602 THE SPIRAL SHELLS [ch. 



Our problem, then, becomes reduced to that of investigating 

 the possible configurations which may be derived from the succes- 

 sive symmetrical apposition of similar bodies whose magnitudes 

 are in continued proportion ; and it is obvious, mathematically 

 speaking, that the various possible arrangements all come under 

 the head of the logarithmic spiral, together with the limiting cases 

 which it includes. Since the difference between one such form 

 and another depends upon the numerical value of certain 

 coefiicients of magnitude, it is plain that any one must tend to 

 pass into any other by small and continuous gradations ; in 

 other words, that a classification of these forms must (like any 

 classification whatsoever of logarithmic spirals or of any other 

 mathematical curves), be theoretic or "artificial." But we may 

 easily make such an artificial classification, and shall probably 

 find it to agree, more or less, with the usual methods of classification 

 recognised by biological students of the Foraminifera. 



Firstly we have the typically spiral shells, which occur in 

 great variety, and which (for our present purpose) we need hardly 

 describe further. We may merely notice how in certain cases, 

 for instance Globigerina, the individual chambers are httle removed 

 from spheres; in other words, the area of contact between the 

 adjacent chambers is small. In such forms as Cyclammina and 

 Pulvinuhna, on the other hand, each chamber is greatly over- 

 lapped by its successor, and the spherical form of each is lost in 

 a marked asymmetry. Furthermore, in Globigerina and some 

 others we have a tendency to the development of a helicoid spiral 

 in space, as in so many of our univalve molluscan shells. The 

 mathematical problem of how a shell should grow, under the 

 assumptions which we have made, would probably find its most 

 general statement in such a case as that of Globigerina, where the 

 whole organism fives and grows freely poised in a medium whose 

 density is little different from its own. 



The majority of spiral forms, on the other hand, are plane 

 or discoid spirals, and we may take it that in these cases some 

 force has exercised a controlUng influence, so as to keep all the 

 chambers in a plane. This is especially the case in forms like 

 Rotaha or Discorbina (Fig. 316), where the organism lives attached 

 to a rock or a frond of sea- weed ; for here (just as in the case of 



