ion 2 and 1. This limitation, as we have already seen, is shared by
the Botocudos, the Chiquitos, and many of the other native races of South
America. But the Abipones, in seeking for words with which to enable
themselves to pass beyond the limit 3, invented the singular terms just
given for 4 and 5. The ostrich, having three toes in front and one behind
on each foot presented them with a living example of 3 + 1; hence "toes of
an ostrich" became their numeral for 4. Similarly, the number of colours in
a certain hide being five, the name for that hide was adopted as their next
numeral. At this point they began to resort to digital numeration also; and
any higher number is expressed by that method.

In the sense in which the word is defined by mathematicians, _number_ is a
pure, abstract concept. But a moment's reflection will show that, as it
originates among savage races, number is, and from the limitations of their
intellect must be, entirely concrete. An abstract conception is something
quite foreign to the essentially primitive mind, as missionaries and
explorers have found to their chagrin. The savage can form no mental
concept of what civilized man means by such a word as "soul"; nor would his
idea of the abstract number 5 be much clearer. When he says _five_, he
uses, in many cases at least, the same word that serves him when he wishes
to say _hand_; and his mental concept when he says _five_ is of a hand. The
concrete idea of a closed fist or an open hand with outstretched fingers,
is what is upper-most in his mind. He knows no more and cares no more about
the pure number 5 than he does about the law of the conservation of energy.
He sees in his mental picture only the real, material image, and his only
comprehension of the number is, "these objects are as many as the fingers
on my hand." Then, in the lapse of the long interval of centuries which
intervene between lowest barbarism and highest civilization, the abstract
and the concrete become slowly dissociated, the one from the other. First
the actual hand picture fades away, and the number is recognized without
the original assistance furnished by the derivation of the word. But the
number is still for a long time a certain number _of objects_, and not an
independent concept. It is only when the savage ceases to be wholly an
animal, and becomes a thinking human being, that number in the abstract can
come within the grasp of his mind. It is at this point that mere