DISCUSSION AND COMPARISONS

Jiflhri ⁽ Tithiiniiiicfiit triht'. ('osln }\ir(i] ^{^} 4

et. 5 skang.

but. ^{• ti} terl.

miiyat. 7 kugu.

keng, ka. S dsohtan, pai,pa.

Bruiira(Talaiaancari,tribe, CostaRica)^''

etsik. 5 kchLsskan.

bug. ^li teschan.

mang. 7 kuchk.

bachkan. ^.S ochtan.

Carrizo {nearMonclova, Coahaila) ^{^}

pequeteu. 4 uaiye.

acequeteu. 5 maguele.

guiye.

•.tU)

DISCUSSION AND COMPARISONS

Before I disc-u.s.s the^e listis ;uul uttenipt to

draw

conclusions

from

thoni. there is one point whicli deserves notice. Itis this:

To what

extent can these

number

lists be considered reliable? I do not

by

this inquiry wish to ([ue.stion the veracity of any author

whose works

I have quoted or used, but to refer to the

method

by which the lists

were

obtained, especially the portions relating to the high numbers.

Did

the

Maya.

Aztec, and other tribes

make

use in actual count or computation oi thousands, tens of thousands, hundred.s of thousands, and even millions as given in theselists, orhave they been

filled out. in part, by the authors according to the systems found in

vogueⁱ That implicit reliar)ce can lie placed on the

judgment

and accuracy of the

more

^recent authorities who, as is

known,

derived their information direct

from

the natives, as StoU, Gatschet. etc., is

conceded, but the lists given ^l)V these authors seldom if ever reach

beyond

the thousand.

Most

of the lists

from

the tribes of

Mexico

and Central America, which run into high numbers, are given l)y the early authors (chiefly Spanish) or are based on their staten)ents.

When

the Mexicans spoke of caxtol-tz(mtl!=lb tzoiitU (6,000); ^cein- jH)((l-,riq (dj) ////'—20 ^{.v/qiiJj)ill J (ItiO.}()()0)^; anda-m-jioal-tzuii-xiqui.j)!Jli^=

20 times 4-00 .riquiplJJ! ((14.000.(100

—

^{seelist),did they have in thought}

the actual

numbers

given as equivalents of these terms, or merely measures?

When,

forexample, they said, " 15 tzontlP- {tzontll signi- fying bundle or package) did they intend to signify 15x4:Ot>, or simply 15 bundles or packages^ In other words, did the reference

1AdolphUhle,inCompteRenduCong.Araericanistes. Berlin.^1S.S8,p. 474.

^Iliici..p.47.i.

^Clik',DieLiincieramunlern KidBruvodi-lNorte,p. 120,(jnotedIjyBrinton, .Vmeriruiiiiaee.]>.y3.

they rtvkoned thorn l\v sacks, every saok havintr l>een rei^koned to contain ^{:^} .ri<p/>jr/?h\ or ^{:*4.«^Vi} nuts."

Now.

an^

wo

to suju^s*^ that in oHintinjrthe sacks tho nunilvrof nuts

was

kept in view^.'

Did

tho nion^hant

who

purchased af3<»>tJ>'of sacks ^(44>(l) have in

mind

or pur-

}X^s«^ buyinsr '.'.(><>i\<VK^ nuts.' This will suffitv to

make

«-lear the thoujrht intended to 1>e pivsonted. and ^{will, it} stM>ms. justify theques- tion

—

^{liave the hijrh}

^numlxTs

^{in these listsIven sidded in} attxirdance with tho i"omputation of tho nxvirder. or

wen^

they in actual use amouirtho native Moxicsins^.'

As

contactwith

Europeans

and their decimalsystem for

m^rly

four centuries has m<xiilie<i to a srroateror loss extent the ori»Tinal native

method

ofc«iuntiniT.it isdoubtful

whether

dinx't n^'ferenco tothosur- viviu«^nativesof the presentday

would

settletJiequestion.

The Mava

^)rV has. as

we

have s«vn. Ixx^n chansrini fr^im S.otM to 1.<xhi. and the

sisrnifiin\tionofother nunjoral tonus hasIxvn ohanp^d in similar

man-

ner.

Our

only apix\il is thor»^fore to tho native records,

and

here.

jx^.-v>ibly

fwni

our inability to intorpn^t tho Mexirtin syni1x>ls.

wo

are linuttxi to the

Max^n

txxiitvs

and

inscriptions. Here, however, as has Tx?en clearly

shown

in another paper,

and

as hasl>een proved

by Forstemann

and (t«xximan. tho

evidemv

^isclear that tho

Ma\

a. orat least tho priests or authors of the Dresdenctxlox

and

the inscriptions, ix>uUi and actually did carry their tx>iuputations to the millions, in terms

where

tho numlxn- element >\-5is n«vessarily rotaino<^.

where

the primary^unit

—

^{in thost^ instancesthe day}

—

^{had tolx^kept in view.}

^Of

<xHirsothey

made

use of the hisrhor units to facilitate ixnintinsr. as

we do

at the ptvsent day. If the

Maya

wer*^ t^jiabloof i-«^uniinsr intel- lisrontly to this tigure. it is not unreasonable to suppose that the

more

advancedatnousrthesurixnindinsx triK^s

may

have

made

similar,though

jx^ssilily not sogn^t. pn\«m^ss in their numorit-al systems. That tho Mexi»^ns had symlxils for high

nvuuVrs

^isasserto<l

by

tho early his- torians,

and

is evident

from

their rtMuainingixxiiiv^ but

no

mrtins of testing these, as the

Maya

uianuscripts and inscriptions have Ixxmi testixi. has yet Ixx^n found; however, the explanation of symlxils carryingthecount tothe tens of thousands has ^Ixx'n given.

Notwithstandingthis conclusion, it is apjwrtMit that tho intluonivof the F.ur>>ix\in decimal sy>toni ha> Ixvn felt in

some

of the native

THOMAS] niSl'l'SSlON ANI> ^Ci)M

TAUISONS

'.>'_M

fountsliiMvin yivt'ii. Tliis. for oxaiiipU'. ^is prohulily true of tlio IIiuis- toc!iM count,wlioro tlic siiupli' tiMMu ,(/ is usod to doiioto 1jxto. miuI also in tlir count t'roui I'OO to ;Hio in tiiis system aiul in sonic otiicrs.

All till' prcccdinii' lists showinji' tlio count from ^lo

upwanl

which

liclony to the Mi>xican and

Mayan

^si'roups. except that of the Taralui- mari. pt>rtain to the viijesinial system and in

method

of counting'Ix^ar a >tioiie- oeneial resemblanceoiu>to another, yet wiienthey are^closely examined

minor

ditVerenci^sare fomul wliich havean important beur- ine- on the question of the origin and rtdationship of tiiese systems.

Of

^thcsi> variations

we

notice the following:

The Nahuatl count lx)llow.s strii-tly the (piinary-xigesiiual system, as has been already stated, 5 and 15. as well as ^_'(>. lieing basal nuinl)ers.

'l"he count ^is always from a lower nmnber. that ^is to say. the luinor iuuub(M's are always adihnl to a nundter ])assed: thus 41 and 4i.* ar(>

formed ^tty adding 1 and ^•_' to 40. and not by counting the ¹ and ^_'on the next or third setMV. as

we

have seen was the rule

among some

of the

Mayan

trib(>s. as the ^laya ]>rop(M- oi- Yucatec. the (^uicht\

t'akciu(|uel. I'oUonchi. t^iickclii.

Mam.

^Ixil. ami probal)iy most of tht^ soutliern tribes of the grou]>. but not

among

^{tiie lluasteca.}

who

formed the northern otfslioot. TIh^ count of the latter, though, like tlieothersof the

Mayan

group, fundamentally ^\igcsimal to^{'.•ou. is.} like the Nsihuatl. by additions of the

minor

nnudiers to a inuubei']iassed as -JO-i-lO to

form

HO and

2X*J0+U)

to

form

^.")0.

The

numeral system of the

Mayan

^{tribes generally ditl'ered} from the Nahuatl. Zapotec.

Mazatec. Trike. ^lixe. and

Zoque

syst(MUs all of which are regulai'ly (|uinary-vigesinial. and generally add the minoi' luuubers to the pre- ceding base

—

^{in being}

^more

nearly decimal-vigesimal, and in adding the nuuib(>rs abo\i^4o to the following basi>, as ¹ on thethird seoi-(\ oi'

third ^1*0, to

form

41. In the

Mayan

dialects the count ^is never bas(>d

on ^.') except, as ha> heretofore be(Mi suggested,

from

^t! to ^s. anil in

om^ dialect fi'om ^t! to ^;•.

So

far. therefore, as these difl'ercnccs are concerned, they tiMul toward grouping together tiie systems of the Nahuallan. Zajiotecan. and ^{Zoipican tribes,} as contrasted with tlw

Mayan:

but the teiiu Nahuatlan ^is used here as refei-ring oidy to tlx' stock in its limited stMisc

—

tlie .Vztecan branch as ^{tln^ I'ule} does not hold

good

thi'oughout, wluMi

we

pass into the Sonoran l>ranch.

llowe\(M'. the gron])ing on tlu>se points is int(M'(>sting- ;is it is in

harmony

with otherdala.

In one jieculiarity. however, the Zaimtcc count ^diti'ers from the Nahuatl and aj>jiroaches tiie

Mayan

^systems.

From

.>.')-,)!•, 7.')-7!t,and

!I5-I>lt the

numbers

are obt.ained l)y subtraction from the next higher

bas(>

—

^{tlius, for} .>.'"i they say <v-<'(?rt qul<m<i oret-caoijn<i)n<i)i<i: that is,

5

from

^(io.

For

.)()-.")!i. 7()-79, and !t5-9i> they have two methods of

counting

—

^{thus for ;")(! they}.say re-ciiin/o <j>uintii-hl-fiiht; tiiat is, 5 from

the tlecimal system.

For

example, in the Cahita count the (juinary- vigesimal rule prevails; ^tl. 7, and 10 are based on ^;"); S on 4; 11 to lit

on 10, or,rather, twice five.

From

20

upward

the count is vigesimal,

!(•

when

used retaining throughout its

form

of

2X5. The

contaet, however, in this region with the decimal svstem is clearly indicated by the followingstatement of the author of the

Arte Lengua

Cahita, given above:

"Some

^nationssaj' xenutacaua orsesencher<'for 20;others say for 10 si'mwhere and ^follow

up

the countthus:11, sfxiiva/ieix' ((iikoi Kciiu; 12,xcHavehereamaii ^tioi.,etc.

For

20 they say itosaccherc,which ^is

two

times 10.

The Yaqui

saj'for 5saxaveJiere,

and

counting

from

5 to 5 say U(>mi'i'lii'ri\ 10[

=

2

X

5]; vahiDehere,15

[=3 X

5]. These alsosay for

20^{.^I'liufdcaiia} [1

X

20] ornut'quiocheiv [4X5],andfor 25srsarelierc (this

particidai- count is of this nationonly), and for 100 ^sa}' taamnitacaua

|5x20|or

tiicaiircliri'i; which is 20 fives." In the paragraph which follows he states in general terms that

some

of the triljescount by

fives, others by tens, both using the

same

term, vehere, prefixing the

"numeral

abverbs" ^{.svw^}

"one

^{time." nosa,}

"two

times," etc.

The

••nations" alluded^toareprobal)lythe Cahita tribes, such asthe Tehu- eco, Zuaque,

Mayo,

^Yaqui. and other related or neighboringtribes.

This change ⁱⁿ the appii<ation of a gi\-en tei-m in closely related dia- lects is not only interesting. ^I)ut

somewhat

remarkable; and addedto the fact that the closely related

Tarahumari

of the

same

section use the decimal system, indicates that the latterand thevigesimal system here

came

^{into contat^t.}

Do

^the data furnish evidence as to which was the s])readingor aggressiveand which^{th(^}yieldingone^'.

Without

entering into adiscussion of the (piestion the following facts are presented for the benefit of those desiring to look further ^{into this sultject.}

The

similarit\' of the

number names

^of the Cahita

and Tarahumari

^to

tho.se of the Naliuatl ^is too apparent to jjass unobserved even ^liy the

mere

cursory glance. Include tiie allied

Opata

and ^take for

example

the

numbers

1 to 5 and ^{10. as follow:}

DISCUSSION

AND COMPARISONS 923 The

resemblance between the

names

in each cohimn, except />//<, 1 inTarahumai'i (forwhich Charenceysays he finds the alternatesi/icjj/',

which

would

be in harmouj' with the others), and uomatrml (2x5), 10 in Cahita,is atonceapparent. This,however, ismerelyⁱⁿ accoi-dance with the recognized athnit}'of the first three idioms with flicNahuatl.

It seems, however, ^that

we

lookin vain tothe Nahuatl

names

for the vehere{vehe-n') as it can not be derived

from maauMll

(5), mathictJi (10), orp<iaJH(20), nor

from

the

names

for 5, 10, or 20 in the Opata, Cahita, orTarahumari.

The name

for20in

Opata

^is wri {ne-un),which

signifies

"man;"

ⁱⁿ Cahita, tucaun; inTarahumari.'>.SY/-///^^rr';/(2

X

10).

la these languages the only

number name

which resembles ^it is that for 3, which isnot a divisor.

Turning

to the Shoshonean

group we

notice the following facts.

Whether

they are svxfiicient to justify a decision on the point is very doubtful; ^this, however, ^{is left} for the reader to determine.

The

following list of the

names

for 2. 5, 10, and 20 is

from

Gatschefs Forty Vocabularies.

'

Sonora to the southern

boundary

of Guatemala, and ^to souie extent as far as the isthnms. There

seem

to have been hut few, if any, trilies in this area as far south asthe southern houndary of (luatc'iuala that did not

make

use of this system; at lea.stthedata ()btainal)le bear out this conclusion. North of the northern l)oundarv of this area this .system is found. act'ordin<i' to C'onant.' '"in the northern regiuns of North America, inwesternCanada, andin noi'thwesternUnitedStates";

however, the only examples he yives arethe .systems of the '"Alaskan F^skinios," "•Tchiulit," ••Tlinyit." ••Nootka," and '-'Tsimshian."

As

•a genei'al rule the systems of the tribes of the western part of tlie Unit(>d States,

from

the south(>ru boundary to the

Cohuubia

river, were decimal or(juinai'v-decimal; however, instances of the vig-esimal system appearhere and theic in this area.

As

one

example we

call attention to the numerals of the

Huchnon

diali'ct of the Yukiaii family obtained ^liy

Mr

Stephen

Powers

at

Round

Valley reservation.

California, given in the preceding chapter.

That a count referring the

minor numbers

to the next higher base, which ^is. as

we

have seen, contin(>d in the southern regions almost exclusively to the dialects ^f)f the

more

southern sections, ch.ietly to tho.se of the

Mayan

^group, should be found in California ^is. to say the least, interesting; however, it is not the only examj)le

from

this sectioTi, aswill a])pear. It is

somewhat

singulai' that two otluM- idioms of the

same

family, thi' vocabulariesof wdiichare given by

Mr

Powers, follow the decimal instead of thi^ vigi'simal system. Other examples of this system are found south of the C'olumhia ^ri\'er. as in the

Pomo

diale<'t

(Round

\'alley reser\ation. Califoinia);^'• the 'I'uohumie dialei-t

(Tuohnune

ri\-er. California);'^ the Koid<au and Nishinaui dialects.'

and the .Vchomawi dialect.'

The

^first, third, and foui'th of these appear to refer the count to the following ^score, while in the last (Achonuiwi) it is a})plied to the preceding^.scoi-e.

The

Tuolunnie sys- tem ^is .somewhat doubtfid, asthere are but

two

muuiiers (I'uand ^Itii|

on which ^{to base} a decision. Accoi'iling to

Major

Powcdl's classilica- tion (7tli

Ann.

Rept. Bur. Ethnology), the

Pomo

^{are included in the}

1NumberConcept,p.in."). <I'owers, op.eit.,p. 596.

-Power-s.Tribesof Ciiliforniii. p..t02. ''Ibid., p. UOC.

^GiblK, op.eit., p.MH.

THOMAS] DISCUSSION

AND CoMTARISONS 925

Kuhuiiipan family; the

Achomawi

ⁱⁿ the Falaihniluui tainiiy, and ^tlip

Konkau

and

Nishinam

inthe

Pujunan

family.

AVithout ivferring to other examples ^it

may

^{lie stated in} general terms thatwhile the vigesimal s_vstem has notbeen found in use east of the

Rocky

mountains, except in (rreenland and among-

some

tiilies in the northwestern cis-montane portion of British Columbia, it pre- vailedto a eonsideral)le extentonthe Paeitic slope

from Mexico

north-

ward

to the Arctic ocean, and it

may

alsobe added that it is found iimong- the eastern tribes of Siberia and was the

method

adojjted by the Aino. Conant^' says that the Tschukschi and

Aino

systems are

"among

the best illustrations of counting

by

twenties that areto he found anywhei-e in the

Old World."

These have been given in the precedingchapter ^foi' comparison.

The

countof the

minor numbers

inthe

Aino

is based,as will beseen, on the following score, as in the

Mayan

group.

Whether

the ecpiiva- lentsadded are correctly given ^is

somewhat

doubtful, as the pi-oper interpretation of the

name

foryo

may

be 10on the second score; that for 50, 10 on the third score, etc., as

we

have indicatedin parenthesis.

In the Tschukschi the addition ^is to the preceding score

—

^{thus :'>() is}

formed

bj^adding 10 to20.

These andadditional facts (jf the

same

character tend to .show that in North

America

the vigesimal system of counting, like

some

other customs,

was

confined almost exclusively to that area which ^I have

in a previous

work"

designated the "Paeitic section." which includes the Paeitic slope north of

Mexico

and ^all of

Mexico

and ^Central America. This fact and the additional fact that the system prevails in noi'theastern Asia, while ^{it is} rare in other parts of that grand division, except an area in theCaucasus region, and ^is wanting in the Atlantic slope of

North

America, are interesting and of considerable importance in the study of the ethnology of our continetit.

It

would

be interesting in this connection to iniiuire into the rang'^

of this numeral system in South America, but

we

have not the dataat hand necessary for this purjjo.se.

Conant

says in genei'al terms that

it prevailed in the northern and western portions of the t'ontinent.

though it is

known

that on the Paeitic slope it did not extend south-

ward

farther than the bordiM-s of Peru,

where

the decimal system prevailed. It appears to have been in use

among

^{the Chibchasor} Muyscas, a

group

extending both northand south of the Isthmus. It isor was in use

among some

of the triiies on the Orinoco, in eastcrti Brazil, and in Pai-aguay.

However,

the range of the >ystem in South

America

isasyet unascertained.'^

1Number^Concept,p. 191.

^TwelfthAnn.Rep.Bur.Ethn.. pp.T>i->i.

'Professor

W

^.TMcGeesuggests that itmaypossiblyholdtrue ina generalsensetliatthe barfidot orsandal-wearing habitaecompaniedtheuse of thissystemofcounting.

It is not to be understood thatanygeographical law of distriluitionhas ever been observedwhichgovernsthis, butmerelythatcertain families of raceshaveshown a preferencefortheoneortheothermethodofcounting. Thesefamilies,disseminat- ingtheir characteristicsthroughtheirvariousbranches,have producedcertaingroups ofraceswhich exhibita well-marked tendency,here toward thedecimal and there towardthe vigesimal form ^of numeration. Asfarascan be a.scertained,thechoice oftheoneor the otherscaleisdeternunedbynoexternal circumstances,Imtdepends solelyon thementalcharacteristics ofthetribesthemselves. Enviromnentdoesnot exertanyappreciable influenceeither. Bothdecimal andvigesimal numerationare found indifferently in warm and incold countries; in fruitfuland in barren lands;

inmaritimeand in inland regions; andamong highlycivilized or deeplydegraded peoples.

Whetheror not theprincii)alnundierba.seofanytribeistobe20seems todepend

entirelyupona singleconsideration; are tliefingersaloneusedasanaidtocounting, or are both fingersand toesused? If oidy the fingers are employed, the resulting scalemust becomedecimal if sutlicientlyextended. If use is made of the toes in additiontothefingers, tlieoutcome mustinevitablybea vigesimalsystem. Subor- dinatetoeitheroneof these the quinarymayand often does appear. ^{It is} never the principalbaseinanyextendedsystem.

Tothe statementjustmaderespectingtheorigin of vigesimal counting,exception may, ^of course, be taken. In the case of numeral scales like the Welsh, the Nalniatl,and manyotherswhere the exact meanings of the numerals can not be ascertained, noproofexiststhatthe ancestorsofthese peopleseverused eitherfinger ortoecounting; andthesweepingstatement thatanyvig<'simalscale^istheoutgrowth ofthe use^ofthes(^naturalcountersisnotsuscejitiljleofjiroof. But^.10manyexam])les aremetwithinwhich theoriginisclearly of this nature that no hesitation is felt in jiuttingtheabove forwardasa general explanation ^forthe existenceof this kind of counting. Any otherorigin isditficult to reconcile with observed facts, andstill

moreditticult toreconcilewithany rational theory- of numbersystem development.

I note

some

faet.s, tal<en in part

from

the

work

quoted, in order that the I'eader

may

see thehearingtiieyhaveon theopinions expressed

in this ([notation. Aeeording to the data furnished hy this writer it sei'Uis that this .systemt)ceui'red in

Kurope

onlyalong the western sea- coast and thatalmost exclusively

among

^{tlie Celts, the} only

group

of the

Aryan

stock which seemstohaveused ^it. in .Vsia ithasheenfound toany extentotdyinthe t'aucasic

groupand

in thenortheastern part of ofthe continent, that^is. inwhat Hrintontermsthe "Arctic

droup"

of hisSihericlirancii. Nota single

example

isnoted

from

the Sinitic

group

or from the Semitic branch. In Africanone have been rejxjrted

from

the llamitic gi'oiip. and but few

from

the nc'grodialects. l)iit the latter field hasbeen oidy superficially exaiuined in thisrespect. Nota single

1N'umlKT Concept,p.171>-J^.