CHAPTER XIII

THE SHAPES OF HORNS, AND OF TEETH OR TUSKS: WITH A NOTE ON TORSION

We have had so much to say on the subject of shell-spirals that we must deal briefly with the analogous problems which are presented by the horns of sheep, goats, antelopes and other horned quadrupeds; and all the more, because these horn-spirals are on the whole less symmetrical, less easy of measurement than those of the shell, and in other ways also are less easy of investigation. Let us dispense altogether in this case with mathematics; and be content with a very simple account of the configuration of a horn.

There are three types of horn which deserve separate consideration: firstly, the horn of the rhinoceros; secondly the horns of the sheep, the goat, the ox or the antelope, that is to say, of the so-called hollow-horned ruminants; and thirdly, the solid bony horns, or “antlers,” which are characteristic of the deer.

The horn of the rhinoceros presents no difficulty. It is physiologically equivalent to a mass of consolidated hairs, and, like ordinary hair, it consists of non-living or “formed” material, continually added to by the living tissues at its base. In section, that is to say in the form of its “generating curve,” the horn is approximately elliptical, with the long axis fore-and-aft, or, in some species, nearly circular. Its longitudinal growth proceeds with a maximum velocity anteriorly, and a minimum posteriorly; and the ratio of these velocities being constant, the horn curves into the form of a logarithmic spiral in the manner that we have already studied. The spiral is of small angle, but in the longer-horned species, such as the great white rhinoceros (Ceratorhinus), the spiral form is distinctly to be recognised. As the horn {613} occupies a median position on the head,—a position, that is to say, of symmetry in respect to the field of force on either side,—there is no tendency towards a lateral twist, and the horn accordingly develops as a _plane_ logarithmic spiral. When two horns coexist, the hinder one is much the smaller of the two: which is as much as to say that the force, or rate, of growth diminishes as we pass backwards, just as it does within the limits of the single horn. And accordingly, while both horns have _essentially_ the same shape, the spiral curvature is less manifest in the second one, simply by reason of its comparative shortness.

The paired horns of the ordinary hollow-horned ruminants, such as the sheep or the goat, grow under conditions which are in some respects similar, but which differ in other and important respects from the conditions under which the horn grows in the rhinoceros. As regards its structure, the entire horn now consists of a bony core with a covering of skin; the inner, or dermal, layer of the latter is richly supplied with nutrient blood-vessels, while the outer layer, or epidermis, develops the fibrous or chitinous material, chemically and morphologically akin to a mass of cemented or consolidated hairs, which constitutes the “sheath” of the horn. A zone of active growth at the base of the horn keeps adding to this sheath, ring by ring, and the specific form of this annular zone is, accordingly, that of the “generating curve” of the horn. Each horn no longer lies, as it does in the rhinoceros, in the plane of symmetry of the animal of which it forms a part; and the limited field of force concerned in the genesis and growth of the horn is bound, accordingly, to be more or less laterally asymmetrical. But the two horns are in symmetry one with another; they form “conjugate” spirals, one being the “mirror-image” of the other. Just as in the hairy coat of the animal each hair, on either side of the median “parting,” tends to have a certain definite direction of its own axis, inclined away from the median axial plane of the whole system, so is it both with the bony core of the horn and with the consolidated mass of hairs or hair-like substance which constitutes its sheath; the primary axis of the horn is more or less inclined to, and may even be nearly perpendicular to, the axial plane of the animal.

The growth of the horny sheath is not continuous, but more or {614} less definitely periodic: sometimes, as in the sheep, this periodicity is particularly well-marked, and causes the horny sheath to be composed of a series of all but separate rings, which are supposed to be formed year by year, and so to record the age of the animal[560].

Just as we sought for the true generating curve in the orifice, or “lip,” of the molluscan shell, so we might be apt to assume that in the spiral horn the generating curve corresponded to the lip or margin of one of the horny rings or annuli. This annular margin, or boundary of the ring, is usually a sinuous curve, not lying in a plane, but such as would form the boundary of an anticlastic surface of great complexity: to the meaning and origin of which phenomenon we shall return presently. But, as we have already seen in the case of the molluscan shell, the complexities of the lip itself, or of the corresponding lines of growth upon the shell, need not concern us in our study of the development of the spiral: inasmuch as we may substitute for these actual boundary lines, their “trace,” or projection on a plane perpendicular to the axis—in other words the simple outline of a transverse section of the whorl. In the horn, this transverse section is often circular or nearly so, as in the oxen and many antelopes: it now and then becomes of somewhat complicated polygonal outline, as in a highland ram; but in many antelopes, and in most of the sheep, the outline is that of an isosceles, or sometimes nearly equilateral triangle, a form which is typically displayed, for instance, in _Ovis Ammon_. The horn in this latter case is a trihedral prism, whose three faces are, (1) an upper, or frontal face, in continuation of the plane of the frontal bone; (2) an outer, or orbital, starting from the upper margin of the orbit; and (3) an inner, or “nuchal,” abutting on the parietal bone[561]. Along these three faces, and their corresponding angles or edges, we can trace in the fibrous substance of the horn a series of homologous spirals, such as we {615} have called in a preceding chapter the “_ensemble_ of generating spirals” which constitute the surface.

In some few cases, of which the male musk ox is one of the most notable, the horn is not developed in a continuous spiral curve. It changes its shape as growth proceeds; and this, as we have seen, is enough to show that it does not constitute a logarithmic spiral. The reason is that the bony exostoses, or horn-cores, about which the horny sheath is shaped and moulded, neither grow continuously nor even remain of constant size after attaining their full growth. But as the horns grow heavy the bony core is bent downwards by their weight, and so guides the growth of the horn in a new direction. Moreover as age advances, the horn-core is further weakened and to a great extent absorbed: and the horny sheath or horn proper, deprived of its support, continues to grow, but in a flattened curve very different from its original spiral[562]. The chamois is a somewhat analogous case. Here the terminal, or oldest, part of the horn is curved; it tends to assume a spiral form, though from its comparative shortness it seems merely to be bent into a hook. But later on, the bony core within, as it grows and strengthens, stiffens the horn, and guides it into a straighter course or form. The same phenomenon {616} of change of curvature, manifesting itself at the time when, or the place where, the horn is freed from the support of the internal core, is seen in a good many other antelopes (such as the hartebeest) and in many buffaloes; and the cases where it is most manifest appear to be those where the bony core is relatively short, or relatively weak.

But in the great majority of horns, we have no difficulty in recognising a continuous logarithmic spiral, nor in referring it, as before, to an unequal rate of growth (parallel to the axis) on two opposite sides of the horn, the inequality maintaining a constant ratio as long as growth proceeds. In certain antelopes, such as the gemsbok, the spiral angle is very small, or in other words the horn is very nearly straight; in other species of the same genus Oryx, such as the Beisa antelope and the Leucoryx, a gentle {617} curve (not unlike though generally less than that of a Dentalium shell) is evident; and the spiral angle, according to the few measurements I have made, is found to measure from about 20° to nearly 40°. In some of the large wild goats, such as the Scinde wild goat, we have a beautiful logarithmic spiral, with a constant angle of rather less than 70°; and we may easily arrange a series of forms, such for example as the Siberian ibex, the moufflon, _Ovis Ammon_, etc., and ending with the long-horned Highland ram: in which, as we pass from one to another, we recognise precisely homologous spirals, with an increasing angular constant, the spiral angle being, for instance, about 75° or rather less in _Ovis Ammon_, and in the Highland ram a very little more. We have already seen that in the neighbourhood of 70° or 80° a small change of angle makes a marked difference in the appearance of the spire; and we know also that the actual length of the horn makes a very striking difference, for the spiral becomes especially conspicuous to the eye when a horn or shell is long enough to shew several whorls, or at least a considerable part of one entire whorl.

Even in the simplest cases, such as the wild goats, the spiral is never (strictly speaking) a plane or discoid spiral: but in greater or less degree there is always superposed upon the plane logarithmic spiral a helical spiral in space. Sometimes the latter is scarcely apparent, for the helical curvature is comparatively small, and the horn (though long, as in the said wild goats) is not nearly long enough to shew a complete convolution: at other times, as in the ram, and still better in many antelopes, such as the koodoo, the helicoid or corkscrew curve of the horn is its most characteristic feature.

Accordingly we may study, as in the molluscan shell, the helicoid component of the spire—in other words the variation in what we have called (on p. 555) the angle _θ_. This factor it is which, more than the constant angle of the logarithmic spiral, imparts a characteristic appearance to the various species of sheep, for instance to the various closely allied species of Asiatic wild sheep, or Argali. In all of these the constant angle of the logarithmic spiral is very much the same, but the shearing component differs greatly. And thus the long drawn out horns of {618} _Ovis Poli_, four feet or more from tip to tip, differ conspicuously from those of _Ovis Ammon_ or _O. hodgsoni_, in which a very similar logarithmic spiral is wound (as it were) round a much blunter cone.

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The ram’s horn then, like the snail’s shell, is a curve of double curvature, in which one component has imposed upon the structure a plane logarithmic spiral, and the other has produced a continuous displacement, or “shear,” proportionate in magnitude to, and perpendicular or otherwise inclined in direction to, the axis of the former spiral curvature. The result is precisely analogous to that which we have studied in the snail and other spiral univalves; but while the form, and therefore the resultant forces, are similar, the original distribution of force is not the same: for we have not here, as we had in the snail-shell, a “columellar” muscle, to introduce the component acting in the direction of the axis. We have, it is true, the central bony core, which in part performs an analogous function; but the main phenomenon here is apparently a complex distribution of rates of growth, perpendicular to the plane of the generating curve.

Let us continue to dispense with mathematics, for the mathematical treatment of a curve of double curvature is never very simple, and let us deal with the matter by experiment. We have seen that the generating curve, or transverse section, of a typical ram’s horn is triangular in form. Measuring (along the curve of the horn) the length of the three edges of the trihedral structure in a specimen of _Ovis Ammon_, and calling them respectively the outer, inner, and hinder edges (from their position at the base of the horn, relatively to the skull), I find the outer edge to measure 80 cm., the inner 74 cm., and the posterior 45 cm.; let us say that, roughly, they are in the ratio of 9 : 8 : 5. Then, if we make a number of little cardboard triangles, equip each with three little legs (I make them of cork), whose relative lengths are as 9 : 8 : 5, and pile them up and stick them all together, we straightway build up a curve of double curvature precisely analogous to the ram’s horn: except only that, in this first approximation, we have not allowed for the gradual increment (or decrement) of the triangular surfaces, that is to say, for the _tapering_ of the horn due to the growth in its own plane of the generating curve. {619}

In this case then, and in most other trihedral or three-sided horns, one of the three components, or three unequal velocities of growth, is of relatively small magnitude, but the other two are nearly equal one to the other. It would involve but little change for these latter to become precisely equal; and again but little to turn the balance of inequality the other way. But the immediate consequence of this altered ratio of growth would be that the horn would appear to wind the other way, as it does in the antelopes, and also in certain goats, e.g. the markhor, _Capra falconeri_.

For these two opposite directions of twist Dr Wherry has introduced a convenient nomenclature. When the horn winds so that we follow it from base to apex in the direction of the hands of a watch, it is customary to call it a “left-handed” spiral. Such a spiral we have in the horn on the left-hand side of a ram’s head. Accordingly, Dr Wherry calls the condition _homonymous_, where, as in the sheep, a right-handed spiral is on the right side of the head, and a left-handed spiral on the left side; while he calls the opposite condition _heteronymous_, as we have it in the antelopes, where the right-handed twist is on the left side of the head, and the left-handed twist on the right-hand side. Among the goats, we may have either condition. Thus the domestic and most of the wild goats agree with the sheep; but in the markhor the twisted horns are heteronymous, as in the antelopes. The difference, as we have seen, is easily explained; and (very much as in the case of our opposite spirals in the apple-snail, referred to on p. 560), it has no very deep importance.

Summarised then, in a very few words, the argument by which we account for the spiral conformation of the horn is as follows: The horn elongates by dint of continual growth within a narrow zone, or annulus, at its base. If the rate of growth be identical on all sides of this zone, the horn will grow straight; if it be greater on one side than on the other, the horn will become curved: and it probably _will_ be greater on one side than on the other, because each single horn occupies an unsymmetrical field with reference to the plane of symmetry of the animal. If the maximal and minimal velocities of growth be precisely at opposite sides of the zone of growth, the resultant spiral will be a plane spiral; but if they be not precisely or diametrically opposite, then the spiral will be a spiral in space, with a winding or helical component; and it is by no means likely that the maximum and minimum _will_ occur at precisely opposite ends of a diameter, for {620} no such plane of symmetry is manifested in the field of force to which the growing annulus corresponds or appertains.

Now we must carefully remember that the rates of growth of which we are here speaking are the net rates of longitudinal increment, in which increment the activity of the living cells in the zone of growth at the base of the horn is only one (though it is the fundamental) factor. In other words, if the horny sheath were continually being added to with equal rapidity all round its zone of active growth, but at the same time had its elongation more retarded on one side than the other (prior to its complete solidification) by varying degrees of adhesion or membranous attachment to the bone core within, then the net result would be a spiral curve precisely such as would have arisen from initial inequalities in the rate of growth itself. It seems highly probable that this is a very important factor, and sometimes even the chief factor in the case. The same phenomenon of attachment to the bony core, and the consequent friction or retardation with which the sheath slides over its surface, will lead to various subsidiary phenomena: among others to the presence of transverse folds or corrugations upon the horn, and to their unequal distribution upon its several faces or edges. And while it is perfectly true that nearly all the characters of the horn can be accounted for by unequal velocities of longitudinal growth upon its different sides, it is also plain that the actual field of force is a very complicated one indeed. For example, we can easily see that (at least in the great majority of cases) the direction of growth of the horny fibres of the sheath is by no means parallel to the axis of the core within; accordingly these fibres will tend to wind in a system of helicoid curves around the core, and not only this helicoid twist but any other tendency to spiral curvature on the part of the sheath will tend to be opposed or modified by the resistance of the core within. But on the other hand living bone is a very plastic structure, and yields easily though slowly to any forces tending to its deformation; and so, to a considerable extent, the bony core itself will tend to be modelled by the curvature which the growing sheath assumes, and the final result will be determined by an equilibrium between these two systems of forces. {621}

While it is not very safe, perhaps, to lay down any general rule as to what horns are more, and what are less spirally curved, I think it may be said that, on the whole, the thicker the horn, the greater is its spiral curvature. It is the slender horns, of such forms as the Beisa antelope, which are gently curved, and it is the robust horns of goats or of sheep in which the curvature is more pronounced. Other things being the same, this is what we should expect to find; for it is where the transverse section of the horn is large that we may expect to find the more marked differences in the intensity of the field of force, whether of active growth or of retardation, on opposite sides or in different sectors thereof.

But there is yet another and a very remarkable phenomenon which we may discern in the growth of a horn, when it takes the form of a curve of double curvature, namely, an effect of torsional strain; and this it is which gives rise to the sinuous “lines of growth,” or sinuous boundaries of the separate horny rings, of which we have already spoken. It is not at first sight obvious that a mechanical strain of torsion is necessarily involved in the growth of the horn. In our experimental illustration (p. 618), we built up a twisted coil of separate elements, and no torsional strain attended the development of the system. So would it be if the horny sheath grew by successive annular increments, free save for their relation to one another, and having no attachment to the solid core within. But as a matter of fact there is {622} such an attachment, by subcutaneous connective tissue, to the bony core; and accordingly a torsional strain will be set up in the growing horny sheath, again provided that the forces of growth therein be directed more or less obliquely to the axis of the core; for a “couple” is thus introduced, giving rise to a strain which the sheath would not experience were it free (so to speak) to slip along, impelled only by the pressure of its own growth from below. And furthermore, the successive small increments of the growing horn (that is to say, of the horny sheath) are not instantaneously converted from living to solid and rigid substance; but there is an intermediate stage, probably long-continued, during which the new-formed horny substance in the neighbourhood of the zone of active growth is still plastic and capable of deformation.

Now we know, from the celebrated experiments of St Venant[563], that in the torsion of an elastic body, other than a cylinder of circular section, a very remarkable state of strain is introduced. If the body be thus cylindrical (whether solid or hollow), then a twist leaves each circular section unchanged, in dimensions and in figure. But in all other cases, such as an elliptic rod or a prism of any particular sectional form, forces are introduced which act parallel to the axis of the structure, and which warp each section into a complex anticlastic surface. Thus in the case of a triangular and equilateral prism, such as is shewn in section in Fig. 321, if the part of the rod represented in the section be twisted by a force acting in the direction of the arrow, then the originally plane section will be warped as indicated in the diagram:—where the full contour-lines represent elevation above, and the dotted lines represent depression below, the original level. On the external surface of the prism, then, contour-lines which were originally parallel and horizontal, will be found warped into sinuous 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 _Ovis 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 {623} 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[564]. 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 {624} 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[565].

_A further Note upon Torsion._

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 books[566], 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[567]. 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, in 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 growing 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 {625} ascribed, such as contact (thigmotaxis), exposure to light (heliotropism), and so forth, need not be discussed here[568].

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 _torsion_ 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 {626} 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 {627} of _torsion_, so manifest in the former case, will be absent in the latter.

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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-pointed 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 considerable 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 {628} 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.

_Of Deer’s Antlers._

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[569]. 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[570]. 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: {629} 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-vessels run. 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_[571]. 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 {630} 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 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 {631} 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 surprising 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 {632} 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[572].” But all these many variations lie 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.

_Of Teeth, and of Beak and Claw._

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; {633} 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, it is 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 {634} 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.

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