiliaries be finally 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 being, either the successive powers of the same primitive infinitely small quantity, or as being quantities which may be regarded as having finite ratios with 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 prover 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 simple to treat, only observing the condition that the new elements shall differ from the preceding only by quantities infinitely small in relation to them. It is thus that it becomes possible in geometry to treat curve: 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 motions, 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 estate fish, 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