Fig. 238. A Foraminiferal shell (Globigerina).
Again, there is a manifest and not unimportant difference between such a spiral conformation as is built up by the separate and successive florets in the sunflower, and that which, in the snail or Nautilus shell, is apparently a single and indivisible unit. And a similar, if not identical difference is apparent between the Nautilus shell and the minute shells of the Foraminifera, which so closely simulate it; inasmuch as the spiral shells of these latter are essentially composite structures, combined out of successive and separate chambers, while the molluscan shell, though it may (as in Nautilus) become secondarily subdivided, has grown as one continuous tube. It follows from all this that there cannot {496} possibly be a physical or dynamical, though there may well be a mathematical Law of Growth, which is common to, and which defines, the spiral form in the Nautilus, in the Globigerina, in the ram’s horn, and in the disc of the sunflower.
Of the spiral forms which we have now mentioned, every one (with the single exception of the outline of the cordate leaf) is an example of the remarkable curve known as the Logarithmic Spiral. But before we enter upon the mathematics of the logarithmic spiral, let us carefully observe that the whole of the organic forms in which it is clearly and permanently exhibited, however different they may be from one another in outward appearance, in nature and in origin, nevertheless all belong, in a certain sense, to one particular class of conformations. In the great majority of cases, when we consider an organism in part or whole, when we look (for instance) at our own hand or foot, or contemplate an insect or a worm, we have no reason (or very little) to consider one part of the existing structure as older than another; through and through, the newer particles have been merged and commingled, by intussusception, among the old; the whole outline, such as it is, is due to forces which for the most part are still at work to shape it, and which in shaping it have shaped it as a whole. But the horn, or the snail-shell, is curiously different; for in each of these, the presently existing structure is, so to speak, partly old and partly new; it has been conformed by successive and continuous increments; and each successive stage of growth, starting from the origin, remains as an integral and unchanging portion of the still growing structure, and so continues to represent what at some earlier epoch constituted for the time being the structure in its entirety.
In a slightly different, but closely cognate way, the same is true of the spirally arranged florets of the sunflower. For here again we are regarding serially arranged portions of a composite structure, which portions, similar to one another in form, differ in age; and they differ also in magnitude in a strict ratio according to their age. Somehow or other, in the logarithmic spiral the time-element always enters in; and to this important fact, full of curious biological as well as mathematical significance, we shall afterwards return. {497}
It is, as we have so often seen, an essential part of our whole problem, to try to understand what distribution of forces is capable of producing this or that organic form,—to give, in short, a dynamical expression to our descriptive morphology. Now the general distribution of forces which lead to the formation of a spiral (whether logarithmic or other) is very easily understood; and need not carry us beyond the use of very elementary mathematics.
Fig. 239.
If we imagine growth to act in a perpendicular direction, as for example the upward force of growth in a growing stem (OA), then, in the absence of other forces, elongation will as a matter of course proceed in an unchanging direction, that is to say the stem will grow straight upwards. Suppose now that there be some constant external force, such as the wind, impinging on the growing stem; and suppose (for simplicity’s sake) that this external force be in a constant direction (AB) perpendicular to the intrinsic force of growth. The direction of actual growth will be in the line of the resultant of the two forces: and, since the external force is (by hypothesis) constant in direction, while the internal force tends always to act in the line of actual growth, it is obvious that our growing organism will tend to be bent into a curve, to which, for the time being, {498} the actual force of growth will be acting at a tangent. So long as the two forces continue to act, the curve will approach, but will never attain, the direction of AB, perpendicular to the original direction OA. If the external force be constant in amount the curve will approximate to the form of a hyperbola; and, at any rate, it is obvious that it will never tend to assume a spiral form.
In like manner, if we consider a horizontal beam, fixed at one end, the imposition of a weight at the other will bend the beam into a curve, which, as the beam elongates or the weight increases, will bring the weighted end nearer and nearer to the vertical. But such a force, constant in direction, will obviously never curve the beam into a spiral,—a fact so patent and obvious that it would be superfluous to state it, were it not that some naturalists have been in the habit of invoking gravity as the force to which may be attributed the spiral flexure of the shell.