556 THE LOGARITHMIC SPIRAL [ch. 



the inner as compared with the outer part of each whorl, and 

 therefore measures the extent to which one whorl overlaps, or the 

 extent to which it is separated from, another. 



The spiral angle (a) is very small in a limpet, where it is usually 

 taken as = 0° ; but it is evidently of a significant amount, though 

 obscured by the shortness of the tubular shell. In Dentalium 

 it is still small, but sufficient to give the appearance of a regular 

 curve ; it amounts here probably to about 30° to 40°. In Haliotis 

 it is from about 70° to 75° ; in Nautilus about 80° ; and it lies 

 between 80° and 85°, or even more, in the majority of Gastropods. 



The case of Fissurella is curious. Here we have, apparently, 

 a conical shell with no trace of spiral curvature, or (in other 

 words) with a spiral angle which approximates to 0° ; but in the 

 minute embryonic shell (as in that of the limpet) a spiral convolution 

 is distinctly to be seen. It would seem, then, that what we have 

 to do with here is an unusually large growth-factor in the generating 

 curve, which causes the shell to dilate into a cone of very wide 

 angle, the apical portion of which has become lost or absorbed, 

 and the remaining part of which is too short to show clearly its 

 intrinsic curvature. In the closely allied Emarginula, there is 

 likewise a well-marked spiral in the embryo, which however is 

 still manifested in the curvature of the adult, nearly conical, shell. 

 In both cases we have to do with a very wide-angled cone, and 

 with a high retardation-factor for its inner, or posterior, border. 

 The series is continued, from the apparently simple cone to the 

 complete spiral, through such forms as Calyptraea. 



The angle a, as we have seen, is not always, nor rigorously, 

 a constant angle. In some Ammonites it may increase with age, 

 the whorls becoming closer and closer ; in others it may decrease 

 rapidly, and even fall to zero, the coiled shell then straightening 

 out, as in Lituites and similar forms. It diminishes somewhat, 

 also, in many Orthocerata, which are sHghtly curved in youth, 

 but straight in age. It tends to increase notably in some common 

 land-shells, the Pupae and Bulimi ; and it decreases in Succinea. 



Directly related to the angle a is the ratio which subsists 

 between the breadths of successive whorls. The following table 

 gives a few illustrations of this ratio in particular cases, in addition 

 to those which we have already studied. 



