their resj^ective masses, their mean densities, and the lead- ing circumstances of the fall of heavy bodies on their respective surfaces, etc. Through the power of mathema- tical theories, all this and very much more has been obtained by means of a very small number of straight lines, properly chosen, and a larger number of angles. We might even say, to describe the general bearing of the science in a sentence, that, but for the fear of multiplying mathematical operations unnecessarily, and for the conse- quent necessity of reserving them for the determination of quantities which could not be measured directly, the know- ledge of all magnitudes susceptible of precise estimate which can be offered by the various orders of phenomena, would be finally reducible to the immediate measure- ment of a single straight line, and of a suitable number of angles.
True defini- We can now define Mathematical science tion of mathe- with precision. It has for its object the matics. indirect measurement of magnitudes, and it ACTUAL SCOPE OF MATHEMATICS. 41 proposes to determine magnitudes by each other, according to the precise relations tvhich exist between them. Preceding definitions Lave given to Mathematics the character of an Art; this raises it at once to the rank of a true Science. According to this definition, the spirit of Mathematics con- sists in regarding as mutually connected all the quantities which can be presented by any phenomenon whatsoever, in order to deduce all from each other. Now, there is evi- dently no phenomenon which may not be regarded as affoi'ding such considerations. Hence results the naturally indefinite extent, and the rigorous logical universality of Mathematical science. As for its actual practical extent, we shall see what that is hereafter.
These explanations justify the name of Mathematics, applied to the science we are considering. By itself it signifies Science. The G-reeks had no other, and we may call it the science; for its definition is neither more nor less (if we omit the specific notion of magnitudes) than the definition of all science whatsoever. All science cousists in the co-ordination of facts; and no science could exist among isolated observations. It might even be said that Mathe- matics might enable us to dispense with all direct observa- tion, by empowering us to deduce from the smallest possible number of immediate data the largest possible amount of results. Is not this the real use, both in speculation and in action, of the laws which we discover among natural pheno- mena? If so, Mathematics merely urges to the ultimate degree, in its own way, researches which every real science pursues, in various inferior degrees, in its own sphere. Thus it is only through Mathematics that we can thoroughly understand what true science is. Here alone can we find in the highest degree simplicity and severity of scientific law, and such abstraction as the human mind can attain. Any scientific education setting forth from any other point, is faulty in its basis.
Thus far, we have viewed the science as a whole. We must now consider its primary division. The secondary divisions will be laid down afterwards.
Every mathematical solution spontaneously tt-stwop\rts separates into two parts. The inquiry being, as we have seen, the determination of unknown magnitudes.
42 POSITIVE PHILOSOPHY.
through their relation to the known, the student must, in „,.,.™, the first phice, ascertain what these relations Ineir dmerent • xi i i • a- mi • ^ i obiects ^^®' ^-"^ '^^^ ^^^*^ under his notice, iliis first is the Concrete part of the inquiry. When it is accomplished, what remains is a pure question of num- hers, consisting simply in the determination of unknown numbers, when we know by what relation they are con- nected with known numbers. This second operation is the Abstract jjart of the inquiry. The primary division of Mathematics is therefore into two great sciences: — Abstract Mathematics, and Concrete Mathematics. This divi- sion exists in all complete mathematical questions what- ever, whether more or less simple.
Recurring to the simplest case of a falling body, we must begin by learning the relation between the height from which it falls and the time occupied in falling. As Geometers say, we must find the equation which exists be- tween them. Till this is done, there is no basis for a com- putation. This ascertainment may be extremely difficult, and it is incomparably the superior part of the problem. The true scientific spirit is so modern, that as far as we know, no one before Galileo had remarked the acceleration of velocity in a falling body, the natural supposition having been that the height was in uniform proportion to the time. This first inquiry issued in the discovery of the law of Galileo. The Concrete part being accomplished, the Abstract remains. We have ascertained that the spaces traversed in each second increase as the series of odd num- bers, and we now have only the task of the computation of the height from the time, or of the time from the height; and this consists in finding that, by the established law, the first of these two quantities is a known multiple of the second power of the other; whence we may finally deter- mine the value of the one when that of the other is given. In this instance the concrete question is the more difficult of the two. If the same phenomenon were taken in its greatest generality, the reverse would be the case. Take the two together, and they may be regarded as exactly equivalent in difficulty. The mathematical law may be easy to ascertain, and difficult to work; or it may be diffi- cult to ascertain, and easy to work. In imi^ortance, in ex- CHARACTER OF THE TWO DIVISIONS. 43 tent, and in difficulty, these two great sections of Mathe- matical Science will be seen hereafter to be equivalent.
We have seen the difference in their Their difterent objects. They are no less different in their natures, nature.
The Concrete must depend on the character of the objects examined, and must vary when new phenomena present themselves: whereas, the Abstract is wholly inde- pendent of the nature of the objects, and is concerned only with their numerical relations. Thus, a great variety of phenomena may be brought under one geometrical solution. Cases which appear as unlike each other as possible may stand for one another under the Abstract pro- cess, which thus serves for all, while the Concrete process must be new in each case. Thus the Concrete process is Special, and the Abstract is General. The character of the Concrete is experimental, physical, phenomenal: while the Abstract is purely logical, rational. The Concrete pai't of every mathematical question is necessarily founded on con- sideration of the external world; while the Abstract part consists of a series of logical deductions. The equations being once found, in any case, it is for the understanding, without external aid, to educe the results which these equations contain.
We see how natural and complete this main division is. We will briefly j^rescribe the limits of each section.
As it is the business of Concrete Mathe-,, matics to discover the equations of pheno- ^Mathematics mena, we might suppose that it must com- prehend as many distinct sciences as there are distinct categories of phenomena; bvit we are very far indeed from having discovered mathematical laws in all orders of phenomena. In fact, there are as yet only two great categories of phenomena whose equations are constantly known: — Geometrical and Mechanical phenomena. Thus, the Concrete part of Mathematics consists of Geometry and Kational Mechanics.
There is a point of view from which all phenomena might be included under these two divisions. All natural effects, considered statically or dynamically, might be re- ferred to laws of extension or laws of motion. But this 44 POSITIVE PHILOSOPHY.
jjoint of view is too high for us at present; and it is only in the regions of Astronomy, and, partially, of terrestrial Physics, that this vast transformation has taken place. We will then proceed on the supposition that Geometry and Mechanics are the constituents of Conci'ete Mathe- matics.
., ^ ^ The nature of Abstract Mathematics is Mathematics precisely determined. It is composed of what is called the Calctdus, taking this word in its widest extension, which reaches from the simplest numerical operations to the highest combinations of trans- cendental analysis. Its proper object is to resolve all questions of numbers. Its starting-point is that which is the limit of Concrete Mathematics, — the knowledge of the precise relations — that is, the equations — between different magnitudes which are considered simultaneously. The object of the Calculus, however indirect or complicated the relations may be, is to discover unknown quantities by the known. This science, though more advanced than any other, is, in reality, only at its beginning yet; but it is necessary, in order to define the nature of any science, to suppose it perfect. And the true character of the Calculus is what we have said.
From an historical point of view. Mathematical Analysis appeal's to have arisen out of the contemplation of geome- trical and mechanical facts; but it is not the less indepen- dent of these sciences, logically speaking. Analytical ideas are, above all others, universal, abstract, and simj^le; and geometrical and mechanical conceptions are necessarily founded on them. Mathematical Analysis is therefore the true rational basis of the whole system of our positive knowledge. We can now also explain why it not only gives precisiou to our actual knowledge, but establishes a far more perfect co-ordination in the study of phenomena which allow of such an application. If a single analytical question, brought to an abstract solution, involves the implicit solution of a multitude of physical questions, the mind is enabled to perceive relations between i)henomena apparently isolated, and to extract from them the quality which they have in common. To the wonder of the student, unsuspected relations arise between problems which, instead EXTENT OF THE DOMAIN OF MATHEMATICS. 45 of being, as they appeared before, wholly unconnected, turn out to be identical. There appears to be no connection between the determination of the direction of a curve at each of its points and that of the velocity of a body at each moment of its variable motion; yet, in the eyes of the geometer, these questions are but one.
Wlien we have seized the true general character of Mathe- matical Analysis, we easily see how perfect it is, in com- parison with all other branches of our positive science. The 2)erfection consists in the simjilicity of the ideas con- templated; and not, as Condillac and others have supposed, to the conciseness and generality of the signs used as instru- ments of reasoning. The signs are of admirable use to work out the ideas, when once obtained; but, in fact, all the great analytical conceptions were formed without any essential aid from the signs. Subjects which are by their nature inferior in simplicity and generality cannot be raised to logical perfection by any artifice of scientific language.
We have now seen what is the object and what is the character of Mathematical Science.,t^„^II„'^ ^ ^ It remains tor us to consider the extent or its domain.
We must first admit that, in a logical view, r± • i-^. ,1...-1 -, '^. 1 Itsiiniversahty, this science is necessarily and. rigorously universal. There is no inquiry which is not finally reducible to a question of Numbers; for there is none which may not be conceived of as consisting in the determination of quantities by each other, according to certain relations.
The fact is, we are always endeavouring to arrive at numbers, at fixed quantities, whatever may be our subject, however uncertain our methods, and however rough our results. Nothing can appear less like a mathematical inquiry than the study of living bodies in a state of disease; yet, in studying the cure of disease, we are endeavouring to ascertain the quantities of the different agents which are to modify the organism, in order to bring it to its natural state, admitting, as geometers do, for some of these quantities, in certain cases, values which are equal to zero, negative, or even contradictory. It is not meant that such a method can be actually followed in the case of complicated phenomena; but the logical extension of the science, which 46 POSITIVE PHILOSOPHY, is what we are now considering, conij)reliends such instances as this.
Kant has divided human ideas into the two categories of quantity and quahty, which, if true, would destroy the universality of Mathematics; but Descartes' fundamental conception of the relation of the concrete to the abstract in Mathematics abolishes this division, and proves that all ideas of quality are reducible to ideas of cj^uantity. He had m view geometrical phenomena only; but his successors have included in this generalization, first, mechanical phenomena, and, more recently, those of heat. There are now no geometers who do not consider it of universal ap- j)lication, and admit that every phenomenon may be as Jogically capable of being represented by an equation as a curve or a motion, if only we were always capable (whicli we are very far from being) of first discovering, and then resolving it.
.,,..,,. _, The limitations of Mathematical science are not, then, in its nature. The limitations are in our intelligence: and by these we find the domain of the science remarkably restricted, in proportion as pheno- meua, in becoming special, become complex.
Though, as we have seen, every question may be con- ceived of as reducible to numbers, the reduction cannot be made by us except in the case of the simplest and most general phenomena. The difiiculty of finding the equation in the case of special, and therefore complex phenomena, soon becomes insurmountable, so that, at the utmost, it is only the phenomena of the first three classes, — that is, only those of Inorganic Physics, — that we can even hope to subject to the process. The properties of inorganic bodies are nearly invariable; and therefore, with regard to them, the first condition of mathematical inquiry can be fulfilled: the different quantities which they present may be resolved into fixed numbers; but the variableness of the properties of organic bodies is beyond our management. An inorganic body, possessing solidity, form, consistency, specific gravity, elasticity, etc., presents qualities which are within our estimate, and can be treated mathematically; but the case is altered when Chemical action is added to these. Comjilications and variations then enter into the THE LIMITATIONS OF MxVl'HEMATICS. 47 question which at present baffle mathematical analysis. Hereafter, it may be discovered what fixed numbers exist in chemical combinations: but we are as yet very far from having any practical knowledge of them. Still further are we from being able to foi'm such computations amidst the continual agitation of atoms which constitutes what we call life, and therefore from being able to carry mathe- matical analysis into the study of Physiology. By the rapidity of their changes, and their incessant numerical variations, vital phenomena are, practically, placed in opposition to mathematical jirocesses. If we should desire to compute, in a single case, the most simple facts of a living body, — such as its mean density, its teinperature, the velocity of its circulation, the proportion of elements which at any moment compose its solids or its fluids, the quantity of oxygen which it consumes in a given time, the amount of its absorptions or its exhalations, — and, yet more, the energy of its muscular force, the intensity of its impressions, etc., we must make as many observations as there are sjjecies or races, and varieties in each; we must measure the changes which take place ia passing from one individual to another, and in the same individual, according to age, health, interior condition, surrounding circum- stances perpetually varying, such as the constitution of the atmosphere, etc. It is clear that no mathematical precision can be attained amidst a complexity like this. Social phenomena, being more complicated still, are even more out 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 clear- ness, 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 48 POSITIVE PHILOSOPHY.
not grasp and wield sucli an aggregate of conditions, how- ever 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 com- plex problem which they j)resent is the influence of a third body acting in the same way on two which are tending to- wards each other in virtue of gravitation; and this is a more simple question than any terrestrial problem what- ever. We have, however, attained only approximate solu- tions 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 constitu- tion 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 conse- quence, 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 occa- sioning perceptible derangements. If any of the condi- tions 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 investi- gate 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 estab- lished. 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 intro- ducing 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 in- troduced 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 Mathema- tical 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.
50 CHAPTER II.
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 pass- ing 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 im- l)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 mag- nitudes all the auxiliary magnitudes which mav be con- nected 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, some- times 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 ques- tion of any determinate function being abstract, and thei*e- 52 POSITIVE PHILOSOPHY.
fore 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 analy- tical 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 pro- ceed -very far.
It is the insufficiency of this very small number of analy- tical 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 con- crete 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 pro- posed equations, so as to exhibit the mode of formation of unknown quantities by the known. This.,, constitutes the «Z(/e&ratc question. Then en- sues the task of finding the values of the formulas thus ob- tained. 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 alge- braic and the arithmetical calculus differ in their object. They differ also in their view of quantities, — Algebra con- sidering quantities in regard to their relations, and Arith- metic 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.