792 THE EQUIANGULAR SPIRAL [ch. 



angles the apparent form of the spiral is greatly altered, and the 

 very fact of its being a spiral soon ceases to be apparent (Figs. 379, 

 380). Suppose one whorl to be an inch in breadth, then, if the 

 angle of the spiral were 80°, the next whorl would (as we have just 

 seen) be about three inches broad; if it were 70°, the next whprl 

 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 spme 15,000 miles broad. 



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

 constailt angle, such as Dentalium or Cristellaria, are true equi- 

 angular spirals, just as are those of Nautilus or Rotalia: from 



Fig. 379. 



Fig. 380. 



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 a single whorl. 



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

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

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

 curvature had already greatly diminished. That is to say we might suppose 

 that, however small the angle a, and however rapidly the whorls accordingly 

 increased, there would nevertheless be a manifest spiral convolution in the 

 immediate neighbourhood of the pole, as the starting point of the curve. 

 But it is easy to see that it is not so. It is not that there cease to be con- 

 volutions of the, spiral round the pole when a is a small angle ; on the contrary, 

 there are infinitely many, mathematically speaking. But as a diminishes, 

 and cot a increases towards infinity, the ratio between the breadth of one 

 whorl and the next increases very rapidly. Our taTble shews us that even 

 when a is no less than 40°, and our shell stiU looks strongly curved, one whorl 

 is a thousandt'i part of the breadth of the next, and a thousandfold that 



