By restricting ourselves to this special form of the line-element we are enabled to use the measure laws of Euclidean geometry in all our space-measurements.
But this particular assumption concerning the metrical constitution of space contains the hypothesis, as Helmholtz has shown in a detailed discussion, that finite rigid point-systems, i.e. finite fixed distances, are capable of unrestrained motion in space, and can be made (by superposition) to coincide with other (congruent) point-systems. With respect to the postulate of continuity, this hypothesis seems inconsistent, in so far as it introduces implicit statements about finite distances into purely differential laws, in which only line-elements occur; but it does not contradict the postulate.
The postulate of the relativity of all motion adopts a different attitude towards the possibility of giving the line-element the Euclidean form in particular.[6]
[6]Strictly speaking, I should at this juncture state in anticipation that the above investigations can manifestly also be so generalized as to be valid for the four-dimensional space-time manifold, in which all events actually take place, and that the transformation-formulæ apply to the four variables of this manifold. In these general remarks the neglect of the fourth dimension is of no importance. This statement will be justified later in [§ 3(b)].
According to the principle of the relativity of all motions, all systems, which come about owing to relative motions of bodies towards one another, may be regarded as fully equivalent. The laws of physics must, therefore, preserve their form in passing from one such system to another; i.e. the transformation-formulæ of the variables
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