568 



THE LOGARITHMIC SPIRAL 



[CH. 



majority of univalves, or to let us measure or estimate the 

 magnitude of the apical angle (6) of the enveloping cone. This 

 however we can do in forms like Isocardia and Diceras ; while in 

 Caprinella we see that the whorls lie in a plane perpendicular to 

 the axis, forming a discoidal spire. As in the latter shell, so also 

 universally among the Brachiopods, there is no lateral asymmetry 

 in the plane of the generating curve such as to lead to the develop- 

 ment of a helix; but in the majority of the Lamellibranchiata 

 it is obvious, from the obhquity of the lines of growth, that the 

 angle 6 is significant in amount. 



The so-called "spiral arms" of Spirifer and many other 

 Brachiopods are not difficult to explain. They begin as a single 



structure, in the form of a loop of 

 shelly substance, attached to the 

 dorsal valve of the shell, in the 

 neighbourhood of the hinge. This 

 loop has a curvature of its own, similar 

 to but not necessarily identical with 

 that of the valve to which it is 

 attached ; and this curvature will tend 

 to be developed, by continuous and 

 symmetrical growth, into a fully 

 formed logarithmic spiral, so far as 

 it is permitted to do so under the 

 constraint of the shell in which it is 

 contained. In various Terebratulae we see the spiral growth of 

 the loop, -more or less flattened and distorted by the restraining 

 pressure of the ventral valve. In a number of cases the loop 

 remains small, but gives off two nearly parallel branches or off- 

 shoots, which continue to grow. And these, starting with just 

 such a sHght curvature as the loop itself possessed, grow on and 

 on till they may form close-wound spirals, always provided that 

 the "spiral angle" of the curve is such that the resulting spire 

 can be freely contained within the cavity of the shell. Owing to 

 the bilateral symmetry of the whole system, the case will be rare, 

 and unlikely to occur, in which each separate arm will coil strictly 

 in a plane, so as to constitute a discoid spiral ; for the original 



Fig. 292. Skeletal loop of 

 Terehratula. (From Woods.) 



