We see, by this brief general comparison, how natural and profound is our fundamental division of mathematical science.

We have now to circumscribe, as exactly as we can in this first sketch, each of these two great sections.

CONCRETE MATHEMATICS.

Concrete Mathematics having for its object the discovery of the equations of phenomena, it would seem at first that it must be composed of as many distinct sciences as we find really distinct categories among natural phenomena. But we are yet very far from having discovered mathematical laws in all kinds of phenomena; we shall even see, presently, that the greater part will very probably always hide themselves from our investigations. In reality, in the present condition of the human mind, there are directly but two great general classes of phenomena, whose equations we constantly know; these are, firstly, geometrical, and, secondly, mechanical phenomena. Thus, then, the concrete part of mathematics is composed of Geometry and Rational Mechanics.

This is sufficient, it is true, to give to it a complete character of logical universality, when we consider all phenomena from the most elevated point of view of natural philosophy. In fact, if all the parts of the universe were conceived as immovable, we should evidently have only geometrical phenomena to observe, since all would be reduced to relations of form, magnitude, and position; then, having regard to the motions which take place in it, we would have also to consider mechanical phenomena. Hence the universe, in the statical point of view, presents only geometrical phenomena; and, considered dynamically, only mechanical phenomena. Thus geometry and mechanics constitute the two fundamental natural sciences, in this sense, that all natural effects may be conceived as simple necessary results, either of the laws of extension or of the laws of motion.

But although this conception is always logically possible, the difficulty is to specialize it with the necessary precision, and to follow it exactly in each of the general cases offered to us by the study of nature; that is, to effectually reduce each principal question of natural philosophy, for a certain determinate order of phenomena, to the question of geometry or mechanics, to which we might rationally suppose it should be brought. This transformation, which requires great progress to have been previously made in the study of each class of phenomena, has thus far been really executed only for those of astronomy, and for a part of those considered by terrestrial physics, properly so called. It is thus that astronomy, acoustics, optics, &c., have finally become applications of mathematical science to certain orders of observations.[1] But these applications not being by their nature rigorously circumscribed, to confound them with the science would be to assign to it a vague and indefinite domain; and this is done in the usual division, so faulty in so many other respects, of the mathematics into "Pure" and "Applied."

ABSTRACT MATHEMATICS.

The nature of abstract mathematics (the general division of which will be examined in the following chapter) is clearly and exactly determined. It is composed of what is called the Calculus,[2] taking this word in its greatest extent, which reaches from the most simple numerical operations to the most sublime combinations of transcendental analysis. The Calculus has the solution of all questions relating to numbers for its peculiar object. Its starting point is, constantly and necessarily, the knowledge of the precise relations, i.e., of the equations, between the different magnitudes which are simultaneously considered; that which is, on the contrary, the stopping point of concrete mathematics. However complicated, or however indirect these relations may be, the final object of the calculus always is to obtain from them the values of the unknown quantities by means of those which are known. This science, although nearer perfection than any other, is really little advanced as yet, so that this object is rarely attained in a manner completely satisfactory.

Mathematical analysis is, then, the true rational basis of the entire system of our actual knowledge. It constitutes the first and the most perfect of all the fundamental sciences. The ideas with which it occupies itself are the most universal, the most abstract, and the most simple which it is possible for us to conceive.

This peculiar nature of mathematical analysis enables us easily to explain why, when it is properly employed, it is such a powerful instrument, not only to give more precision to our real knowledge, which is self-evident, but especially to establish an infinitely more perfect co-ordination in the study of the phenomena which admit of that application; for, our conceptions having been so generalized and simplified that a single analytical question, abstractly resolved, contains the implicit solution of a great number of diverse physical questions, the human mind must necessarily acquire by these means a greater facility in perceiving relations between phenomena which at first appeared entirely distinct from one another. We thus naturally see arise, through the medium of analysis, the most frequent and the most unexpected approximations between problems which at first offered no apparent connection, and which we often end in viewing as identical. Could we, for example, without the aid of analysis, perceive the least resemblance between the determination of the direction of a curve at each of its points and that of the velocity acquired by a body at every instant of its variable motion? and yet these questions, however different they may be, compose but one in the eyes of the geometer.

The high relative perfection of mathematical analysis is as easily perceptible. This perfection is not due, as some have thought, to the nature of the signs which are employed as instruments of reasoning, eminently concise and general as they are. In reality, all great analytical ideas have been formed without the algebraic signs having been of any essential aid, except for working them out after the mind had conceived them. The superior perfection of the science of the calculus is due principally to the extreme simplicity of the ideas which it considers, by whatever signs they may be expressed; so that there is not the least hope, by any artifice of scientific language, of perfecting to the same degree theories which refer to more complex subjects, and which are necessarily condemned by their nature to a greater or less logical inferiority.

THE EXTENT OF ITS FIELD.

Our examination of the philosophical character of mathematical science would remain incomplete, if, after having viewed its object and composition, we did not examine the real extent of its domain.

Its Universality. For this purpose it is indispensable to perceive, first of all, that, in the purely logical point of view, this science is by itself necessarily and rigorously universal; for there is no question whatever which may not be finally conceived as consisting in determining certain quantities from others by means of certain relations, and consequently as admitting of reduction, in final analysis, to a simple question of numbers. In all our researches, indeed, on whatever subject, our object is to arrive at numbers, at quantities, though often in a very imperfect manner and by very uncertain methods. Thus, taking an example in the class of subjects the least accessible to mathematics, the phenomena of living bodies, even when considered (to take the most complicated case) in the state of disease, is it not manifest that all the questions of therapeutics may be viewed as consisting in determining the quantities of the different agents which modify the organism, and which must act upon it to bring it to its normal state, admitting, for some of these quantities in certain cases, values which are equal to zero, or negative, or even contradictory?

The fundamental idea of Descartes on the relation of the concrete to the abstract in mathematics, has proven, in opposition to the superficial distinction of metaphysics, that all ideas of quality may be reduced to those of quantity. This conception, established at first by its immortal author in relation to geometrical phenomena only, has since been effectually extended to mechanical phenomena, and in our days to those of heat. As a result of this gradual generalization, there are now no geometers who do not consider it, in a purely theoretical sense, as capable of being applied to all our real ideas of every sort, so that every phenomenon is logically susceptible of being represented by an equation; as much so, indeed, as is a curve or a motion, excepting the difficulty of discovering it, and then of resolving it, which may be, and oftentimes are, superior to the greatest powers of the human mind.

Its Limitations. Important as it is to comprehend the rigorous universality, in a logical point of view, of mathematical science, it is no less indispensable to consider now the great real limitations, which, through the feebleness of our intellect, narrow in a remarkable degree its actual domain, in proportion as phenomena, in becoming special, become complicated.

Every question may be conceived as capable of being reduced to a pure question of numbers; but the difficulty of effecting such a transformation increases so much with the complication of the phenomena of natural philosophy, that it soon becomes insurmountable.

This will be easily seen, if we consider that to bring a question within the field of mathematical analysis, we must first have discovered the precise relations which exist between the quantities which are found in the phenomenon under examination, the establishment of these equations being the necessary starting point of all analytical labours. This must evidently be so much the more difficult as we have to do with phenomena which are more special, and therefore more complicated. We shall thus find that it is only in inorganic physics, at the most, that we can justly hope ever to obtain that high degree of scientific perfection.

The first condition which is necessary in order that phenomena may admit of mathematical laws, susceptible of being discovered, evidently is, that their different quantities should admit of being expressed by fixed numbers. We soon find that in this respect the whole of organic physics, and probably also the most complicated parts of inorganic physics, are necessarily inaccessible, by their nature, to our mathematical analysis, by reason of the extreme numerical variability of the corresponding phenomena. Every precise idea of fixed numbers is truly out of place in the phenomena of living bodies, when we wish to employ it otherwise than as a means of relieving the attention, and when we attach any importance to the exact relations of the values assigned.

We ought not, however, on this account, to cease to conceive all phenomena as being necessarily subject to mathematical laws, which we are condemned to be ignorant of, only because of the too great complication of the phenomena. The most complex phenomena of living bodies are doubtless essentially of no other special nature than the simplest phenomena of unorganized matter. If it were possible to isolate rigorously each of the simple causes which concur in producing a single physiological phenomenon, every thing leads us to believe that it would show itself endowed, in determinate circumstances, with a kind of influence and with a quantity of action as exactly fixed as we see it in universal gravitation, a veritable type of the fundamental laws of nature.

There is a second reason why we cannot bring complicated phenomena under the dominion of mathematical analysis. Even if we could ascertain the mathematical law which governs each agent, taken by itself, the combination of so great a number of conditions would render the corresponding mathematical problem so far above our feeble means, that the question would remain in most cases incapable of solution.

To appreciate this difficulty, let us consider how complicated mathematical questions become, even those relating to the most simple phenomena of unorganized bodies, when we desire to bring sufficiently near together the abstract and the concrete state, having regard to all the principal conditions which can exercise a real influence over the effect produced. We know, for example, that the very simple phenomenon of the flow of a fluid through a given orifice, by virtue of its gravity alone, has not as yet any complete mathematical solution, when we take into the account all the essential circumstances. It is the same even with the still more simple motion of a solid projectile in a resisting medium.

Why has mathematical analysis been able to adapt itself with such admirable success to the most profound study of celestial phenomena? Because they are, in spite of popular appearances, much more simple than any others. The most complicated problem which they present, that of the modification produced in the motions of two bodies tending towards each other by virtue of their gravitation, by the influence of a third body acting on both of them in the same manner, is much less complex than the most simple terrestrial problem. And, nevertheless, even it presents difficulties so great that we yet possess only approximate solutions of it. It is even easy to see that the high perfection to which solar astronomy has been able to elevate itself by the employment of mathematical science is, besides, essentially due to our having skilfully profited by all the particular, and, so to say, accidental facilities presented by the peculiarly favourable constitution of our planetary system. The planets which compose it are quite few in number, and their masses are in general very unequal, and much less than that of the sun; they are, besides, very distant from one another; they have forms almost spherical; their orbits are nearly circular, and only slightly inclined to each other, and so on. It results from all these circumstances that the perturbations are generally inconsiderable, and that to calculate them it is usually sufficient to take into the account, in connexion with the action of the sun on each particular planet, the influence of only one other planet, capable, by its size and its proximity, of causing perceptible derangements.

If, however, instead of such a state of things, our solar system had been composed of a greater number of planets concentrated into a less space, and nearly equal in mass; if their orbits had presented very different inclinations, and considerable eccentricities; if these bodies had been of a more complicated form, such as very eccentric ellipsoids, it is certain that, supposing the same law of gravitation to exist, we should not yet have succeeded in subjecting the study of the celestial phenomena to our mathematical analysis, and probably we should not even have been able to disentangle the present principal law.

These hypothetical conditions would find themselves exactly realized in the highest degree in chemical phenomena, if we attempted to calculate them by the theory of general gravitation.

On properly weighing the preceding considerations, the reader will be convinced, I think, that in reducing the future extension of the great applications of mathematical analysis, which are really possible, to the field comprised in the different departments of inorganic physics, I have rather exaggerated than contracted the extent of its actual domain. Important as it was to render apparent the rigorous logical universality of mathematical science, it was equally so to indicate the conditions which limit for us its real extension, so as not to contribute to lead the human mind astray from the true scientific direction in the study of the most complicated phenomena, by the chimerical search after an impossible perfection.

Having thus exhibited the essential object and the principal composition of mathematical science, as well as its general relations with the whole body of natural philosophy, we have now to pass to the special examination of the great sciences of which it is composed.

BOOK I.

ANALYSIS.

BOOK I.

ANALYSIS.

CHAPTER I.

GENERAL VIEW OF MATHEMATICAL ANALYSIS.

In the historical development of mathematical science since the time of Descartes, the advances of its abstract portion have always been determined by those of its concrete portion; but it is none the less necessary, in order to conceive the science in a manner truly logical, to consider the Calculus in all its principal branches before proceeding to the philosophical study of Geometry and Mechanics. Its analytical theories, more simple and more general than those of concrete mathematics, are in themselves essentially independent of the latter; while these, on the contrary, have, by their nature, a continual need of the former, without the aid of which they could make scarcely any progress. Although the principal conceptions of analysis retain at present some very perceptible traces of their geometrical or mechanical origin, they are now, however, mainly freed from that primitive character, which no longer manifests itself except in some secondary points; so that it is possible (especially since the labours of Lagrange) to present them in a dogmatic exposition, by a purely abstract method, in a single and continuous system. It is this which will be undertaken in the present and the five following chapters, limiting our investigations to the most general considerations upon each principal branch of the science of the calculus.

The definite object of our researches in concrete mathematics being the discovery of the equations which express the mathematical laws of the phenomenon under consideration, and these equations constituting the true starting point of the calculus, which has for its object to obtain from them the determination of certain quantities by means of others, I think it indispensable, before proceeding any farther, to go more deeply than has been customary into that fundamental idea of equation, the continual subject, either as end or as beginning, of all mathematical labours. Besides the advantage of circumscribing more definitely the true field of analysis, there will result from it the important consequence of tracing in a more exact manner the real line of demarcation between the concrete and the abstract part of mathematics, which will complete the general exposition of the fundamental division established in the introductory chapter.

THE TRUE IDEA OF AN EQUATION.

We usually form much too vague an idea of what an equation is, when we give that name to every kind of relation of equality between any two functions of the magnitudes which we are considering. For, though every equation is evidently a relation of equality, it is far from being true that, reciprocally, every relation of equality is a veritable equation, of the kind of those to which, by their nature, the methods of analysis are applicable.

This want of precision in the logical consideration of an idea which is so fundamental in mathematics, brings with it the serious inconvenience of rendering it almost impossible to explain, in general terms, the great and fundamental difficulty which we find in establishing the relation between the concrete and the abstract, and which stands out so prominently in each great mathematical question taken by itself. If the meaning of the word equation was truly as extended as we habitually suppose it to be in our definition of it, it is not apparent what great difficulty there could really be, in general, in establishing the equations of any problem whatsoever; for the whole would thus appear to consist in a simple question of form, which ought never even to exact any great intellectual efforts, seeing that we can hardly conceive of any precise relation which is not immediately a certain relation of equality, or which cannot be readily brought thereto by some very easy transformations.

Thus, when we admit every species of functions into the definition of equations, we do not at all account for the extreme difficulty which we almost always experience in putting a problem into an equation, and which so often may be compared to the efforts required by the analytical elaboration of the equation when once obtained. In a word, the ordinary abstract and general idea of an equation does not at all correspond to the real meaning which geometers attach to that expression in the actual development of the science. Here, then, is a logical fault, a defect of correlation, which it is very important to rectify.

Division of Functions into Abstract and Concrete. To succeed in doing so, I begin by distinguishing two sorts of functions, abstract or analytical functions, and concrete functions. The first alone can enter into veritable equations. We may, therefore, henceforth define every equation, in an exact and sufficiently profound manner, as a relation of equality between two abstract functions of the magnitudes under consideration. In order not to have to return again to this fundamental definition, I must add here, as an indispensable complement, without which the idea would not be sufficiently general, that these abstract functions may refer not only to the magnitudes which the problem presents of itself, but also to all the other auxiliary magnitudes which are connected with it, and which we will often be able to introduce, simply as a mathematical artifice, with the sole object of facilitating the discovery of the equations of the phenomena. I here anticipate summarily the result of a general discussion of the highest importance, which will be found at the end of this chapter. We will now return to the essential distinction of functions as abstract and concrete.

This distinction may be established in two ways, essentially different, but complementary of each other, à priori and à posteriori; that is to say, by characterizing in a general manner the peculiar nature of each species of functions, and then by making the actual enumeration of all the abstract functions at present known, at least so far as relates to the elements of which they are composed.

À priori, the functions which I call abstract are those which express a manner of dependence between magnitudes, which can be conceived between numbers alone, without there being need of indicating any phenomenon whatever in which it is realized. I name, on the other hand, concrete functions, those for which the mode of dependence expressed cannot be defined or conceived except by assigning a determinate case of physics, geometry, mechanics, &c., in which it actually exists.

Most functions in their origin, even those which are at present the most purely abstract, have begun by being concrete; so that it is easy to make the preceding distinction understood, by citing only the successive different points of view under which, in proportion as the science has become formed, geometers have considered the most simple analytical functions. I will indicate powers, for example, which have in general become abstract functions only since the labours of Vieta and Descartes. The functions x², x³, which in our present analysis are so well conceived as simply abstract, were, for the geometers of antiquity, perfectly concrete functions, expressing the relation of the superficies of a square, or the volume of a cube to the length of their side. These had in their eyes such a character so exclusively, that it was only by means of the geometrical definitions that they discovered the elementary algebraic properties of these functions, relating to the decomposition of the variable into two parts, properties which were at that epoch only real theorems of geometry, to which a numerical meaning was not attached until long afterward.

I shall have occasion to cite presently, for another reason, a new example, very suitable to make apparent the fundamental distinction which I have just exhibited; it is that of circular functions, both direct and inverse, which at the present time are still sometimes concrete, sometimes abstract, according to the point of view under which they are regarded.

À posteriori, the general character which renders a function abstract or concrete having been established, the question as to whether a certain determinate function is veritably abstract, and therefore susceptible of entering into true analytical equations, becomes a simple question of fact, inasmuch as we are going to enumerate all the functions of this species.

Enumeration of Abstract Functions. At first view this enumeration seems impossible, the distinct analytical functions being infinite in number. But when we divide them into simple and compound, the difficulty disappears; for, though the number of the different functions considered in mathematical analysis is really infinite, they are, on the contrary, even at the present day, composed of a very small number of elementary functions, which can be easily assigned, and which are evidently sufficient for deciding the abstract or concrete character of any given function; which will be of the one or the other nature, according as it shall be composed exclusively of these simple abstract functions, or as it shall include others.

We evidently have to consider, for this purpose, only the functions of a single variable, since those relative to several independent variables are constantly, by their nature, more or less compound.

Let x be the independent variable, y the correlative variable which depends upon it. The different simple modes of abstract dependence, which we can now conceive between y and x, are expressed by the ten following elementary formulas, in which each function is coupled with its inverse, that is, with that which would be obtained from the direct function by referring x to y, instead of referring y to x.

Such are the elements, very few in number, which directly compose all the abstract functions known at the present day. Few as they are, they are evidently sufficient to give rise to an infinite number of analytical combinations.

No rational consideration rigorously circumscribes, à priori, the preceding table, which is only the actual expression of the present state of the science. Our analytical elements are at the present day more numerous than they were for Descartes, and even for Newton and Leibnitz: it is only a century since the last two couples have been introduced into analysis by the labours of John Bernouilli and Euler. Doubtless new ones will be hereafter admitted; but, as I shall show towards the end of this chapter, we cannot hope that they will ever be greatly multiplied, their real augmentation giving rise to very great difficulties.

We can now form a definite, and, at the same time, sufficiently extended idea of what geometers understand by a veritable equation. This explanation is especially suited to make us understand how difficult it must be really to establish the equations of phenomena, since we have effectually succeeded in so doing only when we have been able to conceive the mathematical laws of these phenomena by the aid of functions entirely composed of only the mathematical elements which I have just enumerated. It is clear, in fact, that it is then only that the problem becomes truly abstract, and is reduced to a pure question of numbers, these functions being the only simple relations which we can conceive between numbers, considered by themselves. Up to this period of the solution, whatever the appearances may be, the question is still essentially concrete, and does not come within the domain of the calculus. Now the fundamental difficulty of this passage from the concrete to the abstract in general consists especially in the insufficiency of this very small number of analytical elements which we possess, and by means of which, nevertheless, in spite of the little real variety which they offer us, we must succeed in representing all the precise relations which all the different natural phenomena can manifest to us. Considering the infinite diversity which must necessarily exist in this respect in the external world, we easily understand how far below the true difficulty our conceptions must frequently be found, especially if we add that as these elements of our analysis have been in the first place furnished to us by the mathematical consideration of the simplest phenomena, we have, à priori, no rational guarantee of their necessary suitableness to represent the mathematical law of every other class of phenomena. I will explain presently the general artifice, so profoundly ingenious, by which the human mind has succeeded in diminishing, in a remarkable degree, this fundamental difficulty which is presented by the relation of the concrete to the abstract in mathematics, without, however, its having been necessary to multiply the number of these analytical elements.

THE TWO PRINCIPAL DIVISIONS OF THE CALCULUS.

The preceding explanations determine with precision the true object and the real field of abstract mathematics. I must now pass to the examination of its principal divisions, for thus far we have considered the calculus as a whole.

The first direct consideration to be presented on the composition of the science of the calculus consists in dividing it, in the first place, into two principal branches, to which, for want of more suitable denominations, I will give the names of Algebraic calculus, or Algebra, and of Arithmetical calculus, or Arithmetic; but with the caution to take these two expressions in their most extended logical acceptation, in the place of the by far too restricted meaning which is usually attached to them.

The complete solution of every question of the calculus, from the most elementary up to the most transcendental, is necessarily composed of two successive parts, whose nature is essentially distinct. In the first, the object is to transform the proposed equations, so as to make apparent the manner in which the unknown quantities are formed by the known ones: it is this which constitutes the algebraic question. In the second, our object is to find the values of the formulas thus obtained; that is, to determine directly the values of the numbers sought, which are already represented by certain explicit functions of given numbers: this is the arithmetical question.[4] It is apparent that, in every solution which is truly rational, it necessarily follows the algebraical question, of which it forms the indispensable complement, since it is evidently necessary to know the mode of generation of the numbers sought for before determining their actual values for each particular case. Thus the stopping-place of the algebraic part of the solution becomes the starting point of the arithmetical part.

We thus see that the algebraic calculus and the arithmetical calculus differ essentially in their object. They differ no less in the point of view under which they regard quantities; which are considered in the first as to their relations, and in the second as to their values. The true spirit of the calculus, in general, requires this distinction to be maintained with the most severe exactitude, and the line of demarcation between the two periods of the solution to be rendered as clear and distinct as the proposed question permits. The attentive observation of this precept, which is too much neglected, may be of much assistance, in each particular question, in directing the efforts of our mind, at any moment of the solution, towards the real corresponding difficulty. In truth, the imperfection of the science of the calculus obliges us very often (as will be explained in the next chapter) to intermingle algebraic and arithmetical considerations in the solution of the same question. But, however impossible it may be to separate clearly the two parts of the labour, yet the preceding indications will always enable us to avoid confounding them.

In endeavouring to sum up as succinctly as possible the distinction just established, we see that Algebra may be defined, in general, as having for its object the resolution of equations; taking this expression in its full logical meaning, which signifies the transformation of implicit functions into equivalent explicit ones. In the same way, Arithmetic may be defined as destined to the determination of the values of functions. Henceforth, therefore, we will briefly say that Algebra is the Calculus of Functions, and Arithmetic the Calculus of Values.

We can now perceive how insufficient and even erroneous are the ordinary definitions. Most generally, the exaggerated importance attributed to Signs has led to the distinguishing the two fundamental branches of the science of the Calculus by the manner of designating in each the subjects of discussion, an idea which is evidently absurd in principle and false in fact. Even the celebrated definition given by Newton, characterizing Algebra as Universal Arithmetic, gives certainly a very false idea of the nature of algebra and of that of arithmetic.[5]

Having thus established the fundamental division of the calculus into two principal branches, I have now to compare in general terms the extent, the importance, and the difficulty of these two sorts of calculus, so as to have hereafter to consider only the Calculus of Functions, which is to be the principal subject of our study.

THE CALCULUS OF VALUES, OR ARITHMETIC.

Its Extent. The Calculus of Values, or Arithmetic, would appear, at first view, to present a field as vast as that of algebra, since it would seem to admit as many distinct questions as we can conceive different algebraic formulas whose values are to be determined. But a very simple reflection will show the difference. Dividing functions into simple and compound, it is evident that when we know how to determine the value of simple functions, the consideration of compound functions will no longer present any difficulty. In the algebraic point of view, a compound function plays a very different part from that of the elementary functions of which it consists, and from this, indeed, proceed all the principal difficulties of analysis. But it is very different with the Arithmetical Calculus. Thus the number of truly distinct arithmetical operations is only that determined by the number of the elementary abstract functions, the very limited list of which has been given above. The determination of the values of these ten functions necessarily gives that of all the functions, infinite in number, which are considered in the whole of mathematical analysis, such at least as it exists at present. There can be no new arithmetical operations without the creation of really new analytical elements, the number of which must always be extremely small. The field of arithmetic is, then, by its nature, exceedingly restricted, while that of algebra is rigorously indefinite.

It is, however, important to remark, that the domain of the calculus of values is, in reality, much more extensive than it is commonly represented; for several questions truly arithmetical, since they consist of determinations of values, are not ordinarily classed as such, because we are accustomed to treat them only as incidental in the midst of a body of analytical researches more or less elevated, the too high opinion commonly formed of the influence of signs being again the principal cause of this confusion of ideas. Thus not only the construction of a table of logarithms, but also the calculation of trigonometrical tables, are true arithmetical operations of a higher kind. We may also cite as being in the same class, although in a very distinct and more elevated order, all the methods by which we determine directly the value of any function for each particular system of values attributed to the quantities on which it depends, when we cannot express in general terms the explicit form of that function. In this point of view the numerical solution of questions which we cannot resolve algebraically, and even the calculation of "Definite Integrals," whose general integrals we do not know, really make a part, in spite of all appearances, of the domain of arithmetic, in which we must necessarily comprise all that which has for its object the determination of the values of functions. The considerations relative to this object are, in fact, constantly homogeneous, whatever the determinations in question, and are always very distinct from truly algebraic considerations.

To complete a just idea of the real extent of the calculus of values, we must include in it likewise that part of the general science of the calculus which now bears the name of the Theory of Numbers, and which is yet so little advanced. This branch, very extensive by its nature, but whose importance in the general system of science is not very great, has for its object the discovery of the properties inherent in different numbers by virtue of their values, and independent of any particular system of numeration. It forms, then, a sort of transcendental arithmetic; and to it would really apply the definition proposed by Newton for algebra.

The entire domain of arithmetic is, then, much more extended than is commonly supposed; but this calculus of values will still never be more than a point, so to speak, in comparison with the calculus of functions, of which mathematical science essentially consists. This comparative estimate will be still more apparent from some considerations which I have now to indicate respecting the true nature of arithmetical questions in general, when they are more profoundly examined.

Its true Nature. In seeking to determine with precision in what determinations of values properly consist, we easily recognize that they are nothing else but veritable transformations of the functions to be valued; transformations which, in spite of their special end, are none the less essentially of the same nature as all those taught by analysis. In this point of view, the calculus of values might be simply conceived as an appendix, and a particular application of the calculus of functions, so that arithmetic would disappear, so to say, as a distinct section in the whole body of abstract mathematics.

In order thoroughly to comprehend this consideration, we must observe that, when we propose to determine the value of an unknown number whose mode of formation is given, it is, by the mere enunciation of the arithmetical question, already defined and expressed under a certain form; and that in determining its value we only put its expression under another determinate form, to which we are accustomed to refer the exact notion of each particular number by making it re-enter into the regular system of numeration. The determination of values consists so completely of a simple transformation, that when the primitive expression of the number is found to be already conformed to the regular system of numeration, there is no longer any determination of value, properly speaking, or, rather, the question is answered by the question itself. Let the question be to add the two numbers one and twenty, we answer it by merely repeating the enunciation of the question,[6] and nevertheless we think that we have determined the value of the sum. This signifies that in this case the first expression of the function had no need of being transformed, while it would not be thus in adding twenty-three and fourteen, for then the sum would not be immediately expressed in a manner conformed to the rank which it occupies in the fixed and general scale of numeration.

To sum up as comprehensively as possible the preceding views, we may say, that to determine the value of a number is nothing else than putting its primitive expression under the form

a + bz + cz² + dz³ + ez⁴ . . . . . + pz^m,

z being generally equal to 10, and the coefficients a, b, c, d, &c., being subjected to the conditions of being whole numbers less than z; capable of becoming equal to zero; but never negative. Every arithmetical question may thus be stated as consisting in putting under such a form any abstract function whatever of different quantities, which are supposed to have themselves a similar form already. We might then see in the different operations of arithmetic only simple particular cases of certain algebraic transformations, excepting the special difficulties belonging to conditions relating to the nature of the coefficients.

It clearly follows that abstract mathematics is essentially composed of the Calculus of Functions, which had been already seen to be its most important, most extended, and most difficult part. It will henceforth be the exclusive subject of our analytical investigations. I will therefore no longer delay on the Calculus of Values, but pass immediately to the examination of the fundamental division of the Calculus of Functions.

THE CALCULUS OF FUNCTIONS, OR ALGEBRA.

Principle of its Fundamental Division. We have determined, at the beginning of this chapter, wherein properly consists the difficulty which we experience in putting mathematical questions into equations. It is essentially because of the insufficiency of the very small number of analytical elements which we possess, that the relation of the concrete to the abstract is usually so difficult to establish. Let us endeavour now to appreciate in a philosophical manner the general process by which the human mind has succeeded, in so great a number of important cases, in overcoming this fundamental obstacle to The establishment of Equations.

1. By the Creation of new Functions. In looking at this important question from the most general point of view, we are led at once to the conception of one means of facilitating the establishment of the equations of phenomena. Since the principal obstacle in this matter comes from the too small number of our analytical elements, the whole question would seem to be reduced to creating new ones. But this means, though natural, is really illusory; and though it might be useful, it is certainly insufficient.

In fact, the creation of an elementary abstract function, which shall be veritably new, presents in itself the greatest difficulties. There is even something contradictory in such an idea; for a new analytical element would evidently not fulfil its essential and appropriate conditions, if we could not immediately determine its value. Now, on the other hand, how are we to determine the value of a new function which is truly simple, that is, which is not formed by a combination of those already known? That appears almost impossible. The introduction into analysis of another elementary abstract function, or rather of another couple of functions (for each would be always accompanied by its inverse), supposes then, of necessity, the simultaneous creation of a new arithmetical operation, which is certainly very difficult.

If we endeavour to obtain an idea of the means which the human mind employs for inventing new analytical elements, by the examination of the procedures by the aid of which it has actually conceived those which we already possess, our observations leave us in that respect in an entire uncertainty, for the artifices which it has already made use of for that purpose are evidently exhausted. To convince ourselves of it, let us consider the last couple of simple functions which has been introduced into analysis, and at the formation of which we have been present, so to speak, namely, the fourth couple; for, as I have explained, the fifth couple does not strictly give veritable new analytical elements. The function a^x, and, consequently, its inverse, have been formed by conceiving, under a new point of view, a function which had been a long time known, namely, powers—when the idea of them had become sufficiently generalized. The consideration of a power relatively to the variation of its exponent, instead of to the variation of its base, was sufficient to give rise to a truly novel simple function, the variation following then an entirely different route. But this artifice, as simple as ingenious, can furnish nothing more; for, in turning over in the same manner all our present analytical elements, we end in only making them return into one another.

We have, then, no idea as to how we could proceed to the creation of new elementary abstract functions which would properly satisfy all the necessary conditions. This is not to say, however, that we have at present attained the effectual limit established in that respect by the bounds of our intelligence. It is even certain that the last special improvements in mathematical analysis have contributed to extend our resources in that respect, by introducing within the domain of the calculus certain definite integrals, which in some respects supply the place of new simple functions, although they are far from fulfilling all the necessary conditions, which has prevented me from inserting them in the table of true analytical elements. But, on the whole, I think it unquestionable that the number of these elements cannot increase except with extreme slowness. It is therefore not from these sources that the human mind has drawn its most powerful means of facilitating, as much as is possible, the establishment of equations.

2. By the Conception of Equations between certain auxiliary Quantities. This first method being set aside, there remains evidently but one other: it is, seeing the impossibility of finding directly the equations between the quantities under consideration, to 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 return to that between the primitive magnitudes. Such is, in substance, the eminently fruitful conception, which the human mind has succeeded in establishing, and which constitutes its most admirable instrument for the mathematical explanation of natural phenomena; the analysis, called transcendental.

As a general philosophical principle, the auxiliary quantities, which are introduced in the place of the primitive magnitudes, or concurrently with them, in order to facilitate the establishment of equations, might be derived according to any law whatever from the immediate elements of the question. This conception has thus a much more extensive reach than has been commonly attributed to it by even the most profound geometers. It is extremely important for us to view it in its whole logical extent, for it will perhaps be by establishing a general mode of derivation different from that to which we have thus far confined ourselves (although it is evidently very far from being the only possible one) that we shall one day succeed in essentially perfecting mathematical analysis as a whole, and consequently in establishing more powerful means of investigating the laws of nature than our present processes, which are unquestionably susceptible of becoming exhausted.

But, regarding merely the present constitution of the science, the only auxiliary quantities habitually introduced in the place of the primitive quantities in the Transcendental Analysis are what are called, 1^o, infinitely small elements, the differentials (of different orders) of those quantities, if we regard this analysis in the manner of Leibnitz; or, 2^o, 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, 3^o, the derivatives, properly so called, of those quantities, that is, the coefficients of the different terms of their respective increments, according to the conception of Lagrange.

These three principal methods of viewing our present transcendental analysis, and all the other less distinctly characterized ones which have been successively proposed, are, by their nature, necessarily identical, whether in the calculation or in the application, as will be explained in a general manner in the third chapter. As to their relative value, we shall there see that the conception of Leibnitz has thus far, in practice, an incontestable superiority, but that its logical character is exceedingly vicious; while that the conception of Lagrange, admirable by its simplicity, by its logical perfection, by the philosophical unity which it has established in mathematical analysis (till then separated into two almost entirely independent worlds), presents, as yet, serious inconveniences in the applications, by retarding the progress of the mind. The conception of Newton occupies nearly middle ground in these various relations, being less rapid, but more rational than that of Leibnitz; less philosophical, but more applicable than that of Lagrange.

This is not the place to explain the advantages of the introduction of this kind of auxiliary quantities in the place of the primitive magnitudes. The third chapter is devoted to this subject. At present I limit myself to consider this conception in the most general manner, in order to deduce therefrom the fundamental division of the calculus of functions into two systems essentially distinct, whose dependence, for the complete solution of any one mathematical question, is invariably determinate.

In this connexion, and in the logical order of ideas, the transcendental analysis presents itself as being necessarily the first, since its general object is to facilitate the establishment of equations, an operation which must evidently precede the resolution of those equations, which is the object of the ordinary analysis. But though it is exceedingly important to conceive in this way the true relations of these two systems of analysis, it is none the less proper, in conformity with the regular usage, to study the transcendental analysis after ordinary analysis; for though the former is, at bottom, by itself logically independent of the latter, or, at least, may be essentially disengaged from it, yet it is clear that, since its employment in the solution of questions has always more or less need of being completed by the use of the ordinary analysis, we would be constrained to leave the questions in suspense if this latter had not been previously studied.

Corresponding Divisions of the Calculus of Functions. It follows from the preceding considerations that the Calculus of Functions, or Algebra (taking this word in its most extended meaning), is composed of two distinct fundamental branches, one of which has for its immediate object the resolution of equations, when they are directly established between the magnitudes themselves which are under consideration; and the other, starting from equations (generally much easier to form) between quantities indirectly connected with those of the problem, has for its peculiar and constant destination the deduction, by invariable analytical methods, of the corresponding equations between the direct magnitudes which we are considering; which brings the question within the domain of the preceding calculus.

The former calculus bears most frequently the name of Ordinary Analysis, or of Algebra, properly so called. The second constitutes what is called the Transcendental Analysis, which has been designated by the different denominations of Infinitesimal Calculus, Calculus of Fluxions and of Fluents, Calculus of Vanishing Quantities, the Differential and Integral Calculus, &c., according to the point of view in which it has been conceived.

In order to remove every foreign consideration, I will propose to name it Calculus of Indirect Functions, giving to ordinary analysis the title of Calculus of Direct Functions. These expressions, which I form essentially by generalizing and epitomizing the ideas of Lagrange, are simply intended to indicate with precision the true general character belonging to each of these two forms of analysis.

Having now established the fundamental division of mathematical analysis, I have next to consider separately each of its two parts, commencing with the Calculus of Direct Functions, and reserving more extended developments for the different branches of the Calculus of Indirect Functions.

CHAPTER II.

ORDINARY ANALYSIS, OR ALGEBRA.

The Calculus of direct Functions, or Algebra, is (as was shown at the end of the preceding chapter) entirely sufficient for the solution of mathematical questions, when they are so simple that we can form directly the equations between the magnitudes themselves which we are considering, without its being necessary to introduce in their place, or conjointly with them, any system of auxiliary quantities derived from the first. It is true that in the greatest number of important cases its use requires to be preceded and prepared by that of the Calculus of indirect Functions, which is intended to facilitate the establishment of equations. But, although algebra has then only a secondary office to perform, it has none the less a necessary part in the complete solution of the question, so that the Calculus of direct Functions must continue to be, by its nature, the fundamental base of all mathematical analysis. We must therefore, before going any further, consider in a general manner the logical composition of this calculus, and the degree of development to which it has at the present day arrived.

Its Object. The final object of this calculus being the resolution (properly so called) of equations, that is, the discovery of the manner in which the unknown quantities are formed from the known quantities, in accordance with the equations which exist between them, it naturally presents as many different departments as we can conceive truly distinct classes of equations. Its appropriate extent is consequently rigorously indefinite, the number of analytical functions susceptible of entering into equations being in itself quite unlimited, although they are composed of only a very small number of primitive elements.

Classification of Equations. The rational classification of equations must evidently be determined by the nature of the analytical elements of which their numbers are composed; every other classification would be essentially arbitrary. Accordingly, analysts begin by dividing equations with one or more variables into two principal classes, according as they contain functions of only the first three couples (see the table in chapter i., page 51), or as they include also exponential or circular functions. The names of Algebraic functions and Transcendental functions, commonly given to these two principal groups of analytical elements, are undoubtedly very inappropriate. But the universally established division between the corresponding equations is none the less very real in this sense, that the resolution of equations containing the functions called transcendental necessarily presents more difficulties than those of the equations called algebraic. Hence the study of the former is as yet exceedingly imperfect, so that frequently the resolution of the most simple of them is still unknown to us,[7] and our analytical methods have almost exclusive reference to the elaboration of the latter.

ALGEBRAIC EQUATIONS.

Considering now only these Algebraic equations, we must observe, in the first place, that although they may often contain irrational functions of the unknown quantities as well as rational functions, we can always, by more or less easy transformations, make the first case come under the second, so that it is with this last that analysts have had to occupy themselves exclusively in order to resolve all sorts of algebraic equations.

Their Classification. In the infancy of algebra, these equations were classed according to the number of their terms. But this classification was evidently faulty, since it separated cases which were really similar, and brought together others which had nothing in common besides this unimportant characteristic.[8] It has been retained only for equations with two terms, which are, in fact, capable of being resolved in a manner peculiar to themselves.

The classification of equations by what is called their degrees, is, on the other hand, eminently natural, for this distinction rigorously determines the greater or less difficulty of their resolution. This gradation is apparent in the cases of all the equations which can be resolved; but it may be indicated in a general manner independently of the fact of the resolution. We need only consider that 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. Consequently, however slight we may suppose the difficulty peculiar to the degree which we are considering, since it is inevitably complicated in the execution with those presented by all the preceding degrees, the resolution really offers more and more obstacles, in proportion as the degree of the equation is elevated.

ALGEBRAIC RESOLUTION OF EQUATIONS.

Its Limits. The resolution of algebraic equations is as yet known to us only in the four first degrees, such is the increase of difficulty noticed above. In this respect, algebra has made no considerable progress since the labours of Descartes and the Italian analysts of the sixteenth century, although in the last two centuries there has been perhaps scarcely a single geometer who has not busied himself in trying to advance the resolution of equations. The general equation of the fifth degree itself has thus far resisted all attacks.

The constantly increasing complication which the formulas for resolving equations must necessarily present, in proportion as the degree increases (the difficulty of using the formula of the fourth degree rendering it almost inapplicable), has determined analysts to renounce, by a tacit agreement, the pursuit of such researches, although they are far from regarding it as impossible to obtain the resolution of equations of the fifth degree, and of several other higher ones.

General Solution. The only question of this kind which would be really of great importance, at least in its logical relations, would be the general resolution of algebraic equations of any degree whatsoever. Now, the more we meditate on this subject, the more we are led to think, with Lagrange, that it really surpasses the scope of our intelligence. We must besides observe that the formula which would express the root of an equation of the m^th degree would necessarily include radicals of the m^th order (or functions of an equivalent multiplicity), because of the m determinations which it must admit. Since we have seen, besides, that this formula must also embrace, as a particular case, that formula which corresponds to every lower degree, it follows that it would inevitably also contain radicals of the next lower degree, the next lower to that, &c., so that, even if it were possible to discover it, it would almost always present too great a complication to be capable of being usefully employed, unless we could succeed in simplifying it, at the same time retaining all its generality, by the introduction of a new class of analytical elements of which we yet have no idea. We have, then, reason to believe that, without having already here arrived at the limits imposed by the feeble extent of our intelligence, we should not be long in reaching them if we actively and earnestly prolonged this series of investigations.

It is, besides, important to observe that, even supposing we had obtained the resolution of algebraic equations of any degree whatever, we would still have treated only a very small part of algebra, properly so called, that is, of the calculus of direct functions, including the resolution of all the equations which can be formed by the known analytical functions.

Finally, we must remember that, by an undeniable law of human nature, our means for conceiving new questions being much more powerful than our resources for resolving them, or, in other words, the human mind being much more ready to inquire than to reason, we shall necessarily always remain below the difficulty, no matter to what degree of development our intellectual labour may arrive. Thus, even though we should some day discover the complete resolution of all the analytical equations at present known, chimerical as the supposition is, there can be no doubt that, before attaining this end, and probably even as a subsidiary means, we would have already overcome the difficulty (a much smaller one, though still very great) of conceiving new analytical elements, the introduction of which would give rise to classes of equations of which, at present, we are completely ignorant; so that a similar imperfection in algebraic science would be continually reproduced, in spite of the real and very important increase of the absolute mass of our knowledge.

What we know in Algebra. In the present condition of algebra, the complete resolution of the equations of the first four degrees, of any binomial equations, of certain particular equations of the higher degrees, and of a very small number of exponential, logarithmic, or circular equations, constitute the fundamental methods which are presented by the calculus of direct functions for the solution of mathematical problems. But, limited as these elements are, geometers have nevertheless succeeded in treating, in a truly admirable manner, a very great number of important questions, as we shall find in the course of the volume. The general improvements introduced within a century into the total system of mathematical analysis, have had for their principal object to make immeasurably useful this little knowledge which we have, instead of tending to increase it. This result has been so fully obtained, that most frequently this calculus has no real share in the complete solution of the question, except by its most simple parts; those which have reference to equations of the two first degrees, with one or more variables.