i3. Newton's "laws of motion" are postulated relations between forces and their effects. If we were concerned merely to formulate an abstract scheme of dynamics, having no necessary relation to the material universe, we could postulate any relations that are self-consistent. Newton was con cerned to "explain" (i.e., to describe) the observed motions of the heavenly bodies, and the special significance of his relations lies in the fact that they lead to deductions which are confirmed by ex periment. This fact, however, does not compel us to appeal to direct experiment for a justification of his "laws": as stated al ready, we are quite at liberty to postulate them arbitrarily, as fundamental axioms of an abstract dynamical scheme, and to leave for subsequent investigation the question whether this scheme does in fact correspond with "reality." Newton's First Law.—i4. The first question which presents itself is, what is the behaviour of a body which is not affected by force? Newton's answer to this question is contained in his first law : "Every body remains in a state of rest or of uniform motion in a straight line, unless it is compelled by impressed forces to change that state." This law, sometimes called the "law of inertia," had been propounded by Galileo in 1638, nearly fifty years before the appearance of Newton's Principia.
To give precision to the foregoing statement, we must explain what we mean by "a state of rest." The position of a body in space can be defined (§ 8) by specifying the values of three co ordinates x, y, z, relative to axes Ox, Oy, Oz. When x, y and z have constant values (not varying with time), we may say that the body is at rest in relation to Ox, Oy, Oz; if these axes are moving in space, the body will move with them : if they are fixed in space, the body may be said to be in a state of absolute rest. The difficulty in this description (if we keep in mind the question of an ultimate appeal to experiment) is to state what we mean by "axes fixed in space." From a logical standpoint, this is a serious diffi culty in the Newtonian scheme. For practical purposes however we may be content with the assertion, that we are at liberty to postulate the existence of some fixed system of axes, when we formulate our abstract scheme of dynamics; and that we shall be involved in no sensible error, when we come to a comparison with the real universe, if we assume that these axes are at rest in rela tion to the "fixed" stars. By "uniform motion in a straight line" Newton means, in our language, "velocity constant in magnitude and direction." So the answer to our first question is definite : the motion of a body not affected by force may be described, in accordance with the kinematical principles of §§ 5-8, as a motion, referred to a system of axes fixed in space, in which the component velocities u, v, w do not vary with time.
i5. It follows from this discussion that, if u, v, w vary with time (i.e., if the body has acceleration in relation to the fixed axes), then forces must be assumed to act. The next question is, how do forces affect the motion of a body? Newton's second law states : "Change of motion is proportional to the impressed force, and takes place in the direction of the straight line in which that force is impressed." "Change of mo tion," in our language, may be taken to mean acceleration : we deduce that the possession by a given body of an acceleration F in any direction implies that a force P, proportional to F, must be acting in that direction ; i.e., I 2. The general problem of dynamics is to investigate the motions of two or more bodies, as affected by interaction. Such interaction may be caused by collision, as of two billiard balls, or it may be due to mutual attraction, as of the earth and the sun. Following Newton, we shall describe it as operating by the exertion of force; when two bodies A and B interact, we say that A exerts a force on B, and that B exerts a force on A. From a strictly logi si] in We postulate that two or more forces acting simultaneously produce an acceleration which is a combination (i.e., the resultant) of the accelerations which they would produce when acting sep arately; and since F is the resultant of the accelerations f., fy, f., we deduce that the forces X, Y and Z, acting together, produce the same effect as the single force P. So P may be regarded as the resultant of three component forces X, V, Z; and since these four forces are proportional to the accelerations F, f., fy,
it follows that forces, like accelerations, can be resolved or com pounded by the vector law.
i7. This last result will be made the basis of the science of statics (§ 37). If now we imagine the body to be changed, it is clear that, whilst Newton's second law will still hold, the constant of proportionality M in equation (22) may be differ ent. We must regard M as a quantity associated with a given body : it is termed inertia (i.e., sluggishness), because, according to equation (22), the acceleration produced by a given force will be less, the greater the value of M. Inertia, regarded as a meas urable quantity, is commonly designated by the term mass, and defined as the quantity of matter in the body.
When the mass of a body is specified, and the magnitude and direction of the force which acts upon it, the acceleration F can be deduced, and we are left with a problem in kinematics; e.g., if we know that the earth exerts on any body a constant down ward force, we may conclude that it gives to the body a constant downward acceleration, denoted by the symbol g; so the prob lem of a projectile in vacuo may be treated as an example in kinematics (§§ 9—i I).