Thus, suppose that m n, Fig. 77, be the level of the water in the boiler

a

m

B

Fig. 77.

of a steam-engine; on its surface let there float a body, B, attached by means of a rod, C, a, to a lever, a cb, which works on the fulcrum e; on the other side of the lever, at b, let there be attached, by the rod b V, a valve, V, allowing water to flow into the boiler, through the feed-pipe, V O. Now, as the level of the water, m n, in the boiler lowers through evaporation, the float, B, sinks with it, and depresses the end, a, of the lever; but the end, b, rising, lifts the valve, V, and allows the water to go down the feed-pipe; and as the level again rises in the boiler the valve, V, again shuts. Instead of a piece of wood or hollow copper-ball, a flat piece of stone, B, is commonly used: and to make it float it is counterpoised by a weight, W,

on the opposite arm of the lever.

SECTION III.-OF REST AND MOTION.-MECHANICS.

CHAPTER XV.

MOTION AND REST.

Causes of Motion-Classification of Forces-Estimate of Forces—Direction and Intensity-Uniform and Variable Motions-Initial and Final Velocities-Direct, Rotatory, and Vibratory Motions.

ALL objects around us are necessarily in a condition either of motion or of rest. [Were there no motion in the universe it would be dead. It would be without the rising or setting of sun, or river-flows, or morning winds, or sound or light, or animal existence. To understand the nature and laws of the motions or changes which are going on around him, is to man of the greatest importance, as it enables him to adapt his actions to what is coming in futurity, and often to interfere so as to control and direct futurity for his special purposes.-Dr. Arnott's "Elements of Physics."] We shall soon find that matter has not of itself a predisposition for one or other of these states; and it is the business of natural philosophy to assign the particular causes which determine it to either in any special instance. A very superficial investigation soon puts us on our guard against deception. Things may appear in motion which are at rest, or at rest when in reality they are in motion. A passenger in a railroad-car sees the houses and trees in rapid motion, though he is well assured that this is a deception-a deception like that which occurs on a greater scale in the apparent revolution of the stars from east to west every night-the true motion not being in them, but in the earth, which is turning in the opposite direction on its axis.

If deceptions thus take place as respects the state of motion, the same holds good as respects the state of rest. On the surface of the earth even

those objects which seem to us to be quite stationary are not so in reality. Motion is described in any particular case by referring to centre objects and certain standards of velocity. A man sitting on the deck of a sailing-ship has common motion with the ship; if walking on the deck, he has relative motion to the ship; but if he be walking towards the stern, just as fast as the ship advances he is at rest relatively to the bottom or shore. A ship sailing against the tide, just as fast as the tide runs, has rest relatively both to the earth and water. Absolute motion is that which is relative to the whole universe, or to the space in which the universe exists. We have no means of ascertaining such: for, although we know how fast our globe whirls upon its axis, and round the sun; we have no measure of the motion of the sun himself—revolving, probably, round some more distant centre, and carrying all the planets along with him.

[Motion is called rapid, as that of lightning-slow, as that of the sundial shadow; both terms having reference to ordinary intermediate velocities. It is called straight or rectilineal, in the observed path of a falling bodybent, or curvilinear, in the track of a bullet shot obliquely-accelerated, in a stone falling to the earth-retarded, in a stone thrown upwards while rising to the point where it stops before again descending.—Dr. Arnott's "Elements of Physics."] Natural objects, as mountains and the various works of man, though they seem to maintain an unchangeable relation as respects position with all the world for centuries together, are but in a condition of RELATIVE REST. They are, of course, affected by the daily revolution of the earth on its axis, and accompany it in its annual movements round the sun. Indeed, as respects themselves, their parts are continually changing position. Whatever has been affected by the warmth of summer shrinks into smaller space through the cold of winter. Two objects which maintain their position toward each other are said to be at relative and absolute rest; but we make a wide distinction between this and absolute rest. All philosophy leads us to suppose that throughout the universe, there is not a solitary particle which is in reality in the latter state.

The

Whenever an object, from a state of apparent rest, commences to move, a cause for the motion may always be assigned. And inasmuch as such causes are of different kinds, they may be classified as primary or secondary motive powers. The primary motive powers are universal in their action. Such, for instance, as the general attractive force of matter, or GRAVITY. secondary are transient in their effects. The action of animals, of elastic springs, of gunpowder, are examples. Of the secondary forces, some are momentary and others more permanent, some giving rise to a blow or shock, and some to effects of a continued duration.

Forces may be compared together as respects their intensities by numbers or by lines. Thus one force may be five, ten, or a hundred times the intensity of another, and that relation be expressed by the appropriate figures. In the same manner, by lines drawn of appropriate length, we may exhibit the relation of forces; and that not only as respects their relative intensity, but also in other particulars. The direction of motion resulting from the application of a given force may always be represented by a straight line drawn from the point at which the motion commences toward the point to which the moving body is impelled. The point at which the force takes effect upon the body is termed the point of application; and the direction of motion is the path in which the body moves. To this special

designations are given appropriate to the nature of the case, such as curvilinear, rectilinear, &c.

Moving bodies pass over their paths with different degrees of speed.

One

may pass through ten feet in a second of time, and another through a thousand in the sane interval. We say, therefore, that they have different velocities. Such estimates of velocity are obviously obtained by comparing the spaces passed over in a given unit of time. The unit of time selected in natural philosophy is one second.

A moving body may be in a state of either uniform or variable motion. In the former case its velocity continually remains unchanged, and it passes over equal distances in equal times. In the latter its velocity undergoes alterations, and the spaces over which it passes in equal times are different. If the velocity is on the increase, it is spoken of as a uniformly accelerated motion. If on the decrease, as a uniformly retarded motion. In these cases we mean, by the term initial velocity, the velocity which the body had when it commenced moving, as measured by the space it would then have passed over in one second; and, by the final velocity, that which it possessed at the moment we are considering if measured in the same way. The flight of bomb-shells upward in the air is an instance of retarded motion; their descent downward of accelerated motion. The movement of the hands of a clock is an example of uniform motion.

There are motions of different kinds: 1st, direct; 2nd, rotatory; 3rd, vibratory.

1st. By direct motion we mean that in which all the parts of the whole body are advancing in the same direction with the same velocity.

2nd. By rotatory motion we imply that some parts of the body are going in opposite directions to others. The axis of rotation is an imaginary line, round which the parts of the body turn, it being itself at rest.

3rd. By vibratory movement we mean that the body which changes its original position with a motion in the opposite direction. Thus, the particles of water which form waves alternately rise and sink, and the pendulum of a clock beats backward and forward. These are examples of vibratory or oscillatory movement.

CHAPTER XVI.

OF THE COMPOSITION AND RESOLUTION or FORCES.

Compound Motion — Equilibrium · - Resultant - The Parallelogram of Forces-Case where there are more Forces than Two-Parallel Forces Resolution of Forces · Equilibrium of three Forces — Curvilinear Motions.

WHEN several forces act simultaneously on a body so as to put it in motion, that motion is said to be compound.

In cases of compound motion, if the component or constituent forces all act in the same direction, the resulting effect will be equal to the sum of all those forces taken together.

e

a

If the constituent forces act in opposite directions, the resulting effect will be equal to their difference, and its direction will be that of the greater force. Thus, if to a knot, a, Fig. 78, we attach several weights, bc, by means of a string passing over a pulley, these weights will evidently tend to pull the knot from a to e. But if to the same knot we attach a weight, f, by a string passing over the pulley g, this tends to draw it in the opposite direction. When the weights on each side of the knot act conjointly, they tend to draw it opposite ways, and it moves in the direction of the greater force.

Fig. 78.

If two forces of equal intensity, but in opposite directions, act upon a given point, that point remains motionless, and the forces are said to be in equilibrio. When there are many forces acting upon a point in equilibrio, the sum of all those acting on one side must be equal to the sum of all the rest which act in the opposite direction.

By the resultant of forces we mean a single force which would represent in intensity and direction, the conjoint action of those forces.

If the constituent forces neither act in the same nor in opposite directions, but at an angle to each other, their resultant can be found in the following manner :-Let a be the point on which the forces act; let one of them be represented in intensity and direction by the line a b, and the other likewise in intensity and direction by the line, a c. Draw the lines bd, cd, so as to complete the parallelogram, abcd; draw also the diagonal, a d. This diagonal will be the resultant of the two forces, and will, therefore, represent their conjoint action in intensity and direction.

Fig. 79.

The operation of pairs of forces upon a point is readily understood. Thus, 1st, on a point a, Fig. 80, let two forces, a b, a c, act. Complete the parallelogram a b, d c, and draw its diagonal, a d. This line will represent in

d

Fig. 81.

intensity and direction the resultant force. 2nd-On a point, a, Fig. 81, let two forces again represented in intensity and direction by the lines, a b, a c, act. Complete the parallelogram, a b c d, draw its diagonal, a, d, which is the resultant, as before. Now, on comparing Fig. 80 with Fig. 81, it readily appears that the resultant of two forces is greater as those forces act more nearly in the same direction, and less as those forces act more nearly in opposite directions.

Many popular illustrations of the parallelogram of forces might be cited. The following may, however, suffice. If a boat be rowed across a river when

there is no current, it will pass in a straight line from bank to bank perpendicularly; but this will not take place if there is a current; for as the boat crosses, it is drifted by the stream, and makes the opposite bank at a point which is lower according as the stream is more rapid. It moves in a diagonal direction.

On the same principles we can determine the common resultant of many forces acting on a point. Two of the forces are first taken and their resultant found. This resultant is combined with the third force, and a second resultant found.

Fig. 82.

This again is combined with the fourth force, and

so on, until the forces are exhausted. The final resultant represents the conjoint action of all."

Thus, let there be three forces applied to the point a, represented in intensity and direction by the lines, a b, a c, a d, Fig. 82, respectively; if a b and a c be combined they give as their resultant a e, and if this resultant, a e, be combined with the third force, a d, it yields the resultant, a f, which, therefore, represents the common action of all three forces.

The resultant of two parallel forces applied to a line, and on the same side of it, is equal to their sum and parallel to their direction. Thus the forces a b, a'b', applied to the line a a', give a resultant pr, parallel to their common direction and equal to their sum. (Fig. 83.)

Fig. 83.

But when parallel forces are applied on opposite sides of a line, the resultant is equal to their difference, and its direction is parallel to theirs. In this, as also in the foregoing case, the point at which the resultant acts is at a distance from the point at which the two forces act, inversely proportional to their intensities. In the foregoing case this point falls between the directions of the two forces, and in the latter on the outside of the direction of the greater force.

The parallelogram of forces not only serves to effect the composition of

Fig. 84.

d

several forces, but also the resolution of any given force; that is, to assign several forces which in their intensities and directions shall be equivalent to it. Thus let a f, Fig. 84, be the given force; by making it the diagonal of a parallelogram it may be resolved into its components, ad, a e; in the same manner, a e may be resolved into its components, a c, a b. Thus, therefore the original force is resolved into three components, a b, a c, a d.

Upon similar principles it may be readily proved that two forces acting at any angle upon a point can never maintain that point in equilibrio; but three forces may; and in this instance, they will be represented in intensity and direction by the three sides of a triangle, perpendicular to their respective directions.

If two forces act upon a point in the direction of and in magnitude proportional to the sides of a parallelogram, that point will be kept in