THROUGH CUMULATIVE SEGREGATION. 
251 
crossing with each other, but also upon their Positive Segrega- 
tion. 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 Segregation cannot by itself 
prevent the fusion of species. It therefore follows that in order 
to account for the continuance of divergent races we must sup- 
pose either that the Segregation is complete, or that the divergent 
evolution is strong enough to more than counterbalance the 
influence of the occasional crossing, or that the partial Segre- 
gation is aided by Segregate Fecundity or Segregate Vigour. 
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 
explanation 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 partial Segrega- 
tion and mutual sterility, that is not complete. Here, again, 
mathematical analysis will help us in understanding the subject. 
Though I have not succeeded in constructing a complete mathe- 
matical 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 breed 
pure as contrasted with the proportion of first crosses and their 
descendants that will be produced, in any case in which the 
degree of Segregation and the ratios of fertility for the pure and 
crossed breeds are known. As my object is simply to show 
under what conditions the pure races will continue without 
being swamped by crossing, it is not necessary that I should 
follow the action and reaction between the three-quarter-breeds. 
I wish, however, to call attention to the fact that when the 
number of the pure forms and of the half-breeds is constantly 
decreasing, without a general decrease in the sum of the de- 
scendants, it is evident that the three-quarter-breeds and their 
descendants are increasing ; and when a three-quarter-breed on 
one side crosses with a three-quarter-breed on the other side, the 
offspring will usually be about intermediate between the two 
species ; therefore, where the two species are equally numerous, 
if we find that the pure forms will disappear through fusion, we 
