566 THE LOGARITHMIC SPIRAL [ch. 



lamellibranchiate mollusc, and bearing lines of growth in all 

 respects analogous to or even identical with those of the latter. 

 The explanation is very curious and interesting. In ordinary 

 Crustacea the carapace, like the rest of the chitinised and calcified 

 integument, is shed of? in successive moults, and is restored again 

 as a whole. But in Estheria (and one or two other small Crustacea) 

 the moult is incomplete : the old carapace is retained, and the 

 new, growing up underneath it, adheres to it like a lining, and 

 projects beyond its edge: so that in course of time the margins 

 of successive old carapaces appear as "lines of growth" upon the 

 surface of the shell. In this mode of formation, then (but not 

 in the usual one), we obtain a structure which "is partly old and 

 partly new," and whose successive increments are all similar, 

 similarly situated, and enlarged in a continued progression. We 

 have, in short, all the conditions appropria,te and necessary for 

 the development of a logarithmic spiral ; and this logarithmic 

 spiral (though it is one of small angle) gives its own character to 

 the structure, and causes the little carapace to partake of the 

 characteristic conformation of the molluscan shell. 



The essential simplicity, as well as the great regularity of the 

 "curves of growth" which result in the familiar configurations of 

 our bivalve shells, sufficiently explain, in a general way, the ease 

 with which they may be imitated, as for instance in the so-called 

 "artificial shells" which Kappers has produced from the conchoidal 

 form and lamination of lumps of melted and quickly cooled 

 parafiin*. 



In the above account of the mathematical form of the bivalve shell, we 

 have supposed, for simplicity's sake, that the pole or origin of the system is 

 at a point where all the successive curves touch one another. But such an 

 arrangement is neither theoretically probable, nor is it actually the case; 

 for it would mean that in a certain direction growth fell, not merely to a 

 minimum, but to zero. As a matter of fact, the centre of the system (the 

 "umbo" of the conchologists) Ues not at the edge of the system, but very 

 near to it; in other words, there is a certain amount of growth all round. 

 But to take account of this condition would involve more troublesome mathe- 

 matics, and it is obvious that the foregoing illustrations are a sufficiently near 

 approximation to the actual case. 



* Kappers, C. U. A., Die Bildung kiinstlicher Molluskenschalen, Zeitschr. f. 

 allg. Physiol, vn, p. 166, 1908. 



