1899] ANIMAL SYMMETRY 61 



certain geometrical centre. In the former three systems the units were 

 arranged with respect to (1) a point, (2) a straight line, (3) a plane. Each 

 of these two last geometrical expressions is formed by the motion of the 

 former, thus conferring upon the centre of symmetry an additional dimen- 

 sion, and lastly, the motion of the third centre of symmetry, or the plane, 

 results in a centre of symmetry of three dimensions. In the same way, in 

 centro-symmetry we found that organs with secondary centres were re- 

 peated in three dimensions (6), those in axo-symmetry in two dimensions 

 (4), and those in piano-symmetry were repeated in one dimension (2). 

 Similarly in this last type (stereo-symmetry), as 

 the centre of symmetry is tri-dimensional, the 

 organs will not be repeated at all (Fig. 8). This y C.' 



is evidently an entirely different condition from 



asymmetry, in which there is no " system of a. ? 



arrangement of the parts " found in the amorphic 

 Protozoa and others (Fig. 9). c ■'' 



In the first three forms of symmetry one 

 can notice how the organism, by its gradual B - 



differentiation, becomes as it were the deter- F f-j ^7^71^7' 



(A + A+B + B+C + C.) 



mining factor in its relations to the environment. 



Thus it is mainly the mode of locomotion which is adopted (or the 

 introduction of voluntary movement) that results in the production of 

 a piano-symmetric organism, and in a precisely analogous manner in 

 stereo -symmetry the organism becomes more independent and pre- 

 dominant over its own fate in determining the way in which its 

 environment shall affect it. 



The gradual evolution in the animal kingdom results in a graduated 

 succession of organisms becoming more and more independent of their 

 environment, in so far as that environment has power to mould the 

 organismal form into harmony with itself. 



In the preceding cases the organisms exhibiting those types of 

 symmetry are to that extent in equilibrium with the same type of 

 environment, and in asymmetry the organism rapidly changes its form 

 to adapt itself to a change of environment, but in a stereo-symmetric form 

 the organism has so far brought itself to be partially equilibrated to 

 any change of environment without actual change in general form on 

 its own part, that it may be said practically to have released itself from 

 its environmental thraldom, as far as symmetry is concerned. Hence its 

 organs will never be repeated in cases where the pair are not absolutely 

 inter-dependent (e.g. the paired eyes), and the form of the organism will be 

 determined solely by the inter-relations of its parts. This ideal is not as 

 yet reached in the animal kingdom, for all the piano-symmetric animals 

 still show marked traces of their " paired " condition. On the other 

 hand, stereo-symmetry is continually being evolved in them by a division 

 of labour between the two " paired " elements, so that in the highest 

 types the " bilateral" or paired arrangement is to some extent effaced. 



