But here comes in a very beautiful illustration of that property of the logarithmic spiral which causes its whole shape to remain unchanged, in spite of its apparently unsymmetrical, or unilateral, mode of growth. For the mouth of the tubular shell, into which the operculum has to fit, is growing or widening on all sides: while the operculum is increasing, not by additions made at the same time all round its margin, but by additions made only on one side of it at each successive stage. One edge of the operculum thus remains unaltered as it is advanced into each new position, and as it is placed in a newly formed section of the tube, similar to but greater than the last. Nevertheless, the two apposed structures, the chamber and its plug, at all times fit one another to perfection. The mechanical problem (by no means an easy one), is thus solved: “How to shape a tube of a variable section, so that a piston driven along it shall, by one side of its margin, coincide continually with its surface as it advances, provided only that the piston be made at the same time continually to revolve in its own plane.”
As Moseley puts it: “That the same edge which fitted a portion of the first less section should be capable of adjustment, so as to fit a portion of the next similar but greater section, supposes a geometrical provision in the curved form of the chamber of great apparent complication and difficulty. But God hath bestowed upon this humble architect the practical skill of a learned geometrician, and he makes this provision with admirable precision in that curvature of the logarithmic spiral which he gives to the section of the shell. This curvature obtaining, he has only to turn his operculum slightly round in its own plane as he advances it into each newly formed portion of his chamber, to adapt one margin of it to a new and larger surface and a different curvature, leaving the space to be filled up by increasing the operculum wholly on the other margin.”
But in many, and indeed more numerous Gastropod mollusca, the operculum does not grow in this remarkable spiral fashion, but by the apparently much simpler process of accretion by concentric rings. This suggests to us another mathematical {524} feature of the logarithmic spiral. We have already seen that the logarithmic spiral has a number of “limiting cases,” apparently very diverse from one another. Thus the right cone is a logarithmic spiral in which the revolution of the radius vector is infinitely slow; and, in the same sense, the straight line 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 elementary 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 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 characteristic point within this closed curve, such as its centre of gravity. Then, starting from a fixed {525} origin, let this centre of gravity describe an equiangular spiral in space, about a fixed axis (namely the axis of the shell), while at the same time the generating curve grows, with each angular increment of rotation, in such a way as to preserve the symmetry of the entire figure, with or without a simultaneous movement of translation along the axis.
Fig. 265. Melo ethiopicus, L.
It is plain that the entire resulting shell may now be looked upon in either of two ways. It is, on the one hand, an ensemble of similar closed curves spirally arranged in space, gradually increasing in dimensions, in proportion to the increase of their vectorial angle from the pole. In other words, we can imagine our shell cut up into a system of rings, following one another in continuous spiral succession from that terminal and largest one, which constitutes the lip of the orifice of the shell. Or, on the other hand, we may figure to ourselves the whole shell as made up of an ensemble of spiral lines in space, each spiral having been {526} traced out by the gradual growth and revolution of a radius vector from the pole to a given point of the generating curve.