On Growth and Form

In the case of the conical egg with spherical ends, as is niore or less the case in the plover’s and the guillemot’s, then at either end of the egg r and 7” are identical, and they are greater at the blunt anterior end than at the other. If we may assume that p,, vanishes at the poles of the egg, then it is plain that 7 varies in the neighbourhood of these poles, and, further, that the tension T is greatest at and near the small end of the egg. It is here, in short, that the egg is most likely to be irregularly distorted or

even to burst, and it is here that we most commonly find irregularities of shape in abnormal] eggs. If one portion of the envelope were to become practically stiff before p ceases to vary, that would be tantamount to a sudden variation of 7’, and would introduce asymmetry by the imposition of a boundary condition in addition to the above equation. Within the egg lies the yolk, and the yolk is invariably spherical or very nearly so, whatever be the form of the entire egg. The reason is simple, and lies in the fact that the yolk is itself enclosed in another membrane, between which and the outer membrane hes a fluid the presence of which makes »,, for the inner membrane practically constant. The smallness of friction is indicated by the well-known fact that the “germinal spot” on the surface of the yolk is always found uppermost, however we may place and wherever we may open the egg; that is to say, the yolk easily rotates within the egg, bringing its lighter pole uppermost. So, owing to this lack of friction in the outer fluid, or white, whatever shear is produced within the egg will not be easily transmitted to the yolk, and, moreover, owing to the same fluidity, the yolk will easily recover its normal sphericity after the egg-shell is formed and the unequal pressure relieved.

These, then, are the general principles involved in, and illustrated by, the configuration of an egg; and they take us as far as we can safely go without actual quantitative determinations, in each particular case, of the forces concerned. In certain cases among the invertebrates, we again find instances of hard-shelled eggs which have obviously been moulded by the oviduct, or so-called “ootype,” in which they have lain: and not merely in such a way as to shew the effects of peristaltic pressure upon a uniform elastic envelope, but so as to impress upon the egg the more or less irregular form of the cavity, within which it had been for a time contained and compressed. After this fashion Dr Looss* of Cairo has

* Journal of Tropical Medicine, 15th June, 1911. I leave this paragraph as it was written, though it is now once more asserted that the terminal and lateralspined eggs belong to separate and distinct species of Bilharzia (Leiper, Brit. Med. Journ., 18th March, 1916, p. 411). explained the curious form of the egg in Bilharzia (Schistosoma) haematobium, a formidable parasitic worm to which is due a disease wide-spread in Africa and Arabia, and an especial scourge of the Mecca pilgrims. The egg in this worm is provided at one end with a little spine; which now and then is found to be placed not terminally but laterally or ventrally, and which when so placed has been looked upon as the mark of a supposed new species, S. Mansoni. As Looss has now shewn, the little spine must be explained as having been moulded within a little funnel-shaped expansion of the uterus, just where it communicates with the common duct leading from the ovary and yolk-gland; by the accumulation of eggs in the ootype, the one last formed is crowded into a sideways position, and then, where the side-wall of the egg bulges in the funnel-shaped orifice of the duct, a little lateral “spine” is formed. In another species, S. japonicum, the egg is described as bulging into a so-called “calotte,” or bubble-like convexity at the end opposite to the spine. This, I think, may, with very little doubt, be ascribed to hardening of the egg-shell having taken place just at the period when partial relief from pressure was being experienced by the egg in the neighbourhood of the dilated orifice of the oviduct.

This case of Bilharzia is not, from our present point of view, a very important one, but nevertheless it is interesting. It ascribes to a mechanical cause a curious peculiarity of form; it shews, by reference to this mechanical principle, that two conditions which were very different to the systematic naturalist’s eye, were really only two simple mechanical modifications of the same thing; and it destroys the chief evidence for the existence of a supposed new species of worm, a continued belief in which, among worms of such great pathogenic importance, might lead to gravely erroneous pathological deductions.

As a corollary to the problem of the bird’s egg, we may consider for a moment the forms assumed by the shells of the sea-urchins. These latter are commonly divided into two classes, the Regular and the Irregular Echinids. The regular sea-urchins, save in slight details which do not affect our problem, have a complete radial symmetry. The axis of the animal’s body is vertical, with mouth below and the intestinal outlet above; and around this axis the shell is built as a symmetrical system. It follows that in horizontal section the shell is everywhere circular, and we shall have only to consider its form as seen in vertical section or projection. The irregular urchins (very inaccurately so-called) have the anal extremity of the body removed from its central, dorsal situation; and it follows that they have now a single plane of symmetry, about which the organism, shell and all, is bilaterally symmetrical. We need not concern ourselves in detail with the shapes of their shells, which may be very simply interpreted, by the help of radial co-ordinates, as deformations of the circular or “regular” type.

The sea-urchin shell consists of a membrane, stiffened into rigidity by calcareous deposits, which constitute a beautiful skeleton of separate, neatly fitting “ossicles.” The rigidity of the shell is more apparent than real, for the entire structure is, in a sluggish way, plastic; inasmuch as each little ossicle is capable of growth, and the entire shell grows by increments to each and all of these multitudinous elements, whose individual growth involves a certain amount of freedom to move relatively to one another; in a few cases the ossicles are so little developed that the whole shell appears soft and flexible. The viscera of the animal occupy but a small part of the space within the shell, the cavity being mainly filled by a large quantity of watery fluid, whose density must be very near to that of the external sea-water.

Apart from the fact that the sea-urchin continues to grow, it is plain that we have here the same general conditions as in the ego-shell, and that the form of the sea-urchin is subject to a similar equilibrium of forces. * But there is this important difference, that an external muscular pressure (such as the oviduct administers during the consolidation of egg-shell), is now lacking. In its place we have the steady continuous influence of gravity, and there is yet another force which in all probability we require to take into consideration.

While the sea-urchin is alive, an immense number of delicate “tube-feet,” with suckers at their tips, pass through minute pores in the shell, and, like so many long cables, moor the animal to the ground. They constitute a symmetrical system of forces, with one resultant downwards, in the direction of gravity, and another outwards in a radial direction; and if we look upon the shell as originally spherical, both will tend to depress the sphere into a flattened cake. We need not consider the radial component, but may treat the case as that of a spherical shell symmetrically depressed under the influence of gravity. This is precisely the condition which we have to deal with in a drop of liquid lying on a plate; the form of which is determined by its own uniform — surface-tension, plus gravity, acting against the uniform internal hydrostatic pressure. Simple as this system is, the full mathematical investigation of the form of a drop is not easy, and we can scarcely hope that the systematic study of the Echinodermata will ever be conducted by methods based on Laplace’s differential equation*; but we have no difficulty in seeing that the various forms represented in a series of sea-urchin shells are no other than those which we may easily and perfectly imitate in drops.

In the case of the drop of water (or of any other particular liquid) the specific surface-tension is always constant, and the pressure varies inversely as the radius of curvature; therefore the smaller the drop the more nearly is it able to conserve the spherical form, and the larger the drop the more does it become flattened under gravity. We can represent the phenomenon by using india-rubber balls filled with water, of different sizes; the little ones will remain very nearly spherical, but the larger will fall down “of their own weight,” into the form of more and more flattened cakes; and we see the same thing when we let drops of heavy oil (such as the orthotoluidene spoken of on p. 219), fall through a tall column of water, the little ones remaining round, and the big ones getting more and more flattened as they sink. In the case of the sea-urchin, the same series of forms may be assumed to occur, irrespective of size, through variations in 7, the specific tension, or “strength,” of the enveloping shell. Accordingly we may study, entirely from this point of view, such a series as the following (Fig. 328). In a very few cases, such as the fossil Palaeechinus, we have an approximately spherical

* Cf. Bashforth and Adams, Theoretical Forms of Drops, etc., Cambridge, 1883. shell, that is to say a shell so strong that the influence of gravity becomes negligible as a cause of deformation. The ordinary species of Echinus begin to display a pronounced depression, and this reaches its maximum in such soft-shelled flexible forms as Phormosoma. On the general question I took the opportunity of consulting Mr C. R. Darling, who is an acknowledged expert in drops, and he at once agreed with me that such forms as are represented in Fig. 328 are no other than diagrammatic illustrations of various kinds of drops, “most of which can easily be reproduced

Fig. 328. Diagrammatic vertical outlines of various Sea-urchins: A, Palaeechinus; B, Echinus acutus; C, Cidaris; D, D’ Coelopleurus; E, E’ Genicopatagus; F, Phormosoma luculenter; G, P. tenuis; H. Asthenosoma; I, Urechinus. in outline by the aid of liquids of approximately equal density to water, although some of them are fugitive.’ He found a difficulty in the case of the outline which represents Asthenosoma, but the reason for the anomaly is obvious; the flexible shell has flattened down until it has come in contact with the hard skeleton of the jaws, or “Aristotle’s lantern,” within, and the curvature of the outlne is accordingly disturbed. The elevated, conical shells such as those of Urechinus and Coelopleurus evidently call for some further explanation; for there is here some cause at work

to elevate, rather than to depress the shell. Mr Darling tells me that these forms “are nearly identical in shape with globules I have frequently obtained, in which, on standing, bubbles of gas rose to the summit and pressed the skin upwards, without being able to escape.” The same condition may be at work in the sea-urchin; but a similar tendency would also be manifested by the presence in the upper part of the shell of any accumulation of substance lighter than water, such as is actually present in the masses of fatty, oily eggs.

Passing to what may seem a very different subject, we may investigate a number of interesting points in connection with the form and structure of the blood-vessels, on the same principle and by help of the same equations as those we have used, for instance, in studying the egg-shell. We know that the fluid pressure (P) within the vessel is balanced by (1) the tension (7') of the wall, divided by the radius of curvature, and (2) the external pressure (p,), normal to the wall: according to our formula

If we neglect the external pressure, that is to say any support which may be given to the vessel by the surrounding tissues, and if we deal only with a cylindrical vein or artery, this formula becomes simplified to the form P= T7/R. That is to say, under constant pressure, the tension varies as the radius. But the tension, per unit area of the vessel, depends upon the thickness of the wall, that is to say on the amount of membranous and especially of muscular tissue of which it is composed.

Therefore, so long as the pressure is constant, the thickness of the wall should vary as the radius, or as the diameter, of the blood-vessel. But it is not the case that the pressure is constant, for it gradually falls off, by loss through friction, as we pass from the large arteries to the small; and accordingly we find that while, for a time, the cross-sections of the larger and smaller vessels are symmetrical figures, with the wall-thickness proportional to the size of the tube, this proportion is gradually lost, and the walls

of the small arteries, and still more of the capillaries, become exceedingly thin, and more so than in strict proportion to the narrowing of the tube. In the case of the heart we have, within each of its cavities, a pressure which, at any given moment, is constant over the whole wall-area, but the thickness of the wall varies very considerably. For instance, in the left ventricle, the apex is by much the thinnest portion, as it is also that with the greatest curvature. We may assume, therefore (or at least suspect), that the formula, t (1/r + 1/r’) =C, holds good; that is to say, that the thickness (¢é) of the wall varies inversely as the mean curvature. This may be tested experimentally, by dilating a heart with alcohol under a known pressure, and then measuring the thickness of the walls in various parts after the whole organ has become hardened. By this means it is found that, for each of the cavities, the law holds good with great accuracy*. Moreover, if we begin by dilating the right ventricle and then dilate the left in like manner, until the whole heart is equally and symmetrically dilated, we find (1) that we have had to use a pressure in the left ventricle from six to seven times as great as in the right ventricle, and (2) that the thickness of the walls is just in the same proportion 7.

A great many other problems of a mechanical or hydrodynamical kind arise in connection with the blood-vesselst, and while these are chiefly interesting to the physiologist they have also their interest for the morphologist in so far as they bear upon structure and form. As an example of such mechanical problems * Woods, R. H., On a Physical Theorem applied to tense Membranes, Journ. of Anat. and Phys. xxvi, pp. 362-371, 1892. A similar investigation of the tensions in the uterine wall, and of the varying thickness of its muscles, was attempted by Haughton in his Animal Mechanics, pp. 151-158, 1873.

+ This corresponds with a determination of the normal pressures (in systole) by Krohl, as being in the ratio of 1: 6°8. t Cf. Schwalbe, G., Ueber Wechselbeziehungen und ihr Einfluss auf die Gestaltung des Arteriensystem, Jen. Zeitschr. xu, p. 267, 1878; Roux, Ueber die Verzweigungen der Blutgefassen des Menschen, zhid. xm, p. 205, 1878; Ueber die Bedeutung der Ablenkung des Arterienstimmen bei der Astaufgabe, ibid. xu, p- 301, 1879; Hess, Walter, Eine mechanisch bedingte Gesetzmissigkeit im Bau des Blutgefasssystems, A. f. Entw. Mech. xvi, p. 632, 1903; Thoma, R., Ueber die Histogenese und Histomechanik des Blutgefdsssystems, 1893.

we may take the conditions which determine or help to determine the manner of branching of an artery, or the angle at which its branches are given off; for, as John Hunter said*, “To keep up a circulation sufficient for the part, and no more, Nature has varied the angle of the origin of the arteries accordingly.”” The general principle is that the form and arrangement of the blood-vessels is such that the circulation proceeds with a minimum of effort, and with a minimum of wall-surface, the latter condition leading to a minimum of friction and being therefore included in the first. What, then, should be the angle of branching, such that there shall be the least possible loss of energy in the course of the circulation? In order to solve this problem in any particular case we should obviously require to know (1) how the loss of energy depends upon the distance travelled, and (2) how the loss of energy varies with the diameter of the vessel. The loss of energy is evidently greater in a narrow tube than in a wide one, and greater, obviously, in a long journey than a short. If the large artery, AB, give off a comparatively narrow branch leading to P (such-as CP, or DP), the route ACP is evidently shorter than ADP, but on the other i hand, by the latter path, the blood has A tarried longer in the wide vessel AB,

and has had a shorter course in the Cf D0’ narrow branch. The relative advantage of energy in the portion CD, as com- A pared with that in the alternative portion CD’, the latter being short and narrow, the former long and wide. If we ask, then, which factor is the more important, length or width, we may safely take it that the question is one of degree: and that the factor of width will become much the more important wherever the artery and its branch are markedly unequal in size. In other words, it would seem that for small branches a large angle of bifurcation, and for large branches a small one, is always the better. Roux has laid down certain rules in regard to the branching of arteries, which correspond with the general con-

clusions which we have just arrived at. The most important of these are as follows: (1) If an artery bifurcate into two equal branches, these branches come off at equal angles to the main stem. (2) If one of the two branches be smaller than the other, then the main branch, or continuation of the original artery, makes with the latter a smaller angle than does the smaller or “lateral” branch. And (3) all branches which are so small that they scarcely seem to weaken or diminish the main stem come off from it at a large angle, from about 70° to 90°.

We may follow Hess in a further investigation of this phenomenon. Let AB be an artery, from which a branch has to be given off so as to reach P, and let ACP, ADP, etc., be alternative courses which the branch may follow: CD, DE, etc., in the diagram, being equal distances (=1) along AB. Let us call the angles PCD, PCE, 2, 7, etc.: and the distances CD’, DE’, by which each branch exceeds the next in length, we shall call 7,, /,, ete. Now it is evident that, of the courses shewn, ACP is the shortest which the blood can take, but it is also that by which its transit through the narrow branch is the longest. We may reduce its transit through the narrow branch more and more, till we come to CGP, or rather to a point where the branch comes off at right angles to the main

travelled. We may take it that there will be some intermediate point which will strike the balance of advantage. Now it is easy to shew that if, in Fig. 330, the route ADP and AEP (two contiguous routes) be equally favourable, then any other route on either side of these, such as ACP or AFP, must be less favourable than either. Let ADP and AEP, then, be equally favourable; that is to say, let the loss of energy which the blood suffers in its passage along these two routes be equal.

Then, if we make the distance DE very small, the angles x, and “3 are nearly equal, and may be so treated. And again, if DH be very small, then DH’E becomes a right angle, and Jl, (or DE) =T cosa. But if LZ be the loss of energy per unit distance in the wide tube AB, and L’ be the corresponding loss of energy in the narrow tube DP, etc., then /Z = 1,L’, because, as we have assumed, the loss of energy on the route DP is equal to that on the whole route DEP. Therefore IL = IL’ cosz,, and cos z, = L/L’. That is to say, the most favourable angle of branching will be such that the cosine of the angle is equal to the ratio of the loss of energy which the blood undergoes, per unit of length, in the main vessel, as compared with that which it undergoes in the branch.

While these statements are so far true, and while they undoubtedly cover a great number of observed facts, yet it is plain that, as in all such cases, we must regard them not as a complete explanation, but as factors in a complicated phenomenon : not forgetting that (as the most learned of all students of the heart and arteries, Dr Thomas Young, said in his Croonian lecture*) all such questions as these, and all matters connected with the muscular and elastic powers of the blood-vessels, “belong to the most refined departments of hydraulics.” Some other explanation must be sought in order to account for a phenomenon which particularly impressed John Hunter’s mind, namely the gradually altering angle at which the successive intercostal arteries are given off from the thoracic aorta: the special interest of this case arising from the regularity and symmetry of the series, for “there is not another set of arteries in the body whose origins are so much the same, whose offices are so much the same, whose distances from their origin to the place of use, and whose uses [?sizes|+ are so much the same.”

* On the Functions of the Heart and Arteries. Phil. Trans. 1809, pp. 1-31, cf. 1808, pp. 164-186; Collected Works, 1, pp. 511-534, 1855. The same lesson is conveyed by all such work as that of Volkmann, E. H. Weber and Poiseuille. Cf. Stephen Hales’ Statical Essays, u, Introduction: ‘* Especially considering that they [i.e. animal Bodies] are in a manner framed of one continued Maze of innumerable Canals, in which Fiuids*are incessantly circulating, some with great Force and Rapidity, others with very different Degrees of rebated Velocity: Hence, etc.”

+ “Sizes” is Owen’s editorial emendation, which seems amply justified. There is a certain large class of morphological problems of which we have not yet spoken, and of which we shall be able to say but little. Nevertheless they are so important, so full of deep theoretical significance, and are so bound up with the general question of form and of its determination asa result of growth, that an essay on growth and form is bound to take account of them, however imperfectly and briefly. The phenomena which I have in mind are just those many cases where adaptation, in the strictest sense, is obviously present, in the clearly demonstrable form of mechanical fitness for the exercise of some particular function or action which has become inseparable from the life and well-being of the organism.

When we discuss certain so-called “adaptations” to outward circumstance, in the way of form, colour and so forth, we are often apt to use illustrations convincing enough to certain minds but unsatisfying to others—in other words, incapable of demonstration. With regard to colouration, for instance, it is by colours “eryptic,’ “warning,” “signalling,” “mimetic,” and so on*, that we prosaically expound, and slavishly profess to justify, the vast Aristotelian synthesis that Nature makes all things with a purpose and “does nothing in vain.” Only for a moment let us glance at some few instances by which the modern teleologist accounts for this or that manifestation of colour, and is led on and on to beliefs and doctrines to which it becomes more and more difficult to subscribe.

* For a more elaborate classification, into colours cryptic, procryptic, anticryptic, apatetic, epigamic, sematic, episematic, aposematic, etc., see Poulton’s Colours of Animals (Int. Scientific Series, Lxvm), 1890; cf. also Meldola, R., Variable Protective Colouring in Insects, P.Z.S. 1873, pp. 153-162, ete. Some dangerous and malignant animals are said (in sober earnest) to wear a perpetual war-paint, in order to “remind their enemies that they had better leave them alone*.”’ The wasp and the hornet, in gallant black and gold, are terrible as an army with banners; and the Gila Monster (the poison-lizard of the Arizona desert) is splashed with scarlet—its dread and_ black complexion stained With heraldry more dismal. But the wasplike livery of the noisy, idle hover-flies and drone-flies is but stage armour, and in their tinsel suits the little counterfeit cowardly knaves mimic the -fighting crew.