Chapter XVII (2)

In Figs. 377–380 I have represented another series of Acanthopterygian fishes, not very distantly related to the foregoing. If we start this series with the figure of _Polyprion_, in Fig. 377, we see that the outlines of _Pseudopriacanthus_ (Fig. 378) and of _Sebastes_ or _Scorpaena_ (Fig. 379) are easily derived by substituting a system of triangular, or radial, co-ordinates for the rectangular ones in {750} which we had inscribed _Polyprion_. The very curious fish _Antigonia capros_, an oceanic relative of our own “boar-fish,” conforms closely to the peculiar deformation represented in Fig. 380.

Fig. 381 is a common, typical _Diodon_ or porcupine-fish, and in Fig. 382 I have deformed its vertical co-ordinates into a system of concentric circles, and its horizontal co-ordinates into a system of curves which, approximately and provisionally, are made to resemble a system of hyperbolas[657]. The old outline, transferred {751} in its integrity to the new network, appears as a manifest representation of the closely allied, but very different looking, sunfish, _Orthagoriscus mola_. This is a particularly instructive case of deformation or transformation. It is true that, in a mathematical sense, it is not a perfectly satisfactory or perfectly regular deformation, for the system is no longer isogonal; but nevertheless, it is symmetrical to the eye, and obviously approaches to an isogonal system under certain conditions of friction or constraint. And as such it accounts, by one single integral transformation, for all the apparently separate and distinct external differences between the two fishes. It leaves the parts near to the origin of the system, the whole region of the head, the opercular orifice and the pectoral fin, practically unchanged {752} in form, size and position; and it shews a greater and greater apparent modification of size and form as we pass from the origin towards the periphery of the system.

In a word, it is sufficient to account for the new and striking contour in all its essential details, of rounded body, exaggerated dorsal and ventral fins, and truncated tail. In like manner, and using precisely the same co-ordinate networks, it appears to me possible to shew the relations, almost bone for bone, of the skeletons of the two fishes; in other words, to reconstruct the skeleton of the one from our knowledge of the skeleton of the other, under the guidance of the same correspondence as is indicated in their external configuration.

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The family of the crocodiles has had a special interest for the evolutionist ever since Huxley pointed out that, in a degree only second to the horse and its ancestors, it furnishes us with a close and almost unbroken series of transitional forms, running down in continuous succession from one geological formation to another. I should be inclined to transpose this general statement into other terms, and to say that the Crocodilia constitute a case in which, with unusually little complication from the presence of independent variants, the trend of one particular mode of transformation is visibly manifested. If we exclude meanwhile from our comparison a few of the oldest of the crocodiles, such as _Belodon_, which differ more fundamentally from the rest, we shall find a long series of genera in which we can refer not only the changing contours of the skull, but even the shape and size of the many constituent bones and their intervening spaces or “vacuities,” to one and the same simple system of transformed co-ordinates. The manner in which the skulls of various Crocodilians differ from one another may be sufficiently illustrated by three or four examples.

Let us take one of the typical modern crocodiles as our standard of form, e.g. _C. porosus_, and inscribe it, as in Fig. 383, _a_, in the usual Cartesian co-ordinates. By deforming the rectangular network into a triangular system, with the apex of the triangle a little way in front of the snout, as in _b_, we pass to such a form as _C. americanus_. By an exaggeration of the same process we at once get an approximation to the form of one of the sharp-snouted, {753} or longirostrine, crocodiles, such as the genus _Tomistoma_; and, in the species figured, the oblique position of the orbits, the arched contour of the occipital border, and certain other characters suggest a certain amount of curvature, such as I have represented in the diagram (Fig. 383, _b_), on the part of the horizontal co-ordinates. In the still more elongated skull of such a form as the Indian Gavial, the whole skull has undergone a great longitudinal extension, or, in other words, the ratio of _x_/_y_ is greatly diminished; and this extension is not uniform, but is at a maximum in the region of the nasal and maxillary bones. This especially elongated region is at the same time narrowed in an exceptional degree, and its excessive narrowing is represented by a curvature, convex towards the median axis, on the part of the vertical ordinates. Let us take as a last illustration one of the Mesozoic crocodiles, the little _Notosuchus_, from the Cretaceous formation. This little crocodile is very different from our type in the proportions of its skull. The region of the snout, in front of and including the frontal bones, is greatly shortened; from constituting fully two-thirds of the whole length of the skull in _Crocodilus_, it now constitutes less than half, or, say, three-sevenths of the whole; and the whole skull, and especially its posterior part, is curiously compact, broad, and squat. The orbit is unusually large. If in the diagram of this skull we select a number of points obviously corresponding {754} to points where our rectangular co-ordinates intersect particular bones or other recognisable features in our typical crocodile, we shall easily discover that the lines joining these points in _Notosuchus_ fall into such a co-ordinate network as that which is represented in Fig. 383, _c_. To all intents and purposes, then, this not very complex system, representing one harmonious “deformation,” accounts for _all_ the differences between the two figures, and is sufficient to enable one at any time to reconstruct a detailed drawing, bone for bone, of the skull of _Notosuchus_ from the model furnished by the common crocodile.

The many diverse forms of Dinosaurian reptiles, all of which manifest a strong family likeness underlying much superficial diversity, furnish us with plentiful material for comparison by the method of transformations. As an instance, I have figured the pelvic bones of _Stegosaurus_ and of _Camptosaurus_ (Fig. 384, _a_, _b_) to show that, when the former is taken as our Cartesian type, a slight curvature and an approximately logarithmic extension of the _x_-axis brings us easily to the configuration of the other. In the original specimen of _Camptosaurus_ described by Marsh[658], the anterior portion of the iliac bone is missing; and in Marsh’s restoration this part of the bone is drawn as though it came somewhat abruptly to a sharp point. In my figure I {755} have completed this missing part of the bone in harmony with the general co-ordinate network which is suggested by our comparison of the two entire pelves; and I venture to think that the result is more natural in appearance, and more likely to be correct than was Marsh’s conjectural restoration. It would seem, in fact, that there is an obvious field for the employment of the method of co-ordinates in this task of reproducing missing portions of a structure to the proper scale and in harmony with related types. To this subject we shall presently return.

In Fig. 385, _a_, _b_, I have drawn the shoulder-girdle of _Cryptocleidus_, a Plesiosaurian reptile, half-grown in the one case and full-grown in the other. The change of form during growth in this region of the body is very considerable, and its nature is well brought out by the two co-ordinate systems. In Fig. 386 I have drawn the shoulder-girdle of an Ichthyosaur, referring it to _Cryptocleidus_ as a standard of comparison. The interclavicle, which is present in _Ichthyosaurus_, is minute and hidden in _Cryptocleidus_; but the numerous other differences between the two {756} forms, chief among which is the great elongation in _Ichthyosaurus_ of the two clavicles, are all seen by our diagrams to be part and parcel of one general and systematic deformation.

Before we leave the group of reptiles we may glance at the very strangely modified skull of _Pteranodon_, one of the extinct flying reptiles, or Pterosauria. In this very curious skull the region of the jaws, or beak, is greatly elongated and pointed; the occipital bone is drawn out into an enormous backwardly-directed crest; the posterior part of the lower jaw is similarly produced backwards; the orbit is small; and the quadrate bone is strongly inclined downwards and forwards. The whole skull has a configuration which stands, apparently, in the strongest possible contrast to that of a more normal Ornithosaurian such as _Dimorphodon_. But if we inscribe the latter in Cartesian coordinates (Fig. 387, _a_), and refer our _Pteranodon_ to a system of oblique co-ordinates (_b_), in which the two co-ordinate systems of parallel lines become each a pencil of diverging rays, we make manifest a correspondence which extends uniformly throughout all parts of these very different-looking skulls.

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We have dealt so far, and for the most part we shall continue to deal, with our co-ordinate method as a means of comparing one known structure with another. But it is obvious, as I have said, {757} that it may also be employed for drawing hypothetical structures, on the assumption that they have varied from a known form in some definite way. And this process may be especially useful, and will be most obviously legitimate, when we apply it to the particular case of representing intermediate stages between two forms which are actually known to exist, in other words, of reconstructing the transitional stages through which the course of evolution must have successively travelled if it has brought about the change from some ancestral type to its presumed descendant. Some little time ago I sent to my friend, Mr Gerhard Heilmann of Copenhagen, a few of my own rough co-ordinate diagrams, including some in which the pelves of certain ancient and primitive birds were compared one with another. Mr Heilmann, who is both a skilled draughtsman and an able morphologist, returned me a set of diagrams which are a vast improvement on my own, {758} and which are reproduced in Figs. 388–393. Here we have, as extreme cases, the pelvis of _Archaeopteryx_, the most ancient of known birds, and that of _Apatornis_, one of the fossil “toothed”

birds from the North American Cretaceous formations—a bird shewing some resemblance to the modern terns. The pelvis of _Archaeopteryx_ is taken as our type, and referred accordingly to {759} Cartesian co-ordinates (Fig. 388); while the corresponding coordinates of the very different pelvis of _Apatornis_ are represented in Fig. 389. In Fig. 390 the outlines of these two co-ordinate systems are superposed upon one another, and those of three intermediate and equidistant co-ordinate systems are interpolated between them. From each of these latter systems, so determined by direct interpolation, a complete co-ordinate diagram is drawn, and the corresponding outline of a pelvis is found from each of these systems of co-ordinates, as in Figs. 391, 392. Finally, in Fig. 393 the complete series is represented, beginning with the known pelvis of _Archaeopteryx_, and leading up by our three intermediate hypothetical types to the known pelvis of _Apatornis_.

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Among mammalian skulls I will take two illustrations only, one drawn from a comparison of the human skull with that of the higher apes, and another from the group of Perissodactyle {760} Ungulates, the group which includes the rhinoceros, the tapir, and the horse.

Let us begin by choosing as our type the skull of _Hyrachyus agrarius_, Cope, from the Middle Eocene of North America, as figured by Osborn in his Monograph of the Extinct Rhinoceroses[659] (Fig. 394).

The many other forms of primitive rhinoceros described in the monograph differ from _Hyrachyus_ in various details—in the characters of the teeth, sometimes in the number of the toes, and so forth; and they also differ very considerably in the general {761} appearance of the skull. But these differences in the conformation of the skull, conspicuous as they are at first sight, will be found easy to bring under the conception of a simple and homogeneous transformation, such as would result from the application of some not very complicated stress. For instance, the corresponding co-ordinates of _Aceratherium tridactylum_, as shown in Fig. 395, indicate that the essential difference between this skull and the former one may be summed up by saying that the long axis of the skull of _Aceratherium_ has undergone a slight double curvature, while the upper parts of the skull have at the same time been {762} subject to a vertical expansion, or to growth in somewhat greater proportion than the lower parts. Precisely the same changes, on a somewhat greater scale, give us the skull of an existing rhinoceros.

Among the species of _Aceratherium_, the posterior, or occipital, view of the skull presents specific differences which are perhaps more conspicuous than those furnished by the side view; and these differences are very strikingly brought out by the series of conformal transformations which I have represented in Fig. 396. In this case it will perhaps be noticed that the correspondence is not always quite accurate in small details. It could easily have been made much more accurate by giving a slightly sinuous curvature to certain of the co-ordinates. But as they stand, the correspondence indicated is very close, and the simplicity of the figures illustrates all the better the general character of the transformation.

By similar and not more violent changes we pass easily to such allied forms as the Titanotheres (Fig. 397); and the well-known series of species of _Titanotherium_, by which Professor Osborn has {763} illustrated the evolution of this genus, constitutes a simple and suitable case for the application of our method.

But our method enables us to pass over greater gaps than these, and to discern the general, and to a very large extent even the detailed, resemblances between the skull of the rhinoceros and those of the tapir or the horse. From the Cartesian co-ordinates in which we have begun by inscribing the skull of a primitive rhinoceros, we pass to the tapir’s skull (Fig. 398), firstly, by converting the rectangular into a triangular network, by which we represent the depression of the anterior and the progressively increasing elevation of the posterior part of the skull; and secondly, by giving to the vertical ordinates a curvature such as to bring about a certain longitudinal compression, or condensation, in the forepart of the skull, especially in the nasal and orbital regions.

The conformation of the horse’s skull departs from that of our primitive Perissodactyle (that is to say our early type of rhinoceros, _Hyrachyus_) in a direction that is nearly the opposite of that taken by _Titanotherium_ and by the recent species of rhinoceros. For we perceive, by Fig. 399, that the horizontal co-ordinates, which in these latter cases became transformed into curves with the concavity upwards, are curved, in the case of the horse, in the opposite direction. And the vertical ordinates, which are also curved, somewhat in the same fashion as in the tapir, are very nearly equidistant, instead of being, as in that animal, crowded together anteriorly. Ordinates and abscissae form an oblique {764} system, as is shown in the figure. In this case I have attempted to produce the network beyond the region which is actually required to include the diagram of the horse’s skull, in order to show better the form of the general transformation, with a part only of which we have actually to deal.

It is at first sight not a little surprising to find that we can pass, by a cognate and even simpler transformation, from our Perissodactyle skulls to that of the rabbit; but the fact that we can easily do so is a simple illustration of the undoubted affinity which exists between the Rodentia, especially the family of the Leporidae, and the more primitive Ungulates. For my part, I would go further; for I think there is strong reason to believe that the Perissodactyles are more closely related to the Leporidae than the former are to the other Ungulates, or than the Leporidae are to the rest of the Rodentia. Be that as it may, it is obvious from Fig. 400 that the rabbit’s skull conforms to a system of {765} co-ordinates corresponding to the Cartesian co-ordinates in which we have inscribed the skull of _Hyrachyus_, with the difference, firstly, that the horizontal ordinates of the latter are transformed into equidistant curved lines, approximately arcs of circles, with their concavity directed downwards; and secondly, that the vertical ordinates are transformed into a pencil of rays approximately orthogonal to the circular arcs. In short, the configuration of the rabbit’s skull is derived from that of our primitive rhinoceros by the unexpectedly simple process of submitting the latter to a

strong and uniform flexure in the downward direction (cf. Fig. 358, p. 731). In the case of the rabbit the configuration of the individual bones does not conform quite so well to the general transformation as it does when we are comparing the several Perissodactyles one with another; and the chief departures from conformity will be found in the size of the orbit and in the outline of the immediately surrounding bones. The simple fact is that the relatively enormous eye of the rabbit constitutes an independent variation, which cannot be brought into the general and fundamental transformation, but must be dealt with {768} separately. The enlargement of the eye, like the modification in form and number of the teeth, is a separate phenomenon, which supplements but in no way contradicts our general comparison of the skulls taken in their entirety.

Fig. 402. _A_, skull of _Hyracotherium_, from the Eocene, after W. B. Scott; _H_, skull of horse, represented as a co-ordinate transformation of that of _Hyracotherium_, and to the same scale of magnitude; _B_–_G_, various artificial or imaginary types, reconstructed as intermediate stages between _A_ and _H_; _M_, skull of _Mesohippus_, from the Oligocene, after Scott, for comparison with _C_; _P_, skull of _Protohippus_, from the Miocene, after Cope, for comparison with _E_; _Pp_, lower jaw of _Protohippus placidus_ (after Matthew and Gidley), for comparison with _F_; _Mi_, _Miohippus_ (after Osborn), _Pa_, _Parahippus_ (after Peterson), shewing resemblance, but less perfect agreement, with _C_ and _D_.]

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Before we leave the Perissodactyla and their allies, let us look a little more closely into the case of the horse and its immediate relations or ancestors, doing so with the help of a set of diagrams which I again owe to Mr Gerard Heilmann[660]. Here we start afresh, with the skull (Fig. 402, _A_) of _Hyracotherium_ (or _Eohippus_), inscribed in a simple Cartesian network. At the other end of the series (_H_) is a skull of Equus, in its own corresponding network; and the intermediate stages (_B_–_G_) are all drawn by direct and simple interpolation, as in Mr Heilmann’s former series of drawings of _Archaeopteryx_ and _Apatornis_. In this present case, the relative magnitudes are shewn, as well as the forms, of the several skulls. Alongside of these reconstructed diagrams, are set figures of certain extinct “horses” (Equidae or Palaeotheriidae), and in two cases, viz. _Mesohippus_ and _Protohippus_ (_M_, _P_), it will be seen that the actual fossil skull coincides in the most perfect fashion with one of the hypothetical forms or stages which our method shews to be implicitly involved in the transition from _Hyracotherium_ to _Equus_. In a third case, that of _Parahippus_ (_Pa_), the correspondence (as Mr Heilmann points out) is by no means exact. The outline of this skull comes nearest to that of the hypothetical transition stage _D_, but the “fit” is now a bad one; for the skull of _Parahippus_ is evidently a longer, straighter and narrower skull, and differs in other minor characters besides. In short, though some writers have placed _Parahippus_ in the direct line of descent between _Equus_ and _Eohippus_, we see at once that there is no place for it there, and that it must, accordingly, represent a somewhat divergent branch or offshoot of the Equidae[661]. It may be noticed, especially in the case of _Protohippus_ {769} (_P_), that the configuration of the angle of the jaw does not tally quite so accurately with that of our hypothetical diagrams as do other parts of the skull. As a matter of fact, this region is somewhat variable, in different species of a genus, and even in different individuals of the same species; in the small figure (_Pp_) of _Protohippus placidus_ the correspondence is more exact.

In considering this series of figures we cannot but be struck, not only with the regularity of the succession of “transformations,” but also with the slight and inconsiderable differences which separate the known and recorded stages, and even the two extremes of the whole series. These differences are no greater (save in regard to actual magnitude) than those between one human skull and another, at least if we take into account the older or remoter races; and they are again no greater, but if anything less, than the range of variation, racial and individual, in certain other human bones, for instance the scapula[662].

The variability of this latter bone is great, but it is neither {770} surprising nor peculiar; for it is linked with all the considerations of mechanical efficiency and functional modification which we dealt with in our last chapter. The scapula occupies, as it were, a focus in a very important field of force; and the lines of force converging on it will be very greatly modified by the varying development of the muscles over a large area of the body and of the uses to which they are habitually put.

Let us now inscribe in our Cartesian co-ordinates the outline of a human skull (Fig. 404), for the purpose of comparing it with the skulls of some of the higher apes. We know beforehand that the main differences between the human and the simian types depend upon the enlargement or expansion of the brain and braincase in man, and the relative diminution or enfeeblement of his jaws. Together with these changes, the “facial angle” increases from an oblique angle to nearly a right angle in man, {771} and the configuration of every constituent bone of the face and skull undergoes an alteration. We do not know to begin with, and we are not shewn by the ordinary methods of comparison, how far these various changes form part of one harmonious and congruent transformation, or whether we are to look, for instance, upon the changes undergone by the frontal, the occipital, the maxillary, and the mandibular regions as a congeries of separate modifications or independent variants. But as soon as we have marked out a number of points in the gorilla’s or chimpanzee’s skull, corresponding with those which our co-ordinate network intersected in the human skull, we find that these corresponding points may be at once linked up by smoothly curved lines of intersection, which form a new system of co-ordinates and constitute a simple “projection” of our human skull. The network represented in Fig. 405 constitutes such a projection of the human skull on what we may call, figuratively speaking, the “plane” of the chimpanzee; and the full diagram in Fig. 406 demonstrates the correspondence. In Fig. 407 I have shewn the similar deformation in the case of a baboon, and it is obvious that the transformation is of precisely the same order, and differs only in an increased intensity or degree of deformation.

In both dimensions, as we pass from above downwards and from behind forwards, the corresponding areas of the network are seen to increase in a gradual and approximately logarithmic order in the lower as compared with the higher type of skull; and, in short, it becomes at once manifest that the modifications of jaws, braincase, and the regions between are all portions of one continuous and integral process. It is of course easy to draw the {772} inverse diagrams, by which the Cartesian co-ordinates of the ape are transformed into curvilinear and non-equidistant co-ordinates in man.

From this comparison of the gorilla’s or chimpanzee’s with the human skull we realise that an inherent weakness underlies the anthropologist’s method of comparing skulls by reference to a small number of axes. The most important of these are the “facial” and “basicranial” axes, which include between them the “facial angle.” But it is, in the first place, evident that these axes are merely the principal axes of a system of co-ordinates, and that their restricted and isolated use neglects all that can be learned from the filling in of the rest of the co-ordinate network. And, in the second place, the “facial axis,” for instance, as ordinarily used in the anthropological comparison of one human skull with another, or of the human skull with the gorilla’s, is in all cases treated as a straight line; but our investigation has shewn that rectilinear axes only meet the case in the simplest and most closely related transformations; and that, for instance, in the anthropoid skull no rectilinear axis is homologous with a rectilinear axis in a man’s skull, but what is a straight line in the one has become a certain definite curve in the other.

Mr Heilmann tells me that he has tried, but without success, to obtain a transitional series between the human skull and some prehuman, anthropoid type, which series (as in the case of the Equidae) should be found to contain other known types in direct linear sequence. It appears impossible, however, to obtain such a series, or to pass by successive and continuous gradations through such forms as Mesopithecus, Pithecanthropus, _Homo neanderthalensis_, and the lower or higher races of modern man. The failure is not the fault of our method. It merely indicates that no one straight line of descent, or of consecutive transformation, exists; but on the contrary, that among human and anthropoid types, recent and extinct, we have to do with a complex problem of divergent, rather than of continuous, variation. And in like manner, easy as it is to correlate the baboon’s and chimpanzee’s skulls severally with that of man, and easy as it is to see that the chimpanzee’s skull is much nearer to the human type than is the baboon’s, it is also not difficult to perceive that the series is not, {773} strictly speaking, continuous, and that neither of our two apes lies _precisely_ on the same direct line or sequence of deformation by which we may hypothetically connect the other with man.

As a final illustration I have drawn the outline of a dog’s skull (Fig. 408), and inscribed it in a network comparable with the Cartesian network of the human skull in Fig. 404. Here we attempt to bridge over a wider gulf than we have crossed in any of our former comparisons. But, nevertheless, it is obvious that our method still holds good, in spite of the fact that there are various specific differences, such as the open or closed orbit, etc., which have to be separately described and accounted for. We see that the chief essential differences in plan between the dog’s skull and the man’s lie in the fact that, relatively speaking, the former tapers away in front, a triangular taking the place of a rectangular conformation; secondly, that, coincident with the tapering off, there is a progressive elongation, or pulling out, of the whole forepart of the skull; and lastly, as a minor difference, that the straight vertical ordinates of the human skull become curved, with their convexity directed forwards, in the dog. While the net result is that in the dog, just as in the chimpanzee, the brain-pan is smaller and the jaws are larger than in man, it is now conspicuously evident that the co-ordinate network of the ape is by no means intermediate between those which fit the other two. The mode of deformation is on different lines; and, while it may be correct to say that the chimpanzee and the baboon are more brute-like, it would be by no means accurate to assert that they are more dog-like, than man. {774}

In this brief account of co-ordinate transformations and of their morphological utility I have dealt with plane co-ordinates only, and have made no mention of the less elementary subject of co-ordinates in three-dimensional space. In theory there is no difficulty whatsoever in such an extension of our method; it is just as easy to refer the form of our fish or of our skull to the rectangular co-ordinates _x_, _y_, _z_, or to the polar co-ordinates ξ, η, ζ, as it is to refer their plane projections to the two axes to which our investigation has been confined. And that it would be advantageous to do so goes without saying; for it is the shape of the solid object, not that of the mere drawing of the object, that we want to understand; and already we have found some of our easy problems in solid geometry leading us (as in the case of the form of the bivalve and even of the univalve shell) quickly in the direction of co-ordinate analysis and the theory of conformal transformations. But this extended theme I have not attempted to pursue, and it must be left to other times, and to other hands. Nevertheless, let us glance for a moment at the sort of simple cases, the simplest possible cases, with which such an investigation might begin; and we have found our plane co-ordinate systems so easily and effectively applicable to certain fishes that we may seek among them for our first and tentative introduction to the three-dimensional field.

It is obvious enough that the same method of description and analysis which we have applied to one plane, we may apply to another: drawing by observation, and by a process of trial and error, our various cross-sections and the co-ordinate systems which seem best to correspond. But the new and important problem which now emerges is to _correlate_ the deformation or transformation which we discover in one plane with that which we have observed in another: and at length, perhaps, after grasping the general principles of such correlation, to forecast approximately what is likely to take place in the other two planes of reference when we are acquainted with one, that is to say, to determine the values along one axis in terms of the other two.

Let us imagine a common “round” fish, and a common “flat” fish, such as a haddock and a plaice. These two fishes are not as nicely adapted for comparison by means of plane co-ordinates as {775} some which we have studied, owing to the presence of essentially unimportant, but yet conspicuous differences in the position of the eyes, or in the number of the fins,—that is to say in the manner in which the continuous dorsal fin of the plaice appears in the haddock to be cut or scolloped into a number of separate fins. But speaking broadly, and apart from such minor differences as these, it is manifest that the chief factor in the case (so far as we at present see) is simply the broadening out of the plaice’s body, as compared with the haddock’s, in the dorso-ventral direction, that is to say, along the _y_ axis; in other words, the ratio _x_/_y_ is much less, (and indeed little more than half as great), in the haddock than in the plaice. But we also recognise at once that while the plaice (as compared with the haddock) is expanded in one direction, it is also flattened, or thinned out, in the other: _y_ increases, but _z_ diminishes, relatively to _x_. And furthermore, we soon see that this is a common or even a general phenomenon. The high, expanded body in our Antigonia or in our sun-fish is at the same time flattened or _compressed_ from side to side, in comparison with the related fishes which we have chosen as standards of reference or comparison; and conversely, such a fish as the skate, while it is expanded from side to side in comparison with a shark or dogfish, is at the same time flattened or _depressed_ in its vertical section. We proceed then, to enquire whether there be any simple relation of _magnitude_ discernible between these twin factors of expansion and compression; and the very fact that the two dimensions tend to vary _inversely_ already assures us that, in the general process of deformation, the _volume_ is less affected than are the _linear dimensions_. Some years ago, when I was studying the length-weight co-efficient in fishes (of which we have already spoken in Chap. III, p. 98), that is to say the coefficient _k_ in the formula _W_ = _k_ _L_^3, or _k_ = _W_/_L_^3, I was not a little surprised to find that _k_ was all but identical in two such different looking fishes as our haddock and our plaice: thus indicating that these two fishes, little as they resemble one another externally (though they belong to two closely related families), have approximately the same _volume_ when they are equal in _length_; or, in other words, that the extent to which the plaice’s body has become expanded or broadened is _just about {776} compensated for_ by the extent to which it has also got flattened or thinned. In short, if we could permit ourselves to conceive of a haddock being directly transformed into a plaice, a very large part of the change would be simply accounted for by supposing the former fish to be “rolled out,” as a baker rolls a piece of dough. This is, as it were, an extreme case of the _balancement des organes_, or “compensation of parts.”

Simple Cartesian co-ordinates will not suffice very well to compare the haddock with the plaice, for the deformation undergone by the former in comparison with the latter is more on the lines of that by which we have compared our Antigonia with our Polyprion; that is to say, the expansion is greater towards the middle of the fish’s length, and dwindles away towards either end. But again simplifying our illustration to the utmost, and being content with a rough comparison, we may assert that, when haddock and plaice are brought to the same standard of length, we can inscribe them both (approximately) in rectangular co-ordinate networks, such that _Y_ in the plaice is about twice as great as _y_ in the haddock. But if the volumes of the two fishes be equal, this is as much as to say that _xyz_ in the one case (or rather the summation of all these values) is equal to _XYZ_ in the other; and therefore (since _X_ = _x_, and _Y_ = 2_y_), it follows that _Z_ = _z_/2. When we have drawn our vertical transverse section of the haddock (or projected that fish in the _yz_ plane), we have reason accordingly to anticipate that we can draw a similar projection (or section) of the plaice by simply doubling the _y_’s and halving the _z_’s: and, very approximately, this turns out to be the case. The plaice is (in round numbers) just about twice as broad and also just about half as thick as the haddock; and therefore the ratio of breadth to thickness (or _y_ to _z_) is just about four times as great in the one case as in the other.

It is true that this simple, or simplified, illustration carries us but a very little way, and only half prepares us for much greater complications. For instance, we have no right or reason to presume that the equality of weights, or volumes, is a common, much less a general rule. And again, in all cases of more complex deformation, such as that by which we have compared Diodon with the sunfish, we must be prepared for very much more {777} recondite methods of comparison and analysis, leading doubtless to very much more complicated results. In this last case, of Diodon and the sunfish, we have seen that the vertical _expansion_ of the latter as compared with the former fish, increases rapidly as we go backwards towards the tail; but we can by no means say that the lateral _compression_ increases in like proportion. If anything, it would seem that the said expansion and compression tend to vary inversely; for the Diodon is very thick in front and greatly thinned away behind, while the flattened sunfish is more nearly of the same thickness all the way along. Interesting as the whole subject is we must meanwhile leave it alone; recognising, however, that if the difficulties of description and representation could be overcome, it is by means of such co-ordinates in space that we should at last obtain an adequate and satisfying picture of the processes of deformation and of the directions of growth[663].

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