Galton's law of ancestral heredity

Galton's ancestral law states that the two parent contribute between them on average one-half of the total heritage of the offspring, the four grandparents one-quarter, and so on. He interpreted this law both as a representation of the separate contributions of each ancestor to the heritage of the offspring and as a multiple regression formula for predicting the value of a trait from ancestral values. Logical reconstruction of the law is presented based on formalizing Galton's model of heredity outlined in Natural Inheritance (Galton, 1889). The resulting law has a free parameter to be empirically estimated which represents the frequency of latent hereditary elements that are not expressed in a particular individual but are capable of transmission to the next generation. The equation representing ancestral contributions to the heritage of the offspring differs from the multiple regression equation for predicting the value of a trait from ancestral values. The former equation reduces to Galton's ancestral law when the proportion of latent elements is 0.5, the latter when this proportion 0.6. Galton's rather different derivations of the law in 1885 and 1897 are described, and their shortcomings are discussed in the light of these results (Galton, 1885, 1897).