Cloud, circa.1984
Excerpts from Albert Einstein: "Relativity, The Special
& General Theory" 1916
"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 "Trafalgar 6- Square, London," I arrive
at the following result.
The earth is the rigid body to which the specification of place
refers; "Trafalgar Square, London," is a well-defined
point, to which a name has been assigned, and with which the
event coincides in space.
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 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 7- 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 as 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 perpendiculars can be
determined by a series of manipulations with rigid measuring-rods
performed according to the rules and methods laid down by
Euclidean geometry."
Baird's electronic version of Albert Einstein's "Relativity: The Special and General Theory"
Baird's electronic version of Albert Einstein's "Sidelights on Relativity"
Visit John Walker, founder of Autodesk, Inc. and co-author of AutoCAD, at Fourmi Lab
AutoCad drawing of the asymmetric field based on Pythagoras' Windmill Diagram (field-1.dwg)
AutoCad drawing of the symmetric field based on the Star of David (field-2.dwg)
Copyright 1998
Michael Robinett