I choose this case (Figure 11-4) because D'Arcy Thompson invoked the conformity of so many tiny organisms with the well-known and easily-produced unduloid to draw an explicit contrast between his explanatory preferences and the conventional Darwinian account of adaptive design (pp. 248–249) (note, especially, how he highlighted these differences for resolving both the adaptive status of the basic form itself, and the genesis of a rich set of taxonomically designated variants upon the basic form):
Here we have an excellent illustration of the contrast between the different ways in which such a structure may be regarded and interpreted. The teleological explanation is that it is developed for the sake of protection, as a domicile and shelter for the little organism within. The mechanical explanation of the physicist (seeking only after the “efficient,” and not the “final” cause), is that it is present, and has its actual conformation, by reason of certain chemicophysical conditions: that it was inevitable, under the given conditions, that certain constituent substances actually present in the protoplasm should be aggregated by molecular forces in its surface layer; that under this adsorptive process, the conditions continuing favorable, the particles should accumulate and concentrate till they formed a membrane, thicker or thinner as the case might be; that this
11-4. Single celled protists assuming the form of unduloids — and taken by D'Arcy Thompson as proof of immediate physical construction rather than genetic encoding. See text for details. From D'Arcy Thompson, 1917.
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membrane was inevitably bound, by molecular forces, to become a surface of the least possible area which the circumstances permitted; that in the present case, the symmetry and “freedom” of the system permitted, and ipso facto caused, this surface to be a surface of revolution; and that of the few surfaces of revolution which, as being also surfaces minimae areae, were available, the unduloid was manifestly the one permitted, and ipso facto caused, by the dimensions of the organisms. We also see that the actual outline of this or that particular unduloid is also a very subordinate matter, such as physico-chemical variants of a minute kind would suffice to bring about; for between the various unduloids which the various species of Vorticella represent, there is no more real difference than that difference of ratio or degree which exists between two circles of different diameter, or two lines of unequal length.
In cases like this, strict selectionists conventionally assert that no genuine problem exists, but only a conceptual or terminological confusion. After all, any devotee of natural selection knows that adaptive shapes must be explained both in terms of survival value and immediate mode of construction. Therefore, the selectionist would continue, “I am happy to suppose that natural selection built the adaptive unduloid by fostering the differential reproductive success of growth variants that could attain the advantageous form along a single dimension of selection, rather than by having to construct each property in a piecemeal fashion, character by character.”
In fact, selectionists can even cite a terminology to bolster their understanding that any adaptation requires, for its full explication, an account of both survival value and mode of construction (which, in truth, only reflects Aristotle's old distinction of final and efficient causes, and might as well bear these names rather than their currently favored neologisms). Selectionists generally refer to these two complementary modalities as “ultimate” and “proximate” causes — often supposing that they have, by this terminology, won some preciously new insight to clear away the conceptual fog of centuries. However, as stated above, the distinction only codifies a particular expression of Aristotle's argument on the multiple meanings and aspects of causality.
Nonetheless, even if not new, this argument about the complementarity and non-oppositional nature of ultimate and proximate causation cannot be gainsaid — and Darwinians advance this point with complete justice. That is, when selectionists cite the adaptive advantage of a form, they surely do not deny the need for a different statement about the immediate mode of genetic and developmental origin in any individual as well. However, we also need to recognize that this legitimate defense of adaptationist language does not apply to D'Arcy Thompson's point of genuine contention (logically genuine that is, not necessarily empirically correct in any given instance).
D'Arcy Thompson does not merely argue that he has found the mode by which natural selection worked to build adaptive unduloids. Rather, he advances the radically different, and truly oppositional, argument that natural [Page 1195] selection need not be invoked at all, and as any kind of cause in this case. For he holds that physical forces shaped the unduloid directly, without any selection of favored forms from a range of variants. In other words, he believes that the efficient cause of mechanical imposition constructs the final cause automatically, thus obviating the need for any separate and explicitly biological or functional explanation to fashion the adaptive shape of the unduloid. (In selectionist jargon, D'Arcy Thompson argues that the proximate cause fashions the ultimate cause all by itself, thus explaining two properties for the price of one mechanism. I also happen to think that D'Arcy Thompson was probably wrong in this case, and that the traditional Darwinian scheme, with different forms of explanation needed for ultimate and proximate causes, probably applies to this case. But, the logic of D'Arcy Thompson's argument remains sound.)
2. At intermediary sizes, the automatically realized forms of inorganic objects often map the “conflicting” expressions of surficial forces holding things up and volumetric forces pulling them down. In a fascinating section added to the 2nd edition of 1942, D'Arcy Thompson studied drops of more viscid material falling through water. He compares the resulting forms (balancing surface tensions that retard descent and spread out the drops, with gravitational forces that pull the dense drops towards the bottom of the vessel) to the strikingly similar (and often quite complex) radially symmetrical shapes of jellyfishes (Figure 11-5). D'Arcy Thompson wrote (1942, pp. 397-398): “Not only do we recognize in a vorticoid drop a 'schema' or analog of medusoid form, but we seem able to discover various actual phases of the splash or drop in the all but innumerable living types of jellyfish ... It is hard indeed to say how much or little all these analogies imply. But they indicate, at the very least, how certain simple organic forms might be naturally assumed by one fluid mass within another, when gravity, surface tension and fluid friction play their part.”
3. At still larger sizes, surface tension becomes so negligible that rigid hard parts become necessary to maintain shape, lest gravity create a world of pancakes.
11-5. The jellyfish as a map of physical forces for a creature of intermediary size, and therefore subject both to forces that act on its surface and on its volume. The dense protoplasm is pulled down by gravity, but sufficiently retarded in its fall to spread out under forces of surface tension. From D'Arcy Thompson, 1942.
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In what may be his most famous example, D'Arcy Thompson proved a case of direct construction in response to immediate gravitational forces by showing that internal trabeculae in the head of the human femur strengthen the bone along the precise lines of its greatest need for buttressing against compressive forces — for when bones break and heal improperly, the trabeculae are absorbed and then reform along lines of stress dictated by the limping walk of suboptimal reknitting. No one, in this case, could make the usual claim for phyletic determination by natural selection (at least not for these particular trabeculae in these unfavorable circumstances, although one might identify selection as the basis for this underlying lability in trabecular formation). D'Arcy Thompson writes (p. 687):
Our bone is not only a living, but also a highly plastic structure; the little trabeculae are constantly being formed and deformed, demolished and formed anew. Here, for once, it is safe to say that “heredity” need not and cannot be invoked to account for the configuration and arrangement of the trabeculae: for we can see them, at any time of life, in the ma
king, under the direct action and control of the forces to which the system is exposed . . . Herein then lies, so far as we can discern it, a great part at least of the physical causation of what at first sight strikes us as a purely functional adaptation.
The admitted limitation and ultimate failure of an argument
As a common theme in the tragedies of human literature and history, entities of all sorts (from bodies, to cities, to structures of ideas) often unravel at the height of their apparent triumph, for the surface of success may fail to anchor any roots in the general substrate below. Embodied within the very undeniability of D'Arcy Thompson's explanation for the direct mechanical shaping of optimally positioned trabeculae in the human femur, we can also locate the source of a strictly limited applicability that D'Arcy Thompson himself eventually had to own.
After all, the trabeculae can be explained as direct consequences of immediate mechanical forces because they cannot be construed as inherited aspects of a phenotype that might be subject to natural selection, or to any process of truly evolutionary modification for that matter. They represent labile responses of the ontogenetic moment, and will therefore be subject to specification by immediate forces (and, consequentially and crucially, not candidates for hereditary transmission in our non-Lamarckian earthly biology). But when we study stable and inherited features with equal claim to adaptive optimality — the main kinds of characters that theories of adaptive evolution try to explain — how can we make an equally strong case (or even a case of any plausibility at all) for their immediate construction by physical forces acting upon the organism during growth? Immediate physical forces may build my trabeculae, but how can they shape a set of stable and inherited traits that made a first appearance when I was a tiny embryo in utero, long before the [Page 1197] gravitational forces to which these features are adapted could play any role in building my body by direct imposition?
D'Arcy Thompson had to face this critical problem head on — and he did so in a forthright manner by putting the best possible face upon adversity via two arguments that fatally compromised his dream of winning generality for his idiosyncratic theory of structuralist evolution from the outside. First and foremost, he simply admitted that his principle of direct imposition couldn't explain the complex forms of the multicellular phyla that, for however parochial a reason, have always defined the central subject and puzzlement of morphology. D'Arcy Thompson continued to maintain — and he may well have been right in some cases — that good matches between simple organic conformations (primarily the outward forms of unicellular creatures) and geometric shapes of well known mathematical definition and easily accomplished mechanical construction probably illustrate his favored principle of direct imposition by physical forces. But he had to admit that he could not apply this line of reasoning to the basic form of a horse or a tuna.
Interestingly, he “came clean” on this point right after his strong argument about the production of unicellular unduloids by forces of surface tension (cited on p. 1193). He begins by quoting a conventional defense of phylogenetic reasoning by E. Ray Lankester, then refutes the argument for his unduloids, but cannot deny its application to large and complex multicellular forms (pp. 251-252):
“The fact that we are able to classify organisms at all in accordance with the structural characteristics which they present is due to the fact of their being related by descent.” But this great generalisation is apt in my opinion, to carry us too far. It may be safe and sure and helpful and illuminating when we apply it to such complex entities, — such thousandfold resultants of the combination and permutation of many variable characters, — as a horse, a lion or an eagle; but (to my mind) it has a very different look, and a far less firm foundation, when we attempt to extend it to minute organisms whose specific characters are few and simple, whose simplicity becomes much more manifest when we regard it from the point of view of physical and mathematical description and analysis, and whose form is referable, or (to say the least of it) is very largely referable, to the direct and immediate action of a particular physical force.
But D'Arcy Thompson truly throws in the towel during the most poignant and appropriate round — right at the end of his last empirical chapter, as he reaches the apex of complexity in his analysis of vertebrate skeletons (chapter 16 “on form and mechanical efficiency”), and just before he recoups relevance in his brilliant final chapter on the theory of transformed coordinates. He begins by admitting that he cannot describe the skeleton as “a resultant of immediate and direct physical or mechanical conditions” precisely because the very biological principle that he has tried to deny (or at least to underplay) throughout the book — the phyletic inheritance, rather than immediate [Page 1198] construction, of the underlying Bauplan — cannot be gainsaid at this level of complexity. (With refreshing candor, D'Arcy Thompson admits that he had tried his best “to circumscribe the employment of the latter [that is, of heredity] as a working hypothesis in morphology.”) D'Arcy Thompson writes, in his key statement (p. 715):
It would, I dare say, be a gross exaggeration to see in every bone nothing more than a resultant of immediate and direct physical or mechanical conditions; for to do so would be to deny the existence, in this connection, of a principle of heredity. And though I have tried throughout this book to lay emphasis on the direct action of causes other than heredity, in short to circumscribe the employment of the latter as a working hypothesis in morphology, there can still be no question whatsoever but that heredity is a vastly important as well as a mysterious thing; it is one of the great factors in biology . . . But I maintain that it is no less an exaggeration if we tend to neglect these direct physical and mechanical modes of causation altogether, and to see in the characters of a bone merely the results of variation and of heredity, and to trust, in consequence, to those characters as a sure and certain and unquestioned guide to affinity and phylogeny.
This admission then leads to a recovery of relevance via the second argument, presented in his last and, in the judgment of most biologists, his most important chapter “on the theory of transformations, or the comparison of related forms.” All professional evolutionists know D'Arcy Thompson's famous diagrams of related organisms compared by imposing a Cartesian grid upon one form, treated as a reference, and then rendering other forms as results of simple distortions and transformations of the grid lines (see Figure 11-6 for an example). But I think that most of us have not understood the logical and theoretical reasons behind D'Arcy Thompson's invention, largely because (as for the chapter “on magnitude”) we read this section out of context, and do not grasp its intimate relation (as an apotheosis, given the limitations he had to admit) to his general and idiosyncratic theory of form.
That is, we tend to interpret these Thompsonian transformed coordinates as a crude, and ultimately failed, attempt to operationalize (by pictorialization) a good intuition about the multivariate nature of evolutionary change before the development of appropriate statistical techniques, and the invention of computers, permitted us to apply genuine multivariate mathematics to problems of form. Most of us, I think, envisage the deformed coordinate grid as a mere residuum of a qualitative analysis focused on the transformed bodies themselves — just a set of guidelines needed to make a crude map of the organisms under consideration.
In so doing, we misunderstand D'Arcy Thompson's intention in a precisely backwards manner. His interest lay primarily in the lines of the stretched and deformed grids, for he had remained true to his theory that physical forces shape organisms directly. He had made a painful and necessary surrender of that theory — by bowing to conventional evolutionary resolutions in terms of [Page 1199] heredity and phylogeny — for explaining the complex and intricately multivariate Bauplane of complex metazoan animals. In other words, he accepted that the basic designs must be admitted as “primitive terms” or “background conditions” within his theory — as “givens” to be acknowledged (and attrib
uted to other kinds of causes), and as basic inputs before any further analysis could be conducted in his favored terms.
In making such an admission D'Arcy Thompson swallowed a bitter pill. He had to accept the existence and contrary construction of “hipponess” or “eagleness” at the outset, and then to determine what field might be left to his favored causes of direct mapping by physical forces. The theory of transformed coordinates presents his positive approach to this dilemma, his attempt to keep his theory maximally relevant in the light of his enforced concession to historicism in general, and to Darwinism in particular. He could not lay claim to the basic forms themselves, but he would still make a play for the taxonomic variety produced by their transformations.
If the differences among related species could be expressed as simple distortions
11-6. An example of D'Arcy Thompson's theory of transformed coordinates. To understand his view, we must recognize that these figures are meant to feature the transformation grid lines themselves — as indications of physical forces that directly impose phyletic changes upon the organisms. From D'Arcy Thompson, 1917.
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of a grid, then the transformed grid itself would become a picture of the lines of forces responsible for the evolutionary deformation. Thus, D'Arcy Thompson valued the lines of the transformed grids above the altered organisms themselves, for he hoped that his pictures of simple and coordinated transformation would revivify his theory in a more limited domain. The lines of transformation would map the forces that converted the initial form into its descendants or relatives — and by D'Arcy Thompson's theory of direct imposition, those lines would then identify the geometric operation of the actual forces that caused the evolutionary changes by direct imposition. He would not win hipponess (or full happiness) for his theory, but he might encompass the set of realized variations upon hipponess. Put another way, perhaps D'Arcy Thompson could “have it all” for the simple forms of some unicells; but he would have to settle for the variations (leaving the fundamental configurations to history and heredity) when he treated the complexities of vertebrate organization.
The Structure of Evolutionary Theory Page 190