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.
[Illustration: _H. Adlard, sc._
_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.