Chapter VIII: THE FORMS OF TISSUES OR CELL-AGGREGATES (continued) (1)

The problems which we have been considering, and especially that of the bee’s cell, belong to a class of “isoperimetrical” problems, which deal with figures whose surface is a minimum for a definite content or volume. Such problems soon become difficult, but we may find many easy examples which lead us towards the explanation of biological phenomena; and the particular subject which we shall find most easy of approach is that of the division, in definite proportions, of some definite portion of space, by a partition-wall of minimal area. The theoretical principles so arrived at we shall then attempt to apply, after the manner of Berthold and Errera, to the actual biological phenomena of cell-division.

This investigation we may approach in two ways: by considering, namely, the partitioning off from some given space or area of one-half (or some other fraction) of its content; or again, by dealing simultaneously with the partitions necessary for the breaking up of a given space into a definite number of compartments.

If we take, to begin with, the simple case of a cubical cell, it is obvious that, to divide it into two halves, the smallest possible partition-wall is one which runs parallel to, and midway between, two of its opposite sides. If we call _a_ the length of one of the edges of the cube, then _a_^2 is the area, alike of one of its sides, and of the partition which we have interposed parallel, or normal, thereto. But if we now consider the bisected cube, and wish to divide the one-half of it again, it is obvious that another partition parallel to the first, so far from being the smallest possible, is precisely twice the size of a cross-partition perpendicular to it; {347} for the area of this new partition is _a_ × _a_/2. And again, for a third bisection, our next partition must be perpendicular to the other two, and it is obviously a little square, with an area of (½_a_)^2 = ¼(_a_^2).

From this we may draw the simple rule that, for a rectangular body or parallelopiped to be divided equally by means of a partition of minimal area, (1) the partition must cut across the longest axis of the figure; and (2) in the event of successive bisections, each partition must run at right angles to its immediate predecessor.

We have already spoken of “Sachs’s Rules,” which are an empirical statement of the method of cell-division in plant-tissues; and we may now set them forth in full.

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

(2) Each new plane of division tends to intersect at right angles the preceding plane of division.

The first of these rules is a statement of physiological fact, not without its exceptions, but so generally true that it will justify us in limiting our enquiry, for the most part, to cases of equal subdivision. That it is by no means universally true for cells generally is shewn, for instance, by such well-known cases {348} as the unequal segmentation of the frog’s egg. It is true when the dividing cell is homogeneous, and under the influence of symmetrical forces; but it ceases to be true when the field is no longer dynamically symmetrical, for instance, when the parts differ in surface tension or internal pressure. This latter condition, of asymmetry of field, is frequent in segmenting eggs[386], and is then equivalent to the principle upon which Balfour laid stress, as leading to “unequal” or to “partial” segmentation of the egg,—viz. the unequal or asymmetrical distribution of protoplasm and of food-yolk.

The second rule, which also has its exceptions, is true in a large number of cases; and it owes its validity, as we may judge from the illustration of the repeatedly bisected cube, solely to the guiding principle of minimal areas. It is in short subordinate to, and covers certain cases included under, a much more important and fundamental rule, due not to Sachs but to Errera; that (3) the incipient partition-wall of a dividing cell tends to be such that its area is the least possible by which the given space-content can be enclosed.

――――――――――

Let us return to the case of our cube, and let us suppose that, instead of bisecting it, we desire to shut off some small portion only of its volume. It is found in the course of experiments upon soap-films, that if we try to bring a partition-film too near to one side of a cubical (or rectangular) space, it becomes unstable; and is easily shifted to a totally new position, in which it constitutes a curved cylindrical wall, cutting off one corner of the cube. It meets the sides of the cube at right angles (for reasons which we have already considered); and, as we may see from the symmetry {349} of the case, it constitutes precisely one-quarter of a cylinder. Our plane transverse partition, wherever it was placed, had always the same area, viz. _a_^2; and it is obvious that a cylindrical wall, if it cut off a small corner, may be much less than this. We want, accordingly, to determine what is the particular volume which might be partitioned off with equal economy of wall-space in one way as the other, that is to say, what area of cylindrical wall would be neither more nor less than the area _a_^2. The calculation is very easy.

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

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

Now precisely this same volume is, obviously, shut off by a transverse partition of area _a_^2, if the third side of the rectangular space be equal to _a_/π. And this fraction, if we take _a_ = 1, is equal to 0·318..., or rather less than one-third. And, as we have just seen, the radius, or side, of the corresponding quarter-cylinder will be twice that fraction, or equal to ·636 times the side of the cubical cell.

If then, in the process of division of a cubical cell, it so divide that the two portions be not equal in volume but that one portion by anything less than about three-tenths of the whole, or three-sevenths of the other portion, there will be a tendency for the cell to divide, not by means of a plane transverse partition, but by means of a curved, cylindrical wall cutting off one corner of the original cell; and the part so cut off will be one-quarter of a cylinder.

By a similar calculation we can shew that a _spherical_ wall, cutting off one solid angle of the cube, and constituting an octant of a sphere, would likewise be of less area than a plane partition as soon as the volume to be enclosed was not greater than about {350} one-quarter of the original cell[387]. But while both the cylindrical wall and the spherical wall would be of less area than the plane transverse partition after that limit (of one-quarter volume) was passed, the cylindrical would still be the better of the two up to a further limit. It is only when the volume to be partitioned off {351} is no greater than about 0·15, or somewhere about one-seventh, of the whole, that the spherical cell-wall in an angle of the cubical cell, that is to say the octant of a sphere, is definitely of less area than the quarter-cylinder. In the accompanying diagram (Fig. 138) the relative areas of the three partitions are shewn for all fractions, less than one-half, of the divided cell.

In this figure, we see that the plane transverse partition, whatever fraction of the cube it cut off, is always of the same dimensions, that is to say is always equal to _a_^2, or = 1. If one-half of the cube have to be cut off, this plane transverse partition is much the best, for we see by the diagram that a cylindrical partition cutting off an equal volume would have an area about 25%, and a spherical partition would have an area about 50% greater. The point _A_ in the diagram corresponds to the point where the cylindrical partition would begin to have an advantage over the plane, that is to say (as we have seen) when the fraction to be cut off is about one-third, or ·318 of the whole. In like manner, at _B_ the spherical octant begins to have an advantage over the plane; and it is not till we reach the point _C_ that the spherical octant becomes of less area than the quarter-cylinder.

The case we have dealt with is of little practical importance to the biologist, because the cases in which a cubical, or rectangular, cell divides unequally, and unsymmetrically, are apparently few; but we can find, as Berthold pointed out, a few examples, for instance in the hairs within the reproductive “conceptacles” of certain Fuci (Sphacelaria, etc., Fig. 139), or in the “paraphyses” of mosses (Fig. 142). But it is of great theoretical importance: as serving to introduce us to a large class of cases, in which the shape and the relative dimensions of the original cavity lead, according to the principle of minimal areas, to cell-division in very definite and sometimes unexpected ways. It is not easy, nor indeed possible, to give a generalised account of these cases, for the limiting conditions are somewhat complex, and the mathematical treatment soon becomes difficult. But it is easy to comprehend a few simple cases, which of themselves will carry us a good long way; and which will go far to convince the student that, in other cases {352} which we cannot fully master, the same guiding principle is at the root of the matter.

――――――――――

The bisection of a solid (or the subdivision of its volume in other definite proportions) soon leads us into a geometry which, if not necessarily difficult, is apt to be unfamiliar; but in such problems we can go a long way, and often far enough for our particular purpose, if we merely consider the plane geometry of a side or section of our figure. For instance, in the case of the cube which we have been just considering, and in the case of the plane and cylindrical partitions by which it has been divided, it is obvious that, since these two partitions extend symmetrically from top to bottom of our cube, that we need only consider (so far as they are concerned) the manner in which they subdivide the _base_ of the cube. The whole problem of the solid, up to a certain point, is contained in our plane diagram of Fig. 138. And when our particular solid is a solid of revolution, then it is obvious that a study of its plane of symmetry (that is to say any plane passing through its axis of rotation) gives us the solution of the whole problem. The right cone is a case in point, for here the investigation of its modes of symmetrical subdivision is completely met by an examination of the isosceles triangle which constitutes its plane of symmetry.

The bisection of an isosceles triangle by a line which shall be the shortest possible is a very easy problem. Let _ABC_ be such a triangle of which _A_ is the apex; it may be shewn that, for its shortest line of bisection, we are limited to three cases: viz. to a vertical line _AD_, bisecting the angle at _A_ and the side _BC_; to a transverse line parallel to the base _BC_; or to an oblique line parallel to _AB_ or to _AC_. The respective magnitudes, or lengths, of these partition lines follow at once from the magnitudes of the angles of our triangle. For we know, to begin with, since the areas of similar figures vary as the squares of their linear dimensions, that, in order to bisect the area, a line parallel to one side of our triangle must always have a length equal to 1/√2 of that side. If then, we take our base, _BC_, in all cases of a length = 2, the transverse partition drawn parallel to it will always have a length equal to 2/√2, or = √2. The vertical {353} partition, _AD_, since _BD_ = 1, will always equal tan β (β being the angle _ABC_). And the oblique partition, _GH_, being equal to _AB_/√2 = 1/(√2 cos β). If then we call our vertical, transverse

and oblique partitions, _V_, _T_, and _O_, we have _V_ = tan β; _T_ = √2; and _O_ = 1/(√2 cos β), or

_V_ : _T_ : _O_ = tan β/√2 : 1 : 1/(2 cos β).

And, working out these equations for various values of β, we very soon see that the vertical partition (_V_) is the least of the three until β = 45°, at which limit _V_ and _O_ are each equal to 1/√2 = ·707; and that again, when β = 60°, _O_ and _T_ are each = 1, after which _T_ (whose value always = 1) is the shortest of the three partitions. And, as we have seen, these results are at once applicable, not only to the case of the plane triangle, but also to that of the conical cell.

In like manner, if we have a spheroidal body, less than a hemisphere, such for instance as a low, watch-glass shaped cell (Fig. 141, _a_), it is obvious that the smallest possible partition by which we can divide it into two equal halves {354} is (as in our flattened disc) a median vertical one. And likewise, the hemisphere itself can be bisected by no smaller partition meeting the walls at right angles than that median one which divides it into two similar quadrants of a sphere. But if we produce our hemisphere into a more elevated, conical body, or into a cylinder with spherical cap, it is obvious that there comes a point where a transverse, horizontal partition will bisect the figure with less area of partition-wall than a median vertical one (_c_). And furthermore, there will be an intermediate region, a region where height and base have their relative dimensions nearly equal (as in _b_), where an oblique partition will be better than either the vertical or the transverse, though here the analogy of our triangle does not suffice to give us the precise limiting values. We need not examine these limitations in detail, but we must look at the curvatures which accompany the several conditions. We have seen that a film tends to set itself at equal angles to the surface which it meets, and therefore, when that surface is a solid, to meet it (or its tangent if it be a curved surface) at right angles. Our _vertical_ partition is, therefore, everywhere normal to the original cell-walls, and constitutes a plane surface.

But in the taller, conical cell with transverse partition, the latter still meets the opposite sides of the cell at right angles, and it follows that it must itself be curved; moreover, since the tension, and therefore the curvature, of the partition is everywhere uniform, it follows that its curved surface must be a portion of a sphere, concave towards the apex of the original, now divided, cell. In the intermediate case, where we have an oblique partition, meeting both the base and the curved sides of the mother-cell, the contact must still be everywhere at right angles: provided we continue to suppose that the walls of the mother-cell (like those of our diagrammatic cube) have become practically rigid before the partition appears, and are therefore not affected and deformed by the tension of the latter. In such a case, and especially when the cell is elliptical in cross-section, or is still more complicated in form, it is evident that the partition, in adapting itself to circumstances and in maintaining itself as a surface of minimal area subject to all the conditions of the case, may have to assume a complex curvature. {355}

While in very many cases the partitions (like the walls of the original cell) will be either plane or spherical, a more complex curvature will be assumed under a variety of conditions. It will be apt to occur, for instance, when the mother-cell is irregular in shape, and one particular case of such asymmetry will be that in which (as in Fig. 143) the cell has begun to branch, or give off a diverticulum, before division takes place. A very complicated case of a different kind, though not without its analogies to the cases we are considering, will occur in the partitions of minimal area which subdivide the spiral tube of a nautilus, as we shall presently see. And again, whenever we have a marked internal asymmetry of the cell, leading to irregular and anomalous modes of division, in which the cell is not necessarily divided into two equal halves and in which the partition-wall may assume an oblique position, then apparently anomalous curvatures will tend to make their appearance[388].

Suppose that a more or less oblong cell have a tendency to divide by means of an oblique partition (as may happen through various causes or conditions of asymmetry), such a partition will still have a tendency to set itself at right angles to the rigid walls {356} of the mother-cell: and it will at once follow that our oblique partition, throughout its whole extent, will assume the form of a complex, saddle-shaped or anticlastic surface.

Many such cases of partitions with complex or double curvature exist, but they are not always easy of recognition, nor is the particular case where they appear in a _terminal_ cell a common one. We may see them, for instance, in the roots (or rhizoids) of Mosses, especially at the point of development of a new rootlet (Fig. 142, C); and again among Mosses, in the “paraphyses” of the male prothalli (e.g. in _Polytrichum_), we find more or less similar partitions (D). They are frequent also among many Fuci, as in the hairs or paraphyses of Fucus itself (B). In _Taonia atomaria_, as figured in Reinke’s memoir on the Dictyotaceae of the Gulf of Naples[389], we see, in like manner, _oblique_ partitions, which on more careful examination are seen to be curves of double curvature (Fig. 142, A).

The physical cause and origin of these S-shaped partitions is somewhat obscure, but we may attempt a tentative explanation. When we assert a tendency for the cell to divide transversely to its long axis, we are not only stating empirically that the partition tends to appear in a small, rather than a large cross-section of the cell: but we are also implicitly ascribing to the cell a longitudinal _polarity_ (Fig. 143, A), and implicitly asserting that it tends to {357} divide (just as the segmenting egg does), by a partition transverse to its polar axis. Such a polarity may conceivably be due to a chemical asymmetry, or anisotropy, such as we have learned of (from Professor Macallum’s experiments) in our chapter on Adsorption. Now if the chemical concentration, on which this anisotropy or polarity (by hypothesis) depends, be unsymmetrical, one of its poles being as it were deflected to one side, where a little branch or bud is being (or about to be) given off,—all in precise accordance with the adsorption phenomena described on p. 289,—then our “polar axis” would necessarily be a curved axis, and the partition, being constrained (again _ex hypothesi_) to arise transversely to the polar axis, would lie obliquely to the _apparent_ axis of the cell (Fig. 143, B, C). And if the oblique partition be so situated that it has to meet the _opposite_ walls (as in C), then, in order to do so symmetrically (i.e. either perpendicularly, as when the cell-wall is already solidified, or at least at equal angles on either side), it is evident that the partition, in its course from one side of the cell to the other, must necessarily assume a more or less S-shaped curvature (Fig. 143, D).

As a matter of fact, while we have abundant simple illustrations of the principles which we have now begun to study, apparent exceptions to this simplicity, due to an asymmetry of the cell itself, or of the system of which the single cell is but a part, are by no means rare. For example, we know that in cambium-cells, division frequently takes place parallel to the long axis of the cell, when a partition of much less area would suffice if it were set cross-ways: and it is only when a considerable disproportion has been set up between the length and breadth of the cell, that the balance is in part redressed by the appearance of a transverse partition. It was owing to such exceptions that Berthold was led to qualify and even to depreciate the importance of the law of minimal areas as a factor in cell-division, after he himself had done so much to demonstrate and elucidate it[390]. He was deeply and rightly impressed by the fact that other forces besides surface {358} tension, both external and internal to the cell, play their part in the determination of its partitions, and that the answer to our problem is not to be given in a word. How fundamentally important it is, however, in spite of all conflicting tendencies and apparent exceptions, we shall see better and better as we proceed.

――――――――――

But let us leave the exceptions and return to a consideration of the simpler and more general phenomena. And in so doing, let us leave the case of the cubical, quadrangular or cylindrical cell, and examine the case of a spherical cell and of its successive divisions, or the still simpler case of a circular, discoidal cell.

When we attempt to investigate mathematically the position and form of a partition of minimal area, it is plain that we shall be dealing with comparatively simple cases wherever even one dimension of the cell is much less than the other two. Where two dimensions are small compared with the third, as in a thin cylindrical filament like that of Spirogyra, we have the problem at its simplest; for it is at once obvious, then, that the partition must lie transversely to the long axis of the thread. But even where one dimension only is relatively small, as for instance in a flattened plate, our problem is so far simplified that we see at once that the partition cannot be parallel to the extended plane, but must cut the cell, somehow, at right angles to that plane. In short, the problem of dividing a much flattened solid becomes identical with that of dividing a simple _surface_ of the same form.

There are a number of small Algae, growing in the form of small flattened discs, consisting (for a time at any rate) of but a single layer of cells, which, as Berthold shewed, exemplify this comparatively simple problem; and we shall find presently that it is also admirably illustrated in the cell-divisions which occur in the egg of a frog or a sea-urchin, when the egg for the sake of experiment is flattened out under artificial pressure.

Fig. 144 (taken from Berthold’s _Monograph of the Naples Bangiaciae_) represents younger and older discs of the little alga _Erythrotrichia discigera_; and it will be seen that, in all stages save the first, we have an arrangement of cell-partitions which looks somewhat complex, but into which we must attempt to throw some light and order. Starting with the original single, and flattened, {359} cell, we have no difficulty with the first two cell-divisions; for we know that no bisecting partitions can possibly be shorter than the two diameters, which divide the cell into halves and into quarters. We have only to remember that, for the sum total of partitions to be a minimum, three only must meet in a point; and therefore, the four quadrantal walls must shift a little, producing the usual little median partition, or cross-furrow, instead of one common, central point of junction. This little intermediate wall, however, will be very small, and to all intents and purposes we may deal with the case as though we had now to do with four equal cells, each one of them a perfect quadrant. And so our problem is, to find the shortest line which shall divide the quadrant of a circle into two halves of equal area. A radial partition (Fig. 145, A), starting from the apex of the quadrant, is at once excluded, for a reason similar to that just referred to; our choice must lie therefore between two modes of division such as are illustrated in Fig. 145, where the partition is either (as in B) {360} concentric with the outer border of the cell, or else (as in C) cuts that outer border; in other words, our partition may (B) cut _both_ radial walls, or (C) may cut _one_ radial wall and the periphery. These are the two methods of division which Sachs called, respectively, (B) _periclinal_, and (C) _anticlinal_[391]. We may either treat the walls of the dividing quadrant as already solidified, or at least as having a tension compared with which that of the incipient partition film is inconsiderable. In either case the partition must meet the cell-wall, on either side, at right angles, and (its own tension and curvature being everywhere uniform) it must take the form of a circular arc.

Now we find that a flattened cell which is approximately a quadrant of a circle invariably divides after the manner of Fig. 145, C, that is to say, by an approximately circular, _anticlinal_ wall, such as we now recognise in the eight-celled stage of Erythrotrichia (Fig. 144); let us then consider that Nature has solved our problem for us, and let us work out the actual geometric conditions.

Let the quadrant _OAB_ (in Fig. 146) be divided into two parts of equal area, by the circular arc _MP_. It is required to determine (1) the position of _P_ upon the arc of the quadrant, that is to say the angle _BOP_; (2) the position of the point _M_ on the side _OA_; and (3) the length of the arc _MP_ in terms of a radius of the quadrant.

(1) Draw _OP_; also _PC_ a tangent, meeting _OA_ in _C_; and _PN_, perpendicular to _OA_. Let us call _a_ a radius; and θ the angle at _C_, which is obviously equal to _OPN_, or _POB_. Then

_CP_ = _a_ cot θ; _PN_ = _a_ cos θ; _NC_ = _CP_ cos θ = _a_ ⋅ (cos^2 θ)/(sin θ).

The area of the portion _PMN_

= ½_C_(_P_^2)θ − ½_PN_ ⋅ _NC_

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

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

{361}

And the area of the portion _PNA_

= ½_a_^2(π/2 − θ) − ½_ON_ ⋅ _NP_

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

= ½_a_^2(π/2 − θ − sin θ ⋅ cos θ).

Therefore the area of the whole portion _PMA_

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

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

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

Hence θ is defined by the equation

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

or π/4 − θ + θ cot^2 θ − cot θ = 0.

We may solve this equation by constructing a table (of which the following is a small portion) for various values of θ.

θ π/4 − θ − cot θ + θ cot^2 θ = _x_
34° 34′ ·7854 − ·6033 − 1·4514  + 1·2709 = ·0016
35′ ·7854 ·6036 1·4505 1·2700 ·0013
36′ ·7854 ·6039 1·4496 1·2690 ·0009
37′ ·7854 ·6042 1·4487 1·2680 ·0005
38′ ·7854 ·6045 1·4478 1·2671 ·0002
39′ ·7854 ·6048 1·4469 1·2661 −·0002
40′ ·7854 ·6051 1·4460 1·2652 −·0005

{362}

We see accordingly that the equation is solved (as accurately as need be) when θ is an angle somewhat over 34° 38′, or say 34° 38½′. That is to say, a quadrant of a circle is bisected by a circular arc cutting the side and the periphery of the quadrant at right angles, when the arc is such as to include (90° − 34° 38′), i.e. 55° 22′ of the quadrantal arc.

This determination of ours is practically identical with that which Berthold arrived at by a rough and ready method, without the use of mathematics. He simply tried various ways of dividing a quadrant of paper by means of a circular arc, and went on doing so till he got the weights of his two pieces of paper approximately equal. The angle, as he thus determined it, was 34·6°, or say 34° 36′.

(2) The position of _M_ on the side of the quadrant _OA_ is given by the equation _OM_ = _a_ cosec θ − _a_ cot θ; the value of which expression, for the angle which we have just discovered, is ·3028. That is to say, the radius (or side) of the quadrant will be divided by the new partition into two parts, in the proportions of nearly three to seven.

(3) The length of the arc _MP_ is equal to _a_ θ cot θ; and the value of this for the given angle is ·8751. This is as much as to say that the curved partition-wall which we are considering is shorter than a radial partition in the proportion of 8¾ to 10, or seven-eights almost exactly.

But we must also compare the length of this curved “anticlinal” partition-wall (_MP_) with that of the concentric, or periclinal, one (_RS_, Fig. 147) by which the quadrant might also be bisected. The length of this partition is obviously equal to the arc of the quadrant (i.e. the peripheral wall of the cell) divided by √2; or, in terms of the radius, = π/2√2 = 1·111. So that, not only is the anticlinal partition (such as we actually find in nature) notably the best, but the periclinal one, when it comes to dividing an entire quadrant, is very considerably larger even than a radial partition.

The two cells into which our original quadrant is now divided, while they are equal in volume, are of very different shapes; the {363} one is a triangle (_MAP_) with two sides formed of circular arcs, and the other is a four-sided figure (_MOBP_), which we may call approximately oblong. We cannot say as yet how the triangular portion ought to divide; but it is obvious that the least possible partition-wall which shall bisect the other must run across the long axis of the oblong, that is to say periclinally. This, also, is precisely what tends actually to take place. In the following diagrams (Fig. 148) of a frog’s egg dividing under pressure, that is to say when reduced to the form of a flattened plate, we see, firstly, the division into four quadrants (by the partitions 1, 2); secondly, the division of each quadrant by means of an anticlinal circular arc (3, 3), cutting the peripheral wall of the quadrant approximately in the proportions of three to seven; and thirdly, we see that of the eight cells (four triangular and four oblong) into which the whole egg is now divided, the four which we have called oblong now proceed to divide by partitions transverse to their long axes, or roughly parallel to the periphery of the egg.

――――――――――

The question how the other, or triangular, portion of the divided quadrant will next divide leads us to another well-defined problem, which is only a slight extension, making allowance for the circular arcs, of that elementary problem of the triangle we have already considered. We know now that an entire quadrant must divide (so that its bisecting wall shall have the least possible area) by means of an anticlinal partition, but how about any smaller sectors of circles? It is obvious in the case of a small prismatic {364} sector, such as that shewn in Fig. 149, that a _periclinal_ partition is the smallest by which we can possibly bisect the cell; we want, accordingly, to know the limits below which the periclinal partition is always the best, and above which the anticlinal arc, as in the case of the whole quadrant, has the advantage in regard to smallness of surface area.

This may be easily determined; for the preceding investigation is a perfectly general one, and the results hold good for sectors of any other arc, as well as for the quadrant, or arc of 90°. That is to say, the length of the partition-wall _MP_ is always determined by the angle θ, according to our equation _MP_ = _a_θ cot θ; and the angle θ has a definite relation to α, the angle of arc.

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

_RS_ = _a_ ⋅ α/√2.

Accordingly, the arc _RS_ will be just equal to the arc _MP_ when

θ cot θ = α/√2.

When θ cot θ > α/√2 or _MP_ > _RS_,

then division will take place as in _RS_.

When θ cot θ < α/√2, or _MP_ < _RS_,

then division will take place as in _MP_.

In the accompanying diagram (Fig. 150), I have plotted the various magnitudes with which we are concerned, in order to exhibit the several limiting values. Here we see, in the first place, the curve marked α, which shews on the (left-hand) vertical scale the various possible magnitudes of that angle (viz. the angle {365} of arc of the whole sector which we wish to divide), and on the horizontal scale the corresponding values of θ, or the angle which

determines the point on the periphery where it is cut by the partition-wall, _MP_. Two limiting cases are to be noticed here: (1) at 90° (point _A_ in diagram), because we are at present only {366} dealing with arcs no greater than a quadrant; and (2), the point (_B_) where the angle θ comes to equal the angle α, for after that point the construction becomes impossible, since an anticlinal bisecting partition-wall would be partly outside the cell. The only partition which, after the point, can possibly exist, is a periclinal one. This point, as our diagram shews us, occurs when the angles (α and θ) are each rather under 52°.

Next I have plotted, on the same diagram, and in relation to the same scales of angles, the corresponding lengths of the two partitions, viz. _RS_ and _MP_, their lengths being expressed (on the right-hand side of the diagram) in relation to the radius of the circle (_a_), that is to say the side wall, _OA_, of our cell.

The limiting values here are (1), _C_, _C′_, where the angle of arc is 90°, and where, as we have already seen, the two partition-walls have the relative magnitudes of _MP_ : _RS_ = 0·875 : 1·111; (2) the point _D_, where _RS_ equals unity, that is to say where the periclinal partition has the same length as a radial one; this occurs when α is rather under 82° (cf. the points _D_, _D′_); (3) the point _E_, where _RS_ and _MP_ intersect; that is to say the point at which the two partitions, periclinal and anticlinal, are of the same magnitude; this is the case, according to our diagram, when the angle of arc is just over 62½°. We see from this, then, that what we have called an anticlinal partition, as _MP_, is only likely to occur in a triangular or prismatic cell whose angle of arc lies between 90° and 62½°. In all narrower or more tapering cells, the periclinal partition will be of less area, and will therefore be more and more likely to occur.

The case (_F_) where the angle α is just 60° is of some interest. Here, owing to the curvature of the peripheral border, and the consequent fact that the peripheral angles are somewhat greater than the apical angle α, the periclinal partition has a very slight and almost imperceptible advantage over the anticlinal, the relative proportions being about as _MP_ : _RS_ = 0·73 : 0·72. But if the equilateral triangle be a plane spherical triangle, i.e. a plane triangle bounded by circular arcs, then we see that there is no longer any distinction at all between our two partitions; _MP_ and _RS_ are now identical.

On the same diagram, I have inserted the curve for values of {367} cosec θ − cot θ = _OM_, that is to say the distances from the centre, along the side of the cell, of the starting-point (_M_) of the anticlinal partition. The point _C″_ represents its position in the case of a quadrant, and shews it to be (as we have already said) about 3/10 of the length of the radius from the centre. If, on the other hand, our cell be an equilateral triangle, then we have to read off the point on this curve corresponding to α = 60°, and we find it at the point _F‴_ (vertically under _F_), which tells us that the partition now starts 4·5/10, or nearly halfway, along the radial wall.

――――――――――

The foregoing considerations carry us a long way in our investigations of many of the simpler forms of cell-division. Strictly speaking they are limited to the case of flattened cells, in which we can treat the problem as though we were simply partitioning a plane surface. But it is obvious that, though they do not teach us the whole conformation of the partition which divides a more complicated solid into two halves, yet they do, even in such a case, enlighten us so far, that they tell us the appearance presented in one plane of the actual solid. And as this is all that we see in a microscopic section, it follows that the results we have arrived at will greatly help us in the interpretation of microscopic appearances, even in comparatively complex cases of cell-division.

Let us now return to our quadrant cell (_OAPB_), which we have found to be divided into a triangular and a quadrilateral portion, as in Fig. 147 or Fig. 151; and let us now suppose the whole system to grow, in a uniform fashion, as a prelude to further subdivision. The whole quadrant, growing uniformly (or with equal radial increments), will still remain a quadrant, and it is obvious, therefore, that for every new increment of size, more will be added to the margin of its triangular portion than to the {368} narrower margin of its quadrilateral portion; and these increments will be in proportion to the angles of arc, viz. 55° 22′ : 34° 38′, or as ·96 : ·60, i.e. as 8 : 5. And accordingly, if we may assume (and the assumption is a very plausible one), that, just as the quadrant itself divided into two halves after it got to a certain size, so each of its two halves will reach the same size before again dividing, it is obvious that the triangular portion will be doubled in size, and therefore ready to divide, a considerable time before the quadrilateral part. To work out the problem in detail would lead us into troublesome mathematics; but if we simply assume that the increments are proportional to the increasing radii of the circle, we have the following equations:―

Let us call the triangular cell _T_, and the quadrilateral, _Q_ (Fig. 151); let the radius, _OA_, of the original quadrantal cell = _a_ = 1; and let the increment which is required to add on a portion equal to _T_ (such as _PP′A′A_) be called _x_, and let that required, similarly, for the doubling of _Q_ be called _x′_.

Then we see that the area of the original quadrant

= _T_ + _Q_ = ¼π_a_^2 = ·7854_a_^2,

while the area of _T_ = _Q_ = ·3927_a_^2.

The area of the enlarged sector, _p′OA′_,

= (_a_ + _x_)^2 × (55° 22′) ÷ 2 = ·4831(_a_ + _x_)^2,

and the area _OPA_

= _a_^2 × (55° 22′) ÷ 2 = ·4831_a_^2.

Therefore the area of the added portion, _T′_,

= ·4831 ((_a_ + _x_)^2 − _a_^2).

And this, by hypothesis,

= _T_ = ·3927_a_^2.

We get, accordingly, since _a_ = 1,

_x_^2 + 2_x_ = ·3927/·4831 = ·810,

and, solving,

_x_ + 1 = √1·81 = 1·345, or _x_ = 0·345.

Working out _x′_ in the same way, we arrive at the approximate value, _x′_ + 1 = 1·517. {369}

This is as much as to say that, supposing each cell tends to divide into two halves when (and not before) its original size is doubled, then, in our flattened disc, the triangular cell _T_ will tend to divide when the radius of the disc has increased by about a third (from 1 to 1·345), but the quadrilateral cell, _Q_, will not tend to divide until the linear dimensions of the disc have increased by about a half (from 1 to 1·517).

The case here illustrated is of no small general importance. For it shews us that a uniform and symmetrical growth of the organism (symmetrical, that is to say, under the limitations of a plane surface, or plane section) by no means involves a uniform or symmetrical growth of the individual cells, but may, under certain conditions, actually lead to inequality among these; and this inequality may be further emphasised by differences which arise out of it, in regard to the order of frequency of further subdivision. This phenomenon (or to be quite candid, this hypothesis, which is due to Berthold) is entirely independent of any change or variation in individual surface tensions; and accordingly it is essentially different from the phenomenon of unequal segmentation (as studied by Balfour), to which we have referred on p. 348.

In this fashion, we might go on to consider the manner, and the order of succession, in which the subsequent cell-divisions would tend to take place, as governed by the principle of minimal areas. But the calculations would grow more difficult, or the results got by simple methods would grow less and less exact. At the same time, some of these results would be of great interest, and well worth the trouble of obtaining. For instance, the precise manner in which our triangular cell, _T_, would next divide would be interesting to know, and a general solution of this problem is certainly troublesome to calculate. But in this particular case we can see that the width of the triangular cell near _P_ is so obviously less than that near either of the other two angles, that a circular arc cutting off that angle is bound to be the shortest possible bisecting line; and that, in short, our triangular cell will tend to subdivide, just like the original quadrant, into a triangular and a quadrilateral portion.

But the case will be different next time, because in this new {370} triangle, _PRQ_, the least width is near the innermost angle, that at _Q_; and the bisecting circular arc will therefore be opposite to _Q_, or (approximately) parallel to _PR_. The importance of this fact is at once evident; for it means to say that there soon comes a time when, whether by the division of triangles or of quadrilaterals, we find only quadrilateral cells adjoining the periphery of our circular disc. In the subsequent division of these quadrilaterals, the partitions will arise transversely to their long axes, that is to say, _radially_ (as _U_, _V_); and we shall consequently have a superficial or peripheral layer of quadrilateral cells, with sides approximately parallel, that is to say what we are accustomed to call _an epidermis_. And this epidermis or superficial layer will be in clear contrast with the more irregularly shaped cells, the products of triangles and quadrilaterals, which make up the deeper, underlying layers of tissue.

In following out these theoretic principles and others like to them, in the actual division of living cells, we must always bear in mind certain conditions and qualifications. In the first place, the law of minimal area and the other rules which we have arrived at are not absolute but relative: they are links, and very important links, in a chain of physical causation; they are always at work, but their effects may be overridden and concealed by the operation of other forces. Secondly, we must remember that, in the great majority of cases, the cell-system which we have in view is constantly increasing in magnitude by active growth; and by this means the form and also the proportions of the cells are continually liable to alteration, of which phenomenon we have already had an example. Thirdly, we must carefully remember that, until our cell-walls become absolutely solid and rigid, they are always apt to be modified in form owing to the tension of the adjacent {371} walls; and again, that so long as our partition films are fluid or semifluid, their points and lines of contact with one another may shift, like the shifting outlines of a system of soap-bubbles. This is the physical cause of the movements frequently seen among segmenting cells, like those to which Rauber called attention in the segmenting ovum of the frog, and like those more striking movements or accommodations which give rise to a so-called “spiral” type of segmentation.

――――――――――

Bearing in mind, then, these considerations, let us see what our flattened disc is likely to look like, after a few successive divisions into component cells. In Fig. 153, _a_, we have a diagrammatic representation of our disc, after it has divided into four quadrants, and each of these in turn into a triangular and a quadrilateral portion; but as yet, this figure scarcely suggests to us anything like the normal look of an aggregate of living cells. But let us go a little further, still limiting ourselves, however, to the consideration of the eight-celled stage. Wherever one of our radiating partitions meets the peripheral wall, there will (as we know) be a mutual tension between the three convergent films, which will tend to set their edges at equal angles to one another, angles that is to say of 120°. In consequence of this, the outer wall of each individual cell will (in this surface view of our disc) {372} be an arc of a circle of which we can determine the centre by the method used on p. 307; and, furthermore, the narrower cells, that is to say the quadrilaterals, will have this outer border somewhat more curved than their broader neighbours. We arrive, then, at the condition shewn in Fig. 153, _b_. Within the cell, also, wherever wall meets wall, the angle of contact must tend, in every case, to be an angle of 120°; and in no case may more than three films (as seen in section) meet in a point (_c_); and this condition, of the partitions meeting three by three, and at co-equal angles, will obviously involve the curvature of some, if not all, of the partitions (_d_) which in our preliminary investigation we treated as plane. To solve this problem in a general way is no easy matter; but it is a problem which Nature solves in every case where, as in the case we are considering, eight bubbles, or eight cells, meet together in a (plane or curved) surface. An approximate solution has been given in Fig. 153, _d_; and it will now at once be recognised that this figure has vastly more resemblance to an aggregate of living cells than had the diagram of Fig. 153, _a_ with which we began.

Just as we have constructed in this case a series of purely diagrammatic or schematic figures, so it will be as a rule possible to diagrammatise, with but little alteration, the complicated appearances presented by any ordinary aggregate of cells. The accompanying little figure (Fig. 154), of a germinating spore of a Liverwort (Riccia), after a drawing of Professor Campbell’s, scarcely needs further explanation: for it is well-nigh a typical diagram of the method of space-partitioning which we are now considering. Let us look again at our figures (on p. 359) of the disc of Erythrotrichia, from Berthold’s _Monograph of the Bangiaceae_ and redraw the earlier stages in diagrammatic fashion. In the following series of diagrams the new partitions, or those just about to form, are in each case outlined; and in the next succeeding stage they are shewn after settling down into position, and after exercising their respective tractions on the walls previously laid down. It is clear, I think, that these four diagrammatic figures represent all that is shewn in the first five stages drawn by Berthold from the plant itself; but the correspondence cannot {373} in this case be precisely accurate, for the simple reason that Berthold’s figures are taken from different individuals, and are therefore only approximately consecutive and not strictly continuous. The last of the six drawings in Fig. 144 is already too

complicated for diagrammatisation, that is to say it is too complicated for us to decipher with certainty the precise order of appearance of the numerous partitions which it contains. But in Fig. 156 I shew one more diagrammatic figure, of a disc which

has divided, according to the theoretical plan, into about sixty-four cells; and making due allowance for the successive changes which the mutual tensions and tractions of the partitions must {374} bring about, increasing in complexity with each succeeding stage, we can see, even at this advanced and complicated stage, a very considerable resemblance between the actual picture (Fig. 144) and the diagram which we have here constructed in obedience to a few simple rules.

In like manner, in the annexed figures, representing sections through a young embryo of a Moss, we have very little difficulty in discerning the successive stages that must have intervened between the two stages shewn: so as to lead from the just divided quadrants (one of which, by the way, has not yet divided in our figure (_a_)) to the stage (_b_) in which a well-marked epidermal layer surrounds an at first sight irregular agglomeration of “fundamental” tissue.

In the last paragraph but one, I have spoken of the difficulty of so arranging the meeting-places of a number of cells that at each junction only three cell-walls shall meet in a line, and all three shall meet it at equal angles of 120°. As a matter of fact, the problem is soluble in a number of ways; that is to say, when we have a number of cells, say eight as in the case considered, enclosed in a common boundary, there are various ways in which their walls can be made to meet internally, three by three, at equal angles; and these differences will entail differences also in the curvature of the walls, and consequently in the shape of the cells. The question is somewhat complex; it has been dealt with by Plateau, and treated mathematically by M. Van Rees[392].

If within our boundary we have three cells all meeting {375} internally, they must meet in a point; furthermore, they tend to do so at equal angles of 120°, and there is an end of the matter. If we have four cells, then, as we have already seen, the conditions are satisfied by interposing a little intermediate wall, the two extremities of which constitute the meeting-points of three cells each, and the upper edge of which marks the “polar furrow.” Similarly, in the case of five cells, we require _two_ little intermediate walls, and two polar furrows; and we soon arrive at the rule that, for _n_ cells, we require _n_ − 3 little longitudinal partitions (and corresponding polar furrows), connecting the triple junctions of the cells; and these little walls, like all the rest within the system, must be inclined to one another at angles of 120°. Where we have only one such wall (as in the case of four cells), or only two (as in the case of five cells), there is no room for ambiguity. But where we have three little connecting-walls, as in the case of six cells, it is obvious that we can arrange them in three different ways, as in the annexed Fig. 159. In the system of seven cells, the four partitions can be arranged in four ways; and the five partitions required in the case of eight cells can be arranged in no less than thirteen different ways, of which Fig. 158 shews some half-dozen only. It does not follow that, so to speak, these various {376} arrangements are all equally good; some are known to be much more stable than others, and some have never yet been realised in actual experiment.

The conditions which lead to the presence of any one of them, in preference to another, are as yet, so far as I am aware, undetermined, but to this point we shall return.

――――――――――

Examples of these various arrangements meet us at every turn, and not only in cell-aggregates, but in various cases where non-rigid and semi-fluid partitions (or partitions that were so to begin with) meet together. And it is a necessary consequence of this physical phenomenon, and of the limited and very small number of possible arrangements, that we get similar appearances, capable of representation by the same diagram, in the most diverse fields of biology[393].