rigid bodies and clocks, which shall be of such a nature that
measuring-rods and clocks, arranged rigidly with respect to one
another, shall indicate position and time directly. Such was the
essence of the difficulty with which we were confronted in Section
XXIII.
But the considerations of Sections XXV and XXVI show us the way to
surmount this difficulty. We refer the four-dimensional space-time
continuum in an arbitrary manner to Gauss co-ordinates. We assign to
every point of the continuum (event) four numbers, _x_1, _x_2, _x_3,
_x_4 (co-ordinates), which have not the least direct physical
significance, but only serve the purpose of numbering the points of the
continuum in a definite but arbitrary manner. This arrangement does not
even need to be of such a kind that we must regard _x_1, _x_2, _x_3, as
space co-ordinates and _x_4, as a time co-ordinate.
The reader may think that such a description of the world would be
quite inadequate. What does it mean to assign to an event the
particular co-ordinates _x_1, _x_2, _x_3, _x_4, if in themselves these
co-ordinates have no significance? More careful consideration shows,
however, that this anxiety is unfounded. Let us consider, for instance,
a material point with any kind of motion. If this point had only a
momentary existence without duration, then it would to described in
space-time by a single system of values _x_1, _x_2, _x_3, _x_4. Thus
its permanent existence must be characterised by an infinitely large
number of such systems of values, the co-ordinate values of which are
so close together as to give continuity; corresponding to the material
point, we thus have a (uni-dimensional) line in the four-dimensional
continuum. In the same way, any such lines in our continuum correspond
to many points in motion. The only statements having regard to these
points which can claim a physical existence are in reality the
statements about their encounters. In our mathematical treatment, such
an encounter is expressed in the fact that the two lines which
represent the motions of the points in question have a particular
system of co-ordinate values, _x_1, _x_2, _x_3, _x_4, in common. After
mature consideration the reader will doubtless admit that in reality
such encounters constitute the only actual evidence of a time-space
nature with which we meet in physical statements.
When we were describing the motion of a material point relative to a
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