On Growth and Form

On the external surface of the prism, then, contour-lines which were originally parallel and horizontal, will be found warped into smuous curves, such that, on each of the three faces, the curve will be convex upwards on one half, and concave upwards on the other half of the face. The ram’s horn, and still better that of Ovs Ammon, is comparable to such a prism, save that in section it is not quite equilateral, and that its three faces are not plane. The warping is therefore not precisely identical on the three faces

* St Venant, De la torsion des prismes, avec des considérations sur leur flexion, etc., Mém. des Savants Btrangers, Paris, XIv, pp. 233-560, 1856. of the horn; but, in the general distribution of the curves, it is in complete accordance with theory. Similar anticlastic curves are well seen in many antelopes; but they are conspicuous by their absence in the cylindrical horns of oxen. The better to illustrate this phenomenon, the nature of which is indeed obvious enough from a superficial examination of the horn, I made a plaster cast of one of the horny rings in a horn of Ovis Ammon, so as to get an accurate pattern of its sinuous edge: and then, filling the mould up with wet clay, I modelled an anticlastic surface, such as to correspond as nearly as possible with the sinuous outline*. Finally, after making a plaster cast of this sectional surface, I drew its contour-lines (as shewn in Fig. 322), with the help of a simple form of spherometer. It will be seen that in great part this diagram is precisely similar to St Venant’s

diagram of the cross-section of a twisted triangular prism; and this is especially the case in the neighbourhood of the sharp angle of our prismatic section. That in parts the diagram is somewhat asymmetrical is not to be wondered at: and (apart from inaccuracies due to the somewhat rough means by which it was made) this asymmetry can be sufficiently accounted for by anisotropy of the material, by inequalities in thickness of different parts of the horny sheath, and especially (I think) by unequal distributions of rigidity due to the presence of the smaller corrugations of the

—* This is not difficult to do, with considerable accuracy, if the clay be kept well wetted, or semi-fluid, and the smoothing be done with a large wet brush. horn. It is apparently on account of these minor corrugations that, in such horns as the Highland ram’s, where they are strongly marked, the main St Venant effect is not nearly so well shewn as in the smoother horns such as those of O. Ammon and its immediate congeners*. The phenomenon of torsion, to which we have been thus introduced, opens up many wide questions in connection with form. Some of the associated phenomena are admirably illustrated in the case of climbing plants; but we can only deal with these still more briefly and parenthetically.

The subject of climbing plants has been elaborately dealt with not only in Darwin’s books7, but also by a very large number of earlier and later writers. In “twining” plants, which constitute the greater number of “climbers,” the essential phenomenon is a tendency of the growing shoot to revolve about a vertical axis— a tendency long ago discussed and investigated by such writers as Palm, H. von Mohl and Dutrochet{. This tendency to revolution—“ circumvolution,” as Darwin calls it, “revolving nutation,” as Sachs puts it—is very closely comparable to the process by which an antelope’s horn (such as the koodoo’s) grows into its spiral or rather helicoid form; and it is simply due, m like manner, to inequalities in the rate of growth on different sides of the growing stem. There is only this difference between the two cases, that in the antelope’s horn the zone of active growth is confined to the base of the horn, while in the climbing stem the same phenomenon is at work throughout the whole length of the growmg structure. This growth is in the main due to “turgescence,” that is to the extension, or elongation, of ready-formed cells through the imbibition of water; it is a phenomenon due to osmotic pressure. The particular stimuli to which these movements (that is to say, these inequalities of growth) have been

* The curves are well shewn in most of Sir V. Brooke’s figures of the various species of Argali, in the paper quoted on p. 614. ~ 6 Ff Climbing Plants, 1865 (2nd edit. 1875); Power of Movement in Plants, 1880. t Palm, Ueber das Winden der Pflanzen, 1827; von Mohl, Bau und Winden der Ranken, ete., 1827; Dutrochet, Mouvements révolutifs spontanés, C.R. 1843, etc. ascribed, such as contact (thigmotaxis), exposure to light (heliotropism), and so forth, need not be discussed here*.

A simple stem growing upright in the dark, or in uniformly diffused light, would be in a position of equilibrium to a field of force radially symmetrical about its vertical axis. But this complete radial symmetry will not often occur; and the radial anomalies may be such as arise intrinsically from structural peculiarities in the stem itself, or externally to it by reason of unequal illumination or through various other localised forces. The essential fact, so far as we are concerned, is that in twining plants we have a very marked tendency to inequalities in longitudinal growth on different aspects of the stem—a tendency which is but an exaggerated manifestation of one which is more or less present, under certain conditions, in all plants whatsoever. Just as in the case of the ruminants’ horns so we find here, that this inequality may be, so to speak, positive or negative, the maximum lying to the one side or the other of the twining stem; and so it comes to pass that some climbers twine to the one side and some to the other: the hop and the honeysuckle following the sun, and the field-convolvulus twining in the reverse direction; there are also some, like the woody nightshade (Solanum Dulcamara) which twine indifferently either way.

Together with this circumnutatory movement, there is very generally to be seen an actual forsion of the twining stem—a twist, that is to say, about its own axis; and Mohl made the curious observation, confirmed by Darwin, that when a stem twines around a smooth cylindrical stick the torsion does not take place, save “only in that degree which follows as a mechanical necessity from the spiral winding”: but that stems which had climbed around a rough stick were all more or less, and generally much, twisted. Here Darwin did not refrain from introducing that teleological argument which pervades his whole train of reasoning: “The stem,” he says, “probably gains rigidity by being twisted (on the same principle that a much twisted rope

is stiffer than a slackly twisted one), and is thus indirectly benefited so as to be able to pass over inequalities in its spiral ascent, and to carry its own weight when allowed to revolve freely.” The mechanical explanation would appear to be very simple, and such as to render the teleological hypothesis unnecessary. In the case of the roughened support, there is a temporary adhesion or “clinging” between it and the growing stem which twines around it; and a system of forces is thus set up, producing a “couple,” just as it was in the case of the ram’s or antelope’s horn through direct adhesion of the bony core to the surrounding sheath. The twist is the direct result of this couple, and it disappears when the support is so smooth that no such force comes to be exerted.

Another important class of climbers includes the so-called “leaf-climbers.” In these, some portion of the leaf, generally the petiole, sometimes (as in the fumitory) the elongated midrib, curls round a support; and a phenomenon of like nature occurs in many, though not all, of the so-called “tendril-bearers.”’ Except that a different part of the plant, leaf or tendril instead of stem, is concerned in the twining process, the phenomenon here is strictly analogous to our former case; but in the resulting helix there is, as a rule, this obvious difference, that, while the twining stem, for instance of the hop, makes a slow revolution about its support, the typical leaf-climber makes a close, firm coil: the axis of the latter is nearly perpendicular and parallel to the axis of its support, while in the twining stem the angle between the two axes is comparatively small. Mathematically speaking, the difference merely amounts to this, that the component in the direction of the vertical axis is large in the one case, and the corresponding component is small, if not absent, in the other; in other words, we have in the climbing stem a considerable vertical component, due to its own tendency to grow in height, while this longitudinal or vertical extension of the whole system is not apparent, or little apparent, in the other cases. But from the fact that the twining stem tends to run - obliquely to its support, and the coiling petiole of the leaf-climber tends to run transversely to the axis of its support, there immediately follows this marked difference, that the phenomenon

of torsion, so manifest in the former case, will be absent in the latter. There is one other phenomenon which meets us in the twining and twisted stem, and which is doubtless illustrated also, though not so well, in the antelope’s horn; it is a phenomenon which forms the subject of a second chapter of St Venant’s researches on. the effects of torsional strain in elastic bodies. We have already seen how. one effect of torsion, in for instance a prism, is to produce strains parallel to the axis, elevating parts and depressing other parts of each transverse section. But in addition to this, the same torsion has the effect of materially altering the form of the section itself, as we may easily see by twisting a square or oblong piece of india-rubber. If we start with a cylinder, such as a round piece of catapult india-rubber, and twist it on its own long axis, we have already seen that it suffers no other distortion ; it still remains a cylinder, that is to say, it is still in section everywhere circular. But if it be of any other shape than cylindrical the case is quite different, for now the sectional shape tends to alter under the strain of torsion. Thus, if our rod be elliptical in section to begin with, it will, under torsion, become a more elongated ellipse; if it be square, its angles will become more prominent, and its sides will curve. inwards, till at length the square assumes the appearance of a four-poited star, with rounded angles. Furthermore, looking at the results of this process of modification, we find experimentally that the resultant figures are more easily twisted, less resistant to torsion, than were those from which we evolved them; and this is a very curious physical or mathematical fact.

So a cylinder, which is especially resistant to torsion, is very easily bent or flexed; while projecting ribs or angles, such as an engineer makes in a bar or pillar of iron for the purpose of greatly increasing its strength in the way of resistance to bending, actually make it much weaker than before (for the same amount of metal per unit length) in the way of resistance to torsion. In the hop itself, and in a very Eoneiieeea number of other twining and twisting stems, the ribbed or channelled form of the stem is a conspicuous feature. We may safely take it, (1) that

such stems are especially susceptible of torsion; and (2) that the effect of torsion will be to intensify any such peculiarities of sectional outline which they may possess, though not to initiate them in an originally cylindrical structure. In the leaf-climbers the case does not present itself, for there, as we have seen, torsion itself is not, or is very slightly, manifested. There are very distinct traces of the phenomenon in the horns of certain antelopes, but the reason why it is not a more conspicuous feature of the antelope’s horn or of the ram’s is apparently a very simple one: namely, that the presence of the bony core within tends to check that deformation which is perpendicular, while it permits that which is parallel, to the axis of the horn.

But’ let us return to our subject of the shapes of horns, and consider briefly our last class of these structures, namely the bony antlers of the various species of elk and deer*. The problems which these present to us are very different from those which we have had to do with in the antelope or the sheep. With regard to its structure, it is plain that the bony antler corresponds, upon the whole, to the bony core of the antelope’s horn; while in place of the hard horny sheath of the latter, we have the soft “velvet,” which every season covers the new growing antler, and protects the large nutrient blood-vessels by help of which the antler grows+. The main difference lies in the fact that, in the one case, the bony core, imprisoned within its sheath, is rendered incapable of branching and incapable also of lateral expansion, and the whole horn is only permitted to grow in length, while retaining a sectional contour that is identical with (or but little altered from) that which it possesses at its growing base:

7+ Cf. Rhumbler, L., Ueber die Abhaingigkeit des Geweihwachstums der Hirsche, speziell des Edelhirsches, vom Verlauf der Blutgefaisse im Kolbengeweih, Zeitschr. f. Forst. und Jagdwesen, 1911, pp. 295-314. but in the antler, on the other hand, no such restraint is imposed, and the living, growing fabric of bone may expand into a broad flattened plate over which the blood-vesselsrun. In the immediate neighbourhood of the main blood-vessels growth will be most active; in the interspaces between, it may wholly fail: with the result that we may have great notches cut out of the flattened plate, or may at length find it reduced to the form of a simple branching structure. The main point, as it seems to me, is that the “horn” is essentially an axial rod, while the “antler” is essentially an outspread surface*. In other words, I believe that

the whole configuration of an antler is more easily understood by conceiving it as a plate or a surface, more and more notched and scolloped till but a slender skeleton may remain, than to look upon it the other way, namely as an axial stem (or beam) giving * The fact that in one very small deer, the little South American Coassus, the antler is reduced to a simple short spike, does not preclude the general distinction which I have drawn. In Coassus we have the beginnings of an antler, which has not yet manifested its tendency to expand; and in the many allied species of the American genus Cariacus, we find the expansion manifested in various simple modes of ramification or bifurcation. (Cf. Sir V. Brooke, Classification of the Cervidae, p. 897.)

off branches (or tines), the interspaces between which latter may sometimes be filled up to form a continuous plate. In a sense it matters very little whether we regard the broad plate-like antlers of the elk or the slender branching antlers of the stag as the more primitive type; for we are not concerned here with the question of hypothetical phylogeny. And even from the mathematical point of view it makes little or no difference whether we describe the plate as constituted by the interconnection of

Fig. 324. Head and antlers of a Stag (Cervus Duvauceli). (After Lydekker, from P.Z.S.) the branches, or the branches derived by a process of notching or incision from the plate. The important point for us is to recognise that (save for occasional slight irregularities) the branching system in the one conforms essentially to the curved plate or surface which we see plainly in the other. In short the arrangement of the branches is more or less comparable to that of the veins in a leaf, or to that of the blood-vessels as they course over the curved surface of an organ. It is a process of ramification, not, like that of a tree, in various planes, but strictly limited

to a single surface. And just as the veins within a leaf are not necessarily confined (as they happen to be in most ordinary leaves) to a plane surface, but, as in the petal of a tulip or the capsule of a. poppy, may have to run their course within a curved surface, so does the analogy of the leaf lead us directly to the mode of branching which is characteristic of the antler. The surface to which the branches of the antler tend to be confined is a more or less spheroidal, or occasionally an ellipsoidal one; and furthermore, when we inspect any well-developed pair of antlers, such as those of a red deer, a sambur or a wapiti, we have no difficulty in seeing that the two antlers make up between them a single surface, and constitute a symmetrical figure, each half being the mirror-image of the other.

To put the case in another way, a pair of antlers (apart from occasional slight irregularities) tends to constitute a figure such that we could conceive an elastic sheet stretched over or round the entire system, so as to form one continuous and even surface ; and not only would the surface curvature be on the whole smooth and even, but the boundary of the surface would also tend to be an even curve: that is to say the tips of all the tines would approximately have their locus in a continuous. curve.

It follows from this that if we want to make a simple model of a set of antlers, we shall be very greatly helped by taking some appropriate spheroidal surface as our groundwork or scaffolding. The best form of surface is a matter for trial and investigation in each particular case; but even in a sphere, by selecting appropriate areas thereof, we can obtain sufficient varieties of surface to meet all ordinary cases. With merely a bit of sculptor’s clay or plasticine, we should be put hard to it to model the horns of a wapiti or a reindeer: but if we start with an orange (or a round florence flask) and lay our little tapered rolls of plasticine upon it, in simple natural curves, it is surprismg to see how quickly and successfully we can imitate one type of antler after another. In doing so, we shall be struck by the fact that our model may vary in its mode of branching within very considerable limits, and yet look perfectly natural. For the same wide range of variation is characteristic of the natural antlers themselves. As Sir V. Brooke says (op. cit. p. 892), “No two antlers are ever exactly alike; and the

variation to which the antlers are subject is so great that in the absence of a large series they would be held to be indicative of several distinct species*.” But all these many variations le within a limited range, for they are all subject to our general rule that the entire structure is essentially confined to a single curved surface. It is plain that in the curvatures both of the beam and of its tines, in the angles by which these latter meet the beam, and in the contours of the entire system, there are involved many elegant mathematical problems with which we cannot at present attempt to deal. Nor must we attempt meanwhile to enquire into the physical meaning or origin of these phenomena, for as yet the clue seems to be lacking and we should only heap one hypothesis upon another. That there is a complete contrast of mathematical properties between the horn and the antler is the main lesson with which, in the meantime, we must rest content.

In a fashion similar to that manifested in the shell or the horn, we find the logarithmic spiral to be implicit in a great many other organic structures where the phenomena of growth proceed in a similar way: that is to say, where about an axis there is some ~ asymmetry leading to unequal rates of longitudinal growth, and where the structure is of such a kind that each new increment is added on as a permanent and unchanging part of the entire conformation. Nail and claw, beak and tooth, all come under this category. The logarithmic spiral always tends to manifest itself in such structures as these, though it usually only attracts our attention in elongated structures, where (that is to say) the radius vector has described a considerable angle. When the canary-bird’s claws grow long from lack of use, or when the incisor tooth of a rabbit or a rat grows long by reason of an injury to the opponent tooth against which it was wont to bite, we know that the tooth or claw tends to grow into a spiral curve, and we speak of it as a malformation. But there has been no fundamental change of form, save only an abnormal increase in length;

the elongated tooth or claw has the selfsame curvature that it had when it was short, but the spiral curvature becomes more and more ‘manifest the longer it grows. A curious analogous case is that of the New Zealand huia bird, in which the beak of the female is described as being comparatively short and straight, while that of the male is long and curved; it is easy to see that there is a slight curvature also in the beak of the female, and that the beak of the male shows nothing but the same curve produced. In the case of the more curved beaks, such as those of an eagle or a parrot, we may, if we please, determine the constant angle of the logarithmic spiral, just as we have done in the case of the Nautilus shell; and here again, as the bird grows older or the beak longer, the spiral nature of the curve becomes more and more apparent, as in the hooked beak of an old eagle, or as in the great beak of some large parrot such as a hyacinthine macaw.

Let us glance at one or two instances to illustrate the spiral curvature of teeth. A dentist knows that every tooth has a curvature of its own, and that in pulling the tooth he must follow the direction of the curve; but in an ordinary tooth this curvature is scarcely visible, and is least so when the diameter of the tooth is large compared with its length. In the simply formed, more or less conical teeth, such as are those of the dolphin, and in the more or less similarly shaped canines and incisors of mammals in general, the curvature of the tooth is particularly well seen. We see it in the little teeth of a hedgehog, and in the canines of a dog or a cat it is very obvious indeed. When the great canine of the carnivore becomes still further enlarged or elongated, as in Machairodus, it grows into the strongly curved sabre-tooth of that great extinct tiger. In rodents, itis the incisors which undergo a great elongation; their rate of growth differs, though but slightly, on the two sides, anterior and posterior, of the axis, and by summation of these slight differences in the rapid growth of the tooth an unmistakeable logarithmic spiral is gradually built up. We see it admirably in the beaver, or in the great ground-rat, Geomys. The elephant is a similar case, save that the tooth, or tusk, remains, owing to comparative lack of wear, in a more perfect condition. In the rodent (save only in those abnormal cases mentioned on the last page) the

anterior, first-formed, part of the tooth wears away as fast as it is added to from behind; and in the grown animal, all those portions of the tooth near to the pole of the logarithmic spiral have long disappeared. In the elephant, on the other hand, we see, practically speaking, the whole unworn tooth, from point to root; and its actual tip nearly coincides with the pole of the spiral. If we assume (as with no great inaccuracy we may do) that the tip actually coincides with the pole, then we may very easily construct the continuous spiral of which the existing tusk constitutes a part; and by so doing, we see the short, gently curved tusk of our ordinary elephant growing gradually into the spiral tusk of the mammoth. No doubt, just as in the case of our molluscan shells, we have a tendency to variation, both individual and specific, in the constant angle of the spiral; some elephants, and some species of elephant, undoubtedly have a higher spiral angle than others. But in most cases, the angle would seem to be such that a spiral configuration would become very manifest indeed if only the tusk pursued its steady growth, unchanged otherwise in form, till it attained the dimensions which we meet with in the mammoth. In a species such as Mastodon angustidens, or M. arvernensis, the specific angle is low and the tusk comparatively straight; but the American mastodons and the existing species of elephant have tusks which do not differ appreciably, except in size, from the great spiral tusks of the mammoth, though from their comparative shortness the spiral is little developed and only appears to the eye as a gentle curve.

Wherever the tooth is very long indeed, as in the mammoth or the beaver, the effect of some slight and all but inevitable lateral asymmetry in the rate of growth begins to shew itself: in other words, the spiral is seen to lie not absolutely in a plane, but to be a curve of double curvature, like a twisted horn. We see this condition very well in the huge canine tusks of the Babirussa; it is a conspicuous feature in the mammoth, and it is more or less perceptible in any large tusk of the ordinary elephants.

The form of a molar tooth, which is essentially a branching or budding system, and in which such longitudinal growth as gives rise to a spiral curve is but little manifest, constitutes an entirely different problem with which I shall not at present attempt to deal. The beautiful configurations produced by the orderly arrangement of leaves or florets on a stem have long been an object of admiration and curiosity. Leonardo da Vinci would seem, as Sir Theodore Cook tells us, to have been the first to record his thoughts upon this subject; but the old Greek and Egyptian geometers are not likely to have left unstudied or unobserved the spiral traces of the leaves upon a palm-stem, or the spiral curves of the petals of a lotus or the florets in a sunflower.

The spiral leaf-order has been regarded by many learned botanists as involving a fundamental law of growth, of the deepest and most far-reaching importance; while others, such as Sachs, have looked upon the whole doctrine of “ phyllotaxis” as ‘‘a sort of geometrical or arithmetical playing with ideas,” and “the spiral theory as a mode of view gratuitously introduced into the plant.” Sachs even goes so far as to declare this doctrine “in direct opposition to scientific investigation, and based upon the idealistic direction of the Naturphilosophie,’—the mystical biology of Oken and his school.

The essential facts of the case are not difficult to understand ; but the theories built upon them are so varied, so conflicting, and sometimes so obscure, that we must not attempt to submit them to detailed analysis and criticism. There are two chief ways by which we may approach the question, according to whether we regard, as the more fundamental and typical, one or other of the two chief modes in which the phenomenon presents itself. That is to say, we may hold that the phenomenon is displayed in its essential simplicity by the corkscrew spirals, or helices, which -mark the position of the leaves upon a cylindrical stem or on an