48 Life and Letters of Francis Galton 



let A> A> etc. be the means of all the deviations, including their signs, of the ancestors in the 

 1st, 2nd, etc. degrees respectively; then 



£ (M + A) + 1 {M+ A) + etc. = M + (» A + £ A + etc.)." 



This is sufficient evidence that Galton had not at the time under con- 

 sideration 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 regres- 

 sion 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 modi- 

 fied 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 



t supposing it to arise from measuring deviations from the means of their 

 groups.^In this way a rather mysterious entity "the racial centre of regresO 

 sion " was created, which was given biological significance, when it really s 

 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." 



