“space,” of which, we must honestly acknowledge, we cannot form the slightest conception, and we replace it by “motion relative to a practically rigid body of reference.”  The positions relative to the body of reference (railway carriage or embankment) have already been defined in detail in the preceding section.  If instead of " body of reference " we insert " system of co-ordinates,” which is a useful idea for mathematical description, we are in a position to say :  The stone traverses a straight line relative to a system of co-ordinates rigidly attached to the carriage, but relative to a system of co-ordinates rigidly attached to the ground (embankment) it describes a parabola.  With the aid of this example it is clearly seen that there is no such thing as an independently existing trajectory (lit. “path-curve"*), but only a trajectory relative to a particular body of reference.

In order to have a complete description of the motion, we must specify how the body alters its position with time ; i.e. for every point on the trajectory it must be stated at what time the body is situated there.  These data must be supplemented by such a definition of time that, in virtue of this definition, these time-values can be regarded essentially as magnitudes (results of measurements) capable of observation.  If we take our stand on the ground of classical mechanics, we can satisfy this requirement for our illustration in the following manner.  We imagine two clocks of identical construction ; the man at the railway-carriage window is holding one of them, and the man on the footpath the other.  Each of the observers determines the position on his own reference-body occupied by the stone at each tick of the clock he is holding in his hand.  In this connection we have not taken account of the inaccuracy involved by the finiteness of the velocity of propagation of light.  With this and with a second difficulty prevailing here we shall have to deal in detail later.

  Notes

*) That is, a curve along which the body moves.

THE GALILEIAN SYSTEM OF CO-ORDINATES

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 Galflei-Newton can be regarded as valid only for a Galileian system of co-ordinates.