552 THE LOGARITHMIC SPIRAL [ch. 



length in whole numbers, in terms of the radius, we have as 

 follows : 



Accordingly, we see that (1), when the constant angle of the 

 spiral is small, the spiral itself is scarcely distinguishable from 

 a straight line, and its length is but very little greater than that 

 of its own radius vector. This remains pretty much the case for 

 a considerable increase of angle, say from 0° to 20° or more; 

 (2) for a very considerably greater increase of the constant angle, 

 say to 50° or more, the shell would only have the appearance of 

 a gentle curve; (3) the characteristic close coils of the Nautilus 

 or Ammonite would be typically represented only when the 

 constant angle lies ^\^thin a few degrees on either side of about 

 80°. The coiled up spiral of a Nautilus, with a constant angle 

 of about 80°, is about six times the length of its radius vector, 

 or rather more than three times its own diameter ; while that of 

 an Ammonite, with, a constant angle of, say, from 85° to 88°, is 

 from about six to fifteen times as long as its own diameter. And 

 (4) as we approach an angle of 90° (at which point the spiral 

 vanishes in a circle), the length of the coil increases with enormous 

 rapidity. Our spiral would soon assume the appearance of the 

 close coils of a Nummulite, and the successive increments of 

 breadth in the successive whorls would become inappreciable to 

 the eye. The logarithmic spiral of high constant angle would. 

 as we have already seen, tend to become indistinguishable, without 

 the most careful measurement, from an Archimedean spiral. 

 And it is obvious, moreover, that our ordinary methods of 



