of the question, as subjects for mathematical analysis.
It is not that a mathematical basis does not exist in these cases, as truly as in phenomena which exhibit, in all clearness, the law of gravitation: but that our faculties are toa limited for the working of problems so intricate. We are baffled by various phenomena of inorganic bodies, when they are very complex. For instance, no one doubts tiiat meteorological phenomena are subject to mathematical laws, however little we yet know about them; but their multiplicity renders their observed results as variable and irregular as if each cause were free of all such conditions.
We find a second limitation in the number of conditions to be studied, even if we were sure of the mathematical law which governs each accent. Our feeble faculties could not grasp and wield sucli an aggregate of conditions, however certain might be our knowledge of each. In the simplest cases in which we desire to approximate the abstract to the concrete conditions, with any completeness, ■ — as in the phenomenon of the flow of a fluid from a given orifice, by virtue of its gravity alone, — the difiiculty is such that we are, as yet, without any mathematical solution of this very problem. The same is the case with the yet more simple instance of the movement of a solid projectile through a resisting medium.
To the popular mind it may appear strange, considering these facts, that we know so much as we do about the planets. But in reality, that class of phenomena is the most simple of all within our cognizance. The most complex problem which they j)resent is the influence of a third body acting in the same way on two which are tending towards each other in virtue of gravitation; and this is a more simple question than any terrestrial problem whatever. We have, however, attained only approximate solutions in this case. And the high perfection to which solar astronomy has been brought by the use of mathematical science is owing to our having profited by those facilities that we may call accidental, which the favourable constitution of our planetary system presents. The planets which compose it are few; their masses are very unequal, and mvich less than that of the sun; they are far distant from each other; their forms are nearly spherical; their orbits are nearly circular, and only slightly inclined in relation to each other; and so on. Their perturbations are, in consequence, inconsiderable, for the most part; and all we have to do is usually to take into the account, together with the influence of the sun on each planet, the influence of one other planet, capable, by its size and its nearness, of occasioning perceptible derangements. If any of the conditions mentioned above had been different, though the law of gravitatii.m had existed as it is, we might not at this day have discovered it. And if we were now to try to investigate Chemical phenomena by the same law, we should find a solution as imjiossible as it would be in astronomy, if the conditions of the heavenly bodies were such as we could not reduce to an analysis.
JIATHEMATICS: FINAL CONSIDERATIONS. 49 In showing that Mathematical analysis can be applied only to Inorganic Physics, we are not restricting its domain. Its rigorous universality, in a logical view, has been established. To pretend that it is practically applicable to the same extent would be merely to lead away the human mind from the true direction of scientific study, in pursuit of an impossible perfection. The most difficult sciences must remain, for an indefinite time, in that preliminary state which prepares for the others the time when they too may become capable of mathematical treatment. Our business is to study phenomena, in the characters and relations in which they present themselves to us, abstaining from introducing considerations of quantities, and mathematical laws, which it is beyond our power to apply.
We owe to Mathematics both the origin of Positive Philosophy and its Method. When this method was introduced into the other sciences, it was natural that it should be urged too far. But each science modified the method by the operation of its own peculiar phenomena. Thus only could that true definitive character be brought out, which must prevent its being ever confounded with that of any other fundamental science.
The aim, character, and general relations of Mathematical Science have now been exhibited as fully as they could be in such a sketch as this. We must next pass in review the three great sciences of which it is composed, — the Calculus, G-eometry, and Rational Mechanics.
GENERAL VIEW OF MATHEMATICAL ANALYSIS.
. '' I "HE historical develoinneut of the Ab- -L stract portion of Mathematical science has, since the time of Descartes, been for the most part determined by that of the Concrete. Yet the Calculus in all its principal branches must be understood before passing on to Geometry and Mechanics. The Concrete portions of the science depend on the Abstract, which are wholly independent of them. We will now therefore proceed to a rapid review of the leading conceptions of the Analysis.
., ^ First, however, we must take some notice an equation '^^ ^^® general idea of an equation, and see how far it is from being the true one on which geometers proceed in practice; for without settling tliis point we cannot determine, with any precision, the real aim and extent of abstract mathematics.
The business of concrete mathematics is to discover the equations which express the mathematical laws of the phenomenon under consideration; and these equations are the starting-point of the calculus, which must obtain from them certain quantities by means of others. It is only by forming a true idea of an equation that we can lay down the real line of separation between the concrete and the abstract part of mathematics.
It is giving much too extended a sense to the notion of an equation to suppose that it means every kind of relation of equality between any two functions of the magnitudes under consideration; for, if every equation is a relation of equality, it is far from being the case that, reciprocally, every relation of equality must be an equation of the kind to which analysis is, by the nature of the case, applicable. It is evident that this confusion must render it almost iml)ossible to explain the difficulty we find in establishing the FUKCTIOXS OF A TWO-FOLD NATURE. 51 relation of tlie concrete to the abstract whicli meets us in every great mathematical question, taken by itself. If the word equation meant what we are apt to suppose, it is not easy to see what difficulty there could be, in general, in establishing the equations of any problem whatever. This ordinary notion of an equation is widely unlike what geometers understand in the actual working of the science.
According to my view, functions must themselves be divided into Abstract and Concrete; the first of which alone can enter into true equations. Every equation is a relation of equality between two abstract functions of the magnitudes in question, including with the primarv magnitudes all the auxiliary magnitudes which mav be connected with the problem, and the introduction of which may facilitate the discovery of the ec|uations sought.
This distinction may be established by both the a priori and a posteriori methods; by characterizing each kind of function, and by enumerating all the abstract functions yet known, — at least with regard to their elements.
A priori; Abstract functions express a mode of dependence between magnitudes which may tioS'^^* be conceived between numbers alone, without the need of pointing out any phenomena in whichitmay be found reahzed; while Concrete functions are „.
those whose expression requires a sjjecified ^ions^^ ^ ^^ actual case of physics, geometry, mechanics, etc.
Most functions were concrete in their origin, — even those which are at present the most purely abstract; and the ancients discovered only through geometrical definitions elementary algebraic properties of functions, to which a numerical value was not attached till long afterwards, rendering abstract to us what was concrete to the old geometers. There is another example which well exhibits the distinction just made— that of circular functions, both direct and inverse, which are still sometimes concrete, sometimes abstract, according to the point of view from which they are regarded.
A posteriori; the distinguishing character, abstract or concrete, of a function having been established, the question of any determinate function being abstract, and thei*efore able to euter into true analytical equations, becomes a simple question of fact, as we are acquainted with the elements which compose all the abstract functions at present known. We say we know them all, though analytical functions are infinite in number, because we are here speaking, it must be remembered, — of the elements — of the simple, not of the compound. We have ten elementary formulas; and, few as they are, they may give rise to an infinite number of analytical combinations. There is no reason for supposing that there can never be more. We have more than Descartes had, and even Newton and Leibnitz; and our successors will doubtless introduce additions, though there is so much difficulty attending their augmentation, that we cannot hope that it will proceed -very far.
It is the insufficiency of this very small number of analytical elements which constitutes our difficulty in passing from the concrete to the abstract. In order to establish the equations of phenomena, we must conceive of their mathematical laws by the aid of functions composed of these few elements. Up to this point the question has been essentially concrete, not coming within the domain of the calculus. The difficulty of the passage from the concrete to the abstract in general consists in our having only these few analytical elements with which to represent all the precise relations which the whole range of natural phenomena afford to us. Amidst their infinite variety, our conceptions must be far below the real difficulty; and especially because these elements of our analysis have been supplied to us by the mathematical consideration of the simplest phenomena of a geometrical origin, which can afford us a priori no rational guarantee of their fitness to represent the mathematical laws of all other classes of phenomena. We shall hereafter see how this difficulty of the relation of the concrete to the abstract has been diminished, without its being necessary to multiply the number of analytical elements.
Thus far we have considered the Calculus as a whole.
™. We must now consider its divisions. These the Calculus divisions we must call the Algebraic Calculus, or Algebra, and the Arithmetical Calctdus, or ALGEBRAIC AND ARITHMETICAL CALCULUS. 63 Arithmetic, taking care to give them the most extended logical sense, and not the restricted one in which the terms are usually received.
It is clear that every question of Mathematical Analysis presents two successive parts, perfectly distinct in their nature. The first stage is the transformation of the proposed equations, so as to exhibit the mode of formation of unknown quantities by the known. This.,, constitutes the «Z(/e&ratc question. Then ensues the task of finding the values of the formulas thus obtained. The values of the numbers sought are already represented by certain explicit functions of given numbers: these values must be determined; and this.,.,,,. is the arithmetical question. Thus the algebraic and the arithmetical calculus differ in their object. They differ also in their view of quantities, — Algebra considering quantities in regard to their relations, and Arithmetic in regard to their values. In practice, it is not always possible, owing to the imperfection of the science of the calculus, to separate the processes entirely in obtaining a solution; but the radical difference of the two operations should never be lost sight of. Algebra, then, is the Calctilus of Functions, and Arithmetic the Calculus of Values.
We have seen that the division of the Calculus is into two branches. It remains for us to compare the two, in order to learn their respective extent, importance, and difficulty.
TheCalculus of Values, Ai-ithmetic, appears...,,. at first to have as wide a field as Algebra, since as many cjuestions might seem to arise from it as we can conceive different algebraic formulas to be valued. But a very simple reflection will show that it is not so. Functions being divided into simple and compound, it is evident that when we become able to deter- j.. p,,, mine the value of simple functions, there will be no difficulty with the compound. In the algebraic relation, a compound function plays a very different part from that of the elementary functions which constitute it; and this is the source of our chief analytical difficulties. But it is quite otherwise with the Arithmetical Calculus.
Thus, the number of distinct arithmetical operations is indicated by that of the abstract elementary functions, which we have seen to be very few. The determination of the values of these ten functions necessarily affords that of all the infinite niamber comprehended in the whole of mathematical analysis: and there can be no new arithmetical operations otherwise than by the creation of new analytical elements, which must, in any case, for ever be extremely small. The domain of arithmetic then is, by its nature, narrowly restricted, while that of algebra is rigorously indefinite. Still, the domain of arithmetic is more extensive than is commonly represented; for there are many questions treated as incidental in the midst of a body of analytical researches, which, consisting of determinations of values, are truly arithmetical. Of this kind are the construction of a table of logarithms, and the calculation of trigonometrical tables, and some distinct and higher procedures; in short, every operation which has for its object the determination of the values of functions. And we must also include that part of the science of the Calculus which we call the Theory of Numbers, the object of which is to discover the properties inherent in different numbei'S, in virtue of their values, independent of any particular system of numeration. It constitutes a sort of transcendental arithmetic. Though the domain of arithmetic is thus larger than is commonly supposed, this Calculus of values will yet never be more than a point, as it were, in comparison with the calculus of functions, of which mathematical science essentially consists. This is evident, when we look into the real nature of arithmetical questions.
Determinations of values are, in fact, nothing else than y,,, real transformations of the functions to be valued. These transformations have a special end; but they are essentially of the same nature as all taught by analysis. In this view, the Calculus of vahies may be regarded as a supplement, and a particular application of the Calculus of functions, so that arithmetic disappears, as it were, as a distinct section in the body of abstract mathematics. To make this evident, we must observe that when we desire to determine the value of an unknown number whose mode of formation is given, we define ALGEBRA. 55 ALGEBRA. 55 and express that value iu merely announcing the arithmetical question, already defined and expressed under a certain form; and that, in determining its value, we merely express it under another determinate form, to which we are in the habit of referring the idea of each particular number by making it re-enter into the regular system of numeration. This is made clear by what happens when the mode of numeration is such that the question is its own answer; as, for instance, when we want to add together seven and thirty, and call the result seven-and-thirty. In adding other numbers, the terms are not so ready, and we transform the question; as when we add together twenty-three and fourteen: but not the less is the operation merelv one of transformation of a question already defined and expressed. In this view, the calculus of values might be regarded as a particular application of the calculus of functions, arithmetic thereby disappearing, as a distinct section, from the domain of abstract mathematics. — And here we have done with the Calculus of values, and pass to the Calculus of functions, of which abstract mathematics is essentially composed.
We have seen that the difiiculty of esta-.,, blishing the relation of the concrete to the abstract is owing to the insufiicieucy of the very small number of analytical elements that we are in possession of. The obstacle has been surmounted in a great number of important cases: and we will now see how the establishment of the equations of phenomena has been achieved.
The first means of remedying the difiiculty of the small number of analytical elements newfunctions seems to be to create new ones. But a little consideration will show that this resource is illusory. A new analytical element would not serve unless we could immediately determine its value: but how can we determine the value of a function which is simple; that is, which is not formed by a combination of those alreadv known? This appears almost impossible: but the introduction of another elementary abstract function into analysis supposes the simultaneous creation of a new arithmetical operation; which is certainly extremely difficult. If we try to proceed according to the method which procured us the elements we possess, we are left in entire uncertainty; for the artifices thus employed are evidently exhausted. We have thus no idea how to proceed to create new elementary abstract functions. Yet, we must not therefore conclude that we have reached the limit appointed by the powers of our understanding. Special imjjrovements in mathematical analysis have yielded us some ]»artial substitutes, which have increased our resources: but it is clear that the augmentation of these elements cannot proceed but with extreme slowness. It is not in this direction, then, that the human mind has found its means of facilitating the establishment of equations.
Findin<^enua- This first method being discarded, there tions between remains only one other. As it is impossible auxiliary to find the equations directly, we must seek quantities. for corresponding ones between other auxiliary (quantities, connected with the first according to a certain determinate law, and from the relation between which we may ascend to that of the primitive magnitudes. This is the fertile conception which we term the transcendental analysis, and use as our finest instrument for the mathematical exploration of natural phenomena.
This conception has a much larger scope than even profound geometers have hitherto supposed; for the auxiliary quantities resorted to might be derived, according to any law whatever, from the immediate elements of the question. It is well to notice this; becaiise our future improved analytical resources may perhaps be found in a new mode of derivation. But, at present, the only auxiliary quantities habitually substituted for the primitive quantities in transcendental analysis are what are called — • 1st, infinitely small elements, the differentials of different oi'ders of those quantities, if we conceive of this analysis in the manner of Leibnitz: or 2nd, the fluxions, the limits of the ratios of the simultaneous increments of the primitive quantities, compared with one another; or, more briefly, the prime and ultimate ratios of these increments, if we adopt the conception of Newton: or 3rd, the derivatives, properly so called, of these quanti- THE TWO BRANCHES OF ALGEBRA. 57 ties; that is, the coefficients of the different terms of their respective increments, according to the conception of Lagrange.
These conceptions, and all others that have been proposed, are by their nature identical. The various grounds of preference of each of them will be exhibited hereafter.
We now see that the Calculus of f unc- Division of tions, or Algebra, must consist of two distinct the Calculus l>ranches. The one has for its object the of functions. resolution of equations when they are directly established between the magnitudes in question: the other, setting out from equations (generally much more easy to form) between quantities indirectly connected with those of the problem, has to deduce, by invariable analytical procedures, the corresponding equations between the direct magnitudes in question; — bringing the problem within the domain of the preceding calculus. — It might seem that the transcendental analysis ought to be studied before the ordinary, as it provides the equations which the other has to resolve. But, though the transcendental is logically indej^endent of the ordinary, it is best to follow the usual method of study, taking the ordinary first; for, the proposed questions always requiring to be completed by ordinary analysis, they must be left in suspense if the instrument of resolution had not been studied beforehand.
To ordinary analysis I propose to give the name of Cal-CTTLIJS OF DiKECT FUNCTIONS. To transcendental analysis, (which is known by the names of Infinitesimal Calculus, Calculus of fluxions and of fluents, Calculus of Vanishing quantities, the Differential and Integral Calculus, etc., according to the view in which it has been conceived,) I shall give the title of Calculus of Indikect Functions. I obtain these terms by generalizing and giving precision to the ideas of Lagrange, and employ them to indicate the exact character of the two forms of analysis.
ORDINARY ANALYSIS, OR CALCULUS OF DIRECT FUNCTIONS.
Algebra is adequate to the solution of mathematical questions which are so simple that we can form directly the equations between the magnitudes considered, without its being necessary to bring into the problem, either in substitution or alliance, any system of auxiliary quantities derived from the primary. It is true, in the majority of important cases, its use requires to be preceded and prepared for by that of the calculus of indirect functions, by which the establishment of equations is facilitated: but though algebra then takes the second place, it is not the less a necessary agent in the solution of the question; so that the Calculus of direct functions must continue to be, by its nature, the basis of mathematical analysis. We must now, then, notice the rational composition of this calculus, and the degree of development it has attained. J. -I ■ f Its object being the resolution of equations (that is, the discovery of the mode of formation of unknown quantities by the known, according to the equations which exist between them), it presents as many parts as we can imagine distinct classes of equations; and its extent is therefore rigorously indefinite, because the number of analytical functions susceptible of entering into equations is illimitable, though, as we have seen, composed of a very small number of primitive elements.
„,.„,. The rational classification of equations of Eouations ^m^^t evidently be determiued by the nature of the analytical elements of which their members are comjiosed. Accordingly, analysts first divide equations Avith one or more variables into two principal classes, according as they contain functions of only the first three of the ten couples, or as they include also either exponential or circular functions. Though the names of algebraic and transcendental functions given to these principal groups are inapt, the division between the corresponding equations is real enough, insofar as that the resolution of equations containing the transcendental func- RESOLUTION OF ALGEBRAIC EQUATIONS. 59 tions is more difficult than that of algebraic equations. Hence the study of the first is extremely imperfect, and our analytical methods relate almost exclusively to the elaboration of the second.
Our business now is with these Algebraic AUr i " equations only. In the first place, we must equations, observe that, though they may often contain irrational functions of the unknown quantities, as well as rational functions, the first case can always be brought under the second, by transformations more or less easy; so that it is only with the latter that analysts have had to occupy themselves, to resolve all the algebraic equations. As to their classification, the early method of classing them according to the number of their terms has been retained only for equations with two terms, which are, in fact, susceptible of a resolution proper to themselves. The classification by their degrees, long universally established, is eminently natural; for this distinction rigorously determines the greater or less difficulty of their resolution. The gradation can be independently, as well as practically exhibited: for the most general equation of each degree necessarily comprehends all those of the diifereut inferior degrees, as miist also the formula which determines the unknown quantity: and therefore, however slight we may, d priori, suppose the difficulty to be of the degree under notice, it must offer more and more obstacles, in proportion to the rank of the degree, because it is complicated in the execution with those of all the preceding degrees.
This increase of difficulty is so great, that Algebraic rethe resolution of algebraic equations is as yet solution of known to us only in the four first degrees, equations. In this respect, algebra has advanced but little since the labours of Descartes and the Italian analysts of the sixteenth century; though there has probably not been a single geometer for two centuries past who has not sti'iven to advance the resolution of equations. The general equation of the fifth degree has itself, thus far, resisted all attempts. The formula of the fourth degree is so difficult as to be almost inapplicable; and analysts, while by no means despairing of the resolution of equations of the fifth, and even higher degrees, being obtained, have tacitly agreed to give up such researches.
The only question of this kind which would be of eminent importance, at least in its logical relations, would be the general resolution of algebraic equations of any degree whatever. But the more we ponder this subject, the more we are led to suppose, with Lagrange, that it exceeds the scope of our understandings. Even if the requisite formula could be obtained, it could not lie usefully applied, unless we could simplify it, without impairing its generality, by the introduction of a new class of analytical elements, of which we have as yet no idea. And, besides, if we had obtained the resolution of algebraic equations of any degree whatever, we should still have ti'eated only a very small part of algebra, properly so called; that is, of the calculus of dii'ect functions, comprehending the resolution of all the equations that can be formed by the analytical functions known to us at this day. Again, we must remember that by a law of our nature, we shall always remain below the difficulty of science, our means of conceiving of new questions being always more powerful than our resources for resolving them; in other words, the human mind being more apt at imagining than at reasoning. Thus, if we had resolved all the analytical equations now knowu, and if, to do this, we had found new analytical elements, these again would introduce classes of equations of which we now know nothing: and so, however great might be the increase of our knowledge, the imperfection of our algebraic