26

LAWS OF MOTION.

of a force, by the velocity it generates in a given time, in no way involves any consideration of the mass, and must therefore differ from our previous measure of force.

Force, measured by the velocity generated in a given time, is called accelerating force; force, measured by weight or pressure, is termed moving force. Now, though we can conceive, as a consequence of what we have said, that two equal accelerating forces, acting separately on two different masses, would cause them to acquire the same velocity, at the end of a given time, it does not follow that these different bodies would produce the same effect on any body which might oppose their motion. In order that they should do so, it is necessary that the product of the mass and the acquired velocity, should be the same for both moving bodies. Thus a ball of 2 lb. weight, moving with a velocity of 50 feet per second, will cause a ballistic pendulum, when struck by it, to vibrate through the same arc as when struck by a ball of 50 lb. weight, with a velocity of 2 feet per second, or a ball of 100 lb. with a velocity of 1 foot per second. The product of a body's mass, and its velocity, is called its momentum, or quantity of motion. If we conceive two equal weights, W and W', suspended from the extremities of a string passing over a pully P, supposed to be destitute of friction, the weights will remain at rest. If we place ever so small a weight, x, on the weight W, the weight on which. we place it must immediately descend; and, as long as x is placed on W, by the first law of motion, the velocity of its descent will continually increase. If we remove x the weight W will still descend, but with the velocity constant, which it had acquired at the instant of x's removal. Now in this case the weight x is called the moving force, or pressure producing motion; the two weights W and W', together with x, the mass moved: the velocity of W's motion will be a measure of the accelerating force produced by x. Now it is found, by numerous careful experiments, that this accelerating force, multiplied by the mass moved, 2W+x, is always proportional to the pressure-producing motion x: and this is the third law of motion.

W

FIG. 11.

P

W

The laws of motion cannot be proved by any series of experiments, however extensive-these experiments only suggest the laws; and perhaps our firmest conviction of their truth arises from the wonderful manner in which, by combining these laws with the principle of gravitation, Astronomers have been able to predict the motions of the heavenly bodies with such marvellous exactness, and even to point out with certainty the precise spot in the heavens

HOW MATHEMATICAL AND PHYSICAL LAWS DIFFER. 27

where a planet hitherto unknown would be found. To some minds these laws may appear objects of intuitive belief, when once we have acquired correct ideas of matter, force, and motion; but on this point some metaphysical difficulties clearly exist. Our natural belief in the laws of motion certainly differs from that which prevails in regard to mathematical truths; for the opposite of mathematical truths at once presents a contradiction, while the opposite of the laws of motion may not exhibit itself at first as a contradiction to every mind.

Moreover the human mind cannot conceive that even Omnipotence can make two and two anything but four. Nevertheless, if we contemplate a heavenly body at perfect rest, on the assumption that it is for the time the only body in space, that heavenly body, in the language of the first law of motion, will remain at rest for ever, unless some cause of motion come into operation.

In this case who will dare to say that it is impossible for Omnipotence to move that heavenly body without applying a cause of motion? Such an assertion would be wholly inadmissible, unless, among the causes of motion, it is understood that the Fiat of the Almighty is included.

The Balance. The principle of the Balance seems at first sight self-evident; for it is self-evident—at least to a person of ordinary

TI

intelligence that if a rod of uniform material and dimension be fixed by its middle point on a pivot, and two bodies equal in weight be suspended one from either extremity, they will be in equilibrium. But to render this proposition intelligible, the nature of gravity, as a property of bodies at the earth's surface, must be clearly seen.

That being understood, the proposition will then stand thus:Equal causes, applied exactly in the same manner, must produce equal effects; the causes being the like number of particles tending downwards on either side of the fulcrum. And, by an easy de

28 LAW OF GRAVITATION DISCOVERED BY OBSERVATION.

monstration referable to self-evident principles, it can be shown that when the weights differ, there is, nevertheless, an equilibrium, if the fulcrum be at the point in the rod which divides it inversely in the ratio of the weights.

This case, however, plainly differs from the convictions afforded by the necessary truths of mathematics, since the reasoning is mixed up with principles ascertained by experience, the gravity of bodies, for example. And the same thing may be said of the demonstrations respecting the mechanical powers in general, the lever, the wheel and axle, the pulley, the inclined plane, the wedge, and the

screw.

In Hydrostatics it is self-evident that a solid and insoluble body, immersed in a liquid, must displace a quantity of the liquid equal to its bulk. The discovery of this fact cost Archimedes a great effort; but the moment it occurred to his mind, it was self-evident, and required no proof to obtain universal assent. That a solid floating body, like a ship of the line, displaces a quantity of water equal to its weight, is equally true, but not, at first sight, quite so obvious.

The refraction of light, to which so many phenomena can be rcferred, admits of no explanation. The evidence of the truth of this law is as yet derived solely from observation; and a wholly opposite condition of the law could be as readily received upon the same evidence as its actual form.

Gravitation. The law of Universal Gravitation rests ultimately on observation. It is the greatest achievement of Inductive Science. It is expressed in the language of Mathematics; but it has nothing of the character of a mathematical truth. This law declares the mutual gravitation of all bodies, with forces directly as their quantities of matter, and inversely as the squares of their distances.

In the expression of this law bodies are conceived to consist of minute particles, more or less closely aggregated or packed together. In Physics, all such component particles of matter (differing from the laws on Chemistry) are regarded as made up of the same small portions of matter; that is to say, it is a part of the law that any two particles, at whatever distance from each other, exert the same mutual attraction. Thus the attraction of one body or mass of matter for another if the sum of the attractions of all the particles of the one towards the sum of all the particles in the other; and if the attraction be equal on both sides, that is, if the attraction exerted by the one be as great as the attraction by the other, it is determined in the abstract, that the number of particles in the one is exactly the same as the number of particles in the other. But these two bodies, which are thus conceived to contain equal quan

ATTRACTION OF MATTER.

29

tities of matter, may be either of the same magnitude, or may considerably differ in magnitude. A cubic foot (that is 1728 cubic inches of water,) contains no more matter than 128 cubic inches of mercury, which is the same thing as to say there is the same number of particles of matter in 128 cubic inches of mercury as in 1728 cubic inches of water.

The law of Gravitation is expressed in its simplest form, as respects particles of this kind—namely, the particles of matter attract each other inversely as the squares of their distances. For example, to make the violent supposition that there is previously no matter in the universe, let two particles of matter be called into existence, and observed first at the distance of five miles, and then at three miles from each other. Their attraction for each other is greater at the distance of three miles than at the distance of five miles; but the greater attraction is not represented by 5 and the less by 3, but by the squares of those numbers, that is, by the one and the other of these numbers mnltiplied each into itself, the products of which multiplication are 25 and 9. Thus the attraction between these two particles at five miles' distance is represented by 9, and at the distance of three miles by 25. The law does not indicate the velocity with which two such particles will approach each other; but did we know what proportion each bore to the whole mass of the earth, then it might be discovered by reference to the velocity of bodies falling near its surface -sixteen feet in the first second. We may here remark how the laws of motion mix themselves up with the law of gravitation, --the same supposition being continued as to the absence of all other matter in space. If, after these two particles had approached to within three miles of each other, one of them were annihilated, all attraction would of course cease; but the other particle, in accordance with the first law of motion, would continue to move onwards in a straight line with the velocity which it had acquired at the moment of the extinction of the other.

Attraction. To express the attraction exercised by the particles of the sun over the particles composing each of the planets, numbers must be fixed upon which express, in some kind of dimension, the distances of each of these from the sun, and these numbers being squared we shall obtain a series denoting their relative attractions. To keep down the number of figures, it is best to choose some large measure, for example, the distance of the moon from the earth, or 240,000 miles.

In the following table are set down the squares of the distances of the old planets from the sun, expressed in numbers, denoting how many times each planet is more distant from the sun than the moon is from the earth.

These numbers, however, do not express the actual attraction between the sun and these several planets; but only what their relative attractions would be, if each contained the same number of particles. But where an estimate is already formed of the quantity of matter in any planet, and that quantity is considered in connection with the estimate of the quantity of matter in the sun, and the actual velocity in bodies falling near the surface of the earth, then the elements are afforded for calculating the actual force of gravity between the sun and that planet. The roots corresponding to the numbers in the above table denote the actual distances of the planets from the sun, as measured by the distance of the moon from the earth, — namely, for Mercury, 160; for Venus, 280; for the Earth, 400; for Mars, 600; for Jupiter, 2,100; for Saturn, 3,600; for Uranus, 7,600; or nearly as 1, 2, 3, 4, 15, 28, 54.

It is easy to see that the law of gravitation is sufficiently stated, when made to refer to particles of matter, by simply saying that the particles attract each other inversely as the squares of the distances. For it follows, as a necessary consequence, when a number of particles are collected into one mass, and a less number of particles into another mass, that the sum of the attractions in the one shall be to the sum of the attractions. in the other directly as the number of particles in the one is to the number of particles in the other. Again, when two bodies of the same bulk exhibit exactly the same attraction the one for the other, and under the same circumstances, we conclude that the number of particles in each is the same; and this is what is signified when it is said that two bodies have the same density. Moreover it can be proved that the attraction between the centres of two spherical masses of matter is the same as if the whole particles of each mass were collected within their respective central points.

The attraction between two bodies, or masses of particles, is measured not by the mere velocity acquired by each, but by the amount of motion, or the momentum which each exhibits. When two masses of matter, different in the number of their particles, are supposed to come into existence in free space at some distance from each other, the quantity of motion produced in each is the same. That which contains the greatest number of particles would move with less velocity; that which contains the less number of particles with greater velocity; but the momentum, or quantity of motion, in each will be the same.

It is easy, then, to understand, on the principle of gravitation, why two bodies -for example, a pillow and a piece of lead equal to the pillow in weight-were there no atmosphere, would fall to the