and, for the present, be left in the field of
pure speculation.

A most inviting field for research is, however, furnished by the primitive
methods of counting and of giving visible expression to the idea of number.
Our starting-point must, of course, be the sign language, which always
precedes intelligible speech; and which is so convenient and so expressive
a method of communication that the human family, even in its most highly
developed branches, never wholly lays it aside. It may, indeed, be stated
as a universal law, that some practical method of numeration has, in the
childhood of every nation or tribe, preceded the formation of numeral
words.

Practical methods of numeration are many in number and diverse in kind. But
the one primitive method of counting which seems to have been almost
universal throughout all time is the finger method. It is a matter of
common experience and observation that every child, when he begins to
count, turns instinctively to his fingers; and, with these convenient aids
as counters, tallies off the little number he has in mind. This method is
at once so natural and obvious that there can be no doubt that it has
always been employed by savage tribes, since the first appearance of the
human race in remote antiquity. All research among uncivilized peoples has
tended to confirm this view, were confirmation needed of anything so
patent. Occasionally some exception to this rule is found; or some
variation, such as is presented by the forest tribes of Brazil, who,
instead of counting on the fingers themselves, count on the joints of their
fingers.[5] As the entire number system of these tribes appears to be
limited to _three_, this variation is no cause for surprise.

The variety in practical methods of numeration observed among savage races,
and among civilized peoples as well, is so great that any detailed account
of them would be almost impossible. In one region we find sticks or splints
used; in another, pebbles or shells; in another, simple scratches, or
notches cut in a stick, Robinson Crusoe fashion; in another, kernels or
little heaps of grain; in another, knots on a string; and so on, in
diversity of method almost endless. Such are the devices which have been,
and still are, to be found in the daily habit of great numbers of Indian,
negro, Mongolian, and Malay tribes; while, to pass at a single step to the
other extremity of intellectual development, the German student keeps his
b