848 THE EQUIANGULAK SPIRAL [ch. 



In the spiral shell we have a problem, or a phenomenon, of growth, 

 immensely simplified by the fact that each successive increment is 

 no sooner formed than it is fixed irrevocably, instead of remaining 

 in a state of flux and sharing in the further changes which the 

 organism undergoes. In such a structure, then, we have certain 

 primary phenomena of growth manifested in their original simplicity, 

 undisturbed by secondary and conflicting phenomena. What actually 

 grows is merely the lip of an orifice, where there is produced a ring 

 of solid material, whose form we have discussed under the name of 

 the generating curve ; and this generating curve grows in magnitude 

 without alteration of its form. Besides its increase in areal magnitude, 

 the growing curve has certain strictly limited degrees of freedom, 

 which define its motions in space. And, though we may know nothing 

 whatsoever about the actual velocities of any of these motions, we 

 do know that they are so correlated together that their relative 

 velocities remain constant, and accordingly the form and symmetry 

 of the whole system remain in general unchanged. 



But there is a vast range of possibilities in regard to every one 

 of these factors : the generating curve may be of various forms, and 

 even when of simple form, such as an elhpse, its axes may be s4t 

 at various angles to the system; the plane also in which it lies 

 may vary, almost indefinitely, in its angle relatively to that of any 

 plane of reference in the system; and in the several velocities of 

 growth, of rotation and of translation, and therefore in the ratios 

 between all these, we have again a vast range of possibihties. We 

 have then a certain definite type, or group of forms, mathematically 

 isomorphous, but presenting infinite diversities of outward appear- 

 ance: which diversities, as Swammerdam said, ex sola nascuntur 

 diver sitate gyrationum ; and which accordingly are seen to have their 

 origin in differences of rate, or of magnitude, and so to be, essentially, 

 neither more nor less than differences, of degree. 



In nature, we find these forms presenting themselves with but little 

 relation to the character of the creature by which they are produced. 

 Spiral forms of certain particular kinds are common to Gastropods and 

 to Cephalopods, and to diverse famihes of each ; while outside the class 

 of molluscs altogether, among the Foraminifera and among the worms 

 (as in Spirorbis, Spirographis, and in the Dentalium -like shell of 

 Ditrupa), we again meet with similar and corresponding spirals. 



