Thompson wrote On Growth and Form in the maturity of a career that lay somewhat outside the mainstream of the biological sciences of his day. The book is in large part a contribution to morphology, the study of organic form. This branch of the biological sciences had a flowering in the early and mid-nineteenth century but subsequently receded from the forefront of influential research. As an aspect of natural philosophy, morphology can be traced back at least to the Greek philosopher Aristotle, although the term itself is credited to the German poet-philosopher Johann Wolfgang von Goethe. In broad terms, the study was concerned with relationships of structure between diverse plant and animal species and proposed various theories to explain the observed continuity of organic forms. For example, Goethe, whose scientific and poetic work D’Arcy Thompson knew well, had asserted that all plants were modifications of an archetypal plant and that all plant organs were variations of a single fundamental organ, the leaf. The most arresting issues in morphology, however, lay in zoology and paleontology; with the appearance of Charles Darwin’s On the Origin of Species in 1859 the fundamental problems of morphology seemed to have been resolved by reference to paths of evolutionary descent within the animal kingdom.
Thompson’s early education took place during the widespread debate over the merits of the Darwinian evolutionary theories, and his scientific career, though not running altogether counter to concepts of evolution and natural selection, had a pronounced bias toward alternative perspectives on many issues. On Growth and Form is essentially an attempt to establish a theory of organic form based upon the physical and mathematical laws governing the development and function of organisms instead of upon the operation of the mechanisms of heredity and natural selection. In its progression of topics, On Growth and Form demonstrates organic form as principally determined by material forces operating upon biological structures. In the first two chapters, titled “Introductory” and “On Magnitude,” the foundation of the book is set solidly upon the laws of Newtonian physics, yet Thompson carefully avoids giving the reader the impression that his personal philosophy is mechanistic:The waves of the sea, the little ripples on the shore, the sweeping curve of the sandy bay between the headlands, the outline of the hills, the shape of the clouds, all these are so many riddles of form, so many problems of morphology, and all of them the physicist can more or less easily read and adequately solve: solving them by reference to their antecedent phenomena, in the material systems of mechanical forces to which they belong, and to which we interpret them as being due. They have also, doubtless, their immanent teleological significance; but it is on another plane of thought from the physicist’s that we contemplate their intrinsic harmony and perfection, and “see that they are good.”
Thompson’s appreciation of natural beauty is clearly attuned to his love for mathematics, an affinity which appeared early in his career and set him apart from the majority of his colleagues in the biological sciences.
Despite his reputation as a zealous geometer, Thompson avoids mere mathematical abstraction in his approach by emphasizing process in living form. Biological entities, he writes, “can never act as matter alone, but only as seats of energy and as centres of force.” The author’s conception of the vitality of biological form is implicit in his use of the word “growth” in the title of the book; he understands that the idea of “form” by itself can be too easily detached from the “dynamical” aspect of objects and phenomena. Thompson’s prose often exhibits...
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a dynamism uniquely attuned to his subject, as in the following discussion of scale:A certain range, and a narrow one, contains mouse and elephant, and all whose business it is to walk and run; this is our own world, with whose dimensions our lives, our limbs, our senses are in tune. The great whales grow out of this range by throwing the burden of their bulk upon the waters; the dinosaurs wallowed in the swamp, and the hippopotamus, the sea-elephant and Steller’s great sea-cow pass or passed their lives in the rivers or the sea. The things which fly are smaller than the things which walk and run; the flying birds are never as large as the larger mammals, the lesser birds and mammals are much of a muchness, but insects come down a step in the scale and more.
Thompson here evokes not only the energy and movement of organisms but also the dynamism of time on complementary scales of individual and historical development.
The third chapter of On Growth and Form, “The Rate of Growth,” introduces the notion that “the form of an organism is determined by its rate of growth in various directions.” After a promising but brief exposition of this concept, Thompson is diverted into a somewhat superfluous argument regarding a passage from Darwin and then launches informative but pedestrian accounts of human growth and population statistics, the growth rates of insects and animal organs, and environmental factors affecting growth. This long chapter, alone among the seventeen which make up the 1942 edition, concludes with a one-page summary of its contents; its next-to-last sentence, a resigned “But enough of this discussion,” lends credibility to John Tyler Bonner’s editing of this chapter for the revised edition of 1961.
The fourth and fifth chapters of On Growth and Form concern the cell. Chapter 4, “On the Internal Form and Structure of the Cell,” begins with a brief historical review of the morphological approach to cell structure, beginning with the first decades of the nineteenth century. After observing that the visible structure of the cell has been studied more exhaustively than the “purely dynamic problems” associated with cells, Thompson goes on to state, with somewhat uncharacteristic dogmatism, that the “mere study” of cell structure is essentially exhausted—a judgment that was extremely premature. In the absence of sufficiently intricate observations by which he might account for the complexity of cellular activity, he reassures the reader that “very great and wonderful things are done by means of a mechanism (whether natural or artificial) of extreme simplicity.” His aim in this chapter is to emphasize the operations of “physical forces” in cells and thereby to avoid a static, mechanical model of the cell, but Thompson is here writing from a position of weakness. Historically, the study of cell structure and physiology was on the verge of far-reaching developments which might well have inspired a fresh treatment of the topic by Thompson only a few years later; on the professional side, however, his background as a naturalist with virtually no experience in experimental biology left him particularly vulnerable to the obsolescence inherent in this more specialized and more quickly advancing field of study. Thompson seems to be aware of this possibility when he states, “But our sole object meanwhile, as I have said more than once, is to demonstrate, by such illustrations as these, that, whatever be the actual and as yet unknown modus operandi, there are physical conditions and distributions of force which could produce just such phenomena of movement as we see taking place within the living cell.”
There is a logical step from the chapter on the internal structure of cells to that concerning their external forms, but it should be noted that the abridged edition of 1961 eliminates “The Internal Form and Structure of the Cell” as “completely out of date” without sacrificing the continuity of Thompson’s text. The conceptual progression from the molecular scale to the mammalian is such a strong element of On Growth and Form that the reader of the revised edition will miss much detail but no clarity of argument. Thompson’s underlying hypothesis that organic form rises in great part from the action of purely physical forces upon developing biological structures is paralleled by his gradual expansion of the field of view; to the degree that he supports his arguments at each level of physical scale the structure of his argument as a whole is bolstered.
Thompson’s sense of delight in the forms of organic and inorganic nature reaches a high point in chapter 5, where the general reader will appreciate the text’s extensive illustrations. The mathematics which is central to the author’s scientific perspective finally is given concrete significance and appeal as well. Most of the pictorial material consists of line drawings, but in this chapter there are several high-speed photographs of liquid splashes which are significant footnotes to the history of scientific illustration. Three of these photographs, reproduced within the text as “figures,” first appeared in 1908 in a paper by A.M. Worthington; the others are provided as “plates” of much higher quality and were made in the 1930’s by Harold Edgerton of the Massachusetts Institute of Technology (and thus are additions to the original edition). These celebrated images show phases of a splash in which a crownlike rim of liquid is in the process of dividing into droplets. Thompson uses these photographs to demonstrate analogies between various biological forms and the forms caused by the surface tension of liquids.
The last, and briefest, chapter of the first volume, “On Adsorption,” is essentially an extension of aspects of the material in chapter 5, and can be thought of as an extended, digressive footnote to it. This section was justly cut from the revised edition, though it should be noted that in explaining its deletion Bonner refers to it as “a note on absorption” and somewhat misconstrues its content. The second volume begins with “The Forms of Tissues: Or, Cell-Aggregates,” actually two chapters, the second of which is referred to in the table of contents as “The Same (continued).” This arbitrary division of a continuous topic can only have been intended to keep the chapters as close to a uniform size as possible, and with the exception of parenthetical sections the remainder of the book follows this pattern.
It is probably only in the second volume of Growth and Form that the contemporary reader will begin to feel fully acclimated to the author’s style and to appreciate the degree to which the work transcends the need to impart facts and theories. Thompson seemingly tries to create a vision of the unity of the world of forms, tying together organic and inorganic structure by reference both to sense experience and to mathematical analysis. The effect of the measured progression from small to large forms, and from elements of form to the living structures they support, creates a sense of synthesis balancing the process of analysis necessary to the scientific enterprise. Chapter 6, “The Forms of Tissues: Or, Cell-Aggregates,” shows Thompson at his best, integrating mathematics and natural history by the use of commonplace as well as esoteric observations. The central issue of these connected chapters is the explanation of the grouping of cells on the basis of fundamental geometric relationships, many of which are commonly seen in the arrangement of the thin films of soap bubbles. Plane and polyhedral structures found in physical systems are related to similar organic forms, and the concept of the “close packing” of volumes is given a clear and diverse treatment. Where the mathematics of the discussion might be unclear to general readers, illustrations bridge the gaps.
Chapter 9, “On Concretions, Spicules, and Spicular Skeletons,” concerns forms arising from the deposition within living bodies—principally invertebrates—of inorganic materials such as calcium carbonate. Thompson proceeds from brief theoretical comments to specific considerations, such as how the forms of spicules can be artificially imitated or theoretically explained. A series of chemical experiments producing forms analogous to organic structures is described and illustrated, then the discussion is directed toward the mechanical causation of organic forms. Thompson’s objection to Darwinian hypotheses comes to the fore in the midst of this chapter, where he draws attention to “the fundamental difference between the Darwinian conception of the causation and determination of Form, and that which is based on, and characteristic of, the physical sciences.” He writes that “a graduated or consecutive series of forms may be based on physical causes” and that “forms mathematically akin may belong to organisms biologically remote, and . . . in general, mere formal likeness may be a fallacious guide to evolution.” This is Thompson’s position on evolutionary matters: Hereditary hypotheses often leave much to be desired. His glancing blows against Darwinian theory occur at such sporadic intervals, however, that their significance in Thompson’s overall scheme might easily be underestimated.
For aesthetic appeal, “The Equiangular Spiral” surpasses all the others. The mathematical elements of this section are among the least formidable in the book, and the visual material—principally drawings, diagrams, and photographs of shells— is inherently fascinating. Thompson’s discussion originates with a distinction between the “spiral of Archimedes,” which resembles a coiled rope and the equiangular spiral, a form seen with greater or lesser clarity in shells. Thompson presents a remarkable variety of explanations of the significance of the spiral in mathematical, physical, and biological terms; each of his examples seems complementary rather than redundant. Thompson’s essential point is that the growth of horns, shells, and other organic forms in which an equiangular spiral can be recognized demonstrates an increase in size coupled with an identity, or near identity, of shape or volume. The shell “does not alter as it grows; each increment is similar to its predecessor, and the whole, after every spurt of growth, is just like what it was before.”
Having established a theoretical framework for the study of equiangular organic forms, Thompson proceeds to increasingly complex examples of mollusks, foraminifera, and worms for which the basic mathematics must be elaborated. In concluding this chapter, he once again brings his discussion to bear on his reasons for discounting the hypothesis of natural selection: “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.” In his opinion, the sense of natural selection can be nothing more than a vague “nexus of causes” with which “to differentiate between the likely and the unlikely, the scarce and the frequent, the easy and the hard.”
The next four short sections are of varying significance. Chapter 12, “The Spiral Shells of the Foraminifera,” is largely a continuation of the previous chapter on spirals. “The Shapes of Horns, and of Teeth or Tusks: With a Note on Torsion” takes the discussion into the realm of large mammalian structure in anticipation of the concluding chapters. Interjected before these important sections are a fine but unessential chapter on leaf arrangement and “On the Shapes of Eggs, and of Certain Other Hollow Structures,” which is flawed by an erroneous deduction Thompson might have been expected to correct in the second edition.
“On Form and Mechanical Efficiency” is a treatment of the topic of structural form in birds, fishes, and mammals, with an emphasis on the resemblance of skeletal forms to engineered structures. Dealing with matters which are far less esoteric than most of Thompson’s topics, it has the feeling of a fine, self-contained piece of virtuoso public speaking transposed to the printed page; one can easily imagine the illustrations as lantern slides for a lecture. A photograph of a bison, for example, is described parenthetically as “an unusually well-mounted skeleton, of American workmanship, now in the Anatomical Museum of Edinburgh”—a genteel flourish which would seem out of place elsewhere in the book.
The final substantive chapter of On Growth and Form, preceding a short poetic epilogue, is the source of most of the influential excerpts from the book found dispersed among texts on various scientific and cultural topics. The essence of “On the Theory of Transformations: Or, The Comparison of Related Forms” is simple, direct, and elegantly visual. It is a conceptualization of the transformation of a shape, or family of shapes (and volumes, as the issue becomes more refined), into related forms by means of the stretching of a system of coordinates. For example, if the outlines of the human skull are placed on a rectangular grid, a series of regular alterations of the grid can be carried out which transpose the coordinate points of the original drawing so that the drawing resembles that of a chimpanzee’s skull; further warping of the grid changes the drawing to approximate the skull of a baboon. Examples of similar transformations of the shapes of leaves, crustaceans, and fishes elaborate the methodology in a convincing fashion.
Thompson uses the model of the “deformation” of biological form in two senses: to demonstrate how one form can be seen to be related to another and to show how differential rates of growth produce forms of greater or lesser levels of perceived relationship. Ultimately, however, he gravitates to the issue of discontinuity of form across the spectrum of creation, concluding that, since there is an absolute discontinuity between types of geometric forms, there are inevitable gaps in categories of organic forms. “Our geometric analogies weigh heavily against Darwin’s conception of endless small continuous variations,” the author states, though he admits that “this is no argument against the theory of evolutionary descent.” “Physico-mathematical possibility” is for Thompson an essential and perhaps the paramount component of an adequate account of the origin as well as the limits of biological diversity.