896 



THE SHAPES OF TEETH 



[CH. 



with ; but it is wonderfully improved as we separate the two antlers 

 and give a twist to each, turning antler, tines and all, into the 

 appropriate curved or twisted surface. 



It is probable that in the curvatures both of the beam and of its 

 tines, in the angles by which these latter meet the beam, and in 

 the contours of the entire system, there are involved many elegant 

 mathematical problems with which we cannot attempt to deal. 

 Nor must we attempt meanwhile to enquire into the physical 

 meaning or origin of these phenomena, for as yet the clue seems to 

 be lacking and we should only heap one hypothesis upon another. 

 That there is a complete contrast of mathematical properties be- 

 tween the horn and the antler is the main lesson with which, in the 

 meantime, we must rest content. 



A ---^ B 



Fig. 44(T. Diagrams of antlers, before twisting into shape. 

 A, Red-deer; B, Swamp-deer. 



Of teeth, and of beak mid clciw 



In a fashion similar to that manifested in the shell or the horn, 

 we find the equiangular spiral to be implicit in a great many other 

 organic structures where the phenomena of growth proceed in a 

 similar way: that is to say, where about an axis there is some 

 asymmetry leading to unequal rates of longitudinal growth, and 

 where the structure is of such a kind that each new increment is 

 added on as a permanent and unchanging part of the entire con- 



