SCORPION

Tylor, Edward Burnett, Sir, 1832-1917

"Primitive culture : researches into the development of mythology, philosophy, religion, language, art, and custom"

CHAPTER VII.

THE ART OF COUNTING.

Ideas of Number derived from experience State of Arithmetic among
uncivilized races Small extent of Numeral-words among low tribes
Counting by fingers and toes Hand-numerals show derivation of
Verbal reckoning from Gesture-counting Etymology of Numerals
Quinary, Decimal, and Vigesimal notations of the world derived
from counting on fingers and toes Adoption of foreign Numeral-
words Evidence of development of Arithmetic from a low original
level of Culture ....... 240

 CHAPTER VII.

THE ART OF COUNTING.

Ideas of Number derived from experience State of Arithmetic among un-
civilized races Small extent of Numeral-words among low tribes
Counting by fingers and toes Hand-numerals show derivation of Verbal
reckoning from Gesture-counting Etymology of Numerals Quinary,
Decimal, and Vigesimal notations of the world derived from counting
on fingers and toes Adoption of foreign Numeral-words Evidence of
development of Arithmetic from a low original level of Culture.

MR. J. S. MILL, in his ' System of Logic,' takes occa-
sion to examine the foundations of the art of arithmetic.
Against Dr. Whewell, who had maintained that such pro-
positions as that two and three make five are ' necessary
truths,' containing in them an element of certainty beyond
that which mere experience can give, Mr. Mill asserts that
' two and one are equal to three ' expresses merely ' a
truth known to us by early and constant experience : an
inductive truth ; and such truths are the foundation of
the science of Number. The fundamental truths of that
science all rest on the evidence of sense ; they are proved
by showing to our eyes and our fingers that any given
number of objects, ten balls for example, may by sepa-
ration and re-arrangement exhibit to our senses all the
different sets of numbers the sum of which is equal to ten.
All the improved methods of teaching arithmetic to chil-
dren proceed on a knowledge of this fact. All who wish
to carry the child's mind along with them in learning
arithmetic ; all who wish to teach numbers, and not mere
ciphers now teach it through the evidence of the senses,

240

NUMERATION DERIVED FROM EXPERIENCE. 24!

in the manner we have described/ Mr. Mill's argument is
taken from the mental conditions of people among whom
there exists a highly advanced arithmetic. The subject
is also one to be advantageously studied from the eth-
nographer's point of view. The examination of the
methods of numeration in use among the lower races not
only fully bears out Mr. Mill's view, that our knowledge
of the relations of numbers is based on actual experi-
ment, but it enables us to trace the art of counting to
its source, and to ascertain by what steps it arose in
the world among particular races, and probably among
all mankind.

In our advanced system of numeration, no limit is known
either to largeness or smallness. The philosopher cannot
conceive the formation of any quantity so large or of any
atom so small but the arithmetician can keep pace with
him, and can define it in a simple combination of written
signs. But as we go downwards in the scale of culture, we
find that even where the current language has terms for
hundreds and thousands, there is less and less power of
forming a distinct notion of large numbers, the reckoner is
sooner driven to his fingers, and there increases among
the most intelligent that numerical indefiniteness that we
notice among children if there were not a thousand people
in the street there were certainly a hundred, at any rate
there were twenty. Strength in arithmetic does not, it is
true, vary regularly with the level of general culture.
Some savage or barbaric peoples are exceptionally skilled
in numeration. The Tonga Islanders really have native
numerals up to 100,000. Not content even with this, the
French explorer Labillardire pressed them farther and
obtained numerals up to 1000 billions, which were duly
printed, but proved on later examination to be partly non-
sense-words and partly indelicate expressions, 1 so that the
supposed series of high numerals forms at once a little
vocabulary of Tongan indecency, and a warning as to the

1 Mariner, ' Tonga Islands,' vol. ii. p. 390.

242 THE ART OF COUNTING.

probable results of taking down unchecked answers from
question- worried savages. In West Africa, a lively and
continual habit of bargaining has developed a great power
of arithmetic, and little children already do feats of compu-
tation with their heaps of cowries. Among the Yorubas of
Abeokuta, to say ' you don't know nine times nine ' is
actually an insulting way of saying ' you are a dunce.' 1
This is an extraordinary proverb, when we compare it with
the standard which our corresponding European sayings set
for the limits of stupidity : the German says, ' he can
scarce count five ' ; the Spaniard, ' I will tell you how
many make five ' (cuantos son cinco) ; and we have the
same saw in England :

'. . . as sure as I'm alive,
And knows how many beans make five.'

A Siamese law-court will not take the evidence of a witness
who cannot count or reckon figures up to ten ; a rule which
reminds us of the ancient custom of Shrewsbury, , where a
person was deemed of age when he knew how to count up to
twelve pence. 8

Among the lowest living men, the savages of the South
American forests and the deserts of Australia, 5 is actually
found to be a number which the languages of some tribes do
not know by a special word. Not only have travellers
failed to get from them names for numbers above 2, 3, or
4, but the opinion that these are the real limits of their
numeral series is strengthened by the use of their highest
known number as an indefinite term for a great many.
Spix and Martius say of the low tribes of Brazil, ' They
count commonly by their finger joints, so up to three only.
Any larger number they express by the word " many." '*

1 Crowther, ' Yoruba Vocab.' ; Burton, ' W. & W. from W. Africa,' p. 253.
' O daju danu, o ko mo essan messan. You (may seem) very clever, (but)
you can't tell 9 X 9.'

* Low in ' Journ. Ind. Archip.' vol. i. p. 408 ; ' Year-Books Edw I.'
(xx.-i.) ed. Horwood, p. 220.

9 Spix and Martius, ' Reise in Brazilian,' p. 387.

ARITHMETIC OF UNCULTURED RACES. 243

In a Puri vocabulary the numerals are given as i. ami;
2. curiri ; 3. prica, ' many ' : in a Botocudo vocabulary,

1. mokenam ; 2. uruhu, ' many.' The numeration of
the Tasmanians is, according to Jorgensen, i. farmery ;

2. calabawa ; more than 2, cardia ; as Backhouse puts it,
they count 'one, two, plenty/ but an observer who
had specially good opportunities, Dr. Milligan, gives their
numerals up to 5. puggana, which we shall recur to. 1 Mr*
Oldfield (writing especially of Western tribes) says, ' The
New Hollanders have no names for numbers beyond two.
The Watchandie scale of notation is co-ote-on (one), u-tau-
ra (two), bool-tha (many), and bool-tha-bat (very many).
If absolutely required to express the numbers three or four,
they say u-tar-ra coo-te-oo to indicate the former number,
and u-tar-ra u-tar-ra to denote the latter.' That is to say,
their names for one, two, three, and four, are equivalent to
' one,' ' two,' ' two-one/ ' two-two.' Dr. Lang's numerals
from Queensland are just the same in principle, though the
words are different : i. ganar ; 2. burla; 3. burla-ganar,
1 two-one ' ; 4. burla-burla, ' two-two ' ; korumba, ' more than
four, much, great.' The Kamilaroi dialect, though with
the same 2 as the last, improves upon it by having an
independent 3, and with the aid of this it reckons as far as
6 : i. mal ; 2. bularr ; 3. guliba ; 4. bularr-bularr , ' two-
two ' ; 5. bulaguliba, ' two-three ' ; 6. guliba-guliba ' three-
three.' These Australian examples are at least evidence of
a very scanty as well as clumsy numeral system among
certain tribes. 2 Yet here again higher forms will have to
be noticed, which in one district at least carry the native
numerals up to 15 or 20.

It is not to be supposed, because a savage tribe has
no current words for numbers above 3 or 5 or so, that
therefore they cannot count beyond this. It appears that

1 ' Tasmanian Journal/ vol. i. ; Backhouse, ' Narr.' p. 104 ; Milligan in
* Papers, &c., Roy. Soc. Tasmania,' vol. iii. part ii. 1859.

* Oldfield in ' Tr. Eth. Soc.' ; vol. iii. p. 291 ; Lang, ' Queensland,' p. 433 j
'Latham, Comp. Phil.' p. 352. Other terms in Bonwick, 1. c.

244 THE ART OF COUNTING.

they can and do count considerably farther, but it is by
falling back on a lower and ruder method of expression
than speech the gesture-language. The place in in-
tellectual development held by the art of counting on
one's fingers, is well marked in the description which
Massieu, the Abb6 Sicard's deaf-and-dumb pupil, gives of
his notion of numbers in his comparatively untaught
childhood: 'I knew the numbers before my instruction,
my fingers had taught me them. I did not know the
ciphers ; I counted on my fingers, and when the number
passed 10 I made notches on a bit of wood.' 1 It is thus
that all savage tribes have been taught arithmetic by their
iingers. Mr. Oldfield, after giving the account just quoted
of the capability of the Watchandie language to reach 4
"by numerals, goes on to describe the means by which the
tribe contrive to deal with a harder problem in numeration.
I once wished to ascertain the exact number of natives
who had been slain on a certain occasion. The individual
of whom I made the enquiry, began to think over the
names . . . assigning one of his fingers to each, and it
was not until after many failures, and consequent fresh
starts, that he was able 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.' Of the aborigines of
Victoria, Mr. Stanbridge says : ' They have no name for
numerals above two, but by repetition they count to five ;
they also record the days of the moon by means of the
fingers, the bones and joints of the arms and the head.'*
The Bororos of Brazil reckon: i. couai ; 2. macouai ;
3. OIMI ; and then go on counting on their fingers, re-
peating this ouai* Of course it no more follows among
savages than among ourselves that, because a man counts

1 Sicard, * The'orie des Signes pour PInstruction des Sourds-Muets,' vol.
ii. p. 634.

* Stanbridge in ' Tr. Eth. Soc.' vol. i. p. 304.

8 Martius, ' Gloss. Brasil,' p. 15.

ARITHMETIC OF UNCULTURED RACES. 245

on his fingers, his language must be wanting in words to
express the number he wishes to reckon. For example it
was noticed that when natives of Kamchatka were set to
count, they would reckon all their fingers, and then all
their toes, so getting up to 20, and then would ask, ' What
are we to do next ? ' Yet it was found on examination
that numbers up to 100 existed in their language. 1 Travel-
lers notice the use of finger-counting among tribes who can,
if they choose, speak the number, and who either silently
count it upon their fingers, or very usually accompany the
word with the action ; nor indeed are either of these modes
at all unfamiliar in modern Europe. Let Father Gumilla,
one of the early Jesuit missionaries in South America,
describe for us the relation of gesture to speech in count-
ing, and at the same time bring to our minds very remark-
able examples (to be paralleled elsewhere) of the action
of consensus, whereby conventional rules become fixed
among societies of men, even in so simple an art as that of
counting on one's fingers. 'Nobody among ourselves/
he remarks, ' except incidentally, would say for instance
" one," " two," &c., and give the number on his fingers as
well, by touching them with the other hand. Exactly
the contrary happens among Indians. They say, for in-
stance, " give me one pair of scissors," and forthwith they
raise one finger ; " give me two," and at once they raise
two, and so on. They would never say " five " without
showing a hand, never " ten " without holding out both,
never " twenty " without adding up the fingers, placed
opposite to the toes. Moreover, the mode of showing
the numbers with the fingers differs in each nation.
To avoid prolixity, I give as an example the number
" three." The Otomacs to say " three " unite the thumb,
forefinger, and middle finger, keeping the others down.
The Tamanacs show the little finger, the ring finger, and
the middle finger, and close the other two. The Mai-
pures, lastly, raise the fore, middle, and ring fingers,

1 Kracheninnikow, ' Kamtchatka,' p. 17.
I. R

246 t THE ART OF COUNTING.

keeping the other two hidden.' l Throughout the world,
the general relation between finger-counting and word-
counting may be stated as follows. For readiness and
for ease and apprehension of numbers, a palpable arith-
metic, such as is worked on finger-joints or fingers,* or
heaps of pebbles or beans, or the more artificial contri-
vances of the rosary or the abacus, has so great an ad-
vantage over reckoning in words as almost necessarily
to precede it. Thus not only do we find finger-counting
among savages and uneducated men, carrying on a part of
their mental operations where language is only partly able
to follow it, but it also retains a place and an undoubted
use among the most cultured nations, as a preparation for
and means of acquiring higher arithmetical methods.

Now there exists valid evidence to prove that a child
learning to count upon its fingers does in a way reproduce
a process of the mental history of the human race ; that in
fact men counted upon their fingers before they found
words for the numbers they thus expressed ; that in this
department of culture, Word-language not only followed
Gesture-language, but actually grew out of it. The evi-
dence in question is principally that of language itself,
which shows that, among many and distant tribes, men
wanting to express 5 in words called it simply by their
name for the hand which they held up to denote it, that in
like manner they said two hands or half a man to denote
10, that the word foot carried on the reckoning up to 15,

1 Gumilla, ' Historia del Orenoco,' vol. iii. ch. xlv. ; Pott, ' Zahlmethode,'
p. 1 6.

2 The Eastern brokers have used for ages, and still use, the method of
secretly indicating numbers to one another in bargaining, ' by snipping
fingers under a cloth.' ' Every joynt and every finger hath his significa-
tion,' as an old traveller says, and the system seems a more or less artificial
development of ordinary finger-counting, the thumb and little finger
stretched out, and the other fingers closed, standing for 6 or 60, the ad-
dition of the fourth finger making 7 or 70, and so on. It is said that
between two brokers settling a price by thus snipping with the fingers,
cleverness in bargaining, offering a little more, hesitating, expressing
an obstinate refusal to go farther, &c., comes out just as in chaffering in
words.

COUNTING BY FINGERS AND TOES. 247

and to 20, which they described in words as in gesture by
the hands and feet together, or as one man, and that
lastly, by various expressions referring directly to the
gestures of counting on the fingers and toes, they gave
names to these and intermediate numerals. As a definite
term is wanted to describe significant numerals of this class,
it may be convenient to call them ' hand-numerals ' or
' digit-numerals.' A selection of typical instances will
serve to make it probable that this ingenious device was not,
at any rate generally, copied from one tribe by another or
1 inherited from a common source, but that its working out
with original character and curiously varying detail displays
the recurrence of a similar but independent process of
mental development among various races of man.

Father Gilij, describing the arithmetic of the Tamanacs
on the Orinoco, gives their numerals up to 4 : when they
come to 5, they express it by the word amgnaitone, which
being translated means ' a whole hand ; ' 6 is expressed by
a term which translates the proper gesture into words,
itacono amgnapond tevinitpe ' one of the other hand,' and
so on up to 9. Coming to 10, they give it in words as
amgna acepondre ' both hands.' To denote n they stretch
out both the hands, and adding the foot they say puitta-
Pond tevinitpe ' one to the foot,' and thus up to 15, which
is iptaitone 'a whole foot.' Next follows 16, 'one to the
other foot,' and so on to 20, tevin itoto, ' one Indian ; ' 21,
itacono itoto jamgndr bond tevinitpe ' one to the hands of the
other Indian ;' 40, acciache itoto, ' two Indians ; ' thence on
to 60, 80, 100, ' three, four, five Indians,' and beyond if
needful. South America is remarkably rich in such evi-
dence of an early condition of finger-counting recorded in
spoken language. Among its many other languages which
have recognizable digit-numerals, the Cayriri, Tupi, Abi-
pone, and Carib rival the Tamanac in their systematic way
of working out ' hand,' ' hands,' ' foot,' ' feet,' &c. Others
show slighter traces of the same process, where, for
instance, the numerals 5 or 10 are found to be connected

248 THE ART OF COUNTING.

with words for ' hand/ &c., as when the Omagua uses pua,
4 hand/ for 5, and reduplicates this into upapua for 10. In
some South American languages a man is reckoned by
fingers and toes up to 20, while in contrast to this, there are
two languages which display a miserably low mental state,
the man counting only one hand, thus stopping short at 5 ;
the Juri ghomen apa ' one man/ stands for 5 ; the Cayriri
ibichd is used to mean both ' person ' and 5. Digit-
numerals are not confined to tribes standing, like these, low
or high within the limits of savagery. The Muyscas of Bogota
were among the more civilized native races of America,
ranking with the Peruvians in their culture, yet the same
method of formation which appears in the language of the
rude Tamanacs is to be traced in that of the Muyscas, who,
when they came to n, 12, 13, counted quihicha ata, bosa,
mica, i.e., ' foot one, two, three/ l To turn to North
America, Cranz, the Moravian missionary, thus describes
about a century ago the numeration of the Greenlanders.
' Their numerals/ he says, ' go not far, and with them the
proverb holds that they can scarce count five, for they
reckon by the five fingers and then get the help of the toes
on their feet, and so with labour bring out twenty/ The
modern Greenland grammar gives the numerals much as
Cranz does, but more fully. The word for 5 is tatdlimat,
which there is some ground for supposing to have once
meant ' hand ; ' 6 is arfinek-attausek, ' on the other hand
one/ or more shortly arfinigdlit, ' those which have on the
other hand ; ' 7 is arfinek-mardluk, ' on the other hand
two ; ' 13 is arkanck-pingasut, ' on the first foot three ; '
1 8 is arfersanek-pingasut, ' on the other foot three ; ' when
they reach 20, they can say inuk ndvdlugo, ' a man ended/
or inup avatai ndvdlugit, ' the man's outer members ended /
in this way by counting several men they reach higher

1 Gilij ; ' Saggio di Storia Americana,' vol. ii. p. 332 (Tamanac, Maypure).
Martius, ' Gloss, Brasil,' (Cayriri, Tupi, Carib, Omagua, Juri, Guachi, Coretu,
Cherentes, Maxuruna, Caripuna, Cauixana, Carajas, Coroado, &c.) ; Dobriz-
hoffer, 'Abipones,' vol. ii. p. 168; Humboldt, 'Monumens,'pl. xliv. (Muysca).

HAND AND FOOT NUMERALS. 249

numbers, thus expressing, for example, 53 as inup pinga-
jugsdne arkanek-pingasut, ' on the third man on the first foot
three.' 1 If we pass from the rude Greenlanders to the com-
paratively civilized Aztecs, we shall find on the Northern as
on the Southern continent traces of early finger-numeration
surviving among higher races. The Mexican names for the
first four numerals are as obscure in etymology as our own.
But when we come to 5 we find this expressed by macuilli ;
and as ma (ma-itl) means ' hand,' and cuiloa ' to paint or
depict,' it is likely that the word for 5 may have meant
something like ' hand-depicting.' In 10, matlactli, the
word ma, 'hand,' appears again, while tlactli means half, and
is represented in the Mexico picture-writings by the figure
of half a man from the waist upward ; thus it appears that
the Aztec 10 means the ' hand-half ' of a man, just as
among the Towka Indians of South America 10 is expressed
as ' half a man,' a whole man being 20. When the Aztecs
reach 20 they call it cempoalli, ' one counting,' with evi-
dently the same meaning as elsewhere, one whole man,
fingers and toes.

Among races of the lower culture elsewhere, similar facts
are to be observed. The Tasmanian language again shows
the man stopping short at the reckoning of himself when he
has held up one hand and counted its fingers ; this appears
by Milligan's list before mentioned, which ends with puggana,
' man,' standing for 5. Some of the West Australian tribes
have done much better than this, using their word for
' hand,' marh-ra ; marh-jin-bang-ga, ' half the hands,' is
5 ; marh-jin-bang-ga-gudjir-gyn, ' half the hands and one/
is 6, and so on ; marh-fin-belli-belli-giidjir-iina-bang-ga,
' the hand on either side and half the feet,' is 15. * As an ex-
ample from the Melanesian languages the Mare will serve ;
it reckons 10 as ome re rue tubenine, apparently ' the two

1 Cranz, ' Gronland/ p. 286 ; Kleinschmidt, * Gr. der Gronl. Spr. ;' Rae
in ' Tr, Eth. Soc.' vol. iv. p. 14.5.

2 Milligan, 1. c. ; G. F. Moore, ' Vocab. W. Australia.' Compare a series
of quinary numerals to 9, from Sydney, in Pott. * Zahlmcthode,' p. 46.

250 t THE ART OF COUNTING.

sides ' (i.e. both hands), 20 as sa re ngome, ' one man/ &c. ;
thus in John v. 5 ' which had an infirmity thirty and eight
years,' the numeral 38 is expressed by the phrase, 'one
man and both sides five and three.' 1 In the Malayo-
Polynesian languages, the typical word for 5 is lima or rima,
* hand,' and the connexion is not lost by the phonetic
variations among different branches of this family of lan-
guages, as in Malagasy dimy, Marquesan fima, Tongan
nima, but while lima and its varieties mean 5 in almost all
Malay o-Polynesian dialects, its meaning of ' hand ' is con-
fined to a much narrower district, showing that the word
became more permanent by passing into the condition of a
traditional numeral. In languages of the Malayo-Polynesian
family, it is usually found that 6, &c., are carried on with
words whose etymology is no longer obvious, but the forms
lima-sa, lima-zua ' hand-one,' ' hand- two,' have been found
doing duty for 6 and 7.* In West Africa, Kolle's account of
the Vei language gives a case in point. These negroes are
so dependent on their fingers that some can hardly count
without , and their toes are convenient as the calculator squats
on the ground. The Vei people and many other African
tribes, when counting, first count the fingers of their left
hand, beginning, be it remembered, from the little one, then
in the same manner those of the right hand, and afterwards
the toes. The Vei numeral for 20, mo bdnde, means obvi-
ously ' a person (mo) is finished (bande)/ and similarly
40, 60, 80, &c. ' two men, three men, four men, &c., are
finished.' It is an interesting point that the negroes who
used these phrases had lost their original descriptive sense
the words have become mere numerals to them. 3 Lastly,
for bringing before our minds a picture of a man counting
upon his fingers, and being struck by the idea that if ho
describes his gestures in words, these words may become an

1 Gabelentz, ' Melanesiche Sprachen,' p. 183.

2 W. v. Humboldt, * Kawi-Spr.' vol. ii. p. 308 ; corroborated by ' As.
Res.' vol. vi. p. 90 ; * Journ. Ind. Archip.' vol. iii. p. 182, &c.

3 Kolle, ' Gr. of Vei Lang.' p. 27.

HAND AND FOOT NUMERALS. 351

actual name for the number, perhaps no language in the
world surpasses the Zulu. The Zulu counting on his
fingers begins in general with the little finger of his left
hand. When he comes to 5, this he may call edesanta
' finish hand ; ' then he goes on to the thumb of the right
hand, and so the word tatisitupa ' taking the thumb '
becomes a numeral for 6. Then the verb komba ' to point/
indicating the forefinger, or ' pointer,' makes the next
numeral, 7. Thus, answering the question ' How much
did your master give you ? ' a Zulu would say ' U kombile '
' He pointed with his forefinger/ i.e., ' He gave me
seven/ and this curious way of using the numeral verb is
shown in such an example as ' amahasi akombile ' ' the
horses have pointed/ i.e., ' there were seven of them/ In
like manner, Kijangalobili ' keep back two fingers/ i.e. 8,
and * Kijangalolunje ' keep back one finger/ i.e. 9, lead on
to kumi, 10 ; at the completion of each ten the two hands
with open fingers are clapped together. 1

The theory that man's primitive mode of counting was
palpable reckoning on his hands, and the proof that many
numerals in present use are actually derived from such a
state of things, is a great step towards discovering the origin
of numerals in general. Can we go farther, and state
broadly the mental process by which savage men, having no
numerals as yet in their language, came to invent them ?
What was the origin of numerals not named with reference
to hands and feet, and especially of the numerals below five,
to which such a derivation is hardly appropriate ? The
subject is a peculiarly difficult one. Yet as to principle it
is not altogether obscure, for some evidence is forth-
coming as to the actual formation of new numeral words,
these being made by simply pressing into the service
names of objects or actions in some way appropriate to the
purpose.

People possessing full sets of inherited numerals in their

1 Schreuder, ' Gr. for Zulu Sproget,' p. 30 ; Dohne, ' Zulu Die.' ; Grout,
4 Zulu Gr.' See Hahn, ' Gr. des Herero.'

252 THE ART OF COUNTING.

own languages have nevertheless sometimes found it con-
venient to invent new ones. Thus the scholars of India,
ages ago, selected a set of words from a memoria technica in
order to record dates and numbers. These words they chose
for reasons which are still in great measure evident ; thus
' moon ' or ' earth ' expressed i, 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 7, from the seven sages and the seven
vowels ; and so on with higher numbers, ' sun ' for 12,
because of his twelve annual denominations, or ' zodiac '
from its twelve signs, and ' nail ' for 20, a word incidentally
bringing in a finger notation. As Sanskrit is very rich in
synonyms, and as even the numerals themselves might be
used, it becomes very easy to draw up phrases or nonsense-
verses to record series of numbers by this system of arti-
ficial memory. The following is a Hindu astronomical
formula, a list of numbers referring to the stars of the lunar
constellations. Each word stands as the mnemonic equi-
valent of the number placed over it in the English trans-
lation. The general principle on which the words are
chosen to denote the numbers is evident without further
explanation :

' Vahni tri rtvishu gunendu kritagnibhuta
Banasvinetra ?ara bhuku yugabdhi ramah
Rudrabdhiramagunavedacata dviyugma
Danta budhairabhihitah kramago bhatarah.

.336531 4

i.e., ' Fire, three, season, arrow, quality, moon, four-side of die,

3 5

fire, element,

5 2251144 3

Arrow, Asvm, eye, arrow, earth, earth, age, ocean, Rama,

INVENTED NUMERALS. 253

ii 4 33 4 ioo 22

Rudra, ocean, Rama, quality, Veda, hundred, two, couple,

32

Teeth: by the wise have been set forth in order the mighty
lords/ 1

It occurred to Wilhelm von Humboldt, in studying this
curious system of numeration, that he had before his eyes
the evidence of a process very like that which actually pro-
duced the regular numeral words denoting one, two, three,
&c., in the various languages of the world. The following
passage in which, more than sixty years ago, he set forth
this view, seems to me to contain a nearly perfect key to
the theory of numeral words. ' If we take into considera-
tion the origin of actual numerals, the process of their
formation appears evidently to have been the same as that
here described. The latter is nothing else than a wider
extension of the former. For when 5 is expressed, as in
several languages of the Malay family, by " hand " (lima),
this is precisely the same thing as when in the description
of numbers by words, 2 is denoted by " wing." Indisput-
ably there lie at the root of all numerals such metaphors
as these, though they cannot always be now traced. But
people seem early to have felt that the multiplicity of such
signs for the same number was superfluous, too clumsy, and
leading to misunderstandings.' Therefore, he goes on to
argue, synonyms of numerals are very rare. And to
nations with a deep sense of language, the feeling must
soon have been present, though perhaps without rising to
distinct consciousness, that recollections of the original
etymology and descriptive meaning of numerals had best be
allowed to disappear, so as to leave the numerals themselves
to become mere conventional terms.

1 Sir W. Jones in 'As. Res.' vol. ii. 1790, p. 296 ; E. Jacquet in * Nouv.
Journ. Asiat.' 1835 > W. v. Humboldt, ' Kawi-Spr.' vol. i. p. 19. This
system of recording dates, &c., extended as far as Tibet and the Indian
Archipelago. Many important points of Oriental chronology depend on
such formulas. Unfortunately their evidence is more or less vitiated by
inconsistencies in the use of words for numbers.

254 THE ART F COUNTING.

The most instructive evidence I have found bearing on
the formation of numerals, other than digit-numerals,
among the lower races, appears in the use on both sides of
the globe of what may be called numeral-names for children.
In Australia a well-marked case occurs. With all the
poverty of the aboriginal languages in numerals, 3 being
commonly used as meaning ' several or many,' the natives
in the Adelaide district have for a particular purpose gone
far beyond this narrow limit, and possess what is to all
intents a special numeral system, extending perhaps to 9.
They give fixed names to their children in order of age,
which are set down as follows by Mr. Eyre : i. Kertameru ;
2. Warritya ; 3. Kudnutya ; 4. Monaitya ; 5. Milaitya ; 6.
Marrutya; 7. Wangutya ; 8. Ngarlaitya ; 9. Pouarna.
These are the male names, from which the female differ in
termination. They are given at birth, more distinctive
appellations being soon afterwards chosen. 1 A similar
habit makes its appearance among the Malays, who in some
districts are reported to use a series of seven names in order
of age, beginning with i. Sulung ('eldest'); 2. Awang
(' friend, companion '), and ending with Kechil (' little
one '), or Bongsu (' youngest '). These are for sons ;
daughters have Meh prefixed, and nicknames have to be
used for practical distinction. 2 In Madagascar, the Malay
connexion manifests itself in the appearance of a similar set
of appellations given to children in lieu of proper names,
which are, however, often substituted in after years.
Males ; Lahimatoa ( first male '), Lah-ivo ( intermediate
male ') ; Ra-fara-lahy (' last born male '). Females ;
Ramatoa ('eldest female'), Ra-ivo ('intermediate'), Ra-
fara-vavy ('last born female '). The system exists in

1 Eyre, ' Australia,' vol. ii. p. 324 : Shurmann, ' Vocab. of Parnkalla
Lang,' gives forms partially corresponding.

*'Journ. Ind. Archip.' New Ser. vol. ii. 1858, p. 118 [Sulong, Awang,
Itam (' black '), Puteh (' white '), Allang, Pendeh, Kechil or Bongsu] ; Bat-
tian, ' Oestl. Asien,' vol. ii. p. 494. The details are imperfectly given, and
seem not all correct.

3 Ellis, 'Madagascar,' vol. i. p. 154. Also Andriampaivo, or Lahi-Zan-

NUMERAL PERSONAL NAMES. 255

North America. There have been found in use among
the Dacotas the following two series of names for sons
and daughters in order of birth. Eldest son, Chaske ;
second, Haparm; third, Ha-pe-dah; fourth, Chatun ; fifth
Harka. Eldest daughter, Wenonah ; second, Harpen;
third, Harpstenah; fourth, Waska; fifth, We-harka. These
mere numeral appellations they retain through childhood,
till their relations or friends find occasion to replace them
by bestowing some more distinctive personal name. 1 Africa
affords further examples. 2

As to numerals in the ordinary sense, Polynesia shows
remarkable cases of new formation. Besides the well-
known system of numeral words prevalent in Polynesia,
exceptional terms have from time to time grown up. Thus
the habit of altering words which sounded too nearly like a
king's name, has led the Tahitians on the accession of new
chiefs to make several new words for numbers. Thus,
wanting a new term for 2 instead of the ordinary rua, they
for obvious reasons took up the word piti, ' together,' and
made it a numeral, while to get a new word for 5 instead of
rima, 'hand,' which had to be discontinued, they substi-
tuted pae, ' part, division,' meaning probably division of
the two hands. Such words as these, introduced in
Polynesia for ceremonial reasons, are expected to be
dropped again and the old ones replaced, when the reason
for their temporary exclusion ceases, yet the new 2 and 5,
piti and pae, became so positively the proper numerals of
the language, that they stand instead of rua and rima in the
Tahitian translation of the Gospel of St. John made at
the time. Again, various special habits of counting in the
South Sea Islands have had their effect on language. The
Marquesans, counting fish or fruit by one in each hand,

drina, for last male ; Andrianivo for intermediate male. Malagasy laby
4 male' = Malay laki ; Malagasy vavy, 4 female '= Tongan fafine, Maori
waking, ' woman ; ' comp. Malay batina, ' female.'

1 M. Eastman, ' Dahcotah ; or, Life and Legends of the Sioux,' p. xxv.

* ' Journ. Ethnol. Soc.' vol. iv. (Akra) ; Ploss, 4 Das Kind,' vol. i. p. 139
(Elmina).

256 THE ART OF COUNTING.

have come to use a system of counting by pairs instead of
by units. They start with tauna, ' a pair,' which thus
becomes a numeral equivalent to 2 ; then they count
onward by pairs, so that when they talk of takau or 10, they
really mean 10 pair or 20. For bread-fruit, as they are
accustomed to tie them up in knots of four, they begin with
the word pona, ' knot/ which thus becomes a real numeral
for 4, and here again they go on counting by knots, so that
when they say takau or 10, they mean 10 knots or 40.
The philological mystification thus caused in Polynesian
vocabularies is extraordinary ; in Tahitian, &c., ran and
mano, properly meaning 100 and 1,000, have come to
signify 200 and 2,000, while in Hawaii a second doubling
in their sense makes them equivalent to 400 and 4,000.
Moreover, it seems possible to trace the transfer of suitable
names of objects still farther in Polynesia in the Tongan
and Maori word tekau, 10, which seems to have been a
word for ' parcel ' or * bunch,' used in counting yams and
fish, as also in tefuhi, 100, derived from fuhi, ' sheaf or
bundle/ 1

In Africa, also, special numeral formations are to be
noticed. In the Yoruba language, 40 is called ogodzi, ' a
string/ because cowries are strung by forties, and 200 is
igba, ' a heap/ meaning again a heap of cowries. Among
the Dahomans in like manner, 40 cowries make a kade or
' string/ 50 strings make one afo or ' head ; ' these words
becoming numerals for 40 and 2,000, When the king of
Dahome attacked Abeokuta, it is on record that he was
repulsed with the heavy loss of ' two heads, twenty strings ,
and twenty cowries ' of men, that is to say, 4,820. 2

Among cultured nations, whose languages are most
tightly bound to the conventional arid unintelligible

1 H. Hale, ' Ethnography and Philology,' vol. vi. of Wilkes, U.S. Explor-
ing Exp., Philadelphia, 1846, pp. 172, 289. (N.B. The ordinary editions
do not contain this important volume.)

2 Bowen, ' Gr. and Die. of Yoruba.' Burton in ' Mem. Anthrop. Soc.,'
vol. i. p. 314.

VARIOUS NUMERAL TERMS. 257

numerals of their ancestors, it is likewise usual to find
other terms existing which are practically numerals already,
1 and might drop at once into the recognized places of such, if
I by any chance a gap were made for them in the traditional
i series. Had we room, for instance, for a new word instead
I of two, then either pair (Latin par, ' equal ') or couple
I (Latin copula, ' bond or tie/) is ready to fill its place.
1 Instead of twenty, the good English word score, ' notch/
i will serve our turn, while, for the same purpose, German
I can use stiege, possibly with the original sense of ' a stall
t full of cattle, a sty ; ' Old Norse drott, ' a company/
Danish, snees. A list of such words used, but not gram-
matically classed as numerals in European languages, shows
great variety : examples are, Old Norse, flockr (flock), 5 ;
sveit, 6 ; drott (party), 20 ; thiodh (people), 30 ; folk
(people), 40 ; old (people), 80 ; her (army), 100 ; Sleswig,
schilk, 12 (as though we were to make a numeral out of
' shilling') ; Middle High-German, rotte, 4 ; New High-
German, mandel, 15 ; schock (sheaf), 60. The Letts give a
curious parallel to Polynesian cases just cited. They
throw crabs and little fish three at a time in counting them,
and therefore the word mettens, ' a throw/ has come to
mean 3 ; while flounders being fastened in lots of thirty,
the word kahlis, ' a cord/ becomes a term to express this
number. x