therefore the forces ought to be supposed to act on material particles: we can consequently make approximate experiments only, from a number of which we draw our conclusions regarding the laws. Their truth is established in the following manner: (1) an hypothesis is taken which is a priori probable, as far as we can judge; (2) experiments are made in which the conditions of the hypothesis are approximately satisfied, and observations are taken by which we see that the more nearly the above conditions are satisfied, the more nearly does the result agree with what we should expect it to be; (3) calculations are made of complicated cases of motion, under the assumption of the truth of the hypothesis, and the result of calculation is found to agree with the case of nature. The next chapter will be devoted to the discussion of these laws.

CHAPTER VI.

OF THE DYNAMICAL LAWS OF FORCE, COMMONLY CALLED, THE LAWS OF MOTION.

100. IN investigating the connection between forces and the motions produced, it will be clearly necessary and sufficient to obtain definite statements on the three following points: (1) the general effect of a force on a particle: (2) what that effect is: (3) when there are several particles influencing each other's motion, what connection there is between the forces called into action. We shall therefore have three laws of motion. It will be necessary to have a law to satisfy us on the first point, because force does not admit of a satisfactory definition, as the idea of it is got from experiment. The formal statement on this point must then be taken together with our notions of force, thus supplying the place of a definition.

The idea of force was that it caused or changed motion. The first law of motion then will be for the purpose of setting forth this idea definitely, i. e. stating a necessary connection (under the present state of nature) between force and motion in material bodies: this is best done negatively, by considering the state of the case where no force acts. If there is such a necessary connection, no motion will then be caused, or if the particle be in motion, its motion will not be changed; therefore it will either be at rest, or move in a straight line with uniform velocity. Since forces that statically balance each other have no resultant, the same ought to be true when the particle is supposed to be acted upon by forces in statical equilibrium.

101. THE FIRST LAW OF MOTION then will be:-"If a material particle be acted upon by no external force, or by

forces which statically balance each other, it will either be at rest, or be moving uniformly in a straight line."

102. We shall now have to test the truth of this by experiments.

(1) If the particle be at rest it will remain at rest.

This is a reasonable hypothesis from the ideas we have of the passive nature or inertia of matter; for there seems to be no reason why a body at rest should begin to move in any one direction rather than any other: moreover, in all cases when a body at rest begins to move, we find from experience that some force has always acted to cause this motion.

(2) If the particle be in motion it will proceed in a straight line.

There is a priori no reason why it should deviate on one side rather than another. And in all cases of curvilinear motion we find that external forces act. If a stone be thrown in a direction inclined to the vertical, its path is curved; but the force of weight has been continually acting which would make the position of the stone at any instant lower than what it would otherwise be. When a stone is thrown along the ground, supposed horizontal, its path is nearly straight; and considering the asperities and unevennesses of the surface, there will be a number of forces called into action at the stone's contact with them, that may account for its deviation from a rectilinear path: moreover, the more we do away with these forces, which is done by making the surface smoother, the more nearly do we find the path become a straight line, thus leading us to suppose that the above forces do account for the deviation. When a carriage in motion is suddenly turned to one side, a person in it feels a tendency towards the other side of the carriage, i. e. to proceed in space in the same direction as before.

(3) If the particle be in motion its velocity is constant.

There is a priori no reason why of itself the particle should increase its velocity rather than diminish it, or vice versa. And we find from experience that when the velocity is changed, forces have acted: for example, in the case of the stone thrown

along the ground, the velocity certainly is diminished, but this may be due to the friction and the resistance of the air: and it is also found that the more these are diminished, as in the case of a smooth level sheet of ice, the less is the diminution of the velocity. If a carriage in motion be suddenly stopped, a person in it is thrown forward, i. e. his body has a tendency to proceed with its previous motion.

From such experiments as the above we conclude that the first Law of Motion is true.

103. Having settled that the invariable effect of a force is motion, we proceed to inquire what motion does a force cause? To obtain an answer that we may expect to be true, we must consider that the idea of force is one in itself, whether force be considered statically or dynamically: therefore the special properties of force must be intrinsically the same in both subjects. These have been investigated in Statics: therefore we must state them now in a dynamical form. The special properties are the following: that a force is independent of any particular point in its line of action; that it is independent, as far as regards its line of action, of any other force (this latter is got from the parallelogram of forces): and as time is not involved in Statics, because we have there not instantaneous but permanent rest, that force is also independent of time.

From this we should infer that the special dynamical properties of a force would be that its effect is independent of any velocity already existing (for that only depends on space and time); and independent, as far as regards its own direction, of any other force. If with these we combine the consideration of the measure of a force arising from the addition of units, we should expect the effect of a force, i. e. the acceleration it produces, to be proportional to its magnitude.

104. We state all this in the following manner as

THE SECOND LAW OF MOTION.

"When any number of forces act on a material particle, the acceleration which any one of them produces on the motion is the same, both in direction and magnitude, as if it had acted on

the particle at rest, and the other forces had not acted at all, being proportional to the intensity of the force."

105. In establishing the truth of this, we find (1) that a force acting constantly in the same direction and with the same intensity on a particle at rest, will produce an uniformly accelerated motion in its own direction. As an example of such a force we may take the weight of the body, which acts always vertically downwards, and is the same for all moderate heights above the earth's surface; therefore if our statement is true, the motion of a body dropped from rest ought to be uniformly accelerated. If observations be made on such motions, it is found that they very nearly agree with what they ought to be, and the resistance of the air seems quite sufficient to account for the discrepancy.

If a body were let fall down an inclined plane, the force on it would be its weight x sine of the inclination of the plane to the horizon, and therefore is constant, and the motion ought to be of the same kind as before. This motion is easier to observe than the preceding, because by diminishing the inclination of the plane the acceleration may be made small, but there is friction as well as the resistance of the air to cause a discrepancy.

(2) The same is true if the particle have an initial motion in the direction in which the force acts.

This may be tested by observing the motion of a body thrown vertically upwards, or that of a body projected directly up or down an inclined plane; and we can also investigate whether the acceleration in this case is the same as in the preceding: we find that it is so, and hence conclude that the effect of a constant force is independent of any velocity in its own direction.

(3) This is also true if the body have a velocity in any

other direction.

As an example of such motion we may take the case of a body projected in a direction not vertical: then we ought to have the case of a point whose motion is affected by a constant