Fig. 234. The same, diagrammatic.
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Within any polyhedron we may always inscribe another polyhedron, whose corners correspond in number to the sides or facets of the original figure, or (in alternative cases) to a certain number of these sides; and a similar result is obtained by bevelling off the corners of the original polyhedron. We may obtain a precisely similar symmetrical result if (in such a case as these Radiolarians which we are describing), we imagine the radial spines to be interconnected by tangential rods, instead of by the complete facets which we have just been dealing with. In our complicated polyhedron with its twenty radial spines arranged in the manner described there are various symmetrical ways in which we may imagine these interconnecting bars to be arranged. The most symmetrical of these is one in which the whole surface is divided into eighteen rhomboidal areas, obtained by systematically connecting each group of four adjacent radii. This figure has eighteen faces (F), twenty corners (C), and therefore thirty-six edges (E), in conformity with Euler’s theorem, F + C = E + 2.
Fig. 235. Phractaspis prototypus, Hkl.
Another symmetrical arrangement will divide the surface into fourteen rhombs and eight triangles. This latter arrangement is obtained by linking up the radial rods as follows: aaaa, aba, abcb, bcdc, etc. Here we have again twenty corners, but we have twenty-two faces; the number of edges, or tangential spicular bars, will be found, therefore, by the above formula, to be forty. In Haeckel’s figure of Phractaspis prototypus we have a spicular skeleton which appears to be constructed precisely upon this plan, and to be derivable from the faceted polyhedron precisely after this manner.
In all these latter cases it is the arrangement of the axial rods, or in other words the “polar symmetry” of the entire organism, which lies at the root of the matter, and which, if only {485} we could account for it, would make it comparatively easy to explain the superficial configuration. But there are no obvious mechanical forces by which we can so explain this peculiar polarity. This at least is evident, that it arises in the central mass of protoplasm, which is the essential living portion of the organism as distinguished from that frothy peripheral mass whose structure has helped us to explain so many phenomena of the superficial or external skeleton. To say that the arrangement depends upon a specific polarisation of the cell is merely to refer the problem to other terms, and to set it aside for future solution. But it is possible that we may learn something about the lines in which to seek for such a solution by considering the case of Lehmann’s “fluid crystals,” and the light which they throw upon the phenomena of molecular aggregation.
The phenomenon of “fluid crystallisation” is found in a number of chemical bodies; it is exhibited at a specific temperature for each substance; and it would seem to be limited to bodies in which there is a more or less elongated, or “chain-like” arrangement of the atoms in the molecule. Such bodies, at the appropriate temperature, tend to aggregate themselves into masses, which are sometimes spherical drops or globules (the so-called “spherulites”), and sometimes have the definite form of needle-like or prismatic crystals. In either case they remain liquid, and are also doubly refractive, polarising light in brilliant colours. Together with them are formed ordinary solid crystals, also with characteristic polarisation, and into such solid crystals all the fluid material ultimately turns. It is evident that in these liquid crystals, though the molecules are freely mobile, just as are those of water, they are yet subject to, or endowed with, a “directive force,” a force which confers upon them a definite configuration or “polarity,” the Gestaltungskraft of Lehmann.
Such an hypothesis as this had been gradually extruded from the theories of mathematical crystallography[491]; and it had come to be believed that the symmetrical conformation of a homogeneous crystalline structure was sufficiently explained by the mere mechanical fitting together of appropriate structural units along the easiest and simplest lines of “close packing”: just as {486} a pile of oranges becomes definite, both in outward form and inward structural arrangement, without the play of any specific directive force. But while our conceptions of the tactical arrangement of crystalline molecules remain the same as before, and our hypotheses of “modes of packing” or of “space-lattices” remain as useful as ever for the definition and explanation of the molecular arrangements, an entirely new theoretical conception is introduced when we find such space-lattices maintained in what has hitherto been considered the molecular freedom of a liquid field; and we are constrained, accordingly, to postulate a specific molecular force, or “Gestaltungskraft” (not unlike Kepler’s “facultas formatrix”), to account for the phenomenon.
Now just as some sort of specific “Gestaltungskraft” had been of old the deus ex machina accounting for all crystalline phenomena (gnara totius geometriæ, et in ea exercita, as Kepler said), and as such an hypothesis, after being dethroned and repudiated, has now fought its way back and has made good its right to be heard, so it may be also in biology. We begin by an easy and general assumption of specific properties, by which each organism assumes its own specific form; we learn later (as it is the purpose of this book to shew) that throughout the whole range of organic morphology there are innumerable phenomena of form which are not peculiar to living things, but which are more or less simple manifestations of ordinary physical law. But every now and then we come to certain deep-seated signs of protoplasmic symmetry or polarisation, which seem to lie beyond the reach of the ordinary physical forces. It by no means follows that the forces in question are not essentially physical forces, more obscure and less familiar to us than the rest; and this would seem to be the crucial lesson for us to draw from Lehmann’s surprising and most beautiful discovery. For Lehmann seems actually to have demonstrated, in non-living, chemical bodies, the existence of just such a determinant, just such a “Gestaltungskraft,” as would be of infinite help to us if we might postulate it for the explanation (for instance) of our Radiolarian’s axial symmetry. But further than this we cannot go; for such analogy as we seem to see in the Lehmann phenomenon soon evades us, and refuses to be pressed home. Not only is it the case, as we have already {487} seen, that certain of the geometric forms assumed by the symmetrical Radiolarian shells are just such as the “space-lattice” theory would seem to be inapplicable to, but it is in other ways obvious that symmetry of crystallisation, whether liquid or solid, has no close parallel, but only a series of analogies, in the protoplasmic symmetry of the living cell.