,
,
.
Hitherto we have been led to adopt this much more general attitude towards the questions of the metric laws involved in physical formulæ merely by the desire not to introduce, from the very outset, more assumptions into the formulations of physical laws than are compatible with both postulates, and to bring about a deeper appreciation of the points of view, to which the special theory of relativity has led us.
We can briefly summarize by saying: the adoption of Euclidean metric-conditions (measure-relations) is compatible with the postulate of continuity; though the special assumptions thereby involved appear as restrictive or limiting hypotheses, which need not be made. But the second postulate, the reduction of all motions to relative motions, compels us to abandon the Euclidean measure-determination (cf. [p. 43]). A description of the difficulties still remaining in mechanics will make this step clear.
§ 4
THE DIFFICULTIES IN THE PRINCIPLES OF CLASSICAL MECHANICS
THE foundations of classical mechanics cannot be exhaustively described in a narrow space. I can only bring the unfavourable side of the theory into prominent view for the present purpose, without being able to do justice to its great achievements in the past. All doubts about classical mechanics set in at the very commencement with the formulation of the law which Newton places at its head, the formulation of the law of inertia.
As has already been emphasized on [page 31], the assertion that a point-mass which is left to itself moves with uniform velocity in a straight line, omits all reference to a definite co-ordinate system. An insurmountable difficulty here arises: Nature gives us actually no co-ordinate system, with reference to which a uniform rectilinear motion would be possible. For as soon as we connect a co-ordinate system with any body such as the earth, sun, or any other body—and this alone gives it a physical meaning—the first condition of the law of inertia (viz. freedom from external influences) is no longer fulfilled, on account of the mutual gravitational effects of the bodies. One must accordingly either assign to the motion of the body a meaning in itself, i.e. grant the existence of motions relative to "absolute" space, or have recourse to mental experiments by following the example of C. Neumann and introducing a hypothetical body