926 NUMERAL SYSTEMS (ETH. ANN. 19 
Before proceeding I wish to quote some remarks by Conant in regard 
to the origin and spread of the vigesimal system, which I will then 
refer to.’ 
In its ordinary development the quinary system is almost sure to merge into either 
the decimal or the yigesimal system, and to form, with one or the other or both of 
these, a mixed system of counting. In Africa, Oceanica, and parts of North America, 
the union is almost always with the decimal scale; while in other parts of the world 
the quinary and the vigesimal systems have shown a decided affinity for each other. 
It is 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, disseminat- 
ing 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 yvigesimal 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. Subor- 
dinate 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 can not 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. Butso 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. 
I note some facts, taken in part from the work quoted, in order 
that the reader may see the bearing they have on the opinions expressed 
in this quotation. According to the data furnished by this writer it 
seems that this system occurred in Europe only along the western sea- 
coast and that almost exclusively among the Celts, the only group of 
the Aryan stock which seems to have used it. In Asia it has been found 
toany extent only in the Caucasic group and in the northeastern part of 
of the continent, that is, in what Brinton terms the **Arctic Group” of 
his Siberic branch. Notasingle example is noted from the Sinitic group 
or from the Semitic branch. In Africa none have been reported from 
the Hamitic group, and but few from the negro dialects, but the latter 
field has been only superticially examined in this respect. Nota single 
1 Number Concept, p. 176-8. 
