812 THE EQUIANGULAR SPIRAL [ch 



very helpful to the palaeontologist; for, as Blake himself has shewn, 

 it is easy to mistake a symmetrically distorted specimen of (for 

 instance) an Ammonite for a new and distinct species of the same 

 genus. But it is evident that to deal fully with the mathematical 

 problems contained in, or suggested by, the spiral shell, would require 

 a whole treatise, rather than a single chapter of this elementary book. 

 Let us then, leaving mathematics aside, attempt to summarise, and 

 perhaps to extend, what has been said about the general possibilities 

 of form in this class of organisms. 



The univalve shell: a summary 



The surface of any shell, whether discoid or turbinate, may be 

 imagined to be generated by the revolution about a fixed axis of 

 a closed curve, which, remaining always geometrically similar to 

 itself, increases its dimensions continually: and, since the scale of 

 the figure increases in geometrical progression while the angle 

 of rotation increases in arithmetical, and the centre of similitude 

 remains fixed, the curve traced in space by corresponding points 

 in the generating curve is, in all such cases, an equiangular spiral. In 

 discoid shells, the generating figure revolves in a plane perpendicular 

 to the axis, as in the Nautilus, the Argonaut and the Ammonite. In 

 turbinate shells, it follows a skew path with respect to the axis of 

 revolution, and the curve in space generated by any given point makes 

 a constant angle to the axis of the enveloping cone, and partakes, 

 therefore, of the character of a helix, as well as of a logarithmic spiral.; 

 it may be strictly entitled a helico-spiral. Such turbinate or helico- 

 spiral shells include the snail, the periwinkle and all the common 

 typical Gastropods. 



When the envelope of the shell is a right cone — and it is seldom far from 

 being so — then our helico-spiral is a loxodromic curve, and is obviously 

 identical with a projection, parallel with the axis, of the logarithmic spiral 

 of the base. As this spiral cuts all radii at a constant angle, so its orthogonal 

 projection on the surface intersects all generatrices, and consequently all 

 parallel circles, under a constant angle* this being the definition of a loxodromic 

 curve on a surface of revolution. Guido Grandi describes this curve for the 

 first time in a letter to Ceva, printed at the end of his Demonstratio theorematum 

 Hugenianorum circa. . .logarithmicam lineam, 1701 *. 



* See R. C. Archibald, op. cit. 1918. Ohvier discussed it again {Rev. de geom. 

 descriptive, 1843) calling it a "conical equiangular" or "conical logarithmic" 



