632 THE SHAPES OF TEETH [ch. 



variation to which the antlers are subject is so great that in the 

 absence of a large series they would be held to be indicative of 

 several distinct species*." But all these many variations lie 

 within a limited range, for they are all subject to our general 

 rule that the entire structure is essentially confined to a single 

 curved surface. 



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

 tines, in the angles by w^hich these latter meet the beam, and in 

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

 mathematical problems with which we cannot at present 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 

 ])roperties between the horn and the antler is the main lesson with 

 which, in the meantime, we must rest content. 



Of Teeth, and of Beak and Claw. 



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

 horn, we find the logarithmic 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 

 conformation. Nail and claw, beak and tooth, all come under 

 this category. The logarithmic spiral always tends to manifest 

 itself in such structures as these, though it usually only attracts 

 our attention in elongated structures, where (that is to say) the 

 radius vector has described a considerable angle. When the 

 canary-bird's claws grow long from lack of use, or when the 

 incisor tooth of a rabbit or a rat grows long by reason of an injury 

 to the opponent tooth against which it was wont to bite, we know 

 that the tooth or claw tends to grow into a spiral curve, and we 

 speak of it as a malformation. But there has been no funda- 

 mental change of form, save only an abnormal increase in length ; 



* Cf. also the immense range of variation in elks' horns, as described by 

 Lonnberg, P.Z.S. ii, pp. 352-360, 1902. 



