Primitive culture, vol. 1 (of 2)

It occurred to Wilhelm von Humboldt, in studying this curious system of numeration, that he had before his eyes the evidence of a process very like that which actually produced the regular numeral words denoting one, two, three, &c., in the various languages of the world. The following passage in which, more than sixty years ago, he set forth this view, seems to me to contain a nearly perfect key to the theory of numeral words. ‘If we take into consideration the origin of actual numerals, the process of their formation appears evidently to have been the same as that here described. The latter is nothing else than a wider extension of the former. For when 5 is expressed, as in several languages of the Malay family, by “hand” (lima), this is precisely the same thing as when in the description of numbers by words, 2 is denoted by “wing.” Indisputably there lie at the root of all numerals such metaphors as these, though they cannot always be now traced. But people seem early to have felt that the multiplicity of such signs for the same number was superfluous, too clumsy, and leading to misunderstandings.’ Therefore, he goes on to argue, synonyms of numerals are very rare. And to nations with a deep sense of language, the feeling must soon have been present, though perhaps without rising to distinct consciousness, that recollections of the original etymology and descriptive meaning of numerals had best be allowed to disappear, so as to leave the numerals themselves to become mere conventional terms.

The most instructive evidence I have found bearing on the formation of numerals, other than digit-numerals, among the lower races, appears in the use on both sides of the globe of what may be called numeral-names for children. In Australia a well-marked case occurs. With all the poverty of the aboriginal languages in numerals, 3 being commonly used as meaning ‘several or many,’ the natives in the Adelaide district have for a particular purpose gone far beyond this narrow limit, and possess what is to all intents a special numeral system, extending perhaps to 9. They give fixed names to their children in order of age, which are set down as follows by Mr. Eyre: 1. Kertameru; 2. Warritya; 3. Kudnutya; 4. Monaitya; 5. Milaitya; 6. Marrutya; 7. Wangutya; 8. Ngarlaitya; 9. Pouarna. These are the male names, from which the female differ in termination. They are given at birth, more distinctive appellations being soon afterwards chosen.[319] A similar habit makes its appearance among the Malays, who in some districts are reported to use a series of seven names in order of age, beginning with 1. Sulung (‘eldest’); 2. Awang (‘friend, companion’), and ending with Kechil (‘little one’), or Bongsu (‘youngest’). These are for sons; daughters have Meh prefixed, and nicknames have to be used for practical distinction.[320] In Madagascar, the Malay connexion manifests itself in the appearance of a similar set of appellations given to children in lieu of proper names, which are, however, often substituted in after years. Males; Lahimatoa (‘first male’), Lah-ivo (‘intermediate male’);

Ra-fara-lahy (‘last born male’). Females; Ramatoa (‘eldest female’), Ra-ivo (‘intermediate’), Ra-fara-vavy (‘last born female’).[321] The system exists in North America. There have been found in use among the Dacotas the following two series of names for sons and daughters in order of birth. Eldest son, Chaské; second, Haparm; third, Ha-pe-dah; fourth, Chatun; fifth Harka. Eldest daughter, Wenonah; second, Harpen; third, Harpstenah; fourth, Waska; fifth, We-harka. These mere numeral appellations they retain through childhood, till their relations or friends find occasion to replace them by bestowing some more distinctive personal name.[322] Africa affords further examples.[323]

As to numerals in the ordinary sense, Polynesia shows remarkable cases of new formation. Besides the well-known system of numeral words prevalent in Polynesia, exceptional terms have from time to time grown up. Thus the habit of altering words which sounded too nearly like a king’s name, has led the Tahitians on the accession of new chiefs to make several new words for numbers. Thus, wanting a new term for 2 instead of the ordinary rua, they for obvious reasons took up the word piti, ‘together,’ and made it a numeral, while to get a new word for 5 instead of rima, ‘hand,’ which had to be discontinued, they substituted pae, ‘part, division,’ meaning probably division of the two hands. Such words as these, introduced in Polynesia for ceremonial reasons, are expected to be dropped again and the old ones replaced, when the reason for their temporary exclusion ceases, yet the new 2 and 5, piti and pae, became so positively the proper numerals of the language, that they stand instead of rua and rima in the Tahitian translation of the Gospel of St. John made at the time. Again, various special habits of counting in the South Sea Islands have had their effect on language. The Marquesans, counting fish or fruit by one in each hand, have come to use a system of counting by pairs instead of by units. They start with tauna, ‘a pair,’ which thus becomes a numeral equivalent to 2; then they count onward by pairs, so that when they talk of takau or 10, they really mean 10 pair or 20.

For bread-fruit, as they are accustomed to tie them up in knots of four, they begin with the word pona, ‘knot,’ which thus becomes a real numeral for 4, and here again they go on counting by knots, so that when they say takau or 10, they mean 10 knots or 40. The philological mystification thus caused in Polynesian vocabularies is extraordinary; in Tahitian, &c., rau and mano, properly meaning 100 and 1,000, have come to signify 200 and 2,000, while in Hawaii a second doubling in their sense makes them equivalent to 400 and 4,000. Moreover, it seems possible to trace the transfer of suitable names of objects still farther in Polynesia in the Tongan and Maori word tekau, 10, which seems to have been a word for ‘parcel’ or ‘bunch,’ used in counting yams and fish, as also in tefuhi, 100, derived from fuhi, ‘sheaf or bundle.’[324]

In Africa, also, special numeral formations are to be noticed. In the Yoruba language, 40 is called ogodzi, ‘a string,’ because cowries are strung by forties, and 200 is igba, ‘a heap,’ meaning again a heap of cowries. Among the Dahomans in like manner, 40 cowries make a kade or ‘string,’ 50 strings make one afo or ‘head;’ these words becoming numerals for 40 and 2,000. When the king of Dahome attacked Abeokuta, it is on record that he was repulsed with the heavy loss of ‘two heads, twenty strings, and twenty cowries’ of men, that is to say, 4,820.[325]

Among cultured nations, whose languages are most tightly bound to the conventional and unintelligible numerals of their ancestors, it is likewise usual to find other terms existing which are practically numerals already, and might drop at once into the recognized places of such, if by any chance a gap were made for them in the traditional series. Had we room, for instance, for a new word instead of two, then either pair (Latin par, ‘equal’) or couple (Latin copula, ‘bond or tie,’) is ready to fill its place. Instead of twenty, the good English word score, ‘notch,’ will serve our turn, while, for the same purpose, German can use stiege, possibly with the original sense of ‘a stall full of cattle, a sty;’ Old Norse drôtt, ‘a company,’ Danish, snees. A list of such words used, but not grammatically classed as numerals in European languages, shows great variety: examples are, Old Norse, flockr (flock), 5; sveit, 6; drôtt (party), 20; thiodh (people), 30; fölk (people), 40; öld (people), 80; her (army), 100; Sleswig, schilk, 12 (as though we were to make a numeral out of ‘shilling’); Middle High-German, rotte, 4; New High-German, mandel, 15; schock (sheaf), 60. The Letts give a curious parallel to Polynesian cases just cited. They throw crabs and little fish three at a time in counting them, and therefore the word mettens, ‘a throw,’ has come to mean 3; while flounders being fastened in lots of thirty, the word kahlis, ‘a cord,’ becomes a term to express this number.[326]

In two other ways, the production of numerals from merely descriptive words may be observed both among lower and higher races. The Gallas have no numerical fractional terms, but they make an equivalent set of terms from the division of the cakes of salt which they use as money. Thus tchabnana, ‘a broken piece’ (from tchaba, ‘to break,’ as we say ‘a fraction’), receives the meaning of one-half; a term which we may compare with Latin dimidium, French demi. Ordinal numbers are generally derived from cardinal numbers, as third, fourth, fifth, from three, four, five. But among the very low ones there is to be seen evidence of independent formation quite unconnected with a conventional system of numerals already existing. Thus the Greenlander did not use his ‘one’ to make ‘first,’ but calls it sujugdlek, ‘foremost,’ nor ‘two’ to make ‘second,’ which he calls aipâ, ‘his companion;’ it is only at ‘third’ that he takes to his cardinals, and forms pingajuat in connexion with pingasut, 3. So, in Indo-European languages, the ordinal prathamas, πρῶτος, primus, first, has nothing to do with a numerical ‘one,’ but with the preposition pra, ‘before,’ as meaning simply ‘foremost;’ and although Greeks and Germans call the next ordinal δεύτερος, zweite, from δυό, zwei, we call it second, Latin secundus, ‘the following’ (sequi), which is again a descriptive sense-word.

If we allow ourselves to mix for a moment what is with what might be, we can see how unlimited is the field of possible growth of numerals by mere adoption of the names of familiar things. Following the example of the Sleswigers we might make shilling a numeral for 12, and go on to express 4 by groat; week would provide us with a name for 7, and clover for 3. But this simple method of description is not the only available one for the purpose of making numerals. The moment any series of names is arranged in regular order in our minds, it becomes a counting-machine. I have read of a little girl who was set to count cards, and she counted them accordingly, January, February, March, April. She might, of course, have reckoned them as Monday, Tuesday, Wednesday. It is interesting to find a case coming under the same class in the language of grown people. We know that the numerical value of the Hebrew letters is given with reference to their place in the alphabet, which was arranged for reasons that can hardly have had anything to do with arithmetic. The Greek alphabet is modified from a Semitic one, but instead of letting the numeral value of their letters follow throughout their newly-arranged alphabet, they reckon α, β, γ, δ, ε, properly, as 1, 2, 3, 4, 5, then put in σ for 6, and so manage to let ι stand for 10, as י does in Hebrew, where it is really the 10th letter.

Now, having this conventional arrangement of letters made, it is evident that a Greek who had to give up the regular 1, 2, 3,—εἷς, δύο, τρεῖς, could supply their places at once by adopting the names of the letters which had been settled to stand for them, thus calling 1 alpha, 2 bēta, 3 gamma, and so onward. The thing has actually happened; a remarkable slang dialect of Albania, which is Greek in structure, though full of borrowed and mystified words and metaphors and epithets understood only by the initiated, has, as its equivalent for ‘four’ and ‘ten,’ the words δέλτα and ἰῶτα.[327]

While insisting on the value of such evidence as this in making out the general principles of the formation of numerals, I have not found it profitable to undertake the task of etymologizing the actual numerals of the languages of the world, outside the safe limits of the systems of digit-numerals among the lower races, already discussed. There may be in the languages of the lower races other relics of the etymology of numerals, giving the clue to the ideas according to which they were selected for an arithmetical purpose, but such relics seem scanty and indistinct.[328] There may even exist vestiges of a growth of numerals from descriptive words in our Indo-European languages, in Hebrew and Arabic, in Chinese. Such etymologies have been brought forward,[329] and they are consistent with what is known of the principles on which numerals or quasi-numerals are really formed. But so far as I have been able to examine the evidence, the cases all seem so philologically doubtful, that I cannot bring them forward in aid of the theory before us, and, indeed, think that if they succeed in establishing themselves, it will be by the theory supporting them, rather than by their supporting the theory. This state of things, indeed, fits perfectly with the view here adopted, that when a word has once been taken up to serve as a numeral, and is thenceforth wanted as a mere symbol, it becomes the interest of language to allow it to break down into an apparent nonsense-word, from which all traces of original etymology have disappeared.

Etymological research into the derivation of numeral words thus hardly goes with safety beyond showing in the languages of the lower culture frequent instances of digit-numerals, words taken from direct description of the gestures of counting on fingers and toes. Beyond this, another strong argument is available, which indeed covers almost the whole range of the problem. The numerical systems of the world, by the actual schemes of their arrangement, extend and confirm the opinion that counting on fingers and toes was man’s original method of reckoning, taken up and represented in language. To count the fingers on one hand up to 5, and then go on with a second five, is a notation by fives, or as it is called, a quinary notation. To count by the use of both hands to 10, and thence to reckon by tens, is a decimal notation. To go on by hands and feet to 20, and thence to reckon by twenties, is a vigesimal notation. Now though in the larger proportion of known languages, no distinct mention of fingers and toes, hands and feet, is observable in the numerals themselves, yet the very schemes of quinary, decimal, and vigesimal notation remain to vouch for such hand-and-foot-counting having been the original method on which they were founded. There seems no doubt that the number of the fingers led to the adoption of the not especially suitable number 10 as a period in reckoning, so that decimal arithmetic is based on human anatomy.

In surveying the languages of the world at large, it is found that among tribes or nations far enough advanced in arithmetic to count up to five in words, there prevails, with scarcely an exception, a method founded on hand-counting, quinary, decimal, vigesimal, or combined of these. For perfect examples of the quinary method, we may take a Polynesian series which runs 1, 2, 3, 4, 5, 5·1, 5·2, &c.; or a Melanesian series which may be rendered as 1, 2, 3, 4, 5, 2nd 1, 2nd 2, &c. Quinary leading into decimal is well shown in the Fellata series 1 ... 5, 5·1 ... 10, 10·1 ...10·5, 10·5·1 ... 20, ... 30, ... 40, &c. Pure decimal may be instanced from Hebrew 1, 2 ... 10, 10·1 ... 20, 20·1 ... &c. Pure vigesimal is not usual, for the obvious reason that a set of independent numerals to 20 would be inconvenient; but it takes on from quinary, as in Aztec, which may be analyzed as 1, 2 ... 5, 5·1 ... 10, 10·1 ... 10·5, 10·5·1 ... 20, 20·1 ... 20·10, 20·10·1 ... 40, &c.; or from decimal, as in Basque, 1 ... 10, 10·1 ... 20, 20·1 ... 20·10, 20·10·1 ... 40 &c.[331] It seems unnecessary to bring forward here the mass of linguistic details required for any general demonstration of these principles of numeration among the races of the world. Prof. Pott, of Halle, has treated the subject on elaborate philological evidence, in a special monograph,[332] which is incidentally the most extensive collection of details relating to numerals, indispensable to students occupied with such enquiries.

For the present purpose the following rough generalization may suffice, that the quinary system is frequent among the lower races, among whom also the vigesimal system is considerably developed, but the tendency of the higher nations has been to avoid the one as too scanty, and the other as too cumbrous, and to use the intermediate decimal system. These differences in the usage of various tribes and nations do not interfere with, but rather confirm, the general principle which is their common cause, that man originally learnt to reckon from his fingers and toes, and in various ways stereotyped in language the result of this primitive method.

Some curious points as to the relation of these systems may be noticed in Europe. It was observed of a certain deaf-and-dumb boy, Oliver Caswell, that he learnt to count as high as 50 on his fingers, but always ‘fived,’ reckoning, for instance, 18 objects as ‘both hands, one hand, three fingers.’[333] The suggestion has been made that the Greek use of πεμπάζειν, ‘to five,’ as an expression for counting, is a trace of rude old quinary numeration (compare Finnish lokket ‘to count,’ from lokke ‘ten’). Certainly, the Roman numerals I, II, ... V, VI ... X, XI ... XV, XVI, &c., form a remarkably well-defined written quinary system. Remains of vigesimal counting are still more instructive. Counting by twenties is a strongly marked Keltic characteristic. The cumbrous vigesimal notation could hardly be brought more strongly into view in any savage race than in such examples as Gaelic aon deug is da fhichead ‘one, ten, and two twenties,’ i.e., 51; or Welsh unarbymtheg ar ugain ‘one and fifteen over twenty,’ i.e., 36; or Breton unnek ha triugent ‘eleven and three twenties,’ i.e., 71. Now French, being a Romance language, has a regular system of Latin tens up to 100; cinquante, soixante, septante, huitante, nonante, which are to be found still in use in districts within the limits of the French language, as in Belgium. Nevertheless, the clumsy system of reckoning by twenties has broken out through the decimal system in France.

The septante is to a great extent suppressed, soixante-quatorze, for instance, standing for 74; quatre-vingts has fairly established itself for 80, and its use continues into the nineties, quatre-vingt-treize for 93; in numbers above 100 we find six-vingts, sept-vingts, huit-vingts, for 120, 140, 160, and a certain hospital has its name of Les Quinze-vingts from its 300 inmates. It is, perhaps, the most reasonable explanation of this curious phenomenon, to suppose the earlier Keltic system of France to have held its ground, modelling the later French into its own ruder shape. In England, the Anglo-Saxon numeration is decimal, hund-seofontig, 70; hund-eahtatig, 80; hund-nigontig, 90; hund-teontig, 100; hund-enlufontig, 110; hund-twelftig, 120. It may be here also by Keltic survival that the vigesimal reckoning by the ‘score,’ threescore and ten, fourscore and thirteen, &c., gained a position in English which it has not yet totally lost.[334]

From some minor details in numeration, ethnological hints may be gained. Among rude tribes with scanty series of numerals, combination to make out new numbers is very soon resorted to. Among Australian tribes addition makes ‘two-one,’ ‘two-two,’ express 3 and 4; in Guachi ‘two-two’ is 4; in San Antonio ‘four and two-one’ is 7. The plan of making numerals by subtraction is known in North America, and is well shown in the Aino language of Yesso, where the words for 8 and 9 obviously mean ‘two from ten,’ ‘one from ten.’ Multiplication appears, as in San Antonio, ‘two-and-one-two,’ and in a Tupi dialect ‘two-three,’ to express 6. Division seems not known for such purposes among the lower races, and quite exceptional among the higher. Facts of this class show variety in the inventive devices of mankind, and independence in their formation of language. They are consistent at the same time with the general principles of hand-counting. The traces of what might be called binary, ternary, quaternary, senary reckoning, which turn on 2, 3, 4, 6, are mere varieties, leading up to, or lapsing into, quinary and decimal methods.

The contrast is a striking one between the educated European, with his easy use of his boundless numeral series, and the Tasmanian, who reckons 3, or anything beyond 2, as ‘many,’ and makes shift by his whole hand to reach the limit of ‘man,’ that is to say, 5. This contrast is due to arrest of development in the savage, whose mind remains in the childish state which the beginning of one of our nursery number-rhymes illustrates curiously. It runs—

To notice this state of things among savages and children raises interesting points as to the early history of grammar. W. von Humboldt suggested the analogy between the savage notion of 3 as ‘many’ and the grammatical use of 3 to form a kind of superlative, in forms of which ‘trismegistus,’ ‘ter felix,’ ‘thrice blest,’ are familiar instances. The relation of single, dual, and plural is well shown pictorially in the Egyptian hieroglyphics, where the picture of an object, a horse for instance, is marked by a single line | if but one is meant, by two lines | | if two are meant, by three lines | | | if three or an indefinite plural number are meant. The scheme of grammatical number in some of the most ancient and important languages of the world is laid down on the same savage principle. Egyptian, Arabic, Hebrew, Sanskrit, Greek, Gothic, are examples of languages using singular, dual, and plural number; but the tendency of higher intellectual culture has been to discard the plan as inconvenient and unprofitable, and only to distinguish singular and plural. No doubt the dual held its place by inheritance from an early period of culture, and Dr. D. Wilson seems justified in his opinion that it ‘preserves to us the memorial of that stage of thought when all beyond two was an idea of indefinite number.’[335]

When two races at different levels of culture come into contact, the ruder people adopt new art and knowledge, but at the same time their own special culture usually comes to a standstill, and even falls off. It is thus with the art of counting. We may be able to prove that the lower race had actually been making great and independent progress in it, but when the higher race comes with a convenient and unlimited means of not only naming all imaginable numbers, but of writing them down and reckoning with them by means of a few simple figures, what likelihood is there that the barbarian’s clumsy methods should be farther worked out? As to the ways in which the numerals of the superior race are grafted on the language of the inferior, Captain Grant describes the native slaves of Equatorial Africa occupying their lounging hours in learning the numerals of their Arab masters.[336] Father Dobrizhoffer’s account of the arithmetical relations between the native Brazilians and the Jesuits is a good description of the intellectual contact between savages and missionaries. The Guaranis, it appears, counted up to 4 with their native numerals, and when they got beyond, they would say ‘innumerable.’ ‘But as counting is both of manifold use in common life, and in the confessional absolutely indispensable in making a complete confession, the Indians were daily taught at the public catechising in the church to count in Spanish. On Sundays the whole people used to count with a loud voice in Spanish, from 1 to 1,000.’

The missionary, it is true, did not find the natives use the numbers thus learnt very accurately—‘We were washing at a blackamoor,’ he says.[337] If, however, we examine the modern vocabularies of savage or low barbarian tribes, they will be found to afford interesting evidence how really effective the influence of higher on lower civilization has been in this matter. So far as the ruder system is complete and moderately convenient, it may stand, but where it ceases or grows cumbrous, and sometimes at a lower limit than this, we can see the cleverer foreigner taking it into his own hands, supplementing or supplanting the scanty numerals of the lower race by his own. The higher race, though advanced enough to act thus on the lower, need not be itself at an extremely high level. Markham observes that the Jivaras of the Marañon, with native numerals up to 5, adopt for higher numbers those of the Quichua, the language of the Peruvian Incas.[338] The cases of the indigenes of India are instructive. The Khonds reckon 1 and 2 in native words, and then take to borrowed Hindi numerals.

The Oraon tribes, while belonging to a race of the Dravidian stock, and having had a series of native numerals accordingly, appear to have given up their use beyond 4, or sometimes even 2, and adopted Hindi numerals in their place.[339] The South American Conibos were observed to count 1 and 2 with their own words, and then to borrow Spanish numerals, much as a Brazilian dialect of the Tupi family is noticed in the last century as having lost the native 5, and settled down into using the old native numerals up to 3, and then continuing in Portuguese.[340] In Melanesia, the Annatom language can only count in its own numerals to 5, and then borrows English siks, seven, eet, nain, &c. In some Polynesian islands, though the native numerals are extensive enough, the confusion arising from reckoning by pairs and fours as well as units, has induced the natives to escape from perplexity by adopting huneri and tausani.[341] And though the Esquimaux counting by hands, feet, and whole men, is capable of expressing high numbers, it becomes practically clumsy even when it gets among the scores, and the Greenlander has done well to adopt untrîte and tusinte from his Danish teachers. Similarity of numerals in two languages is a point to which philologists attach great and deserved importance in the question whether they are to be considered as sprung from a common stock. But it is clear that so far as one race may have borrowed numerals from another, this evidence breaks down.

At the other end of the scale of civilization, the adoption of numerals from nation to nation still presents interesting philological points. Our own language gives curious instances, as second and million. The manner in which English, in common with German, Dutch, Danish, and even Russian, has adopted Mediæval Latin dozena (from duodecim) shows how convenient an arrangement it was found to buy and sell by the dozen, and how necessary it was to have a special word for it. But the borrowing process has gone farther than this. If it were asked how many sets of numerals are now in use among English-speaking people in England, the probable reply would be one set, the regular one, two, three, &c. There exist, however, two borrowed sets as well. One is the well-known dicing-set, ace, deuce, tray, cater, cinque, size; thus size-ace is ‘6 and one,’ cinques or sinks, ‘double five.’ These came to us from France, and correspond with the common French numerals, except ace, which is Latin as, a word of great philological interest, meaning ‘one.’ The other borrowed set is to be found in the Slang Dictionary. It appears that the English street-folk have adopted as a means of secret communication a set of Italian numerals from the organ-grinders and image-sellers, or by other ways through which Italian or Lingua Franca is brought into the low neighbourhoods of London. In so doing, they have performed a philological operation not only curious, but instructive.