← On Growth and Form

Of Septa

Before we leave the subject of the molluscan shell, we have still another problem to deal with, in regard to the form and arrangement of the septa which divide up the tubular shell into chambers, in the Nautilus, the Ammonite and their allies (Fig. , etc.).

The existence of septa in a Nautiloid shell may probably be accounted for as follows. We have seen that it is a property of a cone that, while growing by increments at one end only, it conserves its original shape: therefore the animal within, which (though growing by a different law) also conserves its shape, will continue to fill the shell if it actually fills it to begin with: as does a snail or other Gastropod. But suppose that our mollusc fills a part only of a conical shell (as it does in the case of Nautilus); then, unless it alter its shape, it must move upward as it grows in the growing cone, until it come to occupy a space similar in form {578} to that which it occupied before: just, indeed, as a little ball drops far down into the cone, but a big one must stay farther up. Then, when the animal after a period of growth has moved farther up in the shell, the mantle-surface continues its normal secretory activity, and that portion which had been in contact with the former septum secretes a septum anew. In short, at any given epoch, the creature is not secreting a tube and a septum by separate operations, but is secreting a shelly case about its rounded body, of which case one part appears to us as the continuation of the tube, and the other part, merging with it by in­dis­tin­guish­able boundaries, appears to us as the septum.

The various forms assumed by the septa in spiral shells present us with a number of problems of great beauty, simple in their essence, but whose full in­ves­ti­ga­tion would soon lead us into mathematics of a very high order.

We do not know in great detail how these septa are laid down; but the essential facts are clear. The septum begins as a very thin cuticular membrane (composed apparently of a substance called conchyolin), which is secreted by the skin, or mantle-surface, of the animal; and upon this membrane nacreous matter is gradually laid down on the mantle-side (that is to say between the animal’s body and the cuticular membrane which has been thrown off from it), so that the membrane remains as a thin pellicle over the hinder surface of the septum, and so that, to begin with, the membranous septum is moulded on the flexible and elastic surface of the animal, within which the fluids of the body must exercise a uniform, or nearly uniform pressure.

Let us think, then, of the septa as they would appear in their uncalcified condition, formed of, or at least superposed upon, an {579} elastic membrane. They must then follow the general law, applicable to all elastic membranes under uniform pressure, that the tension varies inversely as the radius of curvature; and we come back once more to our old equation of Laplace, that

P = T(1 ⁄ r + 1 ⁄ r ′).

Fig. 302.

Moreover, since the cavity below the septum is practically closed, and is filled either with air or with water, P will be constant over the whole area of the septum. And further, we must assume, at least to begin with, that the membrane constituting the incipient septum is homogeneous or isotropic.

Let us take first the case of a straight cone, of circular section, more or less like an Orthoceras; and let us suppose that the septum is attached to the shell in a plane perpendicular to its axis. The septum itself must then obviously be spherical. Moreover the extent of the spherical surface is constant, and easily determined. For obviously, in Fig. , the angle LCL ′ equals the supplement of the angle (LOL ′) of the cone; that is to say, the circle of contact subtends an angle at the centre of the spherical surface, which is constant, and which is equal to π − 2θ. The case is not excluded where, owing to an asymmetry of tensions, the septum meets the side walls of the cone at other than a right angle,

Fig. 303.

as in Fig. ; and here, while the septa still remain portions of spheres, the geometrical construction for the position of their centres is equally easy.

If, on the other hand, the attachment of the septum to the inner walls of the cone be in a plane oblique to the axis, then it is evident that the outline of the septum will be an ellipse, and its surface an {580} ellipsoid. If the attachment of the septum be not in one plane, but form a sinuous line of contact with the cone, then the septum will be a saddle-shaped surface, of great complexity and beauty. In all cases, provided only that the membrane be isotropic, the form assumed will be precisely that of a soap-bubble under similar conditions of attachment: that is to say, it will be (with the usual limitations or conditions) a surface of minimal area.

If our cone be no longer straight, but curved, then the septa will be symmetrically deformed in consequence. A beautiful and interesting case is afforded us by Nautilus itself. Here the outline of the septum, referred to a plane, is ap­prox­i­mate­ly bounded by two elliptic curves, similar and similarly situated, whose areas are to one another in a definite ratio, namely as

A 1 ⁄ A 2 = (r 1 r ′ 1) ⁄ (r 2 r ′ 2) = ε −4π cot α ,

and a similar ratio exists in Ammonites and all other close-whorled spirals, in which however we cannot always make the simple assumption of elliptical form. In a median section of Nautilus, we see each septum forming a tangent to the inner and to the outer wall, just as it did in a section of the straight Orthoceras; but the curvatures in the neighbourhood of these two points of contact are not identical, for they now vary inversely as the radii, drawn from the pole of the spiral shell. The contour of the septum in this median plane is a spiral curve identical with the original logarithmic spiral. Of this it is the “invert,” and the fact that the original curve and its invert are both identical is one of the most beautiful properties of the logarithmic spiral .

But while the outline of the septum in median section is simple and easy to determine, the curved surface of the septum in its entirety is a very complicated matter, even in Nautilus which is one of the simplest of actual cases. For, in the first place, since the form of the septum, as seen in median section, is that of a logarithmic spiral, and as therefore its curvature is constantly altering, it follows that, in successive transverse sections, the {581} curvature is also constantly altering. But in the case of Nautilus, there are other aspects of the phenomenon, which we can illustrate, but only in part, in the following simple manner. Let us imagine

Fig. 304. Section of Nautilus, shewing the contour of the septa in the median plane: the septa being (in this plane) logarithmic spirals, of which the shell-spiral is the evolute.

a pack of cards, in which we have cut out of each card a similar concave arc of a logarithmic spiral, such as we actually see in the median section of the septum of a Nautilus. Then, while we hold the cards together, foursquare, in the ordinary position of the {582} pack, we have a simple “ruled” surface, which in any longitudinal section has the form of a logarithmic spiral but in any transverse section is a straight horizontal line. If we shear or slide the cards upon one another, thrusting the middle cards of the pack forward in advance of the others, till the one end of the pack is a convex, and the other a concave, ellipse, the cut edges which combine to represent our septum will now form a curved surface

Fig. 305. Cast of the interior of Nautilus: to shew the contours of the septa at their junction with the shell-wall.

of much greater complexity; and this is part, but not by any means all, of the deformation produced as a direct consequence of the form in Nautilus of the section of the tube within which the septum has to lie. And the complex curvature of the surface will be manifested in a sinuous outline of the edge, or line of attachment of the septum to the tube, and will vary according to the configuration of the latter. In the case of Nautilus, it is easy to shew empirically (though not perhaps easy to demonstrate {583} math­e­mat­i­cally) that the sinuous or saddle-shaped form of the “suture” (or line of attachment of the septum to the tube) is such as can be precisely accounted for in this manner. It is also easy to see that, when the section of the tube (or “generating curve”) is more complicated in form, when it is flattened, grooved, or otherwise ornamented, the curvature of the septum and the outline of its sutural attachment will become very complicated indeed; but it will be comparatively simple in the case of the first few sutures of the young shell, laid down before any overlapping of whorls has taken place, and this comparative simplicity of the first-formed sutures is a marked feature among Ammonites.

We have other sources of complication, besides those which are at once introduced by the sectional form of the tube. For instance, the siphuncle, or little inner tube which perforates the septa, exercises a certain amount of tension, sometimes evidently considerable, upon the latter; so that we can no longer consider each septum as an isotropic surface, under uniform pressure; and there may be other structural modifications, or inequalities, in that portion of the animal’s body with which the septum is in contact, and by which it is conformed. It is hardly likely, for all these reasons, that we shall ever attain to a full and particular explanation of the septal surfaces and their sutural outlines throughout the whole range of Cephalopod shells; but in general terms, the problem is probably not beyond the reach of math­e­mat­i­cal analysis. The problem might be approached experimentally, after the manner of Plateau’s experiments, by bending {584} a wire into the complicated form of the suture-line, and studying the form of the liquid film which constitutes the cor­re­spon­ding surface minimae areae.

Fig. 306. Ammonites (Sonninia) Sowerbyi. (From Zittel, after Steinmann and Döderlein.)

In certain Ammonites the septal outline is further complicated in another way. Superposed upon the usual sinuous outline, with its “lobes” and “saddles,” we have here a minutely ramified, or arborescent outline,

Fig. 307. Suture-line of a Triassic Ammonite (Pinacoceras). (From Zittel, after Hauer.)

in which all the branches terminate in wavy, more or less circular arcs,—looking just like the ‘landscape marble’ from the Bristol Rhaetic. We have no difficulty in recognising in this a surface-tension phenomenon. The figures are precisely such as we can imitate (for instance) by pouring a {585} few drops of milk upon a greasy plate, or of oil upon an alkaline solution.

We have very far from exhausted, we have perhaps little more than begun, the study of the logarithmic spiral and the associated curves which find exemplification in the multitudinous diversities of molluscan shells. But, with a closing word or two, we must now bring this chapter to an end.

In the spiral shell we have a problem, or a phenomenon, of growth, immensely simplified by the fact that each successive increment is irrevocably fixed in regard to magnitude and position, instead, of remaining in a state of flux and sharing in the further changes which the organism undergoes. In such a structure, then, we have certain primary phenomena of growth manifested in their original simplicity, undisturbed by secondary and conflicting phenomena. What actually grows is merely the lip of an orifice, where there is produced a ring of solid material, whose form we have treated of under the name of the generating curve; and this generating curve grows in magnitude without alteration of its form. Besides its increase in areal magnitude, the growing curve has certain strictly limited degrees of freedom, which define its motions in space: that is to say, it has a vector motion at right angles to the axis of the shell; and it has a sliding motion along that axis. And, though we may know nothing whatsoever about the actual velocities of any of these motions, we do know that they are so correlated together that their relative velocities remain constant, and accordingly the form and symmetry of the whole system remain in general unchanged.

But there is a vast range of possibilities in regard to every one of these factors: the generating curve may be of various forms, and even when of simple form, such as an ellipse, its axes may be set at various angles to the system; the plane also in which it lies may vary, almost indefinitely, in its angle relatively to that of any plane of reference in the system; and in the several velocities of growth, of rotation and of translation, and therefore in the ratios between all these, we have again a vast range of possibilities. We have then a certain definite type, or group of forms, math­e­mat­i­cally isomorphous, but presenting infinite diversities of outward appearance: which diversities, as Swammerdam {586} said, ex sola nascuntur diversitate gyrationum; and which accordingly are seen to have their origin in differences of rate, or of magnitude, and so to be, essentially, neither more nor less than differences of degree.

In nature, we find these forms presenting themselves with but little relation to the character of the creature by which they are produced. Spiral forms of certain particular kinds are common to Gastropods and to Cephalopods, and to diverse families of each; while outside the class of molluscs altogether, among the Foraminifera and among the worms (as in Spirorbis, Spirographis, and in the Dentalium-like shell of Ditrupa), we again meet with similar and cor­re­spon­ding forms.

Again, we find the same forms, or forms which (save for external ornament) are math­e­mat­i­cally identical, repeating themselves in all periods of the world’s geological history; and, irrespective of climate or local conditions, we see them mixed up, one with another, in the depths and on the shores of every sea. It is hard indeed (to my mind) to see where Natural Selection necessarily enters in, or to admit that it has had any share whatsoever in the production of these varied conformations. Unless indeed we use the term Natural Selection in a sense so wide as to deprive it of any purely biological significance; and so recognise as a sort of natural selection whatsoever nexus of causes suffices to differentiate between the likely and the unlikely, the scarce and the frequent, the easy and the hard: and leads accordingly, under the peculiar conditions, limitations and restraints which we call “ordinary circumstances,” one type of crystal, one form of cloud, one chemical compound, to be of frequent occurrence and another to be rare.