Page:The Meaning of Relativity - Albert Einstein (1922).djvu/16

life in judging the relative positions of bodies that it has led to an abstract conception of space which certainly cannot be defended. In order to free ourselves from this fatal error we shall speak only of "bodies of reference," or "space of reference." It was only through the theory of general relativity that refinement of these concepts became necessary, as we shall see later.

I shall not go into detail concerning those properties of the space of reference which lead to our conceiving points as elements of space, and space as a continuum. Nor shall I attempt to analyse further the properties of space which justify the conception of continuous series of points, or lines. If these concepts are assumed, together with their relation to the solid bodies of experience, then it is easy to say what we mean by the three-dimensionality of space; to each point three numbers, $x_{1},x_{2},x_{3}$ (co-ordinates), may be associated, in such a way that this association is uniquely reciprocal, and that $x_{1},x_{2},x_{3}$ vary continuously when the point describes a continuous series of points (a line).

It is assumed in pre-relativity physics that the laws of the orientation of ideal rigid bodies are consistent with Euclidean geometry. What this means may be expressed as follows: Two points marked on a rigid body form an interval. Such an interval can be oriented at rest, relatively to our space of reference, in a multiplicity of ways. If, now, the points of this space can be referred to co-ordinates $x_{1},x_{2},x_{3}$, in such a way that the differences of the co-ordinates, $\Delta x_{1},\Delta x_{2},\Delta x_{3}$ of the two ends of the interval furnish the same sum of squares,

$s^{2}=\Delta x_{1}^{2}+\Delta x_{2}^{2}+\Delta x_{3}^{2}$

(1)