The Number Concept: Its Origin and Development Part 5

8. pargen. 8. tolu.

9. parbai. 9. fa.

10. panim. 10. lima.

Such examples are, I believe, entirely unique among primitive number systems.

In numeral scales where the formative process has been of the general nature just exhibited, irregularities of various kinds are of frequent occurrence. Hand numerals may appear, and then suddenly disappear, just where we should look for them with the greatest degree of certainty. In the Ende,[101] a dialect of the Flores Islands, 5, 6, and 7 are of hand formation, while 8 and 9 are of entirely different origin, as the scale shows.

1. sa.

2. zua.

3. telu.

4. wutu.

5. lima 6. lima sa = hand 1.

7. lima zua = hand 2.

8. rua butu = 2 4.

9. trasa = 10 - 1?

10. sabulu.

One special point to be noticed in this scale is the irregularity that prevails between 7, 8, 9. The formation of 7 is of the most ordinary kind; 8 is 2 fours--common enough duplication; while 9 appears to be 10 - 1. All of these modes of compounding are, in their own way, regular; but the irregularity consists in using all three of them in connective numerals in the same system. But, odd as this jumble seems, it is more than matched by that found in the scale of the Karankawa Indians,[102] an extinct tribe formerly inhabiting the coast region of Texas. The first ten numerals of this singular array are:

1. natsa.

2. haikia.

3. kachayi.

4. hayo hakn = 2 2.

5. natsa behema = 1 father, _i.e._ of the fingers.

6. hayo haikia = 3 2?

7. haikia natsa = 2 + 5?

8. haikia behema = 2 fathers?

9. haikia doatn = 2d from 10?

10. doatn habe.

Systems like the above, where chaos instead of order seems to be the ruling principle, are of occasional occurrence, but they are decidedly the exception.

In some of the cases that have been adduced for ill.u.s.tration it is to be noticed that the process of combination begins with 7 instead of with 6.

Among others, the scale of the Pigmies of Central Africa[103] and that of the Mosquitos[104] of Central America show this tendency. In the Pigmy scale the words for 1 and 6 are so closely akin that one cannot resist the impression that 6 was to them a new 1, and was thus named.

MOSQUITO. PIGMY.

1. k.u.mi. ujju.

2. wal. ibari.

3. niupa. ikaro.

4. wal-wal = 2-2. ikw.a.n.ganya.

5. mata-sip = fingers of 1 hand. b.u.muti.

6. matlalkabe. ijju.

7. matlalkabe pura k.u.mi = 6 and 1. b.u.mutti-na-ibali = 5 and 2.

8. matlalkabe pura wal = 6 and 2. b.u.mutti-na-ikaro = 5 and 3.

9. matlalkabe pura niupa = 6 and 3. b.u.mutti-na-ikw.a.n.ganya = 5 and 4.

10. mata wal sip = fingers of 2 hands. mabo = half man.

The Mosquito scale is quite exceptional in forming 7, 8, and 9 from 6, instead of from 5. The usual method, where combinations appear between 6 and 10, is exhibited by the Pigmy scale. Still another species of numeral form, quite different from any that have already been noticed, is found in the Yoruba[105] scale, which is in many respects one of the most peculiar in existence. Here the words for 11, 12, etc., are formed by adding the suffix _-la_, great, to the words for 1, 2, etc., thus:

1. eni, or okan.

2. edzi.

3. eta.

4. erin.

5. arun.

6. efa.

7. edze.

8. edzo.

9. esan.

10. ewa.

11. okanla = great 1.

12. edzila = great 2.

13. etala = great 3.

14. erinla = great 4, etc.

40. oG.o.dzi = string.

200. igba = heap.

The word for 40 was adopted because cowrie sh.e.l.ls, which are used for counting, were strung by forties; and _igba_, 200, because a heap of 200 sh.e.l.ls was five strings, and thus formed a convenient higher unit for reckoning. Proceeding in this curious manner,[106] they called 50 strings 1 _afo_ or head; and to ill.u.s.trate their singular mode of reckoning--the king of the Dahomans, having made war on the Yorubans, and attacked their army, was repulsed and defeated with a loss of "two heads, twenty strings, and twenty cowries" of men, or 4820.

The number scale of the Abipones,[107] one of the low tribes of the Paraguay region, contains two genuine curiosities, and by reason of those it deserves a place among any collection of numeral scales designed to exhibit the formation of this cla.s.s of words. It is:

1. initara = 1 alone.

2. inoaka.

3. inoaka yekaini = 2 and 1.

4. geyenknate = toes of an ostrich.

5. neenhalek = a five coloured, spotted hide, or hanambegen = fingers of 1 hand.

10. lanamrihegem = fingers of both hands.

20. lanamrihegem cat gracherhaka anamichirihegem = fingers of both hands together with toes of both feet.

That the number sense of the Abipones is but little, if at all, above that of the native Australian tribes, is shown by their expressing 3 by the combination 2 and 1. This limitation, as we have already seen, is shared by the Botocudos, the Chiquitos, and many of the other native races of South America. But the Abipones, in seeking for words with which to enable themselves to pa.s.s beyond the limit 3, invented the singular terms just given for 4 and 5. The ostrich, having three toes in front and one behind on each foot presented them with a living example of 3 + 1; hence "toes of an ostrich" became their numeral for 4. Similarly, the number of colours in a certain hide being five, the name for that hide was adopted as their next numeral. At this point they began to resort to digital numeration also; and any higher number is expressed by that method.

In the sense in which the word is defined by mathematicians, _number_ is a pure, abstract concept. But a moment's reflection will show that, as it originates among savage races, number is, and from the limitations of their intellect must be, entirely concrete. An abstract conception is something quite foreign to the essentially primitive mind, as missionaries and explorers have found to their chagrin. The savage can form no mental concept of what civilized man means by such a word as "soul"; nor would his idea of the abstract number 5 be much clearer. When he says _five_, he uses, in many cases at least, the same word that serves him when he wishes to say _hand_; and his mental concept when he says _five_ is of a hand. The concrete idea of a closed fist or an open hand with outstretched fingers, is what is upper-most in his mind. He knows no more and cares no more about the pure number 5 than he does about the law of the conservation of energy.

He sees in his mental picture only the real, material image, and his only comprehension of the number is, "these objects are as many as the fingers on my hand." Then, in the lapse of the long interval of centuries which intervene between lowest barbarism and highest civilization, the abstract and the concrete become slowly dissociated, the one from the other. First the actual hand picture fades away, and the number is recognized without the original a.s.sistance furnished by the derivation of the word. But the number is still for a long time a certain number _of objects_, and not an independent concept. It is only when the savage ceases to be wholly an animal, and becomes a thinking human being, that number in the abstract can come within the grasp of his mind. It is at this point that mere reckoning ceases, and arithmetic begins.

CHAPTER IV.

THE ORIGIN OF NUMBER WORDS.

(_CONTINUED_.)

By the slow, and often painful, process incident to the extension and development of any mental conception in a mind wholly unused to abstractions, the savage gropes his way onward in his counting from 1, or more probably from 2, to the various higher numbers required to form his scale. The perception of unity offers no difficulty to his mind, though he is conscious at first of the object itself rather than of any idea of number a.s.sociated with it. The concept of duality, also, is grasped with perfect readiness. This concept is, in its simplest form, presented to the mind as soon as the individual distinguishes himself from another person, though the idea is still essentially concrete. Perhaps the first glimmering of any real number thought in connection with 2 comes when the savage contrasts one single object with another--or, in other words, when he first recognizes the _pair_. At first the individuals composing the pair are simply "this one," and "that one," or "this and that"; and his number system now halts for a time at the stage when he can, rudely enough it may be, count 1, 2, many. There are certain cases where the forms of 1 and 2 are so similar than one may readily imagine that these numbers really were "this" and "that" in the savage's original conception of them; and the same likeness also occurs in the words for 3 and 4, which may readily enough have been a second "this" and a second "that." In the Lushu tongue the words for 1 and 2 are _tizi_ and _tazi_ respectively. In Koriak we find _ngroka_, 3, and _ngraka_, 4; in Kolyma, _niyokh_, 3, and _niyakh_, 4; and in Kamtschatkan, _tsuk_, 3, and _tsaak_, 4.[108] Sometimes, as in the case of the Australian races, the entire extent of the count is carried through by means of pairs. But the natural theory one would form is, that 2 is the halting place for a very long time; that up to this point the fingers may or may not have been used--probably not; and that when the next start is made, and 3, 4, 5, and so on are counted, the fingers first come into requisition. If the grammatical structure of the earlier languages of the world's history is examined, the student is struck with the prevalence of the dual number in them--something which tends to disappear as language undergoes extended development. The dual number points unequivocally to the time when 1 and 2 were _the_ numbers at mankind's disposal; to the time when his three numeral concepts, 1, 2, many, each demanded distinct expression. With increasing knowledge the necessity for this differentiatuin would pa.s.s away, and but two numbers, singular and plural, would remain. Incidentally it is to be noticed that the Indo-European words for 3--_three_, _trois_, _drei_, _tres_, _tri,_ etc., have the same root as the Latin _trans_, beyond, and give us a hint of the time when our Aryan ancestors counted in the manner I have just described.

The first real difficulty which the savage experiences in counting, the difficulty which comes when he attempts to pa.s.s beyond 2, and to count 3, 4, and 5, is of course but slight; and these numbers are commonly used and readily understood by almost all tribes, no matter how deeply sunk in barbarism we find them. But the instances that have already been cited must not be forgotten. The Chiquitos do not, in their primitive state, properly count at all; the Andamans, the Veddas, and many of the Australian tribes have no numerals higher than 2; others of the Australians and many of the South Americans stop with 3 or 4; and tribes which make 5 their limit are still more numerous. Hence it is safe to a.s.sert that even this insignificant number is not always reached with perfect ease. Beyond 5 primitive man often proceeds with the greatest difficulty. Most savages, even those of the tribes just mentioned, can really count above here, even though they have no words with which to express their thought. But they do it with reluctance, and as they go on they quickly lose all sense of accuracy. This has already been commented on, but to emphasize it afresh the well-known example given by Mr. Oldfield from his own experience among the Watchandies may be quoted.[109] "I once wished to ascertain the exact number of natives who had been slain on a certain occasion. The individual of whom I made the inquiry began to think over the names ... a.s.signing one of his fingers to each, and it was not until after many failures, and consequent fresh starts, that he was able to express so high a number, which he at length did by holding up his hand three times, thus giving me to understand that fifteen was the answer to this most difficult arithmetical question." This meagreness of knowledge in all things pertaining to numbers is often found to be sharply emphasized in the names adopted by savages for their numeral words. While discussing in a previous chapter the limits of number systems, we found many instances where anything above 2 or 3 was designated by some one of the comprehensive terms _much_, _many_, _very many_; these words, or such equivalents as _lot_, _heap_, or _plenty_, serving as an aid to the finger pantomime necessary to indicate numbers for which they have no real names. The low degree of intelligence and civilization revealed by such words is brought quite as sharply into prominence by the word occasionally found for 5. Whenever the fingers and hands are used at all, it would seem natural to expect for 5 some general expression signifying _hand_, for 10 _both hands_, and for 20 _man_. Such is, as we have already seen, the ordinary method of progression, but it is not universal. A drop in the scale of civilization takes us to a point where 10, instead of 20, becomes the whole man. The Kusaies,[110] of Strong's Island, call 10 _sie-nul_, 1 man, 30 _tol-nul_, 3 men, 40 _a naul_, 4 men, etc.; and the Ku-Mb.u.t.ti[111] of central Africa have _mukko_, 10, and _moku_, man. If 10 is to be expressed by reference to the man, instead of his hands, it might appear more natural to employ some such expression as that adopted by the African Pigmies,[112] who call 10 _mabo_, and man _mabo-mabo_. With them, then, 10 is perhaps "half a man,"