This may be called the Galileo-Newton theory. It has been adopted ever since as the foundation of dynamics, except so far as its scope has been restricted by the theory of relativity (q.v.).
It was expounded by Newton in the Principia (1687), the link con necting the acceleration of falling bodies with celestial motions being supplied by the law of gravitation. The Galileo-Newton theory, in combination with the law of gravitation, has been veri fied within the solar system to a high degree of accuracy, though not flawlessly; and it is inferred to be more generally valid. It assumes a space with Euclidean geometrical properties. This is a proper and useful assumption, so far as it will carry us.
For the purpose of the Galileo-Newton theory of motion, mass is the property of a body which is exhibited by inertia. It is shown, for example, by the difficulty of stopping a rapidly rotating grind stone, compared with a similar body made of wood. Presum ably this would be the same on the moon as on the earth. Newton conceived mass to be an inalienable property of every portion of matter, such that the mass of any body is equal to the sum of the masses of its parts, and is expressible arithmetically by its ratio to the mass of a standard body. He regarded this prop erty of terrestrial bodies as having been established consistently. The ratios to one another of the masses of celestial bodies were to be chosen to fit the theory of their motions. Newton called his measure of time an absolute measure, and it may be defined as the measure of time which all practical clocks aim at recording.
A clock is an instrument which counts the repetitions of some operation, such as the swing of a pendulum, which is repeated under conditions as nearly as possible identical; and it is assumed that all operations give the same measure of time. (See TIME MEASUREMENT.) The rotation of the earth is found to be more accurate than any clock, and it is therefore referred to as a standard.
Newton required, for the measurement of motion, the specifi cation of a base relative to which it is to be reckoned. Such a base might be the walls and floor of a room, or of a cabin of a ship, or any other rigid frame with regard to which all points have distinct measurable positions. A bead moving uniformly on a cord, stretched between two legs of a table, has uniform veloc ity in a straight line relative to the table. But if the table is
being moved about a room, the bead has a quite different motion relative to the walls and floor of the room. Newton's procedure practically amounted to choosing a base such that the motions of bodies, relative to it, would conform to his laws. Their ob served motions were to be reduced to order in this way. He called motion relative to this base absolute motion. The base is sometimes called a Galilean base. The laws of motion deal only with changes of velocities ; therefore if a Galilean base has been found, any other base which moves relative to this with uniform velocity, without rotation, is also a Galilean base. The key to the discovery of such a base is that the mutual forces, which are now to be defined, between any two particles, are to be equal and in opposite directions. The base adopted is a base attached to the centre of mass of the solar system, and without rotation relative to the directions of the fixed stars.
The laws governing motion relative to a Galilean base may be stated as follows: A material system is conceived to consist of particles, represented geometrically by points, to each of which a mass is assigned. Any given particle, A, of mass m, has a cer tain acceleration, f, in a certain direction. To account for this acceleration, it must be the resultant of a number of accelera tions, a, 0, . . . , each along the line from A to some other particle. Multiply each of these accelerations by m, so that we have quantities mf, ma, in13, . each of them associated with the direction of the corresponding acceleration, and such that the first of them is the resultant of all the others, according to the law of composition followed by accelerations, that is to say the parallelogram law. The first is called the resultant force acting on A, and the others are called component forces in the lines joining A to other particles. All other particles, B, C, . . . are to be treated in the same way. And the theory is that this can be done so that the component force applied to A, in the line AB, is equal and opposite to the component force applied to B in this line, and the same for other pairs of particles. Masses being suitably assigned to the bodies in question, all accelerations relative to a Galilean base are to be attributable, in this way, to equal and opposite forces between pairs of particles. This im plies that the centre of mass is a point whose acceleration is zero. In general, any actual calculation deals only with a por tion of the whole system; so that the forces to be taken into account are the mutual forces between the bodies in question, and so called external forces representing the influence of external bodies.