778 THE EQUIANGULAR SPIRAL ' [ch. 



a triangle OPQ be drawn similar to a given triangle, the locus of 

 the vertex Q will* be a spiral similar to the original spiral. We may 

 extend this proposition (as given by Whitworth) from the simple 

 case of the triangle to any similar figures whatsoever ; and see from 

 it how every spot or ridge or tubercle repeated symmetrically from 

 one radius vector (or one generating curve) to another becomes 

 part of a spiral pattern on the shell. 



Viewed in regard to its own fundamental properties and to those 

 of its hmiting cases, the equiangular spiral is one of the simplest 

 of all known curves ; aiid the rigid uniformity of the simple laws by 

 which it is developed sufficiently account for its frequent manifesta- 

 tion in the structures built up by the slow and steady growth of 

 organisms. 



In order to translate into precise terms the whole form and 

 growth of a spiral shell, we should have to employ a mathematical 

 notation considerably more complicated than any that I have 

 attempted to make use of in this book. But we may at least try 

 to describe in elementary language the general method, and some of 

 the variations, of the mathematical development of the shell. But 

 here it is high time to observe that, while we have been speaking of 

 the shell (which is a surface) as a logarithmic spiral (which is a line), 

 we have been simphfying the case, in a provisional or preparatory 

 way. The logarithmic spiral is but one factor in the case, albeit 

 the chief or dominating one. The problem is one not of plane but 

 of sohd geometry, and the solid in question is described by the 

 movement in space of a certain area, or closed curve*. 



Let us imagine a closed curve in space, whether circular or 



eUiptical or of some other and more complex specific form, not 



necessarily in a plane: such a curve as we see before us when we 



consider the mouth, or terminal orifice, of our tubular shell. Let 



* For a more advanced study of the family of surfaces of which the Nautilus 

 is a simple case, see M. Haton de la Goupilliere {op. cit.). The turbinate shells 

 represent a sub-family, which may be called that of the "surfaces cerithioides " ; 

 and "surfaces a front generateur" is a short title of the whole family. The form of 

 the generating curve, its rate of expansion, the direction of its advance, and the 

 angle which the generating front makes with the directrix, define, and give a wide 

 extension to, the family. These parameters are all severally to be recognised in the 

 growth of the living object; and they make of a collection of shells an unusually 

 beautiful materialisation of the rigorous definitions of geometry. 



