If everything has really gone smoothly, then I say that the points of
the marble slab constitute a Euclidean continuum with respect to the
little rod, which has been used as a distance (line-interval). By
choosing one corner of a square as origin I can characterise every
other corner of a square with reference to this origin by means of two
numbers. I only need state how many rods I must pass over when,
starting from the origin, I proceed towards the right and then
upwards, in order to arrive at the corner of the square under
consideration. These two numbers are then the Cartesian co-ordinates
of this corner with reference to the Cartesian co-ordinate system
which is determined by the arrangement of little rods.
By making use of the following modification of this abstract
experiment, we recognise that there must also be cases in which the
experiment would be unsuccessful. We shall suppose that the rods
expand by in amount proportional to the increase of temperature. We
heat the central part of the marble slab, but not the periphery, in
which case two of our little rods can still be brought into coincidence
at every position on the table. But our construction of squares must
necessarily come into disorder during the heating, because the little
rods on the central region of the table expand, whereas those on the
outer part do not.
With reference to our little rods defined as unit lengths the marble
slab is no longer a Euclidean continuum, and we are also no longer in
the position of defining Cartesian co-ordinates directly with their
aid, since the above construction can no longer be carried out. But
since there are other things which are not influenced in a similar
manner to the little rods (or perhaps not at all) by the temperature of
the table, it is possible quite naturally to maintain the point of view
that the marble slab is a Euclidean continuum. This can be done in a
satisfactory manner by making a more subtle stipulation about the
measurement or the comparison of lengths.
But if rods of every kind (_i.e._ of every material) were to behave _in
the same way_ as regards the influence of temperature when they are on
the variably heated marble slab, and if we had no other means of
detecting the effect of temperature than the geometrical behaviour of
our rods in experiments analogous to the one described above, then our
best plan would be to assign the distance one to two points on the
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