This is the most celebrated chapter of the book and it has been widely commented upon in biological literature. I have not made any careful survey, but I suspect that the well-known diagrams of transformations have been reproduced in sundry scientific writings a large number of times.

The comments almost invariably have a few points in common, which I shall briefly summarise here. In the first place it is surprising that despite their fame, the Cartesian transformations have been used very little. This is because, to use Medawar's term, they are ‘analytically unwieldy’. The few times the method has been applied is in the development or change of form in a single system, as for instance, Richards and Riley's study of developing amphibians under different conditions, and Medawar's analysis of tissue culture growth.

A far more significant result in terms of practical application is that the system of transformations of D'Arcy Thompson stimulated and contributed to the much simpler method of analysis of allometric growth, which has found widespread use, mainly through the work of J. S. Huxley. Here instead of attempting to analyse a whole structure in two (or three) divisions, two factors are isolated and compared on a logarithmic scale. In this way it is possible to discover the ratio of the growth-rates of different structures, a method which has found application in embryology, taxonomy, palaeontology and even ecology.