equilibrio by a third force opposed to them in the direction of the diagonal and proportional to it. On the table, a d, Fig. 85, place a circular piece of paper, on which there is drawn any triangle, a b c, c coinciding

Fig. 85.

with the centre of the table; and let us suppose that the sides of this triangle are, as shown in the figure, in the proportion to on another, as 23 4. Draw upon the paper, c e, parallel to a b, and prolong a c to d. Take three strings, making a knot at the point, c, and by means of the moveable pulley, ttt, stretch the strings over the lines cb, cd, ce; at the end of c d, suspend a weight of four pounds, at the end of ce one of three pounds, and at the end of cb one of two pounds. The knot will remain in equilibrio, proving therefore the proposi

tion.

In the composition of forces power must always be lost. Thus, in this experiment, we see that a weight of three pounds and one of two pounds equipoise a weight of four pounds only.

b

If of two forces acting upon a point, one is momentary and the other constant, the point may move in a curve. Thus, if in Fig. 86, a shot be projected obliquely upward from a gun, it is under the action of two forces-the momentary force of the explosion of the gunpowder, and the constant effect of the attraction of the earth. It describes, therefore, a curvilinear path, a b c, the direction of which continually declines towards the direction of the constant force.

Fig. 86, It is only when a force acts in a direction perpendicular to a body that its full effect is obtained. This is easily proved by resolving an oblique force into two others, one of which is perpendicular, and the other parallel to the side of the body acted upon. This latter force is, of course, lost.

CHAPTER XVII.

INERTIA.

Inertia a Property of Matter-Indifference to Motion and Rest-Moving Masses are Motive Powers-Determination of the Quantity of MotionMomentum-Action and Reaction-Newton's Laws of Motion-Bohnenberger's Machine.

ALL bodies have a tendency to maintain their present condition, whether it be of motion or rest. It is only by the exertion of force that that condition can be changed. A mass of any kind, when at rest, resists the application of force to put it in motion, and when in motion resists any attempt to bring it to rest. This property is termed INERTIA. [It is,

therefore, clear that the action exercised upon the condition of motion of a body must depend, on the one hand, upon the intensity of the force, and, on the other, upon the degree of inertia in the body.

[The larger the quantity of matter-that is to say, the greater the mass is on which a force acts-so much greater will be the resistance it offers; and we judge of the mass of a body by the amount of resistance which it can oppose by its inertia to an accelerating or retarding force.-Professor Müller on Physics and Meteorology, Lecture 1.]

It is illustrated by many familar instances: thus, loaded carriages require the exertion of far more force to put them in motion than is subsequently required to keep them going, and a train of railroad cars will run for a great distance after the locomotive is detached.

Universal experience shows that inanimate bodies have no power to produce spontaneous changes in their condition. They are wholly inactive. Even when in motion they exhibit no tendency whatever to alter their state. Thus, the earth rotates on its axis at the same rate which it did thousands of years ago, and the planetary bodies pursue their orbits with an unchange-able velocity. A moving mass can neither increase nor diminish its rate of speed; for if it could do the former it must necessarily have the power spontaneously to put itself in motion if it were in a condition of rest. Nor can such a mass, if in motion, change the direction of its movements any more than it can change its velocity. Such a change of direction would imply the operation of some innate force, which of itself could have put the mass in movement.

Fig. 87.

Whenever, therefore, we discover in a moving body changes in direction or changes in velocity, cwe at once impute them to the agency of acting forces, and not to any innate power of the moving body itself.

If an ivory ball, a, Fig. 87, be laid upon a sheet of paper, bc, on the table, and the paper suddenly pulled away, the ball does not accompany the movement, but remains in the same place on the table.

A person jumping from a carriage in rapid motion falls down, because his body, still participating in the motion of the carriage, follows its direction after his feet have struck the earth.

By the MASS of a body we mean the quantity of matter contained in it— that is, the sum of all its particles. The mass of a body depends on its volume and density.

In consequence of their inertia, masses in motion are themselves motive powers. Such a mass impinging on a second tends to set it in motion. Thus, if a ball, a, Fig. 88, moving towards c, impinge upon a second ball, b, of equal weight, с the two will move together toward c, with a Fig. 88. velocity one half of that which a originally had. In this case, therefore, a has acted as a motive force upon b, and it is obvious that the intensity of this action must depend on the magnitude and velocity of a, increasing as they increase, and diminishing as they diminish. The ball, a, is said, therefore, to have a certain momentum or moment, which depends partly upon its mass and partly upon its velocity; and the moments of any two bodies may be compared by multiplying together the mass and velocity of each. Thus, if a body, A, has twice the mass of another, B, and

moves with the same velocity, the momentum of A will be twice that of B; but if A, having twice the mass of B, has only half its velocity, the moments of the two will be equal.

It is upon this principle that heavy masses moving very slowly exert a great force, and that bodies comparatively light, moving with great speed, produce striking effects. The battering-rams of the ancients, which were heavy masses moving slowly, did not produce more powerful effects than cannon-shot, which, though comparatively light, move with prodigious speed. From the foregoing considerations, it therefore appears that the amount of motion depends neither upon the mass alone nor the velocity alone. A certain mass, A, moving with a given velocity, has a certain momentum or quantity of motion. If to A a second equal mass, B, with a similar velocity, be added, the two conjointly will, of course, possess double the momentum of the first-the mass has doubled, though the speed is the same, and therefore the quantity of motion has doubled. Again, if a certain mass, A, moves with a given speed, and a second one, B, moves with a double speed, it is obvious that this last will have twice the quantity of motion of the former-here the masses are the same, but the velocities are different. The quantity of motion or momentum which a body possesses is, therefore, obtained by multiplying together the numbers which express its mass and its velocity.

Action and Reaction are always equal to each other. The resistance

I

a

which a given body exhibits is equal to the effect of any force operating upon it. This equality of action and reaction may be shown by an aparatus represented in Fig. 89, in which two balls of clay or putty, a b, are suspended by strings so as to move over a graduated arc. If one of the balls be allowed to fall upon the other, through a given number of degrees, it will communicate to it a part of its motion, and the following facts may be observed: 1st. The bodies, after collision, move on together, and Fig. 89. therefore have the same velocity. 2nd. The quantity of motion remains unchanged, the one having gained as much as the other has lost, so that if the two are equal they will have half the velocity after impact that the moving one had when alone. 3rd. If equal, and moving in opposite directions with equal velocities, they will destroy each other's motions and come to rest. 4th. If unequal, and moving in opposite directions, they will come to rest when their velocities are inversely as their masses.

The following three propositions are called "Newton's laws of motion." They contain the results depending on inertia :—

I. Every body must persevere in its state of rest or of uniform motion in a straight line, unless it be compelled to change that state by forces impressed upon it.

II. Every change of motion must be proportional to the impressed force, and must be in the direction of that straight line in which the force is impressed.

III. Action must always be equal, and contrary to re-action; or the action of two bodies upon each other must be equal, and directed to contrary sides.

As an example of the operation of inertia, and illustrating the invariability of position of the axis of the earth during its revolution, I here describe Bohnenberger's machine. It consists of three moveable rings, A A A, Fig, 90, placed at right angles to each other, and in the smallest ring there is a heavy metal ball, B, supported on an axis, which also bears a little roller, C. A thread, being wound round this roller and any particular position being given to the axis, by quickly pulling the thread the ball may be set in rapid rotation. It is now immaterial in what position the instrument is placed; its axis continually maintains the same direction, and the ring which supports it will resist a considerable pressure tending to displace it.

Fig. 90.

CHAPTER XVIII.

GRAVITATION.

Preliminary Ideas of Motions of Attraction-The Earth and Falling Bodies -Laws of Attraction, as respects Mass and Distance-Nature of Weight -Absolute and Specific Weight-The Plumb Line-Convergence of such Lines towards the Earth's Centre-Action of Mountain Masses.

ALL material substances exert upon each other an attractive force. To this the designation of Gravity, or Gravitation, has been given. It was the great discovery of Sir Isaac Newton, that the same force which produces the descent of a stone to the ground holds together the planets and other celestial bodies.

To obtain a preliminary idea of the nature and operation of this force, let us suppose that two balls of equal weight be placed in presence of each other, and under such circumstances that no extraneous agency supervene to interfere with their mutual action. Under these circumstances, all the phenomena of Nature prove that the two balls will commence moving toward each other with equal speed, their velocity continually increasing until they come in contact. Inasmuch, therefore, as their masses are equal, and their velocities equal, the quantities of motion they respectively possess will also be equal, as is proved in Chapter 17.

B

Again, let there be other balls situated as before, but let one of them, B, be twice as large as A. Motion will again ensue by reason of their mutual attraction, and they will approach each other with a velocity continually increasing. In this instance, however, their speed will not be equal, the larger body, B, Fig. 91. having a correspondingly less velocity than the smaller one, A. If, as we have supposed, it is twice as large, its velocity will be only one-half. But in this, as in the former case, the quantity of motion that each possesses is the same, for that depends on velocity and mass conjointly.

Further, if of the two bodies one becomes infinitely less as respects the other, then it is obvious that the little one alone will appear to move. This

condition is what actually obtains in the case of our earth, and bodies subjected to its influence. A mass of any kind, the support of which is suddenly removed, falls at once to the ground, and though in reality the earth moves to meet it just as much as it moves to meet the earth, the difference in these masses is so immeasurably great that the earth's motion is imperceptible, and may be wholly neglected.

The force by which bodies are thus solicited to move to the earth is called terrestrial gravity, or gravitation.

[Gravity is the reciprocal attraction of matter on matter; gravitation is the difference between gravity and the centrifugal force induced by the velocity of rotation or revolution.-Somerville's" Connection of the Physical Sciences," page 437.]

The force of gravity depends on two different conditions: 1st, the mass; 2nd, the distance.

That is

1st. The intensity of the force of gravity is directly as the mass. to say, that, for example, in the case of the earth, if its mass were twice as large, its force of attraction would be twice as great; or if it were only half as large, its attraction would only be half as much as it is.

:

2nd. In common with all other central forces, gravity diminishes as the distance increases. The law which determines this is expressed as follows:"The force of gravity is inversely as the square of the distance;" that is to say, if a body be placed two, three, four, five times its original distance from another, the force attracting it will continually diminish, and in those different instances will successively be four, nine, sixteen, twenty-five times less than at first.

When a body, instead of being allowed to fall freely to the earth, is supported, its tendency to descend is not annihilated, but it exerts upon the supporting surface a degree of pressure. This pressure we speak of as WEIGHT. And inasmuch as the attractive force upon a body depends on its mass, it is obvious that if the mass is doubled, the weight is doubled; if the mass is tripled, the weight is tripled. Or, in other words, the weight of bodies is always proportional to their mass.

The absolute weight of a given body at the same place on the earth's surface is always the same; for the mass, and therefore the attractive force of the earth, never changes. If by any means the attractive influence of the earth could be doubled, the weight of every object would change, and be doubled correspondingly.

The absolute weight of bodies is determined by balances, springs, steelyards, and other such contrivances, as will be explained in their proper place. Different units of weight are adopted in different countries, and for different purposes, as the grain, ounce, pound, gramme, &c.

In bodies of the same nature the absolute weight is proportional to the volume. Thus a mass of iron which is twice the volume of another mass will also have twice its weight.

But when we examine dissimilar bodies the result is very different. A globe of water compared with one of copper, or lead, or wood of the same size, will have a very different weight. The lead will weigh more than the water, and the wood less.

This fact we have already pointed out by the term "specific gravity," or specific weight of bodies. And, inasmuch as it is obviously a relative thing, or a matter of comparison, it is necessary to select some substance which