lOO ANALYSIS OF THB FOUR PRINCIPLES. 



numerators are found in the seventh line of a table of figures 

 which I call the "permutational triangle."* If we have ten species, 

 the probability that in any one trial no species will match truly and 

 be propagated is i^; that i species will match truly and prop- 

 agate is jIS ; that lo will is ^^^. This means that if 3,628,800 trials 

 are made, i of them will probably be a case in which each male 

 pairs with the female of the same species, while 1,334,961 will be 

 cases in which none are so matched, and 1,334,960 will be cases in 

 which one pair is so matched. It therefore appears that more than 

 ^ of the probabilities are against the continuance of more than one 

 of the ten species. 



It is not, however, necessary to have a complete solution of this 

 problem in order to reach the conclusion that the origin of separate 

 races and species depends not only upon their adaptation to the en- 

 vironment and their mutual sterility when crossing with each other, 

 but also upon their positive segregation. We can further see (when 

 considering an extreme case, like either of the above-supposed cases) 

 that segregate fecundity, without the aid of positive segregation, must 

 lead to extinction. We have already seen that partial positive segre- 

 gation can not by itself prevent the fusion of species. It therefore 

 follows that in order to account for the continuance of divergent races 

 we must suppose either that the positive segregation is complete or 

 that the divergent evolution is strong enough to more than counter- 

 balance the influence of the occasional crossing, or that the partial 

 positive segregation is aided by segregate fecundity, or by some other 

 form of segregate survival. 



(6) Partial positive segregation unaided by negative segregation can not 

 prevent fusion. — Between the members of species belonging to different 

 orders we find not only complete segregation, but complete sterility 

 when attempts at crossing are made ; but hope of gaining an explana- 

 tion of how these characteristics have arisen is found, not in the study 

 of those cases in which the process has been completed, but in the 

 study of the relations to each other of species and varieties that are 

 characterized by segregation and mutual sterility, that is, not com- 

 plete. Here, again, mathematical analysis will help us in understand- 

 ing the subject. Though I have not succeeded in constructing a com- 

 plete mathematical representation of all the grades of intermingling 

 that will take place, I have found a general formula that gives a close 

 approximation to the proportion in which two species will produce 

 pure-breeds as contrasted with the proportion of cross-breeds that 



♦ See Appendix 11 of this volume. 



