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.
10 mano = 1 tini = 40,000.
10 tini = 1 tufa = 400,000.
10 tufa = 1 pohi = 4,000,000.
In the Hawaiian dialect this scale is, with slight modification, the universal scale, used not only in counting breadfruit, but any other objects as well. The result is a complete decimal-quaternary system, such as is found nowhere else in the world except in this and a few of the neighbouring dialects of the Pacific. This scale, which is almost identical with the Nukuhivan, is[203]
4 units = 1 ha or tauna = 4.
10 tauna = 1 tanaha = 40.
10 tanaha = 1 lau = 400.
10 lau = 1 mano = 4000.
10 mano = 1 tini = 40,000.
10 tini = 1 lehu = 400,000.
The quaternary element thus introduced has modified the entire structure of the Hawaiian number system. Fifty is _tanaha me ta umi_, 40 + 10; 76 is 40 + 20 + 10 + 6; 100 is _ua tanaha ma tekau_, 2 40 + 10; 200 is _lima tanaha_, 5 40; and 864,895 is 2 400,000 + 40,000 + 6 4000 + 2 400 + 2 40 + 10 + 5.[204] Such examples show that this secondary influence, entering and incorporating itself as a part of a well-developed decimal system, has radically changed it by the establishment of 4 as the primary number base. The role which 10 now plays is peculiar. In the natural formation of a quaternary scale new units would be introduced at 16, 64, 256, etc.; that is, at the square, the cube, and each successive power of the base. But, instead of this, the new units are introduced at 10 4, 100 4, 1000 4, etc.; that is, at the products of 4 by each successive power of the old base. This leaves the scale a decimal scale still, even while it may justly be called quaternary; and produces one of the most singular and interesting instances of number-system formation that has ever been observed. In this connection it is worth noting that these Pacific island number scales have been developed to very high limits--in some cases into the millions. The numerals for these large numbers do not seem in any way indefinite, but rather to convey to the mind of the native an idea as clear as can well be conveyed by numbers of such magnitude. Beyond the limits given, the islanders have indefinite expressions, but as far as can be ascertained these are only used when the limits given above have actually been pa.s.sed. To quote one more example, the Hervey Islanders, who have a binary-decimal scale, count as follows:
5 kaviri (bunches of cocoanuts) = 1 takau = 20.
10 takau = 1 rau = 200.
10 rau = 1 mano = 2000.
10 mano = 1 kiu = 20,000.
10 kiu = 1 tini = 200,000.
Anything above this they speak of in an uncertain way, as _mano mano_ or _tini tini_, which may, perhaps, be paralleled by our English phrases "myriads upon myriads," and "millions of millions."[205] It is most remarkable that the same quarter of the globe should present us with the stunted number sense of the Australians, and, side by side with it, so extended and intelligent an appreciation of numerical values as that possessed by many of the lesser tribes of Polynesia.
The Luli of Paraguay[206] show a decided preference for the base 4. This preference gives way only when they reach the number 10, which is an ordinary digit numeral. All numbers above that point belong rather to decimal than to quaternary numeration. Their numerals are:
1. alapea.
2. tamop.
3. tamlip.
4. lokep.
5. lokep moile alapea = 4 with 1, or is-alapea = hand 1.
6. lokep moile tamop = 4 with 2.
7. lokep moile tamlip = 4 with 3.
8. lokep moile lokep = 4 with 4.
9. lokep moile lokep alapea = 4 with 4-1.
10. is yaoum = all the fingers of hand.
11. is yaoum moile alapea = all the fingers of hand with 1.
20. is elu yaoum = all the fingers of hand and foot.
30. is elu yaoum moile is-yaoum = all the fingers of hand and foot with all the fingers of hand.
Still another instance of quaternary counting, this time carrying with it a suggestion of binary influence, is furnished by the Mocobi[207] of the Parana region. Their scale is exceedingly rude, and they use the fingers and toes almost exclusively in counting; only using their spoken numerals when, for any reason, they wish to dispense with the aid of their hands and feet. Their first eight numerals are:
1. iniateda.
2. inabaca.
3. inabacao caini = 2 above.
4. inabacao cainiba = 2 above 2; or natolatata.
5. inibacao cainiba iniateda = 2 above 2-1; or natolatata iniateda = 4-1.
6. natolatatata inibaca = 4-2.