48 Life and Letters of Francis Galton
let Dl, D21 etc. be the means of all the deviations, including their signs, of the ancestors in the 1st, 2nd, etc. degrees respectively; then
(M + Dl) + 4 (M+ D$) + etc. = M + (I D., + I Da + etc. )."
This is sufficient evidence that Galton had not at the time under consideration reached the full meaning of multiple regression. The Ancestral Law is nothing but the principle of multiple regression applied to ancestral inheritance, but in this case the deviations must all be measured not from a general mean, but from the mean of the corresponding generation. The Law of Ancestral Heredity is therefore independent of the change of type, if such is taking place; it can tell us nothing of the laws ruling that change of type, which is something wholly independent of it. Galton's statements that the law may be applied either to total values or deviations is only true for a population stable through the whole ancestry, whereas the application to deviations (with the proper ancestral coefficients, i.e. the multiple regression coefficients) is generally true, and if Galton had recognised this, it would have saved him from doubts as to the compatibility of his law with evolutionary changes.
That Galton recognised the difference between the Quartile of the single brood and the Quartile of the clubbed broods of like parents shows that he fully appreciated the difference between R and r. I do not think, however, that he recognised that his Ancestral Law, i.e. the values he had chosen for his coefficients, actually enforced a definite relation between R and r. But he fully realised the relation between the regression coefficient and r, his "index of correlation*." We can now continue our citation from the Entomological Society paper, which brings out Galton's difficulty
" The laws in which these constants play a part give calculated results that prove to be closely true to observation in the ordinary cases of simple heredity, where there has been no long-continued selection, but it does not at all follow that they will hold true for the descendants
of a long succession of widely divergent parents. It is this that I want to test. The point towards which Regression tends cannot, as the history of Evolution shows, be really fixed. Then the vexed question arises whether it varies slowly or by abrupt changes, coincident with changes of organic equilibrium which may be transmitted hereditarily; in other words, with small or large changes of type. Moreover the values of the Quartile in (3) and (4) cannot be strictly constant and are probably connected in part with the value of the Median and require a modified treatment by using the geometrical law of error instead of the arithmetical one (Proc. Royal Soc. 1879). Again the diminution of fertility and of vitality that accompany wide divergence from racial mediocrity have yet to be measured, by comparing the A [selected large size] and Z [selected small size] broods with the M [mediocre size] broods. It was assumed not to vary in the approximate theory of which I spoke." (p. 28.)
These words bring out the difficulty which arose in Galton's mind from treating regression as taking place towards a fixed racial value, instead of supposing it to arise from measuring deviations from the means of their groups. In this way a rather mysterious entity "the racial centre of regression " was created, which was given biological significance, when it really was only a factor in the purely statistical description of mass phenomena. Once recognise that in each generation the deviation is measured from the
* He speaks of his five constants being connected by "an equation."