and 2 were _the_ numbers at mankind's disposal; to the time
when his three numeral concepts, 1, 2, many, each demanded distinct
expression. With increasing knowledge the necessity for this
differentiatuin would pass away, and but two numbers, singular and plural,
would remain. Incidentally it is to be noticed that the Indo-European words
for 3--_three_, _trois_, _drei_, _tres_, _tri,_ etc., have the same root as
the Latin _trans_, beyond, and give us a hint of the time when our Aryan
ancestors counted in the manner I have just described.

The first real difficulty which the savage experiences in counting, the
difficulty which comes when he attempts to pass beyond 2, and to count 3,
4, and 5, is of course but slight; and these numbers are commonly used and
readily understood by almost all tribes, no matter how deeply sunk in
barbarism we find them. But the instances that have already been cited must
not be forgotten. The Chiquitos do not, in their primitive state, properly
count at all; the Andamans, the Veddas, and many of the Australian tribes
have no numerals higher than 2; others of the Australians and many of the
South Americans stop with 3 or 4; and tribes which make 5 their limit are
still more numerous. Hence it is safe to assert that even this
insignificant number is not always reached with perfect ease. Beyond 5
primitive man often proceeds with the greatest difficulty. Most savages,
even those of the tribes just mentioned, can really count above here, even
though they have no words with which to express their thought. But they do
it with reluctance, and as they go on they quickly lose all sense of
accuracy. This has already been commented on, but to emphasize it afresh
the well-known example given by Mr. Oldfield from his own experience among
the Watchandies may be quoted.[109] "I once wished to ascertain the exact
number of natives who had been slain on a certain occasion. The individual
of whom I made the inquiry began to think over the names ... assigning one
of his fingers to each, and it was not until after many failures, and
consequent fresh starts, that he was able to express so high a number,
which he at length did by holding up his hand three times, thus giving me
to understand that fifteen was the answer to this most difficult
arithmetical question." This meagreness of knowledge in all things
pertaining to numbers is often found to be sharply emphasized in the names
adopted by savages for their numeral