I. Mathematical contributions to the theory of evolution. —XI. On the influence of natural selection on the variability and correlation of organs

Abstract At an earlier stage in the development of the statistical theory of evolution it was suggested that the coefficient of correlation (Galton’s function) might be found constant for all races of the same species—in fact, it was considered possible that this coefficient might be the long-sought-for criterion of identity in species. Professor Weldon, following up this suggestion of Mr. Galton’s, then made the elaborate series of measurements on crabs with which his name will always be closely associated. To a first approximation these researches seemed to confirm the possibility of Galton’s function being a true criterion of species. When, however, a finer mathematical test was applied to Professor Weldon’s observations as well as to other statistical series for organs in man,* it became clear that the coefficient of correlation varied from local race to local race, and could not be used as a criterion of species. A slight investigation undertaken in the summer of 1896 convinced me that the coefficient of correlation between any two organs, is just as much peculiar and characteristic of a local race as the means and variations of those organs. In fact, if local races be the outcome of natural selection, then their coefficients of correlation must in general differ. The object of the present paper is to show, not only that natural selection must determine the amount of correlation, but that it is probably the chief factor in the production of correlation. If selection, natural or artificial, be capable of producing correlation, then it seems impossible to regard all correlation as evidence of a causal nexus, although the converse proposition that all causal nexus denotes correlation, is undoubtedly the most philosophical method of regarding causality. In dealing with the influence of selection on correlation, I shall suppose the distribution of complex groups of organs to follow the normal correlation surface—the generalised Gaussian law of frequency. I shall further assume the selection surfaces to be normal in character. Neither of these assumptions is absolutely true, but the Gaussian law in a good many cases describes the frequency sufficiently closely to enable us to obtain fair numerical results by its application. Probably in all cases it will enable us to reach qualitative if not accurate quantitative theoretical deductions. I have the less hesitation in asserting this, as Mr. G. U. Yule has recently succeeded in deducing the chief formulæ for correlation and regression as given by the Gaussian law from general principles, which make no appeal to a special law of frequency.