1899] ANIMAL SYMMETRY 55 



Thus we may connect the occurrence and origin of centro- 

 symmetry in organisms with two special conditions of existence : — 

 Firstly, the free rotation, active or passive, of an organism about a 

 central point ; secondly, the free encysted or encased condition in which 

 the living matter is more or less removed from the influence of its 

 environment. The spherical form implies homogeneity, whereas the 

 polyhedral types imply heterogeneity, as limited by certain definite 

 numbers of secondary centres of symmetry. 



Haeckel, taking the geometrical types of the sphere and the regular 

 polyhedra as enumerated above, shows how each type is represented 

 by certain of the Eadiolaria. A similar classification could probably 

 be effected in each group of the Protozoa, though perhaps not quite so 

 completely. 



The experiments of Quincke, Butschli, and others with emulsion 

 " foams," and of Roux with oil-drops, tend to emphasise the great 

 importance of the physical environment in determining the mor- 

 phology of the asymmetric and centro-symmetric Protozoa, and the 

 early ontogenetic stages of Metazoa respectively. 



2. Axo-symmetry. — This consists essentially of repetition of parts 

 in two dimensions, and hence the centre of symmetry is formed by 

 the third dimension, which is the axis of symmetry. 



The minimum number of secondary centres of symmetry is reduced 

 to three, and the dominant number will be four, corresponding to twice 

 the number of the like dimensions. Above this the number is 

 practically unlimited, as represented geometrically, from an equilateral 

 triangle and a square through any regular polygon to the circle. 



This type of symmetry is called " radial " by Spencer, but the term 

 has been applied generally to include both this type and the preceding, 

 as according to the meaning it might. On the other hand, if 

 "spherical" symmetry is retained for the preceding, the equivalent 

 term for " axo-symmetry " would be " circular symmetry." Haeckel 

 classed this type with the next as Protaxonia, whilst Hatschek limits 

 this term to the present type. Haeckel's later term is Centraxonia. 

 Haeckel's types do not exactly correspond to axo-symmetry, for he 

 includes those forms in which the two transverse axes differ from each 

 other, whilst these are here included in the Piano-symmetric. 



There are two conceivable types of axo-symmetry, namely, true 

 or mon-axo-symmetry, and di-axo-symmetry (see Figs. 2 and 3). 



In true axo-symmetry there is heterogeneity between the two 

 poles of the dimension of symmetry ; in di-axo-symmetry they are 

 homogeneous. 



True axo-symmetry is geometrically represented by a right regular 

 pyramid and cone, di-polar by a right regular prism and cylinder. 



The di-axo-symmetry is a rare phenomenon, mainly because 

 axo-symmetric animals have their dimensional axis perpendicular, and 

 hence this axis is subjected to the perpetual heterogeneity of environ- 



