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2   NATURAL INHERITANCE.   [Case.

generations. The large do not always beget the large, nor the small the small, and yet the observed proportions between the large _and the small in each degree of size and in every quality, hardly varies from one generation to another.

A second problem regards the average share contributed to the personal features of the offspring by each ancestor severally. Though one half of every child may be said to be derived from either parent, yet he may receive a- heritage from a distant progenitor that neither of his parents possessed as personal characteristics. Therefore the child does not ' on the average receive so much as one half of his personal qualities from each parent, but something less than a half. The question I have to solve, in a reasonable and not merely inn a statistical way, is, how much less ?

The last of the problems that I need mention now, concerns the nearness of kinship in different degrees. We are all agreed that a brother is nearer akin. than a nephew, and a nephew than a cousin, and` so on, but how much nearer are they in the precise language of numerical statement ?

These and many other problems are all fundamentally connected, and I have worked them out to a first degree of approximation, with some completeness. The conclusions cannot however be intelligibly presented in an introductory chapter. They depend on ideas that must first be well comprehended, and which are now novel to the large majority of readers and unfamiliar to all. But those who care to brace themselves to a