GENERAL CONSIDERATIONS.

Although Mathematical Science is the most ancient and the most perfect of all, yet the general idea which we ought to form of it has not yet been clearly determined. Its definition and its principal divisions have remained till now vague and uncertain. Indeed the plural name—"The Mathematics"—by which we commonly designate it, would alone suffice to indicate the want of unity in the common conception of it.

In truth, it was not till the commencement of the last century that the different fundamental conceptions which constitute this great science were each of them sufficiently developed to permit the true spirit of the whole to manifest itself with clearness. Since that epoch the attention of geometers has been too exclusively absorbed by the special perfecting of the different branches, and by the application which they have made of them to the most important laws of the universe, to allow them to give due attention to the general system of the science.

But at the present time the progress of the special departments is no longer so rapid as to forbid the contemplation of the whole. The science of mathematics is now sufficiently developed, both in itself and as to its most essential application, to have arrived at that state of consistency in which we ought to strive to arrange its different parts in a single system, in order to prepare for new advances. We may even observe that the last important improvements of the science have directly paved the way for this important philosophical operation, by impressing on its principal parts a character of unity which did not previously exist.

To form a just idea of the object of mathematical science, we may start from the indefinite and meaningless definition of it usually given, in calling it "The science of magnitudes," or, which is more definite, "The science which has for its object the measurement of magnitudes." Let us see how we can rise from this rough sketch (which is singularly deficient in precision and depth, though, at bottom, just) to a veritable definition, worthy of the importance, the extent, and the difficulty of the science.

THE OBJECT OF MATHEMATICS.

Measuring Magnitudes. The question of measuring a magnitude in itself presents to the mind no other idea than that of the simple direct comparison of this magnitude with another similar magnitude, supposed to be known, which it takes for the unit of comparison among all others of the same kind. According to this definition, then, the science of mathematics—vast and profound as it is with reason reputed to be—instead of being an immense concatenation of prolonged mental labours, which offer inexhaustible occupation to our intellectual activity, would seem to consist of a simple series of mechanical processes for obtaining directly the ratios of the quantities to be measured to those by which we wish to measure them, by the aid of operations of similar character to the superposition of lines, as practiced by the carpenter with his rule.

The error of this definition consists in presenting as direct an object which is almost always, on the contrary, very indirect. The direct measurement of a magnitude, by superposition or any similar process, is most frequently an operation quite impossible for us to perform; so that if we had no other means for determining magnitudes than direct comparisons, we should be obliged to renounce the knowledge of most of those which interest us.

Difficulties. The force of this general observation will be understood if we limit ourselves to consider specially the particular case which evidently offers the most facility—that of the measurement of one straight line by another. This comparison, which is certainly the most simple which we can conceive, can nevertheless scarcely ever be effected directly. In reflecting on the whole of the conditions necessary to render a line susceptible of a direct measurement, we see that most frequently they cannot be all fulfilled at the same time. The first and the most palpable of these conditions—that of being able to pass over the line from one end of it to the other, in order to apply the unit of measurement to its whole length—evidently excludes at once by far the greater part of the distances which interest us the most; in the first place, all the distances between the celestial bodies, or from any one of them to the earth; and then, too, even the greater number of terrestrial distances, which are so frequently inaccessible. But even if this first condition be found to be fulfilled, it is still farther necessary that the length be neither too great nor too small, which would render a direct measurement equally impossible. The line must also be suitably situated; for let it be one which we could measure with the greatest facility, if it were horizontal, but conceive it to be turned up vertically, and it becomes impossible to measure it.

The difficulties which we have indicated in reference to measuring lines, exist in a very much greater degree in the measurement of surfaces, volumes, velocities, times, forces, &c. It is this fact which makes necessary the formation of mathematical science, as we are going to see; for the human mind has been compelled to renounce, in almost all cases, the direct measurement of magnitudes, and to seek to determine them indirectly, and it is thus that it has been led to the creation of mathematics.

General Method. The general method which is constantly employed, and evidently the only one conceivable, to ascertain magnitudes which do not admit of a direct measurement, consists in connecting them with others which are susceptible of being determined immediately, and by means of which we succeed in discovering the first through the relations which subsist between the two. Such is the precise object of mathematical science viewed as a whole. In order to form a sufficiently extended idea of it, we must consider that this indirect determination of magnitudes may be indirect in very different degrees. In a great number of cases, which are often the most important, the magnitudes, by means of which the principal magnitudes sought are to be determined, cannot themselves be measured directly, and must therefore, in their turn, become the subject of a similar question, and so on; so that on many occasions the human mind is obliged to establish a long series of intermediates between the system of unknown magnitudes which are the final objects of its researches, and the system of magnitudes susceptible of direct measurement, by whose means we finally determine the first, with which at first they appear to have no connexion.

Illustrations. Some examples will make clear any thing which may seem too abstract in the preceding generalities.

1. Falling Bodies. Let us consider, in the first place, a natural phenomenon, very simple, indeed, but which may nevertheless give rise to a mathematical question, really existing, and susceptible of actual applications—the phenomenon of the vertical fall of heavy bodies.

The mind the most unused to mathematical conceptions, in observing this phenomenon, perceives at once that the two quantities which it presents—namely, the height from which a body has fallen, and the time of its fall—are necessarily connected with each other, since they vary together, and simultaneously remain fixed; or, in the language of geometers, that they are "functions" of each other. The phenomenon, considered under this point of view, gives rise then to a mathematical question, which consists in substituting for the direct measurement of one of these two magnitudes, when it is impossible, the measurement of the other. It is thus, for example, that we may determine indirectly the depth of a precipice, by merely measuring the time that a heavy body would occupy in falling to its bottom, and by suitable procedures this inaccessible depth will be known with as much precision as if it was a horizontal line placed in the most favourable circumstances for easy and exact measurement. On other occasions it is the height from which a body has fallen which it will be easy to ascertain, while the time of the fall could not be observed directly; then the same phenomenon would give rise to the inverse question, namely, to determine the time from the height; as, for example, if we wished to ascertain what would be the duration of the vertical fall of a body falling from the moon to the earth.

In this example the mathematical question is very simple, at least when we do not pay attention to the variation in the intensity of gravity, or the resistance of the fluid which the body passes through in its fall. But, to extend the question, we have only to consider the same phenomenon in its greatest generality, in supposing the fall oblique, and in taking into the account all the principal circumstances. Then, instead of offering simply two variable quantities connected with each other by a relation easy to follow, the phenomenon will present a much greater number; namely, the space 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; even the intensity and the direction of its primitive impulse, which may also be viewed as variables; and finally, in certain cases (to take every thing into the account), the resistance of the medium and the intensity of gravity. All these different quantities will be connected with one another, in such a way that each in its turn may be indirectly determined by means of the others; and this will present as many distinct mathematical questions as there may be co-existing magnitudes in the phenomenon under consideration. Such a very slight change in the physical conditions of a problem may cause (as in the above example) a mathematical research, at first very elementary, to be placed at once in the rank of the most difficult questions, whose complete and rigorous solution surpasses as yet the utmost power of the human intellect.

2. Inaccessible Distances. Let us take a second example from geometrical phenomena. Let it be proposed to determine a distance which is not susceptible of direct measurement; it will be generally conceived as making part of a figure, or certain system of lines, chosen in such a way that all its other parts may be observed directly; thus, in the case which is most simple, and to which all the others may be finally reduced, the proposed distance will be considered as belonging to a triangle, in which we can determine directly either another side and two angles, or two sides and one angle. Thence-forward, the knowledge of the desired distance, instead of being obtained directly, will be the result of a mathematical calculation, which will consist in deducing it from the observed elements by means of the relation which connects it with them. This calculation will become successively more and more complicated, if the parts which we have supposed to be known cannot themselves be determined (as is most frequently the case) except in an indirect manner, by the aid of new auxiliary systems, the number of which, in great operations of this kind, finally becomes very considerable. The distance being once determined, the knowledge of it will frequently be sufficient for obtaining new quantities, which will become the subject of new mathematical questions. Thus, when we know at what distance any object is situated, the simple observation of its apparent diameter will evidently permit us to determine indirectly its real dimensions, however inaccessible it may be, and, by a series of analogous investigations, its surface, its volume, even its weight, and a number of other properties, a knowledge of which seemed forbidden to us.

3. Astronomical Facts. It is by such calculations that man has been able to ascertain, not only the distances from the planets to the earth, and, consequently, from each other, but their actual magnitude, their true figure, even to the inequalities of their surface; and, what seemed still more completely hidden from us, their respective masses, their mean densities, the principal circumstances of the fall of heavy bodies on the surface of each of them, &c.

By the power of mathematical theories, all these different results, and many others relative to the different classes of mathematical phenomena, have required no other direct measurements than those of a very small number of straight lines, suitably chosen, and of a greater number of angles. We may even say, with perfect truth, so as to indicate in a word the general range of the science, that if we did not fear to multiply calculations unnecessarily, and if we had not, in consequence, to reserve them for the determination of the quantities which could not be measured directly, the determination of all the magnitudes susceptible of precise estimation, which the various orders of phenomena can offer us, could be finally reduced to the direct measurement of a single straight line and of a suitable number of angles.

TRUE DEFINITION OF MATHEMATICS.

We are now able to define mathematical science with precision, by assigning to it as its object the indirect measurement of magnitudes, and by saying it constantly proposes to determine certain magnitudes from others by means of the precise relations existing between them.

This enunciation, instead of giving the idea of only an art, as do all the ordinary definitions, characterizes immediately a true science, and shows it at once to be composed of an immense chain of intellectual operations, which may evidently become very complicated, because of the series of intermediate links which it will be necessary to establish between the unknown quantities and those which admit of a direct measurement; of the number of variables coexistent in the proposed question; and of the nature of the relations between all these different magnitudes furnished by the phenomena under consideration. According to such a definition, the spirit of mathematics consists in always regarding all the quantities which any phenomenon can present, as connected and interwoven with one another, with the view of deducing them from one another. Now there is evidently no phenomenon which cannot give rise to considerations of this kind; whence results the naturally indefinite extent and even the rigorous logical universality of mathematical science. We shall seek farther on to circumscribe as exactly as possible its real extension.

The preceding explanations establish clearly the propriety of the name employed to designate the science which we are considering. This denomination, which has taken to-day so definite a meaning by itself signifies simply science in general. Such a designation, rigorously exact for the Greeks, who had no other real science, could be retained by the moderns only to indicate the mathematics as the science, beyond all others—the science of sciences.

Indeed, every true science has for its object the determination of certain phenomena by means of others, in accordance with the relations which exist between them. Every science consists in the co-ordination of facts; if the different observations were entirely isolated, there would be no science. We may even say, in general terms, that science is essentially destined to dispense, so far as the different phenomena permit it, with all direct observation, by enabling us to deduce from the smallest possible number of immediate data the greatest possible number of results. Is not this the real use, whether in speculation or in action, of the laws which we succeed in discovering among natural phenomena? Mathematical science, in this point of view, merely pushes to the highest possible degree the same kind of researches which are pursued, in degrees more or less inferior, by every real science in its respective sphere.

ITS TWO FUNDAMENTAL DIVISIONS.

We have thus far viewed mathematical science only as a whole, without paying any regard to its divisions. We must now, in order to complete this general view, and to form a just idea of the philosophical character of the science, consider its fundamental division. The secondary divisions will be examined in the following chapters.

This principal division, which we are about to investigate, can be truly rational, and derived from the real nature of the subject, only so far as it spontaneously presents itself to us, in making the exact analysis of a complete mathematical question. We will, therefore, having determined above what is the general object of mathematical labours, now characterize with precision the principal different orders of inquiries, of which they are constantly composed.

Their different Objects. The complete solution of every mathematical question divides itself necessarily into two parts, of natures essentially distinct, and with relations invariably determinate. We have seen that every mathematical inquiry has for its object to determine unknown magnitudes, according to the relations between them and known magnitudes. Now for this object, it is evidently necessary, in the first place, to ascertain with precision the relations which exist between the quantities which we are considering. This first branch of inquiries constitutes that which I call the concrete part of the solution. When it is finished, the question changes; it is now reduced to a pure question of numbers, consisting simply in determining unknown numbers, when we know what precise relations connect them with known numbers. This second branch of inquiries is what I call the abstract part of the solution. Hence follows the fundamental division of general mathematical science into two great sciences—ABSTRACT MATHEMATICS, and CONCRETE MATHEMATICS.

This analysis may be observed in every complete mathematical question, however simple or complicated it may be. A single example will suffice to make it intelligible.

Taking up again the phenomenon of the vertical fall of a heavy body, and considering the simplest case, we see that in order to succeed in determining, by means of one another, the height whence the body has fallen, and the duration of its fall, we must commence by discovering the exact relation of these two quantities, or, to use the language of geometers, the equation which exists between them. Before this first research is completed, every attempt to determine numerically the value of one of these two magnitudes from the other would evidently be premature, for it would have no basis. It is not enough to know vaguely that they depend on one another—which every one at once perceives—but it is necessary to determine in what this dependence consists. This inquiry may be very difficult, and in fact, in the present case, constitutes incomparably the greater part of the problem. The true scientific spirit is so modern, that no one, perhaps, before Galileo, had ever remarked the increase of velocity which a body experiences in its fall: a circumstance which excludes the hypothesis, towards which our mind (always involuntarily inclined to suppose in every phenomenon the most simple functions, without any other motive than its greater facility in conceiving them) would be naturally led, that the height was proportional to the time. In a word, this first inquiry terminated in the discovery of the law of Galileo.

When this concrete part is completed, the inquiry becomes one of quite another nature. Knowing that the spaces passed through by the body in each successive second of its fall increase as the series of odd numbers, we have then a problem purely numerical and abstract; to deduce the height from the time, or the time from the height; and this consists in finding that the first of these two quantities, according to the law which has been established, is a known multiple of the second power of the other; from which, finally, we have to calculate the value of the one when that of the other is given.

In this example the concrete question is more difficult than the abstract one. The reverse would be the case if we considered the same phenomenon in its greatest generality, as I have done above for another object. According to the circumstances, sometimes the first, sometimes the second, of these two parts will constitute the principal difficulty of the whole question; for the mathematical law of the phenomenon may be very simple, but very difficult to obtain, or it may be easy to discover, but very complicated; so that the two great sections of mathematical science, when we compare them as wholes, must be regarded as exactly equivalent in extent and in difficulty, as well as in importance, as we shall show farther on, in considering each of them separately.

Their different Natures. These two parts, essentially distinct in their object, as we have just seen, are no less so with regard to the nature of the inquiries of which they are composed.

The first should be called concrete, since it evidently depends on the character of the phenomena considered, and must necessarily vary when we examine new phenomena; while the second is completely independent of the nature of the objects examined, and is concerned with only the numerical relations which they present, for which reason it should be called abstract. The same relations may exist in a great number of different phenomena, which, in spite of their extreme diversity, will be viewed by the geometer as offering an analytical question susceptible, when studied by itself, of being resolved once for all. Thus, for instance, the same law which exists between the space and the time of the vertical fall of a body in a vacuum, is found again in many other phenomena which offer no analogy with the first nor with each other; for it expresses the relation between the surface of a spherical body and the length of its diameter; it determines, in like manner, the decrease of the intensity of light or of heat in relation to the distance of the objects lighted or heated, &c. The abstract part, common to these different mathematical questions, having been treated in reference to one of these, will thus have been treated for all; while the concrete part will have necessarily to be again taken up for each question separately, without the solution of any one of them being able to give any direct aid, in that connexion, for the solution of the rest.

The abstract part of mathematics is, then, general in its nature; the concrete part, special.

To present this comparison under a new point of view, we may say concrete mathematics has a philosophical character, which is essentially experimental, physical, phenomenal; while that of abstract mathematics is purely logical, rational. The concrete part of every mathematical question is necessarily founded on the consideration of the external world, and could never be resolved by a simple series of intellectual combinations. The abstract part, on the contrary, when it has been very completely separated, can consist only of a series of logical deductions, more or less prolonged; for if we have once found the equations of a phenomenon, the determination of the quantities therein considered, by means of one another, is a matter for reasoning only, whatever the difficulties may be. It belongs to the understanding alone to deduce from these equations results which are evidently contained in them, although perhaps in a very involved manner, without there being occasion to consult anew the external world; the consideration of which, having become thenceforth foreign to the subject, ought even to be carefully set aside in order to reduce the labour to its true peculiar difficulty. The abstract part of mathematics is then purely instrumental, and is only an immense and admirable extension of natural logic to a certain class of deductions. On the other hand, geometry and mechanics, which, as we shall see presently, constitute the concrete part, must be viewed as real natural sciences, founded on observation, like all the rest, although the extreme simplicity of their phenomena permits an infinitely greater degree of systematization, which has sometimes caused a misconception of the experimental character of their first principles.

We see, by this brief general comparison, how natural and profound is our fundamental division of mathematical science.

We have now to circumscribe, as exactly as we can in this first sketch, each of these two great sections.

Concrete Mathematics having for its object the discovery of the equations of phenomena, it would seem at first that it must be composed of as many distinct sciences as we find really distinct categories among natural phenomena. But we are yet very far from having discovered mathematical laws in all kinds of phenomena; we shall even see, presently, that the greater part will very probably always hide themselves from our investigations. In reality, in the present condition of the human mind, there are directly but two great general classes of phenomena, whose equations we constantly know; these are, firstly, geometrical, and, secondly, mechanical phenomena. Thus, then, the concrete part of mathematics is composed of Geometry and Rational Mechanics.

This is sufficient, it is true, to give to it a complete character of logical universality, when we consider all phenomena from the most elevated point of view of natural philosophy. In fact, if all the parts of the universe were conceived as immovable, we should evidently have only geometrical phenomena to observe, since all would be reduced to relations of form, magnitude, and position; then, having regard to the motions which take place in it, we would have also to consider mechanical phenomena. Hence the universe, in the statical point of view, presents only geometrical phenomena; and, considered dynamically, only mechanical phenomena. Thus geometry and mechanics constitute the two fundamental natural sciences, in this sense, that all natural effects may be conceived as simple necessary results, either of the laws of extension or of the laws of motion.

But although this conception is always logically possible, the difficulty is to specialize it with the necessary precision, and to follow it exactly in each of the general cases offered to us by the study of nature; that is, to effectually reduce each principal question of natural philosophy, for a certain determinate order of phenomena, to the question of geometry or mechanics, to which we might rationally suppose it should be brought. This transformation, which requires great progress to have been previously made in the study of each class of phenomena, has thus far been really executed only for those of astronomy, and for a part of those considered by terrestrial physics, properly so called. It is thus that astronomy, acoustics, optics, &c., have finally become applications of mathematical science to certain orders of observations.[1] But these applications not being by their nature rigorously circumscribed, to confound them with the science would be to assign to it a vague and indefinite domain; and this is done in the usual division, so faulty in so many other respects, of the mathematics into "Pure" and "Applied."

ABSTRACT MATHEMATICS.

The nature of abstract mathematics (the general division of which will be examined in the following chapter) is clearly and exactly determined. It is composed of what is called the Calculus,[2] taking this word in its greatest extent, which reaches from the most simple numerical operations to the most sublime combinations of transcendental analysis. The Calculus has the solution of all questions relating to numbers for its peculiar object. Its starting point is, constantly and necessarily, the knowledge of the precise relations, i.e., of the equations, between the different magnitudes which are simultaneously considered; that which is, on the contrary, the stopping point of concrete mathematics. However complicated, or however indirect these relations may be, the final object of the calculus always is to obtain from them the values of the unknown quantities by means of those which are known. This science, although nearer perfection than any other, is really little advanced as yet, so that this object is rarely attained in a manner completely satisfactory.

Mathematical analysis is, then, the true rational basis of the entire system of our actual knowledge. It constitutes the first and the most perfect of all the fundamental sciences. The ideas with which it occupies itself are the most universal, the most abstract, and the most simple which it is possible for us to conceive.

This peculiar nature of mathematical analysis enables us easily to explain why, when it is properly employed, it is such a powerful instrument, not only to give more precision to our real knowledge, which is self-evident, but especially to establish an infinitely more perfect co-ordination in the study of the phenomena which admit of that application; for, our conceptions having been so generalized and simplified that a single analytical question, abstractly resolved, contains the implicit solution of a great number of diverse physical questions, the human mind must necessarily acquire by these means a greater facility in perceiving relations between phenomena which at first appeared entirely distinct from one another. We thus naturally see arise, through the medium of analysis, the most frequent and the most unexpected approximations between problems which at first offered no apparent connection, and which we often end in viewing as identical. Could we, for example, without the aid of analysis, perceive the least resemblance between the determination of the direction of a curve at each of its points and that of the velocity acquired by a body at every instant of its variable motion? and yet these questions, however different they may be, compose but one in the eyes of the geometer.

The high relative perfection of mathematical analysis is as easily perceptible. This perfection is not due, as some have thought, to the nature of the signs which are employed as instruments of reasoning, eminently concise and general as they are. In reality, all great analytical ideas have been formed without the algebraic signs having been of any essential aid, except for working them out after the mind had conceived them. The superior perfection of the science of the calculus is due principally to the extreme simplicity of the ideas which it considers, by whatever signs they may be expressed; so that there is not the least hope, by any artifice of scientific language, of perfecting to the same degree theories which refer to more complex subjects, and which are necessarily condemned by their nature to a greater or less logical inferiority.

THE EXTENT OF ITS FIELD.

Our examination of the philosophical character of mathematical science would remain incomplete, if, after having viewed its object and composition, we did not examine the real extent of its domain.

Its Universality. For this purpose it is indispensable to perceive, first of all, that, in the purely logical point of view, this science is by itself necessarily and rigorously universal; for there is no question whatever which may not be finally conceived as consisting in determining certain quantities from others by means of certain relations, and consequently as admitting of reduction, in final analysis, to a simple question of numbers. In all our researches, indeed, on whatever subject, our object is to arrive at numbers, at quantities, though often in a very imperfect manner and by very uncertain methods. Thus, taking an example in the class of subjects the least accessible to mathematics, the phenomena of living bodies, even when considered (to take the most complicated case) in the state of disease, is it not manifest that all the questions of therapeutics may be viewed as consisting in determining the quantities of the different agents which modify the organism, and which must act upon it to bring it to its normal state, admitting, for some of these quantities in certain cases, values which are equal to zero, or negative, or even contradictory?

The fundamental idea of Descartes on the relation of the concrete to the abstract in mathematics, has proven, in opposition to the superficial distinction of metaphysics, that all ideas of quality may be reduced to those of quantity. This conception, established at first by its immortal author in relation to geometrical phenomena only, has since been effectually extended to mechanical phenomena, and in our days to those of heat. As a result of this gradual generalization, there are now no geometers who do not consider it, in a purely theoretical sense, as capable of being applied to all our real ideas of every sort, so that every phenomenon is logically susceptible of being represented by an equation; as much so, indeed, as is a curve or a motion, excepting the difficulty of discovering it, and then of resolving it, which may be, and oftentimes are, superior to the greatest powers of the human mind.

Its Limitations. Important as it is to comprehend the rigorous universality, in a logical point of view, of mathematical science, it is no less indispensable to consider now the great real limitations, which, through the feebleness of our intellect, narrow in a remarkable degree its actual domain, in proportion as phenomena, in becoming special, become complicated.

Every question may be conceived as capable of being reduced to a pure question of numbers; but the difficulty of effecting such a transformation increases so much with the complication of the phenomena of natural philosophy, that it soon becomes insurmountable.

This will be easily seen, if we consider that to bring a question within the field of mathematical analysis, we must first have discovered the precise relations which exist between the quantities which are found in the phenomenon under examination, the establishment of these equations being the necessary starting point of all analytical labours. This must evidently be so much the more difficult as we have to do with phenomena which are more special, and therefore more complicated. We shall thus find that it is only in inorganic physics, at the most, that we can justly hope ever to obtain that high degree of scientific perfection.

The first condition which is necessary in order that phenomena may admit of mathematical laws, susceptible of being discovered, evidently is, that their different quantities should admit of being expressed by fixed numbers. We soon find that in this respect the whole of organic physics, and probably also the most complicated parts of inorganic physics, are necessarily inaccessible, by their nature, to our mathematical analysis, by reason of the extreme numerical variability of the corresponding phenomena. Every precise idea of fixed numbers is truly out of place in the phenomena of living bodies, when we wish to employ it otherwise than as a means of relieving the attention, and when we attach any importance to the exact relations of the values assigned.

We ought not, however, on this account, to cease to conceive all phenomena as being necessarily subject to mathematical laws, which we are condemned to be ignorant of, only because of the too great complication of the phenomena. The most complex phenomena of living bodies are doubtless essentially of no other special nature than the simplest phenomena of unorganized matter. If it were possible to isolate rigorously each of the simple causes which concur in producing a single physiological phenomenon, every thing leads us to believe that it would show itself endowed, in determinate circumstances, with a kind of influence and with a quantity of action as exactly fixed as we see it in universal gravitation, a veritable type of the fundamental laws of nature.

There is a second reason why we cannot bring complicated phenomena under the dominion of mathematical analysis. Even if we could ascertain the mathematical law which governs each agent, taken by itself, the combination of so great a number of conditions would render the corresponding mathematical problem so far above our feeble means, that the question would remain in most cases incapable of solution.

To appreciate this difficulty, let us consider how complicated mathematical questions become, even those relating to the most simple phenomena of unorganized bodies, when we desire to bring sufficiently near together the abstract and the concrete state, having regard to all the principal conditions which can exercise a real influence over the effect produced. We know, for example, that the very simple phenomenon of the flow of a fluid through a given orifice, by virtue of its gravity alone, has not as yet any complete mathematical solution, when we take into the account all the essential circumstances. It is the same even with the still more simple motion of a solid projectile in a resisting medium.

Why has mathematical analysis been able to adapt itself with such admirable success to the most profound study of celestial phenomena? Because they are, in spite of popular appearances, much more simple than any others. The most complicated problem which they present, that of the modification produced in the motions of two bodies tending towards each other by virtue of their gravitation, by the influence of a third body acting on both of them in the same manner, is much less complex than the most simple terrestrial problem. And, nevertheless, even it presents difficulties so great that we yet possess only approximate solutions of it. It is even easy to see that the high perfection to which solar astronomy has been able to elevate itself by the employment of mathematical science is, besides, essentially due to our having skilfully profited by all the particular, and, so to say, accidental facilities presented by the peculiarly favourable constitution of our planetary system. The planets which compose it are quite few in number, and their masses are in general very unequal, and much less than that of the sun; they are, besides, very distant from one another; they have forms almost spherical; their orbits are nearly circular, and only slightly inclined to each other, and so on. It results from all these circumstances that the perturbations are generally inconsiderable, and that to calculate them it is usually sufficient to take into the account, in connexion with the action of the sun on each particular planet, the influence of only one other planet, capable, by its size and its proximity, of causing perceptible derangements.

If, however, instead of such a state of things, our solar system had been composed of a greater number of planets concentrated into a less space, and nearly equal in mass; if their orbits had presented very different inclinations, and considerable eccentricities; if these bodies had been of a more complicated form, such as very eccentric ellipsoids, it is certain that, supposing the same law of gravitation to exist, we should not yet have succeeded in subjecting the study of the celestial phenomena to our mathematical analysis, and probably we should not even have been able to disentangle the present principal law.

These hypothetical conditions would find themselves exactly realized in the highest degree in chemical phenomena, if we attempted to calculate them by the theory of general gravitation.

On properly weighing the preceding considerations, the reader will be convinced, I think, that in reducing the future extension of the great applications of mathematical analysis, which are really possible, to the field comprised in the different departments of inorganic physics, I have rather exaggerated than contracted the extent of its actual domain. Important as it was to render apparent the rigorous logical universality of mathematical science, it was equally so to indicate the conditions which limit for us its real extension, so as not to contribute to lead the human mind astray from the true scientific direction in the study of the most complicated phenomena, by the chimerical search after an impossible perfection.

Having thus exhibited the essential object and the principal composition of mathematical science, as well as its general relations with the whole body of natural philosophy, we have now to pass to the special examination of the great sciences of which it is composed.

BOOK I.

ANALYSIS.

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BOOK I.

ANALYSIS.