8 



observation of these objects, we have formed an ideal image of 

 the completely developed and perfect leaf or crystal, by abstraction 

 of all that is accidental ; and we have learned to consider the observed 

 forms as only more or less perfect approximations to that ideal 

 form. For we are convinced that if circumstances are more and 

 more favorable to free and undisturbed development during the 

 growth of the oak-leaf or the alum-crystal, we shall find a closer 

 approximation also to the standard-form mentioned. It is only 

 to this imaginary standard- form, that we can extend our consi- 

 derations regarding the symmetry-principle; it is again an idea- 

 listic scheme of nature only, to which these mathematical reaso- 

 nings are to be applied. In common parlance we say that 

 the leaf of the oak or the poplar is "intended by nature" to 

 be bilaterally symmetrical, or that the crystals of alum "of their 

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

 world of abstraction, the intellectual image of this imperfect 

 visible world. With respect to our mathematical scheme of forms 

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

 of a Plato or Aristotle. It may here be mentioned also 

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

 we have succeeded in giving a rational explanation of the con- 

 nection between the internal structure and the characteristic 

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

 can hardly be hoped within a measurable space of time to 

 connect their intimate nature with the constant occurrence of 

 their typical external forms in any direct way, although that 

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

 talline substances. 



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

 paragraphs our views regarding the principle of symmetry can 

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

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

 the observed forms should more and more closely approach, as 

 the circumstances during the growth of these natural objects are 

 becoming more favorable. 



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

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

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

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

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



