[232] It
was, no doubt, owing to the absence of a sound system of numeration that
the mathematical talent of the Greeks was directed chiefly to geometry,
in which science Euclid, Archimedes, and others made such brilliant
discoveries. It is, however, during the last three centuries only that
the civilised world appears to have become conscious of the possession
of a marvellous faculty which, when supplied with the necessary tools in
the decimal notation, the elements of algebra and geometry, and the
power of rapidly communicating discoveries and ideas by the art of
printing, has developed to an extent, the full grandeur of which can be
appreciated only by those who have devoted some time (even if
unsuccessfully) to the study.

The facts now set forth as to the almost total absence of mathematical
faculty in savages and its wonderful development in quite recent times,
are exceedingly suggestive, and in regard to them we are limited to two
possible theories. Either prehistoric and savage man did not possess
this faculty at all (or only in its merest rudiments); or they did
possess it, but had neither the means nor the incitements for its
exercise. In the former case we have to ask by what means has this
faculty been so rapidly developed in all civilised races, many of which
a few centuries back were, in this respect, almost savages themselves;
while in the latter case the difficulty is still greater, for we have to
assume the existence of a faculty which had never been used either by
the supposed possessors of it or by their ancestors.

Let us take, then, the least difficult supposition--that savages
possessed only the mere rudiments of the faculty, such as their ability
to count, sometimes up to ten, but with an utter inability to perform
the very simplest processes of arithmetic or of geometry--and inquire
how this rudimentary faculty became rapidly developed into that of a
Newton, a La Place, a Gauss, or a Cayley. We will admit that there is
every possible gradation between these extremes, and that there has been
perfect continuity in the development of the faculty; but we ask, What
motive power caused its development?

It must be remembered we are here dealing solely with the capability of
the Darwinian theory to account for the origin of the _mind_, as well as
it accounts for the origin of the _body_ of man, and we must, therefore,
recall the essential features of that theory. These are, the
preservation of useful