tin- Itat of the oak or the poplar is "intended by natui- 

 Iv bilatrr.illv symmetrical, or that the crystals of alum "of tin-M- 

 own nature" represent octahedra. But this only bears upon a 

 world of abstraction, the intellectual image of this imp< : 

 visible world. With respect to our mathematical scheme of forms 

 natural objects, we are indeed still very close to the idealism 

 a Plato or Aristotle. It may here be mentioned also 

 t only in some cases, e. g. in that of crystalline matter, 

 have succeeded in giving a rational explanation of the con- 

 ction between the internal structure and the characteristic 

 trrnal form of a thing. But as regards living organisms, it 

 n hardly be hoped within a measurable space of time to 

 nnect their intimate nature with the constant occurrence of 

 ir typical external forms in any direct way, although that 

 is typical in no less a degree of them, than it is of crys- 

 ine substances. 



In every case it must be remembered here that in the following 

 ragraphs our views regarding the principle of symmetry can 

 ly be applied to objects in the sense mentioned; only the ideal 

 mis, the "standards" of them, are taken into account, to which 

 e observed forms should more and more closely approach, as 

 .e circumstances during the growth of these natural objects are 

 coming more favorable. 



5. It has already been said ( 2), that symmetrical figures 

 n be brought to self-coincidence in several ways; they are 

 ual to themselves in more than one respect. Indeed fig. 2 

 resentents such a "symmetrical" figure, because it takes a new 

 sition always congruent with the initial one, when it is repeatedly 

 tated through an angle of 72 round the axis A before mentioned ; 

 and this can be done five times in the same direction. After the fifth 

 otion the figure is again in exactly the same position as it was at 

 .e beginning. It seems to be adequate in this case to define the 

 ical symmetry of the figure by these characteristic rotations. 

 though in the case considered this will really appear to be justified, 

 we have however still to modify our definition of symmetrical figures 

 with respect to another particular, before it can seem complete. 

 In fig. j? a complex of dots, just like those in fig. 2, is drawn in the 

 same mutual positions and of the same magnitude; the figure 

 evidently possesses the same symmetry as the original one; but 

 notwithstanding all details and properties are the same as in fig. 2, 





