XI] CONCLUSION 585 



few drops of milk upon a greasy plate, or of oil upon an alkaline 

 solution. 



We have very far from exhausted, we have perhaps little 

 more than begun, the study of the logarithmic spiral and the 

 associated curves which find exemplification in the multitudinous 

 diversities of molluscan shells. But, with a closing word or two, 

 we must now bring this chapter to an end. 



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

 growth, immensely simplified by the fact that each successive 

 increment is irrevocably fixed in regard to magnitude and position, 

 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 grotvs is merely the lip of an orifice, 

 w^here there is produced a ring of solid material, whose form we 

 have treated of 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 : that is to say, it has a vector motion at 

 right angles to the axis of the shell ; and it has a sliding motion 

 along that axis. 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 ellipse, its axes 

 may be set 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 diver- 

 sities of outward appearance : which diversities, as Swammerdam 



