VI. A NOTE ON ADSORPTION…

VI. A NOTE ON ADSORPTION

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[324]. 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. {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 {279} or “adsorption” represents the work done, equivalent to the diminished potential energy of the system[325]. 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[326]. And this is all as {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[327].

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 {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.

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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[328]. 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[329].

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 {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[330].” 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[331], 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. {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[332]. 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.

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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 {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 {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[333].

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 {286} unexplained way with the mechanism of ciliary movement. Meves looked upon the chondriosomes as the actual carriers or transmitters of heredity[334]. 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[335]. 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 {287} to the phenomena of surface energy and of adsorption[336]. 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.

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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 {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[337]. 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[338]. {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[339]; 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[340].

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 {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.”

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, {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 CO_{2} 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.

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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 {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.

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VII. THE FORMS OF TISSUES OR CELL-AGGREGATES

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 {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 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 _E__{bc} 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

_E__{bc} < _E__{ab} + _E__{ac}.

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.

_E__{wa} > _E__{oa} + _E__{ow}.

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. {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[341]. 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, 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), {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.

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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

P : Q : R : = sin QOR : sin POR : sin POQ.

or P/sin QOR = Q/sin ROP = R/sin POQ.

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:

P^2 = Q^2 + R^2 + 2_Q_ R cos(QOR), etc.

From this and the foregoing, we learn the following important and useful deductions:

(1) 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. {297}

(2) 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.

(3) 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.

(4) 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[342]; and this little fillet, or “bourrelet,” {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[343].

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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.

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 {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.

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 {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.

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.”

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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, {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[344].

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 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

2 sin 60° = 1·732

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 {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.

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 {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.

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 {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.

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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:

(1) If T be equal to T′, and t be relatively insignificant, the angles α, β will be of 90°.

(2) 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°.

(3) 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.

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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[345]. Ten {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[346]. Years before, Schwendener had found, in the final results of cell-division, a universal system of “orthogonal trajectories[347]”; 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[348].

Within the next few years, a number of botanical writers were content to point out further exceptions to Sachs’s Rule[349]; 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[350].

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” {306} had been pointed out long before, by Melsens, who had made an “artificial tissue” by blowing into a solution of white of egg[351].

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[352] 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[353] 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[354]; and the same line of investigation and thought were pursued and developed by Robert, in connection with the embryology of the Mollusca[355]. Driesch again, in a series of papers, continued to draw attention to the presence of capillary phenomena in the segmenting cells {307} of various embryos, and came to the conclusion that the mode of segmentation was of little importance as regards the final result[356].

Lastly de Wildeman[357], 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.”

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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 {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 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 {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.

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°. {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[358], 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 {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.

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[359]. 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 {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[360].

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, {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 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.

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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 {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[361], 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 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 {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.

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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 {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, {317} with six planes, meeting symmetrically in a point, and constituting there four equal solid angles.

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) {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

op = oa/3 = od/3.

Therefore cos dop = 1/3 and cos aod = − 1/3.

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.

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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[362].

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 {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.

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, {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 {321} 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.

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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 {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.

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.

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 {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[363]. 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[364].” 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.

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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 {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 δ^2 × (√3)/2, the area of a circle being d^2 ⋅ 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 n^2 × π/4 × 2/(√3) = 0·907_n_^2, or (9/10) ⋅ n^2, 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[365]. 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, = 3_N_(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[366].

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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.” {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 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). {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.).

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 {327} known as triasters, polyasters, etc., whose relation to a field of force Hartog has explained[367]. 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.

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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 {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 {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[368] (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[369]. “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 {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[370] and Lhuilier[371], 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[372]) “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.

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. {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.

Then, AC = 2_a_ cos 30° = a√3.

And VX = a/sin θ (from inspection of the triangle LXB)

Therefore the area of the rhombus VAXC = (a^2 √3)/(2 sin θ).

And the area of AabX = (a/2)(2_h_ − ½_VX_ cos θ)

= (a/2)(2_h_ − a/2 ⋅ cot θ).

Therefore the total area of the figure

= hexagon abcdef + 3_a_(2_h_ − (a/2) cot θ) + 3((a^2 √3)/(2 sin θ)).

Therefore d(Area)/_d_θ = (3_a_^2/2)((1/sin^2 θ) − (√3 cos θ)/(sin^2 θ)).

But this expression vanishes, that is to say, d(Area)/_d_θ = 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.

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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; {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[373].” 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[374]. But it is very doubtful {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[375]. 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, {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[376]. 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[377].” {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.

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 {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.

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A few years after the publication of Plateau’s book, Lord Kelvin shewed, in a short but very beautiful paper[378], 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 {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.

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[379], 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 {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° [380].

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. {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.

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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 {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.

M. 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. {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[381]; 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 {342} in molluscs and annelids ‘stands in obvious relation to the different size of the cells produced at the two poles[382].’ ”

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)[383]. 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, {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.

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Before we leave for the present the subject of the segmenting {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[384]. 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[385], 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 {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.

{346}

VIII. 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 a^2 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; {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 (½_a_)^2 = ¼(a^2).

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.

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.

(1) The cell typically tends to divide into two co-equal parts.

(2) 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 {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[386], 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.

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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 {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. a^2; 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 a^2. The calculation is very easy.

The surface-area of a cylinder of length a is 2π_r_ ⋅ a, and that of our quarter-cylinder is, therefore, a ⋅ π_r_/2; and this being, by hypothesis, = a^2, we have a = π_r_/2, or r = 2_a_/π.

The volume of a cylinder, of length a, is _a_π_r_^2, and that of our quarter-cylinder is (a ⋅ π_r_^2)/4, which (by substituting the value of r) is equal to (a^3)/π.

Now precisely this same volume is, obviously, shut off by a transverse partition of area a^2, 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.

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 {350} one-quarter of the original cell[387]. 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 {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.

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 a^2, 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.

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 {352} which we cannot fully master, the same guiding principle is at the root of the matter.

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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 {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 and oblique partitions, V, T, and O, we have V = tan β; T = √2; and O = 1/(√2 cos β), or

V : T : O = tan β/√2 : 1 : 1/(2 cos β).

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.

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 {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. {355}

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[388].

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 {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.

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[389], 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 {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[390]. He was deeply and rightly impressed by the fact that other forces besides surface {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.

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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.

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, {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) {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[391]. 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.

(1) 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

CP = a cot θ; PN = a cos θ; NC = CP cos θ = a ⋅ (cos^2 θ)/(sin θ).

The area of the portion PMN

= ½_C_(P^2)θ − ½_PN_ ⋅ NC

= ½_a_^2(cot^2 θ) − ½_a_(cos θ) ⋅ a(cos^2 θ)/(sin θ)

= ½_a_^2(cot^2 θ − (cos^3 θ)/(sin θ)).

{361}

And the area of the portion PNA

= ½_a_^2(π/2 − θ) − ½_ON_ ⋅ NP

= ½_a_^2(π/2 − θ) − ½_a_(sin θ) ⋅ a(cos θ)

= ½_a_^2(π/2 − θ − sin θ ⋅ cos θ).

Therefore the area of the whole portion PMA

= (a^2)/2 (π/2 − θ + θ cot^2 θ − (cos^3 θ)/sin θ − sin θ ⋅ cos θ)

= (a^2)/2 (π/2 − θ + θ cot^2 θ − cot θ),

and also, by hypothesis, = ½ ⋅ area of the quadrant, = (π_a_^2)/8.

Hence θ is defined by the equation

a^2/2 (π/2 − θ + θ cot^2 θ − cot θ) = (π_a_^2)/8,

or π/4 − θ + θ cot^2 θ − cot θ = 0.

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

{362}

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′.

(2) 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.

(3) 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.

The two cells into which our original quadrant is now divided, while they are equal in volume, are of very different shapes; the {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.

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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 {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 = _a_θ cot θ; and the angle θ has a definite relation to α, the angle of arc.

Moreover, in the case of the periclinal boundary, RS (Fig. 147) (or ab, Fig. 149), we know that, if it bisect the cell,

RS = a ⋅ α/√2.

Accordingly, the arc RS will be just equal to the arc MP when

θ cot θ = α/√2.

When θ cot θ > α/√2 or MP > RS,

then division will take place as in RS.

When θ cot θ < α/√2, or MP < RS,

then division will take place as in MP.

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 {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 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 {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 {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.

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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.

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 {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

= T + Q = ¼π_a_^2 = ·7854_a_^2,

while the area of T = Q = ·3927_a_^2.

The area of the enlarged sector, p′OA′,

= (a + x)^2 × (55° 22′) ÷ 2 = ·4831(a + x)^2,

and the area OPA

= a^2 × (55° 22′) ÷ 2 = ·4831_a_^2.

Therefore the area of the added portion, T′,

= ·4831 ((a + x)^2 − a^2).

And this, by hypothesis,

= T = ·3927_a_^2.

We get, accordingly, since a = 1,

x^2 + 2_x_ = ·3927/·4831 = ·810,

and, solving,

x + 1 = √1·81 = 1·345, or x = 0·345.

Working out x′ in the same way, we arrive at the approximate value, x′ + 1 = 1·517. {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 {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.

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 {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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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) {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.

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 {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 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 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 {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.

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[392].

If within our boundary we have three cells all meeting {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 {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.

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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[393].

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 {377} considered) is responsible for many inaccurate or incomplete representations of the mutual arrangement of aggregated cells.

In the accompanying series of figures (Figs. 160–167) I have {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 {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.

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.

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It is obvious enough, without more ado, that these phenomena are in the strictest and completest way common to both plants {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[395]. {381}

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 {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[396].

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.

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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 {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 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 2_n_ − 3; or the number of what we call the internal or interfacial contacts is 2(2_n_ − 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 {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[397]. 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.

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. {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[398]. 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.

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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 {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.

Another case, geometrically akin but biologically very different, is to be found in the little diatoms of the genus Asterolampra, and their immediate congeners[399]. 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 {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[400] (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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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 {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.

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[401], 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[402], described them as “paradoxical in their anatomy.”

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 {389} case of the other two we may fairly assume that their proper and original arrangement was that of our type 6_b_ (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.” {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 {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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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 {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.

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 {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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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) {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 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 {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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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 {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 {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, {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.

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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[403]. 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 {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[404]. {400}

(1) 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.

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. {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.

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 {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.

(2) 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, {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[405]. 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, 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.

The method of division by means of oblique partitions is a common one in the case of ‘growing points’; for it evidently {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, {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.

(3) 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.

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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 {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.

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 {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.

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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.

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 {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.

A curious and beautiful case, apparently aberrant but which would doubtless be found conforming strictly to physical laws, if {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.

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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 {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.

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