Levi Leonard Conant - The Number Concept

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CHAPTER III.

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
Babusesse 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 _fuenf 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 _fuenf 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 x 10; just as "three hundred" means 3 x 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 x 8 + 100 x 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 Zuni scale, given by Lieutenant Gushing[68] leaves nothing to be
desired:

1. toepinte = 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. oepte = the notched off.

This finishes the list of original simple numerals, the Zuni stopping, or
"notching off," when he finishes the fingers of one hand. Compounding now
begins.

6. topalik'ya = another brought to add to the done with.
7. kwillilik'ya = two brought to and held up with the rest.
8. hailik'ye = three brought to and held up with the rest.
9. tenalik'ya = all but all are held up with the rest.
10. aestem'thila = all the fingers.
11. aestem'thla topayae'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'yenaestem'thlan = two times all the fingers.
100. aessiaestem'thlak'ya = the fingers all the fingers.
1000. aessiaestem'thlanak'yenaestem'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 Zuni, 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 Zuni 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. abbatekkabe = 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. biamantekabbe = 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 Zuni 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 Zuni, 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:

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