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point of the border of the moon and finally that of the observer's eye. A similar remark may be made when the moment of reappearance is read on a clock. Let us suppose that the light-vibration itself lights the dial-plate, reaching it when the hand is at the point $a$; then we may say that three world-lines, viz. that of the light-vibration, that of the hand and that of the point $a$ intersect.

§ 3. We may imagine that, in order to investigate a gravitation field as e.g. that of the sun, a great number of material points, moving in all directions and with different velocities, are thrown into it, that light-beams are also made to traverse the field and that all coincidences are noted^[1]. It would be possible to represent the results of these observations by world-lines in a four-dimensional figure — let us say in a "field-figure" — the lines being drawn in such a way that each observed coincidence is represented by an intersection of two lines and that the points of intersection of one line with a number of the others succeed each other in the right order.

Now, as we have to attend only to the intersections, we have a great degree of liberty in the construction of the "field-figure". If, independently of each other, two persons were to describe the same observations, their figures would probably look quite different and if these figures were deformed in an arbitrary way, without break of continuity, they would not cease to serve the purpose.

After having constructed a field-figure $F$ we may introduce "coordinates", by which we mean that to each point $P$ we ascribe four numbers $x_{1},x_{2},x_{3},x_{4}$, in such a way that along any line in the field-figure these numbers change continuously and that never two different points get the same four numbers. Having done this we may for each point $P$ seek a point $P'$ in a four-dimensional extension $R'_{4}$, in which the numbers $x_{1},\dots x_{4}$ ascribed to $P$ are the Cartesian coordinates of the point $P'$. In this way we obtain in $R'_{4}$ a figure $F'$, which just as well as $F$ can serve as field-figure and which of course may be quite different according to the choice of the numbers $x_{1},\dots x_{4}$, that have been ascribed to the points of $F$.

If now it is true that the coincidences only are of importance it must be possible to express the fundamental laws of the phenomena by geometric considerations referring to the field-figure, in such a way that this mode of expression is the same for all possible field-figures; from our point of view all these figures can be considered as being the same. In such a geometric treatment the introduction of

1. In other terms, that the data procured by astronomical observations can be extended arbitrarily and unboundedly.