808 THE EQUIANGULAR SPIRAL [ch. 



which was the foundation of the Theory of Evolution, aUke to 

 Lamarck and to Darwin and Wallace; and which we see to exist 

 whatever be our ideas of the "origin of species," or of the nature 

 and origin of "functional adaptations." And to my mind, the 

 mathematical (as distinguished from the purely physical) study of 

 morphology bids fair to help us to recognise this phenomenon of 

 orthogenesis in many cases where it is not at once patent to the 

 eye; and, on the other hand, to warn us in many other cases that 

 even strong and apparently complex resemblances in form may be 

 capable of arising independently, and may sometimes signify no 

 more than the equally accidental numerical coincidences which are 

 manifested in identity of length or weight or any other simple 

 magnitudes. 



I have already referred to the fact that, while in general a very 

 great and remarkable regularity of form is characteristic of the 

 molluscan shell, yet that complete regularity is apt to be departed 

 from. We have clear cases of such a departure in Pujpa, Clausilia 

 and various Bulimi, where the spire is not conical, but its -sides are 

 curved and narrow in. 



The following measurements of three specimens of Clausilia shew 

 a gradual change in the ratio to one another of successive whorls, or 

 in other words a marked departure from the logarithmic law: 



Clausilia lamellosa. (From Chr. Petersen*.) 



In many ammonites, where the helicoid factor does not enter into 

 the case, we have a clear illustration of how gradual and marked 



* From Chr. Petersen, Das Quotientengesetz, p. 36. After making a careful 

 statistical study of 1000 Clausilias, Peterson found the following mean ratios of the 

 successive whorls, aJb, b/c, etc.: 1-37, 1-33, 1-27, 1-24, 1-22, 119. 



