Part 11 (1/2)

If the existence of number systems like the above are to be accounted for simply on the ground of low civilization, one might reasonably expect to find ternary and and quaternary scales, as well as binary. Such scales actually exist, though not in such numbers as the binary. An example of the former is the Betoya scale,[195] which runs thus:

1. edoyoyoi.

2. edoi = another.

3. ibutu = beyond.

4. ibutu-edoyoyoi = beyond 1, or 3-1.

5. ru-mocoso = hand.

The Kamilaroi scale, given as an example of binary formation, is partly ternary; and its word for 6, _guliba guliba_, 3-3, is purely ternary. An occasional ternary trace is also found in number systems otherwise decimal or quinary vigesimal; as the _dlkunoutl_, second 3, of the Haida Indians of British Columbia. The Karens of India[196] in a system otherwise strictly decimal, exhibit the following binary-ternary-quaternary vagary:

6. then tho = 3 2.

7. then tho ta = 3 2-1.

8. lwie tho = 4 2.

9. lwie tho ta = 4 2-1.

In the Wokka dialect,[197] found on the Burnett River, Australia, a single ternary numeral is found, thus:

1. karboon.

2. wombura.

3. chrommunda.

4. chrommuda karboon = 3-1.

Instances of quaternary numeration are less rare than are those of ternary, and there is reason to believe that this method of counting has been practised more extensively than any other, except the binary and the three natural methods, the quinary, the decimal, and the vigesimal. The number of fingers on one hand is, excluding the thumb, four. Possibly there have been tribes among which counting by fours arose as a legitimate, though unusual, result of finger counting; just as there are, now and then, individuals who count on their fingers with the forefinger as a starting-point. But no such practice has ever been observed among savages, and such theorizing is the merest guess-work. Still a definite tendency to count by fours is sometimes met with, whatever be its origin. Quaternary traces are repeatedly to be found among the Indian languages of British Columbia. In describing the Columbians, Bancroft says: ”Systems of numeration are simple, proceeding by fours, fives, or tens, according to the different languages....”[198] The same preference for four is said to have existed in primitive times in the languages of Central Asia, and that this form of numeration, resulting in scores of 16 and 64, was a development of finger counting.[199]

In the Hawaiian and a few other languages of the islands of the central Pacific, where in general the number systems employed are decimal, we find a most interesting case of the development, within number scales already well established, of both binary and quaternary systems. Their origin seems to have been perfectly natural, but the systems themselves must have been perfected very slowly. In Tahitian, Rarotongan, Mangarevan, and other dialects found in the neighbouring islands of those southern lat.i.tudes, certain of the higher units, _tekau_, _rau_, _mano_, which originally signified 10, 100, 1000, have become doubled in value, and now stand for 20, 200, 2000. In Hawaiian and other dialects they have again been doubled, and there they stand for 40, 400, 4000.[200] In the Marquesas group both forms are found, the former in the southern, the latter in the northern, part of the archipelago; and it seems probable that one or both of these methods of numeration are scattered somewhat widely throughout that region.

The origin of these methods is probably to be found in the fact that, after the migration from the west toward the east, nearly all the objects the natives would ever count in any great numbers were small,--as yams, cocoanuts, fish, etc.,--and would be most conveniently counted by pairs.

Hence the native, as he counted one pair, two pairs, etc., might readily say _one_, _two_, and so on, omitting the word ”pair” altogether. Having much more frequent occasion to employ this secondary than the primary meaning of his numerals, the native would easily allow the original significations to fall into disuse, and in the lapse of time to be entirely forgotten. With a subsequent migration to the northward a second duplication might take place, and so produce the singular effect of giving to the same numeral word three different meanings in different parts of Oceania. To ill.u.s.trate the former or binary method of numeration, the Tahuatan, one of the southern dialects of the Marquesas group, may be employed.[201] Here the ordinary numerals are:

1. tahi, 10. onohuu.

20. takau.

200. au.

2,000. mano.

20,000. tini.

20,000. tufa.

2,000,000. pohi.

In counting fish, and all kinds of fruit, except breadfruit, the scale begins with _tauna_, pair, and then, omitting _onohuu_, they employ the same words again, but in a modified sense. _Takau_ becomes 10, _au_ 100, etc.; but as the word ”pair” is understood in each case, the value is the same as before. The table formed on this basis would be:

2 (units) = 1 tauna = 2.

10 tauna = 1 takau = 20.

10 takau = 1 au = 200.

10 au = 1 mano = 2000.

10 mano = 1 tini = 20,000.

10 tini = 1 tufa = 200,000.

10 tufa = 1 pohi = 2,000,000.

For counting breadfruit they use _pona_, knot, as their unit, breadfruit usually being tied up in knots of four. _Takau_ now takes its third signification, 40, and becomes the base of their breadfruit system, so to speak. For some unknown reason the next unit, 400, is expressed by _tauau_, while _au_, which is the term that would regularly stand for that number, has, by a second duplication, come to signify 800. The next unit, _mano_, has in a similar manner been twisted out of its original sense, and in counting breadfruit is made to serve for 8000. In the northern, or Nukuhivan Islands, the decimal-quaternary system is more regular. It is in the counting of breadfruit only,[202]

4 breadfruits = 1 pona = 4.

10 pona = 1 toha = 40.

10 toha = 1 au = 400.

10 au = 1 mano = 4000.