XI] 



OF THE NAUTILUS SHELL 



771 



Fig. 367), ab is one-third of 6c, de of e/, gh of hi, and kl of Im. The 

 curve is therefore an equiangular spiral." 



The numerical ratio in the case of the Nautilus happens to be 

 one of unusual simplicity. Let us take, with ^oseley, a somewhat 

 more complicated example. 



Fig. 367. Spiral of the Naviilua. 



Fig. 368. Turritella dupli- 

 cata (L.), Moseley's 



. Turbo duplicatus. From 

 Chenu. x ^. 



From the apex of a large Turritella {Turbo) duplicata* a line 

 was drawn across its whorls, and their widths were measured upon 

 it in succession, beginning with the last but one. The measure- 



* In the case of ''Turbo'", and all other turbinate shells, we are dealing not with 

 a plane logarithmic spiral, as in Nautilus, but with a "gauche" spiral, such that 

 the radius vector no longer revolves in a plane perpendicular to the axis of the 

 system, but is inclined to that axis at some constant angle (j3). The figure still 

 preserves its continued similarity, and may be called a logarithmic spiral in space; 

 indeed it is commonly spoken of as a logarithmic spiral wrapped upon a cone, its pole 

 coinciding with the apex of the cone. It follows that the distances of successive 

 whorls of the spiral measured on the same straight Line passing through the apex 

 of the cone are in geometrical progression, and conversely; just as in the former 

 case. But the ratio between any two consecutive interspaces (i.e. ^3 - R2IR2 - Ri) 

 is now equal to e^'^^i^^cota^^ being the semi-angle of the enveloping cone. (Cf. 

 Moseley, Phil. Mag. xxi, p. 300, 1842.) 



