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 scales come into being.  After the completion of one count of the fingers the savage announces his result in some form which definitely states to his mind the fact that the end of a well-marked series has been reached.  Beginning again, he now repeats his count of 10, either on his own fingers or on the fingers of another.  With the completion of the second 10 the result is announced, not in a new unit, but by means of a duplication of the term already used.  It is scarcely credible that the unit unconsciously adopted at the termination of the first count should now be dropped, and a new one substituted in its place.  When the method here described is employed, 20 is not a natural unit to which higher numbers may be referred.  It is wholly artificial; and it would be most surprising if it were adopted.  But if the count of the second 10 is made on the toes in place of the fingers, the element of repetition which entered into the previous method is now wanting.  Instead of referring each new number to the 10 already completed, the savage is still feeling his way along, designating his new terms by such phrases as “1 on the foot,” “2 on the other foot,” etc.  And now, when 20 is reached, a single series is finished instead of a double series as before; and the result is expressed in one of the many methods already noticed—­“one man,” “hands and feet,” “the feet finished,” “all the fingers of hands and feet,” or some equivalent formula.  Ten is no longer the natural base.  The number from which the new start is made is 20, and the resulting scale is inevitably vigesimal.  If pebbles or sticks are used instead of fingers, the system will probably be decimal.  But back of the stick and pebble counting the 10 natural counters always exist, and to them we must always look for the origin of this scale.