The Number Concept: Its Origin and Development

inary

the numerous methods illustrated in earlier chapters, he passes from 5 to 10, using here the fingers of his second hand. He now has two fives; and, just as we say "twenty," i.e. two tens, he says "two hands," "the second hand finished," "all the fingers," "the fingers of both hands," "all the fingers come to an end," or, much more rarely, "one man." That is, he is, in one of the many ways at his command, say

ution, and its digital origin is usually traced without difficulty. A consistent formation would require the expression of 10 by some phrase meaning "two fives," 15 by "three fives,"

nte =

, or caya hu

mba-ente

a-ente =

in many more of which mention might be made. Collecting the significant numerals from a few such scales, and ta

. amnaiton

tse ponare

a. 5. abba te

ntekkabe

alac?teglad

gladu =

iam 5.

auwe

= the end (of the

ine = 2 series

28 5. li

lima =

29 5. li

lim = 2

5. e-li

a-lim = t

230 5.

, or wtya

5.

u?k =

ri 5

egu =

San Antonio.23

amauj ajt

32 5.

-lima =

go 5. s

ru-lim

lumbia.233 5. kedji

kat = bo

d. It is a fact, as will be fully illustrated in the following pages, that quinary number systems, when extended, usually merge into either the decimal or the vigesimal. The result is, of course, a compound of two, and sometimes of three, systems in one scale. A pure quinary or vigesimal number system is exceedingly rare; but quinary scales certainly do exi

lled to depart from his strict reckoning by fives, and to assume a new basis of

= the no

hla = all t

ire count of the fingers has taken its place. The division of the fingers into two sets of five each is still in his mind, but it is no longer the leading idea. As the count proceeds furthe

atoneigne oiet

toue = the

ret oupoume =

= twice the

5. ace popet

pomocoi

epiabe = han

5. lima

ima = 2

a rua = (

mibika mis

misa sai =

ibi sai = both hands

na yimana-ite

mana-die = en

iri-die = en

ul 5.

ra = belonging

= 5 toes added on

nna = belongin

kani-iktsi-mo

tsi-bo = al

ao-mo = 1

i-pume

passes unconsciously from the quinary into the decimal scale. Again, the summing up of the 10 fingers and 10 toes often results in the concept of a single whole, a lump sum, so to speak, and the savage then says "one man," or something that gives utterance to this thought of a new unit. This leads the quinary into the vigesimal scale, and produces the combination so often found in certain parts of the world. Thus the inevitable tendency of any number system of quinary origin is toward the establishment of another and larger base, and the formation of a number system in which both are used

his chapter. In the simplicity and regularity of its construction it is so noteworthy that it is worth repeating, as the first of the long list of quinary systems given i

t

cay

oazu

2 with plural

ente

te tey =

e cayapa

e toazumb

e caesea

, or caya hue

nte-tey =

mba-ente

a-ente-tey

ea ente

ce can, from this point onward, be quinary only in the subordinate sense to which allusion has just been made. Examples of this are furnished in a more or less perfect manner by nearly a

.240

d

t

ped

p

chw

sai

w

n

.

r ddeg =

ddeg =

ar ddeg

r ar ddeg

theg =

bymtheg =

bymtheg =

bymtheg =

ar bymtheg

ug

l.241

o

y

n

acui

uacen =

ome = [

uey = [

cnaui =

matl

ctli oce

ctli omom

ctli omey

tli onnaui

caxt

olli oce

lli omome

lli omey

lli onnau

ualli =

ew Caledonia

car

car

cab

c

on-chagui

mon-caro

on-careri

on-caboue

panr

-mon-chagui

e-mon-caro

e-mon-care

e-mon-cabo

e-mon-cani

-cani-mon-chagu

n-cani-mon-car

-cani-mon-carer

-cani-mon-cabou

uemo =

243 1.

dou

c

dek

qui

icaira = 1

aicaira = 2

caira = 3 o

icaira = 4

noi-da

-caibo = 1 +

i-ai-caibo

ai-caibo

-ai-caibo

n-oibo

ai-quacoib

-ai-quacoib

i-quacoibo

ai-quacoib

-ai-quacoib

viously mean 1, 2, 3, 4, taken a second time, and as the meanings I have given

lty Island

l

koe

e

tji

gemen =

gemen =

ngemen

gemen =

e pi =

ca

lu

koen

ek

ni pi =

a hua

ua hu

eni hu

ke hu

atj =

245 1.

ngo

mot

neh

m

otu =

orr = [

tta = [

eo = [

.

kpo kotu

kpo ngorr

kpo motta

kpo neheo

okpo mui

mui okpo kotu = 10 +

ui okpo ngorr = 10 +

ui okpo motta = 10 +

ui okpo nehea = 10 +

baba

systems are vigesimal, so that th

d from this point to 20, the numeral words in both scales are such as to show that the notion of counting by fives is quite as prominent as the notion of referring to 10 as a base. Above 20 the systems become vigesimal, with a quinary or decimal structure appearing in all numerals except multiples of 20. Thus, in Welsh, 36 is unarbymtheg ar ugain, 1

quinary scale will be found, with a few exceptions like those just instanced, to have the following structure or one similar to it in all essential details: 1, 2, 3, 4, 5, 5-1, 5-2, 5-3, 5-4, 10,

f the "Dark Continent." In some cases the numerals of certain tribes, as given by one writer, are found to differ widely from the same numerals as reported by another. N

.246 1.

aba, or

sis

ibak

foo

uck-eno

ck-cooka

ck-sisaj

ck-sibak

ibank

247 1.

m

n

i

ngu

m-pum

m-miu

ommag

enu-io

.

ee.248

n

s

n

t

ata =

ifeenoo

ifeessa

ifeena

no

249 1

s

t

hin

h

le-do =

le-so =

le-ta =

e-hinyo

bla-

s.250 1.

fid

sar

n

soo

s

ma fiddin

ma sarr

ma nani

nu

1.

de-

de-

de-

de-

e-du

son =

tan =

adu =

.

s.251

y

y

yan

jud

om-wea

om-yar

om-yat

m yanet

fo

52 1.

b

bit

ban

zon

tongbali

i tobisi

i tobitt

to banda

ni

h.253

dee

tet

n

jou

ego

eeddee

tettee

-nee

sa

.254 1.

fir

sar

n

sou

s

o-fere

mazarkan

-manani

.

.255 1

t

r

h

m

n-bul

n-tin

n-ra

-hyul

.

56 1.

f

sag

n

s

-dondo

-fera

-sagba

-nani

.

.257

r

d

n

wdy

etem

erou

, bed

enuan

yer =

e. 1

r

s

a

tr-

at rok-i

at de ra

at re sa

t ro n-an

tr-o

.258 1

b

dot

a

i

kili

-bone

-dotta

ashe

ch

a.259 1

s

m

s

m

-ga =

tsi

ta = 5

so = [

duk-

260 1.

a

ott

e

att

att

uwe =

atong =

enne =

.

261 1

i

i

i

iti

o-kiet

ia-ba

a-eta

kiet =

du

262 1

gu-

gu-

gu-

gu-

ua-yin

ua-ba =

tu-ta

ua-ni =

gu

.263

i

i

i

ütt

üekee

iaba =

eiata

usch

.

.264 1

n

yas

d

u

sge =

tul

sge =

leg

egu =

.265

b

l

n

t

hu =

bi =

ra =

(tanin?

.

266 1.

rom

tot

sos

s

obaia =

rommu =

totto =

sosso =

.

.267 1

v

t

n

t

u-mue =

u-vari =

u-tatu =

ou-ne =

0

.268 1

waw

wat

mch

msa

o na jum

o na wir

na watat

na mchec

ik

Po.269

mem

m

mie

mim

o na mul

na mempa

o na met

na miene

emieu

sa 1. k

vi-

vi-

vin

vis

na kimod

na vi-wi

na vitat

na vinye

chik

.270 1.

iba

r

i

ita

na guevoh

na ibar

na raro

na ina?

um, or n

.271 1

b

sad

sal

kus

'-ella =

e-bare =

e-sadde

e-salle

ol-la

72 1.

n

n

t

non

oum = [

tie = [

tai =

ina = [

es

.273

are

s

s

o

d

ariga

-sena

-mada = w

le

i.274 1

p

t

t

t

a mosa

a pili

a taru

l

oco n

.275

v

d

n

lol

da = [

ile = [

haba =

nan =

.

1. m

bev

bet

ben

bet

ni moue

ni beva

ni beta

ni bena

nchi

New Quinea.2

rou

tou

f

r

-samos

-rouet

-touro

-faat

ou

Flores.

z

t

w

ma =

a = 5-1,

a-zua

-butu

sa = [

sa

o.278 1.

e

e

eba

ere

kaee =

y = [5

rey =

bats =

sene

ll Islands.2

d

chi

e

lai

chinu =

chime =

thuk = [

ejuwou =

iu

ty Island.

l

t

f

l

t

l

t

f

li

ther dialect

l

k

tha

tha

acha =

-alo

kuun =

thack

lebe

Pines.28

b

b

b

ta-

-ta =

-bo =

beti =

-beu

de

anks Islands.2

vo

vo

vo

liem =

jea =

ro = o

tol =

vet =

owul =

Islands.28

nir

nit

niv

lima =

atea =

arua =

atol =

avat =

avul =

onia.283

par

par

par

pan

im-gha

im-roo

im-ghe

im-bai

paro

ew Cal.28

hel

hey

pob

im =

m-wet

-weluk

-weyen

-pobit

pain

m.285 1

e

ese

anoh

nik

cled et et

cled et or

cled et es

led et mano

n lep ikm

a 1.

k

kah

k

kri

um riti

um karu

um kaha

um kefa

.

nga 1

d

dis

div

lim =

kai = o

im naru

im disi

mindivat

olim =

Heb.286

r

t

b

ma =

esa = o

rua =

olu = o

iti = o

ima = 2

w Heb.

l

t

v

ma =

ai = o

ua = o

olo =

ari =

lima =

New Heb.

d

d

p

ma =

esa = o

dua =

olu = o

eti = o

lima =

ew Heb.

e l

e t

e h

lime

tai = o

u = ot

olu =

ati =

ua lim =

New Heb.

i r

i t

i v

lima =

tea =

rua =

tol =

vat =

wulu =

287 1

g

y

u

im = 1

or = o

ic = o

a = ot

u = ot

ya

and.288

yak

tet

tar

l mala =

Victoria.2

bul

its kia

ts bulai

munnar

bulaits bul

ts munnar

nd the former is strongly binary, as are so many others of that continent

ia.289

p

b

b

p

m muy

m pil

m bey

m buon

.

chi.290

nir

n'r

n'r

igen =

milige

h milige

nwro

ona t

tken = b

h291 1.

i

t

s

che

lutsa

lina =

ltona

naga =

ha

W. Alaska.292

ah, or

ingi

esai

tal

or ahchegaret

alronik = 5

hu-okving

gotalia

ko

, South.293

kas

tsc

scha

kum

y'lk

tyk =

ookotuk

uaktuk

echtuk

294 1.

alj

qan

sit

n = my

un =

un =

tsin =

sin =

ha

kenzie R.295

k, or m

ak, or

?ita

alle

venel

oerit-aypa

oerit-illa

erit-t?itam

kr

ez Perces).

lap

mit

lapt =

pac

aks = [

apt = [

atat =

koi

puti

d.297 1.

achd

pin

sis

adli

-atauseq =

machdluq = o

-pinasut =

-sisamat =

qu

-atauseq = f

-machdluq =

-pinasut = f

-sisamat = f

chfec

neq-atauseq =

neq-machdlup

neq-pinasut =

neq-sisamat =

vdlucho = a

shows that the system will from now on be vigesimal. This scale is one of the most interesting of which we have any record, and will be noticed again in the next chapter. In many respects it is like the scale of the Point Barrow Es

ay.298

n

nis

niw

nan

twasswi

asswi =

asswi =

asswi =

asswi =

etts.299

nee

n

y

on one side

tatash =

usuk =

osuk =

= it comes ne

pu

egoimegon.30

nee

nis

new

nan

dwaswe =

waswe =

swe = 3

gaswe =

aswe =

1. nin

nin

nis

niw

nan

twaswi =

waswi =

waswi =

sha

kw

e. 1. n

nis

nak

n

kin to pale

= 1 on the

= 2 on the

= 3 on the

onk = co

len = n

e. 1.

nes

nit

n

linwe

wathwe =

athwe =

kswa = 3

akin to chagis

thwe = n

301 1.

tah

see

nai

n

soo-

oo-ig

-gumo

scoon

mt

od, tense, person, and number. The forms given above are not those that would be used in counting, but are for specific use, being varied according to the thought it was intended to express. For examp

pers. tahboosee-ek

osee-yok = the

o-sijik = ther

pers. tahboosee-egup

see-yogup = the

ee-sibunik = the

tahboosee-dak = there

tahboo-seekw, there are not 2 of them; mah tahboo-seekw, there will not be 2 of t

nquin. 1

nin

nis

ney

hran

soo = 1 on th

oo = 2 on th

o = 3 on th

[akin to chagi

ssoo = n

1. meea

nom

abee

too

i.e. all the fi

ppai =

mba = fi

eenee = f

only 1 finge

baira =

w. 1.

tuk

tuc

ush

= the first

han

klo = a

china =

on the end; i.e

po

. 1. k

beh

d

heh

ihse

dun

ekah =

sehka

hsehka =

ehneh

ay. 1.

nee

nee

n

man =

woy = 1 on t

woy = 2 on th

= 3 on the

[akin to chagis

swoy = no

. 1. n

n

col

tac

eppa

nancus

aness

alcon

ish = han

ne

e. 1.

pee

tou

hkee

oksh = h

shabish

sheeshabis

etshabis

heereewa

ksheere

. 1.

nis

nak

n

lan =

ash = 1

oash =

ash =

now

wi

chen.

t

n

taa

ejet

uschu

te uschu

t uschu

chok =

tsch

t.303

d

nat

k'on

djin

durcu =

durcu =

a durcu

goc

kat = bo

all, Indian

nee

nar

n

yau

artuce

rtuce =

artuce =

etwartuce

rtuce = n

uk.304

m

y

m

sky

kat

aaus =

uaus =

ene =

aiky

.305 1

a

kat

m

sut

o = ot

po = o

kutl =

wakutl

ha

an.306

tep

gua

tqa

(from an

lt =

pqalt

ndalt

kct

gy

306 1. (

tln

asm

m

tse

otl =

tlnos

tlnos

k'e

skchl

.307 1

lap

mut

p

p

itka =

itka =

pitka

instsh

nawi

tpu.30

lep

mat

pip

taw

na = [

lip =

mat =

auiais

ning

.307 1.

lap

nta

won

ton

ishuptan

shuptane

ishuptane

aiakish

ta

sta).309 1

h

hat

ira

ets

tah

ikinis

ikikiri

hariki-

etse

o.310 1

m

mep

ewit

omek

un-supl

-munwi

-munpa

munwits

noma

a.311

yut

h

tse

mar

reka

itsa =

inahu

-tsaket

tu

312 1

yoc

c

goc

k

to =

to =

ato =

to =

re

o.313

dzi

tan

t

y

kui

ziman =

animo =

tamu =

te

ncan.314

ina

iny

nkun

ink

-towi =

-towi

ukunowi

tadahata

inda

15 1.

hua

hua

moa

anx

evi =

apoa =

aeica =

acua =

(akin to moa

.316 1

p

kim

p

pis

tso

kalko

a-kalko

-kalko

tu

equibo, Guian

o

oro

o-bai

etanee

puimapo

uimapo =

-puimapo

ema-puimapo

een-ab

ucouyenne?) 1

bia

ele

bouri =

apourcou-ab

yagone-ouac

yagone-ouaca

oyagone-ouac

.

hon n

in undoubtedly in certain gestures or finger motions. The numerals obtained from this region, and from the tribes to the south and east of the Carib country, are especially rich in digital terms, and

.319 1

wat

tsani

mara

che misa

i bu-bich

i watsani

watsandik

i sumara

misa sai =

ba320

mbe

kim

p

pis

suk

aluku =

-kaluku =

kaluku =

tu

na320 1.

mit

kur

tsa

a (from a

rirobo = 1

irobo = 2

rirobo = 3

irobo = 4

rutse =

.321 1

arep

omep

guem

ueam

hueapueh

hueatare

eatameapue

apueh =

go

322 1.

nan

mun

ojuino

ten

natea =

i-natea =

-natea =

ino-natea

jejuino

Roman notation, the familiar I., II., III., IV. (originally IIII.), V., VI., etc., with equal certainty suggests quinary counting, but the Latin language contains no vestige of anything of the kind, and the whole range of Latin literature is silent on this point, though it contains numerous references to finger counting. It is quite within the bounds of possibility that the prehistoric nations of Europe possessed and used a quinary numeration. But of these races the modern world knows nothing save the few scanty facts that can be gathered from the stone implements which have now and then been brought to light. Their languages have perished as utterly as have the races themselves, and speculation concerning them is useless. Whatever their form of numeration may have been, it has left no perceptible trace on t

of the latter a mixed system.323 In support of this theory it is urged that extensive regions which now show nothing but decimal counting were, beyond all reasonable doubt, quinary. It is well known, for example, that the decimal system of the Malays has spread over almost the entire Polynesian region, di

ll his new number 5-1, or, with equal probability, give it an entirely new name, independent in all respects of any that have preceded it. With the use of this new name there may be associated the conception of "5 and 1 more"; but in such multitudes of instances the words employed show no trace of any such meaning, that it is impossible for any one to draw, with any degree of safety, the inference that the signification was originally there, but that the changes of time had wrought changes in verbal form so great as to bury i

. 1. m

onne

ahgh

t

sat

chap

enna

hdagh

chun

edeh

.325 1.

kro

rom

rob

asch

ter

kog

wong

chka

dwow

n 1. n

nee

nog

nau

nun

gwit

upou

ghu

aune

mtan

7 and 8 have little or no claim to relationship with 2 and 3. In some scales a single word only is found in the second quinate to indicate that 5 was originally the base on which the system rested. It

mitive peoples is by no means established. In Chapter II, examples were given of races which had no number base. Later on it was observed that in Australia and South America many tribes used 2 as their number base; in some cases counting on past 5 without showing any tendency to use that as a new unit. Again, through the habit of counting upon the finger joints, instead of the fingers themselves, the use of 3 as a base is brought into prominence, and 6 and 9 become 2 threes and 3 threes,

and by the repetition of the same word. Occasionally the same word is used for two successive numbers, some gesture undoubtedly serving to distinguish the one from the other in the savage's mind. Examples of this are not infrequent among the forest tribes of South America. In the Tariana dialect 9 and 10 are expressed by the same word, paihipawalianuda; in Cobeu, 8 and 9 by pepelicoloblicouilini; in Barre, 4, 5, and 9 by ualibucubi.326 In other languages the change from one numeral

cchae

hai

igunh

hgtsch

chihating

-stchihat

-tschihath

-tschihati

ny argument which might tend to show that the quinary, or any other scale, was ever the sole number scale of prim

inl

nak

t'a

din

e-su

e-t'are

-oyertan

s dinri

e-dinri

ye-oyerta

oner