with it. Indeed, it \\v imagine the minor-image rotated round a 

 \eitiral a.\i> through 180", and than shifted parallel to itself until it 



as tar in imnt of the mirror, as it is now behind it, the b 

 11 appear to coincide absolutely with the body itseli I ' 

 ive altered nothing oi the original mirror-image during thi> ojM-r- 

 :ion. tlu- coincidence of both proves undeniably that the human 

 ly i- an object which does not differ from its mirror-image. \V. 

 ea>ily test this moreover, if we think for a moment of the body 

 re tier ted in a vertical mirror-plane, coinciding with the meridian 

 u . which would divide the body in two symmetrical halves. 1 i 



would appear to be each others mirror-images also, but the 

 ly as a whole is just the same as the original object. Afterwards 

 shall see that just because this meridian plane is characteristic 

 the special symmetry of the human body, this has the property 

 being congruent with its mirror- 



. ') 



Another instance of such a figure 

 lieh is in different ways similar 

 its mirror-image, is the cube 



"fc 4)- 

 From fig. 4. it appears, that 



le cube, reflected in each plane 

 brought through two opposite 

 Iges, will coincide with its 

 -iginal position; and evidently 

 lere are six of such mirror-planes 

 resent. In the same way the cube 

 ill coincide with itself if reflected at one of the three possible planes 



1 ). In this connection it may be mentioned that the difference between 

 two kinds of operations here considered, by which a figure is brought 

 ito coincidence either with itself or with its mirror-image, has a simple 

 lytical expression too. For in reality .ve are dealing here only with 

 linary orthogonal substitutions of coordinates. Now if the position of the 

 new coordinate-axes X', Y', Z', with respect to the old ones A". V, / is -ivrn 

 In nine directional cosines Cj-s', Cyx', C = j-' } etc., the relations C-rx' f C*j/x' -4- 

 = 1 and Cxx Cxy' -f Cyx . C yu ' -f C- x ' . C~y' = o etc., have always 

 validity. From this it is readily seen that the square of the determinant 

 I'irincil from these nine cosines, must have the value -- 1. And from this 

 relation: A- = 1, it follows, that A itself can be + 1 or - - /. The case 

 of A = + 1 corresponds to the transformation of the system to a posi- 

 tion* in which it remains congruent with itself; the case: A = 1 however, 

 to that in wich it is the mirror-image of the initial system. 



D 



Fig. 4. 





