Fig. 257. The same in a system of hexagons.
structure be built up of successive parts, similar and similarly situated, we can always trace through corresponding points a series of logarithmic spirals (Figs. [258], 259, etc.); (2) it is characteristic of the growth of the horn, of the shell, and of all other organic forms in which a logarithmic spiral can be recognised, that each successive increment of growth is a gnomon to the entire pre-existing structure. And conversely (3) it follows obviously, that in the logarithmic spiral outline of the shell or of the horn we can always inscribe an endless variety of other gnomonic figures, having no necessary relation, save as a {514} mathematical accident, to the nature or mode of development of the actual structure[504]. {515}
Fig. 258. A shell of Haliotis, with two of the many lines of growth, or generating curves, marked out in black: the areas bounded by these lines of growth being in all cases “gnomons” to the pre-existing shell.
Fig. 259. A spiral foraminifer (Pulvinulina), to show how each successive chamber continues the symmetry of, or constitutes a gnomon to, the rest of the structure.
Of these three propositions, the second is of very great use and advantage for our easy understanding and simple description of the molluscan shell, and of a great variety of other structures whose mode of growth is analogous, and whose mathematical properties are therefore identical. We see at once that the successive chambers of a spiral Nautilus (Fig. [237]) or of a straight Orthoceras (Fig. [300]), each whorl or part of a whorl of a periwinkle or other gastropod (Fig. [258]), each new increment of the operculum of a gastropod (Fig. [263]), each additional increment of
Fig. 260. Another spiral foraminifer, Cristellaria.
an elephant’s tusk, or each new chamber of a spiral foraminifer (Figs. [259] and 260), has its leading characteristic at once described and its form so far explained by the simple statement that it constitutes a gnomon to the whole previously existing structure. And herein lies the explanation of that “time-element” in the development of organic spirals of which we have spoken already, in a preliminary and empirical way. For it follows as a simple corollary to this theorem of gnomons that we must not expect to find the logarithmic spiral manifested in a structure whose parts are simultaneously produced, as for instance in the margin of a leaf, or among the many curves that make the contour of a fish. But we must rather look for it wherever the organism retains for us, and still presents to us at a single view, the successive phases of preceding growth, the successive magnitudes attained, the successive outlines occupied, as the organism or a part thereof pursued the even tenour of its growth, year by year and day by day. And it easily follows from this, that it is in the hard parts of organisms, and not the soft, fleshy, actively growing parts, that this spiral is commonly and characteristically found; not in the fresh mobile tissues whose form is constrained merely by the active forces of the moment; but in things like shell and tusk, and horn and claw, where the object is visibly composed of parts {516} successively, and permanently, laid down. In the main, the logarithmic spiral is characteristic, not of the living tissues, but of the dead. And for the same reason, it will always or nearly always be accompanied, and adorned, by a pattern formed of “lines of growth,” the lasting record of earlier and successive stages of form and magnitude.