Page:Nature (v104).djvu/677

February 12, 1920] 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, ab, cd, and ef are the images of straight

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 produced in the same point at 90°, 45°, and 22$1⁄2$° respectively. 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 f 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 conservation 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 constructed, 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 corresponds to the increase of mass according to the theory of relativity when the velocity of a body increases, becoming infinite as the velocity of light is approached. According to the theory of relativity, 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 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 inhabitants 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 dimensional term to be introduced into its dynamical equations. 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 unable 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- NO. 2624, VOL. 104]