On Growth and Form

* von Moller, V., Die spiral-gewundenen Foraminifera des russischen Kohlenkalks, Mém. de l Acad. Imp. Sci., St Pétersbourg (7), xxv, 1878. 86°, or just such a spiral as we commonly meet with in the Ammonites* (cf. p. 539). In Fusulina, and in some few other Foraminifera (cf. Fig. 310, a), the spire seems to wind evenly on, with little or no external sign of the successive periods of growth, or successive chambers of the shell. The septa which mark off the chambers, and correspond to retardations or cessations in the periodicity of growth, are still to be found in sections of the shell of Fusulna; but they are somewhat irregular and comparatively inconspicuous ; the measurements we have just spoken of are taken without

Fig. 310. A, Cornuspira foliacea, Phil.; B, Operculina complanata, Defr. reference to the segments or chambers, but only with reference to the whorls, or in other words with direct reference to the vectorial angle. The linear dimensions of successive chambers have been * As von Miller is careful to explain, Naumann’s formula for the “cyclocentric conchospiral” is appropriate to this and other spiral Foraminifera, since we have in all these cases a central or initial chamber, approximately spherical, about which the logarithmic spiral is coiled (cf. Fig. 309). In species where the central chamber is especially large, Naumann’s formula is all the more advantageous. But it is plain that it is only required when we are dealing with diameters, or with radii; so long as we are merely comparing the breadths of successive whorls, the two formulae come to the same thing.

measured in a number of cases. Van Iterson* has done so in various Miholinidae, with such results as the following: Here the mean ratio of breadth of consecutive chambers may be taken as 1-323 (that is to say, the eighth root of 319/34); and the calculated values, as given above, are based on this determination. Again, Rhumbler has measured the lnear dimensions of a number of rotaline forms, for instance Pulvinulina menardi (Fig. 259): in which common species he finds the mean linear ratio of consecutive chambers to be about 1-187. In both cases, and especially in the latter, the ratio is not strictly constant from chamber to chamber, but is subject to a small secondary fluctuation f.

When the linear dimensions of successive chambers are in continued proportion, then, in order that the whole shell may constitute a logarithmic spiral, it is necessary that the several chambers should subtend equal angles of revolut@on at the pole. In the case of the Miliolidae this is obviously the case (Fig. 311); for in this family the chambers lie in two rows (Biloculina), or three rows (Triloculina), or in some other small number of series: so that the angles subtended by them are large, simple fractions of the circular arc, such as 180° or 120°. In many of the nautiloid forms, such as Cyclammina (Fig. 312), the angles subtended, though of less magnitude, are still remarkably constant, as we

* Van Iterson, G., Mathem. u. mikrosk.-anat. Studien iiber Blattstellungen, nebst Betrachtungen iiber den Schalenbau der Miliolinen, 331 pp., Jena, 1907. + Hans Przibram asserts that the linear ratio of successive chambers tends in many Foraminifera to approximate to 1-26, which =./2; in other words, that the volumes of successive chambers tend to double. This Przibram would bring into relation with another law, viz. that insects and other arthropods tend to moult, or to metamorphose, just when they double their weights, or increase their linear dimensions in the ratio of 1 : i) 2. (Die Kammerprogression der Foraminiferen als Parallele zur Hiutungsprogression der Mantiden, Arch. f. Entw. Mech. xxx1v p- 680, 1813.) Neither rule seems to me to be well grounded.

may see by Fig. 313; where the angle subtended by each chamber is made equal to 20°, and this diagrammatic figure is not perceptibly different from the other. In some cases the subtended angle is less constant; and in these it would be necessary to equate the several linear dimensions with the corresponding vector angles, according to our equation r= e’°°*, It is probable that, by so taking account of variations of 8, such variations of r as (according to Rhumbler’s measurements) Pulvinulina and other genera appear to shew, would be found to diminish or even to disappear.

The law of increase by which each chamber bears a constant ratio of magnitude to the next may be looked upon as a simple consequence of the structural uniformity or homogeneity of the organism; we have merely to suppose (as this uniformity would naturally lead us to do) that the rate of increase is at each instant proportional to the whole existing mass. For if Vy, V,, etc., be the volumes of the successive chambers, let V, bear a constant proportion to Vy, so that V,;=qVo, and let V, bear the same proportion to the whole pre-existing volume: then

This ratio of 1/(1 + q) is easily shewn to be the constant ratio running through the whole series, from chamber to chamber; and if this ratio of volumes be constant, so also are the ratios of corresponding surfaces, and of corresponding linear dimensions, provided always that the successive increments, or successive chambers, are similar in form. We have still to discuss the similarity of form and the symmetry of position which characterise the successive chambers, and which, together with the law of continued proportionality of size, are the distinctive characters and the indispensable conditions of a series of “gnomons.”

The minute size of the foraminiferal shell or at least of each successive increment thereof, taken in connection with the fluid or semi-fluid nature of the protoplasmic substance, is enough to suggest that the molecular forces, and especially the force of surface-tension, must exercise a controlling influence over the form of the whole structure; and this suggestion, or belief, is already implied in our statement that each successive increment of growing protoplasm constitutes a separate drop. These “drops,” partially concealed by their successors, but still shewing in part their rounded outlines, are easily recognisable in the various foraminiferal shells which are illustrated in this chapter.

The accompanying figure represents, to begin with, the spherical shell characteristic of the common, floating, oceanic Orbulina. In the specimen illustrated, a second chamber, superadded to the first, has arisen as a drop of protoplasm which exuded through the pores of the first chamber, accumulated on its surface, and spread over the latter till it came to rest in a position of equilibrium. We may take it that this position of equilibrium is determined, at least in the first instance, by the “law of the constant angle,” which holds, or tends to hold, in all cases where the free surface of a given liquid is in contact with a given solid, in presence of another liquid or a gas. The corresponding equations are precisely the same as those which we have used in discussing the form of a drop (on p. 294); though some slight modification must be made in our definitions, inasmuch as the consideration of surface-tension is no longer appropriate at the solid surfaces, and the concept of surface-energy must take its place. Be that as it may, it is enough for us to observe that, in such a case as ours, when a given fluid (namely protoplasm) is in surface contact with a solid (viz. a calcareous shell), in presence of another fluid (seawater), then the angle of contact, or angle by which the common surface (or interface) of the two liquids abuts against the solid wall, tends to be constant: and that being so, the drop will have a certain definite form, depending (¢nter alia) on the form of the surface with which it is in contact. After a period of rest, during which the surface of our second drop becomes rigid by calcification, a new period of growth will recur and a new drop of protoplasm be accumulated. Circumstances remaining the same, this new drop will meet the solid surface of the shell at the same angle as did the former one; and, the other forces at work on the system remaining the same, the form of the whole drop, or chamber, will be the same as before.

According to Rhumbler, this “law of the constant angle” is the fundamental principle in the mechanical conformation of the foraminiferal shell, and provides for the symmetry of form as well as of position in each succeeding drop of protoplasm: which form and position, once acquired, become rigid and fixed with the onset of calcification. But Rhumbler’s explanation brings with it its own difficulties. It is by no means easy of verification, for on the very complicated curved surfaces of the shell it seems to me extraordinarily difficult to measure, or even to recognise, the actual angle of contact: of which angle of contact, by the way,

but little is known, save only in the particular case where one of the three bodies is air, as when a surface of water is exposed to air and in contact with glass. It is easy moreover to see that in many of our Foraminifera the angle of contact, though it may be constant in homologous positions from chamber to chamber, is by no means constant at all points along the boundary of each chamber. In Cristellaria, for instance (Fig. 315), 1t would seem to be (and Rhumbler asserts that it actually is) about 90° on the outer side and only about 50° on the inner side of each septal partition; in Pulvinulina (Fig. 259), according to Rhumbler, the angles adjacent to the mouth are of 90°, and the opposite angles

are of 60°, in each chamber. For these and other similar discrepancies Rhumbler would account by simply invoking the heterogeneity of the protoplasmic drop: that is to say, by assuming that the protoplasm has a different composition and different properties (including a very different distribution of surface-energy), at points near to and remote from the mouth of the shell. Whether the differences in angle of contact be as great as Rhumbler takes them to be, whether marked heterogeneities of the protoplasm occur, and whether these be enough to account for the differences of angle, I cannot tell. But it seems to me that we had better rest content with a general statement, and that Rhumbler has taken too precise and narrow a view.

In the molecular growth of a crystal, although we must of necessity assume that each molecule settles down in a position of minimum potential energy, we find it very hard indeed to explain precisely, even in simple cases and after all the labours of modern erystallographers, why this or that position is actually a place of minimum potential. In the case of our little Foraminifer (just as in the case of the crystal), let us then be content to assert that each drop or bead of protoplasm takes up a position of minimum potential energy, in relation to all the circumstances of the case ; and let us not attempt, in the present state of our knowledge, to define that position of minimum potential by reference to angle of contact or any other particular condition of equlibrium. In most cases the whole exposed surface, on some portion of which the drop must come to rest, is an extremely complicated one, and the forces involved constitute a system which, in its entirety, is more complicated still; but from the symmetry of the case and the continuity of the whole phenomenon, we are entitled to believe that the conditions are just the same, or very nearly the same, time after time, from one chamber to another: as the one chamber is conformed so will the next tend to be, and as the one is situated relatively to the system so will its successor tend to be situated in turn. The physical law of minimum potential (including also the law of minimal area) is all that we need in order to explain, in general terms, the continued similarity of one chamber to another ; and the physiological law of growth, by which a continued proportionality of size tends to run through the series of successive chambers, impresses upon this series of similar increments the form of a logarithmic spiral.

In each particular case the nature of the logarithmic spiral, as defined by its constant angle, will be chiefly determined by the rate of growth; that is to say by the particular ratio in which each new chamber exceeds its predecessor in magnitude. But shells having the same constant angle.(a) may still differ from one another in many ways—in the general form and relative position of the chambers, in their extent of overlap, and hence in the actual contour and appearance of the shell; and these variations must correspond to particular distributions of energy within the system, which is governed as a whole by the law of minimum potential.

Our problem, then, becomes reduced to that of investigating the possible configurations which may be derived from the successive symmetrical apposition of similar bodies whose magnitudes are in continued proportion; and it is obvious, mathematically speaking, that the various possible arrangements all come under the head of the logarithmic spiral, together with the limiting cases which it includes. Since the difference between one such form and another depends upon the numerical value of certain coefficients of magnitude, it is plain that any one must tend to pass into any other by small and continuous gradations; in other words, that a classification of these forms must (like any classification whatsoever of logarithmic spirals or of any other mathematical curves), be theoretic or “artificial.” But we may easily make such an artificial classification, and shall probably find it to agree, more or less, with the usual methods of classification recognised by biological students of the Foraminifera.

Firstly we have the typically spiral shells, which occur in great variety, and which (for our present purpose) we need hardly describe further. We may merely notice how in certain cases, for instance Globigerina, the individual chambers are little removed from spheres; in other words, the area of contact between the adjacent chambers is small. In such forms as Cyclammina and Pulvinulina, on the other hand, each chamber is greatly overlapped by its successor, and the spherical form of each is lost in a marked asymmetry. Furthermore, in Globigerina and some others we have a tendency to the development of a helicoid spiral in space, as In so many of our univalve molluscan shells. The mathematical problem of how a shell should grow, under the assumptions which we have made, would probably find its most general statement in such a case as that of Globigerina, where the whole organism lives and grows freely poised in a medium whose density is little different from its own.

The majority of spiral forms, on the other hand, are plane or discoid spirals, and we may take it that in these cases some force has exercised a controlling influence, so as to keep all the chambers in a plane. This is especially the case in forms like Rotalhia or Discorbina (Fig. 316), where the organism lives attached to a rock or a frond of sea-weed; for here (just as in the case of the coiled tubes which little worms such as Serpula and Spirorbis make, under similar conditions) the spiral disc is itself asymmetrical, its whorls being markedly flattened on their attached surfaces.

We may also conceive, among other conditions, the very curious case in which the protoplasm may entirely overspread the surface of the shell without reaching a position of equilibrium; in which case a new shell will be formed enclosing the old one, whether the old one be in the form of a single, solitary chamber, © or have already attained to the form of a chambered or spiral shell. This is precisely what often happens in the case of Orbulina, when within the spherical shell we find a small, but perfectly formed, spiral “ Globigerina*.”

The various Miliolidae (Fig. 311), only differ from the typical spiral, or rotaline forms, in the large angle subtended by each chamber, and the consequent abruptness of their inclination to each other. In these cases the outward appearance of a spiral tends to be lost; and it behoves us to recollect, all the more, that our spiral curve is not necessarily identical with the outline of the shell, but is always a line drawn through corresponding points in the successive chambers of the latter.

We reach a limiting case of the logarithmic spiral when the chambers are arranged in a straight line; and the eye will tend to associate with this limiting case the much more numerous forms in which the spiral angle is small, and the shell only exhibits a gentle curve with no succession of enveloping whorls. This constitutes the Nodosarian type (Fig. 87, p. 262); and here again, we must postulate some force which has tended to keep the chambers in a rectilinear series: such for instance as gravity, acting on a system of “hanging drops.”

In Textularia and its allies (Fig. 317), we have a_ precise parallel to the helicoid cyme of the botanists (cf. p. 502): that is to say we have a screw translation, perpendicular to the plane of the underlying logarithmic spiral. In other words, in tracing a genetic spiral through the whole succession of chambers, we do so by a continuous vector rotation, through successive angles of 180° (or 120° in some cases), while the pole moves along an axis perpendicular to the original plane of the spiral. |

Another type is furnished by the “cyclic” shells of the Orbitolitidae, where small and numerous chambers tend to be added on round and round the system, so building up a circular flattened disc. This again we perceive to be, mathematically, a hmiting case of the logarithmic spiral, where the spiral has become a circle and the constant angle is now an angle of 90°. Lastly there are a certain number of Foraminifera in which, without more ado, we may simply say that the arrangement of the chambers is irregular, neither the law of constant ratio of magnitude nor that of constant form being obeyed. The chambers are heaped pell-mell upon one another, and such forms are known to naturalists ‘as the Acervularidae.

While in these last we have an extreme lack of regularity, we must not exaggerate the regularity or constancy which the more ordinary forms display. We may think it hard to believe that the simple causes, or simple laws, which we have described should operate, and operate again and again, in millions of individuals to produce the same delicate and complex conformations. But we are taking a good deal for granted if we assert that they do so, and in particular we are assuming, with very little proof, the “constancy of species” in this group of animals. Just as Verworn has shewn that the typical Amoeba proteus, when a trace of alkali is added to the water in which it lives, tends, by alteration of surface tensions, to protrude the more delicate pseudopodia characteristic of A. radiosa,—and again when the water is rendered a little more alkaline, to turn apparently into the so-called A. limax,—so it is evident that a very slight modification in ,the surface-energies concerned, might tend to turn one so-called species into another among the Foraminifera. To what extent this process actually occurs, we do not know.

But that this, or something of the kind, does actually occur we can scarcely doubt. For example in the genus Peneroplis, the first portion of the shell consists of a series of chambers arranged in a spiral or nautiloid series; but as age advances the spiral is apt to be modified in various ways*. Sometimes the successive chambers grow rapidly broader, the whole shell becoming fanshaped. Sometimes the chambers become narrower, till they no longer enfold the earlier chambers but only come in contact each with its immediate predecessor: the result being that the shell straightens out, and (taking into account the earlier spiral portion) may be described as crozier-shaped. Between these extremes of shape, and in regard to other variations of thickness or thinness, roughness or smoothness, and so on, there are innumerable gradations passing one into another and intermixed without regard to geographical distribution :—‘wherever Peneroplides abound this wide variation exists, and nothing can be more easy than to pick out a number of striking specimens and give to each a distinctive name, but 7m no other way can they be divided into ‘species. +’? Some writers have wondered at the peculiar variability of this particular shellt; but for all we know of the life-history of the Foraminifera, it may well be that a great number of the other forms which we distinguish as separate species and even genera are no more than temporary manifestations of the same variability §.

+ Brady, op. cit., p. 206; Batsch, one of the earliest writers on Foraminifera. had already noticed that this whole series of ear-shaped and crozier-shaped shells was filled in by gradational forms; Conchylien des Seesandes, 1791, p. 4, pl. v1, fig. 15 a-f. See also, in particular, Dreyer, Peneroplis; eine Studie zur biologischen Morphologie und zur Speciesfrage, Leipzig, 1898; also Eimer und Fickert, Artbildung und Verwandschaft bei den Foraminiferen, Tiibinger zool. Arbeiten, m1, p. 35,

t Doflein, Protozoenkunde, 1911, p. 263; “Was diese Art veranlasst in dieser Weise gelegentlich zu variiren, ist vorlaufig noch ganz rathselhaft.” § In the case of Globigerina, some fourteen species (out of a very much larger number of described forms) were allowed by Brady (in 1884) to be distinct; and this list has been, I believe, rather added to than diminished. But these so-called species depend for the most part on slight differences of degree, differences in the angle of the spiral, in the ratio of magnitude of the segments, or in their area of contact one with another. Moreover with the exception of one or two “dwarf” forms, said to be limited to Arctic and Antarctic waters, there is no principle of geographical distribution to be discerned amongst them. A species found fossil

If we can comprehend and interpret on some such lines as these the form and mode of growth of the foraminiferal shell, we may also begin to understand two striking features of the group, namely, on the one hand the large number of diverse types or families which exist and the large number of species and varieties within each, and on the other the persistence of forms which in many cases seem to have undergone little change or none at all from the Cretaceous or even from earlier periods to the present . day. In few other groups, perhaps only among the Radiolaria, do we seem to possess so nearly complete a picture of all possible transitions between form and form, and of the whole branching system of the evolutionary tree: as though little or nothing of it had ever perished, and the whole web of life, past and present, were as complete as ever. It leads one to imagine that these shells have grown according to laws so simple, so much in harmony with their material, with their environment, and with all the forces internal and external to which they are exposed, that none is better than another and none fitter or less fit to survive. It invites one also to contemplate the possibility of the lines of possible variation being here so narrow and determinate that identical forms may have come independently into being again and again. .

While we can trace in the most complete and beautiful manner the passage of one form into another among these little shells, and ascribe them all at last (if we please) to a series which starts with the simple sphere of Orbulina or with the amoeboid body of Astrorhiza, the question stares us in the face whether this be an “evolution” which we have any right to correlate with historic time. The mathematician can trace one conic section into another, and “evolve” for example, through innumerable graded ellipses, the circle from the straight line: which tracing of continuous steps is a true “evolution,” though time has no part therein. It was after this fashion that Hegel, and for that matter Aristotle himself, was an evolutionist—to whom evolution was

in New Britain turns up in the North Atlantic: a species described from the West Indies is rediscovered at the ice-barrier. of the Antarctic. a mental concept, involving order and continuity in thought, but not an actual sequence of events in time. Such a conception of evolution is not easy for the modern biologist to grasp, and harder still to appreciate. And so it is that even those who, like Dreyer* and like Rhumbler, study the foraminiferal shell as a physical system, who recognise that its whole plan and mode of growth is closely akin to the phenomena exhibited by fluid drops under particular conditions, and who explain the conformation of the shell by help of the same physical principles and mathematical laws—yet all the while abate no jot or tittle of the ordinary postulates of modern biology, nor doubt the validity and universal applicability of the concepts of Darwinian evolution. For these writers the biogenetisches Grundgesetz remains impregnable. The Foraminifera remain for them a great family tree, whose actual, pedigree is traceable to the remotest ages; in which historical evolution has coincided with progressive change; and in which structyral fitness for a particular function (or functions) has exercised its selective action and ensured “the survival of the fittest.” By successive stages of historic evolution we are supposed to pass from the irregular Astrorhiza to a Rhabdammina with its more concentrated disc; to the forms of the same genus which consist of but a single tube with central chamber; to those where this chamber is more and more distinctly segmented; so to the typical many-chambered Nodosariae; and from these, by another definite advance and later evolution to the spiral Trochamminae. After this fashion, throughout the whole varied series of the Foraminifera, Dreyer and Rhumbler (following Neumayr) recognise so many successions of related forms, one passing into another, and standing towards it in a definite relationship of ancestry or descent.