February 12, 1920] 



NATURE 



629 



over a sphere of half the radius of the mirror, and this 

 spherical surface is infinity to all the dwellers in the 

 mirror space. The image of an object which subtends 

 a large angle at the centre of the mirror will be bent. 

 In Fig. 1, ah, cd, and ef are the images of straight 



Fig. I. 



lines all passing through the same point distant half 

 the radius from the face of the mirror. These lines are 

 all curved and concave to the centre of the mirror, but 

 they are straight lines in convex mirror space, and pass 

 through the smallest number of spatial points of any 

 line joining the extreme fKjints. They are the paths 

 which would be taken by rays of light in space in 

 which the spatial points were packed as in convex 

 mirror space. In every case the light is refracted 

 towards the portion of space in which the point density 

 is greatest. In the figure PQ represents the mirror, 

 RS the focal sphere of half the radius, while the 

 images correspond to straight lines cutting FA pro- 

 duced in the same point at 90°, 45°, and 22^° respec- 

 tively. It will be seen that the curvature of ab enables 

 it to pass through a region in which the points are less 

 closely packed than along the line joining a and b, 

 which appears to the external observer to be straight. 

 On the Einstein theory, light passing a -gravitating 

 body like the sun is refracted in the same way. In 

 convex mirror space strings stretched between the 

 points a and b, c and d, and e and / would take the 

 forms shown. A person in a hurry and endeavouring 

 to pass through a crowd will make a detour to avoid 

 the more densely packed portions of the crowd. 



According to the theory of relativity, motion and 

 force, involving time, change the properties of space. 

 In convex looking-glass space position and direction 

 only are involved, so that the problem is much 

 simpler, while many of the results are very similar. 



If the two great mechanical principles of the con- 

 servation of momentum and the conservation of 

 energy are applied to the movement of bodies in B's 

 space a consistent system of dynamics can be con- 

 structed, and B with his measuring instruments will 

 be quite unable to detect any divergence from Newton's 

 laws of motion. To A, however, the laws will appear 

 very different. For example, a body under the action 

 of no external force moving along the axis of the 

 mirror will move with a velocity varying as the square 

 of its distance from F. This means that the apparent 

 mass will vary inversely as the square of the distance 

 of the body from F, and as the body approaches F the 

 mass appears to increase indefinitely. This corre- 

 sponds to the increase of mass according to the theory 

 of relativitv when the velocity of a body increases, 

 becoming infinite as the velocity of light is approached. 

 According to the theory of relativitv, the mass of a 

 body is greater in the direction of its motion than in 

 directions at right angles to its direction of motion. In 

 convex looking-glass space the mass is greater, when 

 measured bv the accelerative effect of a force, in the 

 NO. 2624, VOL. 104] 



direction of the axis than in directions at right angles 

 to the axis, and greater the nearer the focus. The 

 reason why B cannot detect any of these changes is 

 that all his standard units change in the same way; 

 and, as all physical measurements ultimately reduce 

 themselves to a comparison with standard units, if the 

 units change a corresponding change in the quantity 

 measured cannot be detected. We cannot, for 

 instance, detect the variation in the weight of a body 

 between the equator and the poles by means of 

 standard weights and a pair of scales, though we may 

 detect it by a spring-balance or a pendulum. It is 

 always the looker-on, A, who sees most of the game. 



Some thirty or more years ago a little jeu d'esprit 

 was written by Dr. Edwin Abbott entitled "Flatland." 

 At the time of its publication it did not attract as 

 much attention as it deserved. Dr. Abbott pictures 

 intelligent beings whose whole experience is confined to 

 a plane, or other space of two dimensions, who have 

 no faculties by which they can become conscious of 

 anything outside that space and no means of moving 

 off the surface on which they live. He then asks the 

 reader, who has consciousness of the third dimension, 

 to imagine a sphere descending upon the plane of 

 Flatland and passing through it. How will the in- 

 habitants regard this phenomenon? They will not see 

 the approaching sphere and will have no conception 

 of its solidity. They will only be conscious of the 

 circle in which it cuts their plane. This circle, at 

 first a point, will gradually increase in diameter, 

 driving the inhabitants of Flatland outwards from its 

 circumference, and this will go on until half the sphere 

 has passed through the plane, when the circle will 

 gradually contract to a point and then vanish, leaving 

 the Flatlanders in undisturbed possession of their 

 country (supposing the wound in the plane to have 

 healed). Their experience will be that of a circular 

 obstacle gradually expanding or growing, and then 

 contracting, and they will attribute to growth in time 

 what the external observer in three dimensions assigns 

 to motion in the third dimension. Transfer this 

 analogy to a movement of the fourth dimension 

 through three-dimensional space. Assume the past 

 and future of the universe to be all depicted in 

 four-dimensional space and visible to any being who 

 has consciousness of the fourth dimension. If there 

 is motion of our three-dimensional space relative to the 

 fourth dimension, all the changes we experience and 

 assign to the flow of time will be due simply to this 

 movement, the whole of the future as well as the 

 past always existing in the fourth dimension. 



The theory of relativity requires a fourth dimen- 

 sional term to be introduced into its dynamical equa- 

 tions. This term involves time and the velocity of 

 light. Generally, the easiest method of expressing 

 algebraically position and motion in three-dimensional 

 space is by reference to three directions mutually at 

 right angles, like the edges of a cube which meet at 

 one corner. These lines may, for example, be drawn 

 through the observer north and south and east and 

 west, like the reference lines on a map, while the 

 third line is up and down. The observer's point of 

 reference is where these three lines meet. In four- 

 dimensional geometry there is a fourth direction at 

 right angles to each of the three. Most of us are un- 

 able to form any clear picture of such a direction as a 

 purely geometrical conception. To us the only figure 

 which is at right angles to every straight line drawn 

 through a point O is a sphere, or any number of 

 spheres, having O as centre. As stated above, the 

 fourth co-ordinate involves time and the velocity of 

 light together. Imagine these spheres to be always 

 moving inwards towards O with the velocity of Ii£<ht, 

 and then to expand again from O with the same 

 velocity, and this to take place quite uniformly, how- 



