524 THE LOGARITHMIC SPIRAL [ch. 



feature of the logarithmic spiral. We have alreadj' seen that the 

 logarithmic spiral has a number of "limiting cases," apparently 

 very diverse from one another. Thus the right cone is a logarith- 

 mic spiral in which the revolution of the radius vector is infinitely 

 slow ; and, in the same sense, the straight Hne itself is a limiting 

 case of the logarithmic spiral. The spiral of Archimedes, though 

 not a limiting case of the logarithmic spiral, closely resembles 

 one in which the angle of the spiral is very near to 90°, and the 

 spiral is coiled around a central core. But if the angle of the 

 spiral were actually 90°, the radius vector would describe a circle, 

 identical with the "core" of which we have just spoken; and 

 accordingly it may be said that the circle is, in this sense, a true 

 limiting case of the logarithmic spiral. In this sense, then, the 

 circular concentric operculum, for instance of Turritella or 

 Littorina, does not represent a breach of continuity, but a " limiting 

 case "' of the spiral operculum of Turbo ; the successive " gnomons " 

 are now not lateral or terminal additions, but complete concentric 

 rings. 



Viewed in regard to its own fundamental properties and to 

 those of its limiting cases, the logarithmic spiral is the simplest 

 of all known curves ; and the rigid uniformity of the simple laws, 

 or forces, by which it is developed sufficiently account for its 

 frequent manifestation in the structures built up by the slow and 

 steady growth of organisms. 



In order to translate into precise terms the whole form and 

 growth of a spiral shell, we should have to employ a mathematical 

 notation, considerably more complicated than any that I have 

 attempted to make use of in this book. But, in the most ele- 

 mentary language, we may now at least attempt to describe the 

 general method, and some of the variations, of the mathematical 

 development of the shell. 



Let us imagine a closed curve in space, whether circular or 

 elhptical or of some other and more complex specific form, not 

 necessarily in a plane : such a curve as we see before us when we 

 consider the mouth, or terminal orifice, of our tubular shell ; and 

 let us imagine some one characteristic point within this closed 

 curve, such as its centre of gravity. Then, starting from a fixed 



