518 THE LOGARITHMIC SPIRAL [ch. 



It was by observing, with the help of very careful measure- 

 ment, this continued proportionahty, that Moseley was enabled 

 to verify his first assumption, based on the general appearance of 

 the shell, that the shell of Nautilus was actually a logarithmic 

 spiral, and this demonstration he was immediately afterwards 

 in a position to generalise by extending it to all the spiral 

 Ammonitoid and Gastropod mollusca*. 



For, taking a median transverse section of a Nautilus pompilius, 

 and carefully measuring the successive breadths of the whorls 

 (from the dark hne which marks what was originally the outer 

 surface, before it was covered up by fresh deposits on the part 

 of the growing and advancing shell), Moseley found that "the 

 distance of any two of its whorls measured upon a radius vector 

 is one-third that of the two next whorls measured upon the same 

 radius vector f. Thus (in Fig. 262), ab is one-third of be, de of 

 e/, gh of hi, and M of Im. The curve is therefore a logarithmic 

 spiral." 



The numerical ratio in the case of the Nautilus happens to 

 be one of unusual simplicity. Let us take, with Moseley, a 

 somewhat more complicated example. 



From the apex of a large specimen of Turbo duplicatus% a 



* The Rev. H. Moseley, On the Geometrical Forms of Turbinated and Discoid 

 Shells, Phil. Trans, pp. 351-370. 1838. 



f It will be observed that here Moseley, speaking as a mathematician and 

 considering the linear spiral, speaks of whorls when he means the linear bomidaries, 

 or lines traced by the revolving radius vector; while the conchologist usually 

 applies the term whorl to the whole space between the two boundaries. As con- 

 chologists, therefore, we call the breadth of a whorl what Moseley looked upon as 

 the distance between two consecutive whorls. But this latter nomenclature Moseley 

 himself often uses. 



J 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 (6). The figure 

 still preserves its continued similarity, and may with sti-ict accuracy be called a 

 logarithmic spiral in space. It is evident that its envelope will be a right circular 

 cone ; and ind sed 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 lin" 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. 

 i?3 - B^/R^ - Bj) is now equal to e "^'^ ^"^ ^ ^^^ ", 6 being the semi-angle of the enveloping 

 cone. "(Of. Moseley, Phil. Mag. xxi, p. 300, 1842.) 



