XIII] AND OF BEAK AND CLAW 897 



formation. 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 disease or of injury of 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 mal- 

 formation! . But there has been no fundamental change of form, 

 only an abnormal increase in length; the elongated tooth or 

 claw has the selfsame curvature which it had when it was short, 

 but the spiral becomes more and more manifest the longer it grows. 

 It is only natural, but nevertheless it is curious to see, how 

 closely a rabbit's abnormally overgrown teeth come to resemble 

 the tusks of swine or elephants, of which the normal state is one 

 of hypertrophy. A curiously analogous case is that of the New 

 Zealand Huia bird, in which the beak of the male is comparatively 

 short and straight, while that of the female is long and curved; 

 it is easy to see that there is a shght but identical curve also in the 

 beak of the male, and that the beak of the female shews nothing 

 but an extension or prolongation of the same. In the case of the 

 more curved beaks, such as those of an eagle or a parrot, we may, 

 if we please, determine the constant angle of the logarithmic spiral, 

 just as we have done in the case of the Nautilus shell; and here 

 again, as the bird grows older or the beak longer, the spiral nature 

 of the curve becomes more and more apparent, as in the hooked 

 beak of an old eagle, or in the great beak of a hyacinthine macaw. 

 Let us glance at one or two instances to illustrate the spiral 

 curvature of teeth. 



* Cf. John Hunter, Natural History of the Human Teeth (3rd ed.), 1808, p. 110: 

 "Where a tooth has lost its opposite, it will in time become really so much longer 

 than the rest as the others grow shorter by abrasion". Cf. James Murie, Notes 

 on some diseased dental conditions in animals, Tr. Odontol. Soc. 1867-8, pp. 37-69, 

 257-298. We now know that a Coenurus-cjat in a rabbit's masseter muscle may 

 twist the jaw sideways, so that the incisors fail to meet, and grow accordingly: 

 H. A. Baylis, Trans. B. Soc. Trop. Medicine, xxxiii, p. 4, 1939. 



t See Professor W. C. Mcintosh's paper on "Abnormal teeth in certain mammals, 

 especially in the rabbit," Trans. B.S.E. lvi, pp. 333-407, for a large collection of 

 instances admirably illustrated. 



