550 



THE LOGARITHMIC SPIRAL 



[CH. 



translation along the axis, that is to say in the helicoid which, 

 in all turbinate shells, is superposed upon the spiral. Very careful 

 measurements will be necessary to determine to which of these 

 factors, or in what proportions to each, the phenomenon is due. 

 But in many Ammonitoidea where the helicoid factor does not 

 enter into the case, we have a clear illustration of gradual and 

 marked changes in the spiral angle itself, that is to say of the ratio 

 of growth corresponding to increase of vectorial angle. We have 

 seen from some of Naumann's and Grabau's measurements that 

 such a tendency to vary, such an acceleration or retardation, 

 may be detected even in Ammonites which present nothing 

 abnormal to the eye. But let us suppose that the spiral angle 

 increases somewhat rapidly ; we shall then get a spiral with 

 gradually narrowing whorls, and this condition is characteristic 



Fig. 281. An ammonitoid shell (Macroscaphites) to shew change of 

 curvature. 



of Oekotraustes, a subgenus of Ammonites. If on the other hand, 

 the angle a gradually diminishes, and even falls away to zero, we 

 shall have the spiral curve opening out, as it does in Scaphites, 

 Ancyloceras and Lituites, until the spiral coil is replaced by a spiral 

 curve so gentle as to seem all but straight. Lastly, there are a 

 few cases, such as Bellero'jjJion expansus and some Goniatites, 

 where the outer spiral does not perceptibly change, but the whorls 

 become more "embracing" or the whole shell more involute. 

 Here it is the angle of retardation, the ratio of growth between 

 the outer and inner parts of the whorl, which undergoes a gradual 

 change. 



In order to understand the relation of a close-coiled shell to 

 one of its straighter congeners, to compare (for example) an 



