508 



THE LOGARITHMIC SPIRAL 



[CH. 



metrical expansion as to preserve its form unchanged; though 

 even there, as we have already seen, the unchanging form denotes 

 a nice balance between the rates of growth in various directions, 

 which is but seldom accurately maintained for long. But the 

 shell retains its unchanging form in spite of its asymmetrical 

 growth; it grows at one end only, and so does the horn. And 

 this remarkable property of increasing by terminal growth, but 

 nevertheless retaining unchanged the form of the entire figure, is 

 characteristic of the logarithmic spiral, and of no other mathe- 

 matical curve. 



We may at once illustrate this curious phenomenon by drawing 

 the outhne of a Uttle Nautilus shell within a big one. We know, 

 or we may see at once, that they are of precisely the same shape ; 

 so that, if we look at the little shell through a magnifying glass, 

 it becomes identical with the big one. But we know, on the other 



Fig. 248. 



hand, that the httle Nautilus shell grows into the big one, not by 

 uniform growth or magnification in all directions, as is (though 

 only approximately) the case when the boy grows into the man, 

 but by growing at one end only. 



Though of all curves, this property of continued similarity is 

 found only in the logarithmic spiral, there are very many rectilinear 

 figures in which it may be observed. For instance, as we may 

 easily see, it holds good of any right cone ; for evidently, in Fig. 248, 

 the Httle inner cone (represented in its triangular section) may 

 become identical with the larger one either by magnification all 

 round (as in a), or simply by an increment at one end (as in b): 

 indeed, in the case of the cone, we have yet a third possibihty, 

 for the same result is attained when it increases all round, save 

 only at the base, that is to say when the triangular section increases 



