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Relativity: The Special and General Theory

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. 1)
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.2)
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 rigidly attached to a
rigid body. Referred to a system of co-ordinates, the scene of any event will be
determined (for the main part) by the specification of the lengths of the three