On Growth and Form

In certain little Crustacea (of the genus Estheria) the carapace takes the form of a bivalve shell, closely simulating that of a lamellibranchiate mollusc, and bearing lines of growth in all respects analogous to or even identical with those of the latter. The explanation is very curious and interesting. In ordinary Crustacea the carapace, like the rest of the chitinised and calcified integument, is shed off in successive moults, and is restored again asa whole. But in Estheria (and one or two other small crustacea) the moult is incomplete: the old carapace is retained, and the new, growing up underneath it, adheres to it like a liming, and projects beyond its edge: so that in course of time the margins of successive old carapaces appear as “lines of growth” upon the surface of the shell. In this mode of formation, then (but not in the usual one), we obtain a structure which “is partly old and partly new,” and whose successive increments are all similar, similarly situated, and enlarged in a continued progression. We have, in short, all the conditions appropriate and necessary for the development of a logarithmic spiral; and this logarithmic spiral (though it is one of small angle) gives its own character to the structure, and causes the little carapace to partake of the characteristic conformation of the molluscan shell.

The essential simplicity, as well as the great regularity of the “curves of growth” which result in the familiar configurations of our bivalve shells, sufficiently explain, in a general way, the ease with which they may be imitated, as for instance in the so-called “artificial shells’? which Kappers has produced from the conchoidal form and lamination of lumps of melted and quickly cooled paraffin *. In the above account of the mathematical form of the bivalve shell, we have supposed, for simplicity’s sake, that the pole or origin of the system is at a point where all the successive curves touch one another. But such an arrangement is neither theoretically probable, nor is it actually the case; for it would mean that in a certain direction growth fell, not merely to a minimum, but to zero. As a matter of fact, the centre of the system (the “umbo” of the conchologists) lies not at the edge of the system, but very near to it; in other words, there is a certain amount of growth all round. But to take account of this condition would involve more troublesome mathematics, and it is obvious that the foregoing illustrations are a sufficiently near approximation to the actual case.

Among the bivalves the spiral angle (a) is very small in the flattened shells, such as Orthis, Lingula or Anomia. It is larger, as a rule, in the Lamellibranchs than in the Brachiopods, but in the latter it is of considerable magnitude among the Pentameri. Among the Lamellibranchs it is largest in such forms as Isocardia and Diceras, and in the very curious genus Caprinella; in all of these last-named genera its magnitude leads to the production of a spiral shell of several whorls, precisely as in the univalves. The angle is usually equal, but of opposite sign, in the two valves of the Lamellibranch, and usually of opposite sign but unequal in

the two valves of the Brachiopod. It is very unequal in many Ostreidae, and especially in such forms as Gryphaea, or in Caprinella, which is a kind of exaggerated Gryphaea. Occasionally it is of the same sign in both valves (that is to say, both valves curve the same way) as we see sometimes in Anomia, and much better in Productus or Strophomena. Owing to the large growth-factor of the generating curve, and the comparatively small angle of the spiral, the whole shell seldom assumes a spiral form so conspicuous as to manifest in a typical way the helical twist or shear which is so conspicuous in the

majority of univalves, or to let us measure or estimate the magnitude of the apical angle (@) of the enveloping cone. This however we can do in forms like Isocardia and Diceras; while in Caprinella we see that the whorls lie in a plane perpendicular to the axis, forming a discoidal spire. As in the latter shell, so also universally among the Brachiopods, there is no lateral asymmetry in the plane of the generating curve such as to lead to the development of a helix; but in the majority of the Lamellibranchiata it is obvious, from the obliquity of the lines of growth, that the angle @ is significant in amount.

The so-called “spiral arms” of Spirifer and many other Brachiopods are not difficult to explain. They begin as a single structure, in the form of a loop of shelly substance, attached to the dorsal valve of the shell, in the neighbourhood of the hinge. This loop has a curvature of its own, similar to but not necessarily identical with that of. the valve to which it is attached; and this curvature will tend to be developed, by continuous and symmetrical growth, into a_ fully

; ae ae formed logarithmic spiral, so far as Big. 292: Skeletal loop of . - : Terebratula. (From Woods.) it 18 permitted to do so under the constraint of the shell in which it is contained. In various Terebratulae we see the spiral growth of the loop,-more or less flattened and distorted by the restraining pressure of the ventral valve. In a number of cases the loop remains small, but gives off two nearly parallel branches or offshoots, which continue to grow. And these, starting with just such a shght curvature as the loop itself possessed, grow on and on till they may form close-wound spirals, always provided that the “spiral angle” of the curve is such that the resulting spire can be freely contained within the cavity of the shell. Owing to. the bilateral symmetry of the whole system, the case will be rare, and unlikely to occur, in which each separate arm will coil strictly mm a plane, so as to constitute a discoid spiral; for the original

direction of each of the two branches, parallel to the valve (or nearly so) and outwards from the middle line, will tend to constitute a curve of double curvature, and so, on further growth, to develop into a helicoid. This is what actually occurs, in the great majority of cases. But the curvature may be such that the helicoid grows outwards from the middle line, or inwards towards the middle line, a very slight difference in the initial curvature being sufficient to direct the spire the one way or the other; the middle course of an undeviating discoid spire will be rare, from the usual lack of any obvious controlling force to prevent its deviation. The cases in which the helicoid spires point towards, or point away from, the middle line are ascribed, in zoological classification, to particular “families” of Brachiopods, the former

Fig. 293. Spiral arms of Fig. 294. Inwardly directed Spirifer. (From Woods.) spiral arms of Atrypa. condition defining (or helping to define) the Atrypidae and the latter the Spiriferidae and Athyridae. It is obvious that the incipient curvature of the arms, and consequently the form and direction of the spirals, will be influenced by the surrounding pressures, and these in turn by the general shape of the shell. We shall expect, accordingly, to find the long outwardly directed spirals associated with shells which are transversely elongated, as Spirifer is; while the more rounded Atrypas will tend to the opposite condition. In a few cases, as in Cyrtina or Reticularia, where the shell is comparatively narrow but long, and where the uncoiled basal support of the arms is long also, the spiral coils into which the latter grow are turned backwards, in the direction where there is room for them. And in the few cases where the shell is very considerably flattened, the spirals (if they find room

to grow at all) will be constrained to do so in a discoid or nearly discoid fashion, and this is actually the case in such flattened forms as Koninckina or Thecidium. While mathematically speaking we are entitled to look upon the bivalve shell of the Lamellibranch as consisting of two distinct elements, each comparable to the entire shell of the univalve, we have no biological grounds for such a statement; for the shell arises from a single embryonic origin, and afterwards becomes split into portions which constitute the two separate valves. We can perhaps throw some indirect light upon this phenomenon, and upon several other phenomena connected with shell-growth, by a consideration of the simple conical or tubular shells of the Pteropods. The shells of the latter are in few cases suitable for simple mathematical investigation, but nevertheless they are of very considerable interest in connection with our general problem.

The morphology of the Pteropods is by no means well understood, and in speaking of them I will assume that there are still — grounds for believing (in spite of Boas’ —S and Pelseneer’s arguments) that they are The simplest shells among the Pteropods have the form of a tube, more or less cylindrical (Cuvierina), more often conical (Creseis, Clio); and this tubular shell (as we have already had occasion to remark, on p. 258), frequently tends, when it is very small and delicate, to assume the Fis. 295. Pteropod shells; Character of an unduloid. (In such a case

(1) Cuxterina columnella; it 1s more than likely that the tiny shell, (2) Cleodora chierchiaes 4» that portion of it which constitutes the (3) C. pygmaea. (After _ i Boas.) unduloid, has not grown by successive * We need not assume a close relationship, nor indeed any more than such a one as permits us to compare the shell of a Nautilus with that of a Gastropod. increments or “rings of growth,” but has developed as a whole.) A thickened “rib” is often, perhaps generally, present on the dorsal side of the little conical shell. In a few cases (Limacina,

Fig. 296. Diagrammatic transverse sections, or outlines of the mouth, in certain Pteropod shells: A, B, Cleodora australis; C, C. pyramidalis; D, C. balantium ; E, C. cuspidata. (After Boas.) Fig. 297. Shells of thecosome Pteropods (after Boas). (1) Cleodora Peraclis) the tube becomes spirally coiled, in a normal logarithmic spiral or helico-spiral. In certain cases (e.g. Cleodora, Hyalaea) the tube or cone is curiously modified. In the first place, its cross-section, originally

circular or nearly so, becomes flattened or compressed dorsoventrally ; and the angle, or rather edge, where dorsal and ventral walls meet, becomes more and more drawn out into a ridge or keel. Along the free margin, both of the dorsal and the ventral portion of the shell, growth proceeds with a regularly varying velocity, so that these margins, or lips, of the shell become regularly curved or markedly sinuous. At the same time, growth in a transverse direction proceeds with an acceleration which manifests itself in a curvature of the sides, replacing the straight borders of the original cone. In other words, the cross-section of the cone, or what we have been calling the generating curve, increases its dimensions more rapidly than its distance from the pole.

In the above figures, for instance in that of Cleodora cuspidata, the markings of the shell which represent the successive edges of the lip at former stages of growth, furnish us at once with a “graph” of the varying velocities of growth as measured, radially, from the apex. We can reveal more clearly the nature of these variations in the following way which is simply tantamount to converting our radial into rectangular coordinates. Neglecting curvature (if any) of the sides and treating the shell (for simplicity’s sake) as a right cone, we lay off equal angles from the apex O, along the radii Oa, Ob, etc. If we then plot, as vertical equidistant ordinates, the magnitudes Oa, Ob...OY, and again on to Oa’, we obtain a diagram such as the following (Fig. 299); by

help of which we not only see more clearly the way in which the growth-rate varies from point to point, but we also recognise much better than before, the similar nature of the law which governs this variation in the different species. Fig. 299. Curves obtained by transforming radial ordinates, as in Fig. 298, into vertical equidistant ordinates. 1, Hyalaea trispinosa; 2, Cleodora cuspidata. Furthermore, the young shell having become differentiated into a dorsal and a ventral part, marked off from one another by a lateral edge or keel, and the inequality of growth being such as to cause

Fig. 300. Development of the shell of Hyalaea (Cavolinia) tridentata, Forskal: the earlier stages being the “ Pleuropus longifilis”’ of Troschel. (After Tesch.) each portion to increase most rapidly in the median line, it follows that the entire shell will appear to have been split into a dorsal and a ventral plate, both connected with, and projecting from, what remains of the original undivided cone. Putting the same thing in other words, we may say that the generating figure, which lay at first in a plane perpendicular to the axis of the cone, has now, by unequal growth, been sharply bent or folded, so as to lie approximately in two planes, parallel to the anterior and posterior faces of the cone. We have only to imagine the apical connecting portion to be further reduced, and finally to disappear or rupture, and we should have a bivalve shell developed out of the original simple cone.

In its outer and growing portion, the shell of our Pteropod now consists of two parts which, though still connected together at the apex, may be treated as growing practically independently. The shell is no longer a simple tube, or simple cone, in which regular inequalities of growth will lead to the development of a spiral; and this for the simple reason that we have now two opposite maxima of growth, instead of a maximum on the one side and a minimum on the other side of our tubular shell. As a matter of fact, the dorsal and the ventral plate tend to curve in opposite directions, towards the middle line, the dorsal curving ventrally and the ventral curving towards the dorsal side.

In the case of the Lamellibranch or the Brachiopod, it is quite possible for both valves to grow into more or less pronounced spirals, for the simple reason that they are hinged upon one another ; and each growing edge, instead of being brought to a standstill by the growth of its opposite neighbour, is free to move out of the way, by the rotation about the hinge of the plane in which it lies. But where, as in the Pteropod, there is no such hinge, the dorsal and ventral halves of the shell (or dorsal and ventral valves, if we may call them so), if they curved towards one another (as they do in a cockle), would soon interfere with one another’s progress, and the development of a pair of conjugate spirals would become impossible. Nevertheless, there is obviously, in both dorsal and ventral valve, a tendency to the development of a spiral curve, that of the ventral valve being more marked than that of the larger and overlapping dorsal one, exactly as in the two unequal valves of Terebratula. In many cases (e.g. Cleodora cuspidata), the dorsal valve or plate,

strengthened and stiffened by its midrib, is nearly straight, while the curvature of the other is well displayed. But the case will be materially altered and simplified if growth be arrested or retarded in either half of the shell. Suppose for instance that the dorsal valve grew so slowly that after a while, in comparison with the other, we might speak of it as being absent altogether: or suppose that it merely became so reduced in relative size as to form no impediment to the continued growth of the ventral one; the latter would continue to grow in the direction of its natural curvature, and would end by forming a complete and coiled logarithmic spiral. It would be precisely analogous to the spiral shell of Nautilus, and, in regard to its ventral position, concave towards the dorsal side, it would even deserve to be called directly

homologous with it. Suppose, on the other hand, that the ventral valve were to be greatly reduced, and even to disappear, the dorsal valve would then pursue its unopposed growth; and, were it to be markedly curved, it would come to form a logarithmic spiral, concave towards the ventral side, as is the case in the shell of Spirula*. Were the dorsal valve to be destitute of any marked curvature (or in other words, to have but a low spiral angle), it would form a simple plate, as in the shells of Sepia or Loligo. In- deed, in the shells of these latter, and especially in that of Sepia, we seem to recognise a manifest resemblance to the dorsal plate of the Pteropod shell, as we have it (e.g.) in Cleodora or Hyalaea;

* Cf. Owen, “These shells [Nautilus and Ammonites] are revolutely spiral or the little “rostrum” of Sepia is but the apex of the primitive cone, and the rounded anterior extremity has grown according to a law precisely such as that which has produced the curved margin of the dorsal valve in the Pteropod.- The ventral portion of the ~ original cone is nearly, but not wholly, wanting. It is represented by the so-called posterior wall of the “siphuncular space.” In many decapod cuttle-fishes also (e.g. Todarodes, Illex, etc.) we still see at the posterior end of the “pen,” a vestige of the primitive cone, whose dorsal margin only has continued to grow; and the same phenomenon, on an exaggerated scale, is represented in the Belemnites.

It is not at all impossible that we may explain on the same lines the development of the curious “operculum” of the Ammonites. This consists of a single horny plate (Anaptychus), or of a thicker, more calcified plate divided into two symmetrical halves (Aptychi), often found inside the terminal chamber of the Ammonite,-and occasionally to be seen lying in situ, as an operculum which partially closes the mouth of the shell; this structure is known to exist even in connection with the early embryonic shell. In form the Anaptychus, or the pair of conjomed Aptychi, shew an upper and a lower border, the latter strongly convex, the former sometimes slightly concave, sometimes slightly convex, and usually shewing a median projection or slightly developed rostrum. From this “rostral” border the curves of growth start, and course round parallel to, finally constituting, the convex border. It is this convex border which fits into the free margin of the mouth of the Ammonite’s shell, while the other is applied to and overlaps the preceding whorl of the spire. Now this relationship is precisely what we should expect, were we to imagine as our starting-point a shell similar to that of Hyalaea, in which however the dorsal part of the split cone had become separate from the ventral half, had remained flat, and had grown comparatively slowly, while at the same time it kept slipping forward over the growing and coiling spire into which the ventral half of the original shell develops*. In short, I think there is reason to believe, or at least to suspect, that we

* The case of Terebratula or of Gryphaea would be closely analogous, if the smaller valve were less closely connected and co-articulated with the larger. have in the shell and Aptychus of the Ammonites, two portions of a once united structure; of which other Cephalopods retain not both parts but only one or other, one as the ventrally situated shell of Nautilus, the other as the dorsally placed shell for example of Sepia or of Spirula. In the case of the bivalve shells of the Lamellibranchs or of the Brachiopods, we have to deal with a phenomenon precisely analogous to the split and flattened cone of our Pteropods, save only that the primitive cone has been split into two portions, not incompletely as in the Pteropod (Hyalaea), but completely, so as to form two separate valves. Though somewhat greater freedom is given to growth now that the two valves are separate and hinged, yet still the two valves oppose and hamper one another, so that in the longitudinal direction each is capable of only a moderate curvature. This curvature, as we have seen, is recognisable as a logarithmic spiral, but only now and then does the growth of the spiral continue so far as to develop successive coils: as it does in a few symmetrical forms such as [socardia cor ; and as it does still more conspicuously in a few others, such as Gryphaea and Caprinella, where one of the two valves is stunted, and the growth of the other is (relatively speaking) unopposed.

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 chambets, in the Nautilus, the Ammonite and their allies (Fig. 304, 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

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 shellst 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 cleart. The septum begins as a very thin cuticular membrane (composed apparently of a substance called conchyolin), which is secreted by the skin, or mantlesurface, 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 * “Tt has been suggested, and I think in some quarters adopted as a dogma, that the formation of successive septa [in Nautilus] is correlated with the recurrence of reproductive periods. This is not the case, since, according to my observations, propagation only takes place after the last septum is formed;” Willey, Zoological Results, p. 746, 1902.

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 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, 302, 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 w-—20. 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, as in Fig. 303; and here, while the septa still remain portions of spheres, the geometrical construction for the position of their centres is equally easy.