XI] 



ITS MATHEMATICAL PROPERTIES 



535 



next whorl would (as we have just seen) be about three inches 

 broad ; if it were 70°, the next whorl would be nearly ten inches, 

 and if it "were 60°, the next whorl would be nearly four feet 

 broad. If the angle were 28°, the next whorl would be a mile 

 and a half in breadth; and if it were 17°, the next would be 

 some 15,000 miles broad. 



Fig. 272. 



In other words, the spiral shells of gentle curvature, or of 

 small constant angle, such as Dentalium or Nodosaria, are true 

 logarithmic spirals, just as are those of Nautilus or Rotalia: 

 from which they differ only in degree, in the magnitude of an 

 angular constant. But this diminished magnitude of the angle 

 causes the spiral to dilate with such immense rapidity that, so 

 to speak, "it never comes round"; and so, in such a shell as 

 Dentalium, we never see but a small portion of the initial whorl. 



Fio^. 273. 



We might perhaps be incUned to suppose that, in such a shell as Dentalium, 

 the lack of a visible spiral convolution was only clue to our seeing but a small 

 portion of the curve, at a distance from the pole, and when, therefore, its 



