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(229.) In all cases where there is a direct and simple relation between the phenomenon observed and a single _datum_ on which it depends, every single observation will give a value of this quantity, and the average of all (under certain restrictions) will be its exact value. We say, under certain restrictions; for, if the circumstances under which the observations are made be not alike, they may not all be equally favourable to exactness, and it would be doing injustice to those most advantageous, to class them with the rest. In such cases as these, as well as in cases where the _data_ are numerous and complicated together, so as not to admit of single, separate determination (a thing of continual occurrence), we have to enter into very nice, and often not a little intricate, considerations respecting the _probable_ accuracy of our results, or the limits of error within which it is _probable_ they lie. In so doing we are obliged to have recourse to a refined and curious branch of mathematical enquiry, called the doctrine of probabilities, the object of which (as its name imports) is to reduce our estimation of the probability of any conclusion to calculation, so as to be able to give more than a mere guess at the degree of reliance which ought to be placed in it.

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(230.) To give some general idea of the considerations which such computations involve, let us imagine a person firing with a pistol at a wafer on a wall ten yards distant: we might, in a general way, take it for granted, that he would hit the wall, but not the wafer, at the first shot; but if we would form any thing like a probable conjecture of _how near_ he would come to it, we must first have an idea of his skill. No better way of judging could be devised than by letting him fire a hundred shots at it, and marking where they all struck. Suppose this done,--suppose the wafer has been hit once or twice, that a certain number of balls have hit the wall within an inch of it, a certain number between one and two inches, and so on, and that one or two have been some feet wide of the mark. Still the question arises, what estimate are we thence to form of his skill? how _near_ (or nearer) may we, after this experience, safely, or at least not unfairly, bet that he will come to the mark the next subsequent shot? This the laws of probability enable us on such data to say. Again, suppose, _before_ we were allowed to measure the distances, the wafer were to have been taken away, and we were called upon, on the mere evidence of the marks on the wall, to say where it had been placed; it is clear that no reasoning would enable any one to say with certainty; yet there is assuredly one place which we may fix on with greater probability of being right than any other. Now, this is a very similar case to that of an observer--an astronomer for example--who would determine the exact place of a heavenly body. He points to it his telescope, and obtains a series of results disagreeing among themselves, but yet all agreeing within certain limits, and only a comparatively small number of them deviating considerably from the mean of all; and from these he is called upon to say, definitively, what he shall consider to have been the most probable place of his star at the moment. Just so in the calculation of physical _data_; where no two results agree exactly, and where all come within limits, some wide, some close, what have we to guide us when we would make up our minds what to conclude respecting them? It is evident that any system of calculation that can be shown to lead of necessity to the most probable conclusion where certainty is not to be had must be valuable. However, as this doctrine is one of the most difficult and delicate among the applications of mathematics to natural philosophy, this slight mention of it must suffice at present.

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(231.) In the foregoing pages we have endeavoured to explain the spirit of the methods to which, since the revival of philosophy, natural science has been indebted for the great and splendid advances it has made. What we have all along most earnestly desired to impress on the student is, that natural philosophy is essentially united in all its departments, through all which one spirit reigns and one method of enquiry applies. It cannot, however, be studied as a whole, without subdivision into parts; and, in the remainder of this discourse, we shall therefore take a summary view of the progress which has been made in the different branches into which it may be most advantageously so subdivided, and endeavour to give a general idea of the nature of each, and of its relations to the rest. In the course of this, we shall have frequent opportunity to point out the influence of those general principles we have above endeavoured to explain, on the progress of discovery. But this we shall only do as cases arise, without entering into any regular analysis of the history of each department with that view. Such an analysis would, indeed, be a most useful and valuable work, but would far exceed our present limits. We are not, however, without a hope that this great desideratum in science will, ere long, be supplied from a quarter every way calculated to do it justice.

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PART III.

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OF THE SUBDIVISION OF PHYSICS INTO DISTINCT BRANCHES, AND THEIR MUTUAL RELATIONS.

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CHAPTER I.

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OF THE PHENOMENA OF FORCE, AND OF THE CONSTITUTION OF NATURAL BODIES.

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(232.) Natural History may be considered in two very different lights: either, 1st, as a collection of facts and objects presented by nature, from the examination, analysis, and combination of which we acquire whatever knowledge we are capable of attaining both of the order of nature, and of the agents she employs for producing her ends, and from which, therefore, all sciences arise; or, 2dly, as an assemblage of phenomena to be explained; of effects to be deduced from causes; and of materials prepared to our hands, for the application of our principles to useful purposes. Natural history, therefore, considered in the one or the other of these points of view, is either the beginning or the end of physical science. As it offers to us, in a confused and interwoven mass, the elements of all our knowledge, our business is to disentangle, to arrange, and to present them in a separate and distinct state: and to this end we are called upon to resolve the important but complicated problem,--Given the effect, or assemblage of effects, to find the causes. The principles on which this enquiry relies are those which constitute the relation of cause and effect, as it exists with reference to our minds; and their rules and mode of application have been attempted to be sketched out, (though in far less detail than the intrinsic interest of the subject, both in a logical and practical point of view, would demand,) in the foregoing pages. It remains now to bring together, in a summary statement, the results of the general examination of nature, so far as it has been prosecuted to the discovery of natural agents, and the mode in which they act.

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(233.) The first great agent which the analysis of natural phenomena offers to our consideration, more frequently and prominently than any other, is force. Its effects are either, 1st, to counteract the exertion of opposing force, and thereby to maintain _equilibrium_; or, 2dly, to produce _motion_ in matter.

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(234.) Matter, or that, whatever it be, of which all the objects in nature which manifest themselves directly to our senses consist, presents us with two general qualities, which at first sight appear to stand in contradiction to each other--activity and inertness. Its activity is proved by its power of spontaneously setting other matter in motion, and of itself obeying their mutual impulse, and moving under the influence of its own and other force; inertness, in refusing to move unless obliged to do so by a force impressed externally, or mutually exerted between itself and other matter, and by persisting in its state of motion or rest unless disturbed by some external cause. Yet in reality this contradiction is only apparent. Force being the cause, and motion the effect produced by it on matter, to say that matter is inert, or has _inertia_, as it is termed, is only to say that the cause is expended in producing its effect, and that the same cause cannot (without renewal) produce double or triple its own proper effect. In this point of view, equilibrium may be conceived as a continual production of two opposite effects, each undoing at every instant what the other has done.

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(235.) However, if this should appear too metaphysical, at all events this difference of effects gives rise to two great divisions of the science of force, which are commonly known by the names of STATICS and DYNAMICS; the latter term, which is general, and has been used by us before in its general sense, being usually confined to the doctrine of motion, as produced and modified by force. Each of these great divisions again branches out into distinct subdivisions, according as we consider the equilibrium or motion of matter in the three distinct states in which it is presented to us in nature, the solid, liquid, and aëriform state, to which, perhaps, ought to be added the _viscous_, as a state intermediate between that of solidity and fluidity, the consideration of which, though very obscure and difficult, offers a high degree of interest on a variety of accounts.

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_Statics and Dynamics._

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(236.) The principles have been definitively fixed by Galileo and his successors, down to Newton, on a basis of sound induction; and as they are perfectly general, and apply to every case, they are competent, as we have already before observed, to the solution of every problem that can occur in the deductive processes, by which phenomena are to be explained, or effects calculated. Hence, they include every question that can arise respecting the motions and rest of the smallest particles of matter, as well as of the largest masses. But the mode of reasoning from these general principles differs materially, whether we consider them as applied to masses of matter of a sensible size, or to those excessively minute, and perhaps indivisible, molecules of which such masses are composed. The investigations which relate to the latter subject are extremely intricate, as they necessarily involve the consideration of the hypotheses we may form respecting the intimate constitution of the several sorts of bodies above enumerated.

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(237.) On the other hand, those which respect the equilibrium and motions of sensible masses of matter are happily capable of being so managed as to render unnecessary the adoption of any particular hypothesis of structure. Thus, in reasoning respecting the application of forces to a solid mass, we suppose its parts indissolubly and unalterably connected; it matters not by what tie, provided this condition be satisfied, that one point of it cannot be moved without setting all the rest in motion, so that the relative situation of the parts one among another be not changed. This is the abstract notion of a solid which the mechanician employs in his reasonings. And their conclusions will apply to natural bodies, of course, only so far as they conform to such a definition. In strictness of speaking, however, there are no bodies which absolutely conform to it. No substance is known whose parts are absolutely incapable of yielding one among another; but the amount by which they do yield is so excessively small as to be demonstrably incapable, in most cases, of having any influence on the results: and in those where it has such influence, an especial investigation of its amount can always be made. This gives rise to two subdivisions of the application of mechanical reasonings to solid masses. Those which refer to the action of forces on flexible or elastic, and on inflexible or rigid, bodies, comprehending under the latter all such whose resistance to flexure or fracture is so very great as to permit our adoption of the language and ideas of the extreme case without fear of material error.

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(238.) In like manner, when we reason respecting the action of forces on a fluid mass, all we have occasion to assume is, that its parts are freely moveable one among the other. If, besides this, we choose to regard a fluid as incompressible, and deduce conclusions on this supposition, they will hold good only so far as there may be found such fluids in nature. Now, in strictness, there are none such; but, practically speaking, in the greater number of cases their resistance to compression is so very great that the result of the reasoning so carried on is not sensibly vitiated; and, in the remaining cases, the same general principles enable us to enter on a special enquiry directed to this point: and hence the division of fluids, in mechanical language, into compressible and incompressible, the latter being only the extreme or limiting case of the former.

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(239.) As we propose here, however, only to consider what is the actual constitution of nature, we shall regard all bodies, as they really are, more or less flexible and yielding. We know for certain, that the space which any material body appears to occupy is not entirely filled by it; because there is none which by the application of a sufficient force may not be _compressed_ or forced into a smaller space, and which, either wholly, as in air or liquids, or in part, as in the greater number of solids, will not recover its former dimensions when the force is taken off. In the case of air, this condensation may be urged to almost any extent; and not only does a mass of air so condensed completely recover its original bulk, when the applied pressure is removed, but if that ordinary pressure under which it exists at the earth’s surface (and which arises from the weight of the atmosphere) be also removed by an air-pump, it will still further dilate itself without limit so far as we have yet been able to try it. Hence we are led to the conclusion that the particles of air are mutually elastic, and have a _tendency to recede from one another_, which can only be counteracted by _force_, and therefore is itself a force of the repulsive kind. Nevertheless, as air is heavy, and as gravitation is a universal property of matter, there is no doubt that this repulsive tendency must have a limit, and that there is a distance to which, if the particles of the air could be removed from each other, their mutual repulsion would cease, and an attraction take its place. This limit is probably attained at some very great height above the earth’s surface, beyond which, of course, its atmosphere cannot extend.

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(240.) What, however, we can only conclude by this or similar reasoning respecting air, we see distinctly in liquids. They are all, though in a small degree, compressible, and recover their former dimensions completely when the pressure is removed; but they cannot be dilated (by mechanical means), and have no tendency, while they remain liquids, to enlarge themselves beyond a certain limit, and therefore they assume a determinate _surface_ while at rest, and their parts actually resist further separation with a considerable force, thus giving rise to the phenomenon of the _cohesion of liquids_.

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(241.) Both in air and in liquids, however, the most perfect freedom of motion of the parts among each other subsists, which could hardly be the case if they were not separate and independent of each other. And from this, combined with the foregoing considerations, it has been concluded that they do not actually touch, but are kept asunder at determinate distances from each other, by the constant action of the two forces of attraction and repulsion, which are supposed to balance and counteract each other at the ordinary distances of the particles, but to prevail, the one, or the other, according as they are forcibly urged together or pulled asunder.

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(242.) In solids, however, the case is very different. The mutual free motion of their parts _inter se_ is powerfully impeded, and in some almost destroyed. In some, a slow and gradual change of figure may be produced to a great extent, by pressure or blows, as for instance in the metals, clay, butter, &c.; in others, fracture is the consequence of any attempt to change the figure by violence beyond a certain very small limit. In solids, then, it is evident, that the consideration of their intimate structure has a very great influence in modifying the general results of the action of such attractive and repulsive forces as may be assumed to account for the phenomena they present; yet the general facts that their parts _cohere_ with a certain energy, and that they resist displacement or intrusion on the part of other bodies, are sufficient to demonstrate at least the existence of such forces, whatever obscurity may subsist as to their mode of action.

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(243.) This division of bodies into airs, liquids, and solids, gives rise, then, to three distinct branches of mechanical science, in each of which the general principles of equilibrium and motion have their peculiar mode of application; viz. pneumatics, hydrostatics, and what might, without impropriety, be termed stereostatics.

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_Pneumatics._

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(244.) Pneumatics relates to the equilibrium or movements of aërial fluids under all circumstances of pressure, density, and elasticity. The weight of the air, and its pressure on all the bodies on the earth’s surface, were quite unknown to the ancients, and only first perceived by Galileo, on the occasion of a sucking-pump refusing to draw water above a certain height. Before his time it had always been supposed that water rose by suction in a pipe, in consequence of a certain natural _abhorrence of a vacuum_ or empty space, which obliged the water to enter by way of supplying the place of the air sucked out. But if any such abhorrence existed, and had the force of an _acting cause_, which could urge water a single foot into a pipe, there is no reason why the same principle should not carry it up two, three, or any number of feet; none why it should suddenly stop short at a certain height, and refuse to rise higher, however violent the suction might be, nay, even fall back, if purposely forced up too high.

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(245.) Galileo, however, at first contented himself with the conclusion, that the natural abhorrence of a vacuum was not strong enough to sustain the water more than about thirty-two feet above its level; and, although the true cause of the phenomenon at length occurred to him, in the pressure of the air on the general surface, it was not satisfactorily demonstrated till his pupil, Torricelli, conceived the happy idea of instituting an experiment on a small scale by the use of a much heavier liquid, mercury, instead of water, and, in place of sucking out the air from above, employing the much more effectual method of filling a long glass tube with mercury, and inverting it into a basin of the same metal. It was then at once seen, as by a _glaring instance_, that the maintenance of the mercury in the tube (which is nothing else than the common barometer) was the effect of a perfectly definite external cause, while its fluctuations from day to day, with the varying state of the atmosphere, strongly corroborated the notion of its being due to the pressure of the external air on the surface of the mercury in the reservoir.

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(246.) The discovery of Torricelli was, however, at first much misconceived, and even disputed, till the question was finally decided by appeal to a _crucial instance_, one of the first, if not the very first on record in physics, and for which we are indebted to the celebrated Pascal. His acuteness perceived that if the weight of the incumbent air be the direct cause of the elevation of the mercury, it must be measured by the amount of that elevation, and therefore that, by carrying a barometer up a high mountain, and so ascending into the atmosphere _above_ a large portion of the incumbent air, the pressure, as well as the length of the column sustained by it, must be diminished; while, on the other hand, if the phenomenon were due to the cause originally assigned, no difference could be expected to take place, whether the observation were made on a mountain or on the plain. Perhaps the decisive effect of the experiment which he caused to be instituted for the purpose, on the Puy de Dôme, a high mountain in Auvergne, while it convinced every one of the truth of Torricelli’s views, tended more powerfully than any thing which had previously been done in science to confirm, in the minds of men, that disposition to experimental verification which had scarcely yet taken full and secure root.

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(247.) Immediately on this discovery followed that of the air-pump, by Otto von Guericke of Magdeburgh, whose aim seems to have been to decide the question, whether a vacuum could or could not exist, by endeavouring to make one. The imperfection of his mechanism enabled him only to diminish the aërial contents of his receivers, not entirely to empty them; but the curious effects produced by even a partial exhaustion of air speedily excited attention, and induced our illustrious countryman, Robert Boyle, to the prosecution of those experiments which terminated in his hands, and in those of Hauksbee, Hooke, Mariotte, and others, in a satisfactory knowledge of the general law of the equilibrium of the air under the influence of greater or less pressures. These discoveries have since been extended to all the various descriptions of aërial fluids which chemistry has shown to exist, and to maintain their aëriform state under artificial pressure, and even to those which may be produced from liquids reduced to a state of vapour by heat, so long as they retain that state.

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(248.) The manner in which the observed law of equilibrium of an elastic fluid, like air, may be considered to originate in the mutual repulsion of its particles, has been investigated by Newton, and the actual statement of the law itself, as announced by Mariotte, “that the density of the air, or the quantity of it contained in the same space, is, _cæteris paribus_, proportional to the pressure it supports,” has recently been verified within very extensive limits by direct experiment, by a committee of the Royal Academy of Paris. This law contains the principle of solution of every dynamical question that can occur relative to the equilibrium of elastic fluids, and is therefore to be regarded as one of the highest _axioms_ in the science of pneumatics.

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_Hydrostatics._

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(249.) The principles of the equilibrium of liquids, understanding by this word such fluids as do not, though quite at liberty, attempt to dilate themselves beyond a certain point, are at once few and simple. The first steps towards a knowledge of them were made by Archimedes, who established the general fact, that a solid immersed in a liquid loses a portion of its weight equal to that of the liquid it displaces. It seems very astonishing, after this, that it should not have been at once concluded that the weight thus said to be _lost_ is only _counteracted_ by the upward pressure of the liquid, and that, therefore, a portion of any liquid, surrounded on all sides by a liquid of the same kind, does really exert its weight in keeping its place. Yet the prejudice that “liquids do not gravitate in their natural place” kept its ground, and was only dispelled with the mass of error and absurdity which the introduction of a rational and experimental philosophy by Galileo swept away.

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(250.) The hydrostatical law of _the equal pressure of liquids in all directions_, with its train of curious and important consequences, is an immediate conclusion from the perfect mobility of their parts among one another, in consequence of which each of them tends to recede from an excess of pressure on one side, and thus bears upon the rest, and distributes the pressure among its neighbours. In this form it was laid down by Newton, and has proved one of the most useful and fertile principles of physico-mathematical reasoning on the equilibrium of fluid masses, as affording a means of tracing the action of a force applied at any point of a liquid through its whole extent. It applies, too, without any modification, to expansible fluids as well as to liquids; and, in the applications of geometry to this subject, enables us to dispense with any minute and intricate enquiries as to the mode in which individual particles act on each other.

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(251.) In a practical point of view, this law is remarkable for the directness of its application to useful purposes. The immediate and perfect distribution of a pressure applied on any one part, however small, of a fluid surface through the whole mass, enables us to communicate _at one instant_ the same pressure to any number of such parts by merely increasing the surface of the fluid, which may be done by enlarging the containing vessel; and if the vessel be so constructed that a large portion of its surface shall be moveable together, the pressures on all the similar parts of this portion will be united into one consentient force, which may thus be increased to any extent we please. The hydraulic press, invented by Bramah, (or rather applied by him after a much more ancient inventor, Stevin,) is constructed on this principle. A small quantity of water is driven by sufficient pressure into a vessel _already full_, and provided with a moveable surface or piston of great size. Under such circumstances something must give way; the great surface of the piston accumulates the pressure on it to such an extent that nothing can resist its violence. Thus, trees are torn up by the roots; piles extracted from the earth; woollen and cotton goods compressed into the most portable dimensions; and even hay, for military service, reduced to such a state of coercion as to be easily packed on board transports.
