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38   Life and Letters of Francis Galton

namely, that if an individual has h of a certain character, the most probable value of the character in his parent will be 1h, and in his grand

parent 32h and his great grandparent h and so on.

Consequently, if we know nothing beyond the one parent of character h, the expected heritage is

h{4+2(3x2,)+4(9x28~- ...j=hx0'30.

When one grandparent only is known to have h then the corresponding parent has 1 h, and the two great grandparents 1 h, the four great great

grandparents 32 h and soon. Thus A he formula is

h{(3x229+1x(2,)-~-2(3x2g)-+-4(32x28)+...j=hx(2 40)=hx0.16,

i.e. actually 0.1583 h.

If a parent and the corresponding two grandparents be known Galton says the parent will contribute 4 of his character and the two grandparents and their ancestry : 0o as above. But I do not think this is correct, even on Galton's assumptions. In the previous case we predicted the great grandparents and higher ascendants from a knowledge of the grandparents only. But in this case we have not only these two grandparents, but also the knowledge of their offspring, the parent, to predict from, and accordingly Galton's -~U for the rest of the ancestry is not satisfactory. As he is working in round numbers, Galton puts ; (= -075) as equal to •08.

Three cases are now dealt with : I, both parents only known; II, four grandparents only known; and III, both parents and four grandparents known. I gives 2 x •30 = •60 of heritage with a residue of '40 undetermined. Galton distributes this residue in the general population proportions of light to dark eyes after distributing the hazel eyes 3 to light and 1 to dark eyes, which give 70'/. and 30 °/, of those eyes. Thus the residue -40 is to be given 28 to light and '12 to dark eyes. The corresponding residues for cases II and III are -36 and •18, which Galton distributes as •25 and •11, •12 and •06* respectively.

Galton now combines all these results in a table from which with knowledge of the ancestry as far as parents and grandparents are concerned he considers prediction of eye-colour in offspring can be ascertained (p. 39).

Let me illustrate the use of this table. A family of 12 given by Galton had both predict light-eyed,. 3 grandparents light-eyed and 1 hazel-eyed. If we predict from parents only we should have

12 x (2 x •30 +,2 8) = 12 x •88 =10.56 light-eyed.

If f we predict from grandparents only we should have

12 x (3 x •16 + 1 x '10 + •25) = 9.96 light-eyed.

* More accurately the latter pair should be -13 and •05.