We have already emphasized that the space of astronomy is not a physical space of meter sticks, but is a space of light waves. We may, therefore, have different kinds of space, depending on the fundamental operations. The space of meter sticks we have called "tactual space", and the space of light beams "optical space". If we ask whether astronomical space is Euclidean, we mean merely to ask whether those features of optical space which are within the reach of astronomical measurement are Euclidean. The only possible attitude with respect to this question, or such related questions as whether the total volume of space is finite, or whether space has curvature, is that it is entirely for experiment to decide, and that we have no right to form any preconceived notion whatever. It is therefore beyond the scope of this discussion. It is interesting to notice that the restricted theory of relativity virtually assumes, although often without making the explicit statement, that tactual and optical space are the same. This equivalence results from the properties assumed for light beams. The distance of a mirror may be found equally well by measuring it with meter sticks, or by determining the time required by a light signal to travel there and back. This situation is, however, logically unsatisfying, because it must be assumed that the operations for measuring time are independently defined, and we shall see that they are not. It is a consequence of the assumed equivalence of tactual and optical space that the path of a beam of light is a straight line, as a straight line is determined by operations with meter sticks. When we come to astronomical phenomena, the physical operations with meter sticks can no longer be carried out, and it is meaningless to ascribe to beams of light on an astronomical scale the same geometrical properties that we do on a small scale. THE CONCEPT OF TIME