4 8o SCIENCE PROGRESS 



manifold, we can form, not quite a physical concept, but what may be called a 

 mathematical concept, of absolute position in space, without the aid of the ether. 

 Consider a straight line of length a as traced out by a moving point, giving a one- 

 dimensional point-manifold. Let the line trace out a square by moving through 

 a distance a at right angles to its direction and in one plane, giving a two-dimensional 

 point-manifold. Let the square trace out a cube by moving through a distance a 

 at right angles to its plane, giving a three-dimensional point-manifold. So far all 

 the points may be considered as existing simultaneously, for no point has been 

 required to move into a position already occupied. And this simultaneous existence 

 is characteristic of the conception of space. The division between two successive 

 positions at every stage in the motion is a division between here and there. But 

 in forming the line as a manifold of points, the plane as a manifold of parallel 

 straight lines, or the cube as a manifold of parallel planes, we may alternatively 

 regard the division as between before and after, each thus giving a different 

 representation of the one-dimensional time-flux. Now mathematical analysis, 

 which is nothing but a highly developed system of formal logic, shows beyond the 

 possibility of question that the obstacle to the indefinite continuation of the former 

 process is due entirely to the limitation of our power of visualisation of the relations, 

 and not to anything inherent in the relations themselves. It would therefore be 

 quite legitimate to represent what we call the present state of the universe, not as 

 a division between a past which has ceased to exist and a future not yet existing, 

 but as a division between two continuously existing systems. The three- 

 dimensional space p in which our minds picture the external world, by the 

 correlation of the impressions derived from our senses, a process which begins in 

 earliest infancy, would then form, at any given instant, a division between two 

 portions of four-dimensional space. To an intelligence capable of such a visualisa- 

 tion, they might be pictured as existing simultaneously. If two observers at 

 differents points on the earth's surface desired to compare the results of the study 

 of a moving point, say an agreed point on the moon's surface, their space frame- 

 works would differ, for the axes would in general differ in direction, and be drawn 

 from different origins. Therefore calculations would have to be made, and would 

 be made by Euclid's geometry, which we know by experience to be reliable so far 

 as ordinary astronomical observations are concerned. But whether this would be 

 the case for calculations of the minute accuracy requisite in such a case as the 

 Michelson-Morley experiment we cannot be sure. For the Euclidian geometry 

 involves axioms and implicit assumptions founded on ordinary observation, and 

 these may be only approximately true. Such is Euclid's axiom of parallel straight 

 lines, which leads to the conclusion that the sum of the three interior angles of a 

 plane triangle is equal to two right angles, and Lobachefsky has shown that two 

 different and perfectly self-consistent systems of geometry are derivable from the 

 alternative assumptions that the sum of these three angles is, respectively, greater 

 or less than two right angles. If the difference were small enough, either of the 

 resulting spaces would be indistinguishable from Euclidian space, but in neither 

 of them could a straight line be moved from one position to another without 

 change in size or shape, as Euclid tacitly assumes in the fourth proposition of his 

 first book. Indeed, a straight line, as we conceive it, could not exist, and would 

 be more properly described as a straightest line. The time would be the same 

 for the two observers, but this would not be the case for observers on different 

 planets, and the correlation of the two time systems would depend on the speed of 

 transmission of a light signal through the ether, from one to the other — that is to 

 say, on the very question which it is sought to determine, 



