On Growth and Form

Precisely the same series of unduloid forms may be traced in even greater variety among various other families or genera of the Fig. 77. Various species of Tintinnus, Dinobryon and Codonella. (After Saville Kent and others.) Infusoria. Sometimes, as in Vorticella itself, the unduloid is seen merely in the contour of the soft semifluid body of the living animal. At other times, as in Salpingoeca, Tintinnus, and many other genera, we have a distinct. _ membranous cup, separate from the animal, but originally secreted by, and moulded upon, its semifluid living surface. Here we have an excellent illustration of the contrast between the different ways in which such a structure may be regarded and interpreted. The teleological explanation is that it is developed for the sake of protection, as a domicile and shelter for the little organism within. The mechanical explanation of the physicist (seeking only after the “efficient,’”’ and not the “final” cause), is that it is

present, and has its actual conformation, by reason of certain chemico-physical conditions: that it was inevitable, under the given conditions, that certain constituent substances actually present in the proto- | plasm should be aggregated by molecular forces in its surface layer; that under this adsorptive process, the conditions continuing favourable, the particles should _ , accumulate and concentrate till they \ formed (with the help of the surrounding ‘ medium) a pellicle or membrane, thicker or thinner as the case might be; that this surface pellicle or membrane was inevitably bound, by molecular forces, to become a surface of the least possible area which the circumstances permitted; that in the present case, the symmetry and “freedom” of the system permitted, and ipso facto caused, this surface to be a surface of revolution; and that of the few surfaces of revolution which, as being also surfaces minimae areae, were available, the unduloid was manifestly the one permitted, and ipso facto caused, by the dimensions of the organisms and other circumstances of the case. And just as the thickness or thinness of the pellicle was obviously a subordinate matter, a mere matter of degree, so we also see that the actual outline of this or that particular unduloid is also a very subordinate matter, such as physico-chemical variants of a minute kind would suffice to bring about; for between the various unduloids which the various species of Vorticella represent, there is no more real difference than that difference li of ratio or degree which exists between Fig. 80. Trachelophyllum. two circles of different diameter, or two (After Wreszniowski.) lines of unequal length.

In very many cases (of which Fig. 80 is an example), we have an unduloid form exhibited, not by a surrounding pellicle or shell, but by the soft, protoplasmic body of a ciliated organism. In such cases the form is mobile, and continually changes from one to another unduloid contour, according to the movements of the animal. We have here, apparently, to deal with an unstable equilibrium, and also sometimes with the more complicated problem of “stream-lines,” as in the difficult problems suggested by the form of a fish. But this whole class of cases, and of problems, we can merely take note of in passing, for their treatment is too hard for us.

In considering such series of forms as the various unduloids which we have just been regarding, we are brought sharply - up (as in the case of our Bacteria or Micrococci) against the biological concept of organic species. In the intense classificatory activity of the last hundred years, it has come about that every form which is apparently characteristic, that is to say which is capable of being described or portrayed, and capable of being recognised when met with again, has been recorded as a species,— for we need not concern ourselves with the occasional discussions, or individual opinions, as to whether such and such a form deserve “specific rank,” or be “only a variety.” And this secular labour is pursued in direct obedience to the precept of the Systema Naturae,—“ut sic in summa confusione rerum apparenti, summus conspiciatur Natwrae ordo.”’ In lke manner the physicist records, and is entitled to record, his many hundred “species” of snow- ‘ erystals*, or of crystals of calcium carbonate. But regarding these latter species, the physicist makes no assumptions: he records them s¢mpliciter, as specific “forms”; he notes, as best he can, the circumstances, (such as temperature or humidity) under which they occur, in the hope of elucidating the conditions _ determining their formation; but above all, he does not introduce |

* The case of the snow-crystals is a particularly interesting one; for their . “distribution” is in some ways analogous to what we find, for instance, among our microscopic skeletons of Radiolarians. That is to say, we may one day meet with myriads of some one particular form or species only, and another day with myriads of another; while at another time and place we may find species intermingled in inexhaustible variety. (Cf. e.g. J. Glaisher, [1]. London News, Feb. 17, 1855; Q.J.M.S. m1, pp. 179-185, 1855).

the element of time, and of succession, or discuss their origin and affiliation as an historical sequence of events. But in biology, the term species carries with it many large, though often vague assumptions. Though the doctrine or concept of the “ permanence " of species” is dead and gone, yet a certain definite value, or sort of quasi-permanency, is still connoted by the term. Thus if a tiny foraminiferal shell, a Lagena for instance, be found living to-day, and a shell indistinguishable from it to the eye be found fossil in the Chalk or some other remote geological formation, the assumption is deemed legitimate that that species has “survived,” and has handed down its minute specific character or characters, from generation to generation, unchanged for untold myriads of years*. Or if the ancient forms be like to, rather than identical with the recent, we still assume an unbroken descent, accompanied by the hereditary transmission of common characters and progressive variations. And if two identical forms be discovered at the ends of the earth, still (with occasional shght reservations on the score of possible “homoplasy”’), we build hypotheses on this fact of identity, taking it for granted that the two appertain to a common stock, whose dispersal in space must somehow be accounted for, its route traced, its epoch determined, and its causes discussed or discovered. In short, the naturalist admits no exception to the rule that a “natural classification” can only be a genealogical one, nor ever doubts that “The fact that we are able to classify organisms at all in accordance with the structural characteristics which they present, 1s due to the fact of the being related by descent}. But this great generalisation is apt in my opinion, to carry us too far.

It may be safe and sure and helpful and illuminating when we apply it to such complex entities,— such thousand-fold resultants of the combination and permutation of many variable characters,—as a horse, a lon or an eagle; but (to my mind) it has a very different look, and a far less firm foundation, when we attempt to extend it to minute organisms whose specific characters are few and simple, whose simplicity * Cf. Bergson, Creative Evolution, p. 107: “Certain Foraminifera have not varied since the Silurian epoch. Unmoved witnesses of the innumerable revolutions that have upheaved our planet, the Lingulae are today what they were at the remotest times of the palaeozoic era.”

becomes much more manifest when we regard it from the point of view of physical and mathematical description and analysis, and whose form is referable, or (to. say the least of it) is very largely referable, to the direct and immediate action of a particular physical force. When we come to deal with the minute skeletons — of the Radiolaria we shall again find ourselves dealing with endless modifications of form, in which it becomes still more difficult to discern, or to apply, the guiding principle of affiliation or genealogy.

Among the more aberrant forms of Infusoria is a little species known as Trichodina pediculus, a parasite on the Hydra, or freshwater polype(Fig.81.) This Trichodina has the form of a more or less flattened circular disc, with a ring of cilia around both its upper and lower margins. The salient ridge \ Pe | be taken, as we have already said, JE aan to play the part of a strengthening po Cor ae “fillet.” The circular base of the WEES animal is flattened, in contact with the flattened surface of the Hydra over which it creeps, and the opposite, upper surface may be flattened nearly to a plane, or may at other times appear slightly convex or slightly concave. The sides of the little organism are contracted, forming a symmetrical equatorial groove between the upper and lower discs; and, on account of the minute size of the animal and its constant movements, we cannot submit the curvature of this concavity to measurement, nor recognise by the eye its exact contour. But it is evident that the conditions are precisely similar to those described on p. 223, where we were considering the conditions of stability of the catenoid. And it is further evident that, when the upper disc is actually plane, the equatorial groove is strictly a catenoid surface of revolution; and when on the other hand it is depressed, then the equatorial groove will tend to assume the form of a nodoidal surface. Another curious type is the flattened spiral of Dinenympha*

which reminds us of the cylindrical spiral of a Spirillum among the bacteria. In Dinenympha we have a symmetrical figure, whose two opposite surfaces each constitute a surface of constant mean curvature; it is evidently a figure of equilibrium under certain special conditions of restraint. The cylindrical coil of the Spirillum, on the other hand, is a surface of constant mean curvature, and therefore of equilibrium, as truly, and in the same sense, as the cylinder itself.

A very curious conformation is that of the vibratile “collar,” found in Codosiga and the other ‘‘Choanoflagellates,’” and which we also meet with in the “collar-cells” which line the interior cavities of a sponge. Such collar-cells are always very minute, and the collar is constituted of a very delicate film, which shews an undulatory or rippling motion. It is a surface of revolution, and as it maintains itself in equilibrium (though a somewhat unstable and fluctuating one), it must be, under the restricted circumstances of its case, a surface of minimal area. But it is not so easy to see what these special circumstances are ; and it is obvious that the collar, if left to itself, must at once

contract downwards towards its base, and become confluent with the general surface of the cell; for it has no longitudinal supports and no strengthening ring at its periphery. But in all these collar-cells, there stands within the annulus of the collar a large and powerful cilium or flagellum, in constant movement; and by the action of this flagellum, and doubtless in part also by the intrinsic vibrations of the collar itself, there is set up a constant steady current in the surrounding water, whose direction would seem to be such that it passes up the outside of the collar, down its inner side, and out in the middle in the direction of the flagellum; and there is a gy. 83. distinct eddy, in which foreign particles tend to be caught, around the peripheral margin of the collar. When the cell dies, that is to say when motion ceases, the collar immediately shrivels away and disappears. It is notable, by the way, that the edge of this little mobile cup is always smooth, never notched or lobed as in the cases we have discussed on p. 236: this latter condition being the outcome of a definite instability, marking the close of a period of equilibrium; while in the vibratile collar of Codosiga the equilibrium, such as it is, is being constantly: renewed and perpetuated like that of a juggler’s pole, by the motions of the system. I take it that, somehow, its existence (in a state of partial equilibrium) is due to the current motions, and to the traction exerted upon it through the friction of the stream which is constantly passing by. I think, in short, that it is formed very much in the same way as the cup-like ring of streaming ribbons, which we see fluttering and vibrating in the air-current of a ventilating fan.

It is likely enough, however, that a different and much better explanation may yet be found: and if we turn once more to Mr Worthington’s Study of Splashes, we may find a curious suggestion of analogy in the beautiful craters encircling a central jet (as the collar of Codosiga encircles the flagellum), which we see produced in the later stages of the splash of a pebble*. Among the Foraminifera we have an immense variety of forms, which, in the light of surface tension and of the principle of minimal area, are capable of explanation and of reduction to a small number of characteristic types. Many of the Foraminifera are composite structures, formed by the successive imposition of cell upon cell, and these we shall deal with later on; let us glance here at the simpler conformations exhibited by the single chambered or “‘monothalamic”’ genera, and perhaps one or two of the simplest composites.

We begin with forms, like Astrorhiza (Fig. 219, p. 464), which are in a high degree irregular, and end with others which manifest a perfect and mathematical regularity. The broad difference between these two types is that the former are characterised, like Amoeba, by a variable surface tension, and consequently by unstable equilibrium ; but the strong contrast between these and the regular forms is bridged over by various transition-stages, or differences of degree. Indeed, as in all other Rhizopods, the very fact of the emission of pseudopodia, which reach their highest development in this group of animals, is a sign of unstable surface-equilibrium ; and we must therefore consider that those forms which indicate symmetry and equilibrium in their shells have secreted these during periods when rest and uniformity of surface conditions alternated with the phases of pseudopodial activity. The irregular forms are in almost all cases arenaceous, that is to say they have no solid shells formed by steady adsorptive secretion, but only a looser covering of sand grains with which the protoplasmic body has come in contact and cohered. Sometimes, as in Ramulina, we have a calcareous shell combined with irregularity of form; but here we can easily see a partial and as it were a broken regularity, the recular forms of sphere and cylinder being repeated in various parts of the ramified mass. When we look more closely at the arenaceous forms, we find that the same thing is true of them; they represent, either in whole or part, approximations to the form of surfaces of equilibrium, spheres, cylinders and so forth. In Aschemonella we have a precise replica of the calcareous Ramulina ; and in Astrorhiza itself, in the forms distinguished by naturalists as A. crassatina, what is described as the “subsegmented interior *”’

seems to shew the natural, physical tendency of the long semifluid cylinder of protoplasm to contract, at its limit of stability, mto unduloid constrictions, as a step towards the breaking up into separate spheres: the completion of which process is restrained or prevented by the rigidity and friction of the arenaceous covering. Passing to the typical, calcareous-shelled Foraminifera, we have the most symmetrical of all possible types in the perfect sphere of Orbulina; this is a pelagic organism, whose floating habitat places.

it in a position of perfect symmetry towards all external forces. Save for one or two other forms which are also spherical, or approximately so, like Thurammina, the rest of the monothalamic caleareous Foraminifera are all comprised by naturalists within the genus Lagena. This large and varied genus consists of “flaskshaped” shells, whose surface is simply that of an unduloid, or more frequently, like that of a flask itself, an unduloid combined with a portion of a sphere. We do not know the circumstances

under which the shell of Lagena is formed, nor the nature of the force by which, during its formation, the surface is stretched out into the unduloid form; but we may be pretty sure that it is suspended vertically in the sea, that is to say in a position of symmetry as regards its vertical axis, about which the unduloid surface of revolution is symmetrically formed. At the same time we have other types of the same shell in which the form is more or less flattened; and these are doubtless the cases in which such symmetry of position was not present, or was replaced by a broader, lateral contact with the surface pellicle*.

While Orbulina is a simple spherical drop, Lagena suggests to our minds a “hanging drop,” drawn out to a long and slender neck by its own weight, aided by the viscosity of the material. Indeed the various hanging drops, such as Mr C. R. Darling shews us, are the most beautiful and perfect unduloids, with spherical ends, that it is possible to conceive. A suitable liquid, a little denser than water and incapable of mixing with it (such as ethyl benzoate), is poured on a surface of water. It spreads

* That the Foraminifera not only can but do hang from the surface of the water is confirmed by the following apt quotation which I owe to Mr EK. Heron- Allen: “Quand on place, comme il a été dit, le dép6t provenant du lavage des fucus dans un flacon que l’on remplit de nouvelle eau, on voit au bout d’une heure environ les animaux [Gromia dujardinii] se mettre en mouvement et commencer a grimper. Six heures apres ils tapissent l’extérieur du flacon, de sorte que les plus élevés sont a trente-six ou quarante-deux millimétres du fond; le lendemain beaucoup d’entre eux, aprés avoir atteint le niveau du liquide, ont continué a ramper @ sa surface, en se laissant pendre au-dessous comme certains mollusques gastéropodes.” (Dujardin, F., Observations nouvelles sur les prétendus céphalopodes microscopiques, Ann. des Sci. Nat. (2), m1, p. 312, 1835.)

over the surface and gradually forms a hanging drop, approximately hemispherical; but as more liquid is added the drop sinks or rather grows downwards, still adhering to the surface film; and the balance of forces between gravity and surface tension results in the unduloid contour, as the increasing weight of the drop tends to stretch it out and finally break it in two. At the moment of rupture, by the way, a tiny droplet is formed in the attenuated neck, such as we described in the normal division of a cylindrical thread (p. 233).

To pass to a much more highly organised class of animals, we find the unduloid beautifully exemplified in the little flask-shaped shells of certain Pteropod mollusca, e.g. Cuvierina*. Here again the symmetry of the figure would at once lead us to suspect that the creature lived in a position of symmetry to the surrounding forces, as for instance if it floated in the ocean in an erect position, that is to say with its long axis coincident with the direction of gravity; and this we know to be actually the mode of life of the little Pteropod.

Many species of Lagena are complicated and beautified by a pattern, and some by the superaddition to the shell of plane extensions or “wings.” These latter give a secondary, bilateral symmetry to the little shell, and are strongly suggestive of a phase or period of growth in which it lay horizontally on the surface, instead of hanging vertically from the surface-film: in which, that is to say, it was a floating and not a hanging drop. The pattern is of two kinds. Sometimes it consists of a sort of fine reticulation, with rounded or more or less hexagonal interspaces: in other cases it is produced by a symmetrical series of ridges or folds, usually longitudinal, on the body of the flask-shaped cell, but occasionally transversely arranged upon the narrow neck. The reticulated and folded patterns we may consider separately. The netted pattern is very similar to the wrinkled surface of a dried pea, or to the more regular wrinkled patterns upon many other seeds and even pollen-grains. If a spherical body after developing a “skin” begin to shrink a little, and if the skin have so far lost its elasticity as to be unable to keep pace with the shrinkage of the inner mass, it will tend to fold or wrinkle; and if the shrinkage be uniform, and the elasticity and flexibility of the skin be also uniform, then the amount of

folding will be uniformly distributed over the surface. Little concave depressions will appear, regularly interspaced, and separated by convex folds. The little concavities being of equal size (unless the system be otherwise perturbed) each one will tend to be surrounded by six others; and when the process has reached its limit, the intermediate boundary-walls, or raised folds, will be found converted into a regular pattern of hexagons. But the analogy of the mechanical wrinkling of the coat of a seed is but a rough and distant one; for we are evidently dealing with molecular rather than with mechanical forces. In one of Darling’s experiments, a little heavy tar-oil is dropped onto a saucer of water, over which it spreads in a thin film showing beautiful interference colours after the fashion of those of a soapbubble. Presently tiny holes appear in the film, which gradually increase in size till they form a cellular pattern or honeycomb, the oil gathering together in the meshes or walls of the cellular net. Some action of this sort is in all probability at work in a surface-film. of protoplasm covering the shell. As a physical phenomenon the actions involved are by no means fully understood, but surface-tension, diffusion and cohesion doubtless play their respective parts therein*. The very perfect cellular patterns obtained by Leduc (to which we shall have occasion to refer in a subsequent chapter) are diffusion patterns on a larger scale, but not essentially different.

The folded or pleated pattern is doubtless to be explained, in a general way, by the shrinkage of a surface-film under certain * This cellular pattern would seem to be related to the ‘“‘cohesion figures’ described by Tomlinson in various surface-films (Phil. Mag. 1861 to 1870); to the “tesselated structure” in liquids described by Professor James Thomson in 1882 (Collected Papers, p. 136); and to the tourbillons cellulaiies of Prof. H. Bénard (Ann. de Chimie (7), xxmm, pp. 62-144, 1901, (8), xxv, pp. 563-566, 1911), Rev. génér. des Sci. xt. p. 1268, 1900; cf. also E. H. Weber. Pcggend. Ann. xcrv, p. 452, 1855, etc.). The phenomenon is of great interest and various appearances have been referred to it, in biology, geology, metallurgy and even astronomy: for the flocculent clouds in the solar photosphere shew an analogous configuration. (See letters by Kerr Grant, Larmor, Wager and others, in Nature, April 16 to June 11, 1914.) In many instances, marked by strict symmetry or regularity, it is very possible that the interference of waves or ripples may play its part in the phenomenon. But in the majority of cases, it is fairly certain that localised centres of action, or of diminished tension, are present, such as might be provided by dust-particles in the case of Darling’s experiment (cf. infra, p. 590).

conditions of viscous or frictional restraint. A case which (as it seems to me) is closely analogous to that of our foraminiferal shells is described by Quincke*, who let a film of albumin or of resin set and harden upon a surface of quicksilver, and found that the little solid pellicle had been thrown into a pattern of symmetrical folds. If the surface thus thrown into folds be that of a cylinder, or any other figure with one principal axis of symmetry, such as an ellipsoid or unduloid, . the direction of the folds will tend to be related to the axis of symmetry, and we might expect accordingly to find regular longitudinal, or regular transverse wrinkling. Now as a matter of fact we almost invariably find in the Lagena the tormer condition: that is to say, in our ellipsoid or unduloid cell, the puckering takes the form of the vertical fluting on a column, rather than that of the transverse pleating of an accordion. And further, there is often a tendency for such longitudinal flutings to be more or less localised at the end of the ellipsoid, or in the region where the unduloid merges into its spherical base. In this latter region we often meet with a regular series of short longitudinal folds, as we do in the forms of Lagena denominated L. semistriata. All these various forms of surface can be imitated, or rather can be precisely reproduced, by the art of the glass-blower f.

Furthermore, they remind one, in a striking way, of the regular ribs or flutings in the film or sheath which splashes up to envelop a smooth ball which has been dropped into a liquid, as Mr Wer hington has so beautifully shewn f. + Certain palaeontologists (e.g. Haeusler and Spandel) have maintained that in each family or genus the plain smooth-shelled forms are the primitive and ancient ones, and that the ribbed and otherwise ornamented shells make their appearance at later dates in the course of a definite evolution (cf. Rhumbler, Foraminiferen der Plavkton-Expedition, 1911, 1, p. 21). If this were true it would be of funda-— mental importance: but this book of mine would not deserve to be written.

In Mr Worthington’s experiment, there appears to be something of the nature of a viscous drag in the surface-pellicle; but whatever be the actual cause of variation of tension, it is not difficult to see that there must be in general a tendency towards longitudinal puckering or “‘fluting” in the case of a thin-walled cylindrical or other elongated body, rather than a tendency towards transverse puckering, or “pleating.” For let us suppose that some change takes place involving an increase of surface-tension in some small area of the curved wall, and leading therefore to an increase of pressure: that is to say let T become 7 + t, and P become P + p. Our new equation of equilibrium, then, in place of P= T/r + T/r’ becomes

Therefore, in order to produce the small increment of pressure p, it is easier to do so by increasing ¢t/7 than t/r’; that is to say, the easier way is to alter, or diminish r. And the same will hold good if the tension and pressure be diminished instead of increased. This is as much as to say that, when corrugation or “rippling” of the walls takes place owing to small changes of surface-tension, and consequently of pressure, such corrugation is more likely to take place in the plane of 7,—that is to say, in the plane of greatest curvature. And it follows that in such a figure as an ellipsoid, wrinkling will be most likely to take place not only in a longitudinal direction but near the extremities of the figure, that is to say again in the region of greatest curvature.

The longitudinal wrinkling of the flask-shaped bodies of our Lagenae, and of the more or less cylindrical cells of many other Foraminifera (Fig. 87), is in complete accord with the above theoretical considerations; but nevertheless, we soon find that our result is not a general one, but is defined by certain limiting conditions, and is accordingly subject to what are, at first sight, important exceptions. For instance, when we turn to the narrow neck of the Lagena we see at once that our theory no longer holds; for

the wrinkling which was invariably longitudinal in the body of the cell is as invariably transverse in the narrow neck. The reason for the difference is not far to seek. The conditions in the neck are very different from those in the expanded portion of the cell: the main difference being that the thickness of the wall is no longer insignificant, but is of considerable magnitude as compared with the diameter, or circumference, of the neck. We must accordingly take it into account in considering the bending moments at any point in this region of the shell-wall. And it is at once obvious that, in any portion of the narrow neck, flerwre of a wall in a