NUMERAL SYSTEMS. There is no language without some numerals ; the notion of unity and plurality is expressed at least in the formation of "one" and "two," though "two" is often equal to "much," thus concluding a numeration that has just only start ed. It is doubtful whether even systemless numeration really exists, as it is mostly reported of peoples who are but vaguely known. The eastern languages of Australia, in spite of the occur rence of numerals for "three" and even "four," and in a less degree the western languages, Yuin-Kuri, Wiradhuri, Kamilaroi, and the southern central languages have been suspected of it. It remains doubtful whether Tasmania practised systemless or pair numera tion. The Pygmies of the Andaman islands and Malacca form nu merals for "one" and "two," yet sum up only with units, not with pairs. So with the Chiquito in South America.
The pair system has numerals for "one" and "two" and forms the following numerals by addition to the "pair": 3=2+1, 4=2+2, 5=2+2+1, etc. It is found in Aus tralia among tribes ethnologically the oldest—the Kulin-Kurnai of the south-east, the Narrinyeri of the south ; several of them count up to "ten" in this manner. A pure pair system still occurs in many Papuan languages of Torres straits and the adjacent coast of New Guinea. In Africa it is practised by the Bushmen. In South America it is found among the ethnologically oldest tribes—the Fuegian tribes: Yamana and Halakwulup, the Guayaki and Shipaya, and the Ges-Tapuya tribes.
The pair system starts from the parts of the human body that exist in pairs, like eyes, ears, hands, feet. The pair system is also found in various associations with later numeration systems. The formation of a dual with the personal pronoun (and substantive) can be traced back to those times.
The quaternary system forms the numerals above "four" by composition: 5=4+1, 7=4+3, 8=4+4 (or
16=4X4. In this consequent type, however, it is but seldom met, e.g., in California with the Salina and traces of it with the Chumash. In California the four
quarters of the sky play an important part in religion, mythology and custom.
When further devel oped to a duodecimal system this is the most useful of all as it per mits of more divisions without fractional numbers than any other system, and in later stages it has been repeatedly introduced espe cially for astronomy or in metrical or monetary systems. In primi tive numeration it has a rather limited dispersion in north-west Africa, e.g., in the Huka, the Bulanda, the Apko; traces of it are to be met with among the Bube on Fernando Po.
The quinary system in its pure form, where for instance 10=2 hands, 25=5 hands, is found only in Saraweka, a South American Arowak language. Everywhere else it is combined either with the decimal or the vigesimal system.
The vigesimal system takes 20 as a basis, so as to make 2oX2o the numeral next in height, if the numeration goes as far as that. It may combine either with the quinary system forming a quinary vigesimal system or more rarely and only in younger forms with the decimal system, and then be comes a decimal vigesimal system. The quinary vigesimal system is frequently combined with the pair system, so that the numbers 3 and 4, often also 6, 7, 8, 9, and further 12 and 13, are formed according to the pair system.
These systems start with the fingers and the toes. Therefore "five" often means "hand," "ten" means "the two hands," I I "one at the foot." 20 "both the feet (and hands)" or "the whole man." The quinary vigesimal system is found sporadically in almost all the Australian linguistic groups; in nearly all the Papuan languages of the (north) east coast and exterior of New Guinea, in the oldest Melanesian languages of New Caledonia, the Loyalty islands, etc. In Asia-Europe it occurs in the border languages : Ainu, Chukchi, Koryak, in Burushaski (q.v.), and in the Himalaya group of the Tibeto-Burman languages.