Page:Eddington A. Space Time and Gravitation. 1920.djvu/184

motion relative to $B$ and wishes to hand his standards to $B$ to check his measures, he must stop their motion; this means in practice that he must bombard his standards with material molecules until they come to rest. Is it fair to assume that no alteration of the standard is caused by this process? Or if $A$ measures time by the vibrations of a hydrogen atom, and space by the wave-length of the vibration, still it is necessary to stop the atom by a collision in which electrical forces are involved.

The standard of length in physics is the length in the year 1799 of a bar deposited at Paris. Obviously no interval is ever compared directly with that length; there must be a continuous chain of intermediate steps extending like a geodetic triangulation through space and time, first along the past history of the scale actually used, then through intermediate standards, and finally along the history of the Paris metre itself. It may be that these intermediate steps are of no importance—that the same result is reached by whatever route we approach the standard; but clearly we ought not to make that assumption without due consideration. We ought to construct our geometry in such a way as to show that there are intermediate steps, and that the comparison of the interval with the ultimate standard is not a kind of action at a distance.

To compare intervals in different directions at a point in space and time does not require this comparison with a distant standard. The physicist's method of describing phenomena near a point $P$ is to lay down for comparison (1) a mesh-system, (2) a unit of length (some kind of material standard), which can also be used for measuring time, the velocity of light being unity. With this system of reference he can measure in terms of his unit small intervals $PP^{\prime }$ running in any direction from $P$, summarising the results in the fundamental formula $ds^{2}=g_{11}\,{dx_{1}}^{2}+g_{22}\,{dx_{2}}^{2}+\ldots +2g_{12}\,dx_{1}\,dx_{2}+\ldots$ If now he wishes to measure intervals near a distant point $Q$, he must lay down a mesh-system and a unit of measure there. He naturally tries to simplify matters by using what he would call the same unit of measure at $P$ and $Q$, either by transporting a material rod or some equivalent device. If it is immaterial by what route the unit is carried from $P$ to $Q$, and replicas of the