Fig. 408. Skull of dog, compared with the human skull of Fig. [404].
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 = kL3 , or k = W ⁄ L3 , 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 = 2y), 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].
EPILOGUE.
In the beginning of this book I said that its scope and treatment were of so prefatory a kind that of other preface it had no need; and now, for the same reason, with no formal and elaborate conclusion do I bring it to a close. The fact that I set little store by certain postulates (often deemed to be fundamental) of our present-day biology the reader will have discovered and I have not endeavoured to conceal. But it is not for the sake of polemical argument that I have written, and the doctrines which I do not subscribe to I have only spoken of by the way. My task is finished if I have been able to shew that a certain mathematical aspect of morphology, to which as yet the morphologist gives little heed, is interwoven with his problems, complementary to his descriptive task, and helpful, nay essential, to his proper study and comprehension of Form. Hic artem remumque repono.