What justifies us in dispensing with the preference for inertial systems over all other co-ordinate systems, a preference that seems so securely established by experiment based upon the principle of inertia? The weakness of the principle of inertia lies in this, that it involves an argument in a circle: a mass moves without acceleration if it is sufficiently far from other bodies; we know that it is sufficiently far from other bodies only by the fact that it moves without acceleration. Are there, in general, any inertial systems for very extended portions of the space-time continuum, or, indeed, for the whole universe? We may look upon the principle of inertia as established, to a high degree of approximation, for the space of our planetary system, provided that we neglect the perturbations due to the sun and planets. Stated more exactly, there are finite regions, where, with respect to a suitably chosen space of reference, material particles move freely without acceleration, and in which the laws of the special theory of relativity, which have been developed above, hold with remarkable accuracy. Such regions we shall call "Galilean regions." We shall proceed from the consideration of such regions [Pg 62] as a special case of known properties. The principle of equivalence demands that in dealing with Galilean regions we may equally well make use of non-inertial systems, that is, such co-ordinate systems as, relatively to inertial systems, are not free from acceleration and rotation. If, further, we are going to do away completely with the difficult question as to the objective reason for the preference of certain systems of co-ordinates, then we must allow the use of arbitrarily moving systems of co-ordinates. As soon as we make this attempt seriously we come into conflict with that physical interpretation of space and time to which we were led by the special theory of relativity. For let ' be a system of co-ordinates whose '-axis coincides with the -axis of , and which rotates about the latter axis with constant angular velocity. Are the configurations of rigid bodies, at rest relatively to ', in accordance with the laws of Euclidean geometry? Since ' is not an inertial system, we do not know directly the laws of configuration of rigid bodies with respect to ', nor the laws of nature, in general. But we do know these laws with respect to the inertial system , and we can therefore estimate them with respect to '. Imagine a circle drawn about the origin in the plane of ' and a diameter of this circle. Imagine, further, that we have given a large number of rigid rods, all equal to each other. We suppose these laid in series along the periphery and the diameter of the circle, at rest relatively to '. If is the number of these rods along the periphery, the number along the diameter, then, if ' does not rotate relatively to , we shall have