THE ORIGIN OF NUMBER WORDS.
In the comparison of languages and the search for primitive root forms, no
class of expressions has been subjected to closer scrutiny than the little
cluster of words, found in each language, which constitutes a part of the
daily vocabulary of almost every human being--the words with which we begin
our counting. It is assumed, and with good reason, that these are among the
earlier words to appear in any language; and in the mutations of human
speech, they are found to suffer less than almost any other portion of a
language. Kinship between tongues remote from each other has in many
instances been detected by the similarity found to exist among the
every-day words of each; and among these words one may look with a good
degree of certainty for the 1, 2, 3, etc., of the number scale. So fruitful
has been this line of research, that the attempt has been made, even, to
establish a common origin for all the races of mankind by means of a
comparison of numeral words.[51] But in this instance, as in so many others
that will readily occur to the mind, the result has been that the theory
has finally taken possession of the author and reduced him to complete
subjugation, instead of remaining his servant and submitting to the
legitimate results of patient and careful investigation. Linguistic
research is so full of snares and pitfalls that the student must needs
employ the greatest degree of discrimination before asserting kinship of
race because of resemblances in vocabulary; or even relationship between
words in the same language because of some chance likeness of form that may
exist between them. Probably no one would argue that the English and the
Babusessé of Central Africa were of the same primitive stock simply because
in the language of the latter _five atano_ means 5, and _ten kumi_ means
10.[52] But, on the other hand, many will argue that, because the German
_zehn_ means 10, and _zehen_ means toes, the ancestors of the Germans
counted on their toes; and that with them, 10 was the complete count of the
toes. It may be so. We certainly have no evidence with which to disprove
this; but, before accepting it as a fact, or even as a reasonable
hypothesis, we may be pardoned for demanding some evidence aside from the
mere resemblance in the form of the words. If, in the study of numeral
words, form is to constitute our chief guide, we must expect now and then
to be confronted with facts which are not easily reconciled with any pet
theory.
The scope of the present work will admit of no more than a hasty
examination of numeral forms, in which only actual and well ascertained
meanings will be considered. But here we are at the outset confronted with
a class of words whose original meanings appear to be entirely lost. They
are what may be termed the numerals proper--the native, uncompounded words
used to signify number. Such words are the one, two, three, etc., of
English; the eins, zwei, drei, etc., of German; words which must at some
time, in some prehistoric language, have had definite meanings entirely
apart from those which they now convey to our minds. In savage languages it
is sometimes possible to detect these meanings, and thus to obtain
possession of the clue that leads to the development, in the barbarian's
rude mind, of a count scale--a number system. But in languages like those
of modern Europe, the pedigree claimed by numerals is so long that, in the
successive changes through which they have passed, all trace of their
origin seems to have been lost.
The actual number of such words is, however, surprisingly small in any
language. In English we count by simple words only to 10. From this point
onward all our numerals except "hundred" and "thousand" are compounds and
combinations of the names of smaller numbers. The words we employ to
designate the higher orders of units, as million, billion, trillion, etc.,
are appropriated bodily from the Italian; and the native words _pair_,
_tale_, _brace_, _dozen_, _gross_, and _score_, can hardly be classed as
numerals in the strict sense of the word. German possesses exactly the same
number of native words in its numeral scale as English; and the same may be
said of the Teutonic languages generally, as well as of the Celtic, the
Latin, the Slavonic, and the Basque. This is, in fact, the universal method
observed in the formation of any numeral scale, though the actual number of
simple words may vary. The Chiquito language has but one numeral of any
kind whatever; English contains twelve simple terms; Sanskrit has
twenty-seven, while Japanese possesses twenty-four, and the Chinese a
number almost equally great. Very many languages, as might be expected,
contain special numeral expressions, such as the German _dutzend_ and the
French _dizaine_; but these, like the English _dozen_ and _score_, are not
to be regarded as numerals proper.
The formation of numeral words shows at a glance the general method in
which any number scale has been built up. The primitive savage counts on
his fingers until he has reached the end of one, or more probably of both,
hands. Then, if he wishes to proceed farther, some mark is made, a pebble
is laid aside, a knot tied, or some similar device employed to signify that
all the counters at his disposal have been used. Then the count begins
anew, and to avoid multiplication of words, as well as to assist the
memory, the terms already used are again resorted to; and the name by which
the first halting-place was designated is repeated with each new numeral.
Hence the thirteen, fourteen, fifteen, etc., which are contractions of the
fuller expressions three-and-ten, four-and-ten, five-and-ten, etc. The
specific method of combination may not always be the same, as witness the
_eighteen_, or eight-ten, in English, and _dix-huit,_ or ten-eight, in
French; _forty-five_, or four-tens-five, in English, and _fünf und
vierzig_, or five and four tens in German. But the general method is the
same the world over, presenting us with nothing but local variations, which
are, relatively speaking, entirely unimportant. With this fact in mind, we
can cease to wonder at the small number of simple numerals in any language.
It might, indeed, be queried, why do any languages, English and German, for
example, have unusual compounds for 11 and 12? It would seem as though the
regular method of compounding should begin with 10 and 1, instead of 10 and
3, in any language using a system with 10 as a base. An examination of
several hundred numeral scales shows that the Teutonic languages are
somewhat exceptional in this respect. The words _eleven_ and _twelve_ are
undoubtedly combinations, but not in the same direct sense as _thirteen_,
_twenty-five_, etc. The same may be said of the French _onze_, _douze_,
_treize_, _quatorze_, _quinze_, and _seize_, which are obvious compounds,
but not formed in the same manner as the numerals above that point. Almost
all civilized languages, however, except the Teutonic, and practically all
uncivilized languages, begin their direct numeral combinations as soon as
they have passed their number base, whatever that may be. To give an
illustration, selected quite at random from among the barbarous tribes of
Africa, the Ki-Swahili numeral scale runs as follows:[53]
1. moyyi,
2. mbiri,
3. tato,
4. ena,
5. tano,
6. seta,
7. saba,
8. nani,
9. kenda,
10. kumi,
11. kumi na moyyi,
12. kumi na mbiri,
13. kumi na tato,
etc.
The words for 11, 12, and 13, are seen at a glance to signify ten-and-one,
ten-and-two, ten-and-three, and the count proceeds, as might be inferred,
in a similar manner as far as the number system extends. Our English
combinations are a little closer than these, and the combinations found in
certain other languages are, in turn, closer than those of the English; as
witness the _once_, 11, _doce_, 12, _trece_, 13, etc., of Spanish. But the
process is essentially the same, and the law may be accepted as practically
invariable, that all numerals greater than the base of a system are
expressed by compound words, except such as are necessary to establish some
new order of unit, as hundred or thousand.
In the scale just given, it will be noticed that the larger number precedes
the smaller, giving 10 + 1, 10 + 2, etc., instead of 1 + 10, 2 + 10, etc.
This seems entirely natural, and hardly calls for any comment whatever. But
we have only to consider the formation of our English "teens" to see that
our own method is, at its inception, just the reverse of this. Thirteen,
14, and the remaining numerals up to 19 are formed by prefixing the smaller
number to the base; and it is only when we pass 20 that we return to the
more direct and obvious method of giving precedence to the larger. In
German and other Teutonic languages the inverse method is continued still
further. Here 25 is _fünf und zwanzig_, 5 and 20; 92 is _zwei und neunzig_,
2 and 90, and so on to 99. Above 100 the order is made direct, as in
English. Of course, this mode of formation between 20 and 100 is
permissible in English, where "five and twenty" is just as correct a form
as twenty-five. But it is archaic, and would soon pass out of the language
altogether, were it not for the influence of some of the older writings
which have had a strong influence in preserving for us many of older and
more essentially Saxon forms of expression.
Both the methods described above are found in all parts of the world, but
what I have called the direct is far more common than the other. In
general, where the smaller number precedes the larger it signifies
multiplication instead of addition. Thus, when we say "thirty," _i.e._
three-ten, we mean 3 × 10; just as "three hundred" means 3 × 100. When the
larger precedes the smaller, we must usually understand addition. But to
both these rules there are very many exceptions. Among higher numbers the
inverse order is very rarely used; though even here an occasional exception
is found. The Taensa Indians, for example, place the smaller numbers before
the larger, no matter how far their scale may extend. To say 1881 they make
a complete inversion of our own order, beginning with 1 and ending with
1000. Their full numeral for this is _yeha av wabki mar-u-wab mar-u-haki_,
which means, literally, 1 + 80 + 100 × 8 + 100 × 10.[54] Such exceptions
are, however, quite rare.
One other method of combination, that of subtraction, remains to be
considered. Every student of Latin will recall at once the _duodeviginti_,
2 from 20, and _undeviginti_, 1 from 20, which in that language are the
regular forms of expression for 18 and 19. At first they seem decidedly
odd; but familiarity soon accustoms one to them, and they cease entirely to
attract any special attention. This principle of subtraction, which, in the
formation of numeral words, is quite foreign to the genius of English, is
still of such common occurrence in other languages that the Latin examples
just given cease to be solitary instances.
The origin of numerals of this class is to be found in the idea of
reference, not necessarily to the last, but to the nearest, halting-point
in the scale. Many tribes seem to regard 9 as "almost 10," and to give it a
name which conveys this thought. In the Mississaga, one of the numerous
Algonquin languages, we have, for example, the word _cangaswi_, "incomplete
10," for 9.[55] In the Kwakiutl of British Columbia, 8 as well as 9 is
formed in this way; these two numbers being _matlguanatl_, 10 - 2, and
_nanema_, 10 - 1, respectively.[56] In many of the languages of British
Columbia we find a similar formation for 8 and 9, or for 9 alone. The same
formation occurs in Malay, resulting in the numerals _delapan_, 10 - 2, and
_sambilan_ 10 - 1.[57] In Green Island, one of the New Ireland group, these
become simply _andra-lua_, "less 2," and _andra-si_, "less 1."[58] In the
Admiralty Islands this formation is carried back one step further, and not
only gives us _shua-luea_, "less 2," and _shu-ri_, "less 1," but also makes
7 appear as _sua-tolu_, "less 3."[59] Surprising as this numeral is, it is
more than matched by the Ainu scale, which carries subtraction back still
another step, and calls 6, 10 - 4. The four numerals from 6 to 9 in this
scale are respectively, _iwa_, 10 - 4, _arawa_, 10 - 3, _tupe-san_, 10 - 2,
and _sinepe-san_, 10 - 1.[60] Numerous examples of this kind of formation
will be found in later chapters of this work; but they will usually be
found to occur in one or both of the numerals, 8 and 9. Occasionally they
appear among the higher numbers; as in the Maya languages, where, for
example, 99 years is "one single year lacking from five score years,"[61]
and in the Arikara dialects, where 98 and 99 are "5 men minus" and "5 men 1
not."[62] The Welsh, Danish, and other languages less easily accessible
than these to the general student, also furnish interesting examples of a
similar character.
More rarely yet are instances met with of languages which make use of
subtraction almost as freely as addition, in the composition of numerals.
Within the past few years such an instance has been noticed in the case of
the Bellacoola language of British Columbia. In their numeral scale 15,
"one foot," is followed by 16, "one man less 4"; 17, "one man less 3"; 18,
"one man less 2"; 19, "one man less 1"; and 20, one man. Twenty-five is
"one man and one hand"; 26, "one man and two hands less 4"; 36, "two men
less 4"; and so on. This method of formation prevails throughout the entire
numeral scale.[63]
One of the best known and most interesting examples of subtraction as
a well-defined principle of formation is found in the Maya scale. Up
to 40 no special peculiarity appears; but as the count progresses beyond
that point we find a succession of numerals which one is almost tempted
to call 60 - 19, 60 - 18, 60 - 17, etc. Literally translated the meanings
seem to be 1 to 60, 2 to 60, 3 to 60, etc. The point of reference is 60,
and the thought underlying the words may probably be expressed by the
paraphrases, "1 on the third score, 2 on the third score, 3 on the third
score," etc. Similarly, 61 is 1 on the fourth score, 81 is one on the
fifth score, 381 is 1 on the nineteenth score, and so on to 400. At 441
the same formation reappears; and it continues to characterize the system
in a regular and consistent manner, no matter how far it is extended.[64]
The Yoruba language of Africa is another example of most lavish use of
subtraction; but it here results in a system much less consistent and
natural than that just considered. Here we find not only 5, 10, and 20
subtracted from the next higher unit, but also 40, and even 100. For
example, 360 is 400 - 40; 460 is 500 - 40; 500 is 600 - 100; 1300 is
1400 - 100, etc. One of the Yoruba units is 200; and all the odd hundreds
up to 2000, the next higher unit, are formed by subtracting 100 from the
next higher multiple of 200. The system is quite complex, and very
artificial; and seems to have been developed by intercourse with
traders.[65]
It has already been stated that the primitive meanings of our own simple
numerals have been lost. This is also true of the languages of nearly all
other civilized peoples, and of numerous savage races as well. We are at
liberty to suppose, and we do suppose, that in very many cases these words
once expressed meanings closely connected with the names of the fingers, or
with the fingers themselves, or both. Now and then a case is met with in
which the numeral word frankly avows its meaning--as in the Botocudo
language, where 1 is expressed by _podzik_, finger, and 2 by _kripo_,
double finger;[66] and in the Eskimo dialect of Hudson's Bay, where
_eerkitkoka_ means both 10 and little finger.[67] Such cases are, however,
somewhat exceptional.
In a few noteworthy instances, the words composing the numeral scale of a
language have been carefully investigated and their original meanings
accurately determined. The simple structure of many of the rude languages
of the world should render this possible in a multitude of cases; but
investigators are too often content with the mere numerals themselves, and
make no inquiry respecting their meanings. But the following exposition of
the Zuñi scale, given by Lieutenant Gushing[68] leaves nothing to be
desired:
1. töpinte = taken to start with.
2. kwilli = put down together with.
3. ha'[=i] = the equally dividing finger.
4. awite = all the fingers all but done with.
5. öpte = the notched off.
This finishes the list of original simple numerals, the Zuñi stopping, or
"notching off," when he finishes the fingers of one hand. Compounding now
begins.
6. topalïk'ya = another brought to add to the done with.
7. kwillilïk'ya = two brought to and held up with the rest.
8. hailïk'ye = three brought to and held up with the rest.
9. tenalïk'ya = all but all are held up with the rest.
10. ästem'thila = all the fingers.
11. ästem'thla topayä'thl'tona = all the fingers and another over
above held.
The process of formation indicated in 11 is used in the succeeding numerals
up to 19.
20. kwillik'yënästem'thlan = two times all the fingers.
100. ässiästem'thlak'ya = the fingers all the fingers.
1000. ässiästem'thlanak'yënästem'thla = the fingers all the fingers
times all the fingers.
The only numerals calling for any special note are those for 11 and 9. For
9 we should naturally expect a word corresponding in structure and meaning
to the words for 7 and 8. But instead of the "four brought to and held up
with the rest," for which we naturally look, the Zuñi, to show that he has
used all of his fingers but one, says "all but all are held up with the
rest." To express 11 he cannot use a similar form of composition, since he
has already used it in constructing his word for 6, so he says "all the
fingers and another over above held."
The one remarkable point to be noted about the Zuñi scale is, after all,
the formation of the words for 1 and 2. While the savage almost always
counts on his fingers, it does not seem at all certain that these words
would necessarily be of finger formation. The savage can always distinguish
between one object and two objects, and it is hardly reasonable to believe
that any external aid is needed to arrive at a distinct perception of this
difference. The numerals for 1 and 2 would be the earliest to be formed in
any language, and in most, if not all, cases they would be formed long
before the need would be felt for terms to describe any higher number. If
this theory be correct, we should expect to find finger names for numerals
beginning not lower than 3, and oftener with 5 than with any other number.
The highest authority has ventured the assertion that all numeral words
have their origin in the names of the fingers;[69] substantially the same
conclusion was reached by Professor Pott, of Halle, whose work on numeral
nomenclature led him deeply into the study of the origin of these words.
But we have abundant evidence at hand to show that, universal as finger
counting has been, finger origin for numeral words has by no means been
universal. That it is more frequently met with than any other origin is
unquestionably true; but in many instances, which will be more fully
considered in the following chapter, we find strictly non-digital
derivations, especially in the case of the lowest members of the scale. But
in nearly all languages the origin of the words for 1, 2, 3, and 4 are so
entirely unknown that speculation respecting them is almost useless.
An excellent illustration of the ordinary method of formation which obtains
among number scales is furnished by the Eskimos of Point Barrow,[70] who
have pure numeral words up to 5, and then begin a systematic course of word
formation from the names of their fingers. If the names of the first five
numerals are of finger origin, they have so completely lost their original
form, or else the names of the fingers themselves have so changed, that no
resemblance is now to be detected between them. This scale is so
interesting that it is given with considerable fulness, as follows:
1. atauzik.
2. madro.
3. pinasun.
4. sisaman.
5. tudlemut.
6. atautyimin akbinigin [tudlimu(t)] = 5 and 1 on the next.
7. madronin akbinigin = twice on the next.
8. pinasunin akbinigin = three times on the next.
9. kodlinotaila = that which has not its 10.
10. kodlin = the upper part--_i.e._ the fingers.
14. akimiaxotaityuna = I have not 15.
15. akimia. [This seems to be a real numeral word.]
20. inyuina = a man come to an end.
25. inyuina tudlimunin akbinidigin = a man come to an end and 5 on the
next.
30. inyuina kodlinin akbinidigin = a man come to an end and 10 on the
next.
35. inyuina akimiamin aipalin = a man come to an end accompanied by 1
fifteen times.
40. madro inyuina = 2 men come to an end.
In this scale we find the finger origin appearing so clearly and so
repeatedly that one feels some degree of surprise at finding 5 expressed by
a pure numeral instead of by some word meaning _hand_ or _fingers of one
hand_. In this respect the Eskimo dialects are somewhat exceptional among
scales built up of digital words. The system of the Greenland Eskimos,
though differing slightly from that of their Point Barrow cousins, shows
the same peculiarity. The first ten numerals of this scale are:[71]
1. atausek.
2. mardluk.
3. pingasut.
4. sisamat.
5. tatdlimat.
6. arfinek-atausek = to the other hand 1.
7. arfinek-mardluk = to the other hand 2.
8. arfinek-pingasut = to the other hand 3.
9. arfinek-sisamat = to the other hand 4.
10. kulit.
The same process is now repeated, only the feet instead of the hands are
used; and the completion of the second 10 is marked by the word _innuk_,
man. It may be that the Eskimo word for 5 is, originally, a digital word,
but if so, the fact has not yet been detected. From the analogy furnished
by other languages we are justified in suspecting that this may be the
case; for whenever a number system contains digital words, we expect them
to begin with _five_, as, for example, in the Arawak scale,[72] which runs:
1. abba.
2. biama.
3. kabbuhin.
4. bibiti.
5. abbatekkábe = 1 hand.
6. abbatiman = 1 of the other.
7. biamattiman = 2 of the other.
8. kabbuhintiman = 3 of the other.
9. bibitiman = 4 of the other.
10. biamantekábbe = 2 hands.
11. abba kutihibena = 1 from the feet.
20. abba lukku = hands feet.
The four sets of numerals just given may be regarded as typifying one of
the most common forms of primitive counting; and the words they contain
serve as illustrations of the means which go to make up the number scales
of savage races. Frequently the finger and toe origin of numerals is
perfectly apparent, as in the Arawak system just given, which exhibits the
simplest and clearest possible method of formation. Another even more
interesting system is that of the Montagnais of northern Canada.[73] Here,
as in the Zuñi scale, the words are digital from the outset.
1. inl'are = the end is bent.
2. nak'e = another is bent.
3. t'are = the middle is bent.
4. dinri = there are no more except this.
5. se-sunla-re = the row on the hand.
6. elkke-t'are = 3 from each side.
7.{ t'a-ye-oyertan = there are still 3 of them.
{ inl'as dinri = on one side there are 4 of them.
8. elkke-dinri = 4 on each side.
9. inl'a-ye-oyert'an = there is still 1 more.
10. onernan = finished on each side.
11. onernan inl'are ttcharidhel = 1 complete and 1.
12. onernan nak'e ttcharidhel = 1 complete and 2, etc.
The formation of 6, 7, and 8 of this scale is somewhat different from that
ordinarily found. To express 6, the Montagnais separates the thumb and
forefinger from the three remaining fingers of the left hand, and bringing
the thumb of the right hand close to them, says: "3 from each side." For 7
he either subtracts from 10, saying: "there are still 3 of them," or he
brings the thumb and forefinger of the right hand up to the thumb of the
left, and says: "on one side there are 4 of them." He calls 8 by the same
name as many of the other Canadian tribes, that is, two 4's; and to show
the proper number of fingers, he closes the thumb and little finger of the
right hand, and then puts the three remaining fingers beside the thumb of
the left hand. This method is, in some of these particulars, different from
any other I have ever examined.
It often happens that the composition of numeral words is less easily
understood, and the original meanings more difficult to recover, than in
the examples already given. But in searching for number systems which show
in the formation of their words the influence of finger counting, it is not
unusual to find those in which the derivation from native words signifying
_finger, hand, toe, foot_, and _man_, is just as frankly obvious as in the
case of the Zuñi, the Arawak, the Eskimo, or the Montagnais scale. Among
the Tamanacs,[74] one of the numerous Indian tribes of the Orinoco, the
numerals are as strictly digital as in any of the systems already examined.
The general structure of the Tamanac scale is shown by the following
numerals:
5. amgnaitone = 1 hand complete.
6. itacono amgna pona tevinitpe = 1 on the other hand.
10. amgna aceponare = all of the 2 hands.
11. puitta pona tevinitpe = 1 on the foot.
16. itacono puitta pona tevinitpe = 1 on the other foot.
20. tevin itoto = 1 man.
21. itacono itoto jamgnar bona tevinitpe = 1 on the hands of another
man.
In the Guarani[75] language of Paraguay the same method is found, with a
different form of expression for 20. Here the numerals in question are
5. asepopetei = one hand.
10. asepomokoi = two hands.
20. asepo asepi abe = hands and feet.
Another slight variation is furnished by the Kiriri language,[76] which is
also one of the numerous South American Indian forms of speech, where we
find the words to be
5. mi biche misa = one hand.
10. mikriba misa sai = both hands.
20. mikriba misa idecho ibi sai = both hands together with the feet.
Illustrations of this kind might be multiplied almost indefinitely; and it
is well to note that they may be drawn from all parts of the world. South
America is peculiarly rich in native numeral words of this kind; and, as
the examples above cited show, it is the field to which one instinctively
turns when this subject is under discussion. The Zamuco numerals are, among
others, exceedingly interesting, giving us still a new variation in method.
They are[77]
1. tsomara.
2. gar.
3. gadiok.
4. gahagani.
5. tsuena yimana-ite = ended 1 hand.
6. tsomara-hi = 1 on the other.
7. gari-hi = 2 on the other.
8. gadiog-ihi = 3 on the other.
9. gahagani-hi = 4 on the other.
10. tsuena yimana-die = ended both hands.
11. tsomara yiri-tie = 1 on the foot.
12. gar yiritie = 2 on the foot.
20. tsuena yiri-die = ended both feet.
As is here indicated, the form of progression from 5 to 10, which we should
expect to be "hand-1," or "hand-and-1," or some kindred expression,
signifying that one hand had been completed, is simply "1 on the other."
Again, the expressions for 11, 12, etc., are merely "1 on the foot," "2 on
the foot," etc., while 20 is "both feet ended."
An equally interesting scale is furnished by the language of the
Maipures[78] of the Orinoco, who count
1. papita.
2. avanume.
3. apekiva.
4. apekipaki.
5. papitaerri capiti = 1 only hand.
6. papita yana pauria capiti purena = 1 of the other hand we take.
10. apanumerri capiti = 2 hands.
11. papita yana kiti purena = 1 of the toes we take.
20. papita camonee = 1 man.
40. avanume camonee = 2 men.
60. apekiva camonee = 3 men, etc.
In all the examples thus far given, 20 is expressed either by the
equivalent of "man" or by some formula introducing the word "feet." Both
these modes of expressing what our own ancestors termed a "score," are so
common that one hesitates to say which is of the more frequent use. The
following scale, from one of the Betoya dialects[79] of South America, is
quite remarkable among digital scales, making no use of either "man" or
"foot," but reckoning solely by fives, or hands, as the numerals indicate.
1. tey.
2. cayapa.
3. toazumba.
4. cajezea = 2 with plural termination.
5. teente = hand.
6. teyentetey = hand + 1.
7. teyente cayapa = hand + 2.
8. teyente toazumba = hand + 3.
9. teyente caesea = hand + 4.
10. caya ente, or caya huena = 2 hands.
11. caya ente-tey = 2 hands + 1.
15. toazumba-ente = 3 hands.
16. toazumba-ente-tey = 3 hands + 1.
20. caesea ente = 4 hands.
In the last chapter mention was made of the scanty numeral systems of the
Australian tribes, but a single scale was alluded to as reaching the
comparatively high limit of 20. This system is that belonging to the
Pikumbuls,[80] and the count runs thus:
1. mal.
2. bular.
3. guliba.
4. bularbular = 2-2.
5. mulanbu.
6. malmulanbu mummi = 1 and 5 added on.
7. bularmulanbu mummi = 2 and 5 added on.
8. gulibamulanbu mummi = 3 and 5 added on.
9. bularbularmulanbu mummi = 4 and 5 added on.
10. bularin murra = belonging to the 2 hands.
11. maldinna mummi = 1 of the toes added on (to the 10 fingers).
12. bular dinna mummi = 2 of the toes added on.
13. guliba dinna mummi = 3 of the toes added on.
14. bular bular dinna mummi = 4 of the toes added on.
15. mulanba dinna = 5 of the toes added on.
16. mal dinna mulanbu = 1 and 5 toes.
17. bular dinna mulanbu = 2 and 5 toes.
18. guliba dinna mulanbu = 3 and 5 toes.
19. bular bular dinna mulanbu = 4 and 5 toes.
20. bularin dinna = belonging to the 2 feet.
As has already been stated, there is good ground for believing that this
system was originally as limited as those obtained from other Australian
tribes, and that its extension from 4, or perhaps from 5 onward, is of
comparatively recent date.
A somewhat peculiar numeral nomenclature is found in the language of the
Klamath Indians of Oregon. The first ten words in the Klamath scale
are:[81]
1. nash, or nas.
2. lap = hand.
3. ndan.
4. vunep = hand up.
5. tunep = hand away.
6. nadshkshapta = 1 I have bent over.
7. lapkshapta = 2 I have bent over.
8. ndankshapta = 3 I have bent over.
9. nadshskeksh = 1 left over.
10. taunep = hand hand?
In describing this system Mr. Gatschet says: "If the origin of the Klamath
numerals is thus correctly traced, their inventors must have counted only
the four long fingers without the thumb, and 5 was counted while saying
_hand away! hand off!_ The 'four,' or _hand high! hand up!_ intimates that
the hand was held up high after counting its four digits; and some term
expressing this gesture was, in the case of _nine_, substituted by 'one
left over' ... which means to say, 'only one is left until all the fingers
are counted.'" It will be observed that the Klamath introduces not only the
ordinary finger manipulation, but a gesture of the entire hand as well. It
is a common thing to find something of the kind to indicate the completion
of 5 or 10, and in one or two instances it has already been alluded to.
Sometimes one or both of the closed fists are held up; sometimes the open
hand, with all the fingers extended, is used; and sometimes an entirely
independent gesture is introduced. These are, in general, of no special
importance; but one custom in vogue among some of the prairie tribes of
Indians, to which my attention was called by Dr. J. Owen Dorsey,[82] should
be mentioned. It is a gesture which signifies multiplication, and is
performed by throwing the hand to the left. Thus, after counting 5, a wave
of the hand to the left means 50. As multiplication is rather unusual among
savage tribes, this is noteworthy, and would seem to indicate on the part
of the Indian a higher degree of intelligence than is ordinarily possessed
by uncivilized races.
In the numeral scale as we possess it in English, we find it necessary to
retain the name of the last unit of each kind used, in order to describe
definitely any numeral employed. Thus, fifteen, one hundred forty-two, six
thousand seven hundred twenty-seven, give in full detail the numbers they
are intended to describe. In primitive scales this is not always considered
necessary; thus, the Zamucos express their teens without using their word
for 10 at all. They say simply, 1 on the foot, 2 on the foot, etc.
Corresponding abbreviations are often met; so often, indeed, that no
further mention of them is needed. They mark one extreme, the extreme of
brevity, found in the savage method of building up hand, foot, and finger
names for numerals; while the Zuñi scale marks the extreme of prolixity in
the formation of such words. A somewhat ruder composition than any yet
noticed is shown in the numerals of the Vilelo scale,[83] which are:
1. agit, or yaagit.
2. uke.
3. nipetuei.
4. yepkatalet.
5. isig-nisle-yaagit = hand fingers 1.
6. isig-teet-yaagit = hand with 1.
7. isig-teet-uke = hand with 2.
8. isig-teet-nipetuei = hand with 3.
9. isig-teet-yepkatalet = hand with 4.
10. isig-uke-nisle = second hand fingers (lit. hand-two-fingers).
11. isig-uke-nisle-teet-yaagit = second hand fingers with 1.
20. isig-ape-nisle-lauel = hand foot fingers all.
In the examples thus far given, it will be noticed that the actual names of
individual fingers do not appear. In general, such words as thumb,
forefinger, little finger, are not found, but rather the hand-1, 1 on the
next, or 1 over and above, which we have already seen, are the type forms
for which we are to look. Individual finger names do occur, however, as in
the scale of the Hudson's Bay Eskimos,[84] where the three following words
are used both as numerals and as finger names:
8. kittukleemoot = middle finger.
9. mikkeelukkamoot = fourth finger.
10. eerkitkoka = little finger.
Words of similar origin are found in the original Jiviro scale,[85] where
the native numerals are:
1. ala.
2. catu.
3. cala.
4. encatu.
5. alacötegladu = 1 hand.
6. intimutu = thumb (of second hand).
7. tannituna = index finger.
8. tannituna cabiasu = the finger next the index finger.
9. bitin ötegla cabiasu = hand next to complete.
10. catögladu = 2 hands.
As if to emphasize the rarity of this method of forming numerals, the
Jiviros afterward discarded the last five of the above scale, replacing
them by words borrowed from the Quichuas, or ancient Peruvians. The same
process may have been followed by other tribes, and in this way numerals
which were originally digital may have disappeared. But we have no evidence
that this has ever happened in any extensive manner. We are, rather,
impelled to accept the occasional numerals of this class as exceptions to
the general rule, until we have at our disposal further evidence of an
exact and critical nature, which would cause us to modify this opinion. An
elaborate philological study by Dr. J.H. Trumbull[86] of the numerals used
by many of the North American Indian tribes reveals the presence in the
languages of these tribes of a few, but only a few, finger names which are
used without change as numeral expressions also. Sometimes the finger gives
a name not its own to the numeral with which it is associated in
counting--as in the Chippeway dialect, which has _nawi-nindj_, middle of
the hand, and _nisswi_, 3; and the Cheyenne, where _notoyos_, middle
finger, and _na-nohhtu_, 8, are closely related. In other parts of the
world isolated examples of the transference of finger names to numerals are
also found. Of these a well-known example is furnished by the Zulu
numerals, where "_tatisitupa_, taking the thumb, becomes a numeral for six.
Then the verb _komba_, to point, indicating the forefinger, or 'pointer,'
makes the next numeral, seven. 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 also 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._ eight, and
_Kijangalolunje_, 'keep back one finger,' _i.e._ nine, lead on to _kumi_,
ten."[87]
Returning for a moment to the consideration of number systems in the
formation of which the influence of the hand has been paramount, we find
still further variations of the method already noticed of constructing
names for the fives, tens, and twenties, as well as for the intermediate
numbers. Instead of the simple words "hand," "foot," etc., we not
infrequently meet with some paraphrase for one or for all these terms, the
derivation of which is unmistakable. The Nengones,[88] an island tribe of
the Indian Ocean, though using the word "man" for 20, do not employ
explicit hand or foot words, but count
1. sa.
2. rewe.
3. tini.
4. etse.
5. se dono = the end (of the first hand).
6. dono ne sa = end and 1.
7. dono ne rewe = end and 2.
8. dono ne tini = end and 3.
9. dono ne etse = end and 4.
10. rewe tubenine = 2 series (of fingers).
11. rewe tubenine ne sa re tsemene = 2 series and 1 on the next?
20. sa re nome = 1 man.
30. sa re nome ne rewe tubenine = 1 man and 2 series.
40. rewe ne nome = 2 men.
Examples like the above are not infrequent. The Aztecs used for 10 the word
_matlactli_, hand-half, _i.e._ the hand half of a man, and for 20
_cempoalli_, one counting.[89] The Point Barrow Eskimos call 10 _kodlin_,
the upper part, _i.e._ of a man. One of the Ewe dialects of Western
Africa[90] has _ewo_, done, for 10; while, curiously enough, 9, _asieke_,
is a digital word, meaning "to part (from) the hand."
In numerous instances also some characteristic word not of hand derivation
is found, like the Yoruba _ogodzi_, string, which becomes a numeral for 40,
because 40 cowries made a "string"; and the Maori _tekau_, bunch, which
signifies 10. The origin of this seems to have been the custom of counting
yams and fish by "bunches" of ten each.[91]
Another method of forming numeral words above 5 or 10 is found in the
presence of such expressions as second 1, second 2, etc. In languages of
rude construction and incomplete development the simple numeral scale is
often found to end with 5, and all succeeding numerals to be formed from
the first 5. The progression from that point may be 5-1, 5-2, etc., as in
the numerous quinary scales to be noticed later, or it may be second 1,
second 2, etc., as in the Niam Niam dialect of Central Africa, where the
scale is[92]
1. sa.
2. uwi.
3. biata.
4. biama.
5. biswi.
6. batissa = 2d 1.
7. batiwwi = 2d 2.
8. batti-biata = 2d 3.
9. batti-biama = 2d 4.
10. bauwé = 2d 5.
That this method of progression is not confined to the least developed
languages, however, is shown by a most cursory examination of the numerals
of our American Indian tribes, where numeral formation like that exhibited
above is exceedingly common. In the Kootenay dialect,[93] of British
Columbia, _qaetsa_, 4, and _wo-qaetsa,_ 8, are obviously related, the
latter word probably meaning a second 4. Most of the native languages of
British Columbia form their words for 7 and 8 from those which signify 2
and 3; as, for example, the Heiltsuk,[94] which shows in the following
words a most obvious correspondence:
2. matl. 7. matlaaus.
3. yutq. 8. yutquaus.
In the Choctaw language[95] the relation between 2 and 7, and 3 and 8, is
no less clear. Here the words are:
2. tuklo. 7. untuklo.
3. tuchina. 8. untuchina.
The Nez Percés[96] repeat the first three words of their scale in their 6,
7, and 8 respectively, as a comparison of these numerals will show.
1. naks. 6. oilaks.
2. lapit. 7. oinapt.
3. mitat. 8. oimatat.
In all these cases the essential point of the method is contained in the
repetition, in one way or another, of the numerals of the second quinate,
without the use with each one of the word for 5. This may make 6, 7, 8, and
9 appear as second 1, second 2, etc., or another 1, another 2, etc.; or,
more simply still, as 1 more, 2 more, etc. It is the method which was
briefly discussed in the early part of the present chapter, and is by no
means uncommon. In a decimal scale this repetition would begin with 11
instead of 6; as in the system found in use in Tagala and Pampanaga, two of
the Philippine Islands, where, for example, 11, 12, and 13 are:[97]
11. labi-n-isa = over 1.
12. labi-n-dalaua = over 2.
13. labi-n-tatlo = over 3.
A precisely similar method of numeral building is used by some of our
Western Indian tribes. Selecting a few of the Assiniboine numerals[98] as
an illustration, we have
11. ak kai washe = more 1.
12. ak kai noom pah = more 2.
13. ak kai yam me nee = more 3.
14. ak kai to pah = more 4.
15. ak kai zap tah = more 5.
16. ak kai shak pah = more 6, etc.
A still more primitive structure is shown in the numerals of the
Mboushas[99] of Equatorial Africa. Instead of using 5-1, 5-2, 5-3, 5-4, or
2d 1, 2d 2, 2d 3, 2d 4, in forming their numerals from 6 to 9, they proceed
in the following remarkable and, at first thought, inexplicable manner to
form their compound numerals:
1. ivoco.
2. beba.
3. belalo.
4. benai.
5. betano.
6. ivoco beba = 1-2.
7. ivoco belalo = 1-3.
8. ivoco benai = 1-4.
9. ivoco betano = 1-5.
10. dioum.
No explanation is given by Mr. du Chaillu for such an apparently
incomprehensible form of expression as, for example, 1-3, for 7. Some
peculiar finger pantomime may accompany the counting, which, were it known,
would enlighten us on the Mbousha's method of arriving at so anomalous a
scale. Mere repetition in the second quinate of the words used in the first
might readily be explained by supposing the use of fingers absolutely
indispensable as an aid to counting, and that a certain word would have one
meaning when associated with a certain finger of the left hand, and another
meaning when associated with one of the fingers of the right. Such scales
are, if the following are correct, actually in existence among the islands
of the Pacific.
BALAD.[100] UEA.[100]
1. parai. 1. tahi.
2. paroo. 2. lua.
3. pargen. 3. tolu.
4. parbai. 4. fa.
5. panim. 5. lima.
6. parai. 6. tahi.
7. paroo. 7. lua.
8. pargen. 8. tolu.
9. parbai. 9. fa.
10. panim. 10. lima.
Such examples are, I believe, entirely unique among primitive number
systems.
In numeral scales where the formative process has been of the general
nature just exhibited, irregularities of various kinds are of frequent
occurrence. Hand numerals may appear, and then suddenly disappear, just
where we should look for them with the greatest degree of certainty. In the
Ende,[101] a dialect of the Flores Islands, 5, 6, and 7 are of hand
formation, while 8 and 9 are of entirely different origin, as the scale
shows.
1. sa.
2. zua.
3. telu.
4. wutu.
5. lima
6. lima sa = hand 1.
7. lima zua = hand 2.
8. rua butu = 2 × 4.
9. trasa = 10 - 1?
10. sabulu.
One special point to be noticed in this scale is the irregularity that
prevails between 7, 8, 9. The formation of 7 is of the most ordinary kind;
8 is 2 fours--common enough duplication; while 9 appears to be 10 - 1. All
of these modes of compounding are, in their own way, regular; but the
irregularity consists in using all three of them in connective numerals in
the same system. But, odd as this jumble seems, it is more than matched by
that found in the scale of the Karankawa Indians,[102] an extinct tribe
formerly inhabiting the coast region of Texas. The first ten numerals of
this singular array are:
1. natsa.
2. haikia.
3. kachayi.
4. hayo hakn = 2 × 2.
5. natsa behema = 1 father, _i.e._ of the fingers.
6. hayo haikia = 3 × 2?
7. haikia natsa = 2 + 5?
8. haikia behema = 2 fathers?
9. haikia doatn = 2d from 10?
10. doatn habe.
Systems like the above, where chaos instead of order seems to be the ruling
principle, are of occasional occurrence, but they are decidedly the
exception.
In some of the cases that have been adduced for illustration it is to be
noticed that the process of combination begins with 7 instead of with 6.
Among others, the scale of the Pigmies of Central Africa[103] and that of
the Mosquitos[104] of Central America show this tendency. In the Pigmy
scale the words for 1 and 6 are so closely akin that one cannot resist the
impression that 6 was to them a new 1, and was thus named.
MOSQUITO. PIGMY.
1. kumi. ujju.
2. wal. ibari.
3. niupa. ikaro.
4. wal-wal = 2-2. ikwanganya.
5. mata-sip = fingers of 1 hand. bumuti.
6. matlalkabe. ijju.
7. matlalkabe pura kumi = 6 and 1. bumutti-na-ibali = 5 and 2.
8. matlalkabe pura wal = 6 and 2. bumutti-na-ikaro = 5 and 3.
9. matlalkabe pura niupa = 6 and 3. bumutti-na-ikwanganya = 5 and 4.
10. mata wal sip = fingers of 2 hands. mabo = half man.
The Mosquito scale is quite exceptional in forming 7, 8, and 9 from 6,
instead of from 5. The usual method, where combinations appear between 6
and 10, is exhibited by the Pigmy scale. Still another species of numeral
form, quite different from any that have already been noticed, is found in
the Yoruba[105] scale, which is in many respects one of the most peculiar
in existence. Here the words for 11, 12, etc., are formed by adding the
suffix _-la_, great, to the words for 1, 2, etc., thus:
1. eni, or okan.
2. edzi.
3. eta.
4. erin.
5. arun.
6. efa.
7. edze.
8. edzo.
9. esan.
10. ewa.
11. okanla = great 1.
12. edzila = great 2.
13. etala = great 3.
14. erinla = great 4, etc.
40. ogodzi = string.
200. igba = heap.
The word for 40 was adopted because cowrie shells, which are used for
counting, were strung by forties; and _igba_, 200, because a heap of 200
shells was five strings, and thus formed a convenient higher unit for
reckoning. Proceeding in this curious manner,[106] they called 50 strings 1
_afo_ or head; and to illustrate their singular mode of reckoning--the king
of the Dahomans, having made war on the Yorubans, and attacked their army,
was repulsed and defeated with a loss of "two heads, twenty strings, and
twenty cowries" of men, or 4820.
The number scale of the Abipones,[107] one of the low tribes of the
Paraguay region, contains two genuine curiosities, and by reason of those
it deserves a place among any collection of numeral scales designed to
exhibit the formation of this class of words. It is:
1. initara = 1 alone.
2. inoaka.
3. inoaka yekaini = 2 and 1.
4. geyenknate = toes of an ostrich.
5. neenhalek = a five coloured, spotted hide,
or hanambegen = fingers of 1 hand.
10. lanamrihegem = fingers of both hands.
20. lanamrihegem cat gracherhaka anamichirihegem = fingers of both
hands together with toes of both feet.
That the number sense of the Abipones is but little, if at all, above that
of the native Australian tribes, is shown by their expressing 3 by the
combination 2 and 1. This limitation, as we have already seen, is shared by
the Botocudos, the Chiquitos, and many of the other native races of South
America. But the Abipones, in seeking for words with which to enable
themselves to pass beyond the limit 3, invented the singular terms just
given for 4 and 5. The ostrich, having three toes in front and one behind
on each foot presented them with a living example of 3 + 1; hence "toes of
an ostrich" became their numeral for 4. Similarly, the number of colours in
a certain hide being five, the name for that hide was adopted as their next
numeral. At this point they began to resort to digital numeration also; and
any higher number is expressed by that method.
In the sense in which the word is defined by mathematicians, _number_ is a
pure, abstract concept. But a moment's reflection will show that, as it
originates among savage races, number is, and from the limitations of their
intellect must be, entirely concrete. An abstract conception is something
quite foreign to the essentially primitive mind, as missionaries and
explorers have found to their chagrin. The savage can form no mental
concept of what civilized man means by such a word as "soul"; nor would his
idea of the abstract number 5 be much clearer. When he says _five_, he
uses, in many cases at least, the same word that serves him when he wishes
to say _hand_; and his mental concept when he says _five_ is of a hand. The
concrete idea of a closed fist or an open hand with outstretched fingers,
is what is upper-most in his mind. He knows no more and cares no more about
the pure number 5 than he does about the law of the conservation of energy.
He sees in his mental picture only the real, material image, and his only
comprehension of the number is, "these objects are as many as the fingers
on my hand." Then, in the lapse of the long interval of centuries which
intervene between lowest barbarism and highest civilization, the abstract
and the concrete become slowly dissociated, the one from the other. First
the actual hand picture fades away, and the number is recognized without
the original assistance furnished by the derivation of the word. But the
number is still for a long time a certain number _of objects_, and not an
independent concept. It is only when the savage ceases to be wholly an
animal, and becomes a thinking human being, that number in the abstract can
come within the grasp of his mind. It is at this point that mere reckoning
ceases, and arithmetic begins.