On Growth and Form is a book by the Scottish mathematical biologist D'Arcy Wentworth Thompson (1860–1948). The book is 793 pages in the first edition of 1917, 1116 pages in the second edition of 1942.
The book covers many topics including the effects of scale on the shape of animals and plants, large ones necessarily being relatively thick in shape; the effects of surface tension in shaping soap films and similar structures such as cells; the logarithmic spiral as seen in mollusc shells and ruminant horns; the arrangement of leaves and other plant parts (phyllotaxis); and Thompson's own method of transformations, showing the changes in shape of animal skulls and other structures on a Cartesian grid.
The work is widely admired by biologists, anthropologists and architects among others, but is often not read by people who cite it. Peter Medawar explains this as being because it clearly pioneered the use of mathematics in biology, and helped to defeat mystical ideas of vitalism; but that the book is weakened by Thompson's failure to understand the role of evolution and evolutionary history in shaping living structures. Philip Ball and Michael Ruse, on the other hand, suspect that while Thompson argued for physical mechanisms, his rejection of natural selection bordered on vitalism.
Overview
D'Arcy Wentworth Thompson was a Scottish biologist and pioneer of mathematical biology. His most famous work, On Growth and Form was written in Dundee, mostly in 1915, but publication was put off until 1917 because of the delays of wartime and Thompson's many late alterations to the text. The central theme of the book is that biologists of its author's day overemphasized evolution as the fundamental determinant of the form and structure of living organisms, and underemphasized the roles of physical laws and mechanics. At a time when vitalism was still being considered as a biological theory, he advocated structuralism as an alternative to natural selection in governing the form of species, with the smallest hint of vitalism as the unseen driving force.
Thompson had previously criticized Darwinism in his paper Some Difficulties of Darwinism. On Growth and Form explained in detail why he believed Darwinism to be an inadequate explanation for the origin of new species. He did not reject natural selection, but regarded it as secondary to physical influences on biological form.
Using a mass of examples, Thompson pointed out correlations between biological forms and mechanical phenomena. He showed the similarity in the forms of jellyfish and the forms of drops of liquid falling into viscous fluid, and between the internal supporting structures in the hollow bones of birds and well-known engineering truss designs. He described phyllotaxis (numerical relationships between spiral structures in plants) and its relationship to the Fibonacci sequence.
Perhaps the most famous part of the book is Chapter 17, "The Comparison of Related Forms," where Thompson explored the degree to which differences in the forms of related animals could be described, in work inspired by the German engraver Albrecht Dürer (1471–1528), by mathematical transformations.
The book is descriptive rather than experimental science: Thompson did not articulate his insights in the form of hypotheses that can be tested. He was aware of this, saying that "This book of mine has little need of preface, for indeed it is 'all preface' from beginning to end."Thompson, 1917. 'Prefatory Note', first paragraph.
Editions
The first edition appeared in 1917 in a single volume of 793 pages published by Cambridge University Press. A second edition, enlarged to 1116 pages, was published in two volumes in 1942. Thompson wrote in the preface to the 1942 edition that he had written "this book in wartime, and its revision has employed me during another war. It gave me solace and occupation, when service was debarred me by my years. Few are left of the friends who helped me write it." An edition of 346 pages was abridged by John Tyler Bonner, and is widely published under the same title. The book, often in the abridged edition, has been reprinted more than 40 times, and has been translated into Chinese, French, German, Greek, Italian, and Spanish.
Contents
The contents of the chapters in the first edition are summarized below. All but Chapter 11 have the same titles in the second edition, but many are longer, as indicated by the page numbering of the start of each chapter. Bonner's abridgment shortened all the chapters, and removed some completely, again as indicated at the start of each chapter's entry below.
1. Introductory
(1st edition p. 1 – 2nd edition p. 1 – Bonner p. 1) Thompson names the progress of chemistry towards Kant's goal of a mathematical science able to explain reactions by molecular mechanics, and points out that zoology has been slow to look to mathematics. He agrees that zoologists rightly seek for reasons in animals' adaptations, and reminds readers of the related but far older philosophical search for teleology, explanation by some Aristotelian final cause. His analysis of "growth and form" will try to show how these can be explained with ordinary physical laws.
2. On Magnitude
(1st p. 16 – 2nd p. 22 – Bonner p. 15) Thompson begins by showing that an animal's surface and volume (or weight) increase with the square and cube of its length, respectively, and deducing simple rules for how bodies will change with size. He shows in a few short equations that the speed of a fish or ship rises with the square root of its length. He then derives the slightly more complex scaling laws for birds or aircraft in flight. He shows that an organism thousands of times smaller than a bacterium is essentially impossible.
3. The Rate of Growth
(1st p. 50 – 2nd p. 78 – Bonner removed) Thompson points out that all changes of form are phenomena of growth. He analyses growth curves for man, noting rapid growth before birth and again in the teens; and then curves for other animals. In plants, growth is often in pulses, as in Spirogyra, peaks at a specific temperature, and below that value roughly doubles every 10 degrees Celsius. Tree growth varies cyclically with season (less strongly in evergreens), preserving a record of historic climates. Tadpole tails regenerate rapidly at first, slowing exponentially.
4. On the Internal Form and Structure of the Cell
(1st p. 156 – 2nd p. 286 – Bonner removed) Thompson argues for the need to study cells with physical methods, as morphology alone had little explanatory value. He notes that in mitosis the dividing cells look like iron filings between the poles of a magnet, in other words like a force field.
5. The Forms of Cells
(1st p. 201 – 2nd p. 346 – Bonner p. 49) He considers the forces such as surface tension acting on cells, and Plateau's experiments on soap films. He illustrates the way a splash breaks into droplets and compares this to the shapes of Campanularian zoophytes (Hydrozoa). He looks at the flask-like shapes of single-celled organisms such as species of Vorticella, considering teleological and physical explanations of their having minimal areas; and at the hanging drop shapes of some Foraminifera such as Lagena. He argues that the cells of trypanosomes are similarly shaped by surface tension.
6. A Note on Adsorption
(1st p. 277 – 2nd p. 444 – Bonner removed) Thompson notes that surface tension in living cells is reduced by substances resembling oils and soaps; where the concentrations of these vary locally, the shapes of cells are affected. In the green alga Pleurocarpus (Zygnematales), potassium is concentrated near growing points in the cell.
7. The Forms of Tissues, or Cell-aggregates
(1st p. 293 – 2nd p. 465 – Bonner p. 88) Thompson observes that in multicellular organisms, cells influence each other's shapes with triangles of forces. He analyses parenchyma and the cells in a frog's egg as soap films, and considers the symmetries bubbles meeting at points and edges. He compares the shapes of living and fossil corals such as Cyathophyllum and Comoseris, and the hexagonal structure of honeycomb, to such soap bubble structures.
8. The same (continued) (1st p. 346 – 2nd p. 566 – Bonner merged with previous chapter) Thompson considers the laws governing the shapes of cells, at least in simple cases such as the fine hairs (a cell thick) in the rhizoids of mosses. He analyses the geometry of cells in a frog's egg when it has divided into 4, 8 and even 64 cells. He shows that uniform growth can lead to unequal cell sizes, and argues that the way cells divide is driven by the shape of the dividing structure (and not vice versa).
9. On Concretions, Spicules, and Spicular Skeletons
(1st p. 411 – 2nd p. 645 – Bonner p. 132) Thompson considers the skeletal structures of diatoms, radiolarians, foraminifera and sponges, many of which contain hard spicules with geometric shapes. He notes that these structures form outside living cells, so that physical forces must be involved.
10. A Parenthetic Note on Geodetics (1st p. 488 – 2nd p. 741 – Bonner removed) Thompson applies the use of the geodetic line, "the shortest distance between two points on the surface of a solid of revolution", to the spiral thickening of plant cell walls and other cases.
11. The Logarithmic Spiral ['The Equiangular Spiral' in 2nd Ed.]
(1st p. 493 – 2nd p. 748 – Bonner p. 172) Thompson observes that there are many spirals in nature, from the horns of ruminants to the shells of molluscs; other spirals are found among the florets of the sunflower. He notes that the mathematics of these are similar but the biology differs. He describes the spiral of Archimedes before moving on to the logarithmic spiral, which has the property of never changing its shape: it is equiangular and is continually self-similar. Shells as diverse as Haliotis, Triton, Terebra and Nautilus (illustrated with a halved shell and a radiograph) have this property; different shapes are generated by sweeping out curves (or arbitrary shapes) by rotation, and if desired also by moving downwards. Thompson analyses both living molluscs and fossils such as ammonites.
12. The Spiral Shells of the Foraminifera (1st p. 587 – 2nd p. 850 – Bonner merged with previous chapter) Thompson analyses diverse forms of minute spiral shells of the foraminifera, many of which are logarithmic, others irregular, in a manner similar to the previous chapter.
13. The Shapes of Horns, and of Teeth or Tusks: with A Note on Torsion
(1st p. 612 – 2nd p. 874 – Bonner p. 202) Thompson considers the three types of horn that occur in quadrupeds: the keratin horn of the rhinoceros; the paired horns of sheep or goats; and the bony antlers of deer. In a note on torsion, Thompson mentions Charles Darwin's treatment of climbing plants which often spiral around a support, noting that Darwin also observed that the spiralling stems were themselves twisted. Thompson disagrees with Darwin's teleological explanation, that the twisting makes the stems stiffer in the same way as the twisting of a rope; Thompson's view is that the mechanical adhesion of the climbing stem to the support sets up a system of forces which act as a 'couple' offset from the centre of the stem, making it twist.
14. On Leaf-arrangement, or Phyllotaxis
(1st p. 635 – 2nd p. 912 – Bonner removed) Thompson analyses phyllotaxis, the arrangement of plant parts around an axis. He notes that such parts include leaves around a stem; fir cones made of scales; sunflower florets forming an elaborate crisscrossing pattern of different spirals (parastichies). He recognises their beauty but dismisses any mystical notions; instead he remarks that
The numbers that result from such spiral arrangements are the Fibonacci sequence of ratios 1/2, 2/3, 3.5 ... converging on 0.61803..., the golden ratio which is
15. On the Shapes of Eggs, and of certain other Hollow Structures (1st p. 652 – 2nd p. 934 – Bonner removed)