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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 reappear- 
ance 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 
tield 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 *). 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 # we may introduce “coor- 
dinates”, by which we mean that to each point P we ascribe four 
numbers #,, @, #, v,, 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 A’, in which the numbers v,,...a, ascribed to P are 
the Cartesian coordinates of the point ’. In this way we obtain in 
h’, a figure #”, which just as well as Fean serve as field-figure and 
which of course may be quite different according to the choice of 
the numbers 2,...a, 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 eonsidered 
as being the same. In such a geometric treatment the introduction of 
') In other terms, that the data procured by astronomical observations can be 
extended arbitrarily and unboundedly. 
