It may seem at first sight surprising that a brother and a sister should each have the same average number of brothers. It puzzled me until I had thought the matter out, and when the results were published in “Nature,” it also seems to have puzzled an able mathematician, and gave rise to some newspaper controversy, which need not be recapitulated. The essence of the problem is that the sex of one child is supposed to give no clue of any practical importance to that of any other child in the same family. Therefore, if one child be selected out of a family of brothers and sisters, the proportion of males to females in those that remain will be, on the average, identical with that of males to females in the population at large. It makes no difference whether the selected child be a boy or a girl. Of course, if the conditions were “given a family of three boys and three girls,” each boy would have only two brothers and three sisters, and each girl would have three brothers and two sisters, but that is not the problem.
Subject to this explanation, the general accuracy of the observed figures which attest the truth of the above conclusion cannot be gainsaid on theoretical grounds, nor can the conclusions be ignored to which they lead. They enable us to make calculations concerning the average number of kinsfolk in each and every specified degree in a stationary population, or, if desired, in one that increases or decreases at a specified rate. It will here be supposed for convenience that the average number of males and females are equal, but any other proportion may be substituted. The calculations only regard its fertile members; they show that every person has, on the average, about one male fertile relative in each and every form of specific kinship.
Kinsfolk may be divided into direct ancestry, collaterals of all kinds, and direct descendants. As regards the direct ancestry, each person has one and only one ancestor in each specific degree, one fa, one fa fa, one me fa, and so on, although in each generic degree it is otherwise; he has two grandfathers, four great-grandfathers, etc. With collaterals and descendants the average number of fertile relatives in each specified degree must be stationary in a stationary population, and calculation shows that number is approximately one. The calculation takes no cognizance of infertile relatives, and so its results are unaffected by the detail whether the population is kept stationary by an increased birth-rate of children or other infertiles, accompanied by an increased death-rate among them, or contrariwise.
The exact conclusions were (“Nature,” September 29, 1904, p. 529), that if 2d be the number of children in a family, half of them on the average being male, and if the population be stationary, the number of fertile males in each specific ancestral kinship would be one, in each collateral it would be d - 1/2, in each descending kinship d. If 2d = 5 (which is a common size of family), one of these on the average would be a fertile son, one a fertile daughter, and the three that remained would leave no issue. They would either die as boys or girls or they would remain unmarried, or, if married, would have no children.
The reasonable and approximate assumption I now propose to make is that the number of fertile individuals is not grossly different to that of those who live long enough to have an opportunity of distinguishing themselves. Consequently, the calculations that apply to fertile persons will be held to apply very roughly to those who were in a position, so far as age is concerned, to achieve noteworthiness, whether they did so or not. Thus, if a group of 100 men had between them 20 noteworthy paternal uncles, it will be assumed that the total number of their paternal uncles who reached mature age was about 100, making the intensity of success as 20 to 100, or as 1 to 5. This method of roughly evading the serious difficulty arising from ignorance of the true values in the individual cases is quite legitimate, and close enough for present purposes.
Chapter VIII.—Number of Noteworthy Kinsmen in each Degree.
The materials with which I am dealing do not admit of adequately discussing noteworthiness in women, whose opportunities of achieving distinction are far fewer than those of men, and whose energies are more severely taxed by domestic and social duties. Women have sometimes been accredited in these returns by a member of their own family circle, as being gifted with powers at least equal to those of their distinguished brothers, but definite facts in corroboration of such estimates were rarely supplied.
The same absence of solid evidence is more or less true of gifted youths whose scholastic successes, unless of the highest order, are a doubtful indication of future power and performance, these depending much on the length of time during which their minds will continue to develop. Only a few of the Subjects of the pedigrees in the following pages have sons in the full maturity of their powers, so it seemed safer to exclude all relatives who were of a lower generation than themselves from the statistical inquiry. This will therefore be confined to the successes of fathers, brothers, grandfathers, uncles, great-uncles, great-grandfathers, and male first cousins.
Only 207 persons out of the 467 who were addressed sent serviceable replies, and these cannot be considered a fair sample of the whole. Abstention might have been due to dislike of publicity, to inertia, or to pure ignorance, none of which would have much affected the values as a sample; but an unquestionably common motive does so seriously—namely, when the person addressed had no noteworthy kinsfolk to write about. On the latter ground the 260 who did not reply would, as a whole, be poorer in noteworthy kinsmen than the 207 who did. The true percentages for the 467 lie between two limits: the upper limit supposes the richness of the 207 to be shared by the 260; the lower limit supposes it to be concentrated in the 207, the remaining 260 being utterly barren of it. Consequently, the upper limit is found by multiplying the number of observations by 100 and dividing by 207, the lower by multiplying by 100 and dividing by 467. These limits are unreasonably wide; I cannot guess which is the more remote from the truth, but it cannot be far removed from their mean values, and this may be accepted as roughly approximate. The observations and conclusions from them are given in [Table VII.], p. [xl].