Page:Relativity (1931).djvu/133

 The reader may think that such a description of the world would be quite inadequate. What does it mean to assign to an event the particular co-ordinates $$x_1$$, $$x_2$$, $$x_3$$, $$x_4$$, if in themselves these co-ordinates have no significance? More careful consideration shows, however, that this anxiety is unfounded. Let us consider, for instance, a material point with any kind of motion. If this point had only a momentary existence without duration, then it would be described in spacetime by a single system of values $$x_1$$, $$x_2$$, $$x_3$$, $$x_4$$. Thus its permanent existence must be characterised by an infinitely large number of such systems of values, the co-ordinate values of which are so close together as to give continuity; corresponding to the material point, we thus have a (uni-dimensional) line in the four-dimensional continuum. In the same way, any such lines in our continuum correspond to many points in motion. The only statements having regard to these points which can claim a physical existence are in reality the statements about their encounters. In our mathematical treatment, such an encounter is expressed in the fact that the two lines which represent the motions of the points in question have a particular system of co-ordinate values, $$x_1$$, $$x_2$$, $$x_3$$, $$x_4$$, in common. After mature consideration the reader will doubtless admit that in reality such encounters con-