Relativity - The Special and General Theory | Page 3

relative position of practically rigid
bodies.* 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.
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.
Notes
*) 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.

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, and which we employ as a standard measure. If, now, A and B are two
points on a rigid body, we can construct the line joining them according to the rules of

geometry ; then, starting from A, we can mark off the distance S time after time until we
reach B. The number of these operations required is the numerical measure of the
distance AB. This is the basis of all measurement of length. *
Every description of the scene of an event or of the position of an object in space is based
on the specification of the point on a rigid body (body of reference) with which that event
or object coincides. This applies not only to scientific description, but also to everyday
life. If I analyse the place specification " Times Square, New York," **A I arrive at the
following result. The earth is the rigid body to which the specification of place refers; "
Times Square, New York," is a well-defined point, to which a name has been assigned,
and with which the event coincides in space.**B
This primitive method of place specification deals only with places on the surface of rigid
bodies, and is dependent on the existence of points on this surface which are
distinguishable from each other. But we can free ourselves from both of these limitations
without altering the nature of our specification of position. If, for instance, a cloud is
hovering over Times Square, then we can determine its position relative to the surface of
the earth by erecting a pole perpendicularly on the Square, so that it reaches the cloud.
The length of the pole measured with the standard measuring-rod, combined with the
specification of the position of the foot of the pole, supplies us with a complete place
specification. On the basis of this illustration, we are able to see the manner in which a
refinement of the conception of position has been developed.
(a) We imagine the rigid body, to which the place specification is referred, supplemented
in such a manner that the object whose position we require is reached by. the completed
rigid body.
(b) In locating the position of the object, we make use of a number (here the length of the
pole measured with the measuring-rod) instead of designated points of reference.
(c) We speak of the height of the cloud even when the pole which reaches the cloud has
not been erected. By means of optical observations of the cloud from different positions
on the ground, and taking into account the properties of the propagation of light, we
determine the length of the pole we should have required in order to reach the cloud.
From this consideration we see that it will be advantageous if, in the description of
position, it should be possible by means of numerical measures to make ourselves
independent of the existence of marked positions (possessing names) on the rigid body of
reference. In the physics of measurement this is attained by the application of the
Cartesian system of co-ordinates.
This consists of three plane surfaces perpendicular to each other and