not to be
understood that any geographical law of distribution has ever been observed
which governs this, but merely that certain families of races have shown a
preference for the one or the other method of counting. These families,
disseminating their characteristics through their various branches, have
produced certain groups of races which exhibit a well-marked tendency, here
toward the decimal, and there toward the vigesimal form of numeration. As
far as can be ascertained, the choice of the one or the other scale is
determined by no external circumstances, but depends solely on the mental
characteristics of the tribes themselves. Environment does not exert any
appreciable influence either. Both decimal and vigesimal numeration are
found indifferently in warm and in cold countries; in fruitful and in
barren lands; in maritime and in inland regions; and among highly civilized
or deeply degraded peoples.

Whether or not the principal number base of any tribe is to be 20 seems to
depend entirely upon a single consideration; are the fingers alone used as
an aid to counting, or are both fingers and toes used? If only the fingers
are employed, the resulting scale must become decimal if sufficiently
extended. If use is made of the toes in addition to the fingers, the
outcome must inevitably be a vigesimal system. Subordinate to either one of
these the quinary may and often does appear. It is never the principal base
in any extended system.

To the statement just made respecting the origin of vigesimal counting,
exception may, of course, be taken. In the case of numeral scales like the
Welsh, the Nahuatl, and many others where the exact meanings of the
numerals cannot be ascertained, no proof exists that the ancestors of these
peoples ever used either finger or toe counting; and the sweeping statement
that any vigesimal scale is the outgrowth of the use of these natural
counters is not susceptible of proof. But so many examples are met with in
which the origin is clearly of this nature, that no hesitation is felt in
putting the above forward as a general explanation for the existence of
this kind of counting. Any other origin is difficult to reconcile with
observed facts, and still more difficult to reconcile with any rational
theory of number system development. Dismissing from consideration the
quinary scale, let us briefly examine once more the natural process of
evolution through which the decimal and the vigesimal scal