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images. Such mirror-images, although built up by the same geometrical 

 elements, are evidently not congruent and they can never be made to 

 coincide by mere motions. 



The same is observed in tridime'nsional space : there are numerous 

 objects, e.g. "right" and "left" hand or foot, screwthreads and 

 tendrils, etc., which are wellknown instances of this kind. They 

 are related to each other as mirror-images, and they can never be 

 brought to coincidence by mere rotations or shifting. Only the 

 "mirror-image" of each of them will coincide with the other object 

 in the way described above. This is commonly expressed by saying 

 that the right and left extremities, or the screwthreads, etc., are 

 objects which are different from their mirror-images. 



It must, however, be kept in mind that a number of objects are 

 not at all different from their mirror- images : our own body is a 

 good example of this. If we look into a mirror, we soon come to 

 the conviction that the mirror-image of our body is really congruent 

 with it. Indeed, if we imagine the mirror-image rotated round a 

 vertical axis through 1 80 and than shifted parallel to itself until it 

 is just as far in front of the mirror, as it is now behind it, the image 

 will appear to coincide absolutely with the body itself. Because we 

 have altered nothing of the original mirror-image during this opera- 

 tion, the coincidence of both proves undeniably that the human 

 body is an object which does not differ from its mirror-image. We 

 can easily test this, moreover, if we think for a moment of the body 

 as reflected in a vertical mirror-plane, coinciding with the meridian 

 plane which would divide the body in two symmetrical halves. These 

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

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

 we shall see that just because this meridian plane is characteristic 

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

 of being congruent with its mirror-image. l ) 



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

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

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

 analytical expression. For in reality we are dealing here only with ordinary 

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

 coordinate-axes X' ', Y', Z', with respect to the old ones X, Y, Z is given 

 by nine directional cosines C xx ' > Cy X '> Czx'> etc., the relations C 2 XX ' -f- C 2 yx ' + 

 &zx' = * and Cxx- c xy + Cyx'- Cyy + c zx ^zy = o, etc., have always 

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

 formed from these nine cosines, must have the value 1. And from this 



