As is well known, the fundamental law of the mechanics of Galilei-Newton, which is known as the _law of inertia_, can be stated thus: A body removed sufficiently far from other bodies continues in a state of rest or of uniform motion in a straight line. This law not only says something about the motion of the bodies, but it also indicates the reference-bodies or systems of coordinates, permissible in mechanics, which can be used in mechanical description. The visible fixed stars are bodies for which the law of inertia certainly holds to a high degree of approximation. Now if we use a system of co-ordinates which is rigidly attached to the earth, then, relative to this system, every fixed star describes a circle of immense radius in the course of an astronomical day, a result which is opposed to the statement of the law of inertia. So that if we adhere to this law we must refer these motions only to systems of coordinates relative to which the fixed stars do not move in a circle. A system of co-ordinates of which the state of motion is such that the law of inertia holds relative to it is called a “Galileian system of co-ordinates.” The laws of the mechanics of Galilei-Newton can be regarded as valid only for a Galileian system of co-ordinates. V. THE PRINCIPLE OF RELATIVITY (IN THE RESTRICTED SENSE) In order to attain the greatest possible clearness, let us return to our example of the railway carriage supposed to be travelling uniformly. We call its motion a uniform translation (“uniform” because it is of constant velocity and direction, “translation” because although the carriage changes its position relative to the embankment yet it does not rotate in so doing). Let us imagine a raven flying through the air in such a manner that its motion, as observed from the embankment, is uniform and in a straight line. If we were to observe the flying raven from the moving railway carriage. we should find that the motion of the raven would be one of different velocity and direction, but that it would still be uniform and in a straight line. Expressed in an abstract manner we may say: If a mass _m_ is moving uniformly in a straight line with respect to a co-ordinate system _K_, then it will also be moving uniformly and in a straight line relative to a second co-ordinate system _K′_ provided that the latter is executing a uniform translatory motion with respect to _K_. In accordance with the discussion contained in the preceding section, it follows that: If _K_ is a Galileian co-ordinate system. then every other co-ordinate system _K′_ is a Galileian one, when, in relation to _K_, it is in a condition of uniform motion of translation. Relative to _K′_ the mechanical laws of Galilei-Newton hold good exactly as they do with respect to _K_.