in these sciences, should be finally abandoned for a rational sys- 

 tem of description, in which the doctrine of symmetry is the 

 trustworthy guide. 



For our purpose it is only necessary for the moment to keep 

 in mind that the "symmetry" of a figure consists in some 

 regular repetition of definite parts of it. Thus such figures can be 

 made to coincide with themselves in several ways, either by 

 superimposing or by some other operation. 



3. With respect to the aesthetic value 1 ) of the symmetry- 

 principle some few suggestions may be made here. The aesthetic 

 action of symmetrical arrangement is really established beyond 

 all doubt. Now E. Mach. (Pop.Wiss. Vorlesungen. (1903), p. 100) 

 has drawn attention to the remarkable fact that the symmetry 

 of a figure with respect to a single plane, will immediately be 

 noticed, if the plane of symmetry is a vertical one ; that in the event 

 of its position being horizontal, however, the symmetry of the 

 figure does not make a very strong impression: we can walk for 

 many hours by the side of a lake, before our attention is drawn 

 to the fact that the image in the water is the replica of the 

 scenery itself. Vertical bilateral symmetry appears to be the 

 one naturally adapted to us, while apparently horizontal is 

 almost imperceptible to the observer. Mach tries to give an 

 explanation of this fact by drawing attention to the other, that 

 our visual apparatus itself possesses a vertical plane of sym- 

 metry. The right and the left eye are in their internal structure 

 each other's mirror-images: the function of the one can 

 therefore not be substituted for that of the other, as appears if 

 one transposes the pictures of a stereoscopical photo. If we now 

 look through the stereoscope, a strange world is observed, in 

 which concave and convex are interchanged, and in which all 

 that should stand out, now recedes into the background, and 

 vice versa. The impression of a vertical symmetrical figure thus 

 would result from the circumstance that the right eye sees 

 the right part of the figure in just the same way as the left eye 

 sees the left part of it. The effect of vertical symmetry of this 

 kind would really be an intensifying of the single impression, and 

 would therefore be immediately noticed by us. 



However the question seems to be much more complicated than 



i). Cf. H. N. Day, Aesthetics, p. 76, 72, (1872): "Akin to this beauty of 

 proportion is the beauty of symmetry", etc. 





