On Growth and Form

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, ¢. 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, 6) to show that, when the former is taken as our Cartesian type, a slight curvature and an approximately logarithmic extension of the z-axis brings us easily to the configuration of the other. In the original specimen of Camptosaurus described by Marsh*, 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

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

forms, chief among which is the great elongation in [chthyosaurus 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 les become each a pencil of diverging rays, we make manifest a correspondence which extends uniformly pate all parts of these very different-looking skulls.

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, 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 1 2 3 parole | Ha aN al US Ae | me ele a aa 7 7a i (Zl a FES Sa eg ee

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,

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” Fig. 390. The co-ordinate systems of Figs. 388 and 389. with three intermediate systems interpolated, Fig. 391. The first intermediate co-ordinate network, with its 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

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

Fig. 392. The second and third intermediate co-ordinate networks, with their corresponding inscribed pelves. 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. 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

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 Fig. 393. The pelves of Archaeopteryx and of Apatornis, with three transitional types interpolated between them. figured by Osborn in his Monograph of the Extinct Rhinoceroses* (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

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 Fig. 395. Skull of Aceratherium tridactylum. (After Osborn.) 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 Aceratherjum has undergone a slight double curvature, while the upper parts of the skull have at the same time been

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 Acerathervum, 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 F have represented in Fig. 396.

Fig. 396. Occipital view of the skulls of various extinct rhinoceroses (Aceratherium spp.). (After Osborn.) 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 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 Titanothericum 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

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

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

Fig. 401. A, outline diagram of the Cartesian co-ordinates of the skull of Hyracotherium or Eohippus, as shewn in Fig. 402, A. H, outline of the corresponding projection of the horse’s skull. B-G, intermediate, or interpolated, outlines. 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

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; WM, skull of Mesohippus, from the Oligocene, after Scott, for comparison with C; P, skull of Protohippus, from the Miocene, after Cope. for comparison with H; 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.

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. 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*. Here we start afresh,’

with the skull (Fig. 402, 4) of Hyracotherium (or Eohippus), inscribed in a simple Cartesian network. At the other end of the series (1) 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 Archaeop'eryx 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 comcides 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 Hyracothervum 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, Hohippus, 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 Equidaet. It may be noticed, especially in the case of Pro ohippus * These and also other coordinate diagrams will be found in Mr G. Heilmann’s book Fuglenes Afstamning, 398 pp., Copenhagen, 1916; see especially pp. 368-380. + Cf. Zittel, Grundziige d. Palaeontologie, p. 463, 1911.

t Cf. W. B. Scott (Amer. Journ. of Science, xivim, pp. 335-374, 1894), “We find that any mammalian series at all complete, such as that of the horses, is remarkably continuous, and that the progress of discovery is steadily fillmg up — (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

Fig. 403. Human scapulae (after Dwight), A, Caucasian; B, Negro; C, North American Indian (from Kentucky Mountains). 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*. The variability of this latter bone is great, but it is neither what few gaps remain. So closely do successive stages follow upon one another that it is sometimes extremely difficult to arrange them all in order, and to distinguish clearly those members which belong in the main line of descent, and those which represent incipient branches. Some phylogenies actually suffer from an embarrassment of riches.”

* Cf. Dwight, T., The Range of Variation of the Human Scapula, Amer. Nat. XXI, pp. 627-638, 1887. Cf. also Turner, Challenger Rep. XLvIr, on Human Skeletons, p. 86, 1886: “I gather both from my own measurements, and those of other observers, that the range of variation in the relative length and breadth of the scapula is very considerable in the same race, so that it needs a large number of bones to enable one to obtain an accurate idea of the mean of the race.”

surprising nor peculiar; for it is inked 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. Fig. 405. Co-ordinates of chimpanzee’s skull, as a projection of the Cartesian co-ordinates of Fig. 404.