February 12, 1920] 



NATURE 



627 



LETTERS TO THE EDITOR. 



[The Editor does not hold himself responsible for 

 opinions expressed by his correspondents. Neither 

 can he undertake to return, or to correspond with 

 the writers of, rejected manuscripts intended for 

 this or any other part of Nature. No notice is 

 taken of anonymous communications.] 



Euclid, Newton, and Einstein. 



Since the results of the EcHpse Expedition of May 

 last have been made public a very great deal of general 

 interest has been displayed in a theory which, until 

 a few weeks ago, was known only to mathematicians 

 and physicists. Even among these, not many could 

 offer any adequate explanation of the new view of space 

 and time and their mutual relations, while some 

 regarded the whole question as a mathematical joke 

 which led to interesting results of no practical 

 value ; and probably not a few thought that a non- 

 Euclidean system of geometry was inadmissible in 

 any physical theory of the universe. On the other 

 hand, there are some who have gone so far as to 

 advocate that non-Euclidean geometry should be 

 taught to boys and girls in secondary schools. The 

 published books on this subject do not come into 

 touch with any ordinary experience, and the whole 

 subject, consequently, has been regarded as a 

 mathematical fiction. So far from this being so, 

 most people have actually seen the ordinary operations 

 of life proceeding in non-Euclidean space, though they 

 have not realised the meaning of all they have seen. 

 In the space behind a plane mirror objects are 

 reversed right and left (perverted), though in all other 

 respects they correspond precisely to the real objects 

 in front of the mirror of which they are the images, 

 but in the space behind a convex mirror this is not 

 the case. The geometry of this space and the be- 

 haviour of moving bodies therein, as viewed by the 

 external observer and as studied by an intelligent 

 being within the image space, say, the image of the 

 external observer, who applies to the images and their 

 movements the same standards of measurement as 

 the external observer applies to the real objects in his 

 own space, introduce us to a non-Euclidean space 

 which is the subject of common observation, and pre- 

 pare the mind for the reception of manv of the con- 

 clusions of the now famous theory of relativity. In 

 the discussion of that theory two observers are sup- 

 posed to be moving relatively to one another, each with 

 his own set of measuring instruments and each living 

 in his own world or system, and the differences between 

 the phenomena which occur in each system as 

 measured by the dweller in that system and by the 

 external observer form the basis of the theory. Cor- 

 responding to these two observers we propose to con- 

 sider the actual observer outside the convex mirror 

 and his supposed intelligent image behind the mirror, 

 and to consider how the images behind the mirror, 

 treated as real objects, appear to behave to both 

 observers. 



In the first place, it is necessary to consider the 

 size and shape of the objects, or, in other words, 

 the geometry of the space. To save repetition it will 

 be convenient to call the external observer .\ and his 

 intelligent image B. The line joining the middle point 

 of the mirror with the centre of the sphere of which 

 the surface of the mirror is a part is the axis of the 

 mirror, and may be supposed to be extended in- 

 definitely outside the mirror. The image of an 

 infinitely distant star on the axis of the mirror will 

 be formed at a point half-way between the surface of 

 the mirror and the centre of the sphere. This point is 

 called the principal focus, and its distance from the 

 mirror is the focal length, which is half the radius. 



NO. 2624, VOL. 104] 



It will be convenient to call this point F. A series 

 of lines drawn from the circumference of the mirror 

 outwards and all parallel to the axis encloses a cylin- 

 drical space to which the external objects considered 

 are to be confined. .AH these lines produced indefinitely 

 will at length meet the star on the axis of the mirror. 

 Their images will, therefore, all converge to the 

 principal focus F, and the whole of the infinite cylinder 

 in the external world will correspond to a cone behind 

 the mirror having F for its vertex and the rnirror for 

 its curved base. If an object outside moves away to 

 infinity its image will never get beyond F, and the 

 images of straight lines meeting the mirror and ex- 

 tending parallel to the axi.s as far as the distant star 

 will all meet at F. We shall suppose the radius of 

 curvature of the mirror to be very large as compared 

 with the dimensions of the mirror itself or of the 

 observer. 



There is a very simple geometrical law connecting 

 the distance of an object from the mirror and the 

 distance of its image from F. This law need not con- 

 cern us except to point out that as the object recedes 

 from the mirror its image approaches F, and, as seen 

 by the external observer, the dimensions of the image 

 in all directions at right angles to the axis are pro- 

 portional to its distance from F, but the dimensions 

 parallel to the axis are proportional to the square of 

 the distance from F of the image. This is the 

 peculiar property of convex mirror space. If a cricket- 

 ball is placed in front of the mirror at a distance 

 equal to the focal length, its image will be half-way 

 between the mirror and F, but the image will not be 

 spherical. In all directions at right angles to the axis 

 the dimensions will be reduced to one-half, but along 

 the axis they will be reduced to one-quarter, so that 

 the sphere will be represented by an oblate spheroid 

 (an orange) with a polar axis one-half of the equa- 

 torial diameter. If the ball moves farther from the 

 mirror the oblateness of the spheroid will be increased, 

 and when the image is three-quarters of the way 

 between the mirror and F the polar axis will be only 

 one-quarter of the equatorial diameter of the spheroid, 

 which will itself be only one-quarter of the diameter 

 of the cricket-ball. If a circular hoop is placed with 

 its plane at right angles to the axis its image will be 

 circular, but if it is turned round so that its plane 

 is parallel to the axis the image will be an ellipse, 

 which will become more and more eccentric as the 

 hoop recedes from the mirror and the image diminishes 

 on approaching F. A top set spinning with its axis 

 perpendicular to the axis of the mirror will appear in 

 its image to the external observer to be elliptic, with 

 its axes fixed in space, so that as any line of particles 

 in the top approach parallelism to the axis of the 

 mirror thev will be squeezed together and expand again 

 as they recede from parallelism. Midway between 

 the mirror and F the density of the top will appear 

 to A to be twice as great in the direction of the axis 

 as in any direction at right angles to the axis, for 

 the same number of particles will be squeezed into 

 half the length. 



.Ml this has been written from the point of view 

 of A, the external observer. But how will all 

 these things appear to B, who is living and 

 moving in the mirror space? Like A. the observer B 

 mav use a foot-rule for measuring length, breadth, 

 and thickness, and a protractor for measuring angles. 

 .\s A oroceeds to measure the real object, B proceeds 

 to measure the imaee, but as he nporoaches the focus 

 his foot-rule, like himself and the image he is going 

 to measure, gets smaller and in precisely the same 

 proportion, so that if the image mea.sured 6 in. in 

 height when close to the mirror, it would always 

 appear to measure 6 in. in height, for, as seen by A, 

 the foot-rule would contract just as the image con- 



