On Growth and Form

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 otherf. 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 dwellt. 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 + Delight in beauty is one of the pleasures of the imagination; there is no limit to its indulgence, and no end to the results which we may ascribe to its exercise. But as for the particular “standard of beauty” which the bird (for instance) admires and selects (as Darwin says in the Origin, p. 70, edit. 1884), we are very much in the dark, and we run the risk of arguing in a circle: for wellnigh all we can safely say is what Addison says (in the 412th Spectator)—that each different species “is most affected with the beauties of its own kind....Hine merula in nigro se oblectat nigra marito;...hinc noctua tetram Canitiem alarum et glaucos miratur ocellos.”

had brown spots over his eyes that he might seem awake when he was sleeping*: so that an enemy might let the sleeping dog he, 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 incarnadinef. 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 theologiant, “The free use of final causes to: explain what seems obscure was temptingly easy....Hence the finalist was often the man who made a hberal use of the zgnava 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

+ They are “wonderfully fitted for “vanishment’ against the flushed, richcoloured skies of early morning and evening,...their chief feeding-times”; and ‘‘look like a real sunset or dawn, repeated on the opposite side of the heavens,— either east or west as the case may be”: Thayer, Concealing-coloration in the Animal Kingdom, New York, 1909, pp. 154-155. This hypothesis, like the rest, is not free from difficulty. Twilight is apt to be short in the homes of the flamingo: and moreover. Mr Abel Chapman, who watched them on the Guadalquivir, tells

us that they feed by day. { Principal Galloway, Philosophy of Religion, p. 344, 1914. 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*.

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, in * Cf. Professor Flint, in his Preface to Affleck’s translation of Janet’s Causes finales: “ We are, no doubt, still a long way from a mechanical theory of organic growth, but it may be said to be the quaesitwm of modern science, and no one can say that it is a chimaera.” + Cf. Sir Donald MacAlister, How a Bone is Built, Hngl. Ill. Mag. 1884.

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 imorganic 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 familar 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*.”’ However, be that as it may, we know, roughly, that when the beam is

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 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 “compressionstrains” 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*. 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 le 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 togetherf.

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 hmit in a few substances. ;

* In preparing or ‘“‘macerating”’ a skeleton, the naturalist nowadays carries on the process till nothing is left but the whitened bones. But the old anatomists, whose object was not the study of “comparative” morphology but the wider theme of comparative physiology, were wont to macerate by easy stages; and in many of their most instructive preparations, the ligaments were intentionally left in connection with the bones, and as part of the “skeleton.”

+ In a few anatomical diagrams, for instance in some of the drawings in Schmaltz’s Atlas der Anatomie des Pferdes, we may see the system of “ties” diagrammatically inserted in the figure of the skeleton. Cf. Gregory, On the principles of Quadrupedal Locomotion, Ann. N. Y. Acad. of Sciences, xxu, p. 289, 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 ZL.

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.

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 wheatstraw bearing its heavy burden in the ear, are all illustrations which Galileo used in his account of this mechanical principle*.

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. °

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 Fig. 332. where the bracket is fastened to the tally 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 “mirrorimage’; 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.

-Schwendener* 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.

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

* Das mechanische Prinzip im anatomischen Bau der Monocotylen, Leipzig, stronger bundles; sometimes with these bundles further strengthened by radial balks or ridges; sometimes with all the fibres set close together in a continuous hollow cylinder. In the case figured in Fig. _ 333 Schwendener calculated that the resistance to bending was at least twenty-five times as great as it would have been had the six main bundles been brought close together in a solid core. In many cases the centre of the stem is altogether empty; in all other cases it is filled with soft tissue, suitable for the ascent of sap or other functions, but never such as to confer mechanical rigidity. Ina 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*.

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.

* For further botanical illustrations, see (int. al.) Hegler, Einfluss der Zugkraften auf die Festigkeit und die Ausbildung mechanischer Gewebe in Pflanzen, SB. stichs. Ges. d. Wiss. p. 638, 1891; Kny, L,, Einfluss von Zug und Druck auf die Richtung der Scheidewande in sich teilenden Pflanzenzellen, Ber. d. bot. Gesellsch. xtv, 1896; Sachs, Mechanomorphose und Phylogenie, Flora, LXXVII, 1894; cf. also Pfliiger, Einwirkung der Schwerkraft, etc., iiber die Richtung der Zelltheilung, Archiv, xxxiv, 1884.

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, bloodvessels and other tissues; and among these living tissues lies a fine lattice-work of little interlaced “trabeculae” of bone, forming

Fig. 334. Head of the human femur in section. (After Schafer, from a photo by Prof. A. Robinson.) 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

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 Ziirich (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

<u << =a [] Sh Wylie SSS AANA {/ x9 iM, OSES TT TGS OS ogwweeD: YLYZ XS 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

or inner side, overhung by the loaded head, is the “compressionmember’; 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*. * Among other works on the mechanical construction of bone see: Bourgery, Traité de Vanatomie (I. Ostéologie), 1832 (with admirable illustrations of trabecular structure); Fick, L., Die Ursachen der Knochenformen, Gottingen, 1857; Meyer. H., Die Architektur der Spongiosa, Archiv f. Anat. und Physiol. xuvu, pp. 615-628, 1867; Statik uw.

Mechanik des menschlichen Knochengeriistes, Leipzig, 1873; Wolff, J., Die innere Architektur der Knochen, Arch. f. Anat. und Phys. L, 1870; Das Gesetz der Transformation bei Knochen, 1892; von Ebner, V., Der feinere Bau der Knochensubstanz, Wiener Bericht, uxxi1, 1875; Rauber, Anton, Elastizitat und Festigkeit der Knochen, Leipzig, 1876; O. Meserer, Elast. u. Festigk. d. menschlichen Knochen, Stuttgart, 1880; MacAlister, Sir Donald, How a Bone is Built, English Illustr. Mag. pp. 640-649, 1884; Rasumowsky, Architektonik des Fussskelets, Int. Monatsschr. f. Anat. p. 197, 1889; Zschokke, Weitere Unters. iiber das Verhdltniss der Knochenbildung zur Statik und Mechanik des Vertebratenskelets, Ziirich, 1892; Roux, W., Ges. Abhandlungen iiber Entwicklungsmechanik der Organismen, Bd. I, Funktionelle Anpassung, Leipzig, 1895; Triepel. H., Die Stossfestigkeit der Knochen, Arch. f. Anat. vu. Phys. 1900; Gebhardt, Funktionell