As we remember, Dr. Whitehead’s definition informs the experimenter that he may recognise the critical speed when his measurements have succeeded in detecting a velocity which is invariant. Failing this, he may also fix its value as soon as his measurements succeed in detecting a trace of non-Euclideanism in the law of composition of velocities. But this is obviously a very roundabout way to help him. The experimenter will now be compelled to fire bullets with various speeds and endeavour to detect traces of non-Euclideanism. If his attempts are unsuccessful, he can draw no conclusions, since it might always be supposed that the invariant velocity was too great to exert perceptible effects on the velocities at his disposal. Even were he to detect what might appear to be non-Euclideanism, how would he know that the effect had not arisen from secondary causes? We remember, in this connection, that neither Fresnel nor Lorentz ever thought it necessary to attribute the result of Fizeau’s experiment to a non-Euclidean composition of velocities.[69] Fresnel ascribed this result to interactions between ether and matter, whereas Lorentz attributed it to the electronic constitution of matter. In short, Dr. Whitehead’s definition places the experimenter in the unenviable position of a man looking for a needle in a haystack without having the consolation of knowing that the needle is really there. From this we see that the definition, when interpreted in a purely formal sense, is ambiguous and tells us nothing, and, when interpreted in a physical sense, is of very little use to the physicist.
And now let us consider Einstein’s definition. First we may note that had the bare invariance of the critical or maximum velocity represented a proper definition in his opinion, he would have given it twenty years ago, seeing that this invariance constitutes the A B C of his theory. Yet, as we know, though he often refers to the maximum velocity as the invariant velocity, and vice versa, he considered it necessary to complete this partial description by a proper definition.
For Einstein, the important point was not so much to furnish a physical illustration of this invariant velocity, which might conceivably never be found to be realised in nature as a physical existent. (In the same way, the infinite velocity of classical science was never given by any physical propagation.) The important point was that he should justify his belief that the world-structure demanded the existence of a critical velocity of this sort; and by its existence we mean that were it ever to be realised, it should prove to be an unsurpassable maximum and an invariant.
We are now in a position to understand the advantage of Einstein’s definition. First, he states that the electromagnetic experiments have established the relativity of velocity. This in turn entails the maintenance of form of all natural laws—in particular, Maxwell’s laws of electromagnetics. Hence it follows that there must exist a critical invariant velocity that is given by the constant c which enters into Maxwell’s equations. Inasmuch as Maxwell had proved that
and the velocity of light in vacuo were one and the same, the invariant velocity is defined by this velocity of light. Thus, with Einstein’s definition, we know that the world-structure demands the existence of a finite invariant velocity, and not of an infinite one, as was believed by classical science. And, in addition, we know that this velocity is given by Maxwell’s celebrated constant
, a constant whose value had been determined with accuracy long before the advent of the relativity theory. In other words, Einstein defines the unknown in terms of the known, not in terms of the ambiguous.
To return to our illustration of the needle in the haystack, not only has Einstein given us the assurance that the needle is there, but he also tells us exactly where to look for it. For this reason his definition, in contradistinction to Dr. Whitehead’s, satisfies the requirements demanded by physical science.
In these pages we have discussed Whitehead’s definition at some length because it illustrates the danger there is in confusing mathematics and physics. Thus, whereas in mathematics we may postulate anything we please (with certain reservations) and then proceed to reach our conclusions deductively, in physics this procedure is impossible. We must take our cue from experiment and formulate our premises accordingly. The result is that whereas in mathematics we are concerned with formal possibilities, in physics we are limited to an analysis of actual facts. And in every case the transition from the possibilities of mathematics to the actualities of physics necessitates the introduction of physical measurements.