CHAPTER 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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