Sweet indeed are the uses of adversity, as D'Arcy Thompson put his best possible spin upon a positive ending to a most unwelcome admission (p. 723):
But in a very large part of morphology, our essential task lies in the comparison of related forms rather than in the precise definition of each; and the deformation of a complicated figure may be a phenomenon easy of comprehension, though the figure itself have to be left unanalysed and undefined. This process of comparison, of recognising in one form a definite permutation or deformation of another, apart altogether from a precise and adequate understanding of the original “type” or standard of comparison, lies within the immediate province of mathematics, and finds its solution in the elementary use of a certain method of the mathematician. This method is the Method of Co-ordinates, on which is based the Theory of Transformations.
Odd Man In (D'Arcy Thompson's structuralist critique of
Darwinism) and Odd Man Out (his disparagement of historicism)
I premised this discussion by arguing that D'Arcy Thompson's showy singularity in both style and substance — the quirks and anachronisms that seem to place him “out of time” — must not blind us to the fact that these uniquenesses overlie a rather standard early 20th century structuralist critique of Darwinian functionalism. D'Arcy Thompson's residence within his own time therefore becomes as informative as his idiosyncrasies.
In particular, I have emphasized, throughout this book, the logical and almost ineluctable linkage in structuralist thought between defenses of internally channeled directionality and saltationist mechanics (as particularly exemplified in the definitive model of Galton's Polyhedron — see pp. 342–351). I have also stressed the equally tight relationship of these views to a critique of adaptations — not of their preeminent existence (which D'Arcy Thompson embraces and celebrates), but of their necessary construction “for” utility by [Page 1201] a functionalist mechanism like natural selection (see my previous discussion of D'Arcy Thompson's contemporary, William Bateson, pp. 396–415).
On the theme of constraint, D'Arcy Thompson frequently argues that once one accepts a more than analogical comparison between the good designs, based on idealized geometry, of organic and inorganic objects, one must also defend (not apologetically, but for the utility of the conclusion in understanding anatomical order and taxonomic interrelationships) the limitation of organic morphology to channels of transformation that the physical causes of morphogenesis must follow (p. 137): “The world of things living, like the world of things inanimate, grows of itself, and pursues its ceaseless course of creative evolution. It has room, wide but not unbounded, for variety of living form and structure, as these tend towards their seemingly endless, but yet strictly limited, possibilities of permutation and degree.”
Once these directing channels of evolution have been established by physical forces governing the changes, we have no reason to restrict our concept of movement down these channels to the slow and continuous pacing of Darwinian gradualism. Relevant physical laws will regulate the path, “be the journey taken fast or slow.” In the following passage (p. 155), D'Arcy Thompson begins to link his model of channeled discontinuity to critiques of specific principles within evolutionary theory — in this case to the prevalent idea that similarities between ontogeny and phylogeny must arise for phyletic or historical reasons. Why introduce this “extra” and superfluous hypothesis, D'Arcy Thompson asks, if the same physical law, “caring little” whether it works during the growth of individuals or the evolution of lineages, mandates the same changes in any separate case:
The differences of form, and changes of form, which are brought about by varying rates (or “laws”) of growth, are essentially the same phenomenon whether they be, so to speak, episodes in the life-history of the individual, or manifest themselves as the normal and distinctive characteristics of what we call separate species of the race. From one form, or ratio of magnitude, to another there is but one straight and direct road of transformation, be the journey taken fast or slow; and if the transformation take place at all, it will in all likelihood proceed in the self-same way, whether it occur within the life-time of an individual or during the long ancestral history of a race.
But for sheer iconoclasm, nothing in all of D'Arcy Thompson's prose matches the section that he added as “conclusion” to the chapter on transformed coordinates in the second (1942) edition of Growth and Form — a “great quotation” well worth citing in extenso. Here he makes his most incisive and forthright linkage between the two great structuralist themes of channeling and saltational discontinuity. Taxonomic groups of organisms may be compared (truly “homologized” in the conceptual sense of being regulated by the same causal principle) with families of mathematical curves generated by definite parameters of construction. Complete continuity may reign within a family, but the gaps between can only be crossed per saltum. D'Arcy [Page 1202] Thompson emphasizes that “there is no argument against the theory of evolutionary descent” (1942, p. 1094) in such a concept, but he also maintains that this view of life's conformity to ordinary physical principles suggests a wide and radical range of non-Darwinian implications, including the rejection of Darwin's views on imperfection of the geological record, a defense of saltational transition between Bauplane (contrasted with potential continuity of taxonomic order within), and a solution to the problem of life's inhomogeneous occupation of potential morphospace as an organic incarnation of real mathematical discontinuities in the geometry of nature.
There is one last lesson, which coordinate geometry helps us to learn.... In the study of evolution, and in all attempts to trace the descent of the animal kingdom, fourscore years' study of the Origin of Species has had an unlooked-for and disappointing result. ...
This failure to solve the cardinal problem of evolutionary biology is a very curious thing . . . We used to be told, and were content to believe, that the old record was of necessity imperfect — we could not expect it to be otherwise; the story was hard to read because every here and there a page had been lost or torn away . . . But there is a deeper reason. When we begin to draw comparisons between our algebraic curves and attempt to transform one into another, we find ourselves limited by the very nature of the case . . .
An algebraic curve has its fundamental formula, which defines the family to which it belongs ... With some extension of the meaning of parameters, we may say the same of the families, or genera, or other classificatory groups of plants and animals . . . We never think of “transforming” a helicoid into an ellipsoid, or a circle into a frequency-curve. So it is with the forms of animals. We cannot transform an invertebrate into a vertebrate, nor a coelenterate into a worm, by any simple and legitimate deformation...
A “principle of discontinuity,” then, is inherent in all our classifications, whether mathematical, physical or biological; and the infinitude of possible forms, always limited, may be further reduced and discontinuity further revealed ... The lines of the spectrum, the six families of crystals, Dalton's atomic law, the chemical elements themselves, all illustrate this principle of discontinuity. In short, nature proceeds from one type to another among organic as well as inorganic forms; and these types vary according to their own parameters, and are defined by physico-mathematical conditions of possibility. In natural history Cuvier's “types” may not be perfectly chosen nor numerous enough, but types they are; and to seek for stepping-stones across the gaps between is to seek in vain, forever. ...
Our geometrical analogies weigh heavily against Darwin's conception of endless small continuous variations; they help to show that discontinuous variations are a natural thing, that “mutations” — or sudden changes, greater or less — are bound to have taken place, and new [Page 1203] “types” to have arisen, now and then. Our argument indicates, if it does not prove, that such mutations [occur] on comparatively few definite lines, or plain alternatives, of physico-mathematical possibility.
If D'
Arcy Thompson ended the final chapter on transformed coordinates with his most trenchant critique of Darwinian functionalism, he also lets us know, within the body of the chapter, that he built his entire theory as an alternative to the major implication of Darwinism for the daily practice of evolutionary biology and our conventional manner of conceptualizing organisms — for Darwinism implies separability of traits, and the subsequent potential for their independent optimization by natural selection, whereas the very possibility of relating creatures by such simple transformation grids points to structural channeling by overarching laws of growth (p. 727):
This independent variability of parts and organs . . . would appear to be implicit in our ordinary accepted notions regarding variation; and, unless I am greatly mistaken, it is precisely on such a conception of the easy, frequent, and normal independent variability of parts that our conception of the process of natural selection is fundamentally based ... But if, on the other hand, diverse and dissimilar fishes can be referred as a whole to identical functions of very different coordinate systems, this fact will of itself constitute a proof that variation has proceeded on definite and orderly lines, that a comprehensive “law of growth” has pervaded the whole structure in its integrity, and that some more or less simple and recognisable system of forces has been at work.
In discussing morphological variety among Radiolaria, where he suspects that realized taxonomic diversity most closely approaches the filling of all slots permitted by generating laws of form, D'Arcy Thompson extends his critique to the ultimate step of even doubting whether many configurations, as occupants of geometrically attainable positions in a predictable series, even need to be interpreted as adaptive at all (p. 607):
In few other groups do we seem to possess so nearly complete a picture of all possible transitions between form and form, and of the whole branching system of the evolutionary tree: as though little or nothing of it had ever perished, and the whole web of life, past and present, were as complete as ever. It leads one to imagine that these shells have grown according to laws so simple, so much in harmony with their material, with their environment, and with all the forces internal and external to which they are exposed, that none is better than another and none fitter or less fit to survive. It invites one also to contemplate the possibility of the lines of possible variation being here so narrow and determinate that identical forms may have come independently into being again and again.
D'Arcy Thompson bases these critiques of natural selection directly upon his own idiosyncratic theory of form. But, in his more conventional participation in the general debate of his time, he also presents the standard anti-Darwinian [Page 1204] arguments that prevailed within evolutionary biology before the Modern Synthesis reinstated natural selection at the center of the subject. In a brief commentary on the extinction of dinosaurs, for example, D'Arcy Thompson reiterates the common claim that natural selection, although a genuine force in evolution, can only play the minor and negative role of eliminating the unfit, and not the central part of creating the fit. “We begin to see,” he states (p. 137), “that it is in order to account, not for the appearance, but for the disappearance of such forms as these that natural selection must be invoked.” He then embellishes the argument with one of his characteristic prose flourishes (pp. 137-138):
But there comes a time when “variation,” in form, dimensions, or other qualities of the organism, goes farther than is compatible with all the means at hand of health and welfare for the individual and the stock; when, under the active and creative stimulus of forces from within and from without, the active and creative energies of growth pass the bounds of physical and physiological equilibrium: Then, at last, we are entitled to use the customary metaphor, and to see in natural selection an inexorable force, whose function is not to create but to destroy, — to weed, to prune, to cut down, and to cast into the fire.
But if this general and multifaceted critique of Darwinian functionalism — arising from his contrary structuralist account of the direct production of adaptive form by physical forces — harmonized well with major trends of thought in the evolutionary biology of his generation, the second prominent implication that he drew from his idiosyncratic theory of form could not have stood in more oppositional relationship to an even deeper, and even more general, assumption of the evolutionary sciences, both in D'Arcy Thompson's day and in our own. For D'Arcy Thompson used his theory to disparage, indeed virtually to abolish, phylogenetic reasoning and historical explanations in general. He did not, of course, deny that phylogeny happened, and that a tree of life existed; and he did not challenge the fact that each species stands atop a historical series of ancestral forms. But he did argue that these common features of biological reality possessed virtually no explanatory value for understanding either the morphology of individual species, or the anatomical relationships among species, as depicted in his own diagrams of transformed coordinates.
For if immediate physical forces shape the adaptive configurations of each modern species directly, then what relevance can be found in extinct ancestral shapes that responded to different physical forces of prior times, and that do not constrain the current forms of their descendants? The sensible, rational, and constrained order of related living forms in taxonomic groups represents a set of realizable positions within the mathematical boundaries of laws that currently shape the relevant organisms, not a set of phyletic constraints inherited from the past and now resident within organisms in the form of inherited genetic and developmental patterns that limit and channel the taxonomic structure of the living world.
For these reasons, I have labeled D'Arcy Thompson's “physicalist” and [Page 1205] “externalist” form of structuralism as deeply out of harmony with the internalist and historically based theories of structural constraint that are now enjoying such a revival within evolutionary theory, and that constitute the subject of the rest of this chapter. I have also not hidden my general approbation for these more popular internalist and historicist versions. Paleontologists are, after all, historians at heart and by profession, and I regard historical causation as the most powerful and distinctive mode of reasoning — indeed the raison d'etre — of the evolutionary sciences. But I must also confess my lifelong attraction to the prose, and the chutzpah, behind D'Arcy Thompson's iconoclasm. His work has much to teach us, although I do not think that his general theory of form can be validated as more than a peripheral aid and secondary shaper within a primary nexus of historicism.
In any case, D'Arcy Thompson wrote many of his stylistically strongest and philosophically most interesting disquisitions in a form that can only be labeled as “riffs” for different taxonomic groups on the common subject of the irrelevance of phylogeny in general, and the uselessness of Darwinian mechanisms in particular, for explaining either their anatomical ground plan or the ordered array of forms defining their taxonomic structure. He presents variants on the same riff for unduloid protists, radiolarians, and the hard parts of foraminifers, sponge spicules, bird eggs, the multifarious variations in shape among species of Mesozoic ammonoids, and the spiral patterns of stems in climbing plants.
For example, in the final paragraph of his chapter on logarithmic spirals, D'Arcy Thompson notes that the same set of varieties upon this universal curve populate the seas throughout Phanerozoic times. Therefore, he can't help wondering whether molluscan shells of this form can be meaningfully parsed into historical series of descent, or even accorded particular adaptive values (p. 586):
Again, we find the same forms, or forms which (save for external ornament) are mathematically identical, repeating themselves in all periods of the world's geological history; and, irrespective of climate or local conditions, we see them mixed up, one with another, in the depths and on the shores of every sea. It is hard indeed (to my mind) to see where Natural Selection necessarily enters in, or to admit that it has had any share whatsoever in the production of these varied conformations. Un
less indeed we use the term Natural Selection in a sense so wide as to deprive it of any purely biological significance; and so recognise as a sort of natural selection whatsoever nexus of causes suffices to differentiate between the likely and the unlikely, the scarce and the frequent, the easy and the hard: and leads accordingly, under the peculiar conditions, limitations and restraints which we call “ordinary circumstances,” one type of crystal, one form of cloud, one chemical compound, to be of frequent occurrence and another to be rare.
In my opinion, D'Arcy Thompson's most powerful statement (also most likely to include important elements of validity) occupies several pages at the end of his next chapter 12 “on the spiral shells of the foraminifera.” In this [Page 1206] “conclusion” (pp. 607-611), he decries the standard tendency of the time (now happily abandoned, but mostly for reasons other than D'Arcy Thompson's critique) to establish speculative phylogenies not so much on stratigraphic successions, but on idealized series moving from simple forms like the spherical Orbulina to ever more complex designs, and with attributions of increasing adaptive value (usually for greater test strength) to these more elaborate forms. Instead, D'Arcy Thompson views the diversity of foram shells as so many incarnations of permitted designs under the shaping rules that actually direct the building of tests. Since these rules do not change through time, the potential forms remain immanent in nature, and become occupied again and again, throughout the history of the entire lineage, by various clades that fall under the relevant set of forces. These forms are therefore no more arrangeable from worse to better, and no more subject to specific accounts of historical filiation, than are the varied shapes of snowflakes or quartz crystals. (I particularly like this “riff” because D'Arcy Thompson admits that phylogenies for complex metazoans may be better founded, and because he may well be right that, for these simplest creatures of the fossil record, repeated shapes in widely separated times and places may represent massive homoplasy based on the ease of revolving certain basic designs. After all, a hippo is only a hippo and will evolve just once, but the form of a microscopic calcareous sphere, floating in the ocean, may be attainable along many routes — pp. 610-611):
The Structure of Evolutionary Theory Page 191