angle of the spiral is very near to 90ô¯, and the spiral is coiled around a central core. But if the angle of the spiral were actually 90ô¯, the radius vector would describe a circle, identical with the ãcoreã 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 ãlimiting caseã of the spiral operculum of Turbo; the successive ãgnomonsã 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 mathôÙeôÙmatôÙiôÙcal notation, considerably more complicated than any that I have attempted to make use of in this book. But, in the most elementary language, we may now at least attempt to describe the general method, and some of the variations, of the mathôÙeôÙmatôÙiôÙcal development of the shell.
Let us imagine a closed curve in space, whether circular or elliptical 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 charôÙacôÙterôÙisôÙtic point within this closed curve, such as its centre of gravity. Then, starting from a fixed {525} origin, let this centre