628 



NATURE 



[February 12, 1920 



Iracted, though B would be unconscious of the con- 

 traction. Moreover, half-way between the mirror and 

 the focus B's foot-rule will appear to A to be only 

 b 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. lonj^, 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 un- 

 able 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, bul 

 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 tlie 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 .\ to be reduced to one-half, the 

 length of his step ap|x>ars to be reduced to one-quarter, 

 .-ind 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 .•\, 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 nerpendicular 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 sauare 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 anpear to B to 

 be parallel are cut by a third line, and the figure is 

 examined by -X, the two interior angles on the same 

 side of the cutting line do not apnear to be equal to 

 two right angles, and the exterior angle does not 

 appear to be equal to the interior and opposite angle. 

 NO. 2624, VOL. 104] 



This is the essential feature of convex looking-glass 

 space, but B will not agree with .\ 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 ap- 

 proached, or, in technical phrase, it is not homoloidal, 

 and it has different projicrties in different directions, 

 like a uniaxial crystal — that is. It is not isotropic, 

 but differs from the crystal since its lack of iso- 

 tiopism increases as the focus is approached. Tin- 

 image of a metre rod nine-tenths of the distance 

 from the mirror to the focus will appear to the ex- 

 ternal 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 neighbour- 

 hood 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, coo 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, an' 

 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 dis- 

 tributed 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 t 

 Prof. Eddington. Light always goes from on 

 point to another in the shortest possible time. Th 

 I>rincip!e lerds to the ordinary laws of reflec- 

 tion and refraction. In passing through space 

 in the presence of gravitation it will take 

 the path which necessitates passing through thfe 

 smallest number of spatial points, and this means rev_ 

 fraction similar to that produced when it passes into r? 

 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 

 diminishiui? in density as the distance from the sun 

 is increased. This will act like a convex lens refract- 

 ing the light, which will travel more slowly as i 

 approaches the sun. K comet approaching the su 

 with the velocity of light would, according to the laws 

 of Newton, travel more quickly as it ap|3roached, 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-eifrhths of a second of 

 arc. On the theory of relativity it would be deflected 

 through ij 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 

 b? very small as compared with its radius of curvature, 

 and the obiects and images much smaller still. If a 

 complete spherical mirror is suspended in free sp,ace 

 the geometrical images of the stars will be distributed 



