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_.

FUNCTION. ITS NAME.

1st couple {1° _y_ = _a_ + _x_ _Sum._
{2° _y_ = _a_ - _x_ _Difference._

2d couple {1° _y_ = _ax_ _Product._
{2° _y_ = _a/x_ _Quotient._

3d couple {1° _y_ = _x^a_ _Power._
{2° _y_ = _[ath root]x_ _Root._

4th couple {1° _y_ = _a^x_ _Exponential._
{2° _y_ = _[log a]x_ _Logarithmic._

5th couple {1° _y_ = sin. _x_ _Direct Circular._
{2° _y_ = arc(sin. = _x_). _Inverse Circular._[3]

[Footnote 3: With the view of increasing as much as possible the
resources and the extent (now so insufficient) of mathematical
analysis, geometers count this last couple of functions among the
analytical elements. Although this inscription is strictly
legitimate, it is important to remark that circular functions are
not exactly in the same situation as the other abstract elementary
functions. There is this very essential difference, that the
functions of the four first couples are at the same time simple and
abstract, while the circular functions, which may manifest each
character in succession, according to the point of view under which
they are considered and the manner in which they are employed,
never present these two properties simultaneously.

Some other concrete functions may be usefully introduced into the
number of analytical elements, certain conditions being fulfilled.
It is thus, for example, that the labours of M. Legendre and of M.
Jacobi on _elliptical_ functions have truly enlarged the field of
analysis; and the same is true of some definite integrals obtained
by M. Fourier in the theory of heat.]

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.

[Footnote 4: Suppose, for example, that a question gives the
following equation between an unknown magnitude x, and two known
magnitudes, _a_ and _b_,

_x³_ + 3_ax_ = 2_b_,

as is the case in the problem of the trisection of an angle. We see
at once that the dependence between _x_ on the one side, and _ab_ on
the other, is completely determined; but, so long as the equation
preserves its primitive form, we do not at all perceive in what
manner the unknown quantity is derived from the data. This must be
discovered, however, before we can think of determining its value.
Such is the object of the algebraic part of the solution. When, by a
series of transformations which have successively rendered that
derivation more and more apparent, we have arrived at presenting the
proposed equation under the form

_x_ = ∛(_b_ + √(_b²_ + _a³_)) + ∛(_b_ - √(_b²_ + _a³_)),

the work of _algebra_ is finished; and even if we could not perform
the arithmetical operations indicated by that formula, we would
nevertheless have obtained a knowledge very real, and often very
important. The work of _arithmetic_ will now consist in taking that
formula for its starting point, and finding the number _x_ when the
values of the numbers _a_ and _b_ are given.]

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]

[Footnote 5: I have thought that I ought to specially notice this
definition, because it serves as the basis of the opinion which
many intelligent persons, unacquainted with mathematical science,
form of its abstract part, without considering that at the time of
this definition mathematical analysis was not sufficiently
developed to enable the general character of each of its principal
parts to be properly apprehended, which explains why Newton could
at that time propose a definition which at the present day he would
certainly reject.]

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.

[Footnote 6: This is less strictly true in the English system of
numeration than in the French, since "twenty-one" is our more usual
mode of expressing this number.]

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, 1⁰, _infinitely small_ elements, the _differentials_ (of different orders) of those quantities, if we regard this analysis in the manner of LEIBNITZ; or, 2⁰, 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, 3⁰, 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_.