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The Number Concept - Its Origin And Development
by
LEVI LEONARD CONANT, Ph.D.
ASSOCIATE PROFESSOR OF MATHEMATICS IN THE WORCESTER
POLYTECHNIC INSTITUTE
New York
MACMILLAN AND CO.
AND LONDON
1931
Copyright, 1896, Copyright, 1924, All rights reserved—no part of this
book may be reproduced in any form
without permission in writing from
the publisher.
Set up and electrotyped. Published July, 1896.
Norwood Press Worcester, Mass., Nov. 12, 1895.
The process of formation indicated in 11 is used in the
succeeding numerals up to 19.
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.”
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
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:
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.
By THE MACMILLAN COMPANY.
By EMMA B. CONANT.
J. S. Cushing Co.—Berwick & Smith Co.
Norwood, Mass., U.S.A.
Preface.: Properties and Meanings
In the selection of authorities which have been consulted
in the preparation of this work, and to which
reference is made in the following pages, great care has
been taken. Original sources have been drawn upon in
the majority of cases, and nearly all of these are the most
recent attainable. Whenever it has not been possible to
cite original and recent works, the author has quoted only
such as are most standard and trustworthy. In the choice
of orthography of proper names and numeral words, the
forms have, in almost all cases, been written as they were
found, with no attempt to reduce them to a systematic
English basis. In many instances this would have been
quite impossible; and, even if possible, it would have been
altogether unimportant. Hence the forms, whether German,
French, Italian, Spanish, or Danish in their transcription,
are left unchanged. Diacritical marks are omitted,
however, since the proper key could hardly be furnished
in a work of this kind.
With the above exceptions, this study will, it is hoped,
be found to be quite complete; and as the subject here
investigated has never before been treated in any thorough
and comprehensive manner, it is hoped that this book may
be found helpful. The collections of numeral systems
illustrating the use of the binary, the quinary, and other
number systems, are, taken together, believed to be the
most extensive now existing in any language. Only the
cardinal numerals have been considered. The ordinals
present no marked peculiarities which would, in a work
of this kind, render a separate discussion necessary. Accordingly
they have, though with some reluctance, been
omitted entirely.
Sincere thanks are due to those who have assisted the
author in the preparation of his materials. Especial
acknowledgment should be made to Horatio Hale, Dr. D. G. Brinton, Frank Hamilton Cushing, and Dr. A. F.
Chamberlain.
Contents.: Properties and Meanings
The Number Concept: Its Origin And Development.
Chapter I.: Properties and Meanings
Counting.
Among the speculative questions which arise in connection
with the study of arithmetic from a historical
standpoint, the origin of number is one that has provoked
much lively discussion, and has led to a great
amount of learned research among the primitive and
savage languages of the human race. A few simple
considerations will, however, show that such research must
necessarily leave this question entirely unsettled, and will
indicate clearly that it is, from the very nature of things,
a question to which no definite and final answer can be given.
Among the barbarous tribes whose languages have been
studied, even in a most cursory manner, none have ever
been discovered which did not show some familiarity with
the number concept. The knowledge thus indicated has
often proved to be most limited; not extending beyond
the numbers 1 and 2, or 1, 2, and 3. Examples of
this poverty of number knowledge are found among
the forest tribes of Brazil, the native races of Australia
and elsewhere, and they are considered in some
detail in the next chapter. At first thought it seems
quite inconceivable that any human being should be
destitute of the power of counting beyond 2. But
such is the case; and in a few instances languages have
been found to be absolutely destitute of pure numeral
words. The Chiquitos of Bolivia had no real numerals
whatever,1 but expressed their idea for “one” by the word
etama, meaning alone. The Tacanas of the same country
have no numerals except those borrowed from Spanish,
or from Aymara or Peno, languages with which they have
long been in contact.2 A few other South American
languages are almost equally destitute of numeral words.
But even here, rudimentary as the number sense undoubtedly
is, it is not wholly lacking; and some indirect
expression, or some form of circumlocution, shows a conception
of the difference between one and two, or at least,
between one and many.
These facts must of necessity deter the mathematician
from seeking to push his investigation too far back
toward the very origin of number. Philosophers have
endeavoured to establish certain propositions concerning
this subject, but, as might have been expected, have failed
to reach any common ground of agreement. Whewell
has maintained that “such propositions as that two and
three make five are necessary truths, containing in them
an element of certainty beyond that which mere experience
can give.” Mill, on the other hand, argues that any
such statement merely expresses a truth derived from
early and constant experience; and in this view he is
heartily supported by Tylor.3 But why this question
should provoke controversy, it is difficult for the
mathematician to understand. Either view would seem to be
correct, according to the standpoint from which the
question is approached. We know of no language in
which the suggestion of number does not appear, and we
must admit that the words which give expression to the
number sense would be among the early words to be
formed in any language. They express ideas which are,
at first, wholly concrete, which are of the greatest
possible simplicity, and which seem in many ways to be
clearly understood, even by the higher orders of the brute
creation. The origin of number would in itself, then,
appear to lie beyond the proper limits of inquiry; and the
primitive conception of number to be fundamental with
human thought.
In connection with the assertion that the idea of number
seems to be understood by the higher orders of
animals, the following brief quotation from a paper by
Sir John Lubbock may not be out of place: “Leroy …
mentions a case in which a man was anxious to
shoot a crow. ‘To deceive this suspicious bird, the plan
was hit upon of sending two men to the watch house,
one of whom passed on, while the other remained; but
the crow counted and kept her distance. The next day
three went, and again she perceived that only two retired.
In fine, it was found necessary to send five or six men to
the watch house to put her out in her calculation. The
crow, thinking that this number of men had passed by,
lost no time in returning.’ From this he inferred that
crows could count up to four. Lichtenberg mentions a
nightingale which was said to count up to three. Every
day he gave it three mealworms, one at a time. When
it had finished one it returned for another, but after the
third it knew that the feast was over.… There is
an amusing and suggestive remark in Mr. Galton's interesting
Narrative of an Explorer in Tropical South
Africa. After describing the Demara's weakness in
calculations, he says: ‘Once while I watched a Demara
floundering hopelessly in a calculation on one side of
me, I observed, “Dinah,” my spaniel, equally embarrassed
on the other; she was overlooking half a dozen of her
new-born puppies, which had been removed two or three
times from her, and her anxiety was excessive, as she
tried to find out if they were all present, or if any were
still missing. She kept puzzling and running her eyes
over them backwards and forwards, but could not satisfy
herself. She evidently had a vague notion of counting,
but the figure was too large for her brain. Taking the
two as they stood, dog and Demara, the comparison
reflected no great honour on the man.…’ According to
my bird-nesting recollections, which I have refreshed
by more recent experience, if a nest contains four eggs,
one may safely be taken; but if two are removed, the
bird generally deserts. Here, then, it would seem as if
we had some reason for supposing that there is sufficient
intelligence to distinguish three from four. An interesting
consideration arises with reference to the number of
the victims allotted to each cell by the solitary wasps.
One species of Ammophila considers one large caterpillar
of Noctua segetum enough; one species of Eumenes
supplies its young with five victims; another 10, 15, and
even up to 24. The number appears to be constant in
each species. How does the insect know when her task
is fulfilled? Not by the cell being filled, for if some be
removed, she does not replace them. When she has
brought her complement she considers her task accomplished,
whether the victims are still there or not. How,
then, does she know when she has made up the number
24? Perhaps it will be said that each species feels some
mysterious and innate tendency to provide a certain number
of victims. This would, under no circumstances, be
any explanation; but it is not in accordance with the
facts. In the genus Eumenes the males are much smaller
than the females.… If the egg is male, she supplies
five; if female, 10 victims. Does she count? Certainly
this seems very like a commencement of arithmetic.”4
Many writers do not agree with the conclusions which
Lubbock reaches; maintaining that there is, in all such
instances, a perception of greater or less quantity rather
than any idea of number. But a careful consideration
of the objections offered fails entirely to weaken the
argument. Example after example of a nature similar
to those just quoted might be given, indicating on the
part of animals a perception of the difference between
1 and 2, or between 2 and 3 and 4; and any reasoning
which tends to show that it is quantity rather
than number which the animal perceives, will apply
with equal force to the Demara, the Chiquito, and
the Australian. Hence the actual origin of number
may safely be excluded from the limits of investigation,
and, for the present, be left in the field of pure
speculation.
A most inviting field for research is, however, furnished
by the primitive methods of counting and of
giving visible expression to the idea of number. Our
starting-point must, of course, be the sign language,
which always precedes intelligible speech; and which
is so convenient and so expressive a method of communication
that the human family, even in its most highly
developed branches, never wholly lays it aside. It may,
indeed, be stated as a universal law, that some practical
method of numeration has, in the childhood of every
nation or tribe, preceded the formation of numeral
words.
Practical methods of numeration are many in number
and diverse in kind. But the one primitive method of
counting which seems to have been almost universal
throughout all time is the finger method. It is a matter
of common experience and observation that every child,
when he begins to count, turns instinctively to his fingers;
and, with these convenient aids as counters, tallies
off the little number he has in mind. This method is
at once so natural and obvious that there can be no
doubt that it has always been employed by savage
tribes, since the first appearance of the human race in
remote antiquity. All research among uncivilized peoples
has tended to confirm this view, were confirmation
needed of anything so patent. Occasionally some exception
to this rule is found; or some variation, such as is
presented by the forest tribes of Brazil, who, instead of
counting on the fingers themselves, count on the joints
of their fingers.5 As the entire number system of these
tribes appears to be limited to three, this variation is
no cause for surprise.
The variety in practical methods of numeration observed
among savage races, and among civilized peoples
as well, is so great that any detailed account of them
would be almost impossible. In one region we find
sticks or splints used; in another, pebbles or shells; in
another, simple scratches, or notches cut in a stick,
Robinson Crusoe fashion; in another, kernels or little
heaps of grain; in another, knots on a string; and so
on, in diversity of method almost endless. Such are the
devices which have been, and still are, to be found in
the daily habit of great numbers of Indian, negro,
Mongolian, and Malay tribes; while, to pass at a single
step to the other extremity of intellectual development,
the German student keeps his beer score by
chalk marks on the table or on the wall. But back of
all these devices, and forming a common origin to which
all may be referred, is the universal finger method; the
method with which all begin, and which all find too
convenient ever to relinquish entirely, even though
their civilization be of the highest type. Any such
mode of counting, whether involving the use of the
fingers or not, is to be regarded simply as an extraneous
aid in the expression or comprehension of an idea which
the mind cannot grasp, or cannot retain, without assistance.
The German student scores his reckoning with
chalk marks because he might otherwise forget; while
the Andaman Islander counts on his fingers because he
has no other method of counting,—or, in other words,
of grasping the idea of number. A single illustration
may be given which typifies all practical methods of
numeration. More than a century ago travellers in
Madagascar observed a curious but simple mode of
ascertaining the number of soldiers in an army.6 Each
soldier was made to go through a passage in the presence
of the principal chiefs; and as he went through,
a pebble was dropped on the ground. This continued
until a heap of 10 was obtained, when one was set aside
and a new heap begun. Upon the completion of 10
heaps, a pebble was set aside to indicate 100; and so
on until the entire army had been numbered. Another
illustration, taken from the very antipodes of Madagascar,
recently found its way into print in an incidental
manner,7 and is so good that it deserves a place
beside de Flacourt's time-honoured example. Mom Cely,
a Southern negro of unknown age, finds herself in debt
to the storekeeper; and, unwilling to believe that the
amount is as great as he represents, she proceeds to
investigate the matter in her own peculiar way. She
had “kept a tally of these purchases by means of a
string, in which she tied commemorative knots.” When
her creditor “undertook to make the matter clear to
Cely's comprehension, he had to proceed upon a system
of her own devising. A small notch was cut in a smooth
white stick for every dime she owed, and a large notch
when the dimes amounted to a dollar; for every five
dollars a string was tied in the fifth big notch, Cely
keeping tally by the knots in her bit of twine; thus,
when two strings were tied about the stick, the ten dollars
were seen to be an indisputable fact.” This interesting
method of computing the amount of her debt,
whether an invention of her own or a survival of the
African life of her parents, served the old negro woman's
purpose perfectly; and it illustrates, as well as a score
of examples could, the methods of numeration to which
the children of barbarism resort when any number is
to be expressed which exceeds the number of counters
with which nature has provided them. The fingers are,
however, often employed in counting numbers far above
the first decade. After giving the Il-Oigob numerals up
to 60, Müller adds:8 “Above 60 all numbers, indicated
by the proper figure pantomime, are expressed by means
of the word ipi.” We know, moreover, that many of the
American Indian tribes count one ten after another on
their fingers; so that, whatever number they are endeavouring
to indicate, we need feel no surprise if the savage
continues to use his fingers throughout the entire extent
of his counts. In rare instances we find tribes which, like
the Mairassis of the interior of New Guinea, appear to
use nothing but finger pantomime.9 This tribe, though
by no means destitute of the number sense, is said to
have no numerals whatever, but to use the single word
awari with each show of fingers, no matter how few or
how many are displayed.
In the methods of finger counting employed by savages
a considerable degree of uniformity has been observed.
Not only does he use his fingers to assist him
in his tally, but he almost always begins with the little
finger of his left hand, thence proceeding towards
the thumb, which is 5. From this point onward the
method varies. Sometimes the second 5 also is told off
on the left hand, the same order being observed as in
the first 5; but oftener the fingers of the right hand
are used, with a reversal of the order previously employed;
i.e. the thumb denotes 6, the index finger 7,
and so on to the little finger, which completes the
count to 10.
At first thought there would seem to be no good
reason for any marked uniformity of method in finger
counting. Observation among children fails to detect
any such thing; the child beginning, with almost entire
indifference, on the thumb or on the little finger of the
left hand. My own observation leads to the conclusion
that very young children have a slight, though not
decided preference for beginning with the thumb.
Experiments in five different primary rooms in the public
schools of Worcester, Mass., showed that out of a
total of 206 children, 57 began with the little finger
and 149 with the thumb. But the fact that nearly
three-fourths of the children began with the thumb,
and but one-fourth with the little finger, is really far
less significant than would appear at first thought.
Children of this age, four to eight years, will count in
either way, and sometimes seem at a loss themselves
to know where to begin. In one school room where
this experiment was tried the teacher incautiously asked
one child to count on his fingers, while all the other
children in the room watched eagerly to see what he
would do. He began with the little finger—and so did
every child in the room after him. In another case
the same error was made by the teacher, and the child
first asked began with the thumb. Every other child
in the room did the same, each following, consciously
or unconsciously, the example of the leader. The results
from these two schools were of course rejected
from the totals which are given above; but they serve
an excellent purpose in showing how slight is the preference
which very young children have in this particular.
So slight is it that no definite law can be
postulated of this age; but the tendency seems to be
to hold the palm of the hand downward, and then
begin with the thumb. The writer once saw a boy
about seven years old trying to multiply 3 by 6; and
his method of procedure was as follows: holding his
left hand with its palm down, he touched with the
forefinger of his right hand the thumb, forefinger, and
middle finger successively of his left hand. Then returning
to his starting-point, he told off a second three
in the same manner. This process he continued until
he had obtained 6 threes, and then he announced his
result correctly. If he had been a few years older, he
might not have turned so readily to his thumb as a
starting-point for any digital count. The indifference
manifested by very young children gradually disappears,
and at the age of twelve or thirteen the tendency is
decidedly in the direction of beginning with the little
finger. Fully three-fourths of all persons above that
age will be found to count from the little finger toward
the thumb, thus reversing the proportion that was found
to obtain in the primary school rooms examined.
With respect to finger counting among civilized
peoples, we fail, then, to find any universal law; the
most that can be said is that more begin with the little
finger than with the thumb. But when we proceed to
the study of this slight but important particular among
savages, we find them employing a certain order of
succession with such substantial uniformity that the
conclusion is inevitable that there must lie back of this
some well-defined reason, or perhaps instinct, which
guides them in their choice. This instinct is undoubtedly
the outgrowth of the almost universal right-handedness
of the human race. In finger counting, whether
among children or adults, the beginning is made on
the left hand, except in the case of left-handed individuals;
and even then the start is almost as likely to
be on the left hand as on the right. Savage tribes, as
might be expected, begin with the left hand. Not
only is this custom almost invariable, when tribes as
a whole are considered, but the little finger is nearly
always called into requisition first. To account for this
uniformity, Lieutenant Gushing gives the following
theory,10 which is well considered, and is based on the
results of careful study and observation among the Zuñi
Indians of the Southwest: “Primitive man when abroad
never lightly quit hold of his weapons. If he wanted to
count, he did as the Zuñi afield does to-day; he tucked
his instrument under his left arm, thus constraining the
latter, but leaving the right hand free, that he might
check off with it the fingers of the rigidly elevated left
hand. From the nature of this position, however, the
palm of the left hand was presented to the face of the
counter, so that he had to begin his score on the little
finger of it, and continue his counting from the right
leftward. An inheritance of this may be detected to-day
in the confirmed habit the Zuñi has of gesticulating
from the right leftward, with the fingers of the
right hand over those of the left, whether he be counting
and summing up, or relating in any orderly manner.”
Here, then, is the reason for this otherwise unaccountable
phenomenon. If savage man is universally right-handed,
he will almost inevitably use the index finger of his right
hand to mark the fingers counted, and he will begin his
count just where it is most convenient. In his case it
is with the little finger of the left hand. In the case
of the child trying to multiply 3 by 6, it was with the
thumb of the same hand. He had nothing to tuck under
his arm; so, in raising his left hand to a position where
both eye and counting finger could readily run over its
fingers, he held the palm turned away from his face.
The same choice of starting-point then followed as with
the savage—the finger nearest his right hand; only in
this case the finger was a thumb. The deaf mute is
sometimes taught in this manner, which is for him an
entirely natural manner. A left-handed child might
be expected to count in a left-to-right manner, beginning,
probably, with the thumb of his right hand.
To the law just given, that savages begin to count
on the little finger of the left hand, there have been
a few exceptions noted; and it has been observed that
the method of progression on the second hand is by no
means as invariable as on the first. The Otomacs11 of
South America began their count with the thumb, and
to express the number 3 would use the thumb, forefinger,
and middle finger. The Maipures,12 oddly enough,
seem to have begun, in some cases at least, with the
forefinger; for they are reported as expressing 3 by
means of the fore, middle, and ring fingers. The Andamans13
begin with the little finger of either hand, tapping
the nose with each finger in succession. If they have
but one to express, they use the forefinger of either hand,
pronouncing at the same time the proper word. The
Bahnars,14 one of the native tribes of the interior of
Cochin China, exhibit no particular order in the sequence
of fingers used, though they employ their digits freely
to assist them in counting. Among certain of the negro
tribes of South Africa15 the little finger of the right hand
is used for 1, and their count proceeds from right to left.
With them, 6 is the thumb of the left hand, 7 the forefinger,
and so on. They hold the palm downward instead
of upward, and thus form a complete and striking exception
to the law which has been found to obtain with such
substantial uniformity in other parts of the uncivilized
world. In Melanesia a few examples of preference for
beginning with the thumb may also be noticed. In the
Banks Islands the natives begin by turning down the
thumb of the right hand, and then the fingers in succession
to the little finger, which is 5. This is followed
by the fingers of the left hand, both hands with closed
fists being held up to show the completed 10. In Lepers'
Island, they begin with the thumb, but, having reached
5 with the little finger, they do not pass to the other
hand, but throw up the fingers they have turned down,
beginning with the forefinger and keeping the thumb
for 10.16 In the use of the single hand this people is
quite peculiar. The second 5 is almost invariably told
off by savage tribes on the second hand, though in
passing from the one to the other primitive man does
not follow any invariable law. He marks 6 with either
the thumb or the little finger. Probably the former is
the more common practice, but the statement cannot be
made with any degree of certainty. Among the Zulus
the sequence is from thumb to thumb, as is the case
among the other South African tribes just mentioned;
while the Veis and numerous other African tribes pass
from thumb to little finger. The Eskimo, and nearly
all the American Indian tribes, use the correspondence
between 6 and the thumb; but this habit is by no means
universal. Respecting progression from right to left or
left to right on the toes, there is no general law with
which the author is familiar. Many tribes never use
the toes in counting, but signify the close of the first 10
by clapping the hands together, by a wave of the right
hand, or by designating some object; after which the
fingers are again used as before.
One other detail in finger counting is worthy of a
moment's notice. It seems to have been the opinion
of earlier investigators that in his passage from one
finger to the next, the savage would invariably bend
down, or close, the last finger used; that is, that the
count began with the fingers open and outspread. This
opinion is, however, erroneous. Several of the Indian
tribes of the West17 begin with the hand clenched, and
open the fingers one by one as they proceed. This
method is much less common than the other, but that
it exists is beyond question.
In the Muralug Island, in the western part of Torres
Strait, a somewhat remarkable method of counting formerly
existed, which grew out of, and is to be regarded
as an extension of, the digital method. Beginning with
the little finger of the left hand, the natives counted
up to 5 in the usual manner, and then, instead of
passing to the other hand, or repeating the count on
the same fingers, they expressed the numbers from 6
to 10 by touching and naming successively the left
wrist, left elbow, left shoulder, left breast, and sternum.
Then the numbers from 11 to 19 were indicated by
the use, in inverse order, of the corresponding portions
of the right side, arm, and hand, the little finger of
the right hand signifying 19. The words used were
in each case the actual names of the parts touched;
the same word, for example, standing for 6 and 14;
but they were never used in the numerical sense
unless accompanied by the proper gesture, and bear no
resemblance to the common numerals, which are but
few in number. This method of counting is rapidly
dying out among the natives of the island, and is at
the present time used only by old people.18 Variations
on this most unusual custom have been found to exist
in others of the neighbouring islands, but none were
exactly similar to it. One is also reminded by it of
a custom19 which has for centuries prevailed among bargainers
in the East, of signifying numbers by touching
the joints of each other's fingers under a cloth. Every
joint has a special signification; and the entire system
is undoubtedly a development from finger counting.
The buyer or seller will by this method express 6
or 60 by stretching out the thumb and little finger
and closing the rest of the fingers. The addition of
the fourth finger to the two thus used signifies 7
or 70; and so on. “It is said that between two brokers
settling a price by thus snipping with the fingers,
cleverness in bargaining, offering a little more, hesitating,
expressing an obstinate refusal to go further, etc.,
are as clearly indicated as though the bargaining were
being carried on in words.
The place occupied, in the intellectual development
of man, by finger counting and by the many other artificial
methods of reckoning,—pebbles, shells, knots, the
abacus, etc.,—seems to be this: The abstract processes
of addition, subtraction, multiplication, division, and even
counting itself, present to the mind a certain degree of
difficulty. To assist in overcoming that difficulty, these
artificial aids are called in; and, among savages of a low
degree of development, like the Australians, they make
counting possible. A little higher in the intellectual
scale, among the American Indians, for example, they
are employed merely as an artificial aid to what could
be done by mental effort alone. Finally, among semi-civilized
and civilized peoples, the same processes are
retained, and form a part of the daily life of almost
every person who has to do with counting, reckoning,
or keeping tally in any manner whatever. They are
no longer necessary, but they are so convenient and
so useful that civilization can never dispense with them.
The use of the abacus, in the form of the ordinary
numeral frame, has increased greatly within the past
few years; and the time may come when the abacus in
its proper form will again find in civilized countries a
use as common as that of five centuries ago.
In the elaborate calculating machines of the present,
such as are used by life insurance actuaries and others
having difficult computations to make, we have the extreme
of development in the direction of artificial aid
to reckoning. But instead of appearing merely as an
extraneous aid to a defective intelligence, it now presents
itself as a machine so complex that a high degree
of intellectual power is required for the mere grasp
of its construction and method of working.
Chapter II.: Properties and Meanings
Number System Limits.
With respect to the limits to which the number
systems of the various uncivilized races of the earth
extend, recent anthropological research has developed
many interesting facts. In the case of the Chiquitos
and a few other native races of Bolivia we found no
distinct number sense at all, as far as could be judged
from the absence, in their language, of numerals in the
proper sense of the word. How they indicated any
number greater than one is a point still requiring
investigation. In all other known instances we find
actual number systems, or what may for the sake of
uniformity be dignified by that name. In many cases,
however, the numerals existing are so few, and the
ability to count is so limited, that the term number
system is really an entire misnomer.
Among the rudest tribes, those whose mode of living
approaches most nearly to utter savagery, we find a
certain uniformity of method. The entire number
system may consist of but two words, one and many;
or of three words, one, two, many. Or, the count may
proceed to 3, 4, 5, 10, 20, or 100; passing always,
or almost always, from the distinct numeral limit
to the indefinite many or several, which serves for the
expression of any number not readily grasped by the
mind. As a matter of fact, most races count as high
as 10; but to this statement the exceptions are so
numerous that they deserve examination in some detail.
In certain parts of the world, notably among the
native races of South America, Australia, and many
of the islands of Polynesia and Melanesia, a surprising
paucity of numeral words has been observed. The Encabellada
of the Rio Napo have but two distinct numerals;
tey, 1, and cayapa, 2.20 The Chaco languages21 of
the Guaycuru stock are also notably poor in this respect.
In the Mbocobi dialect of this language the
only native numerals are yña tvak, 1, and yfioaca, 2.
The Puris22 count omi, 1, curiri, 2, prica, many; and
the Botocudos23 mokenam, 1, uruhu, many. The Fuegans,24
supposed to have been able at one time to count
to 10, have but three numerals,—kaoueli, 1, compaipi, 2,
maten, 3. The Campas of Peru25 possess only three
separate words for the expression of number,—patrio,
1, pitteni, 2, mahuani, 3. Above 3 they proceed by
combinations, as 1 and 3 for 4, 1 and 1 and 3 for 5.
Counting above 10 is, however, entirely inconceivable
to them, and any number beyond that limit they indicate
by tohaine, many. The Conibos,26 of the same
region, had, before their contact with the Spanish, only
atchoupre, 1, and rrabui, 2; though they made some
slight progress above 2 by means of reduplication.
The Orejones, one of the low, degraded tribes of the
Upper Amazon,27 have no names for number except
nayhay, 1, nenacome, 2, feninichacome, 3, ononoeomere, 4.
In the extensive vocabularies given by Von Martins,28
many similar examples are found. For the Bororos he
gives only couai, 1, maeouai, 2, ouai, 3. The last word,
with the proper finger pantomime, serves also for any
higher number which falls within the grasp of their comprehension.
The Guachi manage to reach 5, but their
numeration is of the rudest kind, as the following
scale shows: tamak, 1, eu-echo, 2, eu-echo-kailau, 3, eu-echo-way,
4, localau, 5. The Carajas counted by a
scale equally rude, and their conception of number
seemed equally vague, until contact with the neighbouring
tribes furnished them with the means of going
beyond their original limit. Their scale shows clearly
the uncertain, feeble number sense which is so marked
in the interior of South America. It contains wadewo,
1, wadebothoa, 2, wadeboaheodo, 3, wadebojeodo, 4, wadewajouclay,
5, wadewasori, 6, or many.
Turning to the languages of the extinct, or fast vanishing,
tribes of Australia, we find a still more noteworthy
absence of numeral expressions. In the Gudang
dialect29 but two numerals are found—pirman, 1, and
ilabiu, 2; in the Weedookarry, ekkamurda, 1, and kootera,
2; and in the Queanbeyan, midjemban, 1, and bollan,
2. In a score or more of instances the numerals stop
at 3. The natives of Keppel Bay count webben, 1, booli,
2, koorel, 3; of the Boyne River, karroon, 1, boodla, 2,
numma, 3; of the Flinders River, kooroin, 1, kurto, 2,
kurto kooroin, 3; at the mouth of the Norman River,
lum, 1, buggar, 2, orinch, 3; the Eaw tribe, koothea, 1,
woother, 2, marronoo, 3; the Moree, mal, 1, boolar,
2, kooliba, 3; the Port Essington,30 erad, 1, nargarick,
2, nargarickelerad, 3; the Darnly Islanders,31 netat, 1,
naes, 2, naesa netat, 3; and so on through a long list
of tribes whose numeral scales are equally scanty. A
still larger number of tribes show an ability to count
one step further, to 4; but beyond this limit the majority
of Australian and Tasmanian tribes do not go. It
seems most remarkable that any human being should
possess the ability to count to 4, and not to 5. The
number of fingers on one hand furnishes so obvious a
limit to any of these rudimentary systems, that positive
evidence is needed before one can accept the statement.
A careful examination of the numerals in upwards of a
hundred Australian dialects leaves no doubt, however,
that such is the fact. The Australians in almost all cases
count by pairs; and so pronounced is this tendency that
they pay but little attention to the fingers. Some tribes
do not appear ever to count beyond 2—a single pair.
Many more go one step further; but if they do, they are
as likely as not to designate their next numeral as two-one,
or possibly, one-two. If this step is taken, we may
or may not find one more added to it, thus completing
the second pair. Still, the Australian's capacity for
understanding anything which pertains to number is so
painfully limited that even here there is sometimes an
indefinite expression formed, as many, heap, or plenty,
instead of any distinct numeral; and it is probably true
that no Australian language contains a pure, simple
numeral for 4. Curr, the best authority on this subject,
believes that, where a distinct word for 4 is given,
investigators have been deceived in every case.32 If
counting is carried beyond 4, it is always by means of
reduplication. A few tribes gave expressions for 5,
fewer still for 6, and a very small number appeared
able to reach 7. Possibly the ability to count extended
still further; but if so, it consisted undoubtedly in
reckoning one pair after another, without any consciousness
whatever of the sum total save as a larger
number.
The numerals of a few additional tribes will show
clearly that all distinct perception of number is lost as
soon as these races attempt to count above 3, or at most,
4. The Yuckaburra33 natives can go no further than
wigsin, 1, bullaroo, 2, goolbora, 3. Above here all is
referred to as moorgha, many. The Marachowies34 have
but three distinct numerals,—cooma, 1, cootera, 2, murra,
3. For 4 they say minna, many. At Streaky Bay we
find a similar list, with the same words, kooma and
kootera, for 1 and 2, but entirely different terms, karboo
and yalkata for 3 and many. The same method obtains
in the Minnal Yungar tribe, where the only numerals
are kain, 1, kujal, 2, moa, 3, and bulla, plenty. In the
Pinjarra dialect we find doombart, 1, gugal, 2, murdine,
3, boola, plenty; and in the dialect described as belonging
to “Eyre's Sand Patch,” three definite terms are
given—kean, 1, koojal, 2, yalgatta, 3, while a fourth,
murna, served to describe anything greater. In all
these examples the fourth numeral is indefinite; and
the same statement is true of many other Australian
languages. But more commonly still we find 4, and
perhaps 3 also, expressed by reduplication. In the Port
Mackay dialect35 the latter numeral is compound, the
count being warpur, 1, boolera, 2, boolera warpur, 3. For
4 the term is not given. In the dialect which prevailed
between the Albert and Tweed rivers36 the scale appears
as yaburu, 1, boolaroo, 2, boolaroo yaburu, 3, and gurul for
4 or anything beyond. The Wiraduroi37 have numbai, 1,
bula, 2, bula numbai, 3, bungu, 4, or many, and bungu galan
or bian galan, 5, or very many. The Kamilaroi38 scale
is still more irregular, compounding above 4 with little
apparent method. The numerals are mal, 1, bular, 2,
guliba, 3, bular bular, 4, bular guliba, 5, guliba guliba, 6.
The last two numerals show that 5 is to these natives
simply 2-3, and 6 is 3-3. For additional examples of a
similar nature the extended list of Australian scales
given in Chapter V. may be consulted.
Taken as a whole, the Australian and Tasmanian tribes
seem to have been distinctly inferior to those of South
America in their ability to use and to comprehend
numerals. In all but two or three cases the Tasmanians39
were found to be unable to proceed beyond 2; and as the
foregoing examples have indicated, their Australian
neighbours were but little better off. In one or two instances
we do find Australian numeral scales which reach
10, and perhaps we may safely say 20. One of these is
given in full in a subsequent chapter, and its structure
gives rise to the suspicion that it was originally as limited
as those of kindred tribes, and that it underwent a considerable
development after the natives had come in contact
with the Europeans. There is good reason to believe
that no Australian in his wild state could ever count
intelligently to 7.40
In certain portions of Asia, Africa, Melanesia, Polynesia,
and North America, are to be found races whose
number systems are almost and sometimes quite as limited
as are those of the South. American and Australian
tribes already cited, but nowhere else do we find these
so abundant as in the two continents just mentioned,
where example after example might be cited of tribes
whose ability to count is circumscribed within the narrowest
limits. The Veddas41 of Ceylon have but two
numerals, ekkameī, 1, dekkameï, 2. Beyond this they
count otameekaï, otameekaï, otameekaï, etc.; i.e. “and
one more, and one more, and one more,” and so on indefinitely.
The Andamans,42 inhabitants of a group of
islands in the Bay of Bengal, are equally limited in
their power of counting. They have ubatulda, 1, and
ikporda, 2; but they can go no further, except in a
manner similar to that of the Veddas. Above two they
proceed wholly by means of the fingers, saying as they
tap the nose with each successive finger, anka, “and
this.” Only the more intelligent of the Andamans can
count at all, many of them seeming to be as nearly destitute
of the number sense as it is possible for a human
being to be. The Bushmen43 of South Africa have but
two numerals, the pronunciation of which can hardly be
indicated without other resources than those of the English
alphabet. Their word for 3 means, simply, many,
as in the case of some of the Australian tribes. The
Watchandies44 have but two simple numerals, and their
entire number system is cooteon, 1, utaura, 2, utarra
cooteoo, 3, atarra utarra, 4. Beyond this they can only
say, booltha, many, and booltha bat, very many. Although
they have the expressions here given for 3 and 4, they are
reluctant to use them, and only do so when absolutely
required. The natives of Lower California45 cannot count
above 5. A few of the more intelligent among them
understand the meaning of 2 fives, but this number
seems entirely beyond the comprehension of the ordinary
native. The Comanches, curiously enough, are so reluctant
to employ their number words that they appear to
prefer finger pantomime instead, thus giving rise to the
impression which at one time became current, that they
had no numerals at all for ordinary counting.
Aside from the specific examples already given, a considerable
number of sweeping generalizations may be
made, tending to show how rudimentary the number
sense may be in aboriginal life. Scores of the native
dialects of Australia and South America have been found
containing number systems but little more extensive than
those alluded to above. The negro tribes of Africa give
the same testimony, as do many of the native races of
Central America, Mexico, and the Pacific coast of the
United States and Canada, the northern part of Siberia,
Greenland, Labrador, and the arctic archipelago. In
speaking of the Eskimos of Point Barrow, Murdoch46
says: “It was not easy to obtain any accurate information
about the numeral system of these people, since in
ordinary conversation they are not in the habit of specifying
any numbers above five.” Counting is often carried
higher than this among certain of these northern
tribes, but, save for occasional examples, it is limited at
best. Dr. Franz Boas, who has travelled extensively
among the Eskimos, and whose observations are always
of the most accurate nature, once told the author that
he never met an Eskimo who could count above 15.
Their numerals actually do extend much higher; and
a stray numeral of Danish origin is now and then met
with, showing that the more intelligent among them are
able to comprehend numbers of much greater magnitude
than this. But as Dr. Boas was engaged in active
work among them for three years, we may conclude
that the Eskimo has an arithmetic but little more extended
than that which sufficed for the Australians and
the forest tribes of Brazil. Early Russian explorers
among the northern tribes of Siberia noticed the same
difficulty in ordinary, every-day reckoning among the
natives. At first thought we might, then, state it as
a general law that those races which are lowest in the
scale of civilization, have the feeblest number sense also;
or in other words, the least possible power of grasping
the abstract idea of number.
But to this law there are many and important exceptions.
The concurrent testimony of explorers seems to
be that savage races possess, in the great majority of
cases, the ability to count at least as high as 10. This
limit is often extended to 20, and not infrequently to
100. Again, we find 1000 as the limit; or perhaps
10,000; and sometimes the savage carries his number
system on into the hundreds of thousands or millions.
Indeed, the high limit to which some savage races
carry their numeration is far more worthy of remark
than the entire absence of the number sense exhibited
by others of apparently equal intelligence. If the life
of any tribe is such as to induce trade and barter with
their neighbours, a considerable quickness in reckoning
will be developed among them. Otherwise this power
will remain dormant because there is but little in the
ordinary life of primitive man to call for its exercise.
In giving 1, 2, 3, 5, 10, or any other small number
as a system limit, it must not be overlooked that this
limit mentioned is in all cases the limit of the spoken
numerals at the savage's command. The actual ability
to count is almost always, and one is tempted to say
always, somewhat greater than their vocabularies would
indicate. The Bushman has no number word that
will express for him anything higher than 2; but
with the assistance of his fingers he gropes his way on
as far as 10. The Veddas, the Andamans, the Guachi,
the Botocudos, the Eskimos, and the thousand and one
other tribes which furnish such scanty numeral systems,
almost all proceed with more or less readiness as
far as their fingers will carry them. As a matter of
fact, this limit is frequently extended to 20; the toes,
the fingers of a second man, or a recount of the savage's
own fingers, serving as a tale for the second
10. Allusion is again made to this in a later chapter,
where the subject of counting on the fingers and toes
is examined more in detail.
In saying that a savage can count to 10, to 20, or to
100, but little idea is given of his real mental conception
of any except the smallest numbers. Want of
familiarity with the use of numbers, and lack of convenient
means of comparison, must result in extreme
indefiniteness of mental conception and almost entire
absence of exactness. The experience of Captain
Parry,47 who found that the Eskimos made mistakes
before they reached 7, and of Humboldt,48 who says
that a Chayma might be made to say that his age
was either 18 or 60, has been duplicated by all
investigators who have had actual experience among
savage races. Nor, on the other hand, is the development
of a numeral system an infallible index of mental
power, or of any real approach toward civilization. A
continued use of the trading and bargaining faculties
must and does result in a familiarity with numbers
sufficient to enable savages to perform unexpected feats
in reckoning. Among some of the West African tribes
this has actually been found to be the case; and among
the Yorubas of Abeokuta49 the extraordinary saying,
“You may seem very clever, but you can't tell nine
times nine,” shows how surprisingly this faculty has
been developed, considering the general condition of
savagery in which the tribe lived. There can be no
doubt that, in general, the growth of the number sense
keeps pace with the growth of the intelligence in other
respects. But when it is remembered that the Tonga
Islanders have numerals up to 100,000, and the Tembus,
the Fingoes, the Pondos, and a dozen other South
African tribes go as high as 1,000,000; and that Leigh
Hunt never could learn the multiplication table, one
must confess that this law occasionally presents to our
consideration remarkable exceptions.
While considering the extent of the savage's arithmetical
knowledge, of his ability to count and to grasp the
meaning of number, it may not be amiss to ask ourselves
the question, what is the extent of the development of
our own number sense? To what limit can we absorb
the idea of number, with a complete appreciation of the
idea of the number of units involved in any written or
spoken quantity? Our perfect system of numeration
enables us to express without difficulty any desired number,
no matter how great or how small it be. But how
much of actually clear comprehension does the number
thus expressed convey to the mind? We say that one
place is 100 miles from another; that A paid B 1000
dollars for a certain piece of property; that a given
city contains 10,000 inhabitants; that 100,000 bushels
of wheat were shipped from Duluth or Odessa on such
a day; that 1,000,000 feet of lumber were destroyed by
the fire of yesterday,—and as we pass from the smallest
to the largest of the numbers thus instanced, and from
the largest on to those still larger, we repeat the question
just asked; and we repeat it with a new sense of our
own mental limitation. The number 100 unquestionably
stands for a distinct conception. Perhaps the same
may be said for 1000, though this could not be postulated
with equal certainty. But what of 10,000? If that
number of persons were gathered together into a single
hall or amphitheatre, could an estimate be made by the
average onlooker which would approximate with any
degree of accuracy the size of the assembly? Or if an
observer were stationed at a certain point, and 10,000
persons were to pass him in single file without his counting
them as they passed, what sort of an estimate would
he make of their number? The truth seems to be that
our mental conception of number is much more limited
than is commonly thought, and that we unconsciously
adopt some new unit as a standard of comparison when
we wish to render intelligible to our minds any number
of considerable magnitude. For example, we say that
A has a fortune of $1,000,000. The impression is at once
conveyed of a considerable degree of wealth, but it is
rather from the fact that that fortune represents an
annual income of $40,000 than, from the actual magnitude
of the fortune itself. The number 1,000,000 is, in itself,
so greatly in excess of anything that enters into our daily
experience that we have but a vague conception of it,
except as something very great. We are not, after all,
so very much better off than the child who, with his arms
about his mother's neck, informs her with perfect gravity
and sincerity that he “loves her a million bushels.” His
idea is merely of some very great amount, and our own
is often but little clearer when we use the expressions
which are so easily represented by a few digits. Among
the uneducated portions of civilized communities the
limit of clear comprehension of number is not only relatively,
but absolutely, very low. Travellers in Russia
have informed the writer that the peasants of that
country have no distinct idea of a number consisting of
but a few hundred even. There is no reason to doubt
this testimony. The entire life of a peasant might be
passed without his ever having occasion to use a number
as great as 500, and as a result he might have respecting
that number an idea less distinct than a trained mathematician
would have of the distance from the earth to
the sun. De Quincey50 incidentally mentions this characteristic
in narrating a conversation which occurred
while he was at Carnarvon, a little town in Wales. “It
was on this occasion,” he says, “that I learned how vague
are the ideas of number in unpractised minds. ‘What
number of people do you think,’ I said to an elderly
person, ‘will be assembled this day at Carnarvon?’
‘What number?’ rejoined the person addressed; ‘what
number? Well, really, now, I should reckon—perhaps
a matter of four million.’ Four millions of extra people
in little Carnarvon, that could barely find accommodation
(I should calculate) for an extra four hundred!”
So the Eskimo and the South American Indian are,
after all, not so very far behind the “elderly person”
of Carnarvon, in the distinct perception of a number
which familiarity renders to us absurdly small.
Chapter III.: Properties and Meanings
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
Gushing68 leaves nothing to be desired:
1.töpinte= taken to start with.
2.kwilli= put down together with.
3.ha'ī= 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.
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 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.
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.
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.
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 Guarani75 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 are77
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 Maipures78 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 dialects79 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.
Trumbull86 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
Africa90 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 is92
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 language95 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 Percés96 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 numerals98 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 Mboushas99 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.
1.parai.
2.paroo.
3.pargen.
4.parbai.
5.panim.
6.parai.
7.paroo.
8.pargen.
9.parbai.
10.panim.
1.tahi.
2.lua.
3.tolu.
4.fa.
5.lima.
6.tahi.
7.lua.
8.tolu.
9.fa.
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 Africa103 and that
of the Mosquitos104 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 Yoruba105 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.
Chapter IV.: Properties and Meanings
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 thanthat 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-Mbutti111 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 are119
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 Greenlanders120
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 Mabas121 call 10 atuk, great 1; the
Hottentots122 and the Hidatsa Indians call 100 great 10,
their words being gei disi and pitikitstia respectively.
The Nicaraguans123 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 Dakota124 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”: