Thus would be given by a sum of four squares instead of by a sum of three squares and a difference. This would prove that the space-time continuum was Euclidean, having its four dimensions of the same category, instead of semi-Euclidean, as in Einstein’s theory, with an imaginary dimension for time. Time would now fail to be differentiated from space. It is difficult to realise how a world of this sort would manifest itself to us. The only reason we mention this case is in order to show with greater clarity that the classical belief in the separateness of time and space was equivalent to the classical assumption that the invariant velocity was infinite, and not a finite magnitude, whether real or imaginary. A point that appears to have misled a number of persons relates to the meaning to be given to a four-dimensional space and time continuum. In all the cases mentioned in this chapter, it would be permissible to speak of the world as a four-dimensional continuum of events. But with the separate space and time of classical science the statement was artificial, owing to the absoluteness of time, which stood out by itself as distinct from space; and so the four-dimensional aspect of the world was never stressed. It is different with space-time. For now this aspect can no longer be disregarded, since it becomes impossible to divorce space from time in any absolute way holding for all observers. Thus we see that the four-dimensional nature of the world is a fact which has been disclosed by the theory of relativity. Prior to Einstein’s discoveries, any reference to space and time as one continuum would have been an unjustified extension of the accepted meaning of words. Now let us return to the various types of worlds we have discussed in the preceding paragraphs. It is possible to give an alternative presentation of the results mentioned. Thus, we start from the mathematical expression of : where represents the invariant velocity without any specification as to its value. (It may be infinite, as in classical science, or equal to Maxwell’s constant , as in Einstein’s theory, or even equal to some imaginary number, as in the case last discussed.) Then it can be shown that by dividing this equation by and equating the result to zero, we obtain an equation which corresponds to relations holding in three-dimensional space. This equation is given by where the ’s are the components of this velocity along the three mutually perpendicular directions of space. When a mathematical expression of this kind is given as invariant or absolute, the theory of groups enables us to determine the nature of the geometrical rules according to which the variables entering into the expression may be added together. In the present case these variables represent velocities, so that we are on the way to discover the rules governing the addition of velocities. From this equation the following results are obtained. [Pg 203]