The Number Concept: Its Origin and Development

n of Numb

e of research, that the attempt has been made, even, to establish a common origin for all the races of mankind by means of a comparison of numeral words.51 But in this instance, as in so many others that will readily occur to the mind, the result has been that the theory has finally taken possession of the author and reduced him to complete subjugation, instead of remaining his servant and submitting to the legitimate results of patient and careful investigation. Linguistic research is so full of snares and pitfalls that the student must needs employ the greatest degree of discrimination before asserting kinship of race because of resemblances in vocabulary; or even relationship between words in the same language because of some chance likeness of form that may exist between them. Probably no one would argue

pounded words used to signify number. Such words are the one, two, three, etc., of English; the eins, zwei, drei, etc., of German; words which must at some time, in some prehistoric language, have had definite meanings entirely apart from those which they now convey to our minds. In savage languages it is sometimes possible to detect these mean

e, dozen, gross, and score, can hardly be classed as numerals in the strict sense of the word. German possesses exactly the same number of native words in its numeral scale as English; and the same may be said of the Teutonic languages generally, as well as of the Celtic, the Latin, the Slavonic, and the Basque. This is, in fact, the universal method observed in the formation of any numeral scale, though the actual number of simple words

-ten, etc. The specific method of combination may not always be the same, as witness the eighteen, or eight-ten, in English, and dix-huit, or ten-eight, in French; forty-five, or four-tens-five, in English, and fünf und vierzig, or five and four tens in German. But the general method is the same the world over, presenting us with nothing but local variations, which are, relatively speaking, entirely unimportant. With this fact in mind, we can cease to wonder at the small number of simple numerals in any language. It might, indeed, be queried, why do any languages, English and German, for example, have unusual compounds for 11 and 12? It would seem as though the regular method of compounding should begin with 10 and 1, instead of 10 and 3, in any language using a system with 10 as

moy

mbi

t

e

t

s

s

n

ken

ku

mi na

mi na

umi n

t

e a little closer than these, and the combinations found in certain other languages are, in turn, closer than those of the English; as witness the once, 11, doce, 12, trece, 13, etc., of Spanish. But the process is essentially the sa

up to 19 are formed by prefixing the smaller number to the base; and it is only when we pass 20 that we return to the more direct and obvious method of giving precedence to the larger. In German and other Teutonic languages the inverse method is continued still further. Here 25 is fünf und zwanzig, 5 and 20; 92 is zwei und neunzig, 2 and 90, and so on to 99. Above 100 the order is made direct, as in

dred" means 3 × 100. When the larger precedes the smaller, we must usually understand addition. But to both these rules there are very many exceptions. Among higher numbers the inverse order is very rarely used; though even here an occasional exception is found. The Taensa Indians, for example, place the smaller numbers b

he regular forms of expression for 18 and 19. At first they seem decidedly odd; but familiarity soon accustoms one to them, and they cease entirely to attract any special attention. This principle of subtract

lone. The same formation occurs in Malay, resulting in the numerals delapan, 10 ? 2, and sambilan 10 ? 1.57 In Green Island, one of the New Ireland group, these become simply andra-lua, "less 2," and andra-si, "less 1."58 In the Admiralty Islands this formation is carried back one step further, and not only gives us shua-luea, "less 2," and shu-ri, "less 1," but also makes 7 appear as sua-tolu, "less 3."59 Surprising as this numeral is, it is more than matched by the Ainu scale, which carries subtraction back still another step, and calls 6, 10 ? 4. The four numerals from 6 to 9 in this scale are respectively, iwa, 10 ? 4

ed in the case of the Bellacoola language of British Columbia. In their numeral scale 15, "one foot," is followed by 16, "one man less 4"; 17, "one man less 3"; 18, "one man less 2"; 19, "one man less

call 60 ? 19, 60 ? 18, 60 ? 17, etc. Literally translated the meanings seem to be 1 to 60, 2 to 60, 3 to 60, etc. The point of reference is 60, and the thought underlying the words may probably be expressed by the paraphrases, "1 on the third score, 2 on the third score, 3 on the third

ubtracted from the next higher unit, but also 40, and even 100. For example, 360 is 400 ? 40; 460 is 500 ? 40; 500 is 600 ? 100; 1300 is 1400 ? 100, etc. One of the Yoruba units is 200; and all the odd hundreds up

e do suppose, that in very many cases these words once expressed meanings closely connected with the names of the fingers, or with the fingers themselves, or both. Now and then a case is met with in which the numeral word frankly avows its meani

simple structure of many of the rude languages of the world should render this possible in a multitude of cases; but investigators are too often content with the mere nume

= taken to

put down t

e equally di

the fingers all

= the no

the Zu?i stopping, or "notching off," when he fini

other brought to a

wo brought to and h

e brought to and he

l but all are held

hila = all

'tona = all the fingers a

icated in 11 is used in the

em'thlan = two tim

ak'ya = the finger

em'thla = the fingers all the

ad of the "four brought to and held up with the rest," for which we naturally look, the Zu?i, to show that he has used all of his fingers but one, says "all but all are held up with the

ases they would be formed long before the need would be felt for terms to describe any higher number. If this theory be correct, we should expect to find finger names for numerals beginning not lower than 3, and oftener with 5 than with any other number. The highest authority has ventured the assertion that all numeral words have their origin in the names of the fingers;69 substantially the same conclusion was reached by Professor Pott, of Halle, whose work on numeral nomenclature led him deeply into the study of the origin of these words. But we have

n a systematic course of word formation from the names of their fingers. If the names of the first five numerals are of finger origin, they have so completely lost their original form, or else

ata

mad

pin

sis

udle

igin [tudlimu(t)] =

binigin = twic

inigin = three t

a = that which

e upper part-i.

taityuna = I

seems to be a re

= a man co

binidigin = a man come to

nidigin = a man come to a

n = a man come to an end acc

uina = 2 men

d of by some word meaning hand or fingers of one hand. In this respect the Eskimo dialects are somewhat exceptional among scales built up of digital words. The syste

ata

mar

inga

sis

atdl

ausek = to th

rdluk = to th

gasut = to the

samat = to th

ku

skimo word for 5 is, originally, a digital word, but if so, the fact has not yet been detected. From the analogy furnished by other languages we are justified in suspect

a

bia

abbu

bib

ekkábe =

an = 1 of

iman = 2 o

timan = 3 o

an = 4 of

ntekábbe

ihibena = 1

lukku = h

o to make up the number scales of savage races. Frequently the finger and toe origin of numerals is perfectly apparent, as in the Arawak system just given, which exhibits the simplest and

e = the e

= another

the middl

ere are no mo

re = the row

are = 3 fro

tan = there are

on one side the

inri = 4 o

rt'an = there i

= finished o

re ttcharidhel =

ttcharidhel = 1 co

h side." For 7 he either subtracts from 10, saying: "there are still 3 of them," or he brings the thumb and forefinger of the right hand up to the thumb of the left, and says: "on one side there are 4 of them." He calls 8 by the same name as many of the other Canadian tribes, that

ords the influence of finger counting, it is not unusual to find those in which the derivation from native words signifying finger, hand, toe, foot, and man, is just as frankly obvious as in the case of the Zu?i, the Arawak, the Eskimo, or the M

one = 1 ha

ona tevinitpe = 1

ponare = all

a tevinitpe =

pona tevinitpe =

n itoto

r bona tevinitpe = 1 on

method is found, with a different form of expre

petei =

mokoi = t

epi abe = ha

guage,76 which is also one of the numerous South Americ

he misa =

misa sai =

o ibi sai = both hands

rica is peculiarly rich in native numeral words of this kind; and, as the examples above cited show, it is the field to which one instinctively turns wh

tso

g

gad

ahag

mana-ite = e

-hi = 1 on

i = 2 on

ihi = 3 on

i-hi = 4 o

mana-die = en

yiri-tie = 1

ritie = 2

iri-die = en

," or some kindred expression, signifying that one hand had been completed, is simply "1 on the other." Again, th

urnished by the language of the M

pap

ava

ape

peki

ri capiti =

capiti purena = 1 of t

erri capit

iti purena = 1 of

a camonee

me camone

camonee =

at our own ancestors termed a "score," are so common that one hesitates to say which is of the more frequent use. The following scale, from one of the Betoya dialects7

t

cay

oazu

2 with plural

ente

tetey =

e cayapa

toazumba

e caesea

, or caya hue

nte-tey =

mba-ente

-ente-tey =

ea ente

n tribes, but a single scale was alluded to as reaching the comparatively high limi

m

bul

gul

arbula

mul

mummi = 1 an

bu mummi = 2

bu mummi = 3 a

lanbu mummi = 4

ra = belonging

1 of the toes added o

mummi = 2 of t

mummi = 3 of th

inna mummi = 4 of

nna = 5 of the

a mulanbu =

na mulanbu =

nna mulanbu =

dinna mulanbu

nna = belongin

as originally as limited as those obtained from other Australian tribes, and that

in the language of the Klamath Indians of Oregon

sh, o

ap =

n

ep = h

p = han

pta = 1 I ha

ta = 2 I ha

ta = 3 I hav

keksh = 1

nep = h

.'" It will be observed that the Klamath introduces not only the ordinary finger manipulation, but a gesture of the entire hand as well. It is a common thing to find something of the kind to indicate the completion of 5 or 10, and in one or two instances it has already been alluded to. Sometimes one or both of the closed fists are held up; sometimes the open hand, with all the fingers extended, is used; and sometimes an entirely independent gesture is introduced. These are, in general, of no special importance; but

escribe. In primitive scales this is not always considered necessary; thus, the Zamucos express their teens without using their word for 10 at all. They say simply, 1 on the foot, 2 on the foot, etc. Corresponding abbreviations are often met; so often, indeed, that no further mention of them is needed. They mark

t, or

u

ipet

epka

e-yaagit = h

t-yaagit =

et-uke = h

-nipetuei =

-yepkatalet

second hand fingers (

eet-yaagit = second

le-lauel = hand

finger, are not found, but rather the hand-1, 1 on the next, or 1 over and above, which we have already seen, are the type forms for which we are to look. Indiv

emoot = mid

kkamoot = f

koka = lit

d in the original Jiviro scale,8

a

c

c

enc

tegladu

= thumb (of

una = ind

su = the finger ne

cabiasu = hand n

gladu =

logical study by Dr. J. H. Trumbull86 of the numerals used by many of the North American Indian tribes reveals the presence in the languages of these tribes of a few, but only a few, finger names which are used without change as numeral expressions also. Sometimes the finger gives a name not its own to the numeral with which it is associated in counting-as in the Chippeway dialect, which has nawi-nindj, middle of the hand, and nisswi, 3; and the Cheyenne, where notoyos, middle finger, and na-nohhtu, 8, are closely related. In other parts of the world isolated examples of the transference of finger names to numerals are also found. Of these a well-known example

onstructing names for the fives, tens, and twenties, as well as for the intermediate numbers. Instead of the simple words "hand," "foot," etc., we not infrequently meet with some paraphrase for one or

s

r

t

e

the end (of t

ne sa =

e rewe =

e tini =

e etse =

ine = 2 series

sa re tsemene = 2 ser

re nome

rewe tubenine =

ne nome

for 20 cempoalli, one counting.89 The Point Barrow Eskimos call 10 kodlin, the upper part, i.e. of a man. One of the Ewe dialects o

ring, which becomes a numeral for 40, because 40 cowries made a "string"; and the Maori tekau, bunch, which sig

ete development the simple numeral scale is often found to end with 5, and all succeeding numerals to be formed from the first 5. The progression from that point may be 5-1

s

u

bia

bia

bis

issa =

iwwi =

i-biata

i-biama

auwé

like that exhibited above is exceedingly common. In the Kootenay dialect,93 of British Columbia, qaetsa, 4, and wo-qaetsa, 8, are obviously related, the latter word probably meaning a second 4. Most of

. 7. ma

. 8. yu

ion between 2 and 7, and 3 and 8, i

o. 7. u

na. 8. u

of their scale in their 6, 7, and 8 respective

s. 6.

it. 7.

t. 8. o

, and 9 appear as second 1, second 2, etc., or another 1, another 2, etc.; or, more simply still, as 1 more, 2 more, etc. It is the method which was briefly discussed in the early part of the present chapter, and is b

-n-isa =

n-dalaua

-n-tatlo

by some of our Western Indian tribes. Selecting a few o

ai washe

i noom pa

yam me ne

i to pah

i zap tah

shak pah =

of using 5-1, 5-2, 5-3, 5-4, or 2d 1, 2d 2, 2d 3, 2d 4, in forming their numerals from 6 to 9, they proc

ivo

b

bel

ben

bet

co beb

o belal

o benai

o betan

di

ethod of arriving at so anomalous a scale. Mere repetition in the second quinate of the words used in the first might readily be explained by supposing the use of fingers absolutely indispensable as an aid to counting, and that a certain word would

100 1.

par

par

par

pan

par

par

par

par

pa

00 1.

l

t

f

l

t

l

t

f

li

eve, entirely unique amon

urrence. Hand numerals may appear, and then suddenly disappear, just where we should look for them with the greatest degree of certainty. In the

s

z

t

w

l

a sa =

zua =

butu

sa = 1

sa

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 jumbl

nat

hai

kac

hakn =

= 1 father, i.e.

haikia

a natsa

behema =

doatn =

oatn

r seems to be the ruling principle, are of occasion

th 6. Among others, the scale of the Pigmies of Central Africa103 and that of the Mosquitos104 of Central America show this tendency. In the

ito.

umi.

al.

upa.

l = 2-2.

fingers of 1

lalkab

umi = 6 and 1. bumut

al = 6 and 2. bumutt

pa = 6 and 3. bumutti-

fingers of 2 hand

d by the Pigmy scale. Still another species of numeral form, quite different from any that have already been noticed, is found in the Yoruba105 scale, which is in

i, or

e

e

e

a

e

e

e

e

.

nla = g

ila = g

ala =

la = gre

odzi =

igba

ed a convenient higher unit for reckoning. Proceeding in this curious manner,106 they called 50 strings 1 afo or head; and to illustrate their singular mode of reckoning-the king

ains two genuine curiosities, and by reason of those it deserves a place among any coll

ara = 1

ino

yekaini

te = toes o

a five coloured

en = finger

gem = fingers

amichirihegem = fingers of both han

of South America. But the Abipones, in seeking for words with which to enable themselves to pass 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; h

s 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 low