On Growth and Form

The problem is closely akin to that of the cartographer who transfers identical data to one projection or another; and whose object is to secure (if it be possible) a complete correspondence, in each small unit of area, between the one representation and the other. The morphologist will not seek to draw his organic forms - in a new and artificial projection; but, in the converse aspect of the problem, he will inquire whether two different but more or less obviously related forms can be so analysed and interpreted that each may be shown to be a transformed representation of the other. This once demonstrated, it will be a comparatively easy task (in all probability) to postulate the direction and magnitude of the force capable of effecting the required transformation. Again, if such a simple alteration of the system of forces can be proved adequate to meet the case, we may find ourselves able to dispense with many widely current and more complicated hypotheses of biological causation. For it is a maxim in physics that an effect ought not to be ascribed to

the joint operation of many causes if few are adequate to the production of it: Frustra fit per plura, quod fieri potest per pauciora. It is evident that by the combined action of appropriate forces any material form can be transformed into any other: just as out of a ““shapeless” mass of clay the potter or the sculptor models his artistic product; or just as we attribute to Nature herself the power to effect the gradual and successive transformation of the simplest into the most complex organism. In like manner it is possible, at least theoretically, to cause the outline of any closed curve to appear as a projection of any other whatsoever. But we need not let these theoretical considerations deter us from our method of comparison of related forms. We shall strictly limit ourselves to cases where the transformation necessary to effect a comparison shall be of a simple kind, and where the transformed, as well as the original, co-ordinates shall constitute an harmonious and more or less symmetrical system. We should fall into deserved and inevitable confusion if, whether by the mathematical or any other method, we attempted to compare organisms separated far apart in Nature and in zoological classification. We are limited, not by the nature of our method, but by the whole nature of the case, to the comparison of organisms such as are manifestly related to one another and belong to the same zoological class.

Our inquiry lies, in short, just within the limits which Aristotle himself Jaid down when, in defining a “genus,” he showed that (apart from those superficial characters, such as colour, which he called “accidents’’) the essential differences between one “ species”’ and another are merely differences of proportion, of relative magnitude, or (as he phrased it) of “excess and defect.” “Save only for a difference in the way of excess or defect, the parts are identical in the case of such animals as are of one and the same genus; and by ‘genus’ I mean, for instance, Bird or Fish.” And again: “Within the limits of the same genus, as a general rule, most of the parts exhibit differences...in the way of multitude or fewness, magnitude or parvitude, in short, in the way of excess or defect. For “the more’ and ‘the less’ may be represented as

‘excess’ and ‘defect*.’’’ It is precisely this difference of relative magnitudes, this Aristotelian “excess and defect” in the case of form, which our co-ordinate method .is especially adapted to analyse, and to reveal and demonstrate as the main cause of what (again in the Aristotelian sense) we term “specific” differences. The applicability of our method to particular cases will depend upon, or be further limited by, certain practical considerations or qualifications. Of these the chief, and indeed the essential, condition is, that the form of the entire structure under investigation should be found to vary in a more or less uniform manner, after the fashion of an approximately homogeneous and isotropic body. But an imperfect isotropy, provided always that some “principle of continuity” run through its variations, will not seriously interfere with our method; it will only cause our transformed co-ordinates to be somewhat less regular and harmonious than are those, for instance, by which the physicist depicts the motions of a perfect fluid or a theoretic field of force in a uniform medium.

Again, it is essential that our structyre vary in its entirety, or at least that “independent variants” should be relatively few. That independent variations occur, that localised centres of diminished or exaggerated growth will now and then be found, is not only probable but manifest; and they may even be so pronounced as to appear to constitute new formations altogether. Such independent variants as these Aristotle himself clearly recognised: “It happens further that some have parts that others have not; for instance, some [birds] have spurs and others not, some have crests, or combs, and others not; but, as a general rule, most parts and those that go to make up the bulk of the body are either identical with one another, or differ from one another in the way of contrast and of excess and defect. For ‘the more’ and ‘the less’ may be represented as ‘excess’ or ‘defect.’”

If, in the evolution of a fish, for instance, it be the case that its several and constituent parts—head, body, and tail, or this fin and that fin—represent so many independent variants, then our co-ordinate system will at once become too complex to be intelligible; we shall be making not one comparison but’ several separate comparisons, and our general method will be found inapplicable. Now precisely this independent variability of parts and organs—here, there, and everywhere within the organism —would appear to be implicit in our ordinary accepted notions regarding variation; and, unless I am greatly mistaken, it is precisely on such a conception of the easy, frequent, and normal independent variability of parts that our conception of the process of natural selection is fundamentally based. For the morphologist, when comparing one organism with another, describes the differences between them point by point, and “character” by ‘“character*.” If he is from time to time constrained to admit the existence of “correlation” between characters (as a hundred years ago Cuvier first showed the way), yet all the while he recognises this fact of correlation somewhat vaguely, as a phenomenon due to causes which, except in rare instances, he can hardly hope to trace; and he falls readily into the habit of thinking and talking of evolution as though it had proceeded on the lines of his own descriptions, point by point, and character by charactery. But if, on the other hand, diverse and dissimilar fishes can be referred as a whole to identical functions of very different coordinate systems, this fact will of itself constitute a proof that variation has proceeded on definite and orderly lines, that a comprehensive “law of growth” has pervaded the whole structure in its integrity, and that some more or less simple and recognisable system of forces has been at work. It will not only show how real and deep-seated is the phenomenon of “correlation,” in regard to form, but it will also demonstrate the fact that a correlation which had seemed too complex for analysis or

+ Cf. Osborn, H. F., On the Origin of Single Characters, as observed in fossil and living Animals and Plants, Amer. Nat. xt1x, pp. 193-239, 1915 (and other papers); ibid. p. 194, ‘Each individual is composed of a vast number of somewhat similar new or old characters, each character has its independent and separate history, each character is in a certain stage of evolution, each character is correlated with the other characters of the individual....The real problem has always been that of the origin and development of characters. Since the Origin of Species appeared, the terms variation and variability have always referred to single characters; if a species is said to be variable, we mean that a considerable number of the single characters or groups of characters of which it is composed are variable,” etc.

comprehension is, in many cases, capable of very simple graphic expression. This, after many trials, I believe to be in general the case, bearing always in mind that the occurrence of independent or localised variations must often be considered. We are dealing in this chapter with the forms of related organisms, in order to shew that the differences between them are as a genera! rule simple and symmetrical, and just such as might have been brought about by a slight and simple change in the system of forces to which the living and growing organism was exposed. Mathematically speaking, the phenomenon is identical with one met with by the geologist, when he finds a bed of fossils squeezed flat or otherwise symmetrically deformed by the pressures to which they, and the strata which contain them, have been subjected. In the first step towards fossilisation, when the body of a fish or shellfish is silted over and buried, we may take it that the wet sand or mud exercises, approximately, a hydrostatic pressure— that is to say a pressure which is un form in all directions, and by which the form of the buried object will not be appreciably changed. As the strata consolidate and accumulate, the fossil organisms which they contain will tend to be flattened by the vast superincumbent load, just as the stratum which contains them will also be compressed and will have its molecular arrangement more or less modified*. But the deformation due to direct vertical pressure in a horizontal stratum is not nearly so striking as are the deformations produced by the ob ique or shearing stresses to which inc’ined and folded strata have been exposed, and by which their various “dislocations” have been brought about.

And espec‘ally in mountain regions, where these dislocations are especially numerous and complicated, the contained fossils are apt to be so curiously and yet so symmetrically deformed (usually by a simple shear) that they may easily be interpreted as so many distinct and separate “speciest.” A great number of described species, and here and there a new genus (as the genus Ellipsolithes for an obliquely deformed Goniatite or Nautilus) are said to rest on no other foundation f.

If we begin by drawing a net of rectangular equidistant co-ordinates (about the axes x and y), we may alter or deform this t Thus Ammonites erugatus, when compressed, has been described as 4. planorbis: cf. Blake, J. F., Phil. Mag. (5), v1, p. 260, 1878. Wettstein has shewn that several species of the fish-genus Lepidopus have been based on specimens artificially deformed in various ways: Ueber die Fischfauna des Tertiaren Glarnerschiefers, Abh. Schw. Palaeont. Gesellsch. x11, 1886 (see especially pp. 23-38, pl. 1). The whole subject, interesting as it is, has been little studied; both Blake and Wettstein deal with it mathematically.

network in various ways, several of which are very simple indeed. Thus (1) we may alter the dimensions of our system, extending it along one or other axis, and so converting each little square into a corresponding and directly proportionate oblong (Fig. 353). It follows that any figure which we may have inscribed in the original net, and which we transfer to the new, will thereby be deformed in strict proportion to the deformation of the entire configuration, being still defined by corresponding points in the network and being throughout in conformity with the original figure. For instance, a circle inscribed in the original “Cartesian” net will now, after extension in the y-direction, be found elongated

into an ellipse. In elementary mathematical language, for the original x and y we have substituted x, and cy,, and the equation to our original circle, 22 + y? = a?, becomes that of the ellipse, Le Cy 2. If I draw the cannon-bone of an ox (Fig. 354, A), for instance, within a system of rectangular co-ordinates, and then transfer the same drawing, point for point, to a system in which for the x of the original diagram we substitute x’ = 2z/3, we obtain a drawing (B) which is a very close approximation to the cannonbone of the sheep. In other words, the main (and perhaps the only) difference between the two bones is simply that that of the sheep is elongated, along the vertical axis, as compared with that of the ox in the relation of 3/2. And similarly, the long slender cannon-bone of the giraffe (C) is referable to the same identical type, subject to a reduction of breadth, or increase of length, corresponding to 2” = 2/3. .

(2) The second type is that where extension is not equal or uniform at all distances from the origin: but grows greater or less, as, for instance, when we stretch a tapering elastic band. In such cases, as I have represented it in Fig. 355, the ordinate increases logarithmically, and for y we substitute <”. It is obvious that this logarithmic extension may involve both abscissae and ordinates, x becoming «*, while y becomes «”. The circle in our original figure is now deformed into some such shape as that of Fig. 356. This method of deformation is a common one, and will often be of use to us in our comparison of organic forms.

(3) Our third type is the “simple shear,” where the rectangular co-ordinates become “oblique,” their axes being inclined to one another at a certain angle w. Our original rectangle now becomes such a figure as that of Fig. 357. The system may now be described in terms of the oblique axes X, Y; or may be directly ~ referred to new rectangular co-ordinates €, 7 by the simple transposition x = € — n cota, y = 7 Cosec w. (4) Yet another important class of deformations may be represented by the use of radial co-ordinates, in which one set of lines are represented as radiating from a point or “focus,” while the other set are transformed into circular arcs cutting the radii orthogonally. These radial co-ordinates are especially applicable

to cases where there exists (either within or without the figure) some part which is supposed to suffer no deformation; a simple illustration is afforded by the diagrams which illustrate the flexure of a beam (Fig. 358). In biology these co-ordinates will be especially applicable in cases where the growing structure includes a “node,” or point where growth is absent or at a minimum; and about which node the rate of growth may be assumed to increase symmetrically. Precisely such a case is furnished us in a leaf of an ordinary dicotyledon. The leaf of a

typical monocotyledon—such as a grass or a hyacinth, for instance — grows continuously from its base, and exhibits no node or “ point of arrest.” Its sides taper off gradually from its broad base to its slender tip, according to some law of decrement specific to the plant; and any alteration in the relative velocities of longitudinal and transverse growth will merely make the leaf a little broader or narrower, and will effect no other conspicuous alteration in its contour. But if there once come into existence a node, or “locus of no growth,” about which we may assume the growth— — which in the hyacinth leaf was longitudinal and transverse—to take place radially and transversely to the radii, then we shall

at once see, in the first place, that the sloping and shghtly curved sides of the hyacinth leaf suffer a transformation into what we consider a more typical and “leaf-like” shape, the sides of the figure broadening out to a zone of maximum breadth and then drawing inwards to the pointed apex. If we now alter the ratio between the radial and tangential velocities of growth—in other words, if we increase the angles between corresponding radii— we pass successively through the various configurations which the botanist describes as the lanceolate, the ovate, and finally the cordate leaf. These successive changes may to some extent, and in appropriate cases, be traced as the individual leaf grows

to maturity; but as a much more general rule, the balance of forces, the ratio between radial and tangential velocities of growth, remains so nicely and constantly balanced that the leaf increases in size without conspicuous modification of form. It is rather what we may call a long-period variation, a tendency for the relative velocities to alter from one generation to another, whose result is brought into view by this method of illustration. There are various corollaries to this method of describing the form of a leaf which may be here alluded to, for we shall not return again to the subject of radial co-ordinates. For instance, the so-called unsymmetrical leaf* of a begonia, in which one side of the leaf may be merely ovate while the other has a cordate outline, is seen to be really a case of unequal, and not truly asymmetrical, growth on either side of the midrib. There is nothing more mysterious in its conformation than, for instance, in that of a forked twig in which one limb of the fork has grown longer than the other. The case of the begonia leaf is of sufficient interest to deserve illustration, and in Fig. 360 I have outlined a leaf of the large Begonia daedalea. On the smaller left-hand side of the leaf I have taken at random three points, a, b, c, and have measured the angles, A0a, i etc., which the radii from the Fig. 360. Begonia daedalea. hilus of the leaf to these points make with the median axis. On the other side of the leaf I have marked the points a’, b’, c’, such that the radii drawn to this margin of the leaf are equal to the former, Oa’ to Oa, etc. Now if the two sides of the leaf are

* Cf. Sir Thomas Browne, in The Garden of Cyrus: “But why ofttimes one side of the leaf is unequall unto the other, as in Hazell and Oaks, why on either side the master vein the lesser and derivative channels stand not directly opposite, nor at equall angles, respectively unto the adverse side, but those of one side do often exceed the other, as the Wallnut and many more, deserves another enquiry.” mathematically similar to one another, it is obvious that the respective angles should be in continued proportion, i.e. as AOa is to AOQa’, so should AOb be to AOb’. This proves to be very nearly the case. For I have measured the three angles on one side, and one on the other, and have then compared, as follows, the calculated with the observed values of the other two:

The agreement is very close, and what discrepancy there is may be amply accounted for, firstly, by the slight irregularity of the sinuous margin of the leaf; and secondly, by the fact that the true axis or midrib of the leaf is not straight but slightly curved, and therefore that it is curvilinear and not rectilinear — triangles which we ought to have measured. When we understand these few points regarding the peripheral curvature of the leaf, it is easy to see that its principal veins approximate closely to a beautiful system of isogonal co-ordinates. It is also obvious that we can easily pass, by a process of shearing, from those cases

where the principal veins start from the base of the leaf to those, as in most dicotyledons, where they arise successively from the midrib. It may sometimes happen that the node, or “point of arrest,” is at the upper instead of the lower end of the leaf-blade; and occasionally there may be a node at both ends. In the former case, as we have it in the daisy, the form of the leaf will be, as it were, inverted, the broad, more or less heart-shaped, outline appearing at the upper end, while below the leaf tapers gradually downwards to an ill-defined base. In the latter case, as in Dionaea, we obtain a leaf equally expanded, and similarly ovate or cordate, at both ends. We may notice, lastly, that the shape of a solid fruit, such as an apple or a cherry, is a solid of revolution, developed from similar curves and to be explained on the same principle. In the cherry we have a “ point of arrest” at the base of the berry, where it joins its peduncle, and about this point the fruit (in imaginary section) swells out into a cordate outline; while in the

apple we have two sucli well-marked points of arrest, above and below, and about both of them the same conformation tends to arise. The bean and the human kidney owe their “reniform”’ shape to precisely the same phenomenon, namely, to the existence of a node or “hilus,” about which the forces of growth are radially and symmetrically arranged. Most of the transformations which we have hitherto considered (other than that of the simple shear) are particular cases of a general transformation, obtainable by the method of conjugate functions and equivalent to the projection of the original figure on a new plane. Appropriate transformations, on these general Imes, provide for the cases of a coaxial system where the Cartesian co-ordinates are replaced by coaxial circles, or a confocal system in which they are replaced by confocal ellipses and hyperbolas.

Yet another curious and important transformation, belonging to the same class, is that by which a system of straight lines becomes transformed into a conformal system of logarithmic spirals: the straight line Y—AX-=c corresponding to the logarithmic spiral @ — A logr = (Fig. 361). This beautiful and simple transformation lets us at once convert, for instance, the straight conical shell of the Pteropod or the Orthoceras into the logarithmic spiral of the Nautiloid; it involves a mathematical symbolism which is but a shght extension of that which we have employed in our elementary treatment of the logarithmic spiral.

These various systems of coordinates, which we have now briefly considered, are sometimes called “isothermal co-ordinates,’ from the fact that, when employed in this .particular branch of physics, they perfectly represent the phenomena of the conduction of heat, the contour lines of equal temperature appearing, under appropriate conditions, as the orthogonal lines of the co-ordinate system. And it follows that the “law of growth”? which our biological analysis by means of —

orthogonal co-ordinate systems presupposes, or at least foredevelops along stream lines, which may be defined by a suitable mathematical transformation. of the following illustrations—for instance, that of Orthagoriscus (Fig. 382),—then the transformation is no longer within the reach ~ of comparatively simple mathematical analysis. Such departure | from the typical symmetry of a “stream-line” system is, in the ~ first instance, sufficiently accounted for by the simple fact that

the developing organism is very far from being homogeneous and isotropic, or, in other words, does not behave like a perfect fluid. — But though under such circumstances our co-ordinate systems — may be no longer capable of strict mathematical analysis, they — will still indicate graphically the relation of the new co-ordinate system to the old, and conversely will furnish us with some ~ guidance as to the “law of growth,” or play of forces, by which —

Before we pass from this brief discussion of transformations in general, let us glance at one or two cases in which the forces applied are more or less intelligible, but the resulting transformations are, — from the mathematical point of view, exceedingly complicated. The “marbled papers” of the bookbinder are a beautiful illustration of visible “stream lines.” On a dishful of a sort of — semi-liquid gum the workman dusts a few simple lines or patches of colourmg matter; and then, by passing a comb through the liquid, he draws the colour-bands into the streaks, waves, and spirals which constitute the marbled pattern, and which he then transfers to sheets of paper laid down upon the gum. By some such system of shears, by the effect of unequal traction or unequal growth in various directions and superposed on an originally simple pattern, we may account for the not dissimilar marbled patterns which we recognise, for instance, on a large serpent’s skin. But it must be remarked, in the case of the marbled paper, that though the method of application of the forces is simple, yet in the aggregate the system of forces set up by the many

teeth of the comb is exceedingly complex, and its complexity is revealed in the complicated “diagram of forces” which constitutes To take another and still more instructive illustration. To turn one circle (or sphere) into two circles would be, from the point of view of the mathematician, an extraordinarily difficult transformation; but, physically speaking, its achievement may be extremely simple. The little round gourd grows naturally, by its symmetrical forces of expansive growth, into a big, round, or somewhat oval pumpkin or melon. But the Moorish husbandman ties a rag round its middle, and the same forces of growth, unaltered save for the presence of this trammel, now expand the globular structure into two superposed and connected globes. And again, by varying the position of the encircling band, or by applying several such ligatures instead of one, a great variety of artificial forms of “gourd” may be, and actually are, produced. It is clear, I think, that we may account for many ordinary biological processes of development or transformation of form by the existence of trammels or lines of constraint, which limit and determine the action of the expansive forces of growth that would otherwise be uniform and symmetrical. This case has a close parallel in the operations of the glassblower, to which we have already, more than once, referred in passing*. The glassblower starts his operations with a tube, which he first closes at one end so as to form a hollow vesicle, within which his blast of air exercises a uniform pressure on all sides; but the spherical conformation which this uniform expansive force would naturally tend to produce is modified into all kinds of forms by the trammels or resistances set up as the workman lets one part or another of his bubble be unequally heated or cooled. It was Oliver Wendell Holmes who first shewed this curious parallel between the operations of the glassblower and those of Nature, when she starts, as she so often does, with a simple tubet.

The alimentary canal, * Where gourds are common, the glass-blower is still apt to take them for a prototype, as the prehistoric potter also did. For instance, a tall, annulated Florence oil-flask is an exact but no longer a conscious imitation of a gourd which has been converted into a bottle in the manner described. the arterial system including the heart, the central nervous system of the vertebrate, including the brain itself, all begin as simple tubular structures. And with them Nature does just what the glassblower does, and, we might even say, no more than he. For she can expand the tube here and narrow it there; thicken its walls or thin them; blow off a lateral offshoot or caecal diverticulum; bend the tube, or twist and coil it; and infold or crimp its walls as, so to speak, she pleases. Such a form as that of the human stomach is easily explained when it is regarded from this point of view; it is simply an ill-blown bubble, a bubble that has been rendered lopsided by a trammel or restraint along one side, such as to prevent its symmetrical expansion— such a trammel as is produced if the glassblower lets one side of his bubble get cold, and such as is actually present in the stomach itself in the form of a muscular band.

We may now proceed to consider and illustrate a few permutations or transformations of organic form, out of the vast multitude which are equally open to this method of inquiry. We have already compared in a preliminary fashion the metacarpal or cannon-bone of the ox, the sheep, and the giraffe (Fig. 354); and we have seen that the essential difference in form between these three bones is a matter of relative length and breadth, such that, if we reduce the figures to an identical standard of length (or identical values of y), the breadth (or value of x) will be approximately two-thirds that of the ox in the case of the sheep and one-third that of the ox in the case of the giraffe. We may easily, for the sake of closer comparison, determine these ratios more accurately, for instance, if it be our purpose to compare the different racial varieties within the limits of a single species. And in such cases, by the way, as when we compare with one another various breeds or races of cattle or of horses, the ratios