These space and time transformations permitting us to pass from one Galilean frame to another were the celebrated Galilean transformations of classical science, and they were, as we have seen, the necessary consequences of our belief in absolute duration and distance. They were given by:
or again by
Applying them to particular problems, as, e.g., to the change in the shape of the trajectory of a falling body when viewed successively from a moving train and from the embankment, or to the change in the colour of a monochromatic source of light, or in the pitch of a musical note, the anticipations of the transformations were always found to be verified, to the order of precision of our experiments. Corroborations of this sort, among others, were considered to prove the correctness of the transformations, hence also of our traditional belief in the absoluteness of duration and distance.
Now if, instead of restricting ourselves to any particular phenomenon, we consider all phenomena of the same type, such as all mechanical or all electrodynamic phenomena, we know that these phenomena in their aggregate satisfy the requirements of certain general laws or equations: the laws of mechanics or the laws of electrodynamics, as the case may be. To say, therefore, that Galilean motion or velocity is relative so far as a certain type of phenomenon is concerned means that the general laws governing such types of phenomena remain unchanged in form when we pass from one Galilean frame to another. Expressed mathematically, this means that when the space and time transformations are applied to the general laws, these laws or equations are transformed into equations possessing exactly the same mathematical appearance; they are then said to have been transformed into themselves.
When the transformations were applied to the general laws of mechanics, these laws suffered no change in form; and this fact constituted the mathematical expression of the Newtonian or mechanical principle of the relativity of Galilean motion. But when the transformations were applied to the equations or laws of electrodynamics, a distinct change in form was noted; the laws or equations lost their simple appearance. This suggested that the electrodynamical laws as accepted by classical science held only in their simple form for a privileged Galilean frame, which for reasons of symmetry was naturally assumed to be one at rest in the stagnant ether. But then, the laws changing in form according to the Galilean frame to which they were referred, it followed as a necessary consequence that electromagnetic experiments should pursue a different course and yield different results according to the frame in which the observer and his instruments were placed. It should then be possible to discover the magnitude of the velocity of the frame through the ether by means of electromagnetic or optical experiments, so that once again, independently of any particular ether hypothesis, mathematical reasoning as well as commonplace reasoning confirmed the physical reality of velocity through the ether.
Of course these mathematical anticipations were dependent on a twofold hypothesis: first, that the classical transformation-formulæ based on absolute distance and duration were accurate; and, secondly, that the equations or laws of electrodynamics were correct. To question the first assumption appeared unjustified, and to question the validity of the mathematical expression of the laws led to conflict with experiment. True, we could modify the laws of electrodynamics (by suppressing those mathematical terms which were the cause of their variability), rendering them thereby invariant under the Galilean transformations, but these terms turned out to be those responsible for the phenomenon of electromagnetic induction. To suppress them would have been equivalent to denying the existence of induction, hence of wireless, radio, dynamos and light propagation. Obviously, a solution of this kind could not be accepted, since dynamos, etc., were known to exist.
In view of all these facts, not the slightest doubt was entertained by scientists that experiments sufficiently precise in nature would yield us the velocity of our instruments, hence of our planet, through the stagnant ether.
We must now pass on to a rapid survey of those so-called negative experiments which were primarily responsible for all the trouble. Let us recall once more the point at issue. We wish to ascertain to what extent the velocity of our Galilean frame through the ether is capable of modifying the results of our electromagnetic and optical experiments.