The Number Concept: Its Origin and Development

n of Numb

tinu

e," 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 thanthat 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 use

le 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-Mbutti111 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

y, though not always, carried as far as the system extends; and it sometimes leads to curious terms, of which a single illustration will suffice.

a guena

a pogua

a atala

a nergua

ala guena =

, instead of 20, is expressed by the word "man," the multiples of 10 follow the law just given for multiples of 20.

mo = 10 times

uo = 10 times the who

ess high numbers in this manner the term applied by educated native Greenlanders120

riartut navdlugit = 10 men 5

ntly made use of in the formation of higher numeral words. The African Mabas121 call 10 atuk, great 1; the Ho

language. It is found, rather, in the mental association of the larger with the smaller unit, and the consequent repetition of the name of the smaller. Any unit, whether it be a single thing, a dozen, a score, a hundred, a thousand, or any other unit, is, whenever used, a single and complete group; and where the relation between them is sufficiently close, as in our "gross" and "great gross," this form of nomenclature is natural enough to render it a matter of som

's

's

's a

s a p

little hu

f some higher unit, would, of course, be impossible. The numeral s

e, to an even greater degree, of tribes whose number sense is so defective that they begin almost from the outset to use combinations. If a savage expresses the number 3 by the combination 2-1, it will at once be suspected that his numerals will, by the time he reaches 10 or 20, become so complex and confused that numbers as high as these will be expressed by finger pantomime rather than by words. Such is often the case; and the comment is frequently made by explorers that the tribes they have visited have no words for numbers higher than 3, 4, 5, 10, or 20, but tha

lanesia,131 required for a translation of the numeral 38, which occurs in John v. 5, "had an infirmity thirty and eight years," the circumlocution, "one man and both sides five and three." Such expressions, curious as they seem at first thought, are no more than the natural outgrowth of systems built up by the slow and tedious process which so often obtains among primitive races, where digit numerals are combined in an almost endless variety of ways, and where mere reduplication often serves in place of any independent names for higher units. To what extent this may be carried is shown by the language of the Cayubabi,132 who have for 10 the word tunca, and for 100 and 1000 the compounds tunca tunca, and tunca tunca tunca respectively; or of the Sapibocones, who call 10 bururuche, hand hand, and 100 buruche buruche, hand hand hand hand.133 More remarkable still is the Ojibwa language, which continues its numeral scale without limit, furnishing combinations which are really remarkable; as, e.g., that for 1,000,000,000, which is me das wac me das wac as he me das wac,134 1000 × 1000 × 1000. The Winnebago expression for the same number,135 ho ke he hhuta hhu chen a ho ke he ka ra pa ne za is no less formidable, but it has every appearance o

eek, Latin, etc. It is sufficient in many cases to produce almost an entire change in the form of the word; or to result in compounds which require close scrutiny for the detection of the original root. For example, in the Carrier, one of the Déné dialects of western Canada, the word tha means 3 things; thane, 3 persons; that, 3 times; thatoen, in 3 places; thauh, in 3 ways; thailtoh, all of the 3 things; thahoeltoh, all of the 3 persons; and thahultoh, all of the 3 times.138 In the Tsimshian language of British Columbia we find seven distinct sets of numerals "

ounti

cts

ts Me

Canoes

el k'al k'awuts

upel t'epqadal gaops

le gulal galtskan

q tqalpqdal tqaapskan

c kcenecal k'etoentsk

t k'aldal k'aoltsk

alt t'epqaldal t'epqaltsk

lt yuktleadal ek'tlaedsk

ac kctemacal kctemaestk

pēel kpal kpēets

and other languages of this region exist in several different forms, and can also be modified by any of the innumerable suffixes of these tongues.140 To illustrate the almost illimita

Two.

menok maa

nskam mas

s'ak mats'

aqsa matl

s matlp'enequl

nkh matlp'en

ther. -- mat

emtsmots'utl matltsm

nqtlala matl'a

enqtla matl'

skamala masem

menskam m

mentsake mats'

ments'akis mats

n beach. -

house, etc.

s than among any other race, and it is a more characteristic linguistic peculiarity of this than of any other region, either in the Old World or in the New. It was to some exte

, and so on. The word "times" is, of course, not expressed, any more than in English; but the expressions briefly are, 2 tens, 3 tens, etc. But a singular exception to this method is presented by the

r, 20,

sh, 30,

a, 40,

h, 50, ch

, 60, sh

a, 70,

neh, 80,

a, 90,

f the various branches of this division of the Caucasian race. An analogous method appears in the formation of the tens in the B

the end of the finger count on one hand-as, the Micmac nan, and Mohegan nunon, gone, or spent; the Pawnee sihuks, hands half; the Dakota zaptan, hand turned down; and the Massachusetts napanna, on one side. Ten is the end of the finger count, but is not always expressed by the "both hands" formula so commonly met with. The Cree term for this number is mitatat, no further; and the corresponding word in Delaware is m'tellen, no more. The Dakota 10 is, like its 5, a straightening out of the fingers which have been turned over in counting, or wickchemna, spread out unbent. The same is true of the Hidatsa pitika, which signifies a smoothing out, or straightening. The Pawnee 4, skitiks, is unusual, signifying as it does "all the fingers

ginning. (Cf. the Zu?i t?p

(from the middle,

the oth

parting wi

wo =

eye.149 Still more remote from anything digital in their derivation are the following, taken at random from a very great number of examples that might be cited to illustrate this point. The Assiniboines call 7, shak ko we, or u she nah, the odd number.150 The Crow 1, hamat, signifies "the least";151 the Mississaga 1, pecik, a very small thing.152 In Javanese, Malay, and Manadu, the words for 1, which are respectively siji, satu, and sabuah, signify 1 seed, 1 pebble, and 1 fruit respectively153-words as natural and as much to be expected at the beginning of a number scale as any finger name could possibly be. Among almost all savage races one form or another of palpable arithmetic is found, such as counting by seeds, pebbles, shells, notches, or knots; and the derivation of number words from these sources can constitute no ground for surprise. The Marq

04, m?n = en

ound of the words

indication,

i = capi

term referr

, ow

= extent

ko =

ome kind of a

sai =

sā = s

mit, or more str

, ri

hair (of so

shi =

hanges in word meanings which occur with the lapse of time may have differentiated significations originally alike, until no trace of kinship would appear to the casual observer. Our numerals "score" and "gross" are never thought of as having any original relation to what is conveyed by the other meanings which attach to these words. But the origin of each, which is easily traced, shows that, in the beginning, there existed a well-defined reason for the selection of these, rather than other terms, for the numbers they now describe. Possibly these remarkable Japanese terms may b

the close; and in Eromanga it is narolim narolim (2 × 5)(2 × 5).161 This combination deserves remark only because of the involved form which seems to have been required for the expression of so small a number as 100. A compound instead of a simple term for any higher unit is never to be wondered at, so rude are some of the savage methods of expressing number; but "two fives (times) two fives" is certainly remarkable. Some form like that employed by the Nusqually162 of Puget Sound for 1000, i.

re still in great measure evident; thus 'moon' or 'earth' expressed 1, there being but one of each; 2 might be called 'eye,' 'wing,' 'arm,' 'jaw,' as going in pairs; for 3 they said 'Rama,' 'fire,' or 'quality,' there being considered to be three Ramas, three kinds of fire, three qualities (guna); for 4 were used 'veda,' 'age,' or 'ocean,' there being four of each recognized; 'season' for 6, because they reckoned six seasons; 'sage' or 'vowel,' for

ractice of finger counting. But races as fond of metaphor and of linguistic embellishment as are those of the East, or as are our American Indians even, might readily resort to some other source than that furnished by the members of the human body, when in want of a term with which to describe the 5, 10, or any other number of the numeral scale they were unconsciously forming. That the

d at all, cluster into a little group of significations with which at last we come to associate the idea of unity. Similarly of 2, or 5, or 10, or any one of the little band which does picket duty for the advance guard of the great host of number words which are to follow. A careful examination of the first decade warrants the assertion that

, piece, grou

n, division,

tion, many

two

, group,

, two threes

second two,

cond three, two fo

three threes,

wo fives (hands), h

e, one foot,

s, one man,