distance (line-interval), independently of any changes in position to
which we may subject the body, the propositions of Euclidean geometry
then resolve themselves into propositions on the possible relative
position of practically rigid bodies.[1] Geometry which has been
supplemented in this way is then to be treated as a branch of physics.
We can now legitimately ask as to the truth of geometrical
propositions interpreted in this way, since we are justified in asking
whether these propositions are satisfied for those real things we have
associated with the geometrical ideas. In less exact terms we can
express this by saying that by the truth of a geometrical proposition
in this sense we understand its validity for a construction with rule
and compasses.
[1] It follows that a natural object is associated also with a
straight line. Three points _A, B_ and _C_ on a rigid body thus lie in
a straight line when the points _A_ and _C_ being given, _B_ is chosen
such that the sum of the distances _AB_ and _BC_ is as short as
possible. This incomplete suggestion will suffice for the present
purpose.
Of course the conviction of the truth of geometrical propositions in
this sense is founded exclusively on rather incomplete experience. For
the present we shall assume the truth of the geometrical
propositions, then at a later stage (in the general theory of
relativity) we shall see that this truth is limited, and we shall
consider the extent of its limitation.
II.
THE SYSTEM OF CO-ORDINATES
On the basis of the physical interpretation of distance which has been
indicated, we are also in a position to establish the distance between
two points on a rigid body by means of measurements. For this purpose
we require a distance (rod _S_) which is to be used once and for all,
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