Home > On Growth and Form Summary & Study Guide

On Growth and Form | Introduction

On Growth and Form, first published in 1917 and then republished in 1942 by Cambridge University Press, is D’Arcy Thompson’s radical departure from standard zoology. Looking outside the scope of comparative morphology and evolution, he sought to study nature from a mathematical perspective. He looked to both ancient as well as modern texts to pull together his observations on the development of form and structure in living things. The resulting text is a poetic treatise dedicated to the wonders of nature.

What makes his observations so rich is his careful consideration and blending of philosophy, literature, and mathematical principle. The reader may hope to encounter the philosophy of Goethe or Hegel or perhaps the mention of Darwin (who was a personal acquaintance of Thompson’s) or Galileo, among others. But what makes the text so remarkable is the melding or blending of the literary and the mathematical. In addition to being a brilliant mathematician and scientist, Thompson was a thoughtful teacher. When discussing the relationship of a sphere to its volume, for example, Thompson looks to Swift’s character Gulliver and to the everyday world to illuminate his ideas.

What shines through and adds value to the work as a whole is the author’s love of and deep respect for nature and the pursuit of knowledge. As a zoologist, scientist, and philosopher, Thompson leaves very little to chance in his rich and multifaceted approach to biological study, exploring the concepts of growth and form with objectivity that is persuasive to any audience.

On Growth and Form Summary

Chapter 1: Introduction
The author sets up the basic premise behind the title On Growth and Form in this introductory chapter. Thompson describes the framework for the text, stating that the study of science is not alone dependent on mathematics, nor is it simply to be viewed as unexplainable, divinely created phenomena. His criticism is that both the scientist and the naturalist, among others, attempt to explain the natural world with a limited focus. The intent of his book is to foster a more diverse approach to the study of the concepts of growth and form from a mathematical perspective.

Kant declared that the criterion of true science is in its relation to mathematics. Adds Thompson, ‘‘numerical precision is the very soul of science.’’ Thompson is very careful to point out that he has no interest in reducing the wonders and mystery of the living body to a mechanism (to mathematical formula). He remains an individual who in all creatures is impressed by the beauty manifested in adaptation, that is, the flower for the bee, the berry for the bird. However, he maintains that inquiry into the way in which both living things and physical phenomena grow and take on a specific shape should be approached in the spirit of both scientific theory and mystery. Thompson describes his objective in writing the work as follows:

We want to see how . . . the forms of living things, of the parts of living things, can be explained by physical considerations and to realize that in general no organic forms exist save such as are in conformity with physical and mathematical laws.

More simply put, growth deserves to be studied in relation to form. Both growth and form are necessarily related by mathematical principles, hence the need for a mechanical approach to morphology, or the study of growth and form.

Chapter 2: Magnitude
The form of an object is defined, says Thompson, when we know its magnitude and direction related to the further concept or dimension of time. Growth in length and growth in volume are both parts of one process or phenomenon. Understanding the correlation between length and weight enables us at any time to translate one magnitude to the other by means of mathematical formula.

From a philosophical perspective, the author claims that, concerning magnitude, there is ‘‘no absolute scale of size in the universe, for it is boundless towards the great and also boundless towards the small.’’ Magnitude presents itself conceptually or as an idea in any combination of ways, based on contrasts such as big and small or near and far, for example.

Chapter 3: The Forms of Cells
Surface tension is due to molecular force, arising from the action of one molecule upon another. In the case of a liquid, the molecules of a surface layer are being constantly attracted into the interior by those molecules more deeply situated within the liquid, explains Thompson. He continues, stating, ‘‘the surface shrinks as molecules keep quitting it for the interior, and this surface shrinkage exhibits itself as surface tension. The cell forms explained in the chapter are those forms that a fluid surface can assume under the mere influence of surface tension.

This surface tension accounts, for example, for the spherical form of a raindrop. It is smaller organisms or small cellular elements of larger organisms whose forms will be governed by surface tension. Forms of other larger organisms are dictated by entirely unrelated, non-molecular forces. The author points out, for example, that the surface of a larger body of water is level because it is dictated by gravity, but the surface of water within a narrow tube is curved due to molecular attraction (as in the case of a raindrop).

Thompson discusses various surface tension forms and their surfaces of revolution (surfaces symmetrical about an axis), using soap bubbles to illustrate what happens to create various surface tension forms. For example, if a soap bubble is caught by the end of a pipe and the other side of that same bubble is caught by another pipe, the slow pulling apart of both pipes will cause the bubble to take on a cylindrical form. Eventually, the bubble will break. The fragility of the bubble illustrates Thompson’s next point—that such surfaces realize complete equilibrium within particular dimensional limits or only demonstrate a certain amount of stability, with the exception of the plane or the sphere, whose perfect symmetry allows for perfect stability.

Chapter 4: Forms of Tissues or Cell Aggregates
The movement of the text has been to discuss the nature of solitary cells and then to discuss the action between two cells. This chapter discusses the impact of cell aggregates, or clusters of cells, and the impact of such groupings upon form.

Thompson begins his discussion by restating a general principle underlying the theory of surface tension, namely, that in the case of fluid to fluid or fluid to gas contact, ‘‘a portion of the total energy of the system is proportional to the area of the surface of contact.’’ The author goes on to say that total energy is also ‘‘proportional to a coefficient that is specific for each particular pair of substances and is constant for these,’’ with the exception of changes inspired by temperature or electrical charge. In other words, there are numerical measures (coeffi- cients) that dictate the total energy of a system. Equilibrium, or the state in which minimum potential energy is realized in a system, will be achieved by a reduction in surface contact.

In the case of three bodies, the solution to such a problem becomes a bit more complex. In the example of a substance floating on water, there are three surfaces involved in the process of equilibrium, namely, the water, the air above it, and the substance itself. The condition of equilibrium will be reached by ‘‘contracting those surfaces whose specific energy happens to be large and extending those where it is small.’’ This contraction will result in production of a drop and extension to a spreading film. Turpentine will contract into a drop when it comes in contact with water, as opposed to olive oil, which will instead spread out and form a film on the water’s surface. In the context of either of the previously mentioned examples, a pull for equilibrium on a water surface exists as the result of tensions existing in the other two surfaces of contact. Thompson instructs us to imagine a single particle. If a drop is pulled out by another water particle, without finding yet another providing a counter-pull, ‘‘it will be drawn upon by three different forces, whose directions lie in the three surfaceplanes and whose magnitudes are proportional to the specific tensions characteristic of the three ‘interfacial,’ or smoothly coordinating, surfaces.’’

The chapter then moves from a discussion on the interactions of particles as a function of surface tensions to the forms aggregates of cells tend to take. Thompson elaborates on the tendency not only of cells, ‘‘but of any bodies of uniform size and originally circular outline, close-packed in a plane’’ to adapt the hexagonal pattern, each cell or body being in contact with six others surrounding it ‘‘under widely varying circumstances.’’ Such patterns are evidenced in the bee’s cell as it is formed to create the honeycomb, a circumstance... » Complete On Growth and Form Summary