CHAPTER XI 



b LIBRARY 



<^iS^ MASS .^ ^ 



THE LOGARITHMIC SPIRAL 



The very numerous examples of spiral conformation which we 

 meet with in our studies of organic form are peculiarly adapted 

 to mathematical methods of investigation. But ere we begin to 

 study them, we must take care to define our terms, and we had 

 better also attempt some rough preliminary classification of the 

 objects with which we shall have to deal. 



In general terms, a Spiral Curve is a hue which, starting from 

 a point of origin, continually diminishes in curvature as it recedes 

 from that point; or, in other words, whose radius of curvature 

 continually increases. This definition is wide enough to include 

 a number of different curves, but on the other hand it excludes 

 at least one which in popular speech we are apt to confuse with 

 a true spiral. This latter curve is the simple Screw, or cyhndrical 

 HeUx, which curve, as is very evident, neither starts from a definite 

 origin, nor varies in its curvature as it proceeds. The "spiral"' 

 thickening of a woody plant-cell, the "spiral" thread within an 

 insect's tracheal tube, or the "spiral" twist and twine of a cHmbing 

 stem are not, mathematically speaking, spirals at all, but screivs 

 or helices. They belong to a distinct, though by no means verv 

 remote, family of curves. Some of these helical forms we have 

 just now treated of, briefly and parenthetically, under the subject 

 of Geodetics. 



Of true organic spirals we have no lack*. We think at once 

 of the beautiful spiral curves of the horns of ruminants, and of 

 the still more varied, if not more beautiful, spirals of molluscan 

 shells. Closely related spirals may be traced in the arrangement 



* A great number of spiral forms, both organic and artificial, are described 

 and beautifully illustrated in Sir T. A. Cook's Curves of Life, 1914, and Spirals in 

 Nature and Art, 1903. 



