Chapter XIII…

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.

{635}

XIV. ON LEAF-ARRANGEMENT, OR PHYLLOTAXIS

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 {636} elongated fir-cone; or, on the other hand, we may be more attracted by, and regard as of greater importance, the logarithmic spirals which we trace in the curving rows of florets in the discoidal inflorescence of a sunflower. Whether one way or the other be the better, or even whether one be not positively correct and the other radically wrong, has been vehemently debated. In my judgment they are, both mathematically and biologically, to be regarded as inseparable and correlative phenomena.

The helical arrangement (as in the fir-cone) was carefully studied in the middle of the eighteenth century by the celebrated Bonnet, with the help of Calandrini, the mathematician. Memoirs published about 1835, by Schimper and Braun, greatly amplified Bonnet’s investigations, and introduced a nomenclature which still holds its own in botanical textbooks. Naumann and the brothers Bravais are among those who continued the investigation in the years immediately following, and Hofmeister, in 1868, gave an admirable account and summary of the work of these and many other writers[573].

Starting from some given level and proceeding upwards, let us mark the position of some one leaf (A) upon a cylindrical stem. Another, and a younger leaf (B) will be found standing at a certain distance around the stem, and a certain distance along the stem, {637} from the first. The former distance may be expressed as a fractional “divergence” (such as two-fifths of the circumference of the stem) as the botanists describe it, or by an “angle of azimuth” (such as ϕ = 144°) as the mathematician would be more likely to state it. The position of B relatively to A must be determined, not only by this angle ϕ, in the horizontal plane, but also by an angle (θ) in the vertical plane; for the height of B above the level of A, in comparison with the diameter of the cylinder, will obviously make a great difference in the appearance of the whole system, in short the position of each leaf must be expressed by F(ϕ ⋅ sin θ). But this matter botanical students have not concerned themselves with; in other words, their studies have been limited (or mainly limited) to the relation of the leaves to one another in azimuth.

Whatever relation we have found between A and B, let precisely the same relation subsist between B and C: and so on. Let the growth of the system, that is to say, be continuous and uniform; it is then evident that we have the elementary conditions for the development of a simple cylindrical helix; and this “primary helix” or “genetic spiral” we can now trace, winding round and round the stem, through A, B, C, etc. But if we can trace such a helix through A, B, C, it follows from the symmetry of the system, that we have only to join A to some other leaf to trace another spiral helix, such, for instance, as A, C, E, etc.; parallel to which will run another and similar one, namely in this case B, D, F, etc. And these spirals will run in the opposite direction to the spiral ABC.

In short, the existence of one helical arrangement of points implies and involves the existence of another and then another helical pattern, just as, in the pattern of a wall-paper, our eye travels from one linear series to another.

A modification of the helical system will be introduced when, instead of the leaves appearing, or standing, in singular succession, we get two or more appearing simultaneously upon the same level. If there be two such, then we shall have two generating spirals precisely equivalent to one another; and we may call them A, B, C, etc., and A′, B′, C′, and so on. These are the cases which we call “whorled” leaves, or in the simplest case, where {638} the whorl consists of two opposite leaves only, we call them decussate.

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Among the phenomena of phyllotaxis, two points in particular have been found difficult of explanation, and have aroused discussion. These are (1), the presence of the logarithmic spirals such as we have already spoken of in the sunflower; and (2) the fact that, as regards the number of the helical or spiral rows, certain numerical coincidences are apt to recur again and again, to the exclusion of others, and so to become characteristic features of the phenomenon.

The first of these appears to me to present no difficulty. It is a mere matter of strictly mathematical “deformation.” The stem which we have begun to speak of as a cylinder is not strictly so, inasmuch as it tapers off towards its summit. The curve which winds evenly around this stem is, accordingly, not a true helix, for that term is confined to the curve which winds evenly around the cylinder: it is a curve in space which (like the spiral curve we have studied in our turbinate shells) partakes of the characters of a helix and of a logarithmic spiral, and which is in fact a logarithmic spiral with its pole drawn out of its original plane by a force acting in the direction of the axis. If we imagine a tapering cylinder, or cone, projected, by vertical projection, on a plane, it becomes a circular disc; and a helix described about the cone necessarily becomes in the disc a logarithmic spiral described about a focus which corresponds to the apex of our cone. In like manner we may project an identical spiral in space upon such surfaces as (for instance) a portion of a sphere or of an ellipsoid; and in all these cases we preserve the spiral configuration, which is the more clearly brought into view the more we reduce the vertical component by which it was accompanied. The converse is, of course, equally true, and equally obvious, namely that any logarithmic spiral traced upon a circular disc or spheroidal surface will be transformed into a corresponding spiral helix when the plane or spheroidal disc is extended into an elongated cone approximating to a cylinder. This mathematical conception is translated, in botany, into actual fact. The fir-cone may be looked upon as a cylindrical axis contracted at both ends, until {639} it becomes approximately an ellipsoidal solid of revolution, generated about the long axis of the ellipse; and the semi-ellipsoidal capitulum of the teasel, the more or less hemispherical one of the thistle, and the flattened but still convex one of the sunflower, are all beautiful and successive deformations of what is typically a long, conical, and all but cylindrical stem. On the other hand, every stem as it grows out into its long cylindrical shape is but a deformation of the little spheroidal or ellipsoidal surface, or cone, which was its forerunner in the bud.

This identity of the helical spirals around the stem with spirals projected on a plane was clearly recognised by Hofmeister, who was accustomed to represent his diagrams of leaf-arrangement either in one way or the other, though not in a strictly geometrical projection[574].

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According to Mr A. H. Church[575], who has dealt very carefully and elaborately with the whole question of phyllotaxis, the logarithmic spirals such as we see in the disc of the sunflower have a far greater importance and a far deeper meaning than this brief treatment of mine would accord to them: and Sir Theodore Cook, in his book on the Curves of Life, has adopted and has helped to expound and popularise Mr Church’s investigations.

Mr Church, regarding the problem as one of “uniform growth,” easily arrives at the conclusion that, if this growth can be conceived as taking place symmetrically about a central point or “pole,” the uniform growth would then manifest itself in logarithmic spirals, including of course the limiting cases of the circle and straight line. With this statement I have little fault to find; it is in essence identical with much that I have said in a previous chapter. But other statements of Mr Church’s, and many theories woven about them by Sir T. Cook and himself, I am less able to follow. Mr Church tells us that the essential phenomenon in the sunflower disc is a series of orthogonally intersecting logarithmic spirals. Unless I wholly misapprehend Mr Church’s meaning, I should say that this is very far from essential. The spirals {640} intersect isogonally, but orthogonal intersection would be only one particular case, and in all probability a very infrequent one, in the intersection of logarithmic spirals developed about a common pole. Again on the analogy of the hydrodynamic lines of force in certain vortex movements, and of similar lines of force in certain magnetic phenomena, Mr Church proceeds to argue that the energies of life follow lines comparable to those of electric energy, and that the logarithmic spirals of the sunflower are, so to speak, lines of equipotential[576]. And Sir T. Cook remarks that this “theory, if correct, would be fundamental for all forms of growth, though it would be more easily observed in plant construction than in animals.” The parallel I am not able to follow.

Mr Church sees in phyllotaxis an organic mystery, a something for which we are unable to suggest any precise cause: a phenomenon which is to be referred, somehow, to waves of growth emanating from a centre, but on the other hand not to be explained by the division of an apical cell, or any other histological factor. As Sir T. Cook puts it, “at the growing point of a plant where the new members are being formed, there is simply nothing to see.”

But it is impossible to deal satisfactorily, in brief space, either with Mr Church’s theories, or my own objections to them[577]. Let it suffice to say that I, for my part, see no subtle mystery in the matter, other than what lies in the steady production of similar growing parts, similarly situated, at similar successive intervals of time. If such be the case, then we are bound to have in {641} consequence a series of symmetrical patterns, whose nature will depend upon the form of the entire surface. If the surface be that of a cylinder we shall have a system, or systems, of spiral helices: if it be a plane, with an infinitely distant focus, such as we obtain by “unwrapping” our cylindrical surface, we shall have straight lines; if it be a plane containing the focus within itself, or if it be any other symmetrical surface containing the focus, then we shall have a system of logarithmic spirals. The appearance of these spirals is sometimes spoken of as a “subjective” phenomenon, but the description is inaccurate: it is a purely mathematical phenomenon, an inseparable secondary result of other arrangements which we, for the time being, regard as primary. When the bricklayer builds a factory chimney, he lays his bricks in a certain steady, orderly way, with no thought of the spiral patterns to which this orderly sequence inevitably leads, and which spiral patterns are by no means “subjective.” The designer of a wall-paper not only has no intention of producing a pattern of criss-cross lines, but on the contrary he does his best to avoid them; nevertheless, so long as his design is a symmetrical one, the criss-cross intersections inevitably come.

Let us, however, leave this discussion, and return to the facts of the case.

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Our second question, which relates to the numerical coincidences so familiar to all students of phyllotaxis, is not to be set and answered in a word.

Let us, for simplicity’s sake, avoid consideration of simultaneous or whorled leaf origins, and consider only the more frequent cases where a single “genetic spiral” can be traced throughout the entire system.

It is seldom that this primary, genetic spiral catches the eye, for the leaves which immediately succeed one another in this genetic order are usually far apart on the circumference of the stem, and it is only in close-packed arrangements that the eye readily apprehends the continuous series. Accordingly in such a case as a fir-cone, for instance, it is certain of the secondary spirals or “parastichies” which catch the eye; and among fir-cones, we can easily count these, and we find them to be {642} on the whole very constant in number, according to the species.

Thus in many cones, such as those of the Norway spruce, we can trace five rows of scales winding steeply up the cone in one direction, and three rows winding less steeply the other way; in certain other species, such as the common larch, the normal number is eight rows in the one direction and five in the other; while in the American larch we have again three in the one direction and five in the other. It not seldom happens that two arrangements grade into one another on different parts of one and the same cone. Among other cases in which such spiral series are readily visible we have, for instance, the crowded leaves of the stone-crops and mesembryanthemums, and (as we have said) the crowded florets of the composites. Among these we may find plenty of examples in which the numbers of the serial rows are similar to those of the fir-cones; but in some cases, as in the daisy and others of the smaller composites, we shall be able to trace thirteen rows in one direction and twenty-one in the other, or perhaps twenty-one and thirty-four; while in a great big sunflower we may find (in one and the same species) thirty-four and fifty-five, fifty-five and eighty-nine, or even as many as eighty-nine and one hundred and forty-four. On the other hand, in an ordinary “pentamerous” flower, such as a ranunculus, we may be able to trace, in the arrangement of its sepals, petals and stamens, shorter spiral series, three in one direction and two in the other. It will be at once observed that these arrangements manifest themselves in connection with very different things, in the orderly interspacing of single leaves and of entire florets, and among all kinds of leaf-like structures, foliage-leaves, bracts, cone-scales, and the various parts or members of the flower. Again we must be careful to note that, while the above numerical characters are by much the most common, so much so as to be deemed “normal,” many other combinations are known to occur.

The arrangement, as we have seen, is apt to vary when the entire structure varies greatly in size, as in the disc of the sunflower. It is also subject to less regular variation within one and the same species, as can always be discovered when we examine a sufficiently large sample of fir-cones. For instance, out of 505 {643} cones of the Norway spruce, Beal[578] found 92 per cent. in which the spirals were in five and eight rows; in 6 per cent. the rows were four and seven, and in 4 per cent. they were four and six. In each case they were nearly equally divided as regards direction; for instance of the 467 cones shewing the five-eight arrangement, the five-series ran in right-handed spirals in 224 cases, and in left-handed spirals in 243.

Omitting the “abnormal” cases, such as we have seen to occur in a small percentage of our cones of the spruce, the arrangements which we have just mentioned may be set forth as follows, (the fractional number used being simply an abbreviated symbol for the number of associated helices or parastichies which we can count running in the opposite directions): 2/3, 3/5, 5/8, 8/13, 13/21, 21/34, 34/55, 55/89, 89/144. Now these numbers form a very interesting series, which happens to have a number of curious mathematical properties[579]. We see, for instance, that the denominator of each fraction is the numerator of the next; and further, that each successive numerator, or denominator, is the sum of the preceding two. Our immediate problem, then, is to determine, if possible, how these numerical coincidences come about, and why these particular numbers should be so commonly met with {644} as to be considered “normal” and characteristic features of the general phenomenon of phyllotaxis. The following account is based on a short paper by Professor P. G. Tait[580].

Of the two following diagrams, Fig. 325 represents the general case, and Fig. 326 a particular one, for the sake of possibly greater simplicity. Both diagrams represent a portion of a branch, or fir-cone, regarded as cylindrical, and unwrapped to form a plane surface. A, a, at the two ends of the base-line, represent the same initial leaf or scale: O is a leaf which can be reached from A by m steps in a right-hand spiral (developed into the straight line AO), and by n steps from a in a left-handed spiral aO. Now it is obvious in our fir-cone, that we can include all the scales upon the cone by taking so many spirals in the one direction, and again include them all by so many in the other. Accordingly, in our diagrammatic construction, the spirals AO and aO must, and always can, be so taken that m spirals parallel to aO, and n spirals parallel to AO, shall separately include all the leaves upon the stem or cone.

If m and n have a common factor, l, it can easily be shewn that the arrangement is composite, and that there are l fundamental, or genetic spirals, and l leaves (including A) which are situated exactly on the line Aa. That is to say, we have here a whorled arrangement, which we have agreed to leave unconsidered in favour of the simpler case. We restrict ourselves, accordingly, to the cases where there is but one genetic spiral, and when therefore m and n are prime to one another.

Our fundamental, or genetic, spiral, as we have seen, is that which passes from A (or a) to the leaf which is situated nearest to the base-line Aa. The fundamental spiral will thus be right-handed (A, P, etc.) if P, which is nearer to A than to a, be this leaf—left-handed if it be p. That is to say, we make it a convention that we shall always, for our fundamental spiral, run {645} round the system, from one leaf to the next, by the shortest way.

Now it is obvious, from the symmetry of the figure (as further shewn in Fig. 326), that, besides the spirals running along AO and aO, we have a series running from the steps on aO to the steps on AO. In other words we can find a leaf (S) upon AO, which, like the leaf O, is reached directly by a spiral series from A and from a, such that aS includes n steps, and AS (being part of the old spiral line AO) now includes m − n steps. And, since m and n are prime to one another (for otherwise the system would have been a composite or whorled one), it is evident that we can continue this process of convergence until we come down to a 1, 1 arrangement, that is to say to a leaf which is reached by a single step, in opposite directions from A and from a, which leaf is therefore the first leaf, next to A, of the fundamental or generating spiral. {646}

If our original lines along AO and aO contain, for instance, 13 and 8 steps respectively (i.e. m = 13, n = 8), then our next series, observable in the same cone, will be 8 and (13 − 8) or 5; the next 5 and (8 − 5) or 3; the next 3, 2; and the next 2, 1; leading to the ultimate condition of 1, 1. These are the very series which we have found to be common, or normal; and so far as our investigation has yet gone, it has proved to us that, if one of these exists, it entails, ipso facto, the presence of the rest.

In following down our series, according to the above construction, we have seen that at every step we have changed direction, the longer and the shorter sides of our triangle changing places every time. Let us stop for a moment, when we come to the 1, 2 series, or AT, aT of Fig. 326. It is obvious that there is nothing to prevent us making a new 1, 3 series if we please, by continuing the generating spiral through three leaves, and connecting the leaf so reached directly with our initial one. But in the case represented in Fig. 326, it is obvious that these two series (A, 1, 2, 3, etc., and a, 3, 6, etc.) will be running in the same direction; i.e. they will both be right-handed, or both left-handed spirals. The simple meaning of this is that the third leaf of the generating spiral was distant from our initial leaf by more than the circumference of the cylindrical stem; in other words, that there were more than two, but less than three leaves in a single turn of the fundamental spiral.

Less than two there can obviously never be. When there are exactly two, we have the simplest of all possible arrangements, namely that in which the leaves are placed alternately on opposite sides of the stem. When there are more than two, but less than three, we have the elementary condition for the production of the series which we have been considering, namely 1, 2; 2, 3; 3, 5, etc. To put the latter part of this argument in more precise language, let us say that: If, in our descending series, we come to steps 1 and t, where t is determined by the condition that 1 and t + 1 would give spirals both right-handed, or both left-handed; it follows that there are less than t + 1 leaves in a single turn of the fundamental spiral. And, determined in this manner, it is found in the great majority of cases, in fir-cones and a host of other examples of phyllotaxis, that t = 2. In other words, in the {647} great majority of cases, we have what corresponds to an arrangement next in order of simplicity to the simplest case of all: next, that is to say, to the arrangement which consists of opposite and alternate leaves.

“These simple considerations,” as Tait says, “explain completely the so-called mysterious appearance of terms of the recurring series 1, 2, 3, 5, 8, 13, etc.[581] The other natural series, usually but misleadingly represented by convergents to an infinitely extended continuous fraction, are easily explained, as above, by taking t = 3, 4, 5, etc., etc.” Many examples of these latter series have been given by Dickson[582] and other writers.

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We have now learned, among other elementary facts, that wherever any one system of helical spirals is present, certain others invariably and of necessity accompany it, and are definitely related to it. In any diagram, such as Fig. 326, in which we represent our leaf-arrangement by means of uniform and regularly interspaced dots, we can draw one series of spirals after another, and one as easily as another. But in our fir-cone, for instance, one particular series, or rather two conjugate series, are always conspicuous, while the others are sought and found with comparative difficulty.

The phenomenon is illustrated by Fig. 327, a–d. The ground-plan 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 cone-scales, 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 {648} 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 {649} 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[583].

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°), {650} 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[584], 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[585], 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 acceptably when Sir T. Cook qualifies it by telling us that “all that a plant can do {651} 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 manifest 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 “angle of divergence.” For it is obvious that in F(ϕ ⋅ sin θ) we have a greater range of variation by altering θ than by altering ϕ. 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 selection, are all matters which deserve no place in the plain study of botanical phenomena. As Sachs shrewdly recognised years ago, all such speculations as these hark back to a school of mystical idealism.

{652}

Chapter XV

ON THE SHAPES OF EGGS, AND OF CERTAIN OTHER HOLLOW STRUCTURES

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 Marsigli[586], 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 Marsigli’s footsteps[587]. {653}

We need do no more than mention Aristotle’s belief, doubtless old in his time, that the more pointed egg produces the male chicken, and the blunter egg the hen; though this theory survived into modern times[588] and perhaps still lingers on. Several naturalists, such as Günther (1772) and Bühle (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 Bühle, in their great treatise on eggs[589], and adopted by Des Murs[590] and many other well-known writers.

In a treatise by de Lafresnaye[591], 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[592]” 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[593], 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, {654} 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[594]. 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[595].

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. Günther[596], 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, in 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[597].

(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, {655} 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 {656} 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 Günther 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[598].

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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 inextensible, then deformation can take place under the external radial compression, {657} 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 (b) 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 θ and θ′ respectively, remains in equilibrium, apart from friction, if p cos^2 θ = p′ cos^2 θ′, so that at the more tapered end where θ 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. {658} 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{n}_ + T(1/r + 1/r′) = P, where p{n}_ is the normal component of external pressure at a point where r and r′ are the radii of curvature, T is the tension of the envelope, and P the internal fluid pressure. This is simply the equation of an elastic surface where T 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 T (i.e. the surface-tension of the semi-fluid cell) and had little or nothing to do with the factor of external pressure (p{n}_), which in the case of the egg becomes of chief importance.

The above equation is the equation of equilibrium, 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 T. 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 {659} peristaltic wave which causes the movement, as well as by friction.

The quantity T is the tension of the enclosing capsule—the surrounding membrane. If T 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 T. Not only asymmetry of T, but also asymmetry of p{n}_, 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 p{n}_ + 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{n}_ is everywhere equal. The simplest case is where p{n}_ = 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{n}_ = 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.

In the case of the conical egg with spherical ends, as is more or less the case in the plover’s and the guillemot’s, then at either end of the egg r and r′ are identical, and they are greater at the blunt anterior end than at the other. If we may assume that p{n}_ vanishes at the poles of the egg, then it is plain that T varies in the neighbourhood of these poles, and, further, that the tension T is greatest at and near the small end of the egg. It is here, in short, that the egg is most likely to be irregularly distorted or {660} even to burst, and it is here that we most commonly find irregularities of shape in abnormal eggs.

If one portion of the envelope were to become practically stiff before p ceases to vary, that would be tantamount to a sudden variation of T, and would introduce asymmetry by the imposition of a boundary condition in addition to the above equation.

Within the egg lies the yolk, and the yolk is invariably spherical or very nearly so, whatever be the form of the entire egg. The reason is simple, and lies in the fact that the yolk is itself enclosed in another membrane, between which and the outer membrane lies a fluid the presence of which makes p{n}_ for the inner membrane practically constant. The smallness of friction is indicated by the well-known fact that the “germinal spot” on the surface of the yolk is always found uppermost, however we may place and wherever we may open the egg; that is to say, the yolk easily rotates within the egg, bringing its lighter pole uppermost. So, owing to this lack of friction in the outer fluid, or white, whatever shear is produced within the egg will not be easily transmitted to the yolk, and, moreover, owing to the same fluidity, the yolk will easily recover its normal sphericity after the egg-shell is formed and the unequal pressure relieved.

These, then, are the general principles involved in, and illustrated by, the configuration of an egg; and they take us as far as we can safely go without actual quantitative determinations, in each particular case, of the forces concerned.

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In certain cases among the invertebrates, we again find instances of hard-shelled eggs which have obviously been moulded by the oviduct, or so-called “ootype,” in which they have lain: and not merely in such a way as to shew the effects of peristaltic pressure upon a uniform elastic envelope, but so as to impress upon the egg the more or less irregular form of the cavity, within which it had been for a time contained and compressed. After this fashion Dr Looss[599] of Cairo has {661} explained the curious form of the egg in Bilharzia (Schistosoma) haematobium, a formidable parasitic worm to which is due a disease wide-spread in Africa and Arabia, and an especial scourge of the Mecca pilgrims. The egg in this worm is provided at one end with a little spine, which now and then is found to be placed not terminally but laterally or ventrally, and which when so placed has been looked upon as the mark of a supposed new species, S. Mansoni. As Looss has now shewn, the little spine must be explained as having been moulded within a little funnel-shaped expansion of the uterus, just where it communicates with the common duct leading from the ovary and yolk-gland; by the accumulation of eggs in the ootype, the one last formed is crowded into a sideways position, and then, where the side-wall of the egg bulges in the funnel-shaped orifice of the duct, a little lateral “spine” is formed. In another species, S. japonicum, the egg is described as bulging into a so-called “calotte,” or bubble-like convexity at the end opposite to the spine. This, I think, may, with very little doubt, be ascribed to hardening of the egg-shell having taken place just at the period when partial relief from pressure was being experienced by the egg in the neighbourhood of the dilated orifice of the oviduct.

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

On the Form of Sea-urchins

As a corollary to the problem of the bird’s egg, we may consider for a moment the forms assumed by the shells of the sea-urchins. These latter are commonly divided into two classes, the Regular and the Irregular Echinids. The regular sea-urchins, save in {662} slight details which do not affect our problem, have a complete radial symmetry. The axis of the animal’s body is vertical, with mouth below and the intestinal outlet above; and around this axis the shell is built as a symmetrical system. It follows that in horizontal section the shell is everywhere circular, and we shall have only to consider its form as seen in vertical section or projection. The irregular urchins (very inaccurately so-called) have the anal extremity of the body removed from its central, dorsal situation; and it follows that they have now a single plane of symmetry, about which the organism, shell and all, is bilaterally symmetrical. We need not concern ourselves in detail with the shapes of their shells, which may be very simply interpreted, by the help of radial co-ordinates, as deformations of the circular or “regular” type.

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

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

While the sea-urchin is alive, an immense number of delicate “tube-feet,” with suckers at their tips, pass through minute pores {663} in the shell, and, like so many long cables, moor the animal to the ground. They constitute a symmetrical system of forces, with one resultant downwards, in the direction of gravity, and another outwards in a radial direction; and if we look upon the shell as originally spherical, both will tend to depress the sphere into a flattened cake. We need not consider the radial component, but may treat the case as that of a spherical shell symmetrically depressed under the influence of gravity. This is precisely the condition which we have to deal with in a drop of liquid lying on a plate; the form of which is determined by its own uniform surface-tension, plus gravity, acting against the uniform internal hydrostatic pressure. Simple as this system is, the full mathematical investigation of the form of a drop is not easy, and we can scarcely hope that the systematic study of the Echinodermata will ever be conducted by methods based on Laplace’s differential equation[600]; but we have no difficulty in seeing that the various forms represented in a series of sea-urchin shells are no other than those which we may easily and perfectly imitate in drops.

In the case of the drop of water (or of any other particular liquid) the specific surface-tension is always constant, and the pressure varies inversely as the radius of curvature; therefore the smaller the drop the more nearly is it able to conserve the spherical form, and the larger the drop the more does it become flattened under gravity. We can represent the phenomenon by using india-rubber balls filled with water, of different sizes; the little ones will remain very nearly spherical, but the larger will fall down “of their own weight,” into the form of more and more flattened cakes; and we see the same thing when we let drops of heavy oil (such as the orthotoluidene spoken of on p. 219), fall through a tall column of water, the little ones remaining round, and the big ones getting more and more flattened as they sink. In the case of the sea-urchin, the same series of forms may be assumed to occur, irrespective of size, through variations in T, the specific tension, or “strength,” of the enveloping shell. Accordingly we may study, entirely from this point of view, such a series as the following (Fig. 328). In a very few cases, such as the fossil Palaeechinus, we have an approximately spherical {664} shell, that is to say a shell so strong that the influence of gravity becomes negligible as a cause of deformation. The ordinary species of Echinus begin to display a pronounced depression, and this reaches its maximum in such soft-shelled flexible forms as Phormosoma. On the general question I took the opportunity of consulting Mr C. R. Darling, who is an acknowledged expert in drops, and he at once agreed with me that such forms as are represented in Fig. 328 are no other than diagrammatic illustrations of various kinds of drops, “most of which can easily be reproduced in outline by the aid of liquids of approximately equal density to water, although some of them are fugitive.” He found a difficulty in the case of the outline which represents Asthenosoma, but the reason for the anomaly is obvious; the flexible shell has flattened down until it has come in contact with the hard skeleton of the jaws, or “Aristotle’s lantern,” within, and the curvature of the outline is accordingly disturbed. The elevated, conical shells such as those of Urechinus and Coelopleurus evidently call for some further explanation; for there is here some cause at work {665} to elevate, rather than to depress the shell. Mr Darling tells me that these forms “are nearly identical in shape with globules I have frequently obtained, in which, on standing, bubbles of gas rose to the summit and pressed the skin upwards, without being able to escape.” The same condition may be at work in the sea-urchin; but a similar tendency would also be manifested by the presence in the upper part of the shell of any accumulation of substance lighter than water, such as is actually present in the masses of fatty, oily eggs.

On the Form and Branching of Blood-vessels

Passing to what may seem a very different subject, we may investigate a number of interesting points in connection with the form and structure of the blood-vessels, on the same principle and by help of the same equations as those we have used, for instance, in studying the egg-shell.

We know that the fluid pressure (P) within the vessel is balanced by (1) the tension (T) of the wall, divided by the radius of curvature, and (2) the external pressure (p{n}_), normal to the wall: according to our formula

P = p{n}_ + T(1/r + 1/r′).

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

Therefore, so long as the pressure is constant, the thickness of the wall should vary as the radius, or as the diameter, of the blood-vessel. But it is not the case that the pressure is constant, for it gradually falls off, by loss through friction, as we pass from the large arteries to the small; and accordingly we find that while, for a time, the cross-sections of the larger and smaller vessels are symmetrical figures, with the wall-thickness proportional to the size of the tube, this proportion is gradually lost, and the walls {666} of the small arteries, and still more of the capillaries, become exceedingly thin, and more so than in strict proportion to the narrowing of the tube.

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In the case of the heart we have, within each of its cavities, a pressure which, at any given moment, is constant over the whole wall-area, but the thickness of the wall varies very considerably. For instance, in the left ventricle, the apex is by much the thinnest portion, as it is also that with the greatest curvature. We may assume, therefore (or at least suspect), that the formula, t(1/r + 1/r′) = C, holds good; that is to say, that the thickness (t) of the wall varies inversely as the mean curvature. This may be tested experimentally, by dilating a heart with alcohol under a known pressure, and then measuring the thickness of the walls in various parts after the whole organ has become hardened. By this means it is found that, for each of the cavities, the law holds good with great accuracy[601]. Moreover, if we begin by dilating the right ventricle and then dilate the left in like manner, until the whole heart is equally and symmetrically dilated, we find (1) that we have had to use a pressure in the left ventricle from six to seven times as great as in the right ventricle, and (2) that the thickness of the walls is just in the same proportion[602].

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A great many other problems of a mechanical or hydrodynamical kind arise in connection with the blood-vessels[603], and while these are chiefly interesting to the physiologist they have also their interest for the morphologist in so far as they bear upon structure and form. As an example of such mechanical problems {667} we may take the conditions which determine or help to determine the manner of branching of an artery, or the angle at which its branches are given off; for, as John Hunter said[604], “To keep up a circulation sufficient for the part, and no more, Nature has varied the angle of the origin of the arteries accordingly.” The general principle is that the form and arrangement of the blood-vessels is such that the circulation proceeds with a minimum of effort, and with a minimum of wall-surface, the latter condition leading to a minimum of friction and being therefore included in the first. What, then, should be the angle of branching, such that there shall be the least possible loss of energy in the course of the circulation? In order to solve this problem in any particular case we should obviously require to know (1) how the loss of energy depends upon the distance travelled, and (2) how the loss of energy varies with the diameter of the vessel. The loss of energy is evidently greater in a narrow tube than in a wide one, and greater, obviously, in a long journey than a short. If the large artery, AB, give off a comparatively narrow branch leading to P (such as CP, or DP), the route ACP is evidently shorter than ADP, but on the other hand, by the latter path, the blood has tarried longer in the wide vessel AB, and has had a shorter course in the narrow branch. The relative advantage of the two paths will depend on the loss of energy in the portion CD, as compared with that in the alternative portion CD′, the latter being short and narrow, the former long and wide. If we ask, then, which factor is the more important, length or width, we may safely take it that the question is one of degree: and that the factor of width will become much the more important wherever the artery and its branch are markedly unequal in size. In other words, it would seem that for small branches a large angle of bifurcation, and for large branches a small one, is always the better. Roux has laid down certain rules in regard to the branching of arteries, which correspond with the general {668} conclusions which we have just arrived at. The most important of these are as follows: (1) If an artery bifurcate into two equal branches, these branches come off at equal angles to the main stem. (2) If one of the two branches be smaller than the other, then the main branch, or continuation of the original artery, makes with the latter a smaller angle than does the smaller or “lateral” branch. And (3) all branches which are so small that they scarcely seem to weaken or diminish the main stem come off from it at a large angle, from about 70° to 90°.

We may follow Hess in a further investigation of this phenomenon. Let AB be an artery, from which a branch has to be given off so as to reach P, and let ACP, ADP, etc., be alternative courses which the branch may follow: CD, DE, etc., in the diagram, being equal distances (= l) along AB. Let us call the angles PCD, PCE, _x__{1}, _x__{2}, etc.: and the distances CD′, DE′, by which each branch exceeds the next in length, we shall call _l__{1}, _l__{2}, etc. Now it is evident that, of the courses shewn, ACP is the shortest which the blood can take, but it is also that by which its transit through the narrow branch is the longest. We may reduce its transit through the narrow branch more and more, till we come to CGP, or rather to a point where the branch comes off at right angles to the main stem; but in so doing we very considerably increase the whole distance travelled. We may take it that there will be some intermediate point which will strike the balance of advantage.

Now it is easy to shew that if, in Fig. 330, the route ADP and AEP (two contiguous routes) be equally favourable, then any other route on either side of these, such as ACP or AFP, must be less favourable than either. Let ADP and AEP, then, be equally favourable; that is to say, let the loss of energy which the blood suffers in its passage along these two routes be equal. {669} Then, if we make the distance DE very small, the angles _x__{2} and _x__{3} are nearly equal, and may be so treated. And again, if DE be very small, then DE′E becomes a right angle, and _l__{2} (or DE′) = l cos _x__{2}.

But if L be the loss of energy per unit distance in the wide tube AB, and L′ be the corresponding loss of energy in the narrow tube DP, etc., then lL = _l__{2} L′, because, as we have assumed, the loss of energy on the route DP is equal to that on the whole route DEP. Therefore lL = lL′ cos _x__{2}, and cos _x__{2} = L/L′. That is to say, the most favourable angle of branching will be such that the cosine of the angle is equal to the ratio of the loss of energy which the blood undergoes, per unit of length, in the main vessel, as compared with that which it undergoes in the branch.

While these statements are so far true, and while they undoubtedly cover a great number of observed facts, yet it is plain that, as in all such cases, we must regard them not as a complete explanation, but as factors in a complicated phenomenon: not forgetting that (as the most learned of all students of the heart and arteries, Dr Thomas Young, said in his Croonian lecture[605]) all such questions as these, and all matters connected with the muscular and elastic powers of the blood-vessels, “belong to the most refined departments of hydraulics.” Some other explanation must be sought in order to account for a phenomenon which particularly impressed John Hunter’s mind, namely the gradually altering angle at which the successive intercostal arteries are given off from the thoracic aorta: the special interest of this case arising from the regularity and symmetry of the series, for “there is not another set of arteries in the body whose origins are so much the same, whose offices are so much the same, whose distances from their origin to the place of use, and whose uses [? sizes][606] are so much the same.”

{670}

XVI. ON FORM AND MECHANICAL EFFICIENCY

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

When we discuss certain so-called “adaptations” to outward circumstance, in the way of form, colour and so forth, we are often apt to use illustrations convincing enough to certain minds but unsatisfying to others—in other words, incapable of demonstration. With regard to colouration, for instance, it is by colours “cryptic,” “warning,” “signalling,” “mimetic,” and so on[607], that we prosaically expound, and slavishly profess to justify, the vast Aristotelian synthesis that Nature makes all things with a purpose and “does nothing in vain.” Only for a moment let us glance at some few instances by which the modern teleologist accounts for this or that manifestation of colour, and is led on and on to beliefs and doctrines to which it becomes more and more difficult to subscribe. {671}

Some dangerous and malignant animals are said (in sober earnest) to wear a perpetual war-paint, in order to “remind their enemies that they had better leave them alone[608].” The wasp and the hornet, in gallant black and gold, are terrible as an army with banners; and the Gila Monster (the poison-lizard of the Arizona desert) is splashed with scarlet—its dread and black complexion stained with heraldry more dismal. But the wasp-like livery of the noisy, idle hover-flies and drone-flies is but stage armour, and in their tinsel suits the little counterfeit cowardly knaves mimic the fighting crew.

The jewelled splendour of the peacock and the humming-bird, and the less effulgent glory of the lyre-bird and the Argus pheasant, are ascribed to the unquestioned prevalence of vanity in the one sex and wantonness in the other[609].

The zebra is striped that it may graze unnoticed on the plain, the tiger that it may lurk undiscovered in the jungle; the banded Chaetodont and Pomacentrid fishes are further bedizened to the hues of the coral-reefs in which they dwell[610]. The tawny lion is yellow as the desert sand; but the leopard wears its dappled hide to blend, as it crouches on the branch, with the sun-flecks peeping through the leaves.

The ptarmigan and the snowy owl, the arctic fox and the polar bear, are white among the snows; but go he north or go he south, the raven (like the jackdaw) is boldly and impudently black.

The rabbit has his white scut, and sundry antelopes their piebald flanks, that one timorous fugitive may hie after another, spying the warning signal. The primeval terrier or collie-dog {672} had brown spots over his eyes that he might seem awake when he was sleeping[611]: so that an enemy might let the sleeping dog lie, for the singular reason that he imagined him to be awake. And a flock of flamingos, wearing on rosy breast and crimson wings a garment of invisibility, fades away into the sky at dawn or sunset like a cloud incarnadine[612].

To buttress the theory of natural selection the same instances of “adaptation” (and many more) are used, which in an earlier but not distant age testified to the wisdom of the Creator and revealed to simple piety the high purpose of God. In the words of a certain learned theologian[613], “The free use of final causes to explain what seems obscure was temptingly easy .... Hence the finalist was often the man who made a liberal use of the ignava ratio, or lazy argument: when you failed to explain a thing by the ordinary process of causality, you could “explain” it by reference to some purpose of nature or of its Creator. This method lent itself with dangerous facility to the well-meant endeavours of the older theologians to expound and emphasise the beneficence of the divine purpose.” Mutatis mutandis, the passage carries its plain message to the naturalist.

The fate of such arguments or illustrations is always the same. They attract and captivate for awhile; they go to the building of a creed, which contemporary orthodoxy defends under its severest penalties: but the time comes when they lose their fascination, they somehow cease to satisfy and to convince, their foundations are discovered to be insecure, and in the end no man troubles to controvert them.

But of a very different order from all such “adaptations” as these, are those very perfect adaptations of form which, for instance, fit a fish for swimming or a bird for flight. Here we are {673} far above the region of mere hypothesis, for we have to deal with questions of mechanical efficiency where statical and dynamical considerations can be applied and established in detail. The naval architect learns a great part of his lesson from the investigation of the stream-lines of a fish; and the mathematical study of the stream-lines of a bird, and of the principles underlying the areas and curvatures of its wings and tail, has helped to lay the very foundations of the modern science of aeronautics. When, after attempting to comprehend the exquisite adaptation of the swallow or the albatross to the navigation of the air, we try to pass beyond the empirical study and contemplation of such perfection of mechanical fitness, and to ask how such fitness came to be, then indeed we may be excused if we stand wrapt in wonderment, and if our minds be occupied and even satisfied with the conception of a final cause. And yet all the while, with no loss of wonderment nor lack of reverence, do we find ourselves constrained to believe that somehow or other, in dynamical principles and natural law, there lie hidden the steps and stages of physical causation by which the material structure was so shapen to its ends[614].

But the problems associated with these phenomena are difficult at every stage, even long before we approach to the unsolved secrets of causation; and for my part I readily confess that I lack the requisite knowledge for even an elementary discussion of the form of a fish or of a bird. But in the form of a bone we have a problem of the same kind and order, so far simplified and particularised that we may to some extent deal with it, and may possibly even find, in our partial comprehension of it, a partial clue to the principles of causation underlying this whole class of problems.

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Before we speak of the form of a bone, let us say a word about, the mechanical properties of the material of which it is built[615], in {674} relation to the strength it has to manifest or the forces it has to resist: understanding always that we mean thereby the properties of fresh or living bone, with all its organic as well as inorganic constituents, for dead, dry bone is a very different thing. In all the structures raised by the engineer, in beams, pillars and girders of every kind, provision has to be made, somehow or other, for strength of two kinds, strength to resist compression or crushing, and strength to resist tension or pulling asunder. The evenly loaded column is designed with a view to supporting a downward pressure, the wire-rope, like the tendon of a muscle, is adapted only to resist a tensile stress; but in many or most cases the two functions are very closely inter-related and combined. The case of a loaded beam is a familiar one; though, by the way, we are now told that it is by no means so simple as it looks, and indeed that “the stresses and strains in this log of timber are so complex that the problem has not yet been solved in a manner that reasonably accords with the known strength of the beam as found by actual experiment[616].” However, be that as it may, we know, roughly, that when the beam is loaded in the middle and supported at both ends, it tends to be bent into an arc, in which condition its lower fibres are being stretched, or are undergoing a tensile stress, while its upper fibres are undergoing compression. It follows that in some intermediate layer there is a “neutral zone,” where the fibres of the wood are subject to no stress of either kind. In like manner, a vertical pillar if unevenly loaded (as, for instance, the shaft of our thigh-bone normally is) will tend to bend, and so to endure compression on its concave, and tensile stress upon its convex side. In many cases it is the business of the engineer to separate out, as far as possible, the pressure-lines from the tension-lines, in order to use separate modes of construction, or even different materials for each. In a {675} suspension-bridge, for instance, a great part of the fabric is subject to tensile strain only, and is built throughout of ropes or wires; but the massive piers at either end of the bridge carry the weight of the whole structure and of its load, and endure all the “compression-strains” which are inherent in the system. Very much the same is the case in that wonderful arrangement of struts and ties which constitute, or complete, the skeleton of an animal. The “skeleton,” as we see it in a Museum, is a poor and even a misleading picture of mechanical efficiency[617]. From the engineer’s point of view, it is a diagram showing all the compression-lines, but by no means all of the tension-lines of the construction; it shews all the struts, but few of the ties, and perhaps we might even say none of the principal ones; it falls all to pieces unless we clamp it together, as best we can, in a more or less clumsy and immobilised way. But in life, that fabric of struts is surrounded and interwoven with a complicated system of ties: ligament and membrane, muscle and tendon, run between bone and bone; and the beauty and strength of the mechanical construction lie not in one part or in another, but in the complete fabric which all the parts, soft and hard, rigid and flexible, tension-bearing and pressure-bearing, make up together[618].

However much we may find a tendency, whether in nature or art, to separate these two constituent factors of tension and compression, we cannot do so completely; and accordingly the engineer seeks for a material which shall, as nearly as possible, offer equal resistance to both kinds of strain. In the following table—I borrow it from Sir Donald MacAlister—we see approximately the relative breaking (or tearing) limit and crushing limit in a few substances. {676}

Average Strength of Materials (in kg. per sq. mm.).

Tensile Crushing strength strength Steel 100 145 Wrought Iron 40 20 Cast Iron 12 72 Wood 4 2 Bone 9–12 13–16

At first sight, bone seems weak indeed; but it has the great and unusual advantage that it is very nearly as good for a tie as for a strut, nearly as strong to withstand rupture, or tearing asunder, as to resist crushing. We see that wrought-iron is only half as strong to withstand the former as the latter; while in cast-iron there is a still greater discrepancy the other way, for it makes a good strut but a very bad tie indeed. Cast-steel is not only actually stronger than any of these, but it also possesses, like bone, the two kinds of strength in no very great relative disproportion.

When the engineer constructs an iron or steel girder, to take the place of the primitive wooden beam, we know that he takes advantage of the elementary principle we have spoken of, and saves weight and economises material by leaving out as far as possible all the middle portion, all the parts in the neighbourhood of the “neutral zone”; and in so doing he reduces his girder to an upper and lower “flange,” connected together by a “web,” the whole resembling, in cross-section, an I or an ⌶.

But it is obvious that, if the strains in the two flanges are to be equal as well as opposite, and if the material be such as cast-iron or wrought-iron, one or other flange must be made much thicker than the other in order that it may be equally strong; and if at times the two flanges have, as it were, to change places, or play each other’s parts, then there must be introduced a margin of safety by making both flanges thick enough to meet that kind of stress in regard to which the material happens to be weakest. There is great economy, then, in any material which is, as nearly as possible, equally strong in both ways; and so we see that, from the engineer’s or contractor’s point of view, bone is a very good and suitable material for purposes of construction. {677}

The I or the H-girder or rail is designed to resist bending in one particular direction, but if, as in a tall pillar, it be necessary to resist bending in all directions alike, it is obvious that the tubular or cylindrical construction best meets the case; for it is plain that this hollow tubular pillar is but the I-girder turned round every way, in a “solid of revolution,” so that on any two opposite sides compression and tension are equally met and resisted, and there is now no need for any substance at all in the way of web or “filling” within the hollow core of the tube. And it is not only in the supporting pillar that such a construction is useful; it is appropriate in every case where stiffness is required, where bending has to be resisted. The long bone of a bird’s wing has little or no weight to carry, but it has to withstand powerful bending moments; and in the arm-bone of a long-winged bird, such as an albatross, we see the tubular construction manifested in its perfection, the bony substance being reduced to a thin, perfectly cylindrical, and almost empty shell. The quill of the bird’s feather, the hollow shaft of a reed, the thin tube of the wheat-straw bearing its heavy burden in the ear, are all illustrations which Galileo used in his account of this mechanical principle[619].

Two points, both of considerable importance, present themselves here, and we may deal with them before we go further. In the first place, it is not difficult to see that, in our bending beam, the strain is greatest at its middle; if we press our walking-stick hard against the ground, it will tend to snap midway. Hence, if our cylindrical column be exposed to strong bending stresses, it will be prudent and economical to make its walls thickest in the middle and thinning off gradually towards the ends; and if we look at a longitudinal section of a thigh-bone, we shall see that this is just what nature has done. The thickness of the walls is nothing less than a diagram, or “graph,” of the “bending-moments” from one point to another along the length of the bone.

The second point requires a little more explanation. If we {678} imagine our loaded beam to be supported at one end only (for instance, by being built into a wall), so as to form what is called a “bracket” or “cantilever,” then we can see, without much difficulty, that the lines of stress in the beam run somewhat as in the accompanying diagram. Immediately under the load, the “compression-lines” tend to run vertically downward; but where the bracket is fastened to the wall, there is pressure directed horizontally against the wall in the lower part of the surface of attachment; and the vertical beginning and the horizontal end of these pressure-lines must be continued into one another in the form of some even mathematical curve—which, as it happens, is part of a parabola. The tension-lines are identical in form with the compression-lines, of which they constitute the “mirror-image”; and where the two systems intercross, they do so at right angles, or “orthogonally” to one another. Such systems of stress-lines as these we shall deal with again; but let us take note here of the important, though well-nigh obvious fact, that while in the beam they both unite to carry the load, yet it is always possible to weaken one set of lines at the expense of the other, and in some cases to do altogether away with one set or the other. For example, when we replace our end-supported beam by a curved bracket, bent upwards or downwards as the case may be, we have evidently cut away in the one case the greater part of the tension-lines, and in the other the greater part of the compression-lines. And if instead of bridging a stream with our beam of wood we bridge it with a rope, it is evident that this new construction contains all the tension-lines, but none of the compression-lines of the old. The biological interest connected with this principle lies chiefly in the mechanical construction of the rush or the straw, or any other typically cylindrical stem. The material of which the stalk is constructed is very weak to withstand compression, but parts of it have a very great tensile strength. Schwendener, who was both botanist and engineer, has elaborately investigated the factor of strength in the cylindrical stem, which Galileo was the first to call attention to. {679} Schwendener[620] shewed that the strength was concentrated in the little bundles of “bast-tissue” but that these bast-fibres had a tensile strength per square mm. of section, up to the limit of elasticity, not less than that of steel-wire of such quality as was in use in his day.

For instance, we see in the following table the load which various fibres, and various wires, were found capable of sustaining, not up to the breaking-point, but up to the “elastic limit,” or point beyond which complete recovery to the original length took place no longer after release of the load.

Stress, or load in gms. Strain, or amount per sq. mm., at of stretching, Limit of Elasticity per mille Secale cereale 15–20 4·4 Lilium auratum 19 7·6 Phormium tenax 20 13·0 Papyrus antiquorum 20 15·2 Molinia coerulea 22 11·0 Pincenectia recurvata 25 14·5 Copper wire 12·1 1·0 Brass wire 13·3 1·35 Iron wire 21·9 1·0 Steel wire 24·6 1·2

In other respects, it is true, the plant-fibres were inferior to the wires; for the former broke asunder very soon after the limit of elasticity was passed, while the iron-wire could stand, before snapping, three times the load which was measured by its limit of elasticity: in the language of a modern engineer, the bast-fibres had a low “yield-point,” little above the elastic limit. But nevertheless, within certain limits, plant-fibre and wire were just as good and strong one as the other. And then Schwendener proceeds to shew, in many beautiful diagrams, the various ways in which these strands of strong tensile tissue are arranged in various cases: sometimes, in the simpler cases, forming numerous small bundles arranged in a peripheral ring, not quite at the periphery, for a certain amount of space has to be left for living and active tissue; sometimes in a sparser ring of larger and {680} stronger bundles; sometimes with these bundles further strengthened by radial balks or ridges; sometimes with all the fibres set close together in a continuous hollow cylinder. In the case figured in Fig. 333 Schwendener calculated that the resistance to bending was at least twenty-five times as great as it would have been had the six main bundles been brought close together in a solid core. In many cases the centre of the stem is altogether empty; in all other cases it is filled with soft tissue, suitable for the ascent of sap or other functions, but never such as to confer mechanical rigidity. In a tall conical stem, such as that of a palm-tree, we can see not only these principles in the construction of the cylindrical trunk, but we can observe, towards the apex, the bundles of fibre curving over and intercrossing orthogonally with one another, exactly after the fashion of our stress-lines in Fig. 332; but of course, in this case, we are still dealing with tensile members, the opposite bundles taking on in turn, as the tree sways, the alternate function of resisting tensile strain[621].

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Let us now come, at last, to the mechanical structure of bone, of which we find a well-known and classical illustration in the various bones of the human leg. In the case of the tibia, the bone is somewhat widened out above, and its hollow shaft is capped by an almost flattened roof, on which the weight of the body directly rest. It is obvious that, under these circumstances, the engineer would find it necessary to devise means for supporting this flat roof, and for distributing the vertical pressures which impinge upon it to the cylindrical walls of the shaft. {681}

In the case of the bird’s wing-bone, the hollow of the bone is practically empty, as we have already said, being filled only with air save for a thin layer of living tissue immediately within the cylinder of bone; but in our own bones, and all weight-carrying bones in general, the hollow space is filled with marrow, blood-vessels and other tissues; and among these living tissues lies a fine lattice-work of little interlaced “trabeculae” of bone, forming the so-called “cancellous tissue.” The older anatomists were content to describe this cancellous tissue as a sort of “spongy network,” or irregular honeycomb, until, some fifty years ago, a remarkable discovery was made regarding it. It was found by Hermann Meyer (and afterwards shewn in greater detail by Julius Wolff and others) that the trabeculae, as seen in a longitudinal section of a long bone, were arranged in a very definite and orderly way; in the femur, they spread in beautiful curving {682} lines from the head to the tubular shaft of the bone, and these bundles of lines were crossed by others, with so nice a regularity of arrangement that each intercrossing was as nearly as possible an orthogonal one: that is to say, the one set of fibres crossed the other everywhere at right angles. A great engineer, Professor Culmann of Zürich (to whom, by the way, we owe the whole modern method of “graphic statics”), happened to see some of Meyer’s drawings and preparations, and he recognised in a moment that in the arrangement of the trabeculae we had nothing more nor less than a diagram of the lines of stress, or directions of compression and tension, in the loaded structure: in short, that nature was strengthening the bone in precisely the manner and direction in which strength was needed. In the accompanying diagram of a crane-head, by Culmann, we recognise a slight modification (caused entirely by the curved shape of the structure) of the still simpler lines of tension and compression which we have already seen in our end-supported beam as represented in Fig. 332. In the shaft of the crane, the concave {683} or inner side, overhung by the loaded head, is the “compression-member”; the outer side is the “tension-member”; and the pressure-lines, starting from the loaded surface, gather themselves together, always in the direction of the resultant pressure, till they form a close bundle running down the compressed side of the shaft: while the tension-lines, running upwards along the opposite side of the shaft, spread out through the head, orthogonally to, and linking together, the system of compression-lines. The head of the femur (Fig. 335) is a little more complicated in form and a little less symmetrical than Culmann’s diagrammatic crane, from which it chiefly differs in the fact that the load is divided into two parts, that namely which is borne by the head of the bone, and that smaller portion which rests upon the great trochanter; but this merely amounts to saying that a notch has been cut out of the curved upper surface of the structure, and we have no difficulty in seeing that the anatomical arrangement of the trabeculae follows precisely the mechanical distribution of compressive and tensile stress or, in other words, accords perfectly with the theoretical stress-diagram of the crane. The lines of stress are bundled close together along the sides of the shaft, and lost or concealed there in the substance of the solid wall of bone; but in and near the head of the bone, a peripheral shell of bone does not suffice to contain them, and they spread out through the central mass in the actual concrete form of bony trabeculae[622]. {684}

Mutatis mutandis, the same phenomenon may be traced in any other bone which carries weight and is liable to flexure; and in the os calcis and the tibia, and more or less in all the bones of the lower limb, the arrangement is found to be very simple and clear.

Thus, in the os calcis, the weight resting on the head of the bone has to be transmitted partly through the backward-projecting heel to the ground, and partly forwards through its articulation with the cuboid bone, to the arch of the foot. We thus have, very much as in a triangular roof-tree, two compression-members, sloping apart from one another; and these have to be bound together by a “tie” or tension-member, corresponding to the third, horizontal member of the truss.

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So far, dealing wholly with the stresses and strains due to tension and compression, we have altogether omitted to speak of a third very important factor in the engineer’s calculations, namely what is known as “shearing stress.” A shearing force is one which produces “angular distortion” in a figure, or (what comes to the same thing) which tends to cause its particles to {685} slide over one another. A shearing stress is a somewhat complicated thing, and we must try to illustrate it (however imperfectly) in the simplest possible way. If we build up a pillar, for instance, of a pile of flat horizontal slates, or of a pack of cards, a vertical load placed upon it will produce compression, but will have no tendency to cause one card to slide, or shear, upon another; and in like manner, if we make up a cable of parallel wires and, letting it hang vertically, load it evenly with a weight, again the tensile stress produced has no tendency to cause one wire to slip or shear upon another. But the case would have been very different if we had built up our pillar of cards or slates lying obliquely to the lines of pressure, for then at once there would have been a tendency for the elements of the pile to slip and slide asunder, and to produce what the geologists call “a fault” in the structure.

Somewhat more generally, if AB be a bar, or pillar, of cross-section a under a direct load P, giving a stress per unit area = p, then the whole pressure P = pa. Let CD be an oblique section, inclined at an angle θ to the cross-section; the pressure on CD will evidently be = pa cos θ. But at any point O in CD, the pressure P may be resolved into the force Q acting along CD, and N perpendicular to it: where N = P cos θ, and Q = P sin θ = pa sin θ. The whole force Q upon CD = q ⋅ area of CD, which is = q ⋅ a/(cos θ). {686} Therefore qa/(cos θ) = pa sin θ, therefore q = p sin θ cos θ, = ½_p_ sin 2θ. Therefore when sin 2θ = 1, that is, when θ = 45°, q is a maximum, and = p/2; and when sin 2θ = 0, that is when θ = 0° or 90°, then q vanishes altogether.

This is as much as to say, that a shearing stress vanishes altogether along the lines of maximum compression or tension; it has a definite value in all other positions, and a maximum value when it is inclined at 45° to either, or half-way between the two. This may be further illustrated in various simple ways. When we submit a cubical block of iron to compression in the testing machine, it does not tend to give way by crumbling all to pieces; but as a rule it disrupts by shearing, and along some plane approximately at 45° to the axis of compression. Again, in the beam which we have already considered under a bending moment, we know that if we substitute for it a pack of cards, they will be strongly sheared on one another; and the shearing stress is greatest in the “neutral zone,” where neither tension nor compression is manifested: that is to say in the line which cuts at equal angles of 45° the orthogonally intersecting lines of pressure and tension.

In short we see that, while shearing stresses can by no means be got rid of, the danger of rupture or breaking-down under shearing stress is completely got rid of when we arrange the materials of our construction wholly along the pressure-lines and tension-lines of the system; for along these lines there is no shear.

To apply these principles to the growth and development of our bone, we have only to imagine a little trabecula (or group of trabeculae) being secreted and laid down fortuitously in any direction within the substance of the bone. If it lie in the direction of one of the pressure-lines, for instance, it will be in a position of comparative equilibrium, or minimal disturbance; but if it be inclined obliquely to the pressure-lines, the shearing force will at once tend to act upon it and move it away. This is neither more nor less than what happens when we comb our {687} hair, or card a lock of wool: filaments lying in the direction of the comb’s path remain where they were; but the others, under the influence of an oblique component of pressure, are sheared out of their places till they too come into coincidence with the lines of force. So straws show how the wind blows—or rather how it has been blowing. For every straw that lies askew to the wind’s path tends to be sheared into it; but as soon as it has come to lie the way of the wind it tends to be disturbed no more, save (of course) by a violence such as to hurl it bodily away.

In the biological aspect of the case, we must always remember that our bone is not only a living, but a highly plastic structure; the little trabeculae are constantly being formed and deformed, demolished and formed anew. Here, for once, it is safe to say that “heredity” need not and cannot be invoked to account for the configuration and arrangement of the trabeculae: for we can see them, at any time of life, in the making, under the direct action and control of the forces to which the system is exposed. If a bone be broken and so repaired that its parts lie somewhat out of their former place, so that the pressure-and tension-lines have now a new distribution, before many weeks are over the trabecular system will be found to have been entirely remodelled, so as to fall into line with the new system of forces. And as Wolff pointed out, this process of reconstruction extends a long way off from the seat of injury, and so cannot be looked upon as a mere accident of the physiological process of healing and repair; for instance, it may happen that, after a fracture of the shaft of a long bone, the trabecular meshwork is wholly altered and reconstructed within the distant extremities of the bone. Moreover, in cases of transplantation of bone, for example when a diseased metacarpal is repaired by means of a portion taken from the lower end of the ulna, with astonishing quickness the plastic capabilities of the bony tissue are so manifested that neither in outward form nor inward structure can the old portion be distinguished from the new.

Herein then lies, so far as we can discern it, a great part at least of the physical causation of what at first sight strikes us as a purely functional adaptation: as a phenomenon, in other words, {688} whose physical cause is as obscure as its final cause or end is, apparently, manifest.

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Partly associated with the same phenomenon, and partly to be looked upon (meanwhile at least) as a fact apart, is the very important physiological truth that a condition of strain, the result of a stress, is a direct stimulus to growth itself. This indeed is no less than one of the cardinal facts of theoretical biology. The soles of our boots wear thin, but the soles of our feet grow thick, the more we walk upon them: for it would seem that the living cells are “stimulated” by pressure, or by what we call “exercise,” to increase and multiply. The surgeon knows, when he bandages a broken limb, that his bandage is doing something more than merely keeping the parts together: and that the even, constant pressure which he skilfully applies is a direct encouragement of growth and an active agent in the process of repair. In the classical experiments of Sédillot[623], the greater part of the shaft of the tibia was excised in some young puppies, leaving the whole weight of the body to rest upon the fibula. The latter bone is normally about one-fifth or sixth of the diameter of the tibia; but under the new conditions, and under the “stimulus” of the increased load, it grew till it was as thick or even thicker than the normal bulk of the larger bone. Among plant tissues this phenomenon is very apparent, and in a somewhat remarkable way; for a strain caused by a constant or increasing weight (such as that in the stalk of a pear while the pear is growing and ripening) produces a very marked increase of strength without any necessary increase of bulk, but rather by some histological, or molecular, alteration of the tissues. Hegler, and also Pfeffer, have investigated this subject, by loading the young shoot of a plant nearly to its breaking point, and then redetermining the breaking-strength after a few days. Some young shoots of the sunflower were found to break with a strain of 160 gms.; but when loaded with 150 gms., and retested after two days, they were able to support 250 gms.; and being again loaded with something short of this, by next day they sustained 300 gms., and a few days later even 400 gms. {689}

Such experiments have been amply confirmed, but so far as I am aware, we do not know much more about the matter: we do not know, for instance, how far the change is accompanied by increase in number of the bast-fibres, through transformation of other tissues; or how far it is due to increase in size of these fibres; or whether it be not simply due to strengthening of the original fibres by some molecular change. But I should be much inclined to suspect that the latter had a good deal to do with the phenomenon. We know nowadays that a railway axle, or any other piece of steel, is weakened by a constant succession of frequently interrupted strains; it is said to be “fatigued,” and its strength is restored by a period of rest. The converse effect of continued strain in a uniform direction may be illustrated by a homely example. The confectioner takes a mass of boiled sugar or treacle (in a particular molecular condition determined by the temperature to which it has been exposed), and draws the soft sticky mass out into a rope; and then, folding it up lengthways, he repeats the process again and again. At first the rope is pulled out of the ductile mass without difficulty; but as the work goes on it gets harder to do, until all the man’s force is used to stretch the rope. Here we have the phenomenon of increasing strength, following mechanically on a rearrangement of molecules, as the original isotropic condition is transmuted more and more into molecular asymmetry or anisotropy; and the rope apparently “adapts itself” to the increased strain which it is called on to bear, all after a fashion which at least suggests a parallel to the increasing strength of the stretched and weighted fibre in the plant. For increase of strength by rearrangement of the particles we have already a rough illustration in our lock of wool or hank of tow. The piece of tow will carry but little weight while its fibres are tangled and awry: but as soon as we have carded it out, and brought all its long fibres parallel and side by side, we may at once make of it a strong and useful cord.

In some such ways as these, then, it would seem that we may co-ordinate, or hope to co-ordinate, the phenomenon of growth with certain of the beautiful structural phenomena which present themselves to our eyes as “provisions,” or mechanical adaptations, for the display of strength where strength is most required. {690} That is to say, the origin, or causation, of the phenomenon would seem to lie, partly in the tendency of growth to be accelerated under strain: and partly in the automatic effect of shearing strain, by which it tends to displace parts which grow obliquely to the direct lines of tension and of pressure, while leaving those in place which happen to lie parallel or perpendicular to those lines: an automatic effect which we can probably trace as working on all scales of magnitude, and as accounting therefore for the rearrangement of minute particles in the metal or the fibre, as well as for the bringing into line of the fibres themselves within the plant, or of the little trabeculae within the bone.

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But we may now attempt to pass from the study of the individual bone to the much wider and not less beautiful problems of mechanical construction which are presented to us by the skeleton as a whole. Certain problems of this class are by no means neglected by writers on anatomy, and many have been handed down from Borelli, and even from older writers. For instance, it is an old tradition of anatomical teaching to point out in the human body examples of the three orders of levers[624]; again, the principle that the limb-bones tend to be shortened in order to support the weight of a very heavy animal is well understood by comparative anatomists, in accordance with Euler’s law, that the weight which a column liable to flexure is capable of supporting varies inversely as the square of its length; and again, the statical equilibrium of the body, in relation for instance to the erect posture of man, has long been a favourite theme of the philosophical anatomist. But the general method, based upon that of graphic statics, to which we have been introduced in our study of a bone, has not, so far as I know, been applied to the general fabric of the skeleton. Yet it is plain that each bone plays {691} a part in relation to the whole body, analogous to that which a little trabecula, or a little group of trabeculae, plays within the bone itself: that is to say, in the normal distribution of forces in the body, the bones tend to follow the lines of stress, and especially the pressure-lines. To demonstrate this in a comprehensive way would doubtless be difficult; for we should be dealing with a framework of very great complexity, and should have to take account of a great variety of conditions[625]. This framework is complicated as we see it in the skeleton, where (as we have said) it is only, or chiefly, the struts of the whole fabric which are represented; but to understand the mechanical structure in detail, we should have to follow out the still more complex arrangement of the ties, as represented by the muscles and ligaments, and we should also require much detailed information as to the weights of the various parts and as to the other forces concerned. Without these latter data we can only treat the question in a preliminary and imperfect way. But, to take once again a small and simplified part of a big problem, let us think of a quadruped (for instance, a horse) in a standing posture, and see whether the methods and terminology of the engineer may not help us, as they did in regard to the minute structure of the single bone.

Standing four-square upon its forelegs and hindlegs, with the weight of the body suspended between, the quadruped at once suggests to us the analogy of a bridge, carried by its two piers. And if it occurs to us, as naturalists, that we never look at a standing quadruped without contemplating a bridge, so, conversely, a similar idea has occurred to the engineer; for Professor Fidler, in this Treatise on Bridge-Construction, deals with the chief descriptive part of his subject under the heading of “The Comparative Anatomy of Bridges.” The designation is most just, for in studying the various types of bridges we are studying a series of well-planned skeletons[626]; and (at the cost of a little pedantry) {692} we might go even further, and study (after the fashion of the anatomist) the “osteology” and “desmology” of the structure, that is to say the bones which are represented by “struts,” and the ligaments, etc., which are represented by “ties.” Furthermore after the methods of the comparative anatomist, we may classify the families, genera and species of bridges according to their distinctive mechanical features, which correspond to certain definite conditions and functions.

In more ways than one, the quadrupedal bridge is a remarkable one; and perhaps its most remarkable peculiarity is that it is a jointed and flexible bridge, remaining in equilibrium under considerable and sometimes great modifications of its curvature, such as we see, for instance, when a cat humps or flattens her back. The fact that flexibility is an essential feature in the quadrupedal bridge, while it is the last thing which an engineer desires and the first which he seeks to provide against, will impose certain important limiting conditions upon the design of the skeletal fabric; but to this matter we shall afterwards return. Let us begin by considering the quadruped at rest, when he stands upright and motionless upon his feet, and when his legs exercise no function save only to carry the weight of the whole body. So far as that function is concerned, we might now perhaps compare the horse’s legs with the tall and slender piers of some railway bridge; but it is obvious that these jointed legs are ill-adapted to receive the horizontal thrust of any arch that may be placed atop of them. Hence it follows that the curved backbone of the horse, which appears to cross like an arch the span between his shoulders and his flanks, cannot be regarded as an arch, in the {693} engineer’s sense of the word. It resembles an arch in form, but not in function, for it cannot act as an arch unless it be held back at each end (as every arch is held back) by abutments capable of resisting the horizontal thrust; and these necessary abutments are not present in the structure. But in various ways the engineer can modify his superstructure so as to supply the place of these external reactions, which in the simple arch are obviously indispensable. Thus, for example, we may begin by inserting a straight steel tie, AB (Fig. 339), uniting the ends of the curved rib AaB; and this tie will supply the place of the external reactions, converting the structure into a “tied arch,” such as we may see in the roofs of many railway-stations. Or we may go on to fill in the space between arch and tie by a “web-system,” converting it into what the engineer describes as a “parabolic bowstring girder” (Fig. 339_b_). In either case, the structure becomes an independent “detached girder,” supported at each end but not otherwise fixed, and consisting essentially of an upper compression-member, AaB, and a lower tension-member, AB. But again, in the skeleton of the quadruped, the necessary tie, AB, is not to be found; and it follows that these comparatively simple types of bridge do not correspond to, nor do they help us to understand, the type of bridge which nature has designed in the skeleton of the quadruped. Nevertheless if we try to look, as an engineer would look, at the actual design of the animal skeleton and the actual distribution of its load, we find that, the one is most admirably adapted to the other, according to the strict principles of engineering construction. The structure is not an arch, nor a tied arch, nor a bowstring girder: but it is strictly and beautifully {694} comparable to the main girder of a double-armed cantilever bridge.

Obviously, in our quadrupedal bridge, the superstructure does not terminate (as it did in our former diagram) at the two points of support, but it extends beyond them at each end, carrying the head at one end and the tail at the other, upon a pair of projecting arms or “cantilevers” (Fig. 346).

In a typical cantilever bridge, such as the Forth Bridge (Fig. 345), a certain simplification is introduced. For each pier carries, in this case, its own double-armed cantilever, linked by a short connecting girder to the next, but so jointed to it that no weight is transmitted from one cantilever to another. The bridge in short is cut into separate sections, practically independent of one another; at the joints a certain amount of bending is not precluded, but shearing strain is evaded; and each pier carries only its own load. By this arrangement the engineer finds that design and construction are alike simplified and facilitated. In the case of the horse, it is obvious that the two piers of the bridge, that is to say the fore-legs and the hind-legs, do not bear (as they do in the Forth Bridge) separate and independent loads, but the whole system forms a continuous structure. In this case, the calculation of the loads will be a little more difficult and the corresponding design of the structure a little more complicated. We shall accordingly simplify our problem very considerably if, to begin with, we look upon the quadrupedal skeleton as constituted of two separate systems, that is to say of two balanced cantilevers, one supported on the fore-legs and the other on the hind; and we may deal afterwards with the fact that these two cantilevers are not independent, but are bound up in one common field of force and plan of construction.

In the horse it is plain that the two cantilever systems into which we may thus analyse the quadrupedal bridge are unequal in magnitude and importance. The fore-part of the animal is much bulkier than its hind quarters, and the fact that the fore-legs carry, as they so evidently do, a greater weight than the hind-legs has long been known and is easily proved; we have only to walk a horse onto a weigh-bridge, weigh first his fore-legs and then his hind-legs, to discover that what we may call his front half weighs {695} a good deal more than what is carried on his hind feet, say about three-fifths of the whole weight of the animal.

The great (or anterior) cantilever then, in the horse, is constituted by the heavy head and still heavier neck on one side of the pier which is represented by the fore-legs, and by the dorsal vertebrae carrying a large part of the weight of the trunk upon the other side; and this weight is so balanced over the fore-legs that the cantilever, while “anchored” to the other parts of the structure, transmits but little of its weight to the hind-legs, and the amount so transmitted will vary with the position of the head and with the position of any artificial load[627]. Under certain conditions, as when the head is thrust well forward, it is evident that the hind-legs will be actually relieved of a portion of the comparatively small load which is their normal share.

Our problem now is to discover, in a rough and approximate way, some of the structural details which the balanced load upon the double cantilever will impress upon the fabric.

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Working by the methods of graphic statics, the engineer’s task is, in theory, one of great simplicity. He begins by drawing in outline the structure which he desires to erect; he calculates the stresses and bending-moments necessitated by the dimensions and load on the structure; he draws a new diagram representing these forces, and he designs and builds his fabric on the lines of this statical diagram. He does, in short, precisely what we have seen nature doing in the case of the bone. For if we had begun, as it were, by blocking out the femur roughly, and considering its position and dimensions, its means of support and the load which it has to bear, we could have proceeded at once to draw the system of stress-lines which must occupy the field of force: and to precisely these stress-lines has nature kept in the building of the bone, down to the minute arrangement of its trabeculae.

The essential function of a bridge is to stretch across a certain span, and carry a certain definite load; and this being so, the {696} chief problem in the designing of a bridge is to provide due resistance to the “bending-moments” which result from the load. These bending-moments will vary from point to point along the girder, and taking the simplest case of a uniform load supported at both ends, they will be represented by points on a parabola. If the girder be of uniform depth, that is to say if its two flanges, respectively under tension and compression, be parallel to one another, then the stress upon these flanges will vary as the bending-moments, and will accordingly be very severe in the middle and will dwindle towards the ends. But if we make the depth of the girder everywhere proportional to the bending-moments, that is to say if we copy in the girder the outlines of the bending-moment diagram, then our design will automatically meet the circumstances of the case, for the horizontal stress in each flange will now be uniform throughout the length of the girder. In short, in {697} Professor Fidler’s words, “Every diagram of moments represents the outline of a framed structure which will carry the given load with a uniform horizontal stress in the principal members.”

In the following diagrams (Fig. 342, a, b) (which are taken from the original ones of Culmann), we see at once that the loaded beam or bracket (a) has a “danger-point” close to its fixed base, that is to say at the point remotest from its load. But in the parabolic bracket (b) there is no danger-point at all, for the dimensions of the structure are made to increase pari passu with the bending-moments: stress and resistance vary together. Again in Fig. 340, we have a simple span (A), with its stress diagram (B); and in Fig. 341 we have the corresponding parabolic girder, whose stresses are now uniform throughout. In fact we see that, by a process of conversion, the stress diagram in each case becomes the structural diagram in the other[629]. Now all this is but the modern rendering of one of Galileo’s most famous propositions. In the Dialogue which we have already quoted more than once[630], Sagredo says “It would be a fine thing if one could discover the proper shape to give a solid in order to make it equally resistant at every point, in which case a load placed at the middle would not produce fracture more easily than if placed at any other point.” And Galileo (in the person of Salviati) first puts the problem into its more general form; and then shews us how, by giving a parabolic outline to our beam, we have its simple and comprehensive solution.

In the case of our cantilever bridge, we shew the primitive girder {698} in Fig. 343, A, with its bending-moment diagram (B); and it is evident that, if we turn this diagram upside down, it will still be illustrative, just as before, of the bending-moments from point to point: for as yet it is merely a diagram, or graph, of relative magnitudes.

To either of these two stress diagrams, direct or inverted, we may fit the design of the construction, as in Figs. 343, C and 344.

Now in different animals the amount and distribution of the load differs so greatly that we can expect no single diagram, drawn from the comparative anatomy of bridges, to apply equally well to all the cases met with in the comparative anatomy of quadrupeds; but nevertheless we have already gained an insight into the general principles of “structural design” in the quadrupedal bridge.

In our last diagram the upper member of the cantilever is under {699} tension; it is represented in the quadruped by the ligamentum nuchae on the one side of the cantilever, and by the supraspinous ligaments of the dorsal vertebrae on the other. The compression member is similarly represented, on both sides of the cantilever, by the vertebral column, or rather by the bodies of the vertebrae; while the web, or “filling,” of the girders, that is to say the upright or sloping members which extend from one flange to the other, is represented on the one hand by the spines of the vertebrae, and on the other hand, by the oblique interspinous ligaments and muscles. The high spines over the quadruped’s withers are no other than the high struts which rise over the supporting piers in the parabolic girder, and correspond to the position of the maximal bending-moments. The fact that these tall vertebrae of the withers usually slope backwards, sometimes steeply, in a quadruped, is easily and obviously explained[631]. For each vertebra tends to act as a “hinged lever,” and its spine, acted on by the tensions transmitted by the ligaments on either side, takes up its position as the diagonal of the parallelogram of forces to which it is exposed.

It happens that in these comparatively simple types of cantilever bridge the whole of the parabolic curvature is transferred to one or other of the principal members, either the tension-member or the compression-member as the case may be. But it is of course equally permissible to have both members curved, in opposite directions. This, though not exactly the case in the Forth Bridge, is approximately so; for here the main compression-member is curved or arched, and the main tension-member slopes downwards on either side from its maximal height above the piers. In short, the Forth Bridge is a nearer approach than either of the other cantilever bridges which we have {700} illustrated to the plan of the quadrupedal skeleton; for the main compression-member almost exactly recalls the form of the vertebral column, while the main tension-member, though not so closely similar to the supraspinous and nuchal ligaments, corresponds to the plan of these in a somewhat simplified form.

We may now pass without difficulty from the two-armed cantilever supported on a single pier, as it is in each separate section of the Forth Bridge, or as we have imagined it to be in the forequarters of a horse, to the condition which actually exists in that quadruped, where a two-armed cantilever has its load distributed over two separate piers. This is not precisely what an engineer calls a “continuous” girder, for that term is applied to a girder which, as a continuous structure, crosses two or more spans, while here there is only one. But nevertheless, this girder is effectively continuous from the head to the tip of the tail; and at each point of support (A and B) it is subjected to the negative bending-moment due to the overhanging load on each of the projecting cantilever arms AH and BT. The diagram of bending-moments will (according to the ordinary conventions) lie below {701} the base line (because the moments are negative), and must take some such form as that shown in the diagram: for the girder must suffer its greatest bending stress not at the centre, but at the two points of support A and B, where the moments are measured by the vertical ordinates. It is plain that this figure only differs from a representation of two independent two-armed cantilevers in the fact that there is no point midway in the span where the bending-moment vanishes, but only a region between the two piers in which its magnitude tends to diminish.

The diagram effects a graphic summation of the positive and negative moments, but its form may assume various modifications according to the method of graphic summation which we may choose to adopt; and it is obvious also that the form of the diagram may assume many modifications of detail according to the actual distribution of the load. In all cases the essential points to be observed are these: firstly that the girder which is to resist the bending-moments induced by the load must possess its two principal members—an upper tension-member or tie, represented by ligament, and a lower compression-member represented by bone: these members being united by a web represented by the vertebral spines with their interspinous ligaments, and being placed one above the other in the order named because the moments are negative; secondly we observe that the depth of the web, or distance apart of the principal members,—that is to say the height of the vertebral spines,—must be proportional to the bending-moment at each point along the length of the girder.

In the case of an animal carrying two-thirds of his weight upon his fore-legs and only one-third upon his hind-legs, the bending-moment diagram will be unsymmetrical, after the fashion of Fig. 347, the vertical ordinate at A being thrice the height of that at B. {702}

On the other hand the Dinosaur, with his light head and enormous tail would give us a moment-diagram with the opposite kind of asymmetry, the greatest bending stress being now found over the haunches, at B (Fig. 348). A glance at the skeleton of Diplodocus Carnegii will shew us the high vertebral spines over the loins, in precise correspondence with the requirements of this diagram: just as in the horse, under the opposite conditions of load, the highest vertebral spines are those of the withers, that is to say those of the posterior cervical and anterior dorsal vertebrae.

We have now not only dealt with the general resemblance, both in structure and in function, of the quadrupedal backbone with its associated ligaments to a double-armed cantilever girder, but we have begun to see how the characters of the vertebral system must differ in different quadrupeds, according to the conditions imposed by the varying distribution of the load: and in particular how the height of the vertebral spines which constitute the web will be in a definite relation, as regards magnitude and position, to the bending-moments induced thereby. We should require much detailed information as to the actual weights of the several parts of the body before we could follow out quantitatively the mechanical efficiency of each type of skeleton; but in an approximate way what we have already learnt will enable us to trace many interesting correspondences between structure and function in this particular part of comparative anatomy. We must, however, be careful to note that the great cantilever system is not of necessity constituted by the vertical column and its ligaments alone, but that the pelvis, firmly united as it is to the sacral vertebrae, and stretching backwards far beyond the acetabulum, becomes an intrinsic part of the system; and helping (as it does) to carry the load of the abdominal viscera, {703} constitutes a great portion of the posterior cantilever arm, or even its chief portion in cases where the size and weight of the tail are insignificant, as is the case in the majority of terrestrial mammals.

We may also note here, that just as a bridge is often a “combined” or composite structure, exhibiting a combination of principles in its construction, so in the quadruped we have, as it were, another girder supported by the same piers to carry the viscera; and consisting of an inverted parabolic girder, whose compression-member is again constituted by the backbone, its tension-member by the line of the sternum and the abdominal muscles, while the ribs and intercostal muscles play the part of the web or filling.

A very few instances must suffice to illustrate the chief variations in the load, and therefore in the bending-moment diagram, and therefore also in the plan of construction, of various quadrupeds. But let us begin by setting forth, in a few cases, the actual weights which are borne by the fore-limbs and the hind-limbs, in our quadrupedal bridge[632].

On On Fore- Hind- % on % on Gross weight. feet feet Fore- Hind- ton cwts. cwts. cwts. feet. feet. Camel (Bactrian) — 14·25 9·25 4·5 67·3 32·7 Llama — 2·75 1·75 ·875 66·7 33·3 Elephant (Indian) 1 15·75 20·5 14·75 58·2 41·8 Horse — 8·25 4·75 3·5 57·6 42·4 Horse (large Clydesdale) — 15·5 8·5 7·0 54·8 45·2

It will be observed that in all these animals the load upon the fore-feet preponderates considerably over that upon the hind, the preponderance being rather greater in the elephant than in the horse, and markedly greater in the camel and the llama than in the other two. But while these weights are helpful and suggestive, it is obvious that they do not go nearly far enough to give us a full insight into the constructional diagram to which the animals are conformed. For such a purpose we should {704} require to weigh the total load, not in two portions, but in many; and we should also have to take close account of the general form of the animal, of the relation between that form and the distribution of the load, and of the actual directions of each bone and ligament by which the forces of compression and tension were transmitted. All this lies beyond us for the present; but nevertheless we may consider, very briefly, the principal cases involved in our enquiry, of which the above animals form only a partial and preliminary illustration.

(1) Wherever we have a heavily loaded anterior cantilever arm, that is to say whenever the head and neck represent a considerable fraction of the whole weight of the body, we tend to have large bending-moments over the fore-legs, and correspondingly high spines over the vertebrae of the withers. This is the case in the great majority of four-footed, terrestrial animals, the chief exceptions being found in animals with comparatively small heads but large and heavy tails, such as the anteaters or the Dinosaurian reptiles, and also (very naturally) in animals such as the crocodile, where the “bridge” can scarcely be said to be developed, for the long heavy body sags down to rest upon the ground. The case is sufficiently exemplified by the horse, and still more notably by the stag, the ox, or the pig. It is illustrated in the accompanying diagram of the conditions in the great extinct Titanotherium.

(2) In the elephant and the camel we have similar conditions, but slightly modified. In both cases, and especially in the latter, the weight on the fore-quarters is relatively large; and in both cases the bending-moments are all the larger, by reason of the length and forward extension of the camel’s neck, and the forward {705} position of the heavy tusks of the elephant. In both cases the dorsal spines are large, but they do not strike us as exceptionally so; but in both cases, and especially in the elephant, they slope backwards in a marked degree. Each spine, as already explained, must in all cases assume the position of the diagonal in the parallelogram of forces defined by the tensions acting on it at its extremity; for it constitutes a “hinged lever,” by which the bending-moments on either side are automatically balanced; and it is plain that the more the spine slopes backwards the more it indicates a relatively large strain thrown upon the great ligament of the neck, and a relief of strain upon the more directly acting, but weaker, ligaments of the back and loins. In both cases, the bending-moments would seem to be more evenly distributed over the region of the back than, for instance, in the stag, with its light hind-quarters and heavy load of antlers: and in both cases the high “girder” is considerably prolonged, by an extension of the tall spines backwards in the direction of the loins. When we come to such a case as the mammoth, with its immensely heavy and immensely elongated tusks, we perceive at once that the bending-moments over the fore-legs are now very severe; and we see also that the dorsal spines in this region are much more conspicuously elevated than in the ordinary elephant.

(3) In the case of the giraffe we have, without doubt, a very heavy load upon the fore-legs, though no weighings are at hand to define the ratio; but as far as possible this disproportionate load would seem to be relieved, by help of a downward as well as backward thrust, through the sloping back, to the unusually low hind-quarters. The dorsal spines of the vertebrae are very high and strong, and the whole girder-system very perfectly formed. The elevated, rather than protruding position of the head lessens the anterior bending-moment as far as possible; but it leads to a strong compressional stress transmitted almost directly downwards through the neck: in correlation with which we observe that the bodies of the cervical vertebrae are exceptionally large and strong and steadily increase in size and strength from the head downwards.

(4) In the kangaroo, the fore-limbs are entirely relieved of their load, and accordingly the tall spines over the withers, which {706} were so conspicuous in all heavy-headed quadrupeds, have now completely vanished. The creature has become bipedal, and body and tail form the extremities of a single balanced cantilever, whose maximal bending-moments are marked by strong, high lumbar and sacral vertebrae, and by iliac bones of peculiar form and exceptional strength.

Precisely the same condition is illustrated in the Iguanodon, and better still by reason of the great bulk of the creature, and of the heavy load which falls to be supported by the great cantilever and by the hind-legs which form its piers. The long and heavy body and neck require a balance-weight (as in the kangaroo) in the form of a long heavy tail. And the double-armed cantilever, so constituted, shews a beautiful parabolic curvature in the graded heights of the whole series of vertebral spines, which rise to a maximum over the haunches and die away slowly towards the neck and the tip of the tail.

(5) In the case of some of the great American fossil reptiles, such as Diplodocus, it has always been a more or less disputed question whether or not they assumed, like Iguanodon, an erect, bipedal attitude. In all of these we see an elongated pelvis, and, in still more marked degree, we see elevated spinous processes of the vertebrae over the hind-limbs; in all of them we have a long heavy tail, and in most of them we have a marked reduction in size and weight both of the fore-limb and of the head itself. The great size of these animals is not of itself a proof against the erect attitude; because it might well have been accompanied by an aquatic or partially submerged habitat, and the crushing stress of the creature’s huge bulk proportionately relieved. But we must consider each such case in the whole light of its own evidence; and it is easy to see that, just as the quadrupedal mammal may carry the greater part but not all of its weight upon its fore-limbs, so a heavy-tailed reptile may carry the greater part upon its hind-limbs, without this process going so far as to relieve its fore-limbs of all weight whatsoever. This would seem to be the case in such a form as Diplodocus, and also in Stegosaurus, whose restoration by Marsh is doubtless substantially correct[633]. The fore-limbs, {707} though comparatively small, are obviously fashioned for support, but the weight which they have to carry is far less than that which the hind-limbs bear. The head is small and the neck short, while on the other hand the hind-quarters and the tail are big and massive. The backbone bends into a great, double-armed cantilever, culminating over the pelvis and the hind-limbs, and here furnished with its highest and strongest spines to separate the tension-member from the compression-member of the girder. The fore-legs form a secondary supporting pier to this great cantilever, the greater part of whose weight is poised upon the hind-limbs alone.

(6) In a bird, such as an ostrich or a common fowl, the bipedal habit necessitates the balancing of the load upon a single double-armed cantilever-girder, just as in the Iguanodon and the kangaroo, but the construction is effected in a somewhat different way. The great heavy tail has entirely disappeared; but, though from the skeleton alone it would seem that nearly all the bulk of the animal lay in front of the hind-limbs, yet in the living bird we can easily perceive that the great weight of the abdominal organs lies suspended behind the socket for the thigh-bone, and so hangs from the posterior lever-arm of the cantilever, balancing the head and neck and thorax whose combined weight hangs from {708} the anterior arm. The great cantilever girder appears, accordingly, balanced over the hind-legs. It is now constituted in part by the posterior dorsal or lumbar vertebrae, all traces of special elevation having disappeared from the anterior dorsals; but the greater part of the girder is made up of the great iliac bones, placed side by side, and gripping firmly the sacral vertebrae, often almost to the extinction of these latter. In the form of these iliac bones, the arched curvature of their upper border, in their elongation fore-and-aft to overhang both ways their supporting pier, and in the coincidence of their greatest height with the median line of support over the centre of gravity, we recognise all the characteristic properties of the typical balanced cantilever[634].

(7) We find a highly important corollary in the case of aquatic animals. For here the effect of gravity is neutralised; we have neither piers nor cantilevers; and we find accordingly in all aquatic mammals of whatsoever group—whales, seals or sea-cows—that the high arched vertebral spines over the withers, or corresponding structures over the hind-limbs, have both entirely disappeared.

Just as the cantilever girder tended to become obsolete in the aquatic mammal so does it tend to weaken and disappear in the aquatic bird. There is a very marked contrast between the high-arched strongly-built pelvis in the ostrich or the hen, and the long, thin, comparatively straight and weakly bone which represents it in a diver, a grebe or a penguin.

But in the aquatic mammal, such as a whale or a dolphin (and not less so in the aquatic bird), stiffness must be ensured in order to enable the muscles to act against the resistance of the water in the act of swimming; and accordingly nature must provide against bending-moments irrespective of gravity. In the dolphin, at any rate as regards its tail end, the conditions will be not very different from those of a column or beam with fixed ends, in which, under deflexion, there will be two points of contrary flexure, as at C, D, in Fig. 351. {709}

Here, between C and D we have a varying bending-moment, represented by a continuous curve with its maximal elevation midway between the points of inflexion. And correspondingly, in our dolphin, we have a continuous series of high dorsal spines, rising to a maximum about the middle of the animal’s body, and falling to nil at some distance from the end of the tail. It is their business (as usual) to keep the tension-member, represented by the strong supraspinous ligaments, wide apart from the compression-member, which is as usual represented by the backbone itself. But in our diagram we see that on the further side of C and D we have a negative curve of bending-moments, or bending-moments in a contrary direction. Without inquiring how these stresses are precisely met towards the dolphin’s head (where the coalesced cervical vertebrae suggest themselves as a partial explanation), we see at once that towards the tail they are met by the strong series of chevron-bones, which in the caudal region, where tall dorsal spines are no longer needed, take their place below the vertebrae, in precise correspondence with the bending-moment diagram. In many cases other than these aquatic ones, when we have to deal with animals with long and heavy tails (like the Iguanodon and the kangaroo of which we have already spoken), we are apt to meet with similar, though usually shorter chevron-bones; and in all these cases we may see without difficulty that a negative bending-moment is there to be resisted.

In the dolphin we may find a good illustration of the fact that not only is it necessary to provide for rigidity in the vertical direction, but also in the horizontal, where a tendency to bending must be resisted on either side. This function is effected in part by the ribs with their associated muscles, but they extend but a little way and their efficacy for this purpose can be but small. We have, however, behind the region of the ribs and on either side of the backbone a strong series of elongated and flattened transverse processes, forming a web for the support of a tension-member in the usual form of ligament, and so playing a part precisely analogous to that performed by the dorsal spines in the same {710} animal. In an ordinary fish, such as a cod or a haddock, we see precisely the same thing: the backbone is stiffened by the indispensable help of its three series of ligament-connected processes, the dorsal and the two transverse series. And here we see (as we see partly also among the whales), that these three series of processes, or struts, tend to be arranged well-nigh at equal angles, of 120°, with one another, giving the greatest and most uniform strength of which such a system is capable. On the other hand, in a flat fish, such as a plaice, where from the natural mode of progression it is necessary that the backbone should be flexible in one direction while stiffened in another, we find the whole outline of the fish comparable to that of a double bowstring girder, the compression-member being (as usual) the backbone, the tension-member on either side being constituted by the interspinous ligaments and muscles, while the web or filling is very beautifully represented by the long and evenly graded spines, which spring symmetrically from opposite sides of each individual vertebra.

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The main result at which we have now arrived, in regard to the construction of the vertebral column and its associated parts, is that we may look upon it as a certain type of girder, whose depth, as we cannot help seeing, is everywhere very nearly proportional to the height of the corresponding ordinate in the diagram of moments: just as it is in the girder of a cantilever bridge as designed by a modern engineer. In short, after the nineteenth or twentieth century engineer has done his best in framing the design of a big cantilever, he may find that some of his best ideas bad, so to speak, been anticipated ages ago in the fabric of the great saurians and the larger mammals.

But it is possible that the modern engineer might be disposed to criticise the skeleton girder at two or three points; and in particular he might think the girder, as we see it for instance in Diplodocus or Stegosaurus, not deep enough for carrying the animal’s enormous weight of some twenty tons. If we adopt a much greater depth (or ratio of depth to length) as in the modern cantilever, we shall greatly increase the strength of the structure; but at the same time we should greatly increase its rigidity, and {711} this is precisely what, in the circumstances of the case, it would seem that nature is bound to avoid. We need not suppose that the great saurian was by any means active and limber; but a certain amount of activity and flexibility he was bound to have, and in a thousand ways he would find the need of a backbone that should be flexible as well as strong. Now this opens up a new aspect of the matter and is the beginning of a long, long story, for in every direction this double requirement of strength and flexibility imposes new conditions upon our design. To represent all the correlated quantities we should have to construct not only a diagram of moments but also a diagram of elastic deflexion and its so-called “curvature”; and the engineer would want to know something more about the material of the ligamentous tension-member—its modulus of elasticity in direct tension, its elastic limit, and its safe working stress.

In various ways our structural problem is beset by “limiting conditions.” Not only must rigidity be associated with flexibility, but also stability must be ensured in various positions and attitudes; and the primary function of support or weight-carrying must be combined with the provision of points d’appui for the muscles concerned in locomotion. We cannot hope to arrive at a numerical or quantitative solution of this complicate problem, but we have found it possible to trace it out in part towards a qualitative solution. And speaking broadly we may certainly say that in each case the problem has been solved by nature herself, very much as she solves the difficult problems of minimal areas in a system of soap-bubbles; so that each animal is fitted with a backbone adapted to his own individual needs, or (in other words) corresponding exactly to the mean resultant of the stresses to which as a mechanical system it is exposed.

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Throughout this short discussion of the principles of construction, limited to one part of the skeleton, we see the same general principles at work which we recognise in the plan and construction of an individual bone. That is to say, we see a tendency for material to be laid down just in the lines of stress, and so as to evade thereby the distortions and disruptions due to shear. In these phenomena there lies a definite law of growth, {712} whatever its ultimate expression or explanation may come to be. Let us not press either argument or hypothesis too far: but be content to see that skeletal form, as brought about by growth, is to a very large extent determined by mechanical considerations, and tends to manifest itself as a diagram, or reflected image, of mechanical stress. If we fail, owing to the immense complexity of the case, to unravel all the mathematical principles involved in the construction of the skeleton, we yet gain something, and not a little, by applying this method to the familiar objects of our anatomical study: obvia conspicimus, nubem pellente mathesi[635].

Before we leave this subject of mechanical adaptation, let us dwell once more for a moment upon the considerations which arise from our conception of a field of force, or field of stress, in which tension and compression (for instance) are inevitably combined, and are met by the materials naturally fitted to resist them. It has been remarked over and over again how harmoniously the whole organism hangs together, and how throughout its fabric one part is related and fitted to another in strictly functional correlation. But this conception, though never denied, is sometimes apt to be forgotten in the course of that process of more and more minute analysis by which, for simplicity’s sake, we seek to unravel the intricacies of a complex organism.

We tend, as we analyse a thing into its parts or into its properties, to magnify these, to exaggerate their apparent independence, and to hide from ourselves (at least for a time) the essential integrity and individuality of the composite whole. We divide the body into its organs, the skeleton into its bones, as in very much the same fashion we make a subjective analysis of the mind, according to the teachings of psychology, into component factors: but we know very well that judgment and knowledge, courage or gentleness, love or fear, have no separate existence, but are somehow mere manifestations, or imaginary co-efficients, of a most complex integral. And likewise, as biologists, we may go so far as to say that even the bones themselves are only in a limited and even a deceptive sense, separate and individual things. The skeleton begins as a continuum, and a continuum it remains all life long. The things that link bone with bone, {713} cartilage, ligaments, membranes, are fashioned out of the same primordial tissue, and come into being pari passu, with the bones themselves. The entire fabric has its soft parts and its hard, its rigid and its flexible parts; but until we disrupt and dismember its bony, gristly and fibrous parts, one from another, it exists simply as a “skeleton,” as one integral and individual whole.

A bridge was once upon a time a loose heap of pillars and rods and rivets of steel. But the identity of these is lost, just as if they were fused into a solid mass, when once the bridge is built; their separate functions are only to be recognised and analysed in so far as we can analyse the stresses, the tensions and the pressures, which affect this part of the structure or that; and these forces are not themselves separate entities, but are the resultants of an analysis of the whole field of force. Moreover when the bridge is broken it is no longer a bridge, and all its strength is gone. So is it precisely with the skeleton. In it is reflected a field of force: and keeping pace, as it were, in action and interaction with this field of force, the whole skeleton and every part thereof, down to the minute intrinsic structure of the bones themselves, is related in form and in position to the lines of force, to the resistances it has to encounter; for by one of the mysteries of biology, resistance begets resistance, and where pressure falls there growth springs up in strength to meet it. And, pursuing the same train of thought, we see that all this is true not of the skeleton alone but of the whole fabric of the body. Muscle and bone, for instance, are inseparably associated and connected; they are moulded one with another; they come into being together, and act and react together[636]. We may study them apart, but it is as a concession to our weakness and to the narrow outlook of our minds. We see, dimly perhaps, but yet with all the assurance of conviction, that between muscle and bone there can be no change in the one but it is correlated with changes in the other; that through and through they are linked in indissoluble association; that they are only separate entities {714} in this limited and subordinate sense, that they are parts of a whole which, when it loses its composite integrity, ceases to exist.

The biologist, as well as the philosopher, learns to recognise that the whole is not merely the sum of its parts. It is this, and much more than this. For it is not a bundle of parts but an organisation of parts, of parts in their mutual arrangement, fitting one with another, in what Aristotle calls “a single and indivisible principle of unity”; and this is no merely metaphysical conception, but is in biology the fundamental truth which lies at the basis of Geoffroy’s (or Goethe’s) law of “compensation,” or “balancement of growth.”

Nevertheless Darwin found no difficulty in believing that “natural selection will tend in the long run to reduce any part of the organisation, as soon as, through changed habits, it becomes superfluous: without by any means causing some other part to be largely developed in a corresponding degree. And conversely, that natural selection may perfectly well succeed in largely developing an organ without requiring as a necessary compensation the reduction of some adjoining part[637].” This view has been developed into a doctrine of the “independence of single characters” (not to be confused with the germinal “unit characters” of Mendelism), especially by the palaeontologists. Thus Osborn asserts a “principle of hereditary correlation,” combined with a “principle of hereditary separability whereby the body is a colony, a mosaic, of single individual and separable characters[638].” I cannot think that there is more than a small element of truth in this doctrine. As Kant said, “die Ursache der Art der Existenz bei jedem Theile eines lebenden Körpers ist im Ganzen enthalten.” And, according to the trend or aspect of our thought, we may look upon the co-ordinated parts, now as related and fitted to the end or function of the whole, and now as related to or resulting from the physical causes inherent in the entire system of forces to which the whole has been exposed, and under whose influence it has come into being[639]. {715}

It would seem to me that the mechanical principles and phenomena which we have dealt with in this chapter are of no small importance to the morphologist, all the more when he is inclined to direct his study of the skeleton exclusively to the problem of phylogeny; and especially when, according to the methods of modern comparative morphology, he is apt to take the skeleton to pieces, and to draw from the comparison of a series of scapulae, humeri, or individual vertebrae, conclusions as to the descent and relationship of the animals to which they belong.

It would, I dare say, be a gross exaggeration to see in every bone nothing more than a resultant of immediate and direct physical or mechanical conditions; for to do so would be to deny the existence, in this connection, of a principle of heredity. And though I have tried throughout this book to lay emphasis on the direct action of causes other than heredity, in short to circumscribe the employment of the latter as a working hypothesis in morphology, there can still be no question whatsoever but that heredity is a vastly important as well as a mysterious thing; it is one of the great factors in biology, however we may attempt to figure to ourselves, or howsoever we may fail even to imagine, its underlying physical explanation. But I maintain that it is no less an exaggeration if we tend to neglect these direct physical and mechanical modes of causation altogether, and to see in the characters of a bone merely the results of variation and of heredity, and to trust, in consequence, to those characters as a sure and certain and unquestioned guide to affinity and phylogeny. Comparative anatomy has its physiological side, which filled men’s minds in John Hunter’s day, and in Owen’s day; it has its {716} classificatory and phylogenetic aspect, which has all but filled men’s minds during the last couple of generations; and we can lose sight of neither aspect without risk of error and misconception.

It is certain that the question of phylogeny, always difficult, becomes especially so in cases where a great change of physical or mechanical conditions has come about, and where accordingly the physical and physiological factors in connection with change of form are bound to be large. To discuss these questions at length would be to enter on a discussion of Lamarck’s philosophy of biology, and of many other things besides. But let us take one single illustration.

The affinities of the whales constitute, as will be readily admitted, a very hard problem in phylogenetic classification. We know now that the extinct Zeuglodons are related to the old Creodont carnivores, and thereby (though distantly) to the seals; and it is supposed, but it is by no means so certain, that in turn they are to be considered as representing, or as allied to, the ancestors of the modern toothed whales[640]. The proof of any such a contention becomes, to my mind, extraordinarily difficult and complicated; and the arguments commonly used in such cases may be said (in Bacon’s phrase) to allure, rather than to extort assent. Though the Zeuglodonts were aquatic animals, we do not know, and we have no right to suppose or to assume, that they swam after the fashion of a whale (any more than the seal does), that they dived like a whale, and leaped like a whale. But the fact that the whale does these things, and the way in which he does them, is reflected in many parts of his skeleton—perhaps more or less in all: so much so that the lines of stress which these actions impose are the very plan and working-diagram of great part of his structure. That the Zeuglodon has a scapula like that of a whale is to my mind no necessary argument that he is akin by blood-relationship to a whale: that his dorsal vertebrae are very different from a whale’s is no conclusive argument that {717} such blood-relationship is lacking. The former fact goes a long way to prove that he used his flippers very much as a whale does; the latter goes still farther to prove that his general movements and equilibrium in the water were totally different. The whale may be descended from the Carnivora, or might for that matter, as an older school of naturalists believed, be descended from the Ungulates; but whether or no, we need not expect to find in him the scapula, the pelvis or the vertebral column of the lion or of the cow, for it would be physically impossible that he could live the life he does with any one of them. In short, when we hope to find the missing links between a whale and his terrestrial ancestors, it must be not by means of conclusions drawn from a scapula, an axis, or even from a tooth, but by the discovery of forms so intermediate in their general structure as to indicate an organisation and, ipso facto, a mode of life, intermediate between the terrestrial and the Cetacean form. There is no valid syllogism to the effect that A has a flat curved scapula like a seal’s, and B has a flat, curved scapula like a seal’s: and therefore A and B are related to the seals and to each other; it is merely a flagrant case of an “undistributed middle.” But there is validity in an argument that B shews in its general structure, extending over this bone and that bone, resemblances both to A and to the seals: and that therefore he may be presumed to be related to both, in his hereditary habits of life and in actual kinship by blood. It is cognate to this argument that (as every palaeontologist knows) we find clues to affinity more easily, that is to say with less confusion and perplexity, in certain structures than in others. The deep-seated rhythms of growth which, as I venture to think, are the chief basis of morphological heredity, bring about similarities of form, which endure in the absence of conflicting forces; but a new system of forces, introduced by altered environment and habits, impinging on those particular parts of the fabric which lie within this particular field of force, will assuredly not be long of manifesting itself in notable and inevitable modifications of form. And if this be really so, it will further imply that modifications of form will tend to manifest themselves, not so much in small and isolated phenomena, in this part of the fabric or in that, in a scapula for instance or a humerus: but rather in {718} some slow, general, and more or less uniform or graded modification, spread over a number of correlated parts, and at times extending over the whole, or over great portions, of the body. Whether any such general tendency to widespread and correlated transformation exists, we shall attempt to discuss in the following chapter.

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