(92.) Whatever produces, or tends to produce, a change in the state of a particle or mass of matter with respect to motion or rest, is a force. Rest, or uniform rectilinear motion, are therefore the only states in which any body can exist which is not subject to the present action of some force. We are not, however, entitled to conclude, that because a body is observed in one or other of these states, it is therefore uninfluenced by any forces. It may be under the immediate action of forces which neutralise each other: thus two forces may be acting upon it which are equal, and in opposite directions. In such a case, its state of rest, or of uniform rectilinear motion, will be undisturbed. The state of uniform rectilinear motion declares more with respect to the body than the state of rest; for the former betrays the action of a force upon the body at some antecedent period; this action having been suspended, while its effect continues to be observed in the motion which it has produced.

(93.) When the state of a body is changed from rest to uniform rectilinear motion, the action of the force is only momentary, in which case it is called an impulse. If a body in uniform rectilinear motion receive an impulse in the direction in which it is moving, the effect will be, that it will continue to move uniformly in the same direction, but its velocity will be increased by the amount of speed which the impulse would have given it had it been previously quiescent. Thus, if the previous motion be at the rate of ten feet in a second, and the impulse be such as would move it from a state of rest at five feet in a second, the velocity, after the impulse, will be fifteen feet in a second.

But if the impulse be received in a direction immediately opposed to the previous motion, then it will diminish the speed by that amount of velocity which it would give to the body had it been previously at rest. In the example already given, if the impulse were opposed to the previous motion, the velocity of the body after the impulse would be five feet in a second. If the impulse received in the direction opposed to the motion be such as would give to the body at rest a velocity equal to that with which it is moving, then the effect will be, that after the impulse no motion will exist; and if the impulse would give it a still greater velocity, the body will be moved in the opposite direction with an uniform velocity equal to the excess of that due to the impulse over that which the body previously had.

When a body in a state of uniform motion receives an impulse in a direction not coinciding with that of its motion, it will move uniformly after the impulse in an intermediate direction, which may be determined by the principles established for the composition of motion in the last chapter.

Thus it appears, that whenever the state of a body is changed either from rest to uniform rectilinear motion or vice versa, or from one state of uniform rectilinear motion to another, differing from that either in velocity or direction, or in both, the phenomenon is produced by that peculiar modification of force whose action continues but for a single instant, and which has been called an impulse.

(94.) In most cases, however, the mechanical state of a body is observed to be subject to a continual change or tendency to change. We are surrounded by innumerable examples of this. A body is placed on the table. A continual pressure is excited on the surface of the table. This pressure is only the consequence of the continual tendency of the body to move downwards. If the body were excited by a force of the nature of an impulse, the effect upon the table would be instantaneous, and would immediately cease. It would, in fact, be a blow. But the continuation of the pressure proves the continuation of the action of the force.

If the table be removed from beneath the body, the force which excites it being no longer resisted, will produce motion; it is manifested, not as before, by a tendency to produce motion, but by the actual exhibition of that phenomenon. Now if the exciting force were an impulse, the body would descend to the ground with an uniform velocity. On the other hand, as will hereafter appear, every moment of its fall increases its speed, and that speed is greatest at the instant it meets the ground.

A piece of iron placed at a distance from a magnet approaches it, but not with an uniform velocity. The force of the magnet continues to act during the approach of the iron, and each moment gives it increased motion.

(95.) The forces which are thus in constant operation, proceed from secret agencies which the human mind has never been able to detect. All the analogies of nature prove that they are not the immediate results of the divine will, but are secondary causes, that is, effects of some more remote principles. To ascend to these secondary causes, and thus as it were approach one step nearer to the Creator, is the great business of philosophy; and the most certain means for accomplishing this, is diligently to observe, to compare, and to classify the phenomena, and to avoid assuming the existence of any thing which has not either been directly observed, or which cannot be inferred demonstratively from natural phenomena. Philosophy should follow nature, and not lead her.

While the law of inertia, established by observation and reason, declares the inability of matter, from any principle resident in it, to change its state, all the phenomena of the universe prove that state to be in constant but regular fluctuation. There is not in existence a single instance of the phenomenon of absolute rest, or of motion which is absolutely uniform and rectilinear. In bodies, or the parts of bodies, there is no known instance of simple passive juxtaposition unaccompanied by pressure or tension, or some other “tendency to motion.” Innumerable secret powers are ever at work, compensating, as it were, for inertia, and supplying the material world with a substitute for the principles of action and will, which give such immeasurable superiority to the character of life.

(96.) The forces which are thus in continual operation, whose existence is demonstrated by their observed effects, but whose nature, seat, and mode of operation are unknown to us, are called by the general name attractions. These forces are classified according to the analogies which prevail among their effects, in the same manner, and according to the same principles, as organised beings are grouped in natural history. In that department of natural science, when individuals are distributed in classes, the object is merely to generalise, and thereby promote the enlargement of knowledge; but nothing is or ought to be thus assumed respecting the essence, or real internal constitution of the individuals. According to their external and observable characters and qualities they are classed; and this classification should never be adduced as an evidence of any thing except that similitude of qualities to which it owed its origin.

Phenomena are to the natural philosopher what organised beings are to the naturalist. He groups and classifies them on the same principles, and with a like object. And as the naturalist gives to each species a name applicable to the individual beings which exhibit corresponding qualities, so the philosopher gives to each force or attraction a name corresponding to the phenomena of which it is the cause. The naturalist is ignorant of the real essence or internal constitution of the thing which he nominates, and of the manner in which it comes to possess or exhibit those qualities which form the basis of his classification; and the natural philosopher is equally ignorant of the nature, seat, and mode of operation of the force which he assigns as the cause of an observed class of effects.

These observations respecting the true import of the term “attraction” seem the more necessary to be premised, because the general phraseology of physical science, taken as language is commonly received, will seem to convey something more. The names of the several attractions which we shall have to notice, frequently refer the seat of the cause to specific objects, and seem to imply something respecting its mode of operation. Thus, when we say “the magnet attracts a piece of iron,” the true philosophical import of the words is, “that a piece of iron placed in the vicinity of the magnet, will move towards it, or placed in contact, will adhere to it, so that some force is necessary to separate them.” In the ordinary sense, however, something more than this simple fact is implied. It is insinuated that the magnet is the seat of the force which gives motion to the iron; that in the production of the phenomenon, the magnet is an agent exerting a certain influence, of which the iron is the subject. Of all this, however, there is no proof; on the contrary, since the magnet must move towards the iron with just as much force as the iron moves towards the magnet, there is as much reason to place the seat of the force in the iron, and consider it as an agent affecting the magnet. But, in fact, the influence which produces this phenomenon may not be resident in either the one body or the other. It may be imagined to be a property of a medium in which both are placed, or to arise from some third body, the presence of which is not immediately observed. However attractive these and like speculations may be, they cannot be allowed a place in physical investigations, nor should consequences drawn from such hypotheses be allowed to taint our conclusions with their uncertainty.

The student ought, therefore, to be aware, that whatever may seem to be implied by the language used in this science in relation to attractions, nothing is permitted to form the basis of reasoning respecting them except their effects; and whatever be the common signification of the terms used, it is to these effects, and to these alone, they should be referred.

(97.) Attractions may be primarily distributed into two classes; one consisting of those which exist between the molecules or constituent parts of bodies, and the other between bodies themselves. The former are sometimes called, for distinction, molecular or atomic attractions.

Without the agency of molecular forces, the whole face of nature would be deprived of variety and beauty; the universe would be a confused heap of material atoms dispersed through space, without form, shape, coherence, or motion. Bodies would neither have the forms of solid, liquid, or air; heat and light would no longer produce their wonted effects; organised beings could not exist; life itself, as connected with body, would be extinct. Atoms of matter, whether distant or in juxtaposition, would have no tendency to change their places, and all would be eternal stillness and rest. If, then, we are asked for a proof of the existence of molecular forces, we may point to the earth and to the heavens; we may name every object which can be seen or felt. The whole material world is one great result of the influence of these powerful agents.

(98.) It has been proved (11. et seq.) that the constituent particles of bodies are of inconceivable minuteness, and that they are not in immediate contact (23), but separated from each other by interstitial spaces, which, like the atoms themselves, although too small to be directly observed, yet are incontestably proved to exist, by observable phenomena, from which their existence demonstratively follows. The resistance which every body opposes to compression, proves that a repulsive influence prevails between the particles, and that this repulsion is the cause which keeps the atoms separate, and maintains the interstitial spaces just mentioned. Although this repulsion is found to exist between the molecules of all substances whatever, yet it has different degrees of energy in different bodies. This is proved by the fact, that some substances admit of easy compression, while in others, the exertion of considerable force is necessary to produce the smallest diminution in bulk.

The space around each atom of a body, through which this repulsive influence extends, is generally limited, and immediately beyond it, a force of the opposite kind is manifested, viz. attraction. Thus, in solid bodies, the particles resist separation as well as compression, and the application of force is as necessary to break the body, or divide it into separate parts, as to force its particles into closer aggregation. It is by virtue of this attraction that solid bodies maintain their figure, and that their parts are not separated and scattered like those of fluids, merely by their own weight. This force is called the attraction of cohesion.

The cohesive force acts in different substances with different degrees of energy: in some its intensity is very great; but the sphere of its influence apparently very limited. This is the case with all bodies which are hard, strong, and brittle, which no force can extend or stretch in any perceptible degree, and which require a great force to break or tear them asunder. Such, for example, is cast iron, certain stones, and various other substances. In some bodies the cohesive force is weak, but the sphere of its action considerable. Bodies which are easily extended, without being broken or torn asunder, furnish examples of this. Such are Indian-rubber, or caoutchouc, several animal and vegetable products, and, in general, all solids of a soft and viscid kind.

Between these extremes, the cohesive force may be observed in various degrees. In lead and other soft metals, its sphere of action is greater, and its energy less, than in the former examples; but its sphere less, and energy greater, than in the latter ones. It is from the influence of this force, and that of the repulsion, whose sphere of action is still closer to the component atoms, that all the varieties of texture which we denominate hard, soft, tough, brittle, ductile, pliant, &c. arise.

After having been broken, or otherwise separated, the parts of a solid may be again united by their cohesion, provided any considerable number of points be brought into sufficiently close contact. When this is done by mechanical means, however, the cohesion is not so strong as before their separation, and a comparatively small force will be sufficient again to disunite them. Two pieces of lead freshly cut, with smooth surfaces, will adhere when pressed together, and will require a considerable force to separate them. In the same manner if a piece of Indian-rubber be torn, the parts separated will again cohere, by being brought together with a slight pressure. The union of the parts in such instances is easy, because the sphere through which the influence of cohesion extends is considerable; but even in bodies in which this influence extends through a more limited space, the cohesion of separate pieces will be manifested, provided their surfaces be highly polished, so as to insure the near approach of a great number of their particles. Thus, two polished surfaces of glass, metal, or stone, will adhere when brought into contact.

In all these cases, if the bodies be disunited by mechanical force, they will separate at exactly the parts at which they had been united, so that after their separation no part of the one will adhere to the other; proving that the force of cohesion of the surfaces brought into contact is less than that which naturally held the particles of each together.

(99.) When a body is in the liquid form, the weight of its particles greatly predominates over their mutual cohesion, and consequently if such a body be unconfined it will be scattered by its own weight; if it be placed in any vessel, it will settle itself, by the force of its weight, into the lowest parts, so that no space in the vessel below the upper surface of the liquid will be unoccupied. The particles of a solid body placed in the vessel have exactly the same tendency, by reason of their weight; but this tendency is resisted and prevented from taking effect by their strong cohesion.

Although this cohesion in solids is much greater than in liquids, and productive of more obvious effects, yet the principle is not altogether unobserved in liquids. Water converted into vapour by heat, is divided into inconceivably minute particles, which ascend in the atmosphere. When it is there deprived of a part of that heat which gave it the vaporous form, the particles, in virtue of their cohesive force, collect into round drops, in which form they descend to the earth.

In the same manner, if a liquid be allowed to fall gradually from the lip of a vessel, it will not be dismissed in particles indefinitely small, as if its mass were incoherent, like sand or powder, but will fall in drops of considerable magnitude. In proportion as the cohesive force is greater, these drops affect a greater size. Thus, oil and viscid liquids fall in large drops; ether, alcohol, and others in small ones.

Two drops of rain trickling down a window pane will coalesce when they approach each other; and the same phenomenon is still more remarkable, if a few drops of quicksilver be scattered on an horizontal plate of glass.

It is the cohesive principle which gives rotundity to grains of shot: the liquid metal is allowed to fall like rain from a great elevation. In its descent the drops become truly globular, and before they reach the end of their fall they are hardened by cooling, so that they retain their shape.

It is also, probably, to the cohesive attraction that we should assign the globular forms of all the great bodies of the universe; the sun, planets, satellites, &c., which originally may have been in the liquid state.

(100.) Molecular attraction is also exhibited between the particles of liquids and solids. A drop of water will not descend freely when it is in contact with a perpendicular glass plane: it will adhere to the glass; its descent will be retarded; and if its weight be insufficient to overcome the adhesive force, it will remain suspended.

If a plate of glass be placed upon the surface of water without being permitted to sink, it will require more force to raise it from the water than is sufficient merely to balance the weight of the glass. This shows the adhesion of the water and glass, and also the cohesive force with which the particles of the water resist separation.

If a needle be dipped in certain liquids, a drop will remain suspended at its point when withdrawn from them: and, in general, when a solid body has been immersed in a liquid and withdrawn, it is wet; that is, some of the liquid has adhered to its surfaces. If no attraction existed between the solid and liquid, the solid would be in the same state after immersion as before. This is proved by liquids and solids between which no attraction exists. If a piece of glass be immersed in mercury, it will be in the same state when withdrawn as before it was immersed. No mercury will adhere to it; it will not be wet.

When it rains, the person and vesture are affected only because this attraction exists between them and water. If it rained mercury, none would adhere to them.

(101.) When molecular attraction is exhibited by liquids pervading the interstices of porous bodies, ascending in crevices or in the bores of small tubes, it is called capillary attraction. Instances of this are innumerable. Liquids are thus drawn into the pores of sponge, sugar, lamp-wick, &c. The animal and vegetable kingdom furnish numerous examples of this class of effects.

A weight being suspended by a dry rope, will be drawn upwards through a considerable height, if the rope be moistened with a wet sponge. The attraction of the particles composing the rope for the water is in this case so powerful, that the tension produced by several hundred weight cannot expel them.

A glass tube, of small bore, being dipped in water tinged by mixture with a little ink, will retain a quantity of the liquid suspended when withdrawn. The height of the liquid in the tube will be seen by looking through it. It is found that the less the bore of the tube is, the greater will be the height of the column sustained. A series of such tubes fixed in the same frame, with their lower orifices at the same level, and with bores gradually decreasing, being dipped in the liquid, will exhibit columns gradually increasing.

A capillary syphon is formed of a hank of cotton threads, one end of which is immersed in the vessel containing the liquid, and the other is carried into the vessel into which the liquid is to be transferred. The liquid may be thus drawn from the one vessel into the other. The same effect may be produced by a glass syphon with a small bore.

(102.) It frequently happens that a molecular repulsion is exhibited between a solid and a liquid. If a piece of wood be immersed in quicksilver, the liquid will be depressed at that part of the surface which is near the wood; and in like manner, if it be contained in a glass vessel, it will be depressed at the edges. In a barometer tube, the surface of the mercury is convex, owing partly to the repulsion between the glass and mercury.

All solids, however, do not repel mercury. If any golden trinket be dipped in that liquid, or even be exposed for a moment to contact with it, the gold will be instantly intermingled with particles of quicksilver, the metal changes its colour, and becomes white like silver, and the mercury can only be extricated by a difficult process. Chains, seals, rings, &c. should always be laid aside by those engaged in experiments or other processes in which mercury is used.

(103.) Of all the forms under which molecular force is exhibited, that in which it takes the name of affinity is attended with the most conspicuous effects. Affinity is in chemistry what inertia is in mechanics, the basis of the science. The present treatise is not the proper place for any detailed account of this important class of natural phenomena. Those who seek such knowledge are referred to our treatise on Chemistry. Since, however, affinity sometimes influences the mechanical state of bodies, and affects their mechanical properties, it will be necessary here to state so much respecting it as to render intelligible those references which we may have occasion to make to such effects.

When the particles of different bodies are brought into close contact, and more especially when, being in a fluid state, they are mixed together, their union is frequently observed to produce a compound body, differing in its qualities from either of the component bodies. Thus the bulk of the compound is often greater or less than the united volumes of the component bodies. The component bodies may be of the ordinary temperature of the atmosphere, and yet the compound may be of a much higher or lower temperature. The components may be liquid, and the compound solid. The colour of the compound may bear no resemblance whatever to that of the components. The species of molecular action between the components, which produce these and similar, effects, is called affinity.

(104.) We shall limit ourselves here to the statement of a few examples of these phenomena.

If a pint of water and a pint of sulphuric acid be mixed, the compound will be considerably less than a quart. The density of the mixture is, therefore, greater than that which would result from the mere diffusion of the particles of the one fluid through those of the other. The particles have assumed a greater proximity, and therefore exhibit a mutual attraction.

In this experiment, although the liquids before being mixed be of the temperature of the surrounding air, the mixture will be so intensely hot, that the vessel which contains it cannot be touched without pain.

If the two aeriform fluids, called oxygen and hydrogen, be mixed together in a certain proportion, the compound will be water. In this case, the components are different from the compound, not merely in the one being air and the other liquid, but in other respects not less striking. The compound water extinguishes fire, and yet of the components, hydrogen is one of the most inflammable substances in nature, and the presence of oxygen is indispensably necessary to sustain the phenomenon of combustion.

Oxygen gas, united with quicksilver, produces a compound of a black colour, the quicksilver being white and the gas colourless. When these substances are combined in another proportion, they give a red compound.

(105.) Having noticed the principal molecular forces, we shall now proceed to the consideration of those attractions which are exhibited between bodies existing in masses. The influence of molecular attractions is limited to insensible distances. On the contrary, the forces which are now to be noticed act at considerable distances, and to the influence of some there is no limit, the effect, however, decreasing as the distance increases.

The effect of the loadstone on iron is well known, and is one of this class of forces. For a detailed account of this force, and the various phenomena of which it is the cause, the reader is referred to our treatise on Magnetism.

When glass, wax, amber, and other substances are submitted to friction with silken or woollen cloth, they are observed to attract feathers, and other light bodies placed near them. A like effect is produced in several other ways, and is attended with other phenomena, the discussion of which forms a principal part of physical science. The force thus exhibited is called electricity. For details respecting it, and for its connection with magnetism, the reader is referred to our treatises on Electricity and Electro-magnetism.

(106.) These attractions exist either between bodies of particular kinds, or are developed by reducing the bodies which manifest them to a certain state by friction, or some other means. There is, however, an attraction, which is manifested between bodies of all species, and under all circumstances whatever; an attraction, the intensity of which is wholly independent of the nature of the bodies, and only depends on their masses and mutual distances. Thus, if a mass of metal and a mass of clay be placed in the vast abyss of space, at a mile asunder, they will instantly commence to approach each other with certain velocities. Again, if a mass of stone and of wood respectively equal to the former, be placed at a like distance, they will also commence to approach each other with the same velocities as the former. This universal attraction, which only depends on the quantity of the masses and their mutual distances, is called the “attraction of gravitation.” We shall first explain the “law” of this attraction, and shall then point out some of the principal phenomena by which its existence and its laws are known.

(107.) The “law of gravitation” sometimes from its universality called the “law of nature,” may be explained as follows:

Let us suppose two masses, A and B, placed beyond the influence or attraction of any other bodies, in a state of rest, and at any proposed distance from each other. By their mutual attraction they will approach each other, but not with the same velocity. The velocity of A will be greater than that of B, in the same proportion as its mass is less than that of B. Thus, if the mass of B be twice that of A, while A approaches B through a space of two feet, B will approach A through a space of one foot. Hence it follows, that the force with which A moves towards B is equal to the force with which B moves towards A (68). This is only a consequence of the property of inertia, and is an example of the equality of action and reaction, as explained in Chapter IV. The velocity with which A and B approach each other is estimated by the diminution of their distance, A B, by their mutual approach in a given time. Thus, if in one second A move towards B through a space of two feet, and in the same time B moves towards A through the space of one foot, they will approach each other through a space of three feet in a second, which will be their relative velocity (91).

If the mass of B be doubled, it will attract A with double the former force, or, what is the same, will cause A to approach B with double the former velocity. If the mass of B be trebled, it will attract A with treble the first force, and, in general, while the distance A B remains the same, the attractive force of B upon A will increase or diminish in exactly the same proportion as the mass of B is increased or diminished.

In the same manner, if the mass A be doubled, it will be attracted by B with a double force, because B exerts the same degree of attraction on every part of the mass A, and any addition which it may receive will not diminish or otherwise affect the influence of B on its former mass.

To express this in general arithmetical symbols let a and b express the space through which A and B respectively would be moved towards each other by their mutual attraction. We would then have

A × a = B × b.

Thus, it is a general law of gravitation, that so long as the distance between two bodies remains the same, each will attract and be attracted by the other, in proportion to its mass; and any increase or decrease of the mass will cause a corresponding increase or decrease in the amount of the attraction.

(108.) We shall now explain the law, according to which the attraction is changed, by changing the distance between the bodies. At the distance of one mile the body B attracts A with a certain force. At the distance of two miles, the masses not being changed, the attraction of B upon A will be one-fourth of its amount at the distance of one mile. At the distance of three miles, it will be one-ninth of its original amount; at four miles, it is reduced to a sixteenth, and so on. The following table exhibits the diminution of the attraction corresponding to the successive increase of distance:

In ARITHMETIC, that number which is found by multiplying any proposed number by itself, is called its square. Thus 4, that is, 2 multiplied by 2, is the square of 2; 9 that is, 3 times 3, is the square of 3, and so on. On inspecting the above table, it will be apparent, therefore, that the attraction of gravitation decreases in the same proportion as the square of the distance from the attracting body increases, the mass of both bodies in this case being supposed to remain the same; but if the mass of either be increased or diminished, the attraction will be increased or diminished in the same proportion.

(109.) Hence the law of gravitation may be thus expressed: “The mutual attraction of two bodies increases in the same proportion as their masses are increased, and as the square of their distance is decreased; and it decreases in proportion as their masses are decreased, and as the square of their distance is increased.”

This law may be more clearly expressed by means of general symbols. Let f express the force with which a mass weighing 1 lb. will attract another mass weighing 1 lb., at the distance of 1 foot. The force with which they will mutually attract, when removed to the distance expressed in feet by D, will be

f/D²

that is, the force f divided by the square of the number D.

If one of the bodies, instead of weighing 1 lb., weigh the number of pounds expressed by A, their mutual attraction will be increased A times, and will therefore be expressed by

A × f/D²

In fine, if the other be also the number of pounds expressed by B, their mutual attraction will be

A × B × f/D²

(110.) Having explained the law of gravitation, we shall now proceed to show how the existence of this force is proved, and its law discovered.

The earth is known to be a globular mass of matter, incomparably greater than any of the detached bodies which are found upon its surface. If one of these bodies suspended at any proposed height above the surface of the earth be disengaged, it will be observed to descend perpendicularly to the earth, that is, in the direction of the earth’s centre. The force with which it descends will also be found to be in proportion to the mass, without any regard to the species of the body. These circumstances are consistent with the account which we have given of gravitation. But by that account we should expect, that as the falling body is attracted with a certain force towards the earth, the earth itself should be attracted towards it by the same force; and instead of the falling body moving towards the earth, which is the phenomenon observed, the earth and it should move towards each other, and meet at some intermediate point. This, in fact, is the case, although it is impossible to render the motion of the earth observable, for reasons which will easily be understood.

Since all the bodies around us participate in this motion, it would not be directly observable, even though its quantity were sufficiently great to be perceived under other circumstances. But setting aside this consideration, the space through which the earth moves in such a case is too minute to be the subject of sensible observation. It has been stated (107), that when two bodies attract each other, the space through which the greater approaches the lesser, bears to that through which the lesser approaches the greater, the same proportion as the mass of the lesser bears to the mass of the greater. Now the mass of the earth is more than 1000,000,000,000,000 times the mass of any body which is observed to fall on its surface; and therefore if even the largest body which can come under observation were to fall through an height of 500 feet, the corresponding motion of the earth would be through a space less than the 1000,000,000,000,000th part of 500 feet, which is less than the 100,000,000,000th part of an inch.

The attraction between the earth and detached bodies on its surface is not only exhibited by the descent of these bodies when unsupported, but by their pressure when supported. This pressure is what is called weight. The phenomena of weight, and the descent of heavy bodies, will be fully investigated in the next chapter.

(111.) It is not alone by the direct fall of bodies that the gravitation of the earth is manifested. The curvilinear motion of bodies projected in directions different from the perpendicular, is a combination of the effects of the uniform velocity which has been given to the projectile by the impulse which it has received, and the accelerated velocity which it receives from the earth’s attraction. Suppose a body placed at any point P, fig. 21., above the surface of the earth, and let P C be the direction of the earth’s centre. If the body were allowed to move without receiving any impulse, it would descend to the earth in the direction P A, with an accelerated motion. But suppose that at the moment of its departure from P, it receives an impulse in the direction P B, which would carry it to B in the time the body would fall from P to A, then, by the composition of motion, the body must at the end of that time be found in the line B D, parallel to P A. If the motion in the direction of P A were uniform, the body P would in this case move in the straight line from P to D. But this is not the case. The velocity of the body in the direction P A is at first so small as to produce very little deflection of its motion from the line P B. As the velocity, however, increases, this deflection increases, so that it moves from P to D in a curve, which is convex, towards P B.

The greater the velocity of the projectile in the direction P A, the greater sweep the curve will take. Thus it will successively take the forms P D, P E, P F, &c., and that velocity can be computed, which (setting aside the resistance of the air) would cause the projectile to go completely round the earth, and return to the point P from which it departed. In this case, the body P would continue to revolve round the earth like the moon. Hence it is obvious, that the phenomenon of the revolution of the moon round the earth, is nothing more than the combined effects of the earth’s attraction, and the impulse which it received when launched into space by the hand of its Creator.

(112.) This is a great step in the analysis of the phenomenon of gravitation. We have thus reduced to the same class two effects apparently very dissimilar, the rectilinear descent of a heavy body, and the nearly circular revolution of the moon round the earth. Hence we are conducted to a generalisation still more extensive.

As the moon’s revolution round the earth, in an orbit nearly circular, is caused by the combination of the earth’s attraction, and an original projectile impulse, so also the singular phenomena of the planets’ revolution round the sun in orbits nearly circular, must be considered an effect of the same class, as well as the revolution of the satellites of those planets which are attended by such bodies. Although the orbits in which the comets move deviate very much from circles, yet this does not hinder the application of the same principle to them, their deviation from circles not depending on the sun’s attraction, but only on the direction and force of the original impulse which put them in motion.

(113.) We therefore conclude that gravitation is the principle which, as it were, animates the universe. All the great changes and revolutions of the bodies which compose our system, can be traced to or derived from this principle. It still remains to show how that remarkable law, by which this force is declared to increase or decrease in the same proportion as the square of the distance from the attracting body is decreased or increased, may be verified and established.

It has been shown, that the curvilinear path of a projectile depends on, and can be derived, by mathematical reasoning, from the consideration of the intensity of the earth’s attraction, and the force of the original impulse, or the velocity of projection. In the same manner, by a reverse process, when we know the curve in which a projectile moves, we can infer the amount of the attracting force which gives the curvature to its path. In this way, from our knowledge of the curvature of the moon’s orbit, and the velocity with which she moves, the intensity of the attraction which the earth exerts upon her can be exactly ascertained. Upon comparing this with the force of gravitation at the earth’s surface, it is found that the latter is as many times greater than the former, as the square of the moon’s distance is greater than the square of the distance of a body on the surface of the earth from its centre.

(114.) If this were the only fact which could be brought to establish the law of gravitation, it might be thought to be an accidental relation, not necessarily characterising the attraction of gravitation. Upon examining the orbits and velocities of the several planets, the same result is, however, obtained. It is found that the forces with which they are severally attracted by the sun are great, in exactly the same proportion as the squares of the several numbers expressing their distances are small. The mutual gravitation of bodies on the surface of the earth towards each other is lost in the predominating force exerted by the earth upon all of them. Nevertheless, in some cases, this effect has not only been observed, but actually measured.

A plumb-line, under ordinary circumstances, hangs in a direction truly vertical; but if it be near a large mass of matter, as a mountain, it has been observed to be deflected from the true vertical, towards the mountain. This effect was observed by Dr. Maskeline near the mountain called Skehallien, in Scotland, and by French astronomers near Chimboraco. For particulars of these observations, see our treatise on Geodæsy.

Cavendish succeeded in exhibiting the effects of the mutual gravitation of metallic spheres. Two globes of lead A, B, each about a foot in diameter, were placed at a certain distance asunder. A light rod, to the ends of which were attached small metallic balls C, D, was suspended at its centre E from a fine wire, and the rod was placed as in fig. 22., so that the attractions of each of the leaden globes had a tendency to turn the rod round the centre E in the same direction. A manifest effect was produced upon the balls C, D, by the gravitation of the spheres. In this experiment, care must be taken that no magnetic substance is intermixed with the materials of the balls.

Having so far stated the principles on which the law of gravitation is established, we shall dismiss this subject without further details, since it more properly belongs to the subject of Physical Astronomy; to which we refer the reader for a complete demonstration of the law, and for the detailed development of its various and important consequences.