On Growth and Form

be an arc of a circle of which we can determine the centre by the method used on p. 307; and, furthermore, the narrower cells, that is to say the quadrilaterals, will have this outer border somewhat more curved than their broader neighbours. We arrive, then, at the condition shewn in Fig. 153, b. Within the cell, also, wherever wall meets wall, the angle of contact must tend, in every case, to be an angle of 120°; and in no case may more than three films (as seen in section) meet in a point (c); and this condition, of the partitions meeting three by three, and at co-equal angles, will obviously involve the curvature of some, if not all, of the partitions (d) which in our preliminary investigation we treated as plane. To solve this problem in a general way is no easy matter; but it 1s a problem which Nature solves in every case where, as in the case we are considering, eight bubbles, or eight cells, meet together in a (plane or curved) surface. An approximate solution has been given in Fig. 153, d; and it will now at once be recognised that this figure has vastly more resemblance to an aggregate of living cells than had the diagram of Fig. 153, a with which we began.

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

in this case be precisely accurate, for the simple reason that Berthold’s figures are taken from different individuals, and are therefore only approximately consecutive and not strictly continuous. The last of the six drawings in Fig. 144 is already too Fig. 155. Theoretical arrangement of successive partitions in a discoid cell; for comparison with Fig. 144. complicated for diagrammatisation, that is to say it is too complicated for us to decipher with certainty the precise order of appearance of the numerous partitions which it contains. But in Fig. 156 I shew one more diagrammatic figure, of a dise which

Fig. 156. Theoretical.division of a discoid cell into sixty-four chambers: no allowance being made for the mutual tractions of the cell-walls. has divided, according to the theoretical plan, into about sixtyfour cells; and making due allowance for the successive changes which the mutual tensions and tractions of the partitions must bring about, increasing in complexity with each succeeding stage, we can see, even at this advanced and complicated stage, a very considerable resemblance between the actual picture (Fig. 144) and the diagram which we have here constructed in obedience to a few simple rules.

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

a b Fig. 157. Sections of embryo of a moss. (After Kienitz-Gerloff.) In the last paragraph but one, I have spoken of the difficulty of so arranging the meeting-places of a number of cells that at each junction only three cell-walls shall meet in a line, and all three shall meet it at equal angles of 120°. As a matter of fact, the problem is soluble in a number of ways; that is to say, when we have a number of cells, say eight as in the case considered, enclosed in a common boundary, there are various ways in which their walls can be made to meet internally, three by three, at equal angles; and these differences will entail differences also in the curvature of the walls, and consequently in the shape of the cells. The question is somewhat complex; it has been dealt with by Plateau, and treated mathematically by M. Van Rees*.

internally, they must meet in a point; furthermore, they tend to do so at’ equal angles of 120°, and there is an end of the matter. If we have four cells, then, as we have already seen, the conditions are satisfied by interposing a little intermediate wall, the two extremities of which constitute the meeting-points of three cells each, and the upper edge of which marks the “polar furrow.” Similarly, in the case of five cells, we require two little intermediate walls, and two polar furrows; and we soon arrive at the rule that, for n cells, we require » — 3 little longitudinal partitions (and corresponding polar furrows), connecting the triple junctions of

4 5 6 u 8 Cells ees Bigeye Nad where ' i BNO ea en Nanety OU aor FG eee Te ee eel ie Fig. 158. Various possible arrangements of intermediate partitions, in groups of 4, 5, 6, 7 or 8 cells. the cells; and these little walls, like all the rest within the system, must be inclined to one another at angles of 120°. Where we have only one such wall (as in the case of four cells), or only two (as in the case of five cells), there is no room for ambiguity. But where we have three little connecting-walls, as in the case of six cells, it is obvious that we can arrange them in three different ways, as in the annexed Fig. 159. In the system of seven cells, the four partitions can be arranged in four ways; and the five partitions required in the case of eight cells can be arranged in no less than thirteen different ways, of which Fig. 158 shews some half-dozen only. It does not follow that, so to speak, these various

arrangements are all equally good; some are known to be much more stable than others, and some have never yet been realised in actual experiment. The conditions which lead to the presence of any one of them, in preference to another, are as yet, so far as I am aware, undetermined, but to this point we shall return. Examples of these various arrangements meet us at every turn, and not only in cell-aggregates, but in various cases where non-rigid and semi-fluid partitions (or partitions that were so to begin with) meet together. And it is a necessary consequence of this physical phenomenon, and of the limited and very small number of possible arrangements, that we get similar appearances, — capable of representation by the same diagram, in the most diverse fields of biology*.

Among the published figures of embryonic stages and other cell aggregates, we only discern these little intermediate partitions in cases where the investigator has drawn carefully just what lay before him, without any preconceived notions as to radial or other symmetry; but even in other cases we can generally recognise, without much difficulty, what the actual arrangement was whereby the cell-walls met together in equilibrium. I have a strong suspicion that a leaning towards Sachs’s Rule, that one cell-wall tends to set itself at right angles to another cell-wall (a rule whose strict limitations, and narrow range of application, we have already

* Even in a Protozoon (Huglena viridis), when kept alive under artificial compression, Ryder found a process of cell-division to occur which he compares to the segmenting blastoderm of a fish’s egg, and which corresponds in its essential features with that here described. Contrib. Zool. Lab. Univ. Pennsylvania, 1, pp. 37-50, 1893. considered) is responsible for many inaccurate or incomplete representations of the mutual arrangement of aggregated cells. In the accompanying series of figures (Figs. 160-167) I have

Fig. 160. Segmenting egg Fig. 161. re views of segmenting egg of Cynthia of Trochus. (After Robert.) partita. (After Conklin.) Fig. 162. (a) Section of apical cone of Salvinia. (After Pringsheim~*.) (b) Diagram of probable actual arrangement. i : Fig. 163. Egg of Pyrosoma. Fig. 164. Egg of Hchinus, segmenting (After Korotneff). under pressure. (After Driesch.) * This, like many similar figures, is manifestly drawn under the influence of Sachs’s theoretical views, or assumptions, regarding orthogonal trajectories, coaxial circles, confocal ellipses, etc.

set forth a few aggregates of eight cells, mostly from drawings of segmenting eggs. In some cases they shew clearly the manner in which the cells meet one another, always at angles of 120°, Fig. 165. (a) Part of segmenting egg of Cephalopod (after Watase); (b) probable actual arrangement. a b Fig. 167. (a) Egg of frog, under pressure (after Roux); (6) probable actual arrangement. and always with the help of five intermediate boundary walls within the eight-celled system; in other cases I have added a slightly altered drawing, so as to shew, with as little change as

possible, the arrangement of boundaries which probably actually existed, and gave rise to the appearance which the observer drew. These drawings may be compared with the various diagrams of Fig. 158, in which some seven out of the possible thirteen arrangements of five intermediate partitions (for a system of eight cells) have been already set forth. It will be seen that M. Robert-Tornow’s figure of the segmenting ege of Trochus (Fig. 160) clearly shews the cells grouped after the fashion of Fig. 158, a. In hke manner, Mr Conklin’s figure of the ascidian egg (Cynthia) shews equally clearly the arrangement g.

A sea-urchin egg, segmenting under pressure, as figured by Driesch, scarcely requires any modification of the drawing to appear as a diagram of the type d. Turning for a moment to a botanical illustration, we have a figure of Pringsheim’s shewing an eight-celled stage in the apex of the young cone of Salvinia; it is in all probability referable, as in my modified diagram, to type c. Beside it is figured a very different object, a segmenting egg of the Ascidian Pyrosoma, after Korotneff; it may be that this also is to be referred to type c, but I think it is more easily referable to type b. For there is a difference between this diagram and that of Salvinia, in that here apparently, of the pairs of lateral cells, the upper and the lower cell are alternately the larger, while in the diagram of Salvinia the lower lateral cells both appear much larger than the upper ones; and this difference tallies with the appearance produced if we fill in the eight cells according to the type b or the type c. In the segmenting cuttlefish egg, there is again a slight dubiety as to which type it should be referred to, but it is in all probability referable, like Driesch’s Echinus egg, to d. Lastly, I have copied from Roux a curious figure of the egg of Rana esculenta, viewed from the animal pole, which appears to me referable, in all probability, to type g. Of type f, in which the five partitions form a figure with four re-entrant angles, that is to say a figure representing the five sides of a hexagon, I have found no examples among segmenting eggs, and that arrangement in all probability is a very unstable one.

It is obvious enough, without more ado, that these phenomena are in the strictest and completest way common to both plants and animals. In other words they tally with, and they further extend, the general and fundamental conclusions laid down by Schwann, in his Mikroskopische Untersuchungen iiber die Uebereinstimmung in der Struktur und dem Wachsthum der Thiere und Pflanzen. But now that we have seen how a certain limited number of types of eight-celled segmentation (or of arrangements of eight cell-partitions) appear and reappear, here and there, throughout the whole world of organisms, there still remains the very important question, whether in each particular organism the conditions are such as to lead to one particular arrangement being predominant, characteristic, or even invariable. In short, isa particular arrangement of cell-partitions to be looked upon (as the published figures of the embryologist are apt to suggest) as a specific character, or at least a constant or normal character, of the particular organism ? The answer to this question is a direct negative, but it is only in the work of the most careful and accurate observers that we find it revealed. Rauber (whom we have more than once had occasion to quote) was one of those embryologists who recorded just what he saw, without prejudice or preconception; as Boerhaave said of Swammerdam, quod vidit id asserwit. Now Rauber has put on record a considerable number of variations in the arrangement of the first eight cells, which form a discoid surface about the dorsal (or “animal”’) pole of the frog’s egg. In a certain number of cases these figures are identical with one another in type, identical (that is to say) save for slight differences in magnitude, relative proportions, or orientation. But I have selected (Fig. 168) six diagrammatic figures, which are all essentially different, and these diagrams seem to me to bear intrinsic evidence of their accuracy : the curvatures of the partition-walls, and the angles at which they meet agree closely with the requirements of theory, and when they depart from theoretical symmetry they do so only to the shght extent which we should naturally expect in a material and imperfectly homogeneous system”.

* Such preconceptions as Rauber entertained were all in a direction likely to lead him away from such phenomena as he has faithfully depicted. Rauber had no idea whatsoever of the principles by which we are guided in this discussion, nor does he introduce at all the analogy of surface-tension, or any other purely physical concept. But he was deeply under the influence of Sachs’s rule of rect- Of these six illustrations, two are exceptional. In Fig. 168, 5, we observe that one of the eight cells is surrounded on all sides by the other seven. This is a perfectly natural condition, and represents, like the rest, a phase of partial or conditional equilibrium. But it is not included in the series we are now considering, which is restricted to the case of eight cells extending outwards to a common boundary. The condition shewn in Fig. 168, 6, is again peculiar, and is probably rare; but it is included under the cases considered on p. 312, in which the cells are not in complete

Fig. 168. Various modes of grouping of eight cells, at the dorsal or epiblastic pole of the frog’s egg. (After Rauber.) fluid contact, but are separated by little droplets of extraneous matter; it needs no further comment. But the other four cases are beautiful diagrams of space-partitioning, similar to those we have just been considering, but so exquisitely clear that they need no modification, no “touching-up,” to exhibit their mathematical regularity. It will easily be recognised that in Fig. 168, 1 and 2, we have the arrangements corresponding to a and d of our diagram (Fig. 158): but the other two (i.e. 3 and 4) represent other of the thirteen possible arrangements, which are not included in that

angular intersection; and he was accordingly disposed to look upon the configuration represented above in Fig. 168, 6, as the most typical or most primitive. diagram. It would be a curious and interesting investigation to ascertain, in a large number of frogs’ eggs, all at this stage of development, the percentage of cases in which these yarious arrangements occur, with a view of correlating their frequency with the theoretical conditions (so far as they are known, or can be ascertained) of relative stability. One thing stands out as very certain indeed: that the elementary diagram of the frog’s ege commonly given in text-books of embryology,—in which the cells are depicted as uniformly symmetrical quadrangular bodies,— is entirely inaccurate and grossly misleading*.

We now begin to realise the remarkable fact, which may even appear a startling one to the biologist, that all possible groupings’ or arrangements whatsoever of eight cells (where all take part in the surface of the group, none being submerged or wholly enveloped by the rest) are referable to some one or other of thirteen types or forms. And that all the thousands and thousands of drawings which diligent observers have made of such eight-celled structures, animal or vegetable, anatomical, histological or embryological, are one and all representations of some one or another of these thirteen types :—or rather indeed of somewhat less than the whole thirteen, for there is reason to believe that, out of the total number of possible groupings, a certain small number are essentially unstable, and have at best, in the concrete, but a transitory and evanescent existence.

Before we leave this subject, on which a vast deal more might be said, there are one or two points which we must not omit to consider. Let us note, in the first place, that the appearance which our plane diagrams suggest of inequality of the several cells is apt to be deceptive; for the differences of magnitude apparent in one plane may well be, and probably generally are, balanced by equal and opposite differences in another. Secondly, let us remark that the rule which we are considering refers only

“Ich betone noch, dass unter meinen Figuren diejenige gar nicht enthalten ist, welche zum Typus der Batrachierfurchung gehérig am meisten bekannt ist....Es haben so ausgezeichnete Beobachter sie als vorhanden beschrieben, dass es mir nicht einfallen kann, sie tiberhaupt nicht anzuerkennen.”’ to angles, and to the number, not to the length of the intermediate partitions ; itis toa great extent by variations in the length of these that the magnitudes of the cells may be equalised, or otherwise balanced, and the whole system brought into equilibrium. Lastly, there is a curious point to consider, in regard to the number of actual contacts, in the various cases, between cell and cell. If we inspect the diagrams in Fig. 169 (which represent three out of our thirteen possible arrangements of eight cells) we shall see that, in the case of type b, two cells are each in contact with two others, two cells with three others, and four cells each with four other cells. In type a four cells are each in contact with two, two with four, and two with five. In type f, two are in contact with two, four with three, and one with no less than seven. In all cases the

number of contacts is twenty-six in all; or, in other words, there are thirteen internal partitions, besides the eight peripheral walls. For it is easy to see that, in all cases of n cells with a common external boundary, the number of internal partitions is 2n — 3; or the number of what we call the internal or interfacial contacts is 2(2n — 3). But it would appear that the most stable arrangements are those in which the total number of contacts is most evenly divided, and the least stable are those in which some one cell has, as in type f, a predominant number of contacts. In a well-known series of experiments, Roux has shewn how, by means of oil-drops, various arrangements, or aggregations, of cells can be simulated ; and in Fig. 170 I shew a number of Roux’s figures, and have ascribed them to what seem to be their appropriate “types”? among those which we have just been considering; but

it will be observed that in these figures of Roux’s the drops are not always in complete contact, a little air-bubble often keeping them apart at their apical junctions, so that we see the configuration towards which the system is tending rather than that which it has fully attained*. The type which we have called f was found by Roux to be unstable, the large (or apparently large) drop a’’ quickly passing into the centre of the system, and here taking up a position of equilibrium in which, as usual, three cells meet throughout in a point, at equal angles, and in which, in this case, all the cells have an equal number of “interfacial” contacts.

Fig. 170. Aggregations of oil-drops. (After Roux.) Figs. 4-6 represent successive changes in a single system. We need by no means be surprised to find that, in such arrangements, the commonest and most stable distributions are those in which the cell-contacts are distributed as uniformly as possible between the several cells. We always expect to find some such tendency to equality in cases where we have to do with small oscillations on either side of a symmetrical condition.

* Roux’s experiments were performed with drops of paraffin suspended in dilute alcohol, to which a little calcium acetate was added to form a soapy pellicle over the drops and prevent them from reuniting with one another. The rules and principles which we have arrived at from the point of view of surface tension have a much wider bearing than is at once suggested by the problems to which we have applied them ; for in this elementary study of the cell-boundaries in a segmenting egg or tissue we are on the verge of a difficult and important subject in pure mathematics. It is a subject adumbrated by Leibniz, studied somewhat more deeply by Euler, and greatly developed of recent years. It is the Geometria Situs of Gauss, the Analysis Situs of Riemann, the Theory of Partitions of Cayley, and of Spatial Complexes of Listing*. The crucial point for the biologist to comprehend is, that in a closed surface divided into a number of faces, the arrangement of all the faces, lines and points in the system is capable of analysis, and that, when the number of faces or areas is small, the number of possible arrangements is small also. This is the simple reason why we meet in such a case as we have been discussing (viz. the arrangement of a group or system of eight cells) with the same few types recurring again and again in all sorts of organisms, plants as well as animals, and with no relation to the lines of biological classification: and. why, further, we find similar configurations occurring to mark the symmetry, not of cells merely, but of the parts and organs of entire animals. The phenomena are not “functions,” or specific characters, of this or that tissue or organism, but involve general principles which lie within the province of the mathematician.

The theory of space-partitioning, to which the segmentation of the egg gives us an easy practical introduction, is illustrated in much more complex ways in other fields of natural history. A very beautiful but immensely complicated case is furnished by the “venation”’ of the wings of insects. Here we have sometimes (as in the dragon-flies), a general reticulum of small, more or less hexagonal “cells”: but in most other cases, in flies, bees, butterflies, etc., we have a moderate number of cells, whose partitions always impinge upon one another three by three, and whose arrangement, therefore, includes of necessity a number of small intermediate partitions, analogous to our polar furrows. I think

that a mathematical study of these, including an investigation of the “deformation” of the wing (that is to say, of the changes in shape and changes in the form of its “cells” which it undergoes during the life of the individual, and from one species to another) would be of great interest. In very many cases, the entomologist relies upon this venation, and upon the occurrence of this or that ° intermediate vein, for his classification, and therefore for his hypothetical phylogeny of particular groups; which latter procedure hardly commends itself to the physicist or the mathematician.

Another case, geometrically akin but biologically very different, is to be found in the little diatoms of the genus Asterolampra, and their immediate congeners*. In Asterolampra we have a little disc, in which we see (as it were) radiating spokes of one material, alternating with intervals occupied on the flattened wheel-like disc by another (Fig. 171). The spokes vary in number, but the general appearance is in a high degree suggestive of the Chladni figures produced by the vibration of a circular plate. The spokes broaden out towards the centre, and interlock by visible junctions, which obey the rule of triple intersection, and accordingly exemplify the partition-figures with which we are dealing. But whereas we have found the particular arrangement in which one cell is in contact with all the rest to be unstable, according to Roux’s oil-drop experiments, and to be conspicuous

by its absence from our diagrams of segmenting eggs, here in Asterolampra, on the other hand, it occurs frequently, and is indeed the commonest arrangement* (Fig. 171, B). In all probability, we are entitled to consider this marked difference natural enough. For we may suppose that in Asterolampra (unlike the case of the segmenting egg) the tendency is to perfect radial symmetry, all the spokes emanating from a point in the centre: such a condition would be eminently unstable, and would break down under the least asymmetry. A very simple, perhaps the simplest case, would be that one single spoke should differ slightly from the rest, and should so tend to be drawn in amid the others, these latter remaining similar and symmetrical among themselves. Such a configuration would be vastly less unstable than the original one in which all the boundaries meet in a point; and the fact that further progress is not made towards other configurations of still greater stability may be sufficiently accounted for by viscosity, rapid solidification, or other conditions of restraint. A perfectly stable condition would of course be obtained if, as in the case of Roux’s oil-drop (Fig. 170, 6), one of the cellular spaces passed into the centre of the system, the other partitions radiating outwards from its circular wall to the periphery of the whole system. Precisely such a condition occurs among our diatoms; but when it does so, it is looked