THE FORMS OF TISSUES OR CELL-AGGREGATES
We now pass from the consideration of the solitary cell to that of
cells in contact with one another,—to what we may call in the first
instance “cell-aggregates,”—through which we shall be led ultimately to
the study of complex tissues. In this part of our subject, as in the
preceding chapters, we shall have to give some consideration to the
effects of various forces; but, as in the case of the conformation of
the solitary cell, we shall probably find, and we may at least begin
by assuming, that the agency of surface tension is especially manifest
and important. The effect of this surface tension will chiefly manifest
itself in the production of surfaces _minimae areae_: where, as Plateau
was always careful to point out, we must understand by this expression
not an absolute, but a relative minimum, an area, that is to say, which
approximates to an absolute minimum as nearly as circumstances and the
conditions of the case permit.
There are certain fundamental principles, or fundamental equations,
besides those which we have already considered, which we shall need in
our enquiry. For instance the case which we briefly touched upon (on
p. 265) of the angle of contact between the protoplasm and the axial
filament in a Heliozoan we shall now find to be but a particular case
of a general and elementary theorem.
Let us re-state as follows, in terms of _Energy_, the general principle
which underlies the theory of surface tension or capillarity.
When a fluid is in contact with another fluid, or with a solid or a
gas, a portion of the energy of the system (that, namely, which we call
surface energy), is proportional to the area of the surface of contact:
it is also proportional to a coefficient which is specific for each
particular pair of substances, and which is constant for these, save
only in so far as it may be modified by {294} changes of temperature
or of electric charge. The condition of _minimum potential energy_ in
the system, which is the condition of equilibrium, will accordingly be
obtained by the utmost possible diminution in the area of the surfaces
in contact. When we have _three_ bodies in contact, the case becomes
a little more complex. Suppose for instance we have a drop of some
fluid, _A_, floating on another fluid, _B_, and exposed to air, _C_.
The whole surface energy of the system may now be considered as divided
into two parts, one at the surface of the drop, and the other outside
of the same; the latter portion is inherent in the surface _BC_,
between the mass of fluid _B_ and the superincumbent air, _C_; but the
former portion consists of two parts, for it is divided between the two
surfaces _AB_ and _AC_, that namely which separates the drop from the
surrounding fluid and that which separates it from the atmosphere. So
far as
[Illustration: Fig. 99.]
the drop is concerned, then, equilibrium depends on a proper balance
between the energy, per unit area, which is resident in its own two
surfaces, and that which is external thereto: that is to say, if we
call _E__{_bc_} the energy at the surface between the two fluids, and
so on with the other two pairs of surface energies, the condition of
equilibrium, or of maintenance of the drop, is that
_E__{_bc_} < _E__{_ab_} + _E__{_ac_}.
If, on the other hand, the fluid _A_ happens to be oil and the fluid
_B_, water, then the energy _per unit area_ of the water-air surface
is greater than that of the oil-air surface and that of the oil-water
surface together; i.e.
_E__{_wa_} > _E__{_oa_} + _E__{_ow_}.
Here there is no equilibrium, and in order to obtain it the water-air
surface must always tend to decrease and the other two interfacial
surfaces to increase; which is as much as to say that the water tends
to become covered by a spreading film of oil, and the water-air surface
to be abolished. {295}
The surface energy of which we have here spoken is manifested in that
contractile force, or “tension,” of which we have already had so much
to say[341]. In any part of the free water surface, for instance, one
surface particle attracts another surface particle, and the resultant
of these multitudinous attractions is an equilibrium of tension
throughout this particular surface. In the case of our three bodies
in contact with one another, and within a small area very near to
the point of contact, a water particle (for instance) will be pulled
outwards by another water particle; but on the opposite side, so to
speak, there will be no water surface, and no water particle, to
furnish the counterbalancing pull; this counterpull,
[Illustration: Fig. 100.]
[Illustration: Fig. 101.]
which is necessary for equilibrium, must therefore be provided by
the tensions existing in the _other two_ surfaces of contact. In
short, if we could imagine a single particle placed at the very point
of contact, it would be drawn upon by three different forces, whose
directions would lie in the three surface planes, and whose magnitude
would be proportional to the specific tensions characteristic of
the two bodies which in each case combine to form the “interfacial”
surface. Now for three forces acting at a point to be in equilibrium,
they must be capable of representation, in magnitude and direction, by
the three sides of a triangle, taken in order, in accordance with the
elementary theorem of the Triangle of Forces. So, if we know the form
of our floating drop (Fig. 100), then by drawing tangents from _O_
(the point of mutual contact), {296} we determine the three angles of
our triangle (Fig. 101), and we therefore know the relative magnitudes
of the three surface tensions, which magnitudes are proportional to
its sides; and conversely, if we know the magnitudes, or relative
magnitudes, of the three sides of the triangle, we also know its
angles, and these determine the form of the section of the drop. It is
scarcely necessary to mention that, since all points on the edge of the
drop are under similar conditions, one with another, the form of the
drop, as we look down upon it from above, must be circular, and the
whole drop must be a solid of revolution.
――――――――――
The principle of the Triangle of Forces is expanded, as follows, by
an old seventeenth-century theorem, called Lami’s Theorem: “_If three
forces acting at a point be in equilibrium, each force is proportional
to the sine of the angle contained between the directions of the other
two._” That is to say
_P_ : _Q_ : _R_ : = sin _QOR_ : sin _POR_ : sin _POQ_.
or _P_/sin _QOR_ = _Q_/sin _ROP_ = _R_/sin _POQ_.
And from this, in turn, we derive the equivalent formulae, by which
each force is expressed in terms of the other two, and of the angle
between them:
_P_^2 = _Q_^2 + _R_^2 + 2_Q_ _R_ cos(_QOR_), etc.
From this and the foregoing, we learn the following important and
useful deductions:
(1) The three forces can only be in equilibrium when any one of them
is less than the sum of the other two: for otherwise, the triangle is
impossible. Now in the case of a drop of olive-oil upon a clean water
surface, the relative magnitudes of the three tensions (at 15° C.) have
been determined as follows:
Water-air surface 75
Oil-air surface 32
Oil-water surface 21
No triangle having sides of these relative magnitudes is possible; and
no such drop therefore can remain in equilibrium. {297}
(2) The three surfaces may be all alike: as when a soap-bubble floats
upon soapy water, or when two soap-bubbles are joined together, on
either side of a partition-film. In this case, the three tensions are
all equal, and therefore the three angles are all equal; that is to
say, when three similar liquid surfaces meet together, they always
do so at an angle of 120°. Whether our two conjoined soap-bubbles
be equal or unequal, this is still the invariable rule; because the
specific tension of a particular surface is unaffected by any changes
of magnitude or form.
(3) If two only of the surfaces be alike, then two of the angles will
be alike, and the other will be unlike; and this last will be the
difference between 360° and the sum of the other two. A particular case
is when a film is stretched between solid and parallel walls, like
a soap-film within a cylindrical tube. Here, so long as there is no
external pressure applied to either side, so long as both ends of the
tube are open or closed, the angles on either side of the film will be
equal, that is to say the film will set itself at right angles to the
sides.
Many years ago Sachs laid it down as a principle, which has become
celebrated in botany under the name of Sachs’s Rule, that one cell-wall
always tends to set itself at right angles to another cell-wall.
This rule applies to the case which we have just illustrated; and
such validity as the rule possesses is due to the fact that among
plant-tissues it very frequently happens that one cell-wall has become
solid and rigid before another and later partition-wall is developed in
connection with it.
(4) There is another important principle which arises not out of our
equations but out of the general considerations by which we were
led to them. We have seen that, at and near the point of contact
between our several surfaces, there is a continued balance of forces,
carried, so to speak, across the interval; in other words, there is
_physical continuity_ between one surface and another. It follows
necessarily from this that the surfaces merge one into another by a
continuous curve. Whatever be the form of our surfaces and whatever
the angle between them, this small intervening surface, approximately
spherical, is always there to bridge over the line of contact[342];
and this little fillet, or “bourrelet,” {298} as Plateau called
it, is large enough to be a common and conspicuous feature in the
microscopy of tissues (Fig. 102). For instance, the so-called
“splitting” of the cell-wall, which is conspicuous at the angles of the
large “parenchymatous” cells in the succulent tissues of all higher
plants (Fig. 103), is nothing more than a manifestation of Plateau’s
“bourrelet,” or surface of continuity[343].
――――――――――
We may now illustrate some of the foregoing principles, before we
proceed to the more complex cases in which more bodies than three are
in mutual contact. But in doing so, we must constantly bear in mind
the principles set forth in our chapter on the forms of cells, and
especially those relating to the pressure exercised by a curved film.
[Illustration: Fig. 102. (After Berthold.)]
[Illustration: Fig. 103. Parenchyma of Maize.]
Let us look for a moment at the case presented by the partition-wall
in a double soap-bubble. As we have just seen, the three films in
contact (viz. the outer walls of the two bubbles and the partition-wall
between) being all composed of the same substance {299} and all alike
in contact with air, the three surface tensions must be equal; and the
three films must therefore, in all cases, meet at an angle of 120°.
But, unless the two bubbles be of precisely equal size (and therefore
of equal curvature) it is obvious that the tangents to the spheres
will not meet the plane of their circle of contact at equal angles,
and therefore that the partition-wall must be a _curved_ surface: it
is only plane when it divides two equal and symmetrical cells. It is
also obvious, from the symmetry of the figure, that the centres of
the spheres, the centre of the partition, and the centres of the two
spherical surfaces are all on one and the same straight line.
[Illustration: Fig. 104.]
Now the surfaces of the two bubbles exert a pressure inwards
which is inversely proportional to their radii: that is to say
_p_ : _p′_ :: 1/_r′_ : 1/_r_; and the partition wall must,
for equilibrium, exert a pressure (_P_) which is equal to the
difference between these two pressures, that is to say, _P_ = 1/_R_
= 1/_r′_ − 1/_r_ = (_r_ − _r′_)/_r_ _r′_. It follows that the curvature
of the partition wall must be just such a curvature as is capable of
exerting this pressure, that is to say, _R_ = _r_ _r′_/(_r_ − _r′_).
The partition wall, then, is always a portion of a spherical surface,
whose radius is equal to the product, divided by the difference, of
the radii of the two vesicles. It follows at once from this that if
the two bubbles be equal, the radius of curvature of the partition is
infinitely great, that is to say the partition is (as we have already
seen) a plane surface.
The geometrical construction by which we obtain the position of the
centres of the two spheres and also of the partition surface is
very simple, always provided that the surface tensions are uniform
throughout the system. If _p_ be a point of contact between the two
spheres, and _cp_ be a radius of one of them, then make the angle _cpm_
= 60°, and mark off on _pm_, _pc′_ equal to the {300} radius of the
other sphere; in like manner, make the angle _c′pn_ = 60°, cutting the
line _cc′_ in _c″_; then _c′_ will be the centre of the second sphere,
and _c″_ that of the spherical partition.
[Illustration: Fig. 105.]
[Illustration: Fig. 106.]
Whether the partition be or be not a plane surface, it is obvious that
its _line of junction_ with the rest of the system lies in a plane, and
is at right angles to the axis of symmetry. The actual curvature of
the partition-wall is easily seen in optical section; but in surface
view, the line of junction is _projected_ as a plane (Fig. 106),
perpendicular to the axis, and this appearance has also helped to lend
support and authority to “Sachs’s Rule.”
――――――――――
[Illustration: Fig. 107. Filaments, or chains of cells, in various
lower Algae. (A) _Nostoc_; (B) _Anabaena_; (C) _Rivularia_; (D)
_Oscillatoria_.]
Many spherical cells, such as Protococcus, divide into two equal
halves, which are therefore separated by a plane partition. Among
the other lower Algae, akin to Protococcus, such as the Nostocs
and Oscillatoriae, in which the cells are imbedded in a gelatinous
matrix, we find a series of forms such as are represented in Fig. 107.
Sometimes the cells are solitary or disunited; sometimes they run in
pairs or in rows, separated one from another by flat partitions; and
sometimes the conjoined cells are approximately hemispherical, but
at other times each half is more than a hemisphere. These various
conditions depend, {301} according to what we have already learned,
upon the relative magnitudes of the tensions at the surface of the
cells and at the boundary between them[344].
In the typical case of an equally divided cell, such as a double and
co-equal soap-bubble, where the partition-wall and the outer walls
are similar to one another and in contact with similar substances, we
can easily determine the form of the system. For, at any point of the
boundary of the partition-wall, _O_, the tensions being equal, the
angles _QOP_, _ROP_, _QOR_ are all equal, and each is, therefore, an
angle of 120°. But _OQ_, _OR_ being tangents, the centres of the two
spheres (or circular arcs in the figure) lie on perpendiculars to them;
therefore the radii _CO_, _C′O_ meet at an
[Illustration: Fig. 108.]
angle of 60°, and _COC′_ is an equilateral triangle. That is to say,
the centre of each circle lies on the circumference of the other; the
partition lies midway between the two centres; and the length (i.e. the
diameter) of the partition-wall, _PO_, is
2 sin 60° = 1·732
times the radius, or ·866 times the diameter, of each of the cells.
This gives us, then, the _form_ of an aggregate of two equal cells
under uniform conditions.
As soon as the tensions become unequal, whether from changes in their
own substance or from differences in the substances with which they
are in contact, then the form alters. If the tension {302} along
the partition, _P_, diminishes, the partition itself enlarges, and
the angle _QOR_ increases: until, when the tension _P_ is very small
compared to _Q_ or _R_, the whole figure becomes a circle, and the
partition-wall, dividing it into two hemispheres, stands at right
angles to the outer wall. This is the case when the outer wall of the
cell is practically solid. On the other hand, if _P_ begins to increase
relatively to _Q_ and _R_, then the partition-wall contracts, and the
two adjacent cells become larger and larger segments of a sphere, until
at length the system becomes divided into two separate cells.
[Illustration: Fig. 109. Spore of _Pellia_. (After Campbell.)]
In the spores of Liverworts (such as _Pellia_), the first
partition-wall (the equatorial partition in Fig. 109, _a_) divides the
spore into two equal halves, and is therefore a plane surface, normal
to the surface of the cell; but the next partitions arise near to
either end of the original spherical or elliptical cell. Each of these
latter partitions will (like the first) tend to set itself normally to
the cell-wall; at least the angles on either side of the partition will
be identical, and their magnitude will depend upon the tension existing
between the cell-wall and the surrounding medium. They will only be
right angles if the cell-wall is already practically solid, and in all
probability (rigidity of the cell-wall not being quite attained) they
will be somewhat greater. In either case the partition itself will be
a portion of a sphere, whose curvature will now denote a difference of
pressures in the two chambers or cells, which it serves to separate.
(The later stages of cell-division, represented in the figures _b_ and
_c_, we are not yet in a position to deal with.)
We have innumerable cases, near the tip of a growing filament, where
in like manner the partition-wall which cuts off the terminal {303}
cell constitutes a spherical lens-shaped surface, set normally to the
adjacent walls. At the tips of the branches of many Florideae, for
instance, we find such a lenticular partition. In _Dictyota dichotoma_,
as figured by Reinke, we have a succession of such partitions; and,
by the way, in such cases as these, where the tissues are very
transparent, we have often in optical section a puzzling confusion of
lines; one being the optical section of the curved partition-wall, the
other being the straight linear projection of its outer edge to which
we have already referred. In the conical terminal cell of Chara, we
have the same lens-shaped curve, but a little lower down, where the
sides of the shoot are approximately parallel, we have flat transverse
partitions, at the edges of which, however, we recognise a convexity of
the outer cell-wall and a definite angle of contact, equal on the two
sides of the partition.
[Illustration: Fig. 110. Cells of _Dictyota_. (After Reinke.)]
[Illustration: Fig. 111. Terminal and other cells of _Chara_.]
[Illustration: Fig. 112. Young antheridium of _Chara_.]
In the young antheridia of Chara (Fig. 112), and in the not dissimilar
case of the sporangium (or conidiophore) of Mucor, we easily recognise
the hemispherical form of the septum which shuts off the large
spherical cell from the cylindrical filament. Here, in the first phase
of development, we should have to take into consideration the different
pressures exerted by the single curvature of the cylinder and the
double curvature of its spherical cap (p. 221); and we should find
that the partition would have a somewhat low curvature, with a radius
_less_ than the diameter of the cylinder; which it would have exactly
equalled but for the additional pressure inwards which it receives
{304} from the curvature of the large surrounding sphere. But as the
latter continues to grow, its curvature decreases, and so likewise does
the inward pressure of its surface; and accordingly the little convex
partition bulges out more and more.
――――――――――
In order to epitomise the foregoing facts let the annexed diagrams
(Fig. 113) represent a system of three films, of which one is a
partition-wall between the other two; and let the tensions at the
three surfaces, or the tractions exercised upon a point at their
meeting-place, be proportional to _T_, _T′_ and _t_. Let α, β, γ be, as
in the figure, the opposite angles. Then:
(1) If _T_ be equal to _T′_, and _t_ be relatively insignificant, the
angles α, β will be of 90°.
[Illustration: Fig. 113.]
(2) If _T_ = _T′_, but be a little greater than _t_, then _t_ will
exert an appreciable traction, and α, β will be more than 90°, say, for
instance, 100°.
(3) If _T_ = _T′_ = _t_, then α, β, γ will all equal 120°.
The more complicated cases, when _t_, _T_ and _T′_ are all unequal, are
already sufficiently explained.
――――――――――
The biological facts which the foregoing considerations go a long way
to explain and account for have been the subject of much argument
and discussion, especially on the part of the botanists. Let me
recapitulate, in a very few words, the history of this long discussion.
Some fifty years ago, Hofmeister laid it down as a general law that
“The partition-wall stands always perpendicular to what was previously
the principal direction of growth in the cell,”—or, in other words,
perpendicular to the long axis of the cell[345]. Ten {305} years
later, Sachs formulated his rule, or principle, of “rectangular
section,” declaring that in all tissues, however complex, the
cell-walls cut one another (at the time of their formation) at right
angles[346]. Years before, Schwendener had found, in the final results
of cell-division, a universal system of “orthogonal trajectories[347]”;
and this idea Sachs further developed, introducing complicated systems
of confocal ellipses and hyperbolæ, and distinguishing between
periclinal walls, whose curves approximate to the peripheral contours,
radial partitions, which cut these at an angle of 90°, and finally
anticlines, which stand at right angles to the other two.
Reinke, in 1880, was the first to throw some doubt upon this
explanation. He pointed out various cases where the angle was not
a right angle, but was very definitely an acute one; and he saw,
apparently, in the more common rectangular symmetry merely what he
calls a necessary, but _secondary_, result of growth[348].
Within the next few years, a number of botanical writers were content
to point out further exceptions to Sachs’s Rule[349]; and in some cases
to show that the _curvatures_ of the partition-walls, especially such
cases of lenticular curvature as we have described, were by no means
accounted for by either Hofmeister or Sachs; while within the same
period, Sachs himself, and also Rauber, attempted to extend the main
generalisation to animal tissues[350].
While these writers regarded the form and arrangement of the
cell-walls as a biological phenomenon, with little if any direct
relation to ordinary physical laws, or with but a vague reference to
“mechanical conditions,” the physical side of the case was soon urged
by others, with more or less force and cogency. Indeed the general
resemblance between a cellular tissue and a “froth” {306} had been
pointed out long before, by Melsens, who had made an “artificial
tissue” by blowing into a solution of white of egg[351].
In 1886, Berthold published his _Protoplasmamechanik_, in which he
definitely adopted the principle of “minimal areas,” and, following
on the lines of Plateau, compared the forms of many cell-surfaces and
the arrangement of their partitions with those assumed under surface
tension by a system of “weightless films.” But, as Klebs[352] points
out in reviewing Berthold’s book, Berthold was careful to stop short of
attributing the biological phenomena to a definite mechanical cause.
They remained for him, as they had done for Sachs, so many “phenomena
of growth,” or “properties of protoplasm.”
In the same year, but while still apparently unacquainted with
Berthold’s work, Errera[353] published a short but very lucid article,
in which he definitely ascribed to the cell-wall (as Hofmeister had
already done) the properties of a semi-liquid film and drew from
this as a logical consequence the deduction that it _must_ assume
the various configurations which the law of minimal areas imposes on
the soap-bubble. So what we may call _Errera’s Law_ is formulated as
follows: A cellular membrane, at the moment of its formation, tends to
assume the form which would be assumed, under the same conditions, by a
liquid film destitute of weight.
Soon afterwards Chabry, in discussing the embryology of the Ascidians,
indicated many of the points in which the contacts between cells repeat
the surface-tension phenomena of the soap-bubble, and came to the
conclusion that part, at least, of the embryological phenomena were
purely physical[354]; and the same line of investigation and thought
were pursued and developed by Robert, in connection with the embryology
of the Mollusca[355]. Driesch again, in a series of papers, continued
to draw attention to the presence of capillary phenomena in the
segmenting cells {307} of various embryos, and came to the conclusion
that the mode of segmentation was of little importance as regards the
final result[356].
Lastly de Wildeman[357], in a somewhat wider, but also vaguer
generalisation than Errera’s, declared that “The form of the cellular
framework of vegetables, and also of animals, in its essential
features, depends upon the forces of molecular physics.”
――――――――――
[Illustration: Fig. 114.]
Let us return to our problem of the arrangement of partition films.
When we have three bubbles in contact, instead of two as in the case
already considered, the phenomenon is strictly analogous to our former
case. The three bubbles will be separated by three partition surfaces,
whose curvature will depend upon the relative size of the spheres, and
which will be plane if the latter are all of the same dimensions; but
whether plane or curved, the three partitions will meet one another
at an angle of 120°, in an axial line. Various pretty geometrical
corollaries accompany this arrangement. For instance, if Fig. 114
represent the three associated bubbles in a plane drawn through their
centres, _c_, _c′_, _c″_ (or what is the same thing, if it represent
the base of three bubbles resting on a plane), then the lines _uc_,
_uc″_, or _sc_, _sc′_, etc., drawn to the {308} centres from the
points of intersection of the circular arcs, will always enclose an
angle of 60°. Again (Fig. 115), if we make the angle _c″uf_ equal to
60°, and produce _uf_ to meet _cc″_ in _f_, _f_ will be the centre of
the circular arc which constitutes the partition _Ou_; and further, the
three points _f_, _g_, _h_, successively determined in this
[Illustration: Fig. 115.]
manner, will lie on one and the same straight line. In the case
of coequal bubbles or cells (as in Fig. 114, B), it is obvious that
the lines joining their centres form an equilateral triangle; and
consequently, that the centre of each circle (or sphere) lies on the
circumference of the other two; it is also obvious that _uf_ is now
{309} parallel to _cc″_, and accordingly that the centre of curvature
of the partition is now infinitely distant, or (as we have already
said), that the partition itself is plane.
[Illustration: Fig. 116.]
When we have four bubbles in conjunction, they would seem to be capable
of arrangement in two symmetrical ways: either, as in Fig. 116 (A),
with the four partition-walls meeting at right angles, or, as in (B),
with _five_ partitions meeting, three and three, at angles of 120°.
This latter arrangement is strictly analogous to the arrangement of
three bubbles in Fig. 114. Now, though both of these figures, from
their symmetry, are apparently figures of equilibrium, yet, physically,
the former turns out to be of unstable and the latter of stable
equilibrium. If we try to bring our four bubbles into the form of Fig.
116, A, such an arrangement endures only for an instant; the partitions
glide upon each other, a median wall springs into existence, and the
system at once assumes the form of our second figure (B). This is a
direct consequence of the law of minimal areas: for it can be shewn, by
somewhat difficult mathematics (as was first done by Lamarle), that,
in dividing a closed space into a given number of chambers by means
of partition-walls, the least possible area of these partition-walls,
taken together, can only be attained when they meet together in groups
of three, at equal angles, that is to say at angles of 120°. {310}
Wherever we have a true cellular complex, an arrangement of cells in
actual physical contact by means of a boundary film, we find this
general principle in force; we must only bear in mind that, for its
perfect recognition, we must be able to view the object in a plane
at right angles to the boundary walls. For instance, in any ordinary
section of a vegetable parenchyma, we recognise the appearance of
a “froth,” precisely resembling that which we can construct by
imprisoning a mass of soap-bubbles in a narrow vessel with flat sides
of glass; in both cases we see the cell-walls everywhere meeting, by
threes, at angles of 120°, irrespective of the size of the individual
cells: whose relative size, on the other hand, determines the
_curvature_ of the partition-walls. On the surface of a honey-comb we
have precisely the same conjunction, between cell and cell, of three
boundary walls, meeting at 120°. In embryology, when we examine a
segmenting egg, of four (or more) segments, we find in like manner, in
the great majority of cases, if not in all, that the same principle
is still exemplified; the four segments do not meet in a common
centre, but each cell is in contact with two others, and the three,
and only three, common boundary walls meet at the normal angle of
120°. A so-called _polar furrow_[358], the visible edge of a vertical
partition-wall, joins (or separates) the two triple contacts, precisely
as in Fig. 116, B.
In the four-celled stage of the frog’s egg, Rauber (an exceptionally
careful observer) shews us three alternative modes in which the
four cells may be found to be conjoined (Fig. 117). In (A) we have
the commonest arrangement, which is that which we have just studied
and found to be the simplest theoretical one; that namely where a
straight “polar furrow” intervenes, and where, at its extremities,
the partition-walls are conjoined three by three. In (B), we have
again a polar furrow, which is now seen to be a portion of the first
“segmentation-furrow” (cf. Fig. 155 etc.) by which the egg was
originally divided into two; the four-celled stage being reached by
the appearance of the transverse furrows {311} and their corresponding
partitions. In this case, the polar furrow is seen to be sinuously
curved, and Rauber tells us that its curvature gradually alters: as a
matter of fact, it (or rather the partition-wall corresponding to it)
is gradually setting itself into a position of equilibrium, that is
to say of equiangular contact with its neighbours, which position of
equilibrium is already attained or nearly so in Fig. 117, A. In Fig.
117, C, we have a very different condition, with which we shall deal in
a moment.
[Illustration: Fig. 117. Various ways in which the four cells are
co-arranged in the four-celled stage of the frog’s egg. (After Rauber.)]
According to the relative magnitude of the bodies in contact, this
“polar furrow” may be longer or shorter, and it may be so minute as to
be not easily discernible; but it is quite certain that no simple and
homogeneous system of fluid films such as we are dealing with is in
equilibrium without its presence. In the accounts given, however, by
embryologists of the segmentation of the egg, while the polar furrow
is depicted in the great majority of cases, there are others in which
it has not been seen and some in which its absence is definitely
asserted[359]. The cases where four cells, lying in one plane, meet _in
a point_, such as were frequently figured by the older embryologists,
are very difficult to verify, and I have not come across a single
clear case in recent literature. Considering the physical stability
of the other arrangement, the great preponderance of cases in which
it is known to occur, the difficulty of recognising the polar furrow
in cases where it is very small and unless it be specially looked
for, and the natural tendency of the draughtsman to make an all but
symmetrical structure appear wholly so, I am much inclined to attribute
to {312} error or imperfect observation all those cases where the
junction-lines of four cells are represented (after the manner of Fig.
116, A) as a simple cross[360].
But while a true four-rayed intersection, or simple cross, is
theoretically impossible (save as a transitory and highly unstable
condition), there is another condition which may closely simulate
it, and which is common enough. There are plenty of representations
of segmenting eggs, in which, instead of the triple junction and
polar furrow, the four cells (and in like manner their more numerous
successors) are represented as _rounded off_, and separated from one
another by an empty space, or by a little drop of an extraneous fluid,
evidently not directly miscible with the fluid surfaces of the cells.
Such is the case in the obviously accurate figure which Rauber gives
(Fig. 117, C) of the third mode of conjunction in the four-celled stage
of the frog’s egg. Here Rauber is most careful to point out that the
furrows do not simply “cross,” or meet in a point, but are separated
by a little space, which he calls the _Polgrübchen_, and asserts to be
constantly present whensoever the polar furrow, or _Brechungslinie_, is
not to be discerned. This little interposed space, with its contained
drop of fluid, materially alters the case, and implies a new condition
of theoretical and actual equilibrium. For, on the one hand, we see
that now the four intercellular partitions do not meet _one another
at all_; but really impinge upon four new and separate partitions,
which constitute interfacial contacts, not between cell and cell, but
between the respective cells and the intercalated drop. And secondly,
the angles at which these four little surfaces will meet the four
cell-partitions, will be determined, in the usual way, by the balance
between the respective tensions of these several surfaces. In an
extreme case (as in some pollen-grains) it may be found that the cells
under the observed circumstances are not truly in surface contact:
that they are so many drops which touch but do not “wet” one another,
and which are merely held together by the pressure of the surrounding
envelope. But even supposing, {313} as is in all probability the
actual case, that they are in actual fluid contact, the case from the
point of view of surface tension presents no difficulty. In the case of
the conjoined soap-bubbles, we were dealing with _similar_ contacts and
with _equal_ surface tensions throughout the system; but in the system
of protoplasmic cells which constitute the segmenting egg we must make
allowance for _an inequality_ of tensions, between the surfaces where
cell meets cell, and where on the other hand cell-surface is in contact
with the surrounding medium,—in this case generally water or one of the
fluids of the body. Remember that our general condition is that, in our
[Illustration: Fig. 118.]
entire system, the _sum of the surface energies_ is a minimum; and,
while this is attained by the _sum of the surfaces_ being a minimum
in the case where the energy is uniformly distributed, it is not
necessarily so under non-uniform conditions. In the diagram (Fig. 118)
if the energy per unit area be greater along the contact surface _cc′_,
where cell meets cell, than along _ca_ or _cb_, where cell-surface is
in contact with the surrounding medium, these latter surfaces will
tend to increase and the surface of cell-contact to diminish. In short
there will be the usual balance of forces between the tension along
the surface _cc′_, and the two opposing tensions along _ca_ and _cb_.
If the former be greater than either of the other two, the outside
angle will be less than 120°; and if the tension along the surface
_cc′_ be as much or more than the sum of the other two, then the drops
will stand in contact only, save for the possible effect of external
pressure, at a point. This is the explanation, in general terms, of
the peculiar conditions obtaining in Nostoc and its allies (p. 300),
and it also leads us to a consideration of the general properties and
characters of an “epidermal” layer.
――――――――――
While the inner cells of the honey-comb are symmetrically situated,
sharing with their neighbours in equally distributed pressures or
tensions, and therefore all tending with great accuracy {314} to
identity of form, the case is obviously different with the cells at
the borders of the system. So it is, in like manner, with our froth of
soap-bubbles. The bubbles, or cells, in the interior of the mass are
all alike in general character, and if they be equal in size are alike
in every respect: their sides are uniformly flattened[361], and tend
to meet at equal angles of 120°. But the bubbles which constitute the
outer layer retain their spherical surfaces, which however still tend
to meet the partition-walls connected with them at constant angles
of 120°. This outer layer of bubbles, which forms the surface of our
froth, constitutes after a fashion what we should call in botany an
“epidermal” layer. But in our froth of soap-bubbles we have, as a rule,
the same kind of contact (that is to say, contact with _air_) both
within and without the bubbles; while in our living cell, the outer
wall of the epidermal cell is exposed to air on the one side, but is in
contact with the
[Illustration: Fig. 119.]
protoplasm of the cell on the other: and this involves a difference
of tensions, so that the outer walls and their adjacent partitions
are no longer likely to meet at equal angles of 120°. Moreover, a
chemical change, due for instance to oxidation or possibly also to
adsorption, is very likely to affect the external wall, and may tend
to its consolidation; and this process, as we have seen, is tantamount
to a large increase, and at the same time an equalisation, of tension
in that outer wall, and will lead the adjacent partitions to impinge
upon it at angles more and more nearly approximating to 90°: the
bubble-like, or spherical, surfaces of the individual cells being
more and more flattened in consequence. Lastly, the chemical changes
which affect the outer walls of the superficial cells may extend, in
greater or less degree, to their inner walls also: with the result
that these {315} cells will tend to become more or less rectangular
throughout, and will cease to dovetail into the interstices of the
next subjacent layer. These then are the general characters which
we recognise in an epidermis; and we perceive that the fundamental
character of an epidermis simply is that it lies on the outside, and
that its main physical characteristics follow, as a matter of course,
from the position which it occupies and from the various consequences
which that situation entails. We have however by no means exhausted
the subject in this short account; for the botanist is accustomed to
draw a sharp distinction between a true epidermis and what is called
epidermal tissue. The latter, which is found in such a sea-weed as
Laminaria and in very many other cryptogamic plants, consists, as in
the hypothetical case we have described, of a more or less simple and
direct modification of the general or fundamental tissue. But a “true
epidermis,” such as we have it in the higher plants, is something
with a long morphological history, something which has been laid down
or differentiated in an early stage of the plant’s growth, and which
afterwards retains its separate and independent character. We shall
see presently that a physical reason is again at hand to account,
under certain circumstances, for the early partitioning off, from a
mass of embryonic tissue, of an outer layer of cells which from their
first appearance are marked off from the rest by their rectangular and
flattened form.
――――――――――
We have hitherto considered our cells, or bubbles, as lying in a plane
of symmetry, and further, we have only considered the appearance which
they present as projected on that plane: in simpler words, we have been
considering their appearance in surface or in sectional view. But we
have further to consider them as solids, whether they be still grouped
in relation to a single plane (like the four cells in Fig. 116) or
heaped upon one another, as for instance in a tetrahedral form like
four cannon-balls; and in either case we have to pass from the problems
of plane to those of solid geometry. In short, the further development
of our theme must lead us along two paths of enquiry, which continually
intercross, namely (1) the study of more complex cases of partition and
of contact in a plane, and (2) the whole question of the surfaces {316}
and angles presented by solid figures in symmetrical juxtaposition.
Let us take a simple case of the latter kind, and again afterwards, so
far as possible, let us try to keep the two themes separate.
Where we have three spheres in contact, as in Fig. 114 or in either
half of Fig. 116, B, let us consider the point of contact (_O_, Fig.
114) not as a point in the plane section of the diagram, but as a point
where three _furrows_ meet on the surface of the system. At this point,
_three cells_ meet; but it is also obvious that there meet here _six
surfaces_, namely the outer, spherical walls of the three bubbles,
and the three partition-walls which divide them, two and two. Also,
_four_ lines or _edges_ meet here; viz. the three external arcs which
form the outer boundaries of the partition-walls (and which correspond
to what we commonly call the “furrows” in the segmenting egg); and
as a fourth edge, the “arris” or junction of the three partitions
(perpendicular to the plane of the paper), where they all three meet
together, as we have seen, at equal angles of 120°. Lastly, there meet
at the point _four solid angles_, each bounded by three surfaces: to
wit, within each bubble a solid angle bounded by two partition-walls
and by the surface wall; and (fourthly) an external solid angle bounded
by the outer surfaces of all three bubbles. Now in the case of the
soap-bubbles (whose surfaces are all in contact with air, both outside
and in), the six films meeting at the point, whether surface films
or partition films, are all similar, with similar tensions. In other
words the tensions, or forces, acting at the point are all similar
and symmetrically arranged, and it at once follows from this that the
angles, solid as well as plane, are all equal. It is also obvious that,
as regards the point of contact, the system will still be symmetrical,
and its symmetry will be quite unchanged, if we add a fourth bubble in
contact with the other three: that is to say, if where we had merely
the outer air before, we now replace it by the air in the interior of
another bubble. The only difference will be that the pressure exercised
by the walls of this fourth bubble will alter the curvature of the
surfaces of the others, so far as it encloses them; and, if all four
bubbles be identical in size, these surfaces which formerly we called
external and which have now come to be internal partitions, will,
like the others, be flattened by equal and opposite pressure, into
planes. We are now dealing, in short, {317} with six planes, meeting
symmetrically in a point, and constituting there four equal solid
angles.
[Illustration: Fig. 120.]
If we make a wire cage, in the form of a regular tetrahedron, and dip
it into soap-solution, then when we withdraw it we see that to each
one of the six edges of the tetrahedron, i.e. to each one of the six
wires which constitute the little cage, a film has attached itself; and
these six films meet internally at a point, and constitute in every
respect the symmetrical figure which we have just been describing. In
short, the system of films we have hereby automatically produced is
precisely the system of partition-walls which exist in our tetrahedral
aggregation of four spherical bubbles:—precisely the same, that is to
say, in the neighbourhood of the meeting-point, and only differing in
that we have made the wires of our tetrahedron straight, instead of
imitating the circular arcs which actually form the intersections of
our bubbles. This detail we can easily introduce in our wire model if
we please.
Let us look for a moment at the geometry of our figure. Let _o_ (Fig.
120) be the centre of the tetrahedron, i.e. the centre of symmetry
where our films meet; and let _oa_, _ob_, _oc_, _od_, be lines drawn
to the four corners of the tetrahedron. Produce _ao_ to meet the base
in _p_; then _apd_ is a right-angled triangle. It is not difficult to
prove that in such a figure, _o_ (the centre of gravity of the system)
{318} lies just three-quarters of the way between an apex, _a_, and
a point, _p_, which is the centre of gravity of the opposite base.
Therefore
_op_ = _oa_/3 = _od_/3.
Therefore cos _dop_ = 1/3 and cos _aod_ = − 1/3.
That is to say, the angle _aod_ is just, as nearly as possible,
109° 28′ 16″. This angle, then, of 109° 28′ 16″, or very nearly 109
degrees and a half, is the angle at which, in this and _every other
solid system_ of liquid films, the edges of the partition-walls meet
one another at a point. It is the fundamental angle in the solid
geometry of our systems, just as 120° was the fundamental angle of
symmetry so long as we considered only the plane projection, or plane
section, of three films meeting in an edge.
――――――――――
Out of these two angles, we may construct a great variety of figures,
plane and solid, which become all the more varied and complex when, by
considering the case of unequal as well as equal cells, we admit curved
(e.g. spherical) as well as plane boundary surfaces. Let us consider
some examples and illustrations of these, beginning with those which we
need only consider in reference to a plane.
Let us imagine a system of equal cylinders, or equal spheres, in
contact with one another in a plane, and represented in section by
the equal and contiguous circles of Fig. 121. I borrow my figure, by
the way, from an old Italian naturalist, Bonanni (a contemporary of
Borelli, of Hay and Willoughby and of Martin Lister), who dealt with
this matter in a book chiefly devoted to molluscan shells[362].
It is obvious, as a simple geometrical fact, that each of these equal
circles is in contact with six surrounding circles. Imagine now that
the whole system comes under some uniform stress. It may be of uniform
surface tension at the boundaries of all the cells; it may be of
pressure caused by uniform growth or expansion within the cells; or
it may be due to some uniformly applied constricting pressure from
without. In all of these cases the _points_ of contact between the
circles in the diagram will be extended into {319} _lines_ of contact,
representing _surfaces_ of contact in the actual spheres or cylinders;
and the equal circles of our diagram will be converted into regular and
equal hexagons. The angles of these hexagons, at each of which three
hexagons meet, are of course angles of 120°. So far as the form is
concerned, so long as we are concerned only with a morphological result
and not with a physiological process, the result is precisely the same
whatever be the force which brings the bodies together in symmetrical
apposition; it is by no means necessary for us, in the first instance,
even to enquire whether it be surface tension or mechanical pressure or
some other physical force which is the cause, or the main cause, of the
phenomenon.
[Illustration: Fig. 121. Diagram of hexagonal cells. (After Bonanni.)]
The production by mutual interaction of polyhedral cells, which, under
conditions of perfect symmetry, become regular hexagons, is very
beautifully illustrated by Prof. Bénard’s “_tourbillons cellulaires_”
(cf. p. 259), and also in some of Leduc’s diffusion experiments. A weak
(5 per cent.) solution of gelatine is allowed to set on a plate of
glass, and little drops of a 5 or 10 per cent. solution of ferrocyanide
of potassium are then placed at regular intervals upon the gelatine.
Immediately each little drop becomes the centre, or pole, of a system
of diffusion currents, {320} and the several systems conflict with and
repel one another, so that presently each little area becomes the seat
of a double current system, from its centre outwards and back again;
until at length the concentration of the field becomes equalised and
the currents {321}
[Illustration: Fig. 122. An “artificial tissue,” formed by coloured
drops of sodium chloride solution diffusing in a less dense solution of
the same salt. (After Leduc.)]
[Illustration: Fig. 123. An artificial cellular tissue, formed by the
diffusion in gelatine of drops of a solution of potassium ferrocyanide.
(After Leduc.)]
cease. After equilibrium is attained, and when the gelatinous mass
is permitted to dry, we have an artificial tissue of more or less
regularly hexagonal “cells,” which simulate in the closest way an
organic parenchyma. And by varying the experiment, in ways which Leduc
describes, we may simulate various forms of tissue, and produce cells
with thick walls or with thin, cells in close contact or with wide
intercellular spaces, cells with plane or with curved partitions, and
so forth.
――――――――――
[Illustration: Fig. 124. Epidermis of _Girardia_. (After Goebel.)]
The hexagonal pattern is illustrated among organisms in countless
cases, but those in which the pattern is perfectly regular, by
reason of perfect uniformity of force and perfect equality of the
individual cells, are not so numerous. The hexagonal epithelium-cells
of the pigment layer of the eye, external to the retina, are a good
example. Here we have a single layer of uniform cells, reposing on
the one hand upon a basement membrane, supported behind by the solid
wall of the sclerotic, and exposed on the other hand to the uniform
fluid pressure of the vitreous humour. The conditions all point, and
lead, to a perfectly symmetrical result: that is to say, the cells,
uniform in size, are flattened out to a uniform thickness by the fluid
pressure acting radially; and their reaction on each other converts
the flattened discs into regular hexagons. In an ordinary columnar
epithelium, such as that of the intestine, we see again that the
columnar cells have been compressed into hexagonal prisms; but here as
a rule the cells are less uniform in size, small cells are apt to be
intercalated among the larger, and the perfect symmetry is accordingly
lost. The same is true of ordinary vegetable parenchyma; the
originally spherical cells are approximately equal in size, but only
approximately; and there are accordingly all degrees in the regularity
and symmetry of the resulting tissue. But obviously, wherever we {322}
have, in addition to the forces which tend to produce the regular
hexagonal symmetry, some other asymmetrical component arising from
growth or traction, then our regular hexagons will be distorted in
various simple ways. This condition is illustrated in the accompanying
diagram of the epidermis of Girardia; it also accounts for the more or
less pointed or fusiform cells, each still in contact (as a rule) with
six others, which form the epithelial lining of the blood-vessels: and
other similar, or analogous, instances are very common.
[Illustration: Fig. 125. Soap-froth under pressure. (After Rhumbler.)]
In a soap-froth imprisoned between two glass plates, we have a
symmetrical system of cells, which appear in optical section (as in
Fig. 125, B) as regular hexagons; but if we press the plates a little
closer together, the hexagons become deformed or flattened (Fig. 125,
A). In this case, however, if we cease to apply further pressure, the
tension of the films throughout the system soon adjusts itself again,
and in a short time the system has regained the former symmetry of Fig.
125, B.
[Illustration: Fig. 126. From leaf of _Elodea canadensis_. (After
Berthold.)]
In the growth of an ordinary dicotyledonous leaf, we once more see
reflected in the form of its epidermal cells the tractions, irregular
but on the whole longitudinal, which growth has superposed on the
tensions of the partition-walls (Fig. 126). In the narrow elongated
leaf of a Monocotyledon, such as a hyacinth, the elongated, apparently
quadrangular {323} cells of the epidermis appear as a necessary
consequence of the simpler laws of growth which gave its simple form to
the leaf as a whole. In this last case, however, as in all the others,
the rule still holds that only three partitions (in surface view) meet
in a point; and at their point of meeting the walls are for a short
distance manifestly curved, so as to permit the junction to take place
at or nearly at the normal angle of 120°.
Briefly speaking, wherever we have a system of cylinders or spheres,
associated together with sufficient mutual interaction to bring them
into complete surface contact, there, in section or in surface view, we
tend to get a pattern of hexagons.
While the formation of an hexagonal pattern on the basis of
ready-formed and symmetrically arranged material units is a very
common, and indeed the general way, it does not follow that there are
not others by which such a pattern can be obtained. For instance,
if we take a little triangular dish of mercury and set it vibrating
(either by help of a tuning-fork, or by simply tapping on the sides)
we shall have a series of little waves or ripples starting inwards
from each of the three faces; and the intercrossing, or interference
of these three sets of waves produces crests and hollows, and
intermediate points of no disturbance, _whose loci are seen_ as a
beautiful pattern of minute hexagons. It is possible that the very
minute and astonishingly regular pattern of hexagons which we see,
for instance, on the surface of many diatoms, may be a phenomenon
of this order[363]. The same may be the case also in Arcella, where
an apparently hexagonal pattern is found not to consist of simple
hexagons, but of “straight lines in three sets of parallels, the lines
of each set making an angle of sixty degrees with those of the other
two sets[364].” We must also bear in mind, in the case of the minuter
forms, the large possibilities of optical illusion. For instance, in
one of Abbe’s “diffraction-plates,” a pattern of dots, set at equal
interspaces, is reproduced on a very minute scale by photography; but
under certain conditions of microscopic illumination and focussing,
these isolated dots appear as a pattern of hexagons.
――――――――――
A symmetrical arrangement of hexagons, such as we have just been
studying, suggests various simple geometrical corollaries, of which
the following may perhaps be a useful one.
We may sometimes desire to estimate the number of hexagonal areas or
facets in some structure where these are numerous, such for instance
as the {324} cornea of an insect’s eye, or in the minute pattern of
hexagons on many diatoms. An approximate enumeration is easily made as
follows.
For the area of a hexagon (if we call δ the short diameter, that
namely which bisects two of the opposite sides) is δ^2 × (√3)/2,
the area of a circle being _d_^2 ⋅ π/4. Then, if the diameter (_d_)
of a circular area include _n_ hexagons, the area of that circle
equals (_n_ ⋅ δ)^2 × π/4. And, dividing this number by the area of
a single hexagon, we obtain for the number of areas in the circle,
each equal to a hexagonal facet, the expression _n_^2 × π/4 × 2/(√3)
= 0·907_n_^2, or (9/10) ⋅ _n_^2, nearly.
This calculation deals, not only with the complete facets, but with
the areas of the broken hexagons at the periphery of the circle. If
we neglect these latter, and consider our whole field as consisting
of successive rings of hexagons about a central one, we may obtain a
still simpler rule[365]. For obviously, around our central hexagon
there stands a zone of six, and around these a zone of twelve, and
around these a zone of eighteen, and so on. And the total number,
excluding the central hexagon, is accordingly:
For one zone 6 = 2 × 3 = 3 × 1 × 2,
For two zones 18 = 3 × 6 = 3 × 2 × 3,
For three zones 36 = 4 × 9 = 3 × 3 × 4,
For four zones 60 = 5 × 12 = 3 × 4 × 5,
For five zones 90 = 6 x 15 = 3 × 5 × 6,
and so forth. If _N_ be the number of zones, and if we add one to
the above numbers for the odd central hexagon, the rule evidently
is, that the total number, _H_, = 3_N_(_N_ + 1) + 1. Thus, if in a
preparation of a fly’s cornea, I can count twenty-five facets in a
line from a central one, the total number in the entire circular field
is (3 × 25 × 26) + 1 = 1951[366].
――――――――――
The same principles which account for the development of hexagonal
symmetry hold true, as a matter of course, not only of individual
_cells_ (in the biological sense), but of any close-packed bodies
of uniform size and originally circular outline; and the hexagonal
pattern is therefore of very common occurrence, under widely different
circumstances. The curious reader may consult Sir Thomas Browne’s
quaint and beautiful account, in the _Garden of Cyrus_, of hexagonal
(and also of quincuncial) symmetry in plants and animals, which “doth
neatly declare how nature Geometrizeth, and observeth order in all
things.” {325}
We have many varied examples of this principle among corals, wherever
the polypes are in close juxtaposition, with neither empty space nor
accumulations of matrix between their adjacent walls. _Favosites
gothlandica_, for instance, furnishes us with an excellent example. In
the great genus Lithostrotion we have some species that are “massive”
and others that are “fasciculate”; in other words in some the long
cylindrical corallites are in close contact with one another, and in
others they are separate and loosely bundled (Fig. 127). Accordingly in
the former the corallites are
[Illustration: Fig. 127. _Lithostrotion Martini._ (After Nicholson.)]
[Illustration: Fig. 128. _Cyathophyllum hexagonum._ (From Nicholson,
after Zittel.)]
squeezed into hexagonal prisms, while in the latter they retain
their cylindrical form. Where the polypes are comparatively few, and
so have room to spread, the mutual pressure ceases to work or only
tends to push them asunder, letting them remain circular in outline
(e.g. Thecosmilia). Where they vary gradually in size, as for instance
in _Cyathophyllum hexagonum_, they are more or less hexagonal but are
not regular hexagons; and where there is greater and more irregular
variation in size, the cells will be _on the average_ hexagonal, but
some will have fewer and some more sides than six, as in the annexed
figure of Arachnophyllum (Fig. 129). {326} Where larger and smaller
cells, corresponding to two different kinds of zooids, are mixed
together, we may get various results. If the larger cells are numerous
enough to be more or less in contact with one another (e.g. various
Monticuliporae) they will be irregular hexagons, while the smaller
cells between them will be crushed into all manner of irregular angular
forms. If on the other hand the large cells are comparatively few and
are large and strong-walled compared with their smaller neighbours,
then the latter alone will be squeezed into hexagons, while the larger
ones will tend to retain their circular outline undisturbed (e.g.
Heliopora, Heliolites, etc.).
[Illustration: Fig. 129. _Arachnophyllum pentagonum._ (After
Nicholson.)]
[Illustration: Fig. 130. _Heliolites._ (After Woods.)]
When, as happens in certain corals, the peripheral walls or “thecae”
of the individual polypes remain undeveloped but the radiating
septa are formed and calcified, then we obtain new and beautiful
mathematical configurations (Fig. 131). For the radiating septa are
no longer confined to the circular or hexagonal bounds of a polypite,
but tend to meet and become confluent with their neighbours on every
side; and, tending to assume positions of equilibrium, or of minimal
area, under the restraints to which they are subject, they fall into
congruent curves; and these correspond, in a striking manner, to the
lines of force running, in a common field of force, between a number
of secondary centres. Similar patterns may be produced in various
ways, by the play of osmotic or magnetic forces; and a particular
and very curious case is to be found in those complicated forms of
nuclear division {327} known as triasters, polyasters, etc., whose
relation to a field of force Hartog has explained[367]. It is obvious
that, in our corals, these curving septa are all orthogonal to the
non-existent hexagonal boundaries. As the phenomenon is wholly due to
the imperfect development or non-existence of a thecal wall, it is
not surprising that we find identical configurations among various
corals, or families of corals, not otherwise related to one another;
we find the same or very similar patterns displayed, for instance, in
Synhelia (_Oculinidae_), in Phillipsastraea (_Rugosa_), in Thamnastraea
(_Fungida_), and in many more.
[Illustration: Fig. 131. Surface-views of Corals with undeveloped
thecae and confluent septa. A, _Thamnastraea_; B, _Comoseris_. (From
Nicholson, after Zittel.)]
――――――――――
The most famous of all hexagonal conformations and perhaps the most
beautiful is that of the bee’s cell. Here we have, as in our last
examples, a series of equal cylinders, compressed by symmetrical forces
into regular hexagonal prisms. But in this case we have two rows of
such cylinders, set opposite to one another, end to end; and we have
accordingly to consider also the conformation of their ends. We may
suppose our original cylindrical cells to have spherical ends, which
is their normal and symmetrical mode of termination; and, for closest
packing, it is obvious that the end of any one cylinder will touch, and
fit in between, the ends of three cylinders in the opposite row. It is
just as when we pile round-shot in a heap; each sphere that we {328}
set down fits into its nest between three others, and the four form a
regular tetrahedral arrangement. Just as it was obvious, then, that by
mutual pressure from the six _laterally_ adjacent cells, any one cell
would be squeezed into a hexagonal prism, so is it also obvious that,
by mutual pressure against the three _terminal_ neighbours, the end
of any one cell will be compressed into a solid trihedral angle whose
edges will meet, as in the analogous case already described of a system
of soap-bubbles, at a plane angle of 109° and so many minutes and
seconds. What we have to comprehend, then, is how the _six_ sides of
the cell are to be combined with its _three_ terminal facets. This is
done by bevelling off three alternate angles of the prism, in a uniform
manner, until we have tapered the prism to a point; and by so doing,
we evidently produce three _rhombic_ surfaces, each of which is double
of the triangle formed by joining the apex to the three untouched
angles of the prism. If we experiment, not with cylinders, but with
spheres, if for instance we pile together a mass of bread-pills (or
pills of plasticine), and then submit the whole to a uniform pressure,
it is obvious that each ball (like the seeds in a pomegranate, as
Kepler said), will be in contact with _twelve_ others,—six in its
own plane, three below and three above, and in compression it will
therefore develop twelve plane surfaces. It will in short repeat,
above and below, the conditions to which the bee’s cell is subject at
one end only; and, since the sphere is symmetrically situated towards
its neighbours on all sides, it follows that the twelve plane sides
to which its surface has been reduced will be all similar, equal and
similarly situated. Moreover, since we have produced this result by
squeezing our original spheres close together, it is evident that the
bodies so formed completely fill space. The regular solid which fulfils
all these conditions is the _rhombic dodecahedron_. The bee’s cell,
then, is this figure incompletely formed: it is a hexagonal prism
with one open or unfinished end, and one trihedral apex of a rhombic
dodecahedron.
The geometrical form of the bee’s cell must have attracted the
attention and excited the admiration of mathematicians from time
immemorial. Pappus the Alexandrine has left us (in the introduction to
the Fifth Book of his _Collections_) an account of its hexagonal plan,
and he drew from its mathematical symmetry the {329} conclusion that
the bees were endowed with reason: “There being, then, three figures
which of themselves can fill up the space round a point, viz. the
triangle, the square and the hexagon, the bees have wisely selected
for their structure that which contains most angles, suspecting indeed
that it could hold more honey than either of the other two.” Erasmus
Bartholinus was apparently the first to suggest that this hypothesis
was not warranted, and that the hexagonal form was no more than the
necessary result of equal pressures, each bee striving to make its own
little circle as large as possible.
The investigation of the ends of the cell was a more difficult
matter, and came later, than that of its sides. In general terms this
arrangement was doubtless often studied and described: as for instance,
in the _Garden of Cyrus_: “And the Combes themselves so regularly
contrived that their mutual intersections make three Lozenges at the
bottom of every Cell; which severally regarded make three Rows of
neat Rhomboidall Figures, connected at the angles, and so continue
three several chains throughout the whole comb.” But Maraldi[368]
(Cassini’s nephew) was the first to measure the terminal solid angle or
determine the form of the rhombs in the pyramidal ending of the cell.
He tells us that the angles of the rhomb are 110° and 70°: “Chaque
base d’alvéole est formée par trois rhombes presque toujours égaux et
semblables, qui, suivant les mesures que nous avons prises, ont les
deux angles obtus chacun de 110 degrés, et par conséquent les deux
aigus chacun de 70°.” He also stated that the angles of the trapeziums
which form the sides of the body of the cell were identical angles, of
110° and 70°; but in the same paper he speaks of the angles as being,
respectively, 109° 28′ and 70° 32′. Here a singular confusion at once
arose, and has been perpetuated in the books[369]. “Unfortunately
Réaumur chose to look upon this second determination of Maraldi’s as
being, as well as the first, a direct result of measurement, whereas
it is in reality theoretical. He speaks of it as Maraldi’s more
precise measurement, and this error has been repeated in spite of its
absurdity to the present day; nobody {330} appears to have thought of
the impossibility of measuring such a thing as the end of a bee’s cell
to the nearest minute.” At any rate, it now occurred to Réaumur (as
curiously enough, it had not done to Maraldi) that, just as the closely
packed hexagons gave the minimal extent of boundary in a plane, so the
actual solid figure, as determined by Maraldi, might be that which,
for a given solid content, gives the minimum of surface: or which, in
other words, would hold the most honey for the least wax. He set this
problem before Koenig, and the geometer confirmed his conjecture, the
result of his calculations agreeing within two minutes (109° 26′ and
70° 34′) with Maraldi’s determination. But again, Maclaurin[370] and
Lhuilier[371], by different methods, obtained a result identical with
Maraldi’s; and were able to shew that the discrepancy of 2′ was due to
an error in Koenig’s calculation (of tan θ = √2),—that is to say to the
imperfection of his logarithmic tables,—not (as the books say[372]) “to
a mistake on the part of the Bee.” “Not to a mistake on the part of
Maraldi” is, of course, all that we are entitled to say.
[Illustration: Fig. 132.]
The theorem may be proved as follows:
_ABCDEF_, _abcdef_, is a right prism upon a regular hexagonal base. The
corners _BDF_ are cut off by planes through the lines _AC_, _CE_, _EA_,
meeting in a point _V_ on the axis _VN_ of the prism, and intersecting
_Bb_, _Dd_, _Ff_, at _X_, _Y_, _Z_. It is evident that the volume of
the figure thus formed is the same as that of the original prism with
hexagonal ends. For, if the axis cut the hexagon _ABCDEF_ in _N_, the
volumes _ACVN_, _ACBX_ are equal. {331}
It is required to find the inclination of the faces forming the
trihedral angle at _V_ to the axis, such that the surface of the figure
may be a minimum.
Let the angle _NVX_, which is half the solid angle of the prism, = θ;
the side of the hexagon, as _AB_, = _a_; and the height, as _Aa_, = _h_.
Then, _AC_ = 2_a_ cos 30° = _a_√3.
And _VX_ = _a_/sin θ (from inspection of the triangle _LXB_)
Therefore the area of the rhombus _VAXC_ = (_a_^2 √3)/(2 sin θ).
And the area of _AabX_ = (_a_/2)(2_h_ − ½_VX_ cos θ)
= (_a_/2)(2_h_ − _a_/2 ⋅ cot θ).
Therefore the total area of the figure
= hexagon _abcdef_ + 3_a_(2_h_ − (_a_/2) cot θ)
+ 3((_a_^2 √3)/(2 sin θ)).
Therefore _d_(Area)/_d_θ = (3_a_^2/2)((1/sin^2 θ)
− (√3 cos θ)/(sin^2 θ)).
But this expression vanishes, that is to say, _d_(Area)/_d_θ = 0,
when cos θ = 1/√3, that is when θ = 54° 44′ 8″ = ½(109° 28′ 16″).
This then is the condition under which the total area of the figure has
its minimal value.
――――――――――
That the beautiful regularity of the bee’s architecture is due to some
automatic play of the physical forces, and that it were fantastic to
assume (with Pappus and Réaumur) that the bee intentionally seeks for a
method of economising wax, is certain, but the precise manner of this
automatic action is not so clear. When the hive-bee builds a solitary
cell, or a small cluster of cells, as it does for those eggs which are
to develop into queens, it makes but a rude production. The queen-cells
are lumps of coarse wax hollowed out and roughly bitten into shape,
bearing the marks of the bee’s jaws, like the marks of a blunt adze on
a rough-hewn log. Omitting the simplest of all cases, when (as among
some humble-bees) the old cocoons are used to hold honey, the cells
built by the “solitary” wasps and bees are of various kinds. They may
be formed by partitioning off little chambers in a hollow stem; {332}
they may be rounded or oval capsules, often very neatly constructed,
out of mud, or vegetable _fibre_ or little stones, agglutinated
together with a salivary glue; but they shew, except for their rounded
or tubular form, no mathematical symmetry. The social wasps and many
bees build, usually out of vegetable matter chewed into a paste with
saliva, very beautiful nests of “combs”; and the close-set papery
cells which constitute these combs are just as regularly hexagonal as
are the waxen cells of the hive-bee. But in these cases (or nearly
all of them) the cells are in a single row; their sides are regularly
hexagonal, but their ends, from the want of opponent forces, remain
simply spherical. In _Melipona domestica_ (of which Darwin epitomises
Pierre Huber’s description) “the large waxen honey-cells are nearly
spherical, nearly equal in size, and are aggregated into an irregular
mass.” But the spherical form is only seen on the outside of the mass;
for inwardly each cell is flattened into “two, three or more flat
surfaces, according as the cell adjoins two, three or more other cells.
When one cell rests on three other cells, which from the spheres being
nearly of the same size is very frequently and necessarily the case,
the three flat surfaces are united into a pyramid; and this pyramid, as
Huber has remarked, is manifestly a gross imitation of the three-sided
pyramidal base of the cell of the hive-bee[373].” The question is, to
what particular force are we to ascribe the plane surfaces and definite
angles which define the sides of the cell in all these cases, and the
ends of the cell in cases where one row meets and opposes another. We
have seen that Bartholin suggested, and it is still commonly believed,
that this result is due to simple physical pressure, each bee enlarging
as much as it can the cell which it is a-building, and nudging its wall
outwards till it fills every intervening gap and presses hard against
the similar efforts of its neighbour in the cell next door[374].
But it is very doubtful {333} whether such physical or mechanical
pressure, more or less intermittently exercised, could produce the
all but perfectly smooth, plane surfaces and the all but perfectly
definite and constant angles which characterise the cell, whether it
be constructed of wax or papery pulp. It seems more likely that we
have to do with a true surface-tension effect; in other words, that
the walls assume their configuration when in a semi-fluid state, while
the papery pulp is still liquid, or while the wax is warm under the
high temperature of the crowded hive[375]. Under these circumstances,
the direct efforts of the wasp or bee may be supposed to be limited to
the making of a tubular cell, as thin as the nature of the material
permits, and packing these little cells as close as possible together.
It is then easily conceivable that the symmetrical tensions of the
adjacent films (though somewhat retarded by viscosity) should suffice
to bring the whole system into equilibrium, that is to say into the
precise configuration which the comb actually presents. In short, the
Maraldi pyramids which terminate the bee’s cell are precisely identical
with the facets of a rhombic dodecahedron, such as we have assumed to
constitute (and which doubtless under certain conditions do constitute)
the surfaces of contact in the interior of a mass of soap-bubbles or of
uniform parenchymatous cells; and there is every reason to believe that
the physical explanation is identical, and not merely mathematically
analogous.
The remarkable passage in which Buffon discusses the bee’s cell and the
hexagonal configuration in general is of such historical importance,
and tallies so closely with the whole trend of our enquiry, that I
will quote it in full: “Dirai-je encore un mot; ces cellules des
abeilles, tant vantées, tant admirées, me fournissent une preuve
de plus contre l’enthousiasme et l’admiration; cette figure, toute
géométrique et toute régulière qu’elle nous paraît, et qu’elle est
en effet dans la spéculation, n’est ici qu’un résultat mécanique et
assez imparfait qui se trouve souvent dans la nature, {334} et que
l’on remarque même dans les productions les plus brutes; les cristaux
et plusieurs autres pierres, quelques sels, etc., prennent constamment
cette figure dans leur formation. Qu’on observe les petites écailles
de la peau d’une roussette, on verra qu’elles sont hexagones, parce
que chaque écaille croissant en même temps se fait obstacle, et tend à
occuper le plus d’espace qu’il est possible dans un espace donné: on
voit ces mêmes hexagones dans le second estomac des animaux ruminans,
on les trouve dans les graines, dans leurs capsules, dans certaines
fleurs, etc. Qu’on remplisse un vaisseau de pois, ou plûtot de
quelque autre graine cylindrique, et qu’on le ferme exactement après
y avoir versé autant d’eau que les intervalles qui restent entre ces
graines peuvent en recevoir; qu’on fasse bouillir cette eau, tous
ces cylindres deviendront de colonnes à six pans[376]. On y voit
clairement la raison, qui est purement mécanique; chaque graine, dont
la figure est cylindrique, tend par son renflement à occuper le plus
d’espace possible dans un espace donné, elles deviennent donc toutes
nécessairement hexagones par la compression réciproque. Chaque abeille
cherche à occuper de même le plus d’espace possible dans un espace
donné, il est donc nécessaire aussi, puisque le corps des abeilles est
cylindrique, que leurs cellules sont hexagones,—par la même raison
des obstacles réciproques. On donne plus d’esprit aux mouches dont
les ouvrages sont les plus réguliers; les abeilles sont, dit-on, plus
ingénieuses que les guêpes, que les frélons, etc., qui savent aussi
l’architecture, mais dont les constructions sont plus grossières et
plus irrégulières que celles des abeilles: on ne veut pas voir, ou l’on
ne se doute pas que cette régularité, plus ou moins grande, dépend
uniquement du nombre et de la figure, et nullement de l’intelligence
de ces petites bêtes; plus elles sont nombreuses, plus il y a des
forces qui agissent également et s’opposent de même, plus il y a
par conséquent de contrainte mécanique, de régularité forcée, et de
perfection apparente dans leurs productions[377].” {335}
A very beautiful hexagonal symmetry, as seen in section, or
dodecahedral, as viewed in the solid, is presented by the cells which
form the pith of certain rushes (e.g. _Juncus effusus_), and somewhat
less diagrammatically by those which make the pith of the banana.
These cells are stellate in form, and the tissue presents in section
the appearance of a network of six-rayed stars (Fig. 133, _c_), linked
together by the tips of the rays, and separated by symmetrical,
air-filled, intercellular spaces. In thick sections, the solid
twelve-rayed stars may be very beautifully seen under the binocular
microscope.
[Illustration: Fig. 133. Diagram of development of “stellate cells,” in
pith of _Juncus_. (The dark, or shaded, areas represent the cells; the
light areas being the gradually enlarging “intercellular spaces.”)]
What has happened here is not difficult to understand. Imagine, as
before, a system of equal spheres all in contact, each one therefore
touching six others in an equatorial plane; and let the cells be not
only in contact, but become attached at the points of contact. Then
instead of each cell expanding, so as to encroach on and fill up the
intercellular spaces, let each cell tend to contract or shrivel up,
by the withdrawal of fluid from its interior. The {336} result will
obviously be that the intercellular spaces will increase; the six
equatorial attachments of each cell (Fig. 133, _a_) (or its twelve
attachments in all, to adjacent cells) will remain fixed, and the
portions of cell-wall between these points of attachment will be
withdrawn in a symmetrical fashion (_b_) towards the centre. As the
final result (_c_) we shall have a “dodecahedral star” or star-polygon,
which appears in section as a six-rayed figure. It is obviously
necessary that the pith-cells should not only be attached to one
another, but that the outermost layer should be firmly attached to
a boundary wall, so as to preserve the symmetry of the system. What
actually occurs in the rush is tantamount to this, but not absolutely
identical. Here it is not so much the pith-cells which tend to shrivel
within a boundary of constant size, but rather the boundary wall (that
is, the peripheral ring of woody and other tissues) which continues to
expand after the pith-cells which it encloses have ceased to grow or
to multiply. The twelve points of attachment on the spherical surface
of each little pith-cell are uniformly drawn asunder; but the content,
or volume, of the cell does not increase correspondingly; and the
remaining portions of the surface, accordingly, shrink inwards and
gradually constitute the complicated surface of a twelve-pointed star,
which is still a symmetrical figure and is still also a surface of
minimal area under the new conditions.
――――――――――
A few years after the publication of Plateau’s book, Lord Kelvin
shewed, in a short but very beautiful paper[378], that we must
not hastily assume from such arguments as the foregoing, that a
close-packed assemblage of rhombic dodecahedra will be the true and
general solution of the problem of dividing space with a minimum
partitional area, or will be present in a cellular liquid “foam,” in
which it is manifest that the problem is actually and automatically
solved. The general mathematical solution of the problem (as we have
already indicated) is, that every interface or partition-wall must
have constant curvature throughout; that where such partitions meet
in an edge, they must intersect at angles such that equal forces, in
planes perpendicular to the line {337} of intersection, shall balance;
and finally, that no more than three such interfaces may meet in a
line or edge, whence it follows that the angle of intersection of the
film-surfaces must be exactly 120°. An assemblage of equal and similar
rhombic dodecahedra goes far to meet the case: it completely fills
up space; all its surfaces or interfaces are planes, that is to say,
surfaces of constant curvature throughout; and these surfaces all meet
together at angles of 120°. Nevertheless, the proof that our rhombic
dodecahedron (such as we find exemplified in the bee’s cell) is a
surface of minimal area, is not a comprehensive proof; it is limited to
certain conditions, and practically amounts to no more than this, that
of the regular solids, with all sides plane and similar, this one has
the least surface for its solid content.
[Illustration: Fig. 134.]
The rhombic dodecahedron has six tetrahedral angles, and eight
trihedral angles; and it is obvious, on consideration, that at each of
the former six dodecahedra meet in a point, and that, where the four
tetrahedral facets of each coalesce with their neighbours, we have
twelve plane films, or interfaces, meeting in a point. In a precisely
similar fashion, we may imagine twelve plane films, drawn inwards
from the twelve edges of a cube, to meet at a point in the centre of
the cube. But, as Plateau discovered[379], when we dip a cubical wire
skeleton into soap-solution and take it out again, the twelve films
which are thus generated do _not_ meet in a point, but are grouped
around a small central, plane, quadrilateral film (Fig. 134). In
other words, twelve plane films, meeting in a point, are _essentially
unstable_. If we blow upon our artificial film-system, the little
quadrilateral alters its place, setting itself parallel now to one and
now to another of the paired faces of the cube; but we never get rid
of it. Moreover, the size and shape of the quadrilateral, as of all
the other films in the system, are perfectly definite. Of the twelve
films (which we had {338} expected to find all plane and all similar)
four are plane isosceles triangles, and eight are slightly curved
quadrilateral figures. The former have two curved sides, meeting at an
angle of 109° 28′, and their apices coincide with the corners of the
central quadrilateral, whose sides are also curved, and also meet at
this identical angle;—which (as we observe) is likewise an angle which
we have been dealing with in the simpler case of the bee’s cell, and
indeed in all the regular solids of which we have yet treated.
By completing the assemblage of polyhedra of which Plateau’s
skeleton-cube gives a part, Lord Kelvin shewed that we should
obtain a set of equal and similar fourteen-sided figures, or
“tetrakaidecahedra”; and that by means of an assemblage of these
figures space is homogeneously partitioned—that is to say, into equal,
similar and similarly situated cells—with an economy of surface
in relation to area even greater than in an assemblage of rhombic
dodecahedra.
In the most generalised case, the tetrakaidecahedron is bounded by
three pairs of equal and parallel quadrilateral faces, and four pairs
of equal and parallel hexagonal faces, neither the quadrilaterals nor
the hexagons being necessarily plane. In a certain particular case, the
quadrilaterals are plane surfaces, but the hexagons slightly curved
“anticlastic” surfaces; and these latter have at every point equal
and opposite curvatures, and are surfaces of minimal curvature for a
boundary of six curved edges. The figure has the remarkable property
that, like the plane rhombic dodecahedron, it so partitions space that
three faces meeting in an edge do so everywhere at equal angles of
120° [380].
We may take it as certain that, in a system of _perfectly_ fluid
films, like the interior of a mass of soap-bubbles, where the films
are perfectly free to glide or to rotate over one another, the mass
is actually divided into cells of this remarkable conformation. {339}
And it is quite possible, also, that in the cells of a vegetable
parenchyma, by carefully macerating them apart, the same conformation
may yet be demonstrated under suitable conditions; that is to say when
the whole tissue is highly symmetrical, and the individual cells are
as nearly as possible equal in size. But in an ordinary microscopic
_section_, it would seem practically impossible to distinguish the
fourteen-sided figure from the twelve-sided. Moreover, if we have
anything whatsoever interposed so as to prevent our twelve films
meeting in a point, and (so to speak) to take the place of our little
central quadrilateral,—if we have, for instance, a tiny bead or droplet
in the centre of our artificial system, or even a little thickening,
or “bourrelet” as Plateau called it, of the cell-wall, then it is
no longer necessary that the tetrakaidecahedron should be formed.
Accordingly, it is very probably the case that, in the parenchymatous
tissue, under the actual conditions of restraint and of very imperfect
fluidity, it is after all the rhombic dodecahedral configuration which,
even under perfectly symmetrical conditions, is generally assumed.
――――――――――
It follows from all that we have said, that the problems connected
with the conformation of cells, and with the manner in which a given
space is partitioned by them, soon become exceedingly complex. And
while this is so even when all our cells are equal and symmetrically
placed, it becomes vastly more so when cells varying even slightly in
size, in hardness, rigidity or other qualities, are packed together.
The mathematics of the case very soon become too hard for us; but in
its essence, the phenomenon remains the same. We have little reason to
doubt, and no just cause to disbelieve, that the whole configuration,
for instance of an egg in the advanced stages of segmentation, is
accurately determined by simple physical laws, just as much as in
the early stages of two or four cells, during which early stages we
are able to recognise and demonstrate the forces and their resultant
effects. But when mathematical investigation has become too difficult,
it often happens that physical experiment can reproduce for us the
phenomena which Nature exhibits to us, and which we are striving to
comprehend. For instance, in an admirable research, M. Robert shewed,
some years ago, not only that the early segmentation of {340} the
egg of _Trochus_ (a marine univalve mollusc) proceeded in accordance
with the laws of surface tension, but he also succeeded in imitating
by means of soap-bubbles, several stages, one after another, of the
developing egg.
[Illustration: Fig. 135. Aggregations of four soap-bubbles, to shew
various arrangements of the intermediate partition and polar furrows.
(After Robert.)]
M. Robert carried his experiments as far as the stage of sixteen cells,
or bubbles. It is not easy to carry the artificial system quite so far,
but in the earlier stages the experiment is easy; we have merely to
blow our bubbles in a little dish, adding one to another, and adjusting
their sizes to produce a symmetrical system. One of the simplest and
prettiest parts of his investigation concerned the “polar furrow” of
which we have spoken on p. 310. On blowing four little contiguous
bubbles he found (as we may all find with the greatest ease) that they
form a symmetrical system, two in contact with one another by a laminar
film, and two, which are elevated a little above the others, and which
are separated by the length of the aforesaid lamina. The bubbles are
thus in contact three by three, their partition-walls making with
one another equal angles of 120°. The upper and lower edges of the
intermediate lamina (the lower one visible through the transparent
system) constitute the two polar furrows of the embryologist (Fig.
135, 1–3). The lamina itself is plane when the system is symmetrical,
but it responds by a corresponding curvature to the least inequality
of the bubbles on either side. In the experiment, the upper polar
furrow is usually a little shorter than the lower, but parallel to
it; that is to say, the lamina is of trapezoidal form: this lack of
perfect symmetry being due (in the experimental case) to the lower
portion of the bubbles being somewhat drawn asunder by the tension of
their attachments to the sides of the dish (Fig. 135, 4). A similar
phenomenon is usually found in Trochus, according to Robert, and many
other observers have likewise found the upper furrow to be shorter
than the one below. In the various species of the genus Crepidula,
Conklin asserts that the two furrows are equal in _C. convexa_, that
the upper one is the shorter in _C. fornicata_, and that the upper
one all but disappears in _C. plana_; but we may well be permitted to
doubt, without the evidence of very special investigations, whether
these slight physical differences are actually characteristic of, and
constant in, particular allied _species_. {341} Returning to the
experimental case, Robert found that by withdrawing a little air from,
and so diminishing the bulk of the two terminal bubbles (i.e. those
at the ends of the intermediate lamina), the upper polar furrow was
caused to elongate, till it became equal in length to the lower; and
by continuing the process it became the longer in its turn. These two
conditions have again been described by investigators as characteristic
of this embryo or that; for instance in Unio, Lillie has described the
two furrows as gradually altering their respective lengths[381]; and
Wilson (as Lillie remarks) had already pointed out that “the reduction
of the apical cross-furrow, as compared with that at the vegetative
pole {342} in molluscs and annelids ‘stands in obvious relation to the
different size of the cells produced at the two poles[382].’ ”
When the two lateral bubbles are gradually reduced in size, or the two
terminal ones enlarged, the upper furrow becomes shorter and shorter;
and at the moment when it is about to vanish, a new furrow makes its
instantaneous appearance in a direction perpendicular to the old one;
but the inferior furrow, constrained by its attachment to the base,
remains unchanged, and accordingly our two polar furrows, which were
formerly parallel, are now at right angles to one another. Instead of
a single plane quadrilateral partition, we have now two triangular
ones, meeting in the middle of the system by their apices, and lying in
planes at right angles to one another (Fig. 135, 5–7)[383]. Two such
polar furrows, equal in length and arranged in a cross, have again
been frequently described by the embryologists. Robert himself found
this condition in Trochus, as an occasional or exceptional occurrence:
it has been described as normal in Asterina by Ludwig, in Branchipus
by Spangenberg, and in Podocoryne and Hydractinia by Bunting. It is
evident that it represents a state of unstable equilibrium, only to be
maintained under certain conditions of restraint within the system.
So, by slight and delicate modifications in the relative size of the
cells, we may pass through all the possible arrangements of the median
partition, and of the “furrows” which correspond to its upper and
lower edges; and every one of these arrangements has been frequently
observed in the four-celled stage of various embryos. As the phases
pass one into the other, they are accompanied by changes in the
curvature of the partition, which in like manner correspond precisely
to phenomena which the embryologists have witnessed and described.
And all these configurations belong to that large class of phenomena
whose distribution among embryos, or among organisms in general,
bears no relation to the boundaries of zoological classification;
through molluscs, worms, {343} coelenterates, vertebrates and what
not, we meet with now one and now another, in a medley which defies
classification. They are not “vital phenomena,” or “functions” of the
organism, or special characteristics of this or that organism, but
purely physical phenomena. The kindred but more complicated phenomena
which correspond to the polar furrow when a larger number of cells than
four are associated together, we shall deal with in the next chapter.
Having shewn that the capillary phenomena are patent and unmistakable
during the earlier stages of embryonic development, but soon become
more obscure and incapable of experimental reproduction in the later
stages, when the cells have increased in number, various writers
including Robert himself have been inclined to argue that the physical
phenomena die away, and are overpowered and cancelled by agencies
of a very different order. Here we pass into a region where direct
observation and experiment are not at hand to guide us, and where a
man’s trend of thought, and way of judging the whole evidence in the
case, must shape his philosophy. We must remember that, even in a froth
of soap-bubbles, we can apply an exact analysis only to the simplest
cases and conditions of the phenomenon; we cannot describe, but can
only imagine, the forces which in such a froth control the respective
sizes, positions and curvatures of the innumerable bubbles and films
of which it consists; but our knowledge is enough to leave us assured
that what we have learned by investigation of the simplest cases
includes the principles which determine the most complex. In the case
of the growing embryo we know from the beginning that surface tension
is only one of the physical forces at work; and that other forces,
including those displayed within the interior of each living cell,
play their part in the determination of the system. But we have no
evidence whatsoever that at this point, or that point, or at any, the
dominion of the physical forces over the material system gives place
to a new condition where agencies at present unknown to the physicist
impose themselves on the living matter, and become responsible for the
conformation of its material fabric.
――――――――――
Before we leave for the present the subject of the segmenting {344}
egg, we must take brief note of two associated problems: viz. (1)
the formation and enlargement of the segmentation cavity, or central
interspace around which the cells tend to group themselves in a single
layer, and (2) the formation of the gastrula, that is to say (in a
typical case) the conversion “by invagination,” of the one-layered ball
into a two-layered cup. Neither problem is free from difficulty, and
all we can do meanwhile is to state them in general terms, introducing
some more or less plausible assumptions.
The former problem is comparatively easy, as regards the tendency of a
segmentation cavity to _enlarge_, when once it has been established.
We may then assume that subdivision of the cells is due to the
appearance of a new-formed septum within each cell, that this septum
has a tendency to shrink under surface tension, and that these changes
will be accompanied on the whole by a diminution of surface energy
in the system. This being so, it may be shewn that the volume of the
divided cells must be less than it was prior to division, or in other
words that part of their contents must exude during the process of
segmentation[384]. Accordingly, the case where the segmentation cavity
enlarges and the embryo developes into a hollow blastosphere may, under
the circumstances, be simply described as the case where that outflow
or exudation from the cells of the blastoderm is directed on the whole
inwards.
The physical forces involved in the invagination of the cell-layer to
form the gastrula have been repeatedly discussed[385], but the true
explanation seems as yet to be by no means clear. The case, however,
is probably not a very difficult one, provided that we may assume a
difference of osmotic pressure at the two poles of the blastosphere,
that is to say between the cells which are being differentiated into
outer and inner, into epiblast and hypoblast. It is plain that a
blastosphere, or hollow vesicle bounded by a layer of vesicles, is
under very different physical conditions from a single, simple vesicle
or bubble. The blastosphere has no effective surface tension of its
own, such as to exert pressure on {345} its contents or bring the
whole into a spherical form; nor will local variations of surface
energy be directly capable of affecting the form of the system. But if
the substance of our blastosphere be sufficiently viscous, then osmotic
forces may set up currents which, reacting on the external fluid
pressure, may easily cause modifications of shape; and the particular
case of invagination itself will not be difficult to account for on
this assumption of non-uniform exudation and imbibition.
{346}