XI] OF VARIOUS CEPHALOPODS 811 



Accordingly, we see that (1), when the constant angle of the 

 spiral is small, the shell (or for that matter the tooth, or horn or 

 claw) is scarcely to be distinguished from a straight cone or cylinder; 

 and this remains pretty much the case for a considerable increase of 

 angle, say from 0° to 20° or more; (2) for a considerably greater 

 increase of the constant angle, say to 50° or rnore, the shell would 

 still only have the appearance of a gentle curve; (3) the charac- 

 teristic close coils of the Nautilus or Ammonite would be typically 

 represented only when the constant angle lies within a few degrees 

 on either side of about 80°. The coiled up spiral of a Nautilus, 

 with a constant angle of about 80°, is about six times the length 

 of its radius vector, or rather more than three times its own 

 diameter ; while that of an Ammonite, with a constant angle of, say, 

 from 85° to 88°, is from about six to fifteen times as long as its own 

 diameter. And (4) as we approach an angle of 90° (at which point 

 the spiral vanishes in a circle), the length of the coil increases with 

 enormous rapidity. Our spiral would soon assume the appearance 

 of the close coils of a Nummuhte, and the successive increments 

 of breadth in the successive whorls would become inappreciable to 

 the eye. 



The geometrical form of the shell involves many other beautiful 

 properties, of great interest to the mathematician but which it is 

 not possible to reduce to such simple expressions as we have been 

 content to use. For instance, we may obtain an equation which 

 shall express completely the surface of any shell, in terms of polar 

 or of rectangular coordinates (as has been done by Moseley and 

 by Blake), or in Hamiltonian vector notation*. It is likewise pos- 

 sible (though of little interest to the naturalist) to determine the 

 area of a conchoid al surface or the volume of a conchoidal solid, 

 and to find the centre of gravity of either surface or solid f. And 

 Blake has further shewn, with considerable elaboration, how we may 

 deal with the symmetrical distortion due to pressure which fossil 

 shells are often found to have undergone, and how we may re- 

 constitute by calculation their original undistorted form — a problem 

 which, were the available methods only a Kttle easier, would be 



* Cf. H. W. L. Hime's Outlines of Quaternions, 1894, pp. 171-173. 

 t See Moseley, op. cit. p. 361 seq. Also, for more complete and elaborate treat- 

 ment, Haton de la Goupilliere, op. cit. 1908, pp. 5-46, 69-204. 



