Finally, if this invariant velocity were infinite, we should have the separate space and time of classical science entailing the absoluteness of simultaneity, length and duration. There is nothing in Dr. Whitehead’s mathematical deductions to suggest which of these values the invariant velocity is to have; it might even be an imaginary number. [Pg 208] At all events, we see that the precise magnitude of the invariant velocity is not a matter of any small importance; hence it should inevitably be defined by the definition. Yet this essential piece of information Dr. Whitehead’s definition fails to give, by reason of its being compatible with a whole class of possible invariant velocities ranging from zero to infinity, just as a definition of New York as “a city by the sea” gives us a whole class of possible cities. Definitions of this sort are ambiguous, they are indefinite, and therefore are no definitions at all. The best that can be said of them is that they constitute partial descriptions.[68] Up to this point we have discussed Whitehead’s definition in the light of a self-supporting mathematical definition that would necessitate no appeal to physical measurements in order to acquire a determinate significance. Such appeared to be his intention. Thus interpreted, the definition is a failure, as we have seen. It should therefore be obvious that our only means of rendering it determinate must be to establish the existence, then the magnitude of this elusive critical velocity. But this can be done only through the medium of physical measurements. It matters not whether the measurements be accomplished with rods, clocks, light rays, or by electromagnetic means. In every case they will be physical; hence, when finally the value of the invariant velocity is obtained, this magnitude will be fixed, determined or defined by means of physical measurements. Dr. Whitehead’s purely formal definition cannot emancipate us from this restriction. It seems strange, therefore, that he should criticise in Einstein a procedure which he himself is unable to avoid. At any rate, inasmuch as scientists convinced themselves years ago that in the matter of physical definitions, physical measurements could not be avoided, their aim has always been to select a definition which would define the unknown quantity in terms of magnitudes already known. If this method fails them, they attempt to present their definition in a form which will be of aid to the physicist, directing him towards his goal. When these requirements are considered, it is seen that Einstein’s definition of the invariant velocity as that of light in vacuo cannot easily be improved upon. To understand this point, we shall contrast the situation of an experimenter to whom Dr. Whitehead’s definition, then Einstein’s, would be given. [Pg 209]