At the beginning of this chapter I introduced the idea of 'homology', using the arms of bats and humans as an example. Indulging an idiosyncratic use of language, I said that the skeletons were identical while the bones were different. D'Arcy Thompson's transformations furnish us with a way to make this idea more precise. In this formulation, two organs - for example, bat hand and human hand - are homologous if it is possible to draw one on a sheet of rubber and then distort the rubber to make the other one. Mathematicians have a word for this: 'homeomorphic'.*
Zoologists recognized homology in pre-Darwinian times, and pre-evolutionists would describe, say, bat wings and human hands as homologous. If they had known enough mathematics, they would have been happy to use the word 'homeomorphic'. In post-Darwinian times, when it became generally accepted that bats and humans share a common ancestor, zoologists started to define homology in evolutionary terms. Homologous resemblances are those inherited from the shared ancestor. The word 'analogous' came to be used for resemblances due to shared function, not ancestry. For example, a bat wing and an insect wing would be described as analogous, as opposed to the homologous bat wing and human arm. If we want to use homology as evidence for the fact of evolution, we can't use evolution to define it. For this purpose, therefore, it is convenient to revert to the pre-evolutionary definition of homology. The bat wing and human arm are homeomorphic: you can transform one into the other by distorting the rubber on which it is drawn. You cannot transform a bat wing into an insect wing in this way, because there are no corresponding parts. The widespread existence of homeomorphisms, which are not defined in terms of evolution, can be used as evidence for evolution. It is easy to see how evolution could go to work on any vertebrate arm and transform it into any other vertebrate arm, simply by changing relative rates of growth in the embryo.
Ever since becoming acquainted with computers as a graduate student in the 1960s, I have wondered what D'Arcy Thompson might have done with a computer. The question became pressing in the 1980s, when affordable computers with screens (as opposed to just paper printers) became common. Drawing on stretched rubber and then distorting the drawing surface in a mathematical way - it was just begging for the computer treatment! I suggested that Oxford University should bid for a grant to employ a programmer to put D'Arcy Thompson's transformations on a computer screen, and make them available in a user-friendly manner. We got the money, and employed Will Atkinson, a first-class programmer and biologist, who became a friend and an adviser to me on my own programming projects. Once he had solved the difficult problem of programming a rich repertoire of mathematical distortions of the 'rubber', it was then a relatively simple task for him to incorporate this mathematical wizardry into a biomorph-style artificial selection program, similar to my own 'biomorph' programs, here described in Chapter 2. As with my programs, the 'player' was confronted with a screen full of animal forms, and invited to choose one of them for 'breeding', generation after generation. Once again there were 'genes' that persisted through the generations, and once again the genes influenced the form of the 'animals'. But in this case, the way the genes influenced animal form was by controlling the distortion of the 'rubber' on which an animal's form had been drawn. Theoretically, therefore, it should have been possible to start with, say, an Australopithecus skull drawn on the undistorted 'rubber', and breed your way through creatures with progressively larger braincases and progressively shorter muzzles - progressively more human-like, in other words. In practice it proved very difficult to do anything like that, and I think the fact is, in itself, interesting.
I think one reason it was difficult is, yet again, that D'Arcy Thompson's transformations change one adult form into another adult form. As I emphasized in Chapter 8, that is not how genes in evolution work. Every individual animal has a developmental history. It starts as an embryo and grows, by disproportionate growth of different parts of the body, into an adult. Evolution is not a genetically controlled distortion of one adult form into another; it is a genetically controlled alteration in a developmental program. Julian Huxley (grandson of T.H. and brother of Aldous) recognized this when, soon after publication of the first edition of D'Arcy Thompson's book, he modified the 'method of transformations' to study the way early embryos turn into later embryos or adults. That's all I want to say about D'Arcy Thompson's method of transformations here. I'll return to the topic in the final chapter, to make a related point.
Comparative evidence has always, as I suggested at the beginning of this chapter, told even more compellingly than fossil evidence in favour of the fact of evolution. Darwin himself took a similar view, at the end of his chapter in On the Origin of Species on the 'Mutual Affinities of Organic Beings':
Finally, the several classes of facts which have been considered in this chapter, seem to me to proclaim so plainly, that the innumerable species, genera, and families of organic beings, with which this world is peopled, have all descended, each within its own class or group, from common parents, and have all been modified in the course of descent, that I should without hesitation adopt this view even if it were unsupported by other facts or arguments.
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