On Growth and Form

of length and breadth in this particular bone are extremely significant*. 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.

This summary of values of y’, coupled with the equations for the * This significance is particularly remarkable in connection with the development of speed, for the metacarpal region is the seat of very important leverage in the propulsion of the body. In the Museum of the Royal College of Surgeons in Edinburgh, there stand side by side the skeleton of an immense carthorse (celebrated for having drawn all the stones of the Bell Rock Lighthouse to the shore), and a beautiful skeleton of a racehorse, which (though the fact is disputed) there is good reason to believe is the actual skeleton of Eclipse. When I was a boy my grandfather used to point out to me that the cannon-bone of the little racer is not only relatively, but actually, longer than that of the great Clydesdale.

value of x, will enable us, from any drawing of the ox’s foot, to construct a figure of that of the sheep or ay the giraffe with remarkable accuracy. That underlying the varying amounts of extension to which — the parts or segments of the — hmb 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 7’. 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*; it is fully described and put in practice by Albert Diirer in his Geometry, and especially in his Treatise on Proportiony. In this latter work, the manner in which the + Les quatres livres d’ Albert Diirer de la proportion des parties et pourtraicts des corps humains, Arnheim, 1613, folio (and earlier editions). Cf. also Lavater, Essays on Physiognomy, U1, p. 271, 1799. 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 tapiz’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° (6). 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

toe is a precisely identical function of the form F (e”, y,) = 0, where 21, y, are oblique co-ordinate axes inclined to one another at an angle of 50°. Diirer was acquainted with these oblique co-ordinates also, and I have copied two illustrative figures from his book*. In Fig. 367 I have sketched the common Copepod Oithona nana, * Tt was these very drawings of Diirer’s that gave to Peter Camper his notion of the “facial angle.’’ Camper’s method of comparison was the very same as ours, save that he only drew the axes, without filling in the network, of his coordinate system; he saw clearly the essential fact, that the skull varies as a whole, and that the “‘facial angle” is the index toa general deformation. “The great object was to shew that natural differences might be reduced to rules, of which the direction of the facial line forms the norma or canon; and that these directions and inclinations are always accompanied by correspondent form, size and position of the other parts of the cranium,” etc.; from Dr T. Cogan’s preface to Camper’s work On the Connexion between the Science of Anatomy and the Arts of Drawing, Painting and Sculpture (17687), quoted in Dr R. Hamilton’s Memoir of Camper, in Lives of Eminent Naturalists (Nat. Libr.), Edin. 1840.

and have inscribed it in a rectangular net, with abscissae threefifths 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 yin 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, imstead of seeking for an actual equation, we simply tabulate our values of # 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 Ovthona, we obtain that of Sapphirina without further trouble, e.g.:

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

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 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 Paralonis. 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

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 Fig. 370. 1. Harpinia plumosa Kr. 2. Stegocephalus inflatus Kr. 3. Hyperia galba.

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. 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

by the living animal, but upon the form, size and arrangement of the little cups, or “calycles,” secreted and inhabited by the httle 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 wineglassshaped 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

Fig. 371. a, Campanularia macroscyphus, Allm.; 6, Gonothyraea hyalina, Hineks; c, Clytia Johnstoni, Alder. Aglaopheniidae the calycles are set unilaterally upon a jointed stem, and small cup-lke 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

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. a 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 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.

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 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. J

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*. The old outline, transferred * The co-ordinate system of Fig. 382 is somewhat different from that which I drew and published in my former paper. It is not unlikely that further

investigation will further simplify the comparison, and shew it to involve a still more symmetrical system. 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 in form, size and position; and it shews a greater ahd 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.

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, im 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,

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 z/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

Fig. 383. A, Crocodilus porosus. B, C. americanus. C, Notosuchus terrestris. 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