CONCLUSION 163 



This last feature is sometimes carried to an extreme, as in 

 the lower valve of Pecten shells, where all growth is in the 

 direction of width, with a perfectly flat shell as resultant. 

 Or, as in other forms which habitually lie on one valve, the 

 original direction of the gradient is reversed, and the curva- 

 ture of the two valves becomes similar in sign, instead of 

 opposed (e.g. Productus, occasionally Anomia). 



Still more complexity of detail is shown in many Pteropod 

 shells ; these, as well as the shells of Foraminifera, which 

 achieve logarithmic-spiral form by a somewhat different 

 method (the addition of whole chambers instead of the mere 

 prolongation of a single shell), have been well analysed by 

 D'Arcy Thompson, and need not detain us here. Mention 

 should also be made of the interesting paper of Sporn (1926), 

 who applies a different set of mathematical ideas to the analysis 

 of growth in molluscan shells. These have been related to 

 growth problems in a more general way by Smirnov and 

 Zhelochovtsev (1931). 



§ 4. Conclusion 



We may sum up the most important points of the present 

 chapter as follows : Accretionary growth, in which the new 

 material deposited is not itself capable of further growth, 

 gives rise to structures whose general appearance is radically 

 different from those produced by ordinary intussusceptive or 

 multiplicative growth. But the differences turn out to be due 

 only to this difference in the fate of the new material added 

 by growth, as result of which the fundamental law of accre- 

 tionary growth is one of simple interest, that of multiplicative 

 growth one of compound interest. In other respects, the 

 relative growth obtaining in the two kinds of structures is 

 similar. In both we find growth-centres and growth-gradients, 

 not only major gradients extending through major regions or 

 the whole body, but also minor gradients superposed upon 

 and locally overriding the main gradients. 



The prevalence of the logarithmic-spiral form in nature is 

 due to the fact that a uniform single growth-gradient, com- 

 bined with the method of accretionary growth, must produce 

 a structure in the form of a logarithmic spiral. Departures 

 from the strict logarithmic-spiral form are due to irregularities 

 in the growth-gradients, or to changes, either sudden or pro- 

 gressive, in one or other of the growth-ratios concerned. 



The precise form of the shell or other accretionary structure 



