318 DIVERGENT EVOLUTION THROUGH SEGREGATION. 
The problems that arise in considering the different results produced — 
by different degrees of positive segregation and segregrate fecundity 
are of a nature suitable for mathematical treatment. Before, however, 
computing the effects of segregate fecundity when co-operating with 
positive segregation, it will be in place to show that it is of itself only 
a negative form of segregation, having no power to insure the propaga- 
tion of the varieties thus characterized, though they are fully adapted 
to the environment. This is most easily brought to light by consider- 
ing the effect of a high degree of this quality when positive segregation 
is entmely ve wanting, or when it is sufficient to give simply a chance of 
segregate breeding by bringing each individual near to its natural mate. 
For example, let us suppose, first, that a male and a female each of 
several allied but mutually sterile species are brought together on one 
small island, all other tendencies to positive segregation being removed, 
while mutual sterilty still remains; second, that a male and female when 
once mated remain together for the breeding season; and third, that all 
find mates. Now, if we have seven species, each represented by one 
individual of each sex, what is the probability that all the species will 
be propagated? And what the probability for the propagation of none, 
or of but one, or of but two, or of but three of the species? The answers, 
as I have computed them, are as follows: The probability that none 
will be propagated is 4354; that one species will be is 30404 that two 
species 3040} that three species #13,; that four species =42,; that five 
species =21,; that seven species i These numerators are found in 
the seventh line of a table that 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 1234261; that one species will 
match truly and propagate is 1324269; that ten willis gg54¢9,- ‘This 
means that if 3,628,800 trials are made, one 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 eight-elevenths of the probabilities are against the continu- 
ance of more than one of the ten species. 
There will perhaps be some hesitation in receiving these figures before 
I have given the method by which the results have been reached; but 
the necessary length of this paper, even when restricted to the briefest 
discussion of general principles, induces me to reserve my computa- 
tions for another occasion. 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 environment 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 
