It will be seen that under these circumstances we cannot hope to measure experimentally either the Fitzgerald contraction itself or any other effect of motion through the æther so long as we ourselves move with the moving system, for all these effects of motion form part of a general sympathetic change which takes place in everything with which we can operate, so that no alteration in the mutual relations of the members of the system is recognisable from within. The change, such as it is, is best described as a variation of the standards of length, time, and force.
It is this last-mentioned aspect of the matter which has chiefly occupied attention of late. On what principles are measurements of length, time, and force ultimately based?
It may be remarked that the measurement of space cannot be effected without introducing the idea of time, for of the events with which we deal in natural philosophy each is perceived to happen at some definite location at some definite instant of time, and we have no obvious means of preserving the identity of this location from one moment to another. Experience deals directly, not with a three-dimensional world of space, but with a four-dimensional world of time and space combined, and the choice of a system of measurement really means the choice of a particular projection of this four-dimensional world into a three-dimensional world of space and a one-dimensional world of time. In practice we perform this act of choice in such a way as to simplify the description of natural phenomena as much as possible; thus we lay down the condition that the increment of the time-variable in the interval between any two consecutive beats of a pendulum carried along with the observer shall be the same as its increment in the interval between any other two consecutive beats; by measuring time in this way we are enabled to formulate the laws of dynamics in simple terms.
We have already seen reason to suppose that the fundamental branch of physical science is the theory of the æther, and we are consequently led to measure space, time, and force in such a way as to give the simplest possible form to the laws of æthereal disturbance. Accordingly two philosophers, situated respectively on two stars which are in motion relative to each other, will not choose the same standards of length and time; each of them will in fact choose his standards so as to satisfy the condition that the velocity of propagation of æthereal disturbance, relative to a framework which moves with his own star, is to be reckoned equal in all directions.
The projection of the four-dimensional world of space and time into the three-dimensional world of space and the one-dimensional world of time is therefore really arbitrary, it may be done in an infinite number of ways, no one of which has any absolute primacy over the others; and each observer chooses the way which best suits the circumstances of his own motion. If we wish to describe natural phenomena in a way independent of the bias of the particular observer, we must have recourse to the language of four-dimensional