100 ANALYSIS OF THE 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 3\"'; that 1 species will match truly and prop- 
agate is Swe; that 10 willis ss. This means that if 3,628,800 trials 
are made, 1 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 unarded by negative segregation can not 
prevent fuston.— 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 II of this volume. 
