Levi Leonard Conant - The Number Concept

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CHAPTER IV.

THE ORIGIN OF NUMBER WORDS.
(_CONTINUED_.)

By the slow, and often painful, process incident to the extension and
development of any mental conception in a mind wholly unused to
abstractions, the savage gropes his way onward in his counting from 1, or
more probably from 2, to the various higher numbers required to form his
scale. The perception of unity offers no difficulty to his mind, though he
is conscious at first of the object itself rather than of any idea of
number associated with it. The concept of duality, also, is grasped with
perfect readiness. This concept is, in its simplest form, presented to the
mind as soon as the individual distinguishes himself from another person,
though the idea is still essentially concrete. Perhaps the first glimmering
of any real number thought in connection with 2 comes when the savage
contrasts one single object with another--or, in other words, when he first
recognizes the _pair_. At first the individuals composing the pair are
simply "this one," and "that one," or "this and that"; and his number
system now halts for a time at the stage when he can, rudely enough it may
be, count 1, 2, many. There are certain cases where the forms of 1 and 2
are so similar than one may readily imagine that these numbers really were
"this" and "that" in the savage's original conception of them; and the same
likeness also occurs in the words for 3 and 4, which may readily enough
have been a second "this" and a second "that." In the Lushu tongue the
words for 1 and 2 are _tizi_ and _tazi_ respectively. In Koriak we find
_ngroka_, 3, and _ngraka_, 4; in Kolyma, _niyokh_, 3, and _niyakh_, 4; and
in Kamtschatkan, _tsuk_, 3, and _tsaak_, 4.[108] Sometimes, as in the case
of the Australian races, the entire extent of the count is carried through
by means of pairs. But the natural theory one would form is, that 2 is the
halting place for a very long time; that up to this point the fingers may
or may not have been used--probably not; and that when the next start is
made, and 3, 4, 5, and so on are counted, the fingers first come into
requisition. If the grammatical structure of the earlier languages of the
world's history is examined, the student is struck with the prevalence of
the dual number in them--something which tends to disappear as language
undergoes extended development. The dual number points unequivocally to the
time when 1 and 2 were _the_ numbers at mankind's disposal; to the time
when his three numeral concepts, 1, 2, many, each demanded distinct
expression. With increasing knowledge the necessity for this
differentiatuin would pass away, and but two numbers, singular and plural,
would remain. Incidentally it is to be noticed that the Indo-European words
for 3--_three_, _trois_, _drei_, _tres_, _tri,_ etc., have the same root as
the Latin _trans_, beyond, and give us a hint of the time when our Aryan
ancestors counted in the manner I have just described.

The first real difficulty which the savage experiences in counting, the
difficulty which comes when he attempts to pass beyond 2, and to count 3,
4, and 5, is of course but slight; and these numbers are commonly used and
readily understood by almost all tribes, no matter how deeply sunk in
barbarism we find them. But the instances that have already been cited must
not be forgotten. The Chiquitos do not, in their primitive state, properly
count at all; the Andamans, the Veddas, and many of the Australian tribes
have no numerals higher than 2; others of the Australians and many of the
South Americans stop with 3 or 4; and tribes which make 5 their limit are
still more numerous. Hence it is safe to assert that even this
insignificant number is not always reached with perfect ease. Beyond 5
primitive man often proceeds with the greatest difficulty. Most savages,
even those of the tribes just mentioned, can really count above here, even
though they have no words with which to express their thought. But they do
it with reluctance, and as they go on they quickly lose all sense of
accuracy. This has already been commented on, but to emphasize it afresh
the well-known example given by Mr. Oldfield from his own experience among
the Watchandies may be quoted.[109] "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 inquiry 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." This meagreness of knowledge in all things
pertaining to numbers is often found to be sharply emphasized in the names
adopted by savages for their numeral words. While discussing in a previous
chapter the limits of number systems, we found many instances where
anything above 2 or 3 was designated by some one of the comprehensive terms
_much_, _many_, _very many_; these words, or such equivalents as _lot_,
_heap_, or _plenty_, serving as an aid to the finger pantomime necessary to
indicate numbers for which they have no real names. The low degree of
intelligence and civilization revealed by such words is brought quite as
sharply into prominence by the word occasionally found for 5. Whenever the
fingers and hands are used at all, it would seem natural to expect for 5
some general expression signifying _hand_, for 10 _both hands_, and for 20
_man_. Such is, as we have already seen, the ordinary method of
progression, but it is not universal. A drop in the scale of civilization
takes us to a point where 10, instead of 20, becomes the whole man. The
Kusaies,[110] of Strong's Island, call 10 _sie-nul_, 1 man, 30 _tol-nul_, 3
men, 40 _a naul_, 4 men, etc.; and the Ku-Mbutti[111] of central Africa
have _mukko_, 10, and _moku_, man. If 10 is to be expressed by reference to
the man, instead of his hands, it might appear more natural to employ some
such expression as that adopted by the African Pigmies,[112] who call 10
_mabo_, and man _mabo-mabo_. With them, then, 10 is perhaps "half a man,"
as it actually is among the Towkas of South America; and we have already
seen that with the Aztecs it was _matlactli_, the "hand half" of a
man.[113] The same idea crops out in the expression used by the Nicobar
Islanders for 30--_heam-umdjome ruktei_, 1 man (and a) half.[114] Such
nomenclature is entirely natural, and it accords with the analogy offered
by other words of frequent occurrence in the numeral scales of savage
races. Still, to find 10 expressed by the term _man_ always conveys an
impression of mental poverty; though it may, of course, be urged that this
might arise from the fact that some races never use the toes in counting,
but go over the fingers again, or perhaps bring into requisition the
fingers of a second man to express the second 10. It is not safe to
postulate an extremely low degree of civilization from the presence of
certain peculiarities of numeral formation. Only the most general
statements can be ventured on, and these are always subject to modification
through some circumstance connected with environment, mode of living, or
intercourse with other tribes. Two South American races may be cited, which
seem in this respect to give unmistakable evidence of being sunk in deepest
barbarism. These are the Juri and the Cayriri, who use the same word for
man and for 5. The former express 5 by _ghomen apa_, 1 man,[115] and the
latter by _ibicho_, person.[116] The Tasmanians of Oyster Bay use the
native word of similar meaning, _puggana_, man,[117] for 5.

Wherever the numeral 20 is expressed by the term _man_, it may be expected
that 40 will be 2 men, 60, 3 men, etc. This form of numeration is usually,
though not always, carried as far as the system extends; and it sometimes
leads to curious terms, of which a single illustration will suffice. The
San Blas Indians, like almost all the other Central and South American
tribes, count by digit numerals, and form their twenties as follows:[118]

20. tula guena = man 1.
40. tula pogua = man 2.
100. tula atala = man 5.
120. tula nergua = man 6.
1000. tula wala guena = great 1 man.

The last expression may, perhaps, be translated "great hundred," though the
literal meaning is the one given. If 10, instead of 20, is expressed by the
word "man," the multiples of 10 follow the law just given for multiples of
20. This is sufficiently indicated by the Kusaie scale; or equally well by
the Api words for 100 and 200, which are[119]

_duulimo toromomo_ = 10 times the whole man.

_duulimo toromomo va juo_ = 10 times the whole man taken 2 times.

As an illustration of the legitimate result which is produced by the
attempt to express high numbers in this manner the term applied by educated
native Greenlanders[120] for a thousand may be cited. This numeral, which
is, of course, not in common use, is

_inuit kulit tatdlima nik kuleriartut navdlugit_ = 10 men 5 times 10
times come to an end.

It is worth noting that the word "great," which appears in the scale of the
San Blas Indians, is not infrequently made use of in the formation of
higher numeral words. The African Mabas[121] call 10 _atuk_, great 1; the
Hottentots[122] and the Hidatsa Indians call 100 great 10, their words
being _gei disi_ and _pitikitstia_ respectively.

The Nicaraguans[123] express 100 by _guhamba_, great 10, and 400 by
_dinoamba_, great 20; and our own familiar word "million," which so many
modern languages have borrowed from the Italian, is nothing more nor less
than a derivative of the Latin _mille_, and really means "great thousand."
The Dakota[124] language shows the same origin for its expression of
1,000,000, which is _kick ta opong wa tunkah_, great 1000. The origin of
such terms can hardly be ascribed to poverty of language. It is found,
rather, in the mental association of the larger with the smaller unit, and
the consequent repetition of the name of the smaller. Any unit, whether it
be a single thing, a dozen, a score, a hundred, a thousand, or any other
unit, is, whenever used, a single and complete group; and where the
relation between them is sufficiently close, as in our "gross" and "great
gross," this form of nomenclature is natural enough to render it a matter
of some surprise that it has not been employed more frequently. An old
English nursery rhyme makes use of this association, only in a manner
precisely the reverse of that which appears now and then in numeral terms.
In the latter case the process is always one of enlargement, and the
associative word is "great." In the following rhyme, constructed by the
mature for the amusement of the childish mind, the process is one of
diminution, and the associative word is "little":

One's none,
Two's some,
Three's a many,
Four's a penny,
Five's a little hundred.[125]

Any real numeral formation by the use of "little," with the name of some
higher unit, would, of course, be impossible. The numeral scale must be
complete before the nursery rhyme can be manufactured.

It is not to be supposed from the observations that have been made on the
formation of savage numeral scales that all, or even the majority of
tribes, proceed in the awkward and faltering manner indicated by many of
the examples quoted. Some of the North American Indian tribes have numeral
scales which are, as far as they go, as regular and almost as simple as our
own. But where digital numeration is extensively resorted to, the
expressions for higher numbers are likely to become complex, and to act as
a real bar to the extension of the system. The same thing is true, to an
even greater degree, of tribes whose number sense is so defective that they
begin almost from the outset to use combinations. If a savage expresses the
number 3 by the combination 2-1, it will at once be suspected that his
numerals will, by the time he reaches 10 or 20, become so complex and
confused that numbers as high as these will be expressed by finger
pantomime rather than by words. Such is often the case; and the comment is
frequently made by explorers that the tribes they have visited have no
words for numbers higher than 3, 4, 5, 10, or 20, but that counting is
carried beyond that point by the aid of fingers or other objects. So
reluctant, in many cases, are savages to count by words, that limits have
been assigned for spoken numerals, which subsequent investigation proved to
fall far short of the real extent of the number systems to which they
belonged. One of the south-western Indian tribes of the United States, the
Comanches, was for a time supposed to have no numeral words below 10, but
to count solely by the use of fingers. But the entire scale of this
taciturn tribe was afterward discovered and published.

To illustrate the awkward and inconvenient forms of expression which
abound in primitive numeral nomenclature, one has only to draw from such
scales as those of the Zuni, or the Point Barrow Eskimos, given in the
last chapter. Terms such as are found there may readily be duplicated
from almost any quarter of the globe. The Soussous of Sierra Leone[126]
call 99 _tongo solo manani nun solo manani_, _i.e._ to take (10
understood) 5 + 4 times and 5 + 4. The Malagasy expression for 1832
is[127] _roambistelo polo amby valonjato amby arivo_, 2 + 30 + 800 + 1000.
The Aztec equivalent for 399 is[128] _caxtolli onnauh poalli ipan caxtolli
onnaui_, (15 + 4) x 20 + 15 + 4; and the Sioux require for 29 the
ponderous combination[129] _wick a chimen ne nompah sam pah nep e chu wink
a._ These terms, long and awkward as they seem, are only the legitimate
results which arise from combining the names of the higher and lower
numbers, according to the peculiar genius of each language. From some of
the Australian tribes are derived expressions still more complex, as for
6, _marh-jin-bang-ga-gudjir-gyn_, half the hands and 1; and for 15,
_marh-jin-belli-belli-gudjir-jina-bang-ga_, the hand on either side and
half the feet.[130] The Mare tribe, one of the numerous island tribes of
Melanesia,[131] required for a translation of the numeral 38, which occurs
in John v. 5, "had an infirmity thirty and eight years," the
circumlocution, "one man and both sides five and three." Such expressions,
curious as they seem at first thought, are no more than the natural
outgrowth of systems built up by the slow and tedious process which so
often obtains among primitive races, where digit numerals are combined in
an almost endless variety of ways, and where mere reduplication often
serves in place of any independent names for higher units. To what extent
this may be carried is shown by the language of the Cayubabi,[132] who have
for 10 the word _tunca_, and for 100 and 1000 the compounds _tunca tunca_,
and _tunca tunca tunca_ respectively; or of the Sapibocones, who call 10
_bururuche_, hand hand, and 100 _buruche buruche_, hand hand hand
hand.[133] More remarkable still is the Ojibwa language, which continues
its numeral scale without limit, furnishing combinations which are really
remarkable; as, _e.g._, that for 1,000,000,000, which is _me das wac me das
wac as he me das wac_,[134] 1000 x 1000 x 1000. The Winnebago expression
for the same number,[135] _ho ke he hhuta hhu chen a ho ke he ka ra pa ne
za_ is no less formidable, but it has every appearance of being an honest,
native combination. All such primitive terms for larger numbers must,
however, be received with caution. Savages are sometimes eager to display a
knowledge they do not possess, and have been known to invent numeral words
on the spot for the sake of carrying their scales to as high a limit as
possible. The Choctaw words for million and billion are obvious attempts to
incorporate the corresponding English terms into their own language.[136]
For million they gave the vocabulary-hunter the phrase _mil yan chuffa_,
and for billion, _bil yan chuffa_. The word _chuffa_ signifies 1, hence
these expressions are seen at a glance to be coined solely for the purpose
of gratifying a little harmless Choctaw vanity. But this is innocence
itself compared with the fraud perpetrated on Labillardiere by the Tonga
Islanders, who supplied the astonished and delighted investigator with a
numeral vocabulary up to quadrillions. Their real limit was afterward found
to be 100,000, and above that point they had palmed off as numerals a
tolerably complete list of the obscene words of their language, together
with a few nonsense terms. These were all accepted and printed in good
faith, and the humiliating truth was not discovered until years
afterward.[137]

One noteworthy and interesting fact relating to numeral nomenclature is the
variation in form which words of this class undergo when applied to
different classes of objects. To one accustomed as we are to absolute and
unvarying forms for numerals, this seems at first a novel and almost
unaccountable linguistic freak. But it is not uncommon among uncivilized
races, and is extensively employed by so highly enlightened a people, even,
as the Japanese. This variation in form is in no way analogous to that
produced by inflectional changes, such as occur in Hebrew, Greek, Latin,
etc. It is sufficient in many cases to produce almost an entire change in
the form of the word; or to result in compounds which require close
scrutiny for the detection of the original root. For example, in the
Carrier, one of the Dene dialects of western Canada, the word _tha_ means 3
things; _thane_, 3 persons; _that_, 3 times; _thatoen_, in 3 places;
_thauh_, in 3 ways; _thailtoh_, all of the 3 things; _thahoeltoh_, all of
the 3 persons; and _thahultoh_, all of the 3 times.[138] In the Tsimshian
language of British Columbia we find seven distinct sets of numerals "which
are used for various classes of objects that are counted. The first set is
used in counting where there is no definite object referred to; the second
class is used for counting flat objects and animals; the third for counting
round objects and divisions of time; the fourth for counting men; the fifth
for counting long objects, the numerals being composed with _kan_, tree;
the sixth for counting canoes; and the seventh for measures. The last seem
to be composed with _anon_, hand."[139] The first ten numerals of each of
these classes is given in the following table:

+----+---------+---------+---------+----------+------------+-------------+-------------+
|No. |Counting | Flat | Round | Men | Long | Canoes | Measures |
| | | Objects | Objects | | Objects | | |
+----+---------+---------+---------+----------+------------+-------------+-------------+
| 1 |gyak gak |g'erel |k'al |k'awutskan|k'amaet |k'al | |
| 2 |t'epqat |t'epqat |goupel |t'epqadal |gaopskan |g'alp[=e]eltk|gulbel |
| 3 |guant |guant |gutle |gulal |galtskan |galtskantk |guleont |
| 4 |tqalpq |tqalpq |tqalpq |tqalpqdal |tqaapskan |tqalpqsk |tqalpqalont |
| 5 |kct[=o]nc|kct[=o]nc|kct[=o]nc|kcenecal |k'etoentskan|kct[=o]onsk |kctonsilont |
| 6 |k'alt |k'alt |k'alt |k'aldal |k'aoltskan |k'altk |k'aldelont |
| 7 |t'epqalt |t'epqalt |t'epqalt |t'epqaldal|t'epqaltskan|t'epqaltk |t'epqaldelont|
| 8 |guandalt |yuktalt |yuktalt |yuktleadal|ek'tlaedskan|yuktaltk |yuktaldelont |
| 9 |kctemac |kctemac |kctemac |kctemacal |kctemaestkan|kctemack |kctemasilont |
|10 |gy'ap |gy'ap |kp[=e]el |kpal |kp[=e]etskan|gy'apsk |kpeont |
+----+---------+---------+---------+----------+------------+-------------+-------------+

Remarkable as this list may appear, it is by no means as extensive as that
derived from many of the other British Columbian tribes. The numerals of
the Shushwap, Stlatlumh, Okanaken, and other languages of this region exist
in several different forms, and can also be modified by any of the
innumerable suffixes of these tongues.[140] To illustrate the almost
illimitable number of sets that may be formed, a table is given of "a few
classes, taken from the Heiltsuk dialect.[141] It appears from these
examples that the number of classes is unlimited."

+-----------------------+-------------+--------------+--------------+
| | One. | Two. | Three. |
+-----------------------+-------------+--------------+--------------+
|Animate. |menok |maalok |yutuk |
|Round. |menskam |masem |yutqsem |
|Long. |ments'ak |mats'ak |yututs'ak |
|Flat. |menaqsa |matlqsa |yutqsa |
|Day. |op'enequls |matlp'enequls |yutqp'enequls |
|Fathom. |op'enkh |matlp'enkh |yutqp'enkh |
|Grouped together. |---- |matloutl |yutoutl |
|Groups of objects. |nemtsmots'utl|matltsmots'utl|yutqtsmots'utl|
|Filled cup. |menqtlala |matl'aqtlala |yutqtlala |
|Empty cup. |menqtla |matl'aqtla |yutqtla |
|Full box. |menskamala |masemala |yutqsemala |
|Empty box. |menskam |masem |yutqsem |
|Loaded canoe. |mentsake |mats'ake |yututs'ake |
|Canoe with crew. |ments'akis |mats'akla |yututs'akla |
|Together on beach. |---- |maalis |---- |
|Together in house, etc.|---- |maalitl |---- |
+-----------------------+-------------+--------------+--------------+

Variation in numeral forms such as is exhibited in the above tables is not
confined to any one quarter of the globe; but it is more universal among
the British Columbian Indians than among any other race, and it is a more
characteristic linguistic peculiarity of this than of any other region,
either in the Old World or in the New. It was to some extent employed by
the Aztecs,[142] and its use is current among the Japanese; in whose
language Crawfurd finds fourteen different classes of numerals "without
exhausting the list."[143]

In examining the numerals of different languages it will be found that the
tens of any ordinary decimal scale are formed in the same manner as in
English. Twenty is simply 2 times 10; 30 is 3 times 10, and so on. The word
"times" is, of course, not expressed, any more than in English; but the
expressions briefly are, 2 tens, 3 tens, etc. But a singular exception to
this method is presented by the Hebrew, and other of the Semitic languages.
In Hebrew the word for 20 is the plural of the word for 10; and 30, 40, 50,
etc. to 90 are plurals of 3, 4, 5, 6, 7, 8, 9. These numerals are as
follows:[144]

10, eser, 20, eserim,
3, shalosh, 30, shaloshim,
4, arba, 40, arbaim,
5, chamesh, 50, chamishshim,
6, shesh, 60, sheshshim,
7, sheba, 70, shibim,
8, shemoneh 80, shemonim,
9, tesha, 90, tishim.

The same formation appears in the numerals of the ancient Phoenicians,[145]
and seems, indeed, to be a well-marked characteristic of the various
branches of this division of the Caucasian race. An analogous method
appears in the formation of the tens in the Bisayan,[146] one of the Malay
numeral scales, where 30, 40, ... 90, are constructed from 3, 4, ... 9, by
adding the termination _-an_.

No more interesting contribution has ever been made to the literature of
numeral nomenclature than that in which Dr. Trumbull embodies the results
of his scholarly research among the languages of the native Indian tribes
of this country.[147] As might be expected, we are everywhere confronted
with a digital origin, direct or indirect, in the great body of the words
examined. But it is clearly shown that such a derivation cannot be
established for all numerals; and evidence collected by the most recent
research fully substantiates the position taken by Dr. Trumbull. Nearly all
the derivations established are such as to remind us of the meanings we
have already seen recurring in one form or another in language after
language. Five is the end of the finger count on one hand--as, the Micmac
_nan_, and Mohegan _nunon_, gone, or spent; the Pawnee _sihuks_, hands
half; the Dakota _zaptan_, hand turned down; and the Massachusetts
_napanna_, on one side. Ten is the end of the finger count, but is not
always expressed by the "both hands" formula so commonly met with. The Cree
term for this number is _mitatat_, no further; and the corresponding word
in Delaware is _m'tellen_, no more. The Dakota 10 is, like its 5, a
straightening out of the fingers which have been turned over in counting,
or _wickchemna_, spread out unbent. The same is true of the Hidatsa
_pitika_, which signifies a smoothing out, or straightening. The Pawnee 4,
_skitiks_, is unusual, signifying as it does "all the fingers," or more
properly, "the fingers of the hand." The same meaning attaches to this
numeral in a few other languages also, and reminds one of the habit some
people have of beginning to count on the forefinger and proceeding from
there to the little finger. Can this have been the habit of the tribes in
question? A suggestion of the same nature is made by the Illinois and Miami
words for 8, _parare_ and _polane_, which signify "nearly ended." Six is
almost always digital in origin, though the derivation may be indirect, as
in the Illinois _kakatchui_, passing beyond the middle; and the Dakota
_shakpe_, 1 in addition. Some of these significations are well matched by
numerals from the Ewe scales of western Africa, where we find the
following:[148]

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