548 THE LOGARITHMIC SPIRAL [ch. 



The various forms of straight or spiral shells among the 

 Cephalopods, which we have seen to be capable of complete 

 definition by the help of elementary mathematics, have received 

 a very complicated descriptive nomenclature from the palaeon- 

 tologists. For instance, the straight cones are spoken of as 

 orthoceracones or hactriticones, the loosely coiled forms as gyrocera- 

 cones or mimoceracones, the more closely coiled shells, in which 

 one whorl overlaps the other, as nautilicones or ammoniticones, 

 and so forth. In such a succession of forms the biologist sees 

 undoubted and unquestioned evidence of ancestral descent. For 

 instance we read in Zittel's Palaeontology'^ : "The bactriticone 

 obviously represents the primitive or primary radical of the 

 Ammonoidea, and the mimoceracone the next or secondary radical 

 of this order" ; while precisely the opposite conclusion was drawn 

 by Owen, who supposed that the straight chambered shells of 

 such fossil cephalopods as Orthoceras had been produced by the 

 gradual unwinding of a coiled nautiloid shell |. To such phylogenetic 

 hypotheses the mathematical or dynamical study of the forms of 

 shells lends no valid support. If we have two shells in which the 

 constant angle of the spire be respectively 80° and 60°, that fact 

 in itself does not at all justify an assertion that the one is more 

 primitive, more ancient, or more "ancestral" than the other. 

 Nor, if we find a third in which the angle happens to be 70°, 

 does that fact entitle us to say that this shell is intermediate 

 between the other two, in time, or in blood relationship, or in 

 any other sense whatsoever save only the strictly formal and 

 mathematical one. For it is evident that, though these particular 

 arithmetical constants manifest themselves in visible and recog- 

 nisable differences of form, yet they are not necessarily more 

 deep-seated or significant than are those which manifest them- 

 selves only in difference of magnitude; and the student of 

 phylogeny scarcely ventures to draw conclusions as to the relative 

 antiquity of two allied organisms on the ground that one happens 

 to be bigger or less, or longer or shorter, than the other. 



* English edition, p. 537, 1900. The chapter is revised by Prof. Alpheus 

 Hyatt, to whom the nomenclature is largely due. For a more copious terminology, 

 see Hyatt, Phylogeny of an Acquired Characteristic, p. 422 seq., 1894. 



f This latter conclusion is adopted by Willey, Zoological Results, p. 747, 1902. 



