non-rotating Galileian reference-body $K$ . As judged from this body, the clock at the centre of the disc has no velocity, whereas the clock at the edge of the disc is in motion relative to $K$ in consequence of the rotation.

According to a result obtained in [chapter:XII]Section XII, it follows that the latter clock goes at a rate permanently slower than that of the clock at the centre of the circular disc, i.e. as observed from $K$ . It is obvious that the same effect would be noted by an observer whom we will imagine sitting alongside his clock at the centre of the circular disc. Thus on our circular disc, or, to make the case more general, in every gravitational field, a clock will go more quickly or less quickly, according to the position in which the clock is situated (at rest). For this reason it is not possible to obtain a reasonable definition of time with the aid of clocks which are arranged at rest with

respect to the body of reference. A similar difficulty presents itself when we attempt to apply our earlier definition of simultaneity in such a case, but I do not

wish to go any farther into this question.

Moreover, at this stage the definition of the space

co-ordinates also presents insurmountable difficulties. If the observer applies his standard measuring-rod

(a rod which is short as compared with the radius of the disc) tangentially to the edge of the disc, then, as judged from the Galileian system, the length of this rod will be less than $1$ , since, according to [chapter:XII]Section XII, moving bodies suffer a shortening in the direction of the motion. On the other hand, the measuring-rod will not experience a shortening in length, as judged from $K$ , if it is applied to the disc in the direction of the radius. If, then, the observer first measures the circumference of the disc with his measuring-rod and then the diameter of the

disc, on dividing the one by the other, he will not obtain as quotient the familiar number $π = 3.14 \dots$ , but a larger number,Throughout this consideration we have to use the Galileian (non-rotating) system $K$ as reference-body, since we may only assume the validity of the results of the special theory of relativity relative to $K$ (relative to $K^{'}$ a gravitational field prevails). whereas of course, for a disc which is at rest with respect to $K$ , this operation would yield $π$

exactly. This proves that the propositions of Euclidean

geometry cannot hold exactly on the rotating disc, nor in general in a gravitational field, at least if we attribute the length $1$ to the rod in all positions and in every orientation. Hence the idea of a straight line also loses

its meaning. We are therefore not in a position to define exactly the co-ordinates $x$ , $y$ , $z$ relative to the disc by means of the method used in discussing the special theory, and as long as the co-ordinates and times of events have not been defined, we cannot assign an exact meaning to the natural laws in which these occur.

Thus all our previous conclusions based on general relativity would appear to be called in question. In reality we must make a subtle detour in order to be able to apply the postulate of general relativity exactly. I shall prepare the reader for this in the following paragraphs.