22 Life and Letters of Francis Galton
based on h, for h27 h„ ..., etc. Galton deals only with the latter. He writes as follows
" When we say that the midparent contributes two-thirds of his peculiarity of height to the offspring, it is supposed that nothing is known about the previous ancestor. But though nothing is known, something is implied, and this must be eliminated before we can learn what the parental bequest, pure and simple, may amount to. Let the deviate of the midparent be x (including the sign), then the implied deviate of the midgrandparent will be 3 x, of the midancestor in the next generation I x and so on. Hence the sum of the deviates of all the midgenerations that constitute the heritage of the offspring is x (1 + - +.- + etc.) = x 3.
Now I think this result erroneous because it assumes that the quantities y, ya, y&, ... of the generant above can be found from the simple regression formula of parent on offspring. This we know to be very far from the fact, the multiple regression coefficients have no such simple relations to parental regression. The fallacy lies, I think, in this : we could imply that value of the grandparental from the parental deviate by means of the simple regression formula, but to do this is to assert that all the remaining h's, h,, h4i etc., are put zero, i.e. are given every conceivable value, with the mean value zero. -But we are going to imply other than zero values for these h's, hence our system of implied ancestral values is not consistent and this, I think, is indicated by the total heritage coming out x -,a. To get over this difficulty Galton proceeds "to tax" each contribution to the heritage. He takes as two extreme cases (a) a uniform taxation of all ancestral contributions of 1,
and (b) a taxation geometrical in amount, supposing 1 of the total only to
be transmitted from one generation. to a second. He thus reaches the following expressions
X (m + 3m2 + yM 3 -~ etc....' = x 3m-1
But x being the midparental character the heritage of the offspring is, Galton says, 3 x, thus n = 9, and 1 = 6 . From this he draws the conclusion that both may be taken to be 2 approximately. But here the reasoning
seems at fault, for the offspring heritage of 3 x is based on all the other midparental deviates h„ h,.. taking their average or zero values. The
regression coefficient would not be two-thirds, if they took the values 3 h„ 2hl f etc.
Finally from what Galton has just said it would appear that we might have two series for determining ancestral contributions, the one in n, i.e. 2, 2, 2, ..., or the one in m, i.e. 2, 4, , .... But this is clearly not what he
* Roy. Soc. Proc. Vol. LXII, p. 61, and compare Journ. Anthrop. Instit. Vol. xv, pp. 260 et seq.