CHAPTER 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.
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