574 THE LOGARITHMIC SPIRAL [ch. 



what remains of the original undivided cone. Putting the same 

 thing in other words, we may say that the generating figure, which 

 lay at first in a plane perpendicular to the axis of the cone, has 

 now, by unequal growth, been sharply bent or folded, so as to 

 lie approximately in two planes, parallel to the anterior and 

 posterior faces of the cone. We have only to imagine the apical 

 connecting portion to be further reduced, and finally to disappear 

 or rupture, and we should have a bivalve shell developed out of 

 the original simple cone. 



In its outer and growing portion, the shell of our Pteropod 

 now consists of two parts which, though still connected together 

 at the apex, may be treated as growing practically independently. 

 The shell is no longer a simple tube, or simple cone, in which 

 regular inequalities of growth will lead to the development of a 

 spiral ; and this for the simple reason that we have now two 

 opposite maxima of growth, instead of a maximum on the one side 

 and a minimum on the other side of our tubular shell. As a matter 

 of fact, the dorsal and the ventral plate tend to curve in opposite 

 directions, towards the middle line, the dorsal curving ventrally 

 and the ventral curving towards the dorsal side. 



In the case of the Lamellibranch or the Brachiopod, it is quite 

 possible for both valves to grow into more or less pronounced 

 spirals, for the simple reason that they are hinged upon one another ; 

 and each growing edge, instead of being brought to a standstill 

 by the growth of its opposite neighbour, is free to move out of 

 the way, by the rotation about the hinge of the plane in which 

 it lies. 



But where, as in the Pteropod, there is no such hinge, the 

 dorsal and ventral halves of the shell (or dorsal and ventral 

 valves, if we may call them so), if they curved towards one 

 another (as they do in a cockle), would soon interfere with 

 one another's progress, and the development of a pair of 

 conjugate spirals would become impossible. Nevertheless, there 

 is obviously, in both dorsal and ventral valve, a tendency to 

 the development of a spiral curve, that of the ventral valve 

 being more marked than that of the larger and overlapping 

 dorsal one, exactly as in the two unequal valves of Terebratula. 

 In many cases (e.g. Cleodora cuspidata), the dorsal valve or plate. 



