CHAPTER VIII

THE FORMS OF TISSUES OR CELL-AGGREGATES (_continued_)

The problems which we have been considering, and especially that of

the bee’s cell, belong to a class of “isoperimetrical” problems, which

deal with figures whose surface is a minimum for a definite content or

volume. Such problems soon become difficult, but we may find many easy

examples which lead us towards the explanation of biological phenomena;

and the particular subject which we shall find most easy of approach

is that of the division, in definite proportions, of some definite

portion of space, by a partition-wall of minimal area. The theoretical

principles so arrived at we shall then attempt to apply, after the

manner of Berthold and Errera, to the actual biological phenomena of

cell-division.

This investigation we may approach in two ways: by considering, namely,

the partitioning off from some given space or area of one-half (or some

other fraction) of its content; or again, by dealing simultaneously

with the partitions necessary for the breaking up of a given space into

a definite number of compartments.

If we take, to begin with, the simple case of a cubical cell, it is

obvious that, to divide it into two halves, the smallest possible

partition-wall is one which runs parallel to, and midway between, two

of its opposite sides. If we call _a_ the length of one of the edges of

the cube, then _a_^2 is the area, alike of one of its sides, and of the

partition which we have interposed parallel, or normal, thereto. But if

we now consider the bisected cube, and wish to divide the one-half of

it again, it is obvious that another partition parallel to the first,

so far from being the smallest possible, is precisely twice the size of

a cross-partition perpendicular to it; {347} for the area of this new

partition is _a_ × _a_/2. And again, for a third bisection, our next

partition must be perpendicular to the other two, and it is obviously a

little square, with an area of (½_a_)^2 = ¼(_a_^2).

From this we may draw the simple rule that, for a rectangular body or

parallelopiped to be divided equally by means of a partition of minimal

area, (1) the partition must cut across the longest axis of the figure;

and (2) in the event of successive bisections, each partition must run

at right angles to its immediate predecessor.

[Illustration: Fig. 136. (After Berthold.)]

We have already spoken of “Sachs’s Rules,” which are an empirical

statement of the method of cell-division in plant-tissues; and we may

now set them forth in full.

(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.

[Illustration: Fig. 137.]

If then, in the process of division of a cubical cell, it so divide

that the two portions be not equal in volume but that one portion by

anything less than about three-tenths of the whole, or three-sevenths

of the other portion, there will be a tendency for the cell to divide,

not by means of a plane transverse partition, but by means of a curved,

cylindrical wall cutting off one corner of the original cell; and the

part so cut off will be one-quarter of a cylinder.

By a similar calculation we can shew that a _spherical_ wall, cutting

off one solid angle of the cube, and constituting an octant of a

sphere, would likewise be of less area than a plane partition as

soon as the volume to be enclosed was not greater than about {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.

[Illustration: Fig. 138.]

In this figure, we see that the plane transverse partition, whatever

fraction of the cube it cut off, is always of the same dimensions,

that is to say is always equal to _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.

[Illustration: Fig. 139.]

The case we have dealt with is of little practical importance to the

biologist, because the cases in which a cubical, or rectangular,

cell divides unequally, and unsymmetrically, are apparently few; but

we can find, as Berthold pointed out, a few examples, for instance

in the hairs within the reproductive “conceptacles” of certain Fuci

(Sphacelaria, etc., Fig. 139), or in the “paraphyses” of mosses

(Fig. 142). But it is of great theoretical importance: as serving to

introduce us to a large class of cases, in which the shape and the

relative dimensions of the original cavity lead, according to the

principle of minimal areas, to cell-division in very definite and

sometimes unexpected ways. It is not easy, nor indeed possible, to

give a generalised account of these cases, for the limiting conditions

are somewhat complex, and the mathematical treatment soon becomes

difficult. But it is easy to comprehend a few simple cases, which of

themselves will carry us a good long way; and which will go far to

convince the student that, in other cases {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

[Illustration: Fig. 140.]

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.

[Illustration: Fig. 141.]

In like manner, if we have a spheroidal body, less than a hemisphere,

such for instance as a low, watch-glass shaped cell (Fig. 141, _a_),

it is obvious that the smallest possible partition by which we can

divide it into two equal halves {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}

[Illustration: Fig. 142. S-shaped partitions: _A_, from _Taonia

atomaria_ (after Reinke); _B_, from paraphyses of _Fucus_; _C_, from

rhizoids of Moss; _D_, from paraphyses of _Polytrichum_.]

While in very many cases the partitions (like the walls of the original

cell) will be either plane or spherical, a more complex curvature will

be assumed under a variety of conditions. It will be apt to occur,

for instance, when the mother-cell is irregular in shape, and one

particular case of such asymmetry will be that in which (as in Fig.

143) the cell has begun to branch, or give off a diverticulum, before

division takes place. A very complicated case of a different kind,

though not without its analogies to the cases we are considering, will

occur in the partitions of minimal area which subdivide the spiral

tube of a nautilus, as we shall presently see. And again, whenever we

have a marked internal asymmetry of the cell, leading to irregular

and anomalous modes of division, in which the cell is not necessarily

divided into two equal halves and in which the partition-wall may

assume an oblique position, then apparently anomalous curvatures will

tend to make their appearance[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.

[Illustration: Fig. 143. Diagrammatic explanation of S-shaped

partition.]

Many such cases of partitions with complex or double curvature exist,

but they are not always easy of recognition, nor is the particular

case where they appear in a _terminal_ cell a common one. We may see

them, for instance, in the roots (or rhizoids) of Mosses, especially

at the point of development of a new rootlet (Fig. 142, C); and again

among Mosses, in the “paraphyses” of the male prothalli (e.g. in

_Polytrichum_), we find more or less similar partitions (D). They are

frequent also among many Fuci, as in the hairs or paraphyses of Fucus

itself (B). In _Taonia atomaria_, as figured in Reinke’s memoir on

the Dictyotaceae of the Gulf of Naples[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.

[Illustration: Fig. 144. Development of _Erythrotrichia_. (After

Berthold.)]

[Illustration: Fig. 145.]

Fig. 144 (taken from Berthold’s _Monograph of the Naples Bangiaciae_)

represents younger and older discs of the little alga _Erythrotrichia

discigera_; and it will be seen that, in all stages save the first, we

have an arrangement of cell-partitions which looks somewhat complex,

but into which we must attempt to throw some light and order. Starting

with the original single, and flattened, {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.

[Illustration: Fig. 146.]

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.

[Illustration: Fig. 147.]

The two cells into which our original quadrant is now divided, while

they are equal in volume, are of very different shapes; the {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.

[Illustration: Fig. 148. Segmentation of frog’s egg, under artificial

compression. (After Roux.)]

――――――――――

The question how the other, or triangular, portion of the divided

quadrant will next divide leads us to another well-defined problem,

which is only a slight extension, making allowance for the circular

arcs, of that elementary problem of the triangle we have already

considered. We know now that an entire quadrant must divide (so that

its bisecting wall shall have the least possible area) by means of an

anticlinal partition, but how about any smaller sectors of circles?

It is obvious in the case of a small prismatic {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.

[Illustration: Fig. 149.]

Moreover, in the case of the periclinal boundary, _RS_ (Fig. 147) (or

_ab_, Fig. 149), we know that, if it bisect the cell,

_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

[Illustration: Fig. 150.]

determines the point on the periphery where it is cut by the

partition-wall, _MP_. Two limiting cases are to be noticed here: (1)

at 90° (point _A_ in diagram), because we are at present only {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.

[Illustration: Fig. 151.]

Let us now return to our quadrant cell (_OAPB_), which we have found

to be divided into a triangular and a quadrilateral portion, as in

Fig. 147 or Fig. 151; and let us now suppose the whole system to

grow, in a uniform fashion, as a prelude to further subdivision. The

whole quadrant, growing uniformly (or with equal radial increments),

will still remain a quadrant, and it is obvious, therefore, that

for every new increment of size, more will be added to the margin

of its triangular portion than to the {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.

[Illustration: Fig. 152.]

In following out these theoretic principles and others like to them,

in the actual division of living cells, we must always bear in mind

certain conditions and qualifications. In the first place, the law

of minimal area and the other rules which we have arrived at are not

absolute but relative: they are links, and very important links, in a

chain of physical causation; they are always at work, but their effects

may be overridden and concealed by the operation of other forces.

Secondly, we must remember that, in the great majority of cases, the

cell-system which we have in view is constantly increasing in magnitude

by active growth; and by this means the form and also the proportions

of the cells are continually liable to alteration, of which phenomenon

we have already had an example. Thirdly, we must carefully remember

that, until our cell-walls become absolutely solid and rigid, they are

always apt to be modified in form owing to the tension of the adjacent

{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.

――――――――――

[Illustration: Fig. 153. Diagram of flattened or discoid cell

dividing into octants: to shew gradual tendency towards a position of

equilibrium.]

Bearing in mind, then, these considerations, let us see what our

flattened disc is likely to look like, after a few successive divisions

into component cells. In Fig. 153, _a_, we have a diagrammatic

representation of our disc, after it has divided into four quadrants,

and each of these in turn into a triangular and a quadrilateral

portion; but as yet, this figure scarcely suggests to us anything like

the normal look of an aggregate of living cells. But let us go a little

further, still limiting ourselves, however, to the consideration of the

eight-celled stage. Wherever one of our radiating partitions meets the

peripheral wall, there will (as we know) be a mutual tension between

the three convergent films, which will tend to set their edges at equal

angles to one another, angles that is to say of 120°. In consequence of

this, the outer wall of each individual cell will (in this surface view

of our disc) {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.

[Illustration: Fig. 154.]

Just as we have constructed in this case a series of purely

diagrammatic or schematic figures, so it will be as a rule possible to

diagrammatise, with but little alteration, the complicated appearances

presented by any ordinary aggregate of cells. The accompanying little

figure (Fig. 154), of a germinating spore of a Liverwort (Riccia),

after a drawing of Professor Campbell’s, scarcely needs further

explanation: for it is well-nigh a typical diagram of the method of

space-partitioning which we are now considering. Let us look again

at our figures (on p. 359) of the disc of Erythrotrichia, from

Berthold’s _Monograph of the Bangiaceae_ and redraw the earlier stages

in diagrammatic fashion. In the following series of diagrams the new

partitions, or those just about to form, are in each case outlined;

and in the next succeeding stage they are shewn after settling down

into position, and after exercising their respective tractions on the

walls previously laid down. It is clear, I think, that these four

diagrammatic figures represent all that is shewn in the first five

stages drawn by Berthold from the plant itself; but the correspondence

cannot {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

[Illustration: Fig. 155. Theoretical arrangement of successive

partitions in a discoid cell; for comparison with Fig. 144.]

complicated for diagrammatisation, that is to say it is too

complicated for us to decipher with certainty the precise order of

appearance of the numerous partitions which it contains. But in Fig.

156 I shew one more diagrammatic figure, of a disc which

[Illustration: Fig. 156. Theoretical division of a discoid cell into

sixty-four chambers: no allowance being made for the mutual tractions

of the cell-walls.]

has divided, according to the theoretical plan, into about sixty-four

cells; and making due allowance for the successive changes which the

mutual tensions and tractions of the partitions must {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.

[Illustration: Fig. 157. Sections of embryo of a moss. (After

Kienitz-Gerloff.)]

In the last paragraph but one, I have spoken of the difficulty of so

arranging the meeting-places of a number of cells that at each junction

only three cell-walls shall meet in a line, and all three shall meet it

at equal angles of 120°. As a matter of fact, the problem is soluble in

a number of ways; that is to say, when we have a number of cells, say

eight as in the case considered, enclosed in a common boundary, there

are various ways in which their walls can be made to meet internally,

three by three, at equal angles; and these differences will entail

differences also in the curvature of the walls, and consequently in the

shape of the cells. The question is somewhat complex; it has been dealt

with by Plateau, and treated mathematically by M. Van Rees[392].

[Illustration: Fig. 158. Various possible arrangements of intermediate

partitions, in groups of 4, 5, 6, 7 or 8 cells.]

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

[Illustration: Fig. 159.]

Among the published figures of embryonic stages and other cell

aggregates, we only discern these little intermediate partitions in

cases where the investigator has drawn carefully just what lay before

him, without any preconceived notions as to radial or other symmetry;

but even in other cases we can generally recognise, without much

difficulty, what the actual arrangement was whereby the cell-walls

met together in equilibrium. I have a strong suspicion that a leaning

towards Sachs’s Rule, that one cell-wall tends to set itself at right

angles to another cell-wall (a rule whose strict limitations, and

narrow range of application, we have already {377} considered) is

responsible for many inaccurate or incomplete representations of the

mutual arrangement of aggregated cells.

[Illustration: Fig. 160. Segmenting egg of _Trochus_. (After Robert.)]

[Illustration: Fig. 161. Two views of segmenting egg of _Cynthia

partita_. (After Conklin.)]

[Illustration: Fig. 162. (_a_) Section of apical cone of _Salvinia_.

(After Pringsheim[394].) (_b_) Diagram of probable actual arrangement.]

[Illustration: Fig. 163. Egg of _Pyrosoma_. (After Korotneff).]

[Illustration: Fig. 164. Egg of _Echinus_, segmenting under pressure.

(After Driesch.)]

In the accompanying series of figures (Figs. 160–167) I have {378}

set forth a few aggregates of eight cells, mostly from drawings of

segmenting eggs. In some cases they shew clearly the manner in which

the cells meet one another, always at angles of 120°, and always with

the help of five intermediate boundary walls within the eight-celled

system; in other cases I have added a slightly altered drawing, so as

to shew, with as little change as {379} possible, the arrangement

of boundaries which probably actually existed, and gave rise to the

appearance which the observer drew. These drawings may be compared

with the various diagrams of Fig. 158, in which some seven out of the

possible thirteen arrangements of five intermediate partitions (for a

system of eight cells) have been already set forth.

[Illustration: Fig. 165. (_a_) Part of segmenting egg of Cephalopod

(after Watase); (_b_) probable actual arrangement.]

[Illustration: Fig. 166. (_a_) Egg of _Echinus_; (_b_) do. of _Nereis_,

under pressure. (After Driesch).]

[Illustration: Fig. 167. (_a_) Egg of frog, under pressure (after

Roux); (_b_) probable actual arrangement.]

It will be seen that M. Robert-Tornow’s figure of the segmenting egg of

Trochus (Fig. 160) clearly shews the cells grouped after the fashion of

Fig. 158, _a_. In like manner, Mr Conklin’s figure of the ascidian egg

(_Cynthia_) shews equally clearly the arrangement _g_.

A sea-urchin egg, segmenting under pressure, as figured by Driesch,

scarcely requires any modification of the drawing to appear as

a diagram of the type _d_. Turning for a moment to a botanical

illustration, we have a figure of Pringsheim’s shewing an eight-celled

stage in the apex of the young cone of Salvinia; it is in all

probability referable, as in my modified diagram, to type _c_. Beside

it is figured a very different object, a segmenting egg of the Ascidian

_Pyrosoma_, after Korotneff; it may be that this also is to be referred

to type _c_, but I think it is more easily referable to type _b_. For

there is a difference between this diagram and that of Salvinia, in

that here apparently, of the pairs of lateral cells, the upper and the

lower cell are alternately the larger, while in the diagram of Salvinia

the lower lateral cells both appear much larger than the upper ones;

and this difference tallies with the appearance produced if we fill

in the eight cells according to the type _b_ or the type _c_. In the

segmenting cuttlefish egg, there is again a slight dubiety as to which

type it should be referred to, but it is in all probability referable,

like Driesch’s Echinus egg, to _d_. Lastly, I have copied from Roux a

curious figure of the egg of _Rana esculenta_, viewed from the animal

pole, which appears to me referable, in all probability, to type _g_.

Of type _f_, in which the five partitions form a figure with four

re-entrant angles, that is to say a figure representing the five sides

of a hexagon, I have found no examples among segmenting eggs, and that

arrangement in all probability is a very unstable one.

――――――――――

It is obvious enough, without more ado, that these phenomena are in

the strictest and completest way common to both plants {380} and

animals. In other words they tally with, and they further extend,

the general and fundamental conclusions laid down by Schwann, in

his _Mikroskopische Untersuchungen über die Uebereinstimmung in der

Struktur und dem Wachsthum der Thiere und Pflanzen_.

But now that we have seen how a certain limited number of types of

eight-celled segmentation (or of arrangements of eight cell-partitions)

appear and reappear, here and there, throughout the whole world of

organisms, there still remains the very important question, whether

_in each particular organism_ the conditions are such as to lead to

one particular arrangement being predominant, characteristic, or even

invariable. In short, is a particular arrangement of cell-partitions to

be looked upon (as the published figures of the embryologist are apt to

suggest) as a _specific character_, or at least a constant or normal

character, of the particular organism? The answer to this question

is a direct negative, but it is only in the work of the most careful

and accurate observers that we find it revealed. Rauber (whom we have

more than once had occasion to quote) was one of those embryologists

who recorded just what he saw, without prejudice or preconception; as

Boerhaave said of Swammerdam, _quod vidit id asseruit_. Now Rauber has

put on record a considerable number of variations in the arrangement

of the first eight cells, which form a discoid surface about the

dorsal (or “animal”) pole of the frog’s egg. In a certain number of

cases these figures are identical with one another in type, identical

(that is to say) save for slight differences in magnitude, relative

proportions, or orientation. But I have selected (Fig. 168) six

diagrammatic figures, which are all _essentially different_, and these

diagrams seem to me to bear intrinsic evidence of their accuracy: the

curvatures of the partition-walls, and the angles at which they meet

agree closely with the requirements of theory, and when they depart

from theoretical symmetry they do so only to the slight extent which

we should naturally expect in a material and imperfectly homogeneous

system[395]. {381}

[Illustration: Fig. 168. Various modes of grouping of eight cells, at

the dorsal or epiblastic pole of the frog’s egg. (After Rauber.)]

Of these six illustrations, two are exceptional. In Fig. 168, 5, we

observe that one of the eight cells is surrounded on all sides by the

other seven. This is a perfectly natural condition, and represents,

like the rest, a phase of partial or conditional equilibrium. But it is

not included in the series we are now considering, which is restricted

to the case of eight cells extending outwards to a common boundary.

The condition shewn in Fig. 168, 6, is again peculiar, and is probably

rare; but it is included under the cases considered on p. 312, in

which the cells are not in complete fluid contact, but are separated

by little droplets of extraneous matter; it needs no further comment.

But the other four cases are beautiful diagrams of space-partitioning,

similar to those we have just been considering, but so exquisitely

clear that they need no modification, no “touching-up,” to exhibit

their mathematical regularity. It will easily be recognised that in

Fig. 168, 1 and 2, we have the arrangements corresponding to _a_

and _d_ of our diagram (Fig. 158): but the other two (i.e. 3 and 4)

represent other of the thirteen possible arrangements, which are not

included in that {382} diagram. It would be a curious and interesting

investigation to ascertain, in a large number of frogs’ eggs, all at

this stage of development, the percentage of cases in which these

various arrangements occur, with a view of correlating their frequency

with the theoretical conditions (so far as they are known, or can

be ascertained) of relative stability. One thing stands out as very

certain indeed: that the elementary diagram of the frog’s egg commonly

given in text-books of embryology,—in which the cells are depicted as

uniformly symmetrical quadrangular bodies,—is entirely inaccurate and

grossly misleading[396].

We now begin to realise the remarkable fact, which may even appear

a startling one to the biologist, that all possible groupings or

arrangements whatsoever of eight cells (where all take part in the

_surface_ of the group, none being submerged or wholly enveloped by

the rest) are referable to some one or other of _thirteen_ types or

forms. And that all the thousands and thousands of drawings which

diligent observers have made of such eight-celled structures, animal

or vegetable, anatomical, histological or embryological, are one and

all representations of some one or another of these thirteen types:—or

rather indeed of somewhat less than the whole thirteen, for there is

reason to believe that, out of the total number of possible groupings,

a certain small number are essentially unstable, and have at best, in

the concrete, but a transitory and evanescent existence.

――――――――――

Before we leave this subject, on which a vast deal more might be said,

there are one or two points which we must not omit to consider. Let us

note, in the first place, that the appearance which our plane diagrams

suggest of inequality of the several cells is apt to be deceptive; for

the differences of magnitude apparent in one plane may well be, and

probably generally are, balanced by equal and opposite differences in

another. Secondly, let us remark that the rule which we are considering

refers only {383} to angles, and to the number, not to the length of

the intermediate partitions; it is to a great extent by variations in

the length of these that the magnitudes of the cells may be equalised,

or otherwise balanced, and the whole system brought into equilibrium.

Lastly, there is a curious point to consider, in regard to the number

of actual contacts, in the various cases, between cell and cell. If

we inspect the diagrams in Fig. 169 (which represent three out of our

thirteen possible arrangements of eight cells) we shall see that, in

the case of type _b_, two cells are each in contact with two others,

two cells with three others, and four cells each with four other cells.

In type _a_ four cells are each in contact with two, two with four,

and two with five. In type _f_, two are in contact with two, four with

three, and one with no less than seven. In all cases the

[Illustration: Fig. 169.]

number of contacts is twenty-six in all; or, in other words, there

are thirteen internal partitions, besides the eight peripheral walls.

For it is easy to see that, in all cases of _n_ cells with a common

external boundary, the number of internal partitions is 2_n_ − 3; or

the number of what we call the internal or interfacial contacts is

2(2_n_ − 3). But it would appear that the most stable arrangements are

those in which the total number of contacts is most evenly divided,

and the least stable are those in which some one cell has, as in

type _f_, a predominant number of contacts. In a well-known series

of experiments, Roux has shewn how, by means of oil-drops, various

arrangements, or aggregations, of cells can be simulated; and in Fig.

170 I shew a number of Roux’s figures, and have ascribed them to what

seem to be their appropriate “types” among those which we have just

been considering; but {384} it will be observed that in these figures

of Roux’s the drops are not always in complete contact, a little

air-bubble often keeping them apart at their apical junctions, so that

we see the configuration towards which the system is _tending_ rather

than that which it has fully attained[397]. The type which we have

called _f_ was found by Roux to be unstable, the large (or apparently

large) drop _a″_ quickly passing into the centre of the system, and

here taking up a position of equilibrium in which, as usual, three

cells meet throughout in a point, at equal angles, and in which, in

this case, all the cells have an equal number of “interfacial” contacts.

[Illustration: Fig. 170. Aggregations of oil-drops. (After Roux.) Figs.

4–6 represent successive changes in a single system.]

We need by no means be surprised to find that, in such arrangements,

the commonest and most stable distributions are those in which the

cell-contacts are distributed as uniformly as possible between the

several cells. We always expect to find some such tendency to equality

in cases where we have to do with small oscillations on either side of

a symmetrical condition. {385}

The rules and principles which we have arrived at from the point of

view of surface tension have a much wider bearing than is at once

suggested by the problems to which we have applied them; for in this

elementary study of the cell-boundaries in a segmenting egg or tissue

we are on the verge of a difficult and important subject in pure

mathematics. It is a subject adumbrated by Leibniz, studied somewhat

more deeply by Euler, and greatly developed of recent years. It is the

_Geometria Situs_ of Gauss, the _Analysis Situs_ of Riemann, the Theory

of Partitions of Cayley, and of Spatial Complexes of Listing[398]. The

crucial point for the biologist to comprehend is, that in a closed

surface divided into a number of faces, the arrangement of all the

faces, lines and points in the system is capable of analysis, and that,

when the number of faces or areas is small, the number of possible

arrangements is small also. This is the simple reason why we meet in

such a case as we have been discussing (viz. the arrangement of a group

or system of eight cells) with the same few types recurring again and

again in all sorts of organisms, plants as well as animals, and with no

relation to the lines of biological classification: and why, further,

we find similar configurations occurring to mark the symmetry, not

of cells merely, but of the parts and organs of entire animals. The

phenomena are not “functions,” or specific characters, of this or that

tissue or organism, but involve general principles which lie within the

province of the mathematician.

――――――――――

The theory of space-partitioning, to which the segmentation of the egg

gives us an easy practical introduction, is illustrated in much more

complex ways in other fields of natural history. A very beautiful but

immensely complicated case is furnished by the “venation” of the wings

of insects. Here we have sometimes (as in the dragon-flies), a general

reticulum of small, more or less hexagonal “cells”: but in most other

cases, in flies, bees, butterflies, etc., we have a moderate number of

cells, whose partitions always impinge upon one another three by three,

and whose arrangement, therefore, includes of necessity a number of

small intermediate partitions, analogous to our polar furrows. I think

{386} that a mathematical study of these, including an investigation

of the “deformation” of the wing (that is to say, of the changes in

shape and changes in the form of its “cells” which it undergoes during

the life of the individual, and from one species to another) would be

of great interest. In very many cases, the entomologist relies upon

this venation, and upon the occurrence of this or that intermediate

vein, for his classification, and therefore for his hypothetical

phylogeny of particular groups; which latter procedure hardly commends

itself to the physicist or the mathematician.

[Illustration: Fig. 171. (A) _Asterolampra marylandica_, Ehr.; (B, C)

_A. variabilis_, Grev. (After Greville.)]

Another case, geometrically akin but biologically very different, is

to be found in the little diatoms of the genus Asterolampra, and their

immediate congeners[399]. In Asterolampra we have a little disc, in

which we see (as it were) radiating spokes of one material, alternating

with intervals occupied on the flattened wheel-like disc by another

(Fig. 171). The spokes vary in number, but the general appearance is

in a high degree suggestive of the Chladni figures produced by the

vibration of a circular plate. The spokes broaden out towards the

centre, and interlock by visible junctions, which obey the rule of

triple intersection, and accordingly exemplify the partition-figures

with which we are dealing. But whereas we have found the particular

arrangement in which one cell is in contact with all the rest to

be unstable, according to Roux’s oil-drop experiments, and to be

conspicuous {387} by its absence from our diagrams of segmenting

eggs, here in Asterolampra, on the other hand, it occurs frequently,

and is indeed the commonest arrangement[400] (Fig. 171, B). In all

probability, we are entitled to consider this marked difference natural

enough. For we may suppose that in Asterolampra (unlike the case of the

segmenting egg) the tendency is to perfect radial symmetry, all the

spokes emanating from a point in the centre: such a condition would be

eminently unstable, and would break down under the least asymmetry. A

very simple, perhaps the simplest case, would be that one single spoke

should differ slightly from the rest, and should so tend to be drawn in

amid the others, these latter remaining similar and symmetrical among

themselves. Such a configuration would be vastly less unstable than

the original one in which all the boundaries meet in a point; and the

fact that further progress is not made towards other configurations of

still greater stability may be sufficiently accounted for by viscosity,

rapid solidification, or other conditions of restraint. A perfectly

stable condition would of course be obtained if, as in the case of

Roux’s oil-drop (Fig. 170, 6), one of the cellular spaces passed into

the centre of the system, the other partitions radiating outwards from

its circular wall to the periphery of the whole system. Precisely

such a condition occurs among our diatoms; but when it does so, it is

looked upon as the mark and characterisation of the _allied genus_

Arachnoidiscus.

――――――――――

[Illustration: Fig. 172. Section of Alcyonarian polype.]

In a diagrammatic section of an Alcyonarian polype (Fig. 172), we have

eight chambers set, symmetrically, about a ninth, which constitutes

the “stomach.” In this arrangement there is no difficulty, for it is

obvious that, throughout the system, three boundaries meet (in plane

section) in a point. In many corals we have as {388} simple, or even

simpler conditions, for the radiating calcified partitions either

converge upon a central chamber, or fail to meet it and end freely.

But in a few cases, the partitions or “septa” converge to meet _one

another_, there being no central chamber on which they may impinge; and

here the manner in which contact is effected becomes complicated, and

involves problems identical with those which we are now studying.

[Illustration: Fig. 173. _Heterophyllia angulata_. (After Nicholson.)]

In the great majority of corals we have as simple or even simpler

conditions than those of Alcyonium; for as a rule the calcified

partitions or septa of the coral either converge upon a central chamber

(or central “columella”), or else fail to meet it and end freely. In

the latter case the problem of space-partitioning does not arise; in

the former, however numerous the septa be, their separate contacts

with the wall of the central chamber comply with our fundamental rule

according to which three lines and no more meet in a point, and from

this simple and symmetrical arrangement there is little tendency to

variation. But in a few cases, the septal partitions converge to

meet _one another_, there being no central chamber on which they may

impinge; and here the manner in which contact is effected becomes

complicated, and involves problems of space-partitioning identical

with those which we are now studying. In the genus Heterophyllia and

in a few allied forms we have such conditions, and students of the

Coelenterata have found them very puzzling. McCoy[401], their first

discoverer, pronounced these corals to be “totally unlike” any other

group, recent or fossil; and Professor Martin Duncan, writing a memoir

on Heterophyllia and its allies[402], described them as “paradoxical in

their anatomy.”

[Illustration: Fig. 174. _Heterophyllia_ sp. (After Martin Duncan.)]

The simplest or youngest Heterophylliae known have six septa (as in

Fig. 174, _a_); in the case figured, four of these septa are conjoined

two and two, thus forming the usual triple junctions together with

their intermediate partition-walls: and in the {389} case of the other

two we may fairly assume that their proper and original arrangement

was that of our type 6_b_ (Fig. 158), though the central intermediate

partition has been crowded out by partial coalescence. When with

increasing age the septa become more numerous, their arrangement

becomes exceedingly variable; for the simple reason that, from the

mathematical point of view, the number of possible arrangements, of 10,

12 or more cellular partitions in triple contact, tends to increase

with great rapidity, and there is little to choose between many of

them in regard to symmetry and equilibrium. But while, mathematically

speaking, each particular case among the multitude of possible cases

is an orderly and definite arrangement, from the purely biological

point of view on the other hand no law or order is recognisable; and so

McCoy described the genus as being characterised by the possession of

septa “destitute of any order of arrangement, but irregularly branching

and coalescing in their passage from the solid external walls towards

some indefinite point near the centre where the few main lamellae

irregularly anastomose.” {390}

In the two examples figured (Fig. 174), both comparatively simple ones,

it will be seen that, of the main chambers, one is in each case an

unsymmetrical one; that is to say, there is one chamber which is in

contact with a greater number of its neighbours than any other, and

which at an earlier stage must have had contact with them all; this was

the case of our type _f_, in the eight-celled system (Fig. 158). Such

an asymmetrical chamber (which may occur in a system of any number of

cells greater than six), constitutes what is known to students of the

Coelenterata as a “fossula”; and we may recognise it not only here,

but also in Zaphrentis and its allies, and in a good many other corals

besides. Moreover certain corals are described as having more than one

fossula: this appearance being naturally produced under certain of

the other asymmetrical variations of normal space-partitioning. Where

a single fossula occurs, we are usually told that it is a symptom of

“bilaterality”; and this is in turn interpreted as an indication of

a higher grade of organisation than is implied in the purely “radial

symmetry” of the commoner types of coral. The mathematical aspect of

the case gives no warrant for this interpretation.

Let us carefully notice (lest we run the risk of confusing two

distinct problems) that the space-partitioning of Heterophyllia by

no means agrees with the details of that which we have studied in

(for instance) the case of the developing disc of Erythrotrichia: the

difference simply being that Heterophyllia illustrates the general

case of cell-partitioning as Plateau and Van Rees studied it, while

in Erythrotrichia, and in our other embryological and histological

instances, we have found ourselves justified in making the additional

assumption that each new partition divided a cell into _co-equal

parts_. No such law holds in Heterophyllia, whose case is essentially

different from the others: inasmuch as the chambers whose partition

we are discussing in the coral are mere empty spaces (empty save

for the mere access of sea-water); while in our histological and

embryological instances, we were speaking of the division of a cellular

unit of living protoplasm. Accordingly, among other differences, the

“transverse” or “periclinal” partitions, which were bound to appear at

regular intervals and in definite positions, when co-equal bisection

was a feature of the {391} case, are comparatively few and irregular

in the earlier stages of Heterophyllia, though they begin to appear in

numbers after the main, more or less radial, partitions have become

numerous, and when accordingly these radiating partitions come to

bound narrow and almost parallel-sided interspaces; then it is that

the transverse or periclinal partitions begin to come in, and form

what the student of the Coelenterata calls the “dissepiments” of the

coral. We need go no further into the configuration and anatomy of the

corals; but it seems to me beyond a doubt that the whole question of

the complicated arrangement of septa and dissepiments throughout the

group (including the curious vesicular or bubble-like tissue of the

Cyathophyllidae and the general structural plan of the Tetracoralla,

such as Streptoplasma and its allies) is well worth investigation from

the physical and mathematical point of view, after the fashion which is

here slightly adumbrated.

[Illustration: Fig. 175. Diagrammatic section of a Ctenophore

(_Eucharis_).]

――――――――――

The method of dividing a circular, or spherical, system into eight

parts, equal as to their areas but unequal in their peripheral

boundaries, is probably of wide biological application; that is to say,

without necessarily supposing it to be rigorously followed, the typical

configuration which it yields seems to recur again and again, with

more or less approximation to precision, and under widely different

circumstances. I am inclined to think, for instance, that the unequal

division of the surface of a Ctenophore by its {392} meridian-like

ciliated bands is a case in point (Fig. 175). Here, if we imagine

each quadrant to be twice bisected by a curved anticline, we shall

get what is apparently a close approximation to the actual position

of the ciliated bands. The case however is complicated by the fact

that the sectional plan of the organism is never quite circular, but

always more or less elliptical. One point, at least, is clearly seen

in the symmetry of the Ctenophores; and that is that the radiating

canals which pass outwards to correspond in position with the ciliated

bands, have no common centre, but diverge from one another by repeated

bifurcations, in a manner comparable to the conjunctions of our

cell-walls.

In like manner I am inclined to suggest that the same principle may

help us to understand the apparently complex arrangement of the

skeletal rods of a larval Echinoderm, and the very complex conformation

of the larva which is brought about by the presence of these long,

slender skeletal radii.

[Illustration: Fig. 176. Diagrammatic arrangement of partitions,

represented by skeletal rods, in larval Echinoderm (_Ophiura_).]

In Fig. 176 I have divided a circle into its four quadrants, and have

bisected each quadrant by a circular arc (_BC_), passing from radius to

periphery, as in the foregoing cases of cell-division; and I have again

bisected, in a similar way, the triangular halves of each quadrant

(_DD_). I have also inserted a small circle in the middle of the

figure, concentric with the large one. If now we imagine those lines

in the figure which I have drawn black to be replaced by solid rods we

shall have at once the frame-work of an Ophiurid (Pluteus) larva. Let

us imagine all these arms to be {393} bent symmetrically downwards, so

that the plane of the paper is transformed into a spheroidal surface,

such as that of a hemisphere, or that of a tall conical figure with

curved sides; let a membrane be spread, umbrella-like, between the

outstretched skeletal rods, and let its margin loop from rod to rod in

curves which are possibly catenaries, but are more probably portions

of an “elastic curve,” and the outward resemblance to a Pluteus

larva is now complete. By various slight modifications, by altering

the relative lengths of the rods, by modifying their curvature or

by replacing the curved rod by a tangent to itself, we can ring the

changes which lead us from one known type of Pluteus to another. The

case of the Bipinnaria larvae of Echinids is certainly analogous,

but it becomes very much more complicated; we have to do with a more

complex partitioning of space, and I confess that I am not yet able to

represent the more complicated forms in so simple a way.

[Illustration: Fig. 177. Pluteus-larva of Ophiurid.]

――――――――――

[Illustration: Fig. 178. Diagrammatic development of Stomata in

_Sedum_. (Cf. fig. in Sachs’s _Botany_, 1882, p. 103.)]

There are a few notable exceptions (besides the various unequally

segmenting eggs) to the general rule that in cell-division the

mother-cell tends to divide into equal halves; and one of these

exceptional cases is to be found in connection with the development of

“stomata” in the leaves of plants. The epidermal cells by which the

leaf is covered may be of various shapes; sometimes, as in a hyacinth,

they are oblong, but more often they have an irregular shape in which

we can recognise, more or less clearly, a distorted or imperfect

hexagon. In the case of the oblong cells, a transverse partition

will be the least possible, whether the cell be equally or unequally

divided, unless (as we have already seen) {394} the space to be cut

off be a very small one, not more than about three-tenths the area of

a square based on the _short_ side of the original rectangular cell.

As the portion usually cut off is not nearly so small as this, we

get the form of partition shewn in Fig. 179, and the cell so cut off

is next bisected by a partition at right angles to the first; this

latter partition splits, and the two last-formed cells constitute the

so-called “guard-cells” of the

[Illustration: Fig. 179. Diagrammatic development of stomata in

Hyacinth.]

stoma. In other cases, as in Fig. 178, there will come a point where

the minimal partition necessary to cut off the required fraction of

the cell-content is no longer a transverse one, but is a portion of a

cylindrical wall (2) cutting off one corner of the mother-cell. The

cell so cut off is now a certain segment of a circle, with an arc of

approximately 120°; and its next division will be by means of a curved

wall cutting it into a triangular and a quadrangular portion (3). The

triangular portion will continue to divide in a similar way (4, 5),

and at length (for a reason which is not yet clear) the partition wall

{395} between the new-formed cells splits, and again we have the

phenomenon of a “stoma” with its attendant guard-cells. In Fig. 179 are

shewn the successive stages of division, and the changing curvatures

of the various walls which ensue as each subsequent partition appears,

introducing a new tension into the system.

It is obvious that in the case of the oblong cells of the epidermis in

the hyacinth the stomata will be found arranged in regular rows, while

they will be irregularly distributed over the surface of the leaf in

such a case as we have depicted in Sedum.

While, as I have said, the mechanical cause of the split which

constitutes the orifice of the stoma is not quite clear, yet there

can be little or no doubt that it, like the rest of the phenomenon,

is related to surface tension. It might well be that it is directly

due to the presence underneath this portion of epidermis of the hollow

air-space which the stoma is apparently developed “for the purpose”

of communicating with; this air-surface on both sides of the delicate

epidermis might well cause such an alteration of tensions that the

two halves of the dividing cell would tend to part company. In short,

if the surface-energy in a cell-air contact were half or less than

half that in a contact between cell and cell, then it is obvious that

our partition would tend to split, and give us a two-fold surface

in contact with air, instead of the original boundary or interface

between one cell and the other. In Professor Macallum’s experiments,

which we have briefly discussed in our short chapter on Adsorption, it

was found that large quantities of potassium gathered together along

the outer walls of the guard-cells of the stoma, thereby indicating

a low surface-tension along these outer walls. The tendency of the

guard-cells to bulge outwards is so far explained, and it is possible

that, under the existing conditions of restraint, we may have here a

force tending, or helping, to split the two cells asunder. It is clear

enough, however, that the last stage in the development of a stoma, is,

from the physical point of view, not yet properly understood.

――――――――――

[Illustration: Fig. 180. Various pollen-grains and spores (after

Berthold, Campbell, Goebel and others). (1) _Epilobium_; (2)

_Passiflora_; (3) _Neottia_; (4) _Periploca graeca_; (5) _Apocynum_;

(6) _Erica_; (7) Spore of _Osmunda_; (8) Tetraspore of _Callithamnion_.]

[Illustration: Fig. 181. Dividing spore of _Anthoceros_. (After

Campbell.)]

In all our foregoing examples of the development of a “tissue” we

have seen that the process consists in the _successive_ division of

cells, each act of division being accompanied by the formation {396}

of a boundary-surface, which, whether it become at once a solid or

semi-solid partition or whether it remain semi-fluid, exercises in all

cases an effect on the position and the form of the boundary which

comes into being with the next act of division. In contrast to this

general process stands the phenomenon known as “free cell-formation,”

in which, out of a common mass of protoplasm, a number of separate

cells are _simultaneously_, or all but simultaneously, differentiated.

In a number of cases it happens that, to begin with, a number of

“mother-cells” are formed simultaneously, and each of these divides,

by two successive divisions, into four “daughter-cells.” These

daughter-cells will tend to group themselves, just as would four

soap-bubbles, into a “tetrad,” the four cells corresponding to the

angles of a regular tetrahedron. For the system of four bodies is

evidently here in perfect symmetry; the partition-walls and their

respective edges meet at equal angles: three walls everywhere meeting

in an edge, and the four edges converging to a point in the geometrical

centre of the system. This is the typical mode of development of

pollen-grains, common among Monocotyledons and all but universal among

Dicotyledonous plants. By a loosening of the surrounding tissue and

an expansion of the cavity, or anther-cell, in which {397} they lie,

the pollen-grains afterwards fall apart, and their individual form

will depend upon whether or no their walls have solidified before

this liberation takes place. For if not, then the separate grains

will be free to assume a spherical form as a consequence of their

own individual and unrestricted growth; but if they become solid or

rigid prior to the separation of the tetrad, then they will conserve

more or less completely the plane interfaces and sharp angles of the

elements of the tetrahedron. The latter is the case, for instance, in

the pollen-grains of Epilobium (Fig. 180, 1) and in many others. In

the Passion-flower (2) we have an intermediate condition: where we

can still see an indication of the facets where the grains abutted on

one another in the tetrad, but the plane faces have been swollen by

growth into spheroidal or spherical surfaces. It is obvious that there

may easily be cases where the tetrads of daughter-cells are prevented

from assuming the tetrahedral form: cases, that is to say, where the

four cells are forced and crushed into one plane. The figures given by

Goebel of the development of the pollen of Neottia (3, _a_–_e_: all the

figures referring to grains taken from a single anther), illustrate

this to perfection; and it will be seen that, when the four cells lie

in a plane, they conform exactly to our typical diagram of the first

four cells in a segmenting ovum. Occasionally, though the four cells

lie in a plane, the diagram seems to fail us, for the cells appear to

meet in a simple cross (as in 5); but here we soon perceive that the

cells are not in complete interfacial contact, but are kept apart by a

little intervening drop of fluid or bubble of air. The spores of ferns

(7) develop in very much the same way as pollen-grains; and they also

very often retain traces of the shape which they assumed as members of

a tetrahedral figure. Among the “tetraspores” (8) of the Florideae, or

Red Seaweeds, we have a phenomenon which is in every respect analogous.

Here again it is obvious that, apart from differences in actual

magnitude, and apart from superficial or “accidental” differences

(referable to other physical phenomena) in the way of colour, {398}

texture and minute sculpture or pattern, it comes to pass, through the

laws of surface-tension and the principles of the geometry of position,

that a very small number of diagrammatic figures will sufficiently

represent the outward forms of all the tetraspores, four-celled

pollen-grains, and other four-celled aggregates which are known or are

even capable of existence.

――――――――――

We have been dealing hitherto (save for some slight exceptions) with

the partitioning of cells on the assumption that the system either

remains unaltered in size or else that growth has proceeded uniformly

in all directions. But we extend the scope of our enquiry very greatly

when we begin to deal with _unequal growth_, with growth, that is

to say, which produces a greater extension along some one axis than

another. And here we come close in touch with that great and still (as

I think) insufficiently appreciated generalisation of Sachs, that the

manner in which the cells divide is _the result_, and not the cause, of

the form of the dividing structure: that the form of the mass is caused

by its growth as a whole, and is not a resultant of the growth of the

cells individually considered[403]. Such asymmetry of growth may be

easily imagined, and may conceivably arise from a variety of causes.

In any individual cell, for instance, it may arise from molecular

asymmetry of the structure of the cell-wall, giving it greater rigidity

in one direction than another, while all the while the hydrostatic

pressure within the cell remains constant and uniform. In an aggregate

of cells, it may very well arise from a greater chemical, or osmotic,

activity in one than another, leading to a localised increase in the

fluid pressure, and to a corresponding bulge over a certain area of

the external surface. It might conceivably occur as a direct result

of the preceding cell-divisions, when these are such as to produce

many peripheral or concentric walls in one part and few or none in

another, with the obvious result of strengthening the common boundary

wall and resisting the outward pressure of growth in parts where the

former is the case; that is to say, in our dividing quadrant, if {399}

its quadrangular portion subdivide by periclines, and the triangular

portion by oblique anticlines (as we have seen to be the natural

tendency), then we might expect that external growth would be more

manifest over the latter than over the former areas. As a direct and

immediate consequence of this we might expect a tendency for special

outgrowths, or “buds,” to arise from the triangular rather than from

the quadrangular cells; and this turns out to be not merely a tendency

towards which theoretical considerations point, but a widespread and

important factor in the morphology of the cryptogams. But meanwhile,

without enquiring further into this complicated question, let us simply

take it that, if we start from such a simple case as a round cell which

has divided into two halves, or four quarters (as the case may be),

we shall at once get bilateral symmetry about a main axis, and other

secondary results arising therefrom, as soon as one of the halves, or

one of the quarters, begins to shew a rate of growth in advance of

the others; for the more rapidly growing cell, or the peripheral wall

common to two or more such rapidly growing cells, will bulge out into

an ellipsoid form, and may finally extend into a cylinder with rounded

or ellipsoid end.

This latter very simple case is illustrated in the development of a

pollen-tube, where the rapidly growing cell develops into the elongated

cylindrical tube, and the slow-growing or quiescent part remains behind

as the so-called “vegetative” cell or cells.

Just as we have found it easier to study the segmentation of a circular

disc than that of a spherical cell, so let us begin in the same way, by

enquiring into the divisions which will ensue if the disc tend to grow,

or elongate, in some one particular direction, instead of in radial

symmetry. The figures which we shall then obtain will not only apply

to the disc, but will also represent, in all essential features, a

projection or longitudinal section of a solid body, spherical to begin

with, preserving its symmetry as a solid of revolution, and subject to

the same general laws as we have studied in the disc[404]. {400}

(1) Suppose, in the first place, that the axis of growth lies

symmetrically in one of the original quadrantal cells of a segmenting

disc; and let this growing cell elongate with comparative rapidity

before it subdivides. When it does divide, it will necessarily do so by

a transverse partition, concave towards the apex of the cell: and, as

further elongation takes place, the cylindrical structure which will be

developed thereby will tend to be again and again subdivided by similar

concave transverse partitions. If at any time, through this process

of concurrent elongation and subdivision, the apical cell become

equivalent to, or less than, a hemisphere, it will next divide by means

of a longitudinal, or vertical partition; and similar longitudinal

partitions will arise in the other segments of the cylinder, as soon as

it comes about that their length (in the direction of the axis) is less

than their breadth.

[Illustration: Fig. 182.]

But when we think of this structure in the solid, we at once perceive

that each of these flattened segments of the cylinder, into which our

cylinder has divided, is equivalent to a flattened circular disc;

and its further division will accordingly tend to proceed like any

other flattened disc, namely into four quadrants, and afterwards by

anticlines and periclines in the usual way. {401} A section across the

cylinder, then, will tend to shew us precisely the same arrangements

as we have already so fully studied in connection with the typical

division of a circular cell into quadrants, and of these quadrants into

triangular and quadrangular portions, and so on.

But there are other possibilities to be considered, in regard to the

mode of division of the elongating quasi-cylindrical portion, as it

gradually develops out of the growing and bulging quadrantal cell; for

the manner in which this latter cell divides will simply depend upon

the form it has assumed before each successive act of division takes

place, that is to say upon the ratio between its rate of growth and

the frequency of its successive divisions. For, as we have already

seen, if the growing cell attain a markedly oblong or cylindrical form

before division ensues, then the partition will arise transversely to

the long axis; if it be but a little more than a hemisphere, it will

divide by an oblique partition; and if it be less than a hemisphere

(as it may come to be after successive transverse divisions) it will

divide by a vertical partition, that is to say by one coinciding with

its axis of growth. An immense number of permutations and combinations

may arise in this way, and we must confine our illustrations to a small

number of cases. The important thing is not so much to trace out the

various conformations which may arise, but to grasp the fundamental

principle: which is, that the forces which dominate the _form_ of each

cell regulate the manner of its subdivision, that is to say the form of

the new cells into which it subdivides; or in other words, the form of

the growing organism regulates the form and number of the cells which

eventually constitute it. The complex cell-network is not the cause but

the result of the general configuration, which latter has its essential

cause in whatsoever physical and chemical processes have led to a

varying velocity of growth in one direction as compared with another.

[Illustration: Fig. 183. Development of _Sphagnum_. (After Campbell.)]

In the annexed figure of an embryo of Sphagnum we see a mode of

development almost precisely corresponding to the hypothetical case

which we have just described,—the case, that is to say, where one of

the four original quadrants of the mother-cell is the chief agent in

future growth and development. We see at the base of our first figure

(_a_), the three stationary, or {402} undivided quadrants, one of

which has further slowly divided in the stage _b_. The active quadrant

has grown quickly into a cylindrical structure, which inevitably

divides, in the next place, into a series of transverse partitions; and

accordingly, this mode of development carries with it the presence of a

single “apical cell,” whose lower wall is a spherical surface with its

convexity downwards. Each cell of the subdivided cylinder now appears

as a more or less flattened disc, whose mode of further sub-division

we may prognosticate according to our former investigation, to which

subject we shall presently return.

[Illustration: Fig. 184.]

(2) In the next place, still keeping to the case where only one of the

original quadrant-cells continues to grow and develop, let us suppose

that this growing cell falls to be divided when by growth it has

become just a little greater than a hemisphere; it will then divide,

as in Fig. 184, 2, by an oblique partition, in the usual way, whose

precise position and inclination to the base will depend entirely on

the configuration of the cell itself, save only, of course, that we

may have also to take into account the possibility of the division

being into two unequal halves. By our hypothesis, {403} the growth

of the whole system is mainly in a vertical direction, which is as

much as to say that the more actively growing protoplasm, or at least

the strongest osmotic force, will be found near the apex; where

indeed there is obviously more external surface for osmotic action.

It will therefore be that one of the two cells which contains, or

constitutes, the apex which will grow more rapidly than the other,

and which therefore will be the first to divide, and indeed in any

case, it will usually be this one of the two which will tend to

divide first, inasmuch as the triangular and not the quadrangular

half is bound to constitute the apex[405]. It is obvious that (unless

the act of division be so long postponed that the cell has become

quasi-cylindrical) it will divide by another oblique partition,

starting from, and running at right angles to, the first. And so

division will proceed,

[Illustration: Fig. 185. Gemma of Moss. (After Campbell.)]

by oblique alternate partitions, each one tending to be, at first,

perpendicular to that on which it is based and also to the peripheral

wall; but all these points of contact soon tending, by reason of the

equal tensions of the three films or surfaces which meet there, to

form angles of 120°. There will always be, in such a case, a single

apical cell, of a more or less distinctly triangular form. The annexed

figure of the developing antheridium of a Liverwort (Riccia) is a

typical example of such a case. In Fig. 185 which represents a “gemma”

of a Moss, we see just the same thing; with this addition, that here

the lower of the two original cells has grown even more quickly than

the other, constituting a long cylindrical stalk, and dividing in

accordance with its shape, by means of transverse septa.

In all such cases as these, the cells whose development we have studied

will in turn tend to subdivide, and the manner in which they will do so

must depend upon their own proportions; and in all cases, as we have

already seen, there will sooner or later be a tendency to the formation

of periclinal walls, cutting off an “epidermal layer of cells,” as Fig.

186 illustrates very well.

[Illustration: Fig. 186. Development of antheridium of _Riccia_. (After

Campbell.)]

[Illustration: Fig. 187. Section of growing shoot of Selaginella,

diagrammatic.]

[Illustration: Fig. 188. Embryo of Jungermannia. (After

Kienitz-Gerloff.)]

The method of division by means of oblique partitions is a common one

in the case of ‘growing points’; for it evidently {404} includes all

cases in which the act of cell-division does not lag far behind that

elongation which is determined by the specific rate of growth. And it

is also obvious that, under a common type, there must here be included

a variety of cases which will, at first sight, present a very different

appearance one from another. For instance, in Fig. 187 which represents

a growing shoot of Selaginella, and somewhat less diagrammatically in

the young embryo of Jungermannia (Fig. 188), we have the appearance of

an almost straight vertical partition running up in the axis of the

system, and the primary cell-walls are set almost at right angles to

it,—almost transversely, that is to say to the outer walls and to the

long axis of the structure. We soon recognise, however, {405} that

the difference is merely a difference of degree. The more remote the

partitions are, that is to say the greater the velocity of growth

relatively to division, the less abrupt will be the alternate kinks or

curvatures of the portions which lie in the neighbourhood of the axis,

and the more will these portions appear to constitute a single unbroken

wall.

[Illustration: Fig. 189.]

(3) But an appearance nearly, if not quite, indistinguishable from

this may be got in another way, namely, when the original growing cell

is so nearly hemispherical that it is actually divided by a vertical

partition, into two quadrants; and from this vertical partition, as it

elongates, lateral partition-walls will arise on either side. And by

the tensions exercised by these, the vertical partition will be bent

into little portions set at 120° one to another, and the whole will

come to look just like that which, in the former case, was made up of

portions of many successive oblique partitions.

――――――――――

Let us now, in one or two cases, follow out a little further the

stages of cell-division whose beginning we have studied in the last

paragraphs. In the antheridium of Riccia, after the successive oblique

partitions have produced the longitudinal series of cells shewn in Fig.

186, it is plain that the next partitions will arise periclinally, that

is to say parallel to the outer wall, which in this particular case

represents the short axis of the oblong cells. The effect is at once to

produce an epidermal layer, whose cells will tend to subdivide further

by means of partitions perpendicular to the free surface, that is to

say crossing the flattened cells by their shortest diameter. The inner

mass, beneath the epidermis, consists of cells which are still more or

less oblong, or which become {406} definitely so in process of growth;

and these again divide, parallel to their short axes, into squarish

cells, which as usual, by the mutual tension of their walls, become

hexagonal, as seen in a plane section. There is a clear distinction,

then, in form as well as in position, between the outer covering-cells

and those which lie within this envelope; the latter are reduced to a

condition which merely fulfils the mechanical function of a protective

coat, while the former undergo less modification, and give rise to the

actively living, reproductive elements.

[Illustration: Fig. 190. Development of sporangium of _Osmunda_. (After

Bower.)]

In Fig. 190 is shewn the development of the sporangium of a fern

(Osmunda). We may trace here the common phenomenon of a series of

oblique partitions, built alternately on one another, and cutting off a

conspicuous triangular apical cell. Over the whole system an epidermal

layer has been formed, in the manner we have described; and in this

case it covers the apical cell also, owing to the fact that it was of

such dimensions that, at one stage of growth, a periclinal partition

wall, cutting off its outer end, was indicated as of less area than

an anticlinal one. This periclinal wall cuts down the apical cell to

the proportions, very nearly, of an equilateral triangle, but the

solid form of the cell is obviously that of a tetrahedron with curved

faces; and accordingly, the least possible partitions by which further

subdivision can be effected will run successively parallel to its four

sides (or its three sides when we confine ourselves to the appearances

as seen in {407} section). The effect, as seen in section, is to

cut off on each side a characteristically flattened cell, oblong as

seen in section, still leaving a triangular (or strictly speaking, a

tetrahedral) one in the centre. The former cells, which constitute no

specific structure or perform no specific physiological function, but

which merely represent certain directions in space towards which the

whole system of partitioning has gradually led, are called by botanists

the “tapetum.” The active growing tetrahedral cell which lies between

them, and from which in a sense every other cell in the system has

been either directly or indirectly segmented off, still manifests, as

it were, its vigour and activity, and now, by internal subdivision,

becomes the mother-cell of the spores.

――――――――――

In all these cases, for simplicity’s sake, we have merely considered

the appearances presented in a single, longitudinal, plane of optical

section. But it is not difficult to interpret from these appearances

what would be seen in another plane, for instance in a transverse

section. In our first example, for instance, that of the developing

embryo of Sphagnum (Fig. 183), we can see that, at appropriate levels,

the cells of the original cylindrical row have divided into transverse

rows of four, and then of eight cells. We may be sure that the four

cells represent, approximately, quadrants of a cylindrical disc, the

four cells, as usual, not meeting in a point, but intercepted by a

small intermediate partition. Again, where we have a plate of eight

cells, we may well imagine that the eight octants are arranged in what

we have found to be the way naturally resulting from the division

of four quadrants, that is to say into alternately triangular and

quadrangular portions; and this is found by means of sections to be

the case. The accompanying figure is precisely comparable to our

previous diagrams of the arrangement of an aggregate of eight cells in

a dividing disc, save only that, in two cases, the cells have already

undergone a further subdivision.

[Illustration: Fig. 191. (A, B,) Sections of younger and older embryos

of _Phascum_; (C) do. of _Adiantum_. (After Kienitz-Gerloff.)]

[Illustration: Fig. 192. Section through frond of _Girardia

sphacelaria_. (After Goebel.)]

It follows in like manner, that in a host of cases we meet with this

characteristic figure, in one or other of its possible, and strictly

limited, variations,—in the cross sections of growing embryonic

structures, just as we have already seen that it appears in a host of

cases where the entire system (or a portion of its {408} surface)

consists of eight cells only. For example, in Fig. 191, we have it

again, in a section of a young embryo of a moss (Phascum), and in

a section of an embryo of a fern (Adiantum). In Fig. 192 shewing a

section through a growing frond of a sea-weed (Girardia) we have a

case where the partitions forming the eight octants have conformed to

the usual type; but instead of the usual division by periclines of the

four quadrangular spaces, these latter are dividing by means of oblique

septa, apparently owing to the fact that the cell is not dividing into

two equal, but into two unequal portions. In this last figure we have

a peculiar look of stiffness or formality, such that it appears at

first to bear little resemblance to the rest. The explanation is of

the simplest. The mode of partitioning differs little (except to some

slight extent in the way already mentioned) from the normal type; but

in this case the partition walls are so thick and become so quickly

comparatively solid and rigid, that the secondary curvatures due to

their successive mutual tractions are here imperceptible.

[Illustration: Fig. 193. Development of antheridium of _Pteris_. (After

Strasbürger.)]

A curious and beautiful case, apparently aberrant but which would

doubtless be found conforming strictly to physical laws, if {409}

only we clearly understood the actual conditions, is indicated

in the development of the antheridium of a fern, as described by

Strasbürger. Here the antheridium develops from a single cell, whose

form has grown to be something more than a hemisphere; and the first

partition, instead of stretching transversely across the cell, as

we should expect it to do if the cell were actually spherical, has

as it were sagged down to come in contact with the base, and so to

develop into an annular partition, running round the lower margin of

the cell. The phenomenon is akin to that cutting off of the corner

of a cubical cell by a spherical partition, of which we have spoken

on p. 349, and the annular film is very easy to reproduce by means

of a soap-bubble in the bottom of a cylindrical dish or beaker. The

next partition is a periclinal one, concentric with the outer surface

of the young antheridium; and this in turn is followed by a concave

partition which cuts off the apex of the original cell: but which

becomes connected with the second, or periclinal partition in precisely

the same annular fashion as the first partition did with the base of

the little antheridium. The result is that, at this stage, we have

four cell-cavities in the little antheridium: (1) a central cavity;

(2) an annular space around the lower margin; (3) a narrow annular or

cylindrical space around the sides of the antheridium; and (4) a small

terminal or apical cell. It is evident that the tendency, in the next

place, will be to subdivide the flattened external cells by means of

anticlinal partitions, and so to convert the whole structure into a

single layer of epidermal cells, surrounding a central cell within

which, in course of time, the antherozoids are developed.

――――――――――

The foregoing account deals only with a few elementary phenomena,

and may seem to fall far short of an attempt to deal in general

with “the forms of tissues.” But it is the principle involved, and

not its ultimate and very complex results, that we can alone {410}

attempt to grapple with. The stock-in-trade of mathematical physics,

in all the subjects with which that science deals, is for the most

part made up of simple, or simplified, cases of phenomena which in

their actual and concrete manifestations are usually too complex

for mathematical analysis; and when we attempt to apply its methods

to our biological and histological phenomena, in a preliminary and

elementary way, we need not wonder if we be limited to illustrations

which are obviously of a simple kind, and which cover but a small part

of the phenomena with which the histologist has become familiar. But

it is only relatively that these phenomena to which we have found the

method applicable are to be deemed simple and few. They go already far

beyond the simplest phenomena of all, such as we see in the dividing

Protococcus, and in the first stages, two-celled or four-celled, of the

segmenting egg. They carry us into stages where the cells are already

numerous, and where the whole conformation has become by no means

easy to depict or visualise, without the help and guidance which the

phenomena of surface-tension, the laws of equilibrium and the principle

of minimal areas are at hand to supply. And so far as we have gone,

and so far as we can discern, we see no sign of the guiding principles

failing us, or of the simple laws ceasing to hold good.

{411}