Chapter III: Part 3

For this reason, two persons walking in opposite directions receive from their encounter a more violent shock than might be expected. If they be of nearly equal weight, and one be walking at the rate of three and the other four miles an hour, each sustains the same shock as if he had been at rest, and struck by the other running at the rate of seven miles an hour.

This principle accounts for the destructive effects arising from ships running foul of each other at sea. If two ships of 500 tons burden encounter each other, sailing at ten knots an hour, each sustains the shock which, being at rest, it would receive from a vessel of 1000 tons burden sailing ten knots an hour.

It is a mistake to suppose, that when a large and small body encounter, the small body suffers a greater shock than the large one. The shock which they sustain must be the same; but the large body may be better able to bear it.

When the fist of a pugilist strikes the body of his antagonist, it sustains as great a shock as it gives; but the fist being more fitted to endure the blow, the injury and pain are inflicted on his opponent. This is not the case, however, when fist meets fist. Then the parts in collision are equally sensitive and vulnerable, and the effect is aggravated by both having approached each other with great force. The effect of the blow is the same as if one fist, being held at rest, were struck by the other with the combined force of both.

CHAP. V.

THE COMPOSITION AND RESOLUTION OF FORCE.

(72.) Motion and pressure are terms too familiar to need explanation. It may be observed, generally, that definitions in the first rudiments of a science are seldom, if ever, comprehended. The force of words is learned by their application; and it is not until a definition becomes useless, that we are taught the meaning of the terms in which it is expressed. Moreover, we are perhaps justified in saying, that in the mathematical sciences the fundamental notions are of so uncompounded a character, that definitions, when developed and enlarged upon, often draw us into metaphysical subtleties and distinctions, which, whatever be their merit or importance, would be here altogether misplaced. We shall, therefore, at once take it for granted, that the words _motion_ and _pressure_ express phenomena or effects which are the subjects of constant experience and hourly observation; and if the scientific use of these words be more precise than their general and popular application, that precision will soon be learned by their frequent use in the present treatise.

(73.) FORCE is the name given in mechanics to whatever produces motion or pressure. This word is also often used to express the motion or pressure itself; and when the cause of the motion or pressure is not known, this is the only correct use of the word. Thus, when a piece of iron moves toward a magnet, it is usual to say that the cause of the motion is “the attraction of the magnet;” but in effect we are ignorant of the _cause_ of this phenomenon; and the name _attraction_ would be better applied to the effect of which we have experience. In like manner the _attraction_ and _repulsion_ of electrified bodies should be understood, not as names for unknown causes, but as words expressing observed appearances or effects.

When a certain phraseology has, however, gotten into general use, it is neither easy nor convenient to supersede it. We shall, therefore, be compelled, in speaking of motion and pressure, to use the language of causation; but must advise the student that it is effects and not causes which will be expressed.

(74.) If two forces act upon the same point of a body in different directions, a single force may be assigned, which, acting on that point, will produce the same result as the united effects of the other two.

Let P, _fig. 7._, be the point on which the two forces act, and let their directions be P A and P B. From the point P, upon the line P A, take a length P _a_, consisting of as many inches as there are ounces in the force P A; and, in like manner, take P _b_, in the direction P B, consisting of as many inches as there are ounces in the force P B. Through _a_ draw a line parallel to P B, and through _b_ draw a line parallel to P A, and suppose that these lines meet at _c_. Then draw P C. A single force, acting in the direction P C, and consisting of as many ounces as the line P c consists of inches, will produce upon the point P the same effect as the two forces P A and P B produce acting together.

(75.) The figure P _a c b_ is called in GEOMETRY a _parallelogram_; the lines P _a_, P _b_, are called its _sides_, and the line P _c_ is called its _diagonal_. Thus the method of finding an equivalent for two forces, which we have just explained, is generally called “the parallelogram of forces,” and is usually expressed thus: “If two forces be represented in quantity and direction by the sides of a parallelogram, an equivalent force will be represented in quantity and direction by its diagonal.”

(76.) A single force, which is thus mechanically equivalent to two or more other forces, is called their _resultant_, and relatively to it they are called its _components_. In any mechanical investigation, when the resultant is used for the components, which it always may be, the process is called “the composition of force.” It is, however, frequently expedient to substitute for a single force two or more forces, to which it is mechanically equivalent, or of which it is the resultant. This process is called “the resolution of force.”

(77.) To verify experimentally the theorem of the parallelogram of forces is not difficult. Let two small wheels, M N, _fig. 8._, with grooves in their edges to receive a thread, be attached to an upright board, or to a wall. Let a thread be passed over them, having weights A and B, hooked upon loops at its extremities. From any part P of the thread between the wheels let a weight C be suspended: it will draw the thread downwards, so as to form an angle M P N, and the apparatus will settle itself at rest in some determinate position. In this state it is evident that since the weight C, acting in the direction P C, balances the weights A and B, acting in the directions P M and P N, these two forces must be mechanically equivalent to a force equal to the weight C, and acting directly upwards from P. The weight C is therefore the quantity of the resultant of the forces P M and P N; and the direction of the resultant is that of a line drawn directly upwards from P.

To ascertain how far this is consistent with the theorem of “the parallelogram of forces,” let a line P O be drawn upon the upright board to which the wheels are attached, from the point P upward, in the direction of the thread C P. Also, let lines be drawn upon the board immediately under the threads P M and P N. From the point P, on the line P O, take as many inches as there are ounces in the weight C. Let the part of P O thus measured be P _c_, and from _c_ draw _c a_ parallel to P N, and _c b_ parallel to P M. If the sides P _a_ and P _b_ of the parallelogram thus formed be measured, it will be found that P _a_ will consist of as many inches as there are ounces in the weight A, and P _b_ of as many inches as there are ounces in the weight B.

In this illustration, _ounces_ and _inches_ have been used as the subdivisions of _weight_ and _length_. It is scarcely necessary to state, that any other measures of these quantities would serve as well, only observing that the same denominations must be preserved in all parts of the same investigation.

(78.) Among the philosophical apparatus of the University of London, is a very simple and convenient instrument which I constructed for the experimental illustration of this important theorem. The wheels M N are attached to the tops of two tall stands, the heights of which may be varied at pleasure by an adjusting screw. A jointed parallelogram, A B C D, _fig. 9._, is formed, whose sides are divided into inches, and the joints at A and B are moveable, so as to vary the lengths of the sides at pleasure. The joint C is fixed at the extremity of a ruler, also divided into inches, while the opposite joint A is attached to a brass loop, which surrounds the diagonal ruler loosely, so as to slide freely along it. An adjusting screw is provided in this loop so as to clamp it in any required position.

In making the experiment, the sides A B and A D, C B and C D are adjusted by the joints B and A to the same number of inches respectively as there are ounces in the weights A and B, _fig. 8._ Then the diagonal A C is adjusted by the loop and screw at A, to as many inches as there are ounces in the weight C. This done, the point A is placed behind P, _fig. 8._, and the parallelogram is held upright, so that the diagonal A C shall be in the direction of the vertical thread P C. The sides A B and A D will then be found to take the direction of the threads P M and P N. By changing the weights and the lengths of the diagonal and sides of the parallelogram, the experiment may be easily varied at pleasure.

(79.) In the examples of the composition of forces which we have here given, the effects of the forces are the production of pressures, or, to speak more correctly, the theorem which we have illustrated, is “the composition of pressures.” For the point P is supposed to be at rest, and to be drawn or pressed in the directions P M and P N. In the definition which has been given of the word force, it is declared to include motions as well as pressures. In fact, if motion be resisted, the effect is converted into pressure. The same cause acting upon a body, will either produce motion or pressure, according as the body is free or restrained. If the body be free, motion ensues; if restrained, pressure, or both these effects together. It is therefore consistent with analogy to expect that the same theorems which regulate pressures, will also be applicable to motions; and we find accordingly a most exact correspondence.

(80.) If a body have a motion in the direction A B, and at the point P it receive another motion, such as would carry it in the direction P C, _fig. 10._, were it previously quiescent at P, it is required to determine the direction which the body will take, and the speed with which it will move, under these circumstances.

Let the velocity with which the body is moving from A to B be such, that it would move through a certain space, suppose P N, in one second of time, and let the velocity of the motion impressed upon it at P be such, that if it had no previous motion it would move from P to M in one second. From the point M draw a line parallel to P B, and from N draw a line parallel to P C, and suppose these lines to meet at some point, as O. Then draw the line P O. In consequence of the two motions, which are at the same time impressed upon the body at P, it will move in the straight line from P to O.

Thus the two motions, which are expressed in quantity and direction by the sides of a parallelogram, will, when given to the same body, produce a single motion, expressed in quantity and direction by its diagonal; a theorem which is to motions exactly what the former was to pressures.

There are various methods of illustrating experimentally the composition of motion. An ivory ball, being placed upon a perfectly level square table, at one of the corners, and receiving two equal impulses, in the directions of the sides of the table, will move along the diagonal. Apparatus for this experiment differ from each other only in the way of communicating the impulses to the ball.

(81.) As two motions simultaneously communicated to a body are equivalent to a single motion in an intermediate direction, so also a single motion may be mechanically replaced, by two motions in directions expressed by the sides of any parallelogram, whose diagonal represents the single motion. This process is “the resolution of motion,” and gives considerable clearness and facility to many mechanical investigations.

(82.) It is frequently necessary to express the portion of a given force, which acts in some given direction different from the immediate direction of the force itself. Thus, if a force act from A, _fig. 11._, in the direction A C, we may require to estimate what part of that force acts in the direction A B. If the force be a pressure, take as many inches A P from A, on the line A C, as there are ounces in the force, and from P draw P M perpendicular to A B; then the part of the force which acts along A B will be as many ounces as there are inches in A M. The force A B is mechanically equivalent to two forces, expressed by the sides A M and A N of the parallelogram; but A N, being perpendicular to A B, can have no effect on a body at A, in the direction of A B, and therefore the effective part of the force A P in the direction A B is expressed by A M.

(83.) Any number of forces acting on the same point of a body may be replaced by a single force, which is mechanically equivalent to them, and which is, therefore, their resultant. This composition may be effected by the successive application of the parallelogram of forces. Let the several forces be called A, B, C, D, E, &c. Draw the parallelogram whose sides express the forces A and B, and let its diagonal be A′. The force expressed by A′ will be equivalent to A and B. Then draw the parallelogram whose sides express the forces A′ and C, and let its diagonal be B′. This diagonal will express a force mechanically equivalent to A′ and C. But A′ is mechanically equivalent to A and B, and therefore B′ is mechanically equivalent to A, B, and C. Next construct a parallelogram, whose sides express the forces B′ and D, and let its diagonal be C′. The force expressed by C′ will be mechanically equivalent to the forces B′ and D; but the force B′ is equivalent to A, B, C, and therefore C′ is equivalent to A, B, C, and D. By continuing this process it is evident, that a single force may be found, which will be equivalent to, and may be always substituted for, any number of forces which act upon the same point.

If the forces which act upon the point neutralise each other, so that no motion can ensue, they are said to be in equilibrium.

(84.) Examples of the composition of motion and pressure are continually presenting themselves. They occur in almost every instance of motion or force which falls under our observation. The difficulty is to find an example which, strictly speaking, is a simple motion.

When a boat is rowed across a river, in which there is a current, it will not move in the direction in which it is impelled by the oars. Neither will it take the direction of the stream, but will proceed exactly in that intermediate direction which is determined by the composition of force.

Let A, _fig. 12._, be the place of the boat at starting; and suppose that the oars are so worked as to impel the boat towards B with a force which would carry it to B in one hour, if there were no current in the river. But, on the other hand, suppose the rapidity of the current is such, that without any exertion of the rowers the boat would float down the stream in one hour to C. From C draw C D parallel to A B, and draw the straight line A D diagonally. The combined effect of the oars and the current will be, that the boat will be carried along A D, and will arrive at the opposite bank in one hour, at the point D.

If the object be, therefore, to reach the point B, starting from A, the rowers must calculate, as nearly as possible, the velocity of the current. They must imagine a certain point E at such a distance above B that the boat would be floated by the stream from E to B in the time taken in crossing the river in the direction A E, if there were no current. If they row towards the point E, the boat will arrive at the point B, moving in the line A B.

In this case the boat is impelled by two forces, that of the oars in the direction A E, and that of the current in the direction A C. The result will be, according to the parallelogram of forces, a motion in the diagonal A B.

The wind and tide acting upon a vessel is a case of a similar kind. Suppose that the wind is made to impel the vessel in the direction of the keel; while the tide may be acting in any direction oblique to that of the keel. The course of the vessel is determined exactly in the same manner as that of the boat in the last example.

The action of the oars themselves, in impelling the boat, is an example of the composition of force. Let A, _fig. 13._, be the head, and B the stern of the boat. The boatman presents his face towards B, and places the oars so that their blades press against the water in the directions C E, D F. The resistance of the water produces forces on the side of the boat, in the directions G L and H L, which, by the composition of force, are equivalent to die diagonal force K L, in the direction of the keel.

Similar observations will apply to almost every body impelled by instruments projecting from its sides, and acting against a fluid. The motions of fishes, the act of swimming, the flight of birds, are all instances of the same kind.

(85.) The action of wind upon the sails of a vessel, and the force thereby transmitted to the keel, modified by the rudder, is a problem which is solved by the principles of the composition and resolution of force; but it is of too complicated and difficult a nature to be introduced with all its necessary conditions and limitations in this place. The question may, however, be simplified, if we consider the canvass of the sails to be stretched so completely as to form a plane surface. Let A B, _fig. 14._, be the position of the sail, and let the wind blow in the direction C D. If the line C D be taken to express the force of the wind, let D E C F be a parallelogram, of which it is the diagonal. The force C D is equivalent to two forces, one in the direction F D of the plane of the canvass, and the other E D perpendicular to the sail. The effect, therefore, is the same as if there were _two winds_, one blowing in the direction of F D or B A, that is against the edge of the sail, and the other, E D, blowing full against its face. It is evident that the former will produce no effect whatever upon the sail, and that the latter will urge the vessel in the direction D G.

Let us now consider this force D G as acting in the diagonal of the parallelogram D H G I. It will be equivalent to two forces, D H and D I, acting along the sides. One of these forces, D H, is in the direction of the keel, and the other, D I, at right angles to the length of the vessel, so as to urge it _sideways_. The form of the vessel is evidently such as to offer a great resistance to the latter force, and very little to the former. It consequently proceeds with considerable velocity in the direction D H of its keel, and makes way very slowly in the sideward direction D I. The latter effect is called _lee-way_.

From this explanation it will be easily understood, how a wind which is nearly opposed to the course of a vessel may, nevertheless, be made to impel it by the effect of sails. The angle B D V, formed by the sail and the direction of the keel, may be very oblique, as may also be the angle C D B formed by the direction of the wind and that of the sail. Therefore the angle C D V, made up of these two, and which is that formed by the direction of the wind and that of the keel, may be very oblique. In _fig. 15._ the wind is nearly contrary to the direction of the keel, and yet there is an impelling force expressed by the line D H, the line C D expressing, as before, the whole force of the wind.

In this example there are two successive decompositions of force. First, the original force of the wind C D is resolved into two, E D and F D; and next the element E D, or its equal D G, is resolved into D I and D H; so that the original force is resolved into three, viz. F D, D I, D H, which, taken together, are mechanically equivalent to it. The part F D is entirely ineffectual; it glides off on the surface of the canvass without producing any effect upon the vessel. The part D I produces _lee-way_, and the part D H impels.

_London, Pubd. by Longman & Co._]

(86.) If the wind, however, be directly contrary to the course which it is required that the vessel should take, there is no position which can be given to the sails which will impel the vessel. In this case the required course itself is resolved into two, in which the vessel sails alternately, a process which is called _tacking_. Thus, suppose the vessel is required to move from A to E, _fig. 16._, the wind setting from E to A. The motion A B being resolved into two, by being assumed as the diagonal of a parallelogram, the sides A _a_, _a_ B of the parallelogram are successively sailed over, and the vessel by this means arrives at B, instead of moving along the diagonal A B. In the same manner she moves along B _b_, _b_ C, C _c_, _c_ D, D _d_, _d_ E, and arrives at E. She thus sails continually at a sufficient angle with the wind to obtain an impelling force, yet at a sufficiently small angle to make way in her proposed course.

The consideration of the effect of the rudder, which we have omitted in the preceding illustration, affords another instance of the resolution of force. We shall not, however, pursue this example further.

(87.) A body falling from the top of the mast when the vessel is in full sail, is an example of the composition of motion. It might be expected, that during the descent of the body, the vessel having sailed forward, would leave it behind, and that, therefore, it would fall in the water behind the stern, or at least on the deck, considerably behind the mast. On the other hand, it is found to fall at the foot of the mast, exactly as it would if the vessel were not in motion. To account for this, let A B, _fig. 17._, be the position of the mast when the body at the top is disengaged. The mast is moving onwards with the vessel in the direction A C, so that in the time which the body would take to fall to the deck, the top of the mast would move from A to C. But the body being on the mast at the moment it is disengaged, has this motion A C in common with the mast; and therefore in its descent it is affected by two motions, viz. that of the vessel expressed by A C, and its descending motion expressed by A B. Hence, by the composition of motion, it will be found at the opposite angle D of the parallelogram, at the end of the fall. During the fall, however, the mast has moved with the vessel, and has advanced to C D, so that the body falls at the foot of the mast.

(88.) An instance of the composition of motion, which is worthy of some attention, as it affords a proof of the diurnal motion of the earth, is derived from observing the descent of a body from a very high tower. To render the explanation of this more simple, we shall suppose the tower to be on the equator of the earth. Let E P Q, _fig. 18._, be a section of the earth through the equator, and let P T be the tower. Let us suppose that the earth moves on its axis in the direction E P Q. The foot P of the tower will, therefore, in one day move over the circle E P Q, while the top T moves over the greater circle T T′ R. Hence it is evident, that the top of the tower moves with greater speed than the foot, and therefore in the same time moves through a greater space. Now suppose a body placed at the top; it participates in the motion which the top of the tower has in common with the earth. If it be disengaged, it also receives the descending motion T P. Let us suppose that the body would take five seconds to fall from T to P, and that in the same time the top T is moved by the rotation of the earth from T to T′, the foot being moved from P to P′. The falling body is therefore endued with two motions, one expressed by T T′, and the other by T P. The combined effect of these will be found in the usual way by the parallelogram. Take T _p_ equal to T T′; the body will move from T to _p_ in the time of the fall, and will meet the ground at _p_. But since T T′ is greater than P P′, it follows that the point _p_ must be at a distance from P′ equal to the excess of T T′ above P P′. Hence the body will not fall exactly at the foot of the tower, but at a certain distance from it, in the direction of the earth’s motion, that is, eastward. This is found, by experiment, to be actually the case; and the distance from the foot of the tower, at which the body is observed to fall, agrees with that which is computed from the motion of the earth, to as great a degree of exactness as could be expected from the nature of the experiment.

(89.) The properties of compounded motions cause some of the equestrian feats exhibited at public spectacles to be performed by a kind of exertion very different from that which the spectators generally attribute to the performer. For example, the horseman standing on the saddle leaps over a garter extended over the horse at right angles to his motion; the horse passing under the garter, the rider lights upon the saddle at the opposite side. The exertion of the performer, in this case, is not that which he would use were he to leap from the ground over a garter at the same height. In the latter case, he would make an exertion to rise, and, at the same time, to project his body forward. In the case, however, of the horseman, he merely makes that exertion which is necessary to rise directly upwards to a sufficient height to clear the garter. The motion which he has in common with the horse, compounded with the elevation acquired by his muscular power, accomplishes the leap.

To explain this more fully, let A B C, _fig. 19._, be the direction in which the horse moves, A being the point at which the rider quits the saddle, and C the point at which he returns to it. Let D be the highest point which is to be cleared in the leap. At A the rider makes a leap towards the point E, and this must be done at such a distance from B, that he would rise from B to E in the time in which the horse moves from A to B. On departing from A, the rider has, therefore, two motions, represented by the lines A E and A B, by which he will move from the point A to the opposite angle D of the parallelogram. At D, the exertion of the leap being overcome by the weight of his body, he begins to return downward, and would fall from D to B in the time in which the horse moves from B to C. But at D he still retains the motion which he had in common with the horse; and therefore, in leaving the point D, he has two motions, expressed by the lines D F and D B. The compounded effects of these motions carry him from D to C. Strictly speaking, his motion from A to D, and from D to C, is not in straight lines, but in a curve. It is not necessary here, however, to attend to this circumstance.

(90.) If a billiard-ball strike the cushion of the table obliquely, it will be reflected from it in a certain direction, forming an angle with the direction in which it struck it. This affords an example of the resolution and composition of motion. We shall first consider the effect which would ensue if the ball struck the cushion perpendicularly.

Let A B, _fig. 20._, be the cushion, and C D the direction in which the ball moves towards it. If the ball and the cushion were perfectly inelastic, the resistance of the cushion would destroy the motion of the ball, and it would be reduced to a state of rest at D. If, on the other hand, the ball were perfectly elastic, it would be reflected from the cushion, and would receive as much motion from D to C after the impact, as it had from C to D before it. Perfect elasticity, however, is a quality which is never found in these bodies. They are always elastic, but imperfectly so. Consequently the ball after the impact will be reflected from D towards C, but with a less motion than that with which it approached from C to D.

Now let us suppose that the ball, instead of moving from C to D, moves from E to D. The force with which it strikes D being expressed by D E′, equal to E D, may be resolved into two, D F and D C′. The resistance of the cushion destroys D C′, and the elasticity produces a contrary force in the direction D C, but less than D C or D C′, because that elasticity is imperfect. The line D C expressing the force in the direction C D, let D G (less than D C) express the reflective force in the direction D C. The other element D F, into which the force D E′ is resolved by the impact, is not destroyed or modified by the cushion, and therefore, on leaving the cushion at D, the ball is influenced by two forces, D F (which is equal to C E) and D G. Consequently it will move in the diagonal D H.

(91.) The angle E D C is in this case called the “angle of incidence,” and C D H is called “the angle of reflection.” It is evident, from what has been just inferred, that the ball, being imperfectly elastic, the angle of incidence must always be less than the angle of reflection, and with the same obliquity of incidence, the more imperfect the elasticity is, the less will be the angle of reflection.

In the impact of a perfectly elastic body, the angle of reflection would be equal to the angle of incidence. For then the line D G, expressing the reflective force, would be taken equal to C D, and the angle C D H would be equal to C D E. This is found by experiment to be the case when light is reflected from a polished surface of glass or metal.

Motion is sometimes distinguished into _absolute_ and _relative_. What “relative motion” means is easily explained. If a man walk upon the deck of a ship from stem to stern, he has a relative motion which is measured by the space upon the deck over which he walks in a given time. But while he is thus walking from stem to stern, the ship and its contents, including himself, are impelled through the deep in the opposite direction. If it so happen that the motion of the man, from stem to stern, be exactly equal to the motion of the ship in the contrary way, the man will be, relatively to the surface of the sea and that of the earth, at rest. Thus, relatively to the ship, he is in motion, while, relatively to the surface of the earth, he is at rest. But still this is not absolute rest. The surface itself is moving by the diurnal rotation of the earth upon its axis, as well as by the annual motion in its orbit round the sun. These motions, and others to which the earth is subject, must be all compounded by the theorem of the parallelogram of forces before we can obtain the _absolute state_ of the body with respect to motion or rest.

CHAP. VI.

ATTRACTION.

(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.