476 THE GRAMMAR OF SCIENCE 



breeding, a man's 1024 parents of the tenth generation 

 back are a fair sample of the general population of that 

 date. In other words, if we take the tenth mid-parents 

 of a number of individuals of a race for which pangamic 

 mating is the rule, we should expect them to be sensibly 

 the same, i.e. simply the racial type of that date. Thus, 

 as we go back in ancestry, the variability of mid-parents 

 must become less and less than the racial variability of 

 the same date. Some fairly easy algebra ^ shows us that 



the ratio of mid-parental to racial variability is — r^' 2' 2T/2' 



. for the 1st, 2nd, 3rd, 4th, . . . 9th, 



I I I 



4 ■ ■ ' 16^^/2' 32 ■ ' 

 loth . . . generations of ancestry. Thus while an in- 

 dividual's parents may be removed considerably from the 

 racial type, yet without great selection or much in-and-in 

 breeding, he must be the product in very few generations 

 of a group of individuals whose mean differs very little 

 from the racial type ; for the i oth mid-parents of the race 

 exhibit scarcely any variability. 



Now let H^, H,„ H,, . . . H^^^ ... be the deviations 

 from the mid-parental means of the ist, 2nd, 3rd, . . . 

 I oth . . . mid-parents of a given group of offspring, and 

 let h be the mean deviation of these offspring from the 

 mean of offspring in general, i.e. Ji is the type of the 

 offspring due to a series of mid-parents of given char- 

 acters. Again, let S^, Sg, S3, . . . S^q . • • be the standard 

 deviations or variabilities of the successive generations of 

 mid-parents, and o- the variability of all offspring. Then 

 it may be shown that the type h of the offspring of a 

 given system of mid-parents is determined by 



-'1 -^2 -^3 -'10 



where 7^ 7.,, 7,, . . . 7^0 . . . are numerical quantities 

 depending only on the coefficients of correlation between 

 offspring and mid-parents, and between the mid-parents 

 themselves. 



Now Mr. Francis Galton, as a result of his observa- 



1 Royal Society Proceedings, vol. Ixii. p. 390. 



