be double that of one, of three triple, and so on. But how are we to determine the relative mass of a cube of iron, and a similar cube of lead? It may be answered by their weight, and, as we shall afterwards see, their weight is doubtless a correct representation of their mass; but we cannot accept weight as a fundamental method of estimating mass, for weight is due to the attraction of the Earth, and we might suppose a state of things where there was no large attracting body. Let us, for instance, imagine ourselves carried into empty space, with nothing but a cube of iron, and another of lead; then how are we to determine their relative masses? It is clear we cannot weigh them, for there is no downwards and upwards in such circumstances, there being no earth. We reply, that two different substances are of the same mass when the same force produces in each, after it has acted on it for one second of time, the same velocity. We shall find that the same force will produce at the end of one second the same velocity, if it be applied to set in motion 100 cubic metres of iron, or 69 cubic metres of lead; there is, therefore, the same amount of matter in 69 cubic metres of lead as in 100 cubic metres of iron. As we shall afterwards find weight to be strictly proportional to mass, it is convenient to use weight as a means of estimating mass. We have therefore defined our unit of mass to be the mass of matter contained in one cubic centimetre of pure water at the temperature of 4° centigrade. This definition would, of course, hold good if there were no gravitation, in which case the water would have no weight. 14. Unit of Force.-We are now in a position to define our unit of force. Let this be the force that will impart to unit of mass unit of velocity in unit of time, or, in other words, a force that, if applied during a second to the mass of a gramme, will produce in it a velocity of one metre in a second. It is very easy to see that if we operate on two grammes we shall require the application of a double force in order to produce our unit velocity, for we may suppose the double mass to be made up of two separate grammes placed side by side, and one-half of the force applied to each. It will therefore take one unit of force to produce unit of velocity in the one gramme, and another unit of force to produce the same in the other, and hence we must apply two units of force. It is not, however, equally easy to see that in order to produce double velocity in a mass, we must have a force twice as large as that which produces unit velocity in the same inass in the same time. But the truth of this statement will afterwards be perceived (Art. 23). LESSON II.-FIRST LAW. 15. Having fixed upon our various units, let us now proceed to the laws of motion. The first law of motion asserts that if a body be at rest it will remain so unless acted on by some external force, or if it be in motion it will move in a straight line, and with a uniform velocity, unless acted on by some external force. This law at first sight seems contrary to our every-day experience, for it obviously implies that a body once in motion will continue in motion for ever, unless acted upon by some external force; now we know that all moving bodies on the earth's surface show a tendency to stop. A little reflection, however, will convince us that the law is true enough, but that all bodies in motion on the earth's surface are in reality acted upon by external forces, and that it is impossible to exhibit a body not so acted upon. It will be found that the more we can reduce in amount the external forces acting upon a moving body, the longer will its motion continue, so that in fact this law of motion represents the state of things under an extreme condition, which can be approached but never reached. 16. We find that friction and the resistance of the atmosphere are the two great forces tending to stop all motion at the earth's surface. To illustrate the former let us make a smooth stone slide along the ground: it will soon be brought to rest through friction; now take the same stone to a smooth sheet of ice, and it will slide along it to a much greater distance because the friction is less. In order to illustrate the resistance of the air, set a massive metallic top in rapid rotation in the open air, and it will come to rest in about twenty minutes; but set the same top in motion in vacuo, and it will remain moving for more than an hour. The resistance of the air acts very strongly upon bodies moving with great velocity; were there no air, the range of a cannon-ball would be very much increased. The nearest approach to a perpetual motion, such as is implied in the first law of motion, is that of the earth in its orbit; any resisting medium, like the air, would have the effect of ultimately making the earth approach the sun by a sort of spiral journey, until at last it would be swallowed up by our luminary. We have reason to believe that there is such a medium, but its tenuity is so great, that it would need a long series of ages in order to diminish sensibly the dimensions of the earth's orbit. Thus we see that the first law of motion contemplates a hypothetical state of things which does not really exist, and we shall see further on that the actual state of things may be represented by one of the laws of energy, of which the first law of motion forms an extreme case. 17. Let us now give a few examples in illustration of this law. Example I.-A man is on horseback, and the horse starts off suddenly. In what direction will the man fall? Answer. He will fall backwards, for in order to cause him to change his previous state of rest, and move along with the horse, force must be applied, by the first law of motion. Now, this force can only be applied at those points at which he is in contact with the horse, so that if he be sitting loosely he will fall backwards. Example II.-A man is on horseback, and the horse stops suddenly. In what direction will the man fall? Answer. This is the opposite of Example I. The man has by the first law of motion a tendency to retain that motion which he had before the horse stopped, and this can only be changed by the application of force. This force, as in the previous case, must be applied at the points where he touches the horse; if he sits loosely, he will therefore preserve his previous state of motion, and be thrown forward over the horse's head. Again, the first law of motion serves to explain the phenomena of rotation. Thus if a disk or top be set in rapid Α с B rotation, a particle at the circumference, such as A, is at any moment moving in the direction of a tangent to the circle at that point; that is to say, in the direction of the arrow head, and if left free to itself it would in virtue of the first law of motion, continue to move in this direction A B; but it is constrained, by the cohesion of the other particles to which it is attached, continually to vary its direction. FIG. 2. If, however, the rotation is very rapid, the force of cohesion may be insufficient to accomplish this, and the consequence will be that the particles at the circumference will leave the system, and be scattered about. In the case of a sling, the force which keeps the stone attached to the sling is intentionally withdrawn at the right moment, and the consequence is that the stone, in virtue of the first law of motion, perseveres in that path, which it was following when the central force was withdrawn. LESSON III.-SECOND LAW: ACTION OF A SINGLE FORCE ON A MOVING BODY. 18. We now proceed to the second law of motion, which may be stated as follows: "If any number of forces act together upon a moving body, each force generates the same velocity as it would generate if it acted singly upon the body at rest.” For the sake of clearness we may divide this statement into two, and consider (1) The action of a single force on a moving body; (2) The action of several forces together upon a moving body. Let us at present consider the action of a single force on a moving body. Suppose, for instance, that in a railway carriage which is at rest I throw up a ball with sufficient force to make it reach the roof: if I throw up the ball with the same force when the carriage is in motion, it will equally reach the roof; or if I throw the ball with a force sufficient to strike the side of the carriage with a given velocity when the carriage is at rest, and if when the carriage is in rapid motion I throw the ball with the same force, it will strike the side of the carriage with the same velocity as before. In fact, the motion of the ball relative to the carriage is precisely the same in both cases; but, on the B When the carriage was at rest the ball went from one side A, B A A FIG. 3. B A to another side, B, of the carriage, let us say in one second, and this was also its motion with regard to the ground. But in the moving carriage, while the ball is on its passage from one side to the other, the point A from which it started has in reality travelled over the distance AA', so that when the ball arrives at the opposite side this has attained the position B'; thus the ball has, in reality, so far as the ground is concerned, travelled from A to B'. It has in fact travelled over the diagonal of a parallelogram of which one side represents the motion of the ball by itself, and the other the motion of the carriage by itself. In like manner we know very well that the motion of the earth in its orbit or on its axis does not interfere with the action of forces tending to produce motion at its surface. Thus at the pole there is no motion of rotation, while at the equator there is a motion nearly equal to a mile in three seconds, and yet the same force will produce the same motion |