variations in the struggle for life; that no
creature can be improved beyond its necessities for the time being; that
the law acts by life and death, and by the survival of the fittest. We
have to ask, therefore, what relation the successive stages of
improvement of the mathematical faculty had to the life or death of its
possessors; to the struggles of tribe with tribe, or nation with nation;
or to the ultimate survival of one race and the extinction of another.
If it cannot possibly have had any such effects, then it cannot have
been produced by natural selection.

It is evident that in the struggles of savage man with the elements and
with wild beasts, or of tribe with tribe, this faculty can have had no
influence. It had nothing to do with the early migrations of man, or
with the conquest and extermination of weaker by more powerful peoples.
The Greeks did not successfully resist the Persian invaders by any aid
from their few mathematicians, but by military training, patriotism, and
self-sacrifice. The barbarous conquerors of the East, Timurlane and
Gengkhis Khan, did not owe their success to any superiority of intellect
or of mathematical faculty in themselves or their followers. Even if the
great conquests of the Romans were, in part, due to their systematic
military organisation, and to their skill in making roads and
encampments, which may, perhaps, be imputed to some exercise of the
mathematical faculty, that did not prevent them from being conquered in
turn by barbarians, in whom it was almost entirely absent. And if we
take the most civilised peoples of the ancient world--the Hindoos, the
Arabs, the Greeks, and the Romans, all of whom had some amount of
mathematical talent--we find that it is not these, but the descendants
of the barbarians of those days--the Celts, the Teutons, and the
Slavs--who have proved themselves the fittest to survive in the great
struggle of races, although we cannot trace their steadily growing
success during past centuries either to the possession of any
exceptional mathematical faculty or to its exercise. They have indeed
proved themselves, to-day, to be possessed of a marvellous endowment of
the mathematical faculty; but their success at home and abroad, as
colonists or as conquerors, as individuals or as nations, can in no way
be traced to this faculty, since they were almost the last who devoted
themselves to its exercise. We conclude, then, that the present gigantic
developme