On Growth and Form

The phenomenon is illustrated by Fig. 327, a—d. The groundplan of all these diagrams is identically the same. The generating spiral in each case represents a divergence of 3/8, or 135° of azimuth; and the points succeed one another at the same successional distances parallel to the axis. The rectangular outlines, which correspond to the exposed surface of the leaves or conescales, are of equal area, and of equal number. Nevertheless the appearances presented by these diagrams are very different; for in one the eye catches a 5/8 arrangement, in another a 3/5; and so on, down to an arrangement of 1/1. The mathematical side of this very curious phenomenon I have not attempted to investigate. But it is quite obvious that, in a system within

* The necessary existence of these recurring spirals is also proved, in a somewhat different way, by Leslie Ellis, On the Theory of Vegetable Spirals, in Mathematical and other Writings, 1853, pp. 358-372. which various spirals are implicitly contained, the conspicuousness of one set or another does not depend upon angular divergence. It depends on the relative proportions in length and breadth of the leaves themselves; or, more strictly speaking, on the ratio of

the diagonals of the rhomboidal figure by which each leaf-area is circumscribed. When, as in the fir-cone, the scales by mutual compression conform to these rhomboidal outlines, their inclined edges at once guide the eye in the direction of some one particular spiral; and we shall not fail to notice that in such cases the usual result is to give us arrangements corresponding to the middle diagrams in Fig. 327, which are the configurations in which the quadrilateral outlines approach most nearly to a rectangular form, and give us accordingly the least possible ratio (under the given conditions) of sectional boundary-wall to surface area.

The manner in which one system of spirals may be caused to slide, so to speak, into another, has been ingeniously demonstrated by Schwendener on a mechanical model, consisting essentially of a framework which can be opened or closed to correspond with one after another of the above series of diagrams*. The determination of the precise angle of divergence of two consecutive leaves of the generating spiral does not enter into the above general investigation (though Tait gives, in the same paper, a method by which it may be easily determined); and the very fact that it does not so enter shews it to be essentially unimportant. The determination of so-called “orthostichies,’ or precisely vertical successions of leaves, is also unimportant. We have no means, other than observation, of determining that one leaf is vertically above another, and spiral series such as we have been dealing with will appear, whether such orthostichies exist, whether they be near or remote, or whether the angle of divergence be such that no precise vertical superposition ever occurs. And lastly, the fact that the successional numbers, expressed as fractions, 1/2, 2/3, 3/5, represent a convergent series, whose final term is equal to 0-61803..., the sectio aurea or “golden mean” of unity, is seen to be a mathematical coincidence, devoid of biological significance; it is but a particular case of Lagrange’s theorem that the roots of every numerical equation of the second degree can be expressed by a periodic continued fraction. The same number has a multitude of curious arithmetical properties. It is the final term of all similar series to that with which we have been dealing, such for instance as 1/3, 3/4, 4/7, etc., or 1/4, 4/5, 5/9, etc. It is a number beloved of the circle-squarer, and of all those who seek to find, and then to penetrate, the secrets of the Great Pyramid. It is deep-set in Pythagorean as well as in Euclidean geometry. It enters (as the chord of an angle of 36°),

* A common form of pail-shaped waste-paper basket, with wide rhomboidal meshes of cane, is well-nigh as good a model as is required. into the thrice-isosceles triangle of which we have spoken on p. 511; it is a number which becomes (by the addition of unity) its own reciprocal; its properties never end. To Kepler (as Naber tells us) it was a symbol of Creation, or Generation. Its recent application to biology and art-criticism by Sir Theodore Cook and others is not new. Naber’s book, already quoted, is full of it. Zeising, in 1854, found in it the key to all morphology, — and the same writer, later on*, declared it to dominate both architecture and music. But indeed, to use Sir Thomas Browne’s words (though it was of another number that he spoke): “To enlarge this contemplation into all the mysteries and secrets accommodable unto this number, were inexcusable Pythagorisme.”’ If this number has any serious claim at all to enter into the biological question of phyllotaxis, this must depend on the fact, first emphasized by Chauncey Wright}, that, if the successive leaves of the fundamental spiral be placed at the particular azimuth which divides the circle in this “sectio aurea,” then no two leaves will ever be superposed; and thus we are said to have “the most thorough and rapid distribution of the leaves round the stem, each new or higher leaf falling over the angular space between the two older ones which are nearest in direction, so as to divide it in the same ratio (K), in which the first two or any two successive ones divide the circumference. Now 5/8 and all successive fractions differ inappreciably from K.” To this view there are many simple objections.

In the first place, even 5/8, or -625, is but a moderately close approximation to the “golden mean”; in the second place the arrangements by which a better approximation is got, such as 8/13, 13/21, and the very close approximations such as 34/55, 55/89, 89/144, etc., are comparatively rare, while the much less close approximations of 3/5 or 2/3, or even 1/2, are extremely common. Again, the general type of argument such as that which asserts that the plant is “aiming at” something which we may call an “ideal angle” is one that cannot commend itself to a plain student of physical science: nor is the hypothesis rendered more acceptable, when Sir T. Cook qualifies it by telling us that “all that a plant can do

is to vary, to make blind shots at constructions, or to “mutate: as it is now termed; and the most suitable of these constructions will in the long run be isolated by the action of Natural Selection.” Finally, and this is the most concrete objection of all, the supposed isolation of the leaves, or their most complete “distribution to the action of the surrounding atmosphere” is manifestly very little affected by any conditions which are confined to the angle of azimuth. If we could imagine a case in which all the leaves of the stem, or all the scales of a fir-cone, were crushed down to one and the same level, into a simple ring or whorl of leaves, then indeed they would have their most equable distribution under the condition of the “ideal angle,” that is to say of the “golden mean.” But if it be (so to speak) Nature’s object to set them further apart than they actually are, to give them freer exposure to the air than they actually have, then it is surely maniftst that the simple way to do so is to elongate the axis, and to set the leaves further apart, lengthways on the stem. This has at once a far more potent effect than any nice manipulation of the “angie of divergence.” For it is obvious that in F(¢ . sin 6) we have a greater range of variation by altering 0 than by altering d. We come then, without more ado, to the conclusion that the “ Fibonacci series,” and its supposed usefulness, and the hypothesis of its introduction into plant-structure through natural seiection. are all matters which deserve no place in the plain study oi botanical phenomena. As Sachs shrewdly recognised years ago, all such speculations as these hark back to a school of mystical idealism.

The eggs of birds and all other hard-shelled eggs, such as those of the tortoise and the crocodile, are simple solids of revolution ; but they differ greatly in form, according to the configuration of the plane curve by the revolution of which the egg is, in a mathematical. sense, generated. Some few eggs, such as those of the owl, the penguin, or the tortoise, are spherical or very nearly so; a few more, such as the grebe’s, the cormorant’s or the pelican’s, are approximately ellipsoidal, with symmetrical or nearly symmetrical ends, and somewhat similar are the so-called “cylindrical” eggs of the megapodes and the sand-grouse; the great majority, like the hen’s egg, are ovoid, a little blunter at one end than the other; and some, by an exaggeration of this lack of antero-posterior symmetry, are blunt at one end but characteristically pointed at the other, as is the case with the eggs of the guillemot and puffin, the sandpiper, plover and curlew. It is an obvious but by no means negligible fact that the egg, while often pointed, is never flattened or discoidal; it is a prolate, but never an oblate, spheroid.

The careful study and collection of birds’ eggs would seem to have begun with the Count de Marsigl*, the same celebrated naturalist who first studied the ‘‘ flowers” of the coral, and who wrote the Histoire physique de la mer; and the specific form, as well as the colour and other attributes of the egg have been again and again discussed, and not least by the many dilettanti naturalists of the eighteenth century who soon followed in Marsigh’s footsteps f.

* De avibus circa aquas Danubw vagantibus et de ipsarum Nidis (Vol. v of the Danubius Pannonico-mysicus). Hagae Com., 1726. + Sir Thomas Browne had a collection of eggs at Norwich, according to Evelyn, in 1671. We need do no more than mention Aristotle’s belief, doubtless old in his time, that the more poimted egg produces the male chicken, and the blunter egg the hen; though this theory survived into modern times* and perhaps still lingers on. Several naturalists, such as Giinther (1772) and Biihle (1818), have taken the trouble to disprove it by experiment. A more modern and more generally accepted explanation has been that the form of the egg is in direct relation to that of the bird which has to be hatched within—a view that would seem to have been first set forth by Naumann and Biihle, in their great treatise on eggs}, and adopted by Des Murst and many other well-known writers.

In a treatise by de Lafresnaye§, an elaborate comparison is made between the skeleton and the egg of the various birds, to shew, for instance, how those birds with a deep-keeled sternum laid rounded eggs, which alone could accommodate the form of the young. According to this view, that “Nature had foreseen||”’ the form adapted to and necessary for the growing embryo, it was easy to correlate the owl with its spherical egg, the diver with its elliptical one, and in like manner the round egg of the tortoise and the elongated one of the crocodile with the shape of the creatures which had afterwards to be hatched therein. A few writers, such as Thienemann 4], looked at the same facts the other way, and asserted that the form of the egg was determined by that of the bird by which it was laid, and in whose body it had been conformed.

In more recent times, other theories, based upon the principles of Natural Selection, have been current and very generally accepted, to account for these diversities of form. The pointed, conical egg of the guillemot is generally supposed to be an adaptation, § Lafresnaye, F. de, Comparaison des ceufs des Oiseaux avec leurs squelettes, comme seul moyen de reconnaitre la cause de leurs différentes formes, Rev. Zool.. 1845, pp. 180-187, 239-244. || Cf. Des Murs, p. 67: “Elle devait encore penser au moment ot ce germe aurait besoin de l’espace nécessaire & son accroissement, 2 ce moment ou...il devra remplir exactement l’intervalle circonscrit par sa fragile prison, etc.”

advantageous to the species in the circumstances under which the egg is laid; the pointed egg is less apt than a spherical one to roll off the narrow ledge of rock on which this bird is said to lay its solitary egg, and the more pointed the egg, so much the fitter and likelier is it to survive. The fact that the plover or the sandpiper, breeding in very different situations, lay eggs that are also conical, elicits another explanation, to the effect that here the conical form permits the many large eggs to be packed closely under the mother bird*. Whatever truth there be in these apparent adaptations to existing circumstances, it is only by a very hasty logic that we can accept them as a vera causa, or adequate explanation of the facts; and it is obvious that, in the bird’s egg, we have an admirable case for the direct investigation of the mechanical or physical significance of its form f.

Of all the many naturalists of the eighteenth and nineteenth centuries who wrote on the subject of eggs, one alone (so far as I am aware) ascribed the form of the egg to direct mechanical causes. Giinthert, in 1772, declared that the more or less rounded or pointed form of the egg is a mechanical consequence of the pressure of the oviduct at a time when the shell is yet unformed or unsolidified; and that accordingly, to explain the round egg of the owl or the kingfisher, we have only to admit that the oviduct of these birds is somewhat larger than that of most others, or less subject to violent contractions. This statement contains, im essence, the whole story of the mechanical conformation of the egg.

Let us consider, very briefly, the conditions to which the egg is subject in its passage down the oviduct§. (1) The “egg,” as it enters the oviduct, consists of the yolk only, enclosed in its vitelline membrane. As it passes down the first portion of the oviduct, the white is gradually superadded, + Jacob Steiner suggested a Cartesian oval, r+ mr’ =c, as a general formula fails in such a case as the guillemot), is purely empirical, and has no mechanical foundation.

{ Giinther, F. C., Sammlung von Nestern und Eyern verschiedener Vogel, Niirnb. 1772. Cf. also Raymond Pearl, Morphogenetic Activity of the Oviduct, J. Exp. Zool. vi, pp. 339-359, 1909. § The following account is in part reprinted from Nature, June 4, 1908. and becomes in turn surrounded by the “shell-membrane.” About this latter the shell is secreted, rapidly and at a late period ; the egg having meanwhile passed on into a wider portion of the oviducal tube, called (by loose analogy, as Owen says) the “uterus.” Here the egg assumes its permanent form, here it becomes rigid, and it is to this portion of the “oviduct” that our argument principally refers.

(2) Both the yolk and the entire egg tend to fill completely their respective membranes, and, whether this be due to growth or imbibition on the part of the contents or to contraction on the part of the surrounding membranes, the resulting tendency is for both yolk and egg to be, in the first instance, spherical, unless otherwise distorted by external pressure. (3) The egg is subject to pressure within the oviduct, which is an elastic, muscular tube, along the walls of which pass peristaltic waves of contraction. These muscular contractions may be described as the contraction of successive annuli of muscle, giving annular (or radial) pressure to successive portions of the egg; they drive the egg forward against the frictional resistance of the tube, while tending at the same time to distort its form. While nothing is known, so far as I am aware, of the muscular physiology of the oviduct, it is well known in the case of the intestine that the presence of an obstruction leads to the development of violent contractions in its rear, which waves of contraction die away, and are scarcely if at all propagated in advance of the obstruction.

(4) It is known by observation that a hen’s egg is always laid blunt end foremost. (5) It can be shown, at least as a very common rule, that those eggs which are most unsymmetrical, or most tapered off posteriorly, are also eggs of a large size relatively to the parent bird. The guillemot is a notable case in point, and so also are the curlews, sandpipers, phaleropes and terns. We may accordingly presume that the more pointed eggs are those that are large relatively to the tube or oviduct through which they have to pass, or, in other words, are those which are subject to the greatest pressure while being forced along. So general is this relation that we may go still further, and presume with great plausibility

in the few exceptional cases (of which the apteryx is the most conspicuous) where the egg is relatively large though not markedly unsymmetrical, that in these cases the oviduct itself is in all probability large (as Giinther had suggested) in proportion to the size of the bird. In the case of the common fowl we can trace a direct relation between the size and shape of the egg, for the first eggs laid by a young pullet are usually smaller, and at the same time are much more nearly spherical than the later ones; and, moreover, some breeds of fowls lay proportionately smaller eggs than others, and on the whole the former eggs tend to be rounder than the latter*.

We may now proceed to inquire more particularly how the form of the egg is controlled by the pressures to which it is subjected. The egg, just prior to the formation of the shell, is, as we have seen, a fluid body, tending to a spherical shape and enclosed within a membrane. Our problem, then, is: Given a practically incompressible fluid, contained in a deformable capsule, which is either (a) entirely inextensible, or (b) slightly extensible, and which is placed in a long elastic tube the walls of which are radially contractile, to” determine the shape under pressure.

If the capsule be spherical, inextensible, and completely filled with the fluid, absolutely no deformation can take place. The few eggs that are actually or approximately spherical, such as those of the tortoise or the owl, may thus be alternatively explained as cases where little or no deforming pressure has been applied prior to the solidification of the shell, or else as cases where the capsule was so little capable of extension and so completely filled as to preclude the possibility of deformation.

If the capsule be not spherical, but be imextensible, then deformation can take place under the external radial compression, * In so far as our explanation involves a shaping or moulding of the egg by the uterus or “oviduct” (an agency supplemented by the proper tensions of the egg), it is curious to note that this is very much the same as that old view of Telesius regarding the formation of the embryo (De rerum natura, vi, cc. 4 and 10), which he had inherited from Galen, and of which Bacon speaks (Nov. Ory. cap. 50; cf. Fliis’s note). Bacon expressly remarks thut ‘“‘Telesius should have been able to shew the like formation in the shells of eggs.” This old theory of embryonic. modelling survives only in our usage of the term “‘matrix” for a “mould.”

only provided that the pressure tends to make the shape more nearly spherical, and then only on the further supposition that the capsule is also not entirely filled as the deformation proceeds. In other words, an incompressible fluid contained in an inextensible envelope cannot be deformed without puckering of the envelope taking place. Let us next assume, as the conditions by which this result may be avoided, (a) that the envelope is to some extent extensible, or (6) that the whole structure grows under relatively fixed conditions. The two suppositions are practically identical with one another in effect. It is obvious that, on the presumption that the envelope is only moderately extensible, the whole structure can only be distorted to a moderate degree away from the spherical or spheroidal form.

At all points the shape is determined by the law of the distribution of radial pressure within the given region of the tube, surface friction helping to maintain the egg in position. If the egg be under pressure from the oviduct, but without any marked component either in a forward or backward direction, the egg will be compressed in the middle, and will tend more or less to the form of a cylinder with spherical ends. The eggs of the grebe, cormorant, or crocodile may be supposed to receive their shape in such circumstances.

When the egg is subject to the peristaltic contraction of the oviduct during its formation, then from the nature and direction of motion of the peristaltic wave the pressure will be greatest somewhere behind the middle of the egg; in other words, the tube is converted for the time being into a more conical form, and the simple result follows that the anterior end of the egg becomes the broader and the posterior end the narrower. With a given shape and size of body, equilibrium in the tube - may be maintained under greater radial pressure towards one end than towards the other. For example, a cylinder having conical ends, of semi-angles @ and 6’ respectively, remains in equilibrium, apart from friction, if pcos*@ = p’ cos?0’, so that at the more tapered end where @ is small p is small. Therefore the whole structure might assume such a configuration, or grow under such conditions, finally becoming rigid by solidification of the envelope.

According to the preceding paragraph, we must assume some initial distribution of pressure, some squeeze applied to the posterior part of the egg, in order to give it its tapering form. But, that form once acquired, the egg may remain in equilibrium both as regards form and position within the tube, even after that excess of pressure on the posterior part is relieved. Moreover, the above equation shews that a normal pressure no greater and (within certain limits) actually less acting upon the posterior part than on the anterior part of the egg after the shell is formed will be sufficient to communicate to it a forward motion. This is an important consideration, for it’ shews that the ordinary form of an egg, and even the conical form of an extreme case such as the guillemot’s, is directly favourable to the movement of the egg within the oviduct, blunt end foremost.

The mathematical statement of the whole case is as follows: In our egg, consisting of an extensible membrane filled with an incompressible fluid and under external pressure, the equation of the envelope is p,-+ T (1/r+ 1/r’) = P, where p, is the normal component of external pressure at a point where 7 and 7’ are the radii of curvature, 7' is the tension of the envelope, and P the internal fluid pressure. This is simply the equation of an elastic surface where 7 represents the coefficient of elasticity; in other words, a flexible elastic shell has the same mathematical properties as our fluid, membrane-covered egg. And this is the identical equation which we have already had so frequent occasion to employ in our discussion of the forms of cells; save only that in these latter we had chiefly to study the tension 7 (1.e. the surface-tension of the semi-fluid cell) and had little or nothing to do with the factor of external pressure (p,,), which in the case of the egg becomes of chief importance.

The above equation is the equation of eguilibrium, so that it must be assumed either that the whole body is at rest or that its motion while under pressure is not such as to affect the result. Tangential forces, which have been neglected, could modify the form by alteration of 7. In our case we must, and may very reasonably, assume that any movement of the egg down the oviduct during the period when its form is being impressed upon it is very slow, being possibly balanced by the advance of the

peristaltic wave which causes the movement, as well as by friction. The quantity 7 is the tension of the enclosing capsule—the surrounding membrane. If 7 be constant or symmetrical about the axis of the body, the body is symmetrical. But the abnormal eggs that a hen sometimes lays, cylindrical, annulated, or quite irregular, are due to local weakening of the membrane, in other words, to asymmetry of 7. Not only asymmetry of 7, but also asymmetry of p,, will render the body subject to deformation, and this factor, the unknown but regularly varying, largely radial, pressure applied by successive annuli of the oviduct, is the essential cause of the form, and variations of form, of the egg. In fact, in so far as the postulates correspond near enough to actualities, the above equation is the equation of all eggs in the universe. At least this is so if we generalise it in the form Pn t+ T/r+ T'/r = P in recognition of a possible difference between the principal tensions.

In the case of the spherical egg it is obvious that p, is everywhere equal. The simplest case is where p, = 0, in other words, where the egg is so small as practically to escape deforming pressure from the tube. . But we may also conceive the tube to be so thin-walled and extensible as to press with practically equal force upon all parts of the contained sphere. If while our egg be in process of conformation the envelope be free at any part from external pressure (that is to say, if p, = 0), then it is obvious that that part (if of circular section) will be a portion of a sphere. This is not unlikely to be the case actually or approximately at one or both poles of the egg, and is evidently the case over a considerable portion of the anterior end of the plover’s egg.