THE VIGESIMAL SYSTEM OF ENUMERATION

CYRUS THOMAS

As the vigesimal system is a factor of considerable importance in the study of the ancient civilization of Mexico and Central America, especially in regard to the native calendar of those regions, it is interesting to know to what extent this system of enumeration has prevailed in other parts of the world. As a step toward bringing together the data on this subject, the writer presents the following notes:

Although, as is well known, the people of Malaysia and southeastern Asia use the decimal system, yet there are some indications that the vigesimal system was formerly in use, at least at one point, in the latter region. Aymonier discovered, by an examination of the inscriptions at Bakou and Loley in Cambodia, an account of which is published in the Journal Asiatique for 1883, evidence of two systems of enumeration; one of these, which appeared to be the most recent and generally used, the decimal system; the other and more ancient, the vigesimal system. The examples he gives in the original characters make this so clear as to leave no doubt on the point. There are characters for each of the nine digits, for 20 and for 100. The character for 20 is distinct, and not two tens. In order to indicate 37, there is, first, the character for 20, then for 10, and last for 7. The 40 is two twenties; 50, two twenties and ten; 60, three twenties; 80, four twenties; 98 is four twenties, ten, and eight; for 384, three hundreds, four twenties, and four. A mingling of the two systems is apparent in some of the examples given by Aymonier, but the evidence of the ancient vigesimal system is too clear and distinct to permit of doubt.

Whether further evidence on this point has been obtained from the ruins of Cambodia the writer is unable to say, as he has not had access to the most recent publications on this subject. There are, however, a few facts which indicate the use of the vigesimal system in ancient times in Malaysia or southeastern Asia, or both.

Although the Malayo-Polynesian question is still considerably tangled, it is generally admitted that both the language and

the people of Polynesia were derived from the region of Malaysia and Farther India. It is therefore legitimate to look to Polynesia for echoes of the customs of the pristine home. According to A. Featherman (Oceano-Melanesians), the Marquesans, although using the decimal system, denoted "twenty" by a specific word, all the rest of the numbers being "compounded from ten and twenty with a multiple unit." The Nukahivahs, of the New Marquesas group, "have specific words for the units and ten, for twenty, for forty, for four hundred, and four thousand; all the other numerals are compounded of these with the aid of ten and the units." Thus tekau-onohuu, 20 plus 10 equal 30; etahi-touha, 1 by 40; ua-touha, 2 by 40 equal 80; tou-ao', 3 by 400 equal 1,200, etc. According to the same authority the Hawaiian system of numeration is decimal, but "progresses by forties. There are specific words for the units and ten; eleven is expressed by ten and one over; for 76 they would say 40, 20, 10, and 6, and thus the numbers are counted by forties to four hundred, for which there exists a specific word. In this manner the numbers are expressed by the addition of intervening fractional numbers as high as four thousand and four hundred thousand, each of which is denoted by a specific word." These facts apparently indicate a primary vigesimal system. John Crawfurd also came to the conclusion that there was an older numeral system once in use in Polynesia.

The Maya method of enumeration was very similar to that of the Polynesian nations mentioned. The numbers from one to eleven had specific names, but from twelve to nineteen by the addition of units to ten. There was a specific name for twenty, for four hundred, and for eight thousand. The intermediate numbers from twenty to four hundred are formed mostly by twenty as the multiple, and units, though there was not entire uniformity in this respect; from four hundred to eight thousand progress was made by four hundred as the multiple; yet there is evidence in several places of the use of ten as a multiple. It is apparent, however, in the codices that the count was by units to five, and then by fives to twenty, precisely as stated by Landa.

1 See also Transactions American Ethnological Society, vol. 11, 229.

AUSTRALIAN CLASS SYSTEMS

R. H. MATHEWS

In a paper' contributed to the Queensland Branch of the Royal Geographical Society of Australasia in November, 1894, I briefly showed how Australian savages having the Kamilaroi type of class system are organized into families, totemic groups, classes, tribes, and communities, and made reference to the rules of marriage and descent. I pointed out that all previous investigations respecting Australian class systems had been merely of an elementary character, leaving what appeared to me the most important part of the subject untouched. I also stated that the work which is most necessary, and that should be undertaken while there is still opportunity for doing so, is to trace out and formulate the details of the laws which regulate the intermarriage of the classes and totems, as well as the course of descent in them, together with the distribution of the totems under the four classes.2

In the present article I shall endeavor to follow this line of research, and take as an example some of the Wiradjuri tribes, who occupy the country on the Murrumbidgee river from some distance above Jugiong down as far as Hay, extending southerly to the Murray and stretching northerly up Lachlan river as far as about the effluxion of the Willandra Billabong. These tribes comprise an important section of the great Wiradjuri community, and are included in what has been termed the "Kamilaroi organization." They are divided into four classes, the names of the men being different from those of the women. In one class all the men are called Murri and all the women Matha; in another, Kubbi and Kubbitha; in another, Ippai and Ippatha, and in another the men are Oombi and the women Butha. These four classes have groups of totems corresponding to each.

1 "The Kamilaroi Class System of the Australian Aborigines." Proc. Roy. Geog. Soc. Aust. (Q), x, 17-34.

2 Loc. cit., 26, 27.

3 The Willandra Billabong runs out of Lachlan river and flows away across the level country toward the Lower Darling river.

4 Oombi is the equivalent of the Kamilaroi Kumbo.

From a considerable number of totems, four which can be relied upon have been selected as examples under the head of each class, as set out in the subjoined table, which may be called A. In arranging this table the classes Murri and Kubbi are placed next to each other, and the classes Ippai and Oombi are likewise put side by side. The totems are arranged in a certain order under each pair of classes, Murri Emu being opposite Kubbi Flying Squirrel, and Ippai Mallee-hen opposite Oombi Common Fly, and so on, so that they can be conveniently referred to by and by in describing the rules of descent.

I will endeavor to explain the laws regulating the marriage of the four classes in accordance with their totemic divisions: Murri marries Ippatha, the sister of Ippai; Kubbi marries Butha, the sister of Oombi; Ippai marries Matha, the sister of Murri, and Oombi marries Kubbitha, the sister of Kubbi. Although marriages follow this fundamental class-law to a certain extent, there are irregularities or innovations to which I referred in my former memoir,' under which a man may marry into one or more of the other classes under certain totemic restrictions. These irregularities, as I have termed them, are so interwoven with the class law that they may be said to form part of it. They are so general and so widespread that they are found in nearly all the divisions of the Wiradjuri, Kamilaroi, and other tribes having the "Kamilaroi organization" before referred to, and until they are thoroughly recorded and explained our knowledge of these class systems will necessarily be incomplete.

For the purpose of illustration, I will take the sixteen totems mentioned in Table A, and show separately how each one intermarries with certain others. The four Murri men-Emu, Red

1 Proc. Roy. Geog. Soc. Aust. (Q), x, 24.

Kangaroo, Brown Snake, and Ground Iguana-will be taken first, giving the names of the different women from amongst whom they may lawfully choose a wife. Then the four men of the Kubbi class; next the four Ippais; and, lastly, the four Oombi men will be dealt with in the same manner. The totem names of these sixteen men will be taken in the order in which they appear in Table A.

No. 1, Murri Emu, marries Ippatha Eaglehawk, Ippatha Opossum, Kubbitha Native Bee, and Matha Brown Snake.

No. 2, Murri Red Kangaroo, marries Ippatha Opossum and Ippatha Eaglehawk.

No. 3, Murri Brown Snake, marries Ippatha Opossum, Ippatha Eaglehawk, Matha Emu, and Kubbitha Native Bee.

No. 4, Murri Ground Iguana, marries Ippatha Mallee-hen, Ippatha Jew Lizard, Butha Codfish, Kubbitha Flying Squirrel, and Kubbitha Bandicoot.

No. 5, Kubbi Flying Squirrel, marries Butha Goonhur, Butha Gray Kangaroo, Butha Codfish, and Kubbitha Porcupine.

No. 6, Kubbi Bandicoot, marries Butha Goonhur, Butha Gray Kangaroo, Butha Codfish, and Kubbitha Porcupine.

No. 7, Kubbi Porcupine, marries Butha Goonhur, Butha Gray Kangaroo, Butha Codfish, Kubbitha Flying Squirrel, and Kubbitha Bandicoot.

No. 8, Kubbi Native Bee, marries Butha Common Fly, Ippatha Jew Lizard, Matha Emu, and Matha Brown Snake.

No. 9, Ippai Mallee-hen, marries Matha Ground Iguana, Kubbitha Flying Squirrel, Kubbitha Bandicoot, and Butha Codfish. No. 10, Ippai Jew Lizard, marries Matha Ground Iguana and Kubbitha Native Bee.

No. 11, Ippai Opossum, marries Matha Emu, Matha Brown Snake, Matha Red Kangaroo, and Ippatha Eaglehawk.

No. 12, Ippai Eaglehawk, marries Matha Emu, Matha Brown Snake, and Ippatha Opossum.

No. 13, Oombi Common Fly, marries Kubbitha Native Bee. No. 14, Oombi Goonhur, marries Kubbitha Flying Squirrel, Kubbitha Bandicoot, Kubbitha Porcupine, and Butha Gray Kangaroo.

No. 15, Oombi Gray Kangaroo, marries Kubbitha Flying Squirrel, Kubbitha Bandicoot, Kubbitha Porcupine, Ippatha Malleehen, and Butha Goonhur.