XI] ITS MATHEMATICAL PROPERTIES 791 



As soon, then, as we have prepared tables for these values, the 

 determination of the constant angle a in a particular shell becomes 

 a very simple matter. 



A complete table would be cumbrous, and it will be sufficient 

 to deal with the simple case of the ratio between the breadths of 

 adjacent, or immediately succeeding, whorls. 



Here we have r = e27rcota^ qp log/ = log e x 27r x cot a, from 

 which we obtain the following figures*: 



The shape of a nautiloid spiral 



We learn several interesting things from this short table. We 

 see, in the first place, that where each whorl is about three times 

 the breadth of its neighbour and predecessor, as is the case in 

 Nautilus, the constant angle is in the neighbourhood of 80°; and 

 hence also that, in all the ordinlary ammonitoid shells, and in all 

 the t3rpically spiral shells of the gastropods f, the constant angle 

 is also a large one, being very seldom less than 80°, and usually 

 between 80° and 85°. In the next place, we see that with smaller 



* It is obvious that the ratios of opposite whorls, or of radii 180° apart, are 

 represented by the square roots of these values ; and the ratios of whorls or radii 

 90° apart, by the square roots of these again. 



■j" For the correction to be applied in the case of the helicoid, or "turbinate" 

 shells, see p. 816. 



