AT..,.. r^.^r. It cannot but have been observed that in Mathematics...„.,.,, our enumeration of the sciences there is a prodigious omission. We have said nothing of Mathematical science. The omission was intentional; and the reason is no other than the vast importance of mathematics. This science will be the first of which we shall treat. Meantime, in order not to omit from our sketch a department so prominent, we may indicate here the general results of the study we are about to enter upon.
., In the present stage of our knowledge we '^ ' must regard Mathematics less as a constituent part of natural philosophy than as having been, since the .,. time of Descartes and Newton, the true basis of the whole of natural jihilosophy; though it is, exactly sj^eaking, both the one and the other. To us it is of less value for the knowledge of which it consists, substantial and valuable as that knowledge is, than as..., beinof the most powerful instrument that the human mind can employ m the investigation of the laws of natural phenomena.
In due precision, Mathematics must be science divided into two great sciences, quite distinct from each other — Abstract Mathematics, or the Calculus (taking the word in its most extended sense), and Concrete Mathematics, which is composed of General Geometry and of Rational Mechanics. The Concrete part is necessarily founded on the Abstract, and it becomes in its turn the basis of all natural philosophy; all the phenomena of the universe being regarded, as far as possible, as geometrical or mechanical.
MATHEMATICS A DOUBLE SCIENCE. 35 The Abstract portion is the only one which Abstract mais purely instrumental, it being simply an thematics an immense extension of natural logic to a cer- nistruinent. tain order of deductions. G-eometry and Concrete mamechanics must, on the contrary, be re- tlieniatics a garded as true natural sciences, founded, science, like all others, on observation, though, by the extreme simplicity of their phenomena, they can be systematized to much greater perfection. It is this capacity which has caused the experimental character of their first principles to be too much lost sight of. But these two physical sciences have this peculiarity, that they are now, and will be more and more, employed rather as method than as doctrine.
It needs scarcely be j^ointed out that in placing Mathematics at the head of Positive Philosophy, we are only extending the application of the jsrincijile which has governed our whole Classification. We are simply carrying back our principle to its first manifestation. Geometrical and Mechanical phenomena are the most general, the most simple, the most abstract of all, — the most irreducible to others, the most independent of them; serving, in fact, as a basis to all others. It follows that the study of them is an indispensable preliminary to that of all others. Therefore must Mathematics hold the first place in the Mathematics hierarchy of the sciences, and be the point pre-eminent of departure of all Education, whether general ^" the scale. or special. In an empirical way, this has hitherto been the custom, — a custom which arose from the great antiquity of mathematical science. We now see why it must be renewed on a rational foundation.
We have now considered, in the form of a philosophical problem, the rational plan of the study of the Positive Philosophy. The order that results is this; an order which of all possible arrangements is the only one that accords with the natural manifestation of all phenomena. Mathematics, Astronomy, Physics, Chemistry, Physiology, Social Physics.
BOOK I.
MATHEMATICS.
MATHEMATICS, ABSTRACT AND COKCRETE.
WE are now to enter upon the study of the first of the Six great Sciences; and we begin by establishing the importance of the Positive Philosophy in perfecting the character of each science in itself.
Though Mathematics is the most ancient and the most perfect science of all, the general idea of it is far from being clearly determined. The definition of the science, and its chief divisions, have remained up to this time vague and uncertain. The plural form of the name (grammatically used as singular) indicates the want of unity in its philosophical character, as commonly conceived. In fact, it is only since the beginning of the last century that it could be conceived of as a whole; and since that time geometers have been too much engaged on its different branches, and in apjjlying it to the most important laws of the universe, to have much attention left for the general system of the science. Now however the pursuit of its specialities is no longer so engrossing as to exclude us from the study of Mathematics in its unity. It has now reached a degree of consistency which admits of the effort to reduce its parts into a system, in preparation for further advance. The latest achievements of mathematicians have prepared the way for this by evidencing a character of unity in its principal parts which was not before known to exist. Such is eminently the sjiirit of the great author of the Theory of Eunctions and of Analytical Mechanics.
NATURE AND OBJECT OF MATHEMATICS. o7 The common description of Mathematics, „.. as the science of Magnitudes, or, somewhat niath"matics* more i^ositively, the science which relates to the Measurement of Mag7iitudes, is too vague and unmeaning to have been used but for want of a better. Yet the idea contained in it is just at bottom, and is even sufficiently extensive, if proj^erly understood; but it needs precision and depth. It is important in such matters not to depart unnecessarily from notions generally admitted; and we will therefore see how, from this point of view, we can rise to such a definition of Mathematics as will be adequate to the importance, extent, and difficulty of the science.
Our first idea of measuring a magnitude r^-i ■ i. f is simply that of comparing the magnitude mathematics, in question with another supposed to be known, which is taken for the unit of comparison among all others of the same kind. Thus, when we define mathematics as being the measurement of magnitudes, we give a very imperfect idea of it, and one which seems to bear no relation, in this respect, to any science whatever. We seem to speak only of a series of mechanical procedures, like a superposition of lines, for obtaining the comparison of magnitudes, instead of a vast chain of reasonings, inexhaustible by the intellect. Nevertheless, this definition has no other fault than not being deep enough. It does not mistake the real aim of mathematics, but it presents as direct an object which is usually indirect; and thus it misleads us as to the nature of the science. To rectify this, we must attend to a general fact, which is easily established; that the direct measurement of a magnitude is often an impossible operation; so that if we had no other means of doing what we Avant, we must often forego the knowledge we desire. We can rarely even measure a right line by another right line; and this is the simplest measurement of all. The very first condition of this is that we should be able to traverse the line from one end to the other; and this cannot be done with the greater number of the distances which interest us the most. We cannot do it with the heavenly bodies, nor with the earth and any heavenly body, nor even with many distances on the earth; aud again, the length must be neither too great nor too small, and it must be conveniently situated; and a line which could be easily measured if it were horizontal becomes impracticable if vertical. There are so few lines capable of being directly measured with precision, that we are compelled to resort to artificial lines, created to admit of a direct determination, aud to be the point of refei'ence for all others. If there is difficulty about the m.easurement of lines, the embarrassment is much greater when we have to deal with surfaces, volumes, velocities, times, forces, etc., and in general with all other magnitudes susceptible of estimate, and, by their nature, difficult of direct measurement. It is the general fact of this difficulty, inherent in almost every case, which necessitates the formation of mathematical science; for, finding direct measurement so often impossible, we are compelled to devise means of doing it indirectly. Hence arose Mathematics.
. The general method employed, and the method only conceivable one, is to connect the magnitudes in question with some that can be directly determined, and thus to ascertain the former, through their relations with the latter. Such is the precise object of Mathematics, regarded as a whole. To form anything like a wortliy idea of it, we must remember that the indirect determination of magnitudes may have many degrees of indirectness. It often happens that the magnitudes to which undetermined magnitudes are to be referred cannot themselves be measured directly, and must themselves be made the subject of a prior process, and so on through a whole series; and thus, the mind is often obliged to establish a long course of intermediaries between the one aud the other point of the inquiry — points which may appear at the outset to have no connection whatever. -p., If this appears too abstract, it may become plain by a tew examples. In observmg a falling body, we are aware that two quantities are involved: the height from which the body falls, and the time occupied in its descent. These two quantities are connected, as they vary together, and together remain fixed. In the language of mathematicians, they are functions of each METHOD OF APPLICATION. 39 other. The measurement of one being impracticable, it is supplied by that of the other. By observing the time occupied by a stone in falling down a j^recipice, we can ascertain the height of the precipice as accurately as if we could measure it with a horizontal line. In another case, we may be able to know the height whence a body has fallen, and unable to observe the time with precision, and then we must have recourse to the inverse cjuestion, — to determine the time by the distance; as, for instance, if we were to inquire how long it would take for a body to fall from the moon. In these cases, the question is very simple, supposing we do not complicate it with considerations of intensity of gravity, resistance of a fluid medium, etc. But, to enlarge the question, we must contemplate the phenomenon in its greatest generality by supposing the fall to be oblique, and taking into account all the principal circumstances. Then, instead of two variable quantities, simply connected, the phenomenon will present a considerable number, — the sjDace traversed, whether in a vertical or horizontal direction; the time employed in traversing it: the velocity of the body at each point of its course; and even the intensity and direction of the impulse which sent it forth; and finally, in some cases, the resistance of the medium, and the intensity of gravity. All these quantities are so connected that each in its turn may be determined indirectly by means of the others, and thus we shall have as many mathematical inquiries as there are magnitudes co-existing in the phenomenon considered. Such a very simple change as this in the physical conditions of a problem may place a mathematical question, originally quite elementary, in the rank of those difficult questions whose complete and rigorous solution transcends the power of the human understanding.
Again,— we may take a geometrical example. We want to determine a distance not directly measurable. We shall conceive of it as making a part of some ficjure, or system of lines of some soi't, of which the other parts are directly measurable; let us say a triangle (for this is the simplest, and to it all others are reducible). The distance in question is supposed to form a portion of a triangle, in which we are able to determine directly, either another 40 ' POSITIVE PHILOSOPHY, side and two angles, or two sides and one angle. The knowledge required is obtained by the mathematical labour of deducing the unknown distance from the observed elements, by means of the relation between them. The process may, and commonly does, become highly complicated by the elements supjDosed to be known being themselves determinable only in an indirect manner, by the aid of fresh auxiliary systems, the number of which may be very considerable. The distance, once ascertained, will often enable us to obtain new quantities, which will offer occasion for new mathematical questions. Thus, when we once know the distance of any object, the observation, simple and always possible, of its apparent diameter, may disclose to us, with certainty, however indirectly, its real dimensions; and at length, by a series of analogous inquiries, its surface, its volume, even its weight, and a multitude of other qualities which might have seemed out of the reach of our knowledge for ever. It is by such labours that Man has learned to know, not only the distances of the planets from the earth and from each other, but their actual magnitude,- — their true form, even to the inequalities on their surface, and (what seems much more out of his reach) their resj^ective masses, their mean densities, and the leading circumstances of the fall of heavy bodies on their respective surfaces, etc. Through the power of mathematical 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 consequent necessity of reserving them for the determination of quantities which could not be measured directly, the knowledge of all magnitudes susceptible of precise estimate which can be offered by the various orders of phenomena, would be finally reducible to the immediate measurement 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 consists 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 evidently 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 Mathematics might enable us to dispense with all direct observation, 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 phenomena? 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.
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 numhers, consisting simply in the determination of unknown numbers, when we know by what relation they are connected 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 division exists in all complete mathematical questions whatever, 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 between them. Till this is done, there is no basis for a computation. 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 numbers, 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 determine 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 difficult 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 Mathematical 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 independent 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 process, 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 consideration 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 comprehend 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 referred to laws of extension or laws of motion. But this 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 Mathematics.
., ^ ^ 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 transcendental 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 geometrical and mechanical facts; but it is not the less independent 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 Mathematical Analysis, we easily see how perfect it is, in comparison with all other branches of our positive science. The 2)erfection consists in the simjilicity of the ideas contemplated; and not, as Condillac and others have supposed, to the conciseness and generality of the signs used as instruments 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 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 apj)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 phenomeua, in becoming special, become complex.
Though, as we have seen, every question may be conceived 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 mathematical 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 circumstances 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