The Number Concept: Its Origin and Development

eous Numb

e unhesitatingly passes it by, calling it many, heap, innumerable, as many as the leaves on the trees, or something else equally expressive and equally indefinite. But the time comes at last when a greater degree of exactness is required. Perhaps the number 11 is to be indicated, and indicated precisely. A fresh mental effort is required of the ignorant child of nature; and the result is "all the fingers and one more," "both hands and one more," "one on another count," or some equivalent circumlocution. If he has an independent word for 10, the result will be simply ten-one. When t

d beyond his base, and not the distance from his original starting-point. Some idea may, perhaps, be gained of the nature of this difficulty by imagining the numbers of our ordinary scale to be represented, each one by a single symbol different from that used to denote any other number. How long would it take the average intellect to master the first 50 even, so that each number could without hesitation be indicated by its appropriate symbol? After the first 50 were once mastered, what of the next 50? and the next? and the next? and so on. The acquisition of a sca

her and mathematician, Leibnitz, proposed a binary system of numeration. The only symbols needed in such a system would be 0 and 1. The number which is now symbolized by the figure 2 would be represented by 10; while 3, 4, 5, 6, 7, 8, etc., would appear in the binary notation as 11, 100, 101, 110, 111, 1000, etc. The difficulty with such a system is that it rapidly gro

eus impa

the r

ilo ducendis s

ry endeavour in his power to bring it to the notice of scholars and to urge its claims. But it appears to

ough a binary number system were not to be found elsewhere. This attempt to make out of the rude and unusual method of counting which obtained among the Australians a racial characteristic is hardly justified by fuller investigation. Binary number systems, which are given in full on another page, are found in South America. Some of the Dravidian scales are binary;167 and the marked preference, not infrequently observed among savage races, for counting by pairs, is in itself a sufficient refutation of this theory. Still it is an unquestionable fact that this binary tendency is more pronounced among the Australians than among any other extensive number of kindred races. They seldom count in words above 4, and almost never as high as 7. One of th

en to 10, without becoming exceedingly cumbersome. A binary scale inevitably suggests a wretchedly low degree of mental development, which stands in the way of the formation of any number scale worthy to be dignified

ura

oko

a urapu

a okosa

kosa urapu

okosa oko

ve 6 they ca

y, 4 and 6 are respectively 2 pairs and 3 pairs, while 5 is 1 more than 2 pairs. Five objects, however, they sometimes denote by urapuni-getal, 1 hand. A precisely similar condition is found to prevail respecting the arithmetic of all the Australian tribes. In some cases only two numerals are found, and in others three. But in a very great number of the native languages of that c

show how scanty was the numerical ability possessed by these tri

iver.170

etch

hevalen

val peteh

a. 1.

bar

olo nuk

lo barko

eramana.

mon

arko

oo mondr

lar. 1.

enge

eroganm

ovor benge

a. 1.

pol

it keya

it poll

Bay.

awit

bari-mot

oi.171

bul

gul

rrbular

agulib

bagulib

ngton.172

arga

rikelera

riknarga

o. 1.

bar

na barka

Island.

ori

lkeraro

land.173 1

oca

ra woora

ra ocasa

.174 1

bul

bop

gira bulle

a buller kali

Island.17

bul

oorb

a-bulla

Bay.176

bud

mud

a berdel

Bay.177 1.

nin

epal

o = 2-2, or

kuko ki

kuko kuk

ko kuko ki

e.178 1

aitye,

arnk

-bula =

ula kuma

a purlaitye

i.179 1.

b

-numbai

ngu =

galan = v

rri.180

boo

r moora

ar booll

ngoon

ngun

Creek.181

arko

ola goon

la barkoo

ing River.18

boo

la neec

la bool

, N.W. Bend.

ran

ool mat

ol ranko

a.184

thr

al mo

l thral

in.185 1.

lleti

llick kula

lick kallet

Bay.186 1

ooth

era koot

ra woothe

River.187

oole

roo wogi

oo booler

River. 1

aull

udy onke

dy paullu

River.

ooro

ora yabr

ra booroo

quarie.

bla

vo warc

vo blar

nd. 1.

ulla

agut mi

ut bullag

oo 1.

jala,

la boo

jala

ation.

pel

ige kar

ge pelli

arra. 1

ben

ro kaamb

o on benj

. 1.

arko

lala bor

ala warko

ver. 1.

boo

catha kooto

catha boolo

wro. 1.

pla

r warran

ir plat

tructure too clearly marked to require comment. In a few cases, however, the systems are to be regarded rather as showing a trace of binary structure, than as perfect e

show no important variation on the Australian systems cited abo

188 1. t

asa

e tokal

e asage

189 1.

misci

muckum

ui ckara mait

ara maitacka nuqua

acki ckaramsit

s.190 1

t cr

oudi-psh

ad-acrou

191 1.

i

te-huet

te-seze

-seze-hue

2 1. uni

nigu

ta dugani

ta drinig

uidi

193 1

cay

-teyo

a-ria =

-jente

194 1.

ap-

ara

edyái

chum

n, one might reasonably expect to find ternary and and quaternary scales, as well as binary. Such scales actually

doyo

i = an

tu = b

yoyoi = beyo

mocoso

occasional ternary trace is also found in number systems otherwise decimal or quinary vigesimal; as the dlkunoutl, second 3, of the Haida Ind

n tho

tho ta =

e tho

tho ta =

the Burnett River, Australia, a si

kar

wom

hrom

uda karbo

a legitimate, though unusual, result of finger counting; just as there are, now and then, individuals who count on their fingers with the forefinger as a starting-point. But no such practice has ever been observed among savages, and such theorizing is the merest guess-work. Still a definite tendency to count by fours is sometimes met with, whatever be its origin. Quaternary traces are repeatedly to be found among

dialects they have again been doubled, and there they stand for 40, 400, 4000.200 In the Marquesas group both forms are found, the former in the southern, the latter in the northern, part of the archipelago; and it seems probable that one or both of these methods of numeration are scattered somewhat widely throughout that region. The origin of these methods is probably to be found in the fact that, after the migration from the west toward the east, nearly all the objects the natives would ever count in any great numbers were small,-as yams, cocoanuts, fish, etc.,-and would be most conveniently counted by pairs. Hence the native, as he counted one pair, two pairs, etc., might

tah

on

ta

0.

0. m

00.

000.

,000.

ng onohuu, they employ the same words again, but in a modified sense. Takau becomes 10, au 100, etc.; but as the

) = 1 ta

= 1 tak

u = 1 a

1 mano

= 1 tini

1 tufa =

1 pohi =

e unknown reason the next unit, 400, is expressed by tauau, while au, which is the term that would regularly stand for that number, has, by a second duplication, come to signify 800. The next unit, mano, has in a similar mann

uits = 1

= 1 to

= 1 au

1 mano

= 1 tini

1 tufa =

1 pohi =

any other objects as well. The result is a complete decimal-quaternary system, such as is found nowhere else in the world exce

1 ha or

= 1 tan

a = 1 la

= 1 man

= 1 tini

1 lehu =

al formation of a quaternary scale new units would be introduced at 16, 64, 256, etc.; that is, at the square, the cube, and each successive power of the base. But, instead of this, the new units are introduced at 10 × 4, 100 × 4, 1000 × 4, etc.; that is, at the products of 4 by each successive power of the old base. This leaves the scale a decimal scale still, even while it may justly be called quaternary; and produces one of the most singular and interesting instances of number-system formation that has ever been observed. In this connection it is worth noting that

es of cocoanuts

u = 1 r

= 1 man

= 1 kiu

1 tini

iads," and "millions of millions."205 It is most remarkable that the same quarter of the globe should present us with the stunted number sense of the Austra

ay only when they reach the number 10, which is an ordinary digit numeral. All numbers ab

ala

tam

tam

lok

ile alapea

lapea =

oile tamop

ile tamlip

oile lokep

e lokep alape

= all the fin

alapea = all the fi

= all the fingers

= all the fingers of hand and fo

07 of the Parana region. Their scale is exceedingly rude, and they use the fingers and toes almost exclusively in counting; only using

niat

ina

ao caini

o cainiba

atol

iniba iniateda

tata inia

atata inib

nibacao-caini

ta-natola

fragments is such as to lead us to the border land of the might-have-been, and to cause us to speculate on the possibility of so great a numerical curiosity as a senary or a septenary scale. The Bretons call 18 triouec'h, 3-6, but otherwise their language contains no hint of counting by sixes; and we are left at perfect liberty to theorize at will on the existence of so unusual a number word. Pott remarks208 that the Bolans, of western Africa, appear to make some use of 6 as their number base, but their system, taken as a whole, is really a quinary-decimal. The language of the Sundas,209 or mountaineers of Java, contains traces of senary counting. The Akra words for 7 and 8, paggu and paniu, appear to mean 6-1 a

k

w

niu

l-wal

= fingers

atla

abe pura k

abe pura w

be pura niu

p = fingers of

resumed, and is continued from that point onward. Few number systems contain as many as three numerals which are associated with 6 as their base. In nearly all instances we find such numerals sin

eksha

shabish = 2-

abish = 3-6

herewa

ed to 6, though the meaning of 7 is not given, and it is impossible t

ra-et

pi-hairiwigani

gapi-matscha

ingle trace of senary counting appear

ildj

ji me djuu

Marshall Islands, the following curiously irregula

thino

ilim-thuo

li-dok

-thuon = 1

quite exceptional and outside all ordinary rules of number-system formation. But a closer and more accurate knowledge of the Maori language and customs served to correct the mistake, and to show that this system was a simple decimal system, and that the error arose from the following habit. Sometimes when counting a number of objects the Maoris would put aside 1 to represent each 10, and then those so set aside would afterward be counted to ascertain the number o

d by the term system-as the sexagesimal method of measuring time and angular magnitude; and the duodecimal system of reckoning, so extensivel

orrowed 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 neighbour

tee 1d.

tee 2d.

tee 3d.

altee 4d. q

ltee 5d. c

ee 6d. s

or setter salte

or otter saltee

or nobba saltee

ee, or dacha salte

or dacha oney salte

beo

say salt

r madza caroon 2s. 6d. (

s for no abstract reason that 6 is thus made the turning-point, but simply because the costermonger is adding pence up to the silver sixpence, and then adding pence

t of the needs arising in connection with any special line of work. As is well known, it is the custom in ocean, lake, and river navigation to measure soundings by the fathom. On the Missis

= five

= si

= nine

ess twain; i.e. a quarte

= mar

. = a qua

= a quarter

= mar

. = a qua

= dee

m prevails, only it is extended to meet the requirements of the deeper soundings there found, and instead of "six feet," "mark twain," etc., we find the fuller expressions, "by the mark one," "by the mark two," and so on, as far as the depth requires. This example also suggests the older and far

ns, who are concerned rather with the pure science involved, than with reckoning on any special base. A slightly increased simplicity would appear in the work of stockbrokers, and others who reckon extensively by quarters, eighths, and sixteenths. But such men experience no difficulty whatever in performing their mental computations in the decimal system; and they acquire through constant practice such quickness and accuracy of calculation, that it is difficult to see how octonary reckoning would materially assist them. Altogether, the reasons that have in the past been adduced in favour of this form of arithmetic seem trivial. There is no record of any tribe that ever counted by eights, nor is there the slightest likelihood that such a system could ever meet with any general favour. It is said that the ancient Saxons

t originally have been the base. Pursuing this thought by investigation into different languages, the same resemblance is found there. Hence the theory is strengthened by corroborative evidence. In language after langu

navan = 9.

nuh = 9.

ν?α = 9.

em = 9. no

eun = 9.

nio = 9.

en = 9. ni

ni = 9.

nyr = 9.

nine = 9.

uf = 9. no

ueve = 9.

ove = 9. n

nove = 9.

oi = 9. n

w = 9. ne

ez = 9. nuh

ttle change. Not only are the two words in question akin in each individual language, but they are akin in all the languages. Hence all these resemblances reduce to a single resemblance, or perhaps identity, that between the Aryan words for "nine" and "new." This was probably an accidental resemblance, no more significant than any one of the scores of other similar cases occurring in every language. If there were any further evidence of the former existence of an Aryan octonary scale, the coincidence would possess a certain degree of significance; but not a shred has ever been produced which is worthy of consideration. If our remote ancestors

siti-dat

itjuma =

8. sin-the

hun = a

, from this fact alone, that eith

use and importance in China, India, and central Asia, as well as among some of the islands of the Pacific, and in Central America, leads him to the conclusion that there was a time, long before the beginning of recorded history, when 8 was the common number base of the world. But his conclusion h

l scale, and the substitution of the duodecimal in its stead. It is said that Charles XII. of Sweden was actually contemplating such a change in his dominions at the time of his death. In pursuance of this idea, some writers have gone so far as to suggest symbols for 10 and 11, and to recast our entire numeral nomenclature to conform to the duodecimal base.225 Were such a change made, we should express the first nine numbers as a

odecimal is not a natural scale in the same sense as are the quinary, the decimal, and the vigesimal; but it is a system which is called into being long after the complete development of one of the natural systems, solely because of the simple and familiar fractions into which its base is divided. It is the scale of civilization, just as the three common scales are the scales of nature. But an example of its use was long sought for in vain among the primitive races of the world. Humboldt, in com