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.