The decimal vigesimal system is found in the Munda languages of India and in Tibeto-Chinese groups of the Himalayas, in Nico barese, in the north (and south) Caucasian languages. With the Indo-European and the Semitic languages the dual form for 20 seems to point to a former vigesimal system. The Sumerian, too, shows traces of it. Basque practises a full (decimal) vigesimal sys tem, so that forms like soixante-dix, quatre-vingt, quatre-vingt-dix in French are to be explained.
In Africa the vigesimal system occurs in upper Guinea, and its hinterland, from the Senegal to the Cross river, in a series of Bantu, Mande, Togo and Niger languages, and in some east Bantu languages, in the Hamitic Kunama (and Barea) and in the Bantu languages Konde and Sango.
In North America it is found in Eskimo and in the north-west in Bellacoola and Haida and several languages of California, together with Pawnee (Caddo). This system mostly as decimal vigesimal is dominant in Mexico, Central America, even in the languages of the high civilization of the Aztecs and Maya. In South America the vigesimal system is found both in the progressive civilization of the Chibchas and with a great many primitive tribes belonging to nearly every linguistic group save the Gez-Tapuya and the tribes south of the Gran Chaco.
The quinary vigesimal system, especially its older form com bined with the pair system, appears to have originated in the cul ture cycle (Kulturkreis, culture area) of the totemistic-patriarchal progressive hunters; a culture type in which the human body in art (sculpture), clothing and adornment has gained greatest im portance and is most cared for. According to that the vigesimal system, too, agrees up to 20 with the parts of the body.
In north-western Africa on the coast of upper Guinea a quadra gesimal system has developed, derived from a vigesimal system. Its origin is due to commercial reasons, as the monetary standard in this region is a set of 4o cowries.
bers of the second pentad are formed by composition with "five" (6=5+1, 7=5+2, etc.) or by the pair system (especially 6= 8=4X2) or by subtraction (especially 9— 10— I). It seems to have originated in the matriarchal soil-tilling culture. The great quantity of agricultural products which had to be counted, ar ranged, stored, and brought to market required to be frequently repeated and counted quickly, and thus led to this system. In the languages of this culture numeral unities have developed which spring from the special grouping of certain products : a dozen, a brace, score (three score), etc.
In Australia and in the Papuan languages it is found isolated and in rudimentary form ; with the Melanesians it is fully dis played. In Asia it originally dominated the two great families of the Austroasiatic and the Tibeto-Chinese languages. Whether a quinary or a pure decimal system was the original numeration in the Ural-Altaic languages cannot be ascertained. In Africa the great majority of the languages show originally a quinary decimal system, especially the Bantu and the semi-Bantu languages and the majority of the Nilotic, Wule, Ngo-Nke and Manfu languages. So too, in North America; but neither in Mexico, Central America, nor in South America has the system spread widely.
With the pure decimal system there is no partition of the decad into two pentads and therefore the compound character of the numerals from five to ten is not met with. The whole system becomes simpler and its forms are better fitted for general applica tion. The pure decimal system seems to have originated in the culture-cycle of the nomadic herders, who, in counting their large flocks of horses, cows, camels, sheep, etc., needed to employ high numbers with more facility. From the aristocracies of the nomadic herders it has spread everywhere, and it is now found in all nations of high culture on the whole globe, except those of Mexico and Central America, where the number 20 was used in astronomy, and thus was safe from competition. (W. Sc.) NUMERATION: see NUMERALS ; ARITHMETIC.