tracted, though B would be unconscious of the contraction. Moreover, half-way between the mirror and the focus B's foot-rule will appear to A to be only 3 in. long when held perpendicular to the axis, but when turned parallel to the axis it will appear to A to be only 3 in. long, and if it is turned round it will contract in exactly the same way as the image which it is used to measure. B, therefore, will be quite unable by means of his foot-rule to ascertain that the cricket-ball is no longer spherical, or the top or hoop no longer circular. The judgment of A and that of B will therefore be entirely discordant.
If a circle divided by radii, say 5° apart, into equal angles is held with its plane perpendicular to the axis, the image will appear to both A and B to be circular and the angles equal, but if it is turned with its plane parallel to the axis the image to A will appear an ellipse and the angles in each quadrant unequal, but B will have no means of detecting these inequalities, and he will place implicit faith in the accuracy of his protractor.
The question will naturally be asked: Cannot B see that his circle has become an ellipse? When the plane of the circle is at right angles to the axis and B looks straight at it, the image on B's retina, as it appears to A as well as to B, is circular, but when the circle is turned round and B turns round to look at it, B's retina undergoes precisely the same changes as the circle itself, and still the image occupies the same portion of B's retina as before, and therefore produces the same mental impression of a circle on B, though A recognises the ellipticity of B's retinal image (which A is supposed to see in the mirror).
If A walks straight away from the mirror to an indefinite distance, B will walk towards the focus, but as A can never reach the star, so B, walking, as he thinks, uniformly, can never reach F. In fact, his speed of walking as seen by A appears to diminish in proportion to the square of his distance from F, as all small distances measured along the axis diminish in this ratio, but B can never discover this, for he always appears to walk the same number of feet in a minute, as measured by his own diminishing foot-rule. It is true that when B's height and the length of his legs appear to A to be reduced to one-half, the length of his step appears to be reduced to one-quarter, and the angle between his legs as he walks to be reduced correspondingly; but if B tries to measure this angle, his protractor suffers the same distortion, as recognised by A, and B thinks he is walking always in precisely the same way.
It appears, then, that to B the principal focus F is infinity. He can never reach it, however long or however quickly he walks; and there is nothing in his world beyond it. All straight lines drawn from F to the mirror appear to B to be parallel, for they meet only at infinity, and he can never reach their point of meeting. They correspond to parallel lines in the Euclidean space outside the mirror. The image of a square held with its plane perpendicular to the axis will appear to both A and B to be square, but, held with two of its sides parallel to the axis, the angles of the square will appear to to be unequal, for the two sides parallel to the axis will converge to F, and the dimensions of the square along the axis will be less than its dimensions at right angles, but neither the foot-rule nor the protractor in the hands of B will detect these irregularities. In convex mirror space straight lines which meet at F are parallel.
If two of the straight lines which appear to B to be parallel are cut by a third line, and the figure is examined by A, the two interior angles on the same side of the cutting line do not appear to be equal to two right angles, and the exterior angle does not appear to be equal to the interior and opposite angle. This is the essential feature of convex looking-glass space, but B will not agree with A on either question. To B, Euclid's propositions respecting parallel straight lines will appear to hold. He will think that he is living in Euclidean space, though A knows better, or thinks he knows.
To the external observer, then, convex looking-glass space has different properties as the focus is approached, or, in technical phrase, it is not homoloidal, and it has different properties in different directions, like a uniaxial crystal—that is. It is not isotropic, but differs from the crystal since its lack of isotiopism increases as the focus is approached. The image of a metre rod nine-tenths of the distance from the mirror to the focus will appear to the external observer to measure a decimetre when at right angles to the axis, but only a centimetre when parallel to the axis.
This "distortion" of space is precisely what happens according to the theory of relativity in the neighbourhood of a gravitating body, though the distortion is very small even at the surface of the sun. In the direction of the gravitation pull space is contracted, and a foot-rule is actually shorter than when it lies at right angles to the force to the extent of about 43 parts in 10,000,000 at the sun's surface. The effect is greater the greater the intensity of gravitation, and, consequently, it increases on approaching a gravitating body.
If space is supposed to be occupied by points, and the length of a line to be measured by the number 6 points in it, then in space free from gravitation the points are equally distributed in all directions, but when gravity acts the points are closer together in the direction of gravity than in other directions, as soldiers in column are closer together from right to left than from front to rear, or as the images of evenly distributed points in space are more closely packed along the axis of a convex mirror than in other directions. This representation of the effect of gravity is due to Prof. Eddington. Light always goes from on point to another in the shortest possible time. The principle leads to the ordinary laws of reflection and refraction. In passing through space in the presence of gravitation it will take the path which necessitates passing through the smallest number of spatial points, and this means refraction similar to that produced when it passes into a denser medium in which its velocity is reduced. The effect on light in passing near to the sun will be the same as if the sun were surrounded by an atmosphere extending to a distance of many millions of miles, and diminished in density as the distance from the sun is increased. This will act like a convex lens refracting the light, which will travel more slowly as it approaches the sun. A comet approaching the sun with the velocity of light would, according to the laws of Newton, travel more quickly as it approached, but its orbit would be bent towards the sun as the light is bent, but only to one-half the extent. If light from a star were passing the sun close to its limb, and behaved like a comet under the sun's attraction, it would be deflected about seven-eighths of a second of arc. On the theory of relativity it would be deflected through 134 seconds. It was this deflection which the Eclipse Expedition set out to measure. The behaviour of comets shows that there is no solar atmosphere to account for the refraction at distances from the sun at which the refraction was observed.
In all that has been said respecting the space behind a convex mirror the size of the mirror is supposed to be very small as compared with its radius of curvature, and the objects and images much smaller still. If a complete spherical mirror is suspended in free space the geometrical images of the stars will be distributed
