On Growth and Form

Citation

Thompson, D. W. (1917). On Growth and Form. Cambridge University Press. Free edition (Project Gutenberg): https://www.gutenberg.org/ebooks/55264

Chapter Summary

The 1917 edition has seventeen chapters and a short epilogue, about 780 pages of text with 408 numbered figures. It is organised as a progression: size and growth first, then the physical forces acting on cells, then tissues, skeletons and spirals, and finally mechanical design and the comparison of related forms. The prefatory note calls the book "all preface" and an "easy introduction" to the study of organic form by methods that are commonplace in physical science. Thompson says he claims no mathematical skill and uses only simple mathematics.

Chapter 1: Introductory

Thompson sets the programme. Physiology gained strength by borrowing from physics and chemistry, but the morphologist has been slow to describe form in mathematical language, preferring explanation by purpose or "final cause". He traces that tradition from Aristotle through to Darwin's natural selection, argues for the search for physical causes, and begins to treat the cell as a fluid body shaped by surface tension and other forces. Growth, he proposes, may be considered both as a process and as a force.

Chapter 2: On Magnitude

Form is defined by magnitudes in different directions. Surface increases as the square and volume as the cube of linear size, so a fish that doubles in length multiplies its weight eightfold. Thompson develops the principle of similitude, Galileo's argument that neither a house nor an animal can grow beyond a certain size without changing proportions, with examples from trees, skeletons, ships and the speed and working rate of animals (Froude's law). He also considers very small organisms, for which viscosity and Brownian movement matter more than weight.

Chapter 3: The Rate of Growth

Form is a ratio of magnitudes; growth is magnitude considered over time. The chapter reviews curves of growth in human beings, plants and animals, seasonal and temperature effects, and regeneration, to show that rate of growth is a characteristic property of each species and part.

Chapters 4–6: The Internal Form and Structure of the Cell; The Forms of Cells; A Note on Adsorption

Chapter 4 treats cell division and the structures seen in it (mitosis and the chromosomes) in terms of fields of force and physical analogies. Chapter 5 takes protoplasm as a fluid and explains the shapes of single cells and of unicellular organisms through surface tension, with Plateau's liquid figures (the cylinder, unduloid, nodoid and catenoid) as models. Chapter 6 adds a brief account of adsorption, which alters surface energy.

Chapters 7–8: The Forms of Tissues, or Cell-Aggregates

These chapters move from the single cell to cells in contact. Soap films and partition walls of minimal area explain the shapes of cells in tissues, the configurations of cells in dividing eggs and the geometry of the bee's cell. The second chapter applies the same reasoning to the order in which cells divide.

Chapter 9: On Concretions, Spicules, and Spicular Skeletons

The inorganic deposits and skeletons of sponges, radiolarians and other organisms, whose forms depend either on crystal chemistry or on the shape of the living cells around them.

Chapter 10: A Parenthetic Note on Geodetics

Shortest paths on surfaces of revolution, applied to the spiral thickenings in plant cells and to patterns on shells and the spicules of the preceding chapter.

Chapters 11–12: The Logarithmic Spiral; The Spiral Shells of the Foraminifera

The mathematics of the logarithmic spiral and of the gnomon (the addition that keeps a figure's shape unchanged) is applied to the shells of the Nautilus, the ammonites and other molluscs. The following chapter turns to the chambered shells of the Foraminifera.

Chapters 13–15: The Shapes of Horns, and of Teeth or Tusks; On Leaf-Arrangement, or Phyllotaxis; On the Shapes of Eggs, and of Certain Other Hollow Structures

Chapter 13 applies spiral geometry, without much mathematics, to horns, tusks and antlers, including the effect of torsion. Chapter 14 reviews the spiral arrangements of leaves and florets and doubts that the Fibonacci series, and its supposed usefulness, deserve a place in botany. Chapter 15 analyses the egg as a solid of revolution and extends the discussion to other hollow structures, including the branching of blood vessels.

Chapter 16: On Form and Mechanical Efficiency

Beams, girders and bridges serve as models for skeletons. Thompson shows the arrangement of the bone trabeculae in the femur matching the lines of stress drawn by the engineer Karl Culmann, and compares the backbones of quadrupeds and dinosaurs with bridge structures such as the cantilever.

Chapter 17: On the Theory of Transformations, or the Comparison of Related Forms

The best-known chapter. It shows how the outlines of related species can be inscribed on a Cartesian grid and compared by distorting the grid, with examples from fishes, crabs, skulls and pelvic bones.

Epilogue

A brief closing statement that mathematics and dynamics should be part of the morphologist's work, and that biological form offers the mathematician a field for labour.

Key Concepts

Chapters 1–3

• Final cause versus physical cause: the older habit of explaining organisms by purpose, against which the book sets mathematics and mechanics.
• Principle of similitude: how proportions change with size, because surface grows as the square and volume as the cube.
• Rate of growth: form considered as a function of time.

Chapters 4–10

• Surface tension: the force that shapes drops, cells and films.
• Minimal partitions: the cell walls and divisions that give the smallest surface.
• Geodetic line: the shortest path between two points on a surface.

Chapters 11–17

• Logarithmic spiral and gnomon: the curve of shells and horns, and the increment that preserves form.
• Phyllotaxis: the spiral arrangement of leaves.
• Lines of stress: the directions of tension and compression in a loaded structure, such as bone.
• Transformation by coordinates: comparison of forms by the deformation of a grid.

Critical Analysis

Strengths

• A clear method. Throughout, Thompson asks what shape a physical force would produce and compares it with the organism, and the illustrations (408 numbered figures) make the comparisons concrete.
• Mathematics at an elementary level. The preface says that the methods are the easiest and simplest, and chapters such as the one on horns are written without mathematics.
• Range of examples. The same few principles are applied to cells, sponges, shells, horns, eggs, bones and skulls.
• Prose. The writing is learned and often vivid, with historical and literary references, and can be read as an essay.

Limitations

• Dated biology. The book predates molecular genetics and the modern synthesis; the epilogue says Thompson sets little store by certain postulates of the biology of his day, and Chapter 14 rejects the Fibonacci series and natural selection as explanations of leaf arrangement; a modern reader needs to bring the other half.
• Length and uneven pace. Chapter 3 alone runs to over a hundred pages and Chapters 5 and 7–9 are also long, while others, such as Chapters 6 and 10, take only a few pages.
• Some arguments are analogies rather than proofs. The resemblance between a force diagram and a biological structure shows what physical forces could do, not that they did so.
• Dated science in the details. Authorities and measurements are those of the early twentieth century.

Where It Fits in the Course

This book sits on the further-reading shelf alongside The Selfish Gene, a very different work: Dawkins argues from genes and natural selection, Thompson from physics and geometry, so the two make a deliberate contrast of explanatory styles. The remaining titles on the shelf come from other fields (law, psychology, economics, computing) and have little connection with it.

How to Use This Book

No specialist training is needed beyond school mathematics and some biology, though readers who know calculus and basic physics will get more out of it. Read the Prefatory Note, Chapters 1–2 and the Epilogue first, then the chapters that match your interest: Chapters 5 and 7 for surface tension, Chapter 11 for spirals, Chapter 16 for mechanics and Chapter 17 for transformations. The cell chapters (4–6) and Chapters 8–10 are the most technical and can be skipped on a first pass. Reading the figures alongside the text is essential, and modern work on morphogenesis and biomechanics supplies the developments since 1917.

Edition and Sources

This guide covers the first edition of 1917 (Cambridge University Press), read in the free Project Gutenberg text (eBook #55264). The prefatory note, contents and list of illustrations, the opening of every chapter and the epilogue were read, with the passages on size, similitude and growth, the closing pages of the final chapter, and the sections on phyllotaxis and the trabeculae of bone in full. Later enlarged editions are not covered.