Chapter XVII (1)

ON THE THEORY OF TRANSFORMATIONS, OR THE COMPARISON OF RELATED FORMS[641]

In the foregoing chapters of this book we have attempted to study the inter-relations of growth and form, and the part which certain of the physical forces play in this complex interaction; and, as part of the same enquiry, we have tried in comparatively simple cases to use mathematical methods and mathematical terminology in order to describe and define the forms of organisms. We have learned in so doing that our own study of organic form, which we call by Goethe’s name of Morphology, is but a portion of that wider Science of Form which deals with the forms assumed by matter under all aspects and conditions, and, in a still wider sense, with forms which are theoretically imaginable.

The study of form may be descriptive merely, or it may become analytical. We begin by describing the shape of an object in the simple words of common speech: we end by defining it in the precise language of mathematics; and the one method tends to follow the other in strict scientific order and historical continuity. Thus, for instance, the form of the earth, of a raindrop or a rainbow, the shape of the hanging chain, or the path of a stone thrown up into the air, may all be described, however inadequately, in common words; but when we have learned to comprehend and to define the sphere, the catenary, or the parabola, we have made a wonderful and perhaps a manifold advance. The mathematical definition of a “form” has a quality of precision which was quite lacking in our earlier stage of mere description; it is expressed in few words, or in still briefer symbols, and these {720} words or symbols are so pregnant with meaning that thought itself is economised; we are brought by means of it in touch with Galileo’s aphorism (as old as Plato, as old as Pythagoras, as old perhaps as the wisdom of the Egyptians), that “the Book of Nature is written in characters of Geometry.”

Next, we soon reach through mathematical analysis to mathematical synthesis; we discover homologies or identities which were not obvious before, and which our descriptions obscured rather than revealed: as for instance, when we learn that, however we hold our chain, or however we fire our bullet, the contour of the one or the path of the other is always mathematically homologous. Lastly, and this is the greatest gain of all, we pass quickly and easily from the mathematical conception of form in its statical aspect to form in its dynamical relations: we pass from the conception of form to an understanding of the forces which gave rise to it; and in the representation of form and in the comparison of kindred forms, we see in the one case a diagram of forces in equilibrium, and in the other case we discern the magnitude and the direction of the forces which have sufficed to convert the one form into the other. Here, since a change of material form is only effected by the movement of matter, we have once again the support of the schoolman’s and the philosopher’s axiom, “_Ignorato motu, ignoratur Natura_.”

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In the morphology of living things the use of mathematical methods and symbols has made slow progress; and there are various reasons for this failure to employ a method whose advantages are so obvious in the investigation of other physical forms. To begin with, there would seem to be a psychological reason lying in the fact that the student of living things is by nature and training an observer of concrete objects and phenomena, and the habit of mind which he possesses and cultivates is alien to that of the theoretical mathematician. But this is by no means the only reason; for in the kindred subject of mineralogy, for instance, crystals were still treated in the days of Linnaeus as wholly within the province of the naturalist, and were described by him after the simple methods in use for animals and plants: but as soon as Haüy showed the application of mathematics to {721} the description and classification of crystals, his methods were immediately adopted and a new science came into being.

A large part of the neglect and suspicion of mathematical methods in organic morphology is due (as we have partly seen in our opening chapter) to an ingrained and deep-seated belief that even when we seem to discern a regular mathematical figure in an organism, the sphere, the hexagon, or the spiral which we so recognise merely resembles, but is never entirely explained by, its mathematical analogue; in short, that the details in which the figure differs from its mathematical prototype are more important and more interesting than the features in which it agrees, and even that the peculiar aesthetic pleasure with which we regard a living thing is somehow bound up with the departure from mathematical regularity which it manifests as a peculiar attribute of life. This view seems to me to involve a misapprehension. There is no such essential difference between these phenomena of organic form and those which are manifested in portions of inanimate matter[642]. No chain hangs in a perfect catenary and no raindrop is a perfect sphere: and this for the simple reason that forces and resistances other than the main one are inevitably at work. The same is true of organic form, but it is for the mathematician to unravel the conflicting forces which are at work together. And this process of investigation may lead us on step by step to new phenomena, as it has done in physics, where sometimes a knowledge of form leads us to the interpretation of forces, and at other times a knowledge of the forces at work guides us towards a better insight into form. I would illustrate this by the case of the earth itself. After the fundamental advance had been made which taught us that the world was round, Newton showed that the forces at work upon it must lead to its being imperfectly spherical, and in the course of time its oblate spheroidal shape was actually verified. But now, in turn, it has been shown that its form is still more complicated, and the next step will be to seek for the forces that have deformed the oblate spheroid. {722}

The organic forms which we can define, more or less precisely, in mathematical terms, and afterwards proceed to explain and to account for in terms of force, are of many kinds, as we have seen; but nevertheless they are few in number compared with Nature’s all but infinite variety. The reason for this is not far to seek. The living organism represents, or occupies, a field of force which is never simple, and which as a rule is of immense complexity. And just as in the very simplest of actual cases we meet with a departure from such symmetry as could only exist under conditions of _ideal_ simplicity, so do we pass quickly to cases where the interference of numerous, though still perhaps very simple, causes leads to a resultant which lies far beyond our powers of analysis. Nor must we forget that the biologist is much more exacting in his requirements, as regards form, than the physicist; for the latter is usually content with either an ideal or a general description of form, while the student of living things must needs be specific. The physicist or mathematician can give us perfectly satisfying expressions for the form of a wave, or even of a heap of sand; but we never ask him to define the form of any particular wave of the sea, nor the actual form of any mountain-peak or hill[643]. {723}

For various reasons, then, there are a vast multitude of organic forms which we are unable to account for, or to define, in mathematical terms; and this is not seldom the case even in forms which are apparently of great simplicity and regularity. The curved outline of a leaf, for instance, is such a case; its ovate, lanceolate, or cordate shape is apparently very simple, but the difficulty of finding for it a mathematical expression is very great indeed. To define the complicated outline of a fish, for instance, or of a vertebrate skull, we never even seek for a mathematical formula.

But in a very large part of morphology, our essential task lies in the comparison of related forms rather than in the precise definition of each; and the _deformation_ of a complicated figure may be a phenomenon easy of comprehension, though the figure itself have to be left unanalysed and undefined. This process of comparison, of recognising in one form a definite permutation or _deformation_ of another, apart altogether from a precise and adequate understanding of the original “type” or standard of comparison, lies within the immediate province of mathematics, and finds its solution in the elementary use of a certain method of the mathematician. This method is the Method of Co-ordinates, on which is based the Theory of Transformations.

I imagine that when Descartes conceived the method of co-ordinates, as a generalisation from the proportional diagrams of the artist and the architect, and long before the immense possibilities of this analysis could be foreseen, he had in mind a very simple purpose; it was perhaps no more than to find a way of translating the _form_ of a curve into _numbers_ and into _words_. This is precisely what we do, by the method of co-ordinates, every time we study a statistical curve; and conversely, we translate numbers into form whenever we “plot a curve” to illustrate a table of mortality, a rate of growth, or the daily variation of temperature or barometric pressure. In precisely the same way it is possible to inscribe in a net of rectangular co-ordinates the outline, for instance, of a fish, and so to translate {724} it into a table of numbers, from which again we may at pleasure reconstruct the curve.

But it is the next step in the employment of co-ordinates which is of special interest and use to the morphologist; and this step consists in the alteration, or “transformation,” of our system of co-ordinates and in the study of the corresponding transformation of the curve or figure inscribed in the co-ordinate network.

Let us inscribe in a system of Cartesian co-ordinates the outline of an organism, however complicated, or a part thereof: such as a fish, a crab, or a mammalian skull. We may now treat this complicated figure, in general terms, as a function of _x_, _y_. If we submit our rectangular system to “deformation,” on simple and recognised lines, altering, for instance, the direction of the axes, the ratio of _x_/_y_, or substituting for _x_ and _y_ some more complicated expressions, then we shall obtain a new system of co-ordinates, whose deformation from the original type the inscribed figure will precisely follow. In other words, we obtain a new figure, which represents the old figure _under strain_, and is a function of the new co-ordinates in precisely the same way as the old figure was of the original co-ordinates _x_ and _y_.

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 {725} the joint operation of many causes if few are adequate to the production of it: _Frustra fit per plura, quod fieri potest per pauciora._

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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 laid 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 {726} ‘excess’ and ‘defect[644].’ ” 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 structure 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 {727} 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[645].” 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 character[646].

But if, on the other hand, diverse and dissimilar fishes can be referred as a whole to identical functions of very different co-ordinate 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 {728} 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 general 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 uniform 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[647]. But the deformation due to direct vertical pressure in a horizontal stratum is not nearly so striking as are the deformations produced by the oblique or shearing stresses to which inclined and folded strata have been exposed, and by which their various “dislocations” have been brought about. And especially 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 “species[648].” 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[649].

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If we begin by drawing a net of rectangular equidistant co-ordinates (about the axes _x_ and _y_), we may alter or _deform_ this {729} 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 {730} into an ellipse. In elementary mathematical language, for the original _x_ and _y_ we have substituted _x__{1} and _c_ _y__{1}, and the equation to our original circle, _x_^2 + _y_^2 = _a_^2, becomes that of the ellipse, _x__{1}^2 + _c_^2 _y__{1}^2 = _a_^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′_ = 2_x_/3, we obtain a drawing (B) which is a very close approximation to the cannon-bone 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 _x″_ = _x_/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 ε^{_y_}. It is obvious that this logarithmic extension may involve both abscissae and ordinates, _x_ becoming ε^{_x_}, while _y_ becomes ε^{_y_}. 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 ω. 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 ξ, η by the simple transposition _x_ = ξ − η cot ω, _y_ = η cosec ω.

(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 {731} 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 {732} 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 slightly 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 {733} 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[650] 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, _AOa_, etc., which the radii from the 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 {734} mathematically similar to one another, it is obvious that the respective angles should be in continued proportion, i.e. as _AOa_ is to _AOa′_, 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:

_AOa_ _AOb_ _AOc_ _AOa′_ _AOb′_ _AOc′_ Observed values 12° 28.5° 88° — — 157° Calculated values — — — 21.5° 51.1° — Observed values — — — 20 52 —

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 {735} apple we have two such 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.

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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 lines, 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 θ − _A_ log _r_ = _c_ (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 slight 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 {736} the “law of growth” which our biological analysis by means of orthogonal co-ordinate systems presupposes, or at least foreshadows, is one according to which the organism grows or develops along _stream lines_, which may be defined by a suitable mathematical transformation.

When the system becomes no longer orthogonal, as in many 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 the transformation has been effected.

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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 colouring 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 {737} teeth of the comb is exceedingly complex, and its complexity is revealed in the complicated “diagram of forces” which constitutes the pattern.

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[651]. 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 tube[652]. The alimentary canal, {738} 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.

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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 {739} of length and breadth in this particular bone are extremely significant[653].

If, instead of limiting ourselves to the cannon-bone, we inscribe the entire foot of our several Ungulates in a co-ordinate system, the same ratios of _x_ that served us for the cannon-bones still give us a first approximation to the required comparison; but even in the case of such closely allied forms as the ox and the sheep there is evidently something wanting in the comparison. The reason is that the relative elongation of the several parts, or individual bones, has not proceeded equally or proportionately in all cases; in other words, that the equations for _x_ will not suffice without some simultaneous modification of the values of _y_ (Fig. 362). In such a case it may be found possible to satisfy the varying values of _y_ by some logarithmic or other formula; but, even if that be possible, it will probably be somewhat difficult of discovery or verification in such a case as the present, owing to the fact that we have too few well-marked points of correspondence between the one object and the other, and that especially along the shaft of such long bones as the cannon-bone of the ox, the deer, the llama, or the giraffe there is a complete lack of easily recognisable corresponding points. In such a case a brief tabular statement of apparently corresponding values of _y_, or of those obviously corresponding values which coincide with the boundaries of the several bones of the foot, will, as in the following example, enable us to dispense with a fresh equation.

_a_ _b_ _c_ _d_ _y_ (Ox) 0 18 27 42 100 _y′_ (Sheep) 0 10 19 36 100 _y″_ (Giraffe) 0 5 10 24 100

This summary of values of _y′_, coupled with the equations for the {740} value of _x_, will enable us, from any drawing of the ox’s foot, to construct a figure of that of the sheep or of the giraffe with remarkable accuracy.

That underlying the varying amounts of extension to which the parts or segments of the limb have been subject there is a law, or principle of continuity, may be discerned from such a diagram as the above (Fig. 363), where the values of _y_ in the case of the ox are plotted as a straight line, and the corresponding values for the sheep (extracted from the above table) are seen to form a more or less regular and even curve. This simple graphic result implies the existence of a comparatively simple equation between _y_ and _y′_.

An elementary application of the principle of co-ordinates to the study of proportion, as we have here used it to illustrate the varying proportions of a bone, was in common use in the sixteenth and seventeenth centuries by artists in their study of the human form. The method is probably much more ancient, and may even be classical[654]; it is fully described and put in practice by Albert Dürer in his _Geometry_, and especially in his _Treatise on Proportion_[655]. In this latter work, the manner in which the {741} human figure, features, and facial expression are all transformed and modified by slight variations in the relative magnitude of the parts is admirably and copiously illustrated (Fig. 364).

In a tapir’s foot there is a striking difference, and yet at the same time there is an obvious underlying resemblance, between the middle toe and either of its unsymmetrical lateral neighbours. Let us take the median terminal phalanx and inscribe its outline in a net of rectangular equidistant co-ordinates (Fig. 365, _a_). Let us then make a similar network about axes which are no longer at right angles, but inclined to one another at an angle of about 50° (_b_). If into this new network we fill in, point for point, an outline precisely corresponding to our original drawing of the middle toe, we shall find that we have already represented the main features of the adjacent lateral one. We shall, however, perceive that our new diagram looks a little too bulky on one side, the inner side, of the lateral toe. If now we substitute for our equidistant ordinates, ordinates which get gradually closer and closer together as we pass towards the median side of the toe, then we shall obtain a diagram which differs in no essential respect from an actual outline copy of the lateral toe (_c_). In short, the difference between the outline of the middle toe of the tapir and the next lateral toe may be almost completely expressed by saying that if the one be represented by rectangular equidistant co-ordinates, the other will be represented by oblique co-ordinates, whose axes make an angle of 50°, and in which the abscissal interspaces decrease in a certain logarithmic ratio. We treated our original complex curve or projection of the tapir’s toe as a function of the form _F_ (_x_, _y_) = 0. The figure of the tapir’s lateral {742} toe is a precisely identical function of the form _F_ (_e_^{_x_}, _y__{1}) = 0, where _x__{1}, _y__{1} are oblique co-ordinate axes inclined to one another at an angle of 50°.

Dürer was acquainted with these oblique co-ordinates also, and I have copied two illustrative figures from his book[656].

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In Fig. 367 I have sketched the common Copepod _Oithona nana_, {743} and have inscribed it in a rectangular net, with abscissae three-fifths the length of the ordinates. Side by side (Fig. 368) is drawn a very different Copepod, of the genus _Sapphirina_; and about it is drawn a network such that each co-ordinate passes (as nearly as possible) through points corresponding to those of the former figure. It will be seen that two differences are apparent. (1) The values of _y_ in Fig. 368 are large in the upper part of the figure, and diminish rapidly towards its base. (2) The values of _x_ are very large in the neighbourhood of the origin, but diminish rapidly as we pass towards either side, away from the median vertical axis; and it is probable that they do so according to a definite, but somewhat complicated, ratio. If, instead of seeking for an actual equation, we simply tabulate our values of _x_ and _y_ in the second figure as compared with the first (just as we did in comparing the feet of the Ungulates), we get the dimensions of a net in which, by simply projecting the figure of _Oithona_, we obtain that of _Sapphirina_ without further trouble, e.g.:

_x_ (_Oithona_) 0 3 6 9 12 15 — _x′_ (_Sapphirina_) 0 8 10 12 13 14 —

_y_ (_Oithona_) 0 5 10 15 20 25 30 _y′_ (_Sapphirina_) 0 2 7 3 23 32 40

In this manner, with a single model or type to copy from, we may record in very brief space the data requisite for the production of approximate outlines of a great number of forms. For instance the difference, at first sight immense, between the attenuated body of a _Caprella_ and the thick-set body of a _Cyamus_ is obviously little, and is probably nothing, more than a difference of relative magnitudes, capable of tabulation by numbers and of complete expression by means of rectilinear co-ordinates.

The Crustacea afford innumerable instances of more complex deformations. Thus we may compare various higher Crustacea with one another, even in the case of such dissimilar forms as a lobster and a crab. It is obvious that the whole body of the former is elongated as compared with the latter, and that the crab is relatively broad in the region of the carapace, while it tapers off rapidly towards its attenuated and abbreviated tail. In a general way, the elongated rectangular system of co-ordinates {744} in which we may inscribe the outline of the lobster becomes a shortened triangle in the case of the crab. In a little more detail we may compare the outline of the carapace in various crabs one with another: and the comparison will be found easy and significant, even, in many cases, down to minute details, such as the number and situation of the marginal spines, though these are in other cases subject to independent variability.

If we choose, to begin with, such a crab as _Geryon_ (Fig. 369, 1), and inscribe it in our equidistant rectangular co-ordinates, we shall see that we pass easily to forms more elongated in a transverse {745} direction, such as _Matuta_ or _Lupa_ (5), and conversely, by transverse compression, to such a form as _Corystes_ (2). In certain other cases the carapace conforms to a triangular diagram, more or less curvilinear, as in Fig. 4, which represents the genus _Paralomis_. Here we can easily see that the posterior border is transversely elongated as compared with that of _Geryon_, while at the same time the anterior part is longitudinally extended as compared with the posterior. A system of slightly curved and converging ordinates, with orthogonal and logarithmically interspaced abscissal lines, as shown in the figure, appears to satisfy the conditions.

In an interesting series of cases, such as the genus _Chorinus_, or _Scyramathia_, and in the spider-crabs generally, we appear to have just the converse of this. While the carapace of these crabs presents a somewhat triangular form, which seems at first sight more or less similar to those just described, we soon see that the actual posterior border is now narrow instead of broad, the broadest part of the carapace corresponding precisely, not to that which is broadest in _Paralomis_, but to that which was broadest in _Geryon_; while the most striking difference from the latter lies in an antero-posterior lengthening of the forepart of the carapace, culminating in a great elongation of the frontal region, with its two spines or “horns.” The curved ordinates here converge posteriorly and diverge widely in front (Figs. 3 and 6), while the decremental interspacing of the abscissae is very marked indeed.

We put our method to a severer test when we attempt to sketch an entire and complicated animal than when we simply compare corresponding parts such as the carapaces of various Malacostraca, or related bones as in the case of the tapir’s toes. Nevertheless, up to a certain point, the method stands the test very well. In other words, one particular mode and direction of variation is often (or even usually) so prominent and so paramount throughout the entire organism, that one comprehensive system of co-ordinates suffices to give a fair picture of the actual phenomenon. To take another illustration from the Crustacea, I have drawn roughly in Fig. 370, 1 a little amphipod of the family Phoxocephalidae (_Harpinia_ sp.). Deforming the co-ordinates of the figure into the {746} curved orthogonal system in Fig. 2, we at once obtain a very fair representation of an allied genus, belonging to a different family of amphipods, namely _Stegocephalus_. As we proceed further from our type our co-ordinates will require greater deformation, and the resultant figure will usually be somewhat less accurate. In Fig. 3 I show a network, to which, if we transfer our diagram of _Harpinia_ or of _Stegocephalus_, we shall obtain a tolerable representation of the aberrant genus _Hyperia_, with its narrow abdomen, its reduced pleural lappets, its great eyes, and its inflated head.

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The hydroid zoophytes constitute a “polymorphic” group, within which a vast number of species have already been distinguished; and the labours of the systematic naturalist are constantly adding to the number. The specific distinctions are for the most part based, not upon characters directly presented {747} by the living animal, but upon the form, size and arrangement of the little cups, or “calycles,” secreted and inhabited by the little individual polypes which compose the compound organism. The variations, which are apparently infinite, of these conformations are easily seen to be a question of relative magnitudes, and are capable of complete expression, sometimes by very simple, sometimes by somewhat more complex, co-ordinate networks.

For instance, the varying shapes of the simple wineglass-shaped cups of the Campanularidae are at once sufficiently represented and compared by means of simple Cartesian co-ordinates (Fig. 371). In the two allied families of Plumulariidae and Aglaopheniidae the calycles are set unilaterally upon a jointed stem, and small cup-like structures (holding rudimentary polypes) are associated with the large calycles in definite number and position. These small calyculi are variable in number, but in the great majority of cases they accompany the large calycle in groups of three—two standing by its upper border, and one, which is especially variable in form and magnitude, lying at its base. The stem is liable to flexure and, in a high degree, to extension or compression; and these variations extend, often on an exaggerated scale, to the related calycles. As a result we find that we can draw various systems of curved or sinuous co-ordinates, which express, all but completely, the configuration of the various {748} hydroids which we inscribe therein (Fig. 372). The comparative smoothness or denticulation of the margin of the calycle, and the number of its denticles, constitutes an independent variation, and requires separate description; we have already seen (p. 236) that this denticulation is in all probability due to a particular physical cause.

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Among the fishes we discover a great variety of deformations, some of them of a very simple kind, while others are more striking and more unexpected. A comparatively simple case, involving a simple shear, is illustrated by Figs. 373 and 374. Fig. 373 represents, within Cartesian co-ordinates, a certain little oceanic fish known as _Argyropelecus Olfersi_. Fig. 474 represents precisely the same outline, transferred to a system of oblique co-ordinates whose {749} axes are inclined at an angle of 70°; but this is now (as far as can be seen on the scale of the drawing) a very good figure of an allied fish, assigned to a different genus, under the name of _Sternoptyx diaphana_. The deformation illustrated by this case of _Argyropelecus_ is precisely analogous to the simplest and commonest kind of deformation to which fossils are subject (as we have seen on p. 553) as the result of shearing-stresses in the solid rock.

Fig. 375 is an outline diagram of a typical Scaroid fish. Let us deform its rectilinear co-ordinates into a system of (approximately) coaxial circles, as in Fig. 376, and then filling into the new system, space by space and point by point, our former diagram of _Scarus_, we obtain a very good outline of an allied fish, belonging to a neighbouring family, of the genus _Pomacanthus_. This case is all the more interesting, because upon the body of our _Pomacanthus_ there are striking colour bands, which correspond in direction very closely to the lines of our new curved ordinates. In like manner, the still more bizarre outlines of other fishes of the same family of Chaetodonts will be found to correspond to very slight modifications of similar co-ordinates; in other words, to small variations in the values of the constants of the coaxial curves.