Notebooks

## D'Arcy Wentworth Thompson, 1860--1948

23 Jul 2000 17:03

D'Arcy Thompson was a British biologist and a classical scholar (translator of Aristotle's biological works, author of Greek Birds and Greek Fishes). His masterwork, On Growth and Form, is a profound consideration of the shapes of living things, starting from the simple premiss that "everything is the way it is because it got that way" (*). Hence one must study not only finished forms, but also the forces that moulded them: "the form of an object is a 'diagram of forces', in this sense, at least, that from it we can judge of or deduce the forces that are acting or have acted upon it" (ch. I). Now by "forces" Thompson meant forces, and one of his great themes is the tremendous light cast on living things by using mathematics to describe their shapes and fairly simple physics and chemistry to explain them. In other words, Thompson wrote a thousand page treatise on self-organization long before the word existed. (Ideas get names, not so much when they crystallize into coherent concepts, as when the crystals get big enough to be noticed; retrospectively one can trace how they grew up to that point.) Now, Thompson thought of his physical forces as rivals of natural selection, on the grounds that they are inescapable and unchanging over time --- "A snow-crystal is the same to-day as when the first snows fell" (ch. XII) --- but, given our much better understanding of genetics and development, it's pretty easy to see that what the genes do is control pattern formation, set up mechanisms which work along lines he would have approved of. (It's a shame he didn't live long enough to know about Turing's work on morphogenesis.)

Turn of the century biologists with classical educations who thought they had an alternative to Darwin are a dime a dozen; two things make Thompson worth remembering. One is the sheer brilliance of On Growth and Form, which would deserve an honored place in the history of biology were the writing ever so bad. The other is that the writing is brilliant, that Thompson had one of the best prose styles of any scientist; indeed, an excellent writer, period. A brief sample (from Chapter II, "Of Magnitude") may be in order:

We are accustomed to think of magnitude as a purely relative matter. We call a thing big or little with reference to what it is wont to be, as when we speak of a small elephant or a large rat; and we are apt accordingly to suppose that size makes no other or more essential difference, and that Lilliput and Brobdingnag are all alike, according as we look at them through one end of the glass or the other. Gulliver himself decalred, in Brobdingnag, that 'undoubtedly philosophers are in the right when they tell us that nothing is great and little otherwise than by comparison': and Oliver Heaviside used to say, in like manner, that there is no absolute scale of size in the Universe, for it is boundless towards the great and also boundless towards the small. It is of the essence of the Newtonian philosophy that we should be able to extend our concepts and deductions from the one extreme of magnitude to the other; and Sir John Herschel said that 'the student must lay his account to finding the disction of great and little altogether annihilated in nature.'

All this is true of number, and of relative magnitude. The Universe has its endless gamut of great and small, of near and far, of many and few. Nevertheless, in physical science the scale of absolute magnitude becomes a very real and important thing; and a new and deeper interest arises out of the changing ratio of dimensions when we come to consider the inevitable changes of physical relations with which it is bound up. The effect of scale depends not on a thing in itself, but in relation to its whole environment or milieu; it is in conformity with the thing's 'place in Nature', its field of action and reaction in the Universe. Everywhere Nature works true to scale, and everything has its proper size accordingly. Men and trees, birds and fishes, stars and star-systems, have their appropriate dimensions, and their more or less narrow range of absolute magnitudes. The scale of human observation and experience lies within the narrow bounds of inches, feet or miles, all measured in terms drawn from our own selves or our own doings. Scales which include light-years, parsecs, Angström units, or atomic and sub-atomic magnitudes, belong to other orders of things and other principles of cognition.

A common effect of scale is due to the fact that, of the physical forces, some act either directly at the surface of a body, or otherwise in proportion to its surface or area; while others, and above all gravity, act on all particles, internal and external alike, and exert a force which is proportional to the mass, and so usually to the volume of the body.

A simple case is that of two similar weights hung by two similar wires. The forces exerted by the weights are proportional to their masses, and these to their volumes, and so to the cubes of the several linear dimensions, including the diameters of the wires. But the areas of cross-section of the wires are as the squares of the said linear dimensions; therefore the stresses in the wires per unit area are not identical, but increase in the ratio of the linear dimensions, and the larger the structure the more severe the strain becomes: $\frac{\mathrm{Force}}{\mathrm{Area}} \propto \frac{l^3}{l^2} \propto l$ and the less the wires are capable of supporting it.

In short, it often happens that of the forces in action in a system some vary as one power and some as another, of the masses, distances or other magnitudes involved; the 'dimensions' remain the same in our equations of equilibrium, but the relative values alter with the scale. This is known as the 'Principle of Similitude', or of dynamical similarity, and it and its consequences are of great importance. In handful of matter cohesion, capillarity, chemical affinity, electric charge are all potent; across the solar system gravitation rules supreme; in the mysterious region of the nebulae, it may haply be that gravitation grows negligible again.

From which he goes on to discourse on bridges, and Galileo's use of the Principle of Similitude, and the heights and shapes of tall trees, and so forth.

Given the combination of real intellectual power and great rhetorical force, it's not surprising that On Growth and Form has had great influence outside biology (e.g. on design), and even to some degree within it, though less than one might expect. He had many valuable ideas, but opened up few routes for sustained investigation. This is a key point to posthumous success as a scientist, and one reason why the ancestors of modern developmental biology were the people (like Needham) who were re-arranging sea-urchin and newt embryoes and injecting everything they could find into them, and not Thompson, whose influence has been more indirect and inspirational.

*, addendum, 9 September 2014: Prof. Myong-Hun Chang asked me where, exactly, this passage was located in Thompson's work. I had thought it was next to the bit about every form being a "diagram of forces", but it's not, and searching the texts of both the 1917 and 1942 editions at the Internet Archive shows nothing that matches it, or even comes close. (The OCR isn't great, but I was pretty thorough about searching.) Searching several other book archives shows lots of people attributing the phrase to him, or to the book, but none giving a page or even a chapter. Unless this comes from one of his other works, I believe we have here an example of a paraphrase becoming attributed to the original author.

Recommended:
• DAT, On Growth and Form was originally published in 1917; all currently available editions are based on the second edition of 1942. Dover publishes a complete reprint of that edition, weighing in at over eleven hundred pages (and a couple of pounds). An abridgement, edited by John Tyler Bonner, is published by Cambridge University Press. The older versions of this had an introduction by Bonner; more recent ones have an introduction by Stephen Jay Gould. (Sic transit etc.)
• By others:
• Philip Ball, The Self-Made Tapestry: Pattern Formation in Nature is a superb popular book on the modern understanding of physical pattern formation, and how, if at all, it relates to biology [Review: The Blind Snowflake-Maker]
• Peter Medawar has a fine essay on D'Arcy Thompson, from which I hope I've cribbed no more than half a dozen passages, reprinted in Pluto's Republic.