758 THE EQUIANGULAR SPIRAL [ch. 



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 cnaracteristic of the 

 equiangular spiral, and of no other mathematical curve. It well 

 deserves the name, by which James Bernoulli was wont to call it, 

 of spira mirahilis. 



We may at once illustrate this curious phenomenon by drawing 

 the outline 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 

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

 growth or magnification in all parts and directions, as when the boy 

 grows into the man, but by growing at one end only. 



If we should want further proof or illustration of the fact that the spiral 

 shell remains of the same shape while increasing in magnitude by its terminal 

 growth, we may find it by help of our ratio W : L^, which remains constant 

 so long as the shape remains unchanged. Here are weights and measurements 

 of a series of small land-shells {Clausilia):* 



W (mgm.) L (mm.) </lf/L 



Though of all plane curves, this property of continued similarity 

 is found only in the equiangular spiral, there are many rectilinear 

 figures in which it may be shewn. For instance, it holds good of 



* In 100 specimens of Clausilia the mean value of '^'WjL was found to be 

 2-517, the coefficient of variation 0092, and the standard deviation 3-6. That 

 is to say, over 90 per cent, grouped themselves about a mean value of 2-5 with 

 a deviation of less than 4 per /;ent. Cf. C. Petersen, Das Quotientengesetz, 1921, 

 p. 55. 



