this that in all such cases the whorls must be of excessively small breadth.
There are many other specific properties of the logarithmic spiral, so interrelated to one another that we may choose pretty well any one of them as the basis of our definition, and deduce the others from it either by analytical methods or by the methods of elementary geometry. For instance, the equation r =ã₤aÿ£¢ö¡ may be written in the form logã₤r =ã₤ö¡ã₤logã₤a, or ö¡ =ã₤(logã₤r)ã₤ãã₤(logã₤a), or (since a is a constant), ö¡ =ã₤kã₤logã₤r. Which is as much as to say that the vector angles about the pole are proportional to the logarithms of the successive radii; from which circumstance the name of the ãlogarithmic spiralã is derived.
Let us next regard our logarithmic spiral from the dynamical point of view, as when we consider the forces concerned in the