Unit II: Capillarity, Cellular Foams, the Bee’s Cell, and Spicular Skeletons¶
1917 Cambridge First Edition — Chapters V–X (pp. 201–492)
Chapter V: The Forms of Cells¶
1917 Cambridge First Edition (pp. 201–276)
Protoplasm, as we have already said, is a fluid or rather a semifluid substance, and we need not pause here to attempt to describe the particular properties of the semifluid, colloid, or jelly-like substances to which it is allied; we should find it no easy matter. Nor need we appeal to precise theoretical definitions of fluidity, lest we come into a debateable land. It is in the most general sense that protoplasm is “fluid.” As Graham said (of colloid matter in general), “its softness partakes of fluidity, and enables the colloid to become a vehicle for liquid diffusion, like water itself [1].” When we can deal with protoplasm in sufficient quantity we see it flow; particles move freely through it, air-bubbles and liquid droplets shew round or spherical within it; and we shall have much to say about other phenomena manifested by its own surface, which are those especially characteristic of liquids. It may encompass and contain solid bodies, and it may “secrete” within or around itself solid substances; and very often in the complex living organism these solid substances formed by the living protoplasm, like shell or nail or horn or feather, may remain when the protoplasm which formed them is dead and gone; but the protoplasm itself is fluid or semifluid, and accordingly permits of free (though not necessarily rapid) diffusion and easy convection of particles within itself. This simple fact is of elementary importance in connection with form, and with what appear at first sight to be common characteristics or peculiarities of the forms of living things.
The older naturalists, in discussing the differences between inorganic and organic bodies, laid stress upon the fact or statement that the former grow by “agglutination,” and the latter by [p. 202] what they termed “intussusception.” The contrast is true, rather, of solid as compared with jelly-like bodies of all kinds, living or dead, the great majority of which as it so happens, but by no means all, are of organic origin.
A crystal “grows” by deposition of new molecules, one by one and layer by layer, superimposed or aggregated upon the solid substratum already formed. Each particle would seem to be influenced, practically speaking, only by the particles in its immediate neighbourhood, and to be in a state of freedom and independence from the influence, either direct or indirect, of its remoter neighbours. As Lord Kelvin and others have explained the formation and the resulting forms of crystals, so we believe that each added particle takes up its position in relation to its immediate neighbours already arranged, generally in the holes and corners that their arrangement leaves, and in closest contact with the greatest number [2]. And hence we may repeat or imitate this process of arrangement, with great or apparently even with precise accuracy (in the case of the simpler crystalline systems), by piling up spherical pills or grains of shot. In so doing, we must have regard to the fact that each particle must drop into the place where it can go most easily, or where no easier place offers. In more technical language, each particle is free to take up, and does take up, its position of least potential energy relative to those already deposited; in other words, for each particle motion is induced until the energy of the system is so distributed that no tendency or resultant force remains to move it more. The application of this principle has been shewn to lead to the production of planes [3] (in all cases where by the limitation of material, surfaces must occur); and where we have planes, straight edges and solid angles must obviously also occur; and, if equilibrium is [p. 203] to follow, must occur symmetrically. Our piling up of shot, or manufacture of mimic crystals, gives us visible demonstration that the result is actually to obtain, as in the natural crystal, plane surfaces and sharp angles, symmetrically disposed.
But the living cell grows in a totally different way, very much as a piece of glue swells up in water, by “imbibition,” or by interpenetration into and throughout its entire substance. The semifluid colloid mass takes up water, partly to combine chemically with its individual molecules [4], partly by physical diffusion into the interstices between these molecules, and partly, as it would seem, in other ways; so that the entire phenomenon is a very complex and even an obscure one. But, so far as we are concerned, the net result is a very simple one. For the equilibrium or tendency to equilibrium of fluid pressure in all parts of its interior while the process of imbibition is going on, the constant rearrangement of its fluid mass, the contrast in short with the crystalline method of growth where each particle comes to rest to move (relatively to the whole) no more, lead the mass of jelly to swell up, very much as a bladder into which we blow air, and so, by a graded and harmonious distribution of forces, to assume everywhere a rounded and more or less bubble-like external form [5]. So, when the same school of older naturalists called attention to a new distinction or contrast of form between the organic and inorganic objects, in that the contours of the former tended to roundness and curvature, and those of the latter to be bounded by straight lines, planes and sharp angles, we see that this contrast was not a new and different one, but only another aspect of their former statement, and an immediate consequence of the difference between the processes of agglutination and intussusception.
This common and general contrast between the form of the crystal on the one hand, and of the colloid or of the organism on the other, must by no means be pressed too far. For Lehmann, [p. 204] in his great work on so-called Fluid Crystals [6], to which we shall afterwards return, has shewn how, under certain circumstances, surface-tension phenomena may coexist with crystallisation, and produce a form of minimal potential which is a resultant of both: the fact being that the bonds maintaining the crystalline arrangement are now so much looser than in the solid condition that the tendency to least total surface-area is capable of being satisfied. Thus the phenomenon of “liquid crystallisation” does not destroy the distinction between crystalline and colloidal forms, but gives added unity and continuity to the whole series of phenomena [7]. Lehmann has also demonstrated phenomena within the crystal, known for instance as transcrystallisation, which shew us that we must not speak unguardedly of the growth of crystals as limited to deposition upon a surface, and Bütschli has already pointed out the possible great importance to the biologist of the various phenomena which Lehmann has described [8].
So far then, as growth goes on, unaffected by pressure or other external force, the fluidity of protoplasm, its mobility internal and external, and the manner in which particles move with comparative freedom from place to place within, all manifestly tend to the production of swelling, rounded surfaces, and to their great predominance over plane surfaces in the contour of the organism. These rounded contours will tend to be preserved, for a while, in the case of naked protoplasm by its viscosity, and in the presence of a cell-wall by its very lack of fluidity. In a general way, the presence of curved boundary surfaces will be especially obvious in the unicellular organisms, and still more generally in the external forms of all organisms; and wherever mutual pressure between adjacent cells, or other adjacent parts, has not come into play to flatten the rounded surfaces into planes.
But the rounded contours that are assumed and exhibited by [p. 205] a piece of hard glue, when we throw it into water and see it expand as it sucks the water up, are not nearly so regular or so beautiful as are those which appear when we blow a bubble, or form a drop, or pour water into a more or less elastic bag. For these curving contours depend upon the properties of the bag itself, of the film or membrane that contains the mobile gas, or that contains or bounds the mobile liquid mass. And hereby, in the case of the fluid or semifluid mass, we are introduced to the subject of surface tension: of which indeed we have spoken in the preceding chapter, but which we must now examine with greater care.
Among the forces which determine the forms of cells, whether they be solitary or arranged in contact with one another, this force of surface-tension is certainly of great, and is probably of paramount importance. But while we shall try to separate out the phenomena which are directly due to it, we must not forget that, in each particular case, the actual conformation which we study may be, and usually is, the more or less complex resultant of surface tension acting together with gravity, mechanical pressure, osmosis, or other physical forces.
Surface tension is that force by which we explain the form of a drop or of a bubble, of the surfaces external and internal of a “froth” or collocation of bubbles, and of many other things of like nature and in like circumstances [9]. It is a property of liquids (in the sense at least with which our subject is concerned), and it is manifested at or very near the surface, where the liquid comes into contact with another liquid, a solid or a gas. We note here that the term surface is to be interpreted in a wide sense; for wherever we have solid particles imbedded in a fluid, wherever we have a non-homogeneous fluid or semi-fluid such as a particle [p. 206] of protoplasm, wherever we have the presence of “impurities,” as in a mass of molten metal, there we have always to bear in mind the existence of “surfaces” and of surface tensions, not only on the exterior of the mass but also throughout its interstices, wherever like meets unlike.
Surface tension is due to molecular force, to force that is to say arising from the action of one molecule upon another, and it is accordingly exerted throughout a small thickness of material, comparable to the range of the molecular forces. We imagine that within the interior of the liquid mass such molecular interactions negative one another: but that at and near the free surface, within a layer or film approximately equal to the range of the molecular force, there must be a lack of such equilibrium and consequently a manifestation of force.
The action of the molecular forces has been variously explained. But one simple explanation (or mode of statement) is that the molecules of the surface layer (whose thickness is definite and constant) are being constantly attracted into the interior by those which are more deeply situated, and that consequently, as molecules keep quitting the surface for the interior, the bulk of the latter increases while the surface diminishes; and the process continues till the surface itself has become a minimum, the surface-shrinkage exhibiting itself as a surface-tension. This is a sufficient description of the phenomenon in cases where a portion of liquid is subject to no other than its own molecular forces, and (since the sphere has, of all solids, the smallest surface for a given volume) it accounts for the spherical form of the raindrop, of the grain of shot, or of the living cell in many simple organisms. It accounts also, as we shall presently see, for a great number of much more complicated forms, manifested under less simple conditions.
Let us here briefly note that surface tension is, in itself, a comparatively small force, and easily measurable: for instance that of water is equivalent to but a few grains per linear inch, or a few grammes per metre. But this small tension, when it exists in a curved surface of very great curvature, gives rise to a very great pressure directed towards the centre of curvature. We can easily calculate this pressure, and so satisfy ourselves that, when the radius of curvature is of molecular dimensions, the [p. 207] pressure is of the magnitude of thousands of atmospheres,—a conclusion which is supported by other physical considerations.
The contraction of a liquid surface and other phenomena of surface tension involve the doing of work, and the power to do work is what we call energy. It is obvious, in such a simple case as we have just considered, that the whole energy of the system is diffused throughout its molecules; but of this whole stock of energy it is only that part which comes into play at or very near to the surface which normally manifests itself in work, and hence we may speak (though the term is open to some objections) of a specific surface energy. The consideration of surface energy, and of the manner in which its amount is increased and multiplied by the multiplication of surfaces due to the subdivision of the organism into cells, is of the highest importance to the physiologist; and even the morphologist cannot wholly pass it by, if he desires to study the form of the cell in its relation to the phenomena of surface tension or “capillarity.” The case has been set forth with the utmost possible lucidity by Tait and by Clerk Maxwell, on whose teaching the following paragraphs are based: they having based their teaching upon that of Gauss,—who rested on Laplace.
Let E be the whole potential energy of a mass M of liquid; let e0 be the energy per unit mass of the interior liquid (we may call it the internal energy); and let e be the energy per unit mass for a layer of the skin, of surface S, of thickness t, and density ρ (e being what we call the surface energy). It is obvious that the total energy consists of the internal plus the surface energy, and that the former is distributed through the whole mass, minus its surface layers. That is to say, in mathematical language,
But this is equivalent to writing:
and this is as much as to say that the total energy of the system may be taken to consist of two portions, one uniform throughout the whole mass, and another, which is proportional on the one hand to the amount of surface, and on the other hand is proportional to the difference between e and e0, that is to say to the difference between the unit values of the internal and the surface energy. [p. 208]
It was Gauss who first shewed after this fashion how, from the mutual attractions between all the particles, we are led to an expression which is what we now call the potential energy of the system; and we know, as a fundamental theorem of dynamics, that the potential energy of the system tends to a minimum, and in that minimum finds, as a matter of course, its stable equilibrium.
We see in our last equation that the term M e0 is irreducible, save by a reduction of the mass itself. But the other term may be diminished (1) by a reduction in the area of surface, S, or (2) by a tendency towards equality of e and e0, that is to say by a diminution of the specific surface energy, e.
These then are the two methods by which the energy of the system will manifest itself in work. The one, which is much the more important for our purposes, leads always to a diminution of surface, to the so-called “principle of minimal areas”; the other, which leads to the lowering (under certain circumstances) of surface tension, is the basis of the theory of Adsorption, to which we shall have some occasion to refer as the modus operandi in the development of a cell-wall, and in a variety of other histological phenomena. In the technical phraseology of the day, the “capacity factor” is involved in the one case, and the “intensity factor” in the other.
Inasmuch as we are concerned with the form of the cell it is the former which becomes our main postulate: telling us that the energy equations of the surface of a cell, or of the free surfaces of cells partly in contact, or of the partition-surfaces of cells in contact with one another or with an adjacent solid, all indicate a minimum of potential energy in the system, by which the system is brought, ipso facto, into equilibrium. And we shall not fail to observe, with something more than mere historical interest and curiosity, how deeply and intrinsically there enter into this whole class of problems the “principle of least action” of Maupertuis, the “lineae curvae maximi minimive proprietate gaudentes” of Euler, by which principles these old natural philosophers explained correctly a multitude of phenomena, and drew the lines whereon the foundations of great part of modern physics are well and truly laid. [p. 209]
In all cases where the principle of maxima and minima comes into play, as it conspicuously does in the systems of liquid films which are governed by the laws of surface-tension, the figures and conformations produced are characterised by obvious and remarkable symmetry. Such symmetry is in a high degree characteristic of organic forms, and is rarely absent in living things,—save in such cases as amoeba, where the equilibrium on which symmetry depends is likewise lacking. And if we ask what physical equilibrium has to do with formal symmetry and regularity, the reason is not far to seek; nor can it be put better than in the following words of Mach’s [10]. “In every symmetrical system every deformation that tends to destroy the symmetry is complemented by an equal and opposite deformation that tends to restore it. In each deformation positive and negative work is done. One condition, therefore, though not an absolutely sufficient one, that a maximum or minimum of work corresponds to the form of equilibrium, is thus supplied by symmetry. Regularity is successive symmetry. There is no reason, therefore, to be astonished that the forms of equilibrium are often symmetrical and regular.”
As we proceed in our enquiry, and especially when we approach the subject of tissues, or agglomerations of cells, we shall have from time to time to call in the help of elementary mathematics. But already, with very little mathematical help, we find ourselves in a position to deal with some simple examples of organic forms.
When we melt a stick of sealing-wax in the flame, surface tension (which was ineffectively present in the solid but finds play in the now fluid mass), rounds off its sharp edges into curves, so striving towards a surface of minimal area; and in like manner, by melting the tip of a thin rod of glass, Leeuwenhoek made the little spherical beads which served him for a microscope [11]. When any drop of protoplasm, either over all its surface or at some free end, as at the extremity of the pseudopodium of an amoeba, is [p. 210] seen likewise to “round itself off,” that is not an effect of “vital contractility,” but (as Hofmeister shewed so long ago as 1867) a simple consequence of surface tension; and almost immediately afterwards Engelmann [12] argued on the same lines, that the forces which cause the contraction of protoplasm in general may “be just the same as those which tend to make every non-spherical drop of fluid become spherical!” We are not concerned here with the many theories and speculations which would connect the phenomena of surface tension with contractility, muscular movement or other special physiological functions, but we find ample room to trace the operation of the same cause in producing, under conditions of rest and equilibrium, certain definite and inevitable forms of surface.
It is however of great importance to observe that the living cell is one of those cases where the phenomena of surface tension are by no means limited to the outer surface; for within the heterogeneous substance of the cell, between the protoplasm and its nuclear and other contents, and in the alveolar network of the cytoplasm itself (so far as that “alveolar structure” is actually present in life), we have a multitude of interior surfaces; and, especially among plants, we may have a large inner surface of “interfacial” contact, where the protoplasm contains cavities or “vacuoles” filled with a different and more fluid material, the “cell-sap.” Here we have a great field for the development of surface tension phenomena: and so long ago as 1865, Nägeli and Schwendener shewed that the streaming currents of plant cells might be very plausibly explained by this phenomenon. Even ten years earlier, Weber had remarked upon the resemblance between these protoplasmic streamings and the streamings to be observed in certain inanimate drops, for which no cause but surface tension could be assigned [13].
The case of amoeba, though it is an elementary case, is at the same time a complicated one. While it remains “amoeboid,” it is never at rest or in equilibrium; it is always moving, from one to another of its protean changes of configuration; its surface tension is constantly varying from point to point. Where the [p. 211] surface tension is greater, that portion of the surface will contract into spherical or spheroidal forms; where it is less the surface will correspondingly extend. While generally speaking the surface energy has a minimal value, it is not necessarily constant. It may be diminished by a rise of temperature; it may be altered by contact with adjacent substances [14], by the transport of constituent materials from the interior to the surface, or again by actual chemical and fermentative change. Within the cell, the surface energies developed about its heterogeneous contents will constantly vary as these contents are affected by chemical metabolism. As the colloid materials are broken down and as the particles in suspension are diminished in size the “free surface energy” will be increased, but the osmotic energy will be diminished [15]. Thus arise the various fluctuations of surface tension and the various phenomena of amoeboid form and motion, which Bütschli and others have reproduced or imitated by means of the fine emulsions which constitute their “artificial amoebae.” A multitude of experiments shew how extraordinarily delicate is the adjustment of the surface tension forces, and how sensitive they are to the least change of temperature or chemical state. Thus, on a plate which we have warmed at one side, a drop of alcohol runs towards the warm area, a drop of oil away from it; and a drop of water on the glass plate exhibits lively movements when [p. 212] we bring into its neighbourhood a heated wire, or a glass rod dipped in ether. When we find that a plasmodium of Aethalium, for instance, creeps towards a damp spot, or towards a warm spot, or towards substances that happen to be nutritious, and again creeps away from solutions of sugar or of salt, we seem to be dealing with phenomena every one of which can be paralleled by ordinary phenomena of surface tension [16]. Even the soap-bubble itself is imperfectly in equilibrium, for the reason that its film, like the protoplasm of amoeba or Aethalium, is an excessively heterogeneous substance. Its surface tensions vary from point to point, and chemical changes and changes of temperature increase and magnify the variation. The whole surface of the bubble is in constant movement as the concentrated portions of the soapy fluid make their way outwards from the deeper layers; it thins and it thickens, its colours change, currents are set up in it, and little bubbles glide over it; it continues in this state of constant movement, as its parts strive one with another in all their interactions towards equilibrium [17].
In the case of the naked protoplasmic cell, as the amoeboid phase is emphatically a phase of freedom and activity, of chemical and physiological change, so, on the other hand, is the spherical form indicative of a phase of rest or comparative inactivity. In the one phase we see unequal surface tensions manifested in the creeping movements of the amoeboid body, in the rounding off of the ends of the pseudopodia, in the flowing out of its substance over a particle of “food,” and in the current-motions in the interior of its mass; till finally, in the other phase, when internal homogeneity and equilibrium have been attained and the potential [p. 213] energy of the system is for the time being at a minimum, the cell assumes a rounded or spherical form, passing into a state of “rest,” and (for a reason which we shall presently see) becoming at the same time “encysted.”
1917 Source Illustration
Fig. 60.
In a budding yeast-cell (Fig. 60), we see a more definite and restricted change of surface tension. When a “bud” appears, whether with or without actual growth by osmosis or otherwise of the mass, it does so because at a certain part of the cell-surface the surface tension has more or less suddenly diminished, and the area of that portion expands accordingly; but in turn the surface tension of the expanded area will make itself felt, and the bud will be rounded off into a more or less spherical form.
The yeast-cell with its bud is a simple example of a principle which we shall find to be very important. Our whole treatment of cell-form in relation to surface-tension depends on the fact (which Errera was the first to point out, or to give clear expression to) that the incipient cell-wall retains with but little impairment the properties of a liquid film [18], and that the growing cell, in spite of the membrane by which it has already begun to be surrounded, behaves very much like a fluid drop. But even the ordinary yeast-cell shows, by its ovoid and non-spherical form, that it has acquired its shape under the influence of some force other than that uniform and symmetrical surface-tension which would be productive of a sphere; and this or any other asymmetrical form, once acquired, may be retained by virtue of the solidification and consequent rigidity of the membranous wall of the cell. Unless such rigidity ensue, it is plain that such a conformation as that of the cell with its attached bud could not be long retained, amidst the constantly varying conditions, as a figure of even partial equilibrium. But as a matter of fact, the cell in this case is not in equilibrium at all; it is in process of budding, and is slowly altering its shape by rounding off the bud. It is plain that over its surface the surface-energies are unequally distributed, owing to some heterogeneity of the substance; and to this matter we shall afterwards return. In like manner the developing egg [p. 214] through all its successive phases of form is never in complete equilibrium; but is merely responding to constantly changing conditions, by phases of partial, transitory, unstable and conditional equilibrium.
It is obvious that there are innumerable solitary plant-cells, and unicellular organisms in general, which, like the yeast-cell, do not correspond to any of the simple forms that may be generated under the influence of simple and homogeneous surface-tension; and in many cases these forms, which we should expect to be unstable and transitory, have become fixed and stable by reason of the comparatively sudden or rapid solidification of the envelope. This is the case, for instance, in many of the more complicated forms of diatoms or of desmids, where we are dealing, in a less striking but even more curious way than in the budding yeast-cell, not with one simple act of formation, but with a complicated result of successive stages of localised growth, interrupted by phases of partial consolidation. The original cell has acquired or assumed a certain form, and then, under altering conditions and new distributions of energy, has thickened here or weakened there, and has grown out or tended (as it were) to branch, at particular points. We can often, or indeed generally, trace in each particular stage of growth or at each particular temporary growing point, the laws of surface tension manifesting themselves in what is for the time being a fluid surface; nay more, even in the adult and completed structure, we have little difficulty in tracing and recognising (for instance in the outline of such a desmid as Euastrum) the rounded lobes that have successively grown or flowed out from the original rounded and flattened cell. What we see in a many chambered foraminifer, such as Globigerina or Rotalia, is just the same thing, save that it is carried out in greater completeness and perfection. The little organism as a whole is not a figure of equilibrium or of minimal area; but each new bud or separate chamber is such a figure, conditioned by the forces of surface tension, and superposed upon the complex aggregate of similar bubbles after these latter have become consolidated one by one into a rigid system.
Let us now make some enquiry regarding the various forms [p. 215] which, under the influence of surface tension, a surface can possibly assume. In doing so, we are obviously limited to conditions under which other forces are relatively unimportant, that is to say where the “surface energy” is a considerable fraction of the whole energy of the system; and this in general will be the case when we are dealing with portions of liquid so small that their dimensions come within what we have called the molecular range, or, more generally, in which the “specific surface” is large [19]: in other words it will be small or minute organisms, or the small cellular elements of larger organisms, whose forms will be governed by surface-tension; while the general forms of the larger organisms will be due to other and non-molecular forces. For instance, a large surface of water sets itself level because here gravity is predominant; but the surface of water in a narrow tube is manifestly curved, for the reason that we are here dealing with particles which are mutually within the range of each other’s molecular forces. The same is the case with the cell-surfaces and cell-partitions which we are presently to study, and the effect of gravity will be especially counteracted and concealed when, as in the case of protoplasm in a watery fluid, the object is immersed in a liquid of nearly its own specific gravity.
We have already learned, as a fundamental law of surface-tension phenomena, that a liquid film in equilibrium assumes a form which gives it a minimal area under the conditions to which it is subject. And these conditions include (1) the form of the boundary, if such exist, and (2) the pressure, if any, to which the film is subject; which pressure is closely related to the volume, of air or of liquid, which the film (if it be a closed one) may have to contain. In the simplest of cases, when we take up a soap-film on a plane wire ring, the film is exposed to equal atmospheric pressure on both sides, and it obviously has its minimal area in the form of a plane. So long as our wire ring lies in one plane (however irregular in outline), the film stretched across it will still be in a plane; but if we bend the ring so that it lies no longer in a plane, then our film will become curved into a surface which may be extremely complicated, but is still the smallest possible [p. 216] surface which can be drawn continuously across the uneven boundary.
The question of pressure involves not only external pressures acting on the film, but also that which the film itself is capable of exerting. For we have seen that the film is always contracting to its smallest limits; and when the film is curved, this obviously leads to a pressure directed inwards,—perpendicular, that is to say, to the surface of the film. In the case of the soap-bubble, the uniform contraction of whose surface has led to its spherical form, this pressure is balanced by the pressure of the air within; and if an outlet be given for this air, then the bubble contracts with perceptible force until it stretches across the mouth of the tube, for instance the mouth of the pipe through which we have blown the bubble. A precisely similar pressure, directed inwards, is exercised by the surface layer of a drop of water or a globule of mercury, or by the surface pellicle on a portion or “drop” of protoplasm. Only we must always remember that in the soap-bubble, or the bubble which a glass-blower blows, there is a twofold pressure as compared with that which the surface-film exercises on the drop of liquid of which it is a part; for the bubble consists (unless it be so thin as to consist of a mere layer of molecules [20]) of a liquid layer, with a free surface within and another without, and each of these two surfaces exercises its own independent and coequal tension, and corresponding pressure [21].
If we stretch a tape upon a flat table, whatever be the tension of the tape it obviously exercises no pressure upon the table below. But if we stretch it over a curved surface, a cylinder for instance, it does exercise a downward pressure; and the more curved the surface the greater is this pressure, that is to say the greater is this share of the entire force of tension which is resolved in the downward direction. In mathematical language, the pressure (p) varies directly as the tension (T), and inversely as the radius of curvature (R): that is to say, p = T/R, per unit of surface. [p. 217]
If instead of a cylinder, which is curved only in one direction, we take a case where there are curvatures in two dimensions (as for instance a sphere), then the effects of these must be simply added to one another, and the resulting pressure p is equal to T/R + T/R′ or p = T(1/R + 1/R′) [22].
And if in addition to the pressure p, which is due to surface tension, we have to take into account other pressures, p′, p″, etc., which are due to gravity or other forces, then we may say that the total pressure, P = p′ + p″ + T(1/R + 1/R′). While in some cases, for instance in speaking of the shape of a bird’s egg, we shall have to take account of these extraneous pressures, in the present part of our subject we shall for the most part be able to neglect them.
Our equation is an equation of equilibrium. The resistance to compression,—the pressure outwards,—of our fluid mass, is a constant quantity (P); the pressure inwards, T(1/R + 1/R′), is also constant; and if (unlike the case of the mobile amoeba) the surface be homogeneous, so that T is everywhere equal, it follows that throughout the whole surface 1/R + 1/R′ = C (a constant).
Now equilibrium is attained after the surface contraction has done its utmost, that is to say when it has reduced the surface to the smallest possible area; and so we arrive, from the physical side, at the conclusion that a surface such that 1/R + 1/R′ = C, in other words a surface which has the same mean curvature at all points, is equivalent to a surface of minimal area: and to the same conclusion we may also arrive through purely analytical mathematics. It is obvious that the plane and the sphere are two examples of such surfaces, for in both cases the radius of curvature is everywhere constant, being equal to infinity in the case of the plane, and to some definite magnitude in the case of the sphere.
From the fact that we may extend a soap-film across a ring of wire however fantastically the latter may be bent, we realise that there is no limit to the number of surfaces of minimal area which may be constructed or may be imagined; and while some of these are very complicated indeed, some, for instance a spiral helicoid screw, are relatively very simple. But if we limit ourselves to [p. 218] surfaces of revolution (that is to say, to surfaces symmetrical about an axis), we find, as Plateau was the first to shew, that those which meet the case are very few in number. They are six in all, namely the plane, the sphere, the cylinder, the catenoid, the unduloid, and a curious surface which Plateau called the nodoid.
These several surfaces are all closely related, and the passage from one to another is generally easy. Their mathematical interrelation is expressed by the fact (first shewn by Delaunay [23], in 1841) that the plane curves by whose rotation they are generated are themselves generated as “roulettes” of the conic sections.
Let us imagine a straight line upon which a circle, an ellipse or other conic section rolls; the focus of the conic section will describe a line in some relation to the fixed axis, and this line (or roulette), rotating around the axis, will describe in space one or other of the six surfaces of revolution with which we are dealing.
1917 Source Illustration
Fig. 61.
If we imagine an ellipse so to roll over a line, either of its foci will describe a sinuous or wavy line (Fig. 61B) at a distance alternately maximal and minimal from the axis; and this wavy line, by rotation about the axis, becomes the meridional line of the surface which we call the unduloid. The more unequal the two axes are of our ellipse, the more pronounced will be the sinuosity of the described roulette. If the two axes be equal, then our ellipse becomes a circle, and the path described by its rolling centre is a straight line parallel to the axis (A); and obviously the solid of revolution generated therefrom will be a cylinder. If one axis of our ellipse vanish, while the other remain of finite length, then the ellipse is reduced to a straight line, and its roulette will appear as a succession of semicircles touching one another upon the axis (C); the solid of revolution will be a series of equal spheres. If as before one axis of the ellipse vanish, but the other be infinitely long, then the curve described by the rotation [p. 219] of this latter will be a circle of infinite radius, i.e. a straight line infinitely distant from the axis; and the surface of rotation is now a plane. If we imagine one focus of our ellipse to remain at a given distance from the axis, but the other to become infinitely remote, that is tantamount to saying that the ellipse becomes transformed into a parabola; and by the rolling of this curve along the axis there is described a catenary (D), whose solid of revolution is the catenoid.
Lastly, but this is a little more difficult to imagine, we have the case of the hyperbola.
We cannot well imagine the hyperbola rolling upon a fixed straight line so that its focus shall describe a continuous curve. But let us suppose that the fixed line is, to begin with, asymptotic to one branch of the hyperbola, and that the rolling proceed until the line is now asymptotic to the other branch, that is to say touching it at an infinite distance; there will then be mathematical continuity if we recommence rolling with this second branch, and so in turn with the other, when each has run its course. We shall see, on reflection, that the line traced by one and the same focus will be an “elastic curve” describing a succession of kinks or knots (E), and the solid of revolution described by this meridional line about the axis is the so-called nodoid.
The physical transition of one of these surfaces into another can be experimentally illustrated by means of soap-bubbles, or better still, after the method of Plateau, by means of a large globule of oil, supported when necessary by wire rings, within a fluid of specific gravity equal to its own.
To prepare a mixture of alcohol and water of a density precisely equal to that of the oil-globule is a troublesome matter, and a method devised by Mr C. R. Darling is a great improvement on Plateau’s [24]. Mr Darling uses the oily liquid orthotoluidene, which does not mix with water, has a beautiful and conspicuous red colour, and has precisely the same density as water when both are kept at a temperature of 24° C. We have therefore only to run the liquid into water at this temperature in order to produce beautifully spherical drops of any required size: and by adding [p. 220] a little salt to the lower layers of water, the drop may be made to float or rest upon the denser liquid.
We have already seen that the soap-bubble, spherical to begin with, is transformed into a plane when we relieve its internal pressure and let the film shrink back upon the orifice of the pipe. If we blow a small bubble and then catch it up on a second pipe, so that it stretches between, we may gradually draw the two pipes apart, with the result that the spheroidal surface will be gradually flattened in a longitudinal direction, and the bubble will be transformed into a cylinder. But if we draw the pipes yet farther apart, the cylinder will narrow in the middle into a sort of hourglass form, the increasing curvature of its transverse section being balanced by a gradually increasing negative curvature in the longitudinal section. The cylinder has, in turn, been converted into an unduloid. When we hold a portion of a soft glass tube in the flame, and “draw it out,” we are in the same identical fashion converting a cylinder into an unduloid (Fig. 62A); when on the other hand we stop the end and blow, we again convert the cylinder into an unduloid (B), but into one which is now positively, while the former was negatively curved. The two figures are essentially the same, save that the two halves of the one are reversed in the other.
1917 Source Illustration
Fig. 62.
That spheres, cylinders and unduloids are of the commonest occurrence among the forms of small unicellular organism, or of individual cells in the simpler aggregates, and that in the processes of growth, reproduction and development transitions are frequent from one of these forms to another, is obvious to the naturalist, and we shall deal presently with a few illustrations of these phenomena.
But before we go further in this enquiry, it will be necessary to consider, to some small extent at least, the curvatures of the six different surfaces, that is to say, to determine what modification [p. 221] is required, in each case, of the general equation which applies to them all. We shall find that with this question is closely connected the question of the pressures exercised by, or impinging on the film, and also the very important question of the limitations which, from the nature of the case, exist to prevent the extension of certain of the figures beyond certain bounds. The whole subject is mathematical, and we shall only deal with it in the most elementary way.
We have seen that, in our general formula, the expression 1/R + 1/R′ = C, a constant; and that this is, in all cases, the condition of our surface being one of minimal area. In other words, it is always true for one and all of the six surfaces which we have to consider. But the constant C may have any value, positive, negative, or nil.
In the case of the plane, where R and R′ are both infinite, it is obvious that 1/R + 1/R′ = 0. The expression therefore vanishes, and our dynamical equation of equilibrium becomes P = p. In short, we can only have a plane film, or we shall only find a plane surface in our cell, when on either side thereof we have equal pressures or no pressure at all. A simple case is the plane partition between two equal and similar cells, as in a filament of spirogyra.
In the case of the sphere, the radii are all equal, R = R′; they are also positive, and T (1/R + 1/R′), or 2 T/R, is a positive quantity, involving a positive pressure P, on the other side of the equation.
In the cylinder, one radius of curvature has the finite and positive value R; but the other is infinite. Our formula becomes T/R, to which corresponds a positive pressure P, supplied by the surface-tension as in the case of the sphere, but evidently of just half the magnitude developed in the latter case for a given value of the radius R.
The catenoid has the remarkable property that its curvature in one direction is precisely equal and opposite to its curvature in the other, this property holding good for all points of the surface. That is to say, R = −R′; and the expression becomes
in other words, the surface, as in the case of the plane, has no [p. 222] curvature, and exercises no pressure. There are no other surfaces, save these two, which share this remarkable property; and it follows, as a simple corollary, that we may expect at times to have the catenoid and the plane coexisting, as parts of one and the same boundary system; just as, in a cylindrical drop or cell, the cylinder is capped by portions of spheres, such that the cylindrical and spherical portions of the wall exert equal positive pressures.
In the unduloid, unlike the four surfaces which we have just been considering, it is obvious that the curvatures change from one point to another. At the middle of one of the swollen portions, or “beads,” the two curvatures are both positive; the expression (1/R + 1/R′) is therefore positive, and it is also finite. The film, accordingly, exercises a positive tension inwards, which must be compensated by a finite and positive outward pressure P. At the middle of one of the narrow necks, between two adjacent beads, there is obviously, in the transverse direction, a much stronger curvature than in the former case, and the curvature which balances it is now a negative one. But the sum of the two must remain positive, as well as constant; and we therefore see that the convex or positive curvature must always be greater than the concave or negative curvature at the same point. This is plainly the case in our figure of the unduloid.
The nodoid is, like the unduloid, a continuous curve which keeps altering its curvature as it alters its distance from the axis; but in this case the resultant pressure inwards is negative instead of positive. But this curve is a complicated one, and a full discussion of it would carry us beyond our scope.
1917 Source Illustration
Fig. 63.
In one of Plateau’s experiments, a bubble of oil (protected from gravity by the specific gravity of the surrounding fluid being identical with its own) is balanced between two annuli. It may then be brought to assume the form of Fig. 63, that is to say the form of a cylinder with spherical ends; and there is then everywhere, owing to the convexity of the surface film, a pressure inwards upon the fluid contents of the bubble. If the surrounding liquid be ever so little heavier or lighter than that which constitutes the drop, then the conditions of equilibrium will be accordingly [p. 223] modified, and the cylindrical drop will assume the form of an unduloid (Fig. 64 A, B), with its dilated portion below or above,
1917 Source Illustration
Fig. 64.
as the case may be; and our cylinder may also, of course, be converted into an unduloid either by elongating it further, or by abstracting a portion of its oil, until at length rupture ensues and the cylinder breaks up into two new spherical drops. In all cases alike, the unduloid, like the original cylinder, will be capped by spherical ends, which are the sign, and the consequence, of the positive pressure produced by the curved walls of the unduloid. But if our initial cylinder, instead of being tall, be a flat or dumpy one (with certain definite relations of height to breadth), then new phenomena may be exhibited. For now, if a little oil be cautiously withdrawn from the mass by help of a small syringe, the cylinder may be made to flatten down so that its upper and lower surfaces become plane; which is of itself an indication that the pressure inwards is now nil. But at the very moment when the upper and lower surfaces become plane, it will be found that the sides curve inwards, in the fashion shewn in Fig. 65B. This figure is a catenoid, which, as
1917 Source Illustration
Fig. 65.
we have already seen, is, like the plane itself, a surface exercising no pressure, and which therefore may coexist with the plane as part of one and the same system. We may continue to withdraw more oil from our bubble, drop by drop, and now the upper and lower surfaces dimple down into concave portions of spheres, as the result of the negative internal pressure; and thereupon the peripheral catenoid surface alters its form (perhaps, on this small scale, imperceptibly), and becomes a portion of a nodoid (Fig. 65A). [p. 224] It represents, in fact, that portion of the nodoid, which in Fig. 66 lies between such points as O, P. While it is easy to
1917 Source Illustration
Fig. 66.
draw the outline, or meridional section, of the nodoid (as in Fig. 66), it is obvious that the solid of revolution to be derived from it, can never be realised in its entirety: for one part of the solid figure would cut, or entangle with, another. All that we can ever do, accordingly, is to realise isolated portions of the nodoid.
If, in a sequel to the preceding experiment of Plateau’s, we use solid discs instead of annuli, so as to enable us to exert direct mechanical pressure upon our globule of oil, we again begin by adjusting the pressure of these discs so that the oil assumes the form of a cylinder: our discs, that is to say, are adjusted to exercise a mechanical pressure equal to what in the former case was supplied by the surface-tension of the spherical caps or ends of the bubble. If we now increase the pressure slightly, the peripheral walls will become convexly curved, exercising a precisely corresponding pressure. Under these circumstances the form assumed by the sides of our figure will be that of a portion of an unduloid. If we increase the pressure between the discs, the peripheral surface of oil will bulge out more and more, and will presently constitute a portion of a sphere. But we may continue the process yet further, and within certain limits we shall find that the system remains perfectly stable. What is this new curved surface which has arisen out of the sphere, as the latter was produced from the unduloid? It is no other than a portion of a nodoid, that part which in Fig. 66 lies between such limits as M and N. But this surface, which is concave in both directions towards the surface of the oil within, is exerting a pressure upon the latter, just as did the sphere out of which a moment ago it was transformed; and we had just stated, in considering the previous experiment, that the pressure inwards exerted by the nodoid was a negative one. The explanation of this seeming discrepancy lies in the simple fact that, if we follow the outline [p. 225] of our nodoid curve in Fig. 66 from O, P, the surface concerned in the former case, to M, N, that concerned in the present, we shall see that in the two experiments the surface of the liquid is not homologous, but lies on the positive side of the curve in the one case and on the negative side in the other.
Of all the surfaces which we have been describing, the sphere is the only one which can enclose space; the others can only help to do so, in combination with one another or with the sphere itself. Thus we have seen that, in normal equilibrium, the cylindrical vesicle is closed at either end by a portion of a sphere, and so on. Moreover the sphere is not only the only one of our figures which can enclose a finite space; it is also, of all possible figures, that which encloses the greatest volume with the least area of surface; it is strictly and absolutely the surface of minimal area, and it is therefore the form which will be naturally assumed by a unicellular organism (just as by a raindrop), when it is practically homogeneous and when, like Orbulina floating in the ocean, its surroundings are likewise practically homogeneous and symmetrical. It is only relatively speaking that all the rest are surfaces minimae areae; they are so, that is to say, under the given conditions, which involve various forms of pressure or restraint. Such restraints are imposed, for instance, by the pipes or annuli with the help of which we draw out our cylindrical or unduloid oil-globule or soap-bubble; and in the case of the organic cell, similar restraints are constantly supplied by solidification, partial or complete, local or general, of the cell-wall.
Before we pass to biological illustrations of our surface-tension figures, we have still another preliminary matter to deal with. We have seen from our description of two of Plateau’s classical experiments, that at some particular point one type of surface gives place to another; and again, we know that, when we draw out our soap-bubble into and then beyond a cylinder, there comes a certain definite point at which our bubble breaks in two, and leaves us with two bubbles of which each is a sphere, or a portion of a sphere. In short there are certain definite limits to the dimensions of our figures, within which limits equilibrium is stable but at which it becomes unstable, and above which it [p. 226] breaks down. Moreover in our composite surfaces, when the cylinder for instance is capped by two spherical cups or lenticular discs, there is a well-defined ratio which regulates their respective curvatures, and therefore their respective dimensions. These two matters we may deal with together.
Let us imagine a liquid drop which by appropriate conditions has been made to assume the form of a cylinder; we have already seen that its ends will be terminated by portions of spheres. Since one and the same liquid film covers the sides and ends of the drop (or since one and the same delicate membrane encloses the sides and ends of the cell), we assume the surface-tension (T) to be everywhere identical; and it follows, since the internal fluid-pressure is also everywhere identical, that the expression (1/R + 1/R′) for the cylinder is equal to the corresponding expression, which we may call (1/r + 1/r′), in the case of the terminal spheres. But in the cylinder 1/R′ = 0, and in the sphere 1/r = 1/r′. Therefore our relation of equality becomes 1/R = 2/r, or r = 2 R; that is to say, the sphere in question has just twice the radius of the cylinder of which it forms a cap.
1917 Source Illustration
Fig. 67.
And if Ob, the radius of the sphere, be equal to twice the radius (Oa) of the cylinder, it follows that the angle aOb is an angle of 60°, and bOc is also an angle of 60°; that is to say, the arc bc is equal to (1/3) π. In other words, the spherical disc which (under the given conditions) caps our cylinder, is not a portion taken at haphazard, but is neither more nor less than that portion of a sphere which is subtended by a cone of 60°. Moreover, it is plain that the height of the spherical cap, de,
where R is the radius of our cylinder, or one-half the radius of our spherical cap: in other words the normal height of the spherical cap over the end of the cylindrical cell is just a very little more than one-eighth of the diameter of the cylinder, or of the radius of the [p. 227] sphere. And these are the proportions which we recognise, under normal circumstances, in such a case as the cylindrical cell of Spirogyra where its free end is capped by a portion of a sphere.
Among the many important theoretical discoveries which we owe to Plateau, one to which we have just referred is of peculiar importance: namely that, with the exception of the sphere and the plane, the surfaces with which we have been dealing are only in complete equilibrium within certain dimensional limits, or in other words, have a certain definite limit of stability; only the plane and the sphere, or any portions of a sphere, are perfectly stable, because they are perfectly symmetrical, figures. For experimental demonstration, the case of the cylinder is the simplest. If we produce a liquid film having the form of a cylinder, either by
1917 Source Illustration
Fig. 68.
drawing out a bubble or by supporting between two rings a globule of oil, the experiment proceeds easily until the length of the cylinder becomes just about three times as great as its diameter. But somewhere about this limit the cylinder alters its form; it begins to narrow at the waist, so passing into an unduloid, and the deformation progresses quickly until at last our cylinder breaks in two, and its two halves assume a spherical form. It is found, by theoretical considerations, that the precise limit of stability is at the point when the length of the cylinder is exactly equal to its circumference, that is to say, when L = 2πR, or when the ratio of length to diameter is represented by π.
In the case of the catenoid, Plateau’s experimental procedure was as follows. To support his globule of oil (in, as usual, a mixture of alcohol and water of its own specific gravity), he used [p. 228] a pair of metal rings, which happened to have a diameter of 71 millimetres; and, in a series of experiments, he set these rings apart at distances of 55, 49, 47, 45, and 43 mm. successively. In each case he began by bringing his oil-globule into a cylindrical form, by sucking superfluous oil out of the drop until this result was attained; and always, for the reason with which we are now acquainted, the cylindrical sides were associated with spherical ends to the cylinder. On continuing to withdraw oil in the hope of converting these spherical ends into planes, he found, naturally, that the sides of the cylinder drew in to form a concave surface; but it was by no means easy to get the extremities actually plane: and unless they were so, thus indicating that the surface-pressure of the drop was nil, the curvature of the sides could not be that of a catenoid. For in the first experiment, when the rings were 55 mm. apart, as soon as the convexity of the ends was to a certain extent diminished, it spontaneously increased again; and the transverse constriction of the globule correspondingly deepened, until at a certain point equilibrium set in anew. Indeed, the more oil he removed, the more convex became the ends, until at last the increasing transverse constriction led to the breaking of the oil-globule into two. In the third experiment, when the rings were 47 mm. apart, it was easy to obtain end-surfaces that were actually plane, and they remained so even though more oil was withdrawn, the transverse constriction deepening accordingly. Only after a considerable amount of oil had been sucked up did the plane terminal surface become gradually convex, and presently the narrow waist, narrowing more and more, broke across in the usual way. Finally in the fifth experiment, where the rings were still nearer together, it was again possible to bring the ends of the oil-globule to a plane surface, as in the third and fourth experiments, and to keep this surface plane in spite of some continued withdrawal of oil. But very soon the ends became gradually concave, and the concavity deepened as more and more oil was withdrawn, until at a certain limit, the whole oil-globule broke up in general disruption.
We learn from this that the limiting size of the catenoid was reached when the distance of the supporting rings was to their diameter as 47 to 71, or, as nearly as possible, as two to three; [p. 229] and as a matter of fact it can be shewn that 2/3 is the true theoretical value. Above this limit of 2/3, the inevitable convexity of the end-surfaces shows that a positive pressure inwards is being exerted by the surface film, and this teaches us that the sides of the figure actually constitute not a catenoid but an unduloid, whose spontaneous changes tend to a form of greater stability. Below the 2/3 limit the catenoid surface is essentially unstable, and the form into which it passes under certain conditions of disturbance such as that of the excessive withdrawal of oil, is that of a nodoid (Fig. 65A).
The unduloid has certain peculiar properties as regards its limitations of stability. But as to these we need mention two facts only: (1) that when the unduloid, which we produce with our soap-bubble or our oil-globule, consists of the figure containing a complete constriction, it has somewhat wide limits of stability; but (2) if it contain the swollen portion, then equilibrium is limited to the condition that the figure consists simply of one complete unduloid, that is to say that its ends are constituted by the narrowest portions, and its middle by the widest portion of the entire curve. The theoretical proof of this latter fact is difficult, but if we take the proof for granted, the fact will serve to throw light on what we have learned regarding the stability of the cylinder. For, when we remember that the meridional section of our unduloid is generated by the rolling of an ellipse upon a straight line in its own plane, we shall easily see that the length of the entire unduloid is equal to the circumference of the generating ellipse. As the unduloid becomes less and less sinuous in outline, it gradually approaches, and in time reaches, the form of a cylinder; and correspondingly, the ellipse which generated it has its foci more and more approximated until it passes into a circle. The cylinder of a length equal to the circumference of its generating circle is therefore precisely homologous to an unduloid whose length is equal to the circumference of its generating ellipse; and this is just what we recognise as constituting one complete segment of the unduloid.
While the figures of equilibrium which are at the same time surfaces of revolution are only six in number, there is an infinite [p. 230] number of figures of equilibrium, that is to say of surfaces of constant mean curvature, which are not surfaces of revolution; and it can be shewn mathematically that any given contour can be occupied by a finite portion of some one such surface, in stable equilibrium. The experimental verification of this theorem lies in the simple fact (already noted) that however we may bend a wire into a closed curve, plane or not plane, we may always, under appropriate precautions, fill the entire area with an unbroken film.
Of the regular figures of equilibrium, that is to say surfaces of constant mean curvature, apart from the surfaces of revolution which we have discussed, the helicoid spiral is the most interesting to the biologist. This is a helicoid generated by a straight line perpendicular to an axis, about which it turns at a uniform rate while at the same time it slides, also uniformly, along this same axis. At any point in this surface, the curvatures are equal and of opposite sign, and the sum of the curvatures is accordingly nil. Among what are called “ruled surfaces” (which we may describe as surfaces capable of being defined by a system of stretched strings), the plane and the helicoid are the only two whose mean curvature is null, while the cylinder is the only one whose curvature is finite and constant. As this simplest of helicoids corresponds, in three dimensions, to what in two dimensions is merely a plane (the latter being generated by the rotation of a straight line about an axis without the superadded gliding motion which generates the helicoid), so there are other and much more complicated helicoids which correspond to the sphere, the unduloid and the rest of our figures of revolution, the generating planes of these latter being supposed to wind spirally about an axis. In the case of the cylinder it is obvious that the resulting figure is indistinguishable from the cylinder itself. In the case of the unduloid we obtain a grooved spiral, such as we may meet with in nature (for instance in Spirochætes, Bodo gracilis, etc.), and which accordingly it is of interest to us to be able to recognise as a surface of minimal area or constant curvature.
The foregoing considerations deal with a small part only of the theory of surface tension, or of capillarity: with that part, namely, which relates to the forms of surface which are [p. 231] capable of subsisting in equilibrium under the action of that force, either of itself or subject to certain simple constraints. And as yet we have limited ourselves to the case of a single surface, or of a single drop or bubble, leaving to another occasion a discussion of the forms assumed when such drops or vesicles meet and combine together. In short, what we have said may help us to understand the form of a cell,—considered, as with certain limitations we may legitimately consider it, as a liquid drop or liquid vesicle; the conformation of a tissue or cell-aggregate must be dealt with in the light of another series of theoretical considerations. In both cases, we can do no more than touch upon the fringe of a large and difficult subject. There are many forms capable of realisation under surface tension, and many of them doubtless to be recognised among organisms, which we cannot touch upon in this elementary account. The subject is a very general one; it is, in its essence, more mathematical than physical; it is part of the mathematics of surfaces, and only comes into relation with surface tension, because this physical phenomenon illustrates and exemplifies, in a concrete way, most of the simple and symmetrical conditions with which the general mathematical theory is capable of dealing. And before we pass to illustrate by biological examples the physical phenomena which we have described, we must be careful to remember that the physical conditions which we have hitherto presupposed will never be wholly realised in the organic cell. Its substance will never be a perfect fluid, and hence equilibrium will be more or less slowly reached; its surface will seldom be perfectly homogeneous, and therefore equilibrium will (in the fluid condition) seldom be perfectly attained; it will very often, or generally, be the seat of other forces, symmetrical or unsymmetrical; and all these causes will more or less perturb the effects of surface tension acting by itself. But we shall find that, on the whole, these effects of surface tension though modified are not obliterated nor even masked; and accordingly the phenomena to which I have devoted the foregoing pages will be found manifestly recurring and repeating themselves among the phenomena of the organic cell.
In a spider’s web we find exemplified several of the principles [p. 232] of surface tension which we have now explained. The thread is formed out of the fluid secretion of a gland, and issues from the body as a semi-fluid cylinder, that is to say in the form of a surface of equilibrium, the force of expulsion giving it its elongation and that of surface tension giving it its circular section. It is prevented, by almost immediate solidification on exposure to the air, from breaking up into separate drops or spherules, as it would otherwise tend to do as soon as the length of the cylinder had passed its limit of stability. But it is otherwise with the sticky secretion which, coming from another gland, is simultaneously poured over the issuing thread when it is to form the spiral portion of the web. This latter secretion is more fluid than the first, and retains its fluidity for a very much longer time, finally drying up after several hours. By capillarity it “wets” the thread, spreading itself over it in an even film, which film is now itself a cylinder. But this liquid cylinder has its limit of stability when its length equals its own circumference, and therefore just at the points so defined it tends to disrupt into separate segments: or rather, in the actual case, at points somewhat more distant, owing to the imperfect fluidity of the viscous film, and still more to the frictional drag upon it of the inner solid cylinder, or thread, with which it is in contact. The cylinder disrupts in the usual manner, passing first into the wavy outline of an unduloid, whose swollen portions swell more and more till the contracted parts break asunder, and we arrive at a series of spherical drops or beads, of equal size, strung at equal intervals along the thread. If we try to spread varnish over a thin stretched wire, we produce automatically the same identical result [25]; unless our varnish be such as to dry almost instantaneously, it gathers into beads, and do what we can, we fail to spread it smooth. It follows that, according to the viscidity and drying power of the varnish, the process may stop or seem to stop at any point short of the formation of the perfect spherules; it is quite possible, therefore, that as our final stage we may only obtain half-formed beads, or the wavy outline of an unduloid. The formation of the beads may be facilitated or hastened by jerking the stretched thread, as the spider actually does: the [p. 233] effect of the jerk being to disturb and destroy the unstable equilibrium of the viscid cylinder [26]. Another very curious phenomenon here presents itself.
In Plateau’s experimental separation of a cylinder of oil into two spherical portions, it was noticed that, when contact was nearly broken, that is to say when the narrow neck of the unduloid had become very thin, the two spherical bullae, instead of absorbing the fluid out of the narrow neck into themselves as they had done with the preceding portion, drew out this small remaining part of the liquid into a thin thread as they completed their spherical form and consequently receded from one another: the reason being that, after the thread or “neck” has reached a certain tenuity, the internal friction of the fluid prevents or retards its rapid exit from the little thread to the adjacent spherule. It is for the same reason that we are able to draw a glass rod or tube, which we have heated in the middle, into a long and uniform cylinder or thread, by quickly separating the two ends. But in the case of the glass rod, the long thin intermediate cylinder quickly cools and solidifies, while in the ordinary separation of a liquid cylinder the corresponding intermediate cylinder remains liquid; and therefore, like any other liquid cylinder, it is liable to break up, provided that its dimensions exceed the normal limit of stability. And its length is generally such that it breaks at two points, thus leaving two terminal portions continuous with the spheres and becoming confluent with these, and one median portion which resolves itself into a comparatively tiny spherical drop, midway between the original and larger two. Occasionally, the same process of formation of a connecting thread repeats itself a second time, between the small intermediate spherule and the large spheres; and in this case we obviously obtain two additional spherules, still smaller in size, and lying one on either side of our first little one. This whole phenomenon, of equal and regularly interspaced beads, often with little beads regularly interspaced between the larger ones, and possibly also even a third series of still smaller beads regularly intercalated, may be easily observed in a spider’s web, such as that of Epeira, very often with beautiful regularity,—which [p. 234] naturally, however, is sometimes interrupted and disturbed owing to a slight want of homogeneity in the secreted fluid; and the same phenomenon is repeated on a grosser scale when the web is bespangled with dew, and every thread bestrung with pearls innumerable. To the older naturalists, these regularly arranged and beautifully formed globules on the spider’s web were a cause of great wonder and admiration. Blackwall, counting some twenty globules in a tenth of an inch, calculated that a large garden-spider’s web comprised about 120,000 globules; the net was spun and finished in about forty minutes, and Blackwall was evidently filled with astonishment at the skill and quickness with which the spider manufactured these little beads. And no wonder, for according to the above estimate they had to be made at the rate of about 50 per second [27].
1917 Source Illustration
Fig. 69. Hair of Trianea, in glycerine. (After Berthold.)
The little delicate beads which stud the long thin pseudopodia of a foraminifer, such as Gromia, or which in like manner appear upon the cylindrical film of protoplasm which covers the long radiating spicules of Globigerina, represent an identical phenomenon. Indeed there are many cases, in which we may study in a protoplasmic filament the whole process of formation of such beads. If we squeeze out on to a slide the viscid contents of a mistletoe berry, the long sticky threads into which the substance runs shew the whole phenomenon particularly well. Another way to demonstrate it was noticed many years ago by Hofmeister and afterwards explained by Berthold. The hairs of certain water-plants, such as Hydrocharis or Trianea, constitute very long cylindrical cells, the protoplasm being supported, and maintained in equilibrium by its contact with the cell-wall. But if we immerse the filament in some dense fluid, a little sugar-solution for instance, or dilute glycerine, the cell-sap tends to diffuse outwards, the protoplasm parts company with its surrounding and supporting wall, [p. 235] and lies free as a protoplasmic cylinder in the interior of the cell. Thereupon it immediately shews signs of instability, and commences to disrupt. It tends to gather into spheres, which however, as in our illustration, may be prevented by their narrow quarters from assuming the complete spherical form; and in between these spheres, we have more or less regularly alternate ones, of smaller size [28]. Similar, but less regular, beads or droplets may be caused to appear, under stimulation by an alternating current, in the protoplasmic threads within the living cells of the hairs of Tradescantia. The explanation usually given is, that the viscosity of the protoplasm is reduced, or its fluidity increased; but an increase of the surface tension would seem a more likely reason [29].
1917 Source Illustration
Fig. 70. Phases of a Splash. (From Worthington.)
We may take note here of a remarkable series of phenomena, which, though they seem at first sight to be of a very different order, are closely related to the phenomena which attend and which bring about the breaking-up of a liquid cylinder or thread.
In some of Mr Worthington’s most beautiful experiments on [p. 236] splashes, it was found that the fall of a round pebble into water from a considerable height, caused the rise of a filmy sheet of water in the form of a cup or cylinder; and the edge of this cylindrical film tended to be cut up into alternate lobes and notches, and the prominent lobes or “jets” tended, in more extreme cases, to break off or to break up into spherical beads (Fig. 70) [30]. A precisely similar appearance is seen, on a great scale, in the thin edge of a breaking wave: when the smooth cylindrical edge, at a given moment, shoots out an array of tiny jets which break up into the droplets which constitute “spray” (Fig. 71, a, b). We are at once reminded of the beautifully symmetrical notching on the calycles of many hydroids, which little cups before they became stiff and rigid had begun their existence as liquid or semi-liquid films.
1917 Source Illustration
Fig. 71. A breaking wave. (From Worthington.)
The phenomenon is two-fold. In the first place, the edge of our tubular or crater-like film forms a liquid ring or annulus, which is closely comparable with the liquid thread or cylinder which we have just been considering, if only we conceive the thread to be bent round into the ring. And accordingly, just as the thread spontaneously segments, first into an unduloid, and then into separate spherical drops, so likewise will the edge of our annulus tend to do. This phase of notching, or beading, of the edge of the film is beautifully seen in many of Worthington’s experiments [31]. In the second place, the very fact of the rising of the crater means that liquid is flowing up from below towards the rim; and the segmentation of the rim means that channels of easier flow are [p. 237] created, along which the liquid is led, or is driven, into the protuberances: and these are thus exaggerated into the jets or arms which are sometimes so conspicuous at the edge of the crater. In short, any film or film-like cup, fluid or semi-fluid in its consistency, will, like the straight liquid cylinder, be unstable: and its instability will manifest itself (among other ways) in a tendency to segmentation or notching of the edge; and just such a peripheral notching is a conspicuous feature of many minute organic cup-like structures. In the case of the hydroid calycle (Fig. 72), we are led to the conclusion that the two common and conspicuous features of notching or indentation of the cup, and of constriction or annulation of the long cylindrical stem, are phenomena of the same order and are due to surface-tension in both cases alike.
1917 Source Illustration
Fig. 72. Calycles of Campanularian zoophytes. (A) C. integra; (B) C. groenlandica; (C) C. bispinosa; (D) C. raridentata.
Another phenomenon displayed in the same experiments is the formation of a rope-like or cord-like thickening of the edge of the annulus. This is due to the more or less sudden checking at the rim of the flow of liquid rising from below: and a similar peripheral thickening is frequently seen, not only in some of our hydroid cups, but in many Vorticellas (cf. Fig. 75), and other organic cup-like conformations. A perusal of Mr Worthington’s book will soon suggest that these are not the only manifestations of surface-tension in connection with splashes which present curious resemblances and analogies to phenomena of organic form.
The phenomena of an ordinary liquid splash are so swiftly [p. 238] transitory that their study is only rendered possible by “instantaneous” photography: but this excessive rapidity is not an essential part of the phenomenon. For instance, we can repeat and demonstrate many of the simpler phenomena, in a permanent or quasi-permanent form, by splashing water on to a surface of dry sand, or by firing a bullet into a soft metal target. There is nothing, then, to prevent a slow and lasting manifestation, in a viscous medium such as a protoplasmic organism, of phenomena which appear and disappear with prodigious rapidity in a more mobile liquid. Nor is there anything peculiar in the “splash” itself; it is simply a convenient method of setting up certain motions or currents, and producing certain surface-forms, in a liquid medium,—or even in such an extremely imperfect fluid as is represented (in another series of experiments) by a bed of sand. Accordingly, we have a large range of possible conditions under which the organism might conceivably display configurations analogous to, or identical with, those which Mr Worthington has shewn us how to exhibit by one particular experimental method.
To one who has watched the potter at his wheel, it is plain that the potter’s thumb, like the glass-blower’s blast of air, depends for its efficacy upon the physical properties of the medium on which it operates, which for the time being is essentially a fluid. The cup and the saucer, like the tube and the bulb, display (in their simple and primitive forms) beautiful surfaces of equilibrium as manifested under certain limiting conditions. They are neither more nor less than glorified “splashes,” formed slowly, under conditions of restraint which enhance or reveal their mathematical symmetry. We have seen, and we shall see again before we are done, that the art of the glass-blower is full of lessons for the naturalist as also for the physicist: illustrating as it does the development of a host of mathematical configurations and organic conformations which depend essentially on the establishment of a constant and uniform pressure within a closed elastic shell or fluid envelope. In like manner the potter’s art illustrates the somewhat obscurer and more complex problems (scarcely less frequent in biology) of a figure of equilibrium which is an open surface, or solid, of revolution. It is clear, at the same time, that the two series of problems are closely akin; for the [p. 239] glass-blower can make most things that the potter makes, by cutting off portions of his hollow ware. And besides, when this fails, and the glass-blower, ceasing to blow, begins to use his rod to trim the sides or turn the edges of wineglass or of beaker, he is merely borrowing a trick from the craft of the potter.
It would be venturesome indeed to extend our comparison with these liquid surface-tension phenomena from the cup or calycle of the hydrozoon to the little hydroid polype within: and yet I feel convinced that there is something to be learned by such a comparison, though not without much detailed consideration and mathematical study of the surfaces concerned. The cylindrical body of the tiny polype, the jet-like row of tentacles, the beaded annulations which these tentacles exhibit, the web-like film which sometimes (when they stand a little way apart) conjoins their bases, the thin annular film of tissue which surrounds the little organism’s mouth, and the manner in which this annular “peristome” contracts [32], like a shrinking soap-bubble, to close the aperture, are every one of them features to which we may find a singular and striking parallel in the surface-tension phenomena which Mr Worthington has illustrated and demonstrated in the case of the splash.
Here however, we may freely confess that we are for the present on the uncertain ground of suggestion and conjecture; and so must we remain, in regard to many other simple and symmetrical organic forms, until their form and dynamical stability shall have been investigated by the mathematician: in other words, until the mathematicians shall have become persuaded that there is an immense unworked field wherein they may labour, in the detailed study of organic form.
According to Plateau, the viscidity of the liquid, while it helps to retard the breaking up of the cylinder and so increases the length of the segments beyond that which theory demands, has nevertheless less influence in this direction than we might have expected. On the other hand, any external support or adhesion, such as contact with a solid body, will be equivalent to a reduction of surface-tension and so will very greatly increase the [p. 240] stability of our cylinder. It is for this reason that the mercury in our thermometer tubes does not as a rule separate into drops, though it occasionally does so, much to our inconvenience. And again it is for this reason that the protoplasm in a long and growing tubular or cylindrical cell does not necessarily divide into separate cells and internodes, until the length of these far exceeds the theoretic limits. Of course however and whenever it does so, we must, without ever excluding the agency of surface tension, remember that there may be other forces affecting the latter, and accelerating or retarding that manifestation of surface tension by which the cell is actually rounded off and divided.
In most liquids, Plateau asserts that, on the average, the influence of viscosity is such as to cause the cylinder to segment when its length is about four times, or at most from four to six times that of its diameter: instead of a fraction over three times as, in a perfect fluid, theory would demand. If we take it at four times, it may then be shewn that the resulting spheres would have a diameter of about 1·8 times, and their distance apart would be equal to about 2·2 times the diameter of the original cylinder. The calculation is not difficult which would shew how these numbers are altered in the case of a cylinder formed around a solid core, as in the case of the spider’s web. Plateau has also made the interesting observation that the time taken in the process of division of the cylinder is directly proportional to the diameter of the cylinder, while varying considerably with the nature of the liquid. This question, of the time occupied in the division of a cell or filament, in relation to the dimensions of the latter, has not so far as I know been enquired into by biologists.
From the simple fact that the sphere is of all surfaces that whose surface-area for a given volume is an absolute minimum, we have already seen it to be plain that it is the one and only figure of equilibrium which will be assumed under surface-tension by a drop or vesicle, when no other disturbing factors are present. One of the most important of these disturbing factors will be introduced, in the form of complicated tensions and pressures, when one drop is in contact with another drop and when a system of intermediate films or partition walls is developed between them. [p. 241] This subject we shall discuss later, in connection with cell-aggregates or tissues, and we shall find that further theoretical considerations are needed as a preliminary to any such enquiry. Meanwhile let us consider a few cases of the forms of cells, either solitary, or in such simple aggregates that their individual form is little disturbed thereby.
Let us clearly understand that the cases we are about to consider are those cases where the perfect symmetry of the sphere is replaced by another symmetry, less complete, such as that of an ellipsoidal or cylindrical cell. The cases of asymmetrical deformation or displacement, such as is illustrated in the production of a bud or the development of a lateral branch, are much simpler. For here we need only assume a slight and localised variation of surface-tension, such as may be brought about in various ways through the heterogeneous chemistry of the cell; to this point we shall return in our chapter on Adsorption. But the diffused and graded asymmetry of the system, which brings about for instance the ellipsoidal shape of a yeast-cell, is another matter.
If the sphere be the one surface of complete symmetry and therefore of independent equilibrium, it follows that in every cell which is otherwise conformed there must be some definite force to cause its departure from sphericity; and if this cause be the very simple and obvious one of the resistance offered by a solidified envelope, such as an egg-shell or firm cell-wall, we must still seek for the deforming force which was in action to bring about the given shape, prior to the assumption of rigidity. Such a cause may be either external to, or may lie within, the cell itself. On the one hand it may be due to external pressure or to some form of mechanical restraint: as it is in all our experiments in which we submit our bubble to the partial restraint of discs or rings or more complicated cages of wire; and on the other hand it may be due to intrinsic causes, which must come under the head either of differences of internal pressure, or of lack of homogeneity or isotropy in the surface itself [33]. [p. 242]
Our full formula of equilibrium, or equation to an elastic surface, is P = pe + (T/R + T′/R′), where P is the internal pressure, pe any extraneous pressure normal to the surface, R, R′ the radii of curvature at a point, and T, T′, the corresponding tensions, normal to one another, of the envelope.
Now in any given form which we are seeking to account for, R, R′ are known quantities; but all the other factors of the equation are unknown and subject to enquiry. And somehow or other, by this formula, we must account for the form of any solitary cell whatsoever (provided always that it be not formed by successive stages of solidification), the cylindrical cell of Spirogyra, the ellipsoidal yeast-cell, or (as we shall see in another chapter) the shape of the egg of any bird. In using this formula hitherto, we have taken it in a simplified form, that is to say we have made several limiting assumptions. We have assumed that P was simply the uniform hydrostatic pressure, equal in all directions, of a body of liquid; we have assumed that the tension T was simply due to surface-tension in a homogeneous liquid film, and was therefore equal in all directions, so that T = T′; and we have only dealt with surfaces, or parts of a surface, where extraneous pressure, pn, was non-existent. Now in the case of a bird’s egg, the external pressure pn, that is to say the pressure exercised by the walls of the oviduct, will be found to be a very important factor; but in the case of the yeast-cell or the Spirogyra, wholly immersed in water, no such external pressure comes into play. We are accordingly left, in such cases as these last, with two hypotheses, namely that the departure from a spherical form is due to inequalities in the internal pressure P, or else to inequalities in the tension T, that is to say to a difference between T and T′. In other words, it is theoretically possible that the oval form of a yeast-cell is due to a greater internal pressure, a greater “tendency to grow,” in the direction of the longer axis of the ellipse, or alternatively, that with equal and symmetrical tendencies to growth there is associated a difference of external resistance in [p. 243] respect of the tension of the cell-wall. Now the former hypothesis is not impossible; the protoplasm is far from being a perfect fluid; it is the seat of various internal forces, sometimes manifestly polar; and accordingly it is quite possible that the internal forces, osmotic and other, which lead to an increase of the content of the cell and are manifested in pressure outwardly directed upon its wall may be unsymmetrical, and such as to lead to a deformation of what would otherwise be a simple sphere. But while this hypothesis is not impossible, it is not very easy of acceptance. The protoplasm, though not a perfect fluid, has yet on the whole the properties of a fluid; within the small compass of the cell there is little room for the development of unsymmetrical pressures; and, in such a case as Spirogyra, where a large part of the cavity is filled by a fluid and watery cell-sap, the conditions are still more obviously those under which a uniform hydrostatic pressure is to be expected. But in variations of T, that is to say of the specific surface-tension per unit area, we have an ample field for all the various deformations with which we shall have to deal. Our condition now is, that (T/R + T′/R′) = a constant; but it no longer follows, though it may still often be the case, that this will represent a surface of absolute minimal area. As soon as T and T′ become unequal, it is obvious that we are no longer dealing with a perfectly liquid surface film; but its departure from a perfect fluidity may be of all degrees, from that of a slight non-isotropic viscosity to the state of a firm elastic membrane [34]. And it matters little whether this viscosity or semi-rigidity be manifested in the self-same layer which is still a part of the protoplasm of the cell, or in a layer which is completely differentiated into a distinct and separate membrane. As soon as, by secretion or “adsorption,” the molecular constitution of the surface layer is altered, it is clearly conceivable that the alteration, or the secondary chemical changes which follow it, may be such as to produce an anisotropy, and to render the molecular forces less capable in one direction than another of exerting that contractile force by which they are striving to reduce to an absolute minimum the [p. 244] surface area of the cell. A slight inequality in two opposite directions will produce the ellipsoid cell, and a very great inequality will give rise to the cylindrical cell [35].
I take it therefore, that the cylindrical cell of Spirogyra, or any other cylindrical cell which grows in freedom from any manifest external restraint, has assumed that particular form simply by reason of the molecular constitution of its developing surface-membrane; and that this molecular constitution was anisotropous, in such a way as to render extension easier in one direction than another.
Such a lack of homogeneity or of isotropy, in the cell-wall is often rendered visible, especially in plant-cells, in various ways, in the form of concentric lamellae, annular and spiral striations, and the like.
But this phenomenon, while it brings about a certain departure from complete symmetry, is still compatible with, and coexistent with, many of the phenomena which we have seen to be associated with surface-tension. The symmetry of tensions still leaves the cell a solid of revolution, and its surface is still a surface of equilibrium. The fluid pressure within the cylinder still causes the film or membrane which caps its ends to be of a spherical form. And in the young cell, where the surface pellicle is absent or but little differentiated, as for instance in the oögonium of Achlya, or in the young zygospore of Spirogyra, we always see the tendency of the entire structure towards a spherical form reasserting itself: unless, as in the latter case, it be overcome by direct compression within the cylindrical mother-cell. Moreover, in those cases where the adult filament consists of cylindrical cells, we see that the young, germinating spore, at first spherical, very soon assumes with growth an elliptical or ovoid form: the direct result of an incipient anisotropy of its envelope, which when more developed will convert the ovoid into a cylinder. We may also notice that a truly cylindrical cell is comparatively rare; for in most cases, what we call a cylindrical cell shews a distinct bulging of its sides; it is not truly a cylinder, but a portion of a spheroid or ellipsoid. [p. 245]
Unicellular organisms in general, including the protozoa, the unicellular cryptogams, the various bacteria, and the free, isolated cells, spores, ova, etc. of higher organisms, are referable for the most part to a very small number of typical forms; but besides a certain number of others which may be so referable, though obscurely, there are obviously many others in which either no symmetry is to be recognized, or in which the form is clearly not one of equilibrium. Among these latter we have Amoeba itself, and all manner of amoeboid organisms, and also many curiously shaped cells, such as the Trypanosomes and various other aberrant Infusoria. We shall return to the consideration of these; but in the meanwhile it will suffice to say that, as their surfaces are not equilibrium-surfaces, so neither are the living cells themselves in any stable equilibrium. On the contrary, they are in continual flux and movement, each portion of the surface constantly changing its form, and passing from one phase to another of an equilibrium which is never stable for more than a moment. The former class, which rest in stable equilibrium, must fall (as we have seen) into two classes,—those whose equilibrium arises from liquid surface-tension alone, and those in whose conformation some other pressure or restraint has been superimposed upon ordinary surface-tension.
To the fact that these little organisms belong to an order of magnitude in which form is mainly, if not wholly, conditioned and controlled by molecular forces, is due the limited range of forms which they actually exhibit. These forms vary according to varying physical conditions. Sometimes they do so in so regular and orderly a way that we instinctively explain them merely as “phases of a life-history,” and leave physical properties and physical causation alone: but many of their variations of form we treat as exceptional, abnormal, decadent or morbid, and are apt to pass these over in neglect, while we give our attention to what we suppose to be the typical or “characteristic” form or attitude. In the case of the smallest organisms, the bacteria, micrococci, and so forth, the range of form is especially limited, owing to their minuteness, the powerful pressure which their highly curved surfaces exert, and the comparatively homogeneous nature of their substance. But within their narrow range of possible diversity [p. 246] these minute organisms are protean in their changes of form. A certain species will not only change its shape from stage to stage of its little “cycle” of life; but it will be remarkably different in outward form according to the circumstances under which we find it, or the histological treatment to which we submit it. Hence the pathological student, commencing the study of bacteriology, is early warned to pay little heed to differences of form, for purposes of recognition or specific identification. Whatever grounds we may have for attributing to these organisms a permanent or stable specific identity (after the fashion of the higher plants and animals), we can seldom safely do so on the ground of definite and always recognisable form: we may often be inclined, in short, to ascribe to them a physiological (sometimes a “pathogenic”), rather than a morphological specificity.
1917 Source Illustration
Fig. 73. A flagellate “monad,” Distigma proteus, Ehr. (After Saville Kent.)
1917 Source Illustration
Fig. 74. Noctiluca miliaris.
Among the Infusoria, we have a small number of forms whose symmetry is distinctly spherical, for instance among the small flagellate monads; but even these are seldom actually spherical except when we see them in a non-flagellate and more or less encysted or “resting” stage. In this condition, it need hardly be remarked that the spherical form is common and general among a great variety of unicellular organisms. When our little monad developes a flagellum, that is in itself an indication of “polarity” or symmetrical non-homogeneity of the cell; and accordingly, we [p. 247] usually see signs of an unequal tension of the membrane in the neighbourhood of the base of the flagellum. Here the tension is usually less than elsewhere, and the radius of curvature is accordingly less: in other words that end of the cell is drawn out to a tapering point (Fig. 73). But sometimes it is the other way, as in Noctiluca, where the large flagellum springs from a depression in the otherwise uniformly rounded cell. In this case the explanation seems to lie in the many strands of radiating protoplasm which converge upon this point, and may be supposed to keep it relatively fixed by their viscosity, while the rest of the cell-surface is free to expand (Fig. 74).
1917 Source Illustration
Fig. 75. Various species of Vorticella. (Mostly after Saville Kent.)
A very large number of Infusoria represent unduloids, or portions of unduloids, and this type of surface appears and reappears in a great variety of forms. The cups of the various species of Vorticella (Fig. 75) are nothing in the world but a beautiful series of unduloids, or partial unduloids, in every gradation from a form that is all but cylindrical to one that is all but a perfect sphere. These unduloids are not completely symmetrical, but they are such unduloids as develop themselves when we suspend an oil-globule between two unequal rings, or blow a soap-bubble between two unequal pipes; for, just as in these cases, the surface of our Vorticella bell finds its terminal supports, on the one hand in its attachment to its narrow stalk, and on the other in the thickened ring from which spring its circumoral cilia. And here let me say, that a point or zone from which cilia arise would seem always to have a peculiar relation to the surrounding tensions. It usually forms a sharp salient, a prominent point or ridge, as in our little monads of Fig. 73; shewing that, in its formation, the surface tension had here locally diminished. But if such a ridge or fillet consolidate in the least degree, it becomes a source of strength, and a point d’appui for the adjacent film. We shall deal with this point again in the next chapter. [p. 248]
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Fig. 76. Various species of Salpingoeca.
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Fig. 77. Various species of Tintinnus, Dinobryon and Codonella. (After Saville Kent and others.)
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Fig. 78. Vaginicola.
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Fig. 79. Folliculina.
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Fig. 80. Trachelophyllum. (After Wreszniowski.)
Precisely the same series of unduloid forms may be traced in even greater variety among various other families or genera of the 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 [p. 249] 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 protoplasm 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 of ratio or degree which exists between two circles of different diameter, or two lines of unequal length. [p. 250]
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 Naturae ordo.” In like manner the physicist records, and is entitled to record, his many hundred “species” of snow-crystals [36], or of crystals of calcium carbonate. But regarding these latter species, the physicist makes no assumptions: he records them simpliciter, 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 [p. 251] 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 [37]. 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 slight 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, is due to the fact of their being related by descent [38].” 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 lion 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 [p. 252] 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.
1917 Source Illustration
Fig. 81.
Among the more aberrant forms of Infusoria is a little species known as Trichodina pedicidus, a parasite on the Hydra, or fresh-water 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 from which these cilia spring may be taken, as we have already said, to play the part of a strengthening “fillet.” The circular base of the 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 [39] [p. 253] 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.
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Fig. 82. Dinenympha gracilis, Leidy.
1917 Source Illustration
Fig. 83.
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 [p. 254] 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 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 [40]. [p. 255]
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 regular 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 [41]” [p. 256] seems to shew the natural, physical tendency of the long semifluid cylinder of protoplasm to contract, at its limit of stability, into 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.
1917 Source Illustration
Fig. 84. Various species of Lagena. (After Brady.)
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 calcareous Foraminifera are all comprised by naturalists within the genus Lagena. This large and varied genus consists of “flask-shaped” 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 [p. 257] 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 [42].
1917 Source Illustration
Fig. 85. (After Darling.)
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 [p. 258] 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 [43]. 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 [p. 259] 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 soap-bubble. 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 [44]. 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.
1917 Source Illustration
Fig. 86.
The folded or pleated pattern is doubtless to be explained, in a general way, by the shrinkage of a surface-film under certain [p. 260] 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 [45], 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 former 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 [46].
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 Worthington has so beautifully shewn [47]. [p. 261]
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 T + 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/r 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 r,—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.
1917 Source Illustration
Fig. 87. Nodosaria scalaris, Batsch.
1917 Source Illustration
Fig. 88. Gonangia of Campanularians. (a) C. gracilis; (b) C. grandis. (After Allman.)
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 [p. 262] 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, flexure of a wall in a transverse direction will be very difficult, while flexure in a longitudinal direction will be comparatively easy; just as, in the case of a long narrow strip of iron, we may easily bend it into folds running transversely to its long axis, but not the other way. The manner in which our little Lagena-shell tends to fold or wrinkle, longitudinally in its wider part, and transversely or annularly in its narrow neck, is thus completely and easily explained.
An identical phenomenon is apt to occur in the little flask-shaped gonangia, or reproductive capsules, of some of the hydroid zoophytes. In the annexed drawings of these gonangia in two species of Campanularia, we see that in one case the little vesicle [p. 263] has the flask-shaped or unduloid configuration of a Lagena; and here the walls of the flask are longitudinally fluted, just after the manner we have witnessed in the latter genus. But in the other Campanularian the vesicles are long, narrow and tubular, and here a transverse folding or pleating takes the place of the longitudinally fluted pattern. And the very form of the folds or pleats is enough to suggest that we are not dealing here with a simple phenomenon of surface-tension, but with a condition in which surface-tension and stiffness are both present, and play their parts in the resultant form.
1917 Source Illustration
Fig. 89. Various Foraminifera (after Brady), a, Nodosaria simplex; b, N. pygmaea; c, N. costulata; e, N. hispida; f, N. elata; d, Rheophax (Lituola) distans; g, Sagrina virgata.
Passing from the solitary flask-shaped cell of Lagena, we have, in another series of forms, a constricted cylinder, or succession of unduloids; such as are represented in Fig. 89, illustrating certain species of Nodosaria, Rheophax and Sagrina. In some of these cases, and certainly in that of the arenaceous genus Rheophax, we have to do with the ordinary phenomenon of a segmenting or partially segmenting cylinder. But in others, the structure is not developed out of a continuous protoplasmic cylinder, but as we can see by examining the interior of the shell, it has been formed in successive stages, beginning with a simple unduloid “Lagena,” about whose neck, after its solidification, another drop of protoplasm accumulated, and in turn assumed the unduloid, or lagenoid, form. The chains of interconnected bubbles which [p. 264] Morey and Draper made many years ago of melted resin are a very similar if not identical phenomenon [48].
There now remain for our consideration, among the Protozoa, the great oceanic group of the Radiolaria, and the little group of their freshwater allies, the Heliozoa. In nearly all these forms we have this specific chemical difference from the Foraminifera, that when they secrete, as they generally do secrete, a hard skeleton, it is composed of silica instead of lime. These organisms and the various beautiful and highly complicated skeletal fabrics which they develop give us many interesting illustrations of physical phenomena, among which the manifestations of surface-tension are very prominent. But the chief phenomena connected with their skeletons we shall deal with in another place, under the head of spicular concretions.
In a simple and typical Heliozoan, such as the Sun-animalcule, Actinophrys sol, we have a “drop” of protoplasm, contracted by its surface tension into a spherical form. Within the heterogeneous protoplasmic mass are more fluid portions, and at the surface which separates these from the surrounding protoplasm a similar surface tension causes them also to assume the form of spherical “vacuoles,” which in reality are little clear drops within the big one; unless indeed they become numerous and closely packed, in which case, instead of isolated spheres or droplets they will constitute a “froth,” their mutual pressures and tensions giving rise to regular configurations such as we shall study in the next chapter. One or more of such clear spaces may be what is called a “contractile vacuole”: that is to say, a droplet whose surface tension is in unstable equilibrium and is apt to vanish altogether, so that the definite outline of the vacuole suddenly disappears [49]. Again, within the protoplasm are one or more nuclei, whose own surface tension (at the surface between the nucleus and the surrounding protoplasm), has drawn them in turn into the shape [p. 265] of spheres. Outwards through the protoplasm, and stretching far beyond the spherical surface of the cell, there run stiff linear threads of modified or differentiated protoplasm, replaced or reinforced in some cases by delicate siliceous needles. In either case we know little or nothing about the forces which lead to their production, and we do not hide our ignorance when we ascribe their development to a “radial polarisation” of the cell. In the case of the protoplasmic filament, we may (if we seek for a hypothesis), suppose that it is somehow comparable to a viscid stream, or “liquid vein,” thrust or squirted out from the body of the cell. But when it is once formed, this long and comparatively rigid filament is separated by a distinct surface from the neighbouring protoplasm, that is to say from the more fluid surface-protoplasm of the cell; and the latter begins to creep up the filament, just as water would creep up the interior of a glass tube, or the sides of a glass rod immersed in the liquid. It is the simple case of a balance between three separate tensions: (1) that between the filament and the adjacent protoplasm, (2) that between the filament and the adjacent water, and (3) that between the water and the protoplasm. Calling these tensions respectively Tfp, Tfw, and Twp, equilibrium will be attained when the angle of contact between the fluid protoplasm and the filament is such that cos α = (Tfw − Twp)/Tfp. It is evident in this case that the angle is a very small one. The precise form of the curve is somewhat different from that which, under ordinary circumstances, is assumed by a liquid which creeps up a solid surface, as water in contact with air creeps up a surface of glass; the difference being due to the fact that here, owing to the density of the protoplasm being practically identical with that of the surrounding medium, the whole system is practically immune from gravity. Under normal circumstances the curve is part of the “elastic curve” by which that surface of revolution is generated which we have called, after Plateau, the nodoid; but in the present case it is apparently a catenary. Whatever curve it be, it obviously forms a surface of revolution around the filament.
Since the attraction exercised by this surface tension is symmetrical around the filament, the latter will be pulled equally [p. 266] in all directions; in other words it will tend to be set normally to the surface of the sphere, that is to say radiating directly outwards from the centre. If the distance between two adjacent filaments be considerable, the curve will simply meet the filament at the angle α already referred to; but if they be sufficiently near together, we shall have a continuous catenary curve forming a hanging loop between one filament and the other. And when this is so, and the radial filaments are more or less symmetrically interspaced, we may have a beautiful system of honeycomb-like depressions over the surface of the organism, each cell of the honeycomb having a strictly defined geometric configuration.
1917 Source Illustration
Fig. 90. A, Trypanosoma tineae (after Minchin); B, Spirochaeta anodontae (after Fantham).
In the simpler Radiolaria, the spherical form of the entire organism is equally well-marked; and here, as also in the more complicated Heliozoa (such as Actinosphaerium), the organism is differentiated into several distinct layers, each boundary surface tending to be spherical, and so constituting sphere within sphere. One of these layers at least is close packed with vacuoles, forming an “alveolar meshwork,” with the configurations of which we shall attempt in another chapter to correlate the characteristic structure of certain complex types of skeleton.
An exceptional form of cell, but a beautiful manifestation of surface-tension (or so I take it to be), occurs in Trypanosomes, those tiny parasites of the blood that are associated with sleeping-sickness and many other grave or dire maladies. These tiny organisms consist of elongated solitary cells down one side of which runs a very delicate frill, or “undulating membrane,” the free edge of which is seen to be slightly thickened, and the whole of [p. 267] which undergoes rhythmical and beautiful wavy movements. When certain Trypanosomes are artificially cultivated (for instance T. rotatorium, from the blood of the frog), phases of growth are witnessed in which the organism has no undulating membrane, but possesses a long cilium or “flagellum,” springing from near the front end, and exceeding the whole body in length [50]. Again, in T. lewisii, when it reproduces by “multiple fission,” the products of this division are likewise devoid of an undulating membrane, but are provided with a long free flagellum [51]. It is a plausible assumption to suppose that, as the flagellum waves about, it comes to lie near and parallel to the body of the cell, and that the frill or undulating membrane is formed by the clear, fluid protoplasm of the surface layer springing up in a film to run up and along the flagellum, just as a soap-film would be formed in similar circumstances.
1917 Source Illustration
Fig. 91. A, Trichomonas muris, Hartmann; B, Trichomastix serpentis, Dobell; C, Trichomonas angusta, Alexeieff. (After Kofoid.)
This mode of formation of the undulating membrane or frill appears to be confirmed by the appearances shewn in Fig. 91. [p. 268] Here we have three little organisms closely allied to the ordinary Trypanosomes, of which one, Trichomastix (B), possesses four flagella, and the other two, Trichomonas, apparently three only: the two latter possess the frill, which is lacking in the first [52]. But it is impossible to doubt that when the frill is present (as in A and C), its outer edge is constituted by the apparently missing flagellum (a), which has become attached to the body of the creature at the point c, near its posterior end; and all along its course, the superficial protoplasm has been drawn out into a film, between the flagellum (a) and the adjacent surface or edge of the body (b).
1917 Source Illustration
Fig. 92. Herpetomonas assuming the undulatory membrane of a Trypanosome. (After D. L. Mackinnon.)
Moreover, this mode of formation has been actually witnessed and described, though in a somewhat exceptional case. The little flagellate monad Herpetomonas is normally destitute of an undulating membrane, but possesses a single long terminal flagellum. According to Dr D. L. Mackinnon, the cytoplasm in a certain stage of growth becomes somewhat “sticky,” a phrase which we may in all probability interpret to mean that its surface tension is being reduced. For this stickiness is shewn in two ways. In the first place, the long body, in the course of its various bending movements, is apt to adhere head to tail (so to speak), giving a rounded or sometimes annular form to the organism, such as has also been described in certain species or stages of Trypanosomes. But again, the long flagellum, if it get bent backwards upon the body, tends to adhere to its surface. “Where the flagellum was pretty long and active, its efforts to continue movement under these abnormal conditions resulted in the gradual lifting up from the cytoplasm of the body of a sort of pseudo-undulating membrane (Fig. 92). The movements of this structure were so exactly those of a true undulating membrane that it was [p. 269] difficult to believe one was not dealing with a small, blunt trypanosome [53].” This in short is a precise description of the mode of development which, from theoretical considerations alone, we should conceive to be the natural if not the only possible way in which the undulating membrane could come into existence.
There is a genus closely allied to Trypanosoma, viz. Trypanoplasma, which possesses one free flagellum, together with an undulating membrane; and it resembles the neighbouring genus Bodo, save that the latter has two flagella and no undulating membrane. In like manner, Trypanosoma so closely resembles Herpetomonas that, when individuals ascribed to the former genus exhibit a free flagellum only, they are said to be in the “Herpetomonas stage.” In short all through the order, we have pairs of genera, which are presumed to be separate and distinct, viz. Trypanosoma-Herpetomonas, Trypanoplasma-Bodo, Trichomastix-Trichomonas, in which one differs from the other mainly if not solely in the fact that a free flagellum in the one is replaced by an undulating membrane in the other. We can scarcely doubt that the two structures are essentially one and the same.
The undulating membrane of a Trypanosome, then, according to our interpretation of it, is a liquid film and must obey the law of constant mean curvature. It is under curious limitations of freedom: for by one border it is attached to the comparatively motionless body, while its free border is constituted by a flagellum which retains its activity and is being constantly thrown, like the lash of a whip, into wavy curves. It follows that the membrane, for every alteration of its longitudinal curvature, must at the same instant become curved in a direction perpendicular thereto; it bends, not as a tape bends, but with the accompaniment of beautiful but tiny waves of double curvature, all tending towards the establishment of an “equipotential surface”; and its characteristic undulations are not originated by an active mobility of the membrane but are due to the molecular tensions which produce the very same result in a soap-film under similar circumstances.
In certain Spirochaetes, S. anodontae (Fig. 90) and S. balbiani [p. 270] (which we find in oysters), a very similar undulating membrane exists, but it is coiled in a regular spiral round the body of the cell. It forms a “screw-surface,” or helicoid, and, though we might think that nothing could well be more curved, yet its mathematical properties are such that it constitutes a “ruled surface” whose “mean curvature” is everywhere nil; and this property (as we have seen) it shares with the plane, and with the plane alone. Precisely such a surface, and of exquisite beauty, may be produced by bending a wire upon itself so that part forms an axial rod and part a spiral wrapping round the axis, and then dipping the whole into a soapy solution.
These undulating and helicoid surfaces are exactly reproduced among certain forms of spermatozoa. The tail of a spermatozoon consists normally of an axis surrounded by clearer and more fluid protoplasm, and the axis sometimes splits up into two or more slender filaments. To surface tension operating between these and the surface of the fluid protoplasm (just as in the case of the flagellum of the Trypanosome), I ascribe the formation of the undulating membrane which we find, for instance, in the spermatozoa of the newt or salamander; and of the helicoid membrane, wrapped in a far closer and more beautiful spiral than that which we saw in Spirochaeta, which is characteristic of the spermatozoa of many birds.
Before we pass from the subject of the conformation of the solitary cell we must take some account of certain other exceptional forms, less easy of explanation, and still less perfectly understood. Such is the case, for instance, with the red blood-corpuscles of man and other vertebrates; and among the sperm-cells of the decapod crustacea we find forms still more aberrant and not less perplexing. These are among the comparatively few cells or cell-like structures whose form seems to be incapable of explanation by theories of surface-tension.
In all the mammalia (save a very few) the red blood-corpuscles are flattened circular discs, dimpled in upon their two opposite sides. This configuration closely resembles that of an india-rubber ball when we pinch it tightly between finger and thumb; and we may also compare it with that experiment of Plateau’s [p. 271] (described on p. 223), where a flat cylindrical oil-drop, of certain relative dimensions, can, by sucking away a little of the contained oil, be made to assume the form of a biconcave disc, whose periphery is part of a nodoidal surface. From the relation of the nodoid to the “elastic curve,” we perceive that these two examples are closely akin one to the other.
1917 Source Illustration
Fig. 93.
The form of the corpuscle is symmetrical, and its surface is a surface of revolution; but it is obviously not a surface of constant mean curvature, nor of constant pressure. For we see at once that, in the sectional diagram (Fig. 93), the pressure inwards due to surface tension is positive at A, and negative at C; at B there is no curvature in the plane of the paper, while perpendicular to it the curvature is negative, and the pressure therefore is also negative. Accordingly, from the point of view of surface tension alone, the blood-corpuscle is not a surface of equilibrium; or in other words, it is not a fluid drop suspended in another liquid. It is obvious therefore that some other force or forces must be at work, and the simple effect of mechanical pressure is here excluded, because the blood-corpuscle exhibits its characteristic shape while floating freely in the blood. In the lower vertebrates the blood-corpuscles have the form of a flattened oval disc, with rather sharp edges and ellipsoidal surfaces, and this again is manifestly not a surface of equilibrium.
Two facts are especially noteworthy in connection with the form of the blood-corpuscle. In the first place, its form is only maintained, that is to say it is only in equilibrium, in relation to certain properties of the medium in which it floats. If we add a little water to the blood, the corpuscle quickly loses its characteristic shape and becomes a spherical drop, that is to say a true surface of minimal area and of stable equilibrium. If on the other hand we add a strong solution of salt, or a little glycerine, the corpuscle contracts, and its surface becomes puckered and uneven. In these phenomena it is so far obeying the laws of diffusion and of surface tension. [p. 272]
In the second place, it can be exactly imitated artificially by means of other colloid substances. Many years ago Norris made the very interesting observation that in an emulsion of glue the drops assumed a biconcave form resembling that of the mammalian corpuscles [54]. The glue was impure, and doubtless contained lecithin; and it is possible (as Professor Waymouth Reid tells me) to make a similar emulsion with cerebrosides and cholesterin oleate, in which the same conformation of the drops or particles is beautifully shewn. Now such cholesterin bodies have an important place among those in which Lehmann and others have shewn and studied the formation of fluid crystals, that is to say of bodies in which the forces of crystallisation and the forces of surface tension are battling with one another [55]; and, for want of a better explanation, we may in the meanwhile suggest that some such cause is at the bottom of the conformation the explanation of which presents so many difficulties. But we must not, perhaps, pass from this subject without adding that the case is a difficult and complex one from the physiological point of view. For the surface of a blood-corpuscle consists of a “semi-permeable membrane,” through which certain substances pass freely and not others (for the most part anions and not cations), and it may be, accordingly, that we have in life a continual state of osmotic inequilibrium, of negative osmotic tension within, to which comparatively simple cause the imperfect distension of the corpuscle may be also due [56]. The whole phenomenon would be comparatively easy to understand if we might postulate a stiffer peripheral region to the corpuscle, in the form for instance of a peripheral elastic ring. Such an annular thickening or stiffening, like the “collapse-rings” which an engineer inserts in a boiler, has been actually asserted to exist, but its presence is not authenticated.
But it is not at all improbable that we have still much to learn about the phenomena of osmosis itself, as manifested in the case of minute bodies such as a blood-corpuscle; and (as Professor Peddie suggests to me) it is by no means impossible that curvature [p. 273] of the surface may itself modify the osmotic or perhaps the adsorptive action. If it should be found that osmotic action tended to stop, or to reverse, on change of curvature, it would follow that this phenomenon would give rise to internal currents; and the change of pressure consequent on these would tend to intensify the change of curvature when once started [57].
1917 Source Illustration
Fig. 94. Sperm-cells of Decapod Crustacea (after Koltzoff). a, Inachus scorpio; b, Galathea squamifera; c, do. after maceration, to shew spiral fibrillae.
The sperm-cells of the Decapod crustacea exhibit various singular shapes. In the Crayfish they are flattened cells with stiff curved processes radiating outwards like a St Catherine’s wheel; in Inachus there are two such circles of stiff processes; in Galathea we have a still more complex form, with long and slightly twisted processes. In all these cases, just as in the case of the blood-corpuscle, the structure alters, and finally loses, its characteristic form when the nature or constitution (or as we may assume in particular—the density) of the surrounding medium is changed.
Here again, as in the blood-corpuscle, we have to do with a very important force, which we had not hitherto considered in this connection,—the force of osmosis, manifested under conditions similar to those of Pfeffer’s classical experiments on the plant-cell. The surface of the cell acts as a “semi-permeable membrane,” [p. 274] permitting the passage of certain dissolved substances (or their “ions”) and including or excluding others; and thus rendering manifest and measurable the existence of a definite “osmotic pressure.” In the case of the sperm-cells of Inachus, certain quantitative experiments have been performed [58]. The sperm-cell exhibits its characteristic conformation while lying in the serous fluid of the animal’s body, in ordinary sea-water, or in a 5 per cent. solution of potassium nitrate; these three fluids being all “isotonic” with one another. As we alter the concentration of potassium nitrate, the cell assumes certain definite forms corresponding to definite concentrations of the salt; and, as a further and final proof that the phenomenon is entirely physical, it is found that other salts produce an identical effect when their concentration is proportionate to their molecular weight, and whatever identical effect is produced by various salts in their respective concentrations, a similarly identical effect is produced when these concentrations are doubled or otherwise proportionately changed [59].
1917 Source Illustration
Fig. 95. Sperm-cells of Inachus, as they appear in saline solutions of varying density. (After Koltzoff.)
Thus the following table shews the percentage concentrations of certain salts necessary to bring the cell into the forms a and c of Fig. 95; in each case the quantities are proportional to the molecular weights, and in each case twice the quantity is necessary to produce the effect of Fig. 95c compared with that which gives rise to the all but spherical form of Fig. 95a. [p. 275]
% concentration of salts
in which the sperm-cell
of Inachus assumes the form of
─────────────────────
fig. a fig. c
Sodium chloride 0·6 1·2
Sodium nitrate 0·85 1·7
Potassium nitrate 1·0 2·0
Acetic acid 2·2 4·5
Cane sugar 5·0 10·0
1917 Source Illustration
Fig. 96. Sperm-cell of Dromia. (After Koltzoff.)
If we look then, upon the spherical form of the cell as its true condition of symmetry and of equilibrium, we see that what we call its normal appearance is just one of many intermediate phases of shrinkage, brought about by the abstraction of fluid from its interior as the result of an osmotic pressure greater outside than inside the cell, and where the shrinkage of volume is not kept pace with by a contraction of the surface-area. In the case of the blood-corpuscle, the shrinkage is of no great amount, and the resulting deformation is symmetrical; such structural inequality as may be necessary to account for it need be but small. But in the case of the sperm-cells, we must have, and we actually do find, a somewhat complicated arrangement of more or less rigid or elastic structures in the wall of the cell, which like the wire framework in Plateau’s experiments, restrain and modify the forces acting on the drop. In one form of Plateau’s experiments, instead of supporting his drop on rings or frames of wire, he laid upon its surface one or more elastic coils; and then, on withdrawing oil from the centre of his globule, he saw its uniform shrinkage counteracted by the spiral springs, with the result that the centre of each elastic coil seemed to shoot out into a prominence. Just such spiral coils are figured (after Koltzoff) in Fig. 96; and they may be regarded as precisely akin to those local thickenings, spiral and other, to which we have already ascribed the cylindrical form of the Spirogyra cell. In all probability we must in like manner attribute the peculiar spiral and other forms, for instance of many Infusoria, to the [p. 276] presence, among the multitudinous other differentiations of their protoplasmic substance, of such more or less elastic fibrillae, which play as it were the part of a microscopic skeleton [60].
But these cases which we have just dealt with, lead us to another consideration. In a semi-permeable membrane, through which water passes freely in and out, the conditions of a liquid surface are greatly modified; and, in the ideal or ultimate case, there is neither surface nor surface tension at all. And this would lead us somewhat to reconsider our position, and to enquire whether the true surface tension of a liquid film is actually responsible for all that we have ascribed to it, or whether certain of the phenomena which we have assigned to that cause may not in part be due to the contractility of definite and elastic membranes. But to investigate this question, in particular cases, is rather for the physiologist: and the morphologist may go on his way, paying little heed to what is no doubt a difficulty. In surface tension we have the production of a film with the properties of an elastic membrane, and with the special peculiarity that contraction continues with the same energy however far the process may have already gone; while the ordinary elastic membrane contracts to a certain extent, and contracts no more. But within wide limits the essential phenomena are the same in both cases. Our fundamental equations apply to both cases alike. And accordingly, so long as our purpose is morphological, so long as what we seek to explain is regularity and definiteness of form, it matters little if we should happen, here or there, to confuse surface tension with elasticity, the contractile forces manifested at a liquid surface with those which come into play at the complex internal surfaces of an elastic solid.
Chapter VI: A Note on Adsorption¶
1917 Cambridge First Edition (pp. 277–292)
A very important corollary to, or amplification of the theory of surface tension is to be found in the modern chemico-physical doctrine of Adsorption [61]. In its full statement this subject soon becomes complicated, and involves physical conceptions and mathematical treatment which go beyond our range. But it is necessary for us to take account of the phenomenon, though it be in the most elementary way.
In the brief account of the theory of surface tension with which our last chapter began, it was pointed out that, in a drop of liquid, the potential energy of the system could be diminished, and work manifested accordingly, in two ways. In the first place we saw that, at our liquid surface, surface tension tends to set up an equilibrium of form, in which the surface is reduced or contracted either to the absolute minimum of a sphere, or at any rate to the least possible area which is permitted by the various circumstances and conditions; and if the two bodies which comprise our system, namely the drop of liquid and its surrounding medium, be simple substances, and the system be uncomplicated by other distributions of force, then the energy of the system will have done its work when this equilibrium of form, this minimal area of surface, is once attained. This phenomenon of the production of a minimal surface-area we have now seen to be of fundamental importance in the external morphology of the cell, and especially (so far as we have yet gone) of the solitary cell or unicellular organism. [p. 278]
But we also saw, according to Gauss’s equation, that the potential energy of the system will be diminished (and its diminution will accordingly be manifested in work) if from any cause the specific surface energy be diminished, that is to say if it be brought more nearly to an equality with the specific energy of the molecules in the interior of the liquid mass. This latter is a phenomenon of great moment in modern physiology, and, while we need not attempt to deal with it in detail, it has a bearing on cell-form and cell-structure which we cannot afford to overlook.
In various ways a diminution of the surface energy may be brought about. For instance, it is known that every isolated drop of fluid has, under normal circumstances, a surface-charge of electricity: in such a way that a positive or negative charge (as the case may be) is inherent in the surface of the drop, while a corresponding charge, of contrary sign, is inherent in the immediately adjacent molecular layer of the surrounding medium. Now the effect of this distribution, by which all the surface molecules of our drop are similarly charged, is that by virtue of this charge they tend to repel one another, and possibly also to draw other molecules, of opposite charge, from the interior of the mass; the result being in either case to antagonise or cancel, more or less, that normal tendency of the surface molecules to attract one another which is manifested in surface tension. In other words, an increased electrical charge concentrating at the surface of a drop tends, whether it be positive or negative, to lower the surface tension.
But a still more important case has next to be considered. Let us suppose that our drop consists no longer of a single chemical substance, but contains other substances either in suspension or in solution. Suppose (as a very simple case) that it be a watery fluid, exposed to air, and containing droplets of oil: we know that the specific surface tension of oil in contact with air is much less than that of water, and it follows that, if the watery surface of our drop be replaced by an oily surface the specific surface energy of the system will be notably diminished. Now under these circumstances it is found that (quite apart from gravity, by which the oil might float to the surface) the oil has a tendency to be drawn to the surface; and this phenomenon of molecular attraction [p. 279] or “adsorption” represents the work done, equivalent to the diminished potential energy of the system [62]. In more general terms, if a liquid (or one or other of two adjacent liquids) be a chemical mixture, some one constituent in which, if it entered into or increased in amount in the surface layer, would have the effect of diminishing its surface tension, then that constituent will have a tendency to accumulate or concentrate at the surface: the surface tension may be said, as it were, to exercise an attraction on this constituent substance, drawing it into the surface layer, and this tendency will proceed until at a certain “surface concentration” equilibrium is reached, its opponent being that osmotic force which tends to keep the substance in uniform solution or diffusion.
In the complex mixtures which constitute the protoplasm of the living cell, this phenomenon of “adsorption” has abundant play: for many of these constituents, such as oils, soaps, albumens, etc. possess the required property of diminishing surface tension.
Moreover, the more a substance has the power of lowering the surface tension of the liquid in which it happens to be dissolved, the more will it tend to displace another and less effective substance from the surface layer. Thus we know that protoplasm always contains fats or oils, not only in visible drops, but also in the finest suspension or “colloidal solution.” If under any impulse, such for instance as might arise from the Brownian movement, a droplet of oil be brought close to the surface, it is at once drawn into that surface, and tends to spread itself in a thin layer over the whole surface of the cell. But a soapy surface (for instance) would have in contact with the surrounding water a surface tension even less than that of the film of oil: and consequently, if soap be present in the water it will in turn be adsorbed, and will tend to displace the oil from the surface pellicle [63]. And this is all as [p. 280] much as to say that the molecules of the dissolved or suspended substance or substances will so distribute themselves throughout the drop as to lead towards an equilibrium, for each small unit of volume, between the superficial and internal energy; or so, in other words, as to lead towards a reduction to a minimum of the potential energy of the system. This tendency to concentration at the surface of any substance within the cell by which the surface tension tends to be diminished, or vice versa, constitutes, then, the phenomenon of Adsorption; and the general statement by which it is defined is known as the Willard-Gibbs, or Gibbs-Thomson law [64].
Among the many important physical features or concomitants of this phenomenon, let us take note at present that we need not conceive of a strictly superficial distribution of the adsorbed substance, that is to say of its direct association with the surface layer of molecules such as we imagined in the case of the electrical charge; but rather of a progressive tendency to concentrate, more and more, as the surface is nearly approached. Indeed we may conceive the colloid or gelatinous precipitate in which, in the case of our protoplasmic cell, the dissolved substance tends often to be thrown down, to constitute one boundary layer after another, the general effect being intensified and multiplied by the repeated addition of these new surfaces.
Moreover, it is not less important to observe that the process of adsorption, in the neighbourhood of the surface of a heterogeneous liquid mass, is a process which takes time; the tendency to surface concentration is a gradual and progressive one, and will fluctuate with every minute change in the composition of our substance and with every change in the area of its surface. In other words, it involves (in every heterogeneous substance) a continual instability of equilibrium: and a constant manifestation [p. 281] of motion, sometimes in the mere invisible transfer of molecules but often in the production of visible currents of fluid or manifest alterations in the form or outline of the system.
The physiologist, as we have already remarked, takes account of the general phenomenon of adsorption in many ways: particularly in connection with various results and consequences of osmosis, inasmuch as this process is dependent on the presence of a membrane, or membranes, such as the phenomenon of adsorption brings into existence. For instance it plays a leading part in all modern theories of muscular contraction, in which phenomenon a connection with surface tension was first indicated by FitzGerald and d’Arsonval nearly forty years ago [65]. And, as W. Ostwald was the first to shew, it gives us an entirely new conception of the relation of gases (that is to say, of oxygen and carbon dioxide) to the red corpuscles of the blood [66].
But restricting ourselves, as much as may be, to our morphological aspect of the case, there are several ways in which adsorption begins at once to throw light upon our subject.
In the first place, our preliminary account, such as it is, is already tantamount to a description of the process of development of a cell-membrane, or cell-wall. The so-called “secretion” of this cell-wall is nothing more than a sort of exudation, or striving towards the surface, of certain constituent molecules or particles within the cell; and the Gibbs-Thomson law formulates, in part at least, the conditions under which they do so. The adsorbed material may range from the almost unrecognisable pellicle of a blood-corpuscle to the distinctly differentiated “ectosarc” of a protozoan, and again to the development of a fully formed cell-wall, as in the cellulose partitions of a vegetable tissue. In such cases, the dissolved and adsorbable material has not only the property of lowering the surface tension, and hence [p. 282] of itself accumulating at the surface, but has also the property of increasing the viscosity and mechanical rigidity of the material in which it is dissolved or suspended, and so of constituting a visible and tangible “membrane [67].” The “zoogloea” around a group of bacteria is probably a phenomenon of the same order. In the superficial deposition of inorganic materials we see the same process abundantly exemplified. Not only do we have the simple case of the building of a shell or “test” upon the outward surface of a living cell, as for instance in a Foraminifer, but in a subsequent chapter, when we come to deal with various spicules and spicular skeletons such as those of the sponges and of the Radiolaria, we shall see that it is highly characteristic of the whole process of spicule-formation for the deposits to be laid down just in the “interfacial” boundaries between cells or vacuoles, and that the form of the spicular structures tends in many cases to be regulated and determined by the arrangement of these boundaries.
In physical chemistry, an important distinction is drawn between adsorption and pseudo-adsorption [68], the former being a reversible, the latter an irreversible or permanent phenomenon. That is to say, adsorption, strictly speaking, implies the surface-concentration of a dissolved substance, under circumstances which, if they be altered or reversed, will cause the concentration to diminish or disappear. But pseudo-adsorption includes cases, doubtless originating in adsorption proper, where subsequent changes leave the concentrated substance incapable of re-entering the liquid system. It is obvious that many (though not all) of our biological illustrations, for instance the formation of spicules or of permanent cell-membranes, belong to the class of so-called pseudo-adsorption phenomena. But the apparent contrast between the two is in the main a secondary one, and however important to the chemist is of little consequence to us. [p. 283]
While this brief sketch of the theory of membrane-formation is cursory and inadequate, it is enough to shew that the physical theory of adsorption tends in part to overturn, in part to simplify enormously, the older histological descriptions. We can no longer be content with such statements as that of Strasbürger, that membrane-formation in general is associated with the “activity of the kinoplasm,” or that of Harper that a certain spore-membrane arises directly from the astral rays [69]. In short, we have easily reached the general conclusion that, the formation of a cell-wall or cell-membrane is a chemico-physical phenomenon, which the purely objective methods of the biological microscopist do not suffice to interpret.
If the process of adsorption, on which the formation of a membrane depends, be itself dependent on the power of the adsorbed substance to lower the surface tension, it is obvious that adsorption can only take place when the surface tension already present is greater than zero. It is for this reason that films or threads of creeping protoplasm shew little tendency, or none, to cover themselves with an encysting membrane; and that it is only when, in an altered phase, the protoplasm has developed a positive surface tension, and has accordingly gathered itself up into a more or less spherical body, that the tendency to form a membrane is manifested, and the organism develops its “cyst” or cell-wall.
It is found that a rise of temperature greatly reduces the adsorbability of a substance, and this doubtless comes, either in part or whole, from the fact that a rise of temperature is itself a cause of the lowering of surface tension. We may in all probability ascribe to this fact and to its converse, or at least associate with it, such phenomena as the encystment of unicellular organisms at the approach of winter, or the frequent formation of strong shells or membranous capsules in “winter-eggs.”
Again, since a film or a froth (which is a system of films) can only be maintained by virtue of a certain viscosity or rigidity of [p. 284] the liquid, it may be quickly caused to disappear by the presence in its neighbourhood of some substance capable of reducing the surface tension; for this substance, being adsorbed, may displace from the adsorptive layer a material to which was due the rigidity of the film. In this way a “bathytonic” substance such as ether causes most foams to subside, and the pouring oil on troubled waters not only stills the waves but still more quickly dissipates the foam of the breakers. The process of breaking up an alveolar network, such as occurs at a certain stage in the nuclear division of the cell, may perhaps be ascribed in part to such a cause, as well as to the direct lowering of surface tension by electrical agency.
Our last illustration has led us back to the subject of a previous chapter, namely to the visible configuration of the interior of the cell; and in connection with this wide subject there are many phenomena on which light is apparently thrown by our knowledge of adsorption, and of which we took little or no account in our former discussion. One of these phenomena is that visible or concrete “polarity,” which we have already seen to be in some way associated with a dynamical polarity of the cell.
This morphological polarity may be of a very simple kind, as when, in an epithelial cell, it is manifested by the outward shape of the elongated or columnar cell itself, by the essential difference between its free surface and its attached base, or by the presence in the neighbourhood of the former of mucous or other products of the cell’s activity. But in a great many cases, this “polarised” symmetry is supplemented by the presence of various fibrillae, or of linear arrangements of particles, which in the elongated or “monopolar” cell run parallel with its axis, and which tend to a radial arrangement in the more or less rounded or spherical cell. Of late years especially, an immense importance has been attached to these various linear or fibrillar arrangements, as they occur (after staining) in the cell-substance of intestinal epithelium, of spermatocytes, of ganglion cells, and most abundantly and most frequently of all in gland cells. Various functions, which seem somewhat arbitrarily chosen, have been assigned, and many hard names given to them; for these structures now include your mitochondria and your chondriokonts (both of these being varieties [p. 285] of chondriosomes), your Altmann’s granules, your microsomes, pseudo-chromosomes, epidermal fibrils and basal filaments, your archeoplasm and ergastoplasm, and probably your idiozomes, plasmosomes, and many other histological minutiae [70].
1917 Source Illustration
Fig. 97. A, B, Chondriosomes in kidney-cells, prior to and during secretory activity (after Barratt); C, do. in pancreas of frog (after Mathews).
The position of these bodies with regard to the other cell-structures is carefully described. Sometimes they lie in the neighbourhood of the nucleus itself, that is to say in proximity to the fluid boundary surface which separates the nucleus from the cytoplasm; and in this position they often form a somewhat cloudy sphere which constitutes the Nebenkern. In the majority of cases, as in the epithelial cells, they form filamentous structures, and rows of granules, whose main direction is parallel to the axis of the cell, and which may, in some cases, and in some forms, be conspicuous at the one end, and in some cases at the other end of the cell. But I do not find that the histologists attempt to explain, or to correlate with other phenomena, the tendency of these bodies to lie parallel with the axis, and perpendicular to the extremities of the cell; it is merely noted as a peculiarity, or a specific character, of these particular structures. Extraordinarily complicated and diverse functions have been ascribed to them. Engelmann’s “Fibrillenkonus,” which was almost certainly another aspect of the same phenomenon, was held by him and by cytologists like Breda and Heidenhain, to be an apparatus connected in some [p. 286] unexplained way with the mechanism of ciliary movement. Meves looked upon the chondriosomes as the actual carriers or transmitters of heredity [71]. Altmann invented a new aphorism, Omne granulum e granulo, as a refinement of Virchow’s omnis cellula e cellula; and many other histologists, more or less in accord, accepted the chondriosomes as important entities, sui generis, intermediate in grade between the cell itself and its ultimate molecular components. The extreme cytologists of the Munich school, Popoff, Goldschmidt and others, following Richard Hertwig, declaring these structures to be identical with “chromidia” (under which name Hertwig ranked all extra-nuclear chromatin), would assign them complex functions in maintaining the balance between nuclear and cytoplasmic material; and the “chromidial hypothesis,” as every reader of recent cytological literature knows, has become a very abstruse and complicated thing [72]. With the help of the “binuclearity hypothesis” of Schaudinn and his school, it has given us the chromidial net, the chromidial apparatus, the trophochromidia, idiochromidia, gametochromidia, the protogonoplasm, and many other novel and original conceptions. The names are apt to vary somewhat in significance from one writer to another.
The outstanding fact, as it seems to me, is that physiological science has been heavily burdened in this matter, with a jargon of names and a thick cloud of hypotheses; while, from the physical point of view we are tempted to see but little mystery in the whole phenomenon, and to ascribe it, in all probability and in general terms, to the gathering or “clumping” together, under surface tension, of various constituents of the heterogeneous cell-content, and to the drawing out of these little clumps along the axis of the cell towards one or other of its extremities, in relation to osmotic currents, as these in turn are set up in direct relation [p. 287] to the phenomena of surface energy and of adsorption [73]. And all this implies that the study of these minute structures, if it teach us nothing else, at least surely and certainly reveals to us the presence of a definite “field of force,” and a dynamical polarity within the cell.
Our next and last illustration of the effects of adsorption, which we owe to the investigations of Professor Macallum, is of great importance; for it introduces us to a series of phenomena in regard to which we seem now to stand on firmer ground than in some of the foregoing cases, though we cannot yet consider that the whole story has been told. In our last chapter we were restricted mainly, though not entirely, to a consideration of figures of equilibrium, such as the sphere, the cylinder or the unduloid; and we began at once to find ourselves in difficulties when we were confronted by departures from symmetry, as for instance in the simple case of the ellipsoidal yeast-cell and the production of its bud. We found the cylindrical cell of Spirogyra, with its plane or spherical ends, a comparatively simple matter to understand; but when this uniform cylinder puts out a lateral outgrowth, in the act of conjugation, we have a new and very different system of forces to explain. The analogy of the soap-bubble, or of the simple liquid drop, was apt to lead us to suppose that the surface tension was, on the whole, uniform over the surface of our cell; and that its departures from symmetry of form were therefore likely to be due to variations in external resistance. But if we have been inclined to make such an assumption we must now [p. 288] reconsider it, and be prepared to deal with important localised variations in the surface tension of the cell. For, as a matter of fact, the simple case of a perfectly symmetrical drop, with uniform surface, at which adsorption takes place with similar uniformity, is probably rare in physics, and rarer still (if it exist at all) in the fluid or fluid-containing system which we call in biology a cell. We have mostly to do with cells whose general heterogeneity of substance leads to qualitative differences of surface, and hence to varying distributions of surface tension. We must accordingly investigate the case of a cell which displays some definite and regular heterogeneity of its liquid surface, just as Amoeba displays a heterogeneity which is complex, irregular and continually fluctuating in amount and distribution. Such heterogeneity as we are speaking of must be essentially chemical, and the preliminary problem is to devise methods of “microchemical” analysis, which shall reveal localised accumulations of particular substances within the narrow limits of a cell, in the hope that, their normal effect on surface tension being ascertained, we may then correlate with their presence and distribution the actual indications of varying surface tension which the form or movement of the cell displays. In theory the method is all that we could wish, but in practice we must be content with a very limited application of it; for the substances which may have such action as we are looking for, and which are also actual or possible constituents of the cell, are very numerous, while the means are very seldom at hand to demonstrate their precise distribution and localisation. But in one or two cases we have such means, and the most notable is in connection with the element potassium. As Professor Macallum has shewn, this element can be revealed, in very minute quantities, by means of a certain salt, a nitrite of cobalt and sodium [74]. This salt penetrates readily into the tissues and into the interior of the cell; it combines with potassium to form a sparingly soluble nitrite of cobalt, sodium and potassium; and this, on subsequent treatment with ammonium sulphide, is converted into a characteristic black precipitate of cobaltic sulphide [75]. [p. 289]
By this means Macallum demonstrated some years ago the unexpected presence of accumulations of potassium (i.e. of chloride or other salts of potassium) localised in particular parts of various cells, both solitary cells and tissue cells; and he arrived at the conclusion that the localised accumulations in question were simply evidences of concentration of the dissolved potassium salts, formed and localised in accordance with the Gibbs-Thomson law. In other words, these accumulations, occurring as they actually do in connection with various boundary surfaces, are evidence, when they appear irregularly distributed over such a surface, of inequalities in its surface tension [76]; and we may safely take it that our potassium salts, like inorganic substances in general, tend to raise the surface tension, and will therefore be found concentrating at a portion of the surface whose tension is weak [77].
In Professor Macallum’s figure (Fig. 98, 1) of the little green alga Pleurocarpus, we see that one side of the cell is beginning to bulge out in a wide convexity. This bulge is, in the first place, a sign of weakened surface tension on one side of the cell, which as a whole had hitherto been a symmetrical cylinder; in the second place, we see that the bulging area corresponds to the position of a great concentration of the potassic salt; while in the third place, from the physiological point of view, we call the phenomenon the first stage in the process of conjugation. In Fig. 98, 2, of Mesocarpus (a close ally of Spirogyra), we see the same phenomenon admirably exemplified in a later stage. From the adjacent cells distinct outgrowths are being emitted, where the surface tension has been weakened: just as the glass-blower warms and softens a small part of his tube to blow out the softened area into a bubble or diverticulum; and in our Mesocarpus cells (besides a certain amount of potassium rendered visible over the boundary which [p. 290] separates the green protoplasm from the cell-sap), there is a very large accumulation precisely at the point where the tension of the originally cylindrical cell is weakening to produce the bulge. But in a still later stage, when the boundary between the two conjugating cells is lost and the cytoplasm of the two cells becomes fused together, then the signs of potassic concentration quickly disappear, the salt becoming generally diffused through the now symmetrical and spherical “zygospore.”
1917 Source Illustration
Fig. 98. Adsorptive concentration of potassium salts in (1) cell of Pleurocarpus about to conjugate; (2) conjugating cells of Mesocarpus; (3) sprouting spores of Equisetum. (After Macallum.)
In a spore of Equisetum (Fig. 98, 3), while it is still a single cell, no localised concentration of potassium is to be discerned; but as soon as the spore has divided, by an internal partition, into two cells, the potassium salt is found to be concentrated in the smaller one, and especially towards its outer wall, which is marked by a pronounced convexity. And as this convexity (which corresponds to one pole of the now asymmetrical, or quasi-ellipsoidal spore) grows out into the root-hair, the potassium salt accompanies its growth, and is concentrated under its wall. The concentration is, [p. 291] accordingly, a concomitant of the diminished surface tension which is manifested in the altered configuration of the system.
In the case of ciliate or flagellate cells, there is to be found a characteristic accumulation of potassium at and near the base of the cilia. The relation of ciliary movement to surface tension lies beyond our range, but the fact which we have just mentioned throws light upon the frequent or general presence of a little protuberance of the cell-surface just where a flagellum is given off (cf. p. 247), and of a little projecting ridge or fillet at the base of an isolated row of cilia, such as we find in Vorticella.
Yet another of Professor Macallum’s demonstrations, though its interest is mainly physiological, will help us somewhat further to comprehend what is implied in our phenomenon. In a normal cell of Spirogyra, a concentration of potassium is revealed along the whole surface of the spiral coil of chlorophyll-bearing, or “chromatophoral,” protoplasm, the rest of the cell being wholly destitute of the former substance: the indication being that, at this particular boundary, between chromatophore and cell-sap, the surface tension is small in comparison with any other interfacial surface within the system.
Now as Macallum points out, the presence of potassium is known to be a factor, in connection with the chlorophyll-bearing protoplasm, in the synthetic production of starch from CO2 under the influence of sunlight. But we are left in some doubt as to the consecutive order of the phenomena. For the lowered surface tension, indicated by the presence of the potassium, may be itself a cause of the carbohydrate synthesis; while on the other hand, this synthesis may be attended by the production of substances (e.g. formaldehyde) which lower the surface tension, and so conduce to the concentration of potassium. All we know for certain is that the several phenomena are associated with one another, as apparently inseparable parts or inevitable concomitants of a certain complex action.
And now to return, for a moment, to the question of cell-form. When we assert that the form of a cell (in the absence of mechanical pressure) is essentially dependent on surface tension, and even when we make the preliminary assumption that protoplasm is essentially [p. 292] a fluid, we are resting our belief on a general consensus of evidence, rather than on compliance with any one crucial definition. The simple fact is that the agreement of cell-forms with the forms which physical experiment and mathematical theory assign to liquids under the influence of surface tension, is so frequently and often so typically manifested, that we are led, or driven, to accept the surface tension hypothesis as generally applicable and as equivalent to a universal law. The occasional difficulties or apparent exceptions are such as call for further enquiry, but fall short of throwing doubt upon our hypothesis. Macallum’s researches introduce a new element of certainty, a “nail in a sure place,” when they demonstrate that, in certain movements or changes of form which we should naturally attribute to weakened surface tension, a chemical concentration which would naturally accompany such weakening actually takes place. They further teach us that in the cell a chemical heterogeneity may exist of a very marked kind, certain substances being accumulated here and absent there, within the narrow bounds of the system.
Such localised accumulations can as yet only be demonstrated in the case of a very few substances, and of a single one in particular; and these are substances whose presence does not produce, but whose concentration tends to follow, a weakening of surface tension. The physical cause of the localised inequalities of surface tension remains unknown. We may assume, if we please, that it is due to the prior accumulation, or local production, of chemical bodies which would have this direct effect; though we are by no means limited to this hypothesis.
But in spite of some remaining difficulties and uncertainties, we have arrived at the conclusion, as regards unicellular organisms, that not only their general configuration but also their departures from symmetry may be correlated with the molecular forces manifested in their fluid or semi-fluid surfaces.
Chapter VII: The Forms of Tissues, or Cell-Aggregates¶
1917 Cambridge First Edition (pp. 293–345)
We now pass from the consideration of the solitary cell to that of cells in contact with one another,—to what we may call in the first instance “cell-aggregates,”—through which we shall be led ultimately to the study of complex tissues. In this part of our subject, as in the preceding chapters, we shall have to give some consideration to the effects of various forces; but, as in the case of the conformation of the solitary cell, we shall probably find, and we may at least begin by assuming, that the agency of surface tension is especially manifest and important. The effect of this surface tension will chiefly manifest itself in the production of surfaces minimae areae: where, as Plateau was always careful to point out, we must understand by this expression not an absolute, but a relative minimum, an area, that is to say, which approximates to an absolute minimum as nearly as circumstances and the conditions of the case permit.
There are certain fundamental principles, or fundamental equations, besides those which we have already considered, which we shall need in our enquiry. For instance the case which we briefly touched upon (on p. 265) of the angle of contact between the protoplasm and the axial filament in a Heliozoan we shall now find to be but a particular case of a general and elementary theorem.
Let us re-state as follows, in terms of Energy, the general principle which underlies the theory of surface tension or capillarity.
When a fluid is in contact with another fluid, or with a solid or a gas, a portion of the energy of the system (that, namely, which we call surface energy), is proportional to the area of the surface of contact: it is also proportional to a coefficient which is specific for each particular pair of substances, and which is constant for these, save only in so far as it may be modified by [p. 294] changes of temperature or of electric charge. The condition of minimum potential energy in the system, which is the condition of equilibrium, will accordingly be obtained by the utmost possible diminution in the area of the surfaces in contact. When we have three bodies in contact, the case becomes a little more complex. Suppose for instance we have a drop of some fluid, A, floating on another fluid, B, and exposed to air, C. The whole surface energy of the system may now be considered as divided into two parts, one at the surface of the drop, and the other outside of the same; the latter portion is inherent in the surface BC, between the mass of fluid B and the superincumbent air, C; but the former portion consists of two parts, for it is divided between the two surfaces AB and AC, that namely which separates the drop from the surrounding fluid and that which separates it from the atmosphere. So far as
1917 Source Illustration
Fig. 99.
the drop is concerned, then, equilibrium depends on a proper balance between the energy, per unit area, which is resident in its own two surfaces, and that which is external thereto: that is to say, if we call Ebc the energy at the surface between the two fluids, and so on with the other two pairs of surface energies, the condition of equilibrium, or of maintenance of the drop, is that
If, on the other hand, the fluid A happens to be oil and the fluid B, water, then the energy per unit area of the water-air surface is greater than that of the oil-air surface and that of the oil-water surface together; i.e.
Here there is no equilibrium, and in order to obtain it the water-air surface must always tend to decrease and the other two interfacial surfaces to increase; which is as much as to say that the water tends to become covered by a spreading film of oil, and the water-air surface to be abolished. [p. 295]
The surface energy of which we have here spoken is manifested in that contractile force, or “tension,” of which we have already had so much to say [78]. In any part of the free water surface, for instance, one surface particle attracts another surface particle, and the resultant of these multitudinous attractions is an equilibrium of tension throughout this particular surface. In the case of our three bodies in contact with one another, and within a small area very near to the point of contact, a water particle (for instance) will be pulled outwards by another water particle; but on the opposite side, so to speak, there will be no water surface, and no water particle, to furnish the counterbalancing pull; this counterpull,
1917 Source Illustration
Fig. 100.
1917 Source Illustration
Fig. 101.
which is necessary for equilibrium, must therefore be provided by the tensions existing in the other two surfaces of contact. In short, if we could imagine a single particle placed at the very point of contact, it would be drawn upon by three different forces, whose directions would lie in the three surface planes, and whose magnitude would be proportional to the specific tensions characteristic of the two bodies which in each case combine to form the “interfacial” surface. Now for three forces acting at a point to be in equilibrium, they must be capable of representation, in magnitude and direction, by the three sides of a triangle, taken in order, in accordance with the elementary theorem of the Triangle of Forces. So, if we know the form of our floating drop (Fig. 100), then by drawing tangents from O (the point of mutual contact), [p. 296] we determine the three angles of our triangle (Fig. 101), and we therefore know the relative magnitudes of the three surface tensions, which magnitudes are proportional to its sides; and conversely, if we know the magnitudes, or relative magnitudes, of the three sides of the triangle, we also know its angles, and these determine the form of the section of the drop. It is scarcely necessary to mention that, since all points on the edge of the drop are under similar conditions, one with another, the form of the drop, as we look down upon it from above, must be circular, and the whole drop must be a solid of revolution.
The principle of the Triangle of Forces is expanded, as follows, by an old seventeenth-century theorem, called Lami’s Theorem: “If three forces acting at a point be in equilibrium, each force is proportional to the sine of the angle contained between the directions of the other two.” That is to say
And from this, in turn, we derive the equivalent formulae, by which each force is expressed in terms of the other two, and of the angle between them:
From this and the foregoing, we learn the following important and useful deductions:
The three forces can only be in equilibrium when any one of them is less than the sum of the other two: for otherwise, the triangle is impossible. Now in the case of a drop of olive-oil upon a clean water surface, the relative magnitudes of the three tensions (at 15° C.) have been determined as follows:
Water-air surface 75
Oil-air surface 32
Oil-water surface 21
No triangle having sides of these relative magnitudes is possible; and no such drop therefore can remain in equilibrium. [p. 297]
The three surfaces may be all alike: as when a soap-bubble floats upon soapy water, or when two soap-bubbles are joined together, on either side of a partition-film. In this case, the three tensions are all equal, and therefore the three angles are all equal; that is to say, when three similar liquid surfaces meet together, they always do so at an angle of 120°. Whether our two conjoined soap-bubbles be equal or unequal, this is still the invariable rule; because the specific tension of a particular surface is unaffected by any changes of magnitude or form.
If two only of the surfaces be alike, then two of the angles will be alike, and the other will be unlike; and this last will be the difference between 360° and the sum of the other two. A particular case is when a film is stretched between solid and parallel walls, like a soap-film within a cylindrical tube. Here, so long as there is no external pressure applied to either side, so long as both ends of the tube are open or closed, the angles on either side of the film will be equal, that is to say the film will set itself at right angles to the sides.
Many years ago Sachs laid it down as a principle, which has become celebrated in botany under the name of Sachs’s Rule, that one cell-wall always tends to set itself at right angles to another cell-wall. This rule applies to the case which we have just illustrated; and such validity as the rule possesses is due to the fact that among plant-tissues it very frequently happens that one cell-wall has become solid and rigid before another and later partition-wall is developed in connection with it.
There is another important principle which arises not out of our equations but out of the general considerations by which we were led to them. We have seen that, at and near the point of contact between our several surfaces, there is a continued balance of forces, carried, so to speak, across the interval; in other words, there is physical continuity between one surface and another. It follows necessarily from this that the surfaces merge one into another by a continuous curve. Whatever be the form of our surfaces and whatever the angle between them, this small intervening surface, approximately spherical, is always there to bridge over the line of contact [79]; and this little fillet, or “bourrelet,” [p. 298] as Plateau called it, is large enough to be a common and conspicuous feature in the microscopy of tissues (Fig. 102). For instance, the so-called “splitting” of the cell-wall, which is conspicuous at the angles of the large “parenchymatous” cells in the succulent tissues of all higher plants (Fig. 103), is nothing more than a manifestation of Plateau’s “bourrelet,” or surface of continuity [80].
We may now illustrate some of the foregoing principles, before we proceed to the more complex cases in which more bodies than three are in mutual contact. But in doing so, we must constantly bear in mind the principles set forth in our chapter on the forms of cells, and especially those relating to the pressure exercised by a curved film.
1917 Source Illustration
Fig. 102. (After Berthold.)
1917 Source Illustration
Fig. 103. Parenchyma of Maize.
Let us look for a moment at the case presented by the partition-wall in a double soap-bubble. As we have just seen, the three films in contact (viz. the outer walls of the two bubbles and the partition-wall between) being all composed of the same substance [p. 299] and all alike in contact with air, the three surface tensions must be equal; and the three films must therefore, in all cases, meet at an angle of 120°. But, unless the two bubbles be of precisely equal size (and therefore of equal curvature) it is obvious that the tangents to the spheres will not meet the plane of their circle of contact at equal angles, and therefore that the partition-wall must be a curved surface: it is only plane when it divides two equal and symmetrical cells. It is also obvious, from the symmetry of the figure, that the centres of the spheres, the centre of the partition, and the centres of the two spherical surfaces are all on one and the same straight line.
1917 Source Illustration
Fig. 104.
Now the surfaces of the two bubbles exert a pressure inwards which is inversely proportional to their radii: that is to say p : p′ :: 1/r′ : 1/r; and the partition wall must, for equilibrium, exert a pressure (P) which is equal to the difference between these two pressures, that is to say, P = 1/R = 1/r′ − 1/r = (r − r′)/r r′. It follows that the curvature of the partition wall must be just such a curvature as is capable of exerting this pressure, that is to say, R = r r′/(r − r′). The partition wall, then, is always a portion of a spherical surface, whose radius is equal to the product, divided by the difference, of the radii of the two vesicles. It follows at once from this that if the two bubbles be equal, the radius of curvature of the partition is infinitely great, that is to say the partition is (as we have already seen) a plane surface.
The geometrical construction by which we obtain the position of the centres of the two spheres and also of the partition surface is very simple, always provided that the surface tensions are uniform throughout the system. If p be a point of contact between the two spheres, and cp be a radius of one of them, then make the angle cpm = 60°, and mark off on pm, pc′ equal to the [p. 300] radius of the other sphere; in like manner, make the angle c′pn = 60°, cutting the line cc′ in c″; then c′ will be the centre of the second sphere, and c″ that of the spherical partition.
1917 Source Illustration
Fig. 105.
1917 Source Illustration
Fig. 106.
Whether the partition be or be not a plane surface, it is obvious that its line of junction with the rest of the system lies in a plane, and is at right angles to the axis of symmetry. The actual curvature of the partition-wall is easily seen in optical section; but in surface view, the line of junction is projected as a plane (Fig. 106), perpendicular to the axis, and this appearance has also helped to lend support and authority to “Sachs’s Rule.”
1917 Source Illustration
Fig. 107. Filaments, or chains of cells, in various lower Algae. (A) Nostoc; (B) Anabaena; (C) Rivularia; (D) Oscillatoria.
Many spherical cells, such as Protococcus, divide into two equal halves, which are therefore separated by a plane partition. Among the other lower Algae, akin to Protococcus, such as the Nostocs and Oscillatoriae, in which the cells are imbedded in a gelatinous matrix, we find a series of forms such as are represented in Fig. 107. Sometimes the cells are solitary or disunited; sometimes they run in pairs or in rows, separated one from another by flat partitions; and sometimes the conjoined cells are approximately hemispherical, but at other times each half is more than a hemisphere. These various conditions depend, [p. 301] according to what we have already learned, upon the relative magnitudes of the tensions at the surface of the cells and at the boundary between them [81].
In the typical case of an equally divided cell, such as a double and co-equal soap-bubble, where the partition-wall and the outer walls are similar to one another and in contact with similar substances, we can easily determine the form of the system. For, at any point of the boundary of the partition-wall, O, the tensions being equal, the angles QOP, ROP, QOR are all equal, and each is, therefore, an angle of 120°. But OQ, OR being tangents, the centres of the two spheres (or circular arcs in the figure) lie on perpendiculars to them; therefore the radii CO, C′O meet at an
1917 Source Illustration
Fig. 108.
angle of 60°, and COC′ is an equilateral triangle. That is to say, the centre of each circle lies on the circumference of the other; the partition lies midway between the two centres; and the length (i.e. the diameter) of the partition-wall, PO, is
times the radius, or ·866 times the diameter, of each of the cells. This gives us, then, the form of an aggregate of two equal cells under uniform conditions.
As soon as the tensions become unequal, whether from changes in their own substance or from differences in the substances with which they are in contact, then the form alters. If the tension [p. 302] along the partition, P, diminishes, the partition itself enlarges, and the angle QOR increases: until, when the tension P is very small compared to Q or R, the whole figure becomes a circle, and the partition-wall, dividing it into two hemispheres, stands at right angles to the outer wall. This is the case when the outer wall of the cell is practically solid. On the other hand, if P begins to increase relatively to Q and R, then the partition-wall contracts, and the two adjacent cells become larger and larger segments of a sphere, until at length the system becomes divided into two separate cells.
1917 Source Illustration
Fig. 109. Spore of Pellia. (After Campbell.)
In the spores of Liverworts (such as Pellia), the first partition-wall (the equatorial partition in Fig. 109, a) divides the spore into two equal halves, and is therefore a plane surface, normal to the surface of the cell; but the next partitions arise near to either end of the original spherical or elliptical cell. Each of these latter partitions will (like the first) tend to set itself normally to the cell-wall; at least the angles on either side of the partition will be identical, and their magnitude will depend upon the tension existing between the cell-wall and the surrounding medium. They will only be right angles if the cell-wall is already practically solid, and in all probability (rigidity of the cell-wall not being quite attained) they will be somewhat greater. In either case the partition itself will be a portion of a sphere, whose curvature will now denote a difference of pressures in the two chambers or cells, which it serves to separate. (The later stages of cell-division, represented in the figures b and c, we are not yet in a position to deal with.)
We have innumerable cases, near the tip of a growing filament, where in like manner the partition-wall which cuts off the terminal [p. 303] cell constitutes a spherical lens-shaped surface, set normally to the adjacent walls. At the tips of the branches of many Florideae, for instance, we find such a lenticular partition. In Dictyota dichotoma, as figured by Reinke, we have a succession of such partitions; and, by the way, in such cases as these, where the tissues are very transparent, we have often in optical section a puzzling confusion of lines; one being the optical section of the curved partition-wall, the other being the straight linear projection of its outer edge to which we have already referred. In the conical terminal cell of Chara, we have the same lens-shaped curve, but a little lower down, where the sides of the shoot are approximately parallel, we have flat transverse partitions, at the edges of which, however, we recognise a convexity of the outer cell-wall and a definite angle of contact, equal on the two sides of the partition.
1917 Source Illustration
Fig. 110. Cells of Dictyota. (After Reinke.)
1917 Source Illustration
Fig. 111. Terminal and other cells of Chara.
1917 Source Illustration
Fig. 112. Young antheridium of Chara.
In the young antheridia of Chara (Fig. 112), and in the not dissimilar case of the sporangium (or conidiophore) of Mucor, we easily recognise the hemispherical form of the septum which shuts off the large spherical cell from the cylindrical filament. Here, in the first phase of development, we should have to take into consideration the different pressures exerted by the single curvature of the cylinder and the double curvature of its spherical cap (p. 221); and we should find that the partition would have a somewhat low curvature, with a radius less than the diameter of the cylinder; which it would have exactly equalled but for the additional pressure inwards which it receives [p. 304] from the curvature of the large surrounding sphere. But as the latter continues to grow, its curvature decreases, and so likewise does the inward pressure of its surface; and accordingly the little convex partition bulges out more and more.
In order to epitomise the foregoing facts let the annexed diagrams (Fig. 113) represent a system of three films, of which one is a partition-wall between the other two; and let the tensions at the three surfaces, or the tractions exercised upon a point at their meeting-place, be proportional to T, T′ and t. Let α, β, γ be, as in the figure, the opposite angles. Then:
If T be equal to T′, and t be relatively insignificant, the angles α, β will be of 90°.
1917 Source Illustration
Fig. 113.
If T = T′, but be a little greater than t, then t will exert an appreciable traction, and α, β will be more than 90°, say, for instance, 100°.
If T = T′ = t, then α, β, γ will all equal 120°.
The more complicated cases, when t, T and T′ are all unequal, are already sufficiently explained.
The biological facts which the foregoing considerations go a long way to explain and account for have been the subject of much argument and discussion, especially on the part of the botanists. Let me recapitulate, in a very few words, the history of this long discussion.
Some fifty years ago, Hofmeister laid it down as a general law that “The partition-wall stands always perpendicular to what was previously the principal direction of growth in the cell,”—or, in other words, perpendicular to the long axis of the cell [82]. Ten [p. 305] years later, Sachs formulated his rule, or principle, of “rectangular section,” declaring that in all tissues, however complex, the cell-walls cut one another (at the time of their formation) at right angles [83]. Years before, Schwendener had found, in the final results of cell-division, a universal system of “orthogonal trajectories [84]”; and this idea Sachs further developed, introducing complicated systems of confocal ellipses and hyperbolæ, and distinguishing between periclinal walls, whose curves approximate to the peripheral contours, radial partitions, which cut these at an angle of 90°, and finally anticlines, which stand at right angles to the other two.
Reinke, in 1880, was the first to throw some doubt upon this explanation. He pointed out various cases where the angle was not a right angle, but was very definitely an acute one; and he saw, apparently, in the more common rectangular symmetry merely what he calls a necessary, but secondary, result of growth [85].
Within the next few years, a number of botanical writers were content to point out further exceptions to Sachs’s Rule [86]; and in some cases to show that the curvatures of the partition-walls, especially such cases of lenticular curvature as we have described, were by no means accounted for by either Hofmeister or Sachs; while within the same period, Sachs himself, and also Rauber, attempted to extend the main generalisation to animal tissues [87].
While these writers regarded the form and arrangement of the cell-walls as a biological phenomenon, with little if any direct relation to ordinary physical laws, or with but a vague reference to “mechanical conditions,” the physical side of the case was soon urged by others, with more or less force and cogency. Indeed the general resemblance between a cellular tissue and a “froth” [p. 306] had been pointed out long before, by Melsens, who had made an “artificial tissue” by blowing into a solution of white of egg [88].
In 1886, Berthold published his Protoplasmamechanik, in which he definitely adopted the principle of “minimal areas,” and, following on the lines of Plateau, compared the forms of many cell-surfaces and the arrangement of their partitions with those assumed under surface tension by a system of “weightless films.” But, as Klebs [89] points out in reviewing Berthold’s book, Berthold was careful to stop short of attributing the biological phenomena to a definite mechanical cause. They remained for him, as they had done for Sachs, so many “phenomena of growth,” or “properties of protoplasm.”
In the same year, but while still apparently unacquainted with Berthold’s work, Errera [90] published a short but very lucid article, in which he definitely ascribed to the cell-wall (as Hofmeister had already done) the properties of a semi-liquid film and drew from this as a logical consequence the deduction that it must assume the various configurations which the law of minimal areas imposes on the soap-bubble. So what we may call Errera’s Law is formulated as follows: A cellular membrane, at the moment of its formation, tends to assume the form which would be assumed, under the same conditions, by a liquid film destitute of weight.
Soon afterwards Chabry, in discussing the embryology of the Ascidians, indicated many of the points in which the contacts between cells repeat the surface-tension phenomena of the soap-bubble, and came to the conclusion that part, at least, of the embryological phenomena were purely physical [91]; and the same line of investigation and thought were pursued and developed by Robert, in connection with the embryology of the Mollusca [92]. Driesch again, in a series of papers, continued to draw attention to the presence of capillary phenomena in the segmenting cells [p. 307] of various embryos, and came to the conclusion that the mode of segmentation was of little importance as regards the final result [93].
Lastly de Wildeman [94], in a somewhat wider, but also vaguer generalisation than Errera’s, declared that “The form of the cellular framework of vegetables, and also of animals, in its essential features, depends upon the forces of molecular physics.”
1917 Source Illustration
Fig. 114.
Let us return to our problem of the arrangement of partition films. When we have three bubbles in contact, instead of two as in the case already considered, the phenomenon is strictly analogous to our former case. The three bubbles will be separated by three partition surfaces, whose curvature will depend upon the relative size of the spheres, and which will be plane if the latter are all of the same dimensions; but whether plane or curved, the three partitions will meet one another at an angle of 120°, in an axial line. Various pretty geometrical corollaries accompany this arrangement. For instance, if Fig. 114 represent the three associated bubbles in a plane drawn through their centres, c, c′, c″ (or what is the same thing, if it represent the base of three bubbles resting on a plane), then the lines uc, uc″, or sc, sc′, etc., drawn to the [p. 308] centres from the points of intersection of the circular arcs, will always enclose an angle of 60°. Again (Fig. 115), if we make the angle c″uf equal to 60°, and produce uf to meet cc″ in f, f will be the centre of the circular arc which constitutes the partition Ou; and further, the three points f, g, h, successively determined in this
1917 Source Illustration
Fig. 115.
manner, will lie on one and the same straight line. In the case of coequal bubbles or cells (as in Fig. 114, B), it is obvious that the lines joining their centres form an equilateral triangle; and consequently, that the centre of each circle (or sphere) lies on the circumference of the other two; it is also obvious that uf is now [p. 309] parallel to cc″, and accordingly that the centre of curvature of the partition is now infinitely distant, or (as we have already said), that the partition itself is plane.
1917 Source Illustration
Fig. 116.
When we have four bubbles in conjunction, they would seem to be capable of arrangement in two symmetrical ways: either, as in Fig. 116 (A), with the four partition-walls meeting at right angles, or, as in (B), with five partitions meeting, three and three, at angles of 120°. This latter arrangement is strictly analogous to the arrangement of three bubbles in Fig. 114. Now, though both of these figures, from their symmetry, are apparently figures of equilibrium, yet, physically, the former turns out to be of unstable and the latter of stable equilibrium. If we try to bring our four bubbles into the form of Fig. 116, A, such an arrangement endures only for an instant; the partitions glide upon each other, a median wall springs into existence, and the system at once assumes the form of our second figure (B). This is a direct consequence of the law of minimal areas: for it can be shewn, by somewhat difficult mathematics (as was first done by Lamarle), that, in dividing a closed space into a given number of chambers by means of partition-walls, the least possible area of these partition-walls, taken together, can only be attained when they meet together in groups of three, at equal angles, that is to say at angles of 120°. [p. 310]
Wherever we have a true cellular complex, an arrangement of cells in actual physical contact by means of a boundary film, we find this general principle in force; we must only bear in mind that, for its perfect recognition, we must be able to view the object in a plane at right angles to the boundary walls. For instance, in any ordinary section of a vegetable parenchyma, we recognise the appearance of a “froth,” precisely resembling that which we can construct by imprisoning a mass of soap-bubbles in a narrow vessel with flat sides of glass; in both cases we see the cell-walls everywhere meeting, by threes, at angles of 120°, irrespective of the size of the individual cells: whose relative size, on the other hand, determines the curvature of the partition-walls. On the surface of a honey-comb we have precisely the same conjunction, between cell and cell, of three boundary walls, meeting at 120°. In embryology, when we examine a segmenting egg, of four (or more) segments, we find in like manner, in the great majority of cases, if not in all, that the same principle is still exemplified; the four segments do not meet in a common centre, but each cell is in contact with two others, and the three, and only three, common boundary walls meet at the normal angle of 120°. A so-called polar furrow [95], the visible edge of a vertical partition-wall, joins (or separates) the two triple contacts, precisely as in Fig. 116, B.
In the four-celled stage of the frog’s egg, Rauber (an exceptionally careful observer) shews us three alternative modes in which the four cells may be found to be conjoined (Fig. 117). In (A) we have the commonest arrangement, which is that which we have just studied and found to be the simplest theoretical one; that namely where a straight “polar furrow” intervenes, and where, at its extremities, the partition-walls are conjoined three by three. In (B), we have again a polar furrow, which is now seen to be a portion of the first “segmentation-furrow” (cf. Fig. 155 etc.) by which the egg was originally divided into two; the four-celled stage being reached by the appearance of the transverse furrows [p. 311] and their corresponding partitions. In this case, the polar furrow is seen to be sinuously curved, and Rauber tells us that its curvature gradually alters: as a matter of fact, it (or rather the partition-wall corresponding to it) is gradually setting itself into a position of equilibrium, that is to say of equiangular contact with its neighbours, which position of equilibrium is already attained or nearly so in Fig. 117, A. In Fig. 117, C, we have a very different condition, with which we shall deal in a moment.
1917 Source Illustration
Fig. 117. Various ways in which the four cells are co-arranged in the four-celled stage of the frog’s egg. (After Rauber.)
According to the relative magnitude of the bodies in contact, this “polar furrow” may be longer or shorter, and it may be so minute as to be not easily discernible; but it is quite certain that no simple and homogeneous system of fluid films such as we are dealing with is in equilibrium without its presence. In the accounts given, however, by embryologists of the segmentation of the egg, while the polar furrow is depicted in the great majority of cases, there are others in which it has not been seen and some in which its absence is definitely asserted [96]. The cases where four cells, lying in one plane, meet in a point, such as were frequently figured by the older embryologists, are very difficult to verify, and I have not come across a single clear case in recent literature. Considering the physical stability of the other arrangement, the great preponderance of cases in which it is known to occur, the difficulty of recognising the polar furrow in cases where it is very small and unless it be specially looked for, and the natural tendency of the draughtsman to make an all but symmetrical structure appear wholly so, I am much inclined to attribute to [p. 312] error or imperfect observation all those cases where the junction-lines of four cells are represented (after the manner of Fig. 116, A) as a simple cross [97].
But while a true four-rayed intersection, or simple cross, is theoretically impossible (save as a transitory and highly unstable condition), there is another condition which may closely simulate it, and which is common enough. There are plenty of representations of segmenting eggs, in which, instead of the triple junction and polar furrow, the four cells (and in like manner their more numerous successors) are represented as rounded off, and separated from one another by an empty space, or by a little drop of an extraneous fluid, evidently not directly miscible with the fluid surfaces of the cells. Such is the case in the obviously accurate figure which Rauber gives (Fig. 117, C) of the third mode of conjunction in the four-celled stage of the frog’s egg. Here Rauber is most careful to point out that the furrows do not simply “cross,” or meet in a point, but are separated by a little space, which he calls the Polgrübchen, and asserts to be constantly present whensoever the polar furrow, or Brechungslinie, is not to be discerned. This little interposed space, with its contained drop of fluid, materially alters the case, and implies a new condition of theoretical and actual equilibrium. For, on the one hand, we see that now the four intercellular partitions do not meet one another at all; but really impinge upon four new and separate partitions, which constitute interfacial contacts, not between cell and cell, but between the respective cells and the intercalated drop. And secondly, the angles at which these four little surfaces will meet the four cell-partitions, will be determined, in the usual way, by the balance between the respective tensions of these several surfaces. In an extreme case (as in some pollen-grains) it may be found that the cells under the observed circumstances are not truly in surface contact: that they are so many drops which touch but do not “wet” one another, and which are merely held together by the pressure of the surrounding envelope. But even supposing, [p. 313] as is in all probability the actual case, that they are in actual fluid contact, the case from the point of view of surface tension presents no difficulty. In the case of the conjoined soap-bubbles, we were dealing with similar contacts and with equal surface tensions throughout the system; but in the system of protoplasmic cells which constitute the segmenting egg we must make allowance for an inequality of tensions, between the surfaces where cell meets cell, and where on the other hand cell-surface is in contact with the surrounding medium,—in this case generally water or one of the fluids of the body. Remember that our general condition is that, in our
1917 Source Illustration
Fig. 118.
entire system, the sum of the surface energies is a minimum; and, while this is attained by the sum of the surfaces being a minimum in the case where the energy is uniformly distributed, it is not necessarily so under non-uniform conditions. In the diagram (Fig. 118) if the energy per unit area be greater along the contact surface cc′, where cell meets cell, than along ca or cb, where cell-surface is in contact with the surrounding medium, these latter surfaces will tend to increase and the surface of cell-contact to diminish. In short there will be the usual balance of forces between the tension along the surface cc′, and the two opposing tensions along ca and cb. If the former be greater than either of the other two, the outside angle will be less than 120°; and if the tension along the surface cc′ be as much or more than the sum of the other two, then the drops will stand in contact only, save for the possible effect of external pressure, at a point. This is the explanation, in general terms, of the peculiar conditions obtaining in Nostoc and its allies (p. 300), and it also leads us to a consideration of the general properties and characters of an “epidermal” layer.
While the inner cells of the honey-comb are symmetrically situated, sharing with their neighbours in equally distributed pressures or tensions, and therefore all tending with great accuracy [p. 314] to identity of form, the case is obviously different with the cells at the borders of the system. So it is, in like manner, with our froth of soap-bubbles. The bubbles, or cells, in the interior of the mass are all alike in general character, and if they be equal in size are alike in every respect: their sides are uniformly flattened [98], and tend to meet at equal angles of 120°. But the bubbles which constitute the outer layer retain their spherical surfaces, which however still tend to meet the partition-walls connected with them at constant angles of 120°. This outer layer of bubbles, which forms the surface of our froth, constitutes after a fashion what we should call in botany an “epidermal” layer. But in our froth of soap-bubbles we have, as a rule, the same kind of contact (that is to say, contact with air) both within and without the bubbles; while in our living cell, the outer wall of the epidermal cell is exposed to air on the one side, but is in contact with the
1917 Source Illustration
Fig. 119.
protoplasm of the cell on the other: and this involves a difference of tensions, so that the outer walls and their adjacent partitions are no longer likely to meet at equal angles of 120°. Moreover, a chemical change, due for instance to oxidation or possibly also to adsorption, is very likely to affect the external wall, and may tend to its consolidation; and this process, as we have seen, is tantamount to a large increase, and at the same time an equalisation, of tension in that outer wall, and will lead the adjacent partitions to impinge upon it at angles more and more nearly approximating to 90°: the bubble-like, or spherical, surfaces of the individual cells being more and more flattened in consequence. Lastly, the chemical changes which affect the outer walls of the superficial cells may extend, in greater or less degree, to their inner walls also: with the result that these [p. 315] cells will tend to become more or less rectangular throughout, and will cease to dovetail into the interstices of the next subjacent layer. These then are the general characters which we recognise in an epidermis; and we perceive that the fundamental character of an epidermis simply is that it lies on the outside, and that its main physical characteristics follow, as a matter of course, from the position which it occupies and from the various consequences which that situation entails. We have however by no means exhausted the subject in this short account; for the botanist is accustomed to draw a sharp distinction between a true epidermis and what is called epidermal tissue. The latter, which is found in such a sea-weed as Laminaria and in very many other cryptogamic plants, consists, as in the hypothetical case we have described, of a more or less simple and direct modification of the general or fundamental tissue. But a “true epidermis,” such as we have it in the higher plants, is something with a long morphological history, something which has been laid down or differentiated in an early stage of the plant’s growth, and which afterwards retains its separate and independent character. We shall see presently that a physical reason is again at hand to account, under certain circumstances, for the early partitioning off, from a mass of embryonic tissue, of an outer layer of cells which from their first appearance are marked off from the rest by their rectangular and flattened form.
We have hitherto considered our cells, or bubbles, as lying in a plane of symmetry, and further, we have only considered the appearance which they present as projected on that plane: in simpler words, we have been considering their appearance in surface or in sectional view. But we have further to consider them as solids, whether they be still grouped in relation to a single plane (like the four cells in Fig. 116) or heaped upon one another, as for instance in a tetrahedral form like four cannon-balls; and in either case we have to pass from the problems of plane to those of solid geometry. In short, the further development of our theme must lead us along two paths of enquiry, which continually intercross, namely (1) the study of more complex cases of partition and of contact in a plane, and (2) the whole question of the surfaces [p. 316] and angles presented by solid figures in symmetrical juxtaposition. Let us take a simple case of the latter kind, and again afterwards, so far as possible, let us try to keep the two themes separate.
Where we have three spheres in contact, as in Fig. 114 or in either half of Fig. 116, B, let us consider the point of contact (O, Fig. 114) not as a point in the plane section of the diagram, but as a point where three furrows meet on the surface of the system. At this point, three cells meet; but it is also obvious that there meet here six surfaces, namely the outer, spherical walls of the three bubbles, and the three partition-walls which divide them, two and two. Also, four lines or edges meet here; viz. the three external arcs which form the outer boundaries of the partition-walls (and which correspond to what we commonly call the “furrows” in the segmenting egg); and as a fourth edge, the “arris” or junction of the three partitions (perpendicular to the plane of the paper), where they all three meet together, as we have seen, at equal angles of 120°. Lastly, there meet at the point four solid angles, each bounded by three surfaces: to wit, within each bubble a solid angle bounded by two partition-walls and by the surface wall; and (fourthly) an external solid angle bounded by the outer surfaces of all three bubbles. Now in the case of the soap-bubbles (whose surfaces are all in contact with air, both outside and in), the six films meeting at the point, whether surface films or partition films, are all similar, with similar tensions. In other words the tensions, or forces, acting at the point are all similar and symmetrically arranged, and it at once follows from this that the angles, solid as well as plane, are all equal. It is also obvious that, as regards the point of contact, the system will still be symmetrical, and its symmetry will be quite unchanged, if we add a fourth bubble in contact with the other three: that is to say, if where we had merely the outer air before, we now replace it by the air in the interior of another bubble. The only difference will be that the pressure exercised by the walls of this fourth bubble will alter the curvature of the surfaces of the others, so far as it encloses them; and, if all four bubbles be identical in size, these surfaces which formerly we called external and which have now come to be internal partitions, will, like the others, be flattened by equal and opposite pressure, into planes. We are now dealing, in short, [p. 317] with six planes, meeting symmetrically in a point, and constituting there four equal solid angles.
1917 Source Illustration
Fig. 120.
If we make a wire cage, in the form of a regular tetrahedron, and dip it into soap-solution, then when we withdraw it we see that to each one of the six edges of the tetrahedron, i.e. to each one of the six wires which constitute the little cage, a film has attached itself; and these six films meet internally at a point, and constitute in every respect the symmetrical figure which we have just been describing. In short, the system of films we have hereby automatically produced is precisely the system of partition-walls which exist in our tetrahedral aggregation of four spherical bubbles:—precisely the same, that is to say, in the neighbourhood of the meeting-point, and only differing in that we have made the wires of our tetrahedron straight, instead of imitating the circular arcs which actually form the intersections of our bubbles. This detail we can easily introduce in our wire model if we please.
Let us look for a moment at the geometry of our figure. Let o (Fig. 120) be the centre of the tetrahedron, i.e. the centre of symmetry where our films meet; and let oa, ob, oc, od, be lines drawn to the four corners of the tetrahedron. Produce ao to meet the base in p; then apd is a right-angled triangle. It is not difficult to prove that in such a figure, o (the centre of gravity of the system) [p. 318] lies just three-quarters of the way between an apex, a, and a point, p, which is the centre of gravity of the opposite base. Therefore
That is to say, the angle aod is just, as nearly as possible, 109° 28′ 16″. This angle, then, of 109° 28′ 16″, or very nearly 109 degrees and a half, is the angle at which, in this and every other solid system of liquid films, the edges of the partition-walls meet one another at a point. It is the fundamental angle in the solid geometry of our systems, just as 120° was the fundamental angle of symmetry so long as we considered only the plane projection, or plane section, of three films meeting in an edge.
Out of these two angles, we may construct a great variety of figures, plane and solid, which become all the more varied and complex when, by considering the case of unequal as well as equal cells, we admit curved (e.g. spherical) as well as plane boundary surfaces. Let us consider some examples and illustrations of these, beginning with those which we need only consider in reference to a plane.
Let us imagine a system of equal cylinders, or equal spheres, in contact with one another in a plane, and represented in section by the equal and contiguous circles of Fig. 121. I borrow my figure, by the way, from an old Italian naturalist, Bonanni (a contemporary of Borelli, of Hay and Willoughby and of Martin Lister), who dealt with this matter in a book chiefly devoted to molluscan shells [99].
It is obvious, as a simple geometrical fact, that each of these equal circles is in contact with six surrounding circles. Imagine now that the whole system comes under some uniform stress. It may be of uniform surface tension at the boundaries of all the cells; it may be of pressure caused by uniform growth or expansion within the cells; or it may be due to some uniformly applied constricting pressure from without. In all of these cases the points of contact between the circles in the diagram will be extended into [p. 319] lines of contact, representing surfaces of contact in the actual spheres or cylinders; and the equal circles of our diagram will be converted into regular and equal hexagons. The angles of these hexagons, at each of which three hexagons meet, are of course angles of 120°. So far as the form is concerned, so long as we are concerned only with a morphological result and not with a physiological process, the result is precisely the same whatever be the force which brings the bodies together in symmetrical apposition; it is by no means necessary for us, in the first instance, even to enquire whether it be surface tension or mechanical pressure or some other physical force which is the cause, or the main cause, of the phenomenon.
1917 Source Illustration
Fig. 121. Diagram of hexagonal cells. (After Bonanni.)
The production by mutual interaction of polyhedral cells, which, under conditions of perfect symmetry, become regular hexagons, is very beautifully illustrated by Prof. Bénard’s “tourbillons cellulaires” (cf. p. 259), and also in some of Leduc’s diffusion experiments. A weak (5 per cent.) solution of gelatine is allowed to set on a plate of glass, and little drops of a 5 or 10 per cent. solution of ferrocyanide of potassium are then placed at regular intervals upon the gelatine. Immediately each little drop becomes the centre, or pole, of a system of diffusion currents, [p. 320] and the several systems conflict with and repel one another, so that presently each little area becomes the seat of a double current system, from its centre outwards and back again; until at length the concentration of the field becomes equalised and the currents [p. 321]
1917 Source Illustration
Fig. 122. An “artificial tissue,” formed by coloured drops of sodium chloride solution diffusing in a less dense solution of the same salt. (After Leduc.)
1917 Source Illustration
Fig. 123. An artificial cellular tissue, formed by the diffusion in gelatine of drops of a solution of potassium ferrocyanide. (After Leduc.)
cease. After equilibrium is attained, and when the gelatinous mass is permitted to dry, we have an artificial tissue of more or less regularly hexagonal “cells,” which simulate in the closest way an organic parenchyma. And by varying the experiment, in ways which Leduc describes, we may simulate various forms of tissue, and produce cells with thick walls or with thin, cells in close contact or with wide intercellular spaces, cells with plane or with curved partitions, and so forth.
1917 Source Illustration
Fig. 124. Epidermis of Girardia. (After Goebel.)
The hexagonal pattern is illustrated among organisms in countless cases, but those in which the pattern is perfectly regular, by reason of perfect uniformity of force and perfect equality of the individual cells, are not so numerous. The hexagonal epithelium-cells of the pigment layer of the eye, external to the retina, are a good example. Here we have a single layer of uniform cells, reposing on the one hand upon a basement membrane, supported behind by the solid wall of the sclerotic, and exposed on the other hand to the uniform fluid pressure of the vitreous humour. The conditions all point, and lead, to a perfectly symmetrical result: that is to say, the cells, uniform in size, are flattened out to a uniform thickness by the fluid pressure acting radially; and their reaction on each other converts the flattened discs into regular hexagons. In an ordinary columnar epithelium, such as that of the intestine, we see again that the columnar cells have been compressed into hexagonal prisms; but here as a rule the cells are less uniform in size, small cells are apt to be intercalated among the larger, and the perfect symmetry is accordingly lost. The same is true of ordinary vegetable parenchyma; the originally spherical cells are approximately equal in size, but only approximately; and there are accordingly all degrees in the regularity and symmetry of the resulting tissue. But obviously, wherever we [p. 322] have, in addition to the forces which tend to produce the regular hexagonal symmetry, some other asymmetrical component arising from growth or traction, then our regular hexagons will be distorted in various simple ways. This condition is illustrated in the accompanying diagram of the epidermis of Girardia; it also accounts for the more or less pointed or fusiform cells, each still in contact (as a rule) with six others, which form the epithelial lining of the blood-vessels: and other similar, or analogous, instances are very common.
1917 Source Illustration
Fig. 125. Soap-froth under pressure. (After Rhumbler.)
In a soap-froth imprisoned between two glass plates, we have a symmetrical system of cells, which appear in optical section (as in Fig. 125, B) as regular hexagons; but if we press the plates a little closer together, the hexagons become deformed or flattened (Fig. 125, A). In this case, however, if we cease to apply further pressure, the tension of the films throughout the system soon adjusts itself again, and in a short time the system has regained the former symmetry of Fig. 125, B.
1917 Source Illustration
Fig. 126. From leaf of Elodea canadensis. (After Berthold.)
In the growth of an ordinary dicotyledonous leaf, we once more see reflected in the form of its epidermal cells the tractions, irregular but on the whole longitudinal, which growth has superposed on the tensions of the partition-walls (Fig. 126). In the narrow elongated leaf of a Monocotyledon, such as a hyacinth, the elongated, apparently quadrangular [p. 323] cells of the epidermis appear as a necessary consequence of the simpler laws of growth which gave its simple form to the leaf as a whole. In this last case, however, as in all the others, the rule still holds that only three partitions (in surface view) meet in a point; and at their point of meeting the walls are for a short distance manifestly curved, so as to permit the junction to take place at or nearly at the normal angle of 120°.
Briefly speaking, wherever we have a system of cylinders or spheres, associated together with sufficient mutual interaction to bring them into complete surface contact, there, in section or in surface view, we tend to get a pattern of hexagons.
While the formation of an hexagonal pattern on the basis of ready-formed and symmetrically arranged material units is a very common, and indeed the general way, it does not follow that there are not others by which such a pattern can be obtained. For instance, if we take a little triangular dish of mercury and set it vibrating (either by help of a tuning-fork, or by simply tapping on the sides) we shall have a series of little waves or ripples starting inwards from each of the three faces; and the intercrossing, or interference of these three sets of waves produces crests and hollows, and intermediate points of no disturbance, whose loci are seen as a beautiful pattern of minute hexagons. It is possible that the very minute and astonishingly regular pattern of hexagons which we see, for instance, on the surface of many diatoms, may be a phenomenon of this order [100]. The same may be the case also in Arcella, where an apparently hexagonal pattern is found not to consist of simple hexagons, but of “straight lines in three sets of parallels, the lines of each set making an angle of sixty degrees with those of the other two sets [101].” We must also bear in mind, in the case of the minuter forms, the large possibilities of optical illusion. For instance, in one of Abbe’s “diffraction-plates,” a pattern of dots, set at equal interspaces, is reproduced on a very minute scale by photography; but under certain conditions of microscopic illumination and focussing, these isolated dots appear as a pattern of hexagons.
A symmetrical arrangement of hexagons, such as we have just been studying, suggests various simple geometrical corollaries, of which the following may perhaps be a useful one.
We may sometimes desire to estimate the number of hexagonal areas or facets in some structure where these are numerous, such for instance as the [p. 324] cornea of an insect’s eye, or in the minute pattern of hexagons on many diatoms. An approximate enumeration is easily made as follows.
For the area of a hexagon (if we call δ the short diameter, that namely which bisects two of the opposite sides) is δ2 × (√3)/2, the area of a circle being d2 ⋅ π/4. Then, if the diameter (d) of a circular area include n hexagons, the area of that circle equals (n ⋅ δ)2 × π/4. And, dividing this number by the area of a single hexagon, we obtain for the number of areas in the circle, each equal to a hexagonal facet, the expression n2 × π/4 × 2/(√3) = 0·907n2, or (9/10) ⋅ n2, nearly.
This calculation deals, not only with the complete facets, but with the areas of the broken hexagons at the periphery of the circle. If we neglect these latter, and consider our whole field as consisting of successive rings of hexagons about a central one, we may obtain a still simpler rule [102]. For obviously, around our central hexagon there stands a zone of six, and around these a zone of twelve, and around these a zone of eighteen, and so on. And the total number, excluding the central hexagon, is accordingly:
For one zone 6 = 2 × 3 = 3 × 1 × 2,
For two zones 18 = 3 × 6 = 3 × 2 × 3,
For three zones 36 = 4 × 9 = 3 × 3 × 4,
For four zones 60 = 5 × 12 = 3 × 4 × 5,
For five zones 90 = 6 x 15 = 3 × 5 × 6,
and so forth. If N be the number of zones, and if we add one to the above numbers for the odd central hexagon, the rule evidently is, that the total number, H, = 3N(N + 1) + 1. Thus, if in a preparation of a fly’s cornea, I can count twenty-five facets in a line from a central one, the total number in the entire circular field is (3 × 25 × 26) + 1 = 1951 [103].
The same principles which account for the development of hexagonal symmetry hold true, as a matter of course, not only of individual cells (in the biological sense), but of any close-packed bodies of uniform size and originally circular outline; and the hexagonal pattern is therefore of very common occurrence, under widely different circumstances. The curious reader may consult Sir Thomas Browne’s quaint and beautiful account, in the Garden of Cyrus, of hexagonal (and also of quincuncial) symmetry in plants and animals, which “doth neatly declare how nature Geometrizeth, and observeth order in all things.” [p. 325]
We have many varied examples of this principle among corals, wherever the polypes are in close juxtaposition, with neither empty space nor accumulations of matrix between their adjacent walls. Favosites gothlandica, for instance, furnishes us with an excellent example. In the great genus Lithostrotion we have some species that are “massive” and others that are “fasciculate”; in other words in some the long cylindrical corallites are in close contact with one another, and in others they are separate and loosely bundled (Fig. 127). Accordingly in the former the corallites are
1917 Source Illustration
Fig. 127. Lithostrotion Martini. (After Nicholson.)
1917 Source Illustration
Fig. 128. Cyathophyllum hexagonum. (From Nicholson, after Zittel.)
squeezed into hexagonal prisms, while in the latter they retain their cylindrical form. Where the polypes are comparatively few, and so have room to spread, the mutual pressure ceases to work or only tends to push them asunder, letting them remain circular in outline (e.g. Thecosmilia). Where they vary gradually in size, as for instance in Cyathophyllum hexagonum, they are more or less hexagonal but are not regular hexagons; and where there is greater and more irregular variation in size, the cells will be on the average hexagonal, but some will have fewer and some more sides than six, as in the annexed figure of Arachnophyllum (Fig. 129). [p. 326] Where larger and smaller cells, corresponding to two different kinds of zooids, are mixed together, we may get various results. If the larger cells are numerous enough to be more or less in contact with one another (e.g. various Monticuliporae) they will be irregular hexagons, while the smaller cells between them will be crushed into all manner of irregular angular forms. If on the other hand the large cells are comparatively few and are large and strong-walled compared with their smaller neighbours, then the latter alone will be squeezed into hexagons, while the larger ones will tend to retain their circular outline undisturbed (e.g. Heliopora, Heliolites, etc.).
1917 Source Illustration
Fig. 129. Arachnophyllum pentagonum. (After Nicholson.)
1917 Source Illustration
Fig. 130. Heliolites. (After Woods.)
When, as happens in certain corals, the peripheral walls or “thecae” of the individual polypes remain undeveloped but the radiating septa are formed and calcified, then we obtain new and beautiful mathematical configurations (Fig. 131). For the radiating septa are no longer confined to the circular or hexagonal bounds of a polypite, but tend to meet and become confluent with their neighbours on every side; and, tending to assume positions of equilibrium, or of minimal area, under the restraints to which they are subject, they fall into congruent curves; and these correspond, in a striking manner, to the lines of force running, in a common field of force, between a number of secondary centres. Similar patterns may be produced in various ways, by the play of osmotic or magnetic forces; and a particular and very curious case is to be found in those complicated forms of nuclear division [p. 327] known as triasters, polyasters, etc., whose relation to a field of force Hartog has explained [104]. It is obvious that, in our corals, these curving septa are all orthogonal to the non-existent hexagonal boundaries. As the phenomenon is wholly due to the imperfect development or non-existence of a thecal wall, it is not surprising that we find identical configurations among various corals, or families of corals, not otherwise related to one another; we find the same or very similar patterns displayed, for instance, in Synhelia (Oculinidae), in Phillipsastraea (Rugosa), in Thamnastraea (Fungida), and in many more.
1917 Source Illustration
Fig. 131. Surface-views of Corals with undeveloped thecae and confluent septa. A, Thamnastraea; B, Comoseris. (From Nicholson, after Zittel.)
The most famous of all hexagonal conformations and perhaps the most beautiful is that of the bee’s cell. Here we have, as in our last examples, a series of equal cylinders, compressed by symmetrical forces into regular hexagonal prisms. But in this case we have two rows of such cylinders, set opposite to one another, end to end; and we have accordingly to consider also the conformation of their ends. We may suppose our original cylindrical cells to have spherical ends, which is their normal and symmetrical mode of termination; and, for closest packing, it is obvious that the end of any one cylinder will touch, and fit in between, the ends of three cylinders in the opposite row. It is just as when we pile round-shot in a heap; each sphere that we [p. 328] set down fits into its nest between three others, and the four form a regular tetrahedral arrangement. Just as it was obvious, then, that by mutual pressure from the six laterally adjacent cells, any one cell would be squeezed into a hexagonal prism, so is it also obvious that, by mutual pressure against the three terminal neighbours, the end of any one cell will be compressed into a solid trihedral angle whose edges will meet, as in the analogous case already described of a system of soap-bubbles, at a plane angle of 109° and so many minutes and seconds. What we have to comprehend, then, is how the six sides of the cell are to be combined with its three terminal facets. This is done by bevelling off three alternate angles of the prism, in a uniform manner, until we have tapered the prism to a point; and by so doing, we evidently produce three rhombic surfaces, each of which is double of the triangle formed by joining the apex to the three untouched angles of the prism. If we experiment, not with cylinders, but with spheres, if for instance we pile together a mass of bread-pills (or pills of plasticine), and then submit the whole to a uniform pressure, it is obvious that each ball (like the seeds in a pomegranate, as Kepler said), will be in contact with twelve others,—six in its own plane, three below and three above, and in compression it will therefore develop twelve plane surfaces. It will in short repeat, above and below, the conditions to which the bee’s cell is subject at one end only; and, since the sphere is symmetrically situated towards its neighbours on all sides, it follows that the twelve plane sides to which its surface has been reduced will be all similar, equal and similarly situated. Moreover, since we have produced this result by squeezing our original spheres close together, it is evident that the bodies so formed completely fill space. The regular solid which fulfils all these conditions is the rhombic dodecahedron. The bee’s cell, then, is this figure incompletely formed: it is a hexagonal prism with one open or unfinished end, and one trihedral apex of a rhombic dodecahedron.
The geometrical form of the bee’s cell must have attracted the attention and excited the admiration of mathematicians from time immemorial. Pappus the Alexandrine has left us (in the introduction to the Fifth Book of his Collections) an account of its hexagonal plan, and he drew from its mathematical symmetry the [p. 329] conclusion that the bees were endowed with reason: “There being, then, three figures which of themselves can fill up the space round a point, viz. the triangle, the square and the hexagon, the bees have wisely selected for their structure that which contains most angles, suspecting indeed that it could hold more honey than either of the other two.” Erasmus Bartholinus was apparently the first to suggest that this hypothesis was not warranted, and that the hexagonal form was no more than the necessary result of equal pressures, each bee striving to make its own little circle as large as possible.
The investigation of the ends of the cell was a more difficult matter, and came later, than that of its sides. In general terms this arrangement was doubtless often studied and described: as for instance, in the Garden of Cyrus: “And the Combes themselves so regularly contrived that their mutual intersections make three Lozenges at the bottom of every Cell; which severally regarded make three Rows of neat Rhomboidall Figures, connected at the angles, and so continue three several chains throughout the whole comb.” But Maraldi [105] (Cassini’s nephew) was the first to measure the terminal solid angle or determine the form of the rhombs in the pyramidal ending of the cell. He tells us that the angles of the rhomb are 110° and 70°: “Chaque base d’alvéole est formée par trois rhombes presque toujours égaux et semblables, qui, suivant les mesures que nous avons prises, ont les deux angles obtus chacun de 110 degrés, et par conséquent les deux aigus chacun de 70°.” He also stated that the angles of the trapeziums which form the sides of the body of the cell were identical angles, of 110° and 70°; but in the same paper he speaks of the angles as being, respectively, 109° 28′ and 70° 32′. Here a singular confusion at once arose, and has been perpetuated in the books [106]. “Unfortunately Réaumur chose to look upon this second determination of Maraldi’s as being, as well as the first, a direct result of measurement, whereas it is in reality theoretical. He speaks of it as Maraldi’s more precise measurement, and this error has been repeated in spite of its absurdity to the present day; nobody [p. 330] appears to have thought of the impossibility of measuring such a thing as the end of a bee’s cell to the nearest minute.” At any rate, it now occurred to Réaumur (as curiously enough, it had not done to Maraldi) that, just as the closely packed hexagons gave the minimal extent of boundary in a plane, so the actual solid figure, as determined by Maraldi, might be that which, for a given solid content, gives the minimum of surface: or which, in other words, would hold the most honey for the least wax. He set this problem before Koenig, and the geometer confirmed his conjecture, the result of his calculations agreeing within two minutes (109° 26′ and 70° 34′) with Maraldi’s determination. But again, Maclaurin [107] and Lhuilier [108], by different methods, obtained a result identical with Maraldi’s; and were able to shew that the discrepancy of 2′ was due to an error in Koenig’s calculation (of tan θ = √2),—that is to say to the imperfection of his logarithmic tables,—not (as the books say [109]) “to a mistake on the part of the Bee.” “Not to a mistake on the part of Maraldi” is, of course, all that we are entitled to say.
1917 Source Illustration
Fig. 132.
The theorem may be proved as follows:
ABCDEF, abcdef, is a right prism upon a regular hexagonal base. The corners BDF are cut off by planes through the lines AC, CE, EA, meeting in a point V on the axis VN of the prism, and intersecting Bb, Dd, Ff, at X, Y, Z. It is evident that the volume of the figure thus formed is the same as that of the original prism with hexagonal ends. For, if the axis cut the hexagon ABCDEF in N, the volumes ACVN, ACBX are equal. [p. 331]
It is required to find the inclination of the faces forming the trihedral angle at V to the axis, such that the surface of the figure may be a minimum.
Let the angle NVX, which is half the solid angle of the prism, = θ; the side of the hexagon, as AB, = a; and the height, as Aa, = h.
But this expression vanishes, that is to say, d(Area)/dθ = 0, when cos θ = 1/√3, that is when θ = 54° 44′ 8″ = ½(109° 28′ 16″).
This then is the condition under which the total area of the figure has its minimal value.
That the beautiful regularity of the bee’s architecture is due to some automatic play of the physical forces, and that it were fantastic to assume (with Pappus and Réaumur) that the bee intentionally seeks for a method of economising wax, is certain, but the precise manner of this automatic action is not so clear. When the hive-bee builds a solitary cell, or a small cluster of cells, as it does for those eggs which are to develop into queens, it makes but a rude production. The queen-cells are lumps of coarse wax hollowed out and roughly bitten into shape, bearing the marks of the bee’s jaws, like the marks of a blunt adze on a rough-hewn log. Omitting the simplest of all cases, when (as among some humble-bees) the old cocoons are used to hold honey, the cells built by the “solitary” wasps and bees are of various kinds. They may be formed by partitioning off little chambers in a hollow stem; [p. 332] they may be rounded or oval capsules, often very neatly constructed, out of mud, or vegetable fibre or little stones, agglutinated together with a salivary glue; but they shew, except for their rounded or tubular form, no mathematical symmetry. The social wasps and many bees build, usually out of vegetable matter chewed into a paste with saliva, very beautiful nests of “combs”; and the close-set papery cells which constitute these combs are just as regularly hexagonal as are the waxen cells of the hive-bee. But in these cases (or nearly all of them) the cells are in a single row; their sides are regularly hexagonal, but their ends, from the want of opponent forces, remain simply spherical. In Melipona domestica (of which Darwin epitomises Pierre Huber’s description) “the large waxen honey-cells are nearly spherical, nearly equal in size, and are aggregated into an irregular mass.” But the spherical form is only seen on the outside of the mass; for inwardly each cell is flattened into “two, three or more flat surfaces, according as the cell adjoins two, three or more other cells. When one cell rests on three other cells, which from the spheres being nearly of the same size is very frequently and necessarily the case, the three flat surfaces are united into a pyramid; and this pyramid, as Huber has remarked, is manifestly a gross imitation of the three-sided pyramidal base of the cell of the hive-bee [110].” The question is, to what particular force are we to ascribe the plane surfaces and definite angles which define the sides of the cell in all these cases, and the ends of the cell in cases where one row meets and opposes another. We have seen that Bartholin suggested, and it is still commonly believed, that this result is due to simple physical pressure, each bee enlarging as much as it can the cell which it is a-building, and nudging its wall outwards till it fills every intervening gap and presses hard against the similar efforts of its neighbour in the cell next door [111]. But it is very doubtful [p. 333] whether such physical or mechanical pressure, more or less intermittently exercised, could produce the all but perfectly smooth, plane surfaces and the all but perfectly definite and constant angles which characterise the cell, whether it be constructed of wax or papery pulp. It seems more likely that we have to do with a true surface-tension effect; in other words, that the walls assume their configuration when in a semi-fluid state, while the papery pulp is still liquid, or while the wax is warm under the high temperature of the crowded hive [112]. Under these circumstances, the direct efforts of the wasp or bee may be supposed to be limited to the making of a tubular cell, as thin as the nature of the material permits, and packing these little cells as close as possible together. It is then easily conceivable that the symmetrical tensions of the adjacent films (though somewhat retarded by viscosity) should suffice to bring the whole system into equilibrium, that is to say into the precise configuration which the comb actually presents. In short, the Maraldi pyramids which terminate the bee’s cell are precisely identical with the facets of a rhombic dodecahedron, such as we have assumed to constitute (and which doubtless under certain conditions do constitute) the surfaces of contact in the interior of a mass of soap-bubbles or of uniform parenchymatous cells; and there is every reason to believe that the physical explanation is identical, and not merely mathematically analogous.
The remarkable passage in which Buffon discusses the bee’s cell and the hexagonal configuration in general is of such historical importance, and tallies so closely with the whole trend of our enquiry, that I will quote it in full: “Dirai-je encore un mot; ces cellules des abeilles, tant vantées, tant admirées, me fournissent une preuve de plus contre l’enthousiasme et l’admiration; cette figure, toute géométrique et toute régulière qu’elle nous paraît, et qu’elle est en effet dans la spéculation, n’est ici qu’un résultat mécanique et assez imparfait qui se trouve souvent dans la nature, [p. 334] et que l’on remarque même dans les productions les plus brutes; les cristaux et plusieurs autres pierres, quelques sels, etc., prennent constamment cette figure dans leur formation. Qu’on observe les petites écailles de la peau d’une roussette, on verra qu’elles sont hexagones, parce que chaque écaille croissant en même temps se fait obstacle, et tend à occuper le plus d’espace qu’il est possible dans un espace donné: on voit ces mêmes hexagones dans le second estomac des animaux ruminans, on les trouve dans les graines, dans leurs capsules, dans certaines fleurs, etc. Qu’on remplisse un vaisseau de pois, ou plûtot de quelque autre graine cylindrique, et qu’on le ferme exactement après y avoir versé autant d’eau que les intervalles qui restent entre ces graines peuvent en recevoir; qu’on fasse bouillir cette eau, tous ces cylindres deviendront de colonnes à six pans [113]. On y voit clairement la raison, qui est purement mécanique; chaque graine, dont la figure est cylindrique, tend par son renflement à occuper le plus d’espace possible dans un espace donné, elles deviennent donc toutes nécessairement hexagones par la compression réciproque. Chaque abeille cherche à occuper de même le plus d’espace possible dans un espace donné, il est donc nécessaire aussi, puisque le corps des abeilles est cylindrique, que leurs cellules sont hexagones,—par la même raison des obstacles réciproques. On donne plus d’esprit aux mouches dont les ouvrages sont les plus réguliers; les abeilles sont, dit-on, plus ingénieuses que les guêpes, que les frélons, etc., qui savent aussi l’architecture, mais dont les constructions sont plus grossières et plus irrégulières que celles des abeilles: on ne veut pas voir, ou l’on ne se doute pas que cette régularité, plus ou moins grande, dépend uniquement du nombre et de la figure, et nullement de l’intelligence de ces petites bêtes; plus elles sont nombreuses, plus il y a des forces qui agissent également et s’opposent de même, plus il y a par conséquent de contrainte mécanique, de régularité forcée, et de perfection apparente dans leurs productions [114].” [p. 335]
A very beautiful hexagonal symmetry, as seen in section, or dodecahedral, as viewed in the solid, is presented by the cells which form the pith of certain rushes (e.g. Juncus effusus), and somewhat less diagrammatically by those which make the pith of the banana. These cells are stellate in form, and the tissue presents in section the appearance of a network of six-rayed stars (Fig. 133, c), linked together by the tips of the rays, and separated by symmetrical, air-filled, intercellular spaces. In thick sections, the solid twelve-rayed stars may be very beautifully seen under the binocular microscope.
1917 Source Illustration
Fig. 133. Diagram of development of “stellate cells,” in pith of Juncus. (The dark, or shaded, areas represent the cells; the light areas being the gradually enlarging “intercellular spaces.”)
What has happened here is not difficult to understand. Imagine, as before, a system of equal spheres all in contact, each one therefore touching six others in an equatorial plane; and let the cells be not only in contact, but become attached at the points of contact. Then instead of each cell expanding, so as to encroach on and fill up the intercellular spaces, let each cell tend to contract or shrivel up, by the withdrawal of fluid from its interior. The [p. 336] result will obviously be that the intercellular spaces will increase; the six equatorial attachments of each cell (Fig. 133, a) (or its twelve attachments in all, to adjacent cells) will remain fixed, and the portions of cell-wall between these points of attachment will be withdrawn in a symmetrical fashion (b) towards the centre. As the final result (c) we shall have a “dodecahedral star” or star-polygon, which appears in section as a six-rayed figure. It is obviously necessary that the pith-cells should not only be attached to one another, but that the outermost layer should be firmly attached to a boundary wall, so as to preserve the symmetry of the system. What actually occurs in the rush is tantamount to this, but not absolutely identical. Here it is not so much the pith-cells which tend to shrivel within a boundary of constant size, but rather the boundary wall (that is, the peripheral ring of woody and other tissues) which continues to expand after the pith-cells which it encloses have ceased to grow or to multiply. The twelve points of attachment on the spherical surface of each little pith-cell are uniformly drawn asunder; but the content, or volume, of the cell does not increase correspondingly; and the remaining portions of the surface, accordingly, shrink inwards and gradually constitute the complicated surface of a twelve-pointed star, which is still a symmetrical figure and is still also a surface of minimal area under the new conditions.
A few years after the publication of Plateau’s book, Lord Kelvin shewed, in a short but very beautiful paper [115], that we must not hastily assume from such arguments as the foregoing, that a close-packed assemblage of rhombic dodecahedra will be the true and general solution of the problem of dividing space with a minimum partitional area, or will be present in a cellular liquid “foam,” in which it is manifest that the problem is actually and automatically solved. The general mathematical solution of the problem (as we have already indicated) is, that every interface or partition-wall must have constant curvature throughout; that where such partitions meet in an edge, they must intersect at angles such that equal forces, in planes perpendicular to the line [p. 337] of intersection, shall balance; and finally, that no more than three such interfaces may meet in a line or edge, whence it follows that the angle of intersection of the film-surfaces must be exactly 120°. An assemblage of equal and similar rhombic dodecahedra goes far to meet the case: it completely fills up space; all its surfaces or interfaces are planes, that is to say, surfaces of constant curvature throughout; and these surfaces all meet together at angles of 120°. Nevertheless, the proof that our rhombic dodecahedron (such as we find exemplified in the bee’s cell) is a surface of minimal area, is not a comprehensive proof; it is limited to certain conditions, and practically amounts to no more than this, that of the regular solids, with all sides plane and similar, this one has the least surface for its solid content.
1917 Source Illustration
Fig. 134.
The rhombic dodecahedron has six tetrahedral angles, and eight trihedral angles; and it is obvious, on consideration, that at each of the former six dodecahedra meet in a point, and that, where the four tetrahedral facets of each coalesce with their neighbours, we have twelve plane films, or interfaces, meeting in a point. In a precisely similar fashion, we may imagine twelve plane films, drawn inwards from the twelve edges of a cube, to meet at a point in the centre of the cube. But, as Plateau discovered [116], when we dip a cubical wire skeleton into soap-solution and take it out again, the twelve films which are thus generated do not meet in a point, but are grouped around a small central, plane, quadrilateral film (Fig. 134). In other words, twelve plane films, meeting in a point, are essentially unstable. If we blow upon our artificial film-system, the little quadrilateral alters its place, setting itself parallel now to one and now to another of the paired faces of the cube; but we never get rid of it. Moreover, the size and shape of the quadrilateral, as of all the other films in the system, are perfectly definite. Of the twelve films (which we had [p. 338] expected to find all plane and all similar) four are plane isosceles triangles, and eight are slightly curved quadrilateral figures. The former have two curved sides, meeting at an angle of 109° 28′, and their apices coincide with the corners of the central quadrilateral, whose sides are also curved, and also meet at this identical angle;—which (as we observe) is likewise an angle which we have been dealing with in the simpler case of the bee’s cell, and indeed in all the regular solids of which we have yet treated.
By completing the assemblage of polyhedra of which Plateau’s skeleton-cube gives a part, Lord Kelvin shewed that we should obtain a set of equal and similar fourteen-sided figures, or “tetrakaidecahedra”; and that by means of an assemblage of these figures space is homogeneously partitioned—that is to say, into equal, similar and similarly situated cells—with an economy of surface in relation to area even greater than in an assemblage of rhombic dodecahedra.
In the most generalised case, the tetrakaidecahedron is bounded by three pairs of equal and parallel quadrilateral faces, and four pairs of equal and parallel hexagonal faces, neither the quadrilaterals nor the hexagons being necessarily plane. In a certain particular case, the quadrilaterals are plane surfaces, but the hexagons slightly curved “anticlastic” surfaces; and these latter have at every point equal and opposite curvatures, and are surfaces of minimal curvature for a boundary of six curved edges. The figure has the remarkable property that, like the plane rhombic dodecahedron, it so partitions space that three faces meeting in an edge do so everywhere at equal angles of 120° [117].
We may take it as certain that, in a system of perfectly fluid films, like the interior of a mass of soap-bubbles, where the films are perfectly free to glide or to rotate over one another, the mass is actually divided into cells of this remarkable conformation. [p. 339] And it is quite possible, also, that in the cells of a vegetable parenchyma, by carefully macerating them apart, the same conformation may yet be demonstrated under suitable conditions; that is to say when the whole tissue is highly symmetrical, and the individual cells are as nearly as possible equal in size. But in an ordinary microscopic section, it would seem practically impossible to distinguish the fourteen-sided figure from the twelve-sided. Moreover, if we have anything whatsoever interposed so as to prevent our twelve films meeting in a point, and (so to speak) to take the place of our little central quadrilateral,—if we have, for instance, a tiny bead or droplet in the centre of our artificial system, or even a little thickening, or “bourrelet” as Plateau called it, of the cell-wall, then it is no longer necessary that the tetrakaidecahedron should be formed. Accordingly, it is very probably the case that, in the parenchymatous tissue, under the actual conditions of restraint and of very imperfect fluidity, it is after all the rhombic dodecahedral configuration which, even under perfectly symmetrical conditions, is generally assumed.
It follows from all that we have said, that the problems connected with the conformation of cells, and with the manner in which a given space is partitioned by them, soon become exceedingly complex. And while this is so even when all our cells are equal and symmetrically placed, it becomes vastly more so when cells varying even slightly in size, in hardness, rigidity or other qualities, are packed together. The mathematics of the case very soon become too hard for us; but in its essence, the phenomenon remains the same. We have little reason to doubt, and no just cause to disbelieve, that the whole configuration, for instance of an egg in the advanced stages of segmentation, is accurately determined by simple physical laws, just as much as in the early stages of two or four cells, during which early stages we are able to recognise and demonstrate the forces and their resultant effects. But when mathematical investigation has become too difficult, it often happens that physical experiment can reproduce for us the phenomena which Nature exhibits to us, and which we are striving to comprehend. For instance, in an admirable research, M. Robert shewed, some years ago, not only that the early segmentation of [p. 340] the egg of Trochus (a marine univalve mollusc) proceeded in accordance with the laws of surface tension, but he also succeeded in imitating by means of soap-bubbles, several stages, one after another, of the developing egg.
1917 Source Illustration
Fig. 135. Aggregations of four soap-bubbles, to shew various arrangements of the intermediate partition and polar furrows. (After Robert.)
Robert carried his experiments as far as the stage of sixteen cells, or bubbles. It is not easy to carry the artificial system quite so far, but in the earlier stages the experiment is easy; we have merely to blow our bubbles in a little dish, adding one to another, and adjusting their sizes to produce a symmetrical system. One of the simplest and prettiest parts of his investigation concerned the “polar furrow” of which we have spoken on p. 310. On blowing four little contiguous bubbles he found (as we may all find with the greatest ease) that they form a symmetrical system, two in contact with one another by a laminar film, and two, which are elevated a little above the others, and which are separated by the length of the aforesaid lamina. The bubbles are thus in contact three by three, their partition-walls making with one another equal angles of 120°. The upper and lower edges of the intermediate lamina (the lower one visible through the transparent system) constitute the two polar furrows of the embryologist (Fig. 135, 1–3). The lamina itself is plane when the system is symmetrical, but it responds by a corresponding curvature to the least inequality of the bubbles on either side. In the experiment, the upper polar furrow is usually a little shorter than the lower, but parallel to it; that is to say, the lamina is of trapezoidal form: this lack of perfect symmetry being due (in the experimental case) to the lower portion of the bubbles being somewhat drawn asunder by the tension of their attachments to the sides of the dish (Fig. 135, 4). A similar phenomenon is usually found in Trochus, according to Robert, and many other observers have likewise found the upper furrow to be shorter than the one below. In the various species of the genus Crepidula, Conklin asserts that the two furrows are equal in C. convexa, that the upper one is the shorter in C. fornicata, and that the upper one all but disappears in C. plana; but we may well be permitted to doubt, without the evidence of very special investigations, whether these slight physical differences are actually characteristic of, and constant in, particular allied species. [p. 341] Returning to the experimental case, Robert found that by withdrawing a little air from, and so diminishing the bulk of the two terminal bubbles (i.e. those at the ends of the intermediate lamina), the upper polar furrow was caused to elongate, till it became equal in length to the lower; and by continuing the process it became the longer in its turn. These two conditions have again been described by investigators as characteristic of this embryo or that; for instance in Unio, Lillie has described the two furrows as gradually altering their respective lengths [118]; and Wilson (as Lillie remarks) had already pointed out that “the reduction of the apical cross-furrow, as compared with that at the vegetative pole [p. 342] in molluscs and annelids ‘stands in obvious relation to the different size of the cells produced at the two poles [119].’ ”
When the two lateral bubbles are gradually reduced in size, or the two terminal ones enlarged, the upper furrow becomes shorter and shorter; and at the moment when it is about to vanish, a new furrow makes its instantaneous appearance in a direction perpendicular to the old one; but the inferior furrow, constrained by its attachment to the base, remains unchanged, and accordingly our two polar furrows, which were formerly parallel, are now at right angles to one another. Instead of a single plane quadrilateral partition, we have now two triangular ones, meeting in the middle of the system by their apices, and lying in planes at right angles to one another (Fig. 135, 5–7) [120]. Two such polar furrows, equal in length and arranged in a cross, have again been frequently described by the embryologists. Robert himself found this condition in Trochus, as an occasional or exceptional occurrence: it has been described as normal in Asterina by Ludwig, in Branchipus by Spangenberg, and in Podocoryne and Hydractinia by Bunting. It is evident that it represents a state of unstable equilibrium, only to be maintained under certain conditions of restraint within the system.
So, by slight and delicate modifications in the relative size of the cells, we may pass through all the possible arrangements of the median partition, and of the “furrows” which correspond to its upper and lower edges; and every one of these arrangements has been frequently observed in the four-celled stage of various embryos. As the phases pass one into the other, they are accompanied by changes in the curvature of the partition, which in like manner correspond precisely to phenomena which the embryologists have witnessed and described. And all these configurations belong to that large class of phenomena whose distribution among embryos, or among organisms in general, bears no relation to the boundaries of zoological classification; through molluscs, worms, [p. 343] coelenterates, vertebrates and what not, we meet with now one and now another, in a medley which defies classification. They are not “vital phenomena,” or “functions” of the organism, or special characteristics of this or that organism, but purely physical phenomena. The kindred but more complicated phenomena which correspond to the polar furrow when a larger number of cells than four are associated together, we shall deal with in the next chapter.
Having shewn that the capillary phenomena are patent and unmistakable during the earlier stages of embryonic development, but soon become more obscure and incapable of experimental reproduction in the later stages, when the cells have increased in number, various writers including Robert himself have been inclined to argue that the physical phenomena die away, and are overpowered and cancelled by agencies of a very different order. Here we pass into a region where direct observation and experiment are not at hand to guide us, and where a man’s trend of thought, and way of judging the whole evidence in the case, must shape his philosophy. We must remember that, even in a froth of soap-bubbles, we can apply an exact analysis only to the simplest cases and conditions of the phenomenon; we cannot describe, but can only imagine, the forces which in such a froth control the respective sizes, positions and curvatures of the innumerable bubbles and films of which it consists; but our knowledge is enough to leave us assured that what we have learned by investigation of the simplest cases includes the principles which determine the most complex. In the case of the growing embryo we know from the beginning that surface tension is only one of the physical forces at work; and that other forces, including those displayed within the interior of each living cell, play their part in the determination of the system. But we have no evidence whatsoever that at this point, or that point, or at any, the dominion of the physical forces over the material system gives place to a new condition where agencies at present unknown to the physicist impose themselves on the living matter, and become responsible for the conformation of its material fabric.
Before we leave for the present the subject of the segmenting [p. 344] egg, we must take brief note of two associated problems: viz. (1) the formation and enlargement of the segmentation cavity, or central interspace around which the cells tend to group themselves in a single layer, and (2) the formation of the gastrula, that is to say (in a typical case) the conversion “by invagination,” of the one-layered ball into a two-layered cup. Neither problem is free from difficulty, and all we can do meanwhile is to state them in general terms, introducing some more or less plausible assumptions.
The former problem is comparatively easy, as regards the tendency of a segmentation cavity to enlarge, when once it has been established. We may then assume that subdivision of the cells is due to the appearance of a new-formed septum within each cell, that this septum has a tendency to shrink under surface tension, and that these changes will be accompanied on the whole by a diminution of surface energy in the system. This being so, it may be shewn that the volume of the divided cells must be less than it was prior to division, or in other words that part of their contents must exude during the process of segmentation [121]. Accordingly, the case where the segmentation cavity enlarges and the embryo developes into a hollow blastosphere may, under the circumstances, be simply described as the case where that outflow or exudation from the cells of the blastoderm is directed on the whole inwards.
The physical forces involved in the invagination of the cell-layer to form the gastrula have been repeatedly discussed [122], but the true explanation seems as yet to be by no means clear. The case, however, is probably not a very difficult one, provided that we may assume a difference of osmotic pressure at the two poles of the blastosphere, that is to say between the cells which are being differentiated into outer and inner, into epiblast and hypoblast. It is plain that a blastosphere, or hollow vesicle bounded by a layer of vesicles, is under very different physical conditions from a single, simple vesicle or bubble. The blastosphere has no effective surface tension of its own, such as to exert pressure on [p. 345] its contents or bring the whole into a spherical form; nor will local variations of surface energy be directly capable of affecting the form of the system. But if the substance of our blastosphere be sufficiently viscous, then osmotic forces may set up currents which, reacting on the external fluid pressure, may easily cause modifications of shape; and the particular case of invagination itself will not be difficult to account for on this assumption of non-uniform exudation and imbibition.
Chapter VIII: The Forms of Tissues (continued)¶
1917 Cambridge First Edition (pp. 346–410)
THE FORMS OF TISSUES OR CELL-AGGREGATES (continued)
The problems which we have been considering, and especially that of the bee’s cell, belong to a class of “isoperimetrical” problems, which deal with figures whose surface is a minimum for a definite content or volume. Such problems soon become difficult, but we may find many easy examples which lead us towards the explanation of biological phenomena; and the particular subject which we shall find most easy of approach is that of the division, in definite proportions, of some definite portion of space, by a partition-wall of minimal area. The theoretical principles so arrived at we shall then attempt to apply, after the manner of Berthold and Errera, to the actual biological phenomena of cell-division.
This investigation we may approach in two ways: by considering, namely, the partitioning off from some given space or area of one-half (or some other fraction) of its content; or again, by dealing simultaneously with the partitions necessary for the breaking up of a given space into a definite number of compartments.
If we take, to begin with, the simple case of a cubical cell, it is obvious that, to divide it into two halves, the smallest possible partition-wall is one which runs parallel to, and midway between, two of its opposite sides. If we call a the length of one of the edges of the cube, then a2 is the area, alike of one of its sides, and of the partition which we have interposed parallel, or normal, thereto. But if we now consider the bisected cube, and wish to divide the one-half of it again, it is obvious that another partition parallel to the first, so far from being the smallest possible, is precisely twice the size of a cross-partition perpendicular to it; [p. 347] for the area of this new partition is a × a/2. And again, for a third bisection, our next partition must be perpendicular to the other two, and it is obviously a little square, with an area of (½a)2 = ¼(a2).
From this we may draw the simple rule that, for a rectangular body or parallelopiped to be divided equally by means of a partition of minimal area, (1) the partition must cut across the longest axis of the figure; and (2) in the event of successive bisections, each partition must run at right angles to its immediate predecessor.
1917 Source Illustration
Fig. 136. (After Berthold.)
We have already spoken of “Sachs’s Rules,” which are an empirical statement of the method of cell-division in plant-tissues; and we may now set them forth in full.
The cell typically tends to divide into two co-equal parts.
Each new plane of division tends to intersect at right angles the preceding plane of division.
The first of these rules is a statement of physiological fact, not without its exceptions, but so generally true that it will justify us in limiting our enquiry, for the most part, to cases of equal subdivision. That it is by no means universally true for cells generally is shewn, for instance, by such well-known cases [p. 348] as the unequal segmentation of the frog’s egg. It is true when the dividing cell is homogeneous, and under the influence of symmetrical forces; but it ceases to be true when the field is no longer dynamically symmetrical, for instance, when the parts differ in surface tension or internal pressure. This latter condition, of asymmetry of field, is frequent in segmenting eggs [123], and is then equivalent to the principle upon which Balfour laid stress, as leading to “unequal” or to “partial” segmentation of the egg,—viz. the unequal or asymmetrical distribution of protoplasm and of food-yolk.
The second rule, which also has its exceptions, is true in a large number of cases; and it owes its validity, as we may judge from the illustration of the repeatedly bisected cube, solely to the guiding principle of minimal areas. It is in short subordinate to, and covers certain cases included under, a much more important and fundamental rule, due not to Sachs but to Errera; that (3) the incipient partition-wall of a dividing cell tends to be such that its area is the least possible by which the given space-content can be enclosed.
Let us return to the case of our cube, and let us suppose that, instead of bisecting it, we desire to shut off some small portion only of its volume. It is found in the course of experiments upon soap-films, that if we try to bring a partition-film too near to one side of a cubical (or rectangular) space, it becomes unstable; and is easily shifted to a totally new position, in which it constitutes a curved cylindrical wall, cutting off one corner of the cube. It meets the sides of the cube at right angles (for reasons which we have already considered); and, as we may see from the symmetry [p. 349] of the case, it constitutes precisely one-quarter of a cylinder. Our plane transverse partition, wherever it was placed, had always the same area, viz. a2; and it is obvious that a cylindrical wall, if it cut off a small corner, may be much less than this. We want, accordingly, to determine what is the particular volume which might be partitioned off with equal economy of wall-space in one way as the other, that is to say, what area of cylindrical wall would be neither more nor less than the area a2. The calculation is very easy.
The surface-area of a cylinder of length a is 2πr ⋅ a, and that of our quarter-cylinder is, therefore, a ⋅ πr/2; and this being, by hypothesis, = a2, we have a = πr/2, or r = 2a/π.
The volume of a cylinder, of length a, is aπr2, and that of our quarter-cylinder is (a ⋅ πr2)/4, which (by substituting the value of r) is equal to (a3)/π.
Now precisely this same volume is, obviously, shut off by a transverse partition of area a2, if the third side of the rectangular space be equal to a/π. And this fraction, if we take a = 1, is equal to 0·318…, or rather less than one-third. And, as we have just seen, the radius, or side, of the corresponding quarter-cylinder will be twice that fraction, or equal to ·636 times the side of the cubical cell.
1917 Source Illustration
Fig. 137.
If then, in the process of division of a cubical cell, it so divide that the two portions be not equal in volume but that one portion by anything less than about three-tenths of the whole, or three-sevenths of the other portion, there will be a tendency for the cell to divide, not by means of a plane transverse partition, but by means of a curved, cylindrical wall cutting off one corner of the original cell; and the part so cut off will be one-quarter of a cylinder.
By a similar calculation we can shew that a spherical wall, cutting off one solid angle of the cube, and constituting an octant of a sphere, would likewise be of less area than a plane partition as soon as the volume to be enclosed was not greater than about [p. 350] one-quarter of the original cell [124]. But while both the cylindrical wall and the spherical wall would be of less area than the plane transverse partition after that limit (of one-quarter volume) was passed, the cylindrical would still be the better of the two up to a further limit. It is only when the volume to be partitioned off [p. 351] is no greater than about 0·15, or somewhere about one-seventh, of the whole, that the spherical cell-wall in an angle of the cubical cell, that is to say the octant of a sphere, is definitely of less area than the quarter-cylinder. In the accompanying diagram (Fig. 138) the relative areas of the three partitions are shewn for all fractions, less than one-half, of the divided cell.
1917 Source Illustration
Fig. 138.
In this figure, we see that the plane transverse partition, whatever fraction of the cube it cut off, is always of the same dimensions, that is to say is always equal to a2, or = 1. If one-half of the cube have to be cut off, this plane transverse partition is much the best, for we see by the diagram that a cylindrical partition cutting off an equal volume would have an area about 25%, and a spherical partition would have an area about 50% greater. The point A in the diagram corresponds to the point where the cylindrical partition would begin to have an advantage over the plane, that is to say (as we have seen) when the fraction to be cut off is about one-third, or ·318 of the whole. In like manner, at B the spherical octant begins to have an advantage over the plane; and it is not till we reach the point C that the spherical octant becomes of less area than the quarter-cylinder.
1917 Source Illustration
Fig. 139.
The case we have dealt with is of little practical importance to the biologist, because the cases in which a cubical, or rectangular, cell divides unequally, and unsymmetrically, are apparently few; but we can find, as Berthold pointed out, a few examples, for instance in the hairs within the reproductive “conceptacles” of certain Fuci (Sphacelaria, etc., Fig. 139), or in the “paraphyses” of mosses (Fig. 142). But it is of great theoretical importance: as serving to introduce us to a large class of cases, in which the shape and the relative dimensions of the original cavity lead, according to the principle of minimal areas, to cell-division in very definite and sometimes unexpected ways. It is not easy, nor indeed possible, to give a generalised account of these cases, for the limiting conditions are somewhat complex, and the mathematical treatment soon becomes difficult. But it is easy to comprehend a few simple cases, which of themselves will carry us a good long way; and which will go far to convince the student that, in other cases [p. 352] which we cannot fully master, the same guiding principle is at the root of the matter.
The bisection of a solid (or the subdivision of its volume in other definite proportions) soon leads us into a geometry which, if not necessarily difficult, is apt to be unfamiliar; but in such problems we can go a long way, and often far enough for our particular purpose, if we merely consider the plane geometry of a side or section of our figure. For instance, in the case of the cube which we have been just considering, and in the case of the plane and cylindrical partitions by which it has been divided, it is obvious that, since these two partitions extend symmetrically from top to bottom of our cube, that we need only consider (so far as they are concerned) the manner in which they subdivide the base of the cube. The whole problem of the solid, up to a certain point, is contained in our plane diagram of Fig. 138. And when our particular solid is a solid of revolution, then it is obvious that a study of its plane of symmetry (that is to say any plane passing through its axis of rotation) gives us the solution of the whole problem. The right cone is a case in point, for here the investigation of its modes of symmetrical subdivision is completely met by an examination of the isosceles triangle which constitutes its plane of symmetry.
The bisection of an isosceles triangle by a line which shall be the shortest possible is a very easy problem. Let ABC be such a triangle of which A is the apex; it may be shewn that, for its shortest line of bisection, we are limited to three cases: viz. to a vertical line AD, bisecting the angle at A and the side BC; to a transverse line parallel to the base BC; or to an oblique line parallel to AB or to AC. The respective magnitudes, or lengths, of these partition lines follow at once from the magnitudes of the angles of our triangle. For we know, to begin with, since the areas of similar figures vary as the squares of their linear dimensions, that, in order to bisect the area, a line parallel to one side of our triangle must always have a length equal to 1/√2 of that side. If then, we take our base, BC, in all cases of a length = 2, the transverse partition drawn parallel to it will always have a length equal to 2/√2, or = √2. The vertical [p. 353] partition, AD, since BD = 1, will always equal tan β (β being the angle ABC). And the oblique partition, GH, being equal to AB/√2 = 1/(√2 cos β). If then we call our vertical, transverse
1917 Source Illustration
Fig. 140.
and oblique partitions, V, T, and O, we have V = tan β; T = √2; and O = 1/(√2 cos β), or
And, working out these equations for various values of β, we very soon see that the vertical partition (V) is the least of the three until β = 45°, at which limit V and O are each equal to 1/√2 = ·707; and that again, when β = 60°, O and T are each = 1, after which T (whose value always = 1) is the shortest of the three partitions. And, as we have seen, these results are at once applicable, not only to the case of the plane triangle, but also to that of the conical cell.
1917 Source Illustration
Fig. 141.
In like manner, if we have a spheroidal body, less than a hemisphere, such for instance as a low, watch-glass shaped cell (Fig. 141, a), it is obvious that the smallest possible partition by which we can divide it into two equal halves [p. 354] is (as in our flattened disc) a median vertical one. And likewise, the hemisphere itself can be bisected by no smaller partition meeting the walls at right angles than that median one which divides it into two similar quadrants of a sphere. But if we produce our hemisphere into a more elevated, conical body, or into a cylinder with spherical cap, it is obvious that there comes a point where a transverse, horizontal partition will bisect the figure with less area of partition-wall than a median vertical one (c). And furthermore, there will be an intermediate region, a region where height and base have their relative dimensions nearly equal (as in b), where an oblique partition will be better than either the vertical or the transverse, though here the analogy of our triangle does not suffice to give us the precise limiting values. We need not examine these limitations in detail, but we must look at the curvatures which accompany the several conditions. We have seen that a film tends to set itself at equal angles to the surface which it meets, and therefore, when that surface is a solid, to meet it (or its tangent if it be a curved surface) at right angles. Our vertical partition is, therefore, everywhere normal to the original cell-walls, and constitutes a plane surface.
But in the taller, conical cell with transverse partition, the latter still meets the opposite sides of the cell at right angles, and it follows that it must itself be curved; moreover, since the tension, and therefore the curvature, of the partition is everywhere uniform, it follows that its curved surface must be a portion of a sphere, concave towards the apex of the original, now divided, cell. In the intermediate case, where we have an oblique partition, meeting both the base and the curved sides of the mother-cell, the contact must still be everywhere at right angles: provided we continue to suppose that the walls of the mother-cell (like those of our diagrammatic cube) have become practically rigid before the partition appears, and are therefore not affected and deformed by the tension of the latter. In such a case, and especially when the cell is elliptical in cross-section, or is still more complicated in form, it is evident that the partition, in adapting itself to circumstances and in maintaining itself as a surface of minimal area subject to all the conditions of the case, may have to assume a complex curvature. [p. 355]
1917 Source Illustration
Fig. 142. S-shaped partitions: A, from Taonia atomaria (after Reinke); B, from paraphyses of Fucus; C, from rhizoids of Moss; D, from paraphyses of Polytrichum.
While in very many cases the partitions (like the walls of the original cell) will be either plane or spherical, a more complex curvature will be assumed under a variety of conditions. It will be apt to occur, for instance, when the mother-cell is irregular in shape, and one particular case of such asymmetry will be that in which (as in Fig. 143) the cell has begun to branch, or give off a diverticulum, before division takes place. A very complicated case of a different kind, though not without its analogies to the cases we are considering, will occur in the partitions of minimal area which subdivide the spiral tube of a nautilus, as we shall presently see. And again, whenever we have a marked internal asymmetry of the cell, leading to irregular and anomalous modes of division, in which the cell is not necessarily divided into two equal halves and in which the partition-wall may assume an oblique position, then apparently anomalous curvatures will tend to make their appearance [125].
Suppose that a more or less oblong cell have a tendency to divide by means of an oblique partition (as may happen through various causes or conditions of asymmetry), such a partition will still have a tendency to set itself at right angles to the rigid walls [p. 356] of the mother-cell: and it will at once follow that our oblique partition, throughout its whole extent, will assume the form of a complex, saddle-shaped or anticlastic surface.
1917 Source Illustration
Fig. 143. Diagrammatic explanation of S-shaped partition.
Many such cases of partitions with complex or double curvature exist, but they are not always easy of recognition, nor is the particular case where they appear in a terminal cell a common one. We may see them, for instance, in the roots (or rhizoids) of Mosses, especially at the point of development of a new rootlet (Fig. 142, C); and again among Mosses, in the “paraphyses” of the male prothalli (e.g. in Polytrichum), we find more or less similar partitions (D). They are frequent also among many Fuci, as in the hairs or paraphyses of Fucus itself (B). In Taonia atomaria, as figured in Reinke’s memoir on the Dictyotaceae of the Gulf of Naples [126], we see, in like manner, oblique partitions, which on more careful examination are seen to be curves of double curvature (Fig. 142, A).
The physical cause and origin of these S-shaped partitions is somewhat obscure, but we may attempt a tentative explanation. When we assert a tendency for the cell to divide transversely to its long axis, we are not only stating empirically that the partition tends to appear in a small, rather than a large cross-section of the cell: but we are also implicitly ascribing to the cell a longitudinal polarity (Fig. 143, A), and implicitly asserting that it tends to [p. 357] divide (just as the segmenting egg does), by a partition transverse to its polar axis. Such a polarity may conceivably be due to a chemical asymmetry, or anisotropy, such as we have learned of (from Professor Macallum’s experiments) in our chapter on Adsorption. Now if the chemical concentration, on which this anisotropy or polarity (by hypothesis) depends, be unsymmetrical, one of its poles being as it were deflected to one side, where a little branch or bud is being (or about to be) given off,—all in precise accordance with the adsorption phenomena described on p. 289,—then our “polar axis” would necessarily be a curved axis, and the partition, being constrained (again ex hypothesi) to arise transversely to the polar axis, would lie obliquely to the apparent axis of the cell (Fig. 143, B, C). And if the oblique partition be so situated that it has to meet the opposite walls (as in C), then, in order to do so symmetrically (i.e. either perpendicularly, as when the cell-wall is already solidified, or at least at equal angles on either side), it is evident that the partition, in its course from one side of the cell to the other, must necessarily assume a more or less S-shaped curvature (Fig. 143, D).
As a matter of fact, while we have abundant simple illustrations of the principles which we have now begun to study, apparent exceptions to this simplicity, due to an asymmetry of the cell itself, or of the system of which the single cell is but a part, are by no means rare. For example, we know that in cambium-cells, division frequently takes place parallel to the long axis of the cell, when a partition of much less area would suffice if it were set cross-ways: and it is only when a considerable disproportion has been set up between the length and breadth of the cell, that the balance is in part redressed by the appearance of a transverse partition. It was owing to such exceptions that Berthold was led to qualify and even to depreciate the importance of the law of minimal areas as a factor in cell-division, after he himself had done so much to demonstrate and elucidate it [127]. He was deeply and rightly impressed by the fact that other forces besides surface [p. 358] tension, both external and internal to the cell, play their part in the determination of its partitions, and that the answer to our problem is not to be given in a word. How fundamentally important it is, however, in spite of all conflicting tendencies and apparent exceptions, we shall see better and better as we proceed.
But let us leave the exceptions and return to a consideration of the simpler and more general phenomena. And in so doing, let us leave the case of the cubical, quadrangular or cylindrical cell, and examine the case of a spherical cell and of its successive divisions, or the still simpler case of a circular, discoidal cell.
When we attempt to investigate mathematically the position and form of a partition of minimal area, it is plain that we shall be dealing with comparatively simple cases wherever even one dimension of the cell is much less than the other two. Where two dimensions are small compared with the third, as in a thin cylindrical filament like that of Spirogyra, we have the problem at its simplest; for it is at once obvious, then, that the partition must lie transversely to the long axis of the thread. But even where one dimension only is relatively small, as for instance in a flattened plate, our problem is so far simplified that we see at once that the partition cannot be parallel to the extended plane, but must cut the cell, somehow, at right angles to that plane. In short, the problem of dividing a much flattened solid becomes identical with that of dividing a simple surface of the same form.
There are a number of small Algae, growing in the form of small flattened discs, consisting (for a time at any rate) of but a single layer of cells, which, as Berthold shewed, exemplify this comparatively simple problem; and we shall find presently that it is also admirably illustrated in the cell-divisions which occur in the egg of a frog or a sea-urchin, when the egg for the sake of experiment is flattened out under artificial pressure.
1917 Source Illustration
Fig. 144. Development of Erythrotrichia. (After Berthold.)
1917 Source Illustration
Fig. 145.
Fig. 144 (taken from Berthold’s Monograph of the Naples Bangiaciae) represents younger and older discs of the little alga Erythrotrichia discigera; and it will be seen that, in all stages save the first, we have an arrangement of cell-partitions which looks somewhat complex, but into which we must attempt to throw some light and order. Starting with the original single, and flattened, [p. 359] cell, we have no difficulty with the first two cell-divisions; for we know that no bisecting partitions can possibly be shorter than the two diameters, which divide the cell into halves and into quarters. We have only to remember that, for the sum total of partitions to be a minimum, three only must meet in a point; and therefore, the four quadrantal walls must shift a little, producing the usual little median partition, or cross-furrow, instead of one common, central point of junction. This little intermediate wall, however, will be very small, and to all intents and purposes we may deal with the case as though we had now to do with four equal cells, each one of them a perfect quadrant. And so our problem is, to find the shortest line which shall divide the quadrant of a circle into two halves of equal area. A radial partition (Fig. 145, A), starting from the apex of the quadrant, is at once excluded, for a reason similar to that just referred to; our choice must lie therefore between two modes of division such as are illustrated in Fig. 145, where the partition is either (as in B) [p. 360] concentric with the outer border of the cell, or else (as in C) cuts that outer border; in other words, our partition may (B) cut both radial walls, or (C) may cut one radial wall and the periphery. These are the two methods of division which Sachs called, respectively, (B) periclinal, and (C) anticlinal [128]. We may either treat the walls of the dividing quadrant as already solidified, or at least as having a tension compared with which that of the incipient partition film is inconsiderable. In either case the partition must meet the cell-wall, on either side, at right angles, and (its own tension and curvature being everywhere uniform) it must take the form of a circular arc.
Now we find that a flattened cell which is approximately a quadrant of a circle invariably divides after the manner of Fig. 145, C, that is to say, by an approximately circular, anticlinal wall, such as we now recognise in the eight-celled stage of Erythrotrichia (Fig. 144); let us then consider that Nature has solved our problem for us, and let us work out the actual geometric conditions.
Let the quadrant OAB (in Fig. 146) be divided into two parts of equal area, by the circular arc MP. It is required to determine (1) the position of P upon the arc of the quadrant, that is to say the angle BOP; (2) the position of the point M on the side OA; and (3) the length of the arc MP in terms of a radius of the quadrant.
Draw OP; also PC a tangent, meeting OA in C; and PN, perpendicular to OA. Let us call a a radius; and θ the angle at C, which is obviously equal to OPN, or POB. Then
The area of the portion PMN
And the area of the portion PNA
Therefore the area of the whole portion PMA
and also, by hypothesis, = ½ ⋅ area of the quadrant, = (πa2)/8.
1917 Source Illustration
Fig. 146.
Hence θ is defined by the equation
We may solve this equation by constructing a table (of which the following is a small portion) for various values of θ.
θ π/4 − θ − cot θ + θ cot^2 θ = x
34° 34′ ·7854 − ·6033 − 1·4514 + 1·2709 = ·0016
35′ ·7854 ·6036 1·4505 1·2700 ·0013
36′ ·7854 ·6039 1·4496 1·2690 ·0009
37′ ·7854 ·6042 1·4487 1·2680 ·0005
38′ ·7854 ·6045 1·4478 1·2671 ·0002
39′ ·7854 ·6048 1·4469 1·2661 −·0002
40′ ·7854 ·6051 1·4460 1·2652 −·0005
We see accordingly that the equation is solved (as accurately as need be) when θ is an angle somewhat over 34° 38′, or say 34° 38½′. That is to say, a quadrant of a circle is bisected by a circular arc cutting the side and the periphery of the quadrant at right angles, when the arc is such as to include (90° − 34° 38′), i.e. 55° 22′ of the quadrantal arc.
This determination of ours is practically identical with that which Berthold arrived at by a rough and ready method, without the use of mathematics. He simply tried various ways of dividing a quadrant of paper by means of a circular arc, and went on doing so till he got the weights of his two pieces of paper approximately equal. The angle, as he thus determined it, was 34·6°, or say 34° 36′.
The position of M on the side of the quadrant OA is given by the equation OM = a cosec θ − a cot θ; the value of which expression, for the angle which we have just discovered, is ·3028. That is to say, the radius (or side) of the quadrant will be divided by the new partition into two parts, in the proportions of nearly three to seven.
The length of the arc MP is equal to a θ cot θ; and the value of this for the given angle is ·8751. This is as much as to say that the curved partition-wall which we are considering is shorter than a radial partition in the proportion of 8¾ to 10, or seven-eights almost exactly.
But we must also compare the length of this curved “anticlinal” partition-wall (MP) with that of the concentric, or periclinal, one (RS, Fig. 147) by which the quadrant might also be bisected. The length of this partition is obviously equal to the arc of the quadrant (i.e. the peripheral wall of the cell) divided by √2; or, in terms of the radius, = π/2√2 = 1·111. So that, not only is the anticlinal partition (such as we actually find in nature) notably the best, but the periclinal one, when it comes to dividing an entire quadrant, is very considerably larger even than a radial partition.
1917 Source Illustration
Fig. 147.
The two cells into which our original quadrant is now divided, while they are equal in volume, are of very different shapes; the [p. 363] one is a triangle (MAP) with two sides formed of circular arcs, and the other is a four-sided figure (MOBP), which we may call approximately oblong. We cannot say as yet how the triangular portion ought to divide; but it is obvious that the least possible partition-wall which shall bisect the other must run across the long axis of the oblong, that is to say periclinally. This, also, is precisely what tends actually to take place. In the following diagrams (Fig. 148) of a frog’s egg dividing under pressure, that is to say when reduced to the form of a flattened plate, we see, firstly, the division into four quadrants (by the partitions 1, 2); secondly, the division of each quadrant by means of an anticlinal circular arc (3, 3), cutting the peripheral wall of the quadrant approximately in the proportions of three to seven; and thirdly, we see that of the eight cells (four triangular and four oblong) into which the whole egg is now divided, the four which we have called oblong now proceed to divide by partitions transverse to their long axes, or roughly parallel to the periphery of the egg.
1917 Source Illustration
Fig. 148. Segmentation of frog’s egg, under artificial compression. (After Roux.)
The question how the other, or triangular, portion of the divided quadrant will next divide leads us to another well-defined problem, which is only a slight extension, making allowance for the circular arcs, of that elementary problem of the triangle we have already considered. We know now that an entire quadrant must divide (so that its bisecting wall shall have the least possible area) by means of an anticlinal partition, but how about any smaller sectors of circles? It is obvious in the case of a small prismatic [p. 364] sector, such as that shewn in Fig. 149, that a periclinal partition is the smallest by which we can possibly bisect the cell; we want, accordingly, to know the limits below which the periclinal partition is always the best, and above which the anticlinal arc, as in the case of the whole quadrant, has the advantage in regard to smallness of surface area.
This may be easily determined; for the preceding investigation is a perfectly general one, and the results hold good for sectors of any other arc, as well as for the quadrant, or arc of 90°. That is to say, the length of the partition-wall MP is always determined by the angle θ, according to our equation MP = aθ cot θ; and the angle θ has a definite relation to α, the angle of arc.
1917 Source Illustration
Fig. 149.
Moreover, in the case of the periclinal boundary, RS (Fig. 147) (or ab, Fig. 149), we know that, if it bisect the cell,
Accordingly, the arc RS will be just equal to the arc MP when
In the accompanying diagram (Fig. 150), I have plotted the various magnitudes with which we are concerned, in order to exhibit the several limiting values. Here we see, in the first place, the curve marked α, which shews on the (left-hand) vertical scale the various possible magnitudes of that angle (viz. the angle [p. 365] of arc of the whole sector which we wish to divide), and on the horizontal scale the corresponding values of θ, or the angle which
1917 Source Illustration
Fig. 150.
determines the point on the periphery where it is cut by the partition-wall, MP. Two limiting cases are to be noticed here: (1) at 90° (point A in diagram), because we are at present only [p. 366] dealing with arcs no greater than a quadrant; and (2), the point (B) where the angle θ comes to equal the angle α, for after that point the construction becomes impossible, since an anticlinal bisecting partition-wall would be partly outside the cell. The only partition which, after the point, can possibly exist, is a periclinal one. This point, as our diagram shews us, occurs when the angles (α and θ) are each rather under 52°.
Next I have plotted, on the same diagram, and in relation to the same scales of angles, the corresponding lengths of the two partitions, viz. RS and MP, their lengths being expressed (on the right-hand side of the diagram) in relation to the radius of the circle (a), that is to say the side wall, OA, of our cell.
The limiting values here are (1), C, C′, where the angle of arc is 90°, and where, as we have already seen, the two partition-walls have the relative magnitudes of MP : RS = 0·875 : 1·111; (2) the point D, where RS equals unity, that is to say where the periclinal partition has the same length as a radial one; this occurs when α is rather under 82° (cf. the points D, D′); (3) the point E, where RS and MP intersect; that is to say the point at which the two partitions, periclinal and anticlinal, are of the same magnitude; this is the case, according to our diagram, when the angle of arc is just over 62½°. We see from this, then, that what we have called an anticlinal partition, as MP, is only likely to occur in a triangular or prismatic cell whose angle of arc lies between 90° and 62½°. In all narrower or more tapering cells, the periclinal partition will be of less area, and will therefore be more and more likely to occur.
The case (F) where the angle α is just 60° is of some interest. Here, owing to the curvature of the peripheral border, and the consequent fact that the peripheral angles are somewhat greater than the apical angle α, the periclinal partition has a very slight and almost imperceptible advantage over the anticlinal, the relative proportions being about as MP : RS = 0·73 : 0·72. But if the equilateral triangle be a plane spherical triangle, i.e. a plane triangle bounded by circular arcs, then we see that there is no longer any distinction at all between our two partitions; MP and RS are now identical.
On the same diagram, I have inserted the curve for values of [p. 367] cosec θ − cot θ = OM, that is to say the distances from the centre, along the side of the cell, of the starting-point (M) of the anticlinal partition. The point C″ represents its position in the case of a quadrant, and shews it to be (as we have already said) about 3/10 of the length of the radius from the centre. If, on the other hand, our cell be an equilateral triangle, then we have to read off the point on this curve corresponding to α = 60°, and we find it at the point F‴ (vertically under F), which tells us that the partition now starts 4·5/10, or nearly halfway, along the radial wall.
The foregoing considerations carry us a long way in our investigations of many of the simpler forms of cell-division. Strictly speaking they are limited to the case of flattened cells, in which we can treat the problem as though we were simply partitioning a plane surface. But it is obvious that, though they do not teach us the whole conformation of the partition which divides a more complicated solid into two halves, yet they do, even in such a case, enlighten us so far, that they tell us the appearance presented in one plane of the actual solid. And as this is all that we see in a microscopic section, it follows that the results we have arrived at will greatly help us in the interpretation of microscopic appearances, even in comparatively complex cases of cell-division.
1917 Source Illustration
Fig. 151.
Let us now return to our quadrant cell (OAPB), which we have found to be divided into a triangular and a quadrilateral portion, as in Fig. 147 or Fig. 151; and let us now suppose the whole system to grow, in a uniform fashion, as a prelude to further subdivision. The whole quadrant, growing uniformly (or with equal radial increments), will still remain a quadrant, and it is obvious, therefore, that for every new increment of size, more will be added to the margin of its triangular portion than to the [p. 368] narrower margin of its quadrilateral portion; and these increments will be in proportion to the angles of arc, viz. 55° 22′ : 34° 38′, or as ·96 : ·60, i.e. as 8 : 5. And accordingly, if we may assume (and the assumption is a very plausible one), that, just as the quadrant itself divided into two halves after it got to a certain size, so each of its two halves will reach the same size before again dividing, it is obvious that the triangular portion will be doubled in size, and therefore ready to divide, a considerable time before the quadrilateral part. To work out the problem in detail would lead us into troublesome mathematics; but if we simply assume that the increments are proportional to the increasing radii of the circle, we have the following equations:―
Let us call the triangular cell T, and the quadrilateral, Q (Fig. 151); let the radius, OA, of the original quadrantal cell = a = 1; and let the increment which is required to add on a portion equal to T (such as PP′A′A) be called x, and let that required, similarly, for the doubling of Q be called x′.
Then we see that the area of the original quadrant
The area of the enlarged sector, p′OA′,
We get, accordingly, since a = 1,
Working out x′ in the same way, we arrive at the approximate value, x′ + 1 = 1·517. [p. 369]
This is as much as to say that, supposing each cell tends to divide into two halves when (and not before) its original size is doubled, then, in our flattened disc, the triangular cell T will tend to divide when the radius of the disc has increased by about a third (from 1 to 1·345), but the quadrilateral cell, Q, will not tend to divide until the linear dimensions of the disc have increased by about a half (from 1 to 1·517).
The case here illustrated is of no small general importance. For it shews us that a uniform and symmetrical growth of the organism (symmetrical, that is to say, under the limitations of a plane surface, or plane section) by no means involves a uniform or symmetrical growth of the individual cells, but may, under certain conditions, actually lead to inequality among these; and this inequality may be further emphasised by differences which arise out of it, in regard to the order of frequency of further subdivision. This phenomenon (or to be quite candid, this hypothesis, which is due to Berthold) is entirely independent of any change or variation in individual surface tensions; and accordingly it is essentially different from the phenomenon of unequal segmentation (as studied by Balfour), to which we have referred on p. 348.
In this fashion, we might go on to consider the manner, and the order of succession, in which the subsequent cell-divisions would tend to take place, as governed by the principle of minimal areas. But the calculations would grow more difficult, or the results got by simple methods would grow less and less exact. At the same time, some of these results would be of great interest, and well worth the trouble of obtaining. For instance, the precise manner in which our triangular cell, T, would next divide would be interesting to know, and a general solution of this problem is certainly troublesome to calculate. But in this particular case we can see that the width of the triangular cell near P is so obviously less than that near either of the other two angles, that a circular arc cutting off that angle is bound to be the shortest possible bisecting line; and that, in short, our triangular cell will tend to subdivide, just like the original quadrant, into a triangular and a quadrilateral portion.
But the case will be different next time, because in this new [p. 370] triangle, PRQ, the least width is near the innermost angle, that at Q; and the bisecting circular arc will therefore be opposite to Q, or (approximately) parallel to PR. The importance of this fact is at once evident; for it means to say that there soon comes a time when, whether by the division of triangles or of quadrilaterals, we find only quadrilateral cells adjoining the periphery of our circular disc. In the subsequent division of these quadrilaterals, the partitions will arise transversely to their long axes, that is to say, radially (as U, V); and we shall consequently have a superficial or peripheral layer of quadrilateral cells, with sides approximately parallel, that is to say what we are accustomed to call an epidermis. And this epidermis or superficial layer will be in clear contrast with the more irregularly shaped cells, the products of triangles and quadrilaterals, which make up the deeper, underlying layers of tissue.
1917 Source Illustration
Fig. 152.
In following out these theoretic principles and others like to them, in the actual division of living cells, we must always bear in mind certain conditions and qualifications. In the first place, the law of minimal area and the other rules which we have arrived at are not absolute but relative: they are links, and very important links, in a chain of physical causation; they are always at work, but their effects may be overridden and concealed by the operation of other forces. Secondly, we must remember that, in the great majority of cases, the cell-system which we have in view is constantly increasing in magnitude by active growth; and by this means the form and also the proportions of the cells are continually liable to alteration, of which phenomenon we have already had an example. Thirdly, we must carefully remember that, until our cell-walls become absolutely solid and rigid, they are always apt to be modified in form owing to the tension of the adjacent [p. 371] walls; and again, that so long as our partition films are fluid or semifluid, their points and lines of contact with one another may shift, like the shifting outlines of a system of soap-bubbles. This is the physical cause of the movements frequently seen among segmenting cells, like those to which Rauber called attention in the segmenting ovum of the frog, and like those more striking movements or accommodations which give rise to a so-called “spiral” type of segmentation.
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Fig. 153. Diagram of flattened or discoid cell dividing into octants: to shew gradual tendency towards a position of equilibrium.
Bearing in mind, then, these considerations, let us see what our flattened disc is likely to look like, after a few successive divisions into component cells. In Fig. 153, a, we have a diagrammatic representation of our disc, after it has divided into four quadrants, and each of these in turn into a triangular and a quadrilateral portion; but as yet, this figure scarcely suggests to us anything like the normal look of an aggregate of living cells. But let us go a little further, still limiting ourselves, however, to the consideration of the eight-celled stage. Wherever one of our radiating partitions meets the peripheral wall, there will (as we know) be a mutual tension between the three convergent films, which will tend to set their edges at equal angles to one another, angles that is to say of 120°. In consequence of this, the outer wall of each individual cell will (in this surface view of our disc) [p. 372] be an arc of a circle of which we can determine the centre by the method used on p. 307; and, furthermore, the narrower cells, that is to say the quadrilaterals, will have this outer border somewhat more curved than their broader neighbours. We arrive, then, at the condition shewn in Fig. 153, b. Within the cell, also, wherever wall meets wall, the angle of contact must tend, in every case, to be an angle of 120°; and in no case may more than three films (as seen in section) meet in a point (c); and this condition, of the partitions meeting three by three, and at co-equal angles, will obviously involve the curvature of some, if not all, of the partitions (d) which in our preliminary investigation we treated as plane. To solve this problem in a general way is no easy matter; but it is a problem which Nature solves in every case where, as in the case we are considering, eight bubbles, or eight cells, meet together in a (plane or curved) surface. An approximate solution has been given in Fig. 153, d; and it will now at once be recognised that this figure has vastly more resemblance to an aggregate of living cells than had the diagram of Fig. 153, a with which we began.
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Fig. 154.
Just as we have constructed in this case a series of purely diagrammatic or schematic figures, so it will be as a rule possible to diagrammatise, with but little alteration, the complicated appearances presented by any ordinary aggregate of cells. The accompanying little figure (Fig. 154), of a germinating spore of a Liverwort (Riccia), after a drawing of Professor Campbell’s, scarcely needs further explanation: for it is well-nigh a typical diagram of the method of space-partitioning which we are now considering. Let us look again at our figures (on p. 359) of the disc of Erythrotrichia, from Berthold’s Monograph of the Bangiaceae and redraw the earlier stages in diagrammatic fashion. In the following series of diagrams the new partitions, or those just about to form, are in each case outlined; and in the next succeeding stage they are shewn after settling down into position, and after exercising their respective tractions on the walls previously laid down. It is clear, I think, that these four diagrammatic figures represent all that is shewn in the first five stages drawn by Berthold from the plant itself; but the correspondence cannot [p. 373] in this case be precisely accurate, for the simple reason that Berthold’s figures are taken from different individuals, and are therefore only approximately consecutive and not strictly continuous. The last of the six drawings in Fig. 144 is already too
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Fig. 155. Theoretical arrangement of successive partitions in a discoid cell; for comparison with Fig. 144.
complicated for diagrammatisation, that is to say it is too complicated for us to decipher with certainty the precise order of appearance of the numerous partitions which it contains. But in Fig. 156 I shew one more diagrammatic figure, of a disc which
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Fig. 156. Theoretical division of a discoid cell into sixty-four chambers: no allowance being made for the mutual tractions of the cell-walls.
has divided, according to the theoretical plan, into about sixty-four cells; and making due allowance for the successive changes which the mutual tensions and tractions of the partitions must [p. 374] bring about, increasing in complexity with each succeeding stage, we can see, even at this advanced and complicated stage, a very considerable resemblance between the actual picture (Fig. 144) and the diagram which we have here constructed in obedience to a few simple rules.
In like manner, in the annexed figures, representing sections through a young embryo of a Moss, we have very little difficulty in discerning the successive stages that must have intervened between the two stages shewn: so as to lead from the just divided quadrants (one of which, by the way, has not yet divided in our figure (a)) to the stage (b) in which a well-marked epidermal layer surrounds an at first sight irregular agglomeration of “fundamental” tissue.
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Fig. 157. Sections of embryo of a moss. (After Kienitz-Gerloff.)
In the last paragraph but one, I have spoken of the difficulty of so arranging the meeting-places of a number of cells that at each junction only three cell-walls shall meet in a line, and all three shall meet it at equal angles of 120°. As a matter of fact, the problem is soluble in a number of ways; that is to say, when we have a number of cells, say eight as in the case considered, enclosed in a common boundary, there are various ways in which their walls can be made to meet internally, three by three, at equal angles; and these differences will entail differences also in the curvature of the walls, and consequently in the shape of the cells. The question is somewhat complex; it has been dealt with by Plateau, and treated mathematically by M. Van Rees [129].
1917 Source Illustration
Fig. 158. Various possible arrangements of intermediate partitions, in groups of 4, 5, 6, 7 or 8 cells.
If within our boundary we have three cells all meeting [p. 375] internally, they must meet in a point; furthermore, they tend to do so at equal angles of 120°, and there is an end of the matter. If we have four cells, then, as we have already seen, the conditions are satisfied by interposing a little intermediate wall, the two extremities of which constitute the meeting-points of three cells each, and the upper edge of which marks the “polar furrow.” Similarly, in the case of five cells, we require two little intermediate walls, and two polar furrows; and we soon arrive at the rule that, for n cells, we require n − 3 little longitudinal partitions (and corresponding polar furrows), connecting the triple junctions of the cells; and these little walls, like all the rest within the system, must be inclined to one another at angles of 120°. Where we have only one such wall (as in the case of four cells), or only two (as in the case of five cells), there is no room for ambiguity. But where we have three little connecting-walls, as in the case of six cells, it is obvious that we can arrange them in three different ways, as in the annexed Fig. 159. In the system of seven cells, the four partitions can be arranged in four ways; and the five partitions required in the case of eight cells can be arranged in no less than thirteen different ways, of which Fig. 158 shews some half-dozen only. It does not follow that, so to speak, these various [p. 376] arrangements are all equally good; some are known to be much more stable than others, and some have never yet been realised in actual experiment.
The conditions which lead to the presence of any one of them, in preference to another, are as yet, so far as I am aware, undetermined, but to this point we shall return.
Examples of these various arrangements meet us at every turn, and not only in cell-aggregates, but in various cases where non-rigid and semi-fluid partitions (or partitions that were so to begin with) meet together. And it is a necessary consequence of this physical phenomenon, and of the limited and very small number of possible arrangements, that we get similar appearances, capable of representation by the same diagram, in the most diverse fields of biology [130].
1917 Source Illustration
Fig. 159.
Among the published figures of embryonic stages and other cell aggregates, we only discern these little intermediate partitions in cases where the investigator has drawn carefully just what lay before him, without any preconceived notions as to radial or other symmetry; but even in other cases we can generally recognise, without much difficulty, what the actual arrangement was whereby the cell-walls met together in equilibrium. I have a strong suspicion that a leaning towards Sachs’s Rule, that one cell-wall tends to set itself at right angles to another cell-wall (a rule whose strict limitations, and narrow range of application, we have already [p. 377] considered) is responsible for many inaccurate or incomplete representations of the mutual arrangement of aggregated cells.
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Fig. 160. Segmenting egg of Trochus. (After Robert.)
1917 Source Illustration
Fig. 161. Two views of segmenting egg of Cynthia partita. (After Conklin.)
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Fig. 162. (a) Section of apical cone of Salvinia. (After Pringsheim [131].) (b) Diagram of probable actual arrangement.
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Fig. 163. Egg of Pyrosoma. (After Korotneff).
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Fig. 164. Egg of Echinus, segmenting under pressure. (After Driesch.)
In the accompanying series of figures (Figs. 160–167) I have [p. 378] set forth a few aggregates of eight cells, mostly from drawings of segmenting eggs. In some cases they shew clearly the manner in which the cells meet one another, always at angles of 120°, and always with the help of five intermediate boundary walls within the eight-celled system; in other cases I have added a slightly altered drawing, so as to shew, with as little change as [p. 379] possible, the arrangement of boundaries which probably actually existed, and gave rise to the appearance which the observer drew. These drawings may be compared with the various diagrams of Fig. 158, in which some seven out of the possible thirteen arrangements of five intermediate partitions (for a system of eight cells) have been already set forth.
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Fig. 165. (a) Part of segmenting egg of Cephalopod (after Watase); (b) probable actual arrangement.
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Fig. 166. (a) Egg of Echinus; (b) do. of Nereis, under pressure. (After Driesch).
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Fig. 167. (a) Egg of frog, under pressure (after Roux); (b) probable actual arrangement.
It will be seen that M. Robert-Tornow’s figure of the segmenting egg of Trochus (Fig. 160) clearly shews the cells grouped after the fashion of Fig. 158, a. In like manner, Mr Conklin’s figure of the ascidian egg (Cynthia) shews equally clearly the arrangement g.
A sea-urchin egg, segmenting under pressure, as figured by Driesch, scarcely requires any modification of the drawing to appear as a diagram of the type d. Turning for a moment to a botanical illustration, we have a figure of Pringsheim’s shewing an eight-celled stage in the apex of the young cone of Salvinia; it is in all probability referable, as in my modified diagram, to type c. Beside it is figured a very different object, a segmenting egg of the Ascidian Pyrosoma, after Korotneff; it may be that this also is to be referred to type c, but I think it is more easily referable to type b. For there is a difference between this diagram and that of Salvinia, in that here apparently, of the pairs of lateral cells, the upper and the lower cell are alternately the larger, while in the diagram of Salvinia the lower lateral cells both appear much larger than the upper ones; and this difference tallies with the appearance produced if we fill in the eight cells according to the type b or the type c. In the segmenting cuttlefish egg, there is again a slight dubiety as to which type it should be referred to, but it is in all probability referable, like Driesch’s Echinus egg, to d. Lastly, I have copied from Roux a curious figure of the egg of Rana esculenta, viewed from the animal pole, which appears to me referable, in all probability, to type g. Of type f, in which the five partitions form a figure with four re-entrant angles, that is to say a figure representing the five sides of a hexagon, I have found no examples among segmenting eggs, and that arrangement in all probability is a very unstable one.
It is obvious enough, without more ado, that these phenomena are in the strictest and completest way common to both plants [p. 380] and animals. In other words they tally with, and they further extend, the general and fundamental conclusions laid down by Schwann, in his Mikroskopische Untersuchungen über die Uebereinstimmung in der Struktur und dem Wachsthum der Thiere und Pflanzen.
But now that we have seen how a certain limited number of types of eight-celled segmentation (or of arrangements of eight cell-partitions) appear and reappear, here and there, throughout the whole world of organisms, there still remains the very important question, whether in each particular organism the conditions are such as to lead to one particular arrangement being predominant, characteristic, or even invariable. In short, is a particular arrangement of cell-partitions to be looked upon (as the published figures of the embryologist are apt to suggest) as a specific character, or at least a constant or normal character, of the particular organism? The answer to this question is a direct negative, but it is only in the work of the most careful and accurate observers that we find it revealed. Rauber (whom we have more than once had occasion to quote) was one of those embryologists who recorded just what he saw, without prejudice or preconception; as Boerhaave said of Swammerdam, quod vidit id asseruit. Now Rauber has put on record a considerable number of variations in the arrangement of the first eight cells, which form a discoid surface about the dorsal (or “animal”) pole of the frog’s egg. In a certain number of cases these figures are identical with one another in type, identical (that is to say) save for slight differences in magnitude, relative proportions, or orientation. But I have selected (Fig. 168) six diagrammatic figures, which are all essentially different, and these diagrams seem to me to bear intrinsic evidence of their accuracy: the curvatures of the partition-walls, and the angles at which they meet agree closely with the requirements of theory, and when they depart from theoretical symmetry they do so only to the slight extent which we should naturally expect in a material and imperfectly homogeneous system [132]. [p. 381]
1917 Source Illustration
Fig. 168. Various modes of grouping of eight cells, at the dorsal or epiblastic pole of the frog’s egg. (After Rauber.)
Of these six illustrations, two are exceptional. In Fig. 168, 5, we observe that one of the eight cells is surrounded on all sides by the other seven. This is a perfectly natural condition, and represents, like the rest, a phase of partial or conditional equilibrium. But it is not included in the series we are now considering, which is restricted to the case of eight cells extending outwards to a common boundary. The condition shewn in Fig. 168, 6, is again peculiar, and is probably rare; but it is included under the cases considered on p. 312, in which the cells are not in complete fluid contact, but are separated by little droplets of extraneous matter; it needs no further comment. But the other four cases are beautiful diagrams of space-partitioning, similar to those we have just been considering, but so exquisitely clear that they need no modification, no “touching-up,” to exhibit their mathematical regularity. It will easily be recognised that in Fig. 168, 1 and 2, we have the arrangements corresponding to a and d of our diagram (Fig. 158): but the other two (i.e. 3 and 4) represent other of the thirteen possible arrangements, which are not included in that [p. 382] diagram. It would be a curious and interesting investigation to ascertain, in a large number of frogs’ eggs, all at this stage of development, the percentage of cases in which these various arrangements occur, with a view of correlating their frequency with the theoretical conditions (so far as they are known, or can be ascertained) of relative stability. One thing stands out as very certain indeed: that the elementary diagram of the frog’s egg commonly given in text-books of embryology,—in which the cells are depicted as uniformly symmetrical quadrangular bodies,—is entirely inaccurate and grossly misleading [133].
We now begin to realise the remarkable fact, which may even appear a startling one to the biologist, that all possible groupings or arrangements whatsoever of eight cells (where all take part in the surface of the group, none being submerged or wholly enveloped by the rest) are referable to some one or other of thirteen types or forms. And that all the thousands and thousands of drawings which diligent observers have made of such eight-celled structures, animal or vegetable, anatomical, histological or embryological, are one and all representations of some one or another of these thirteen types:—or rather indeed of somewhat less than the whole thirteen, for there is reason to believe that, out of the total number of possible groupings, a certain small number are essentially unstable, and have at best, in the concrete, but a transitory and evanescent existence.
Before we leave this subject, on which a vast deal more might be said, there are one or two points which we must not omit to consider. Let us note, in the first place, that the appearance which our plane diagrams suggest of inequality of the several cells is apt to be deceptive; for the differences of magnitude apparent in one plane may well be, and probably generally are, balanced by equal and opposite differences in another. Secondly, let us remark that the rule which we are considering refers only [p. 383] to angles, and to the number, not to the length of the intermediate partitions; it is to a great extent by variations in the length of these that the magnitudes of the cells may be equalised, or otherwise balanced, and the whole system brought into equilibrium. Lastly, there is a curious point to consider, in regard to the number of actual contacts, in the various cases, between cell and cell. If we inspect the diagrams in Fig. 169 (which represent three out of our thirteen possible arrangements of eight cells) we shall see that, in the case of type b, two cells are each in contact with two others, two cells with three others, and four cells each with four other cells. In type a four cells are each in contact with two, two with four, and two with five. In type f, two are in contact with two, four with three, and one with no less than seven. In all cases the
1917 Source Illustration
Fig. 169.
number of contacts is twenty-six in all; or, in other words, there are thirteen internal partitions, besides the eight peripheral walls. For it is easy to see that, in all cases of n cells with a common external boundary, the number of internal partitions is 2n − 3; or the number of what we call the internal or interfacial contacts is 2(2n − 3). But it would appear that the most stable arrangements are those in which the total number of contacts is most evenly divided, and the least stable are those in which some one cell has, as in type f, a predominant number of contacts. In a well-known series of experiments, Roux has shewn how, by means of oil-drops, various arrangements, or aggregations, of cells can be simulated; and in Fig. 170 I shew a number of Roux’s figures, and have ascribed them to what seem to be their appropriate “types” among those which we have just been considering; but [p. 384] it will be observed that in these figures of Roux’s the drops are not always in complete contact, a little air-bubble often keeping them apart at their apical junctions, so that we see the configuration towards which the system is tending rather than that which it has fully attained [134]. The type which we have called f was found by Roux to be unstable, the large (or apparently large) drop a″ quickly passing into the centre of the system, and here taking up a position of equilibrium in which, as usual, three cells meet throughout in a point, at equal angles, and in which, in this case, all the cells have an equal number of “interfacial” contacts.
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Fig. 170. Aggregations of oil-drops. (After Roux.) Figs. 4–6 represent successive changes in a single system.
We need by no means be surprised to find that, in such arrangements, the commonest and most stable distributions are those in which the cell-contacts are distributed as uniformly as possible between the several cells. We always expect to find some such tendency to equality in cases where we have to do with small oscillations on either side of a symmetrical condition. [p. 385]
The rules and principles which we have arrived at from the point of view of surface tension have a much wider bearing than is at once suggested by the problems to which we have applied them; for in this elementary study of the cell-boundaries in a segmenting egg or tissue we are on the verge of a difficult and important subject in pure mathematics. It is a subject adumbrated by Leibniz, studied somewhat more deeply by Euler, and greatly developed of recent years. It is the Geometria Situs of Gauss, the Analysis Situs of Riemann, the Theory of Partitions of Cayley, and of Spatial Complexes of Listing [135]. The crucial point for the biologist to comprehend is, that in a closed surface divided into a number of faces, the arrangement of all the faces, lines and points in the system is capable of analysis, and that, when the number of faces or areas is small, the number of possible arrangements is small also. This is the simple reason why we meet in such a case as we have been discussing (viz. the arrangement of a group or system of eight cells) with the same few types recurring again and again in all sorts of organisms, plants as well as animals, and with no relation to the lines of biological classification: and why, further, we find similar configurations occurring to mark the symmetry, not of cells merely, but of the parts and organs of entire animals. The phenomena are not “functions,” or specific characters, of this or that tissue or organism, but involve general principles which lie within the province of the mathematician.
The theory of space-partitioning, to which the segmentation of the egg gives us an easy practical introduction, is illustrated in much more complex ways in other fields of natural history. A very beautiful but immensely complicated case is furnished by the “venation” of the wings of insects. Here we have sometimes (as in the dragon-flies), a general reticulum of small, more or less hexagonal “cells”: but in most other cases, in flies, bees, butterflies, etc., we have a moderate number of cells, whose partitions always impinge upon one another three by three, and whose arrangement, therefore, includes of necessity a number of small intermediate partitions, analogous to our polar furrows. I think [p. 386] that a mathematical study of these, including an investigation of the “deformation” of the wing (that is to say, of the changes in shape and changes in the form of its “cells” which it undergoes during the life of the individual, and from one species to another) would be of great interest. In very many cases, the entomologist relies upon this venation, and upon the occurrence of this or that intermediate vein, for his classification, and therefore for his hypothetical phylogeny of particular groups; which latter procedure hardly commends itself to the physicist or the mathematician.
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Fig. 171. (A) Asterolampra marylandica, Ehr.; (B, C) A. variabilis, Grev. (After Greville.)
Another case, geometrically akin but biologically very different, is to be found in the little diatoms of the genus Asterolampra, and their immediate congeners [136]. In Asterolampra we have a little disc, in which we see (as it were) radiating spokes of one material, alternating with intervals occupied on the flattened wheel-like disc by another (Fig. 171). The spokes vary in number, but the general appearance is in a high degree suggestive of the Chladni figures produced by the vibration of a circular plate. The spokes broaden out towards the centre, and interlock by visible junctions, which obey the rule of triple intersection, and accordingly exemplify the partition-figures with which we are dealing. But whereas we have found the particular arrangement in which one cell is in contact with all the rest to be unstable, according to Roux’s oil-drop experiments, and to be conspicuous [p. 387] by its absence from our diagrams of segmenting eggs, here in Asterolampra, on the other hand, it occurs frequently, and is indeed the commonest arrangement [137] (Fig. 171, B). In all probability, we are entitled to consider this marked difference natural enough. For we may suppose that in Asterolampra (unlike the case of the segmenting egg) the tendency is to perfect radial symmetry, all the spokes emanating from a point in the centre: such a condition would be eminently unstable, and would break down under the least asymmetry. A very simple, perhaps the simplest case, would be that one single spoke should differ slightly from the rest, and should so tend to be drawn in amid the others, these latter remaining similar and symmetrical among themselves. Such a configuration would be vastly less unstable than the original one in which all the boundaries meet in a point; and the fact that further progress is not made towards other configurations of still greater stability may be sufficiently accounted for by viscosity, rapid solidification, or other conditions of restraint. A perfectly stable condition would of course be obtained if, as in the case of Roux’s oil-drop (Fig. 170, 6), one of the cellular spaces passed into the centre of the system, the other partitions radiating outwards from its circular wall to the periphery of the whole system. Precisely such a condition occurs among our diatoms; but when it does so, it is looked upon as the mark and characterisation of the allied genus Arachnoidiscus.
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Fig. 172. Section of Alcyonarian polype.
In a diagrammatic section of an Alcyonarian polype (Fig. 172), we have eight chambers set, symmetrically, about a ninth, which constitutes the “stomach.” In this arrangement there is no difficulty, for it is obvious that, throughout the system, three boundaries meet (in plane section) in a point. In many corals we have as [p. 388] simple, or even simpler conditions, for the radiating calcified partitions either converge upon a central chamber, or fail to meet it and end freely. But in a few cases, the partitions or “septa” converge to meet one another, there being no central chamber on which they may impinge; and here the manner in which contact is effected becomes complicated, and involves problems identical with those which we are now studying.
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Fig. 173. Heterophyllia angulata. (After Nicholson.)
In the great majority of corals we have as simple or even simpler conditions than those of Alcyonium; for as a rule the calcified partitions or septa of the coral either converge upon a central chamber (or central “columella”), or else fail to meet it and end freely. In the latter case the problem of space-partitioning does not arise; in the former, however numerous the septa be, their separate contacts with the wall of the central chamber comply with our fundamental rule according to which three lines and no more meet in a point, and from this simple and symmetrical arrangement there is little tendency to variation. But in a few cases, the septal partitions converge to meet one another, there being no central chamber on which they may impinge; and here the manner in which contact is effected becomes complicated, and involves problems of space-partitioning identical with those which we are now studying. In the genus Heterophyllia and in a few allied forms we have such conditions, and students of the Coelenterata have found them very puzzling. McCoy [138], their first discoverer, pronounced these corals to be “totally unlike” any other group, recent or fossil; and Professor Martin Duncan, writing a memoir on Heterophyllia and its allies [139], described them as “paradoxical in their anatomy.”
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Fig. 174. Heterophyllia sp. (After Martin Duncan.)
The simplest or youngest Heterophylliae known have six septa (as in Fig. 174, a); in the case figured, four of these septa are conjoined two and two, thus forming the usual triple junctions together with their intermediate partition-walls: and in the [p. 389] case of the other two we may fairly assume that their proper and original arrangement was that of our type 6b (Fig. 158), though the central intermediate partition has been crowded out by partial coalescence. When with increasing age the septa become more numerous, their arrangement becomes exceedingly variable; for the simple reason that, from the mathematical point of view, the number of possible arrangements, of 10, 12 or more cellular partitions in triple contact, tends to increase with great rapidity, and there is little to choose between many of them in regard to symmetry and equilibrium. But while, mathematically speaking, each particular case among the multitude of possible cases is an orderly and definite arrangement, from the purely biological point of view on the other hand no law or order is recognisable; and so McCoy described the genus as being characterised by the possession of septa “destitute of any order of arrangement, but irregularly branching and coalescing in their passage from the solid external walls towards some indefinite point near the centre where the few main lamellae irregularly anastomose.” [p. 390]
In the two examples figured (Fig. 174), both comparatively simple ones, it will be seen that, of the main chambers, one is in each case an unsymmetrical one; that is to say, there is one chamber which is in contact with a greater number of its neighbours than any other, and which at an earlier stage must have had contact with them all; this was the case of our type f, in the eight-celled system (Fig. 158). Such an asymmetrical chamber (which may occur in a system of any number of cells greater than six), constitutes what is known to students of the Coelenterata as a “fossula”; and we may recognise it not only here, but also in Zaphrentis and its allies, and in a good many other corals besides. Moreover certain corals are described as having more than one fossula: this appearance being naturally produced under certain of the other asymmetrical variations of normal space-partitioning. Where a single fossula occurs, we are usually told that it is a symptom of “bilaterality”; and this is in turn interpreted as an indication of a higher grade of organisation than is implied in the purely “radial symmetry” of the commoner types of coral. The mathematical aspect of the case gives no warrant for this interpretation.
Let us carefully notice (lest we run the risk of confusing two distinct problems) that the space-partitioning of Heterophyllia by no means agrees with the details of that which we have studied in (for instance) the case of the developing disc of Erythrotrichia: the difference simply being that Heterophyllia illustrates the general case of cell-partitioning as Plateau and Van Rees studied it, while in Erythrotrichia, and in our other embryological and histological instances, we have found ourselves justified in making the additional assumption that each new partition divided a cell into co-equal parts. No such law holds in Heterophyllia, whose case is essentially different from the others: inasmuch as the chambers whose partition we are discussing in the coral are mere empty spaces (empty save for the mere access of sea-water); while in our histological and embryological instances, we were speaking of the division of a cellular unit of living protoplasm. Accordingly, among other differences, the “transverse” or “periclinal” partitions, which were bound to appear at regular intervals and in definite positions, when co-equal bisection was a feature of the [p. 391] case, are comparatively few and irregular in the earlier stages of Heterophyllia, though they begin to appear in numbers after the main, more or less radial, partitions have become numerous, and when accordingly these radiating partitions come to bound narrow and almost parallel-sided interspaces; then it is that the transverse or periclinal partitions begin to come in, and form what the student of the Coelenterata calls the “dissepiments” of the coral. We need go no further into the configuration and anatomy of the corals; but it seems to me beyond a doubt that the whole question of the complicated arrangement of septa and dissepiments throughout the group (including the curious vesicular or bubble-like tissue of the Cyathophyllidae and the general structural plan of the Tetracoralla, such as Streptoplasma and its allies) is well worth investigation from the physical and mathematical point of view, after the fashion which is here slightly adumbrated.
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Fig. 175. Diagrammatic section of a Ctenophore (Eucharis).
The method of dividing a circular, or spherical, system into eight parts, equal as to their areas but unequal in their peripheral boundaries, is probably of wide biological application; that is to say, without necessarily supposing it to be rigorously followed, the typical configuration which it yields seems to recur again and again, with more or less approximation to precision, and under widely different circumstances. I am inclined to think, for instance, that the unequal division of the surface of a Ctenophore by its [p. 392] meridian-like ciliated bands is a case in point (Fig. 175). Here, if we imagine each quadrant to be twice bisected by a curved anticline, we shall get what is apparently a close approximation to the actual position of the ciliated bands. The case however is complicated by the fact that the sectional plan of the organism is never quite circular, but always more or less elliptical. One point, at least, is clearly seen in the symmetry of the Ctenophores; and that is that the radiating canals which pass outwards to correspond in position with the ciliated bands, have no common centre, but diverge from one another by repeated bifurcations, in a manner comparable to the conjunctions of our cell-walls.
In like manner I am inclined to suggest that the same principle may help us to understand the apparently complex arrangement of the skeletal rods of a larval Echinoderm, and the very complex conformation of the larva which is brought about by the presence of these long, slender skeletal radii.
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Fig. 176. Diagrammatic arrangement of partitions, represented by skeletal rods, in larval Echinoderm (Ophiura).
In Fig. 176 I have divided a circle into its four quadrants, and have bisected each quadrant by a circular arc (BC), passing from radius to periphery, as in the foregoing cases of cell-division; and I have again bisected, in a similar way, the triangular halves of each quadrant (DD). I have also inserted a small circle in the middle of the figure, concentric with the large one. If now we imagine those lines in the figure which I have drawn black to be replaced by solid rods we shall have at once the frame-work of an Ophiurid (Pluteus) larva. Let us imagine all these arms to be [p. 393] bent symmetrically downwards, so that the plane of the paper is transformed into a spheroidal surface, such as that of a hemisphere, or that of a tall conical figure with curved sides; let a membrane be spread, umbrella-like, between the outstretched skeletal rods, and let its margin loop from rod to rod in curves which are possibly catenaries, but are more probably portions of an “elastic curve,” and the outward resemblance to a Pluteus larva is now complete. By various slight modifications, by altering the relative lengths of the rods, by modifying their curvature or by replacing the curved rod by a tangent to itself, we can ring the changes which lead us from one known type of Pluteus to another. The case of the Bipinnaria larvae of Echinids is certainly analogous, but it becomes very much more complicated; we have to do with a more complex partitioning of space, and I confess that I am not yet able to represent the more complicated forms in so simple a way.
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Fig. 177. Pluteus-larva of Ophiurid.
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Fig. 178. Diagrammatic development of Stomata in Sedum. (Cf. fig. in Sachs’s Botany, 1882, p. 103.)
There are a few notable exceptions (besides the various unequally segmenting eggs) to the general rule that in cell-division the mother-cell tends to divide into equal halves; and one of these exceptional cases is to be found in connection with the development of “stomata” in the leaves of plants. The epidermal cells by which the leaf is covered may be of various shapes; sometimes, as in a hyacinth, they are oblong, but more often they have an irregular shape in which we can recognise, more or less clearly, a distorted or imperfect hexagon. In the case of the oblong cells, a transverse partition will be the least possible, whether the cell be equally or unequally divided, unless (as we have already seen) [p. 394] the space to be cut off be a very small one, not more than about three-tenths the area of a square based on the short side of the original rectangular cell. As the portion usually cut off is not nearly so small as this, we get the form of partition shewn in Fig. 179, and the cell so cut off is next bisected by a partition at right angles to the first; this latter partition splits, and the two last-formed cells constitute the so-called “guard-cells” of the
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Fig. 179. Diagrammatic development of stomata in Hyacinth.
stoma. In other cases, as in Fig. 178, there will come a point where the minimal partition necessary to cut off the required fraction of the cell-content is no longer a transverse one, but is a portion of a cylindrical wall (2) cutting off one corner of the mother-cell. The cell so cut off is now a certain segment of a circle, with an arc of approximately 120°; and its next division will be by means of a curved wall cutting it into a triangular and a quadrangular portion (3). The triangular portion will continue to divide in a similar way (4, 5), and at length (for a reason which is not yet clear) the partition wall [p. 395] between the new-formed cells splits, and again we have the phenomenon of a “stoma” with its attendant guard-cells. In Fig. 179 are shewn the successive stages of division, and the changing curvatures of the various walls which ensue as each subsequent partition appears, introducing a new tension into the system.
It is obvious that in the case of the oblong cells of the epidermis in the hyacinth the stomata will be found arranged in regular rows, while they will be irregularly distributed over the surface of the leaf in such a case as we have depicted in Sedum.
While, as I have said, the mechanical cause of the split which constitutes the orifice of the stoma is not quite clear, yet there can be little or no doubt that it, like the rest of the phenomenon, is related to surface tension. It might well be that it is directly due to the presence underneath this portion of epidermis of the hollow air-space which the stoma is apparently developed “for the purpose” of communicating with; this air-surface on both sides of the delicate epidermis might well cause such an alteration of tensions that the two halves of the dividing cell would tend to part company. In short, if the surface-energy in a cell-air contact were half or less than half that in a contact between cell and cell, then it is obvious that our partition would tend to split, and give us a two-fold surface in contact with air, instead of the original boundary or interface between one cell and the other. In Professor Macallum’s experiments, which we have briefly discussed in our short chapter on Adsorption, it was found that large quantities of potassium gathered together along the outer walls of the guard-cells of the stoma, thereby indicating a low surface-tension along these outer walls. The tendency of the guard-cells to bulge outwards is so far explained, and it is possible that, under the existing conditions of restraint, we may have here a force tending, or helping, to split the two cells asunder. It is clear enough, however, that the last stage in the development of a stoma, is, from the physical point of view, not yet properly understood.
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Fig. 180. Various pollen-grains and spores (after Berthold, Campbell, Goebel and others). (1) Epilobium; (2) Passiflora; (3) Neottia; (4) Periploca graeca; (5) Apocynum; (6) Erica; (7) Spore of Osmunda; (8) Tetraspore of Callithamnion.
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Fig. 181. Dividing spore of Anthoceros. (After Campbell.)
In all our foregoing examples of the development of a “tissue” we have seen that the process consists in the successive division of cells, each act of division being accompanied by the formation [p. 396] of a boundary-surface, which, whether it become at once a solid or semi-solid partition or whether it remain semi-fluid, exercises in all cases an effect on the position and the form of the boundary which comes into being with the next act of division. In contrast to this general process stands the phenomenon known as “free cell-formation,” in which, out of a common mass of protoplasm, a number of separate cells are simultaneously, or all but simultaneously, differentiated. In a number of cases it happens that, to begin with, a number of “mother-cells” are formed simultaneously, and each of these divides, by two successive divisions, into four “daughter-cells.” These daughter-cells will tend to group themselves, just as would four soap-bubbles, into a “tetrad,” the four cells corresponding to the angles of a regular tetrahedron. For the system of four bodies is evidently here in perfect symmetry; the partition-walls and their respective edges meet at equal angles: three walls everywhere meeting in an edge, and the four edges converging to a point in the geometrical centre of the system. This is the typical mode of development of pollen-grains, common among Monocotyledons and all but universal among Dicotyledonous plants. By a loosening of the surrounding tissue and an expansion of the cavity, or anther-cell, in which [p. 397] they lie, the pollen-grains afterwards fall apart, and their individual form will depend upon whether or no their walls have solidified before this liberation takes place. For if not, then the separate grains will be free to assume a spherical form as a consequence of their own individual and unrestricted growth; but if they become solid or rigid prior to the separation of the tetrad, then they will conserve more or less completely the plane interfaces and sharp angles of the elements of the tetrahedron. The latter is the case, for instance, in the pollen-grains of Epilobium (Fig. 180, 1) and in many others. In the Passion-flower (2) we have an intermediate condition: where we can still see an indication of the facets where the grains abutted on one another in the tetrad, but the plane faces have been swollen by growth into spheroidal or spherical surfaces. It is obvious that there may easily be cases where the tetrads of daughter-cells are prevented from assuming the tetrahedral form: cases, that is to say, where the four cells are forced and crushed into one plane. The figures given by Goebel of the development of the pollen of Neottia (3, a–e: all the figures referring to grains taken from a single anther), illustrate this to perfection; and it will be seen that, when the four cells lie in a plane, they conform exactly to our typical diagram of the first four cells in a segmenting ovum. Occasionally, though the four cells lie in a plane, the diagram seems to fail us, for the cells appear to meet in a simple cross (as in 5); but here we soon perceive that the cells are not in complete interfacial contact, but are kept apart by a little intervening drop of fluid or bubble of air. The spores of ferns (7) develop in very much the same way as pollen-grains; and they also very often retain traces of the shape which they assumed as members of a tetrahedral figure. Among the “tetraspores” (8) of the Florideae, or Red Seaweeds, we have a phenomenon which is in every respect analogous.
Here again it is obvious that, apart from differences in actual magnitude, and apart from superficial or “accidental” differences (referable to other physical phenomena) in the way of colour, [p. 398] texture and minute sculpture or pattern, it comes to pass, through the laws of surface-tension and the principles of the geometry of position, that a very small number of diagrammatic figures will sufficiently represent the outward forms of all the tetraspores, four-celled pollen-grains, and other four-celled aggregates which are known or are even capable of existence.
We have been dealing hitherto (save for some slight exceptions) with the partitioning of cells on the assumption that the system either remains unaltered in size or else that growth has proceeded uniformly in all directions. But we extend the scope of our enquiry very greatly when we begin to deal with unequal growth, with growth, that is to say, which produces a greater extension along some one axis than another. And here we come close in touch with that great and still (as I think) insufficiently appreciated generalisation of Sachs, that the manner in which the cells divide is the result, and not the cause, of the form of the dividing structure: that the form of the mass is caused by its growth as a whole, and is not a resultant of the growth of the cells individually considered [140]. Such asymmetry of growth may be easily imagined, and may conceivably arise from a variety of causes. In any individual cell, for instance, it may arise from molecular asymmetry of the structure of the cell-wall, giving it greater rigidity in one direction than another, while all the while the hydrostatic pressure within the cell remains constant and uniform. In an aggregate of cells, it may very well arise from a greater chemical, or osmotic, activity in one than another, leading to a localised increase in the fluid pressure, and to a corresponding bulge over a certain area of the external surface. It might conceivably occur as a direct result of the preceding cell-divisions, when these are such as to produce many peripheral or concentric walls in one part and few or none in another, with the obvious result of strengthening the common boundary wall and resisting the outward pressure of growth in parts where the former is the case; that is to say, in our dividing quadrant, if [p. 399] its quadrangular portion subdivide by periclines, and the triangular portion by oblique anticlines (as we have seen to be the natural tendency), then we might expect that external growth would be more manifest over the latter than over the former areas. As a direct and immediate consequence of this we might expect a tendency for special outgrowths, or “buds,” to arise from the triangular rather than from the quadrangular cells; and this turns out to be not merely a tendency towards which theoretical considerations point, but a widespread and important factor in the morphology of the cryptogams. But meanwhile, without enquiring further into this complicated question, let us simply take it that, if we start from such a simple case as a round cell which has divided into two halves, or four quarters (as the case may be), we shall at once get bilateral symmetry about a main axis, and other secondary results arising therefrom, as soon as one of the halves, or one of the quarters, begins to shew a rate of growth in advance of the others; for the more rapidly growing cell, or the peripheral wall common to two or more such rapidly growing cells, will bulge out into an ellipsoid form, and may finally extend into a cylinder with rounded or ellipsoid end.
This latter very simple case is illustrated in the development of a pollen-tube, where the rapidly growing cell develops into the elongated cylindrical tube, and the slow-growing or quiescent part remains behind as the so-called “vegetative” cell or cells.
Just as we have found it easier to study the segmentation of a circular disc than that of a spherical cell, so let us begin in the same way, by enquiring into the divisions which will ensue if the disc tend to grow, or elongate, in some one particular direction, instead of in radial symmetry. The figures which we shall then obtain will not only apply to the disc, but will also represent, in all essential features, a projection or longitudinal section of a solid body, spherical to begin with, preserving its symmetry as a solid of revolution, and subject to the same general laws as we have studied in the disc [141]. [p. 400]
Suppose, in the first place, that the axis of growth lies symmetrically in one of the original quadrantal cells of a segmenting disc; and let this growing cell elongate with comparative rapidity before it subdivides. When it does divide, it will necessarily do so by a transverse partition, concave towards the apex of the cell: and, as further elongation takes place, the cylindrical structure which will be developed thereby will tend to be again and again subdivided by similar concave transverse partitions. If at any time, through this process of concurrent elongation and subdivision, the apical cell become equivalent to, or less than, a hemisphere, it will next divide by means of a longitudinal, or vertical partition; and similar longitudinal partitions will arise in the other segments of the cylinder, as soon as it comes about that their length (in the direction of the axis) is less than their breadth.
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Fig. 182.
But when we think of this structure in the solid, we at once perceive that each of these flattened segments of the cylinder, into which our cylinder has divided, is equivalent to a flattened circular disc; and its further division will accordingly tend to proceed like any other flattened disc, namely into four quadrants, and afterwards by anticlines and periclines in the usual way. [p. 401] A section across the cylinder, then, will tend to shew us precisely the same arrangements as we have already so fully studied in connection with the typical division of a circular cell into quadrants, and of these quadrants into triangular and quadrangular portions, and so on.
But there are other possibilities to be considered, in regard to the mode of division of the elongating quasi-cylindrical portion, as it gradually develops out of the growing and bulging quadrantal cell; for the manner in which this latter cell divides will simply depend upon the form it has assumed before each successive act of division takes place, that is to say upon the ratio between its rate of growth and the frequency of its successive divisions. For, as we have already seen, if the growing cell attain a markedly oblong or cylindrical form before division ensues, then the partition will arise transversely to the long axis; if it be but a little more than a hemisphere, it will divide by an oblique partition; and if it be less than a hemisphere (as it may come to be after successive transverse divisions) it will divide by a vertical partition, that is to say by one coinciding with its axis of growth. An immense number of permutations and combinations may arise in this way, and we must confine our illustrations to a small number of cases. The important thing is not so much to trace out the various conformations which may arise, but to grasp the fundamental principle: which is, that the forces which dominate the form of each cell regulate the manner of its subdivision, that is to say the form of the new cells into which it subdivides; or in other words, the form of the growing organism regulates the form and number of the cells which eventually constitute it. The complex cell-network is not the cause but the result of the general configuration, which latter has its essential cause in whatsoever physical and chemical processes have led to a varying velocity of growth in one direction as compared with another.
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Fig. 183. Development of Sphagnum. (After Campbell.)
In the annexed figure of an embryo of Sphagnum we see a mode of development almost precisely corresponding to the hypothetical case which we have just described,—the case, that is to say, where one of the four original quadrants of the mother-cell is the chief agent in future growth and development. We see at the base of our first figure (a), the three stationary, or [p. 402] undivided quadrants, one of which has further slowly divided in the stage b. The active quadrant has grown quickly into a cylindrical structure, which inevitably divides, in the next place, into a series of transverse partitions; and accordingly, this mode of development carries with it the presence of a single “apical cell,” whose lower wall is a spherical surface with its convexity downwards. Each cell of the subdivided cylinder now appears as a more or less flattened disc, whose mode of further sub-division we may prognosticate according to our former investigation, to which subject we shall presently return.
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Fig. 184.
In the next place, still keeping to the case where only one of the original quadrant-cells continues to grow and develop, let us suppose that this growing cell falls to be divided when by growth it has become just a little greater than a hemisphere; it will then divide, as in Fig. 184, 2, by an oblique partition, in the usual way, whose precise position and inclination to the base will depend entirely on the configuration of the cell itself, save only, of course, that we may have also to take into account the possibility of the division being into two unequal halves. By our hypothesis, [p. 403] the growth of the whole system is mainly in a vertical direction, which is as much as to say that the more actively growing protoplasm, or at least the strongest osmotic force, will be found near the apex; where indeed there is obviously more external surface for osmotic action. It will therefore be that one of the two cells which contains, or constitutes, the apex which will grow more rapidly than the other, and which therefore will be the first to divide, and indeed in any case, it will usually be this one of the two which will tend to divide first, inasmuch as the triangular and not the quadrangular half is bound to constitute the apex [142]. It is obvious that (unless the act of division be so long postponed that the cell has become quasi-cylindrical) it will divide by another oblique partition, starting from, and running at right angles to, the first. And so division will proceed,
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Fig. 185. Gemma of Moss. (After Campbell.)
by oblique alternate partitions, each one tending to be, at first, perpendicular to that on which it is based and also to the peripheral wall; but all these points of contact soon tending, by reason of the equal tensions of the three films or surfaces which meet there, to form angles of 120°. There will always be, in such a case, a single apical cell, of a more or less distinctly triangular form. The annexed figure of the developing antheridium of a Liverwort (Riccia) is a typical example of such a case. In Fig. 185 which represents a “gemma” of a Moss, we see just the same thing; with this addition, that here the lower of the two original cells has grown even more quickly than the other, constituting a long cylindrical stalk, and dividing in accordance with its shape, by means of transverse septa.
In all such cases as these, the cells whose development we have studied will in turn tend to subdivide, and the manner in which they will do so must depend upon their own proportions; and in all cases, as we have already seen, there will sooner or later be a tendency to the formation of periclinal walls, cutting off an “epidermal layer of cells,” as Fig. 186 illustrates very well.
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Fig. 186. Development of antheridium of Riccia. (After Campbell.)
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Fig. 187. Section of growing shoot of Selaginella, diagrammatic.
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Fig. 188. Embryo of Jungermannia. (After Kienitz-Gerloff.)
The method of division by means of oblique partitions is a common one in the case of ‘growing points’; for it evidently [p. 404] includes all cases in which the act of cell-division does not lag far behind that elongation which is determined by the specific rate of growth. And it is also obvious that, under a common type, there must here be included a variety of cases which will, at first sight, present a very different appearance one from another. For instance, in Fig. 187 which represents a growing shoot of Selaginella, and somewhat less diagrammatically in the young embryo of Jungermannia (Fig. 188), we have the appearance of an almost straight vertical partition running up in the axis of the system, and the primary cell-walls are set almost at right angles to it,—almost transversely, that is to say to the outer walls and to the long axis of the structure. We soon recognise, however, [p. 405] that the difference is merely a difference of degree. The more remote the partitions are, that is to say the greater the velocity of growth relatively to division, the less abrupt will be the alternate kinks or curvatures of the portions which lie in the neighbourhood of the axis, and the more will these portions appear to constitute a single unbroken wall.
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Fig. 189.
But an appearance nearly, if not quite, indistinguishable from this may be got in another way, namely, when the original growing cell is so nearly hemispherical that it is actually divided by a vertical partition, into two quadrants; and from this vertical partition, as it elongates, lateral partition-walls will arise on either side. And by the tensions exercised by these, the vertical partition will be bent into little portions set at 120° one to another, and the whole will come to look just like that which, in the former case, was made up of portions of many successive oblique partitions.
Let us now, in one or two cases, follow out a little further the stages of cell-division whose beginning we have studied in the last paragraphs. In the antheridium of Riccia, after the successive oblique partitions have produced the longitudinal series of cells shewn in Fig. 186, it is plain that the next partitions will arise periclinally, that is to say parallel to the outer wall, which in this particular case represents the short axis of the oblong cells. The effect is at once to produce an epidermal layer, whose cells will tend to subdivide further by means of partitions perpendicular to the free surface, that is to say crossing the flattened cells by their shortest diameter. The inner mass, beneath the epidermis, consists of cells which are still more or less oblong, or which become [p. 406] definitely so in process of growth; and these again divide, parallel to their short axes, into squarish cells, which as usual, by the mutual tension of their walls, become hexagonal, as seen in a plane section. There is a clear distinction, then, in form as well as in position, between the outer covering-cells and those which lie within this envelope; the latter are reduced to a condition which merely fulfils the mechanical function of a protective coat, while the former undergo less modification, and give rise to the actively living, reproductive elements.
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Fig. 190. Development of sporangium of Osmunda. (After Bower.)
In Fig. 190 is shewn the development of the sporangium of a fern (Osmunda). We may trace here the common phenomenon of a series of oblique partitions, built alternately on one another, and cutting off a conspicuous triangular apical cell. Over the whole system an epidermal layer has been formed, in the manner we have described; and in this case it covers the apical cell also, owing to the fact that it was of such dimensions that, at one stage of growth, a periclinal partition wall, cutting off its outer end, was indicated as of less area than an anticlinal one. This periclinal wall cuts down the apical cell to the proportions, very nearly, of an equilateral triangle, but the solid form of the cell is obviously that of a tetrahedron with curved faces; and accordingly, the least possible partitions by which further subdivision can be effected will run successively parallel to its four sides (or its three sides when we confine ourselves to the appearances as seen in [p. 407] section). The effect, as seen in section, is to cut off on each side a characteristically flattened cell, oblong as seen in section, still leaving a triangular (or strictly speaking, a tetrahedral) one in the centre. The former cells, which constitute no specific structure or perform no specific physiological function, but which merely represent certain directions in space towards which the whole system of partitioning has gradually led, are called by botanists the “tapetum.” The active growing tetrahedral cell which lies between them, and from which in a sense every other cell in the system has been either directly or indirectly segmented off, still manifests, as it were, its vigour and activity, and now, by internal subdivision, becomes the mother-cell of the spores.
In all these cases, for simplicity’s sake, we have merely considered the appearances presented in a single, longitudinal, plane of optical section. But it is not difficult to interpret from these appearances what would be seen in another plane, for instance in a transverse section. In our first example, for instance, that of the developing embryo of Sphagnum (Fig. 183), we can see that, at appropriate levels, the cells of the original cylindrical row have divided into transverse rows of four, and then of eight cells. We may be sure that the four cells represent, approximately, quadrants of a cylindrical disc, the four cells, as usual, not meeting in a point, but intercepted by a small intermediate partition. Again, where we have a plate of eight cells, we may well imagine that the eight octants are arranged in what we have found to be the way naturally resulting from the division of four quadrants, that is to say into alternately triangular and quadrangular portions; and this is found by means of sections to be the case. The accompanying figure is precisely comparable to our previous diagrams of the arrangement of an aggregate of eight cells in a dividing disc, save only that, in two cases, the cells have already undergone a further subdivision.
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Fig. 191. (A, B,) Sections of younger and older embryos of Phascum; (C) do. of Adiantum. (After Kienitz-Gerloff.)
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Fig. 192. Section through frond of Girardia sphacelaria. (After Goebel.)
It follows in like manner, that in a host of cases we meet with this characteristic figure, in one or other of its possible, and strictly limited, variations,—in the cross sections of growing embryonic structures, just as we have already seen that it appears in a host of cases where the entire system (or a portion of its [p. 408] surface) consists of eight cells only. For example, in Fig. 191, we have it again, in a section of a young embryo of a moss (Phascum), and in a section of an embryo of a fern (Adiantum). In Fig. 192 shewing a section through a growing frond of a sea-weed (Girardia) we have a case where the partitions forming the eight octants have conformed to the usual type; but instead of the usual division by periclines of the four quadrangular spaces, these latter are dividing by means of oblique septa, apparently owing to the fact that the cell is not dividing into two equal, but into two unequal portions. In this last figure we have a peculiar look of stiffness or formality, such that it appears at first to bear little resemblance to the rest. The explanation is of the simplest. The mode of partitioning differs little (except to some slight extent in the way already mentioned) from the normal type; but in this case the partition walls are so thick and become so quickly comparatively solid and rigid, that the secondary curvatures due to their successive mutual tractions are here imperceptible.
1917 Source Illustration
Fig. 193. Development of antheridium of Pteris. (After Strasbürger.)
A curious and beautiful case, apparently aberrant but which would doubtless be found conforming strictly to physical laws, if [p. 409] only we clearly understood the actual conditions, is indicated in the development of the antheridium of a fern, as described by Strasbürger. Here the antheridium develops from a single cell, whose form has grown to be something more than a hemisphere; and the first partition, instead of stretching transversely across the cell, as we should expect it to do if the cell were actually spherical, has as it were sagged down to come in contact with the base, and so to develop into an annular partition, running round the lower margin of the cell. The phenomenon is akin to that cutting off of the corner of a cubical cell by a spherical partition, of which we have spoken on p. 349, and the annular film is very easy to reproduce by means of a soap-bubble in the bottom of a cylindrical dish or beaker. The next partition is a periclinal one, concentric with the outer surface of the young antheridium; and this in turn is followed by a concave partition which cuts off the apex of the original cell: but which becomes connected with the second, or periclinal partition in precisely the same annular fashion as the first partition did with the base of the little antheridium. The result is that, at this stage, we have four cell-cavities in the little antheridium: (1) a central cavity; (2) an annular space around the lower margin; (3) a narrow annular or cylindrical space around the sides of the antheridium; and (4) a small terminal or apical cell. It is evident that the tendency, in the next place, will be to subdivide the flattened external cells by means of anticlinal partitions, and so to convert the whole structure into a single layer of epidermal cells, surrounding a central cell within which, in course of time, the antherozoids are developed.
The foregoing account deals only with a few elementary phenomena, and may seem to fall far short of an attempt to deal in general with “the forms of tissues.” But it is the principle involved, and not its ultimate and very complex results, that we can alone [p. 410] attempt to grapple with. The stock-in-trade of mathematical physics, in all the subjects with which that science deals, is for the most part made up of simple, or simplified, cases of phenomena which in their actual and concrete manifestations are usually too complex for mathematical analysis; and when we attempt to apply its methods to our biological and histological phenomena, in a preliminary and elementary way, we need not wonder if we be limited to illustrations which are obviously of a simple kind, and which cover but a small part of the phenomena with which the histologist has become familiar. But it is only relatively that these phenomena to which we have found the method applicable are to be deemed simple and few. They go already far beyond the simplest phenomena of all, such as we see in the dividing Protococcus, and in the first stages, two-celled or four-celled, of the segmenting egg. They carry us into stages where the cells are already numerous, and where the whole conformation has become by no means easy to depict or visualise, without the help and guidance which the phenomena of surface-tension, the laws of equilibrium and the principle of minimal areas are at hand to supply. And so far as we have gone, and so far as we can discern, we see no sign of the guiding principles failing us, or of the simple laws ceasing to hold good.
Chapter IX: On Concretions, Spicules, and Spicular Skeletons¶
1917 Cambridge First Edition (pp. 411–487)
The deposition of inorganic material in the living body, usually in the form of calcium salts or of silica, is a very common and wide-spread phenomenon. It begins in simple ways, by the appearance of small isolated particles, crystalline or non-crystalline, whose form has little relation or sometimes none to the structure of the organism; it culminates in the complex skeletons of the vertebrate animals, in the massive skeletons of the corals, or in the polished, sculptured and mathematically regular molluscan shells. Even among many very simple organisms, such as the Diatoms, the Radiolarians, the Foraminifera, or the Sponges, the skeleton displays extraordinary variety and beauty, whether by reason of the intrinsic form of its elementary constituents or the geometric symmetry with which these are arranged and interconnected.
With regard to the form of these various structures (and this is all that immediately concerns us here), it is plain that we have to do with two distinct problems, which however, though theoretically distinct, may merge with one another. For the form of the spicule or other skeletal element may depend simply upon its chemical nature, as for instance, to take a simple but not the only case, when the form is purely crystalline; or the inorganic solid material may be laid down in conformity with the shapes assumed by the cells, tissues or organs, and so be, as it were, moulded to the shape of the living organism; and again, there may well be intermediate stages in which both phenomena may be simultaneously recognised, the molecular forces playing their part in conjunction with, and under the restraint of, the other forces inherent in the system. [p. 412]
So far as the problem is a purely chemical one, we must deal with it very briefly indeed; and all the more because special investigations regarding it have as yet been few, and even the main facts of the case are very imperfectly known. This at least is evident, that the whole series of phenomena with which we are about to deal go deep into the subject of colloid chemistry, and especially with that branch of the science which deals with the properties of colloids in connection with capillary or surface phenomena. It is to the special student of colloid chemistry that we must ultimately and chiefly look for the elucidation of our problem [143].
In the first and simplest part of our subject, the essential problem is the problem of crystallisation in presence of colloids. In the cells of plants, true crystals are found in comparative abundance, and they consist, in the great majority of cases, of calcium oxalate. In the stem and root of the rhubarb, for instance, in the leaf-stalk of Begonia, and in countless other cases, sometimes within the cell, sometimes in the substance of the cell-wall, we find large and well-formed crystals of this salt; their varieties of form, which are extremely numerous, are simply the crystalline forms proper to the salt itself, and belong to the two systems, cubic and monoclinic, in one or other of which, according to the amount of water of crystallisation, this salt is known to crystallise. When calcium oxalate crystallises according to the latter system (as it does when its molecule is combined with two molecules of water of crystallisation), the microscopic crystals have the form of fine needles, or “raphides,” such as are very common in plants; and it has been found that these are artificially produced when the salt is crystallised out in presence of glucose or of dextrin [144].
1917 Source Illustration
Fig. 194. Alcyonarian spicules: Siphonogorgia and Anthogorgia. (After Studer.)
Calcium carbonate, on the other hand, when it occurs in plant-cells (as it does abundantly, for instance in the “cystoliths” of the Urticaceae and Acanthaceae, and in great quantities in Melobesia [p. 413] and the other calcareous or “stony” algae), appears in the form of fine rounded granules, whose inherent crystalline structure is not outwardly visible, but is only revealed (like that of a molluscan shell) under polarised light. Among animals, a skeleton of carbonate of lime occurs under a multitude of forms, of which we need only mention now a very few of the most conspicuous. The spicules of the calcareous sponges are triradiate, occasionally quadriradiate, bodies, with pointed rays, not crystalline in outward form but with a definitely crystalline internal structure. We shall return again to these, and find for them what would seem to be a satisfactory explanation of their form. Among the Alcyonarian zoophytes we have a great variety of spicules [145], which are sometimes straight and slender rods, sometimes flattened and more or less striated plates, and still more often rounded or branched concretions with rough or knobby surfaces (Figs. 194, 200). A third type, presented by several very different things, such as a pearl, or the ear-bone of a bony fish, consists of a more or less [p. 414] rounded body, sometimes spherical, sometimes flattened, in which the calcareous matter is laid down in concentric zones, denser and clearer layers alternating with one another. In the development of the molluscan shell and in the calcification of a bird’s egg or the shell of a crab, for instance, spheroidal bodies with similar concentric striation make their appearance; but instead of remaining separate they become crowded together, and as they coalesce they combine to form a pattern of hexagons. In some cases, the carbonate of lime on being dissolved away by acid leaves behind it a certain small amount of organic residue; in most cases other salts, such as phosphates of lime, ammonia or magnesia are present in small quantities; and in most cases if not all the developing spicule or concretion is somehow or other so associated with living cells that we are apt to take it for granted that it owes its peculiarities of form to the constructive or plastic agency of these.
The appearance of direct association with living cells, however, is apt to be fallacious; for the actual precipitation takes place, as a rule, not in actively living, but in dead or at least inactive tissue [146]: that is to say in the “formed material” or matrix which (as for instance in cartilage) accumulates round the living cells, in the interspaces between these latter, or at least, as often happens, in connection with the cell-wall or cell-membrane rather than within the substance of the protoplasm itself. We need not go the length of asserting that this is a rule without exception; but, so far as it goes, it is of great importance and to its consideration we shall presently return [147].
Cognate with this is the fact that it is known, at least in some cases, that the organism can go on living and multiplying with apparently unimpaired health, when stinted or even wholly deprived of the material of which it is wont to make its spicules [p. 415] or its shell. Thus, Pouchet and Chabry [148] have shown that the eggs of sea-urchins reared in lime-free water develop in apparent health, into larvae entirely destitute of the usual skeleton of calcareous rods, and in which, accordingly, the long arms of the Pluteus larva, which the rods support and distend, are entirely suppressed. And again, when Foraminifera are kept for generations in water from which they gradually exhaust the lime, their shells grow hyaline and transparent, and seem to consist only of chitinous material. On the other hand, in the presence of excess of lime, the shells become much altered, strengthened with various “ornaments,” and assuming characters described as proper to other varieties and even species [149].
The crucial experiment, then, is to attempt the formation of similar structures or forms, apart from the living organism: but, however feasible the attempt may be in theory, we shall be prepared from the first to encounter difficulties, and to realise that, though the actions involved may be wholly within the range of chemistry and physics, yet the actual conditions of the case may be so complex, subtle and delicate, that only now and then, and in the simplest of cases, shall we find ourselves in a position to imitate them completely and successfully. Such an investigation is only part of that much wider field of enquiry through which Stephane Leduc and many other workers [150] have sought to produce, by synthetic means, forms similar to those of living things; but it is a well-defined and circumscribed part of that wider investigation. When by chemical or physical experiment we obtain configurations similar, for instance, to the phenomena of nuclear division, or conformations similar to a pattern of hexagonal cells, or a group of vesicles which resemble some particular tissue or cell-aggregate, we indeed prove what it is the main object of this book to illustrate, namely, that the physical forces are capable of producing particular organic forms. But it is by no means always that we can feel perfectly assured that the physical forces which we deal with in our experiment are identical with, and not merely analogous to, [p. 416] the physical forces which, at work in nature, are bringing about the result which we have succeeded in imitating. In the present case, however, our enquiry is restricted and apparently simplified; we are seeking in the first instance to obtain by purely chemical means a purely chemical result, and there is little room for ambiguity in our interpretation of the experiment.
When we find ourselves investigating the forms assumed by chemical compounds under the peculiar circumstances of association with a living body, and when we find these forms to be characteristic or recognisable, and somehow different from those which, under other circumstances, the same substance is wont to assume, an analogy presents itself to our minds, captivating though perhaps somewhat remote, between this subject of ours and certain synthetic problems of the organic chemist. There is doubtless an essential difference, as well as a difference of scale, between the visible form of a spicule or concretion and the hypothetical form of an individual molecule; but molecular form is a very important concept; and the chemist has not only succeeded, since the days of Wöhler, in synthesising many substances which are characteristically associated with living matter, but his task has included the attempt to account for the molecular forms of certain “asymmetric” substances, glucose, malic acid and many more, as they occur in nature. These are bodies which, when artificially synthesised, have no optical activity, but which, as we actually find them in organisms, turn (when in solution) the plane of polarised light in one direction or the other; thus dextro-glucose and laevomalic acid are common products of plant metabolism; but dextromalic acid and laevo-glucose do not occur in nature at all. The optical activity of these bodies depends, as Pasteur shewed more than fifty years ago [151], upon the form, right-handed or left-handed, of their molecules, which molecular asymmetry further gives rise to a corresponding right or left-handedness (or enantiomorphism) in the crystalline aggregates. It is a distinct problem in organic or physiological chemistry, [p. 417] and by no means without its interest for the morphologist, to discover how it is that nature, for each particular substance, habitually builds up, or at least selects, its molecules in a one-sided fashion, right-handed or left-handed as the case may be. It will serve us no better to assert that this phenomenon has its origin in “fortuity,” than to repeat the Abbé Galiani’s saying, “les dés de la nature sont pipés.”
The problem is not so closely related to our immediate subject that we need discuss it at length; but at the same time it has its clear relation to the general question of form in relation to vital phenomena, and moreover it has acquired interest as a theme of long-continued discussion and new importance from some comparatively recent discoveries.
According to Pasteur, there lay in the molecular asymmetry of the natural bodies and the symmetry of the artificial products, one of the most deep-seated differences between vital and non-vital phenomena: he went further, and declared that “this was perhaps the only well-marked line of demarcation that can at present [1860] be drawn between the chemistry of dead and of living matter.” Nearly forty years afterwards the same theme was pursued and elaborated by Japp in a celebrated lecture [152], and the distinction still has its weight, I believe, in the minds of many if not most chemists.
“We arrive at the conclusion,” said Professor Japp, “that the production of single asymmetric compounds, or their isolation from the mixture of their enantiomorphs, is, as Pasteur firmly held, the prerogative of life. Only the living organism, or the living intelligence with its conception of asymmetry, can produce this result. Only asymmetry can beget asymmetry.” In these last words (which, so far as the chemist and the biologist are concerned, we may acknowledge to be perfectly true [153]) lies the [p. 418] crux of the difficulty; for they at once bid us enquire whether in nature, external to and antecedent to life, there be not some asymmetry to which we may refer the further propagation or “begetting” of the new asymmetries: or whether in default thereof, we be rigorously confined to the conclusion, from which Japp “saw no escape,” that “at the moment when life first arose, a directive force came into play,—a force of precisely the same character as that which enables the intelligent operator, by the exercise of his will, to select one crystallised enantiomorph and reject its asymmetric opposite [154].”
Observe that it is only the first beginnings of chemical asymmetry that we need to discover; for when asymmetry is once manifested, it is not disputed that it will continue “to beget asymmetry.” A plausible suggestion is now at hand, which if it be confirmed and extended will supply or at least sufficiently illustrate the kind of explanation which is required [155].
We know in the first place that in cases where ordinary non-polarised light acts upon a chemical substance, the amount of chemical action is proportionate to the amount of light absorbed. We know in the second place [156], in certain cases, that light circularly polarised is absorbed in different amounts by the right-handed or left-handed varieties, as the case may be, of an asymmetric substance. And thirdly, we know that a portion of the light which comes to us from the sun is already plane-polarised light, which becomes in part circularly polarised, by reflection (according to Jamin) at the surface of the sea, and then rotated in a particular direction under the influence of terrestrial magnetism. We only require to be assured that the relation between absorption of light and chemical activity will continue to hold good in the case of circularly polarised light; that is to say [p. 419] that the formation of some new substance or other, under the influence of light so polarised, will proceed asymmetrically in consonance with the asymmetry of the light itself; or conversely, that the asymmetrically polarised light will tend to more rapid decomposition of those molecules by which it is chiefly absorbed. This latter proof is now said to be furnished by Byk [157], who asserts that certain tartrates become unsymmetrical under the continued influence of the asymmetric rays. Here then we seem to have an example, of a particular kind and in a particular instance, an example limited but yet crucial (if confirmed), of an asymmetric force, non-vital in its origin, which might conceivably be the starting-point of that asymmetry which is characteristic of so many organic products.
The mysteries of organic chemistry are great, and the differences between its processes or reactions as they are carried out in the organism and in the laboratory are many [158]. The actions, catalytic and other, which go on in the living cell are of extraordinary complexity. But the contention that they are different in kind from what we term ordinary chemical operations, or that in the production of single asymmetric compounds there is actually to be witnessed, as Pasteur maintained, a “prerogative of life,” would seem to be no longer safely tenable. And furthermore, it behoves us to remember that, even though failure continued to attend all artificial attempts to originate the asymmetric or optically active compounds which organic nature produces in abundance, this would only prove that a certain physical force, or mode of physical action, is at work among living things though unknown elsewhere. It is a mode of action which we can easily imagine, though the actual mechanism we cannot set agoing when we please. And it follows that such a difference between living matter and dead would carry us but a little way, for it would still be confined strictly to the physical or mechanical plane.
Our historic interest in the whole question is increased by the [p. 420] fact, or the great probability, that “the tenacity with which Pasteur fought against the doctrine of spontaneous generation was not unconnected with his belief that chemical compounds of one-sided symmetry could not arise save under the influence of life [159].” But the question whether spontaneous generation be a fact or not does not depend upon theoretical considerations; our negative response is based, and is so far soundly based, on repeated failures to demonstrate its occurrence. Many a great law of physical science, not excepting gravitation itself, has no higher claim on our acceptance.
Let us return then, after this digression, to the general subject of the forms assumed by certain chemical bodies when deposited or precipitated within the organism, and to the question of how far these forms may be artificially imitated or theoretically explained.
Mr George Rainey, of St Bartholomew’s Hospital (to whom we have already referred), and Professor P. Harting, of Utrecht, were the first to deal with this specific problem. Mr Rainey published, between 1857 and 1861, a series of valuable and thoughtful papers to shew that shell and bone and certain other organic structures were formed “by a process of molecular coalescence, demonstrable in certain artificially-formed products [160].” Professor Harting, after thirty years of experimental work, published in 1872 a paper, which has become classical, entitled Recherches de Morphologie Synthétique, sur la production artificielle de quelques formations calcaires organiques; his aim was to pave the way for a “morphologie synthétique,” as Wöhler had laid the foundations of a “chimie synthétique,” by his classical discovery forty years before. [p. 421]
1917 Source Illustration
Fig. 195. Calcospherites, or concretions of calcium carbonate, deposited in white of egg. (After Harting.)
1917 Source Illustration
Fig. 196. A single calcospherite, with central “nucleus,” and striated, iridescent border. (After Harting.)
1917 Source Illustration
Fig. 197. Later stages in the same experiment.
1917 Source Illustration
Fig. 198, A. Section of shell of Mya; B. Section of hinge-tooth of do. (After Carpenter.)
Rainey and Harting used similar methods, and these were such as many other workers have continued to employ,—partly with the direct object of explaining the genesis of organic forms and partly as an integral part of what is now known as Colloid Chemistry. The whole gist of the method was to bring some soluble salt of lime, such as the chloride or nitrate, into solution within a colloid medium, such as gum, gelatine or albumin; and then to precipitate it out in the form of some insoluble compound, such as the carbonate or oxalate. Harting found that, when he added a little sodium or potassium carbonate to a concentrated solution of calcium chloride in albumin, he got at first a gelatinous mass, or “colloid precipitate”: which slowly transformed by the appearance of tiny microscopic particles, at first motionless, but afterwards as they grew larger shewing the typical Brownian movement. So far, very much the same phenomena were witnessed whether the solution were albuminous or not, and similar appearances indeed had been witnessed and recorded by Gustav Rose, so far back as 1837 [161]; but in the later stages the presence of albuminoid matter made a great difference. Now, after a few days, the calcium carbonate was seen to be deposited in the form of large rounded concretions, with a more or less distinct central nucleus, and with a surrounding structure at once radiate and [p. 422] concentric; the presence of concentric zones or lamellae, alternately dark and clear, was especially characteristic. These round “calcospherites” shewed a tendency to aggregate together in layers, and then to assume polyhedral, or often regularly hexagonal, outlines. In this latter condition they closely resemble the early stages of calcification in a molluscan (Fig. 198), or still more in a crustacean shell [162]; while in their isolated condition [p. 423] they very closely resemble the little calcareous bodies in the tissues of a trematode or a cestode worm, or in the oesophageal glands of an earthworm [163].
1917 Source Illustration
Fig. 199. Large irregular calcareous concretions, or spicules, deposited in a piece of dead cartilage, in presence of calcium phosphate. (After Harting.)
When the albumin was somewhat scanty, or when it was mixed with gelatine, and especially when a little phosphate of lime was [p. 424] added to the mixture, the spheroidal globules tended to become rough, by an outgrowth of spinous or digitiform projections; and in some cases, but not without the presence of the phosphate, the result was an irregularly shaped knobby spicule, precisely similar to those which are characteristic of the Alcyonaria [164].
1917 Source Illustration
Fig. 200. Additional illustrations of Alcyonarian spicules: Eunicea. (After Studer.)
The rough spicules of the Alcyonaria are extraordinarily variable in shape and size, as, looking at them from the chemist’s or the physicist’s point of view, we should expect them to be. Partly upon the form of these spicules, and partly on the general form or mode of branching of the entire colony of polypes, a vast number of separate “species” have been based by systematic zoologists. But it is now admitted that even in specimens of a single species, from one and the same locality, the spicules may vary immensely in shape and size: and Professor Hickson declares (in a paper published while these sheets are passing through the press) that after many years of laborious work in striving to determine species of these animal colonies, he feels “quite convinced that we have been engaged in a more or less fruitless task [165]”.
The formation of a tooth has very lately been shown to be a phenomenon of the same order. That is to say, “calcification in both dentine and enamel [p. 425] is in great part a physical phenomenon; the actual deposit in both tissues occurs in the form of calcospherites, and the process in mammalian tissue is identical in every point with the same process occurring in lower organisms [166].” The ossification of bone, we may be sure, is in the same sense and to the same extent a physical phenomenon.
The typical structure of a calcospherite is no other than that of a pearl, nor does it differ essentially from that of the otolith of a mollusc or of a bony fish. (The otoliths, by the way, of the elasmobranch fishes, like those of reptiles and birds, are not developed after this fashion, but are true crystals of calc-spar.)
1917 Source Illustration
Fig. 201. A “crust” of close-packed calcareous concretions, precipitated at the surface of an albuminous solution. (After Harting.)
Throughout these phenomena, the effect of surface-tension is manifest. It is by surface-tension that ultra-microscopic particles are brought together in the first floccular precipitate or coagulum; by the same agency, the coarser particles are in turn agglutinated into visible lumps; and the form of the calcospherites, whether it be that of the solitary spheres or that assumed in various stages of aggregation (e.g. Fig. 202) [167], is likewise due to the same agency.
1917 Source Illustration
Fig. 202. Aggregated calcospherites. (After Harting.)
From the point of view of colloid chemistry the whole phenomenon is very important and significant; and not the least significant part is this tendency of the solidified deposits to assume the form of “spherulites,” and other rounded contours. In the phraseology of that science, we are dealing with a two-phase system, which finally consists of solid particles in suspension in a liquid (the former being styled the disperse phase, the latter the [p. 426] dispersion medium). In accordance with a rule first recognised by Ostwald [168], when a substance begins to separate out from a solution, so making its appearance as a new phase, it always makes its appearance first as a liquid [169]. Here is a case in point. The minute quantities of material, on their way from a state of solution to a state of “suspension,” pass through a liquid to a solid form; and their temporary sojourn in the former leaves its impress in the rounded contours which surface-tension brought about while the little aggregate was still labile or fluid: while coincidently with this surface-tension effect upon the surface, crystallisation tended to take place throughout the little liquid mass, or in such portion of it as had not yet consolidated and crystallised.
1917 Source Illustration
Fig. 203. (After Harting.)
Where we have simple aggregates of two or three calcospherites, the resulting figure is precisely that of so many contiguous soap-bubbles. In other cases, composite forms result which are not so easily explained, but which, if we could only account for them, would be of very great interest to the biologist. For instance, when smaller calcospheres seem, as it were, to invade the substance of a larger one, we get curious conformations which in the closest possible way resemble the outlines of certain of the Diatoms (Fig. 203). Another very curious formation, which Harting calls a “conostat,” is of frequent occurrence, and in it we see at least a suggestion of analogy with the configuration which, in a protoplasmic structure, we have spoken of as a “collar-cell.” The [p. 427] conostats, which are formed in the surface layer of the solution, consist of a portion of a spheroidal calcospherite, whose upper part is continued into a thin spheroidal collar, of somewhat larger radius than the solid sphere; but the precise manner in which the collar is formed, possibly around a bubble of gas, possibly about a vortex-like diffusion-current [170] is not obvious.
Among these various phenomena, the concentric striation observed in the calcospherite has acquired a special interest and importance [171]. It is part of a phenomenon now widely known, and recognised as an important factor in colloid chemistry, under the name of “Liesegang’s Rings [172].”
1917 Source Illustration
Fig. 204. Conostats. (After Harting.)
If we dissolve, for instance, a little bichromate of potash in gelatine, pour it on to a glass plate, and after it is set place upon it a drop of silver nitrate solution, there appears in the course of a few hours the phenomenon of Liesegang’s rings. At first the silver forms a central patch of abundant reddish brown chromate precipitate; but around this, as the silver nitrate diffuses slowly through the gelatine, the precipitate no longer comes down in a continuous, uniform layer, but forms a series of zones, beautifully regular, which alternate with clear interspaces of jelly, and which stand farther and farther apart, in logarithmic ratio, as they recede from the centre. For a discussion of the raison d’être of [p. 428] this phenomenon, still somewhat problematic, the student must consult the text-books of physical and colloid chemistry [173].
But, speaking very generally, we may say the appearance of Liesegang’s rings is but a particular and striking case of a more general phenomenon, namely the influence on crystallisation of the presence of foreign bodies or “impurities,” represented in this case by the “gel” or colloid matrix [174]. Faraday shewed long ago that to the presence of slight impurities might be ascribed the banded structure of ice, of banded quartz or agate, onyx, etc.; and Quincke and Tomlinson have added to our scanty knowledge of the same phenomenon [175].
1917 Source Illustration
Fig. 205. Liesegang’s Rings. (After Leduc.)
Besides the tendency to rhythmic action, as manifested in Liesegang’s rings, the association of colloid matter with a crystalloid in solution may lead to other well-marked effects. These, according to Professor J. H. Bowman [176], may be grouped somewhat as follows: (1) total prevention of crystallisation; (2) suppression of certain of the lines of crystalline growth; (3) extension of the crystal to abnormal proportions, with a tendency for it to become a compound crystal; (4) a curving or gyrating of the crystal or its parts. [p. 429]
1917 Source Illustration
Fig. 206. Relay-crystals of common salt. (After Bowman.)
1917 Source Illustration
Fig. 207. Wheel-like crystals in a colloid. (After Bowman.)
For instance, it would seem that, if the supply of material to the growing crystal be not forthcoming in sufficient quantity (as may well happen in a colloid medium, for lack of convection-currents), then growth will follow only the strongest lines of crystallising force, and will be suppressed or partially suppressed along other axes. The crystal will have a tendency to become filiform, or “fibrous”; and the raphides of our plant-cells are a case in point. Again, the long slender crystal so formed, pushing its way into new material, may initiate a new centre of crystallisation: we get the phenomenon known as a “relay,” along the principal lines of force, and sometimes along subordinate axes as well. This phenomenon is illustrated in the accompanying figure of crystallisation in a colloid medium of common salt; and it may possibly be that we have here an explanation, or part of an explanation, of the compound siliceous spicules of the Hexactinellid sponges. Lastly, when the crystallising force is nearly equalled by the resistance of the viscous medium, the crystal takes the line of least resistance, with very various results. One of these results would seem to be a gyratory course, giving to the crystal a curious wheel-like shape, as in Fig. 207; and other results are the feathery, fern-like [p. 430] or arborescent shapes so frequently seen in microscopic crystallisation.
To return to Liesegang’s rings, the typical appearance of concentric rings upon a gelatinous plate may be modified in various experimental ways. For instance, our gelatinous medium may be placed in a capillary tube immersed in a solution of the precipitating salt, and in this case we shall obtain a vertical succession of bands or zones regularly interspaced: the result being very closely comparable to the banded pigmentation which we see in the hair of a rabbit or a rat. In the ordinary plate preparation, the free surface of the gelatine is under different conditions to the lower layers and especially to the lowest layer in contact with the glass; and therefore it often happens that we obtain a double series of rings, one deep and the other superficial, which by occasional blending or interlacing, may produce a netted pattern. In some cases, as when only the inner surface of our capillary tube is covered with a layer of gelatine, there is a tendency for the deposit to take place in a continuous spiral line, rather than in concentric and separate zones. By such means, according to Küster [177] various forms of annular, spiral and reticulated thickenings in the vascular tissue of plants may be closely imitated; and he and certain other writers have of late been inclined to carry the same chemico-physical phenomenon a very long way, in the explanation of various banded, striped, and other rhythmically successional types of structure or pigmentation. For example, the striped pigmentation of the leaves in many plants (such as Eulalia japonica), the striped or clouded colouring of many feathers or of a cat’s skin, the patterns of many fishes, such for instance as the brightly coloured tropical Chaetodonts and the like, are all regarded by him as so many instances of “diffusion-figures” closely related to the typical Liesegang phenomenon. Gebhardt has made a particular study of the same subject in the case of insects [178]. He declares, for instance, that the banded wings of Papilio podalirius are precisely imitated in Liesegang’s experiments; that the finer markings on the wings of the Goatmoth (Cossus ligniperda) shew the double arrangement of larger and of [p. 431] smaller intermediate rhythms, likewise manifested in certain cases of the same kind; that the alternate banding of the antennae (for instance in Sesia spheciformis), a pigmentation not concurrent with the segmented structure of the antenna, is explicable in the same way; and that the “ocelli,” for instance of the Emperor moth, are typical illustrations of the common concentric type. Darwin’s well-known disquisition [179] on the ocellar pattern of the feathers of the Argus Pheasant, as a result of sexual selection, will occur to the reader’s mind, in striking contrast to this or to any other direct physical explanation [180]. To turn from the distribution of pigment to more deeply seated structural characters, Leduc has shewn how, for instance, the laminar structure of the cornea or the lens is again, apparently, a similar phenomenon. In the lens of the fish’s eye, we have a very curious appearance, the consecutive lamellae being roughened or notched by close-set, interlocking sinuosities; and precisely the same appearance, save that it is not quite so regular, is presented in one of Küster’s figures as the effect of precipitating a little sodium phosphate in a gelatinous medium. Biedermann has studied, from the same point of view, the structure and development of the molluscan shell, the problem which Rainey had first attacked more than fifty years before [181]; and Liesegang himself has applied his results to the formation of pearls, and to the development of bone [182]. [p. 432]
Among all the many cases where this phenomenon of Liesegang’s comes to the naturalist’s aid in explanation of rhythmic or zonary configurations in organic forms, it has a special interest where the presence of concentric zones or rings appears, at first sight, as a sure and certain sign of periodicity of growth, depending on the seasons, and capable therefore of serving as a mark and record of the creature’s age. This is the case, for instance, with the scales, bones and otoliths of fishes; and a kindred phenomena in starch-grains has given rise, in like manner, to the belief that they indicate a diurnal and nocturnal periodicity of activity and rest [183].
1917 Source Illustration
Fig. 208.
That this is actually the case in growing starch-grains is generally believed, on the authority of Meyer [184]; but while under certain circumstances a marked alternation of growing and resting periods may occur, and may leave its impress on the structure of the grain, there is now great reason to believe that, apart from such external influences, the internal phenomena of diffusion may, just as in the typical Liesegang experiment, produce the well-known concentric rings. The spherocrystals of inulin, in like manner, shew, like the “calcospherites” of Harting (Fig. 208), a concentric structure which in all likelihood has had no causative impulse save from within.
1917 Source Illustration
Fig. 209. Otoliths of Plaice, showing four zones or “age-rings.” (After Wallace.)
The striation, or concentric lamellation, of the scales and otoliths of fishes has been much employed of recent years as a trustworthy and unmistakeable mark of the fish’s age. There are difficulties in the way of accepting this hypothesis, not the least of which is the fact that the otolith-zones, for instance, are extremely well marked even in the case of some fishes which spend their lives in deep water, [p. 433] where the temperature and other physical conditions shew little or no appreciable fluctuation with the seasons of the year. There are, on the other hand, phenomena which seem strongly confirmatory of the hypothesis: for instance the fact (if it be fully established) that in such a fish as the cod, zones of growth, identical in number, are found both on the scales and in the otoliths [185]. The subject has become a much debated one, and this is not the place for its discussion; but it is at least obvious, with the Liesegang phenomenon in view, that we have no right to assume that an appearance of rhythm and periodicity in structure and growth is necessarily bound up with, and indubitably brought about by, a periodic recurrence of particular external conditions.
But while in the Liesegang phenomenon we have rhythmic precipitation which depends only on forces intrinsic to the system, and is independent of any corresponding rhythmic changes in temperature or other external conditions, we have not far to seek for instances of chemico-physical phenomena where rhythmic alternations of appearance or structure are produced in close relation to periodic fluctuations of temperature. A well-known instance is that of the Stassfurt deposits, where the rock-salt alternates regularly with thin layers of “anhydrite,” or (in another series of beds) with “polyhalite [186]”: and where these zones are commonly regarded as marking years, and their alternate bands as having been formed in connection with the seasons. A discussion, however, of this remarkable and significant phenomenon, and of how the chemist explains it, by help of the “phase-rule,” in connection with temperature conditions, would lead us far beyond our scope [187].
We now see that the methods by which we attempt to study the chemical or chemico-physical phenomena which accompany the development of an inorganic concretion or spicule within the [p. 434] body of an organism soon introduce us to a multitude of kindred phenomena, of which our knowledge is still scanty, and which we must not attempt to discuss at greater length. As regards our main point, namely the formation of spicules and other elementary skeletal forms, we have seen that certain of them may be safely ascribed to simple precipitation or crystallisation of inorganic materials, in ways more or less modified by the presence of albuminous or other colloid substances. The effect of these latter is found to be much greater in the case of some crystallisable bodies than in others. For instance, Harting, and Rainey also, found as a rule that calcium oxalate was much less affected by a colloid medium than was calcium carbonate; it shewed in their hands no tendency to form rounded concretions or “calcospherites” in presence of a colloid, but continued to crystallise, either normally, or with a tendency to form needles or raphides. It is doubtless for this reason that, as we have seen, crystals of calcium oxalate are so common in the tissues of plants, while those of other calcium salts are rare. But true calcospherites, or spherocrystals, of the oxalate are occasionally found, for instance in certain Cacti, and Bütschli [188] has succeeded in making them artificially in Harting’s usual way, that is to say by crystallisation in a colloid medium.
There link on to these latter observations, and to the statement already quoted that calcareous deposits are associated with the dead products rather than with the living cells of the organism, certain very interesting facts in regard to the solubility of salts in colloid media, which have been made known to us of late, and which go far to account for the presence (apart from the form) of calcareous precipitates within the organism [189]. It has been shewn, in the first place, that the presence of albumin has a notable effect on the solubility in a watery solution of calcium salts, increasing the solubility of the phosphate in a marked degree, and that of the carbonate in still greater proportion; but the [p. 435] sulphate is only very little more soluble in presence of albumin than in pure water, and the rarity of its occurrence within the organism is so far accounted for. On the other hand, the bodies derived from the breaking down of the albumins, their “catabolic” products, such as the peptones, etc., dissolve the calcium salts to a much less degree than albumin itself; and in the case of the phosphate, its solubility in them is scarcely greater than in water. The probability is, therefore, that the actual precipitation of the calcium salts is not due to the direct action of carbonic acid, etc. on a more soluble salt (as was at one time believed); but to catabolic changes in the proteids of the organism, which tend to throw down the salts already formed, which had remained hitherto in albuminous solution. The very slight solubility of calcium phosphate under such circumstances accounts for its predominance in, for instance, mammalian bone [190]; and wherever, in short, the supply of this salt has been available to the organism.
To sum up, we see that, whether from food or from sea-water, calcium sulphate will tend to pass but little into solution in the albuminoid substances of the body: calcium carbonate will enter more freely, but a considerable part of it will tend to remain in solution: while calcium phosphate will pass into solution in considerable amount, but will be almost wholly precipitated again, as the albumin becomes broken down in the normal process of metabolism.
We have still to wait for a similar and equally illuminating study of the solution and precipitation of silica, in presence of organic colloids.
From the comparatively small group of inorganic formations which, arising within living organisms, owe their form solely to precipitation or to crystallisation, that is to say to chemical or other molecular forces, we shall presently pass to that other and larger group which appear to be conformed in direct relation to the forms and the arrangement of the cells or other protoplasmic elements [191]. [p. 436] The two principles of conformation are both illustrated in the spicular skeletons of the Sponges.
1917 Source Illustration
Fig. 210. Close-packed calcospherites, or so-called “spicules,” of Astrosclera. (After Lister.)
In a considerable number, but withal a minority of cases, the form of the sponge-spicule may be deemed sufficiently explained on the lines of Harting’s and Rainey’s experiments, that is to say as the direct result of chemical or physical phenomena associated with the deposition of lime or of silica in presence of colloids [192]. This is the case, for instance, with various small spicules of a globular or spheroidal form, formed of amorphous silica, concentrically striated within, and often developing irregular knobs or tiny tubercles over their surfaces. In the aberrant sponge Astrosclera [193], we have, to begin with, rounded, striated discs or globules, which in like manner are nothing more or less than the [p. 437] “calcospherites” of Harting’s experiments; and as these grow they become closely aggregated together (Fig. 210), and assume an angular, polyhedral form, once more in complete accordance with the results of experiment [194]. Again, in many Monaxonid sponges, we have irregularly shaped, or branched spicules, roughened or tuberculated by secondary superficial deposits, and reminding one of the spicules of some Alcyonaria. These also must be looked upon as the simple result of chemical deposition, the form of the deposit being somewhat modified in conformity with the surrounding tissues, just as in the simple experiment the form of the concretionary precipitate is affected by the heterogeneity, visible or invisible, of the matrix. Lastly, the simple needles of amorphous silica, which constitute one of the commonest types of spicule, call for little in the way of explanation; they are accretions or deposits about a linear axis, or fine thread of organic material, just as the ordinary rounded calcospherite is deposited about some minute point or centre of crystallisation, and as ordinary crystallisation is often started by a particle of atmospheric dust; in some cases they also, like the others, are apt to be roughened by more irregular secondary deposits, which probably, as in Harting’s experiments, appear in this irregular form when the supply of material has become relatively scanty.
Our few foregoing examples, diverse as they are in look and kind and ranging from the spicules of Astrosclera or Alcyonium to the otoliths of a fish, seem all to have their free origin in some larger or smaller fluid-containing space, or cavity of the body: pretty much as Harting’s calcospheres made their appearance in the albuminous content of a dish. But we now come at last to a much larger class of spicular and skeletal structures, for whose regular and often complex forms some other explanation than the intrinsic forces of crystallisation or molecular adhesion is manifestly necessary. As we enter on this subject, which is certainly no small or easy one, it may conduce to simplicity, and to brevity, [p. 438] if we try to make a rough classification, by way of forecast, of the chief conditions which we are likely to meet with.
Just as we look upon animals as constituted, some of a vast number of cells, and others of a single cell or of a very few, and just as the shape of the former has no longer a visible relation to the individual shapes of its constituent cells, while in the latter it is cell-form which dominates or is actually equivalent to the form of the organism, so shall we find it to be, with more or less exact analogy, in the case of the skeleton. For example, our own skeleton consists of bones, in the formation of each of which a vast number of minute living cellular elements are necessarily concerned; but the form and even the arrangement of these bone-forming cells or corpuscles are monotonously simple, and we cannot find in these a physical explanation of the outward and visible configuration of the bone. It is as part of a far larger field of force,—in which we must consider gravity, the action of various muscles, the compressions, tensions and bending moments due to variously distributed loads, the whole interaction of a very complex mechanical system,—that we must explain (if we are to explain at all) the configuration of a bone.
In contrast to these massive skeletons, or constituents of a skeleton, we have other skeletal elements whose whole magnitude, or whose magnitude in some dimension or another, is commensurate with the magnitude of a single living cell, or (as comes to very much the same thing) is comparable to the range of action of the molecular forces. Such is the case with the ordinary spicules of a sponge, with the delicate skeleton of a Radiolarian, or with the denser and robuster shells of the Foraminifera. The effect of scale, then, of which we had so much to say in our introductory chapter on Magnitude, is bound to be apparent in the study of skeletal fabrics, and to lead to essential differences between the big and the little, the massive and the minute, in regard to their controlling forces and their resultant forms. And if all this be so, and if the range of action of the molecular forces be in truth the important and fundamental thing, then we may somewhat extend our statement of the case, and include in it not only association with the living cellular elements of the body, but also association with any bubbles, drops, vacuoles or vesicles which [p. 439] may be comprised within the bounds of the organism, and which are (as their names and characters connote) of the order of magnitude of which we are speaking.
Proceeding a little farther in our classification, we may conceive each little skeletal element to be associated, in one case, with a single cell or vesicle, and in another with a cluster or “system” of consociated cells. In either case there are various possibilities. For instance, the calcified or other skeletal material may tend to overspread the entire outer surface of the cell or cluster of cells, and so tend accordingly to assume some configuration comparable to that of a fluid drop or of an aggregation of drops; this, in brief, is the gist and essence of our story of the foraminiferal shell. Another common, but very different condition will arise if, in the case of the cell-aggregates, the skeletal material tends to accumulate in the interstices between the cells, in the partition-walls which separate them, or in the still more restricted distribution indicated by the lines of junction between these partition-walls. Conditions such as these will go a very long way to help us in our understanding of many sponge-spicules and of an immense variety of radiolarian skeletons. And lastly (for the present), there is a possible and very interesting case of a skeletal element associated with the surface of a cell, not so as to cover it like a shell, but only so as to pursue a course of its own within it, and subject to the restraints imposed by such confinement to a curved and limited surface. With this curious condition we shall deal immediately.
This preliminary and much simplified classification of skeletal forms (as is evident enough) does not pretend to completeness. It leaves out of account some kinds of conformation and configuration with which we shall attempt to deal, and others which we must perforce omit. But nevertheless it may help to clear or to mark our way towards the subjects which this chapter has to consider, and the conditions by which they are at least partially defined.
Among the several possible, or conceivable, types of microscopic skeletons let us choose, to begin with, the case of a spicule, more or less simply linear as far as its intrinsic powers of growth are [p. 440] concerned, but which owes its now somewhat complicated form to a restraint imposed by the individual cell to which it is confined, and within whose bounds it is generated. The conception of a spicule developed under such conditions we owe to a distinguished physicist, the late Professor G. F. FitzGerald.
Many years ago, Sollas pointed out that if a spicule begin to grow in some particular way, presumably under the control or constraint imposed by the organism, it continues to grow by further chemical deposition in the same form or direction even after it has got beyond the boundaries of the organism or its cells. This phenomenon is what we see in, and this imperfect explanation goes so far to account for, the continued growth in straight lines of the long calcareous spines of Globigerina or Hastigerina, or the similarly radiating but siliceous spicules of many Radiolaria. In physical language, if our crystalline structure has once begun to be laid down in a definite orientation, further additions tend to accrue in a like regular fashion and in an identical direction; and this corresponds to the phenomenon of so-called “orientirte Adsorption,” as described by Lehmann.
In Globigerina or in Acanthocystis the long needles grow out freely into the surrounding medium, with nothing to impede their rectilinear growth and their approximately radiate distribution. But let us consider some simple cases to illustrate the forms which a spicule will tend to assume when, striving (as it were) to grow straight, it comes under the influence of some simple and constant restraint or compulsion.
If we take any two points on some curved surface, such as that of a sphere or an ellipsoid, and imagine a string stretched between them, we obtain what is known in mathematics as a “geodetic” curve. It is the shortest line which can be traced between the two points, upon the surface itself; and the most familiar of all cases, from which the name is derived, is that curve upon the earth’s surface which the navigator learns to follow in the practice of “great-circle sailing.” Where the surface is spherical, the geodetic is always literally a “great circle,” a circle, that is to say, whose centre is the centre of the sphere. If instead of a sphere we be dealing with an ellipsoid, the geodetic becomes a variable figure, according to the position of our two points. [p. 441] For obviously, if they lie in a line perpendicular to the long axis of the ellipsoid, the geodetic which connects them is a circle, also perpendicular to that axis; and if they lie in a line parallel to the axis, their geodetic is a portion of that ellipse about which the whole figure is a solid of revolution. But if our two points lie, relatively to one another, in any other direction, then their geodetic is part of a spiral curve in space, winding over the surface of the ellipsoid.
To say, as we have done, that the geodetic is the shortest line between two points upon the surface, is as much as to say that it is a projection of some particular straight line upon the surface in question; and it follows that, if any linear body be confined to that surface, while retaining a tendency to grow by successive increments always (save only for its confinement to that surface) in a straight line, the resultant form which it will assume will be that of a geodetic. In mathematical language, it is a property of a geodetic that the plane of any two consecutive elements is a plane perpendicular to that in which the geodetic lies; or, in simpler words, any two consecutive elements lie in a straight line in the plane of the surface, and only diverge from a straight line in space by the actual curvature of the surface to which they are restrained.
1917 Source Illustration
Fig. 211. Sponge and Holothurian spicules.
1917 Source Illustration
Fig. 212.
1917 Source Illustration
Fig. 213. An “amphidisc” of Hyalonema.
Let us now imagine a spicule, whose natural tendency is to grow into a straight linear element, either by reason of its own molecular anisotropy, or because it is deposited about a thread-like axis; and let us suppose that it is confined either within a cell-wall or in adhesion thereto; it at once follows that its line of growth will be simply a geodetic to the surface of the cell. And if the cell be an imperfect sphere, or a more or less regular ellipsoid, the spicule will tend to grow into one or other of three forms: either a plane curve of circular arc; or, more commonly, a plane curve which is a portion of an ellipse; or, most commonly of all, a curve which is a portion of a spiral in space. In the latter case, the number of turns of the spiral will depend, not only on the length of the spicule, but on the relative dimensions of the ellipsoidal cell, as well as upon the angle by which the spicule is inclined to the ellipsoid axes; but a very common case will probably be that in which the spicule looks at first sight to be [p. 442] a plane C-shaped figure, but is discovered, on more careful inspection, to lie not in one plane but in a more complicated spiral twist. This investigation includes a series of forms which are abundantly represented among actual sponge-spicules, as illustrated in Figs. 211 and 212. If the spicule be not restricted to linear growth, but have a tendency to expand, or to branch out from a main axis, we shall obtain a series of more complex figures, all related to the geodetic system of curves. A very simple case will arise where the spicule occupies, in the first instance, the axis of the containing cell, and then, on reaching its boundary, tends to branch or spread outwards. We shall now get various figures, in some of which the spicule will appear as an axis expanding into a disc or wheel at either end; and in other cases, the terminal disc will be replaced, or represented, by a series of rays or spokes, with a reflex curvature, corresponding to the spherical or ellipsoid curvature of the surface of the cell. Such spicules as these are again exceedingly common among various sponges (Fig. 213).
Furthermore, if these mechanical methods of conformation, and others like to these, be the true cause of the shapes which the spicules assume, it is plain that the production of these spicular shapes is not a specific function of sponges or of any particular sponge, but that we should expect [p. 443] the same or very similar phenomena to occur in other organisms, wherever the conditions of inorganic secretion within closed cells was very much the same. As a matter of fact, in the group of Holothuroidea, where the formation of intracellular spicules is a characteristic feature of the group, all the principal types of conformation which we have just described can be closely paralleled. Indeed in many cases, the forms of the Holothurian spicules are identical and indistinguishable from those of the sponges [195]. But the Holothurian spicules are composed of calcium carbonate while those which we have just described in the case of sponges are usually, if not always, siliceous: this being just another proof of the fact that in such cases the form of the spicule is not due to its chemical nature or molecular structure, but to the external forces to which, during its growth, the spicule is submitted.
So much for that comparatively limited class of sponge-spicules whose forms seem capable of explanation on the hypothesis that they are developed within, or under the restraint imposed by, the surface of a cell or vesicle. Such spicules are usually of small size, as well as of comparatively simple form; and they are greatly outstripped in number, in size, and in supposed importance as guides to zoological classification, by another class of spicules. This new class includes such as we have supposed to be capable of explanation on the assumption that they develop in association (of some sort or another) with the lines of junction of contiguous cells. They include the triradiate spicules of the calcareous sponges, the quadriradiate or “tetractinellid” spicules which occur in the same group, but more characteristically in certain siliceous sponges known as the Tetractinellidae, and lastly perhaps (though these last are admittedly somewhat harder to understand) the six-rayed spicules of the Hexactinellids.
The spicules of the calcareous sponges are commonly triradiate, and the three radii are usually inclined to one another at equal, or nearly equal angles; in certain cases, two of the three rays are nearly in a straight line, and at right angles to the [p. 444] third [196]. They are seldom in a plane, but are usually inclined to one another in a solid, trihedral angle, not easy of precise measurement under the microscope. The three rays are very often supplemented by a fourth, which is set tetrahedrally, making, that is to say, coequal angles with the other three. The calcareous spicule consists mainly of carbonate of lime, in the form of calcite, with (according to von Ebner) some admixture of soda and magnesia, of sulphates and of water. According to the same writer (but the fact, though it would seem easy to test, is still disputed) there is no organic matter in the spicule, either in the form of an axial filament or otherwise, and the appearance of stratification, often simulating the presence of an axial fibre, is due to “mixed crystallisation” of the various constituents. The spicule is a true crystal, and therefore its existence and its form are primarily due to the molecular forces of crystallisation; moreover it is a single crystal and not a group of crystals, as is at once seen by its behaviour in polarised light. But its axes are not crystalline axes, and its form neither agrees with, nor in any way resembles, any one of the many polymorphic forms in which calcite is capable of crystallising. It is as though it were carved out of a solid crystal; it is, in fact, a crystal under restraint, a crystal growing, as it were, in an artificial mould; and this mould is constituted by the surrounding cells, or structural vesicles of the sponge.
1917 Source Illustration
Fig. 214. Spicules of Grantia and other calcareous sponges. (After Haeckel.)
We have already studied in an elementary way, but amply for our present purpose, the manner in which three or more cells, or bubbles, tend to meet together under the influence of surface-tension, and also the outwardly similar phenomena which may be brought about by a uniform distribution of mechanical pressure. We have seen that when we confine ourselves to a plane assemblage of such bodies, we find them meeting one another in threes; that in a section or plane projection of such an assemblage we see the partition-walls meeting one another at equal angles of 120°; that when the bodies are uniform in size, the partitions are straight lines, which combine to form regular hexagons; and that when [p. 445] the bodies are unequal in size, the partitions are curved, and combine to form other and less regular polygons. It is plain, accordingly, that in any flattened or stratified assemblage of such cells, a solidified skeletal deposit which originates or accumulates either between the cells or within the thickness of their mutual partitions, will tend to take the form of triradiate bodies, whose rays (in a typical case) will be set at equal angles of 120° (Fig. 214, F). And this latter condition of equality will be open to modification in various ways. It will be modified by any inequality in the specific tensions of adjacent cells; as a special case, it will be apt to be greatly modified at the surface of the system, where a spicule happens to be formed in a plane perpendicular to the cell-layer, so that one of its three rays lies between two adjacent cells and the other two are associated with the surface of contact between the cells and the surrounding medium; in such a case (as in the cases considered in connection with the forms of the cells themselves [p. 446] on p. 314), we shall tend to obtain a spicule with two equal angles and one unequal (Fig. 214, A, C). In the last case, the two outer, or superficial rays, will tend to be markedly curved. Again, the equiangular condition will be departed from, and more or less curvature will be imparted to the rays, wherever the cells of the system cease to be uniform in size, and when the hexagonal symmetry of the system is lost accordingly. Lastly, although we speak of the rays as meeting at certain definite angles, this statement applies to their axes, rather than to the rays themselves. For, if the triradiate spicule be developed in the interspace between three juxtaposed cells, it is obvious that its sides will tend to be concave, for the interspace between our three contiguous equal circles is an equilateral, curvilinear triangle; and even if our spicule be deposited, not in the space between our three cells, but in the thickness of the intervening wall, then we may recollect (from p. 297) that the several partitions never actually meet at sharp angles, but the angle of contact is always bridged over by a small accumulation of material (varying in amount according to its fluidity) whose boundary takes the form of a circular arc, and which constitutes the “bourrelet” of Plateau.
In any sample of the triradiate spicules of Grantia, or in any series of careful drawings, such as those of Haeckel among others, we shall find that all these various configurations are precisely and completely illustrated.
The tetrahedral, or rather tetractinellid, spicule needs no explanation in detail (Fig. 214, D, E). For just as a triradiate spicule corresponds to the case of three cells in mutual contact, so does the four-rayed spicule to that of a solid aggregate of four cells: these latter tending to meet one another in a tetrahedral system, shewing four edges, at each of which four surfaces meet, the edges being inclined to one another at equal angles of about 109°. And even in the case of a single layer, or superficial layer, of cells, if the skeleton originate in connection with all the edges of mutual contact, we shall, in complete and typical cases, have a four-rayed spicule, of which one straight limb will correspond to the line of junction between the three cells, and the other three limbs (which will then be curved limbs) will correspond to the edges where two cells meet one another on the surface of the system. [p. 447]
But if such a physical explanation of the forms of our spicules is to be accepted, we must seek at once for some physical agency by which we may explain the presence of the solid material just at the junctions or interfaces of the cells, and for the forces by which it is confined to, and moulded to the form of, these intercellular or interfacial contacts. It is to Dreyer that we chiefly owe the physical or mechanical theory of spicular conformation which I have just described,—a theory which ultimately rests on the form assumed, under surface-tension, by an aggregation of cells or vesicles. But this fundamental point being granted, we have still several possible alternatives by which to explain the details of the phenomenon.
Dreyer, if I understand him aright, was content to assume that the solid material, secreted or excreted by the organism, accumulated in the interstices between the cells, and was there subjected to mechanical pressure or constraint as the cells got more and more crowded together by their own growth and that of the system generally. As far as the general form of the spicules goes, such explanation is not inadequate, though under it we may have to renounce some of our assumptions as to what takes place at the outer surface of the system.
But in all (or most) cases where, but a few years ago, the concepts of secretion or excretion seemed precise enough, we are now-a-days inclined to turn to the phenomenon of adsorption as a further stage towards the elucidation of our facts. Here we have a case in point. In the tissues of our sponge, wherever two cells meet, there we have a definite surface of contact, and there accordingly we have a manifestation of surface-energy; and the concentration of surface-energy will tend to be a maximum at the lines or edges whereby the three, or four, such surfaces are conjoined. Of the micro-chemistry of the sponge-cells our ignorance is great; but (without venturing on any hypothesis involving the chemical details of the process) we may safely assert that there is an inherent probability that certain substances will tend to be concentrated and ultimately deposited just in these lines of intercellular contact and conjunction. In other words, adsorptive concentration, under osmotic pressure, at and in the surface-film which constitutes the mutual boundary between contiguous [p. 448] cells, emerges as an alternative (and, as it seems to me, a highly preferable alternative) to Dreyer’s conception of an accumulation under mechanical pressure in the vacant spaces left between one cell and another.
But a purely chemical, or purely molecular adsorption, is not the only form of the hypothesis on which we may rely. For from the purely physical point of view, angles and edges of contact between adjacent cells will be loci in the field of distribution of surface-energy, and any material particles whatsoever will tend to undergo a diminution of freedom on entering one of those boundary regions. In a very simple case, let us imagine a couple of soap bubbles in contact with one another. Over the surface of each bubble there glide in every direction, as usual, a multitude of tiny bubbles and droplets; but as soon as these find their way into the groove or re-entrant angle between the two bubbles, there their freedom of movement is so far restrained, and out of that groove they have little or no tendency to emerge. A cognate phenomenon is to be witnessed in microscopic sections of steel or other metals. Here, amid the “crystalline” structure of the metal (where in cooling its imperfectly homogeneous material has developed a cellular structure, shewing (in section) hexagonal or polygonal contours), we can easily observe, as Professor Peddie has shewn me, that the little particles of graphite and other foreign bodies common in the matrix, have tended to aggregate themselves in the walls and at the angles of the polygonal cells—this being a direct result of the diminished freedom which the particles undergo on entering one of these boundary regions [197].
It is by a combination of these two principles, chemical adsorption on the one hand, and physical quasi-adsorption or concentration of grosser particles on the other, that I conceive the substance of the sponge-spicule to be concentrated and aggregated at the cell boundaries; and the forms of the triradiate and tetractinellid spicules are in precise conformity with this hypothesis. A few general matters, and a few particular cases, remain to be considered.
It matters little or not at all, for the phenomenon in question, [p. 449] what is the histological nature or “grade” of the vesicular structures on which it depends. In some cases (apart from sponges), they may be no more than the little alveoli of the intracellular protoplasmic network, and this would seem to be the case at least in one known case, that of the protozoan Entosolenia aspera, in which, within the vesicular protoplasm of the single cell, Möbius has described tiny spicules in the shape of little tetrahedra with concave sides. It is probably also the case in the small beginnings of the Echinoderm spicules, which are likewise intracellular, and are of similar shape. In the case of our sponges we have many varying conditions, which we need not attempt to examine in detail. In some cases there is evidence for believing that the spicule is formed at the boundaries of true cells or histological units. But in the case of the larger triradiate or tetractinellid spicules of the sponge-body, they far surpass in size the actual “cells”; we find them lying, regularly and symmetrically arranged, between the “pore-canals” or “ciliated chambers,” and it is in conformity with the shape and arrangement of these rounded or spheroidal structures that their shape is assumed.
Again, it is not necessarily at variance with our hypothesis to find that, in the adult sponge, the larger spicules may greatly outgrow the bounds not only of actual cells but also of the ciliated chambers, and may even appear to project freely from the surface of the sponge. For we have already seen that the spicule is capable of growing, without marked change of form, by further deposition, or crystallisation, of layer upon layer of calcareous molecules, even in an artificial solution; and we are entitled to believe that the same process may be carried on in the tissues of the sponge, without greatly altering the symmetry of the spicule, long after it has established its characteristic form of a system of slender trihedral or tetrahedral rays.
Neither is it of great importance to our hypothesis whether the rayed spicule necessarily arises as a single structure, or does so from separate minute centres of aggregation. Minchin has shewn that, in some cases at least, the latter is the case; the spicule begins, he tells us, as three tiny rods, separate from one another, each developed in the interspace between two sister-cells, which are themselves the results of the division of one of a [p. 450] little trio of cells; and the little rods meet and fuse together while still very minute, when the whole spicule is only about 1/200 of a millimetre long. At this stage, it is interesting to learn that the spicule is non-crystalline; but the new accretions of calcareous matter are soon deposited in crystalline form.
This observation threw considerable difficulties in the way of former mechanical theories of the conformation of the spicule, and was quite at variance with Dreyer’s theory, according to which the spicule was bound to begin from a central nucleus coinciding with the meeting-place of the three contiguous cells, or rather the interspace between them. But the difficulty is removed when we import the concept of adsorption; for by this agency it is natural enough, or conceivable enough, that the process of deposition should go on at separate parts of a common system of surfaces; and if the cells tend to meet one another by their interfaces before these interfaces extend to the angles and so complete the polygonal cell, it is again conceivable and natural that the spicule should first arise in the form of separate and detached limbs or rays.
1917 Source Illustration
Fig. 215. Spicules of tetractinellid sponges (after Sollas). a–e, anatriaenes; d–f, protriaenes.
Among the tetractinellid sponges, whose spicules are composed of amorphous silica or opal, all or most of the above-described main types of spicule occur, and, as the name of the group implies, the four-rayed, tetrahedral spicules are especially represented. A somewhat frequent type of spicule is one in which one of the four rays is greatly developed, and the other three constitute small prongs diverging at equal angles from the main or axial ray. In all probability, as Dreyer suggests, we have here had to do with a group of four vesicles, of which three were large and co-equal, while a fourth and very much smaller one lay above and between the other three. In certain cases where we have likewise one large and three much smaller [p. 451] rays, the latter are recurved, as in Fig. 215. This type, save for the constancy of the number of rays, and the limitation of the terminal ones to three, and save also for the more important difference that they occur only at one and not at both ends of the long axis, is similar to the type of spicule illustrated in Fig. 213, which we have explained as being probably developed within an oval cell, by whose walls its branches have been conformed to geodetic curves. But it is much more probable that we have here to do with a spicule developed in the midst of a group of three coequal and more or less elongated or cylindrical cells or vesicles, the long axial ray corresponding to their common line of contact, and the three short rays having each lain in the surface furrow between two out of the three adjacent cells.
1917 Source Illustration
Fig. 216. Various holothurian spicules. (After Théel.)
Just as in the case of the little curved or S-shaped spicules, formed apparently within the bounds of a single cell, so also in the case of the larger tetractinellid and analogous types do we find among the Holothuroidea the same configurations reproduced as we have dealt with in the sponges. The holothurian spicules are a little less neatly formed, a little rougher, than the sponge-spicules; and certain forms occur among the former group which do not present themselves among the latter; but for the most part a community of type is obvious and striking (Fig. 216).
A curious and, physically speaking, strictly analogous formation to the tetrahedral spicules of the sponges is found in the [p. 452] spores of a certain little group of parasitic protozoa, the Actinomyxidia. These spores are formed from clusters of six cells, of which three come to constitute the capsule of the spore; and this capsule, always triradiate in its symmetry, is in some species drawn out into long rays, of which one constitutes a straight central axis, while the others, coming off from it at equal angles, are recurved in wide circular arcs. The account given of the development of this structure by its discoverers [198] is somewhat obscure to me, but I think that, on physical grounds, there can be no doubt whatever that the quadriradiate capsule has been somehow modelled upon a group of three surrounding cells, its axis lying between the three, and its three radial arcs occupying the furrows between adjacent pairs.
1917 Source Illustration
Fig. 217. Spicules of hexactinellid sponges. (After F. E. Schultze.)
The typically six-rayed siliceous spicules of the hexactinellid sponges, while they are perhaps the most regular and beautifully formed spicules to be found within the entire group, have been found very difficult to explain, and Dreyer has confessed his complete inability to account for their conformation. But, though it is doubtless only throwing the difficulty a little further back, we may so far account for them by considering that the cells or vesicles by which they are conformed are not arranged in [p. 453] what is known as “closest packing,” but in linear series; so that in their arrangement, and by their mutual compression, we tend to get a pattern, not of hexagons, but of squares: or, looking to the solid, not of dodecahedra but of cubes or parallelopipeda. This indeed appears to be the case, not with the individual cells (in the histological sense), but with the larger units or vesicles which make up the body of the hexactinellid. And this being so, the spicules formed between the linear, or cubical series of vesicles, will have the same tendency towards a “hexactinellid” shape, corresponding to the angles and adjacent edges of a system of cubes, as in our former case they had to a triradiate or a tetractinellid form, when developed in connection with the angles and edges of a system of hexagons, or a system of dodecahedra.
Histologically, the case is illustrated by a well-known phenomenon in embryology. In the segmenting ovum, there is a tendency for the cells to be budded off in linear series; and so they often remain, in rows side by side, at least for a considerable time and during the course of several consecutive cell divisions. Such an arrangement constitutes what the embryologists call the “radial type” of segmentation [199]. But in what is described as the “spiral type” of segmentation, it is stated that, as soon as the first horizontal furrow has divided the cells into an upper and a lower layer, those of “the upper layer are shifted in respect to the lower layer, by means of a rotation about the vertical axis [200].” It is, of course, evident that the whole process is merely that which is familiar to physicists as “close packing.” It is a very simple case of what Lord Kelvin used to call “a problem in tactics.” It is a mere question of the rigidity of the system, of the freedom of movement on the part of its constituent cells, whether or at what stage this tendency to slip into the closest propinquity, or position of minimum potential, will be found to manifest itself.
However the hexactinellid spicules be arranged (and this is [p. 454] not at all easy to determine) in relation to the tissues and chambers of the sponge, it is at least clear that, whether they be separate or be fused together (as often happens) in a composite skeleton, they effect a symmetrical partitioning of space according to the cubical system, in contrast to that closer packing which is represented and effected by the tetrahedral system [201].
This question of the origin and causation of the forms of sponge-spicules, with which we have now briefly dealt, is all the more important and all the more interesting because it has been discussed time and again, from points of view which are characteristic of very different schools of thought in biology. Haeckel found in the form of the sponge-spicule a typical illustration of his theory of “bio-crystallisation”; he considered that these “biocrystals” represented “something midway—ein Mittelding—between an inorganic crystal and an organic secretion”; that there was a “compromise between the crystallising efforts of the calcium carbonate and the formative activity of the fused cells of the syncytium”; and that the semi-crystalline secretions of calcium carbonate “were utilised by natural selection as ‘spicules’ for building up a skeleton, and afterwards, by the interaction of adaptation and heredity, became modified in form and differentiated in a vast variety of ways in the struggle for existence [202].” What Haeckel precisely signified by these words is not clear to me.
Schultze, perceiving that identical forms of spicule were developed whether the material were crystalline or non-crystalline, abandoned all theories based upon crystallisation; he simply saw in the form and arrangement of the spicules something which was “best fitted” for its purpose, that is to say for the support and strengthening of the porous walls of the sponge, and found clear evidence of “utility” in the specific structure of these skeletal elements. [p. 455]
Sollas and Dreyer, as we have seen, introduced in various ways the conception of physical causation,—as indeed Haeckel himself had done in regard to one particular, when he supposed the position of the spicules to be due to the constant passage of the water-currents. Though even here, by the way, if I understand Haeckel aright, he was thinking not merely of a direct or immediate physical causation, but of one manifesting itself through the agency of natural selection [203]. Sollas laid stress upon the “path of least resistance” as determining the direction of growth; while Dreyer dealt in greater detail with the various tensions and pressures to which the growing spicule was exposed, amid the alveolar or vesicular structure which was represented alike by the chambers of the sponge, by the reticulum of constituent cells, or by the minute structure of the intracellular protoplasm. But neither of these writers, so far as I can discover, was inclined to doubt for a moment the received canon of biology, which sees in such structures as these the characteristics of true organic species, and the indications of an hereditary affinity by which blood-relationship and the succession of evolutionary descent throughout geologic time can be ultimately deduced.
Lastly, Minchin, in a well-known paper [204], took sides with Schultze, and gave reasons for dissenting from such mechanical theories as those of Sollas and of Dreyer. For example, after pointing out that all protoplasm contains a number of “granules” or microsomes, contained in the alveolar framework and lodged at the nodes of the reticulum, he argued that these also ought to acquire a form such as the spicules possess, if it were the case that these latter owed their form to their very similar or identical position. “If vesicular tension cannot in any other instance cause the granules at the nodes to assume a tetraxon form, why should it do so for the sclerites?” In all probability the answer to this question is not far to seek. If the force which the “mechanical” hypothesis has in view were simply that of mechanical pressure, [p. 456] as between solid bodies, then indeed we should expect that any substances whatsoever, lying between the impinging spheres, would tend (unless they were infinitely hard) to assume the quadriradiate or “tetraxon” form; but this conclusion does not follow at all, in so far as it is to surface-energy that we ascribe the phenomenon. Here the specific nature of the substances involved makes all the difference. We cannot argue from one substance to another; adsorptive attraction shews its effect on one and not on another; and we have not the least reason to be surprised if we find that the little granules of protoplasmic material, which as they lie bathed in the more fluid protoplasm have (presumably, and as their shape indicates) a strong surface-tension of their own, behave towards the adjacent vesicles in a very different fashion to the incipient aggregations of calcareous or siliceous matter in a colloid medium. “The ontogeny of the spicules,” says Professor Minchin, “points clearly to their regular form being a phylogenetic adaptation, which has become fixed and handed on by heredity, appearing in the ontogeny as a prophetic adaptation.” And again, “The forms of the spicules are the result of adaptation to the requirements of the sponge as a whole, produced by the action of natural selection upon variation in every direction.” It would scarcely be possible to illustrate more briefly and more cogently than by these few words (or the similar words of Haeckel quoted on p. 454), the fundamental difference between the Darwinian conception of the causation and determination of Form, and that which is characteristic of the physical sciences.
If I have dealt comparatively briefly with the inorganic skeleton of sponges, in spite of the obvious importance of this part of our subject from the physical or mechanical point of view, it has been owing to several reasons. In the first place, though the general trend of the phenomena is clear, it must be at once admitted that many points are obscure, and could only be discussed at the cost of a long argument. In the second place, the physical theory is (as I have shewn) in manifest conflict with the accounts given by various embryologists of the development of the spicules, and of the current biological theories which their descriptions embody; it is beyond our scope to deal with such descriptions [p. 457] in detail. Lastly, we find ourselves able to illustrate the same physical principles with greater clearness and greater certitude in another group of animals, namely the Radiolaria. In our description of the skeletons occurring within this group we shall by no means abandon the preliminary classification of microscopic skeletons which we have laid down; but we shall have occasion to blend with it the consideration of certain other more or less correlated phenomena.
The group of microscopic organisms known as the Radiolaria is extraordinarily rich in diverse forms, or “species.” I do not know how many of such species have been described and defined by naturalists, but some thirty years ago the number was said to be over four thousand, arranged in more than seven hundred genera [205]. Of late years there has been a tendency to reduce the number, it being found that some of the earlier species and even genera are but growth-stages of one and the same form, sometimes mere fragments or “fission-products” common to several species, or sometimes forms so similar and so interconnected by intermediate forms that the naturalist denominates them not “species” but “varieties.” It has to be admitted, in short, that the conception of species among the Radiolaria has not hitherto been, and is not yet, on the same footing as that among most other groups of animals. But apart from the extraordinary multiplicity of forms among the Radiolaria, there are certain other features in this multiplicity which arrest our attention. For instance, the distribution of species in space is curious and vague; many species are found all over the world, or at least every here and there, with no evidence of specific limitations of geographical habitat; others occur in the neighbourhood of the two poles; some are confined to warm and others to cold currents of the ocean. In time also their distribution is not less vague: so much so that it has been asserted of them that “from the Cambrian age downwards, the families and even genera appear identical with those now living.” Lastly, except perhaps in the case of a few large “colonial forms,” we seldom if ever find, as is usual [p. 458] in most animals, a local predominance of one particular species. On the contrary, in a little pinch of deep-sea mud or of some fossil “Radiolarian earth,” we shall probably find scores, and it may be even hundreds, of different forms. Moreover, the radiolarian skeletons are of quite extraordinary delicacy and complexity, in spite of their minuteness and the comparative simplicity of the “unicellular” organisms within which they grow; and these complex conformations have a wonderful and unusual appearance of geometric regularity. All these general considerations seem such as to prepare us for the special need of some physical hypothesis of causation. The little skeletal fabrics remind us of such objects as snow-crystals (themselves almost endless in their diversity), rather than of a collection of distinct animals, constructed in apparent accordance with functional needs, and distributed in accordance with their fitness for particular situations. Nevertheless great efforts have been made of recent years to attach “a biological meaning” to these elaborate structures; and “to justify the hope that in time the utilitarian character [of the skeleton] will be more completely recognised [206].”
In the majority of cases, the skeleton of the Radiolaria is composed, like that of so many sponges, of silica; in one large family, the Acantharia (and perhaps in some others), it is composed, in great part at least, of a very unusual constituent, namely strontium sulphate [207]. There is no fundamental or important morphological character in which the shells formed of these two constituents differ from one another; and in no case can the chemical properties of these inorganic materials be said to influence the form of the complex skeleton or shell, save only in this general way that, by their rigidity and toughness, they may give rise to a fabric far more delicate and slender than we find developed among calcareous organisms.
A slight exception to this rule is found in the presence of true crystals, which occur within the central capsules of certain [p. 459] Radiolaria, for instance the genus Collosphaera [208]. Johannes Müller (whose knowledge and insight never fail to astonish us) remarked that these were identical in form with crystals of celestine, a sulphate of strontium and barium; and Bütschli’s discovery of sulphates of strontium and of barium in kindred forms render it all but certain that they are actually true crystals of celestine [209].
In its typical form, the Radiolarian body consists of a spherical mass of protoplasm, around which, and separated from it by some sort of porous “capsule,” lies a frothy mass, composed of protoplasm honeycombed into a multitude of alveoli or vacuoles, filled with a fluid which can scarcely differ much from sea-water [210]. According to their surface-tension conditions, these vacuoles may appear more or less isolated and spherical, or joining together in a “froth” of polygonal cells; and in the latter, which is the commoner condition, the cells tend to be of equal size, and the resulting polygonal meshwork beautifully regular. In many cases, a large number of such simple individual organisms are associated together, forming a floating colony, and it is highly probable that many other forms, with whose scattered skeletons we are alone acquainted, had in life formed part likewise of a colonial organism.
In contradistinction to the sponges, in which the skeleton always begins as a loose mass of isolated spicules, which only in a few exceptional cases (such as Euplectella and Farrea) fuse into a continuous network, the characteristic feature of the Radiolarians lies in the possession of a continuous skeleton, in the form of a netted mesh or perforated lacework, sometimes however replaced by and often associated with minute independent spicules. Before we proceed to treat of the more complex skeletons, we may begin, then, by dealing with these comparatively simple cases where either the entire skeleton or a considerable part of it is represented, not by a continuous fabric, but by a quantity of loose, separate spicules, or aciculae, which seem, like the spicules of Alcyonium, [p. 460] to be developed as free and isolated formations or deposits, precipitated in the colloid matrix, with no relation of form to the cellular or vesicular boundaries. These simple acicular spicules occupy a definite position in the organism. Sometimes, as for instance among the fresh-water Heliozoa (e.g. Raphidiophrys), they lie on the outer surface of the organism, and not infrequently (when the spicules are few in number) they tend to collect round the bases of the pseudopodia, or around the large radiating spicules, or axial rays, in the cases where these latter are present. When the spicules are thus localised around some prominent centre, they tend to take up a position of symmetry in regard to it; instead of forming a tangled or felted layer, they come to lie side by side, in a radiating cluster round the focus. In other cases (as for instance in the well-known Radiolarian Aulacantha scolymantha) the felted layer of aciculae lies at some depth below the surface, forming a sphere concentric with the entire spherical organism. In either case, whether the layer of spicules be deep or be superficial, it tends to mark a “surface of discontinuity,” a meeting place between two distinct layers of protoplasm or between the protoplasm and the water around; and it is obvious that, in either case, there are manifestations of surface-energy at the boundary, which cause the spicules to be retained there, and to take up their position in its plane. The case is somewhat, though not directly, analogous to that of a cirrus cloud, which marks the place of a surface of discontinuity in a stratified atmosphere.
1917 Source Illustration
Fig. 218.
We have, then, to enquire what are the conditions which shall, apart from gravity, confine an extraneous body to a surface-film; and we may do this very simply, by considering the surface-energy of the entire system. In Fig. 218 we have two fluids in contact with one another (let us call them water and protoplasm), and a body (b) which may be immersed in either, or may be restricted to the boundary [p. 461] between. We have here three possible “interfacial contacts” each with its own specific surface-energy, per unit of surface area: namely, that between our particle and the water (let us call it α), that between the particle and the protoplasm (β), and that between water and protoplasm (γ). When the body lies in the boundary of the two fluids, let us say half in one and half in the other, the surface-energies concerned are equivalent to (S/2)α + (S/2)β; but we must also remember that, by the presence of the particle, a small portion (equal to its sectional area s) of the original contact-surface between water and protoplasm has been obliterated, and with it a proportionate quantity of energy, equivalent to sγ, has been set free. When, on the other hand, the body lies entirely within one or other fluid, the surface-energies of the system (so far as we are concerned) are equivalent to Sα + sγ, or Sβ + sγ, as the case may be. According as α be less or greater than β, the particle will have a tendency to remain immersed in the water or in the protoplasm; but if (S/2)(α + β) − sγ be less than either Sα or Sβ, then the condition of minimal potential will be found when the particle lies, as we have said, in the boundary zone, half in one fluid and half in the other; and, if we were to attempt a more general solution of the problem, we should evidently have to deal with possible conditions of equilibrium under which the necessary balance of energies would be attained by the particle rising or sinking in the boundary zone, so as to adjust the relative magnitudes of the surface-areas concerned. It is obvious that this principle may, in certain cases, help us to explain the position even of a radial spicule, which is just a case where the surface of the solid spicule is distributed between the fluids with a minimal disturbance, or minimal replacement, of the original surface of contact between the one fluid and the other.
In like manner we may provide for the case (a common and an important one) where the protoplasm “creeps up” the spicule, covering it with a delicate film. In Acanthocystis we have yet another special case, where the radial spicules plunge only a certain distance into the protoplasm of the cell, being arrested at a boundary-surface between an inner and an outer layer of cytoplasm; here we have only to assume that there is a tension [p. 462] at this surface, between the two layers of protoplasm, sufficient to balance the tensions which act directly on the spicule [211].
In various Acanthometridae, besides such typical characters as the radial symmetry, the concentric layers of protoplasm, and the capillary surfaces in which the outer, vacuolated protoplasm is festooned upon the projecting radii, we have another curious feature. On the surface of the protoplasm where it creeps up the sides of the long radial spicules, we find a number of elongated bodies, forming in each case one or several little groups, and lying neatly arranged in parallel bundles. A Russian naturalist, Schewiakoff, whose views have been accepted in the text-books, tells us that these are muscular structures, serving to raise or lower the conical masses of protoplasm about the radial spicules, which latter serve as so many “tent-poles” or masts, on which the protoplasmic membranes are hoisted up; and the little elongated bodies are dignified with various names, such as “myonemes” or “myophriscs,” in allusion to their supposed muscular nature [212]. This explanation is by no means convincing. To begin with, we have precisely similar festoons of protoplasm in a multitude of other cases where the “myonemes” are lacking; from their minute size (·006–·012 mm.) and the amount of contraction they are said to be capable of, the myonemes can hardly be very efficient instruments of traction; and further, for them to act (as is alleged) for a specific purpose, namely the “hydrostatic regulation” of the organism giving it power to sink or to swim, would seem to imply a mechanism of action and of coordination which is difficult to conceive in these minute and simple organisms. The fact is (as it seems to me), that the whole method of explanation is unnecessary. Just as the supposed “hauling up” of the protoplasmic festoons is at once explained by capillary phenomena, so also, in all probability, is the position and arrangement of the little elongated bodies. Whatever the actual nature of these bodies may be, whether they are truly portions of differentiated protoplasm, or whether they are foreign bodies or spicular structures (as bodies occupying a similar position in other cases undoubtedly are), we can explain their situation on the surface [p. 463] of the protoplasm, and their arrangement around the radial spicules, all on the principles of surface-tension [213].
This last case is not of the simplest; and I do not forget that my explanation of it, which is wholly theoretical, implies a doubt of Schewiakoff’s statements, which are founded on direct personal observation. This I am none too willing to do; but whether it be justly done in this case or not, I hold that it is in principle justifiable to look with great suspicion upon a number of kindred statements where it is obvious that the observer has left out of account the purely physical aspect of the phenomenon, and all the opportunities of simple explanation which the consideration of that aspect might afford.
Whether it be wholly applicable to this particular and complex case or no, our general theorem of the localisation and arrestment of solid particles in a surface-film is of very great biological importance; for on it depends the power displayed by many little naked protoplasmic organisms of covering themselves with an “agglutinated” shell. Sometimes, as in Difflugia, Astrorhiza (Fig. 219) and others, this covering consists of sand-grains picked up from the surrounding medium, and sometimes, on the other hand, as in Quadrula, it consists of solid particles which are said to arise, as inorganic deposits or concretions, within the protoplasm itself, and which find their way outwards to a position of equilibrium in the surface-layer; and in both cases, the mutual capillary attractions between the particles, confined to the boundary-layer but enjoying a certain measure of freedom therein, tends to the orderly arrangement of the particles one with another, and even to the appearance of a regular “pattern” as the result of this arrangement.
1917 Source Illustration
Fig. 219. Arenaceous Foraminifera; Astrorhiza limicola and arenaria. (From Brady’s Challenger Monograph.)
The “picking up” by the protoplasmic organism of a solid particle with which “to build its house” (for it is hard to avoid this customary use of anthropomorphic figures of speech, misleading though they be), is a physical phenomenon kindred to that by which an Amoeba “swallows” a particle of food. This latter process has been reproduced or imitated in various pretty experimental [p. 465] ways. For instance, Rhumbler has shewn that if a thread of glass be covered with shellac and brought near a drop of chloroform suspended in water, the drop takes in the spicule, robs it of its shellac covering, and then passes it out again [214]. It is all a question of relative surface-energies, leading to different degrees of “adhesion” between the chloroform and the glass or its covering. Thus it is that the Amoeba takes in the diatom, dissolves off its proteid covering, and casts out the shell.
Furthermore, as the whole phenomenon depends on a distribution of surface-energy, the amount of which is specific to certain particular substances in contact with one another, we have no difficulty in understanding the selective action, which is very often a conspicuous feature in the phenomenon [215]. Just as some caddis-worms make their houses of twigs, and others of shells and again others of stones, so some Rhizopods construct their agglutinated “test” out of stray sponge-spicules, or frustules of diatoms, or again of tiny mud particles or of larger grains of sand. In all these cases, we have apparently to deal with differences in specific [p. 466] surface-energies, and also doubtless with differences in the total available amount of surface-energy in relation to gravity or other extraneous forces. In my early student days, Wyville Thomson used to tell us that certain deep-sea “Difflugias,” after constructing a shell out of particles of the black volcanic sand common in parts of the North Atlantic, finished it off with “a clean white collar” of little grains of quartz. Even this phenomenon may be accounted for on surface-tension principles, if we assume that the surface-energy ratios have tended to change, either with the growth of the protoplasm or by reason of external variation of temperature or the like; and we are by no means obliged to attribute the phenomenon to a manifestation of volition, or taste, or aesthetic skill, on the part of the microscopic organism. Nor, when certain Radiolaria tend more than others to attract into their own substance diatoms and such-like foreign bodies, is it scientifically correct to speak, as some text-books do, of species “in which diatom selection has become a regular habit.” To do so is an exaggerated misuse of anthropomorphic phraseology.
The formation of an “agglutinated” shell is thus seen to be a purely physical phenomenon, and indeed a special case of a more general physical phenomenon which has many other important consequences in biology. For the shell to assume the solid and permanent character which it acquires, for instance, in Difflugia, we have only to make the additional assumption that some small quantities of a cementing substance are secreted by the animal, and that this substance flows or creeps by capillary attraction between all the interstices of the little quartz grains, and ends by binding them all firmly together. Rhumbler [216] has shewn us how these agglutinated tests, of spicules or of sand-grains, can be precisely imitated, and how they are formed with greater or less ease, and greater or less rapidity, according to the nature of the materials employed, that is to say, according to the specific surface-tensions which are involved. For instance if we mix up a little powdered glass with chloroform, and set a drop of the mixture in water, the glass particles gather neatly round the surface of the drop so quickly that the eye cannot follow the [p. 467] operation. If we perform the same experiment with oil and fine sand, dropped into 70 per cent. alcohol, a still more beautiful artificial Rhizopod shell is formed, but it takes some three hours to do.
It is curious that, just at the very time when Rhumbler was thus demonstrating the purely physical nature of the Difflugian shell, Verworn was studying the same and kindred organisms from the older standpoint of an incipient psychology [217]. But, as Rhumbler himself admits, Verworn was very careful not to overestimate the apparent signs of volition, or selective choice, in the little organism’s use of the material of its dwelling.
This long parenthesis has led us away, for the time being, from the subject of the Radiolarian skeleton, and to that subject we must now return. Leaving aside, then, the loose and scattered spicules, which we have sufficiently discussed, the more perfect Radiolarian skeletons consist of a continuous and regular structure; and the siliceous (or other inorganic) material of which this framework is composed tends to be deposited in one or other of two ways or in both combined: (1) in the form of long spicular axes, usually conjoined at, or emanating from, the centre of the protoplasmic body, and forming a symmetric radial system; (2) in the form of a crust, developed in various ways, either on the outer surface of the organism or in relation to the various internal surfaces which separate its concentric layers or its component vesicles. Not unfrequently, this superficial skeleton comes to constitute a spherical shell, or a system of concentric or otherwise associated spheres.
1917 Source Illustration
Fig. 220. “Reticulum plasmatique.” (After Carnoy.)
We have already learned that a great part of the body of the Radiolarian, and especially that outer portion to which Haeckel has given the name of the “calymma,” is built up of a great mass of “vesicles,” forming a sort of stiff froth, and equivalent in the physical sense (though not necessarily in the biological sense) to “cells,” inasmuch as the little vesicles have their own well-defined boundaries, and their own surface phenomena. In short, all that we have said of cell-surfaces, and cell conformations, in our discussion of cells and of tissues, will apply in like manner, and under appropriate conditions, to these. In certain cases, even in [p. 468] so common and simple a one as the vacuolated substance of an Actinosphaerium, we may see a very close resemblance, or formal analogy, to an ordinary cellular or “parenchymatous” tissue, in the close-packed arrangement and consequent configuration of these vesicles, and even at times in a slight membranous hardening of their walls. Leidy has figured [218] some curious little bodies, like small masses of consolidated froth, which seem to be nothing else than the dead and empty husks, or filmy skeletons, of Actinosphaerium. And Carnoy [219] has demonstrated in certain cell-nuclei an all but precisely similar framework, of extreme delicacy and minuteness, as the result of partial solidification of interstitial matter in a close-packed system of alveoli (Fig. 220).
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Fig. 221. Aulonia hexagona, Hkl.
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Fig. 222. Actinomma arcadophorum, Hkl.
Let us now suppose that, in our Radiolarian, the outer surface of the animal is covered by a layer of froth-like vesicles, uniform or nearly so in size. We know that their tensions will tend to conform them into a “honeycomb,” or regular meshwork of hexagons, and that the free end of each hexagonal prism will be a little spherical cap. Suppose now that it be at the outer surface of the protoplasm (that namely which is in contact with the surrounding sea-water), that the siliceous particles have a tendency to be secreted or adsorbed; it will at once follow that they will show a tendency to aggregate in the grooves which separate the vesicles, and the result will be the development of a most delicate sphere composed of tiny rods arranged in a regular hexagonal network (e.g. Aulonia). Such a conformation is [p. 469] extremely common, and among its many variants may be found cases in which (e.g. Actinomma), the vesicles have been less regular in size, and some in which the hexagonal meshwork has been developed not only on one outer surface, but at successive [p. 470] surfaces, producing a system of concentric spheres. If the siliceous material be not limited to the linear junctions of the cells, but spread over a portion of the outer spherical surfaces or caps, then we shall have the condition represented in Fig. 223 (Ethmosphaera), where the shell appears perforated by circular instead of hexagonal apertures, and the circular pores are set on slight spheroidal eminences; and, interconnected with such types as this, we have others in which the accumulating pellicles of skeletal matter have extended from the edges into the substance of the boundary walls and have so produced a system of films, normal to the surface of the sphere, constituting a very perfect honeycomb, as in Cenosphaera favosa and vesparia [220].
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Fig. 223. Ethmosphaera conosiphonia, Hkl.
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Fig. 224. Portions of shells of two “species” of Cenosphaera: upper figure, C. favosa, lower, C. vesparia, Hkl.
In one or two very simple forms, such as the fresh-water Clathrulina, just such a spherical perforated shell is produced out of some organic, acanthin-like substance; and in some examples of Clathrulina the chitinous lattice-work of the shell is just as [p. 471] regular and delicate, with the meshes just as beautifully hexagonal, as in the siliceous shells of the oceanic Radiolaria. This is only another proof (if proof be needed) that the peculiar conformation of these little skeletons is not due to the material of which they are composed, but to the moulding of that material upon an underlying vesicular structure.
1917 Source Illustration
Fig. 225. Aulastrum triceros, Hkl.
Let us next suppose that, upon some such lattice-work as has just been described, another and external layer of cells or vesicles is developed, and that instead of (or perhaps only in addition to) a second hexagonal lattice-work, which might develop concentrically to the first in the boundary-furrows of this new layer of cells, the siliceous matter now tends to be deposited radially, or normally to the surface of the sphere, just in the lines where the external layer of vesicles meet one another, three by three. The result will be that, when the vesicles themselves are removed, a series of radiating spicules will be revealed, directed outwards from each of the angles of the original hexagon; as is seen in Fig. 225. And it may further happen that these radiating skeletal rods are continued at their distal ends into divergent rays, forming a triple fork, and corresponding (after a fashion [p. 472] which we have already described as occurring in certain sponge-spicules) to the three superficial furrows between the adjacent cells. This last is, as it were, an intermediate stage between the simple rods and the complete formation of another concentric sphere of latticed hexagons. Another possible case is when the large and uniform vesicles of the outer protoplasm are mixed with, or replaced by, much smaller vesicles, piled on one another in more or less concentric layers; in this case the radiating rods will no longer be straight, but will be bent into a zig-zag pattern, with angles in three vertical planes, corresponding to the successive contacts of the groups of cells around the axis (Fig. 226).
1917 Source Illustration
Fig. 226.
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Fig. 227. A Nassellarian skeleton, Callimitra carolotae, Hkl.
Among a certain group called the Nassellaria, we find geometrical forms of peculiar simplicity and beauty,—such for instance as that which I have represented in Fig. 227. It is obvious at a glance that this is such a skeleton as may have been formed [p. 473] (I think we may go so far as to say must have been formed) at the interfaces of a little tetrahedral group of cells, the four equal cells of the tetrahedron being in this particular case supplemented by a little one in the centre of the system. We see, precisely as in the internal boundary-system of an artificial group of four soap-bubbles, the plane surfaces of contact, six in number; the relation to one another of each triple set of interfacial planes, meeting one another at equal angles of 120°; and finally the relation of the four lines or edges of triple contact, which tend (but for the little central vesicle) to meet at co-equal solid angles in the centre of the system, all as we have described on p. 318. In short, each triple-walled re-entrant angle of the little shell has essentially the configuration (or a part thereof) of what we have called a “Maraldi pyramid” in our account of the architecture of the honeycomb, on p. 329 [221].
There are still two or three remarkable or peculiar features in this all but mathematically perfect shell, and they are in part easy and in part they seem more difficult of interpretation.
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Fig. 228. An isolated portion of the skeleton of Dictyocha.
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Fig. 229. Dictyocha stapedia, Hkl.
We notice that the amount of solid matter deposited in the plane interfacial boundaries is greatly increased at the outer margin of each boundary wall, where it merges or coincides with the superficial furrow which separates the free, spherical surfaces of the bubbles from one another; and we may sometimes find that, along these edges, the skeleton remains complete and strong, while it shows signs of imperfect development or of breaking away over great part of the rest of the interfacial surfaces. In this there is nothing anomalous, for we have already recognised that it is at the edges or margins of the interfacial partition-walls that the manifestation of surface-energy will tend to reach its maximum. And just as we have seen that, in certain of our “multicellular” spherical Radiolarians, it is at the superficial [p. 474] edges or borders of the partitions, and here only, that skeletal formation occurs (giving rise to the netted shell with its hexagonal meshes of Fig. 221), so also at times, in the case of such little aggregates of cells or vesicles as the four-celled system of Callimitra, it may happen that about the external boundary-lines, and not in the interior boundary-planes, the whole of the skeletal matter is aggregated. In Fig. 228 we see a curious little skeletal structure or complex spicule, whose conformation is easily accounted for after this fashion. Little spicules such as this form isolated portions of the skeleton in the genus Dictyocha, and occur scattered over the spherical surface of the organism (Fig. 229). The more or less basket-shaped spicule has evidently been developed about a little cluster of four cells or vesicles, lying in or on the plane of the surface of the organism, and therefore arranged, not in the tetrahedral form of Callimitra, but in the manner in which four contiguous cells lying side by side normally set themselves, like the four cells of a segmenting egg: that is to say with an intervening “polar furrow,” whose ends mark the meeting place, at equal angles, of the cells in groups of three.
The little projecting spokes, or spikes, which are set normally to the main basket-work, seem to be incompleted portions of a larger basket, or in other words imperfectly formed elements corresponding to the interfacial contacts in the surrounding parts [p. 475] of the system. Similar but more complex formations, all explicable as basket-like frameworks developed around a cluster of cells, are known in great variety.
In our Nassellarian itself, and in many other cases where the plane interfacial boundary-walls are skeletonised, we see that the siliceous matter is not deposited in an even and continuous layer, like the waxen walls of a bee’s cell, but constitutes a meshwork of fine curvilinear threads; and the curves seem to run, on the whole, isogonally, and to form three main series, one approximately parallel to, or concentric with, the outer or free edge of the partition, and the other two related severally to its two edges of attachment. Sometimes (as may also be seen in our figure), the system is still further complicated by a fourth series of linear elements, which tend to run radially from the centre of the system to the free edge of each partition. As regards the former, their arrangement is such as would result if deposition or solidification had proceeded in waves, starting independently from each of the three boundaries of the little partition-wall; and something of this kind is doubtless what has happened. We are reminded at once of the wave-like periodicity of the Liesegang phenomenon. But apart from this we might conceive of other explanations. For instance, the liquid film which originally constitutes the partition must easily be thrown into vibrations, and (like the dust upon a Chladni’s plate) minute particles of matter in contact with the film would tend to take up their position in a symmetrical arrangement, in direct relation to the nodal points or lines of the vibrating surface [222]. Some such explanation as this (to my thinking) must be invoked to account for the minute and varied and very beautiful patterns upon many diatoms, the resemblance of which patterns (in certain of their simpler cases) to the Chladni figures is sometimes striking and obvious. But the many special problems which the diatom skeleton suggests I have not attempted to consider.
1917 Source Illustration
Fig. 230.
The last peculiarity of our Nassellarian lies in an apparent departure from what we should at first expect in the way of its [p. 476] external symmetry. Were the system actually composed of four spherical vesicles in mutual contact, the outer margin of each of the six interfacial planes would obviously be a circular arc; and accordingly, at each angle of the tetrahedron, we should expect to have a depressed, or re-entrant angle, instead of a prominent cusp. This is all doubtless due to some simple balance of tensions, whose precise nature and distribution is meanwhile a matter of conjecture. But it seems as though an extremely simple explanation would go a long way, and possibly the whole way, to meet this particular case. In our ordinary plane diagram of three cells, or soap-bubbles, in contact, we know (and we have just said) that the tensions of the three partitions draw inwards the outer walls of the system, till at each point of triple contact (P) we tend to get a triradiate, equiangular junction. But if we introduce another bubble into the centre of the system (Fig. 230), then, as Plateau shewed, the tensions of its walls and those of the three partitions by which it is now suspended, again balance one another, and the central bubble appears (in plane projection) as a curvilinear, equilateral triangle. We have only got to convert this plane diagram into that of a tetrahedral solid to obtain almost precisely the configuration which we are seeking to explain. Now we observe that, so far as our figure of Callimitra informs us, this is just the shape of the little bubble which occupies the centre of the tetrahedral system in that Radiolarian skeleton. And I conceive, accordingly, that the entire organism was not limited to the four cells or vesicles (together with the little central [p. 477] fifth) which we have hitherto been imagining, but there must have been an outer tetrahedral system, enclosing the cells which fabricated the skeleton, just as these latter enclosed, and deformed, the little bubble in the centre of all. We have only to suppose that this hypothetical tetrahedral series, forming the outer layer or surface of the whole system, was for some chemico-physical reason incapable of secreting at its interfacial contacts a skeletal fabric [223].
In this hypothetical case, the edges of the skeletal system would be circular arcs, meeting one another at an angle of 120°, or, in the solid pyramid, of 109°: and this latter is very nearly the condition which our little skeleton actually displays. But we observe in Fig. 227 that, in the immediate neighbourhood of the tetrahedral angle, the circular arcs are slightly drawn out into projecting cusps (cf. Fig. 230, B). There is no S-shaped curvature of the tetrahedral edges as a whole, but a very slight one, a very slight change of curvature; close to the apex. This, I conceive, is nothing more than what, in a material system, we are bound to have, to represent a “surface of continuity.” It is a phenomenon precisely analogous to Plateau’s “bourrelet,” which we have already seen to be a constant feature of all cellular systems, rounding off the sharp angular contacts by which (in our more elementary treatment) we expect one film to make its junction with another [224].
In the foregoing examples of Radiolaria, the symmetry which the organism displays would seem to be identical with that symmetry of forces which is due to the assemblage of surface-tensions in the whole system; this symmetry being displayed, in one class of cases, in a complex spherical mass of froth, and in [p. 478] another class in a simpler aggregate of a few, otherwise isolated, vesicles. But among the vast number of other known Radiolaria, there are certain forms (especially among the Phaeodaria and Acantharia) which display a still more remarkable symmetry, the origin of which is by no means clear, though surface-tension doubtless plays a part in its causation. These are cases in which (as in some of those already described) the skeleton consists (1) of radiating spicular rods, definite in number and position, and (2) of interconnecting rods or plates, tangential to the more or less spherical body of the organism, whose form becomes, accordingly, that of a geometric, polyhedral solid. It may be that there is no mathematical difference, save one of degree, between such a hexagonal polyhedron as we have seen in Aulacantha, and those which we are about to describe; but the greater regularity, the numerical symmetry, and the apparent simplicity of these latter, makes of them a class apart, and suggests problems which have not been solved nor even investigated.
1917 Source Illustration
Fig. 231. Skeletons of various Radiolarians, after Haeckel. 1. Circoporus sexfurcus; 2. C. octahedrus; 3. Circogonia icosahedra; 4. Circospathis novena; 5. Circorrhegma dodecahedra.
The matter is sufficiently illustrated by the accompanying figures, all drawn from Haeckel’s Monograph of the Challenger Radiolaria [225]. In one of these we see a regular octahedron, in another a regular, or pentagonal dodecahedron, in a third a regular icosahedron. In all cases the figure appears to be perfectly symmetrical, though neither the triangular facets of the octahedron and icosahedron, nor the pentagonal facets of the dodecahedron, are necessarily plane surfaces. In all of these cases, the radial spicules correspond to the solid angles of the figure; and they are, accordingly, six in number in the octahedron, twenty in the dodecahedron, and twelve in the icosahedron. If we add to these three figures the regular tetrahedron, which we have had frequent occasion to study, and the cube (which is represented, at least in outline, in the skeleton of the hexactinellid sponges), we have completed the series of the five regular polyhedra known to geometers, the Platonic bodies [226] of the older mathematicians. It is at first sight all the more remarkable that we should here meet [p. 480] with the whole five regular polyhedra, when we remember that, among the vast variety of crystalline forms known among minerals, the regular dodecahedron and icosahedron, simple as they are from the mathematical point of view, never occur. Not only do these latter never occur in Crystallography, but (as is explained in text-books of that science) it has been shewn that they cannot occur, owing to the fact that their indices (or numbers expressing the relation of the faces to the three primary axes) involve an irrational quantity: whereas it is a fundamental law of crystallography, involved in the whole theory of space-partitioning, that “the indices of any and every face of a crystal are small whole numbers [227].” At the same time, an imperfect pentagonal dodecahedron, whose pentagonal sides are non-equilateral, is common among crystals. If we may safely judge from Haeckel’s figures, the pentagonal dodecahedron of the Radiolarian is perfectly regular, and we must presume, accordingly, that it is not brought about by principles of space-partitioning similar to those which manifest themselves in the phenomenon of crystallisation. It will be observed that in all these radiolarian polyhedral shells, the surface of each external facet is formed of a minute hexagonal network, whose probable origin, in relation to a vesicular structure, is such as we have already discussed.
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Fig. 232. Dorataspis sp.; diagrammatic.
In certain allied Radiolaria (Fig. 232), which, like the dodecahedral form figured in Fig. 231, 5, have twenty radial spines, these latter are commonly described as being arranged in a certain very singular way. It is stated that their arrangement may be referred [p. 481] to a series of five parallel circles on the sphere, corresponding to the equator (c), the tropics (b, d) and the polar circles (a, e); and that beginning with four equidistant spines in the equator, we have alternating whorls of four, radiating outwards from the sphere in each of the other parallel zones. This rule was laid down by the celebrated Johannes Müller, and has ever since been used and quoted as Müller’s law. The chief point in this alleged arrangement which strikes us at first sight as very curious, is that there is said to be no spine at either pole; and when we come to examine carefully the figure of the organism, we find that the received description does not do justice to the facts. We see, in the first place, from such figures as Figs. 232, 234, that here, unlike our former cases, the radial spines issue through the facets (and through all the facets) of the polyhedron, instead of through its solid angles; and accordingly, that our twenty spines correspond (not, as before, to a dodecahedron) but to some sort of an icosahedron. We see in the next place, that this icosahedron is composed of faces, or plates, of two different kinds, some hexagonal and some pentagonal; and when we look closer, we discover that the whole figure is that of a hexagonal prism, whose twelve solid angles are replaced by pentagonal facets. Both hexagons and pentagons [p. 482] appear to be perfectly equilateral, but if we try to construct a plane-sided polyhedron of this kind, we soon find that it is impossible; for into the angles between the six equatorial hexagons those of the six united pentagons will not fit. The figure however can be easily constructed if we replace the straight edges (or some of them) by curves, and the plane facets by corresponding, slightly curved, surfaces. The true symmetry of this figure, then, is hexagonal, with a polar axis, produced into two polar spicules; with six equatorial spicules, or rays; and with two sets of six spicular rays, interposed between the polar axis and the equatorial rays, and alternating in position with the latter.
Müller’s description was emended by Brandt, and what is now known as “Brandt’s law,” viz. that the symmetry consists of two polar rays, and three whorls of six each, coincides with the above description so far as the spicular axes go: save only that Brandt specifically states that the intermediate whorls stand equidistant between the equator and the poles, i.e. in latitude 45°. While not far from the truth, this statement is not exact; for according to the geometry of the figure, the intermediate cycles obviously stand in a slightly higher latitude, but this latitude I have not attempted to determine; for the calculation seems to be a little troublesome owing to the curvature of the sides of the figure, and the enquiring mathematician will perform it more easily than I. Brandt, if I understand him rightly, did not propose his “law” as a substitute for Müller’s law, but as a second law applicable to a few particular cases. I on the other hand can find no case to which Müller’s law properly applies.
If we construct such a polyhedron, and set it in the position of Fig. 232, B, we shall easily see that it is capable of explanation (though improperly) in accordance with Müller’s law; for the four equatorial rays of Müller (c) now correspond to the two polar and to two opposite equatorial facets of our polyhedron: the four “polar” rays of Müller (a or e) correspond to two adjacent hexagons and two intermediate pentagons of the figure: and Müller’s “tropical” rays (b or d) are those which emanate from the remaining four pentagonal facets, in each half of the figure. In some cases, such as Haeckel’s Phatnaspis cristata (Fig. 233), we have an ellipsoidal body, from which the spines emerge in the order described, but which is not obviously divided by facets. In Fig. 234 I have indicated the facets corresponding to the rays, and dividing the surface in the usual symmetrical way. [p. 483]
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Fig. 233. Phatnaspis cristata, Hkl.
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Fig. 234. The same, diagrammatic.
Within any polyhedron we may always inscribe another polyhedron, whose corners correspond in number to the sides or facets of the original figure, or (in alternative cases) to a certain number of these sides; and a similar result is obtained by bevelling off the corners of the original polyhedron. We may obtain a precisely similar symmetrical result if (in such a case as these Radiolarians which we are describing), we imagine the radial spines to be interconnected by tangential rods, instead of by the complete facets which we have just been dealing with. In our complicated polyhedron with its twenty radial spines arranged in the manner described there are various symmetrical ways in which we may imagine these interconnecting bars to be arranged. The most symmetrical of these is one in which the whole surface is divided into eighteen rhomboidal areas, obtained by systematically connecting each group of four adjacent radii. This figure has eighteen faces (F), twenty corners (C), and therefore thirty-six edges (E), in conformity with Euler’s theorem, F + C = E + 2.
1917 Source Illustration
Fig. 235. Phractaspis prototypus, Hkl.
Another symmetrical arrangement will divide the surface into fourteen rhombs and eight triangles. This latter arrangement is obtained by linking up the radial rods as follows: aaaa, aba, abcb, bcdc, etc. Here we have again twenty corners, but we have twenty-two faces; the number of edges, or tangential spicular bars, will be found, therefore, by the above formula, to be forty. In Haeckel’s figure of Phractaspis prototypus we have a spicular skeleton which appears to be constructed precisely upon this plan, and to be derivable from the faceted polyhedron precisely after this manner.
In all these latter cases it is the arrangement of the axial rods, or in other words the “polar symmetry” of the entire organism, which lies at the root of the matter, and which, if only [p. 485] we could account for it, would make it comparatively easy to explain the superficial configuration. But there are no obvious mechanical forces by which we can so explain this peculiar polarity. This at least is evident, that it arises in the central mass of protoplasm, which is the essential living portion of the organism as distinguished from that frothy peripheral mass whose structure has helped us to explain so many phenomena of the superficial or external skeleton. To say that the arrangement depends upon a specific polarisation of the cell is merely to refer the problem to other terms, and to set it aside for future solution. But it is possible that we may learn something about the lines in which to seek for such a solution by considering the case of Lehmann’s “fluid crystals,” and the light which they throw upon the phenomena of molecular aggregation.
The phenomenon of “fluid crystallisation” is found in a number of chemical bodies; it is exhibited at a specific temperature for each substance; and it would seem to be limited to bodies in which there is a more or less elongated, or “chain-like” arrangement of the atoms in the molecule. Such bodies, at the appropriate temperature, tend to aggregate themselves into masses, which are sometimes spherical drops or globules (the so-called “spherulites”), and sometimes have the definite form of needle-like or prismatic crystals. In either case they remain liquid, and are also doubly refractive, polarising light in brilliant colours. Together with them are formed ordinary solid crystals, also with characteristic polarisation, and into such solid crystals all the fluid material ultimately turns. It is evident that in these liquid crystals, though the molecules are freely mobile, just as are those of water, they are yet subject to, or endowed with, a “directive force,” a force which confers upon them a definite configuration or “polarity,” the Gestaltungskraft of Lehmann.
Such an hypothesis as this had been gradually extruded from the theories of mathematical crystallography [228]; and it had come to be believed that the symmetrical conformation of a homogeneous crystalline structure was sufficiently explained by the mere mechanical fitting together of appropriate structural units along the easiest and simplest lines of “close packing”: just as [p. 486] a pile of oranges becomes definite, both in outward form and inward structural arrangement, without the play of any specific directive force. But while our conceptions of the tactical arrangement of crystalline molecules remain the same as before, and our hypotheses of “modes of packing” or of “space-lattices” remain as useful as ever for the definition and explanation of the molecular arrangements, an entirely new theoretical conception is introduced when we find such space-lattices maintained in what has hitherto been considered the molecular freedom of a liquid field; and we are constrained, accordingly, to postulate a specific molecular force, or “Gestaltungskraft” (not unlike Kepler’s “facultas formatrix”), to account for the phenomenon.
Now just as some sort of specific “Gestaltungskraft” had been of old the deus ex machina accounting for all crystalline phenomena (gnara totius geometriæ, et in ea exercita, as Kepler said), and as such an hypothesis, after being dethroned and repudiated, has now fought its way back and has made good its right to be heard, so it may be also in biology. We begin by an easy and general assumption of specific properties, by which each organism assumes its own specific form; we learn later (as it is the purpose of this book to shew) that throughout the whole range of organic morphology there are innumerable phenomena of form which are not peculiar to living things, but which are more or less simple manifestations of ordinary physical law. But every now and then we come to certain deep-seated signs of protoplasmic symmetry or polarisation, which seem to lie beyond the reach of the ordinary physical forces. It by no means follows that the forces in question are not essentially physical forces, more obscure and less familiar to us than the rest; and this would seem to be the crucial lesson for us to draw from Lehmann’s surprising and most beautiful discovery. For Lehmann seems actually to have demonstrated, in non-living, chemical bodies, the existence of just such a determinant, just such a “Gestaltungskraft,” as would be of infinite help to us if we might postulate it for the explanation (for instance) of our Radiolarian’s axial symmetry. But further than this we cannot go; for such analogy as we seem to see in the Lehmann phenomenon soon evades us, and refuses to be pressed home. Not only is it the case, as we have already [p. 487] seen, that certain of the geometric forms assumed by the symmetrical Radiolarian shells are just such as the “space-lattice” theory would seem to be inapplicable to, but it is in other ways obvious that symmetry of crystallisation, whether liquid or solid, has no close parallel, but only a series of analogies, in the protoplasmic symmetry of the living cell.
Chapter X: A Parenthetic Note on Geodetics¶
1917 Cambridge First Edition (pp. 488–492)
We have made use in the last chapter of the mathematical principle of Geodetics (or Geodesics) in order to explain the conformation of a certain class of sponge-spicules; but the principle is of much wider application in morphology, and would seem to deserve attention which it has not yet received.
1917 Source Illustration
Fig. 236. Annular and spiral thickenings in the walls of plant-cells.
Defining, meanwhile, our geodetic line (as we have already done) as the shortest distance between two points on the surface of a solid of revolution, we find that the geodetics of the cylinder give us one of the simplest of cases. Here it is plain that the geodetics are of three kinds: (1) a series of annuli around the cylinder, that is to say, a system of circles, in planes parallel to one another and at right angles to the axis of the cylinder (Fig. 236, a); (2) a series of straight lines parallel to the axis; and (3) a series of spiral curves winding round the wall of the cylinder (b, c). These three systems are all of frequent occurrence, and are all illustrated in the local thickenings of the wall of the cylindrical cells or vessels of plants.
The spiral, or rather helicoid, geodetic is particularly common in cylindrical structures, and is beautifully shewn for instance in the spiral coil which stiffens the tracheal tubes of an insect, or the so-called “tracheides” of a woody stem. A similar [p. 489] phenomenon is often witnessed in the splitting of a glass tube. If a crack appear in a thin tube, such as a test-tube, it has a tendency to be prolonged in its own direction, and the more perfectly homogeneous and isotropic be the glass the more evenly will the split tend to follow the straight course in which it began. As a result, the crack in our test-tube is often seen to continue till the tube is split into a continuous spiral ribbon.
In a right cone, the spiral geodetic falls into closer and closer coils as the diameter of the cone narrows; and a very beautiful geodetic of this kind is exemplified in the sutural line of a spiral shell, such as Turritella, or in the striations which run parallel with the spiral suture. Similarly, in an ellipsoidal surface, we have a spiral geodetic, whose coils get closer together as we approach the ends of the long axis of the ellipse; in the splitting of the integument of an Equisetum-spore, by which are formed the spiral “elaters” of the spore, we have a case of this kind, though the spiral is not sufficiently prolonged to shew all its features in detail.
We have seen in these various cases, that our original definition of a geodetic requires to be modified; for it is only subject to conditions that it is “the shortest distance between two points on the surface of the solid,” and one of the commonest of these restricting conditions is that our geodetic may be constrained to go twice, or many times, round the surface on its way. In short, we must redefine our geodetic, as a curve drawn upon a surface, such that, if we take any two adjacent points on the curve, the curve gives the shortest distance between them. Again, in the geodetic systems which we meet with in morphology, it sometimes happens that we have two opposite systems of geodetic spirals separate and distinct from one another, as in Fig. 236, c; and it is also common to find the two systems interfering with one another, and forming a criss-cross, or reticulated arrangement. This is a very common source of reticulated patterns.
Among the ciliated Infusoria, we have in the spiral lines along which their cilia are arranged a great variety of beautiful geodetic curves; though it is probable enough that in some complicated cases these are not simple geodetics, but projections of curves other than a straight line upon the surface of the solid. [p. 490]
Lastly, a very instructive case is furnished by the arrangement of the muscular fibres on the surface of a hollow organ, such as the heart or the stomach. Here we may consider the phenomenon from the point of view of mechanical efficiency, as well as from that of purely descriptive or objective anatomy. In fact we have an a priori right to expect that the muscular fibres covering such hollow or tubular organs will coincide with geodetic lines, in the sense in which we are now using the term. For if we imagine a contractile fibre, or elastic band, to be fixed by its two ends upon a curved surface, it is obvious that its first effort of contraction will tend to expend itself in accommodating the band to the form of the surface, in “stretching it tight,” or in other words in causing it to assume a direction which is the shortest possible line upon the surface between the two extremes: and it is only then that further contraction will have the effect of constricting the tube and so exercising pressure on its contents. Thus the muscular fibres, as they wind over the curved surface of an organ, arrange themselves automatically in geodesic curves: in precisely the same manner as we also automatically construct complex systems of geodesics whenever we wind a ball of wool or a spindle of tow, or when the skilful surgeon bandages a limb. In these latter cases we see the production of those “figures-of-eight,” to which, in the case for instance of the heart-muscles, Pettigrew and other anatomists have ascribed peculiar importance. In the case of both heart and stomach we must look upon these organs as developed from a simple cylindrical tube, after the fashion of the glass-blower, as is further discussed on p. 737 of this book, the modification of the simple cylinder consisting of various degrees of dilatation and of twisting. In the primitive undistorted cylinder, as in an artery or in the intestine, the muscular fibres run in geodetic lines, which as a rule are not spiral, but are merely either annular or longitudinal; these are the ordinary “circular and longitudinal coats,” which form the normal musculature of all tubular organs, or of the body-wall of a cylindrical worm [229]. If we consider each muscular fibre as an elastic strand, imbedded in the elastic membrane which constitutes the wall of the organ, it [p. 491] is evident that, whatever be the distortion suffered by the entire organ, the individual fibre will follow the same course, which will still, in a sense, be a geodetic. But if the distortion be considerable, as for instance if the tube become bent upon itself, or if at some point its walls bulge outwards in a diverticulum or pouch, it is obvious that the old system of geodetics will only mark the shortest distance between two points more or less approximate to one another, and that new systems of geodetics will tend to appear, peculiar to the new surface, and linking up points more remote from one another. This is evidently the case in the human stomach. We still have the systems, or their unobliterated remains, of circular and longitudinal muscles; but we also see two new systems of fibres, both obviously geodetic (or rather, when we look more closely, both parts of one and the same geodetic system), in the form of annuli encircling the pouch or diverticulum at the cardiac end of the stomach, and of oblique fibres taking a spiral course from the neighbourhood of the oesophagus over the sides of the organ.
In the heart we have a similar, but more complicated phenomenon. Its musculature consists, in great part, of the original simple system of circular and longitudinal muscles which enveloped the original arterial tubes, which tubes, after a process of local thickening, expansion, and especially twisting, came together to constitute the composite, or double, mammalian heart; and these systems of muscular fibres, geodetic to begin with, remain geodetic (in the sense in which we are using the word) after all the twisting to which the primitive cylindrical tube or tubes have been subjected. That is to say, these fibres still run their shortest possible course, from start to finish, over the complicated curved surface of the organ; and it is only because they do so that their contraction, or longitudinal shortening, is able to produce its direct effect, as Borelli well understood, in the contraction or systole of the heart [230]. [p. 492]
As a parenthetic corollary to the case of the spiral pattern upon the wall of a cylindrical cell, we may consider for a moment the spiral line which many small organisms tend to follow in their path of locomotion [231]. The helicoid spiral, traced around the wall of our cylinder, may be explained as a composition of two velocities, one a uniform velocity in the direction of the axis of the cylinder, the other a uniform velocity in a circle perpendicular to the axis. In a somewhat analogous fashion, the smaller ciliated organisms, such as the ciliate and flagellate Infusoria, the Rotifers, the swarm-spores of various Protists, and so forth, have a tendency to combine a direct with a revolving path in their ordinary locomotion. The means of locomotion which they possess in their cilia are at best somewhat primitive and inefficient; they have no apparent means of steering, or modifying their direction; and, if their course tended to swerve ever so little to one side, the result would be to bring them round and round again in an approximately circular path (such as a man astray on the prairie is said to follow), with little or no progress in a definite longitudinal direction. But as a matter of fact, either through the direct action of their cilia or by reason of a more or less unsymmetrical form of the body, all these creatures tend more or less to rotate about their long axis while they swim. And this axial rotation, just as in the case of a rifle-bullet, causes their natural swerve, which is always in the same direction as regards their own bodies, to be in a continually changing direction as regards space: in short, to make a spiral course around, and more or less approximate to, a straight axial line.