234 PROCEEDINGS OF THE AMERICAN ACADEMY. 



reality, an error lies in the very first step of his calculation, which in- 

 validates all that follows. He says," The 100 pure individuals will give 

 rise to dominant forms in the proportion of 50 pure to 50 hybrids." 

 On the contrary, " pure " dominants bred inter se will produce only pure 

 dominant offspring ; but if they mate at random with any individuals of 

 the entire 300, there are only two chances out of three that they will mate 

 with ?ii/brid dom\ua,nts, which mating alone could yield " dominant forms 

 in the proportion 50 pure to 50 hybrids." Yule accordingly estimates 

 too low the proportion of dominant individuals in the various generations. 



IV. Race Improvement by Selection of Desirable or by 

 Elimination of Undesirable Individuals. 



On the hypotheses, which I understand Yule to adopt, of random mat- 

 ing and equal fertility on the part of all individuals, 300 dominant forms, 

 of which 100 are pure and 200 hybrid, will produce more than 88 per 

 cent of dominant individuals, instead of 83 percent as estimated by Yule. 

 For if we suppose each class to consist of males and females in equal num- 

 bers, the chances are just twice as great that an individual will mate with 

 a hybrid dominant, A (B), as that it will mate with a pure dominant, A. 

 Or, to put the matter in another way, there are, for each individual of 

 the entire 300, 50 possible A mates, and 100 jjossible A (B) mates. This 

 makes the entire number of different matings possible : — 



A. A(B). B. 

 5,000 Ax A, yielding offspring .... 5,000 

 20,000 Ax A (B), yielding offspring . . . 10,000 + 10,000 

 20,000 A (B) X A {B), yielding offspring . 5,000 + 10,000 + 5,000 



Total 20,000 + 20,000 + 5,000 



or 4 ^ : A A (B) : I B. It will be observed that f , or 88.8 per cent, of the 

 offspring have the dominant form, being either ^ or ^ (B) in character. 

 Eliminating the one recessive individual, B, in each nine offs[)ring, the 

 parents of the next generation will consist of 4 ^s (pure dominants) and 

 4 A {B)s (hybrid dominants) ; that is, of equal numbers of individuals 

 A and A (B). The possible matings * in this case will be : — 



* To simplify the calculation, it is well to remember that the numerical 

 proportions of the various matings possible within a population are expressed 

 by the square of that population. Knowing the nature and numerical proportions 

 of the possible matings, one can quickly calculate the numerical proportions of 

 the oflspring. Thus, in a population consisting of equal numbers of individuals 

 A and A (B), the possible matings are expressed by the square of A ■}- A (B), or 



