*Reference: Einstein’s 1920
Book*

*This paper presents Part 1, Chapter 2 from the book **RELATIVITY: THE SPECIAL AND GENERAL
THEORY by A. EINSTEIN.
The contents are from the original publication of this book by Henry
Holt and Company, New York (1920).*

*The paragraphs of the original
material (in black) are accompanied by brief comments (in color) based on the present
understanding. Feedback on these comments is appreciated.*

*The heading below is linked to
the original materials.*

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**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}

^{1} Here we have assumed that there is nothing left over, *i.e.* that the measurement gives a whole
number. This difficulty is got over by the use of divided measuring-rods, the
introduction of which does not demand any fundamentally new method.

*Einstein is defining the basis of all mesurement of length. It requires a rigid body as a reference.*

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 Square, London,”^{2} 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.^{3}

^{2} I have chosen this as being more familiar to the English reader
than the “Potsdamer Platz, Berlin,” which is referred to in the original. (R.
W. L.)

^{3} It is not necessary here to investigate further the significance
of the expression “coincidence in space.” This conception is sufficiently
obvious to ensure that differences of opinion are scarcely likely to arise as
to its applicability in practice.

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 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.

*To measure the height of a position over a point on surface we can erect a rigid pole. Again it requires a rigid body as a reference.*

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 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.

*We simulate the rigid body of reference with the Cartesian system of co-ordinates.*

In
practice, the rigid surfaces which constitute the system of co-ordinates are
generally not available; furthermore, the magnitudes of the co-ordinates are
not actually determined by constructions with rigid rods, but by indirect
means. If the results of physics and astronomy are to maintain their clearness,
the physical meaning of specifications of position must always be sought in
accordance with the above considerations.^{4}

^{4} A refinement and modification of these views does not become
necessary until we come to deal with the general theory of relativity, treated
in the second part of this book.

We thus obtain the following result: Every description of events in space involves the use of a rigid body to which such events have to be referred. The resulting relationship takes for granted that the laws of Euclidean geometry hold for “distances,” the “distance” being represented physically by means of the convention of two marks on a rigid body.

*The means used to measure distances are assumed to comply with the laws of Euclidean geometry.*

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## FINAL COMMENTS

*Our system of measuring distances derives from distances on a rigid flat surface. The Cartesian system of co-ordinates incorporates this system of measurement in all three dimensions.*

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