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.

――――――――――

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,

[Illustration: Fig. 331.]

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.

[Illustration: Fig. 332.]

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

[Illustration: Fig. 333.]

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].

――――――――――

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}

[Illustration: Fig. 334. Head of the human femur in section. (After

Schäfer, from a photo by Prof. A. Robinson.)]

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

[Illustration: Fig. 335. Crane-head and femur. (After Culmann and H.

Meyer.)]

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.

[Illustration: Fig. 336. Diagram of stress-lines in the human foot.

(From Sir D. MacAlister, after H. Meyer.)]

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.

――――――――――

[Illustration: Fig. 337. Trabecular structure of the os calcis. (From

MacAlister.)]

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.

[Illustration: Fig. 338.]

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.

――――――――――

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.

――――――――――

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

[Illustration: Fig. 339.]

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.

――――――――――

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,

[Illustration: Fig. 340. A, Span of proposed bridge. B, Stress diagram,

or diagram of bending-moments[628].]

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

[Illustration: Fig. 341. The bridge constructed, as a parabolic girder.]

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.”

[Illustration: Fig. 342.]

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.

[Illustration: Fig. 343.]

[Illustration: Fig. 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.

[Illustration: Fig. 345. A two-armed cantilever of the Forth Bridge.

Thick lines, compression-members (bones); thin lines, tension-members

(ligaments).]

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

[Illustration: Fig. 346.]

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.

[Illustration: Fig. 347. Stress-diagram of horse’s backbone.]

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.

[Illustration: Fig. 348. Stress-diagram of backbone of Dinosaur.]

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.

[Illustration: Fig. 349. Stress-diagram of Titanotherium.]

(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.

[Illustration: Fig. 350. Diagram of Stegosaurus.]

(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}

[Illustration: Fig. 351.]

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.

――――――――――

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.

――――――――――

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.

{719}