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 Trafalgar 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 perpendiculars or
co-ordinates (_x, y, z_) which can be dropped from the scene of the
event to those three plane surfaces. The lengths of these three
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