their numeral systems which are not homogeneous with the main characteristics of the system show great persistence. The "score" of English is a remnant of old vigesimal counting, and although it has lost its place in the ordinary number system, it is still retained as a semi-poetical form. Still more marked is the "quatre-vingts" of the French. In counting from 61 to 99 they use a purely vigesimal system. If these traces of vigesimal counting still remain, it would seem probable that if the quinary system had ever formed a part of the system it would also somewhere have left its marks, fainter, it is true, on account of its greater antiquity, but still discernible. Now the only indication from a philological source that such a system was ever employed by the Aryan peoples seems to be the Homeric πεμπάςειυ (literally to five), meaning "to count." It is sometimes stated also that the form of the Latin numerals I, II, III, IIII or IV, V, VI, etc., implies the existence of an early form of quinary counting. My own opinion is that evidence derived from written numerals, between which and the formation of the numeral system itself there can be no comparison as to dates, can be of very little weight in deciding what was the scale upon which the system was originally formed. If the Roman V for 5 and VI for 6 were adopted because of a quinary element in the Roman scale at the time these signs were first used, surely the spoken language would have retained some marks of the same system. The evidence all points, therefore, with the one exception quoted above, to the nonexistence of a quinary element in Aryan counting.
The third natural scale, besides the quinary and decimal, is the vigesimal. It is doubtful whether a pure vigesimal scale, unmixed with any quinary and decimal element, occurs in any part of the world. In certain regions, or with certain races, a strong tendency is found to make 20 the principal base of the numeral system. This is so with the Celtic peoples, with some Asiatic and a few African tribes, with some of the Eskimos, and with the peoples who formerly occupied the Central American regions. If a tribe counts up to 20, using their fingers and toes, and then continue their counting beyond this point in a consistent way, a vigesimal system will be the natural result; but on account of the practical difficulty of using the toes in any system of gesture-counting, which, as we have seen, is the second stage in the development of the number system, it seems plausible that most tribes confined themselves to the fingers alone. This would account for the greater predominance of 10 and 5 as number bases. It is true that in the case of many well-developed vigesimal scales we have no positive evidence that they originated in the custom of counting on the fingers and toes, but there is certainly great probability that they did all begin in this way. There seems
