In consequence of the change in our ideas of velocity, there results a change in one of the most widely employed laws of velocity, namely the parallelogram law. Briefly stated, in the relativity mechanics, the composition of velocities by means of the parallelogram law is no longer allowable. This follows evidently from the fact that there is an upper limit for the velocity of a material body, and if the parallelogram law were to hold, it would be easy to imagine two velocities which would combine into a velocity greater than that of light. This failure of the parallelogram law to hold is to the mathematician a very disturbing conclusion, more heretical perhaps than the new doctrines regarding space and time.
Another striking consequence of the relativity theory is that the hypothesis of an ether can now be abandoned. As is well known, there have been two theories advanced in order to explain the phenomena connected with light, the emission theory which asserts that light effect is due to the impinging of particles actually sent out by the source of light, and the wave theory which assumes that the sensation we call light is due to a wave in a hypothetical universal medium, the ether. Needless to say this latter theory is the only one which recently has received any support. And now the relativists assert that the logical thing to do is to abandon the hypothesis of an ether. For they reason that not only has it been impossible to demonstrate the existence of an ether, but we have now arrived at the point where we can safely say that at no time in the future will any one be able to prove its existence. And yet the abandoning of the ether hypothesis places one in a very embarrassing position logically, as the three following statements would indicate:
1. The Michelson and Morley experiment was only possible on the basis of an ether hypothesis.
2. From this experiment, follow the essential principles of the relativity theory.
3. The relativity theory now denies the existence of the ether. Whether there is anything more in this state of affairs than mere filial ingratitude is no question for a mathematician.
It should perhaps be pointed out somewhat more explicitly that these changes in the units of time, space and mass, and in those units depending on them, are changes which are ordinarily looked upon as psychological and not physical. If we imagine that A has a clock and that about him move any number of observers, B, C, D,. . ., in different directions and with different velocities, each one of these observers sees A′s clock running at a different rate. Now the actual physical state of A′s clock, if there is such a state, is not affected by what each observer thinks of it; but the difficulty is that there is no way for any one except A to get at the actual state of A′s clock. We are then driven to one of the two alternatives: Either we must give up all notion of time at all,
